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{-# OPTIONS --without-K --rewriting #-}
open import HoTT
module homotopy.HSpace where
-- This is just an approximation because
-- not all higher cells are killed.
record HSpaceStructure {i} (X : Ptd i) : Type i where
constructor hSpaceStructure
field
⊙μ : (X ⊙× X) ⊙→ X
μ : de⊙ X → de⊙ X → de⊙ X
μ = curry (fst ⊙μ)
field
⊙unit-l : ⊙μ ⊙∘ ⊙×-inr X X ⊙∼ ⊙idf X
⊙unit-r : ⊙μ ⊙∘ ⊙×-inl X X ⊙∼ ⊙idf X
unit-l : ∀ x → μ (pt X) x == x
unit-l = fst ⊙unit-l
unit-r : ∀ x → μ x (pt X) == x
unit-r = fst ⊙unit-r
coh : unit-l (pt X) == unit-r (pt X)
coh = ! (↓-idf=cst-out (snd ⊙unit-l) ∙ ∙-unit-r _)
∙ (↓-idf=cst-out (snd ⊙unit-r) ∙ ∙-unit-r _)
record AlternativeHSpaceStructure {i} (X : Ptd i) : Type i where
constructor hSpaceStructure
field
μ : de⊙ X → de⊙ X → de⊙ X
unit-l : ∀ x → μ (pt X) x == x
unit-r : ∀ x → μ x (pt X) == x
coh : unit-l (pt X) == unit-r (pt X)
⊙μ : (X ⊙× X) ⊙→ X
⊙μ = uncurry μ , unit-l (pt X)
⊙unit-l : ⊙μ ⊙∘ ⊙×-inr X X ⊙∼ ⊙idf X
⊙unit-l = unit-l , ↓-idf=cst-in' idp
⊙unit-r : ⊙μ ⊙∘ ⊙×-inl X X ⊙∼ ⊙idf X
⊙unit-r = unit-r , ↓-idf=cst-in' coh
from-alt-h-space : ∀ {i} {X : Ptd i} → AlternativeHSpaceStructure X → HSpaceStructure X
from-alt-h-space hss = record {AlternativeHSpaceStructure hss}
to-alt-h-space : ∀ {i} {X : Ptd i} → HSpaceStructure X → AlternativeHSpaceStructure X
to-alt-h-space hss = record {HSpaceStructure hss}
module ConnectedHSpace {i} {X : Ptd i} {{_ : is-connected 0 (de⊙ X)}}
(hX : HSpaceStructure X) where
open HSpaceStructure hX public
{-
Given that [X] is 0-connected, to prove that each [μ x] is an equivalence we
only need to prove that one of them is. But for [x] = [pt X], [μ x] is the
identity so we’re done.
-}
l-is-equiv : ∀ x → is-equiv (λ y → μ y x)
l-is-equiv = prop-over-connected {a = pt X}
(λ x → (is-equiv (λ y → μ y x) , is-equiv-is-prop))
(transport! is-equiv (λ= unit-r) (idf-is-equiv _))
r-is-equiv : ∀ x → is-equiv (λ y → μ x y)
r-is-equiv = prop-over-connected {a = pt X}
(λ x → (is-equiv (λ y → μ x y) , is-equiv-is-prop))
(transport! is-equiv (λ= unit-l) (idf-is-equiv _))
l-equiv : de⊙ X → de⊙ X ≃ de⊙ X
l-equiv x = _ , l-is-equiv x
r-equiv : de⊙ X → de⊙ X ≃ de⊙ X
r-equiv x = _ , r-is-equiv x
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module Examples where
data Nat : Set where
zero : Nat
suc : Nat -> Nat
data Bool : Set where
true : Bool
false : Bool
if_then_else_ : {A : Set} -> Bool -> A -> A -> A
if true then x else y = x
if false then x else y = y
not : Bool -> Bool
not x = if x then false else true
isZero : Nat -> Bool
isZero zero = true
isZero (suc _) = false
F : Bool -> Set
F true = Nat
F false = Bool
f : (x : Bool) -> F x -> F (not x)
f true n = isZero n
f false b = if b then zero else suc zero
test : Bool
test = f ? zero
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module Foundation.Level0 where
{-
it : ∀ {a} {A : Set a} {{x : A}} → A
it {{x}} = x
{-# INLINE it #-}
-}
private
module Data where
data ⊥ : Set where
module _ where
open import Foundation.Bottom
open import Foundation.Primitive
instance isBottom-Data : ∀ {a} → IsBottom Data.⊥ a
IsBottom.⊥-elim isBottom-Data ()
instance Bottom-Data : ∀ {a} → Bottom 𝟘 a
Bottom.⊥ Bottom-Data = Data.⊥
Bottom.isBottom Bottom-Data = isBottom-Data
open import Agda.Builtin.Equality
module _ where
open import Foundation.Equivalence
isEquivalence-Builtin : ∀ {a} {A : Set a} → IsEquivalence {A = A} _≡_
IsEquivalence.reflexivity isEquivalence-Builtin x = refl
IsEquivalence.symmetry isEquivalence-Builtin x .x refl = refl
IsEquivalence.transitivity isEquivalence-Builtin x .x z refl x₂ = x₂
instance _ = isEquivalence-Builtin
equivalence-Builtin : ∀ {a} {A : Set a} → Equivalence A a
Equivalence._≈_ equivalence-Builtin = _≡_
instance _ = equivalence-Builtin
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{-# OPTIONS --safe #-}
module Cubical.Algebra.AbGroup where
open import Cubical.Algebra.AbGroup.Base public
open import Cubical.Algebra.AbGroup.Properties public
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open import Agda.Builtin.Equality
module _ (F : Set₁ → Set₁) where
f : (B : Set₁) → B ≡ F Set → Set
f B eq = {!eq!}
-- WAS: splitting on eq produces
-- f .(_ Set) refl = {!!}
-- with unsolved metavariables
-- SHOULD: instead produce
-- f .(F Set) refl = {!!}
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{-# OPTIONS --safe #-}
module CF.Syntax.DeBruijn where
open import Level
open import Data.Bool
open import Data.Product
open import Data.Integer
open import Data.List hiding (null)
open import Data.List.Relation.Unary.All
open import Relation.Unary.PredicateTransformer using (Pt)
open import Relation.Unary hiding (_⊢_)
open import Relation.Binary.Structures using (IsPreorder)
open import Relation.Binary.PropositionalEquality using (isEquivalence)
open import CF.Types
open import CF.Contexts.Lexical using (Ctx; module DeBruijn; Closed) public
open import CF.Syntax using (BinOp; module BinOp) public
open DeBruijn public
open BinOp public
mutual
data Exp : Ty → Pred Ctx 0ℓ where
unit : ∀[ Exp void ]
num : ℤ → ∀[ Exp int ]
bool : Bool → ∀[ Exp bool ]
ifthenelse : ∀[ Exp bool ⇒ Exp a ⇒ Exp a ⇒ Exp a ]
var' : ∀[ Var a ⇒ Exp a ]
bop : BinOp a b c → ∀[ Exp a ⇒ Exp b ⇒ Exp c ]
Exps = λ as Γ → All (λ a → Exp a Γ) as
mutual
data Stmt (r : Ty) : Pred Ctx 0ℓ where
asgn : ∀[ Var a ⇒ Exp a ⇒ Stmt r ]
run : ∀[ Exp a ⇒ Stmt r ]
ifthenelse : ∀[ Exp bool ⇒ Stmt r ⇒ Stmt r ⇒ Stmt r ]
while : ∀[ Exp bool ⇒ Stmt r ⇒ Stmt r ]
block : ∀[ Block r ⇒ Stmt r ]
data Block (r : Ty) : Pred Ctx 0ℓ where
_⍮⍮_ : ∀[ Stmt r ⇒ Block r ⇒ Block r ]
nil : ∀[ Block r ]
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Vectors
------------------------------------------------------------------------
-- This implementation is designed for reasoning about dynamic
-- vectors which may increase or decrease in size.
-- See `Data.Vec.Functional` for an alternative implementation as
-- functions from finite sets, which is more suitable for reasoning
-- about fixed sized vectors and for when ease of retrevial is
-- important.
{-# OPTIONS --without-K --safe #-}
module Data.Vec where
open import Level
open import Data.Bool.Base
import Data.Nat.Properties as ℕₚ
open import Data.Vec.Bounded.Base as Vec≤
using (Vec≤; ≤-cast; fromVec)
open import Relation.Nullary
open import Relation.Unary
private
variable
a p : Level
A : Set a
------------------------------------------------------------------------
-- Publicly re-export the contents of the base module
open import Data.Vec.Base public
------------------------------------------------------------------------
-- Additional operations
module _ {P : A → Set p} (P? : Decidable P) where
filter : ∀ {n} → Vec A n → Vec≤ A n
filter [] = Vec≤.[]
filter (a ∷ as) with does (P? a)
... | true = a Vec≤.∷ filter as
... | false = ≤-cast (ℕₚ.n≤1+n _) (filter as)
takeWhile : ∀ {n} → Vec A n → Vec≤ A n
takeWhile [] = Vec≤.[]
takeWhile (a ∷ as) with does (P? a)
... | true = a Vec≤.∷ takeWhile as
... | false = Vec≤.[]
dropWhile : ∀ {n} → Vec A n → Vec≤ A n
dropWhile Vec.[] = Vec≤.[]
dropWhile (a Vec.∷ as) with does (P? a)
... | true = ≤-cast (ℕₚ.n≤1+n _) (dropWhile as)
... | false = fromVec (a Vec.∷ as)
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open import parse-tree
open import string
module run (ptr : ParseTreeRec) where
open import lib
open import datatypes
open ParseTreeRec ptr
module deriv where
data RunElement : 𝕃 char → Set where
Id : string → RunElement []
InputChar : (c : char) → RunElement (c :: [])
ParseTree : {l : 𝕃 char}{s : string}{pt : ParseTreeT} → isParseTree pt l s → RunElement l
infixr 6 _::'_
data Run : (ls : 𝕃 char) → Set where
[]' : Run []
_::'_ : {lc elc : 𝕃 char} → RunElement elc → Run lc → Run (elc ++ lc)
length-run : {lc : 𝕃 char} → Run lc → ℕ
length-run []' = 0
length-run (x ::' xs) = suc (length-run xs)
RunElement-to-string : {lc : 𝕃 char} → RunElement lc → string
RunElement-to-string (Id s) = ("id:" ^ s)
RunElement-to-string (InputChar c) = "#" ^ (char-to-string c)
RunElement-to-string (ParseTree{pt = pt} ipt) = (ParseTreeToString pt)
Run-to-string : {lc : 𝕃 char} → Run lc → string
Run-to-string []' = "\n"
Run-to-string (e ::' r) = (RunElement-to-string e) ^ " " ^ (Run-to-string r)
assocRun : (ls : 𝕃 (𝕃 char))(lc : 𝕃 char) → Run ((concat ls) ++ lc) × ℕ → Run (foldr _++_ lc ls) × ℕ
assocRun ls lc (r , n) rewrite concat-foldr ls lc = r , n
record rewriteRules : Set where
field
len-dec-rewrite : {lc : 𝕃 char} → (r : Run lc) → maybe (Run lc × ℕ) --(λ r' → length-run r' < length-run r ≡ tt))
module noderiv where
data RunElement : Set where
Id : string → RunElement
InputChar : (c : char) → RunElement
ParseTree : ParseTreeT → RunElement
Posinfo : ℕ → RunElement
Run : Set
Run = 𝕃 RunElement
_::'_ : RunElement → Run → Run
_::'_ = _::_
[]' : Run
[]' = []
length-run : Run → ℕ
length-run = length
RunElement-to-string : RunElement → string
RunElement-to-string (Id s) = ("id:" ^ s)
RunElement-to-string (InputChar c) = "#" ^ (char-to-string c)
RunElement-to-string (ParseTree pt) = (ParseTreeToString pt)
RunElement-to-string (Posinfo n) = "pos:" ^ ℕ-to-string n
Run-to-string : Run → string
Run-to-string [] = "\n"
Run-to-string (e :: r) = (RunElement-to-string e) ^ " " ^ (Run-to-string r)
record rewriteRules : Set where
field
len-dec-rewrite : Run → maybe (Run × ℕ)
empty-string : string
empty-string = ""
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------------------------------------------------------------------------------
-- Miscellaneous properties
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module FOTC.Program.SortList.Properties.MiscellaneousATP where
open import FOTC.Base
open import FOTC.Base.List
open import FOTC.Data.Bool
open import FOTC.Data.Bool.PropertiesATP
open import FOTC.Data.Nat.Inequalities
open import FOTC.Data.Nat.List.PropertiesATP
open import FOTC.Data.Nat.List.Type
open import FOTC.Data.Nat.Type
open import FOTC.Data.List
open import FOTC.Data.List.PropertiesATP
open import FOTC.Program.SortList.Properties.Totality.BoolATP
open import FOTC.Program.SortList.SortList
------------------------------------------------------------------------------
-- This is a weird result but recall that "the relation ≤ between
-- lists is only an ordering if nil is excluded" (Burstall 1969,
-- p. 46).
-- xs≤[] : ∀ {is} → ListN is → OrdList is → ≤-Lists is []
-- xs≤[] nilLN _ = ≤-Lists-[] []
-- xs≤[] (lncons {i} {is} Ni LNis) LOconsL =
-- prf $ xs≤[] LNis (subList-OrdList Ni LNis LOconsL)
-- where
-- postulate prf : ≤-Lists is [] → --IH.
-- ≤-Lists (i ∷ is) []
-- {-# ATP prove prf ≤-ItemList-Bool ordList-Bool x&&y≡true→x≡true #-}
x≤ys→x≤zs→x≤ys++zs : ∀ {i js ks} → N i → ListN js → ListN ks →
≤-ItemList i js →
≤-ItemList i ks →
≤-ItemList i (js ++ ks)
x≤ys→x≤zs→x≤ys++zs {i} {ks = ks} Ni lnnil LNks _ i≤k =
subst (≤-ItemList i) (sym (++-leftIdentity ks)) i≤k
x≤ys→x≤zs→x≤ys++zs {i} {ks = ks} Ni (lncons {j} {js} Nj LNjs) LNks i≤j∷js i≤k =
prf (x≤ys→x≤zs→x≤ys++zs Ni LNjs LNks (&&-list₂-t₂ helper₁ helper₂ helper₃) i≤k)
where
helper₁ : Bool (le i j)
helper₁ = le-Bool Ni Nj
helper₂ : Bool (le-ItemList i js)
helper₂ = le-ItemList-Bool Ni LNjs
helper₃ : le i j && le-ItemList i js ≡ true
helper₃ = trans (sym (le-ItemList-∷ i j js)) i≤j∷js
postulate prf : ≤-ItemList i (js ++ ks) → ≤-ItemList i ((j ∷ js) ++ ks)
{-# ATP prove prf &&-list₂-t helper₁ helper₂ helper₃ #-}
xs≤ys→xs≤zs→xs≤ys++zs : ∀ {is js ks} → ListN is → ListN js → ListN ks →
≤-Lists is js →
≤-Lists is ks →
≤-Lists is (js ++ ks)
xs≤ys→xs≤zs→xs≤ys++zs lnnil LNjs LNks _ _ = le-Lists-[] _
xs≤ys→xs≤zs→xs≤ys++zs {js = js} {ks} (lncons {i} {is} Ni LNis)
LNjs LNks i∷is≤js i∷is≤ks =
prf ((xs≤ys→xs≤zs→xs≤ys++zs LNis LNjs LNks
(&&-list₂-t₂ helper₁ helper₂ helper₃)
(&&-list₂-t₂ helper₄ helper₅ helper₆)))
where
helper₁ = le-ItemList-Bool Ni LNjs
helper₂ = le-Lists-Bool LNis LNjs
helper₃ = trans (sym (le-Lists-∷ i is js)) i∷is≤js
helper₄ = le-ItemList-Bool Ni LNks
helper₅ = le-Lists-Bool LNis LNks
helper₆ = trans (sym (le-Lists-∷ i is ks)) i∷is≤ks
postulate prf : ≤-Lists is (js ++ ks) → ≤-Lists (i ∷ is) (js ++ ks)
{-# ATP prove prf x≤ys→x≤zs→x≤ys++zs &&-list₂-t helper₁ helper₂ helper₃ helper₄ helper₅ helper₆ #-}
xs≤zs→ys≤zs→xs++ys≤zs : ∀ {is js ks} → ListN is → ListN js → ListN ks →
≤-Lists is ks →
≤-Lists js ks →
≤-Lists (is ++ js) ks
xs≤zs→ys≤zs→xs++ys≤zs {js = js} {ks} lnnil LNjs LNks is≤ks js≤ks =
subst (λ t → ≤-Lists t ks)
(sym (++-leftIdentity js))
js≤ks
xs≤zs→ys≤zs→xs++ys≤zs {js = js} {ks}
(lncons {i} {is} Ni LNis) LNjs LNks i∷is≤ks js≤ks =
prf (xs≤zs→ys≤zs→xs++ys≤zs LNis LNjs LNks
(&&-list₂-t₂ helper₁ helper₂ helper₃)
js≤ks)
where
helper₁ = le-ItemList-Bool Ni LNks
helper₂ = le-Lists-Bool LNis LNks
helper₃ = trans (sym (le-Lists-∷ i is ks)) i∷is≤ks
postulate prf : ≤-Lists (is ++ js) ks → ≤-Lists ((i ∷ is) ++ js) ks
{-# ATP prove prf &&-list₂-t helper₁ helper₂ helper₃ #-}
------------------------------------------------------------------------------
-- References
--
-- Burstall, R. M. (1969). Proving properties of programs by
-- structural induction. The Computer Journal 12.1, pp. 41–48.
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------------------------------------------------------------------------
-- Examples/exercises related to CCS from "Enhancements of the
-- bisimulation proof method" by Pous and Sangiorgi
--
-- Implemented using coinductive definitions of strong and weak
-- bisimilarity and expansion.
------------------------------------------------------------------------
{-# OPTIONS --sized-types #-}
open import Prelude hiding (module W; step-→)
module Bisimilarity.Weak.CCS.Examples {ℓ} {Name : Type ℓ} where
open import Equality.Propositional
open import Function-universe equality-with-J hiding (id; _∘_)
open import Bisimilarity.CCS
import Bisimilarity.Equational-reasoning-instances
import Bisimilarity.Weak.CCS as WL
import Bisimilarity.Weak.Equational-reasoning-instances
open import Equational-reasoning
import Expansion.CCS as EL
import Expansion.Equational-reasoning-instances
open import Labelled-transition-system.CCS Name
open import Bisimilarity CCS using (_∼_; ∼:_)
open import Bisimilarity.Weak CCS as W
open import Expansion CCS as E
import Labelled-transition-system.Equational-reasoning-instances CCS
as Dummy
mutual
-- Example 6.5.4.
6-5-4 :
∀ {i a b} →
[ i ] ! name a ∙ (b ∙) ∣ ! co a ∙ ≈ (! a ∙ ∣ ! b ∙) ∣ ! co a ∙
6-5-4 {i} {a} {b} = W.⟨ lr , rl ⟩
where
lemma =
(! a ∙ ∣ ! b ∙) ∣ b ∙ ∼⟨ symmetric ∣-assoc ⟩
! a ∙ ∣ (! b ∙ ∣ b ∙) ∼⟨ symmetric ∣-right-identity ∣-cong 6-1-2 ⟩■
(! a ∙ ∣ ∅) ∣ ! b ∙
lr :
∀ {P μ} →
! name a ∙ (b ∙) ∣ ! co a ∙ [ μ ]⟶ P →
∃ λ Q → (! a ∙ ∣ ! b ∙) ∣ ! co a ∙ [ μ ]⇒̂ Q × [ i ] P ≈′ Q
lr (par-left {P′ = P} tr) = case 6-1-3-2 tr of λ where
(inj₁ (.(b ∙) , action , P∼!ab∣b)) →
P ∣ ! co a ∙ ∼⟨ P∼!ab∣b ∣-cong reflexive ⟩
(! name a ∙ (b ∙) ∣ b ∙) ∣ ! co a ∙ ∼⟨ swap-rightmost ⟩
(! name a ∙ (b ∙) ∣ ! co a ∙) ∣ b ∙ ∼′⟨ 6-5-4′ WL.∣-cong′ reflexive ⟩
((! a ∙ ∣ ! b ∙) ∣ ! co a ∙) ∣ b ∙ ∼⟨ swap-rightmost ⟩ ∼:
((! a ∙ ∣ ! b ∙) ∣ b ∙) ∣ ! co a ∙ ∼⟨ lemma ∣-cong reflexive ⟩■
((! a ∙ ∣ ∅) ∣ ! b ∙) ∣ ! co a ∙
W.⇐̂[ name a ] ←⟨ ⟶: par-left (par-left (replication (par-right action))) ⟩■
(! a ∙ ∣ ! b ∙) ∣ ! co a ∙
(inj₂ (_ , .(b ∙) , _ , .a , action , ab[a̅]⟶ , _)) →
⊥-elim (names-are-not-inverted ab[a̅]⟶)
lr (par-right {Q′ = Q} tr) = case 6-1-3-2 tr of λ where
(inj₁ (.∅ , action , Q∼!a̅∣∅)) →
! name a ∙ (b ∙) ∣ Q ∼⟨ reflexive ∣-cong Q∼!a̅∣∅ ⟩
! name a ∙ (b ∙) ∣ (! co a ∙ ∣ ∅) ∼⟨ ∣-assoc ⟩
(! name a ∙ (b ∙) ∣ ! co a ∙) ∣ ∅ ∼′⟨ 6-5-4′ WL.∣-cong′ reflexive ⟩ ∼:
((! a ∙ ∣ ! b ∙) ∣ ! co a ∙) ∣ ∅ ∼⟨ symmetric ∣-assoc ⟩■
(! a ∙ ∣ ! b ∙) ∣ (! co a ∙ ∣ ∅)
W.⇐̂[ name (co a) ] ←⟨ ⟶: par-right (replication (par-right action)) ⟩■
(! a ∙ ∣ ! b ∙) ∣ ! co a ∙
(inj₂ (_ , .∅ , _ , .(co a) , action , a̅[a̅̅]⟶ , _)) →
⊥-elim (names-are-not-inverted a̅[a̅̅]⟶)
lr (par-τ {P′ = P} {Q′ = Q} tr₁ tr₂) =
case 6-1-3-2 tr₁ ,′ 6-1-3-2 tr₂ of λ where
(inj₁ (.(b ∙) , action , P∼!ab∣b) ,
inj₁ (R , tr , Q∼!a̅∣R)) →
P ∣ Q ∼⟨ P∼!ab∣b ∣-cong Q∼!a̅∣R ⟩
(! name a ∙ (b ∙) ∣ b ∙) ∣ (! co a ∙ ∣ R) ∼≡⟨ cong (λ R → _ ∣ (_ ∣ R)) (·-only⟶ tr) ⟩
(! name a ∙ (b ∙) ∣ b ∙) ∣ (! co a ∙ ∣ ∅) ∼⟨ swap-in-the-middle ⟩
(! name a ∙ (b ∙) ∣ ! co a ∙) ∣ (b ∙ ∣ ∅) ∼′⟨ 6-5-4′ WL.∣-cong′ reflexive ⟩
((! a ∙ ∣ ! b ∙) ∣ ! co a ∙) ∣ (b ∙ ∣ ∅) ∼⟨ swap-in-the-middle ⟩ ∼:
((! a ∙ ∣ ! b ∙) ∣ b ∙) ∣ (! co a ∙ ∣ ∅) ∼⟨ lemma ∣-cong reflexive ⟩■
((! a ∙ ∣ ∅) ∣ ! b ∙) ∣ (! co a ∙ ∣ ∅)
W.⇐̂[ τ ] ←⟨ par-τ (par-left (replication (par-right action)))
(replication (par-right action)) ⟩■
(! a ∙ ∣ ! b ∙) ∣ ! co a ∙
(_ , inj₂ (() , _))
(inj₂ (() , _) , _)
rl-lemma :
∀ {Q μ} →
(! a ∙ ∣ ! b ∙) ∣ ! co a ∙ [ μ ]⟶ Q →
(! a ∙ ∣ ! b ∙) ∣ ! co a ∙ ∼ Q
×
(μ ≡ name a ⊎ μ ≡ name b ⊎ μ ≡ name (co a) ⊎ μ ≡ τ)
rl-lemma (par-left (par-left {P′ = P} tr)) =
case 6-1-3-2 tr of λ where
(inj₁ (.∅ , action , P∼!a∣∅)) →
((! a ∙ ∣ ! b ∙) ∣ ! co a ∙ ∼⟨ (symmetric ∣-right-identity ∣-cong reflexive) ∣-cong reflexive ⟩
((! a ∙ ∣ ∅) ∣ ! b ∙) ∣ ! co a ∙ ∼⟨ (symmetric P∼!a∣∅ ∣-cong reflexive) ∣-cong reflexive ⟩■
(P ∣ ! b ∙) ∣ ! co a ∙)
, inj₁ refl
(inj₂ (refl , .∅ , _ , .a , action , a[a̅]⟶ , _)) →
⊥-elim (names-are-not-inverted a[a̅]⟶)
rl-lemma (par-left (par-right {Q′ = P} tr)) =
case 6-1-3-2 tr of λ where
(inj₁ (.∅ , action , P∼!b∣∅)) →
((! a ∙ ∣ ! b ∙) ∣ ! co a ∙ ∼⟨ (reflexive ∣-cong symmetric ∣-right-identity) ∣-cong reflexive ⟩
(! a ∙ ∣ (! b ∙ ∣ ∅)) ∣ ! co a ∙ ∼⟨ (reflexive ∣-cong symmetric P∼!b∣∅) ∣-cong reflexive ⟩■
(! a ∙ ∣ P) ∣ ! co a ∙)
, inj₂ (inj₁ refl)
(inj₂ (_ , .∅ , _ , .b , action , b[b̅]⟶ , _)) →
⊥-elim (names-are-not-inverted b[b̅]⟶)
rl-lemma (par-left (par-τ {P′ = P} {Q′ = Q} tr₁ tr₂)) =
case 6-1-3-2 tr₁ ,′ 6-1-3-2 tr₂ of λ where
(inj₁ (.∅ , action , P∼!a∣∅) ,
inj₁ (.∅ , action , Q∼!b∣∅)) →
((! a ∙ ∣ ! b ∙) ∣ ! co a ∙ ∼⟨ symmetric (∣-right-identity ∣-cong ∣-right-identity) ∣-cong reflexive ⟩
((! a ∙ ∣ ∅) ∣ (! b ∙ ∣ ∅)) ∣ ! co a ∙ ∼⟨ symmetric (P∼!a∣∅ ∣-cong Q∼!b∣∅) ∣-cong reflexive ⟩■
(P ∣ Q) ∣ ! co a ∙)
, inj₂ (inj₂ (inj₂ refl))
(inj₂ (() , _) , _)
(_ , inj₂ (() , _))
rl-lemma (par-right {Q′ = Q} tr) =
case 6-1-3-2 tr of λ where
(inj₁ (.∅ , action , Q∼!a̅∣∅)) →
((! a ∙ ∣ ! b ∙) ∣ ! co a ∙ ∼⟨ reflexive ∣-cong symmetric ∣-right-identity ⟩
(! a ∙ ∣ ! b ∙) ∣ (! co a ∙ ∣ ∅) ∼⟨ reflexive ∣-cong symmetric Q∼!a̅∣∅ ⟩■
(! a ∙ ∣ ! b ∙) ∣ Q)
, inj₂ (inj₂ (inj₁ refl))
(inj₂ (_ , .∅ , _ , .(co a) , action , a̅[a̅̅]⟶ , _)) →
⊥-elim (names-are-not-inverted a̅[a̅̅]⟶)
rl-lemma (par-τ {Q′ = Q} (par-left {P′ = P} tr₁) tr₂) =
case 6-1-3-2 tr₁ ,′ 6-1-3-2 tr₂ of λ where
(inj₁ (.∅ , action , P∼!a∣∅) ,
inj₁ (R , tr , Q∼!a̅∣R)) →
((! a ∙ ∣ ! b ∙) ∣ ! co a ∙ ∼⟨ symmetric ((∣-right-identity ∣-cong reflexive) ∣-cong ∣-right-identity) ⟩
((! a ∙ ∣ ∅) ∣ ! b ∙) ∣ (! co a ∙ ∣ ∅) ∼≡⟨ cong (λ R → _ ∣ (_ ∣ R)) (sym $ ·-only⟶ tr) ⟩
((! a ∙ ∣ ∅) ∣ ! b ∙) ∣ (! co a ∙ ∣ R) ∼⟨ symmetric ((P∼!a∣∅ ∣-cong reflexive) ∣-cong Q∼!a̅∣R) ⟩■
(P ∣ ! b ∙) ∣ Q)
, inj₂ (inj₂ (inj₂ refl))
(inj₂ (() , _) , _)
(_ , inj₂ (() , _))
rl-lemma (par-τ {Q′ = Q} (par-right {Q′ = P} tr₁) tr₂) =
case 6-1-3-2 tr₁ ,′ 6-1-3-2 tr₂ of λ where
(inj₁ (.∅ , action , P∼!b∣∅) ,
inj₁ (R , tr , Q∼!a̅∣R)) →
((! a ∙ ∣ ! b ∙) ∣ ! co a ∙ ∼⟨ symmetric ((reflexive ∣-cong ∣-right-identity) ∣-cong ∣-right-identity) ⟩
(! a ∙ ∣ (! b ∙ ∣ ∅)) ∣ (! co a ∙ ∣ ∅) ∼≡⟨ cong (λ R → _ ∣ (_ ∣ R)) (sym $ ·-only⟶ tr) ⟩
(! a ∙ ∣ (! b ∙ ∣ ∅)) ∣ (! co a ∙ ∣ R) ∼⟨ symmetric ((reflexive ∣-cong P∼!b∣∅) ∣-cong Q∼!a̅∣R) ⟩■
(! a ∙ ∣ P) ∣ Q)
, inj₂ (inj₂ (inj₂ refl))
(inj₂ (() , _) , _)
(_ , inj₂ (() , _))
rl :
∀ {Q μ} →
(! a ∙ ∣ ! b ∙) ∣ ! co a ∙ [ μ ]⟶ Q →
∃ λ P → ! name a ∙ (b ∙) ∣ ! co a ∙ [ μ ]⇒̂ P × [ i ] P ≈′ Q
rl {Q} tr = case rl-lemma tr of λ where
(!a∣!b∣!a̅∼Q , inj₁ refl) →
! name a ∙ (b ∙) ∣ ! co a ∙ →⟨ ⟶: par-left (replication (par-right action)) ⟩■
W.⇒̂[ name a ]
(! name a ∙ (b ∙) ∣ b ∙) ∣ ! co a ∙ ∼⟨ swap-rightmost ⟩
(! name a ∙ (b ∙) ∣ ! co a ∙) ∣ b ∙ ∼′⟨ 6-5-4′ WL.∣-cong′ reflexive ⟩
((! a ∙ ∣ ! b ∙) ∣ ! co a ∙) ∣ b ∙ ∼⟨ swap-rightmost ⟩
((! a ∙ ∣ ! b ∙) ∣ b ∙) ∣ ! co a ∙ ∼⟨ symmetric ∣-assoc ∣-cong reflexive ⟩
(! a ∙ ∣ (! b ∙ ∣ b ∙)) ∣ ! co a ∙ ∼⟨ (reflexive ∣-cong 6-1-2) ∣-cong reflexive ⟩ ∼:
(! a ∙ ∣ ! b ∙) ∣ ! co a ∙ ∼⟨ !a∣!b∣!a̅∼Q ⟩■
Q
(!a∣!b∣!a̅∼Q , inj₂ (inj₁ refl)) →
! name a ∙ (b ∙) ∣ ! co a ∙ →⟨ par-τ (replication (par-right action))
(replication (par-right action)) ⟩
(! name a ∙ (b ∙) ∣ (b ∙)) ∣ (! co a ∙ ∣ ∅) →⟨ ⟶: par-left (par-right action) ⟩■
W.⇒̂[ name b ]
(! name a ∙ (b ∙) ∣ ∅) ∣ (! co a ∙ ∣ ∅) ∼⟨ ∣-right-identity ∣-cong ∣-right-identity ⟩
(! name a ∙ (b ∙)) ∣ ! co a ∙ ∼′⟨ 6-5-4′ ⟩ ∼:
(! a ∙ ∣ ! b ∙) ∣ ! co a ∙ ∼⟨ !a∣!b∣!a̅∼Q ⟩■
Q
(!a∣!b∣!a̅∼Q , inj₂ (inj₂ (inj₁ refl))) →
! name a ∙ (b ∙) ∣ ! co a ∙ →⟨ ⟶: par-right (replication (par-right action)) ⟩■
W.⇒̂[ name (co a) ]
! name a ∙ (b ∙) ∣ (! co a ∙ ∣ ∅) ∼⟨ reflexive ∣-cong ∣-right-identity ⟩
! name a ∙ (b ∙) ∣ ! co a ∙ ∼′⟨ 6-5-4′ ⟩ ∼:
(! a ∙ ∣ ! b ∙) ∣ ! co a ∙ ∼⟨ !a∣!b∣!a̅∼Q ⟩■
Q
(!a∣!b∣!a̅∼Q , inj₂ (inj₂ (inj₂ refl))) →
! name a ∙ (b ∙) ∣ ! co a ∙ ■
W.⇒̂[ τ ]
! name a ∙ (b ∙) ∣ ! co a ∙ ∼′⟨ 6-5-4′ ⟩ ∼:
(! a ∙ ∣ ! b ∙) ∣ ! co a ∙ ∼⟨ !a∣!b∣!a̅∼Q ⟩■
Q
6-5-4′ :
∀ {i a b} →
[ i ] ! name a ∙ (b ∙) ∣ ! co a ∙ ≈′ (! a ∙ ∣ ! b ∙) ∣ ! co a ∙
force 6-5-4′ = 6-5-4
-- The first part of Exercise 6.5.8.
6-5-8-1 :
∀ a {P Q} →
⟨ν proj₁ a ⟩ (name a ∙ P ∣ name (co a) ∙ Q) ≳ ⟨ν proj₁ a ⟩ (P ∣ Q)
6-5-8-1 a {P} {Q} = E.⟨ lr , rl ⟩
where
lr :
∀ {μ R} →
⟨ν proj₁ a ⟩ (name a ∙ P ∣ name (co a) ∙ Q) [ μ ]⟶ R →
∃ λ R′ → ⟨ν proj₁ a ⟩ (P ∣ Q) [ μ ]⟶̂ R′ × R ≳′ R′
lr (restriction a∉μ (par-left action)) = ⊥-elim (a∉μ refl)
lr (restriction a∉μ (par-right action)) = ⊥-elim (a∉μ refl)
lr (restriction a∉μ (par-τ {Q′ = R} action tr)) =
⟨ν proj₁ a ⟩ (P ∣ R) ∼≡⟨ cong (λ R → ⟨ν _ ⟩ (_ ∣ R)) (·-only⟶ tr) ⟩■
⟨ν proj₁ a ⟩ (P ∣ Q)
⟵̂[ τ ]
⟨ν proj₁ a ⟩ (P ∣ Q) ■
rl :
∀ {μ R′} →
⟨ν proj₁ a ⟩ (P ∣ Q) [ μ ]⟶ R′ →
∃ λ R → ⟨ν proj₁ a ⟩ (name a ∙ P ∣ name (co a) ∙ Q) [ μ ]⇒ R ×
R ≳′ R′
rl {μ} (restriction {P′ = R} a∉μ P∣Q⟶R) =
⟨ν proj₁ a ⟩ (name a ∙ P ∣ name (co a) ∙ Q) →⟨ ⟶: restriction _ (par-τ action action) ⟩
⟨ν proj₁ a ⟩ (P ∣ Q) →⟨ ⟶: restriction a∉μ P∣Q⟶R ⟩■
E.⇒[ μ ]
⟨ν proj₁ a ⟩ R ■
-- One interpretation of the second part of Exercise 6.5.8 is
-- contradictory, assuming that Name is inhabited.
¬-6-5-8-2 :
Name →
¬ (∀ {a b P} →
! ⟨ν proj₁ a ⟩ (name b ∙ (a ∙) ∣ name (co a) ∙ P) ≈
! name b ∙ ⟨ν proj₁ a ⟩ P)
¬-6-5-8-2 x =
(∀ {a b P} →
! ⟨ν proj₁ a ⟩ (name b ∙ (a ∙) ∣ name (co a) ∙ P) ≈
! name b ∙ ⟨ν proj₁ a ⟩ P) ↝⟨ (λ hyp → hyp {a = a} {b = co a}) ⟩
(! ⟨ν proj₁ a ⟩ (name (co a) ∙ (a ∙) ∣ co a ∙) ≈
! name (co a) ∙ ⟨ν proj₁ a ⟩ ∅) ↝⟨ (λ hyp → Σ-map id proj₁ $ W.right-to-left hyp
(replication (par-right action))) ⟩
∃ (! ⟨ν proj₁ a ⟩ (name (co a) ∙ (a ∙) ∣ co a ∙) [ name (co a) ]⇒̂_) ↝⟨ !P⇒̂ ∘ proj₂ ⟩□
⊥ □
where
a = x , true
P = ⟨ν proj₁ a ⟩ (name (co a) ∙ (a ∙) ∣ co a ∙)
P⟶ : ∀ {μ Q} → ¬ P [ μ ]⟶ Q
P⟶ (restriction x≢x (par-left action)) = ⊥-elim (x≢x refl)
P⟶ (restriction x≢x (par-right action)) = ⊥-elim (x≢x refl)
P⟶ (restriction _ (par-τ action tr)) = names-are-not-inverted tr
!P⟶ : ∀ {μ Q} → ¬ ! P [ μ ]⟶ Q
!P⟶ (replication (par-left tr)) = !P⟶ tr
!P⟶ (replication (par-right tr)) = P⟶ tr
!P⟶ (replication (par-τ _ tr)) = P⟶ tr
!P⇒ : ∀ {Q} → ! P ⇒ Q → Q ≡ ! P
!P⇒ done = refl
!P⇒ (step _ tr _) = ⊥-elim (!P⟶ tr)
!P[]⇒ : ∀ {μ Q} → ¬ ! P [ μ ]⇒ Q
!P[]⇒ (steps trs tr _) rewrite !P⇒ trs = !P⟶ tr
!P⇒̂ : ∀ {b Q} → ¬ ! P [ name b ]⇒̂ Q
!P⇒̂ (silent () _)
!P⇒̂ (non-silent _ tr) = !P[]⇒ tr
-- Another interpretation of the second part of Exercise 6.5.8 can be
-- proved.
6-5-8-2 :
∀ {i a b P} →
proj₁ a ≢ proj₁ b →
[ i ] ! ⟨ν proj₁ a ⟩ (name b ∙ (a ∙) ∣ name (co a) ∙ P) ≈
! name b ∙ ⟨ν proj₁ a ⟩ P
6-5-8-2 {i} {a} {b} {P} a≢b = W.⟨ lr , rl ⟩
where
6-5-8-2′ :
[ i ] ! ⟨ν proj₁ a ⟩ (name b ∙ (a ∙) ∣ name (co a) ∙ P) ≈′
! name b ∙ ⟨ν proj₁ a ⟩ P
force 6-5-8-2′ = 6-5-8-2 a≢b
lr :
∀ {Q μ} →
! ⟨ν proj₁ a ⟩ (name b ∙ (a ∙) ∣ name (co a) ∙ P) [ μ ]⟶ Q →
∃ λ Q′ → ! name b ∙ ⟨ν proj₁ a ⟩ P [ μ ]⇒̂ Q′ × [ i ] Q ≈′ Q′
lr {Q} tr = case 6-1-3-2 tr of λ where
(inj₁ (_ , restriction a≢b (par-left action)
, Q∼!νa[ba∣a̅P]∣νa[a∣a̅P])) →
Q ∼⟨ Q∼!νa[ba∣a̅P]∣νa[a∣a̅P] ⟩
! ⟨ν proj₁ a ⟩ (name b ∙ (a ∙) ∣ name (co a) ∙ P) ∣
⟨ν proj₁ a ⟩ (a ∙ ∣ name (co a) ∙ P) ∼⟨ (_ ■) EL.∣-cong 6-5-8-1 _ ⟩
! ⟨ν proj₁ a ⟩ (name b ∙ (a ∙) ∣ name (co a) ∙ P) ∣
⟨ν proj₁ a ⟩ (∅ ∣ P) ∼⟨ 6-5-8-2′ WL.∣-cong′ convert {a = ℓ} (⟨ν refl ⟩-cong ∣-left-identity) ⟩■
! name b ∙ ⟨ν proj₁ a ⟩ P ∣ ⟨ν proj₁ a ⟩ P
W.⇐̂[ name b ] ←⟨ ⟶: replication (par-right action) ⟩■
! name b ∙ ⟨ν proj₁ a ⟩ P
(inj₁ (_ , restriction _ (par-τ action tr) , _)) →
⊥-elim ( $⟨ tr ⟩
name (co a) · _ [ name (co b) ]⟶ _ ↝⟨ ·-only ⟩
name (co a) ≡ name (co b) ↝⟨ cancel-name ⟩
co a ≡ co b ↝⟨ cong proj₁ ⟩
proj₁ a ≡ proj₁ b ↝⟨ a≢b ⟩□
⊥ □)
(inj₁ (_ , restriction a≢a (par-right action) , _)) →
⊥-elim (a≢a refl)
(inj₂ (refl , _ , _ , _
, restriction _ (par-left action)
, restriction _ (par-left tr)
, _)) →
⊥-elim (names-are-not-inverted tr)
(inj₂ (refl , _ , _ , _
, restriction _ (par-right action)
, restriction _ (par-right tr)
, _)) →
⊥-elim (names-are-not-inverted tr)
(inj₂ (refl , _ , _ , _
, restriction a≢b (par-left action)
, restriction _ (par-right tr)
, _)) →
$⟨ tr ⟩
name (co a) ∙ P [ name (co b) ]⟶ _ ↝⟨ cong proj₁ ∘ cancel-name ∘ ·-only ⟩
proj₁ a ≡ proj₁ b ↝⟨ a≢b ⟩
⊥ ↝⟨ ⊥-elim ⟩□
_ □
(inj₂ (refl , _ , _ , c
, restriction a≢c (par-right tr)
, restriction _ (par-left action)
, _)) →
$⟨ tr ⟩
name (co a) ∙ P [ name c ]⟶ _ ↝⟨ cong proj₁ ∘ cancel-name ∘ ·-only ⟩
proj₁ a ≡ proj₁ c ↝⟨ a≢c ⟩
⊥ ↝⟨ ⊥-elim ⟩□
_ □
rl :
∀ {Q′ μ} →
! name b ∙ ⟨ν proj₁ a ⟩ P [ μ ]⟶ Q′ →
∃ λ Q → ! ⟨ν proj₁ a ⟩ (name b ∙ (a ∙) ∣ name (co a) ∙ P) [ μ ]⇒̂ Q ×
[ i ] Q ≈′ Q′
rl {Q′} tr = case 6-1-3-2 tr of λ where
(inj₁ (_ , action , Q′∼!bνaP∣νaP)) →
! ⟨ν proj₁ a ⟩ (name b ∙ (a ∙) ∣ name (co a) ∙ P) →⟨ ⟶: replication (par-right (restriction a≢b (par-left action))) ⟩■
W.⇒̂[ name b ]
! ⟨ν proj₁ a ⟩ (name b ∙ (a ∙) ∣ name (co a) ∙ P) ∣
⟨ν proj₁ a ⟩ (a ∙ ∣ name (co a) ∙ P) ∼′⟨ 6-5-8-2′ WL.∣-cong′ convert {a = ℓ} (6-5-8-1 a) ⟩
! name b ∙ ⟨ν proj₁ a ⟩ P ∣ ⟨ν proj₁ a ⟩ (∅ ∣ P) ∼⟨ (_ ■) ∣-cong ⟨ν refl ⟩-cong ∣-left-identity ⟩ ∼:
! name b ∙ ⟨ν proj₁ a ⟩ P ∣ ⟨ν proj₁ a ⟩ P ∼⟨ symmetric Q′∼!bνaP∣νaP ⟩■
Q′
(inj₂ (refl , _ , _ , _ , action , tr , _)) →
⊥-elim (names-are-not-inverted tr)
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{-# OPTIONS --without-K --rewriting #-}
open import HoTT
open import homotopy.elims.CofPushoutSection
module cohomology.MayerVietoris {i} where
{- Mayer-Vietoris Sequence: Given a span X ←f– Z –g→ Y, the cofiber space
of the natural map [reglue : X ∨ Y → X ⊔_Z Y] (defined below) is equivalent
to the suspension of Z. -}
{- Relevant functions -}
module MayerVietorisFunctions (ps : ⊙Span {i} {i} {i}) where
open ⊙Span ps
module Reglue = ⊙WedgeRec (⊙left ps) (⊙right ps)
reglue : X ∨ Y → de⊙ (⊙Pushout ps)
reglue = Reglue.f
⊙reglue : X ⊙∨ Y ⊙→ ⊙Pushout ps
⊙reglue = Reglue.⊙f
module MVDiff = SuspRec {C = Susp (X ∨ Y)}
north
north
(λ z → merid (winl (fst f z)) ∙ ! (merid (winr (fst g z))))
mv-diff : Susp (de⊙ Z) → Susp (X ∨ Y)
mv-diff = MVDiff.f
⊙mv-diff : ⊙Susp Z ⊙→ ⊙Susp (X ⊙∨ Y)
⊙mv-diff = (mv-diff , idp)
{- We use path induction (via [⊙pushout-J]) to assume that the
basepoint preservation paths of the span maps are [idp]. The module
[MayerVietorisBase] contains the proof of the theorem for this case. -}
module MayerVietorisBase
{A B : Type i} (Z : Ptd i) (f : de⊙ Z → A) (g : de⊙ Z → B) where
X = ⊙[ A , f (pt Z) ]
Y = ⊙[ B , g (pt Z) ]
ps = ⊙span X Y Z (f , idp) (g , idp)
F : Z ⊙→ X
F = (f , idp)
G : Z ⊙→ Y
G = (g , idp)
open MayerVietorisFunctions ps
{- Definition of the maps
into : Cofiber reglue → ΣZ
out : ΣZ → Cofiber reglue
-}
private
into-glue-square :
Square idp idp (ap (extract-glue ∘ reglue) wglue) (merid (pt Z))
into-glue-square =
connection ⊡v∙
! (ap-∘ extract-glue reglue wglue
∙ ap (ap extract-glue) (Reglue.glue-β ∙ !-! (glue (pt Z)))
∙ ExtractGlue.glue-β (pt Z))
module IntoGlue = WedgeElim {P = λ xy → north == extract-glue (reglue xy)}
(λ _ → idp)
(λ _ → merid (pt Z))
(↓-cst=app-from-square into-glue-square)
into-glue = IntoGlue.f
module Into = CofiberRec {f = reglue} north extract-glue into-glue
private
out-glue-and-square : (z : de⊙ Z)
→ Σ (cfbase' reglue == cfbase' reglue)
(λ p → Square (cfglue (winl (f z))) p
(ap cfcod (glue z)) (cfglue (winr (g z))))
out-glue-and-square z = fill-square-top _ _ _
out-glue = fst ∘ out-glue-and-square
out-square = snd ∘ out-glue-and-square
module Out = SuspRec {C = Cofiber reglue}
cfbase
cfbase
out-glue
into = Into.f
out = Out.f
abstract
{- [out] is a right inverse for [into] -}
private
into-out-sq : (z : de⊙ Z) →
Square idp (ap into (ap out (merid z))) (merid z) (merid (pt Z))
into-out-sq z =
(ap (ap into) (Out.merid-β z) ∙v⊡
(! (Into.glue-β (winl (f z)))) ∙h⊡
ap-square into (out-square z)
⊡h∙ (Into.glue-β (winr (g z))))
⊡v∙ (∘-ap into cfcod (glue z) ∙ ExtractGlue.glue-β z)
into-out : ∀ σ → into (out σ) == σ
into-out = Susp-elim
idp
(merid (pt Z))
(λ z → ↓-∘=idf-from-square into out (into-out-sq z))
{- [out] is a left inverse for [into] -}
{- [out] is left inverse on codomain part of cofiber space,
- i.e. [out (into (cfcod γ)) == cfcod γ] -}
private
out-into-cod-square : (z : de⊙ Z) →
Square (cfglue' reglue (winl (f z)))
(ap (out ∘ extract-glue {s = ⊙Span-to-Span ps}) (glue z))
(ap cfcod (glue z)) (cfglue (winr (g z)))
out-into-cod-square z =
(ap-∘ out extract-glue (glue z)
∙ ap (ap out) (ExtractGlue.glue-β z) ∙ Out.merid-β z)
∙v⊡ out-square z
module OutIntoCod = PushoutElim
{d = ⊙Span-to-Span ps} {P = λ γ → out (into (cfcod γ)) == cfcod γ}
(λ x → cfglue (winl x))
(λ y → cfglue (winr y))
(λ z → ↓-='-from-square (out-into-cod-square z))
out-into-cod = OutIntoCod.f
out-into : ∀ κ → out (into κ) == κ
out-into = CofPushoutSection.elim
(λ _ → unit) (λ _ → idp)
idp
out-into-cod
(↓-='-from-square ∘ λ x →
(ap-∘ out into (cfglue (winl x))
∙ ap (ap out) (Into.glue-β (winl x)))
∙v⊡ connection
⊡v∙ ! (ap-idf (glue (winl x))))
(↓-='-from-square ∘ λ y →
(ap-∘ out into (cfglue (winr y))
∙ ap (ap out) (Into.glue-β (winr y))
∙ Out.merid-β (pt Z)
∙ square-top-unique (out-square (pt Z))
(! (ap-cst cfbase wglue) ∙v⊡
natural-square cfglue wglue
⊡v∙ (ap-∘ cfcod reglue wglue
∙ ap (ap cfcod) (Reglue.glue-β ∙ !-! (glue (pt Z))))))
∙v⊡ connection
⊡v∙ ! (ap-idf (glue (winr y))))
{- equivalence and path -}
eq : Cofiber reglue ≃ Susp (de⊙ Z)
eq = equiv into out into-out out-into
⊙eq : ⊙Cofiber ⊙reglue ⊙≃ ⊙Susp Z
⊙eq = ≃-to-⊙≃ eq idp
{- Transporting [cfcod reglue] over the equivalence -}
cfcod-comm-sqr : CommSquare (cfcod' reglue) extract-glue (idf _) into
cfcod-comm-sqr = comm-sqr λ _ → idp
{- Transporting [extract-glue] over the equivalence. -}
ext-comm-sqr : CommSquare extract-glue mv-diff into (idf _)
ext-comm-sqr = comm-sqr $ ! ∘ fn-lemma
where
fn-lemma : ∀ κ → mv-diff (into κ) == extract-glue κ
fn-lemma = CofPushoutSection.elim
(λ _ → unit) (λ _ → idp)
idp
(Pushout-elim
(λ x → merid (winl x))
(λ y → merid (winr y))
(↓-='-from-square ∘ λ z →
(ap-∘ mv-diff extract-glue (glue z)
∙ ap (ap mv-diff) (ExtractGlue.glue-β z)
∙ MVDiff.merid-β z)
∙v⊡ (lt-square (merid (winl (f z)))
⊡h rt-square (merid (winr (g z))))
⊡v∙ ! (ap-cst south (glue z))))
(↓-='-from-square ∘ λ x →
(ap-∘ mv-diff into (cfglue (winl x))
∙ ap (ap mv-diff) (Into.glue-β (winl x)))
∙v⊡ connection
⊡v∙ ! (ExtractGlue.glue-β (winl x)))
(↓-='-from-square ∘ λ y →
(ap-∘ mv-diff into (cfglue (winr y))
∙ ap (ap mv-diff) (Into.glue-β (winr y))
∙ MVDiff.merid-β (pt Z)
∙ ap (λ w → merid w ∙ ! (merid (winr (g (pt Z))))) wglue
∙ !-inv-r (merid (winr (g (pt Z)))))
∙v⊡ connection
⊡v∙ ! (ExtractGlue.glue-β (winr y)))
{- Main results -}
module MayerVietoris (ps : ⊙Span {i} {i} {i}) where
private
record Results (ps : ⊙Span {i} {i} {i}) : Type (lsucc i) where
open ⊙Span ps
open MayerVietorisFunctions ps public
field
⊙eq : ⊙Cofiber ⊙reglue ⊙≃ ⊙Susp Z
cfcod-comm-sqr : CommSquare
(cfcod' reglue) extract-glue
(idf _) (fst (⊙–> ⊙eq))
ext-comm-sqr : CommSquare
extract-glue mv-diff (fst (⊙–> ⊙eq)) (idf _)
results : Results ps
results = ⊙pushout-J Results base-results ps
where
base-results : ∀ {A} {B} Z (f : de⊙ Z → A) (g : de⊙ Z → B) →
Results (⊙span _ _ Z (f , idp) (g , idp))
base-results Z f g = record {
⊙eq = ⊙eq;
cfcod-comm-sqr = cfcod-comm-sqr;
ext-comm-sqr = ext-comm-sqr}
where open MayerVietorisBase Z f g
open Results results public
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{-# OPTIONS --warning=error --safe --without-K #-}
open import LogicalFormulae
open import Lists.Lists
open import Numbers.Naturals.Order
open import Numbers.Naturals.Order.Lemmas
open import Numbers.Naturals.Semiring
open import Numbers.BinaryNaturals.Definition
open import Orders.Partial.Definition
open import Orders.Total.Definition
open import Semirings.Definition
module Numbers.BinaryNaturals.Order where
data Compare : Set where
Equal : Compare
FirstLess : Compare
FirstGreater : Compare
private
badCompare : Equal ≡ FirstLess → False
badCompare ()
badCompare' : Equal ≡ FirstGreater → False
badCompare' ()
badCompare'' : FirstLess ≡ FirstGreater → False
badCompare'' ()
_<BInherited_ : BinNat → BinNat → Compare
a <BInherited b with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
(a <BInherited b) | inl (inl x) = FirstLess
(a <BInherited b) | inl (inr x) = FirstGreater
(a <BInherited b) | inr x = Equal
private
go<Bcomp : Compare → BinNat → BinNat → Compare
go<Bcomp Equal [] [] = Equal
go<Bcomp Equal [] (zero :: b) = go<Bcomp Equal [] b
go<Bcomp Equal [] (one :: b) = FirstLess
go<Bcomp Equal (zero :: a) [] = go<Bcomp Equal a []
go<Bcomp Equal (zero :: a) (zero :: b) = go<Bcomp Equal a b
go<Bcomp Equal (zero :: a) (one :: b) = go<Bcomp FirstLess a b
go<Bcomp Equal (one :: a) [] = FirstGreater
go<Bcomp Equal (one :: a) (zero :: b) = go<Bcomp FirstGreater a b
go<Bcomp Equal (one :: a) (one :: b) = go<Bcomp Equal a b
go<Bcomp FirstGreater [] [] = FirstGreater
go<Bcomp FirstGreater [] (zero :: b) = go<Bcomp FirstGreater [] b
go<Bcomp FirstGreater [] (one :: b) = FirstLess
go<Bcomp FirstGreater (zero :: a) [] = FirstGreater
go<Bcomp FirstGreater (zero :: a) (zero :: b) = go<Bcomp FirstGreater a b
go<Bcomp FirstGreater (zero :: a) (one :: b) = go<Bcomp FirstLess a b
go<Bcomp FirstGreater (one :: a) [] = FirstGreater
go<Bcomp FirstGreater (one :: a) (zero :: b) = go<Bcomp FirstGreater a b
go<Bcomp FirstGreater (one :: a) (one :: b) = go<Bcomp FirstGreater a b
go<Bcomp FirstLess [] b = FirstLess
go<Bcomp FirstLess (zero :: a) [] = go<Bcomp FirstLess a []
go<Bcomp FirstLess (one :: a) [] = FirstGreater
go<Bcomp FirstLess (zero :: a) (zero :: b) = go<Bcomp FirstLess a b
go<Bcomp FirstLess (zero :: a) (one :: b) = go<Bcomp FirstLess a b
go<Bcomp FirstLess (one :: a) (zero :: b) = go<Bcomp FirstGreater a b
go<Bcomp FirstLess (one :: a) (one :: b) = go<Bcomp FirstLess a b
_<Bcomp_ : BinNat → BinNat → Compare
a <Bcomp b = go<Bcomp Equal a b
private
lemma1 : {s : Compare} → (n : BinNat) → go<Bcomp s n n ≡ s
lemma1 {Equal} [] = refl
lemma1 {Equal} (zero :: n) = lemma1 n
lemma1 {Equal} (one :: n) = lemma1 n
lemma1 {FirstLess} [] = refl
lemma1 {FirstLess} (zero :: n) = lemma1 n
lemma1 {FirstLess} (one :: n) = lemma1 n
lemma1 {FirstGreater} [] = refl
lemma1 {FirstGreater} (zero :: n) = lemma1 n
lemma1 {FirstGreater} (one :: n) = lemma1 n
lemma : {s : Compare} → (n : BinNat) → go<Bcomp s (incr n) n ≡ FirstGreater
lemma {Equal} [] = refl
lemma {Equal} (zero :: n) = lemma1 n
lemma {Equal} (one :: n) = lemma {FirstLess} n
lemma {FirstLess} [] = refl
lemma {FirstLess} (zero :: n) = lemma1 n
lemma {FirstLess} (one :: n) = lemma {FirstLess} n
lemma {FirstGreater} [] = refl
lemma {FirstGreater} (zero :: n) = lemma1 {FirstGreater} n
lemma {FirstGreater} (one :: n) = lemma {FirstLess} n
succLess : (n : ℕ) → (NToBinNat (succ n)) <Bcomp (NToBinNat n) ≡ FirstGreater
succLess zero = refl
succLess (succ n) with NToBinNat n
succLess (succ n) | [] = refl
succLess (succ n) | zero :: bl = lemma {FirstLess} bl
succLess (succ n) | one :: bl = lemma1 {FirstGreater} (incr bl)
compareRefl : (n : BinNat) → n <Bcomp n ≡ Equal
compareRefl [] = refl
compareRefl (zero :: n) = compareRefl n
compareRefl (one :: n) = compareRefl n
zeroLess : (n : BinNat) → ((canonical n ≡ []) → False) → [] <Bcomp n ≡ FirstLess
zeroLess [] pr = exFalso (pr refl)
zeroLess (zero :: n) pr with inspect (canonical n)
zeroLess (zero :: n) pr | [] with≡ x rewrite x = exFalso (pr refl)
zeroLess (zero :: n) pr | (x₁ :: y) with≡ x = zeroLess n λ i → nonEmptyNotEmpty (transitivity (equalityCommutative x) i)
zeroLess (one :: n) pr = refl
zeroLess' : (n : BinNat) → ((canonical n ≡ []) → False) → n <Bcomp [] ≡ FirstGreater
zeroLess' [] pr = exFalso (pr refl)
zeroLess' (zero :: n) pr with inspect (canonical n)
zeroLess' (zero :: n) pr | [] with≡ x rewrite x = exFalso (pr refl)
zeroLess' (zero :: n) pr | (x₁ :: y) with≡ x = zeroLess' n (λ i → nonEmptyNotEmpty (transitivity (equalityCommutative x) i))
zeroLess' (one :: n) pr = refl
abstract
canonicalFirst : (n m : BinNat) (state : Compare) → go<Bcomp state n m ≡ go<Bcomp state (canonical n) m
canonicalFirst [] m Equal = refl
canonicalFirst (zero :: n) m Equal with inspect (canonical n)
canonicalFirst (zero :: n) [] Equal | [] with≡ x rewrite x = transitivity (canonicalFirst n [] Equal) (applyEquality (λ i → go<Bcomp Equal i []) {canonical n} x)
canonicalFirst (zero :: n) (zero :: ms) Equal | [] with≡ x rewrite x | canonicalFirst n ms Equal | x = refl
canonicalFirst (zero :: n) (one :: ms) Equal | [] with≡ x rewrite x | canonicalFirst n ms FirstLess | x = refl
canonicalFirst (zero :: n) [] Equal | (x₁ :: y) with≡ x rewrite x | canonicalFirst n [] Equal | x = refl
canonicalFirst (zero :: n) (zero :: ms) Equal | (x₁ :: y) with≡ x rewrite x | canonicalFirst n ms Equal | x = refl
canonicalFirst (zero :: n) (one :: ms) Equal | (x₁ :: y) with≡ x rewrite x | canonicalFirst n ms FirstLess | x = refl
canonicalFirst (one :: n) [] Equal = refl
canonicalFirst (one :: n) (zero :: m) Equal = canonicalFirst n m FirstGreater
canonicalFirst (one :: n) (one :: m) Equal = canonicalFirst n m Equal
canonicalFirst [] m FirstLess = refl
canonicalFirst (zero :: n) [] FirstLess with inspect (canonical n)
canonicalFirst (zero :: n) [] FirstLess | [] with≡ x rewrite x | canonicalFirst n [] FirstLess | x = refl
canonicalFirst (zero :: n) [] FirstLess | (x₁ :: y) with≡ x rewrite x | canonicalFirst n [] FirstLess | x = refl
canonicalFirst (zero :: n) (zero :: m) FirstLess with inspect (canonical n)
canonicalFirst (zero :: n) (zero :: m) FirstLess | [] with≡ x rewrite x | canonicalFirst n m FirstLess | x = refl
canonicalFirst (zero :: n) (zero :: m) FirstLess | (x₁ :: y) with≡ x rewrite x | canonicalFirst n m FirstLess | x = refl
canonicalFirst (zero :: n) (one :: m) FirstLess with inspect (canonical n)
canonicalFirst (zero :: n) (one :: m) FirstLess | [] with≡ x rewrite x | canonicalFirst n m FirstLess | x = refl
canonicalFirst (zero :: n) (one :: m) FirstLess | (x₁ :: y) with≡ x rewrite x | canonicalFirst n m FirstLess | x = refl
canonicalFirst (one :: n) [] FirstLess = refl
canonicalFirst (one :: n) (zero :: m) FirstLess = canonicalFirst n m FirstGreater
canonicalFirst (one :: n) (one :: m) FirstLess = canonicalFirst n m FirstLess
canonicalFirst [] m FirstGreater = refl
canonicalFirst (zero :: n) m FirstGreater with inspect (canonical n)
canonicalFirst (zero :: n) [] FirstGreater | [] with≡ x rewrite x = refl
canonicalFirst (zero :: n) (zero :: ms) FirstGreater | [] with≡ x rewrite x | canonicalFirst n ms FirstGreater | x = refl
canonicalFirst (zero :: n) (one :: ms) FirstGreater | [] with≡ x rewrite x | canonicalFirst n ms FirstLess | x = refl
canonicalFirst (zero :: n) [] FirstGreater | (x₁ :: y) with≡ x rewrite x = refl
canonicalFirst (zero :: n) (zero :: ms) FirstGreater | (x₁ :: y) with≡ x rewrite x | canonicalFirst n ms FirstGreater | x = refl
canonicalFirst (zero :: n) (one :: ms) FirstGreater | (x₁ :: y) with≡ x rewrite x | canonicalFirst n ms FirstLess | x = refl
canonicalFirst (one :: n) [] FirstGreater = refl
canonicalFirst (one :: n) (zero :: m) FirstGreater = canonicalFirst n m FirstGreater
canonicalFirst (one :: n) (one :: m) FirstGreater = canonicalFirst n m FirstGreater
private
greater0Lemma : (n : BinNat) → go<Bcomp FirstGreater n [] ≡ FirstGreater
greater0Lemma [] = refl
greater0Lemma (zero :: n) = refl
greater0Lemma (one :: n) = refl
canonicalSecond : (n m : BinNat) (state : Compare) → go<Bcomp state n m ≡ go<Bcomp state n (canonical m)
canonicalSecond n [] Equal = refl
canonicalSecond [] (zero :: m) Equal with inspect (canonical m)
canonicalSecond [] (zero :: m) Equal | [] with≡ x rewrite x | canonicalSecond [] m Equal | x = refl
canonicalSecond [] (zero :: m) Equal | (x₁ :: y) with≡ x rewrite x | canonicalSecond [] m Equal | x = refl
canonicalSecond (zero :: n) (zero :: m) Equal with inspect (canonical m)
canonicalSecond (zero :: n) (zero :: m) Equal | [] with≡ x rewrite x | canonicalSecond n m Equal | x = refl
canonicalSecond (zero :: n) (zero :: m) Equal | (x₁ :: y) with≡ x rewrite x | canonicalSecond n m Equal | x = refl
canonicalSecond (one :: n) (zero :: m) Equal with inspect (canonical m)
canonicalSecond (one :: n) (zero :: m) Equal | [] with≡ x rewrite x | canonicalSecond n m FirstGreater | x = greater0Lemma n
canonicalSecond (one :: n) (zero :: m) Equal | (x₁ :: y) with≡ x rewrite x | canonicalSecond n m FirstGreater | x = refl
canonicalSecond [] (one :: m) Equal = refl
canonicalSecond (zero :: n) (one :: m) Equal = canonicalSecond n m FirstLess
canonicalSecond (one :: n) (one :: m) Equal = canonicalSecond n m Equal
canonicalSecond n [] FirstLess = refl
canonicalSecond [] (zero :: m) FirstLess = refl
canonicalSecond (x :: n) (zero :: m) FirstLess with inspect (canonical m)
canonicalSecond (zero :: n) (zero :: m) FirstLess | [] with≡ x rewrite x | canonicalSecond n m FirstLess | x = refl
canonicalSecond (one :: n) (zero :: m) FirstLess | [] with≡ x rewrite x | canonicalSecond n m FirstGreater | x = greater0Lemma n
canonicalSecond (zero :: n) (zero :: m) FirstLess | (x₁ :: bl) with≡ pr rewrite pr | canonicalSecond n m FirstLess | pr = refl
canonicalSecond (one :: n) (zero :: m) FirstLess | (x₁ :: bl) with≡ pr rewrite pr | canonicalSecond n m FirstGreater | pr = refl
canonicalSecond [] (one :: m) FirstLess = refl
canonicalSecond (zero :: n) (one :: m) FirstLess = canonicalSecond n m FirstLess
canonicalSecond (one :: n) (one :: m) FirstLess = canonicalSecond n m FirstLess
canonicalSecond n [] FirstGreater = refl
canonicalSecond [] (zero :: m) FirstGreater with inspect (canonical m)
canonicalSecond [] (zero :: m) FirstGreater | [] with≡ x rewrite x | canonicalSecond [] m FirstGreater | x = refl
canonicalSecond [] (zero :: m) FirstGreater | (x₁ :: y) with≡ x rewrite x | canonicalSecond [] m FirstGreater | x = refl
canonicalSecond (zero :: n) (zero :: m) FirstGreater with inspect (canonical m)
canonicalSecond (zero :: n) (zero :: m) FirstGreater | [] with≡ x rewrite x | canonicalSecond n m FirstGreater | x = greater0Lemma n
canonicalSecond (zero :: n) (zero :: m) FirstGreater | (x₁ :: y) with≡ x rewrite x | canonicalSecond n m FirstGreater | x = refl
canonicalSecond (one :: n) (zero :: m) FirstGreater with inspect (canonical m)
canonicalSecond (one :: n) (zero :: m) FirstGreater | [] with≡ x rewrite x | canonicalSecond n m FirstGreater | x = greater0Lemma n
canonicalSecond (one :: n) (zero :: m) FirstGreater | (x₁ :: y) with≡ x rewrite x | canonicalSecond n m FirstGreater | x = refl
canonicalSecond [] (one :: m) FirstGreater = refl
canonicalSecond (zero :: n) (one :: m) FirstGreater = canonicalSecond n m FirstLess
canonicalSecond (one :: n) (one :: m) FirstGreater = canonicalSecond n m FirstGreater
equalContaminated : (n m : BinNat) → go<Bcomp FirstLess n m ≡ Equal → False
equalContaminated' : (n m : BinNat) → go<Bcomp FirstGreater n m ≡ Equal → False
equalContaminated (zero :: n) [] pr = equalContaminated n [] pr
equalContaminated (zero :: n) (zero :: m) pr = equalContaminated n m pr
equalContaminated (zero :: n) (one :: m) pr = equalContaminated n m pr
equalContaminated (one :: n) (zero :: m) pr = equalContaminated' n m pr
equalContaminated (one :: n) (one :: m) pr = equalContaminated n m pr
equalContaminated' [] (zero :: m) pr = equalContaminated' [] m pr
equalContaminated' (zero :: n) (zero :: m) pr = equalContaminated' n m pr
equalContaminated' (zero :: n) (one :: m) pr = equalContaminated n m pr
equalContaminated' (one :: n) (zero :: m) pr = equalContaminated' n m pr
equalContaminated' (one :: n) (one :: m) pr = equalContaminated' n m pr
comparisonEqual : (a b : BinNat) → (a <Bcomp b ≡ Equal) → canonical a ≡ canonical b
comparisonEqual [] [] pr = refl
comparisonEqual [] (zero :: b) pr with inspect (canonical b)
comparisonEqual [] (zero :: b) pr | [] with≡ p rewrite p = refl
comparisonEqual [] (zero :: b) pr | (x :: r) with≡ p rewrite zeroLess b (λ i → nonEmptyNotEmpty (transitivity (equalityCommutative p) i)) = exFalso (badCompare (equalityCommutative pr))
comparisonEqual (zero :: a) [] pr with inspect (canonical a)
comparisonEqual (zero :: a) [] pr | [] with≡ x rewrite x = refl
comparisonEqual (zero :: a) [] pr | (x₁ :: y) with≡ x rewrite zeroLess' a (λ i → nonEmptyNotEmpty (transitivity (equalityCommutative x) i)) = exFalso (badCompare' (equalityCommutative pr))
comparisonEqual (zero :: a) (zero :: b) pr with inspect (canonical a)
comparisonEqual (zero :: a) (zero :: b) pr | [] with≡ x with inspect (canonical b)
comparisonEqual (zero :: a) (zero :: b) pr | [] with≡ pr2 | [] with≡ pr3 rewrite pr2 | pr3 = refl
comparisonEqual (zero :: a) (zero :: b) pr | [] with≡ pr2 | (x₂ :: y) with≡ pr3 rewrite pr2 | pr3 | comparisonEqual a b pr = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative pr3) pr2))
comparisonEqual (zero :: a) (zero :: b) pr | (c :: cs) with≡ pr2 with inspect (canonical b)
comparisonEqual (zero :: a) (zero :: b) pr | (c :: cs) with≡ pr2 | [] with≡ pr3 rewrite pr2 | pr3 | comparisonEqual a b pr = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative pr2) pr3))
comparisonEqual (zero :: a) (zero :: b) pr | (c :: cs) with≡ pr2 | (x₁ :: y) with≡ pr3 rewrite pr2 | pr3 | comparisonEqual a b pr = applyEquality (zero ::_) (transitivity (equalityCommutative pr2) pr3)
comparisonEqual (zero :: a) (one :: b) pr = exFalso (equalContaminated a b pr)
comparisonEqual (one :: a) (zero :: b) pr = exFalso (equalContaminated' a b pr)
comparisonEqual (one :: a) (one :: b) pr = applyEquality (one ::_) (comparisonEqual a b pr)
equalSymmetric : (n m : BinNat) → n <Bcomp m ≡ Equal → m <Bcomp n ≡ Equal
equalSymmetric [] [] n=m = refl
equalSymmetric [] (zero :: m) n=m rewrite equalSymmetric [] m n=m = refl
equalSymmetric (zero :: n) [] n=m rewrite equalSymmetric n [] n=m = refl
equalSymmetric (zero :: n) (zero :: m) n=m = equalSymmetric n m n=m
equalSymmetric (zero :: n) (one :: m) n=m = exFalso (equalContaminated n m n=m)
equalSymmetric (one :: n) (zero :: m) n=m = exFalso (equalContaminated' n m n=m)
equalSymmetric (one :: n) (one :: m) n=m = equalSymmetric n m n=m
equalToFirstGreater : (state : Compare) → (a b : BinNat) → go<Bcomp Equal a b ≡ FirstGreater → go<Bcomp state a b ≡ FirstGreater
equalToFirstGreater FirstGreater [] (zero :: b) pr = equalToFirstGreater FirstGreater [] b pr
equalToFirstGreater FirstGreater (zero :: a) [] pr = refl
equalToFirstGreater FirstGreater (zero :: a) (zero :: b) pr = equalToFirstGreater FirstGreater a b pr
equalToFirstGreater FirstGreater (zero :: a) (one :: b) pr = pr
equalToFirstGreater FirstGreater (one :: a) [] pr = refl
equalToFirstGreater FirstGreater (one :: a) (zero :: b) pr = pr
equalToFirstGreater FirstGreater (one :: a) (one :: b) pr = equalToFirstGreater FirstGreater a b pr
equalToFirstGreater Equal a b pr = pr
equalToFirstGreater FirstLess [] (zero :: b) pr = equalToFirstGreater FirstLess [] b pr
equalToFirstGreater FirstLess (zero :: a) [] pr = equalToFirstGreater FirstLess a [] pr
equalToFirstGreater FirstLess (zero :: a) (zero :: b) pr = equalToFirstGreater FirstLess a b pr
equalToFirstGreater FirstLess (zero :: a) (one :: b) pr = pr
equalToFirstGreater FirstLess (one :: a) [] pr = refl
equalToFirstGreater FirstLess (one :: a) (zero :: b) pr = pr
equalToFirstGreater FirstLess (one :: a) (one :: b) pr = equalToFirstGreater FirstLess a b pr
equalToFirstLess : (state : Compare) → (a b : BinNat) → go<Bcomp Equal a b ≡ FirstLess → go<Bcomp state a b ≡ FirstLess
equalToFirstLess FirstLess [] b pr = refl
equalToFirstLess FirstLess (zero :: a) [] pr = equalToFirstLess FirstLess a [] pr
equalToFirstLess FirstLess (zero :: a) (zero :: b) pr = equalToFirstLess FirstLess a b pr
equalToFirstLess FirstLess (zero :: a) (one :: b) pr = pr
equalToFirstLess FirstLess (one :: a) (zero :: b) pr = pr
equalToFirstLess FirstLess (one :: a) (one :: b) pr = equalToFirstLess FirstLess a b pr
equalToFirstLess Equal a b pr = pr
equalToFirstLess FirstGreater [] (zero :: b) pr = equalToFirstLess FirstGreater [] b pr
equalToFirstLess FirstGreater [] (one :: b) pr = refl
equalToFirstLess FirstGreater (zero :: a) [] pr = transitivity (t a) (equalToFirstLess FirstGreater a [] pr)
where
t : (a : BinNat) → FirstGreater ≡ go<Bcomp FirstGreater a []
t [] = refl
t (zero :: a) = refl
t (one :: a) = refl
equalToFirstLess FirstGreater (zero :: a) (zero :: b) pr = equalToFirstLess FirstGreater a b pr
equalToFirstLess FirstGreater (zero :: a) (one :: b) pr = pr
equalToFirstLess FirstGreater (one :: a) (zero :: b) pr = pr
equalToFirstLess FirstGreater (one :: a) (one :: b) pr = equalToFirstLess FirstGreater a b pr
zeroNotSucc : (n : ℕ) (b : BinNat) → (canonical b ≡ []) → (binNatToN b ≡ succ n) → False
zeroNotSucc n b b=0 b>0 rewrite binNatToNZero' b b=0 = naughtE b>0
chopFirstBit : (m n : BinNat) {b : Bit} (s : Compare) → go<Bcomp s (b :: m) (b :: n) ≡ go<Bcomp s m n
chopFirstBit m n {zero} Equal = refl
chopFirstBit m n {one} Equal = refl
chopFirstBit m n {zero} FirstLess = refl
chopFirstBit m n {one} FirstLess = refl
chopFirstBit m n {zero} FirstGreater = refl
chopFirstBit m n {one} FirstGreater = refl
chopDouble : (a b : BinNat) (i : Bit) → (i :: a) <BInherited (i :: b) ≡ a <BInherited b
chopDouble a b i with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
chopDouble a b zero | inl (inl a<b) with TotalOrder.totality ℕTotalOrder (2 *N binNatToN a) (2 *N binNatToN b)
chopDouble a b zero | inl (inl a<b) | inl (inl x) = refl
chopDouble a b zero | inl (inl a<b) | inl (inr b<a) = exFalso (TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder b<a (lessRespectsMultiplicationLeft (binNatToN a) (binNatToN b) 2 a<b (le 1 refl))))
chopDouble a b zero | inl (inl a<b) | inr a=b rewrite productCancelsLeft 2 (binNatToN a) (binNatToN b) (le 1 refl) a=b = exFalso (TotalOrder.irreflexive ℕTotalOrder a<b)
chopDouble a b one | inl (inl a<b) with TotalOrder.totality ℕTotalOrder (2 *N binNatToN a) (2 *N binNatToN b)
chopDouble a b one | inl (inl a<b) | inl (inl 2a<2b) = refl
chopDouble a b one | inl (inl a<b) | inl (inr 2b<2a) = exFalso (TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive (ℕTotalOrder) a<b (cancelInequalityLeft {2} 2b<2a)))
chopDouble a b one | inl (inl a<b) | inr 2a=2b rewrite productCancelsLeft 2 (binNatToN a) (binNatToN b) (le 1 refl) 2a=2b = exFalso (TotalOrder.irreflexive (ℕTotalOrder) a<b)
chopDouble a b zero | inl (inr b<a) with TotalOrder.totality ℕTotalOrder (2 *N binNatToN a) (2 *N binNatToN b)
chopDouble a b zero | inl (inr b<a) | inl (inl 2a<2b) = exFalso (TotalOrder.irreflexive (ℕTotalOrder) (TotalOrder.<Transitive (ℕTotalOrder) b<a (cancelInequalityLeft {2} {binNatToN a} {binNatToN b} 2a<2b)))
chopDouble a b zero | inl (inr b<a) | inl (inr 2b<2a) = refl
chopDouble a b zero | inl (inr b<a) | inr 2a=2b rewrite productCancelsLeft 2 (binNatToN a) (binNatToN b) (le 1 refl) 2a=2b = exFalso (TotalOrder.irreflexive (ℕTotalOrder) b<a)
chopDouble a b one | inl (inr b<a) with TotalOrder.totality ℕTotalOrder (2 *N binNatToN a) (2 *N binNatToN b)
chopDouble a b one | inl (inr b<a) | inl (inl 2a<2b) = exFalso (TotalOrder.irreflexive (ℕTotalOrder) (TotalOrder.<Transitive (ℕTotalOrder) b<a (cancelInequalityLeft {2} 2a<2b)))
chopDouble a b one | inl (inr b<a) | inl (inr x) = refl
chopDouble a b one | inl (inr b<a) | inr 2a=2b rewrite productCancelsLeft 2 (binNatToN a) (binNatToN b) (le 1 refl) 2a=2b = exFalso (TotalOrder.irreflexive (ℕTotalOrder) b<a)
chopDouble a b i | inr x with TotalOrder.totality ℕTotalOrder (binNatToN (i :: a)) (binNatToN (i :: b))
chopDouble a b zero | inr a=b | inl (inl a<b) rewrite a=b = exFalso (TotalOrder.irreflexive (ℕTotalOrder) a<b)
chopDouble a b one | inr a=b | inl (inl a<b) rewrite a=b = exFalso (TotalOrder.irreflexive (ℕTotalOrder) a<b)
chopDouble a b zero | inr a=b | inl (inr b<a) rewrite a=b = exFalso (TotalOrder.irreflexive (ℕTotalOrder) b<a)
chopDouble a b one | inr a=b | inl (inr b<a) rewrite a=b = exFalso (TotalOrder.irreflexive (ℕTotalOrder) b<a)
chopDouble a b _ | inr a=b | inr x = refl
succNotLess : {n : ℕ} → succ n <N n → False
succNotLess {succ n} (le x proof) = succNotLess {n} (le x (succInjective (transitivity (applyEquality succ (transitivity (Semiring.commutative ℕSemiring (succ x) (succ n)) (transitivity (applyEquality succ (transitivity (Semiring.commutative ℕSemiring n (succ x)) (applyEquality succ (Semiring.commutative ℕSemiring x n)))) (Semiring.commutative ℕSemiring (succ (succ n)) x)))) proof)))
<BIsInherited : (a b : BinNat) → a <BInherited b ≡ a <Bcomp b
<BIsInherited [] b with TotalOrder.totality ℕTotalOrder 0 (binNatToN b)
<BIsInherited [] b | inl (inl x) with inspect (binNatToN b)
<BIsInherited [] b | inl (inl x) | 0 with≡ pr rewrite binNatToNZero b pr | pr = exFalso (TotalOrder.irreflexive (ℕTotalOrder) x)
<BIsInherited [] b | inl (inl x) | (succ bl) with≡ pr rewrite pr = equalityCommutative (zeroLess b λ p → zeroNotSucc bl b p pr)
<BIsInherited [] b | inr 0=b rewrite canonicalSecond [] b Equal | binNatToNZero b (equalityCommutative 0=b) = refl
<BIsInherited (a :: as) [] with TotalOrder.totality ℕTotalOrder (binNatToN (a :: as)) 0
<BIsInherited (a :: as) [] | inl (inr x) with inspect (binNatToN (a :: as))
<BIsInherited (a :: as) [] | inl (inr x) | zero with≡ pr rewrite binNatToNZero (a :: as) pr | pr = exFalso (TotalOrder.irreflexive (ℕTotalOrder) x)
<BIsInherited (a :: as) [] | inl (inr x) | succ y with≡ pr rewrite pr = equalityCommutative (zeroLess' (a :: as) λ i → zeroNotSucc y (a :: as) i pr)
<BIsInherited (a :: as) [] | inr x rewrite canonicalFirst (a :: as) [] Equal | binNatToNZero (a :: as) x = refl
<BIsInherited (zero :: a) (zero :: b) = transitivity (chopDouble a b zero) (<BIsInherited a b)
<BIsInherited (zero :: a) (one :: b) with TotalOrder.totality ℕTotalOrder (binNatToN (zero :: a)) (binNatToN (one :: b))
<BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
<BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) | inl (inl a<b) = equalityCommutative (equalToFirstLess FirstLess a b (equalityCommutative indHyp))
where
t : a <BInherited b ≡ FirstLess
t with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
t | inl (inl x) = refl
t | inl (inr x) = exFalso (TotalOrder.irreflexive (ℕTotalOrder) (TotalOrder.<Transitive (ℕTotalOrder) x a<b))
t | inr x rewrite x = exFalso (TotalOrder.irreflexive (ℕTotalOrder) a<b)
indHyp : FirstLess ≡ go<Bcomp Equal a b
indHyp = transitivity (equalityCommutative t) (<BIsInherited a b)
<BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) | inl (inr b<a) = exFalso (noIntegersBetweenXAndSuccX {2 *N binNatToN a} (2 *N binNatToN b) (lessRespectsMultiplicationLeft (binNatToN b) (binNatToN a) 2 b<a (le 1 refl)) 2a<2b+1)
<BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) | inr a=b rewrite a=b | canonicalFirst a b FirstLess | canonicalSecond (canonical a) b FirstLess | transitivity (equalityCommutative (binToBin a)) (transitivity (applyEquality NToBinNat a=b) (binToBin b)) = equalityCommutative (lemma1 (canonical b))
<BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
<BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) | inl (inl a<b) = exFalso (TotalOrder.irreflexive (ℕTotalOrder) (TotalOrder.<Transitive (ℕTotalOrder) 2b+1<2a (TotalOrder.<Transitive (ℕTotalOrder) (lessRespectsMultiplicationLeft (binNatToN a) (binNatToN b) 2 a<b (le 1 refl)) (le zero refl))))
<BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) | inl (inr b<a) = equalityCommutative (equalToFirstGreater FirstLess a b (equalityCommutative indHyp))
where
t : a <BInherited b ≡ FirstGreater
t with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
t | inl (inl x) = exFalso (TotalOrder.irreflexive (ℕTotalOrder) (TotalOrder.<Transitive (ℕTotalOrder) x b<a))
t | inl (inr x) = refl
t | inr x rewrite x = exFalso (TotalOrder.irreflexive (ℕTotalOrder) b<a)
indHyp : FirstGreater ≡ go<Bcomp Equal a b
indHyp = transitivity (equalityCommutative t) (<BIsInherited a b)
<BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) | inr a=b rewrite a=b = exFalso (succNotLess 2b+1<2a)
<BIsInherited (zero :: a) (one :: b) | inr 2a=2b+1 = exFalso (parity (binNatToN b) (binNatToN a) (equalityCommutative 2a=2b+1))
<BIsInherited (one :: a) (zero :: b) with TotalOrder.totality ℕTotalOrder (binNatToN (one :: a)) (binNatToN (zero :: b))
<BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
<BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) | inl (inl a<b) = equalityCommutative (equalToFirstLess FirstGreater a b (equalityCommutative indHyp))
where
t : a <BInherited b ≡ FirstLess
t with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
t | inl (inr x) = exFalso (TotalOrder.irreflexive (ℕTotalOrder) (TotalOrder.<Transitive (ℕTotalOrder) x a<b))
t | inl (inl x) = refl
t | inr x rewrite x = exFalso (TotalOrder.irreflexive (ℕTotalOrder) a<b)
indHyp : FirstLess ≡ go<Bcomp Equal a b
indHyp = transitivity (equalityCommutative t) (<BIsInherited a b)
<BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) | inl (inr b<a) = exFalso (TotalOrder.irreflexive (ℕTotalOrder) (TotalOrder.<Transitive (ℕTotalOrder) 2a+1<2b (TotalOrder.<Transitive (ℕTotalOrder) (lessRespectsMultiplicationLeft (binNatToN b) (binNatToN a) 2 b<a (le 1 refl)) (le zero refl))))
<BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) | inr a=b rewrite a=b = exFalso (succNotLess 2a+1<2b)
<BIsInherited (one :: a) (zero :: b) | inl (inr 2b<2a+1) with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
<BIsInherited (one :: a) (zero :: b) | inl (inr 2b<2a+1) | inl (inl a<b) = exFalso (noIntegersBetweenXAndSuccX {2 *N binNatToN b} (2 *N binNatToN a) (lessRespectsMultiplicationLeft (binNatToN a) (binNatToN b) 2 a<b (le 1 refl)) 2b<2a+1)
<BIsInherited (one :: a) (zero :: b) | inl (inr 2b<2a+1) | inl (inr b<a) = equalityCommutative (equalToFirstGreater FirstGreater a b (equalityCommutative indHyp))
where
t : a <BInherited b ≡ FirstGreater
t with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
t | inl (inl x) = exFalso (TotalOrder.irreflexive (ℕTotalOrder) (TotalOrder.<Transitive (ℕTotalOrder) x b<a))
t | inl (inr x) = refl
t | inr x rewrite x = exFalso (TotalOrder.irreflexive (ℕTotalOrder) b<a)
indHyp : FirstGreater ≡ go<Bcomp Equal a b
indHyp = transitivity (equalityCommutative t) (<BIsInherited a b)
<BIsInherited (one :: a) (zero :: b) | inl (inr 2b<2a+1) | inr a=b rewrite a=b | canonicalFirst a b FirstGreater | canonicalSecond (canonical a) b FirstGreater | transitivity (equalityCommutative (binToBin a)) (transitivity (applyEquality NToBinNat a=b) (binToBin b)) = equalityCommutative (lemma1 (canonical b))
<BIsInherited (one :: a) (zero :: b) | inr x = exFalso (parity (binNatToN a) (binNatToN b) x)
<BIsInherited (one :: a) (one :: b) = transitivity (chopDouble a b one) (<BIsInherited a b)
_<B_ : BinNat → BinNat → Set
a <B b = (a <Bcomp b) ≡ FirstLess
translate : (a b : BinNat) → (a <B b) → (binNatToN a) <N (binNatToN b)
translate a b a<b with <BIsInherited a b
... | r with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
... | inl (inl x) = x
... | inl (inr x) = exFalso (badCompare'' (transitivity (equalityCommutative a<b) (equalityCommutative r)))
... | inr x = exFalso (badCompare (transitivity r a<b))
private
totality : (a b : BinNat) → ((a <B b) || (b <B a)) || (canonical a ≡ canonical b)
totality [] [] = inr refl
totality [] (zero :: b) with totality [] b
... | inl (inl x) = inl (inl x)
... | inl (inr x) = inl (inr x)
... | inr x with canonical b
... | [] = inr refl
totality [] (one :: b) = inl (inl refl)
totality (zero :: a) [] with totality a []
... | inl (inl x) = inl (inl x)
... | inl (inr x) = inl (inr x)
... | inr x with canonical a
... | [] = inr refl
totality (zero :: a) (zero :: b) with totality a b
... | inl (inl x) = inl (inl x)
... | inl (inr x) = inl (inr x)
... | inr x rewrite x = inr refl
totality (zero :: a) (one :: b) with totality a b
... | inl (inl x) = inl (inl (equalToFirstLess FirstLess a b x))
... | inr x rewrite canonicalSecond a b FirstLess | canonicalFirst a (canonical b) FirstLess | x = inl (inl (lemma1 (canonical b)))
... | inl (inr x) with equalToFirstLess FirstGreater b a x
... | r = inl (inr r)
totality (one :: a) [] = inl (inr refl)
totality (one :: a) (zero :: b) with totality a b
... | inr x rewrite canonicalSecond b a FirstLess | canonicalFirst b (canonical a) FirstLess | x = inl (inr (lemma1 (canonical b)))
... | inl (inr x) = inl (inr (equalToFirstLess FirstLess b a x))
... | inl (inl x) with equalToFirstLess FirstGreater a b x
... | r = inl (inl r)
totality (one :: a) (one :: b) with totality a b
... | inl (inl x) = inl (inl x)
... | inl (inr x) = inl (inr x)
... | inr x rewrite x = inr refl
translate' : (a b : ℕ) → (a <N b) → (NToBinNat a) <B (NToBinNat b)
translate' a b a<b with totality (NToBinNat a) (NToBinNat b)
... | inl (inl x) = x
... | inl (inr x) with translate (NToBinNat b) (NToBinNat a) x
... | m = exFalso (lessIrreflexive (lessTransitive a<b (identityOfIndiscernablesLeft _<N_ (identityOfIndiscernablesRight _<N_ m (nToN a)) (nToN b))))
translate' a b a<b | inr x rewrite NToBinNatInj a b x = exFalso (lessIrreflexive a<b)
private
<BTransitive : (a b c : BinNat) → (a <B b) → (b <B c) → a <B c
<BTransitive a b c a<b b<c with translate' (binNatToN a) (binNatToN c) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) (translate a b a<b) (translate b c b<c))
... | r rewrite binToBin a | binToBin c = transitivity (canonicalFirst a c Equal) (transitivity (canonicalSecond (canonical a) c Equal) r)
-- This order fails to be total because [] is not literally equal to 0::[] .
BinNatOrder : PartialOrder BinNat
PartialOrder._<_ (BinNatOrder) = _<B_
PartialOrder.irreflexive (BinNatOrder) {x} bad = badCompare (transitivity (equalityCommutative (compareRefl x)) bad)
PartialOrder.<Transitive (BinNatOrder) {a} {b} {c} a<b b<c = <BTransitive a b c a<b b<c
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{-# OPTIONS --without-K --safe #-}
module Categories.Adjoint.Instance.Discrete where
-- The Discrete/Forgetful/Codiscrete adjoint triple between Sets and
-- StrictCats.
--
-- We need 'strict' functor equality to prove functionality of the
-- forgetful functor (we cannot extract a propositional equality proof
-- on objects from a natural isomorphism).
open import Level using (Lift; lift)
open import Data.Unit using (⊤; tt)
import Function as Fun
open import Relation.Binary.PropositionalEquality
open import Categories.Adjoint
open import Categories.Category using (Category)
open import Categories.Category.Instance.StrictCats
open import Categories.Category.Instance.Sets
open import Categories.Functor renaming (id to idF)
open import Categories.Functor.Equivalence
import Categories.Category.Discrete as D
open import Categories.Morphism.HeterogeneousIdentity
open import Categories.Morphism.HeterogeneousIdentity.Properties
open import Categories.NaturalTransformation as NT using (NaturalTransformation; ntHelper)
-- The forgetful functor from StrictCats to Sets.
Forgetful : ∀ {o ℓ e} → Functor (Cats o ℓ e) (Sets o)
Forgetful {o} {ℓ} {e} = record
{ F₀ = Obj
; F₁ = F₀
; identity = refl
; homomorphism = refl
; F-resp-≈ = λ F≡G {X} → eq₀ F≡G X
}
where
open Category
open Functor
open _≡F_
-- The discrete functor (strict version)
Discrete : ∀ {o} → Functor (Sets o) (Cats o o o)
Discrete {o} = record
{ F₀ = D.Discrete
; F₁ = λ f → record
{ F₀ = f
; F₁ = cong f
; identity = refl
; homomorphism = λ {_ _ _ p q} → cong-trans f p q
; F-resp-≈ = cong (cong f)
}
; identity = record
{ eq₀ = λ _ → refl
; eq₁ = λ p → let open ≡-Reasoning in begin
trans (cong Fun.id p) refl ≡⟨ trans-reflʳ _ ⟩
cong Fun.id p ≡⟨ cong-id p ⟩
p ∎
}
; homomorphism = λ {_ _ _ f g} → record
{ eq₀ = λ _ → refl
; eq₁ = λ p → let open ≡-Reasoning in begin
trans (cong (g Fun.∘ f) p) refl ≡⟨ trans-reflʳ _ ⟩
cong (g Fun.∘ f) p ≡⟨ cong-∘ p ⟩
cong g (cong f p) ∎
}
; F-resp-≈ = λ {_ _ f g} f≗g → record
{ eq₀ = λ x → f≗g {x}
; eq₁ = λ {x} {y} p → naturality (λ x → subst (f x ≡_) (f≗g {x}) refl)
}
}
where
-- A helper lemma.
-- FIXME: Should probably go into Relation.Binary.PropositionalEquality
cong-trans : ∀ {A B : Set o} (f : A → B) {x y z} (p : x ≡ y) (q : y ≡ z) →
cong f (trans p q) ≡ trans (cong f p) (cong f q)
cong-trans f refl refl = refl
-- The codiscrete functor (strict version)
Codiscrete : ∀ {o} ℓ e → Functor (Sets o) (Cats o ℓ e)
Codiscrete {o} ℓ e = record
{ F₀ = λ A → record
{ Obj = A
; _⇒_ = λ _ _ → Lift ℓ ⊤
; _≈_ = λ _ _ → Lift e ⊤
; id = lift tt
; _∘_ = λ _ _ → lift tt
; assoc = lift tt
; identityˡ = lift tt
; identityʳ = lift tt
; equiv = record
{ refl = lift tt
; sym = λ _ → lift tt
; trans = λ _ _ → lift tt
}
; ∘-resp-≈ = λ _ _ → lift tt
}
; F₁ = λ f → record
{ F₀ = f
; F₁ = λ _ → lift tt
; identity = lift tt
; homomorphism = lift tt
; F-resp-≈ = λ _ → lift tt
}
; identity = Equiv.refl
; homomorphism = Equiv.refl
; F-resp-≈ = λ {_ _ f g} f≗g → record
{ eq₀ = λ x → f≗g {x}
; eq₁ = λ _ → lift tt
}
}
where open Category (Cats o ℓ e)
-- Discrete is left-adjoint to the forgetful functor from StrictCats to Sets
DiscreteLeftAdj : ∀ {o} → Discrete ⊣ Forgetful {o} {o} {o}
DiscreteLeftAdj {o} = record
{ unit = unit
; counit = counit
; zig = zig
; zag = refl
}
where
module U = Functor Forgetful
module Δ = Functor Discrete
unit : NaturalTransformation idF (Forgetful ∘F Discrete)
unit = NT.id
counitFun : ∀ C → Functor (Δ.F₀ (U.F₀ C)) C
counitFun C = let open Category C in record
{ F₀ = Fun.id
; F₁ = hid C
; identity = Equiv.refl
; homomorphism = λ {_ _ _ p q} → Equiv.sym (hid-trans C q p)
; F-resp-≈ = hid-cong C
}
counitComm : ∀ {C D} → (F : Functor C D) →
counitFun D ∘F (Δ.F₁ (U.F₁ F)) ≡F F ∘F counitFun C
counitComm {C} {D} F =
let open Functor F
open Category D
open HomReasoning hiding (refl)
in record
{ eq₀ = λ _ → refl
; eq₁ = λ {X Y} p → begin
id ∘ hid D (cong F₀ p) ≈⟨ identityˡ ⟩
hid D (cong F₀ p) ≈˘⟨ F-hid F p ⟩
F₁ (hid C p) ≈˘⟨ identityʳ ⟩
F₁ (hid C p) ∘ id ∎
}
counit : NaturalTransformation (Discrete ∘F Forgetful) idF
counit = ntHelper record { η = counitFun ; commute = counitComm }
zig : {A : Set o} → counitFun (Δ.F₀ A) ∘F (Δ.F₁ Fun.id) ≡F idF
zig {A} = record { eq₀ = λ _ → refl ; eq₁ = λ{ refl → refl } }
-- Codiscrete is right-adjoint to the forgetful functor from StrictCats to Sets
CodiscreteRightAdj : ∀ {o ℓ e} → Forgetful ⊣ Codiscrete {o} ℓ e
CodiscreteRightAdj {o} {ℓ} {e} = record
{ unit = unit
; counit = counit
; zig = refl
; zag = zag
}
where
module U = Functor Forgetful
module ∇ = Functor (Codiscrete ℓ e)
unitFun : ∀ C → Functor C (∇.F₀ (U.F₀ C))
unitFun C = let open Category C in record
{ F₀ = Fun.id
; F₁ = λ _ → lift tt
; identity = lift tt
; homomorphism = lift tt
; F-resp-≈ = λ _ → lift tt
}
unitComm : ∀ {C D} → (F : Functor C D) →
unitFun D ∘F F ≡F (∇.F₁ (U.F₁ F)) ∘F unitFun C
unitComm {C} {D} F = record { eq₀ = λ _ → refl ; eq₁ = λ _ → lift tt }
unit : NaturalTransformation idF (Codiscrete ℓ e ∘F Forgetful)
unit = ntHelper record { η = unitFun ; commute = unitComm }
counit : NaturalTransformation (Forgetful ∘F Codiscrete ℓ e) idF
counit = NT.id
zag : {B : Set o} → ∇.F₁ Fun.id ∘F unitFun (∇.F₀ B) ≡F idF
zag {B} = record { eq₀ = λ _ → refl ; eq₁ = λ _ → lift tt }
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module list-simplifier where
open import level
open import bool
open import functions
open import eq
open import empty
open import level
open import list
open import list-thms
open import nat
open import neq
open import product
open import product-thms
data 𝕃ʳ : Set → Set lone where
_ʳ : {A : Set} → 𝕃 A → 𝕃ʳ A
_++ʳ_ : {A : Set} → 𝕃ʳ A → 𝕃ʳ A → 𝕃ʳ A
mapʳ : {A B : Set} → (A → B) → 𝕃ʳ A → 𝕃ʳ B
_::ʳ_ : {A : Set} → A → 𝕃ʳ A → 𝕃ʳ A
[]ʳ : {A : Set} → 𝕃ʳ A
𝕃⟦_⟧ : {A : Set} → 𝕃ʳ A → 𝕃 A
𝕃⟦ l ʳ ⟧ = l
𝕃⟦ t1 ++ʳ t2 ⟧ = 𝕃⟦ t1 ⟧ ++ 𝕃⟦ t2 ⟧
𝕃⟦ mapʳ f t ⟧ = map f 𝕃⟦ t ⟧
𝕃⟦ x ::ʳ t ⟧ = x :: 𝕃⟦ t ⟧
𝕃⟦ []ʳ ⟧ = []
is-emptyʳ : {A : Set} → 𝕃ʳ A → 𝔹
is-emptyʳ []ʳ = tt
is-emptyʳ _ = ff
is-emptyʳ-≡ : {A : Set}(t : 𝕃ʳ A) → is-emptyʳ t ≡ tt → t ≡ []ʳ
is-emptyʳ-≡ []ʳ p = refl
is-emptyʳ-≡ (_ ++ʳ _) ()
is-emptyʳ-≡ (mapʳ _ _) ()
is-emptyʳ-≡ (_ ::ʳ _) ()
is-emptyʳ-≡ (_ ʳ) ()
𝕃ʳ-simp-step : {A : Set}(t : 𝕃ʳ A) → 𝕃ʳ A
𝕃ʳ-simp-step ((t1a ++ʳ t1b) ++ʳ t2) = t1a ++ʳ (t1b ++ʳ t2)
𝕃ʳ-simp-step ((x ::ʳ t1) ++ʳ t2) = x ::ʳ (t1 ++ʳ t2)
𝕃ʳ-simp-step ([]ʳ ++ʳ t2) = t2
𝕃ʳ-simp-step ((l ʳ) ++ʳ t2) =
if is-emptyʳ t2 then l ʳ else ((l ʳ) ++ʳ t2)
𝕃ʳ-simp-step ((mapʳ f t1) ++ʳ t2) =
if is-emptyʳ t2 then mapʳ f t1 else ((mapʳ f t1) ++ʳ t2)
𝕃ʳ-simp-step (mapʳ f (t1 ++ʳ t2)) = (mapʳ f t1) ++ʳ (mapʳ f t2)
𝕃ʳ-simp-step (mapʳ f (l ʳ)) = (map f l) ʳ
𝕃ʳ-simp-step (mapʳ f (mapʳ g t)) = mapʳ (f ∘ g) t
𝕃ʳ-simp-step (mapʳ f (x ::ʳ t)) = (f x) ::ʳ (mapʳ f t)
𝕃ʳ-simp-step (mapʳ f []ʳ) = []ʳ
𝕃ʳ-simp-step (l ʳ) = l ʳ
𝕃ʳ-simp-step (x ::ʳ t) = (x ::ʳ t)
𝕃ʳ-simp-step []ʳ = []ʳ
𝕃ʳ-simp-sdev : {A : Set}(t : 𝕃ʳ A) → 𝕃ʳ A
𝕃ʳ-simp-sdev (l ʳ) = (l ʳ)
𝕃ʳ-simp-sdev (t1 ++ʳ t2) = 𝕃ʳ-simp-step ((𝕃ʳ-simp-sdev t1) ++ʳ (𝕃ʳ-simp-sdev t2))
𝕃ʳ-simp-sdev (mapʳ f t1) = 𝕃ʳ-simp-step (mapʳ f (𝕃ʳ-simp-sdev t1))
𝕃ʳ-simp-sdev (x ::ʳ t1) = 𝕃ʳ-simp-step (x ::ʳ (𝕃ʳ-simp-sdev t1))
𝕃ʳ-simp-sdev []ʳ = []ʳ
𝕃ʳ-simp : {A : Set}(t : 𝕃ʳ A) → ℕ → 𝕃ʳ A
𝕃ʳ-simp t 0 = t
𝕃ʳ-simp t (suc n) = 𝕃ʳ-simp-sdev (𝕃ʳ-simp t n)
𝕃ʳ-simp-step-sound : {A : Set}(t : 𝕃ʳ A) → 𝕃⟦ t ⟧ ≡ 𝕃⟦ 𝕃ʳ-simp-step t ⟧
𝕃ʳ-simp-step-sound ((t1a ++ʳ t1b) ++ʳ t2) = ++-assoc 𝕃⟦ t1a ⟧ 𝕃⟦ t1b ⟧ 𝕃⟦ t2 ⟧
𝕃ʳ-simp-step-sound ((x ::ʳ t1) ++ʳ t2) = refl
𝕃ʳ-simp-step-sound ([]ʳ ++ʳ t2) = refl
𝕃ʳ-simp-step-sound ((l ʳ) ++ʳ t2) with keep (is-emptyʳ t2)
𝕃ʳ-simp-step-sound ((l ʳ) ++ʳ t2) | tt , p rewrite p | is-emptyʳ-≡ t2 p | ++[] l = refl
𝕃ʳ-simp-step-sound ((l ʳ) ++ʳ t2) | ff , p rewrite p = refl
𝕃ʳ-simp-step-sound ((mapʳ f t1) ++ʳ t2) with keep (is-emptyʳ t2)
𝕃ʳ-simp-step-sound ((mapʳ f t1) ++ʳ t2) | tt , p rewrite p | is-emptyʳ-≡ t2 p | ++[] (map f 𝕃⟦ t1 ⟧) = refl
𝕃ʳ-simp-step-sound ((mapʳ f t1) ++ʳ t2) | ff , p rewrite p = refl
𝕃ʳ-simp-step-sound (l ʳ) = refl
𝕃ʳ-simp-step-sound (mapʳ f (t1 ++ʳ t2)) = map-append f 𝕃⟦ t1 ⟧ 𝕃⟦ t2 ⟧
𝕃ʳ-simp-step-sound (mapʳ f (l ʳ)) = refl
𝕃ʳ-simp-step-sound (mapʳ f (mapʳ g t)) = map-compose f g 𝕃⟦ t ⟧
𝕃ʳ-simp-step-sound (mapʳ f (x ::ʳ t)) = refl
𝕃ʳ-simp-step-sound (mapʳ f []ʳ) = refl
𝕃ʳ-simp-step-sound (x ::ʳ t) = refl
𝕃ʳ-simp-step-sound []ʳ = refl
𝕃ʳ-simp-sdev-sound : {A : Set}(t : 𝕃ʳ A) → 𝕃⟦ t ⟧ ≡ 𝕃⟦ 𝕃ʳ-simp-sdev t ⟧
𝕃ʳ-simp-sdev-sound (l ʳ) = refl
𝕃ʳ-simp-sdev-sound (t1 ++ʳ t2)
rewrite sym (𝕃ʳ-simp-step-sound ((𝕃ʳ-simp-sdev t1) ++ʳ (𝕃ʳ-simp-sdev t2))) | 𝕃ʳ-simp-sdev-sound t1 | 𝕃ʳ-simp-sdev-sound t2 = refl
𝕃ʳ-simp-sdev-sound (mapʳ f t1)
rewrite sym (𝕃ʳ-simp-step-sound (mapʳ f (𝕃ʳ-simp-sdev t1))) | 𝕃ʳ-simp-sdev-sound t1 = refl
𝕃ʳ-simp-sdev-sound (x ::ʳ t1) rewrite 𝕃ʳ-simp-sdev-sound t1 = refl
𝕃ʳ-simp-sdev-sound []ʳ = refl
𝕃ʳ-simp-sound : {A : Set}(t : 𝕃ʳ A)(n : ℕ) → 𝕃⟦ t ⟧ ≡ 𝕃⟦ 𝕃ʳ-simp t n ⟧
𝕃ʳ-simp-sound t 0 = refl
𝕃ʳ-simp-sound t (suc n) rewrite sym (𝕃ʳ-simp-sdev-sound (𝕃ʳ-simp t n)) = 𝕃ʳ-simp-sound t n
module test1 {A B : Set}(f : A → B)(l1 l2 : 𝕃 A) where
lhs = (mapʳ f (l1 ʳ)) ++ʳ (mapʳ f (l2 ʳ))
rhs = mapʳ f ((l1 ʳ) ++ʳ (l2 ʳ))
test-tp : Set
test-tp = 𝕃⟦ lhs ⟧ ≡ 𝕃⟦ rhs ⟧
test : test-tp
test rewrite (𝕃ʳ-simp-sdev-sound rhs) = refl
module test2 {A B : Set}(f : A → B)(l1 l2 l3 : 𝕃 A) where
lhs = mapʳ f (((l1 ʳ) ++ʳ (l2 ʳ)) ++ʳ (l3 ʳ))
rhs = 𝕃ʳ-simp lhs 3
test-tp : Set
test-tp = 𝕃⟦ lhs ⟧ ≡ 𝕃⟦ rhs ⟧
test : test-tp
test = 𝕃ʳ-simp-sound lhs 3
one-step : 𝕃ʳ B
one-step = 𝕃ʳ-simp-step lhs
sdev : 𝕃ʳ B
sdev = 𝕃ʳ-simp-sdev lhs
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module Sized.SelfRef where
open import Data.Product
open import Data.String.Base
open import Size
open import SizedIO.Object
open import SizedIO.IOObject
open import SizedIO.ConsoleObject
open import SizedIO.Base
open import SizedIO.Console hiding (main)
open import NativeIO
data AMethod : Set where
print : AMethod
m1 : AMethod
m2 : AMethod
AResult : (c : AMethod) → Set
AResult _ = Unit
aI : Interface
Method aI = AMethod
Result aI = AResult
aC : (i : Size) → Set
aC i = ConsoleObject i aI
--
-- Self Referential: method 'm1' calls method 'm2'
--
{-# NON_TERMINATING #-}
aP : ∀{i} (s : String) → aC i
method (aP s) print =
exec1 (putStrLn s) >>
return (_ , aP s)
method (aP s) m1 =
exec1 (putStrLn s) >>
method (aP s) m2 >>= λ{ (_ , c₀) →
return (_ , c₀) }
method (aP s) m2 =
return (_ , aP (s ++ "->m2"))
program : String → IOConsole ∞ Unit
program arg =
let c₀ = aP ("start̄") in
method c₀ m1 >>= λ{ (_ , c₁) → --- ===> m1 called, then m2 prints out text
method c₁ print >>= λ{ (_ , c₂) →
exec1 (putStrLn "end") }}
main : NativeIO Unit
main = translateIOConsole (program "")
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module Formalization.ImperativePL where
open import Data.Boolean
open import Data.List
import Lvl
open import Numeral.Natural
open import String
open import Type
Ident : TYPE
data Lit : TYPE
data Stmt : TYPE
data Expr : TYPE
Ty : TYPE
record Context : TYPE₁
Ident = String
data Lit where
unit : Lit
bool : Bool → Lit
nat : ℕ → Lit
data Expr where
lit : Lit → Expr
ident : Ident → Expr
apply : Expr → Expr → Expr
func : Ident → Ty → Expr → Expr
ifelse : Expr → Expr → Expr → Expr
stmts : List(Stmt) → Expr
Ty = Expr
data Stmt where
expr : Expr → Stmt
decl : Ident → Ty → Stmt
def : Ident → Expr → Stmt
loop : Stmt → Stmt
return : ℕ → Expr → Stmt
record Context where
inductive
field
typing : Ident → Ty → TYPE
open import Logic.Propositional
open import Relator.Equals
add : Ident → Ty → Context → Context
Context.typing (add name ty ctx) a b = (Context.typing ctx a b) ∨ (a ≡ name)
pattern UnitTy = ident "Unit"
pattern BoolTy = ident "Bool"
pattern NatTy = ident "Nat"
pattern FnTy A B = apply (apply (ident "→") A) B
data _⊢ₛ_,_ : Context → Context → Stmt → TYPE₁
data _⊢ₑ_,_=:_ : Context → Context → Expr → Ty → TYPE₁
data _⊢ₛ_,_ where
expr : ∀{Γ₁ Γ₂}{e}{T} → (Γ₁ ⊢ₑ Γ₂ , e =: T) → (Γ₁ ⊢ₛ Γ₂ , expr e)
decl : ∀{Γ}{name}{T} → (Γ ⊢ₛ (add name T Γ) , decl name T)
def : ∀{Γ₁ Γ₂}{name}{e}{T} → Context.typing Γ₁ name T → (Γ₁ ⊢ₑ Γ₂ , e =: T) → (Γ₁ ⊢ₛ Γ₂ , def name e)
loop : ∀{Γ₁ Γ₂}{s} → (Γ₁ ⊢ₛ Γ₂ , s) → (Γ₁ ⊢ₛ Γ₁ , loop s)
data _⊢ₑ_,_=:_ where
unit : ∀{Γ} → (Γ ⊢ₑ Γ , (lit unit) =: UnitTy)
bool : ∀{Γ}{b} → (Γ ⊢ₑ Γ , (lit(bool b)) =: BoolTy)
nat : ∀{Γ}{n} → (Γ ⊢ₑ Γ , (lit(nat n)) =: NatTy)
ident : ∀{Γ}{i}{ty} → Context.typing Γ i ty → (Γ ⊢ₑ Γ , (ident i) =: ty)
apply : ∀{Γ₁ Γ₂ Γ₃}{f x}{A B} → (Γ₁ ⊢ₑ Γ₂ , x =: A) → (Γ₂ ⊢ₑ Γ₃ , f =: FnTy A B) → (Γ₁ ⊢ₑ Γ₃ , (apply f x) =: B)
func : ∀{Γ₁ Γ₂}{A B}{var}{body} → ((add var A Γ₁) ⊢ₑ Γ₂ , body =: B) → (Γ₁ ⊢ₑ Γ₁ , (func var A body) =: FnTy A B)
ifelse : ∀{Γ₁ Γ₂ Γ₃ Γ₄}{b t f}{T} → (Γ₁ ⊢ₑ Γ₂ , b =: BoolTy) → (Γ₂ ⊢ₑ Γ₃ , t =: T) → (Γ₂ ⊢ₑ Γ₄ , f =: T) → (Γ₁ ⊢ₑ Γ₂ , (ifelse b t f) =: T)
-- stmts∅ : ∀{Γ}{n} → (Γ ⊢ₑ Γ , (stmts ) =: NatTy)
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-- Tactic for proving coprimality.
-- Finds Coprime hypotheses in the context.
module Tactic.Nat.Coprime where
import Agda.Builtin.Nat as Builtin
open import Prelude
open import Control.Monad.Zero
open import Control.Monad.State
open import Container.List
open import Container.Traversable
open import Numeric.Nat
open import Tactic.Reflection
open import Tactic.Reflection.Parse
open import Tactic.Reflection.Quote
open import Tactic.Nat.Coprime.Problem
open import Tactic.Nat.Coprime.Decide
open import Tactic.Nat.Coprime.Reflect
private
Proof : Problem → Env → Set
Proof Q ρ with canProve Q
... | true = ⟦ Q ⟧p ρ
... | false = ⊤
erasePrf : ∀ Q {ρ} → ⟦ Q ⟧p ρ → ⟦ Q ⟧p ρ
erasePrf ([] ⊨ a ⋈ b) Ξ = eraseEquality Ξ
erasePrf (ψ ∷ Γ ⊨ φ) Ξ = λ H → erasePrf (Γ ⊨ φ) (Ξ H)
proof : ∀ Q ρ → Proof Q ρ
proof Q ρ with canProve Q | sound Q
... | true | prf = erasePrf Q (prf refl ρ)
... | false | _ = _
-- For the error message
unquoteE : List Term → Exp → Term
unquoteE ρ (atom x) = fromMaybe (lit (nat 0)) (index ρ x)
unquoteE ρ (e ⊗ e₁) = def₂ (quote _*_) (unquoteE ρ e) (unquoteE ρ e₁)
unquoteF : List Term → Formula → Term
unquoteF ρ (a ⋈ b) = def₂ (quote Coprime) (unquoteE ρ a) (unquoteE ρ b)
macro
auto-coprime : Tactic
auto-coprime ?hole = withNormalisation true $ do
goal ← inferType ?hole
ensureNoMetas goal
cxt ← reverse <$> getContext
(_ , Hyps , Q) , ρ ← runParse (parseProblem (map unArg cxt) goal)
unify ?hole (def (quote proof) $ map vArg (` Q ∷ quotedEnv ρ ∷ Hyps))
<|> do
case Q of λ where
(Γ ⊨ φ) → typeError (strErr "Cannot prove" ∷
termErr (unquoteF ρ φ) ∷
strErr "from" ∷
punctuate (strErr "and") (map (termErr ∘ unquoteF ρ) Γ))
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{-# OPTIONS --without-K #-}
module sets.int.core where
open import sum
open import equality
open import function
import sets.nat as N
open N using (ℕ; suc)
open import sets.int.definition
open import sets.int.utils
open import sets.vec
open import hott.level.closure
infixl 7 _*_
infixl 6 _+_
from-nat : ℕ → ℤ
from-nat n = n [-] 0
zero : ℤ
zero = from-nat 0
one : ℤ
one = from-nat 1
private
module _ where
open N
add : ℕ → ℕ → ℕ → ℕ → ℤ
add n n' m m' = (n + m) [-] (n' + m')
add-sound : (n n' d m m' e : ℕ)
→ add n n' m m'
≡ add (d + n) (d + n') (e + m) (e + m')
add-sound n n' d m m' e
= eq-ℤ (n + m) (n' + m') (d + e)
· ap₂ _[-]_ (solve 4 (exp λ d e n m → d :+ e :+ (n :+ m))
(exp λ d e n m → d :+ n :+ (e :+ m))
(d ∷∷ e ∷∷ n ∷∷ m ∷∷ ⟦⟧))
(solve 4 (exp λ d e n m → d :+ e :+ (n :+ m))
(exp λ d e n m → d :+ n :+ (e :+ m))
(d ∷∷ e ∷∷ n' ∷∷ m' ∷∷ ⟦⟧))
mul : ℕ → ℕ → ℕ → ℕ → ℤ
mul n n' m m' = (n * m + n' * m') [-] (n' * m + n * m')
mul-sound : (n n' d m m' e : ℕ)
→ mul n n' m m'
≡ mul (d + n) (d + n') (e + m) (e + m')
mul-sound n n' d m m' e
= eq-ℤ (n * m + n' * m') (n' * m + n * m')
(d * e + d * m + n * e + d * e + d * m' + n' * e)
· ap₂ _[-]_ lem₁ lem₂
where
distr₂ : ∀ a b c d
→ (a + b) * (c + d)
≡ a * c + a * d + b * c + b * d
distr₂ a b c d = right-distr a b (c + d)
· ap₂ _+_ (left-distr a c d) (left-distr b c d)
· sym (+-associativity (a * c + a * d) (b * c) (b * d))
lem₁ : d * e + d * m + n * e + d * e + d * m' + n' * e + (n * m + n' * m')
≡ (d + n) * (e + m) + (d + n') * (e + m')
lem₁ = sym
$ ap₂ _+_ (distr₂ d n e m) (distr₂ d n' e m')
· solve 7 (exp λ a b c d e f g → a :+ b :+ c :+ d :+ (a :+ e :+ f :+ g))
(exp λ a b c d e f g → a :+ b :+ c :+ a :+ e :+ f :+ (d :+ g))
((d * e) ∷∷ (d * m) ∷∷ (n * e) ∷∷ (n * m) ∷∷
(d * m') ∷∷ (n' * e) ∷∷ (n' * m') ∷∷ ⟦⟧)
lem₂ : d * e + d * m + n * e + d * e + d * m' + n' * e + (n' * m + n * m')
≡ (d + n') * (e + m) + (d + n) * (e + m')
lem₂ = sym
$ ap₂ _+_ (distr₂ d n' e m) (distr₂ d n e m')
· solve 7 (exp λ a b c d e f g → a :+ b :+ c :+ d :+ (a :+ e :+ f :+ g))
(exp λ a b c d e f g → a :+ b :+ f :+ a :+ e :+ c :+ (d :+ g))
((d * e) ∷∷ (d * m) ∷∷ (n' * e) ∷∷ (n' * m) ∷∷
(d * m') ∷∷ (n * e) ∷∷ (n * m') ∷∷ ⟦⟧)
neg : ℕ → ℕ → ℤ
neg n n' = n' [-] n
neg-sound : (n n' d : ℕ) → neg n n' ≡ neg (d + n) (d + n')
neg-sound n n' d = eq-ℤ n' n d
_+_ : ℤ → ℤ → ℤ
_+_ = elim₂-ℤ hℤ add add-sound
negate : ℤ → ℤ
negate = elim-ℤ hℤ (neg , neg-sound)
inc : ℤ → ℤ
inc n = one + n
dec : ℤ → ℤ
dec n = negate one + n
_*_ : ℤ → ℤ → ℤ
_*_ = elim₂-ℤ hℤ mul mul-sound
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-- Note that, at the time of writing, Agda does /not/ cache interfaces
-- if an internal error is encountered (at least in some cases).
import Issue2447.Internal-error
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module Error-in-imported-module where
open import Error-in-imported-module.M
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{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9.
Copyright (c) 2021, Oracle and/or its affiliates.
Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl
-}
open import LibraBFT.Base.Types
import LibraBFT.Impl.Consensus.BlockStorage.BlockStore as BlockStore
import LibraBFT.Impl.Consensus.ConsensusTypes.SyncInfo as SyncInfo
import LibraBFT.Impl.Consensus.ConsensusTypes.Vote as Vote
import LibraBFT.Impl.Consensus.PendingVotes as PendingVotes
import LibraBFT.Impl.OBM.ECP-LBFT-OBM-Diff.ECP-LBFT-OBM-Diff-1 as ECP-LBFT-OBM-Diff-1
open import LibraBFT.Impl.OBM.Logging.Logging
open import LibraBFT.Impl.OBM.Rust.Duration as Duration
open import LibraBFT.Impl.OBM.Rust.RustTypes
open import LibraBFT.Impl.OBM.Time
open import LibraBFT.ImplShared.Base.Types
open import LibraBFT.ImplShared.Consensus.Types
open import LibraBFT.ImplShared.Util.Crypto
open import LibraBFT.ImplShared.Util.Dijkstra.All
open import LibraBFT.Abstract.Types.EpochConfig UID NodeId
open import Optics.All
open import Util.Hash
open import Util.ByteString
open import Util.PKCS
open import Util.Prelude
------------------------------------------------------------------------------
open import Agda.Builtin.Float
module LibraBFT.Impl.Consensus.Liveness.RoundState where
------------------------------------------------------------------------------
new : RoundStateTimeInterval → Instant → RoundState
new ti i = mkRoundState
{-_rsTimeInterval =-} ti
{-_rsHighestCommittedRound =-} {-Round-} 0
{-_rsCurrentRound =-} {-Round-} 0
{-_rsCurrentRoundDeadline =-} i
{-_rsPendingVotes =-} PendingVotes∙new
{-_rsVoteSent =-} nothing
------------------------------------------------------------------------------
setupTimeoutM : Instant → LBFT Duration
processLocalTimeoutM : Instant → Epoch → Round → LBFT Bool
processLocalTimeoutM now obmEpoch round = do
currentRound ← use (lRoundState ∙ rsCurrentRound)
ifD round /= currentRound
then pure false
else do
void (setupTimeoutM now) -- setup the next timeout
ECP-LBFT-OBM-Diff-1.e_RoundState_processLocalTimeoutM obmEpoch round
------------------------------------------------------------------------------
maybeAdvanceRound : Round → SyncInfo → Maybe (Round × NewRoundReason)
processCertificatesM : Instant → SyncInfo → LBFT (Maybe NewRoundEvent)
processCertificatesM now syncInfo = do
-- logEE ("RoundState" ∷ "processCertificatesM" {-∷ lsSI syncInfo-} ∷ []) $ do
rshcr <- use (lRoundState ∙ rsHighestCommittedRound)
whenD (syncInfo ^∙ siHighestCommitRound >? rshcr) $ do
lRoundState ∙ rsHighestCommittedRound ∙= (syncInfo ^∙ siHighestCommitRound)
logInfo fakeInfo -- InfoUpdateHighestCommittedRound (syncInfo^.siHighestCommitRound)
rscr ← use (lRoundState ∙ rsCurrentRound)
maybeSD (maybeAdvanceRound rscr syncInfo) (pure nothing) $ λ (pcr' , reason) → do
lRoundState ∙ rsCurrentRound ∙= pcr'
lRoundState ∙ rsPendingVotes ∙= PendingVotes∙new
lRoundState ∙ rsVoteSent ∙= nothing
timeout ← setupTimeoutM now
pure (just (NewRoundEvent∙new pcr' reason timeout))
abstract
processCertificatesM-abs = processCertificatesM
processCertificatesM-abs-≡ : processCertificatesM-abs ≡ processCertificatesM
processCertificatesM-abs-≡ = refl
maybeAdvanceRound currentRound syncInfo =
let newRound = SyncInfo.highestRound syncInfo + 1
in if-dec newRound >? currentRound
then just (newRound , (if is-nothing (syncInfo ^∙ siHighestTimeoutCert) then QCReady else TOReady))
else nothing
------------------------------------------------------------------------------
insertVoteM : Vote → ValidatorVerifier → LBFT VoteReceptionResult
insertVoteM vote verifier = do
currentRound ← use (lRoundState ∙ rsCurrentRound)
ifD vote ^∙ vVoteData ∙ vdProposed ∙ biRound == currentRound
then PendingVotes.insertVoteM vote verifier
else pure (UnexpectedRound (vote ^∙ vVoteData ∙ vdProposed ∙ biRound) currentRound)
------------------------------------------------------------------------------
recordVoteM : Vote → LBFT Unit
recordVoteM v = rsVoteSent-rm ∙= just v
------------------------------------------------------------------------------
setupDeadlineM : Instant → LBFT Duration
roundIndexAfterCommittedRound : Round → Round → Round
getRoundDuration : ExponentialTimeInterval → Round → Duration
setupTimeoutM now = do
timeout ← setupDeadlineM now
r ← use (lRoundState ∙ rsCurrentRound)
-- act (SetTimeout timeout r)
pure timeout
setupDeadlineM now = do
ti ← use (lRoundState ∙ rsTimeInterval)
cr ← use (lRoundState ∙ rsCurrentRound)
hcr ← use (lRoundState ∙ rsHighestCommittedRound)
let timeout = getRoundDuration ti (roundIndexAfterCommittedRound cr hcr)
lRoundState ∙ rsCurrentRoundDeadline ∙= iPlus now timeout
pure timeout
roundIndexAfterCommittedRound currentRound highestCommittedRound =
grd‖ highestCommittedRound == 0 ≔ currentRound ∸ 1
‖ currentRound <?ℕ highestCommittedRound + 3 ≔ 0
‖ otherwise≔ currentRound ∸ highestCommittedRound ∸ 3
postulate -- TODO-1 : _**_, ceiling (for Floats)
_**_ : Float → Float → Float
ceiling : Float → U64
getRoundDuration i r =
let pow = min r (i ^∙ etiMaxExponent) -- TODO/NOTE: cap on max timeout
-- undermines theoretical liveness properties
baseMultiplier = (i ^∙ etiExponentBase) ** {-fromIntegral-} primNatToFloat pow
--durationMs = ceiling (fromIntegral (i^.etiBaseMs) * baseMultiplier)
durationMs = ceiling (primFloatTimes (primNatToFloat (i ^∙ etiBaseMs)) baseMultiplier)
in Duration.fromMillis durationMs
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module Issue2639.DR (Z : Set) where
open import Agda.Builtin.Size
mutual
data D (i : Size) : Set where
a : D i
b : R i → D i
record R (i : Size) : Set where
coinductive
field
force : {j : Size< i} → D j
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-- This code is based on "Parallel Parsing Processes" by Koen
-- Claessen.
-- This module is a variant of Parallel in which Parser uses mixed
-- induction/coinduction.
module Parallel2 {Tok : Set} where
open import Coinduction
open import Data.Bool
import Data.Bool.Properties as Bool
open import Algebra
open import Data.Product
open import Data.Function
import Data.List as List
open List using (List; []; _∷_)
open import Data.Vec using ([]; _∷_)
open import Data.Fin using (#_)
open import Category.Monad.State
open import Relation.Binary.PropositionalEquality
open ≡-Reasoning
------------------------------------------------------------------------
-- Boring lemma
private
lem₁ : ∀ b₁ b₂ → b₁ ∨ b₂ ∧ b₁ ≡ b₁
lem₁ b₁ b₂ = begin
b₁ ∨ b₂ ∧ b₁ ≡⟨ cong (_∨_ b₁) (B.∧-comm b₂ b₁) ⟩
b₁ ∨ b₁ ∧ b₂ ≡⟨ proj₁ B.absorptive b₁ b₂ ⟩
b₁ ∎
where module B = BooleanAlgebra Bool.booleanAlgebra
------------------------------------------------------------------------
-- Parser monad
P : Set → Set
P = StateT (List Tok) List
------------------------------------------------------------------------
-- Basic parsers (no CPS)
-- Note that the recursive argument of symbolBind is coinductive,
-- while that of returnPlus is inductive. An infinite choice is not
-- acceptable, but an infinite tree of potential parsers is fine.
data Parser (R : Set) : Bool → Set where
symbolBind : {e : Tok → Bool} →
(f : (x : Tok) → ∞ (Parser R (e x))) → Parser R false
fail : Parser R false
returnPlus : ∀ {e} (x : R) (p : Parser R e) → Parser R true
parse : ∀ {R e} → Parser R e → P R
parse (symbolBind f) [] = []
parse (symbolBind f) (x ∷ xs) = parse (♭ (f x)) xs
parse fail _ = []
parse (returnPlus x p) xs = (x , xs) ∷ parse p xs
cast : ∀ {R e₁ e₂} → e₁ ≡ e₂ → Parser R e₁ → Parser R e₂
cast refl = id
return : ∀ {R} → R → Parser R true
return x = returnPlus x fail
module DirectImplementations where
-- The definition of _∣_ is fine, but is the definition of _>>=_
-- acceptable?
infixl 1 _>>=_
infixl 0 _∣_
_∣_ : ∀ {R e₁ e₂} → Parser R e₁ → Parser R e₂ → Parser R (e₁ ∨ e₂)
symbolBind f₁ ∣ symbolBind f₂ = symbolBind (λ x → ♯ (♭ (f₁ x) ∣ ♭ (f₂ x)))
fail ∣ p₂ = p₂
symbolBind f₁ ∣ fail = symbolBind f₁
returnPlus x₁ p₁ ∣ fail = returnPlus x₁ p₁
returnPlus x₁ p₁ ∣ p₂ = returnPlus x₁ (p₁ ∣ p₂)
symbolBind f₁ ∣ returnPlus x₂ p₂ = returnPlus x₂ (symbolBind f₁ ∣ p₂)
_>>=_ : ∀ {R₁ R₂ e₁ e₂} →
Parser R₁ e₁ → (R₁ → Parser R₂ e₂) → Parser R₂ (e₁ ∧ e₂)
symbolBind f₁ >>= f₂ = symbolBind (λ x → ♯ (♭ (f₁ x) >>= f₂))
fail >>= f₂ = fail
returnPlus {e} x₁ p₁ >>= f₂ = cast (lem₁ _ e) (f₂ x₁ ∣ p₁ >>= f₂)
-- Implementing _!>>=_ seems tricky.
-- _!>>=_ : ∀ {R₁ R₂} {e₂ : R₁ → Bool} →
-- Parser R₁ false → ((x : R₁) → Parser R₂ (e₂ x)) →
-- Parser R₂ false
-- symbolBind f !>>= p₂ = symbolBind (λ x → ♯ {!♭ (f x) !>>= p₂!})
-- fail !>>= p₂ = fail
module IndirectImplementations where
-- Can _>>=_ be implemented by using the productivity trick?
infixl 1 _>>=_
infixl 0 _∣_
data PProg : Set → Bool → Set1 where
symbolBind : ∀ {R} {e : Tok → Bool} →
(f : (x : Tok) → ∞₁ (PProg R (e x))) → PProg R false
fail : ∀ {R} → PProg R false
returnPlus : ∀ {R e} (x : R) (p : PProg R e) → PProg R true
_∣_ : ∀ {R e₁ e₂}
(p₁ : PProg R e₁) (p₂ : PProg R e₂) → PProg R (e₁ ∨ e₂)
_>>=_ : ∀ {R₁ R₂ e₁ e₂}
(p₁ : PProg R₁ e₁) (f₂ : R₁ → PProg R₂ e₂) →
PProg R₂ (e₁ ∧ e₂)
data PWHNF (R : Set) : Bool → Set1 where
symbolBind : {e : Tok → Bool} →
(f : (x : Tok) → PProg R (e x)) → PWHNF R false
fail : PWHNF R false
returnPlus : ∀ {e} (x : R) (p : PWHNF R e) → PWHNF R true
-- _∣_ is a program, so implementing whnf seems challenging.
whnf : ∀ {R e} → PProg R e → PWHNF R e
whnf (symbolBind f) = symbolBind (λ x → ♭₁ (f x))
whnf fail = fail
whnf (returnPlus x p) = returnPlus x (whnf p)
whnf (p₁ ∣ p₂) with whnf p₁
... | fail = whnf p₂
... | returnPlus x₁ p₁′ = returnPlus x₁ {!(p₁′ ∣ p₂)!} -- (p₁′ ∣ p₂)
... | symbolBind f₁ with whnf p₂
... | symbolBind f₂ = symbolBind (λ x → f₁ x ∣ f₂ x)
... | fail = symbolBind f₁
... | returnPlus x₂ p₂′ = returnPlus x₂ {!!} -- (symbolBind f₁ ∣ p₂′)
whnf (p₁ >>= f₂) with whnf p₁
... | symbolBind f₁ = symbolBind (λ x → f₁ x >>= f₂)
... | fail = fail
... | returnPlus x₁ p₁′ = {!f₂ x₁ ∣ p₁′ >>= f₂!}
mutual
value : ∀ {R e} → PWHNF R e → Parser R e
value (symbolBind f) = symbolBind (λ x → ♯ ⟦ f x ⟧)
value fail = fail
value (returnPlus x p) = returnPlus x (value p)
⟦_⟧ : ∀ {R e} → PProg R e → Parser R e
⟦ p ⟧ = value (whnf p)
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module DerivedProps where
{-
open import Relation.Binary.PropositionalEquality
open import Data.Nat hiding (_>_)
-}
open import StdLibStuff
open import Syntax
open import STT
lem3 : {n : ℕ} {Γ-t : Ctx n} (F G : Form Γ-t $o) → ⊢ ((F => (F => G)) => (F => G))
lem3 F G = inf-V (inf-V ax-4-s ax-1-s) (inf-V (inf-V ax-4-s (inf-V ax-4-s ax-2-s)) ax-3-s)
lem4 : {n : ℕ} {Γ-t : Ctx n} (F G H : Form Γ-t $o) →
⊢ ((F => (G => H)) => (G => (F => H)))
lem4 F G H = inf-V (inf-V ax-4-s ax-1-s) (inf-V (inf-V ax-4-s (inf-V ax-4-s (inf-V (inf-V ax-4-s (inf-V (inf-V ax-4-s ax-3-s) ax-2-s)) ax-2-s))) (inf-V (inf-V ax-4-s ax-3-s) (inf-V ax-4-s (inf-V ax-4-s (inf-V (inf-V ax-4-s ax-3-s) ax-2-s)))))
lem5 : {n : ℕ} {Γ-t : Ctx n} (F G : Form Γ-t $o) →
⊢ ((F => G) => (~ G => ~ F))
lem5 F G = inf-V (inf-V ax-4-s ax-3-s) (inf-V ax-4-s (inf-V ax-3-s (inf-V (inf-V ax-4-s ax-1-s) ax-2-s)))
lem6 : {n : ℕ} {Γ-t : Ctx n} (F G H : Form Γ-t $o) →
⊢ ((F => H) => ((F & G) => H))
lem6 F G H = inf-V (inf-V ax-4-s ax-3-s) (inf-V (inf-V ax-4-s (inf-V ax-4-s (inf-V (inf-V ax-4-s (inf-V ax-3-s (inf-V (inf-V ax-4-s ax-1-s) ax-2-s))) ax-2-s))) ax-3-s)
lem6b : {n : ℕ} {Γ-t : Ctx n} (F G H : Form Γ-t $o) →
⊢ ((G => H) => ((F & G) => H))
lem6b F G H = inf-V (inf-V ax-4-s ax-3-s) (inf-V (inf-V ax-4-s (inf-V ax-4-s (inf-V (inf-V ax-4-s (inf-V ax-3-s (inf-V (inf-V ax-4-s ax-1-s) ax-2-s))) (inf-V (inf-V ax-4-s ax-3-s) ax-2-s)))) ax-3-s)
lemb3 : {n : ℕ} {Γ-t : Ctx n} (F G : Form Γ-t $o) →
⊢ (F => ((F => G) => G))
lemb3 F G = inf-V ax-1-s (inf-V (inf-V ax-4-s ax-3-s) (inf-V (inf-V ax-4-s ax-2-s) (inf-V (inf-V ax-4-s ax-2-s) (inf-V ax-3-s (inf-V ax-4-s (inf-V (inf-V ax-4-s ax-3-s) ax-2-s))))))
lem7 : {n : ℕ} {Γ-t : Ctx n} (F G H : Form Γ-t $o) →
⊢ (F => ((G => H) => ((F => G) => H)))
lem7 F G H = inf-V (inf-V ax-4-s (lem4 (F => G) (G => H) H)) (inf-V (inf-V ax-4-s (inf-V ax-4-s (lemb3 G H))) (lemb3 F G))
lem8h : {n : ℕ} {Γ-t : Ctx n} (F G H : Form Γ-t $o) →
⊢ ((H => (F => G)) => ((H => F) => (H => G)))
lem8h F G H = inf-V (inf-V ax-4-s ax-1-s) (inf-V (inf-V ax-4-s (inf-V ax-4-s (inf-V (inf-V ax-4-s ax-3-s) (inf-V (inf-V ax-4-s ax-2-s) ax-2-s)))) (inf-V (inf-V ax-4-s ax-3-s) (inf-V ax-4-s ax-4-s)))
lem8 : {n : ℕ} {Γ-t : Ctx n} (X F G H : Form Γ-t $o) →
⊢ ((F => (G => H)) => ((X => F) => ((X => G) => (X => H))))
lem8 X F G H = inf-V (inf-V ax-4-s (inf-V ax-4-s (lem8h G H X))) (inf-V (inf-V ax-4-s (lem8h F (G => H) X)) (inf-V (lem4 X (F => (G => H)) (F => (G => H))) (inf-V ax-3-s (inf-V ax-2-s (inf-V (inf-V ax-4-s ax-1-s) ax-2-s)))))
lem8-3 : {n : ℕ} {Γ-t : Ctx n} (X F G H I : Form Γ-t $o) →
⊢ ((F => (G => (H => I))) => ((X => F) => ((X => G) => ((X => H) => (X => I)))))
lem8-3 X F G H I = inf-V (inf-V ax-4-s (inf-V ax-4-s (inf-V ax-4-s (lem8h H I X)))) (inf-V (inf-V ax-4-s (inf-V ax-4-s (lem8h G (H => I) X))) (inf-V (inf-V ax-4-s (lem8h F (G => (H => I)) X)) (inf-V (lem4 X (F => (G => (H => I))) (F => (G => (H => I)))) (inf-V ax-3-s (inf-V ax-2-s (inf-V (inf-V ax-4-s ax-1-s) ax-2-s))))))
lemb2 : {n : ℕ} {Γ-t : Ctx n} (F : Form Γ-t $o) →
⊢ (~ (~ F) => F)
lemb2 F = inf-V ax-3-s (inf-V (inf-V ax-4-s (inf-V ax-3-s (inf-V (inf-V ax-4-s ax-1-s) ax-2-s))) (inf-V ax-3-s (inf-V (inf-V ax-4-s ax-1-s) ax-2-s)))
lem2h2h : {n : ℕ} {Γ-t : Ctx n} (F : Form Γ-t $o) →
⊢ (F => ~ (~ F))
lem2h2h F = inf-V ax-3-s (inf-V (inf-V ax-4-s ax-1-s) ax-2-s)
lem2h2hb : {n : ℕ} {Γ-t : Ctx n} (F : Form Γ-t $o) →
⊢ (F => F)
lem2h2hb F = inf-V (inf-V ax-4-s ax-1-s) ax-2-s
lem2h2 : {n : ℕ} {Γ-t : Ctx n} (F G H : Form Γ-t $o) →
⊢ ((F || (G || H)) => ((F || G) || H))
lem2h2 F G H = inf-V (inf-V ax-4-s (inf-V ax-4-s (lemb2 H))) (inf-V (inf-V ax-4-s ax-3-s) (inf-V (inf-V ax-4-s (inf-V ax-4-s ax-3-s)) (inf-V (inf-V ax-4-s (inf-V ax-4-s (inf-V ax-4-s (lemb2 F)))) (inf-V (inf-V ax-4-s (inf-V ax-4-s ax-3-s)) (inf-V (inf-V ax-4-s (lem4 (~ F) (~ H) G)) (inf-V (inf-V ax-4-s ax-3-s) (inf-V (inf-V ax-4-s (inf-V ax-4-s (lem2h2h F))) (inf-V (inf-V ax-4-s ax-3-s) (inf-V (inf-V ax-4-s (inf-V ax-4-s ax-3-s)) (inf-V (inf-V ax-4-s (inf-V ax-4-s (inf-V ax-4-s (lem2h2h H)))) (lem2h2hb (F || (G || H)))))))))))))
lem2h1 : {n : ℕ} {Γ-t : Ctx n} (F G H : Form Γ-t $o) →
⊢ ((F => (G => H)) => ((F & G) => H))
lem2h1 F G H = inf-V (inf-V ax-4-s ax-3-s) (inf-V (inf-V ax-4-s (inf-V ax-4-s (inf-V ax-3-s (inf-V (inf-V ax-4-s ax-1-s) ax-2-s)))) (inf-V (inf-V ax-4-s ax-3-s) (inf-V (inf-V ax-4-s (lem2h2 (~ F) (~ G) H)) (inf-V (inf-V ax-4-s ax-1-s) ax-2-s))))
lemb1 : {n : ℕ} {Γ-t : Ctx n} (F G : Form Γ-t $o) →
⊢ (F => (G => (F & G)))
lemb1 F G = inf-V ax-1-s (inf-V (inf-V ax-4-s ax-3-s) (inf-V (inf-V ax-4-s ax-2-s) (inf-V (inf-V ax-4-s ax-3-s) (inf-V (inf-V ax-4-s ax-2-s) (inf-V ax-3-s (inf-V ax-4-s ax-2-s))))))
lem2 : {n : ℕ} {Γ-t : Ctx n} (F G H : Form Γ-t $o) → ⊢ (((F => G) & (F => H)) => (F => (G & H)))
lem2 F G H = inf-V (lem2h1 (F => G) (F => H) (F => (G & H))) (inf-V (lem8 F G H (G & H)) (lemb1 G H))
lem9 : {n : ℕ} {Γ-t : Ctx n} (F G H : Form Γ-t $o) →
⊢ ((F => G) => ((G => H) => (F => H)))
lem9 F G H = inf-V (inf-V ax-4-s (lem4 F (G => H) H)) (inf-V (inf-V ax-4-s (inf-V ax-4-s (lemb3 G H))) (inf-V (lem4 F (F => G) G) (lemb3 F G)))
lemb4 : {n : ℕ} {Γ-t : Ctx n} (F G : Form Γ-t $o) →
⊢ (F => (~ F => G))
lemb4 F G = inf-V (inf-V ax-4-s ax-2-s) (inf-V ax-3-s (inf-V (inf-V ax-4-s ax-1-s) ax-2-s))
lem5r : {n : ℕ} {Γ-t : Ctx n} (F G : Form Γ-t $o) →
⊢ ((~ F => ~ G) => (G => F))
lem5r F G = inf-V (inf-V ax-4-s (inf-V ax-4-s (lemb2 F))) (inf-V (inf-V ax-4-s ax-3-s) (lem2h2hb (~ F => ~ G)))
lemb5 : {n : ℕ} {Γ-t : Ctx n} (F G H : Form Γ-t $o) →
⊢ ((F => H) => ((G => H) => ((F || G) => H)))
lemb5 F G H = inf-V (inf-V ax-4-s (inf-V ax-4-s (lem5r H (F || G)))) (inf-V (inf-V ax-4-s ax-3-s) (inf-V (inf-V ax-4-s (inf-V ax-4-s (inf-V (lem5 (G => H) (~ H => ~ G)) (lem5 G H)))) (inf-V (inf-V ax-4-s ax-3-s) (inf-V ax-3-s (inf-V (inf-V ax-4-s (inf-V (lem5 (F => H) (~ H => ~ F)) (lem5 F H))) (inf-V ax-3-s (inf-V (lem8 (~ H) (~ F) (~ G) (~ (F || G))) (inf-V (inf-V ax-4-s (inf-V ax-4-s (inf-V (lem5 (F || G) (~ (~ F) || ~ (~ G))) (inf-V (inf-V ax-4-s (inf-V ax-4-s (lem2h2h G))) (inf-V (inf-V ax-4-s ax-3-s) (inf-V (inf-V ax-4-s (inf-V ax-4-s (lem2h2h F))) (inf-V (inf-V ax-4-s ax-3-s) (lem2h2hb (F || G))))))))) (lemb1 (~ F) (~ G))))))))))
-- ---------------
lem10 : {n : ℕ} {Γ-t : Ctx n} (F : Form Γ-t $o) →
⊢ ((F => $false) => ~ F)
lem10 F = inf-V ax-3-s (inf-V (inf-V ax-4-s (inf-V (lem5 (F => $false) (F => ~ F)) (inf-V ax-4-s (inf-V (inf-V ax-4-s (ax-5-s ($ this {refl}) (~ F))) (lem2h2hb (![ _ ] ($ this {refl}))))))) (inf-V ax-3-s ax-1-s))
lem11 : {n : ℕ} {Γ-t : Ctx n} (F : Form Γ-t $o) →
⊢ ($false => F)
lem11 F = inf-V (inf-V ax-4-s (ax-5-s ($ this {refl}) F)) (lem2h2hb (![ _ ] ($ this {refl})))
lem12 : {n : ℕ} {Γ-t : Ctx n} {t : Type n} (F : Form Γ-t t) →
⊢ (F == F)
lem12 {n} {Γ-t} {t} F = inf-V (inf-III-samectx {n} {Γ-t} {t} {t > $o} (λ z → (z · F)) (^[ t ] (![ t > $o ] (($ this {refl} · $ (next (next this)) {refl}) => ($ this {refl} · $ (next this) {refl})))) F) (inf-V (inf-III-samectx {n} {Γ-t} {t} {$o} (λ z → (z)) (![ t > $o ] (($ this {refl} · weak (weak F)) => ($ this {refl} · $ (next this) {refl}))) F) (inf-VI-s (subst (λ z → ⊢ ((($ this {refl} · z) => ($ this {refl} · weak F)))) (sub-weak-p-1 F F) (inf-V (inf-V ax-4-s ax-1-s) ax-2-s))))
lem13 : {n : ℕ} {Γ-t : Ctx n} (F G : Form Γ-t $o) →
⊢ ((G == F) => (F => G))
lem13 F G =
inf-V (inf-III-samectx (λ z → ((z · F) => (F => G))) (^[ $o ] (![ $o > $o ] (($ this {refl} · $ (next (next this)) {refl}) => ($ this {refl} · $ (next this) {refl})))) G) (
inf-V (inf-III-samectx (λ z → (z => (F => G))) (![ $o > $o ] (($ this {refl} · weak (weak G)) => ($ this {refl} · $ (next this) {refl}))) F) (
subst (λ z → ⊢ ((![ $o > $o ] (($ this {refl} · z) => ($ this {refl} · weak F))) => (F => G))) (sub-weak-p-1 G F) (
inf-V ax-3-s (
inf-V (inf-V ax-4-s (inf-V (lem5 (![ $o > $o ] (($ this {refl} · weak G) => ($ this {refl} · weak F))) (((^[ $o ] ~ ($ this {refl})) · sub (^[ $o ] ~ ($ this {refl})) (weak G)) => ((^[ $o ] ~ ($ this {refl})) · sub (^[ $o ] ~ ($ this {refl})) (weak F))))
(ax-5-s (($ this {refl} · weak G) => ($ this {refl} · weak F)) (^[ $o ] ~ ($ this {refl}))))) (
inf-V ax-3-s (
subst (λ z → ⊢ ((((^[ $o ] ~ ($ this {refl})) · z) => ((^[ $o ] ~ ($ this {refl})) · sub (^[ $o ] ~ ($ this {refl})) (weak F))) => (F => G))) (sub-weak-p-1' G (^[ $o ] ~ ($ this {refl}))) (
subst (λ z → ⊢ ((((^[ $o ] ~ ($ this {refl})) · G) => ((^[ $o ] ~ ($ this {refl})) · z)) => (F => G))) (sub-weak-p-1' F (^[ $o ] ~ ($ this {refl}))) (
inf-V (inf-III-samectx (λ z → ((z => ((^[ $o ] ~ ($ this {refl})) · F)) => (F => G))) (~ ($ this {refl})) G) (
inf-V (inf-III-samectx (λ z → (((~ G) => z) => (F => G))) (~ ($ this {refl})) F) (
lem5r G F
))))))))))
transitivity : {n : ℕ} {Γ-t : Ctx n} {t : Type n} (F G H : Form Γ-t t) →
⊢ ((F == G) => ((G == H) => (F == H)))
transitivity F G H =
inf-V (inf-III-samectx (λ z → ((z · G) => ((G == H) => (F == H)))) (^[ _ ] (![ _ > $o ] (($ this {refl} · $ (next (next this)) {refl}) => ($ this {refl} · $ (next this) {refl})))) F) (
inf-V (inf-III-samectx (λ z → (z => ((G == H) => (F == H)))) (![ _ > $o ] (($ this {refl} · weak (weak F)) => ($ this {refl} · $ (next this) {refl}))) G) (
subst (λ z → ⊢ ((![ _ > $o ] (($ this {refl} · z) => ($ this {refl} · weak G))) => ((G == H) => (F == H)))) (sub-weak-p-1 F G) (
inf-V (inf-III-samectx (λ z → ((![ _ > $o ] (($ this {refl} · weak F) => ($ this {refl} · weak G))) => ((z · H) => (F == H)))) (^[ _ ] (![ _ > $o ] (($ this {refl} · $ (next (next this)) {refl}) => ($ this {refl} · $ (next this) {refl})))) G) (
inf-V (inf-III-samectx (λ z → ((![ _ > $o ] (($ this {refl} · weak F) => ($ this {refl} · weak G))) => (z => (F == H)))) (![ _ > $o ] (($ this {refl} · weak (weak G)) => ($ this {refl} · $ (next this) {refl}))) H) (
subst (λ z → ⊢ ((![ _ > $o ] (($ this {refl} · weak F) => ($ this {refl} · weak G))) => ((![ _ > $o ] (($ this {refl} · z) => ($ this {refl} · weak H))) => (F == H)))) (sub-weak-p-1 G H) (
inf-V (inf-III-samectx (λ z → ((![ _ > $o ] (($ this {refl} · weak F) => ($ this {refl} · weak G))) => ((![ _ > $o ] (($ this {refl} · weak G) => ($ this {refl} · weak H))) => (z · H)))) (^[ _ ] (![ _ > $o ] (($ this {refl} · $ (next (next this)) {refl}) => ($ this {refl} · $ (next this) {refl})))) F) (
inf-V (inf-III-samectx (λ z → ((![ _ > $o ] (($ this {refl} · weak F) => ($ this {refl} · weak G))) => ((![ _ > $o ] (($ this {refl} · weak G) => ($ this {refl} · weak H))) => z))) (![ _ > $o ] (($ this {refl} · weak (weak F)) => ($ this {refl} · $ (next this) {refl}))) H) (
subst (λ z → ⊢ ((![ _ > $o ] (($ this {refl} · weak F) => ($ this {refl} · weak G))) => ((![ _ > $o ] (($ this {refl} · weak G) => ($ this {refl} · weak H))) => (![ _ > $o ] (($ this {refl} · z) => ($ this {refl} · weak H)))))) (sub-weak-p-1 F H) (
inf-V (inf-V ax-4-s (ax-6-s {_} {_} {_} {~ (![ _ ] (($ this {refl} · weak G) => ($ this {refl} · weak H)))} {($ this {refl} · weak F) => ($ this {refl} · weak H)})) (
inf-V (ax-6-s {_} {_} {_} {~ (![ _ ] (($ this {refl} · weak F) => ($ this {refl} · weak G)))} {(weak (![ _ ] (($ this {refl} · weak G) => ($ this {refl} · weak H)))) => (($ this {refl} · weak F) => ($ this {refl} · weak H))}) (
inf-VI-s (
inf-V ax-3-s (
inf-V (inf-V ax-4-s (inf-V (lem5 (![ _ ] (($ this {refl} · weak-i (_ ∷ ε) _ (weak F)) => ($ this {refl} · weak-i (_ ∷ ε) _ (weak G)))) (sub ($ this {refl}) (($ this {refl} · weak-i (_ ∷ ε) _ (weak F)) => ($ this {refl} · weak-i (_ ∷ ε) _ (weak G)))))
(ax-5-s (($ this {refl} · weak-i (_ ∷ ε) _ (weak F)) => ($ this {refl} · weak-i (_ ∷ ε) _ (weak G))) ($ this {refl})))) (
inf-V ax-3-s (
inf-V (inf-V ax-4-s ax-3-s) (
inf-V (inf-V ax-4-s (inf-V ax-4-s (inf-V (lem5 (![ _ ] (($ this {refl} · weak-i (_ ∷ ε) _ (weak G)) => ($ this {refl} · weak-i (_ ∷ ε) _ (weak H)))) (sub ($ this {refl}) (($ this {refl} · weak-i (_ ∷ ε) _ (weak G)) => ($ this {refl} · weak-i (_ ∷ ε) _ (weak H)))))
(ax-5-s (($ this {refl} · weak-i (_ ∷ ε) _ (weak G)) => ($ this {refl} · weak-i (_ ∷ ε) _ (weak H))) ($ this {refl}))))) (
inf-V (inf-V ax-4-s ax-3-s) (
subst (λ z → ⊢ (((($ this {refl}) · z) => (($ this {refl}) · (sub ($ this {refl}) (weak-i (_ ∷ ε) _ (weak G))))) => (((($ this {refl}) · (sub ($ this {refl}) (weak-i (_ ∷ ε) _ (weak G)))) => (($ this {refl}) · (sub ($ this {refl}) (weak-i (_ ∷ ε) _ (weak H))))) => (($ this {refl} · weak F) => ($ this {refl} · weak H))))) (sub-weak-p-23 F ($ this {refl})) (
subst (λ z → ⊢ (((($ this {refl}) · (weak F)) => (($ this {refl}) · z)) => (((($ this {refl}) · (sub ($ this {refl}) (weak-i (_ ∷ ε) _ (weak G)))) => (($ this {refl}) · (sub ($ this {refl}) (weak-i (_ ∷ ε) _ (weak H))))) => (($ this {refl} · weak F) => ($ this {refl} · weak H))))) (sub-weak-p-23 G ($ this {refl})) (
subst (λ z → ⊢ (((($ this {refl}) · (weak F)) => (($ this {refl}) · (weak G))) => (((($ this {refl}) · z) => (($ this {refl}) · (sub ($ this {refl}) (weak-i (_ ∷ ε) _ (weak H))))) => (($ this {refl} · weak F) => ($ this {refl} · weak H))))) (sub-weak-p-23 G ($ this {refl})) (
subst (λ z → ⊢ (((($ this {refl}) · (weak F)) => (($ this {refl}) · (weak G))) => (((($ this {refl}) · (weak G)) => (($ this {refl}) · z)) => (($ this {refl} · weak F) => ($ this {refl} · weak H))))) (sub-weak-p-23 H ($ this {refl})) (
lem9 ($ this {refl} · weak F) ($ this {refl} · weak G) ($ this {refl} · weak H)
))))))))))))))))))))))
lem14 : {n : ℕ} {Γ-t : Ctx n} {t : Type n} (F G H I : Form Γ-t t) →
⊢ ((F == G) => ((G == H) => ((H == I) => (F == I))))
lem14 F G H I = inf-V (inf-V ax-4-s (inf-V ax-4-s (transitivity F H I))) (transitivity F G H)
lem15 : {n : ℕ} {Γ-t : Ctx n} {t : Type n} (F G : Form Γ-t t) →
⊢ ((F == G) => (G == F))
lem15 F G =
inf-V (inf-III-samectx (λ z → ((z · G) => (G == F))) (^[ _ ] (![ _ > $o ] (($ this {refl} · $ (next (next this)) {refl}) => ($ this {refl} · $ (next this) {refl})))) F) (
inf-V (inf-III-samectx (λ z → (z => (G == F))) (![ _ > $o ] (($ this {refl} · weak (weak F)) => ($ this {refl} · $ (next this) {refl}))) G) (
subst (λ z → ⊢ ((![ _ > $o ] (($ this {refl} · z) => ($ this {refl} · weak G))) => (G == F))) (sub-weak-p-1 F G) (
inf-V (inf-III-samectx (λ z → ((![ _ > $o ] (($ this {refl} · weak F) => ($ this {refl} · weak G))) => (z · F))) (^[ _ ] (![ _ > $o ] (($ this {refl} · $ (next (next this)) {refl}) => ($ this {refl} · $ (next this) {refl})))) G) (
inf-V (inf-III-samectx (λ z → ((![ _ > $o ] (($ this {refl} · weak F) => ($ this {refl} · weak G))) => z)) (![ _ > $o ] (($ this {refl} · weak (weak G)) => ($ this {refl} · $ (next this) {refl}))) F) (
subst (λ z → ⊢ ((![ _ > $o ] (($ this {refl} · weak F) => ($ this {refl} · weak G))) => (![ _ > $o ] (($ this {refl} · z) => ($ this {refl} · weak F))))) (sub-weak-p-1 G F) (
inf-V (ax-6-s {_} {_} {_} {~ (![ _ ] (($ this {refl} · weak F) => ($ this {refl} · weak G)))} {($ this {refl} · weak G) => ($ this {refl} · weak F)}) (
inf-VI-s (
inf-V ax-3-s (
inf-V (inf-V ax-4-s (inf-V (lem5 (![ _ ] (($ this {refl} · weak-i (_ ∷ ε) _ (weak F)) => ($ this {refl} · weak-i (_ ∷ ε) _ (weak G)))) (sub (^[ _ ] ~ ($ (next this) {refl} · $ this {refl})) (($ this {refl} · weak-i (_ ∷ ε) _ (weak F)) => ($ this {refl} · weak-i (_ ∷ ε) _ (weak G)))))
(ax-5-s (($ this {refl} · weak-i (_ ∷ ε) _ (weak F)) => ($ this {refl} · weak-i (_ ∷ ε) _ (weak G))) (^[ _ ] ~ ($ (next this) {refl} · $ this {refl}))))) (
inf-V ax-3-s (
subst (λ z → ⊢ ((((^[ _ ] ~ ($ (next this) {refl} · $ this {refl})) · z) => ((^[ _ ] ~ ($ (next this) {refl} · $ this {refl})) · (sub (^[ _ ] ~ ($ (next this) {refl} · $ this {refl})) (weak-i (_ ∷ ε) _ (weak G))))) => (($ this {refl} · weak G) => ($ this {refl} · weak F)))) (sub-weak-p-23 F (^[ _ ] ~ ($ (next this) {refl} · $ this {refl}))) (
subst (λ z → ⊢ ((((^[ _ ] ~ ($ (next this) {refl} · $ this {refl})) · (weak F)) => ((^[ _ ] ~ ($ (next this) {refl} · $ this {refl})) · z)) => (($ this {refl} · weak G) => ($ this {refl} · weak F)))) (sub-weak-p-23 G (^[ _ ] ~ ($ (next this) {refl} · $ this {refl}))) (
inf-V (inf-III-samectx (λ z → ((z => ((^[ _ ] ~ ($ (next this) {refl} · $ this {refl})) · weak G)) => (($ this {refl} · weak G) => ($ this {refl} · weak F)))) (~ ($ (next this) {refl} · $ this {refl})) (weak F)) (
inf-V (inf-III-samectx (λ z → (((~ ($ this {refl} · weak F)) => z) => (($ this {refl} · weak G) => ($ this {refl} · weak F)))) (~ ($ (next this) {refl} · $ this {refl})) (weak G)) (
lem5r ($ this {refl} · weak F) ($ this {refl} · weak G)
)))))))))))))))
substitutivity : {n : ℕ} {Γ-t : Ctx n} {t u : Type n} (F G : Form Γ-t t) (H : Form Γ-t (t > u)) →
⊢ ((F == G) => ((H · F) == (H · G)))
substitutivity F G H =
inf-V (inf-III-samectx (λ z → ((z · G) => ((H · F) == (H · G)))) (^[ _ ] (![ _ > _ ] (($ this {refl} · $ (next (next this)) {refl}) => ($ this {refl} · $ (next this) {refl})))) F) (
inf-V (inf-III-samectx (λ z → (z => ((H · F) == (H · G)))) (![ _ > _ ] (($ this {refl} · weak (weak F)) => ($ this {refl} · $ (next this) {refl}))) G) (
subst (λ z → ⊢ ((![ _ > _ ] (($ this {refl} · z) => ($ this {refl} · weak G))) => ((H · F) == (H · G)))) (sub-weak-p-1 F G) (
inf-V (inf-III-samectx (λ z → ((![ _ > _ ] (($ this {refl} · weak F) => ($ this {refl} · weak G))) => (z · (H · G)))) (^[ _ ] (![ _ > _ ] (($ this {refl} · $ (next (next this)) {refl}) => ($ this {refl} · $ (next this) {refl})))) ((H · F))) (
inf-V (inf-III-samectx (λ z → ((![ _ > _ ] (($ this {refl} · weak F) => ($ this {refl} · weak G))) => z)) (![ _ > _ ] (($ this {refl} · weak (weak ((H · F)))) => ($ this {refl} · $ (next this) {refl}))) ((H · G))) (
subst (λ z → ⊢ ((![ _ > _ ] (($ this {refl} · weak F) => ($ this {refl} · weak G))) => (![ _ > _ ] (($ this {refl} · z) => ($ this {refl} · weak ((H · G))))))) (sub-weak-p-1 ((H · F)) ((H · G))) (
inf-V (ax-6-s {_} {_} {_} {~ (![ _ ] (($ this {refl} · weak F) => ($ this {refl} · weak G)))} {($ this {refl} · weak ((H · F))) => ($ this {refl} · weak ((H · G)))}) (
inf-VI-s (
inf-V ax-3-s (
inf-V (inf-V ax-4-s (inf-V (lem5 (![ _ ] (($ this {refl} · weak-i (_ ∷ ε) _ (weak F)) => ($ this {refl} · weak-i (_ ∷ ε) _ (weak G)))) (sub (^[ _ ] ($ (next this) {refl} · (weak (weak H) · $ this {refl}))) (($ this {refl} · weak-i (_ ∷ ε) _ (weak F)) => ($ this {refl} · weak-i (_ ∷ ε) _ (weak G)))))
(ax-5-s (($ this {refl} · weak-i (_ ∷ ε) _ (weak F)) => ($ this {refl} · weak-i (_ ∷ ε) _ (weak G))) (^[ _ ] ($ (next this) {refl} · (weak (weak H) · $ this {refl})))))) (
inf-V ax-3-s (
subst (λ z → ⊢ ((((^[ _ ] ($ (next this) {refl} · (weak (weak H) · $ this {refl}))) · z) => ((^[ _ ] ($ (next this) {refl} · (weak (weak H) · $ this {refl}))) · (sub (^[ _ ] ($ (next this) {refl} · (weak (weak H) · $ this {refl}))) (weak-i (_ ∷ ε) _ (weak G))))) => (($ this {refl} · weak (H · F)) => ($ this {refl} · weak (H · G))))) {-{weak F} {sub (^[ _ ] ($ (next this) {refl} · (weak (weak H) · $ this {refl}))) (weak-i (_ ∷ ε) _ (weak F))}-} (sub-weak-p-23 F (^[ _ ] ($ (next this) {refl} · (weak (weak H) · $ this {refl})))) (
subst (λ z → ⊢ ((((^[ _ ] ($ (next this) {refl} · (weak (weak H) · $ this {refl}))) · (weak F)) => ((^[ _ ] ($ (next this) {refl} · (weak (weak H) · $ this {refl}))) · z)) => (($ this {refl} · weak (H · F)) => ($ this {refl} · weak (H · G))))) {-{weak G} {sub (^[ _ ] ($ (next this) {refl} · (weak (weak H) · $ this {refl}))) (weak-i (_ ∷ ε) _ (weak G))}-} (sub-weak-p-23 G (^[ _ ] ($ (next this) {refl} · (weak (weak H) · $ this {refl})))) (
inf-V (inf-III-samectx (λ z → ((z => ((^[ _ ] ($ (next this) {refl} · (weak (weak H) · $ this {refl}))) · weak G)) => (($ this {refl} · weak (H · F)) => ($ this {refl} · weak (H · G))))) ($ (next this) {refl} · (weak (weak H) · $ this {refl})) (weak F)) (
subst (λ z → ⊢ ((($ this {refl} · (z · weak F)) => ((^[ _ ] ($ (next this) {refl} · (weak (weak H) · $ this {refl}))) · weak G)) => (($ this {refl} · weak (H · F)) => ($ this {refl} · weak (H · G))))) {-{weak H} {sub (weak F) (weak (weak H))}-} (sub-weak-p-1' (weak H) (weak F)) (
inf-V (inf-III-samectx (λ z → ((($ this {refl} · weak (H · F)) => z) => (($ this {refl} · weak (H · F)) => ($ this {refl} · weak (H · G))))) ($ (next this) {refl} · (weak (weak H) · $ this {refl})) (weak G)) (
subst (λ z → ⊢ ((($ this {refl} · (weak (H · F))) => ($ this {refl} · (z · weak G))) => (($ this {refl} · weak (H · F)) => ($ this {refl} · weak (H · G))))) {-{weak H} {sub (weak G) (weak (weak H))}-} (sub-weak-p-1' (weak H) (weak G)) (
lem2h2hb (($ this {refl} · weak (H · F)) => ($ this {refl} · (weak H · weak G)))
)))))))))))))))))
lem16 : {n : ℕ} {Γ-t : Ctx n} {α β : Type n} (F₁ F₂ : Form Γ-t (α > β)) (G₁ G₂ : Form Γ-t α) →
⊢ ((F₁ == F₂) => ((G₁ == G₂) => ((F₁ · G₁) == (F₂ · G₂))))
lem16 F₁ F₂ G₁ G₂ =
inf-V (inf-V (inf-V (lem8 (F₁ == F₂) ((G₁ == G₂) => ((F₁ · G₁) == (F₂ · G₁))) ((G₁ == G₂) => ((F₂ · G₁) == (F₂ · G₂))) ((G₁ == G₂) => ((F₁ · G₁) == (F₂ · G₂))))
(inf-V (lem8 (G₁ == G₂) ((F₁ · G₁) == (F₂ · G₁)) ((F₂ · G₁) == (F₂ · G₂)) ((F₁ · G₁) == (F₂ · G₂))) (
transitivity (F₁ · G₁) (F₂ · G₁) (F₂ · G₂))))
(inf-V (lem4 (G₁ == G₂) (F₁ == F₂) ((F₁ · G₁) == (F₂ · G₁))) (inf-V ax-3-s (inf-V ax-2-s (
subst (λ z → ⊢ ((F₁ == F₂) => ((F₁ · G₁) == (F₂ · z)))) (sym (sub-weak-p-1' G₁ F₂)) (
inf-V (inf-II-samectx (λ z → ((F₁ == F₂) => ((F₁ · G₁) == z))) (($ this {refl}) · weak G₁) F₂) (
subst (λ z → ⊢ ((F₁ == F₂) => ((F₁ · z) == ((^[ _ ] (($ this {refl}) · weak G₁)) · F₂)))) (sym (sub-weak-p-1' G₁ F₁)) (
inf-V (inf-II-samectx (λ z → ((F₁ == F₂) => (z == ((^[ _ ] (($ this {refl}) · weak G₁)) · F₂)))) (($ this {refl}) · weak G₁) F₁) (
substitutivity F₁ F₂ (^[ _ ] (($ this {refl}) · weak G₁))
)))))))))
(inf-V ax-3-s (inf-V ax-2-s (
subst (λ z → ⊢ ((G₁ == G₂) => ((F₂ · G₁) == (z · G₂)))) (sym (sub-weak-p-1' F₂ G₂)) (
inf-V (inf-II-samectx (λ z → ((G₁ == G₂) => ((F₂ · G₁) == z))) (weak F₂ · ($ this {refl})) G₂) (
subst (λ z → ⊢ ((G₁ == G₂) => ((z · G₁) == ((^[ _ ] (weak F₂ · ($ this {refl}))) · G₂)))) (sym (sub-weak-p-1' F₂ G₁)) (
inf-V (inf-II-samectx (λ z → ((G₁ == G₂) => (z == ((^[ _ ] (weak F₂ · ($ this {refl}))) · G₂)))) (weak F₂ · ($ this {refl})) G₁) (
substitutivity G₁ G₂ (^[ _ ] (weak F₂ · ($ this {refl})))
)))))))
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------------------------------------------------------------------------
-- The Agda standard library
--
-- The Covec type and some operations
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe --sized-types #-}
module Codata.Covec where
open import Size
open import Codata.Thunk using (Thunk; force)
open import Codata.Conat as Conat
open import Codata.Conat.Bisimilarity
open import Codata.Conat.Properties
open import Codata.Cofin as Cofin using (Cofin; zero; suc)
open import Codata.Colist as Colist using (Colist ; [] ; _∷_)
open import Codata.Stream as Stream using (Stream ; _∷_)
open import Function
data Covec {ℓ} (A : Set ℓ) (i : Size) : Conat ∞ → Set ℓ where
[] : Covec A i zero
_∷_ : ∀ {n} → A → Thunk (λ i → Covec A i (n .force)) i → Covec A i (suc n)
module _ {ℓ} {A : Set ℓ} where
head : ∀ {n i} → Covec A i (suc n) → A
head (x ∷ _) = x
tail : ∀ {n} → Covec A ∞ (suc n) → Covec A ∞ (n .force)
tail (_ ∷ xs) = xs .force
lookup : ∀ {n} → Cofin n → Covec A ∞ n → A
lookup zero = head
lookup (suc k) = lookup k ∘′ tail
replicate : ∀ {i} → (n : Conat ∞) → A → Covec A i n
replicate zero a = []
replicate (suc n) a = a ∷ λ where .force → replicate (n .force) a
cotake : ∀ {i} → (n : Conat ∞) → Stream A i → Covec A i n
cotake zero xs = []
cotake (suc n) (x ∷ xs) = x ∷ λ where .force → cotake (n .force) (xs .force)
infixr 5 _++_
_++_ : ∀ {i m n} → Covec A i m → Covec A i n → Covec A i (m + n)
[] ++ ys = ys
(x ∷ xs) ++ ys = x ∷ λ where .force → xs .force ++ ys
fromColist : ∀ {i} → (xs : Colist A ∞) → Covec A i (Colist.length xs)
fromColist [] = []
fromColist (x ∷ xs) = x ∷ λ where .force → fromColist (xs .force)
toColist : ∀ {i n} → Covec A i n → Colist A i
toColist [] = []
toColist (x ∷ xs) = x ∷ λ where .force → toColist (xs .force)
fromStream : ∀ {i} → Stream A i → Covec A i infinity
fromStream = cotake infinity
cast : ∀ {i} {m n} → i ⊢ m ≈ n → Covec A i m → Covec A i n
cast zero [] = []
cast (suc eq) (a ∷ as) = a ∷ λ where .force → cast (eq .force) (as .force)
module _ {a b} {A : Set a} {B : Set b} where
map : ∀ {i n} (f : A → B) → Covec A i n → Covec B i n
map f [] = []
map f (a ∷ as) = f a ∷ λ where .force → map f (as .force)
ap : ∀ {i n} → Covec (A → B) i n → Covec A i n → Covec B i n
ap [] [] = []
ap (f ∷ fs) (a ∷ as) = (f a) ∷ λ where .force → ap (fs .force) (as .force)
scanl : ∀ {i n} → (B → A → B) → B → Covec A i n → Covec B i (1 ℕ+ n)
scanl c n [] = n ∷ λ where .force → []
scanl c n (a ∷ as) = n ∷ λ where
.force → cast (suc λ where .force → 0ℕ+-identity)
(scanl c (c n a) (as .force))
module _ {a b c} {A : Set a} {B : Set b} {C : Set c} where
zipWith : ∀ {i n} → (A → B → C) → Covec A i n → Covec B i n → Covec C i n
zipWith f [] [] = []
zipWith f (a ∷ as) (b ∷ bs) =
f a b ∷ λ where .force → zipWith f (as .force) (bs .force)
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module Structure.Category.Functor.Proofs where
open import Data.Tuple as Tuple using (_,_)
open import Functional using (_$_)
open import Logic.Predicate
import Lvl
open import Structure.Category
open import Structure.Categorical.Properties
open import Structure.Category.Functor
open import Structure.Category.Functor.Equiv
open import Structure.Function
open import Structure.Operator
open import Structure.Relator.Equivalence
open import Structure.Relator.Properties
open import Structure.Setoid
open import Syntax.Transitivity
open import Type
private variable ℓ ℓₗₑ ℓᵣₑ : Lvl.Level
private variable Obj Obj₁ Obj₂ Obj₃ : Type{ℓ}
private variable Morphism Morphism₁ Morphism₂ Morphism₃ : Obj → Obj → Type{ℓ}
module _
⦃ morphism-equiv₁ : ∀{x y} → Equiv{ℓₗₑ}(Morphism₁ x y) ⦄
⦃ morphism-equiv₂ : ∀{x y} → Equiv{ℓᵣₑ}(Morphism₂ x y) ⦄
{cat₁ : Category(Morphism₁)}
{cat₂ : Category(Morphism₂)}
(F : Obj₁ → Obj₂)
⦃ functor : Functor(cat₁)(cat₂)(F) ⦄
where
open Category.ArrowNotation ⦃ … ⦄
open Category ⦃ … ⦄
open Functor(functor)
private open module MorphismEquivₗ {x}{y} = Equiv(morphism-equiv₁{x}{y}) using () renaming (_≡_ to _≡ₗₘ_)
private open module MorphismEquivᵣ {x}{y} = Equiv(morphism-equiv₂{x}{y}) using () renaming (_≡_ to _≡ᵣₘ_)
private instance _ = cat₁
private instance _ = cat₂
private variable x y : Obj₁
isomorphism-preserving : ∀{f : x ⟶ y} → Morphism.Isomorphism ⦃ \{x y} → morphism-equiv₁ {x}{y} ⦄ (_∘_)(id)(f) → Morphism.Isomorphism ⦃ \{x y} → morphism-equiv₂ {x}{y} ⦄ (_∘_)(id)(map f)
∃.witness (isomorphism-preserving ([∃]-intro g)) = map g
∃.proof (isomorphism-preserving {f = f} iso@([∃]-intro g)) =
(Morphism.intro $
map g ∘ map f 🝖-[ op-preserving ]-sym
map(g ∘ f) 🝖-[ congruence₁(map) (inverseₗ(f)(g)) ]
map id 🝖-[ id-preserving ]
id 🝖-end
) , (Morphism.intro $
map f ∘ map g 🝖-[ op-preserving ]-sym
map(f ∘ g) 🝖-[ congruence₁(map) (inverseᵣ(f)(g)) ]
map id 🝖-[ id-preserving ]
id 🝖-end
)
where
open Morphism.OperModule (\{x : Obj₁} → _∘_ {x = x})
open Morphism.IdModule (\{x : Obj₁} → _∘_ {x = x})(id)
open Morphism.Isomorphism(\{x : Obj₁} → _∘_ {x = x})(id)(f)
instance _ = iso
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-- {-# OPTIONS -v tc.lhs:10 -v tc.lhs.split:50 #-}
postulate A : Set
record R : Set where
field f : A
test : _ → A
test record{f = a} = a
-- This could succeed, but Agda currently does not try to guess
-- the type type of the record pattern from its field names.
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module Generic.Test.Eq where
open import Generic.Main hiding (List; []; _∷_)
open import Generic.Test.Data.Fin
open import Generic.Test.Data.List
open import Generic.Test.Data.Vec
xs : Vec (List (Fin 4)) 3
xs = (fsuc fzero ∷ fzero ∷ [])
∷ᵥ (fsuc (fsuc fzero) ∷ [])
∷ᵥ (fzero ∷ fsuc (fsuc (fsuc fzero)) ∷ [])
∷ᵥ []ᵥ
test : xs ≟ xs ≡ yes refl
test = refl
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Relation.Nullary.Base where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function
open import Cubical.Functions.Fixpoint
open import Cubical.Data.Empty as ⊥
open import Cubical.HITs.PropositionalTruncation.Base
private
variable
ℓ : Level
A : Type ℓ
-- Negation
infix 3 ¬_
¬_ : Type ℓ → Type ℓ
¬ A = A → ⊥
-- Decidable types (inspired by standard library)
data Dec (P : Type ℓ) : Type ℓ where
yes : ( p : P) → Dec P
no : (¬p : ¬ P) → Dec P
NonEmpty : Type ℓ → Type ℓ
NonEmpty A = ¬ ¬ A
Stable : Type ℓ → Type ℓ
Stable A = NonEmpty A → A
-- reexport propositional truncation for uniformity
open Cubical.HITs.PropositionalTruncation.Base
using (∥_∥) public
SplitSupport : Type ℓ → Type ℓ
SplitSupport A = ∥ A ∥ → A
Collapsible : Type ℓ → Type ℓ
Collapsible A = Σ[ f ∈ (A → A) ] 2-Constant f
Populated ⟪_⟫ : Type ℓ → Type ℓ
Populated A = (f : Collapsible A) → Fixpoint (f .fst)
⟪_⟫ = Populated
PStable : Type ℓ → Type ℓ
PStable A = ⟪ A ⟫ → A
onAllPaths : (Type ℓ → Type ℓ) → Type ℓ → Type ℓ
onAllPaths S A = (x y : A) → S (x ≡ y)
Separated : Type ℓ → Type ℓ
Separated = onAllPaths Stable
HSeparated : Type ℓ → Type ℓ
HSeparated = onAllPaths SplitSupport
Collapsible≡ : Type ℓ → Type ℓ
Collapsible≡ = onAllPaths Collapsible
PStable≡ : Type ℓ → Type ℓ
PStable≡ = onAllPaths PStable
Discrete : Type ℓ → Type ℓ
Discrete = onAllPaths Dec
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{-# OPTIONS --without-K --safe #-}
module Definition.Conversion.Symmetry where
open import Definition.Untyped
open import Definition.Typed
open import Definition.Typed.Properties
open import Definition.Conversion
open import Definition.Conversion.Stability
open import Definition.Conversion.Soundness
open import Definition.Conversion.Conversion
open import Definition.Typed.Consequences.Syntactic
open import Definition.Typed.Consequences.Equality
open import Definition.Typed.Consequences.Reduction
open import Definition.Typed.Consequences.Injectivity
open import Definition.Typed.Consequences.Substitution
open import Definition.Typed.Consequences.SucCong
open import Tools.Product
import Tools.PropositionalEquality as PE
mutual
-- Symmetry of algorithmic equality of neutrals.
sym~↑ : ∀ {t u A Γ Δ} → ⊢ Γ ≡ Δ
→ Γ ⊢ t ~ u ↑ A
→ ∃ λ B → Γ ⊢ A ≡ B × Δ ⊢ u ~ t ↑ B
sym~↑ Γ≡Δ (var-refl x x≡y) =
let ⊢A = syntacticTerm x
in _ , refl ⊢A
, var-refl (PE.subst (λ y → _ ⊢ var y ∷ _) x≡y (stabilityTerm Γ≡Δ x))
(PE.sym x≡y)
sym~↑ Γ≡Δ (app-cong t~u x) =
let ⊢Γ , ⊢Δ , _ = contextConvSubst Γ≡Δ
B , whnfB , A≡B , u~t = sym~↓ Γ≡Δ t~u
F′ , G′ , ΠF′G′≡B = Π≡A A≡B whnfB
F≡F′ , G≡G′ = injectivity (PE.subst (λ x → _ ⊢ _ ≡ x) ΠF′G′≡B A≡B)
in _ , substTypeEq G≡G′ (soundnessConv↑Term x)
, app-cong (PE.subst (λ x → _ ⊢ _ ~ _ ↓ x) ΠF′G′≡B u~t)
(convConvTerm (symConv↑Term Γ≡Δ x) (stabilityEq Γ≡Δ F≡F′))
sym~↑ Γ≡Δ (fst-cong p~r) =
let B , whnfB , A≡B , r~p = sym~↓ Γ≡Δ p~r
F′ , G′ , Σ≡ = Σ≡A A≡B whnfB
F≡ , G≡ = Σ-injectivity (PE.subst (λ x → _ ⊢ _ ≡ x) Σ≡ A≡B)
in F′ , F≡ , fst-cong (PE.subst (λ x → _ ⊢ _ ~ _ ↓ x) Σ≡ r~p)
sym~↑ Γ≡Δ (snd-cong {p} {r} {F} {G} p~r) =
let fst≡ = soundness~↑ (fst-cong p~r)
B , whnfB , A≡B , r~p = sym~↓ Γ≡Δ p~r
F′ , G′ , Σ≡ = Σ≡A A≡B whnfB
r~p = PE.subst (λ x → _ ⊢ _ ~ _ ↓ x) Σ≡ r~p
F≡ , G≡ = Σ-injectivity (PE.subst (λ x → _ ⊢ _ ≡ x) Σ≡ A≡B)
in G′ [ fst r ] , substTypeEq G≡ fst≡ , snd-cong r~p
sym~↑ Γ≡Δ (natrec-cong x x₁ x₂ t~u) =
let ⊢Γ , ⊢Δ , _ = contextConvSubst Γ≡Δ
B , whnfB , A≡B , u~t = sym~↓ Γ≡Δ t~u
B≡ℕ = ℕ≡A A≡B whnfB
F≡G = stabilityEq (Γ≡Δ ∙ refl (ℕⱼ ⊢Γ)) (soundnessConv↑ x)
F[0]≡G[0] = substTypeEq F≡G (refl (zeroⱼ ⊢Δ))
in _ , substTypeEq (soundnessConv↑ x) (soundness~↓ t~u)
, natrec-cong (symConv↑ (Γ≡Δ ∙ (refl (ℕⱼ ⊢Γ))) x)
(convConvTerm (symConv↑Term Γ≡Δ x₁) F[0]≡G[0])
(convConvTerm (symConv↑Term Γ≡Δ x₂) (sucCong F≡G))
(PE.subst (λ x → _ ⊢ _ ~ _ ↓ x) B≡ℕ u~t)
sym~↑ Γ≡Δ (Emptyrec-cong x t~u) =
let ⊢Γ , ⊢Δ , _ = contextConvSubst Γ≡Δ
B , whnfB , A≡B , u~t = sym~↓ Γ≡Δ t~u
B≡Empty = Empty≡A A≡B whnfB
F≡G = stabilityEq Γ≡Δ (soundnessConv↑ x)
in _ , soundnessConv↑ x
, Emptyrec-cong (symConv↑ Γ≡Δ x)
(PE.subst (λ x₁ → _ ⊢ _ ~ _ ↓ x₁) B≡Empty u~t)
-- Symmetry of algorithmic equality of neutrals of types in WHNF.
sym~↓ : ∀ {t u A Γ Δ} → ⊢ Γ ≡ Δ → Γ ⊢ t ~ u ↓ A
→ ∃ λ B → Whnf B × Γ ⊢ A ≡ B × Δ ⊢ u ~ t ↓ B
sym~↓ Γ≡Δ ([~] A₁ D whnfA k~l) =
let B , A≡B , k~l′ = sym~↑ Γ≡Δ k~l
_ , ⊢B = syntacticEq A≡B
B′ , whnfB′ , D′ = whNorm ⊢B
A≡B′ = trans (sym (subset* D)) (trans A≡B (subset* (red D′)))
in B′ , whnfB′ , A≡B′ , [~] B (stabilityRed* Γ≡Δ (red D′)) whnfB′ k~l′
-- Symmetry of algorithmic equality of types.
symConv↑ : ∀ {A B Γ Δ} → ⊢ Γ ≡ Δ → Γ ⊢ A [conv↑] B → Δ ⊢ B [conv↑] A
symConv↑ Γ≡Δ ([↑] A′ B′ D D′ whnfA′ whnfB′ A′<>B′) =
[↑] B′ A′ (stabilityRed* Γ≡Δ D′) (stabilityRed* Γ≡Δ D) whnfB′ whnfA′
(symConv↓ Γ≡Δ A′<>B′)
-- Symmetry of algorithmic equality of types in WHNF.
symConv↓ : ∀ {A B Γ Δ} → ⊢ Γ ≡ Δ → Γ ⊢ A [conv↓] B → Δ ⊢ B [conv↓] A
symConv↓ Γ≡Δ (U-refl x) =
let _ , ⊢Δ , _ = contextConvSubst Γ≡Δ
in U-refl ⊢Δ
symConv↓ Γ≡Δ (ℕ-refl x) =
let _ , ⊢Δ , _ = contextConvSubst Γ≡Δ
in ℕ-refl ⊢Δ
symConv↓ Γ≡Δ (Empty-refl x) =
let _ , ⊢Δ , _ = contextConvSubst Γ≡Δ
in Empty-refl ⊢Δ
symConv↓ Γ≡Δ (Unit-refl x) =
let _ , ⊢Δ , _ = contextConvSubst Γ≡Δ
in Unit-refl ⊢Δ
symConv↓ Γ≡Δ (ne A~B) =
let B , whnfB , U≡B , B~A = sym~↓ Γ≡Δ A~B
B≡U = U≡A U≡B
in ne (PE.subst (λ x → _ ⊢ _ ~ _ ↓ x) B≡U B~A)
symConv↓ Γ≡Δ (Π-cong x A<>B A<>B₁) =
let F≡H = soundnessConv↑ A<>B
_ , ⊢H = syntacticEq (stabilityEq Γ≡Δ F≡H)
in Π-cong ⊢H (symConv↑ Γ≡Δ A<>B)
(symConv↑ (Γ≡Δ ∙ F≡H) A<>B₁)
symConv↓ Γ≡Δ (Σ-cong x A<>B A<>B₁) =
let F≡H = soundnessConv↑ A<>B
_ , ⊢H = syntacticEq (stabilityEq Γ≡Δ F≡H)
in Σ-cong ⊢H (symConv↑ Γ≡Δ A<>B)
(symConv↑ (Γ≡Δ ∙ F≡H) A<>B₁)
-- Symmetry of algorithmic equality of terms.
symConv↑Term : ∀ {t u A Γ Δ} → ⊢ Γ ≡ Δ → Γ ⊢ t [conv↑] u ∷ A → Δ ⊢ u [conv↑] t ∷ A
symConv↑Term Γ≡Δ ([↑]ₜ B t′ u′ D d d′ whnfB whnft′ whnfu′ t<>u) =
[↑]ₜ B u′ t′ (stabilityRed* Γ≡Δ D) (stabilityRed*Term Γ≡Δ d′)
(stabilityRed*Term Γ≡Δ d) whnfB whnfu′ whnft′ (symConv↓Term Γ≡Δ t<>u)
-- Symmetry of algorithmic equality of terms in WHNF.
symConv↓Term : ∀ {t u A Γ Δ} → ⊢ Γ ≡ Δ → Γ ⊢ t [conv↓] u ∷ A → Δ ⊢ u [conv↓] t ∷ A
symConv↓Term Γ≡Δ (ℕ-ins t~u) =
let B , whnfB , A≡B , u~t = sym~↓ Γ≡Δ t~u
B≡ℕ = ℕ≡A A≡B whnfB
in ℕ-ins (PE.subst (λ x → _ ⊢ _ ~ _ ↓ x) B≡ℕ u~t)
symConv↓Term Γ≡Δ (Empty-ins t~u) =
let B , whnfB , A≡B , u~t = sym~↓ Γ≡Δ t~u
B≡Empty = Empty≡A A≡B whnfB
in Empty-ins (PE.subst (λ x → _ ⊢ _ ~ _ ↓ x) B≡Empty u~t)
symConv↓Term Γ≡Δ (Unit-ins t~u) =
let B , whnfB , A≡B , u~t = sym~↓ Γ≡Δ t~u
B≡Unit = Unit≡A A≡B whnfB
in Unit-ins (PE.subst (λ x → _ ⊢ _ ~ _ ↓ x) B≡Unit u~t)
symConv↓Term Γ≡Δ (ne-ins t u x t~u) =
let B , whnfB , A≡B , u~t = sym~↓ Γ≡Δ t~u
in ne-ins (stabilityTerm Γ≡Δ u) (stabilityTerm Γ≡Δ t) x u~t
symConv↓Term Γ≡Δ (univ x x₁ x₂) =
univ (stabilityTerm Γ≡Δ x₁) (stabilityTerm Γ≡Δ x) (symConv↓ Γ≡Δ x₂)
symConv↓Term Γ≡Δ (zero-refl x) =
let _ , ⊢Δ , _ = contextConvSubst Γ≡Δ
in zero-refl ⊢Δ
symConv↓Term Γ≡Δ (suc-cong t<>u) = suc-cong (symConv↑Term Γ≡Δ t<>u)
symConv↓Term Γ≡Δ (η-eq x₁ x₂ y y₁ t<>u) =
let ⊢F , _ = syntacticΠ (syntacticTerm x₁)
in η-eq (stabilityTerm Γ≡Δ x₂) (stabilityTerm Γ≡Δ x₁)
y₁ y (symConv↑Term (Γ≡Δ ∙ refl ⊢F) t<>u)
symConv↓Term Γ≡Δ (Σ-η ⊢p ⊢r pProd rProd fstConv sndConv) =
let Δ⊢p = stabilityTerm Γ≡Δ ⊢p
Δ⊢r = stabilityTerm Γ≡Δ ⊢r
⊢G = proj₂ (syntacticΣ (syntacticTerm ⊢p))
Δfst≡ = symConv↑Term Γ≡Δ fstConv
Δsnd≡₁ = symConv↑Term Γ≡Δ sndConv
ΔGfstt≡Gfstu = stabilityEq Γ≡Δ (substTypeEq (refl ⊢G)
(soundnessConv↑Term fstConv))
Δsnd≡ = convConvTerm Δsnd≡₁ ΔGfstt≡Gfstu
in Σ-η Δ⊢r Δ⊢p rProd pProd Δfst≡ Δsnd≡
symConv↓Term Γ≡Δ (η-unit [t] [u] tUnit uUnit) =
let [t] = stabilityTerm Γ≡Δ [t]
[u] = stabilityTerm Γ≡Δ [u]
in (η-unit [u] [t] uUnit tUnit)
symConv↓Term′ : ∀ {t u A Γ} → Γ ⊢ t [conv↓] u ∷ A → Γ ⊢ u [conv↓] t ∷ A
symConv↓Term′ tConvU =
symConv↓Term (reflConEq (wfEqTerm (soundnessConv↓Term tConvU))) tConvU
-- Symmetry of algorithmic equality of types with preserved context.
symConv : ∀ {A B Γ} → Γ ⊢ A [conv↑] B → Γ ⊢ B [conv↑] A
symConv A<>B =
let ⊢Γ = wfEq (soundnessConv↑ A<>B)
in symConv↑ (reflConEq ⊢Γ) A<>B
-- Symmetry of algorithmic equality of terms with preserved context.
symConvTerm : ∀ {t u A Γ} → Γ ⊢ t [conv↑] u ∷ A → Γ ⊢ u [conv↑] t ∷ A
symConvTerm t<>u =
let ⊢Γ = wfEqTerm (soundnessConv↑Term t<>u)
in symConv↑Term (reflConEq ⊢Γ) t<>u
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module Haskell.Prim.Maybe where
open import Agda.Builtin.List public
open import Haskell.Prim
open import Haskell.Prim.List
--------------------------------------------------
-- Maybe
data Maybe {ℓ} (a : Set ℓ) : Set ℓ where
Nothing : Maybe a
Just : a -> Maybe a
maybe : ∀ {ℓ₁ ℓ₂} {a : Set ℓ₁} {b : Set ℓ₂} → b → (a → b) → Maybe a → b
maybe n j Nothing = n
maybe n j (Just x) = j x
mapMaybe : (a -> Maybe b) -> List a -> List b
mapMaybe _ [] = []
mapMaybe f (x ∷ xs) =
case f x of λ where
Nothing -> mapMaybe f xs
(Just v) -> v ∷ (mapMaybe f xs) | {
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module Nats where
open import Agda.Builtin.Nat public
renaming (Nat to ℕ; _-_ to _∸_)
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------------------------------------------------------------------------
-- The Agda standard library
--
-- An inductive definition of the sublist relation with respect to a
-- setoid. This is a generalisation of what is commonly known as Order
-- Preserving Embeddings (OPE).
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
{-# OPTIONS --postfix-projections #-}
open import Relation.Binary using (Setoid; Rel)
module Data.List.Relation.Binary.Sublist.Setoid
{c ℓ} (S : Setoid c ℓ) where
open import Level using (_⊔_)
open import Data.List.Base using (List; []; _∷_)
import Data.List.Relation.Binary.Equality.Setoid as SetoidEquality
import Data.List.Relation.Binary.Sublist.Heterogeneous as Heterogeneous
import Data.List.Relation.Binary.Sublist.Heterogeneous.Core
as HeterogeneousCore
import Data.List.Relation.Binary.Sublist.Heterogeneous.Properties
as HeterogeneousProperties
open import Data.Product using (∃; ∃₂; _,_; proj₂)
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as P using (_≡_)
open import Relation.Nullary using (¬_; Dec; yes; no)
open Setoid S renaming (Carrier to A)
open SetoidEquality S
------------------------------------------------------------------------
-- Definition
infix 4 _⊆_ _⊇_ _⊈_ _⊉_
_⊆_ : Rel (List A) (c ⊔ ℓ)
_⊆_ = Heterogeneous.Sublist _≈_
_⊇_ : Rel (List A) (c ⊔ ℓ)
xs ⊇ ys = ys ⊆ xs
_⊈_ : Rel (List A) (c ⊔ ℓ)
xs ⊈ ys = ¬ (xs ⊆ ys)
_⊉_ : Rel (List A) (c ⊔ ℓ)
xs ⊉ ys = ¬ (xs ⊇ ys)
------------------------------------------------------------------------
-- Re-export definitions and operations from heterogeneous sublists
open HeterogeneousCore _≈_ using ([]; _∷_; _∷ʳ_) public
open Heterogeneous {R = _≈_} hiding (Sublist; []; _∷_; _∷ʳ_) public
renaming
(toAny to to∈; fromAny to from∈)
open Disjoint public using ([])
open DisjointUnion public using ([])
------------------------------------------------------------------------
-- Relational properties holding for Setoid case
⊆-reflexive : _≋_ ⇒ _⊆_
⊆-reflexive = HeterogeneousProperties.fromPointwise
open HeterogeneousProperties.Reflexivity {R = _≈_} refl public using ()
renaming (refl to ⊆-refl) -- ⊆-refl : Reflexive _⊆_
open HeterogeneousProperties.Transitivity {R = _≈_} {S = _≈_} {T = _≈_} trans public using ()
renaming (trans to ⊆-trans) -- ⊆-trans : Transitive _⊆_
open HeterogeneousProperties.Antisymmetry {R = _≈_} {S = _≈_} (λ x≈y _ → x≈y) public using ()
renaming (antisym to ⊆-antisym) -- ⊆-antisym : Antisymmetric _≋_ _⊆_
⊆-isPreorder : IsPreorder _≋_ _⊆_
⊆-isPreorder = record
{ isEquivalence = ≋-isEquivalence
; reflexive = ⊆-reflexive
; trans = ⊆-trans
}
⊆-isPartialOrder : IsPartialOrder _≋_ _⊆_
⊆-isPartialOrder = record
{ isPreorder = ⊆-isPreorder
; antisym = ⊆-antisym
}
⊆-preorder : Preorder c (c ⊔ ℓ) (c ⊔ ℓ)
⊆-preorder = record
{ isPreorder = ⊆-isPreorder
}
⊆-poset : Poset c (c ⊔ ℓ) (c ⊔ ℓ)
⊆-poset = record
{ isPartialOrder = ⊆-isPartialOrder
}
------------------------------------------------------------------------
-- Raw pushout
--
-- The category _⊆_ does not have proper pushouts. For instance consider:
--
-- τᵤ : [] ⊆ (u ∷ [])
-- τᵥ : [] ⊆ (v ∷ [])
--
-- Then, there are two unrelated upper bounds (u ∷ v ∷ []) and (v ∷ u ∷ []),
-- since _⊆_ does not include permutations.
--
-- Even though there are no unique least upper bounds, we can merge two
-- extensions of a list, producing a minimial superlist of both.
--
-- For the example, the left-biased merge would produce the pair:
--
-- τᵤ′ : (u ∷ []) ⊆ (u ∷ v ∷ [])
-- τᵥ′ : (v ∷ []) ⊆ (u ∷ v ∷ [])
--
-- We call such a pair a raw pushout. It is then a weak pushout if the
-- resulting square commutes, i.e.:
--
-- ⊆-trans τᵤ τᵤ′ ~ ⊆-trans τᵥ τᵥ′
--
-- This requires a notion of equality _~_ on sublist morphisms.
--
-- Further, commutation requires a similar commutation property
-- for the underlying equality _≈_, namely
--
-- trans x≈y (sym x≈y) == trans x≈z (sym x≈z)
--
-- for some notion of equality _==_ for equality proofs _≈_.
-- Such a property is given e.g. if _≈_ is proof irrelevant
-- or forms a groupoid.
record RawPushout {xs ys zs : List A} (τ : xs ⊆ ys) (σ : xs ⊆ zs) : Set (c ⊔ ℓ) where
field
{upperBound} : List A
leg₁ : ys ⊆ upperBound
leg₂ : zs ⊆ upperBound
open RawPushout
------------------------------------------------------------------------
-- Extending corners of a raw pushout square
-- Extending the right upper corner.
infixr 5 _∷ʳ₁_ _∷ʳ₂_
_∷ʳ₁_ : ∀ {xs ys zs : List A} {τ : xs ⊆ ys} {σ : xs ⊆ zs} →
(y : A) → RawPushout τ σ → RawPushout (y ∷ʳ τ) σ
y ∷ʳ₁ rpo = record
{ leg₁ = refl ∷ leg₁ rpo
; leg₂ = y ∷ʳ leg₂ rpo
}
-- Extending the left lower corner.
_∷ʳ₂_ : ∀ {xs ys zs : List A} {τ : xs ⊆ ys} {σ : xs ⊆ zs} →
(z : A) → RawPushout τ σ → RawPushout τ (z ∷ʳ σ)
z ∷ʳ₂ rpo = record
{ leg₁ = z ∷ʳ leg₁ rpo
; leg₂ = refl ∷ leg₂ rpo
}
-- Extending both of these corners with equal elements.
∷-rpo : ∀ {x y z : A} {xs ys zs : List A} {τ : xs ⊆ ys} {σ : xs ⊆ zs} →
(x≈y : x ≈ y) (x≈z : x ≈ z) → RawPushout τ σ → RawPushout (x≈y ∷ τ) (x≈z ∷ σ)
∷-rpo x≈y x≈z rpo = record
{ leg₁ = sym x≈y ∷ leg₁ rpo
; leg₂ = sym x≈z ∷ leg₂ rpo
}
------------------------------------------------------------------------
-- Left-biased pushout: add elements of left extension first.
⊆-pushoutˡ : ∀ {xs ys zs : List A} →
(τ : xs ⊆ ys) (σ : xs ⊆ zs) → RawPushout τ σ
⊆-pushoutˡ [] σ = record { leg₁ = σ ; leg₂ = ⊆-refl }
⊆-pushoutˡ (y ∷ʳ τ) σ = y ∷ʳ₁ ⊆-pushoutˡ τ σ
⊆-pushoutˡ τ@(_ ∷ _) (z ∷ʳ σ) = z ∷ʳ₂ ⊆-pushoutˡ τ σ
⊆-pushoutˡ (x≈y ∷ τ) (x≈z ∷ σ) = ∷-rpo x≈y x≈z (⊆-pushoutˡ τ σ)
-- Join two extensions, returning the upper bound and the diagonal
-- of the pushout square.
⊆-joinˡ : ∀ {xs ys zs : List A} →
(τ : xs ⊆ ys) (σ : xs ⊆ zs) → ∃ λ us → xs ⊆ us
⊆-joinˡ τ σ = upperBound rpo , ⊆-trans τ (leg₁ rpo)
where
rpo = ⊆-pushoutˡ τ σ
------------------------------------------------------------------------
-- Upper bound of two sublists xs,ys ⊆ zs
record UpperBound {xs ys zs} (τ : xs ⊆ zs) (σ : ys ⊆ zs) : Set (c ⊔ ℓ) where
field
{theUpperBound} : List A
sub : theUpperBound ⊆ zs
inj₁ : xs ⊆ theUpperBound
inj₂ : ys ⊆ theUpperBound
open UpperBound
infixr 5 _∷ₗ-ub_ _∷ᵣ-ub_
∷ₙ-ub : ∀ {xs ys zs} {τ : xs ⊆ zs} {σ : ys ⊆ zs} {x} →
UpperBound τ σ → UpperBound (x ∷ʳ τ) (x ∷ʳ σ)
∷ₙ-ub u = record
{ sub = _ ∷ʳ u .sub
; inj₁ = u .inj₁
; inj₂ = u .inj₂
}
_∷ₗ-ub_ : ∀ {xs ys zs} {τ : xs ⊆ zs} {σ : ys ⊆ zs} {x y} →
(x≈y : x ≈ y) → UpperBound τ σ → UpperBound (x≈y ∷ τ) (y ∷ʳ σ)
x≈y ∷ₗ-ub u = record
{ sub = refl ∷ u .sub
; inj₁ = x≈y ∷ u .inj₁
; inj₂ = _ ∷ʳ u .inj₂
}
_∷ᵣ-ub_ : ∀ {xs ys zs} {τ : xs ⊆ zs} {σ : ys ⊆ zs} {x y} →
(x≈y : x ≈ y) → UpperBound τ σ → UpperBound (y ∷ʳ τ) (x≈y ∷ σ)
x≈y ∷ᵣ-ub u = record
{ sub = refl ∷ u .sub
; inj₁ = _ ∷ʳ u .inj₁
; inj₂ = x≈y ∷ u .inj₂
}
------------------------------------------------------------------------
-- Disjoint union
--
-- Two non-overlapping sublists τ : xs ⊆ zs and σ : ys ⊆ zs
-- can be joined in a unique way if τ and σ are respected.
--
-- For instance, if τ : [x] ⊆ [x,y,x] and σ : [y] ⊆ [x,y,x]
-- then the union will be [x,y] or [y,x], depending on whether
-- τ picks the first x or the second one.
--
-- NB: If the content of τ and σ were ignored then the union would not
-- be unique. Expressing uniqueness would require a notion of equality
-- of sublist proofs, which we do not (yet) have for the setoid case
-- (however, for the propositional case).
⊆-disjoint-union : ∀ {xs ys zs} {τ : xs ⊆ zs} {σ : ys ⊆ zs} →
Disjoint τ σ → UpperBound τ σ
⊆-disjoint-union [] = record { sub = [] ; inj₁ = [] ; inj₂ = [] }
⊆-disjoint-union (x ∷ₙ d) = ∷ₙ-ub (⊆-disjoint-union d)
⊆-disjoint-union (x≈y ∷ₗ d) = x≈y ∷ₗ-ub (⊆-disjoint-union d)
⊆-disjoint-union (x≈y ∷ᵣ d) = x≈y ∷ᵣ-ub (⊆-disjoint-union d)
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module Issue2575.M where
A : Set
A = Set
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------------------------------------------------------------------------------
-- Testing subst using an implicit arguments for the propositional function.
------------------------------------------------------------------------------
{-# OPTIONS --allow-unsolved-metas #-}
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module FOT.FOL.ImplicitArgumentSubst where
infix 7 _≡_
postulate
D : Set
_·_ : D → D → D
zero succ pred : D
succ₁ : D → D
succ₁ n = succ · n
pred₁ : D → D
pred₁ n = pred · n
-- The identity type on the universe of discourse.
data _≡_ (x : D) : D → Set where
refl : x ≡ x
-- The propositional function is not an implicit argument.
subst : (A : D → Set) → ∀ {x y} → x ≡ y → A x → A y
subst A refl Ax = Ax
-- The propositional formula is an implicit argument.
subst' : {A : D → Set} → ∀ {x y} → x ≡ y → A x → A y
subst' refl Ax = Ax
sym : ∀ {x y} → x ≡ y → y ≡ x
sym refl = refl
-- Conversion rules.
postulate
pred-0 : pred₁ zero ≡ zero
pred-S : ∀ n → pred₁ (succ₁ n) ≡ n
-- The FOTC natural numbers type.
data N : D → Set where
nzero : N zero
nsucc : ∀ {n} → N n → N (succ₁ n)
-- Works using subst.
pred-N : ∀ {n} → N n → N (pred₁ n)
pred-N nzero = subst N (sym pred-0) nzero
pred-N (nsucc {n} Nn) = subst N (sym (pred-S n)) Nn
-- Fails using subst'.
pred-N' : ∀ {n} → N n → N (pred₁ n)
pred-N' nzero = subst' (sym pred-0) nzero
pred-N' (nsucc {n} Nn) = subst' (sym (pred-S n)) Nn
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-- A variant of code reported by Andreas Abel.
-- Andreas, 2014-01-07 Issue reported by Dmitriy Traytel
-- Andreas, 2015-04-15 Issue resurrected with Pierre Hyvernat
{-# OPTIONS --guardedness --sized-types #-}
open import Common.Size
open import Common.Prelude hiding (map)
open import Common.Product
record Stream (A : Set) : Set where
coinductive
field
force : A × Stream A
open Stream
map : ∀{A B} → (A → B) → Stream A → Stream B
force (map f s) = let a , as = force s in f a , map f as
-- This type should be empty, as Stream A is isomorphic to ℕ → A.
data D : (i : Size) → Set where
lim : ∀ i → Stream (D i) → D (↑ i)
-- Emptiness witness for D.
empty : ∀ i → D i → ⊥
empty .(↑ i) (lim i s) = empty i (proj₁ (force s))
-- BAD: But we can construct an inhabitant.
inh : Stream (D ∞)
force inh = lim ∞ inh , inh -- Should be rejected by termination checker.
absurd : ⊥
absurd = empty ∞ (lim ∞ inh)
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module Tactic.Reflection.Quote.Class where
open import Builtin.Reflection
record Quotable {a} (A : Set a) : Set a where
field
` : A → Term
open Quotable {{...}} public
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Some properties imply others
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Relation.Binary.Consequences where
open import Relation.Binary.Core
open import Relation.Nullary using (yes; no)
open import Relation.Unary using (∁)
open import Function using (_∘_; flip)
open import Data.Maybe.Base using (just; nothing)
open import Data.Sum using (inj₁; inj₂)
open import Data.Product using (_,_)
open import Data.Empty using (⊥-elim)
------------------------------------------------------------------------
-- Substitutive properties
module _ {a ℓ p} {A : Set a} {_∼_ : Rel A ℓ} (P : Rel A p) where
subst⟶respˡ : Substitutive _∼_ p → P Respectsˡ _∼_
subst⟶respˡ subst {y} x'∼x Px'y = subst (flip P y) x'∼x Px'y
subst⟶respʳ : Substitutive _∼_ p → P Respectsʳ _∼_
subst⟶respʳ subst {x} y'∼y Pxy' = subst (P x) y'∼y Pxy'
subst⟶resp₂ : Substitutive _∼_ p → P Respects₂ _∼_
subst⟶resp₂ subst = subst⟶respʳ subst , subst⟶respˡ subst
module _ {a ℓ p} {A : Set a} {∼ : Rel A ℓ} {P : A → Set p} where
P-resp⟶¬P-resp : Symmetric ∼ → P Respects ∼ → (∁ P) Respects ∼
P-resp⟶¬P-resp sym resp x∼y ¬Px Py = ¬Px (resp (sym x∼y) Py)
------------------------------------------------------------------------
-- Proofs for non-strict orders
module _ {a ℓ₁ ℓ₂} {A : Set a} {_≈_ : Rel A ℓ₁} {_≤_ : Rel A ℓ₂} where
total⟶refl : _≤_ Respects₂ _≈_ → Symmetric _≈_ →
Total _≤_ → _≈_ ⇒ _≤_
total⟶refl (respʳ , respˡ) sym total {x} {y} x≈y with total x y
... | inj₁ x∼y = x∼y
... | inj₂ y∼x = respʳ x≈y (respˡ (sym x≈y) y∼x)
total+dec⟶dec : _≈_ ⇒ _≤_ → Antisymmetric _≈_ _≤_ →
Total _≤_ → Decidable _≈_ → Decidable _≤_
total+dec⟶dec refl antisym total _≟_ x y with total x y
... | inj₁ x≤y = yes x≤y
... | inj₂ y≤x with x ≟ y
... | yes x≈y = yes (refl x≈y)
... | no x≉y = no (λ x≤y → x≉y (antisym x≤y y≤x))
------------------------------------------------------------------------
-- Proofs for strict orders
module _ {a ℓ₁ ℓ₂} {A : Set a} {_≈_ : Rel A ℓ₁} {_<_ : Rel A ℓ₂} where
trans∧irr⟶asym : Reflexive _≈_ → Transitive _<_ →
Irreflexive _≈_ _<_ → Asymmetric _<_
trans∧irr⟶asym refl trans irrefl x<y y<x =
irrefl refl (trans x<y y<x)
irr∧antisym⟶asym : Irreflexive _≈_ _<_ → Antisymmetric _≈_ _<_ →
Asymmetric _<_
irr∧antisym⟶asym irrefl antisym x<y y<x =
irrefl (antisym x<y y<x) x<y
asym⟶antisym : Asymmetric _<_ → Antisymmetric _≈_ _<_
asym⟶antisym asym x<y y<x = ⊥-elim (asym x<y y<x)
asym⟶irr : _<_ Respects₂ _≈_ → Symmetric _≈_ →
Asymmetric _<_ → Irreflexive _≈_ _<_
asym⟶irr (respʳ , respˡ) sym asym {x} {y} x≈y x<y =
asym x<y (respʳ (sym x≈y) (respˡ x≈y x<y))
tri⟶asym : Trichotomous _≈_ _<_ → Asymmetric _<_
tri⟶asym tri {x} {y} x<y x>y with tri x y
... | tri< _ _ x≯y = x≯y x>y
... | tri≈ _ _ x≯y = x≯y x>y
... | tri> x≮y _ _ = x≮y x<y
tri⟶irr : Trichotomous _≈_ _<_ → Irreflexive _≈_ _<_
tri⟶irr compare {x} {y} x≈y x<y with compare x y
... | tri< _ x≉y y≮x = x≉y x≈y
... | tri> x≮y x≉y y<x = x≉y x≈y
... | tri≈ x≮y _ y≮x = x≮y x<y
tri⟶dec≈ : Trichotomous _≈_ _<_ → Decidable _≈_
tri⟶dec≈ compare x y with compare x y
... | tri< _ x≉y _ = no x≉y
... | tri≈ _ x≈y _ = yes x≈y
... | tri> _ x≉y _ = no x≉y
tri⟶dec< : Trichotomous _≈_ _<_ → Decidable _<_
tri⟶dec< compare x y with compare x y
... | tri< x<y _ _ = yes x<y
... | tri≈ x≮y _ _ = no x≮y
... | tri> x≮y _ _ = no x≮y
trans∧tri⟶respʳ≈ : Symmetric _≈_ → Transitive _≈_ →
Transitive _<_ → Trichotomous _≈_ _<_ →
_<_ Respectsʳ _≈_
trans∧tri⟶respʳ≈ sym ≈-tr <-tr tri {x} {y} {z} y≈z x<y with tri x z
... | tri< x<z _ _ = x<z
... | tri≈ _ x≈z _ = ⊥-elim (tri⟶irr tri (≈-tr x≈z (sym y≈z)) x<y)
... | tri> _ _ z<x = ⊥-elim (tri⟶irr tri (sym y≈z) (<-tr z<x x<y))
trans∧tri⟶respˡ≈ : Transitive _≈_ →
Transitive _<_ → Trichotomous _≈_ _<_ →
_<_ Respectsˡ _≈_
trans∧tri⟶respˡ≈ ≈-tr <-tr tri {z} {_} {y} x≈y x<z with tri y z
... | tri< y<z _ _ = y<z
... | tri≈ _ y≈z _ = ⊥-elim (tri⟶irr tri (≈-tr x≈y y≈z) x<z)
... | tri> _ _ z<y = ⊥-elim (tri⟶irr tri x≈y (<-tr x<z z<y))
trans∧tri⟶resp≈ : Symmetric _≈_ → Transitive _≈_ →
Transitive _<_ → Trichotomous _≈_ _<_ →
_<_ Respects₂ _≈_
trans∧tri⟶resp≈ sym ≈-tr <-tr tri =
trans∧tri⟶respʳ≈ sym ≈-tr <-tr tri ,
trans∧tri⟶respˡ≈ ≈-tr <-tr tri
------------------------------------------------------------------------
-- Without Loss of Generality
module _ {a r q} {A : Set a} {_R_ : Rel A r} {Q : Rel A q} where
wlog : Total _R_ → Symmetric Q →
(∀ a b → a R b → Q a b) →
∀ a b → Q a b
wlog r-total q-sym prf a b with r-total a b
... | inj₁ aRb = prf a b aRb
... | inj₂ bRa = q-sym (prf b a bRa)
------------------------------------------------------------------------
-- Other proofs
module _ {a b p} {A : Set a} {B : Set b} {P : REL A B p} where
dec⟶weaklyDec : Decidable P → WeaklyDecidable P
dec⟶weaklyDec dec x y with dec x y
... | yes p = just p
... | no _ = nothing
module _ {a b p q} {A : Set a} {B : Set b }
{P : REL A B p} {Q : REL A B q}
where
map-NonEmpty : P ⇒ Q → NonEmpty P → NonEmpty Q
map-NonEmpty f x = nonEmpty (f (NonEmpty.proof x))
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{-# OPTIONS --without-K --safe #-}
-- Backwards evaluator and the fact that with eval, forms a reversible evaluator
module BackwardsEval where
open import Data.Unit using (tt)
open import Data.Sum using (inj₁; inj₂)
open import Data.Product using (_,_; proj₁; proj₂)
open import Relation.Binary.PropositionalEquality
using (_≡_; refl; sym; trans; cong; cong₂; subst)
open import Singleton
open import PiFrac
bwd : {A B : 𝕌} → (A ⟷ B) → ⟦ B ⟧ → ⟦ A ⟧
bwd-eval : {A B : 𝕌} → (c : A ⟷ B) → (v : ⟦ A ⟧) → bwd c (eval c v) ≡ v
bwd unite₊l v = inj₂ v
bwd uniti₊l (inj₂ v) = v
bwd unite₊r v = inj₁ v
bwd uniti₊r (inj₁ v) = v
bwd swap₊ (inj₁ v) = inj₂ v
bwd swap₊ (inj₂ v) = inj₁ v
bwd assocl₊ (inj₁ (inj₁ v)) = inj₁ v
bwd assocl₊ (inj₁ (inj₂ v)) = inj₂ (inj₁ v)
bwd assocl₊ (inj₂ v) = inj₂ (inj₂ v)
bwd assocr₊ (inj₁ v) = inj₁ (inj₁ v)
bwd assocr₊ (inj₂ (inj₁ v)) = inj₁ (inj₂ v)
bwd assocr₊ (inj₂ (inj₂ v)) = inj₂ v
bwd unite⋆l v = (tt , v)
bwd uniti⋆l (tt , v) = v
bwd unite⋆r v = (v , tt)
bwd uniti⋆r (v , tt) = v
bwd swap⋆ (v₁ , v₂) = (v₂ , v₁)
bwd assocl⋆ ((v₁ , v₂) , v₃) = (v₁ , (v₂ , v₃))
bwd assocr⋆ (v₁ , (v₂ , v₃)) = ((v₁ , v₂) , v₃)
bwd dist (inj₁ (v₁ , v₂)) = (inj₁ v₁ , v₂)
bwd dist (inj₂ (v₂ , v₃)) = (inj₂ v₂ , v₃)
bwd factor (inj₁ v₁ , v₃) = inj₁ (v₁ , v₃)
bwd factor (inj₂ v₂ , v₃) = inj₂ (v₂ , v₃)
bwd distl (inj₁ (v₁ , v₂)) = (v₁ , inj₁ v₂)
bwd distl (inj₂ (v₁ , v₃)) = (v₁ , inj₂ v₃)
bwd factorl (v₁ , inj₁ v₂) = inj₁ (v₁ , v₂)
bwd factorl (v₁ , inj₂ v₃) = inj₂ (v₁ , v₃)
bwd id⟷ v = v
bwd (c₁ ⊚ c₂) v = bwd c₁ (bwd c₂ v)
bwd (c₁ ⊕ c₂) (inj₁ v) = inj₁ (bwd c₁ v)
bwd (c₁ ⊕ c₂) (inj₂ v) = inj₂ (bwd c₂ v)
bwd (c₁ ⊗ c₂) (v₁ , v₂) = (bwd c₁ v₁ , bwd c₂ v₂)
bwd (lift {_} {_} {v₁} c) (●₁ , v≡●₁) = (bwd c ●₁) , (trans (sym (bwd-eval c v₁)) (cong (bwd c) v≡●₁))
bwd tensorl ((w₁ , p₁) , (w₂ , p₂)) = (w₁ , w₂) , (cong₂ _,_ p₁ p₂)
bwd tensorr ((v₁ , v₂) , p) = (v₁ , cong proj₁ p) , (v₂ , cong proj₂ p)
bwd plusll (w , p) = (inj₁ w) , (cong inj₁ p)
bwd pluslr (inj₁ w₁ , refl) = w₁ , refl
bwd plusrl (w , p) = (inj₂ w) , (cong inj₂ p)
bwd plusrr (inj₂ w₂ , refl) = w₂ , refl
bwd fracl (f₁ , f₂) ((w₁ , w₂) , refl) = let _ = f₁ (w₁ , refl) ; _ = f₂ (w₂ , refl) in tt
bwd (fracr {v₁ = v₁} {v₂ = v₂}) f = (λ _ → f ((v₁ , v₂) , refl)) , (λ _ → f ((v₁ , v₂) , refl))
bwd (η v) p = tt
bwd (ε v) tt = (v , refl) , λ _ → tt
bwd (ll {t} {v} {w}) (v , refl) = w , refl
bwd (== c eq) v = bwd c (subst (Singleton ⟦ _ ⟧) (sym eq) v)
bwd-eval unite₊l (inj₂ v) = refl
bwd-eval uniti₊l v = refl
bwd-eval unite₊r (inj₁ v) = refl
bwd-eval uniti₊r v = refl
bwd-eval swap₊ (inj₁ v) = refl
bwd-eval swap₊ (inj₂ v) = refl
bwd-eval assocl₊ (inj₁ v) = refl
bwd-eval assocl₊ (inj₂ (inj₁ v)) = refl
bwd-eval assocl₊ (inj₂ (inj₂ v)) = refl
bwd-eval assocr₊ (inj₁ (inj₁ v)) = refl
bwd-eval assocr₊ (inj₁ (inj₂ v)) = refl
bwd-eval assocr₊ (inj₂ v) = refl
bwd-eval unite⋆l (tt , v) = refl
bwd-eval uniti⋆l v = refl
bwd-eval unite⋆r (v , tt) = refl
bwd-eval uniti⋆r v = refl
bwd-eval swap⋆ (v₁ , v₂) = refl
bwd-eval assocl⋆ (v₁ , (v₂ , v₃)) = refl
bwd-eval assocr⋆ ((v₁ , v₂) , v₃) = refl
bwd-eval dist (inj₁ v₁ , v₃) = refl
bwd-eval dist (inj₂ v₂ , v₃) = refl
bwd-eval factor (inj₁ (v₁ , v₃)) = refl
bwd-eval factor (inj₂ (v₂ , v₃)) = refl
bwd-eval distl (v₁ , inj₁ v₂) = refl
bwd-eval distl (v₁ , inj₂ v₃) = refl
bwd-eval factorl (inj₁ (v₁ , v₂)) = refl
bwd-eval factorl (inj₂ (v₁ , v₃)) = refl
bwd-eval id⟷ v = refl
bwd-eval (c₁ ⊚ c₂) v = trans (cong (bwd c₁) (bwd-eval c₂ (eval c₁ v))) (bwd-eval c₁ v)
bwd-eval (c₁ ⊕ c₂) (inj₁ v₁) = cong inj₁ (bwd-eval c₁ v₁)
bwd-eval (c₁ ⊕ c₂) (inj₂ v₂) = cong inj₂ (bwd-eval c₂ v₂)
bwd-eval (c₁ ⊗ c₂) (v₁ , v₂) = cong₂ _,_ (bwd-eval c₁ v₁) (bwd-eval c₂ v₂)
bwd-eval (lift c) v = pointed-all-paths
bwd-eval tensorl p = pointed-all-paths
bwd-eval tensorr (p₁ , p₂) = cong₂ _,_ pointed-all-paths pointed-all-paths
bwd-eval plusll p = pointed-all-paths
bwd-eval pluslr p = pointed-all-paths
bwd-eval plusrl p = pointed-all-paths
bwd-eval plusrr p = pointed-all-paths
bwd-eval fracl f = {!!} -- needs recip-all-paths
bwd-eval fracr (f₁ , f₂) = {!!}
bwd-eval (η v) tt = refl
bwd-eval (ε v) (p , r) = cong₂ _,_ pointed-all-paths refl
bwd-eval (ll {t} {v} {.w}) (w , refl) = refl
bwd-eval (== c eq) p = pointed-all-paths
eval-bwd : {A B : 𝕌} → (c : A ⟷ B) → (v : ⟦ B ⟧) → eval c (bwd c v) ≡ v
eval-bwd unite₊l v = refl
eval-bwd uniti₊l (inj₂ v) = refl
eval-bwd unite₊r v = refl
eval-bwd uniti₊r (inj₁ v) = refl
eval-bwd swap₊ (inj₁ v) = refl
eval-bwd swap₊ (inj₂ v) = refl
eval-bwd assocl₊ (inj₁ (inj₁ v)) = refl
eval-bwd assocl₊ (inj₁ (inj₂ v)) = refl
eval-bwd assocl₊ (inj₂ v) = refl
eval-bwd assocr₊ (inj₁ v) = refl
eval-bwd assocr₊ (inj₂ (inj₁ v)) = refl
eval-bwd assocr₊ (inj₂ (inj₂ v)) = refl
eval-bwd unite⋆l v = refl
eval-bwd uniti⋆l (tt , v) = refl
eval-bwd unite⋆r v = refl
eval-bwd uniti⋆r (v , tt) = refl
eval-bwd swap⋆ (v₂ , v₁) = refl
eval-bwd assocl⋆ ((v₁ , v₂) , v₃) = refl
eval-bwd assocr⋆ (v₁ , (v₂ , v₃)) = refl
eval-bwd dist (inj₁ (v₁ , v₂)) = refl
eval-bwd dist (inj₂ (v₂ , v₃)) = refl
eval-bwd factor (inj₁ v₁ , v₃) = refl
eval-bwd factor (inj₂ v₂ , v₃) = refl
eval-bwd distl (inj₁ (v₁ , v₂)) = refl
eval-bwd distl (inj₂ (v₁ , v₃)) = refl
eval-bwd factorl (v₁ , inj₁ v₂) = refl
eval-bwd factorl (v₁ , inj₂ v₃) = refl
eval-bwd id⟷ v = refl
eval-bwd (c₁ ⊚ c₂) v = trans (cong (eval c₂) (eval-bwd c₁ (bwd c₂ v))) (eval-bwd c₂ v)
eval-bwd (c₁ ⊕ c₂) (inj₁ v) = cong inj₁ (eval-bwd c₁ v)
eval-bwd (c₁ ⊕ c₂) (inj₂ v) = cong inj₂ (eval-bwd c₂ v)
eval-bwd (c₁ ⊗ c₂) (v₃ , v₄) = cong₂ _,_ (eval-bwd c₁ v₃) (eval-bwd c₂ v₄)
eval-bwd (lift c) x = pointed-all-paths
eval-bwd tensorl p = cong₂ _,_ pointed-all-paths pointed-all-paths
eval-bwd tensorr p = pointed-all-paths
eval-bwd plusll p = pointed-all-paths
eval-bwd pluslr p = pointed-all-paths
eval-bwd plusrl p = pointed-all-paths
eval-bwd plusrr p = pointed-all-paths
eval-bwd fracl (f₁ , f₂) = {!!} -- needs recip-all-paths
eval-bwd fracr f = {!!}
eval-bwd (η v) (p , r) = cong₂ _,_ pointed-all-paths refl
eval-bwd (ε v) tt = refl
eval-bwd (ll {t} {.v} {w}) (v , refl) = refl
eval-bwd (== c eq) p = pointed-all-paths
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{-# OPTIONS --omega-in-omega --no-termination-check --overlapping-instances #-}
module Light.Library.Data.Integer where
open import Light.Level using (Level ; Setω)
open import Light.Library.Data.Empty as Empty using (Empty)
open import Light.Library.Data.Natural as Natural using (ℕ)
open import Light.Subtyping using (DirectSubtyping ; #_)
open import Light.Library.Arithmetic using (Arithmetic)
open import Light.Package using (Package)
open import Light.Library.Relation.Binary
using (SelfTransitive ; SelfSymmetric ; Reflexive)
open import Light.Library.Relation.Binary.Equality as ≈ using (_≈_)
open import Light.Library.Relation.Binary.Equality.Decidable using (DecidableSelfEquality)
import Light.Library.Relation.Decidable
record Dependencies : Setω where
field ⦃ natural‐package ⦄ : Package record { Natural }
record Library (dependencies : Dependencies) : Setω where
field
ℓ ≈ℓ : Level
ℤ : Set ℓ
zero : ℤ
successor : ℤ → ℤ
predecessor : ℤ → ℤ
_+_ : ℤ → ℤ → ℤ
_∗_ : ℤ → ℤ → ℤ
_//_ : ℤ → ℤ → ℤ
_−_ : ℤ → ℤ → ℤ
−_ : ℤ → ℤ
from‐natural : ℕ → ℤ
⦃ equals ⦄ : DecidableSelfEquality ℤ
⦃ ≈‐transitive ⦄ : SelfTransitive (≈.self‐relation ℤ)
⦃ ≈‐symmetric ⦄ : SelfSymmetric (≈.self‐relation ℤ)
⦃ ≈‐reflexive ⦄ : Reflexive (≈.self‐relation ℤ)
instance naturals‐are‐integers : DirectSubtyping ℕ ℤ
naturals‐are‐integers = # from‐natural
instance arithmetic : Arithmetic ℤ
arithmetic = record { _+_ = _+_ ; _∗_ = _∗_ }
open Library ⦃ ... ⦄ public
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{-# OPTIONS --without-K --safe #-}
open import Algebra
module Quasigroup.Properties
{a ℓ} (Q : Quasigroup a ℓ) where
open Quasigroup Q
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Algebra.Magma.Construct.Empty where
open import Cubical.Core.Everything
open import Cubical.Foundations.Prelude
open import Cubical.Algebra.Magma
open import Cubical.Data.Empty
_◯_ : Op₂ ⊥
_◯_ ()
isSet⊥ : isSet ⊥
isSet⊥ = isProp→isSet isProp⊥
⊥-isMagma : IsMagma ⊥ _◯_
⊥-isMagma = record { is-set = isSet⊥ }
⊥-Magma : Magma ℓ-zero
⊥-Magma = record { isMagma = ⊥-isMagma }
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Bundles of parameters for passing to the Ring Solver
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
-- This module packages up all the stuff that's passed to the other
-- modules in a convenient form.
module Tactic.RingSolver.Core.Polynomial.Parameters where
open import Algebra.Bundles using (RawRing)
open import Data.Bool.Base using (Bool; T)
open import Function
open import Level
open import Relation.Unary
open import Tactic.RingSolver.Core.AlmostCommutativeRing
-- This record stores all the stuff we need for the coefficients:
--
-- * A raw ring
-- * A (decidable) predicate on "zeroeness"
--
-- It's used for defining the operations on the Horner normal form.
record RawCoeff ℓ₁ ℓ₂ : Set (suc (ℓ₁ ⊔ ℓ₂)) where
field
rawRing : RawRing ℓ₁ ℓ₂
isZero : RawRing.Carrier rawRing → Bool
open RawRing rawRing public
-- This record stores the full information we need for converting
-- to the final ring.
record Homomorphism ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Set (suc (ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄)) where
field
from : RawCoeff ℓ₁ ℓ₂
to : AlmostCommutativeRing ℓ₃ ℓ₄
module Raw = RawCoeff from
open AlmostCommutativeRing to public
field
morphism : Raw.rawRing -Raw-AlmostCommutative⟶ to
open _-Raw-AlmostCommutative⟶_ morphism renaming (⟦_⟧ to ⟦_⟧ᵣ) public
field
Zero-C⟶Zero-R : ∀ x → T (Raw.isZero x) → 0# ≈ ⟦ x ⟧ᵣ
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{-# OPTIONS --omega-in-omega --no-termination-check --overlapping-instances #-}
module Light.Implementation.Data.Either where
open import Light.Library.Data.Either using (Library ; Dependencies)
open import Light.Level using (_⊔_)
open import Light.Variable.Levels
instance dependencies : Dependencies
dependencies = record {}
instance library : Library dependencies
library = record { Implementation }
where
module Implementation where
ℓf : _
ℓf = _⊔_
data Either (𝕒 : Set aℓ) (𝕓 : Set bℓ) : Set (aℓ ⊔ bℓ) where
left : 𝕒 → Either 𝕒 𝕓
right : 𝕓 → Either 𝕒 𝕓
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module irreflexive where
open import Data.Nat using (ℕ; zero; suc)
open import Negation using (¬_)
infix 4 _<_
data _<_ : ℕ → ℕ → Set where
z<s : ∀ {n : ℕ}
------------
→ zero < suc n
s<s : ∀ {m n : ℕ}
→ m < n
-------------
→ suc m < suc n
-- n が非反射律 ∀ n ∈ ℕ, ¬(n < n) を満たすことの証明
<-irreflexive : ∀ (n : ℕ) → ¬ (n < n)
<-irreflexive zero ()
<-irreflexive (suc n) (s<s n<n) = <-irreflexive n n<n
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{-# OPTIONS --without-K --exact-split --safe #-}
module Identity where
open import Basic_Types
-- ------------------------------------
-- the identity type, or path type
infix 1 _≡_
data _≡_ {A : Set} : A → A → Set where
refl : ∀ (a : A) → a ≡ a
{-# BUILTIN EQUALITY _≡_ #-}
-- path induction
p-ind : ∀ {A : Set} {B : ∀ (x y : A) (p : x ≡ y) → Set} →
(∀ (z : A) → B z z (refl z)) → ∀ (x y : A) (p : x ≡ y) → B x y p
p-ind f z z (refl z) = f z
-- notice here that, in the proof system of agda, the induction principle is
-- builtin for a data type, for their constructor is given expicitly. Hence,
-- one way to prove identity type is to use p-ind, which is itself proved by
-- agda using implicit path induction, or we can use agda's induction expicitly
-- which is the case when we construct apd and ass-p
-- the groupoid structure of path type
concat : ∀ {A : Set} → ∀ (x y : A) (p : x ≡ y) → ∀ (z : A) (q : y ≡ z) → x ≡ z
concat {A} x x (refl x) x (refl x) = refl x
infix 3 _∙_
_∙_ : ∀ {A : Set} {x y z : A} → ∀ (p : x ≡ y) (q : y ≡ z) → x ≡ z
_∙_ {_} {x} {y} {z} p q = concat x y p z q
inv : ∀ {A : Set} {x y : A} → ∀ (p : x ≡ y) → y ≡ x
inv {_} {x} {x} (refl x) = refl x
-- the groupoid structure of identities, viewed as higher paths
p-ass : ∀ {A : Set} {x y z w : A} → ∀ (p : x ≡ y) (q : y ≡ z) (r : z ≡ w) →
p ∙ (q ∙ r) ≡ (p ∙ q) ∙ r
p-ass {_} {x} {x} {x} {x} (refl x) (refl x) (refl x) =
refl (((refl x) ∙ (refl x)) ∙ (refl x))
left_unit : ∀ {A : Set} {x y : A} → ∀ (p : x ≡ y) → (refl x) ∙ p ≡ p
left_unit {_} {x} {x} (refl x) = refl (refl x)
right_unit : ∀ {A : Set} {x y : A} → ∀ (p : x ≡ y) → p ∙ (refl y) ≡ p
right_unit {_} {x} {x} (refl x) = refl (refl x)
rinv-unit : ∀ {A : Set} {x y : A} → ∀ (p : x ≡ y) → (inv p) ∙ p ≡ refl y
rinv-unit {_} {x} {x} (refl x) = refl (refl x)
linv-unit : ∀ {A : Set} {x y : A} → ∀ (p : x ≡ y) → p ∙ (inv p) ≡ refl x
linv-unit {_} {x} {x} (refl x) = refl (refl x)
invinv : ∀ {A : Set} {x y : A} → ∀ (p : x ≡ y) → inv (inv p) ≡ p
invinv {_} {x} {x} (refl x) = refl (refl x)
-- ------------------------------------
-- loop spaces
Ω : ∀ {A : Set} (a : A) → Set
Ω {_} a = a ≡ a
-- commutativity on 2-dimensional loops
-- prove the commutativity by proving a general result in the groupoid
pwr : ∀ {A : Set} {a b c : A} {p q : a ≡ b} → ∀ (r : b ≡ c) → ∀ (α : p ≡ q) →
p ∙ r ≡ q ∙ r
pwr {_} {_} {b} {b} {p} {p} (refl b) (refl p) = refl (p ∙ (refl b))
pwl : ∀ {A : Set} {a b c : A} {q r : b ≡ c} → ∀ (p : a ≡ b) → ∀ (β : q ≡ r) →
p ∙ q ≡ p ∙ r
pwl {_} {_} {_} {_} {q} {q} p (refl q) = refl (p ∙ q)
_*_ : ∀ {A : Set} {a b c : A} {p q : a ≡ b} {r s : b ≡ c} →
∀ (α : p ≡ q) (β : r ≡ s) → p ∙ r ≡ q ∙ s
_*_ {_} {_} {_} {_} {p} {q} {r} {s} α β = α∙r ∙ q∙β
where α∙r : p ∙ r ≡ q ∙ r
α∙r = pwr r α
q∙β : q ∙ r ≡ q ∙ s
q∙β = pwl q β
_*'_ : ∀ {A : Set} {a b c : A} {p q : a ≡ b} {r s : b ≡ c} →
∀ (α : p ≡ q) (β : r ≡ s) → p ∙ r ≡ q ∙ s
_*'_ {_} {_} {_} {_} {p} {q} {r} {s} α β = p∙β ∙ α∙s
where p∙β : p ∙ r ≡ p ∙ s
p∙β = pwl p β
α∙s : p ∙ s ≡ q ∙ s
α∙s = pwr s α
pwr-ru : ∀ {A : Set} {a : A} (p : a ≡ a) (α : refl a ≡ p) →
pwr (refl a) α ≡ α ∙ (inv (right_unit p))
pwr-ru {_} {_} (refl a) (refl (refl a)) = refl (refl (refl a))
pwl-lu : ∀ {A : Set} {a : A} (p : a ≡ a) (α : refl a ≡ p) →
pwl (refl a) α ≡ α ∙ (inv (left_unit p))
pwl-lu {_} {_} (refl a) (refl (refl a)) = refl (refl (refl a))
*≡*' : ∀ {A : Set} {a b c : A} {p q : a ≡ b} {r s : b ≡ c} →
∀ (α : p ≡ q) (β : r ≡ s) → (α * β) ≡ (α *' β)
*≡*' {_} {a} {a} {a} {refl a} {refl a} {refl a} {refl a}
(refl (refl a)) (refl (refl a))
= refl ((refl ((refl a) ∙ (refl a))) ∙ (refl ((refl a) ∙ (refl a))))
∙≡* : ∀ {A : Set} {a : A} → ∀ (α β : Ω (refl a)) → α ∙ β ≡ α * β
∙≡* {_} {a} α β = inv (redua)
where redul : α * β ≡ α ∙ (pwl (refl a) β)
redul = (pwr (pwl (refl a) β) (pwr-ru (refl a) α)) ∙
(pwr (pwl (refl a) β) (right_unit α))
redua : α * β ≡ α ∙ β
redua = (redul ∙ (pwl α (pwl-lu (refl a) β))) ∙ (pwl α (right_unit β))
*'≡∙ : ∀ {A : Set} {a : A} → ∀ (α β : Ω (refl a)) → α *' β ≡ β ∙ α
*'≡∙ {_} {a} α β = redua'
where redul' : α *' β ≡ β ∙ (pwr (refl a) α)
redul' = (pwr (pwr (refl a) α) (pwl-lu (refl a) β)) ∙
(pwr (pwr (refl a) α) (right_unit β))
redua' : α *' β ≡ β ∙ α
redua' = redul' ∙ ((pwl β (pwr-ru (refl a) α)) ∙ (pwl β (right_unit α)))
-- finally, we are able to prove that the higher path group is commutative
Ω²-comm : ∀ {A : Set} {a : A} → ∀ (α β : Ω (refl a)) → α ∙ β ≡ β ∙ α
Ω²-comm α β = ((∙≡* α β) ∙ (*≡*' α β)) ∙ (*'≡∙ α β)
-- ------------------------------------
-- functions behave functorially
-- the action on paths
ap : ∀ {A B : Set} {x y : A} → ∀ (f : A → B) (p : x ≡ y) → f x ≡ f y
ap {_} {_} {x} {x} f (refl x) = refl (f x)
-- ap acts like a functor
ap∙ : ∀ {A B : Set} {x y z : A} → ∀ (f : A → B) (p : x ≡ y) (q : y ≡ z) →
ap f (p ∙ q) ≡ (ap f p) ∙ (ap f q)
ap∙ {_} {_} {x} {x} {x} f (refl x) (refl x) = inv (left_unit (refl (f x)))
apinv : ∀ {A B : Set} {x y : A} → ∀ (f : A → B) (p : x ≡ y) →
ap f (inv p) ≡ inv (ap f p)
apinv {_} {_} {x} {x} f (refl x) = refl (refl (f x))
apcomp : ∀ {A B C : Set} {x y : A} → ∀ (f : A → B) (g : B → C) (p : x ≡ y) →
ap g (ap f p) ≡ ap (comp g f) p
apcomp {_} {_} {_} {x} {x} f g (refl x) = refl (refl (g (f x)))
apid : ∀ {A : Set} {x y : A} → ∀ (p : x ≡ y) →
ap id p ≡ p
apid {_} {x} {x} (refl x) = refl (refl x)
-- the transport action, which is action on dependent types
tr : ∀ {A : Set} {B : A → Set} → ∀ {x y : A} → ∀ (p : x ≡ y) → B x → B y
tr {A} {B} {x} {x} (refl x) = id
<<<<<<< HEAD
=======
trc : ∀ {A B : Set} {x y : A} (p : x ≡ y) (b : B) → tr {A} {λ x → B} p b ≡ b
trc {_} {_} {x} {x} (refl x) b = refl b
>>>>>>> a0dc66dca08059f740d9290f3d1cad1bb007d721
-- If you think about this, this is not obvious from a classical point of view.
-- The existence of the transport function tells us that the dependent type
-- B : A → Set is always going to be continuous, since whenever we have a path
-- in the base A from x to y, and whenever we're given a point in B x, we can
-- construct a value in B y, using the path in the base.
-- path lifting
lift : ∀ {A : Set} {B : A → Set} → ∀ {x y : A} → ∀ (p : x ≡ y) → ∀ (b : B x) →
(x , b) ≡ y , (tr {A} {B} p b)
lift {A} {B} {x} {x} (refl x) b = refl (x , b)
-- the dependent mapping functor
<<<<<<< HEAD
apd : ∀ {A : Set} {B : A → Set} {x y : A} → ∀ (f : (∀ (a : A) → B a)) →
∀ (p : x ≡ y) → tr {A} {B} {x} {y} p (f x) ≡ f y
apd {A} {B} {x} {x} f (refl x) = refl (tr {A} {B} {x} {x} (refl x) (f x))
-- the dependent and non-dependent ap(d) is closely related
trc : ∀ {A B : Set} {x y : A} (p : x ≡ y) (b : B) → tr {A} {λ x → B} p b ≡ b
trc {_} {_} {x} {x} (refl x) b = refl b
-- and now we can prove the following:
apd≡ap : ∀ {A B : Set} {x y : A} →
∀ (f : A → B) (p : x ≡ y) → apd f p ≡ trc p (f x) ∙ (ap f p)
apd≡ap {_} {_} {x} {x} f (refl x) = refl (refl (f x))
-- some useful properties about transport
tr∙ : ∀ {A : Set} {B : A → Set} {x y z : A} → ∀ (p : x ≡ y) (q : y ≡ z) →
∀ (b : B x) → tr {A} {B} q (tr {A} {B} p b) ≡ tr {A} {B} (p ∙ q) b
tr∙ {_} {_} {x} {x} {x} (refl x) (refl x) b = refl b
trcomp : ∀ {A B : Set} {P : B → Set} {x y : A} → ∀ (f : A → B) (p : x ≡ y) →
∀ (b : P (f x)) → tr {A} {λ a → P (f a)} p b ≡ tr {B} {P} (ap f p) b
trcomp {_} {_} {_} {x} {x} f (refl x) b = refl b
trhi : ∀ {A : Set} {P Q : A → Set} {x y : A} → ∀ (f : ∀ (x : A) → P x → Q x) →
∀ (p : x ≡ y) (b : P x) → f y (tr {A} {P} p b) ≡ tr {A} {Q} p (f x b)
trhi {_} {_} {_} {x} {x} f (refl x) b = refl (f x b)
=======
apd : ∀ {A : Set} {B : A → Set} → ∀ (f : (∀ (a : A) → B a)) →
∀ (x y : A) (p : x ≡ y) → tr {A} {B} {x} {y} p (f x) ≡ f y
apd {A} {B} f x x (refl x) = refl (tr {A} {B} {x} {x} (refl x) (f x))
-- apd helps to construct a
>>>>>>> a0dc66dca08059f740d9290f3d1cad1bb007d721
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open import Agda.Builtin.List
open import Agda.Builtin.Reflection
open import Agda.Builtin.Unit
open import Agda.Builtin.Nat
-- setup
infixl 5 _>>=_
_>>=_ = bindTC
defToTerm : Name → Definition → List (Arg Term) → Term
defToTerm _ (function cs) as = pat-lam cs as
defToTerm _ (data-cons d) as = con d as
defToTerm _ _ _ = unknown
derefImmediate : Term → TC Term
derefImmediate (def f args) = getDefinition f >>= λ f' → returnTC (defToTerm f f' args)
derefImmediate x = returnTC x
macro
reflect : ∀ {ℓ} {t : Set ℓ} → t → Term → TC ⊤
reflect x a = quoteTC x >>= derefImmediate >>= quoteTC >>= unify a
unfold : Name → Term → TC ⊤
unfold x a = getDefinition x >>= λ d → unify a (defToTerm x d [])
-- crash
data Vec (A : Set) : Nat → Set where
[] : Vec A zero
_∷_ : ∀ {n} → A → Vec A n → Vec A (suc n)
vid vid' : ∀ n → Vec Nat n → Vec Nat n
vid n [] = []
vid n (x ∷ xs) = x ∷ xs
vid' = unfold vid
crash : Term
crash = reflect vid
len : ∀ {n} → Vec Nat n → Nat
len [] = 0
len (x ∷ xs) = suc (len xs)
-- Forces different order in clause tel and patterns
f f' : ∀ n → Vec Nat n → Vec Nat n → Nat
f n xs [] = n
f n xs (y ∷ ys) = n + y + len xs + len ys
f' = unfold f
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open import Agda.Primitive using (lzero; lsuc; _⊔_)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; setoid; cong; trans)
import Function.Equality
open import Relation.Binary using (Setoid)
import Categories.Category
import Categories.Functor
import Categories.Category.Instance.Setoids
import Categories.Monad.Relative
import Categories.Category.Equivalence
import Categories.Category.Cocartesian
import SecondOrder.Arity
import SecondOrder.Signature
import SecondOrder.Metavariable
import SecondOrder.VRenaming
import SecondOrder.Term
import SecondOrder.IndexedCategory
import SecondOrder.RelativeKleisli
module SecondOrder.Substitution
{ℓ}
{𝔸 : SecondOrder.Arity.Arity}
(Σ : SecondOrder.Signature.Signature ℓ 𝔸)
where
open SecondOrder.Signature.Signature Σ
open SecondOrder.Metavariable Σ
open SecondOrder.Term Σ
open SecondOrder.VRenaming Σ
-- substitution
infix 4 _⊕_⇒ˢ_
_⊕_⇒ˢ_ : ∀ (Θ : MContext) (Γ Δ : VContext) → Set ℓ
Θ ⊕ Γ ⇒ˢ Δ = ∀ {A} (x : A ∈ Γ) → Term Θ Δ A
-- syntactic equality of substitutions
infix 5 _≈ˢ_
_≈ˢ_ : ∀ {Θ} {Γ Δ} (σ τ : Θ ⊕ Γ ⇒ˢ Δ) → Set ℓ
_≈ˢ_ {Θ} {Γ} σ τ = ∀ {A} (x : A ∈ Γ) → σ x ≈ τ x
-- equality of substitutions form a setoid
≈ˢ-refl : ∀ {Θ Γ Δ} {σ : Θ ⊕ Γ ⇒ˢ Δ}
→ σ ≈ˢ σ
≈ˢ-refl x = ≈-refl
≈ˢ-sym : ∀ {Θ Γ Δ} {σ τ : Θ ⊕ Γ ⇒ˢ Δ}
→ σ ≈ˢ τ
→ τ ≈ˢ σ
≈ˢ-sym eq x = ≈-sym (eq x)
≈ˢ-trans : ∀ {Θ Γ Δ} {σ τ μ : Θ ⊕ Γ ⇒ˢ Δ}
→ σ ≈ˢ τ → τ ≈ˢ μ
→ σ ≈ˢ μ
≈ˢ-trans eq1 eq2 x = ≈-trans (eq1 x) (eq2 x)
substitution-setoid : ∀ (Γ Δ : VContext) (Θ : MContext) → Setoid ℓ ℓ
substitution-setoid Γ Δ Θ =
record
{ Carrier = Θ ⊕ Γ ⇒ˢ Δ
; _≈_ = λ σ τ → σ ≈ˢ τ
; isEquivalence =
record
{ refl = λ {σ} x → ≈ˢ-refl {σ = σ} x
; sym = ≈ˢ-sym
; trans = ≈ˢ-trans
}
}
congˢ : ∀ {Θ} {Γ Δ} {A} {σ τ : Θ ⊕ Γ ⇒ˢ Δ} {x : A ∈ Γ} → σ ≈ˢ τ → σ x ≈ τ x
congˢ {x = x} eq = eq x
congˢ-var : ∀ {Θ} {Γ Δ} {A} {σ : Θ ⊕ Γ ⇒ˢ Δ} {x y : A ∈ Γ} → x ≡ y → σ x ≈ σ y
congˢ-var refl = ≈-refl
-- extension of a substitution
⇑ˢ : ∀ {Θ Γ Δ Ξ} → Θ ⊕ Γ ⇒ˢ Δ → Θ ⊕ (Γ ,, Ξ) ⇒ˢ (Δ ,, Ξ)
⇑ˢ σ (var-inl x) = [ var-inl ]ᵛ σ x
⇑ˢ σ (var-inr y) = tm-var (var-inr y)
-- extension respects equality of substitutions
⇑ˢ-resp-≈ˢ : ∀ {Θ Γ Δ Ξ} {σ τ : Θ ⊕ Γ ⇒ˢ Δ} → σ ≈ˢ τ → ⇑ˢ {Ξ = Ξ} σ ≈ˢ ⇑ˢ {Ξ = Ξ} τ
⇑ˢ-resp-≈ˢ ξ (var-inl x) = []ᵛ-resp-≈ (ξ x)
⇑ˢ-resp-≈ˢ ξ (var-inr y) = ≈-refl
-- the action of a renaming on a substitution
infixr 6 _ᵛ∘ˢ_
_ᵛ∘ˢ_ : ∀ {Θ} {Γ Δ Ξ} (ρ : Δ ⇒ᵛ Ξ) (σ : Θ ⊕ Γ ⇒ˢ Δ) → Θ ⊕ Γ ⇒ˢ Ξ
(ρ ᵛ∘ˢ σ) x = [ ρ ]ᵛ (σ x)
infixl 6 _ˢ∘ᵛ_
_ˢ∘ᵛ_ : ∀ {Θ} {Γ Δ Ξ} (σ : Θ ⊕ Δ ⇒ˢ Ξ) (ρ : Γ ⇒ᵛ Δ) → Θ ⊕ Γ ⇒ˢ Ξ
(σ ˢ∘ᵛ ρ) x = σ (ρ x)
-- extension commutes with renaming action
⇑ˢ-resp-ˢ∘ᵛ : ∀ {Θ} {Γ Δ Ξ Ψ} {ρ : Γ ⇒ᵛ Δ} {σ : Θ ⊕ Δ ⇒ˢ Ξ} → ⇑ˢ {Ξ = Ψ} (σ ˢ∘ᵛ ρ) ≈ˢ ⇑ˢ σ ˢ∘ᵛ ⇑ᵛ ρ
⇑ˢ-resp-ˢ∘ᵛ (var-inl x) = ≈-refl
⇑ˢ-resp-ˢ∘ᵛ (var-inr x) = ≈-refl
-- the action of a substitution on a term
infix 6 [_]ˢ_
[_]ˢ_ : ∀ {Θ Γ Δ A} → Θ ⊕ Γ ⇒ˢ Δ → Term Θ Γ A → Term Θ Δ A
[ σ ]ˢ (tm-var x) = σ x
[ σ ]ˢ (tm-meta M ts) = tm-meta M (λ i → [ σ ]ˢ ts i)
[ σ ]ˢ (tm-oper f es) = tm-oper f (λ i → [ ⇑ˢ σ ]ˢ es i)
-- composition of substitutions
infixl 7 _∘ˢ_
_∘ˢ_ : ∀ {Θ} {Γ Δ Ξ} → Θ ⊕ Δ ⇒ˢ Ξ → Θ ⊕ Γ ⇒ˢ Δ → Θ ⊕ Γ ⇒ˢ Ξ
(σ ∘ˢ τ) x = [ σ ]ˢ τ x
-- substitution action respects equality of terms
[]ˢ-resp-≈ : ∀ {Θ} {Γ Δ} {A} (σ : Θ ⊕ Γ ⇒ˢ Δ) {t u : Term Θ Γ A} → t ≈ u → [ σ ]ˢ t ≈ [ σ ]ˢ u
[]ˢ-resp-≈ σ (≈-≡ refl) = ≈-refl
[]ˢ-resp-≈ σ (≈-meta ξ) = ≈-meta (λ i → []ˢ-resp-≈ σ (ξ i))
[]ˢ-resp-≈ σ (≈-oper ξ) = ≈-oper (λ i → []ˢ-resp-≈ (⇑ˢ σ) (ξ i))
-- substitution action respects equality of substitutions
[]ˢ-resp-≈ˢ : ∀ {Θ} {Γ Δ} {A} {σ τ : Θ ⊕ Γ ⇒ˢ Δ} (t : Term Θ Γ A) → σ ≈ˢ τ → [ σ ]ˢ t ≈ [ τ ]ˢ t
[]ˢ-resp-≈ˢ (tm-var x) ξ = ξ x
[]ˢ-resp-≈ˢ (tm-meta M ts) ξ = ≈-meta (λ i → []ˢ-resp-≈ˢ (ts i) ξ)
[]ˢ-resp-≈ˢ (tm-oper f es) ξ = ≈-oper (λ i → []ˢ-resp-≈ˢ (es i) (⇑ˢ-resp-≈ˢ ξ))
-- substitution actions respects both equalities
[]ˢ-resp-≈ˢ-≈ : ∀ {Θ} {Γ Δ} {A} {σ τ : Θ ⊕ Γ ⇒ˢ Δ} {t u : Term Θ Γ A} → σ ≈ˢ τ → t ≈ u → [ σ ]ˢ t ≈ [ τ ]ˢ u
[]ˢ-resp-≈ˢ-≈ {τ = τ} {t = t} ζ ξ = ≈-trans ([]ˢ-resp-≈ˢ t ζ) ([]ˢ-resp-≈ τ ξ)
-- identity substitution
idˢ : ∀ {Θ Γ} → Θ ⊕ Γ ⇒ˢ Γ
idˢ = tm-var
-- extension preserves identity
⇑ˢ-resp-idˢ : ∀ {Θ} {Γ Δ} → ⇑ˢ idˢ ≈ˢ idˢ {Θ = Θ} {Γ = Γ ,, Δ}
⇑ˢ-resp-idˢ (var-inl x) = ≈-refl
⇑ˢ-resp-idˢ (var-inr y) = ≈-refl
-- the identity substution acts trivially
[idˢ] : ∀ {Θ} {Γ} {A} {t : Term Θ Γ A} → [ idˢ ]ˢ t ≈ t
[idˢ] {t = tm-var x} = ≈-refl
[idˢ] {t = tm-meta M ts} = ≈-meta (λ i → [idˢ])
[idˢ] {t = tm-oper f es} = ≈-oper (λ i → ≈-trans ([]ˢ-resp-≈ˢ (es i) ⇑ˢ-resp-idˢ) [idˢ])
-- the identity substitution preserves equality of terms
[idˢ]-resp-≈ : ∀ {Θ} {Γ} {A} {t s : Term Θ Γ A} → t ≈ s → [ idˢ ]ˢ t ≈ s
[idˢ]-resp-≈ t≈s = ≈-trans ([]ˢ-resp-≈ idˢ t≈s) [idˢ]
-- if a substiution is equal to the identity then it acts trivially
≈ˢ-idˢ-[]ˢ : ∀ {Θ} {Γ} {A} {σ : Θ ⊕ Γ ⇒ˢ Γ} {t : Term Θ Γ A} → σ ≈ˢ idˢ → [ σ ]ˢ t ≈ t
≈ˢ-idˢ-[]ˢ {t = t} ξ = ≈-trans ([]ˢ-resp-≈ˢ t ξ) [idˢ]
-- interaction of extension and right renaming action
[⇑ˢ∘ᵛ] : ∀ {Θ} {A} {Γ Δ Ξ Ψ} {σ : Θ ⊕ Δ ⇒ˢ Ξ} {ρ : Γ ⇒ᵛ Δ} (t : Term Θ (Γ ,, Ψ) A) →
[ ⇑ˢ (σ ˢ∘ᵛ ρ) ]ˢ t ≈ [ ⇑ˢ σ ]ˢ [ ⇑ᵛ ρ ]ᵛ t
[⇑ˢ∘ᵛ] (tm-var (var-inl x)) = ≈-refl
[⇑ˢ∘ᵛ] (tm-var (var-inr x)) = ≈-refl
[⇑ˢ∘ᵛ] (tm-meta M ts) = ≈-meta (λ i → [⇑ˢ∘ᵛ] (ts i))
[⇑ˢ∘ᵛ] (tm-oper f es) = ≈-oper (λ i → ≈-trans ([]ˢ-resp-≈ˢ (es i) (⇑ˢ-resp-≈ˢ ⇑ˢ-resp-ˢ∘ᵛ)) ([⇑ˢ∘ᵛ] (es i)))
-- interaction of extension and left renaming action
⇑ˢ-resp-ᵛ∘ˢ : ∀ {Θ} {Γ Δ Ξ Ψ} {σ : Θ ⊕ Γ ⇒ˢ Δ} {ρ : Δ ⇒ᵛ Ξ} →
⇑ˢ {Ξ = Ψ} (ρ ᵛ∘ˢ σ) ≈ˢ ⇑ᵛ ρ ᵛ∘ˢ ⇑ˢ σ
⇑ˢ-resp-ᵛ∘ˢ (var-inl x) = ≈-trans (≈-sym [∘ᵛ]) (≈-trans ([]ᵛ-resp-≡ᵛ (λ _ → refl)) [∘ᵛ])
⇑ˢ-resp-ᵛ∘ˢ (var-inr y) = ≈-refl
[⇑ᵛ∘ˢ] : ∀ {Θ} {A} {Γ Δ Ξ Ψ} {σ : Θ ⊕ Γ ⇒ˢ Δ} {ρ : Δ ⇒ᵛ Ξ} (t : Term Θ (Γ ,, Ψ) A) →
[ ⇑ˢ (ρ ᵛ∘ˢ σ) ]ˢ t ≈ [ ⇑ᵛ ρ ]ᵛ ([ ⇑ˢ σ ]ˢ t)
[⇑ᵛ∘ˢ] (tm-var x) = ⇑ˢ-resp-ᵛ∘ˢ x
[⇑ᵛ∘ˢ] (tm-meta M ts) = ≈-meta (λ i → [⇑ᵛ∘ˢ] (ts i))
[⇑ᵛ∘ˢ] (tm-oper f es) = ≈-oper (λ i → ≈-trans ([]ˢ-resp-≈ˢ (es i) (⇑ˢ-resp-≈ˢ ⇑ˢ-resp-ᵛ∘ˢ)) ([⇑ᵛ∘ˢ] (es i)))
-- functoriality of left renaming action
[ᵛ∘ˢ] : ∀ {Θ} {A} {Γ Δ Ξ} {ρ : Δ ⇒ᵛ Ξ} {σ : Θ ⊕ Γ ⇒ˢ Δ} (t : Term Θ Γ A) →
[ ρ ᵛ∘ˢ σ ]ˢ t ≈ [ ρ ]ᵛ [ σ ]ˢ t
[ᵛ∘ˢ] (tm-var x) = ≈-refl
[ᵛ∘ˢ] (tm-meta M ts) = ≈-meta (λ i → [ᵛ∘ˢ] (ts i))
[ᵛ∘ˢ] (tm-oper f es) = ≈-oper (λ i → [⇑ᵛ∘ˢ] (es i))
-- functoriality of right renaming action
[ˢ∘ᵛ] : ∀ {Θ} {A} {Γ Δ Ξ} {σ : Θ ⊕ Δ ⇒ˢ Ξ} {ρ : Γ ⇒ᵛ Δ} (t : Term Θ Γ A) →
[ σ ˢ∘ᵛ ρ ]ˢ t ≈ [ σ ]ˢ [ ρ ]ᵛ t
[ˢ∘ᵛ] (tm-var x) = ≈-refl
[ˢ∘ᵛ] (tm-meta M ts) = ≈-meta (λ i → [ˢ∘ᵛ] (ts i))
[ˢ∘ᵛ] (tm-oper f es) = ≈-oper (λ i → [⇑ˢ∘ᵛ] (es i))
-- composition commutes with extension
⇑ˢ-resp-∘ˢ : ∀ {Θ} {Γ Δ Ξ Ψ} {σ : Θ ⊕ Γ ⇒ˢ Δ} {τ : Θ ⊕ Δ ⇒ˢ Ξ} →
⇑ˢ {Ξ = Ψ} (τ ∘ˢ σ) ≈ˢ ⇑ˢ τ ∘ˢ ⇑ˢ σ
⇑ˢ-resp-∘ˢ {σ = σ} {τ = τ} (var-inl x) = ≈-trans (≈-sym ([ᵛ∘ˢ] (σ x))) ([ˢ∘ᵛ] (σ x))
⇑ˢ-resp-∘ˢ (var-inr y) = ≈-refl
-- substitition action is functorial
[∘ˢ] : ∀ {Θ} {Γ Δ Ξ} {A} {σ : Θ ⊕ Γ ⇒ˢ Δ} {τ : Θ ⊕ Δ ⇒ˢ Ξ} (t : Term Θ Γ A) →
[ τ ∘ˢ σ ]ˢ t ≈ [ τ ]ˢ ([ σ ]ˢ t)
[∘ˢ] (tm-var x) = ≈-refl
[∘ˢ] (tm-meta M ts) = ≈-meta (λ i → [∘ˢ] (ts i))
[∘ˢ] (tm-oper f es) = ≈-oper (λ i → ≈-trans ([]ˢ-resp-≈ˢ (es i) ⇑ˢ-resp-∘ˢ) ([∘ˢ] (es i)))
-- Terms form a relative monad
module _ where
open Categories.Category
open Categories.Functor using (Functor)
open Categories.Category.Instance.Setoids
open Categories.Monad.Relative
open Function.Equality using () renaming (setoid to Π-setoid)
open Categories.Category.Equivalence using (StrongEquivalence)
open import SecondOrder.IndexedCategory
open import SecondOrder.RelativeKleisli
-- The embedding of contexts into setoids indexed by sorts
slots : Functor VContexts (IndexedCategory sort (Setoids ℓ ℓ))
slots = record
{ F₀ = λ Γ A → setoid (A ∈ Γ)
; F₁ = λ ρ A → record { _⟨$⟩_ = ρ ; cong = cong ρ }
; identity = λ A ξ → ξ
; homomorphism = λ {_} {_} {_} {ρ} {σ} A {_} {_} ξ → cong σ (cong ρ ξ)
; F-resp-≈ = λ ξ A ζ → trans (ξ _) (cong _ ζ)
}
module _ {Θ : MContext} where
open Categories.Category
open Categories.Functor using (Functor)
open Categories.Category.Instance.Setoids
open Categories.Monad.Relative
open Function.Equality using () renaming (setoid to Π-setoid)
open Categories.Category.Equivalence using (StrongEquivalence)
open import SecondOrder.IndexedCategory
open import SecondOrder.RelativeKleisli
-- The relative monad of terms over contexts
Term-Monad : Monad slots
Term-Monad =
let open Function.Equality using (_⟨$⟩_) renaming (cong to func-cong) in
record
{ F₀ = Term-setoid Θ
; unit = λ A → record { _⟨$⟩_ = idˢ ; cong = λ ξ → ≈-≡ (cong idˢ ξ) }
; extend = λ σ A → record { _⟨$⟩_ = [ (σ _ ⟨$⟩_) ]ˢ_ ; cong = []ˢ-resp-≈ (σ _ ⟨$⟩_)}
; identityʳ = λ {_} {_} {σ} A {_} {_} ξ → func-cong (σ A) ξ
; identityˡ = λ A → ≈-trans [idˢ]
; assoc = λ {_} {_} {_} {σ} {ρ} A {_} {t} ξ → ≈-trans ([]ˢ-resp-≈ _ ξ) ([∘ˢ] t)
; extend-≈ = λ {Γ} {Δ} {σ} {ρ} ζ B {s} {t} ξ → []ˢ-resp-≈ˢ-≈ (λ x → ζ _ refl) ξ
}
-- the category of contexts and substitutions
-- we show below that the category of contexts and substitiions is equivalent
-- to the Kleisli category for the Term relative monad. However, we define
-- the category of contexts and substitutions directly, as that it is easier
-- to work with it
Terms : Category ℓ ℓ ℓ
Terms =
record
{ Obj = VContext
; _⇒_ = Θ ⊕_⇒ˢ_
; _≈_ = _≈ˢ_
; id = idˢ
; _∘_ = _∘ˢ_
; assoc = λ {Γ} {Δ} {Ξ} {Ψ} {σ} {τ} {ψ} {A} x → [∘ˢ] (σ x)
; sym-assoc = λ {Γ} {Δ} {Ξ} {Ψ} {σ} {τ} {ψ} {A} x → ≈-sym ([∘ˢ] (σ x))
; identityˡ = λ x → [idˢ]
; identityʳ = λ x → ≈-refl
; identity² = λ x → ≈-refl
; equiv = record { refl = λ {σ} {A} → ≈ˢ-refl {σ = σ} ; sym = ≈ˢ-sym ; trans = ≈ˢ-trans }
; ∘-resp-≈ = λ f≈ˢg g≈ˢi x → []ˢ-resp-≈ˢ-≈ f≈ˢg (g≈ˢi x)
}
Terms-is-Kleisli : StrongEquivalence Terms (Kleisli Term-Monad)
Terms-is-Kleisli =
record
{ F = record
{ F₀ = λ Γ → Γ
; F₁ = λ σ A → record { _⟨$⟩_ = λ x → σ x ; cong = λ i≡j → ≈-≡ (cong σ i≡j) }
; identity = λ A eq → ≈-≡ (cong idˢ eq)
; homomorphism = λ {Γ} {Δ} {Ξ} {σ} {τ} A eq → ≈-≡ (cong (λ x → [ τ ]ˢ σ x) eq)
; F-resp-≈ = λ {Γ} {Δ} {σ} {τ} hom_eq A eq
→ ≈-trans (congˢ hom_eq) (≈-≡ (cong τ eq))
}
; G =
let open Function.Equality using (_⟨$⟩_) renaming (cong to func-cong) in
record
{ F₀ = λ Γ → Γ
; F₁ = λ {Γ} {Δ} σ {A} → λ x → σ A ⟨$⟩ x
; identity = λ x → ≈-refl
; homomorphism = λ x → ≈-refl
; F-resp-≈ = λ {Γ} {Δ} {σ} {τ} σ≈τ {A} x → σ≈τ A refl
}
; weak-inverse =
let open Function.Equality using (_⟨$⟩_) renaming (cong to func-cong) in
record
{ F∘G≈id =
record
{ F⇒G =
record
{ η = λ Γ A → record { _⟨$⟩_ = idˢ
; cong = λ i≡j → ≈-≡ (cong idˢ i≡j)
}
; commute = λ σ A x≡y → [idˢ]-resp-≈ (≈-≡ (cong (λ x → σ A ⟨$⟩ x) x≡y))
; sym-commute = λ σ A x≡y
→ ≈-sym ([idˢ]-resp-≈ (≈-≡ (cong (λ x → σ A ⟨$⟩ x ) (sym x≡y))))
}
; F⇐G =
record
{ η = λ Γ A → record { _⟨$⟩_ = idˢ
; cong = λ i≡j → ≈-≡ (cong idˢ i≡j)
}
; commute = λ σ A x≡y → [idˢ]-resp-≈ (≈-≡ (cong (λ x → σ A ⟨$⟩ x) x≡y))
; sym-commute = λ σ A x≡y
→ ≈-sym ([idˢ]-resp-≈ (≈-≡ (cong (λ x → σ A ⟨$⟩ x ) (sym x≡y))))
}
; iso = λ Γ → record { isoˡ = λ A x≡y → ≈-≡ (cong tm-var x≡y)
; isoʳ = λ A x≡y → ≈-≡ (cong tm-var x≡y)
}
}
; G∘F≈id =
record
{ F⇒G =
record
{ η = λ Γ x → tm-var x
; commute = λ σ x → [idˢ]
; sym-commute = λ σ x → ≈-sym [idˢ]
}
; F⇐G =
record
{ η = λ Γ x → tm-var x
; commute = λ σ x → [idˢ]
; sym-commute = λ σ x → ≈-sym [idˢ]
}
; iso = λ Γ → record { isoˡ = λ x → ≈-refl
; isoʳ = λ x → ≈-refl
}
}
}
}
-- the binary coproduct structure on Terms
infixl 7 [_,_]ˢ
[_,_]ˢ : ∀ {Γ Δ Ξ} (σ : Θ ⊕ Γ ⇒ˢ Ξ) (τ : Θ ⊕ Δ ⇒ˢ Ξ) → Θ ⊕ (Γ ,, Δ) ⇒ˢ Ξ
[ σ , τ ]ˢ (var-inl x) = σ x
[ σ , τ ]ˢ (var-inr y) = τ y
inlˢ : ∀ {Γ Δ} → Θ ⊕ Γ ⇒ˢ Γ ,, Δ
inlˢ x = tm-var (var-inl x)
inrˢ : ∀ {Γ Δ} → Θ ⊕ Δ ⇒ˢ Γ ,, Δ
inrˢ y = tm-var (var-inr y)
[,]ˢ-resp-≈ˢ : ∀ {Γ Δ Ξ} {σ₁ σ₂ : Θ ⊕ Γ ⇒ˢ Ξ} {τ₁ τ₂ : Θ ⊕ Δ ⇒ˢ Ξ} →
σ₁ ≈ˢ σ₂ → τ₁ ≈ˢ τ₂ → [ σ₁ , τ₁ ]ˢ ≈ˢ [ σ₂ , τ₂ ]ˢ
[,]ˢ-resp-≈ˢ ζ ξ (var-inl x) = ζ x
[,]ˢ-resp-≈ˢ ζ ξ (var-inr y) = ξ y
uniqueˢ : ∀ {Γ Δ Ξ} {τ : Θ ⊕ Γ ,, Δ ⇒ˢ Ξ} {ρ : Θ ⊕ Γ ⇒ˢ Ξ} {σ : Θ ⊕ Δ ⇒ˢ Ξ}
→ τ ∘ˢ inlˢ ≈ˢ ρ
→ τ ∘ˢ inrˢ ≈ˢ σ
→ [ ρ , σ ]ˢ ≈ˢ τ
uniqueˢ ξ ζ (var-inl x) = ≈-sym (ξ x)
uniqueˢ ξ ζ (var-inr y) = ≈-sym (ζ y)
Terms-Coproduct : Categories.Category.Cocartesian.BinaryCoproducts Terms
Terms-Coproduct =
let open Function.Equality using (_⟨$⟩_) renaming (cong to func-cong) in
record {
coproduct =
λ {Γ Δ} →
record
{ A+B = Γ ,, Δ
; i₁ = inlˢ
; i₂ = inrˢ
; [_,_] = [_,_]ˢ
; inject₁ = λ x → ≈-≡ refl
; inject₂ = λ x → ≈-≡ refl
; unique = λ {Ξ} {h} p₁ p₂ → uniqueˢ {τ = h} p₁ p₂
}
}
open Categories.Category.Cocartesian.BinaryCoproducts Terms-Coproduct
-- the sum of subsitutions
infixl 6 _+ˢ_
_+ˢ_ : ∀ {Γ₁ Γ₂ Δ₁ Δ₂} (σ : Θ ⊕ Γ₁ ⇒ˢ Δ₁) (τ : Θ ⊕ Γ₂ ⇒ˢ Δ₂) → Θ ⊕ Γ₁ ,, Γ₂ ⇒ˢ Δ₁ ,, Δ₂
σ +ˢ τ = σ +₁ τ
-- reassociations of context sums
assoc-l : ∀ {Γ Δ Ξ} → Θ ⊕ (Γ ,, Δ) ,, Ξ ⇒ˢ Γ ,, (Δ ,, Ξ)
assoc-l (var-inl (var-inl x)) = tm-var (var-inl x)
assoc-l (var-inl (var-inr y)) = tm-var (var-inr (var-inl y))
assoc-l (var-inr z) = tm-var (var-inr (var-inr z))
assoc-r : ∀ {Γ Δ Ξ} → Θ ⊕ Γ ,, (Δ ,, Ξ) ⇒ˢ (Γ ,, Δ) ,, Ξ
assoc-r (var-inl x) = tm-var (var-inl (var-inl x))
assoc-r (var-inr (var-inl y)) = tm-var (var-inl (var-inr y))
assoc-r (var-inr (var-inr z)) = tm-var (var-inr z)
assoc-lr : ∀ {Γ Δ Ξ} → assoc-l {Γ = Γ} {Δ = Δ} {Ξ = Ξ} ∘ˢ assoc-r {Γ = Γ} {Δ = Δ} {Ξ = Ξ} ≈ˢ idˢ
assoc-lr (var-inl x) = ≈-refl
assoc-lr (var-inr (var-inl y)) = ≈-refl
assoc-lr (var-inr (var-inr y)) = ≈-refl
assoc-rl : ∀ {Γ Δ Ξ} → assoc-r {Γ = Γ} {Δ = Δ} {Ξ = Ξ} ∘ˢ assoc-l {Γ = Γ} {Δ = Δ} {Ξ = Ξ} ≈ˢ idˢ
assoc-rl (var-inl (var-inl x)) = ≈-refl
assoc-rl (var-inl (var-inr x)) = ≈-refl
assoc-rl (var-inr z) = ≈-refl
-- summing with the empty context is the unit
sum-ctx-empty-r : ∀ {Γ} → Θ ⊕ Γ ,, ctx-empty ⇒ˢ Γ
sum-ctx-empty-r (var-inl x) = tm-var x
sum-ctx-empty-l : ∀ {Γ} → Θ ⊕ ctx-empty ,, Γ ⇒ˢ Γ
sum-ctx-empty-l (var-inr x) = tm-var x
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Functions.Definition
open import Numbers.Naturals.Semiring
open import Sets.Cardinality.Finite.Definition
module Sets.Cardinality.Countable.Definition where
record CountablyInfiniteSet {a : _} (A : Set a) : Set a where
field
counting : A → ℕ
countingIsBij : Bijection counting
data Countable {a : _} (A : Set a) : Set a where
finite : FiniteSet A → Countable A
infinite : CountablyInfiniteSet A → Countable A
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module L.Base.Unit where
-- Reexport definitions
open import L.Base.Unit.Core public
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{-# OPTIONS --sized-types #-}
open import Relation.Nullary
open import Relation.Binary.PropositionalEquality
module flcagl
(A : Set)
( _≟_ : (a b : A) → Dec ( a ≡ b ) ) where
open import Data.Bool hiding ( _≟_ )
-- open import Data.Maybe
open import Level renaming ( zero to Zero ; suc to succ )
open import Size
module List where
data List (i : Size) (A : Set) : Set where
[] : List i A
_∷_ : {j : Size< i} (x : A) (xs : List j A) → List i A
map : ∀{i A B} → (A → B) → List i A → List i B
map f [] = []
map f ( x ∷ xs)= f x ∷ map f xs
foldr : ∀{i} {A B : Set} → (A → B → B) → B → List i A → B
foldr c n [] = n
foldr c n (x ∷ xs) = c x (foldr c n xs)
any : ∀{i A} → (A → Bool) → List i A → Bool
any p xs = foldr _∨_ false (map p xs)
module Lang where
open List
record Lang (i : Size) : Set where
coinductive
field
ν : Bool
δ : ∀{j : Size< i} → A → Lang j
open Lang
_∋_ : ∀{i} → Lang i → List i A → Bool
l ∋ [] = ν l
l ∋ ( a ∷ as ) = δ l a ∋ as
trie : ∀{i} (f : List i A → Bool) → Lang i
ν (trie f) = f []
δ (trie f) a = trie (λ as → f (a ∷ as))
∅ : ∀{i} → Lang i
ν ∅ = false
δ ∅ x = ∅
ε : ∀{i} → Lang i
ν ε = true
δ ε x = ∅
open import Relation.Nullary.Decidable
char : ∀{i} (a : A) → Lang i
ν (char a) = false
δ (char a) x = if ⌊ a ≟ x ⌋ then ε else ∅
compl : ∀{i} (l : Lang i) → Lang i
ν (compl l) = not (ν l)
δ (compl l) x = compl (δ l x)
_∪_ : ∀{i} (k l : Lang i) → Lang i
ν (k ∪ l) = ν k ∨ ν l
δ (k ∪ l) x = δ k x ∪ δ l x
_·_ : ∀{i} (k l : Lang i) → Lang i
ν (k · l) = ν k ∧ ν l
δ (k · l) x = let k′l = δ k x · l in if ν k then k′l ∪ δ l x else k′l
_*_ : ∀{i} (k l : Lang i ) → Lang i
ν (k * l) = ν k ∧ ν l
δ (_*_ {i} k l) {j} x =
let
k′l : Lang j
k′l = _*_ {j} (δ k {j} x) l
in if ν k then _∪_ {j} k′l (δ l {j} x) else k′l
_* : ∀{i} (l : Lang i) → Lang i
ν (l *) = true
δ (l *) x = δ l x · (l *)
record _≅⟨_⟩≅_ (l : Lang ∞ ) i (k : Lang ∞) : Set where
coinductive
field ≅ν : ν l ≡ ν k
≅δ : ∀ {j : Size< i } (a : A ) → δ l a ≅⟨ j ⟩≅ δ k a
open _≅⟨_⟩≅_
≅refl : ∀{i} {l : Lang ∞} → l ≅⟨ i ⟩≅ l
≅ν ≅refl = refl
≅δ ≅refl a = ≅refl
≅sym : ∀{i} {k l : Lang ∞} (p : l ≅⟨ i ⟩≅ k) → k ≅⟨ i ⟩≅ l
≅ν (≅sym p) = sym (≅ν p)
≅δ (≅sym p) a = ≅sym (≅δ p a)
≅trans : ∀{i} {k l m : Lang ∞}
( p : k ≅⟨ i ⟩≅ l ) ( q : l ≅⟨ i ⟩≅ m ) → k ≅⟨ i ⟩≅ m
≅ν (≅trans p q) = trans (≅ν p) (≅ν q)
≅δ (≅trans p q) a = ≅trans (≅δ p a) (≅δ q a)
open import Relation.Binary
≅isEquivalence : ∀(i : Size) → IsEquivalence _≅⟨ i ⟩≅_
≅isEquivalence i = record { refl = ≅refl; sym = ≅sym; trans = ≅trans }
Bis : ∀(i : Size) → Setoid _ _
Setoid.Carrier (Bis i) = Lang ∞
Setoid._≈_ (Bis i) = _≅⟨ i ⟩≅_
Setoid.isEquivalence (Bis i) = ≅isEquivalence i
-- import Relation.Binary.EqReasoning as EqR
import Relation.Binary.Reasoning.Setoid as EqR
≅trans′ : ∀ i (k l m : Lang ∞)
( p : k ≅⟨ i ⟩≅ l ) ( q : l ≅⟨ i ⟩≅ m ) → k ≅⟨ i ⟩≅ m
≅trans′ i k l m p q = begin
k ≈⟨ p ⟩
l ≈⟨ q ⟩
m ∎ where open EqR (Bis i)
open import Data.Bool.Properties
union-assoc : ∀{i} (k {l m} : Lang ∞) → ((k ∪ l) ∪ m ) ≅⟨ i ⟩≅ ( k ∪ (l ∪ m) )
≅ν (union-assoc k) = ∨-assoc (ν k) _ _
≅δ (union-assoc k) a = union-assoc (δ k a)
union-comm : ∀{i} (l k : Lang ∞) → (l ∪ k ) ≅⟨ i ⟩≅ ( k ∪ l )
≅ν (union-comm l k) = ∨-comm (ν l) _
≅δ (union-comm l k) a = union-comm (δ l a) (δ k a)
union-idem : ∀{i} (l : Lang ∞) → (l ∪ l ) ≅⟨ i ⟩≅ l
≅ν (union-idem l) = ∨-idem _
≅δ (union-idem l) a = union-idem (δ l a)
union-emptyl : ∀{i}{l : Lang ∞} → (∅ ∪ l ) ≅⟨ i ⟩≅ l
≅ν union-emptyl = refl
≅δ union-emptyl a = union-emptyl
union-cong : ∀{i}{k k′ l l′ : Lang ∞}
(p : k ≅⟨ i ⟩≅ k′) (q : l ≅⟨ i ⟩≅ l′ ) → ( k ∪ l ) ≅⟨ i ⟩≅ ( k′ ∪ l′ )
≅ν (union-cong p q) = cong₂ _∨_ (≅ν p) (≅ν q)
≅δ (union-cong p q) a = union-cong (≅δ p a) (≅δ q a)
withExample : (P : Bool → Set) (p : P true) (q : P false) →
{A : Set} (g : A → Bool) (x : A) → P (g x)
withExample P p q g x with g x
... | true = p
... | false = q
rewriteExample : {A : Set} {P : A → Set} {x : A} (p : P x)
{g : A → A} (e : g x ≡ x) → P (g x)
rewriteExample p e rewrite e = p
infixr 6 _∪_
infixr 7 _·_
infix 5 _≅⟨_⟩≅_
union-congl : ∀{i}{k k′ l : Lang ∞}
(p : k ≅⟨ i ⟩≅ k′) → ( k ∪ l ) ≅⟨ i ⟩≅ ( k′ ∪ l )
union-congl eq = union-cong eq ≅refl
union-congr : ∀{i}{k l l′ : Lang ∞}
(p : l ≅⟨ i ⟩≅ l′) → ( k ∪ l ) ≅⟨ i ⟩≅ ( k ∪ l′ )
union-congr eq = union-cong ≅refl eq
union-swap24 : ∀{i} ({x y z w} : Lang ∞) → (x ∪ y) ∪ z ∪ w
≅⟨ i ⟩≅ (x ∪ z) ∪ y ∪ w
union-swap24 {_} {x} {y} {z} {w} = begin
(x ∪ y) ∪ z ∪ w
≈⟨ union-assoc x ⟩
x ∪ y ∪ z ∪ w
≈⟨ union-congr (≅sym ( union-assoc y)) ⟩
x ∪ ((y ∪ z) ∪ w)
≈⟨ ≅sym ( union-assoc x ) ⟩
(x ∪ ( y ∪ z)) ∪ w
≈⟨ union-congl (union-congr (union-comm y z )) ⟩
( x ∪ (z ∪ y)) ∪ w
≈⟨ union-congl (≅sym ( union-assoc x )) ⟩
((x ∪ z) ∪ y) ∪ w
≈⟨ union-assoc (x ∪ z) ⟩
(x ∪ z) ∪ y ∪ w
∎
where open EqR (Bis _)
concat-union-distribr : ∀{i} (k {l m} : Lang ∞) → k · ( l ∪ m ) ≅⟨ i ⟩≅ ( k · l ) ∪ ( k · m )
≅ν (concat-union-distribr k) = ∧-distribˡ-∨ (ν k) _ _
≅δ (concat-union-distribr k) a with ν k
≅δ (concat-union-distribr k {l} {m}) a | true = begin
δ k a · (l ∪ m) ∪ (δ l a ∪ δ m a)
≈⟨ union-congl (concat-union-distribr _) ⟩
(δ k a · l ∪ δ k a · m) ∪ (δ l a ∪ δ m a)
≈⟨ union-swap24 ⟩
(δ k a · l ∪ δ l a) ∪ (δ k a · m ∪ δ m a)
∎
where open EqR (Bis _)
≅δ (concat-union-distribr k) a | false = concat-union-distribr (δ k a)
concat-union-distribl : ∀{i} (k {l m} : Lang ∞) → ( k ∪ l ) · m ≅⟨ i ⟩≅ ( k · m ) ∪ ( l · m )
≅ν (concat-union-distribl k {l} {m}) = ∧-distribʳ-∨ _ (ν k) _
≅δ (concat-union-distribl k {l} {m}) a with ν k | ν l
≅δ (concat-union-distribl k {l} {m}) a | false | false = concat-union-distribl (δ k a)
≅δ (concat-union-distribl k {l} {m}) a | false | true = begin
(if false ∨ true then (δ k a ∪ δ l a) · m ∪ δ m a else (δ k a ∪ δ l a) · m)
≈⟨ ≅refl ⟩
((δ k a ∪ δ l a) · m ) ∪ δ m a
≈⟨ union-congl (concat-union-distribl _) ⟩
(δ k a · m ∪ δ l a · m) ∪ δ m a
≈⟨ union-assoc _ ⟩
(δ k a · m) ∪ ( δ l a · m ∪ δ m a )
≈⟨ ≅refl ⟩
(if false then δ k a · m ∪ δ m a else δ k a · m) ∪ (if true then δ l a · m ∪ δ m a else δ l a · m)
∎
where open EqR (Bis _)
≅δ (concat-union-distribl k {l} {m}) a | true | false = begin
(if true ∨ false then (δ k a ∪ δ l a) · m ∪ δ m a else (δ k a ∪ δ l a) · m) ≈⟨ ≅refl ⟩
((δ k a ∪ δ l a) · m ) ∪ δ m a ≈⟨ union-congl (concat-union-distribl _) ⟩
(δ k a · m ∪ δ l a · m) ∪ δ m a ≈⟨ union-assoc _ ⟩
δ k a · m ∪ ( δ l a · m ∪ δ m a ) ≈⟨ union-congr ( union-comm _ _) ⟩
δ k a · m ∪ δ m a ∪ δ l a · m ≈⟨ ≅sym ( union-assoc _ ) ⟩
(δ k a · m ∪ δ m a) ∪ δ l a · m ≈⟨ ≅refl ⟩
((if true then δ k a · m ∪ δ m a else δ k a · m) ∪ (if false then δ l a · m ∪ δ m a else δ l a · m))
∎
where open EqR (Bis _)
≅δ (concat-union-distribl k {l} {m}) a | true | true = begin
(if true ∨ true then (δ k a ∪ δ l a) · m ∪ δ m a else (δ k a ∪ δ l a) · m) ≈⟨ ≅refl ⟩
(δ k a ∪ δ l a) · m ∪ δ m a ≈⟨ union-congl ( concat-union-distribl _ ) ⟩
(δ k a · m ∪ δ l a · m) ∪ δ m a ≈⟨ union-assoc _ ⟩
δ k a · m ∪ ( δ l a · m ∪ δ m a ) ≈⟨ ≅sym ( union-congr ( union-congr ( union-idem _ ) ) ) ⟩
δ k a · m ∪ ( δ l a · m ∪ (δ m a ∪ δ m a) ) ≈⟨ ≅sym ( union-congr ( union-assoc _ )) ⟩
δ k a · m ∪ ( (δ l a · m ∪ δ m a ) ∪ δ m a ) ≈⟨ union-congr ( union-congl ( union-comm _ _) ) ⟩
δ k a · m ∪ ( (δ m a ∪ δ l a · m ) ∪ δ m a ) ≈⟨ ≅sym ( union-assoc _ ) ⟩
( δ k a · m ∪ (δ m a ∪ δ l a · m )) ∪ δ m a ≈⟨ ≅sym ( union-congl ( union-assoc _ ) ) ⟩
((δ k a · m ∪ δ m a) ∪ δ l a · m) ∪ δ m a ≈⟨ union-assoc _ ⟩
(δ k a · m ∪ δ m a) ∪ δ l a · m ∪ δ m a ≈⟨ ≅refl ⟩
((if true then δ k a · m ∪ δ m a else δ k a · m) ∪ (if true then δ l a · m ∪ δ m a else δ l a · m))
∎
where open EqR (Bis _)
postulate
concat-emptyl : ∀{i} l → ∅ · l ≅⟨ i ⟩≅ ∅
concat-emptyr : ∀{i} l → l · ∅ ≅⟨ i ⟩≅ ∅
concat-unitl : ∀{i} l → ε · l ≅⟨ i ⟩≅ l
concat-unitr : ∀{i} l → l · ε ≅⟨ i ⟩≅ l
star-empty : ∀{i} → ∅ * ≅⟨ i ⟩≅ ε
concat-congl : ∀{i} {m l k : Lang ∞} → l ≅⟨ i ⟩≅ k → l · m ≅⟨ i ⟩≅ k · m
≅ν (concat-congl {i} {m} p ) = cong (λ x → x ∧ ( ν m )) ( ≅ν p )
≅δ (concat-congl {i} {m} {l} {k} p ) a with ν k | ν l | ≅ν p
≅δ (concat-congl {i} {m} {l} {k} p) a | false | false | refl = concat-congl (≅δ p a)
≅δ (concat-congl {i} {m} {l} {k} p) a | true | true | refl = union-congl (concat-congl (≅δ p a))
concat-congr : ∀{i} {m l k : Lang ∞} → l ≅⟨ i ⟩≅ k → m · l ≅⟨ i ⟩≅ m · k
≅ν (concat-congr {i} {m} {_} {k} p ) = cong (λ x → ( ν m ) ∧ x ) ( ≅ν p )
≅δ (concat-congr {i} {m} {l} {k} p ) a with ν m | ν k | ν l | ≅ν p
≅δ (concat-congr {i} {m} {l} {k} p) a | false | x | .x | refl = concat-congr p
≅δ (concat-congr {i} {m} {l} {k} p) a | true | x | .x | refl = union-cong (concat-congr p ) ( ≅δ p a )
concat-assoc : ∀{i} (k {l m} : Lang ∞) → (k · l) · m ≅⟨ i ⟩≅ k · (l · m)
≅ν (concat-assoc {i} k {l} {m} ) = ∧-assoc ( ν k ) ( ν l ) ( ν m )
≅δ (concat-assoc {i} k {l} {m} ) a with ν k
≅δ (concat-assoc {i} k {l} {m}) a | false = concat-assoc _
≅δ (concat-assoc {i} k {l} {m}) a | true with ν l
≅δ (concat-assoc {i} k {l} {m}) a | true | false = begin
( if false then (δ k a · l ∪ δ l a) · m ∪ δ m a else (δ k a · l ∪ δ l a) · m )
≈⟨ ≅refl ⟩
(δ k a · l ∪ δ l a) · m
≈⟨ concat-union-distribl _ ⟩
((δ k a · l) · m ) ∪ ( δ l a · m )
≈⟨ union-congl (concat-assoc _) ⟩
(δ k a · l · m ) ∪ ( δ l a · m )
≈⟨ ≅refl ⟩
δ k a · l · m ∪ (if false then δ l a · m ∪ δ m a else δ l a · m)
∎ where open EqR (Bis _)
≅δ (concat-assoc {i} k {l} {m}) a | true | true = begin
(if true then (δ k a · l ∪ δ l a) · m ∪ δ m a else (δ k a · l ∪ δ l a) · m)
≈⟨ ≅refl ⟩
((( δ k a · l ) ∪ δ l a) · m ) ∪ δ m a
≈⟨ union-congl (concat-union-distribl _ ) ⟩
((δ k a · l) · m ∪ ( δ l a · m )) ∪ δ m a
≈⟨ union-congl ( union-congl (concat-assoc _)) ⟩
(( δ k a · l · m ) ∪ ( δ l a · m )) ∪ δ m a
≈⟨ union-assoc _ ⟩
( δ k a · l · m ) ∪ ( ( δ l a · m ) ∪ δ m a )
≈⟨ ≅refl ⟩
δ k a · l · m ∪ (if true then δ l a · m ∪ δ m a else δ l a · m)
∎ where open EqR (Bis _)
star-concat-idem : ∀{i} (l : Lang ∞) → l * · l * ≅⟨ i ⟩≅ l *
≅ν (star-concat-idem l) = refl
≅δ (star-concat-idem l) a = begin
δ ((l *) · (l *)) a ≈⟨ union-congl (concat-assoc _) ⟩
δ l a · (l * · l *) ∪ δ l a · l * ≈⟨ union-congl (concat-congr (star-concat-idem _)) ⟩
δ l a · l * ∪ δ l a · l * ≈⟨ union-idem _ ⟩
δ (l *) a ∎ where open EqR (Bis _)
star-idem : ∀{i} (l : Lang ∞) → (l *) * ≅⟨ i ⟩≅ l *
≅ν (star-idem l) = refl
≅δ (star-idem l) a = begin
δ ((l *) *) a ≈⟨ concat-assoc (δ l a) ⟩
δ l a · ((l *) · ((l *) *)) ≈⟨ concat-congr ( concat-congr (star-idem l )) ⟩
δ l a · ((l *) · (l *)) ≈⟨ concat-congr (star-concat-idem l ) ⟩
δ l a · l *
∎ where open EqR (Bis _)
postulate
star-rec : ∀{i} (l : Lang ∞) → l * ≅⟨ i ⟩≅ ε ∪ (l · l *)
star-from-rec : ∀{i} (k {l m} : Lang ∞)
→ ν k ≡ false
→ l ≅⟨ i ⟩≅ k · l ∪ m
→ l ≅⟨ i ⟩≅ k * · m
≅ν (star-from-rec k n p) with ≅ν p
... | b rewrite n = b
≅δ (star-from-rec k {l} {m} n p) a with ≅δ p a
... | q rewrite n = begin
(δ l a)
≈⟨ q ⟩
δ k a · l ∪ δ m a
≈⟨ union-congl (concat-congr (star-from-rec k {l} {m} n p)) ⟩
(δ k a · (k * · m) ∪ δ m a)
≈⟨ union-congl (≅sym (concat-assoc _)) ⟩
(δ k a · (k *)) · m ∪ δ m a
∎ where open EqR (Bis _)
open List
record DA (S : Set) : Set where
field ν : (s : S) → Bool
δ : (s : S)(a : A) → S
νs : ∀{i} (ss : List.List i S) → Bool
νs ss = List.any ν ss
δs : ∀{i} (ss : List.List i S) (a : A) → List.List i S
δs ss a = List.map (λ s → δ s a) ss
open Lang
lang : ∀{i} {S} (da : DA S) (s : S) → Lang i
Lang.ν (lang da s) = DA.ν da s
Lang.δ (lang da s) a = lang da (DA.δ da s a)
open import Data.Unit hiding ( _≟_ )
open DA
∅A : DA ⊤
ν ∅A s = false
δ ∅A s a = s
εA : DA Bool
ν εA b = b
δ εA b a = false
open import Relation.Nullary.Decidable
data 3States : Set where
init acc err : 3States
charA : (a : A) → DA 3States
ν (charA a) init = false
ν (charA a) acc = true
ν (charA a) err = false
δ (charA a) init x =
if ⌊ a ≟ x ⌋ then acc else err
δ (charA a) acc x = err
δ (charA a) err x = err
complA : ∀{S} (da : DA S) → DA S
ν (complA da) s = not (ν da s)
δ (complA da) s a = δ da s a
open import Data.Product
_⊕_ : ∀{S1 S2} (da1 : DA S1) (da2 : DA S2) → DA (S1 × S2)
ν (da1 ⊕ da2) (s1 , s2) = ν da1 s1 ∨ ν da2 s2
δ (da1 ⊕ da2) (s1 , s2) a = δ da1 s1 a , δ da2 s2 a
powA : ∀{S} (da : DA S) → DA (List ∞ S)
ν (powA da) ss = νs da ss
δ (powA da) ss a = δs da ss a
open _≅⟨_⟩≅_
powA-nil : ∀{i S} (da : DA S) → lang (powA da) [] ≅⟨ i ⟩≅ ∅
≅ν (powA-nil da) = refl
≅δ (powA-nil da) a = powA-nil da
powA-cons : ∀{i S} (da : DA S) {s : S} {ss : List ∞ S} →
lang (powA da) (s ∷ ss) ≅⟨ i ⟩≅ lang da s ∪ lang (powA da) ss
≅ν (powA-cons da) = refl
≅δ (powA-cons da) a = powA-cons da
composeA : ∀{S1 S2} (da1 : DA S1)(s2 : S2)(da2 : DA S2) → DA (S1 × List ∞ S2)
ν (composeA da1 s2 da2) (s1 , ss2) = (ν da1 s1 ∧ ν da2 s2) ∨ νs da2 ss2
δ (composeA da1 s2 da2) (s1 , ss2) a =
δ da1 s1 a , δs da2 (if ν da1 s1 then s2 ∷ ss2 else ss2) a
-- import Relation.Binary.EqReasoning as EqR
import Relation.Binary.Reasoning.Setoid as EqR
composeA-gen : ∀{i S1 S2} (da1 : DA S1) (da2 : DA S2) → ∀(s1 : S1)(s2 : S2)(ss : List ∞ S2) →
lang (composeA da1 s2 da2) (s1 , ss) ≅⟨ i ⟩≅ lang da1 s1 · lang da2 s2 ∪ lang (powA da2) ss
≅ν (composeA-gen da1 da2 s1 s2 ss) = refl
≅δ (composeA-gen da1 da2 s1 s2 ss) a with ν da1 s1
... | false = composeA-gen da1 da2 (δ da1 s1 a) s2 (δs da2 ss a)
... | true = begin
lang (composeA da1 s2 da2) (δ da1 s1 a , δ da2 s2 a ∷ δs da2 ss a)
≈⟨ composeA-gen da1 da2 (δ da1 s1 a) s2 (δs da2 (s2 ∷ ss) a) ⟩
lang da1 (δ da1 s1 a) · lang da2 s2 ∪ lang (powA da2) (δs da2 (s2 ∷ ss) a)
≈⟨ union-congr (powA-cons da2) ⟩
lang da1 (δ da1 s1 a) · lang da2 s2 ∪
(lang da2 (δ da2 s2 a) ∪ lang (powA da2) (δs da2 ss a))
≈⟨ ≅sym (union-assoc _) ⟩
(lang da1 (δ da1 s1 a) · lang da2 s2 ∪ lang da2 (δ da2 s2 a)) ∪ lang (powA da2) (δs da2 ss a)
∎ where open EqR (Bis _)
postulate
composeA-correct : ∀{i S1 S2} (da1 : DA S1) (da2 : DA S2) s1 s2 →
lang (composeA da1 s2 da2) (s1 , []) ≅⟨ i ⟩≅ lang da1 s1 · lang da2 s2
open import Data.Maybe
acceptingInitial : ∀{S} (s0 : S) (da : DA S) → DA (Maybe S)
ν (acceptingInitial s0 da) (just s) = ν da s
δ (acceptingInitial s0 da) (just s) a = just (δ da s a)
ν (acceptingInitial s0 da) nothing = true
δ (acceptingInitial s0 da) nothing a = just (δ da s0 a)
finalToInitial : ∀{S} (da : DA (Maybe S)) → DA (List ∞ (Maybe S))
ν (finalToInitial da) ss = νs da ss
δ (finalToInitial da) ss a =
let ss′ = δs da ss a
in if νs da ss then δ da nothing a ∷ ss′ else ss′
starA : ∀{S}(s0 : S)(da : DA S) → DA (List ∞(Maybe S))
starA s0 da = finalToInitial (acceptingInitial s0 da)
postulate
acceptingInitial-just : ∀{i S} (s0 : S) (da : DA S) {s : S} →
lang (acceptingInitial s0 da) (just s) ≅⟨ i ⟩≅ lang da s
acceptingInitial-nothing : ∀{i S} (s0 : S) (da : DA S) →
lang (acceptingInitial s0 da) nothing ≅⟨ i ⟩≅ ε ∪ lang da s0
starA-lemma : ∀{i S}(da : DA S)(s0 : S)(ss : List ∞ (Maybe S))→
lang (starA s0 da) ss ≅⟨ i ⟩≅
lang (powA (acceptingInitial s0 da)) ss · (lang da s0) *
starA-correct : ∀{i S} (da : DA S) (s0 : S) →
lang (starA s0 da) (nothing ∷ []) ≅⟨ i ⟩≅ (lang da s0) *
record NAutomaton ( Q : Set ) ( Σ : Set )
: Set where
field
Nδ : Q → Σ → Q → Bool
Nstart : Q → Bool
Nend : Q → Bool
postulate
exists : { S : Set} → ( S → Bool ) → Bool
nlang : ∀{i} {S} (nfa : NAutomaton S A ) (s : S → Bool ) → Lang i
Lang.ν (nlang nfa s) = exists ( λ x → (s x ∧ NAutomaton.Nend nfa x ))
Lang.δ (nlang nfa s) a = nlang nfa (λ x → s x ∧ (NAutomaton.Nδ nfa x a) x)
nlang1 : ∀{i} {S} (nfa : NAutomaton S A ) (s : S → Bool ) → Lang i
Lang.ν (nlang1 nfa s) = NAutomaton.Nend nfa {!!}
Lang.δ (nlang1 nfa s) a = nlang1 nfa (λ x → s x ∧ (NAutomaton.Nδ nfa x a) x)
-- nlang' : ∀{i} {S} (nfa : DA (S → Bool) ) (s : S → Bool ) → Lang i
-- Lang.ν (nlang' nfa s) = DA.ν nfa s
-- Lang.δ (nlang' nfa s) a = nlang' nfa (DA.δ nfa s a)
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-- {-# OPTIONS -v tc.cover.cover:10 -v tc.cover.strategy:20 -v tc.cover.splittree:100 #-}
-- Andreas, 2013-05-15 Reported by jesper.cockx
module Issue849 where
data Bool : Set where
true false : Bool
¬ : Bool → Bool
¬ true = false
¬ false = true
data D : Bool → Set where
d1 : (x : Bool) → D x
d2 : (x : Bool) → D (¬ x)
data ⊥ : Set where
data Test : (x : Bool) → (y : D x) → Set where
c1 : (x : Bool) → Test x (d1 x)
c2 : (y : D true) → Test true y
-- WAS: The following function passes the coverage checker
-- even though the case for "f false (d2 true)" is not covered.
f : (x : Bool) → (y : D x) → Test x y
f x (d1 .x) = c1 x
f true y = c2 y
impossible : Test false (d2 true) → ⊥
impossible ()
error : ⊥
error = impossible (f false (d2 true))
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.ZCohomology.Groups.Wedge where
open import Cubical.ZCohomology.Base
open import Cubical.ZCohomology.Properties
open import Cubical.ZCohomology.MayerVietorisUnreduced
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Pointed
open import Cubical.HITs.Wedge
open import Cubical.HITs.SetTruncation renaming (elim to sElim ; elim2 to sElim2)
open import Cubical.HITs.PropositionalTruncation renaming (rec to pRec ; ∣_∣ to ∣_∣₁)
open import Cubical.Data.Nat
open import Cubical.Data.Prod
open import Cubical.Data.Unit
open import Cubical.Algebra.Group
open import Cubical.ZCohomology.Groups.Unit
open import Cubical.ZCohomology.Groups.Sn
open import Cubical.HITs.Pushout
--- This module contains a proof that Hⁿ(A ⋁ B) ≅ Hⁿ(A) × Hⁿ(B), n ≥ 1
module _ {ℓ ℓ'} (A : Pointed ℓ) (B : Pointed ℓ') where
module I = MV (typ A) (typ B) Unit (λ _ → pt A) (λ _ → pt B)
Hⁿ-⋁ : (n : ℕ) → GroupEquiv (coHomGr (suc n) (A ⋁ B)) (×coHomGr (suc n) (typ A) (typ B))
Hⁿ-⋁ zero =
BijectionIsoToGroupEquiv
(bij-iso
(grouphom
(GroupHom.fun (I.i 1))
(sElim2 (λ _ _ → isOfHLevelPath 2 (isOfHLevelΣ 2 setTruncIsSet λ _ → setTruncIsSet) _ _)
λ a b → GroupHom.isHom (I.i 1) ∣ a ∣₂ ∣ b ∣₂))
(sElim (λ _ → isOfHLevelΠ 2 λ _ → isOfHLevelPath 2 setTruncIsSet _ _)
λ f inker → helper ∣ f ∣₂ (I.Ker-i⊂Im-d 0 ∣ f ∣₂ inker))
(sigmaElim (λ _ → isOfHLevelSuc 1 propTruncIsProp)
λ f g → I.Ker-Δ⊂Im-i 1 (∣ f ∣₂ , g) (isOfHLevelSuc 0 (isContrHⁿ-Unit 0) _ _)))
where
surj-helper : (x : coHom 0 Unit)
→ isInIm _ _ (I.Δ 0) x
surj-helper =
sElim (λ _ → isOfHLevelSuc 1 propTruncIsProp)
λ f → ∣ (∣ (λ _ → f tt) ∣₂ , 0ₕ) , cong ∣_∣₂ (funExt (λ _ → cong ((f tt) +ₖ_) -0ₖ ∙ rUnitₖ (f tt))) ∣₁
helper : (x : coHom 1 (A ⋁ B)) → isInIm _ _ (I.d 0) x
→ x ≡ 0ₕ
helper x inim =
pRec (setTruncIsSet _ _)
(λ p → sym (snd p) ∙
MV.Im-Δ⊂Ker-d _ _ Unit (λ _ → pt A) (λ _ → pt B) 0 (fst p) (surj-helper (fst p)))
inim
Hⁿ-⋁ (suc n) =
vSES→GroupEquiv _ _
(ses (isOfHLevelSuc 0 (isContrHⁿ-Unit n))
(isOfHLevelSuc 0 (isContrHⁿ-Unit (suc n)))
(I.d (suc n))
(I.Δ (suc (suc n)))
(I.i (suc (suc n)))
(I.Ker-i⊂Im-d (suc n))
(I.Ker-Δ⊂Im-i (suc (suc n))))
open import Cubical.Foundations.Isomorphism
wedgeConnected : ((x : typ A) → ∥ pt A ≡ x ∥) → ((x : typ B) → ∥ pt B ≡ x ∥) → (x : A ⋁ B) → ∥ (inl (pt A)) ≡ x ∥
wedgeConnected conA conB =
PushoutToProp (λ _ → propTruncIsProp)
(λ a → pRec propTruncIsProp (λ p → ∣ cong inl p ∣₁) (conA a))
λ b → pRec propTruncIsProp (λ p → ∣ push tt ∙ cong inr p ∣₁) (conB b)
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{-# OPTIONS --rewriting #-}
module Properties.FunctionTypes where
open import FFI.Data.Either using (Either; Left; Right; mapLR; swapLR; cond)
open import Luau.FunctionTypes using (srcⁿ; src; tgt)
open import Luau.Subtyping using (_<:_; _≮:_; Tree; Language; ¬Language; witness; unknown; never; scalar; function; scalar-function; scalar-function-ok; scalar-function-err; scalar-scalar; function-scalar; function-ok; function-err; left; right; _,_)
open import Luau.Type using (Type; Scalar; nil; number; string; boolean; never; unknown; _⇒_; _∪_; _∩_; skalar)
open import Properties.Contradiction using (CONTRADICTION; ¬; ⊥)
open import Properties.Functions using (_∘_)
open import Properties.Subtyping using (<:-refl; ≮:-refl; <:-trans-≮:; skalar-scalar; <:-impl-⊇; skalar-function-ok; language-comp)
open import Properties.TypeNormalization using (FunType; Normal; never; unknown; _∩_; _∪_; _⇒_; normal; <:-normalize; normalize-<:)
-- Properties of src
function-err-srcⁿ : ∀ {T t} → (FunType T) → (¬Language (srcⁿ T) t) → Language T (function-err t)
function-err-srcⁿ (S ⇒ T) p = function-err p
function-err-srcⁿ (S ∩ T) (p₁ , p₂) = (function-err-srcⁿ S p₁ , function-err-srcⁿ T p₂)
¬function-err-srcᶠ : ∀ {T t} → (FunType T) → (Language (srcⁿ T) t) → ¬Language T (function-err t)
¬function-err-srcᶠ (S ⇒ T) p = function-err p
¬function-err-srcᶠ (S ∩ T) (left p) = left (¬function-err-srcᶠ S p)
¬function-err-srcᶠ (S ∩ T) (right p) = right (¬function-err-srcᶠ T p)
¬function-err-srcⁿ : ∀ {T t} → (Normal T) → (Language (srcⁿ T) t) → ¬Language T (function-err t)
¬function-err-srcⁿ never p = never
¬function-err-srcⁿ unknown (scalar ())
¬function-err-srcⁿ (S ⇒ T) p = function-err p
¬function-err-srcⁿ (S ∩ T) (left p) = left (¬function-err-srcᶠ S p)
¬function-err-srcⁿ (S ∩ T) (right p) = right (¬function-err-srcᶠ T p)
¬function-err-srcⁿ (S ∪ T) (scalar ())
¬function-err-src : ∀ {T t} → (Language (src T) t) → ¬Language T (function-err t)
¬function-err-src {T = S ⇒ T} p = function-err p
¬function-err-src {T = nil} p = scalar-function-err nil
¬function-err-src {T = never} p = never
¬function-err-src {T = unknown} (scalar ())
¬function-err-src {T = boolean} p = scalar-function-err boolean
¬function-err-src {T = number} p = scalar-function-err number
¬function-err-src {T = string} p = scalar-function-err string
¬function-err-src {T = S ∪ T} p = <:-impl-⊇ (<:-normalize (S ∪ T)) _ (¬function-err-srcⁿ (normal (S ∪ T)) p)
¬function-err-src {T = S ∩ T} p = <:-impl-⊇ (<:-normalize (S ∩ T)) _ (¬function-err-srcⁿ (normal (S ∩ T)) p)
src-¬function-errᶠ : ∀ {T t} → (FunType T) → Language T (function-err t) → (¬Language (srcⁿ T) t)
src-¬function-errᶠ (S ⇒ T) (function-err p) = p
src-¬function-errᶠ (S ∩ T) (p₁ , p₂) = (src-¬function-errᶠ S p₁ , src-¬function-errᶠ T p₂)
src-¬function-errⁿ : ∀ {T t} → (Normal T) → Language T (function-err t) → (¬Language (srcⁿ T) t)
src-¬function-errⁿ unknown p = never
src-¬function-errⁿ (S ⇒ T) (function-err p) = p
src-¬function-errⁿ (S ∩ T) (p₁ , p₂) = (src-¬function-errᶠ S p₁ , src-¬function-errᶠ T p₂)
src-¬function-errⁿ (S ∪ T) p = never
src-¬function-err : ∀ {T t} → Language T (function-err t) → (¬Language (src T) t)
src-¬function-err {T = S ⇒ T} (function-err p) = p
src-¬function-err {T = unknown} p = never
src-¬function-err {T = S ∪ T} p = src-¬function-errⁿ (normal (S ∪ T)) (<:-normalize (S ∪ T) _ p)
src-¬function-err {T = S ∩ T} p = src-¬function-errⁿ (normal (S ∩ T)) (<:-normalize (S ∩ T) _ p)
fun-¬scalar : ∀ {S T} (s : Scalar S) → FunType T → ¬Language T (scalar s)
fun-¬scalar s (S ⇒ T) = function-scalar s
fun-¬scalar s (S ∩ T) = left (fun-¬scalar s S)
¬fun-scalar : ∀ {S T t} (s : Scalar S) → FunType T → Language T t → ¬Language S t
¬fun-scalar s (S ⇒ T) function = scalar-function s
¬fun-scalar s (S ⇒ T) (function-ok p) = scalar-function-ok s
¬fun-scalar s (S ⇒ T) (function-err p) = scalar-function-err s
¬fun-scalar s (S ∩ T) (p₁ , p₂) = ¬fun-scalar s T p₂
fun-function : ∀ {T} → FunType T → Language T function
fun-function (S ⇒ T) = function
fun-function (S ∩ T) = (fun-function S , fun-function T)
srcⁿ-¬scalar : ∀ {S T t} (s : Scalar S) → Normal T → Language T (scalar s) → (¬Language (srcⁿ T) t)
srcⁿ-¬scalar s never (scalar ())
srcⁿ-¬scalar s unknown p = never
srcⁿ-¬scalar s (S ⇒ T) (scalar ())
srcⁿ-¬scalar s (S ∩ T) (p₁ , p₂) = CONTRADICTION (language-comp (scalar s) (fun-¬scalar s S) p₁)
srcⁿ-¬scalar s (S ∪ T) p = never
src-¬scalar : ∀ {S T t} (s : Scalar S) → Language T (scalar s) → (¬Language (src T) t)
src-¬scalar {T = nil} s p = never
src-¬scalar {T = T ⇒ U} s (scalar ())
src-¬scalar {T = never} s (scalar ())
src-¬scalar {T = unknown} s p = never
src-¬scalar {T = boolean} s p = never
src-¬scalar {T = number} s p = never
src-¬scalar {T = string} s p = never
src-¬scalar {T = T ∪ U} s p = srcⁿ-¬scalar s (normal (T ∪ U)) (<:-normalize (T ∪ U) (scalar s) p)
src-¬scalar {T = T ∩ U} s p = srcⁿ-¬scalar s (normal (T ∩ U)) (<:-normalize (T ∩ U) (scalar s) p)
srcⁿ-unknown-≮: : ∀ {T U} → (Normal U) → (T ≮: srcⁿ U) → (U ≮: (T ⇒ unknown))
srcⁿ-unknown-≮: never (witness t p q) = CONTRADICTION (language-comp t q unknown)
srcⁿ-unknown-≮: unknown (witness t p q) = witness (function-err t) unknown (function-err p)
srcⁿ-unknown-≮: (U ⇒ V) (witness t p q) = witness (function-err t) (function-err q) (function-err p)
srcⁿ-unknown-≮: (U ∩ V) (witness t p q) = witness (function-err t) (function-err-srcⁿ (U ∩ V) q) (function-err p)
srcⁿ-unknown-≮: (U ∪ V) (witness t p q) = witness (scalar V) (right (scalar V)) (function-scalar V)
src-unknown-≮: : ∀ {T U} → (T ≮: src U) → (U ≮: (T ⇒ unknown))
src-unknown-≮: {U = nil} (witness t p q) = witness (scalar nil) (scalar nil) (function-scalar nil)
src-unknown-≮: {U = T ⇒ U} (witness t p q) = witness (function-err t) (function-err q) (function-err p)
src-unknown-≮: {U = never} (witness t p q) = CONTRADICTION (language-comp t q unknown)
src-unknown-≮: {U = unknown} (witness t p q) = witness (function-err t) unknown (function-err p)
src-unknown-≮: {U = boolean} (witness t p q) = witness (scalar boolean) (scalar boolean) (function-scalar boolean)
src-unknown-≮: {U = number} (witness t p q) = witness (scalar number) (scalar number) (function-scalar number)
src-unknown-≮: {U = string} (witness t p q) = witness (scalar string) (scalar string) (function-scalar string)
src-unknown-≮: {U = T ∪ U} p = <:-trans-≮: (normalize-<: (T ∪ U)) (srcⁿ-unknown-≮: (normal (T ∪ U)) p)
src-unknown-≮: {U = T ∩ U} p = <:-trans-≮: (normalize-<: (T ∩ U)) (srcⁿ-unknown-≮: (normal (T ∩ U)) p)
unknown-src-≮: : ∀ {S T U} → (U ≮: S) → (T ≮: (U ⇒ unknown)) → (U ≮: src T)
unknown-src-≮: (witness t x x₁) (witness (scalar s) p (function-scalar s)) = witness t x (src-¬scalar s p)
unknown-src-≮: r (witness (function-ok (scalar s)) p (function-ok (scalar-scalar s () q)))
unknown-src-≮: r (witness (function-ok (function-ok _)) p (function-ok (scalar-function-ok ())))
unknown-src-≮: r (witness (function-err t) p (function-err q)) = witness t q (src-¬function-err p)
-- Properties of tgt
tgt-function-ok : ∀ {T t} → (Language (tgt T) t) → Language T (function-ok t)
tgt-function-ok {T = nil} (scalar ())
tgt-function-ok {T = T₁ ⇒ T₂} p = function-ok p
tgt-function-ok {T = never} (scalar ())
tgt-function-ok {T = unknown} p = unknown
tgt-function-ok {T = boolean} (scalar ())
tgt-function-ok {T = number} (scalar ())
tgt-function-ok {T = string} (scalar ())
tgt-function-ok {T = T₁ ∪ T₂} (left p) = left (tgt-function-ok p)
tgt-function-ok {T = T₁ ∪ T₂} (right p) = right (tgt-function-ok p)
tgt-function-ok {T = T₁ ∩ T₂} (p₁ , p₂) = (tgt-function-ok p₁ , tgt-function-ok p₂)
function-ok-tgt : ∀ {T t} → Language T (function-ok t) → (Language (tgt T) t)
function-ok-tgt (function-ok p) = p
function-ok-tgt (left p) = left (function-ok-tgt p)
function-ok-tgt (right p) = right (function-ok-tgt p)
function-ok-tgt (p₁ , p₂) = (function-ok-tgt p₁ , function-ok-tgt p₂)
function-ok-tgt unknown = unknown
tgt-never-≮: : ∀ {T U} → (tgt T ≮: U) → (T ≮: (skalar ∪ (never ⇒ U)))
tgt-never-≮: (witness t p q) = witness (function-ok t) (tgt-function-ok p) (skalar-function-ok , function-ok q)
never-tgt-≮: : ∀ {T U} → (T ≮: (skalar ∪ (never ⇒ U))) → (tgt T ≮: U)
never-tgt-≮: (witness (scalar s) p (q₁ , q₂)) = CONTRADICTION (≮:-refl (witness (scalar s) (skalar-scalar s) q₁))
never-tgt-≮: (witness function p (q₁ , scalar-function ()))
never-tgt-≮: (witness (function-ok t) p (q₁ , function-ok q₂)) = witness t (function-ok-tgt p) q₂
never-tgt-≮: (witness (function-err (scalar s)) p (q₁ , function-err (scalar ())))
src-tgtᶠ-<: : ∀ {T U V} → (FunType T) → (U <: src T) → (tgt T <: V) → (T <: (U ⇒ V))
src-tgtᶠ-<: T p q (scalar s) r = CONTRADICTION (language-comp (scalar s) (fun-¬scalar s T) r)
src-tgtᶠ-<: T p q function r = function
src-tgtᶠ-<: T p q (function-ok s) r = function-ok (q s (function-ok-tgt r))
src-tgtᶠ-<: T p q (function-err s) r = function-err (<:-impl-⊇ p s (src-¬function-err r))
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{-# OPTIONS --universe-polymorphism #-}
-- Proof that 'Free' is a functor
module Categories.Free.Functor where
open import Categories.Support.PropositionalEquality
open import Categories.Categories
open import Categories.Category
renaming (_[_∼_] to _[_~C_])
open import Categories.Free.Core
open import Categories.Functor
using (Functor)
renaming (_≡_ to _≡F_; _∘_ to _∘F_)
open import Categories.Graphs
open import Data.Product
open import Graphs.Graph
renaming (_[_~_] to _[_~G_]; module Heterogeneous to HeterogeneousG)
open import Graphs.GraphMorphism
renaming (_∘_ to _∘G_)
open import Data.Star
open import Data.Star.Properties
using (gmap-◅◅; gmap-id)
open import Categories.Support.StarEquality
open import Level using (_⊔_)
ε∼ε : ∀{o₁ ℓ₁ e₁}{X : Graph o₁ ℓ₁ e₁}
{o₂ ℓ₂ e₂}{Y : Graph o₂ ℓ₂ e₂}
→ (F G : GraphMorphism X Y)
→ F ≈ G
→ {x : Graph.Obj X}
→ Free₀ Y [ ε {x = GraphMorphism.F₀ F x} ~C ε {x = GraphMorphism.F₀ G x} ]
ε∼ε {Y = Y} F G (F≈G₀ , F≈G₁) {x} = ≣-subst (λ z → Free₀ Y [ ε {x = GraphMorphism.F₀ F x} ~C ε {x = z} ]) (F≈G₀ x) (Heterogeneous.refl (Free₀ Y))
-- the below should probably work, but there's an agda bug
-- XXX bug id? mokus? anybody? bueller?
{-
ε∼ε {Y = Y} F G F≈G {x} rewrite proj₁ F≈G x = refl
where open Heterogeneous (Free₀ Y)
-}
_◅~◅_ :
∀ {o ℓ e}{G : Graph o ℓ e}
{a₀ a₁ b₀ b₁ c₀ c₁ : Graph.Obj G}
{f : G [ a₀ ↝ b₀ ]}
{g : G [ a₁ ↝ b₁ ]}
{fs : Free₀ G [ b₀ , c₀ ]}
{gs : Free₀ G [ b₁ , c₁ ]}
→ G [ f ~G g ]
→ Free₀ G [ fs ~C gs ]
→ Free₀ G [ (f ◅ fs) ~C (g ◅ gs) ]
_◅~◅_ {G = G} (HeterogeneousG.≈⇒~ hd) (Heterogeneous.≡⇒∼ tl)
= ≡⇒∼ (hd ◅-cong tl)
where
open Heterogeneous (Free₀ G)
open PathEquality G
Free : ∀ {o ℓ e} → Functor (Graphs o ℓ e) (Categories o (o ⊔ ℓ) (o ⊔ ℓ ⊔ e))
Free {o}{ℓ}{e} = record
{ F₀ = Free₀
; F₁ = Free₁
; identity = λ {A} f → Heterogeneous.reflexive (Free₀ A) (gmap-id f)
; homomorphism = λ {X}{Y}{Z}{f}{g} → homomorphism {X}{Y}{Z}{f}{g}
; F-resp-≡ = λ {X}{Y}{F G : GraphMorphism X Y} → Free₁-resp-≡ {X}{Y}{F}{G}
}
where
module Graphs = Category (Graphs o ℓ e)
module Categories = Category (Categories o (o ⊔ ℓ) (o ⊔ ℓ ⊔ e))
.homomorphism : ∀ {X Y Z} {f : GraphMorphism X Y} {g : GraphMorphism Y Z}
→ Free₁ (g ∘G f) ≡F (Free₁ g ∘F Free₁ f)
homomorphism ε = Heterogeneous.refl _
homomorphism {X}{Y}{Z}{f}{g}{S}{U} (_◅_ {.S}{T}{.U} h hs) =
HeterogeneousG.refl Z ◅~◅ homomorphism {X}{Y}{Z}{f}{g}{T}{U} hs
.Free₁-resp-≡ : ∀ {X Y} {F G : GraphMorphism X Y}
→ F ≈ G
→ Free₁ F ≡F Free₁ G
Free₁-resp-≡ {X}{Y}{F}{G} F≈G {S}{.S} ε = ε∼ε F G F≈G
Free₁-resp-≡ {X}{Y}{F}{G} F≈G {S}{U} (_◅_ {.S}{T}{.U} h hs)
= proj₂ F≈G h ◅~◅ Free₁-resp-≡ {X}{Y}{F}{G} F≈G {T}{U} hs
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{-# OPTIONS --without-K --rewriting #-}
open import HoTT
open import homotopy.LoopSpaceCircle
open import homotopy.PinSn
module homotopy.SphereEndomorphism where
Sphere-endo : ∀ n → Type₀
Sphere-endo n = Sphere n → Sphere n
⊙Sphere-endo : ∀ n → Type₀
⊙Sphere-endo n = ⊙Sphere n ⊙→ ⊙Sphere n
{-
⊙LiftSphere-endo : ∀ {i} n → Type i
⊙LiftSphere-endo {i} n = ⊙Lift {j = i} (⊙Sphere n) ⊙→ ⊙Lift {j = i} (⊙Sphere n)
-}
Trunc-Sphere-endo : ∀ n → Type₀
Trunc-Sphere-endo = Trunc 0 ∘ Sphere-endo
Trunc-⊙Sphere-endo : ∀ n → Type₀
Trunc-⊙Sphere-endo = Trunc 0 ∘ ⊙Sphere-endo
{-
Trunc-⊙LiftSphere-endo : ∀ {i} n → Type i
Trunc-⊙LiftSphere-endo {i} = Trunc 0 ∘ ⊙LiftSphere-endo {i}
-}
{- Part 0: pointedness is free -}
Trunc-⊙Sphere-endo-out : ∀ n
→ Trunc-⊙Sphere-endo n → Trunc-Sphere-endo n
Trunc-⊙Sphere-endo-out n = Trunc-fmap fst
-- For [S¹], the pointedness is free because of the commutativity of its loop space.
-- favonia: maybe one can simplify the proofs through
-- an intermediate type [Σ S¹ (λ x → x == x)]?
private
⊙S¹-endo-in' : (base* : S¹) (loop* : base* == base*) → (⊙S¹ ⊙→ ⊙S¹)
⊙S¹-endo-in' = S¹-elim
(λ loop* → S¹-rec base loop* , idp)
(↓-app→cst-in λ r → ap (λ loop* → (S¹-rec base loop* , idp)) (lemma₀ r)) where
abstract
lemma₀ : ∀ {loop₁ loop₂}
→ loop₁ == loop₂ [ (λ x → x == x) ↓ loop ]
→ loop₁ == loop₂
lemma₀ {loop₁} {loop₂} p = anti-whisker-right loop $
loop₁ ∙ loop
=⟨ ↓-idf=idf-out' p ⟩
loop ∙' loop₂
=⟨ ∙'=∙ loop loop₂ ⟩
loop ∙ loop₂
=⟨ ΩS¹-is-abelian loop loop₂ ⟩
loop₂ ∙ loop
=∎
⊙S¹-endo-in : (S¹ → S¹) → (⊙S¹ ⊙→ ⊙S¹)
⊙S¹-endo-in f = ⊙S¹-endo-in' (f base) (ap f loop)
Trunc-⊙S¹-endo-in : Trunc 0 (S¹ → S¹) → Trunc 0 (⊙S¹ ⊙→ ⊙S¹)
Trunc-⊙S¹-endo-in = Trunc-fmap ⊙S¹-endo-in
abstract
Trunc-⊙S¹-endo-in-η : ∀ f → Trunc-⊙S¹-endo-in (Trunc-⊙Sphere-endo-out 1 f) == f
Trunc-⊙S¹-endo-in-η = Trunc-elim
λ{(f , pt) → ap [_] $
⊙S¹-endo-in'-shifted pt (ap f loop) ∙ ⊙λ= (S¹-rec-η f , idp)}
where
-- free one end to apply identification elimination
⊙S¹-endo-in'-shifted : {base* : S¹}
(shift : base* == base) (loop* : base* == base*)
→ ⊙S¹-endo-in' base* loop* == (S¹-rec base* loop* , shift)
⊙S¹-endo-in'-shifted idp _ = idp
Trunc-⊙S¹-endo-out-β : ∀ f → Trunc-⊙Sphere-endo-out 1 (Trunc-⊙S¹-endo-in f) == f
Trunc-⊙S¹-endo-out-β = Trunc-elim
λ f → ! (ap (λ f → [ fst (⊙S¹-endo-in f) ]) (λ= $ S¹-rec-η f))
∙ ⊙S¹-endo-out-β (f base) (ap f loop)
∙ ap [_] (λ= $ S¹-rec-η f)
where
-- free [base*] to apply circle elimination
⊙S¹-endo-out-β : (base* : S¹) (loop* : base* == base*)
→ [ fst (⊙S¹-endo-in (S¹-rec base* loop*)) ]
== [ S¹-rec base* loop* ] :> Trunc 0 (S¹ → S¹)
⊙S¹-endo-out-β = S¹-elim
(λ loop* → ap (λ loop* → [ S¹-rec base loop* ]) (S¹Rec.loop-β base loop*))
prop-has-all-paths-↓
Trunc-⊙S¹-endo-out-is-equiv : is-equiv (Trunc-⊙Sphere-endo-out 1)
Trunc-⊙S¹-endo-out-is-equiv = is-eq _ Trunc-⊙S¹-endo-in Trunc-⊙S¹-endo-out-β Trunc-⊙S¹-endo-in-η
-- For [Sphere (S (S n))], the pointedness is free because of its connectivity.
private
SphereSS-conn : ∀ n → is-connected 1 (Sphere (S (S n)))
SphereSS-conn n = connected-≤T (≤T-+2+-l 1 (-2≤T ⟨ n ⟩₋₂))
SphereSS-conn-path : ∀ n (x y : Sphere (S (S n))) → is-connected 0 (x == y)
SphereSS-conn-path n x y = path-conn (SphereSS-conn n)
SphereSS-has-all-trunc-paths : ∀ n (x y : Sphere (S (S n))) → Trunc 0 (x == y)
SphereSS-has-all-trunc-paths n x y = –> (Trunc=-equiv [ x ] [ y ])
(contr-has-all-paths {{SphereSS-conn n}} [ x ] [ y ])
Trunc-⊙SphereSS-endo-in : ∀ n
→ Trunc-Sphere-endo (S (S n)) → Trunc-⊙Sphere-endo (S (S n))
Trunc-⊙SphereSS-endo-in n = Trunc-rec λ f →
Trunc-rec (λ pt → [ f , pt ])
(SphereSS-has-all-trunc-paths n (f north) north)
abstract
Trunc-⊙SphereSS-endo-in-η : ∀ n f → Trunc-⊙SphereSS-endo-in n (Trunc-⊙Sphere-endo-out (S (S n)) f) == f
Trunc-⊙SphereSS-endo-in-η n = Trunc-elim
λ{(f , pt) → ap (Trunc-rec (λ pt → [ f , pt ]))
(contr-has-all-paths {{SphereSS-conn-path n (f north) north}}
(SphereSS-has-all-trunc-paths n (f north) north) [ pt ])}
Trunc-⊙SphereSS-endo-out-β : ∀ n f → Trunc-⊙Sphere-endo-out (S (S n)) (Trunc-⊙SphereSS-endo-in n f) == f
Trunc-⊙SphereSS-endo-out-β n = Trunc-elim
λ f → Trunc-elim
{P = λ pt → Trunc-⊙Sphere-endo-out (S (S n)) (Trunc-rec (λ pt → [ f , pt ]) pt) == [ f ]}
(λ pt → idp) (SphereSS-has-all-trunc-paths n (f north) north)
Trunc-⊙SphereSS-endo-out-is-equiv : ∀ n → is-equiv (Trunc-⊙Sphere-endo-out (S (S n)))
Trunc-⊙SphereSS-endo-out-is-equiv n = is-eq
(Trunc-⊙Sphere-endo-out (S (S n))) (Trunc-⊙SphereSS-endo-in n)
(Trunc-⊙SphereSS-endo-out-β n) (Trunc-⊙SphereSS-endo-in-η n)
-- the unified interface
Trunc-⊙SphereS-endo-out-is-equiv : ∀ n → is-equiv (Trunc-⊙Sphere-endo-out (S n))
Trunc-⊙SphereS-endo-out-is-equiv 0 = Trunc-⊙S¹-endo-out-is-equiv
Trunc-⊙SphereS-endo-out-is-equiv (S n) = Trunc-⊙SphereSS-endo-out-is-equiv n
Trunc-⊙SphereS-endo-in : ∀ n
→ Trunc-Sphere-endo (S n) → Trunc-⊙Sphere-endo (S n)
Trunc-⊙SphereS-endo-in n = is-equiv.g (Trunc-⊙SphereS-endo-out-is-equiv n)
{- Part 1: suspension is an isomorphism -}
private
open import homotopy.Freudenthal
SSSSk≤SSk+2+SSk : ∀ k → S (S (S (S k))) ≤T S (S k) +2+ S (S k)
SSSSk≤SSk+2+SSk k = ≤T-+2+-l 0 $ ≤T-+2+-r (S (S k)) $ -2≤T k
SSn≤n+2+n : ∀ n → S (S ⟨ n ⟩) ≤T ⟨ n ⟩ +2+ ⟨ n ⟩
SSn≤n+2+n n = SSSSk≤SSk+2+SSk ⟨ n ⟩₋₂
module F n = FreudenthalEquiv
⟨ n ⟩₋₁ _ (SSn≤n+2+n n) (⊙Sphere (S (S n))) {{Sphere-conn (S (S n))}}
import homotopy.TruncationLoopLadder as TLL
import homotopy.SuspAdjointLoop as SAL
import homotopy.SuspAdjointLoopLadder as SALL
import homotopy.CircleHSpace as CHS
import homotopy.Pi2HSusp as Pi2
⊙up : ∀ n → ⊙Sphere n ⊙→ ⊙Ω (⊙Sphere (S n))
⊙up n = SAL.η (⊙Sphere n)
Ω^'S-Trunc-up-is-equiv : ∀ n → is-equiv (Ω^'-fmap (S n) (⊙Trunc-fmap {n = ⟨ S n ⟩} (⊙up (S n))))
Ω^'S-Trunc-up-is-equiv O = snd (Ω-emap (Pi2.⊙eq⁻¹ CHS.⊙S¹-hSpace))
Ω^'S-Trunc-up-is-equiv (S n) = snd (Ω^'-emap (S (S n)) (F.⊙eq n))
Trunc-Ω^'S-up-is-equiv : ∀ n → is-equiv (Trunc-fmap {n = 0} (Ω^'-fmap (S n) (⊙up (S n))))
Trunc-Ω^'S-up-is-equiv n = ⊙CommSquareEquiv-preserves-equiv (TLL.ladder (S n) (⊙up (S n))) (Ω^'S-Trunc-up-is-equiv n)
Trunc-post⊙∘-Ω^'S-up-is-equiv : ∀ n →
is-equiv (Trunc-fmap {n = 0} ((⊙Ω^'-fmap (S n) (⊙up (S n)) ⊙∘_) :> (_ → ⊙Bool ⊙→ _)))
Trunc-post⊙∘-Ω^'S-up-is-equiv n = CommSquareEquiv-preserves'-equiv
(Trunc-csemap (⊙Bool→-equiv-idf-nat (⊙Ω^'-fmap (S n) (⊙up (S n)))))
(Trunc-Ω^'S-up-is-equiv n)
Trunc-post⊙∘-upS-is-equiv : ∀ n →
is-equiv (Trunc-fmap {n = 0} ((⊙up (S n) ⊙∘_) :> (_ → ⊙Sphere (S n) ⊙→ _)))
Trunc-post⊙∘-upS-is-equiv n = CommSquareEquiv-preserves'-equiv
(Trunc-csemap (SALL.ladder (S n) (⊙up (S n))))
(Trunc-post⊙∘-Ω^'S-up-is-equiv n)
final-fix : ∀ n →
CommSquareEquiv
(⊙Susp-fmap :> (⊙Sphere-endo (S n) → _))
(SAL.η _ ⊙∘_)
(idf _)
(–> (SAL.eq _ _))
final-fix n = comm-sqr (λ f → ! (SAL.η-natural f)) , idf-is-equiv _ , snd (SAL.eq _ _)
Trunc-⊙SphereS-endo-Susp-is-equiv : ∀ n →
is-equiv (Trunc-fmap ⊙Susp-fmap :> (Trunc-⊙Sphere-endo (S n) → _))
Trunc-⊙SphereS-endo-Susp-is-equiv n =
CommSquareEquiv-preserves'-equiv
(Trunc-csemap (final-fix n))
(Trunc-post⊙∘-upS-is-equiv n)
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-- The debug output should include the text "Termination checking
-- mutual block MutId 0" once, not three times.
{-# OPTIONS -vterm.mutual.id:40 #-}
record R : Set₁ where
constructor c
field
A : Set
_ : A → A
_ = λ x → x
_ : A → A
_ = λ x → x
_ : A → A
_ = λ x → x
-- Included in order to make the code fail to type-check.
Bad : Set
Bad = Set
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Decidable equality over lists parameterised by some setoid
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Relation.Binary
module Data.List.Relation.Binary.Equality.DecSetoid
{a ℓ} (DS : DecSetoid a ℓ) where
import Data.List.Relation.Binary.Equality.Setoid as SetoidEquality
import Data.List.Relation.Binary.Pointwise as PW
open import Level
open import Relation.Binary using (Decidable)
open DecSetoid DS
------------------------------------------------------------------------
-- Make all definitions from setoid equality available
open SetoidEquality setoid public
------------------------------------------------------------------------
-- Additional properties
infix 4 _≋?_
_≋?_ : Decidable _≋_
_≋?_ = PW.decidable _≟_
≋-isDecEquivalence : IsDecEquivalence _≋_
≋-isDecEquivalence = PW.isDecEquivalence isDecEquivalence
≋-decSetoid : DecSetoid a (a ⊔ ℓ)
≋-decSetoid = PW.decSetoid DS
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module Everything where
import Relation.Ternary.Separation
-- The syntax and interpreter of LTLC
import Typed.LTLC
-- The syntax and interpreter of LTLC with strong updatable references
import Typed.LTLCRef
-- The syntax of a session typed language
import Sessions.Syntax
-- ... and its semantics
import Sessions.Semantics.Runtime
import Sessions.Semantics.Commands
import Sessions.Semantics.Expr
import Sessions.Semantics.Communication
import Sessions.Semantics.Process
-- the paper's session-typed program example
import Examples
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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Functions.Definition
open import Lists.Definition
open import Lists.Monad
open import Boolean.Definition
module Lists.Filter.AllTrue where
allTrue : {a b : _} {A : Set a} (f : A → Set b) (l : List A) → Set b
allTrue f [] = True'
allTrue f (x :: l) = f x && allTrue f l
allTrueConcat : {a b : _} {A : Set a} (f : A → Set b) (l m : List A) → allTrue f l → allTrue f m → allTrue f (l ++ m)
allTrueConcat f [] m fl fm = fm
allTrueConcat f (x :: l) m (fst ,, snd) fm = fst ,, allTrueConcat f l m snd fm
allTrueFlatten : {a b : _} {A : Set a} (f : A → Set b) (l : List (List A)) → allTrue (λ i → allTrue f i) l → allTrue f (flatten l)
allTrueFlatten f [] pr = record {}
allTrueFlatten f ([] :: ls) pr = allTrueFlatten f ls (_&&_.snd pr)
allTrueFlatten f ((x :: l) :: ls) ((fx ,, fl) ,, snd) = fx ,, allTrueConcat f l (flatten ls) fl (allTrueFlatten f ls snd)
allTrueMap : {a b c : _} {A : Set a} {B : Set b} (pred : B → Set c) (f : A → B) (l : List A) → allTrue (pred ∘ f) l → allTrue pred (map f l)
allTrueMap pred f [] pr = record {}
allTrueMap pred f (x :: l) pr = _&&_.fst pr ,, allTrueMap pred f l (_&&_.snd pr)
allTrueExtension : {a b : _} {A : Set a} (f g : A → Set b) (l : List A) → ({x : A} → f x → g x) → allTrue f l → allTrue g l
allTrueExtension f g [] pred t = record {}
allTrueExtension f g (x :: l) pred (fg ,, snd) = pred {x} fg ,, allTrueExtension f g l pred snd
allTrueTail : {a b : _} {A : Set a} (pred : A → Set b) (x : A) (l : List A) → allTrue pred (x :: l) → allTrue pred l
allTrueTail pred x l (fst ,, snd) = snd
filter : {a : _} {A : Set a} (l : List A) (f : A → Bool) → List A
filter [] f = []
filter (x :: l) f with f x
filter (x :: l) f | BoolTrue = x :: filter l f
filter (x :: l) f | BoolFalse = filter l f
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module _ (A : Set) where
record R : Set where
field f : A
test : R → R
test r = {!r!}
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module Text.Greek.SBLGNT.Rom where
open import Data.List
open import Text.Greek.Bible
open import Text.Greek.Script
open import Text.Greek.Script.Unicode
ΠΡΟΣ-ΡΩΜΑΙΟΥΣ : List (Word)
ΠΡΟΣ-ΡΩΜΑΙΟΥΣ =
word (Π ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Rom.1.1"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Rom.1.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.1.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.1.1"
∷ word (κ ∷ ∙λ ∷ η ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.1.1"
∷ word (ἀ ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ς ∷ []) "Rom.1.1"
∷ word (ἀ ∷ φ ∷ ω ∷ ρ ∷ ι ∷ σ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "Rom.1.1"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.1.1"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.1.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.1.1"
∷ word (ὃ ∷ []) "Rom.1.2"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ π ∷ η ∷ γ ∷ γ ∷ ε ∷ ί ∷ ∙λ ∷ α ∷ τ ∷ ο ∷ []) "Rom.1.2"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.1.2"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.1.2"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.1.2"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.1.2"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.2"
∷ word (γ ∷ ρ ∷ α ∷ φ ∷ α ∷ ῖ ∷ ς ∷ []) "Rom.1.2"
∷ word (ἁ ∷ γ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Rom.1.2"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "Rom.1.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.1.3"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "Rom.1.3"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.1.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.1.3"
∷ word (γ ∷ ε ∷ ν ∷ ο ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ []) "Rom.1.3"
∷ word (ἐ ∷ κ ∷ []) "Rom.1.3"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.1.3"
∷ word (Δ ∷ α ∷ υ ∷ ὶ ∷ δ ∷ []) "Rom.1.3"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.1.3"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Rom.1.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.1.4"
∷ word (ὁ ∷ ρ ∷ ι ∷ σ ∷ θ ∷ έ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.1.4"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "Rom.1.4"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.1.4"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.4"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ []) "Rom.1.4"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.1.4"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.1.4"
∷ word (ἁ ∷ γ ∷ ι ∷ ω ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "Rom.1.4"
∷ word (ἐ ∷ ξ ∷ []) "Rom.1.4"
∷ word (ἀ ∷ ν ∷ α ∷ σ ∷ τ ∷ ά ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.1.4"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Rom.1.4"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.1.4"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.1.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.1.4"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.1.4"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.1.4"
∷ word (δ ∷ ι ∷ []) "Rom.1.5"
∷ word (ο ∷ ὗ ∷ []) "Rom.1.5"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ β ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.1.5"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Rom.1.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.5"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "Rom.1.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.1.5"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ὴ ∷ ν ∷ []) "Rom.1.5"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.1.5"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.5"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.5"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.5"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.5"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.1.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.1.5"
∷ word (ὀ ∷ ν ∷ ό ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.1.5"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.1.5"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.6"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "Rom.1.6"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "Rom.1.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.6"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.1.6"
∷ word (κ ∷ ∙λ ∷ η ∷ τ ∷ ο ∷ ὶ ∷ []) "Rom.1.6"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.1.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.1.6"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.7"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.7"
∷ word (ο ∷ ὖ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.7"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.7"
∷ word (Ῥ ∷ ώ ∷ μ ∷ ῃ ∷ []) "Rom.1.7"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.1.7"
∷ word (κ ∷ ∙λ ∷ η ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.7"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Rom.1.7"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Rom.1.7"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.1.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.7"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "Rom.1.7"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.1.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.1.7"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.1.7"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.1.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.7"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.1.7"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.1.7"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.1.7"
∷ word (Π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.1.8"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.1.8"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῶ ∷ []) "Rom.1.8"
∷ word (τ ∷ ῷ ∷ []) "Rom.1.8"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.1.8"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.1.8"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.1.8"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.1.8"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.1.8"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "Rom.1.8"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.1.8"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.1.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.1.8"
∷ word (ἡ ∷ []) "Rom.1.8"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "Rom.1.8"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.1.8"
∷ word (κ ∷ α ∷ τ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.1.8"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.8"
∷ word (ὅ ∷ ∙λ ∷ ῳ ∷ []) "Rom.1.8"
∷ word (τ ∷ ῷ ∷ []) "Rom.1.8"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "Rom.1.8"
∷ word (μ ∷ ά ∷ ρ ∷ τ ∷ υ ∷ ς ∷ []) "Rom.1.9"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Rom.1.9"
∷ word (μ ∷ ο ∷ ύ ∷ []) "Rom.1.9"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.1.9"
∷ word (ὁ ∷ []) "Rom.1.9"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "Rom.1.9"
∷ word (ᾧ ∷ []) "Rom.1.9"
∷ word (∙λ ∷ α ∷ τ ∷ ρ ∷ ε ∷ ύ ∷ ω ∷ []) "Rom.1.9"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.9"
∷ word (τ ∷ ῷ ∷ []) "Rom.1.9"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ί ∷ []) "Rom.1.9"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.1.9"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.9"
∷ word (τ ∷ ῷ ∷ []) "Rom.1.9"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ῳ ∷ []) "Rom.1.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.1.9"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "Rom.1.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.1.9"
∷ word (ὡ ∷ ς ∷ []) "Rom.1.9"
∷ word (ἀ ∷ δ ∷ ι ∷ α ∷ ∙λ ∷ ε ∷ ί ∷ π ∷ τ ∷ ω ∷ ς ∷ []) "Rom.1.9"
∷ word (μ ∷ ν ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "Rom.1.9"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.1.9"
∷ word (π ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ μ ∷ α ∷ ι ∷ []) "Rom.1.9"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ο ∷ τ ∷ ε ∷ []) "Rom.1.10"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.1.10"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.1.10"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ ῶ ∷ ν ∷ []) "Rom.1.10"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.1.10"
∷ word (δ ∷ ε ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Rom.1.10"
∷ word (ε ∷ ἴ ∷ []) "Rom.1.10"
∷ word (π ∷ ω ∷ ς ∷ []) "Rom.1.10"
∷ word (ἤ ∷ δ ∷ η ∷ []) "Rom.1.10"
∷ word (π ∷ ο ∷ τ ∷ ὲ ∷ []) "Rom.1.10"
∷ word (ε ∷ ὐ ∷ ο ∷ δ ∷ ω ∷ θ ∷ ή ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.1.10"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.10"
∷ word (τ ∷ ῷ ∷ []) "Rom.1.10"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.1.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.1.10"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.1.10"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.1.10"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.1.10"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.1.10"
∷ word (ἐ ∷ π ∷ ι ∷ π ∷ ο ∷ θ ∷ ῶ ∷ []) "Rom.1.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.1.11"
∷ word (ἰ ∷ δ ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.1.11"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.1.11"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.1.11"
∷ word (τ ∷ ι ∷ []) "Rom.1.11"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ δ ∷ ῶ ∷ []) "Rom.1.11"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ σ ∷ μ ∷ α ∷ []) "Rom.1.11"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.1.11"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ὸ ∷ ν ∷ []) "Rom.1.11"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.1.11"
∷ word (τ ∷ ὸ ∷ []) "Rom.1.11"
∷ word (σ ∷ τ ∷ η ∷ ρ ∷ ι ∷ χ ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "Rom.1.11"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.1.11"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.1.12"
∷ word (δ ∷ έ ∷ []) "Rom.1.12"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.1.12"
∷ word (σ ∷ υ ∷ μ ∷ π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ η ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "Rom.1.12"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.12"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.1.12"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.1.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.1.12"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "Rom.1.12"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.1.12"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.1.12"
∷ word (τ ∷ ε ∷ []) "Rom.1.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.12"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "Rom.1.12"
∷ word (ο ∷ ὐ ∷ []) "Rom.1.13"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "Rom.1.13"
∷ word (δ ∷ ὲ ∷ []) "Rom.1.13"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.1.13"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.1.13"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Rom.1.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.1.13"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ά ∷ κ ∷ ι ∷ ς ∷ []) "Rom.1.13"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ θ ∷ έ ∷ μ ∷ η ∷ ν ∷ []) "Rom.1.13"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.1.13"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.1.13"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.1.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.13"
∷ word (ἐ ∷ κ ∷ ω ∷ ∙λ ∷ ύ ∷ θ ∷ η ∷ ν ∷ []) "Rom.1.13"
∷ word (ἄ ∷ χ ∷ ρ ∷ ι ∷ []) "Rom.1.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.1.13"
∷ word (δ ∷ ε ∷ ῦ ∷ ρ ∷ ο ∷ []) "Rom.1.13"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.1.13"
∷ word (τ ∷ ι ∷ ν ∷ ὰ ∷ []) "Rom.1.13"
∷ word (κ ∷ α ∷ ρ ∷ π ∷ ὸ ∷ ν ∷ []) "Rom.1.13"
∷ word (σ ∷ χ ∷ ῶ ∷ []) "Rom.1.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.13"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.13"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.1.13"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.1.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.13"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.13"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.13"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.13"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.13"
∷ word (Ἕ ∷ ∙λ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ν ∷ []) "Rom.1.14"
∷ word (τ ∷ ε ∷ []) "Rom.1.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.14"
∷ word (β ∷ α ∷ ρ ∷ β ∷ ά ∷ ρ ∷ ο ∷ ι ∷ ς ∷ []) "Rom.1.14"
∷ word (σ ∷ ο ∷ φ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.14"
∷ word (τ ∷ ε ∷ []) "Rom.1.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.14"
∷ word (ἀ ∷ ν ∷ ο ∷ ή ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "Rom.1.14"
∷ word (ὀ ∷ φ ∷ ε ∷ ι ∷ ∙λ ∷ έ ∷ τ ∷ η ∷ ς ∷ []) "Rom.1.14"
∷ word (ε ∷ ἰ ∷ μ ∷ ί ∷ []) "Rom.1.14"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.1.15"
∷ word (τ ∷ ὸ ∷ []) "Rom.1.15"
∷ word (κ ∷ α ∷ τ ∷ []) "Rom.1.15"
∷ word (ἐ ∷ μ ∷ ὲ ∷ []) "Rom.1.15"
∷ word (π ∷ ρ ∷ ό ∷ θ ∷ υ ∷ μ ∷ ο ∷ ν ∷ []) "Rom.1.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.15"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.1.15"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.15"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.15"
∷ word (Ῥ ∷ ώ ∷ μ ∷ ῃ ∷ []) "Rom.1.15"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.1.15"
∷ word (Ο ∷ ὐ ∷ []) "Rom.1.16"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.1.16"
∷ word (ἐ ∷ π ∷ α ∷ ι ∷ σ ∷ χ ∷ ύ ∷ ν ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.1.16"
∷ word (τ ∷ ὸ ∷ []) "Rom.1.16"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.1.16"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ς ∷ []) "Rom.1.16"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.1.16"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.1.16"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.1.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.1.16"
∷ word (σ ∷ ω ∷ τ ∷ η ∷ ρ ∷ ί ∷ α ∷ ν ∷ []) "Rom.1.16"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "Rom.1.16"
∷ word (τ ∷ ῷ ∷ []) "Rom.1.16"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ ι ∷ []) "Rom.1.16"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ῳ ∷ []) "Rom.1.16"
∷ word (τ ∷ ε ∷ []) "Rom.1.16"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.1.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.16"
∷ word (Ἕ ∷ ∙λ ∷ ∙λ ∷ η ∷ ν ∷ ι ∷ []) "Rom.1.16"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ []) "Rom.1.17"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.1.17"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.1.17"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.17"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.1.17"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ π ∷ τ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.1.17"
∷ word (ἐ ∷ κ ∷ []) "Rom.1.17"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.1.17"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.1.17"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.1.17"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.1.17"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.1.17"
∷ word (Ὁ ∷ []) "Rom.1.17"
∷ word (δ ∷ ὲ ∷ []) "Rom.1.17"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ ο ∷ ς ∷ []) "Rom.1.17"
∷ word (ἐ ∷ κ ∷ []) "Rom.1.17"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.1.17"
∷ word (ζ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.1.17"
∷ word (Ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ π ∷ τ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.1.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.1.18"
∷ word (ὀ ∷ ρ ∷ γ ∷ ὴ ∷ []) "Rom.1.18"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.1.18"
∷ word (ἀ ∷ π ∷ []) "Rom.1.18"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ο ∷ ῦ ∷ []) "Rom.1.18"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.1.18"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "Rom.1.18"
∷ word (ἀ ∷ σ ∷ έ ∷ β ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Rom.1.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.18"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ί ∷ α ∷ ν ∷ []) "Rom.1.18"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "Rom.1.18"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.1.18"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.1.18"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Rom.1.18"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.18"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ί ∷ ᾳ ∷ []) "Rom.1.18"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ χ ∷ ό ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.1.18"
∷ word (δ ∷ ι ∷ ό ∷ τ ∷ ι ∷ []) "Rom.1.19"
∷ word (τ ∷ ὸ ∷ []) "Rom.1.19"
∷ word (γ ∷ ν ∷ ω ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.1.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.1.19"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.1.19"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ό ∷ ν ∷ []) "Rom.1.19"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.1.19"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.19"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.19"
∷ word (ὁ ∷ []) "Rom.1.19"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.1.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.1.19"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.19"
∷ word (ἐ ∷ φ ∷ α ∷ ν ∷ έ ∷ ρ ∷ ω ∷ σ ∷ ε ∷ ν ∷ []) "Rom.1.19"
∷ word (τ ∷ ὰ ∷ []) "Rom.1.20"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.1.20"
∷ word (ἀ ∷ ό ∷ ρ ∷ α ∷ τ ∷ α ∷ []) "Rom.1.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.1.20"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.1.20"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.1.20"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "Rom.1.20"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.20"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ μ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.20"
∷ word (ν ∷ ο ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ α ∷ []) "Rom.1.20"
∷ word (κ ∷ α ∷ θ ∷ ο ∷ ρ ∷ ᾶ ∷ τ ∷ α ∷ ι ∷ []) "Rom.1.20"
∷ word (ἥ ∷ []) "Rom.1.20"
∷ word (τ ∷ ε ∷ []) "Rom.1.20"
∷ word (ἀ ∷ ΐ ∷ δ ∷ ι ∷ ο ∷ ς ∷ []) "Rom.1.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.1.20"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ς ∷ []) "Rom.1.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.20"
∷ word (θ ∷ ε ∷ ι ∷ ό ∷ τ ∷ η ∷ ς ∷ []) "Rom.1.20"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.1.20"
∷ word (τ ∷ ὸ ∷ []) "Rom.1.20"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.1.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.1.20"
∷ word (ἀ ∷ ν ∷ α ∷ π ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ή ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.1.20"
∷ word (δ ∷ ι ∷ ό ∷ τ ∷ ι ∷ []) "Rom.1.21"
∷ word (γ ∷ ν ∷ ό ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.1.21"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.1.21"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "Rom.1.21"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "Rom.1.21"
∷ word (ὡ ∷ ς ∷ []) "Rom.1.21"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "Rom.1.21"
∷ word (ἐ ∷ δ ∷ ό ∷ ξ ∷ α ∷ σ ∷ α ∷ ν ∷ []) "Rom.1.21"
∷ word (ἢ ∷ []) "Rom.1.21"
∷ word (η ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ί ∷ σ ∷ τ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.1.21"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.1.21"
∷ word (ἐ ∷ μ ∷ α ∷ τ ∷ α ∷ ι ∷ ώ ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.1.21"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.21"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.21"
∷ word (δ ∷ ι ∷ α ∷ ∙λ ∷ ο ∷ γ ∷ ι ∷ σ ∷ μ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.21"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.1.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.21"
∷ word (ἐ ∷ σ ∷ κ ∷ ο ∷ τ ∷ ί ∷ σ ∷ θ ∷ η ∷ []) "Rom.1.21"
∷ word (ἡ ∷ []) "Rom.1.21"
∷ word (ἀ ∷ σ ∷ ύ ∷ ν ∷ ε ∷ τ ∷ ο ∷ ς ∷ []) "Rom.1.21"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.1.21"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ []) "Rom.1.21"
∷ word (φ ∷ ά ∷ σ ∷ κ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.1.22"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.1.22"
∷ word (σ ∷ ο ∷ φ ∷ ο ∷ ὶ ∷ []) "Rom.1.22"
∷ word (ἐ ∷ μ ∷ ω ∷ ρ ∷ ά ∷ ν ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.1.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.23"
∷ word (ἤ ∷ ∙λ ∷ ∙λ ∷ α ∷ ξ ∷ α ∷ ν ∷ []) "Rom.1.23"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.1.23"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "Rom.1.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.1.23"
∷ word (ἀ ∷ φ ∷ θ ∷ ά ∷ ρ ∷ τ ∷ ο ∷ υ ∷ []) "Rom.1.23"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.1.23"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.23"
∷ word (ὁ ∷ μ ∷ ο ∷ ι ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.1.23"
∷ word (ε ∷ ἰ ∷ κ ∷ ό ∷ ν ∷ ο ∷ ς ∷ []) "Rom.1.23"
∷ word (φ ∷ θ ∷ α ∷ ρ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.1.23"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "Rom.1.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.23"
∷ word (π ∷ ε ∷ τ ∷ ε ∷ ι ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.1.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.23"
∷ word (τ ∷ ε ∷ τ ∷ ρ ∷ α ∷ π ∷ ό ∷ δ ∷ ω ∷ ν ∷ []) "Rom.1.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.23"
∷ word (ἑ ∷ ρ ∷ π ∷ ε ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.1.23"
∷ word (Δ ∷ ι ∷ ὸ ∷ []) "Rom.1.24"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "Rom.1.24"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.1.24"
∷ word (ὁ ∷ []) "Rom.1.24"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.1.24"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.24"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Rom.1.24"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Rom.1.24"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.1.24"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ι ∷ ῶ ∷ ν ∷ []) "Rom.1.24"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.1.24"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.1.24"
∷ word (ἀ ∷ κ ∷ α ∷ θ ∷ α ∷ ρ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Rom.1.24"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.1.24"
∷ word (ἀ ∷ τ ∷ ι ∷ μ ∷ ά ∷ ζ ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.1.24"
∷ word (τ ∷ ὰ ∷ []) "Rom.1.24"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Rom.1.24"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.1.24"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.24"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.24"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "Rom.1.25"
∷ word (μ ∷ ε ∷ τ ∷ ή ∷ ∙λ ∷ ∙λ ∷ α ∷ ξ ∷ α ∷ ν ∷ []) "Rom.1.25"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.1.25"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Rom.1.25"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.1.25"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.1.25"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.25"
∷ word (τ ∷ ῷ ∷ []) "Rom.1.25"
∷ word (ψ ∷ ε ∷ ύ ∷ δ ∷ ε ∷ ι ∷ []) "Rom.1.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.25"
∷ word (ἐ ∷ σ ∷ ε ∷ β ∷ ά ∷ σ ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.1.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.25"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ τ ∷ ρ ∷ ε ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "Rom.1.25"
∷ word (τ ∷ ῇ ∷ []) "Rom.1.25"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "Rom.1.25"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "Rom.1.25"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.1.25"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ α ∷ ν ∷ τ ∷ α ∷ []) "Rom.1.25"
∷ word (ὅ ∷ ς ∷ []) "Rom.1.25"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.1.25"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ η ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.1.25"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.1.25"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.1.25"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ α ∷ ς ∷ []) "Rom.1.25"
∷ word (ἀ ∷ μ ∷ ή ∷ ν ∷ []) "Rom.1.25"
∷ word (Δ ∷ ι ∷ ὰ ∷ []) "Rom.1.26"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.1.26"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "Rom.1.26"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.1.26"
∷ word (ὁ ∷ []) "Rom.1.26"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.1.26"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.1.26"
∷ word (π ∷ ά ∷ θ ∷ η ∷ []) "Rom.1.26"
∷ word (ἀ ∷ τ ∷ ι ∷ μ ∷ ί ∷ α ∷ ς ∷ []) "Rom.1.26"
∷ word (α ∷ ἵ ∷ []) "Rom.1.26"
∷ word (τ ∷ ε ∷ []) "Rom.1.26"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.1.26"
∷ word (θ ∷ ή ∷ ∙λ ∷ ε ∷ ι ∷ α ∷ ι ∷ []) "Rom.1.26"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.1.26"
∷ word (μ ∷ ε ∷ τ ∷ ή ∷ ∙λ ∷ ∙λ ∷ α ∷ ξ ∷ α ∷ ν ∷ []) "Rom.1.26"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.1.26"
∷ word (φ ∷ υ ∷ σ ∷ ι ∷ κ ∷ ὴ ∷ ν ∷ []) "Rom.1.26"
∷ word (χ ∷ ρ ∷ ῆ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.26"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.1.26"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.1.26"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "Rom.1.26"
∷ word (φ ∷ ύ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.26"
∷ word (ὁ ∷ μ ∷ ο ∷ ί ∷ ω ∷ ς ∷ []) "Rom.1.27"
∷ word (τ ∷ ε ∷ []) "Rom.1.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.27"
∷ word (ο ∷ ἱ ∷ []) "Rom.1.27"
∷ word (ἄ ∷ ρ ∷ σ ∷ ε ∷ ν ∷ ε ∷ ς ∷ []) "Rom.1.27"
∷ word (ἀ ∷ φ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.1.27"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.1.27"
∷ word (φ ∷ υ ∷ σ ∷ ι ∷ κ ∷ ὴ ∷ ν ∷ []) "Rom.1.27"
∷ word (χ ∷ ρ ∷ ῆ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.27"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.1.27"
∷ word (θ ∷ η ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Rom.1.27"
∷ word (ἐ ∷ ξ ∷ ε ∷ κ ∷ α ∷ ύ ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.1.27"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.27"
∷ word (τ ∷ ῇ ∷ []) "Rom.1.27"
∷ word (ὀ ∷ ρ ∷ έ ∷ ξ ∷ ε ∷ ι ∷ []) "Rom.1.27"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.1.27"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.1.27"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.1.27"
∷ word (ἄ ∷ ρ ∷ σ ∷ ε ∷ ν ∷ ε ∷ ς ∷ []) "Rom.1.27"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.27"
∷ word (ἄ ∷ ρ ∷ σ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.27"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.1.27"
∷ word (ἀ ∷ σ ∷ χ ∷ η ∷ μ ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.1.27"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ α ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Rom.1.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.27"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.1.27"
∷ word (ἀ ∷ ν ∷ τ ∷ ι ∷ μ ∷ ι ∷ σ ∷ θ ∷ ί ∷ α ∷ ν ∷ []) "Rom.1.27"
∷ word (ἣ ∷ ν ∷ []) "Rom.1.27"
∷ word (ἔ ∷ δ ∷ ε ∷ ι ∷ []) "Rom.1.27"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.1.27"
∷ word (π ∷ ∙λ ∷ ά ∷ ν ∷ η ∷ ς ∷ []) "Rom.1.27"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.1.27"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.27"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.27"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ α ∷ μ ∷ β ∷ ά ∷ ν ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.1.27"
∷ word (Κ ∷ α ∷ ὶ ∷ []) "Rom.1.28"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.1.28"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.1.28"
∷ word (ἐ ∷ δ ∷ ο ∷ κ ∷ ί ∷ μ ∷ α ∷ σ ∷ α ∷ ν ∷ []) "Rom.1.28"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.1.28"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "Rom.1.28"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.1.28"
∷ word (ἐ ∷ ν ∷ []) "Rom.1.28"
∷ word (ἐ ∷ π ∷ ι ∷ γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "Rom.1.28"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "Rom.1.28"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.1.28"
∷ word (ὁ ∷ []) "Rom.1.28"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.1.28"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.1.28"
∷ word (ἀ ∷ δ ∷ ό ∷ κ ∷ ι ∷ μ ∷ ο ∷ ν ∷ []) "Rom.1.28"
∷ word (ν ∷ ο ∷ ῦ ∷ ν ∷ []) "Rom.1.28"
∷ word (π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.1.28"
∷ word (τ ∷ ὰ ∷ []) "Rom.1.28"
∷ word (μ ∷ ὴ ∷ []) "Rom.1.28"
∷ word (κ ∷ α ∷ θ ∷ ή ∷ κ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "Rom.1.28"
∷ word (π ∷ ε ∷ π ∷ ∙λ ∷ η ∷ ρ ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "Rom.1.29"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "Rom.1.29"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ί ∷ ᾳ ∷ []) "Rom.1.29"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ί ∷ ᾳ ∷ []) "Rom.1.29"
∷ word (π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ ε ∷ ξ ∷ ί ∷ ᾳ ∷ []) "Rom.1.29"
∷ word (κ ∷ α ∷ κ ∷ ί ∷ ᾳ ∷ []) "Rom.1.29"
∷ word (μ ∷ ε ∷ σ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.1.29"
∷ word (φ ∷ θ ∷ ό ∷ ν ∷ ο ∷ υ ∷ []) "Rom.1.29"
∷ word (φ ∷ ό ∷ ν ∷ ο ∷ υ ∷ []) "Rom.1.29"
∷ word (ἔ ∷ ρ ∷ ι ∷ δ ∷ ο ∷ ς ∷ []) "Rom.1.29"
∷ word (δ ∷ ό ∷ ∙λ ∷ ο ∷ υ ∷ []) "Rom.1.29"
∷ word (κ ∷ α ∷ κ ∷ ο ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Rom.1.29"
∷ word (ψ ∷ ι ∷ θ ∷ υ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ά ∷ ς ∷ []) "Rom.1.29"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ά ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.1.30"
∷ word (θ ∷ ε ∷ ο ∷ σ ∷ τ ∷ υ ∷ γ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.1.30"
∷ word (ὑ ∷ β ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ά ∷ ς ∷ []) "Rom.1.30"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ η ∷ φ ∷ ά ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "Rom.1.30"
∷ word (ἀ ∷ ∙λ ∷ α ∷ ζ ∷ ό ∷ ν ∷ α ∷ ς ∷ []) "Rom.1.30"
∷ word (ἐ ∷ φ ∷ ε ∷ υ ∷ ρ ∷ ε ∷ τ ∷ ὰ ∷ ς ∷ []) "Rom.1.30"
∷ word (κ ∷ α ∷ κ ∷ ῶ ∷ ν ∷ []) "Rom.1.30"
∷ word (γ ∷ ο ∷ ν ∷ ε ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.30"
∷ word (ἀ ∷ π ∷ ε ∷ ι ∷ θ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.1.30"
∷ word (ἀ ∷ σ ∷ υ ∷ ν ∷ έ ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.1.31"
∷ word (ἀ ∷ σ ∷ υ ∷ ν ∷ θ ∷ έ ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.1.31"
∷ word (ἀ ∷ σ ∷ τ ∷ ό ∷ ρ ∷ γ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.1.31"
∷ word (ἀ ∷ ν ∷ ε ∷ ∙λ ∷ ε ∷ ή ∷ μ ∷ ο ∷ ν ∷ α ∷ ς ∷ []) "Rom.1.31"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "Rom.1.32"
∷ word (τ ∷ ὸ ∷ []) "Rom.1.32"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ί ∷ ω ∷ μ ∷ α ∷ []) "Rom.1.32"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.1.32"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.1.32"
∷ word (ἐ ∷ π ∷ ι ∷ γ ∷ ν ∷ ό ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.1.32"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.1.32"
∷ word (ο ∷ ἱ ∷ []) "Rom.1.32"
∷ word (τ ∷ ὰ ∷ []) "Rom.1.32"
∷ word (τ ∷ ο ∷ ι ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "Rom.1.32"
∷ word (π ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.1.32"
∷ word (ἄ ∷ ξ ∷ ι ∷ ο ∷ ι ∷ []) "Rom.1.32"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "Rom.1.32"
∷ word (ε ∷ ἰ ∷ σ ∷ ί ∷ ν ∷ []) "Rom.1.32"
∷ word (ο ∷ ὐ ∷ []) "Rom.1.32"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Rom.1.32"
∷ word (α ∷ ὐ ∷ τ ∷ ὰ ∷ []) "Rom.1.32"
∷ word (π ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.32"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.1.32"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.1.32"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ υ ∷ δ ∷ ο ∷ κ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.32"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.1.32"
∷ word (π ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.1.32"
∷ word (Δ ∷ ι ∷ ὸ ∷ []) "Rom.2.1"
∷ word (ἀ ∷ ν ∷ α ∷ π ∷ ο ∷ ∙λ ∷ ό ∷ γ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "Rom.2.1"
∷ word (ε ∷ ἶ ∷ []) "Rom.2.1"
∷ word (ὦ ∷ []) "Rom.2.1"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ε ∷ []) "Rom.2.1"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "Rom.2.1"
∷ word (ὁ ∷ []) "Rom.2.1"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ω ∷ ν ∷ []) "Rom.2.1"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.1"
∷ word (ᾧ ∷ []) "Rom.2.1"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.2.1"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ ι ∷ ς ∷ []) "Rom.2.1"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.2.1"
∷ word (ἕ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "Rom.2.1"
∷ word (σ ∷ ε ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.2.1"
∷ word (κ ∷ α ∷ τ ∷ α ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ ι ∷ ς ∷ []) "Rom.2.1"
∷ word (τ ∷ ὰ ∷ []) "Rom.2.1"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.2.1"
∷ word (α ∷ ὐ ∷ τ ∷ ὰ ∷ []) "Rom.2.1"
∷ word (π ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.2.1"
∷ word (ὁ ∷ []) "Rom.2.1"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ω ∷ ν ∷ []) "Rom.2.1"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Rom.2.2"
∷ word (δ ∷ ὲ ∷ []) "Rom.2.2"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.2.2"
∷ word (τ ∷ ὸ ∷ []) "Rom.2.2"
∷ word (κ ∷ ρ ∷ ί ∷ μ ∷ α ∷ []) "Rom.2.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.2.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.2.2"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.2.2"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.2.2"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Rom.2.2"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.2.2"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.2.2"
∷ word (τ ∷ ὰ ∷ []) "Rom.2.2"
∷ word (τ ∷ ο ∷ ι ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "Rom.2.2"
∷ word (π ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.2.2"
∷ word (∙λ ∷ ο ∷ γ ∷ ί ∷ ζ ∷ ῃ ∷ []) "Rom.2.3"
∷ word (δ ∷ ὲ ∷ []) "Rom.2.3"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.2.3"
∷ word (ὦ ∷ []) "Rom.2.3"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ε ∷ []) "Rom.2.3"
∷ word (ὁ ∷ []) "Rom.2.3"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ω ∷ ν ∷ []) "Rom.2.3"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.2.3"
∷ word (τ ∷ ὰ ∷ []) "Rom.2.3"
∷ word (τ ∷ ο ∷ ι ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "Rom.2.3"
∷ word (π ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.2.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.3"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ ν ∷ []) "Rom.2.3"
∷ word (α ∷ ὐ ∷ τ ∷ ά ∷ []) "Rom.2.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.2.3"
∷ word (σ ∷ ὺ ∷ []) "Rom.2.3"
∷ word (ἐ ∷ κ ∷ φ ∷ ε ∷ ύ ∷ ξ ∷ ῃ ∷ []) "Rom.2.3"
∷ word (τ ∷ ὸ ∷ []) "Rom.2.3"
∷ word (κ ∷ ρ ∷ ί ∷ μ ∷ α ∷ []) "Rom.2.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.2.3"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.2.3"
∷ word (ἢ ∷ []) "Rom.2.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.2.4"
∷ word (π ∷ ∙λ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "Rom.2.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.2.4"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ό ∷ τ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "Rom.2.4"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.2.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.2.4"
∷ word (ἀ ∷ ν ∷ ο ∷ χ ∷ ῆ ∷ ς ∷ []) "Rom.2.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.2.4"
∷ word (μ ∷ α ∷ κ ∷ ρ ∷ ο ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ς ∷ []) "Rom.2.4"
∷ word (κ ∷ α ∷ τ ∷ α ∷ φ ∷ ρ ∷ ο ∷ ν ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.2.4"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ῶ ∷ ν ∷ []) "Rom.2.4"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.2.4"
∷ word (τ ∷ ὸ ∷ []) "Rom.2.4"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.2.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.2.4"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.2.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.2.4"
∷ word (μ ∷ ε ∷ τ ∷ ά ∷ ν ∷ ο ∷ ι ∷ ά ∷ ν ∷ []) "Rom.2.4"
∷ word (σ ∷ ε ∷ []) "Rom.2.4"
∷ word (ἄ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.2.4"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.2.5"
∷ word (δ ∷ ὲ ∷ []) "Rom.2.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.2.5"
∷ word (σ ∷ κ ∷ ∙λ ∷ η ∷ ρ ∷ ό ∷ τ ∷ η ∷ τ ∷ ά ∷ []) "Rom.2.5"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.2.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.5"
∷ word (ἀ ∷ μ ∷ ε ∷ τ ∷ α ∷ ν ∷ ό ∷ η ∷ τ ∷ ο ∷ ν ∷ []) "Rom.2.5"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ν ∷ []) "Rom.2.5"
∷ word (θ ∷ η ∷ σ ∷ α ∷ υ ∷ ρ ∷ ί ∷ ζ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.2.5"
∷ word (σ ∷ ε ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "Rom.2.5"
∷ word (ὀ ∷ ρ ∷ γ ∷ ὴ ∷ ν ∷ []) "Rom.2.5"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.5"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "Rom.2.5"
∷ word (ὀ ∷ ρ ∷ γ ∷ ῆ ∷ ς ∷ []) "Rom.2.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.5"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ ψ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.2.5"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ κ ∷ ρ ∷ ι ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Rom.2.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.2.5"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.2.5"
∷ word (ὃ ∷ ς ∷ []) "Rom.2.6"
∷ word (ἀ ∷ π ∷ ο ∷ δ ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "Rom.2.6"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ ῳ ∷ []) "Rom.2.6"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.2.6"
∷ word (τ ∷ ὰ ∷ []) "Rom.2.6"
∷ word (ἔ ∷ ρ ∷ γ ∷ α ∷ []) "Rom.2.6"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.2.6"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.2.7"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.2.7"
∷ word (κ ∷ α ∷ θ ∷ []) "Rom.2.7"
∷ word (ὑ ∷ π ∷ ο ∷ μ ∷ ο ∷ ν ∷ ὴ ∷ ν ∷ []) "Rom.2.7"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ υ ∷ []) "Rom.2.7"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ο ∷ ῦ ∷ []) "Rom.2.7"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "Rom.2.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.7"
∷ word (τ ∷ ι ∷ μ ∷ ὴ ∷ ν ∷ []) "Rom.2.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.7"
∷ word (ἀ ∷ φ ∷ θ ∷ α ∷ ρ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Rom.2.7"
∷ word (ζ ∷ η ∷ τ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.2.7"
∷ word (ζ ∷ ω ∷ ὴ ∷ ν ∷ []) "Rom.2.7"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ν ∷ []) "Rom.2.7"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.2.8"
∷ word (δ ∷ ὲ ∷ []) "Rom.2.8"
∷ word (ἐ ∷ ξ ∷ []) "Rom.2.8"
∷ word (ἐ ∷ ρ ∷ ι ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Rom.2.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.8"
∷ word (ἀ ∷ π ∷ ε ∷ ι ∷ θ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ []) "Rom.2.8"
∷ word (τ ∷ ῇ ∷ []) "Rom.2.8"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ ᾳ ∷ []) "Rom.2.8"
∷ word (π ∷ ε ∷ ι ∷ θ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "Rom.2.8"
∷ word (δ ∷ ὲ ∷ []) "Rom.2.8"
∷ word (τ ∷ ῇ ∷ []) "Rom.2.8"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ί ∷ ᾳ ∷ []) "Rom.2.8"
∷ word (ὀ ∷ ρ ∷ γ ∷ ὴ ∷ []) "Rom.2.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.8"
∷ word (θ ∷ υ ∷ μ ∷ ό ∷ ς ∷ []) "Rom.2.8"
∷ word (θ ∷ ∙λ ∷ ῖ ∷ ψ ∷ ι ∷ ς ∷ []) "Rom.2.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.9"
∷ word (σ ∷ τ ∷ ε ∷ ν ∷ ο ∷ χ ∷ ω ∷ ρ ∷ ί ∷ α ∷ []) "Rom.2.9"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.2.9"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "Rom.2.9"
∷ word (ψ ∷ υ ∷ χ ∷ ὴ ∷ ν ∷ []) "Rom.2.9"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "Rom.2.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.2.9"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ α ∷ ζ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ []) "Rom.2.9"
∷ word (τ ∷ ὸ ∷ []) "Rom.2.9"
∷ word (κ ∷ α ∷ κ ∷ ό ∷ ν ∷ []) "Rom.2.9"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ο ∷ υ ∷ []) "Rom.2.9"
∷ word (τ ∷ ε ∷ []) "Rom.2.9"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.2.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.9"
∷ word (Ἕ ∷ ∙λ ∷ ∙λ ∷ η ∷ ν ∷ ο ∷ ς ∷ []) "Rom.2.9"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "Rom.2.10"
∷ word (δ ∷ ὲ ∷ []) "Rom.2.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.10"
∷ word (τ ∷ ι ∷ μ ∷ ὴ ∷ []) "Rom.2.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.10"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "Rom.2.10"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "Rom.2.10"
∷ word (τ ∷ ῷ ∷ []) "Rom.2.10"
∷ word (ἐ ∷ ρ ∷ γ ∷ α ∷ ζ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ῳ ∷ []) "Rom.2.10"
∷ word (τ ∷ ὸ ∷ []) "Rom.2.10"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ό ∷ ν ∷ []) "Rom.2.10"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ῳ ∷ []) "Rom.2.10"
∷ word (τ ∷ ε ∷ []) "Rom.2.10"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.2.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.10"
∷ word (Ἕ ∷ ∙λ ∷ ∙λ ∷ η ∷ ν ∷ ι ∷ []) "Rom.2.10"
∷ word (ο ∷ ὐ ∷ []) "Rom.2.11"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Rom.2.11"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.2.11"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ω ∷ π ∷ ο ∷ ∙λ ∷ η ∷ μ ∷ ψ ∷ ί ∷ α ∷ []) "Rom.2.11"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "Rom.2.11"
∷ word (τ ∷ ῷ ∷ []) "Rom.2.11"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.2.11"
∷ word (Ὅ ∷ σ ∷ ο ∷ ι ∷ []) "Rom.2.12"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.2.12"
∷ word (ἀ ∷ ν ∷ ό ∷ μ ∷ ω ∷ ς ∷ []) "Rom.2.12"
∷ word (ἥ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.2.12"
∷ word (ἀ ∷ ν ∷ ό ∷ μ ∷ ω ∷ ς ∷ []) "Rom.2.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.12"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.2.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.12"
∷ word (ὅ ∷ σ ∷ ο ∷ ι ∷ []) "Rom.2.12"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.12"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.2.12"
∷ word (ἥ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.2.12"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.2.12"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.2.12"
∷ word (κ ∷ ρ ∷ ι ∷ θ ∷ ή ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.2.12"
∷ word (ο ∷ ὐ ∷ []) "Rom.2.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.2.13"
∷ word (ο ∷ ἱ ∷ []) "Rom.2.13"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ α ∷ τ ∷ α ∷ ὶ ∷ []) "Rom.2.13"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.2.13"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ ο ∷ ι ∷ []) "Rom.2.13"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "Rom.2.13"
∷ word (τ ∷ ῷ ∷ []) "Rom.2.13"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.2.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Rom.2.13"
∷ word (ο ∷ ἱ ∷ []) "Rom.2.13"
∷ word (π ∷ ο ∷ ι ∷ η ∷ τ ∷ α ∷ ὶ ∷ []) "Rom.2.13"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.2.13"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ω ∷ θ ∷ ή ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.2.13"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "Rom.2.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.2.14"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Rom.2.14"
∷ word (τ ∷ ὰ ∷ []) "Rom.2.14"
∷ word (μ ∷ ὴ ∷ []) "Rom.2.14"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.2.14"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "Rom.2.14"
∷ word (φ ∷ ύ ∷ σ ∷ ε ∷ ι ∷ []) "Rom.2.14"
∷ word (τ ∷ ὰ ∷ []) "Rom.2.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.2.14"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.2.14"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.2.14"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ι ∷ []) "Rom.2.14"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.2.14"
∷ word (μ ∷ ὴ ∷ []) "Rom.2.14"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.2.14"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.2.14"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.2.14"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Rom.2.14"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "Rom.2.15"
∷ word (ἐ ∷ ν ∷ δ ∷ ε ∷ ί ∷ κ ∷ ν ∷ υ ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.2.15"
∷ word (τ ∷ ὸ ∷ []) "Rom.2.15"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ν ∷ []) "Rom.2.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.2.15"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.2.15"
∷ word (γ ∷ ρ ∷ α ∷ π ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.2.15"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.15"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Rom.2.15"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Rom.2.15"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.2.15"
∷ word (σ ∷ υ ∷ μ ∷ μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ο ∷ ύ ∷ σ ∷ η ∷ ς ∷ []) "Rom.2.15"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.2.15"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.2.15"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ι ∷ δ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.2.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.15"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ ξ ∷ ὺ ∷ []) "Rom.2.15"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ω ∷ ν ∷ []) "Rom.2.15"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.2.15"
∷ word (∙λ ∷ ο ∷ γ ∷ ι ∷ σ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.2.15"
∷ word (κ ∷ α ∷ τ ∷ η ∷ γ ∷ ο ∷ ρ ∷ ο ∷ ύ ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.2.15"
∷ word (ἢ ∷ []) "Rom.2.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.15"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ ω ∷ ν ∷ []) "Rom.2.15"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.16"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "Rom.2.16"
∷ word (ὅ ∷ τ ∷ ε ∷ []) "Rom.2.16"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ ι ∷ []) "Rom.2.16"
∷ word (ὁ ∷ []) "Rom.2.16"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.2.16"
∷ word (τ ∷ ὰ ∷ []) "Rom.2.16"
∷ word (κ ∷ ρ ∷ υ ∷ π ∷ τ ∷ ὰ ∷ []) "Rom.2.16"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.2.16"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "Rom.2.16"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.2.16"
∷ word (τ ∷ ὸ ∷ []) "Rom.2.16"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ό ∷ ν ∷ []) "Rom.2.16"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.2.16"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.2.16"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.2.16"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.2.16"
∷ word (Ε ∷ ἰ ∷ []) "Rom.2.17"
∷ word (δ ∷ ὲ ∷ []) "Rom.2.17"
∷ word (σ ∷ ὺ ∷ []) "Rom.2.17"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ῖ ∷ ο ∷ ς ∷ []) "Rom.2.17"
∷ word (ἐ ∷ π ∷ ο ∷ ν ∷ ο ∷ μ ∷ ά ∷ ζ ∷ ῃ ∷ []) "Rom.2.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.17"
∷ word (ἐ ∷ π ∷ α ∷ ν ∷ α ∷ π ∷ α ∷ ύ ∷ ῃ ∷ []) "Rom.2.17"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.2.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.17"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ᾶ ∷ σ ∷ α ∷ ι ∷ []) "Rom.2.17"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.17"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.2.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.18"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.2.18"
∷ word (τ ∷ ὸ ∷ []) "Rom.2.18"
∷ word (θ ∷ έ ∷ ∙λ ∷ η ∷ μ ∷ α ∷ []) "Rom.2.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.18"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ά ∷ ζ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.2.18"
∷ word (τ ∷ ὰ ∷ []) "Rom.2.18"
∷ word (δ ∷ ι ∷ α ∷ φ ∷ έ ∷ ρ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "Rom.2.18"
∷ word (κ ∷ α ∷ τ ∷ η ∷ χ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Rom.2.18"
∷ word (ἐ ∷ κ ∷ []) "Rom.2.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.2.18"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.2.18"
∷ word (π ∷ έ ∷ π ∷ ο ∷ ι ∷ θ ∷ ά ∷ ς ∷ []) "Rom.2.19"
∷ word (τ ∷ ε ∷ []) "Rom.2.19"
∷ word (σ ∷ ε ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.2.19"
∷ word (ὁ ∷ δ ∷ η ∷ γ ∷ ὸ ∷ ν ∷ []) "Rom.2.19"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.2.19"
∷ word (τ ∷ υ ∷ φ ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "Rom.2.19"
∷ word (φ ∷ ῶ ∷ ς ∷ []) "Rom.2.19"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.2.19"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.19"
∷ word (σ ∷ κ ∷ ό ∷ τ ∷ ε ∷ ι ∷ []) "Rom.2.19"
∷ word (π ∷ α ∷ ι ∷ δ ∷ ε ∷ υ ∷ τ ∷ ὴ ∷ ν ∷ []) "Rom.2.20"
∷ word (ἀ ∷ φ ∷ ρ ∷ ό ∷ ν ∷ ω ∷ ν ∷ []) "Rom.2.20"
∷ word (δ ∷ ι ∷ δ ∷ ά ∷ σ ∷ κ ∷ α ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.2.20"
∷ word (ν ∷ η ∷ π ∷ ί ∷ ω ∷ ν ∷ []) "Rom.2.20"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "Rom.2.20"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.2.20"
∷ word (μ ∷ ό ∷ ρ ∷ φ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Rom.2.20"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.2.20"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.2.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.20"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.2.20"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Rom.2.20"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.20"
∷ word (τ ∷ ῷ ∷ []) "Rom.2.20"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.2.20"
∷ word (ὁ ∷ []) "Rom.2.21"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.2.21"
∷ word (δ ∷ ι ∷ δ ∷ ά ∷ σ ∷ κ ∷ ω ∷ ν ∷ []) "Rom.2.21"
∷ word (ἕ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "Rom.2.21"
∷ word (σ ∷ ε ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.2.21"
∷ word (ο ∷ ὐ ∷ []) "Rom.2.21"
∷ word (δ ∷ ι ∷ δ ∷ ά ∷ σ ∷ κ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.2.21"
∷ word (ὁ ∷ []) "Rom.2.21"
∷ word (κ ∷ η ∷ ρ ∷ ύ ∷ σ ∷ σ ∷ ω ∷ ν ∷ []) "Rom.2.21"
∷ word (μ ∷ ὴ ∷ []) "Rom.2.21"
∷ word (κ ∷ ∙λ ∷ έ ∷ π ∷ τ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.2.21"
∷ word (κ ∷ ∙λ ∷ έ ∷ π ∷ τ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.2.21"
∷ word (ὁ ∷ []) "Rom.2.22"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ ν ∷ []) "Rom.2.22"
∷ word (μ ∷ ὴ ∷ []) "Rom.2.22"
∷ word (μ ∷ ο ∷ ι ∷ χ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.2.22"
∷ word (μ ∷ ο ∷ ι ∷ χ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.2.22"
∷ word (ὁ ∷ []) "Rom.2.22"
∷ word (β ∷ δ ∷ ε ∷ ∙λ ∷ υ ∷ σ ∷ σ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Rom.2.22"
∷ word (τ ∷ ὰ ∷ []) "Rom.2.22"
∷ word (ε ∷ ἴ ∷ δ ∷ ω ∷ ∙λ ∷ α ∷ []) "Rom.2.22"
∷ word (ἱ ∷ ε ∷ ρ ∷ ο ∷ σ ∷ υ ∷ ∙λ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.2.22"
∷ word (ὃ ∷ ς ∷ []) "Rom.2.23"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.23"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.2.23"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ᾶ ∷ σ ∷ α ∷ ι ∷ []) "Rom.2.23"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.2.23"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.2.23"
∷ word (π ∷ α ∷ ρ ∷ α ∷ β ∷ ά ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.2.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.2.23"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.2.23"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.2.23"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "Rom.2.23"
∷ word (ἀ ∷ τ ∷ ι ∷ μ ∷ ά ∷ ζ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.2.23"
∷ word (τ ∷ ὸ ∷ []) "Rom.2.24"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.2.24"
∷ word (ὄ ∷ ν ∷ ο ∷ μ ∷ α ∷ []) "Rom.2.24"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.2.24"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.2.24"
∷ word (δ ∷ ι ∷ []) "Rom.2.24"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.2.24"
∷ word (β ∷ ∙λ ∷ α ∷ σ ∷ φ ∷ η ∷ μ ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "Rom.2.24"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.24"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.2.24"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Rom.2.24"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.2.24"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.2.24"
∷ word (Π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ὴ ∷ []) "Rom.2.25"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.2.25"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.2.25"
∷ word (ὠ ∷ φ ∷ ε ∷ ∙λ ∷ ε ∷ ῖ ∷ []) "Rom.2.25"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.2.25"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.2.25"
∷ word (π ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ῃ ∷ ς ∷ []) "Rom.2.25"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.2.25"
∷ word (δ ∷ ὲ ∷ []) "Rom.2.25"
∷ word (π ∷ α ∷ ρ ∷ α ∷ β ∷ ά ∷ τ ∷ η ∷ ς ∷ []) "Rom.2.25"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.2.25"
∷ word (ᾖ ∷ ς ∷ []) "Rom.2.25"
∷ word (ἡ ∷ []) "Rom.2.25"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ή ∷ []) "Rom.2.25"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.2.25"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ α ∷ []) "Rom.2.25"
∷ word (γ ∷ έ ∷ γ ∷ ο ∷ ν ∷ ε ∷ ν ∷ []) "Rom.2.25"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.2.26"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.2.26"
∷ word (ἡ ∷ []) "Rom.2.26"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ α ∷ []) "Rom.2.26"
∷ word (τ ∷ ὰ ∷ []) "Rom.2.26"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ώ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Rom.2.26"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.2.26"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.2.26"
∷ word (φ ∷ υ ∷ ∙λ ∷ ά ∷ σ ∷ σ ∷ ῃ ∷ []) "Rom.2.26"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "Rom.2.26"
∷ word (ἡ ∷ []) "Rom.2.26"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ α ∷ []) "Rom.2.26"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.2.26"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.2.26"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "Rom.2.26"
∷ word (∙λ ∷ ο ∷ γ ∷ ι ∷ σ ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.2.26"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.27"
∷ word (κ ∷ ρ ∷ ι ∷ ν ∷ ε ∷ ῖ ∷ []) "Rom.2.27"
∷ word (ἡ ∷ []) "Rom.2.27"
∷ word (ἐ ∷ κ ∷ []) "Rom.2.27"
∷ word (φ ∷ ύ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.2.27"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ α ∷ []) "Rom.2.27"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.2.27"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.2.27"
∷ word (τ ∷ ε ∷ ∙λ ∷ ο ∷ ῦ ∷ σ ∷ α ∷ []) "Rom.2.27"
∷ word (σ ∷ ὲ ∷ []) "Rom.2.27"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.2.27"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.2.27"
∷ word (γ ∷ ρ ∷ ά ∷ μ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.2.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.27"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῆ ∷ ς ∷ []) "Rom.2.27"
∷ word (π ∷ α ∷ ρ ∷ α ∷ β ∷ ά ∷ τ ∷ η ∷ ν ∷ []) "Rom.2.27"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.2.27"
∷ word (ο ∷ ὐ ∷ []) "Rom.2.28"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.2.28"
∷ word (ὁ ∷ []) "Rom.2.28"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.28"
∷ word (τ ∷ ῷ ∷ []) "Rom.2.28"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ῷ ∷ []) "Rom.2.28"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ῖ ∷ ό ∷ ς ∷ []) "Rom.2.28"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.2.28"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Rom.2.28"
∷ word (ἡ ∷ []) "Rom.2.28"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.28"
∷ word (τ ∷ ῷ ∷ []) "Rom.2.28"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ῷ ∷ []) "Rom.2.28"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.28"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "Rom.2.28"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ή ∷ []) "Rom.2.28"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Rom.2.29"
∷ word (ὁ ∷ []) "Rom.2.29"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.29"
∷ word (τ ∷ ῷ ∷ []) "Rom.2.29"
∷ word (κ ∷ ρ ∷ υ ∷ π ∷ τ ∷ ῷ ∷ []) "Rom.2.29"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ῖ ∷ ο ∷ ς ∷ []) "Rom.2.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.2.29"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ὴ ∷ []) "Rom.2.29"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "Rom.2.29"
∷ word (ἐ ∷ ν ∷ []) "Rom.2.29"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.2.29"
∷ word (ο ∷ ὐ ∷ []) "Rom.2.29"
∷ word (γ ∷ ρ ∷ ά ∷ μ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.2.29"
∷ word (ο ∷ ὗ ∷ []) "Rom.2.29"
∷ word (ὁ ∷ []) "Rom.2.29"
∷ word (ἔ ∷ π ∷ α ∷ ι ∷ ν ∷ ο ∷ ς ∷ []) "Rom.2.29"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.2.29"
∷ word (ἐ ∷ ξ ∷ []) "Rom.2.29"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "Rom.2.29"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Rom.2.29"
∷ word (ἐ ∷ κ ∷ []) "Rom.2.29"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.2.29"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.2.29"
∷ word (Τ ∷ ί ∷ []) "Rom.3.1"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.3.1"
∷ word (τ ∷ ὸ ∷ []) "Rom.3.1"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ὸ ∷ ν ∷ []) "Rom.3.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.3.1"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ο ∷ υ ∷ []) "Rom.3.1"
∷ word (ἢ ∷ []) "Rom.3.1"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.3.1"
∷ word (ἡ ∷ []) "Rom.3.1"
∷ word (ὠ ∷ φ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ α ∷ []) "Rom.3.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.3.1"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῆ ∷ ς ∷ []) "Rom.3.1"
∷ word (π ∷ ο ∷ ∙λ ∷ ὺ ∷ []) "Rom.3.2"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.3.2"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Rom.3.2"
∷ word (τ ∷ ρ ∷ ό ∷ π ∷ ο ∷ ν ∷ []) "Rom.3.2"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.3.2"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.3.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.3.2"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.3.2"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.3.2"
∷ word (τ ∷ ὰ ∷ []) "Rom.3.2"
∷ word (∙λ ∷ ό ∷ γ ∷ ι ∷ α ∷ []) "Rom.3.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.3.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.3.2"
∷ word (τ ∷ ί ∷ []) "Rom.3.3"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Rom.3.3"
∷ word (ε ∷ ἰ ∷ []) "Rom.3.3"
∷ word (ἠ ∷ π ∷ ί ∷ σ ∷ τ ∷ η ∷ σ ∷ ά ∷ ν ∷ []) "Rom.3.3"
∷ word (τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "Rom.3.3"
∷ word (μ ∷ ὴ ∷ []) "Rom.3.3"
∷ word (ἡ ∷ []) "Rom.3.3"
∷ word (ἀ ∷ π ∷ ι ∷ σ ∷ τ ∷ ί ∷ α ∷ []) "Rom.3.3"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.3.3"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.3.3"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.3.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.3.3"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.3.3"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "Rom.3.3"
∷ word (μ ∷ ὴ ∷ []) "Rom.3.4"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "Rom.3.4"
∷ word (γ ∷ ι ∷ ν ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "Rom.3.4"
∷ word (δ ∷ ὲ ∷ []) "Rom.3.4"
∷ word (ὁ ∷ []) "Rom.3.4"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.3.4"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ή ∷ ς ∷ []) "Rom.3.4"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "Rom.3.4"
∷ word (δ ∷ ὲ ∷ []) "Rom.3.4"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "Rom.3.4"
∷ word (ψ ∷ ε ∷ ύ ∷ σ ∷ τ ∷ η ∷ ς ∷ []) "Rom.3.4"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.3.4"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.3.4"
∷ word (Ὅ ∷ π ∷ ω ∷ ς ∷ []) "Rom.3.4"
∷ word (ἂ ∷ ν ∷ []) "Rom.3.4"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ω ∷ θ ∷ ῇ ∷ ς ∷ []) "Rom.3.4"
∷ word (ἐ ∷ ν ∷ []) "Rom.3.4"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.3.4"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ι ∷ ς ∷ []) "Rom.3.4"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.3.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.4"
∷ word (ν ∷ ι ∷ κ ∷ ή ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.3.4"
∷ word (ἐ ∷ ν ∷ []) "Rom.3.4"
∷ word (τ ∷ ῷ ∷ []) "Rom.3.4"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ α ∷ ί ∷ []) "Rom.3.4"
∷ word (σ ∷ ε ∷ []) "Rom.3.4"
∷ word (ε ∷ ἰ ∷ []) "Rom.3.5"
∷ word (δ ∷ ὲ ∷ []) "Rom.3.5"
∷ word (ἡ ∷ []) "Rom.3.5"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ί ∷ α ∷ []) "Rom.3.5"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.3.5"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.3.5"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.3.5"
∷ word (σ ∷ υ ∷ ν ∷ ί ∷ σ ∷ τ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "Rom.3.5"
∷ word (τ ∷ ί ∷ []) "Rom.3.5"
∷ word (ἐ ∷ ρ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.3.5"
∷ word (μ ∷ ὴ ∷ []) "Rom.3.5"
∷ word (ἄ ∷ δ ∷ ι ∷ κ ∷ ο ∷ ς ∷ []) "Rom.3.5"
∷ word (ὁ ∷ []) "Rom.3.5"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.3.5"
∷ word (ὁ ∷ []) "Rom.3.5"
∷ word (ἐ ∷ π ∷ ι ∷ φ ∷ έ ∷ ρ ∷ ω ∷ ν ∷ []) "Rom.3.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.3.5"
∷ word (ὀ ∷ ρ ∷ γ ∷ ή ∷ ν ∷ []) "Rom.3.5"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.3.5"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Rom.3.5"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Rom.3.5"
∷ word (μ ∷ ὴ ∷ []) "Rom.3.6"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "Rom.3.6"
∷ word (ἐ ∷ π ∷ ε ∷ ὶ ∷ []) "Rom.3.6"
∷ word (π ∷ ῶ ∷ ς ∷ []) "Rom.3.6"
∷ word (κ ∷ ρ ∷ ι ∷ ν ∷ ε ∷ ῖ ∷ []) "Rom.3.6"
∷ word (ὁ ∷ []) "Rom.3.6"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.3.6"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.3.6"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ν ∷ []) "Rom.3.6"
∷ word (ε ∷ ἰ ∷ []) "Rom.3.7"
∷ word (δ ∷ ὲ ∷ []) "Rom.3.7"
∷ word (ἡ ∷ []) "Rom.3.7"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ []) "Rom.3.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.3.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.3.7"
∷ word (ἐ ∷ ν ∷ []) "Rom.3.7"
∷ word (τ ∷ ῷ ∷ []) "Rom.3.7"
∷ word (ἐ ∷ μ ∷ ῷ ∷ []) "Rom.3.7"
∷ word (ψ ∷ ε ∷ ύ ∷ σ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.3.7"
∷ word (ἐ ∷ π ∷ ε ∷ ρ ∷ ί ∷ σ ∷ σ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "Rom.3.7"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.3.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.3.7"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "Rom.3.7"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.3.7"
∷ word (τ ∷ ί ∷ []) "Rom.3.7"
∷ word (ἔ ∷ τ ∷ ι ∷ []) "Rom.3.7"
∷ word (κ ∷ ἀ ∷ γ ∷ ὼ ∷ []) "Rom.3.7"
∷ word (ὡ ∷ ς ∷ []) "Rom.3.7"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ω ∷ ∙λ ∷ ὸ ∷ ς ∷ []) "Rom.3.7"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.3.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.8"
∷ word (μ ∷ ὴ ∷ []) "Rom.3.8"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.3.8"
∷ word (β ∷ ∙λ ∷ α ∷ σ ∷ φ ∷ η ∷ μ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.3.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.8"
∷ word (κ ∷ α ∷ θ ∷ ώ ∷ ς ∷ []) "Rom.3.8"
∷ word (φ ∷ α ∷ σ ∷ ί ∷ ν ∷ []) "Rom.3.8"
∷ word (τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "Rom.3.8"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.3.8"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.3.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.3.8"
∷ word (Π ∷ ο ∷ ι ∷ ή ∷ σ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Rom.3.8"
∷ word (τ ∷ ὰ ∷ []) "Rom.3.8"
∷ word (κ ∷ α ∷ κ ∷ ὰ ∷ []) "Rom.3.8"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.3.8"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ῃ ∷ []) "Rom.3.8"
∷ word (τ ∷ ὰ ∷ []) "Rom.3.8"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ά ∷ []) "Rom.3.8"
∷ word (ὧ ∷ ν ∷ []) "Rom.3.8"
∷ word (τ ∷ ὸ ∷ []) "Rom.3.8"
∷ word (κ ∷ ρ ∷ ί ∷ μ ∷ α ∷ []) "Rom.3.8"
∷ word (ἔ ∷ ν ∷ δ ∷ ι ∷ κ ∷ ό ∷ ν ∷ []) "Rom.3.8"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.3.8"
∷ word (Τ ∷ ί ∷ []) "Rom.3.9"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.3.9"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ χ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.3.9"
∷ word (ο ∷ ὐ ∷ []) "Rom.3.9"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ς ∷ []) "Rom.3.9"
∷ word (π ∷ ρ ∷ ο ∷ ῃ ∷ τ ∷ ι ∷ α ∷ σ ∷ ά ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.3.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.3.9"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "Rom.3.9"
∷ word (τ ∷ ε ∷ []) "Rom.3.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.9"
∷ word (Ἕ ∷ ∙λ ∷ ∙λ ∷ η ∷ ν ∷ α ∷ ς ∷ []) "Rom.3.9"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.3.9"
∷ word (ὑ ∷ φ ∷ []) "Rom.3.9"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "Rom.3.9"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.3.9"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.3.10"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.3.10"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.3.10"
∷ word (Ο ∷ ὐ ∷ κ ∷ []) "Rom.3.10"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.3.10"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ ο ∷ ς ∷ []) "Rom.3.10"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Rom.3.10"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "Rom.3.10"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.3.11"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.3.11"
∷ word (ὁ ∷ []) "Rom.3.11"
∷ word (σ ∷ υ ∷ ν ∷ ί ∷ ω ∷ ν ∷ []) "Rom.3.11"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.3.11"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.3.11"
∷ word (ὁ ∷ []) "Rom.3.11"
∷ word (ἐ ∷ κ ∷ ζ ∷ η ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.3.11"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.3.11"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "Rom.3.11"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.3.12"
∷ word (ἐ ∷ ξ ∷ έ ∷ κ ∷ ∙λ ∷ ι ∷ ν ∷ α ∷ ν ∷ []) "Rom.3.12"
∷ word (ἅ ∷ μ ∷ α ∷ []) "Rom.3.12"
∷ word (ἠ ∷ χ ∷ ρ ∷ ε ∷ ώ ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.3.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.3.12"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.3.12"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ ν ∷ []) "Rom.3.12"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ό ∷ τ ∷ η ∷ τ ∷ α ∷ []) "Rom.3.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.3.12"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.3.12"
∷ word (ἕ ∷ ω ∷ ς ∷ []) "Rom.3.12"
∷ word (ἑ ∷ ν ∷ ό ∷ ς ∷ []) "Rom.3.12"
∷ word (τ ∷ ά ∷ φ ∷ ο ∷ ς ∷ []) "Rom.3.13"
∷ word (ἀ ∷ ν ∷ ε ∷ ῳ ∷ γ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "Rom.3.13"
∷ word (ὁ ∷ []) "Rom.3.13"
∷ word (∙λ ∷ ά ∷ ρ ∷ υ ∷ γ ∷ ξ ∷ []) "Rom.3.13"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.3.13"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Rom.3.13"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ α ∷ ι ∷ ς ∷ []) "Rom.3.13"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.3.13"
∷ word (ἐ ∷ δ ∷ ο ∷ ∙λ ∷ ι ∷ ο ∷ ῦ ∷ σ ∷ α ∷ ν ∷ []) "Rom.3.13"
∷ word (ἰ ∷ ὸ ∷ ς ∷ []) "Rom.3.13"
∷ word (ἀ ∷ σ ∷ π ∷ ί ∷ δ ∷ ω ∷ ν ∷ []) "Rom.3.13"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Rom.3.13"
∷ word (τ ∷ ὰ ∷ []) "Rom.3.13"
∷ word (χ ∷ ε ∷ ί ∷ ∙λ ∷ η ∷ []) "Rom.3.13"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.3.13"
∷ word (ὧ ∷ ν ∷ []) "Rom.3.14"
∷ word (τ ∷ ὸ ∷ []) "Rom.3.14"
∷ word (σ ∷ τ ∷ ό ∷ μ ∷ α ∷ []) "Rom.3.14"
∷ word (ἀ ∷ ρ ∷ ᾶ ∷ ς ∷ []) "Rom.3.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.14"
∷ word (π ∷ ι ∷ κ ∷ ρ ∷ ί ∷ α ∷ ς ∷ []) "Rom.3.14"
∷ word (γ ∷ έ ∷ μ ∷ ε ∷ ι ∷ []) "Rom.3.14"
∷ word (ὀ ∷ ξ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.3.15"
∷ word (ο ∷ ἱ ∷ []) "Rom.3.15"
∷ word (π ∷ ό ∷ δ ∷ ε ∷ ς ∷ []) "Rom.3.15"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.3.15"
∷ word (ἐ ∷ κ ∷ χ ∷ έ ∷ α ∷ ι ∷ []) "Rom.3.15"
∷ word (α ∷ ἷ ∷ μ ∷ α ∷ []) "Rom.3.15"
∷ word (σ ∷ ύ ∷ ν ∷ τ ∷ ρ ∷ ι ∷ μ ∷ μ ∷ α ∷ []) "Rom.3.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.16"
∷ word (τ ∷ α ∷ ∙λ ∷ α ∷ ι ∷ π ∷ ω ∷ ρ ∷ ί ∷ α ∷ []) "Rom.3.16"
∷ word (ἐ ∷ ν ∷ []) "Rom.3.16"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Rom.3.16"
∷ word (ὁ ∷ δ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.3.16"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.3.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.17"
∷ word (ὁ ∷ δ ∷ ὸ ∷ ν ∷ []) "Rom.3.17"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ ς ∷ []) "Rom.3.17"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.3.17"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "Rom.3.17"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.3.18"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.3.18"
∷ word (φ ∷ ό ∷ β ∷ ο ∷ ς ∷ []) "Rom.3.18"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.3.18"
∷ word (ἀ ∷ π ∷ έ ∷ ν ∷ α ∷ ν ∷ τ ∷ ι ∷ []) "Rom.3.18"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.3.18"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.3.18"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.3.18"
∷ word (Ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Rom.3.19"
∷ word (δ ∷ ὲ ∷ []) "Rom.3.19"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.3.19"
∷ word (ὅ ∷ σ ∷ α ∷ []) "Rom.3.19"
∷ word (ὁ ∷ []) "Rom.3.19"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Rom.3.19"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.3.19"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.3.19"
∷ word (ἐ ∷ ν ∷ []) "Rom.3.19"
∷ word (τ ∷ ῷ ∷ []) "Rom.3.19"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.3.19"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ []) "Rom.3.19"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.3.19"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "Rom.3.19"
∷ word (σ ∷ τ ∷ ό ∷ μ ∷ α ∷ []) "Rom.3.19"
∷ word (φ ∷ ρ ∷ α ∷ γ ∷ ῇ ∷ []) "Rom.3.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.19"
∷ word (ὑ ∷ π ∷ ό ∷ δ ∷ ι ∷ κ ∷ ο ∷ ς ∷ []) "Rom.3.19"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Rom.3.19"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "Rom.3.19"
∷ word (ὁ ∷ []) "Rom.3.19"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ς ∷ []) "Rom.3.19"
∷ word (τ ∷ ῷ ∷ []) "Rom.3.19"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.3.19"
∷ word (δ ∷ ι ∷ ό ∷ τ ∷ ι ∷ []) "Rom.3.20"
∷ word (ἐ ∷ ξ ∷ []) "Rom.3.20"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Rom.3.20"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.3.20"
∷ word (ο ∷ ὐ ∷ []) "Rom.3.20"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ω ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.3.20"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ []) "Rom.3.20"
∷ word (σ ∷ ὰ ∷ ρ ∷ ξ ∷ []) "Rom.3.20"
∷ word (ἐ ∷ ν ∷ ώ ∷ π ∷ ι ∷ ο ∷ ν ∷ []) "Rom.3.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.3.20"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.3.20"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.3.20"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.3.20"
∷ word (ἐ ∷ π ∷ ί ∷ γ ∷ ν ∷ ω ∷ σ ∷ ι ∷ ς ∷ []) "Rom.3.20"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.3.20"
∷ word (Ν ∷ υ ∷ ν ∷ ὶ ∷ []) "Rom.3.21"
∷ word (δ ∷ ὲ ∷ []) "Rom.3.21"
∷ word (χ ∷ ω ∷ ρ ∷ ὶ ∷ ς ∷ []) "Rom.3.21"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.3.21"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ []) "Rom.3.21"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.3.21"
∷ word (π ∷ ε ∷ φ ∷ α ∷ ν ∷ έ ∷ ρ ∷ ω ∷ τ ∷ α ∷ ι ∷ []) "Rom.3.21"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "Rom.3.21"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Rom.3.21"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.3.21"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.3.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.21"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.3.21"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.3.21"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ []) "Rom.3.22"
∷ word (δ ∷ ὲ ∷ []) "Rom.3.22"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.3.22"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.3.22"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.3.22"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.3.22"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.3.22"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.3.22"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.3.22"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.3.22"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.3.22"
∷ word (ο ∷ ὐ ∷ []) "Rom.3.22"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Rom.3.22"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.3.22"
∷ word (δ ∷ ι ∷ α ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ή ∷ []) "Rom.3.22"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.3.23"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.3.23"
∷ word (ἥ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.3.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.23"
∷ word (ὑ ∷ σ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.3.23"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.3.23"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Rom.3.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.3.23"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.3.23"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Rom.3.24"
∷ word (δ ∷ ω ∷ ρ ∷ ε ∷ ὰ ∷ ν ∷ []) "Rom.3.24"
∷ word (τ ∷ ῇ ∷ []) "Rom.3.24"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.3.24"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ι ∷ []) "Rom.3.24"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.3.24"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.3.24"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ υ ∷ τ ∷ ρ ∷ ώ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.3.24"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.3.24"
∷ word (ἐ ∷ ν ∷ []) "Rom.3.24"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.3.24"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.3.24"
∷ word (ὃ ∷ ν ∷ []) "Rom.3.25"
∷ word (π ∷ ρ ∷ ο ∷ έ ∷ θ ∷ ε ∷ τ ∷ ο ∷ []) "Rom.3.25"
∷ word (ὁ ∷ []) "Rom.3.25"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.3.25"
∷ word (ἱ ∷ ∙λ ∷ α ∷ σ ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.3.25"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.3.25"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.3.25"
∷ word (ἐ ∷ ν ∷ []) "Rom.3.25"
∷ word (τ ∷ ῷ ∷ []) "Rom.3.25"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.3.25"
∷ word (α ∷ ἵ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.3.25"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.3.25"
∷ word (ἔ ∷ ν ∷ δ ∷ ε ∷ ι ∷ ξ ∷ ι ∷ ν ∷ []) "Rom.3.25"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.3.25"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "Rom.3.25"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.3.25"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.3.25"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.3.25"
∷ word (π ∷ ά ∷ ρ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Rom.3.25"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.3.25"
∷ word (π ∷ ρ ∷ ο ∷ γ ∷ ε ∷ γ ∷ ο ∷ ν ∷ ό ∷ τ ∷ ω ∷ ν ∷ []) "Rom.3.25"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ η ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "Rom.3.25"
∷ word (ἐ ∷ ν ∷ []) "Rom.3.26"
∷ word (τ ∷ ῇ ∷ []) "Rom.3.26"
∷ word (ἀ ∷ ν ∷ ο ∷ χ ∷ ῇ ∷ []) "Rom.3.26"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.3.26"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.3.26"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.3.26"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.3.26"
∷ word (ἔ ∷ ν ∷ δ ∷ ε ∷ ι ∷ ξ ∷ ι ∷ ν ∷ []) "Rom.3.26"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.3.26"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "Rom.3.26"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.3.26"
∷ word (ἐ ∷ ν ∷ []) "Rom.3.26"
∷ word (τ ∷ ῷ ∷ []) "Rom.3.26"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Rom.3.26"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ῷ ∷ []) "Rom.3.26"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.3.26"
∷ word (τ ∷ ὸ ∷ []) "Rom.3.26"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.3.26"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.3.26"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ ο ∷ ν ∷ []) "Rom.3.26"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.26"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ α ∷ []) "Rom.3.26"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.3.26"
∷ word (ἐ ∷ κ ∷ []) "Rom.3.26"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.3.26"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.3.26"
∷ word (Π ∷ ο ∷ ῦ ∷ []) "Rom.3.27"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.3.27"
∷ word (ἡ ∷ []) "Rom.3.27"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ σ ∷ ι ∷ ς ∷ []) "Rom.3.27"
∷ word (ἐ ∷ ξ ∷ ε ∷ κ ∷ ∙λ ∷ ε ∷ ί ∷ σ ∷ θ ∷ η ∷ []) "Rom.3.27"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.3.27"
∷ word (π ∷ ο ∷ ί ∷ ο ∷ υ ∷ []) "Rom.3.27"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.3.27"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.3.27"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Rom.3.27"
∷ word (ο ∷ ὐ ∷ χ ∷ ί ∷ []) "Rom.3.27"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.3.27"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.3.27"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.3.27"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.3.27"
∷ word (∙λ ∷ ο ∷ γ ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.3.28"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.3.28"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ ῦ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.3.28"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ι ∷ []) "Rom.3.28"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Rom.3.28"
∷ word (χ ∷ ω ∷ ρ ∷ ὶ ∷ ς ∷ []) "Rom.3.28"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Rom.3.28"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.3.28"
∷ word (ἢ ∷ []) "Rom.3.29"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ω ∷ ν ∷ []) "Rom.3.29"
∷ word (ὁ ∷ []) "Rom.3.29"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.3.29"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Rom.3.29"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "Rom.3.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.29"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.3.29"
∷ word (ν ∷ α ∷ ὶ ∷ []) "Rom.3.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.29"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.3.29"
∷ word (ε ∷ ἴ ∷ π ∷ ε ∷ ρ ∷ []) "Rom.3.30"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "Rom.3.30"
∷ word (ὁ ∷ []) "Rom.3.30"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "Rom.3.30"
∷ word (ὃ ∷ ς ∷ []) "Rom.3.30"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "Rom.3.30"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "Rom.3.30"
∷ word (ἐ ∷ κ ∷ []) "Rom.3.30"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.3.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.3.30"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "Rom.3.30"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.3.30"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.3.30"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.3.30"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.3.31"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.3.31"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.3.31"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.3.31"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.3.31"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.3.31"
∷ word (μ ∷ ὴ ∷ []) "Rom.3.31"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "Rom.3.31"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.3.31"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.3.31"
∷ word (ἱ ∷ σ ∷ τ ∷ ά ∷ ν ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.3.31"
∷ word (Τ ∷ ί ∷ []) "Rom.4.1"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.4.1"
∷ word (ἐ ∷ ρ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.4.1"
∷ word (ε ∷ ὑ ∷ ρ ∷ η ∷ κ ∷ έ ∷ ν ∷ α ∷ ι ∷ []) "Rom.4.1"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ὰ ∷ μ ∷ []) "Rom.4.1"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.4.1"
∷ word (π ∷ ρ ∷ ο ∷ π ∷ ά ∷ τ ∷ ο ∷ ρ ∷ α ∷ []) "Rom.4.1"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.4.1"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.4.1"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Rom.4.1"
∷ word (ε ∷ ἰ ∷ []) "Rom.4.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.4.2"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ὰ ∷ μ ∷ []) "Rom.4.2"
∷ word (ἐ ∷ ξ ∷ []) "Rom.4.2"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Rom.4.2"
∷ word (ἐ ∷ δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ώ ∷ θ ∷ η ∷ []) "Rom.4.2"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "Rom.4.2"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ μ ∷ α ∷ []) "Rom.4.2"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Rom.4.2"
∷ word (ο ∷ ὐ ∷ []) "Rom.4.2"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.4.2"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "Rom.4.2"
∷ word (τ ∷ ί ∷ []) "Rom.4.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.4.3"
∷ word (ἡ ∷ []) "Rom.4.3"
∷ word (γ ∷ ρ ∷ α ∷ φ ∷ ὴ ∷ []) "Rom.4.3"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.4.3"
∷ word (Ἐ ∷ π ∷ ί ∷ σ ∷ τ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "Rom.4.3"
∷ word (δ ∷ ὲ ∷ []) "Rom.4.3"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ὰ ∷ μ ∷ []) "Rom.4.3"
∷ word (τ ∷ ῷ ∷ []) "Rom.4.3"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.4.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.3"
∷ word (ἐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ σ ∷ θ ∷ η ∷ []) "Rom.4.3"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.4.3"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.4.3"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.4.3"
∷ word (τ ∷ ῷ ∷ []) "Rom.4.4"
∷ word (δ ∷ ὲ ∷ []) "Rom.4.4"
∷ word (ἐ ∷ ρ ∷ γ ∷ α ∷ ζ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ῳ ∷ []) "Rom.4.4"
∷ word (ὁ ∷ []) "Rom.4.4"
∷ word (μ ∷ ι ∷ σ ∷ θ ∷ ὸ ∷ ς ∷ []) "Rom.4.4"
∷ word (ο ∷ ὐ ∷ []) "Rom.4.4"
∷ word (∙λ ∷ ο ∷ γ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.4.4"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.4.4"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Rom.4.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.4.4"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.4.4"
∷ word (ὀ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ η ∷ μ ∷ α ∷ []) "Rom.4.4"
∷ word (τ ∷ ῷ ∷ []) "Rom.4.5"
∷ word (δ ∷ ὲ ∷ []) "Rom.4.5"
∷ word (μ ∷ ὴ ∷ []) "Rom.4.5"
∷ word (ἐ ∷ ρ ∷ γ ∷ α ∷ ζ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ῳ ∷ []) "Rom.4.5"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ ι ∷ []) "Rom.4.5"
∷ word (δ ∷ ὲ ∷ []) "Rom.4.5"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.4.5"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.4.5"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ α ∷ []) "Rom.4.5"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.4.5"
∷ word (ἀ ∷ σ ∷ ε ∷ β ∷ ῆ ∷ []) "Rom.4.5"
∷ word (∙λ ∷ ο ∷ γ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.4.5"
∷ word (ἡ ∷ []) "Rom.4.5"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "Rom.4.5"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.4.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.4.5"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.4.5"
∷ word (κ ∷ α ∷ θ ∷ ά ∷ π ∷ ε ∷ ρ ∷ []) "Rom.4.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.6"
∷ word (Δ ∷ α ∷ υ ∷ ὶ ∷ δ ∷ []) "Rom.4.6"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.4.6"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.4.6"
∷ word (μ ∷ α ∷ κ ∷ α ∷ ρ ∷ ι ∷ σ ∷ μ ∷ ὸ ∷ ν ∷ []) "Rom.4.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.4.6"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "Rom.4.6"
∷ word (ᾧ ∷ []) "Rom.4.6"
∷ word (ὁ ∷ []) "Rom.4.6"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.4.6"
∷ word (∙λ ∷ ο ∷ γ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.4.6"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.4.6"
∷ word (χ ∷ ω ∷ ρ ∷ ὶ ∷ ς ∷ []) "Rom.4.6"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Rom.4.6"
∷ word (Μ ∷ α ∷ κ ∷ ά ∷ ρ ∷ ι ∷ ο ∷ ι ∷ []) "Rom.4.7"
∷ word (ὧ ∷ ν ∷ []) "Rom.4.7"
∷ word (ἀ ∷ φ ∷ έ ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.4.7"
∷ word (α ∷ ἱ ∷ []) "Rom.4.7"
∷ word (ἀ ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ ι ∷ []) "Rom.4.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.7"
∷ word (ὧ ∷ ν ∷ []) "Rom.4.7"
∷ word (ἐ ∷ π ∷ ε ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ φ ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.4.7"
∷ word (α ∷ ἱ ∷ []) "Rom.4.7"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ι ∷ []) "Rom.4.7"
∷ word (μ ∷ α ∷ κ ∷ ά ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "Rom.4.8"
∷ word (ἀ ∷ ν ∷ ὴ ∷ ρ ∷ []) "Rom.4.8"
∷ word (ο ∷ ὗ ∷ []) "Rom.4.8"
∷ word (ο ∷ ὐ ∷ []) "Rom.4.8"
∷ word (μ ∷ ὴ ∷ []) "Rom.4.8"
∷ word (∙λ ∷ ο ∷ γ ∷ ί ∷ σ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Rom.4.8"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "Rom.4.8"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "Rom.4.8"
∷ word (Ὁ ∷ []) "Rom.4.9"
∷ word (μ ∷ α ∷ κ ∷ α ∷ ρ ∷ ι ∷ σ ∷ μ ∷ ὸ ∷ ς ∷ []) "Rom.4.9"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.4.9"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.4.9"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.4.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.4.9"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "Rom.4.9"
∷ word (ἢ ∷ []) "Rom.4.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.9"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.4.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.4.9"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "Rom.4.9"
∷ word (∙λ ∷ έ ∷ γ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.4.9"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Rom.4.9"
∷ word (Ἐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ σ ∷ θ ∷ η ∷ []) "Rom.4.9"
∷ word (τ ∷ ῷ ∷ []) "Rom.4.9"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ὰ ∷ μ ∷ []) "Rom.4.9"
∷ word (ἡ ∷ []) "Rom.4.9"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "Rom.4.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.4.9"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.4.9"
∷ word (π ∷ ῶ ∷ ς ∷ []) "Rom.4.10"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.4.10"
∷ word (ἐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ σ ∷ θ ∷ η ∷ []) "Rom.4.10"
∷ word (ἐ ∷ ν ∷ []) "Rom.4.10"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῇ ∷ []) "Rom.4.10"
∷ word (ὄ ∷ ν ∷ τ ∷ ι ∷ []) "Rom.4.10"
∷ word (ἢ ∷ []) "Rom.4.10"
∷ word (ἐ ∷ ν ∷ []) "Rom.4.10"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Rom.4.10"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.4.10"
∷ word (ἐ ∷ ν ∷ []) "Rom.4.10"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῇ ∷ []) "Rom.4.10"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Rom.4.10"
∷ word (ἐ ∷ ν ∷ []) "Rom.4.10"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Rom.4.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.11"
∷ word (σ ∷ η ∷ μ ∷ ε ∷ ῖ ∷ ο ∷ ν ∷ []) "Rom.4.11"
∷ word (ἔ ∷ ∙λ ∷ α ∷ β ∷ ε ∷ ν ∷ []) "Rom.4.11"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῆ ∷ ς ∷ []) "Rom.4.11"
∷ word (σ ∷ φ ∷ ρ ∷ α ∷ γ ∷ ῖ ∷ δ ∷ α ∷ []) "Rom.4.11"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.4.11"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "Rom.4.11"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.4.11"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.4.11"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.4.11"
∷ word (ἐ ∷ ν ∷ []) "Rom.4.11"
∷ word (τ ∷ ῇ ∷ []) "Rom.4.11"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Rom.4.11"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.4.11"
∷ word (τ ∷ ὸ ∷ []) "Rom.4.11"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.4.11"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.4.11"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Rom.4.11"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.4.11"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.4.11"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ υ ∷ ό ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.4.11"
∷ word (δ ∷ ι ∷ []) "Rom.4.11"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.4.11"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.4.11"
∷ word (τ ∷ ὸ ∷ []) "Rom.4.11"
∷ word (∙λ ∷ ο ∷ γ ∷ ι ∷ σ ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "Rom.4.11"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.4.11"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.4.11"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.4.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.12"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Rom.4.12"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῆ ∷ ς ∷ []) "Rom.4.12"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.4.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.4.12"
∷ word (ἐ ∷ κ ∷ []) "Rom.4.12"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῆ ∷ ς ∷ []) "Rom.4.12"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Rom.4.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.4.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.12"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.4.12"
∷ word (σ ∷ τ ∷ ο ∷ ι ∷ χ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.4.12"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.4.12"
∷ word (ἴ ∷ χ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Rom.4.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.4.12"
∷ word (ἐ ∷ ν ∷ []) "Rom.4.12"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Rom.4.12"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.4.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.4.12"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.4.12"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.4.12"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ά ∷ μ ∷ []) "Rom.4.12"
∷ word (Ο ∷ ὐ ∷ []) "Rom.4.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.4.13"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.4.13"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.4.13"
∷ word (ἡ ∷ []) "Rom.4.13"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ []) "Rom.4.13"
∷ word (τ ∷ ῷ ∷ []) "Rom.4.13"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ὰ ∷ μ ∷ []) "Rom.4.13"
∷ word (ἢ ∷ []) "Rom.4.13"
∷ word (τ ∷ ῷ ∷ []) "Rom.4.13"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.4.13"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.4.13"
∷ word (τ ∷ ὸ ∷ []) "Rom.4.13"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.4.13"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.4.13"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.4.13"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "Rom.4.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.4.13"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.4.13"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "Rom.4.13"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.4.13"
∷ word (ε ∷ ἰ ∷ []) "Rom.4.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.4.14"
∷ word (ο ∷ ἱ ∷ []) "Rom.4.14"
∷ word (ἐ ∷ κ ∷ []) "Rom.4.14"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.4.14"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ό ∷ μ ∷ ο ∷ ι ∷ []) "Rom.4.14"
∷ word (κ ∷ ε ∷ κ ∷ έ ∷ ν ∷ ω ∷ τ ∷ α ∷ ι ∷ []) "Rom.4.14"
∷ word (ἡ ∷ []) "Rom.4.14"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "Rom.4.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.14"
∷ word (κ ∷ α ∷ τ ∷ ή ∷ ρ ∷ γ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Rom.4.14"
∷ word (ἡ ∷ []) "Rom.4.14"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ []) "Rom.4.14"
∷ word (ὁ ∷ []) "Rom.4.15"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.4.15"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Rom.4.15"
∷ word (ὀ ∷ ρ ∷ γ ∷ ὴ ∷ ν ∷ []) "Rom.4.15"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.4.15"
∷ word (ο ∷ ὗ ∷ []) "Rom.4.15"
∷ word (δ ∷ ὲ ∷ []) "Rom.4.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.4.15"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.4.15"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Rom.4.15"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Rom.4.15"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ β ∷ α ∷ σ ∷ ι ∷ ς ∷ []) "Rom.4.15"
∷ word (Δ ∷ ι ∷ ὰ ∷ []) "Rom.4.16"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.4.16"
∷ word (ἐ ∷ κ ∷ []) "Rom.4.16"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.4.16"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.4.16"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.4.16"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Rom.4.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.4.16"
∷ word (τ ∷ ὸ ∷ []) "Rom.4.16"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.4.16"
∷ word (β ∷ ε ∷ β ∷ α ∷ ί ∷ α ∷ ν ∷ []) "Rom.4.16"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.4.16"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ν ∷ []) "Rom.4.16"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "Rom.4.16"
∷ word (τ ∷ ῷ ∷ []) "Rom.4.16"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.4.16"
∷ word (ο ∷ ὐ ∷ []) "Rom.4.16"
∷ word (τ ∷ ῷ ∷ []) "Rom.4.16"
∷ word (ἐ ∷ κ ∷ []) "Rom.4.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.4.16"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.4.16"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Rom.4.16"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.4.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.16"
∷ word (τ ∷ ῷ ∷ []) "Rom.4.16"
∷ word (ἐ ∷ κ ∷ []) "Rom.4.16"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.4.16"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ά ∷ μ ∷ []) "Rom.4.16"
∷ word (ὅ ∷ ς ∷ []) "Rom.4.16"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.4.16"
∷ word (π ∷ α ∷ τ ∷ ὴ ∷ ρ ∷ []) "Rom.4.16"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.4.16"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.4.16"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.4.17"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.4.17"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.4.17"
∷ word (Π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Rom.4.17"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "Rom.4.17"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.4.17"
∷ word (τ ∷ έ ∷ θ ∷ ε ∷ ι ∷ κ ∷ ά ∷ []) "Rom.4.17"
∷ word (σ ∷ ε ∷ []) "Rom.4.17"
∷ word (κ ∷ α ∷ τ ∷ έ ∷ ν ∷ α ∷ ν ∷ τ ∷ ι ∷ []) "Rom.4.17"
∷ word (ο ∷ ὗ ∷ []) "Rom.4.17"
∷ word (ἐ ∷ π ∷ ί ∷ σ ∷ τ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "Rom.4.17"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.4.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.4.17"
∷ word (ζ ∷ ῳ ∷ ο ∷ π ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.4.17"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.4.17"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.4.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.17"
∷ word (κ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.4.17"
∷ word (τ ∷ ὰ ∷ []) "Rom.4.17"
∷ word (μ ∷ ὴ ∷ []) "Rom.4.17"
∷ word (ὄ ∷ ν ∷ τ ∷ α ∷ []) "Rom.4.17"
∷ word (ὡ ∷ ς ∷ []) "Rom.4.17"
∷ word (ὄ ∷ ν ∷ τ ∷ α ∷ []) "Rom.4.17"
∷ word (ὃ ∷ ς ∷ []) "Rom.4.18"
∷ word (π ∷ α ∷ ρ ∷ []) "Rom.4.18"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ α ∷ []) "Rom.4.18"
∷ word (ἐ ∷ π ∷ []) "Rom.4.18"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ ι ∷ []) "Rom.4.18"
∷ word (ἐ ∷ π ∷ ί ∷ σ ∷ τ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "Rom.4.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.4.18"
∷ word (τ ∷ ὸ ∷ []) "Rom.4.18"
∷ word (γ ∷ ε ∷ ν ∷ έ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.4.18"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.4.18"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Rom.4.18"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "Rom.4.18"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.4.18"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.4.18"
∷ word (τ ∷ ὸ ∷ []) "Rom.4.18"
∷ word (ε ∷ ἰ ∷ ρ ∷ η ∷ μ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "Rom.4.18"
∷ word (Ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.4.18"
∷ word (ἔ ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "Rom.4.18"
∷ word (τ ∷ ὸ ∷ []) "Rom.4.18"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ []) "Rom.4.18"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.4.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.19"
∷ word (μ ∷ ὴ ∷ []) "Rom.4.19"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ή ∷ σ ∷ α ∷ ς ∷ []) "Rom.4.19"
∷ word (τ ∷ ῇ ∷ []) "Rom.4.19"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ι ∷ []) "Rom.4.19"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ν ∷ ό ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Rom.4.19"
∷ word (τ ∷ ὸ ∷ []) "Rom.4.19"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.4.19"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "Rom.4.19"
∷ word (ν ∷ ε ∷ ν ∷ ε ∷ κ ∷ ρ ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "Rom.4.19"
∷ word (ἑ ∷ κ ∷ α ∷ τ ∷ ο ∷ ν ∷ τ ∷ α ∷ ε ∷ τ ∷ ή ∷ ς ∷ []) "Rom.4.19"
∷ word (π ∷ ο ∷ υ ∷ []) "Rom.4.19"
∷ word (ὑ ∷ π ∷ ά ∷ ρ ∷ χ ∷ ω ∷ ν ∷ []) "Rom.4.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.19"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.4.19"
∷ word (ν ∷ έ ∷ κ ∷ ρ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Rom.4.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.4.19"
∷ word (μ ∷ ή ∷ τ ∷ ρ ∷ α ∷ ς ∷ []) "Rom.4.19"
∷ word (Σ ∷ ά ∷ ρ ∷ ρ ∷ α ∷ ς ∷ []) "Rom.4.19"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.4.20"
∷ word (δ ∷ ὲ ∷ []) "Rom.4.20"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.4.20"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ν ∷ []) "Rom.4.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.4.20"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.4.20"
∷ word (ο ∷ ὐ ∷ []) "Rom.4.20"
∷ word (δ ∷ ι ∷ ε ∷ κ ∷ ρ ∷ ί ∷ θ ∷ η ∷ []) "Rom.4.20"
∷ word (τ ∷ ῇ ∷ []) "Rom.4.20"
∷ word (ἀ ∷ π ∷ ι ∷ σ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Rom.4.20"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.4.20"
∷ word (ἐ ∷ ν ∷ ε ∷ δ ∷ υ ∷ ν ∷ α ∷ μ ∷ ώ ∷ θ ∷ η ∷ []) "Rom.4.20"
∷ word (τ ∷ ῇ ∷ []) "Rom.4.20"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ι ∷ []) "Rom.4.20"
∷ word (δ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.4.20"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "Rom.4.20"
∷ word (τ ∷ ῷ ∷ []) "Rom.4.20"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.4.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.21"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ φ ∷ ο ∷ ρ ∷ η ∷ θ ∷ ε ∷ ὶ ∷ ς ∷ []) "Rom.4.21"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.4.21"
∷ word (ὃ ∷ []) "Rom.4.21"
∷ word (ἐ ∷ π ∷ ή ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ τ ∷ α ∷ ι ∷ []) "Rom.4.21"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ τ ∷ ό ∷ ς ∷ []) "Rom.4.21"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.4.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.21"
∷ word (π ∷ ο ∷ ι ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Rom.4.21"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "Rom.4.22"
∷ word (ἐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ σ ∷ θ ∷ η ∷ []) "Rom.4.22"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.4.22"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.4.22"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.4.22"
∷ word (Ο ∷ ὐ ∷ κ ∷ []) "Rom.4.23"
∷ word (ἐ ∷ γ ∷ ρ ∷ ά ∷ φ ∷ η ∷ []) "Rom.4.23"
∷ word (δ ∷ ὲ ∷ []) "Rom.4.23"
∷ word (δ ∷ ι ∷ []) "Rom.4.23"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.4.23"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Rom.4.23"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.4.23"
∷ word (ἐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ σ ∷ θ ∷ η ∷ []) "Rom.4.23"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.4.23"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.4.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.24"
∷ word (δ ∷ ι ∷ []) "Rom.4.24"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.4.24"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "Rom.4.24"
∷ word (μ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ε ∷ ι ∷ []) "Rom.4.24"
∷ word (∙λ ∷ ο ∷ γ ∷ ί ∷ ζ ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.4.24"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.4.24"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.4.24"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.4.24"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.4.24"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ α ∷ ν ∷ τ ∷ α ∷ []) "Rom.4.24"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "Rom.4.24"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.4.24"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.4.24"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.4.24"
∷ word (ἐ ∷ κ ∷ []) "Rom.4.24"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Rom.4.24"
∷ word (ὃ ∷ ς ∷ []) "Rom.4.25"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ δ ∷ ό ∷ θ ∷ η ∷ []) "Rom.4.25"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.4.25"
∷ word (τ ∷ ὰ ∷ []) "Rom.4.25"
∷ word (π ∷ α ∷ ρ ∷ α ∷ π ∷ τ ∷ ώ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Rom.4.25"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.4.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.4.25"
∷ word (ἠ ∷ γ ∷ έ ∷ ρ ∷ θ ∷ η ∷ []) "Rom.4.25"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.4.25"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.4.25"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ί ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Rom.4.25"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.4.25"
∷ word (Δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ω ∷ θ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.5.1"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.5.1"
∷ word (ἐ ∷ κ ∷ []) "Rom.5.1"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.5.1"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ ν ∷ []) "Rom.5.1"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.5.1"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.5.1"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.5.1"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "Rom.5.1"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.5.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.1"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.5.1"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.5.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.5.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.5.1"
∷ word (δ ∷ ι ∷ []) "Rom.5.2"
∷ word (ο ∷ ὗ ∷ []) "Rom.5.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.2"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.5.2"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ α ∷ γ ∷ ω ∷ γ ∷ ὴ ∷ ν ∷ []) "Rom.5.2"
∷ word (ἐ ∷ σ ∷ χ ∷ ή ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Rom.5.2"
∷ word (τ ∷ ῇ ∷ []) "Rom.5.2"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ι ∷ []) "Rom.5.2"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.5.2"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.5.2"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Rom.5.2"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ η ∷ ν ∷ []) "Rom.5.2"
∷ word (ἐ ∷ ν ∷ []) "Rom.5.2"
∷ word (ᾗ ∷ []) "Rom.5.2"
∷ word (ἑ ∷ σ ∷ τ ∷ ή ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Rom.5.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.2"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ώ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.5.2"
∷ word (ἐ ∷ π ∷ []) "Rom.5.2"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ ι ∷ []) "Rom.5.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.5.2"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Rom.5.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.5.2"
∷ word (ο ∷ ὐ ∷ []) "Rom.5.3"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Rom.5.3"
∷ word (δ ∷ έ ∷ []) "Rom.5.3"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.5.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.3"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ώ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.5.3"
∷ word (ἐ ∷ ν ∷ []) "Rom.5.3"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Rom.5.3"
∷ word (θ ∷ ∙λ ∷ ί ∷ ψ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Rom.5.3"
∷ word (ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "Rom.5.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.5.3"
∷ word (ἡ ∷ []) "Rom.5.3"
∷ word (θ ∷ ∙λ ∷ ῖ ∷ ψ ∷ ι ∷ ς ∷ []) "Rom.5.3"
∷ word (ὑ ∷ π ∷ ο ∷ μ ∷ ο ∷ ν ∷ ὴ ∷ ν ∷ []) "Rom.5.3"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.5.3"
∷ word (ἡ ∷ []) "Rom.5.4"
∷ word (δ ∷ ὲ ∷ []) "Rom.5.4"
∷ word (ὑ ∷ π ∷ ο ∷ μ ∷ ο ∷ ν ∷ ὴ ∷ []) "Rom.5.4"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ή ∷ ν ∷ []) "Rom.5.4"
∷ word (ἡ ∷ []) "Rom.5.4"
∷ word (δ ∷ ὲ ∷ []) "Rom.5.4"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ὴ ∷ []) "Rom.5.4"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ α ∷ []) "Rom.5.4"
∷ word (ἡ ∷ []) "Rom.5.5"
∷ word (δ ∷ ὲ ∷ []) "Rom.5.5"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ὶ ∷ ς ∷ []) "Rom.5.5"
∷ word (ο ∷ ὐ ∷ []) "Rom.5.5"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ι ∷ σ ∷ χ ∷ ύ ∷ ν ∷ ε ∷ ι ∷ []) "Rom.5.5"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.5.5"
∷ word (ἡ ∷ []) "Rom.5.5"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "Rom.5.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.5"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.5.5"
∷ word (ἐ ∷ κ ∷ κ ∷ έ ∷ χ ∷ υ ∷ τ ∷ α ∷ ι ∷ []) "Rom.5.5"
∷ word (ἐ ∷ ν ∷ []) "Rom.5.5"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Rom.5.5"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Rom.5.5"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.5.5"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.5.5"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.5.5"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.5.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.5"
∷ word (δ ∷ ο ∷ θ ∷ έ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.5.5"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.5.5"
∷ word (Ἔ ∷ τ ∷ ι ∷ []) "Rom.5.6"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.5.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.5.6"
∷ word (ὄ ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.5.6"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.5.6"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.5.6"
∷ word (ἔ ∷ τ ∷ ι ∷ []) "Rom.5.6"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.5.6"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ὸ ∷ ν ∷ []) "Rom.5.6"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.5.6"
∷ word (ἀ ∷ σ ∷ ε ∷ β ∷ ῶ ∷ ν ∷ []) "Rom.5.6"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "Rom.5.6"
∷ word (μ ∷ ό ∷ ∙λ ∷ ι ∷ ς ∷ []) "Rom.5.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.5.7"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.5.7"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ί ∷ ο ∷ υ ∷ []) "Rom.5.7"
∷ word (τ ∷ ι ∷ ς ∷ []) "Rom.5.7"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ α ∷ ν ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "Rom.5.7"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.5.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.5.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.7"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ο ∷ ῦ ∷ []) "Rom.5.7"
∷ word (τ ∷ ά ∷ χ ∷ α ∷ []) "Rom.5.7"
∷ word (τ ∷ ι ∷ ς ∷ []) "Rom.5.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.7"
∷ word (τ ∷ ο ∷ ∙λ ∷ μ ∷ ᾷ ∷ []) "Rom.5.7"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ α ∷ ν ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.5.7"
∷ word (σ ∷ υ ∷ ν ∷ ί ∷ σ ∷ τ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "Rom.5.8"
∷ word (δ ∷ ὲ ∷ []) "Rom.5.8"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.5.8"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.5.8"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "Rom.5.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.5.8"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.5.8"
∷ word (ὁ ∷ []) "Rom.5.8"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.5.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.5.8"
∷ word (ἔ ∷ τ ∷ ι ∷ []) "Rom.5.8"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ω ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "Rom.5.8"
∷ word (ὄ ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.5.8"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.5.8"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.5.8"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.5.8"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.5.8"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "Rom.5.8"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῷ ∷ []) "Rom.5.9"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.5.9"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.5.9"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ω ∷ θ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.5.9"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Rom.5.9"
∷ word (ἐ ∷ ν ∷ []) "Rom.5.9"
∷ word (τ ∷ ῷ ∷ []) "Rom.5.9"
∷ word (α ∷ ἵ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.5.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.5.9"
∷ word (σ ∷ ω ∷ θ ∷ η ∷ σ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.5.9"
∷ word (δ ∷ ι ∷ []) "Rom.5.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.5.9"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.5.9"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.5.9"
∷ word (ὀ ∷ ρ ∷ γ ∷ ῆ ∷ ς ∷ []) "Rom.5.9"
∷ word (ε ∷ ἰ ∷ []) "Rom.5.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.5.10"
∷ word (ἐ ∷ χ ∷ θ ∷ ρ ∷ ο ∷ ὶ ∷ []) "Rom.5.10"
∷ word (ὄ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.5.10"
∷ word (κ ∷ α ∷ τ ∷ η ∷ ∙λ ∷ ∙λ ∷ ά ∷ γ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "Rom.5.10"
∷ word (τ ∷ ῷ ∷ []) "Rom.5.10"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.5.10"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.5.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.10"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "Rom.5.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.10"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "Rom.5.10"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.5.10"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῷ ∷ []) "Rom.5.10"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.5.10"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ∙λ ∷ α ∷ γ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.5.10"
∷ word (σ ∷ ω ∷ θ ∷ η ∷ σ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.5.10"
∷ word (ἐ ∷ ν ∷ []) "Rom.5.10"
∷ word (τ ∷ ῇ ∷ []) "Rom.5.10"
∷ word (ζ ∷ ω ∷ ῇ ∷ []) "Rom.5.10"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.5.10"
∷ word (ο ∷ ὐ ∷ []) "Rom.5.11"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Rom.5.11"
∷ word (δ ∷ έ ∷ []) "Rom.5.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.5.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.11"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ώ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Rom.5.11"
∷ word (ἐ ∷ ν ∷ []) "Rom.5.11"
∷ word (τ ∷ ῷ ∷ []) "Rom.5.11"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.5.11"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.5.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.11"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.5.11"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.5.11"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.5.11"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.5.11"
∷ word (δ ∷ ι ∷ []) "Rom.5.11"
∷ word (ο ∷ ὗ ∷ []) "Rom.5.11"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Rom.5.11"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.5.11"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ∙λ ∷ α ∷ γ ∷ ὴ ∷ ν ∷ []) "Rom.5.11"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ β ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.5.11"
∷ word (Δ ∷ ι ∷ ὰ ∷ []) "Rom.5.12"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.5.12"
∷ word (ὥ ∷ σ ∷ π ∷ ε ∷ ρ ∷ []) "Rom.5.12"
∷ word (δ ∷ ι ∷ []) "Rom.5.12"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Rom.5.12"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "Rom.5.12"
∷ word (ἡ ∷ []) "Rom.5.12"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.5.12"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.5.12"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.5.12"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ν ∷ []) "Rom.5.12"
∷ word (ε ∷ ἰ ∷ σ ∷ ῆ ∷ ∙λ ∷ θ ∷ ε ∷ ν ∷ []) "Rom.5.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.12"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.5.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.5.12"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.5.12"
∷ word (ὁ ∷ []) "Rom.5.12"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.5.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.12"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.5.12"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.5.12"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.5.12"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ ς ∷ []) "Rom.5.12"
∷ word (ὁ ∷ []) "Rom.5.12"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.5.12"
∷ word (δ ∷ ι ∷ ῆ ∷ ∙λ ∷ θ ∷ ε ∷ ν ∷ []) "Rom.5.12"
∷ word (ἐ ∷ φ ∷ []) "Rom.5.12"
∷ word (ᾧ ∷ []) "Rom.5.12"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.5.12"
∷ word (ἥ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.5.12"
∷ word (ἄ ∷ χ ∷ ρ ∷ ι ∷ []) "Rom.5.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.5.13"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.5.13"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.5.13"
∷ word (ἦ ∷ ν ∷ []) "Rom.5.13"
∷ word (ἐ ∷ ν ∷ []) "Rom.5.13"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "Rom.5.13"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.5.13"
∷ word (δ ∷ ὲ ∷ []) "Rom.5.13"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.5.13"
∷ word (ἐ ∷ ∙λ ∷ ∙λ ∷ ο ∷ γ ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "Rom.5.13"
∷ word (μ ∷ ὴ ∷ []) "Rom.5.13"
∷ word (ὄ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.5.13"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.5.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.5.14"
∷ word (ἐ ∷ β ∷ α ∷ σ ∷ ί ∷ ∙λ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "Rom.5.14"
∷ word (ὁ ∷ []) "Rom.5.14"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.5.14"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.5.14"
∷ word (Ἀ ∷ δ ∷ ὰ ∷ μ ∷ []) "Rom.5.14"
∷ word (μ ∷ έ ∷ χ ∷ ρ ∷ ι ∷ []) "Rom.5.14"
∷ word (Μ ∷ ω ∷ ϋ ∷ σ ∷ έ ∷ ω ∷ ς ∷ []) "Rom.5.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.14"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.5.14"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.5.14"
∷ word (μ ∷ ὴ ∷ []) "Rom.5.14"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ή ∷ σ ∷ α ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.5.14"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.5.14"
∷ word (τ ∷ ῷ ∷ []) "Rom.5.14"
∷ word (ὁ ∷ μ ∷ ο ∷ ι ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.5.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.5.14"
∷ word (π ∷ α ∷ ρ ∷ α ∷ β ∷ ά ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.5.14"
∷ word (Ἀ ∷ δ ∷ ά ∷ μ ∷ []) "Rom.5.14"
∷ word (ὅ ∷ ς ∷ []) "Rom.5.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.5.14"
∷ word (τ ∷ ύ ∷ π ∷ ο ∷ ς ∷ []) "Rom.5.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.14"
∷ word (μ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.5.14"
∷ word (Ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Rom.5.15"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "Rom.5.15"
∷ word (ὡ ∷ ς ∷ []) "Rom.5.15"
∷ word (τ ∷ ὸ ∷ []) "Rom.5.15"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ π ∷ τ ∷ ω ∷ μ ∷ α ∷ []) "Rom.5.15"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.5.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.15"
∷ word (τ ∷ ὸ ∷ []) "Rom.5.15"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ σ ∷ μ ∷ α ∷ []) "Rom.5.15"
∷ word (ε ∷ ἰ ∷ []) "Rom.5.15"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.5.15"
∷ word (τ ∷ ῷ ∷ []) "Rom.5.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.15"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Rom.5.15"
∷ word (π ∷ α ∷ ρ ∷ α ∷ π ∷ τ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.5.15"
∷ word (ο ∷ ἱ ∷ []) "Rom.5.15"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὶ ∷ []) "Rom.5.15"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ο ∷ ν ∷ []) "Rom.5.15"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῷ ∷ []) "Rom.5.15"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.5.15"
∷ word (ἡ ∷ []) "Rom.5.15"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Rom.5.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.15"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.5.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.15"
∷ word (ἡ ∷ []) "Rom.5.15"
∷ word (δ ∷ ω ∷ ρ ∷ ε ∷ ὰ ∷ []) "Rom.5.15"
∷ word (ἐ ∷ ν ∷ []) "Rom.5.15"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ι ∷ []) "Rom.5.15"
∷ word (τ ∷ ῇ ∷ []) "Rom.5.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.15"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Rom.5.15"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "Rom.5.15"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.5.15"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.5.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.5.15"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.5.15"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.5.15"
∷ word (ἐ ∷ π ∷ ε ∷ ρ ∷ ί ∷ σ ∷ σ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "Rom.5.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.16"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "Rom.5.16"
∷ word (ὡ ∷ ς ∷ []) "Rom.5.16"
∷ word (δ ∷ ι ∷ []) "Rom.5.16"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Rom.5.16"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ή ∷ σ ∷ α ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.5.16"
∷ word (τ ∷ ὸ ∷ []) "Rom.5.16"
∷ word (δ ∷ ώ ∷ ρ ∷ η ∷ μ ∷ α ∷ []) "Rom.5.16"
∷ word (τ ∷ ὸ ∷ []) "Rom.5.16"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.5.16"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.5.16"
∷ word (κ ∷ ρ ∷ ί ∷ μ ∷ α ∷ []) "Rom.5.16"
∷ word (ἐ ∷ ξ ∷ []) "Rom.5.16"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Rom.5.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.5.16"
∷ word (κ ∷ α ∷ τ ∷ ά ∷ κ ∷ ρ ∷ ι ∷ μ ∷ α ∷ []) "Rom.5.16"
∷ word (τ ∷ ὸ ∷ []) "Rom.5.16"
∷ word (δ ∷ ὲ ∷ []) "Rom.5.16"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ σ ∷ μ ∷ α ∷ []) "Rom.5.16"
∷ word (ἐ ∷ κ ∷ []) "Rom.5.16"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "Rom.5.16"
∷ word (π ∷ α ∷ ρ ∷ α ∷ π ∷ τ ∷ ω ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "Rom.5.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.5.16"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ί ∷ ω ∷ μ ∷ α ∷ []) "Rom.5.16"
∷ word (ε ∷ ἰ ∷ []) "Rom.5.17"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.5.17"
∷ word (τ ∷ ῷ ∷ []) "Rom.5.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.17"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Rom.5.17"
∷ word (π ∷ α ∷ ρ ∷ α ∷ π ∷ τ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.5.17"
∷ word (ὁ ∷ []) "Rom.5.17"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.5.17"
∷ word (ἐ ∷ β ∷ α ∷ σ ∷ ί ∷ ∙λ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "Rom.5.17"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.5.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.17"
∷ word (ἑ ∷ ν ∷ ό ∷ ς ∷ []) "Rom.5.17"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῷ ∷ []) "Rom.5.17"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.5.17"
∷ word (ο ∷ ἱ ∷ []) "Rom.5.17"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.5.17"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "Rom.5.17"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.5.17"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ο ∷ ς ∷ []) "Rom.5.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.17"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.5.17"
∷ word (δ ∷ ω ∷ ρ ∷ ε ∷ ᾶ ∷ ς ∷ []) "Rom.5.17"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.5.17"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "Rom.5.17"
∷ word (∙λ ∷ α ∷ μ ∷ β ∷ ά ∷ ν ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.5.17"
∷ word (ἐ ∷ ν ∷ []) "Rom.5.17"
∷ word (ζ ∷ ω ∷ ῇ ∷ []) "Rom.5.17"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.5.17"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.5.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.17"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Rom.5.17"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.5.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.5.17"
∷ word (Ἄ ∷ ρ ∷ α ∷ []) "Rom.5.18"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.5.18"
∷ word (ὡ ∷ ς ∷ []) "Rom.5.18"
∷ word (δ ∷ ι ∷ []) "Rom.5.18"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Rom.5.18"
∷ word (π ∷ α ∷ ρ ∷ α ∷ π ∷ τ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.5.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.5.18"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.5.18"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ ς ∷ []) "Rom.5.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.5.18"
∷ word (κ ∷ α ∷ τ ∷ ά ∷ κ ∷ ρ ∷ ι ∷ μ ∷ α ∷ []) "Rom.5.18"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.5.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.18"
∷ word (δ ∷ ι ∷ []) "Rom.5.18"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Rom.5.18"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.5.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.5.18"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.5.18"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ ς ∷ []) "Rom.5.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.5.18"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ί ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Rom.5.18"
∷ word (ζ ∷ ω ∷ ῆ ∷ ς ∷ []) "Rom.5.18"
∷ word (ὥ ∷ σ ∷ π ∷ ε ∷ ρ ∷ []) "Rom.5.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.5.19"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.5.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.5.19"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ο ∷ ῆ ∷ ς ∷ []) "Rom.5.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.19"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Rom.5.19"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "Rom.5.19"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ω ∷ ∙λ ∷ ο ∷ ὶ ∷ []) "Rom.5.19"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ σ ∷ τ ∷ ά ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.5.19"
∷ word (ο ∷ ἱ ∷ []) "Rom.5.19"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ί ∷ []) "Rom.5.19"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.5.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.19"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.5.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.5.19"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ῆ ∷ ς ∷ []) "Rom.5.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.19"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Rom.5.19"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ ο ∷ ι ∷ []) "Rom.5.19"
∷ word (κ ∷ α ∷ τ ∷ α ∷ σ ∷ τ ∷ α ∷ θ ∷ ή ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.5.19"
∷ word (ο ∷ ἱ ∷ []) "Rom.5.19"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ί ∷ []) "Rom.5.19"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Rom.5.20"
∷ word (δ ∷ ὲ ∷ []) "Rom.5.20"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ ι ∷ σ ∷ ῆ ∷ ∙λ ∷ θ ∷ ε ∷ ν ∷ []) "Rom.5.20"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.5.20"
∷ word (π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ ά ∷ σ ∷ ῃ ∷ []) "Rom.5.20"
∷ word (τ ∷ ὸ ∷ []) "Rom.5.20"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ π ∷ τ ∷ ω ∷ μ ∷ α ∷ []) "Rom.5.20"
∷ word (ο ∷ ὗ ∷ []) "Rom.5.20"
∷ word (δ ∷ ὲ ∷ []) "Rom.5.20"
∷ word (ἐ ∷ π ∷ ∙λ ∷ ε ∷ ό ∷ ν ∷ α ∷ σ ∷ ε ∷ ν ∷ []) "Rom.5.20"
∷ word (ἡ ∷ []) "Rom.5.20"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.5.20"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ ε ∷ π ∷ ε ∷ ρ ∷ ί ∷ σ ∷ σ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "Rom.5.20"
∷ word (ἡ ∷ []) "Rom.5.20"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Rom.5.20"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.5.21"
∷ word (ὥ ∷ σ ∷ π ∷ ε ∷ ρ ∷ []) "Rom.5.21"
∷ word (ἐ ∷ β ∷ α ∷ σ ∷ ί ∷ ∙λ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "Rom.5.21"
∷ word (ἡ ∷ []) "Rom.5.21"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.5.21"
∷ word (ἐ ∷ ν ∷ []) "Rom.5.21"
∷ word (τ ∷ ῷ ∷ []) "Rom.5.21"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ῳ ∷ []) "Rom.5.21"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.5.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.5.21"
∷ word (ἡ ∷ []) "Rom.5.21"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Rom.5.21"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ ῃ ∷ []) "Rom.5.21"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.5.21"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "Rom.5.21"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.5.21"
∷ word (ζ ∷ ω ∷ ὴ ∷ ν ∷ []) "Rom.5.21"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ν ∷ []) "Rom.5.21"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.5.21"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.5.21"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.5.21"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.5.21"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.5.21"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.5.21"
∷ word (Τ ∷ ί ∷ []) "Rom.6.1"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.6.1"
∷ word (ἐ ∷ ρ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.6.1"
∷ word (ἐ ∷ π ∷ ι ∷ μ ∷ έ ∷ ν ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Rom.6.1"
∷ word (τ ∷ ῇ ∷ []) "Rom.6.1"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Rom.6.1"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.6.1"
∷ word (ἡ ∷ []) "Rom.6.1"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Rom.6.1"
∷ word (π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ ά ∷ σ ∷ ῃ ∷ []) "Rom.6.1"
∷ word (μ ∷ ὴ ∷ []) "Rom.6.2"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "Rom.6.2"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "Rom.6.2"
∷ word (ἀ ∷ π ∷ ε ∷ θ ∷ ά ∷ ν ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.6.2"
∷ word (τ ∷ ῇ ∷ []) "Rom.6.2"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Rom.6.2"
∷ word (π ∷ ῶ ∷ ς ∷ []) "Rom.6.2"
∷ word (ἔ ∷ τ ∷ ι ∷ []) "Rom.6.2"
∷ word (ζ ∷ ή ∷ σ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.6.2"
∷ word (ἐ ∷ ν ∷ []) "Rom.6.2"
∷ word (α ∷ ὐ ∷ τ ∷ ῇ ∷ []) "Rom.6.2"
∷ word (ἢ ∷ []) "Rom.6.3"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Rom.6.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.6.3"
∷ word (ὅ ∷ σ ∷ ο ∷ ι ∷ []) "Rom.6.3"
∷ word (ἐ ∷ β ∷ α ∷ π ∷ τ ∷ ί ∷ σ ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "Rom.6.3"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.6.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.6.3"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "Rom.6.3"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.6.3"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.6.3"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "Rom.6.3"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.6.3"
∷ word (ἐ ∷ β ∷ α ∷ π ∷ τ ∷ ί ∷ σ ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "Rom.6.3"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ τ ∷ ά ∷ φ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "Rom.6.4"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.6.4"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.6.4"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.6.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.6.4"
∷ word (β ∷ α ∷ π ∷ τ ∷ ί ∷ σ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.6.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.6.4"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.6.4"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "Rom.6.4"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.6.4"
∷ word (ὥ ∷ σ ∷ π ∷ ε ∷ ρ ∷ []) "Rom.6.4"
∷ word (ἠ ∷ γ ∷ έ ∷ ρ ∷ θ ∷ η ∷ []) "Rom.6.4"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.6.4"
∷ word (ἐ ∷ κ ∷ []) "Rom.6.4"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Rom.6.4"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.6.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.6.4"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Rom.6.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.6.4"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ό ∷ ς ∷ []) "Rom.6.4"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.6.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.6.4"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.6.4"
∷ word (ἐ ∷ ν ∷ []) "Rom.6.4"
∷ word (κ ∷ α ∷ ι ∷ ν ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "Rom.6.4"
∷ word (ζ ∷ ω ∷ ῆ ∷ ς ∷ []) "Rom.6.4"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ή ∷ σ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Rom.6.4"
∷ word (Ε ∷ ἰ ∷ []) "Rom.6.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.6.5"
∷ word (σ ∷ ύ ∷ μ ∷ φ ∷ υ ∷ τ ∷ ο ∷ ι ∷ []) "Rom.6.5"
∷ word (γ ∷ ε ∷ γ ∷ ό ∷ ν ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Rom.6.5"
∷ word (τ ∷ ῷ ∷ []) "Rom.6.5"
∷ word (ὁ ∷ μ ∷ ο ∷ ι ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.6.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.6.5"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "Rom.6.5"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.6.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.6.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.6.5"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.6.5"
∷ word (ἀ ∷ ν ∷ α ∷ σ ∷ τ ∷ ά ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.6.5"
∷ word (ἐ ∷ σ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.6.5"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.6.6"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.6.6"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.6.6"
∷ word (ὁ ∷ []) "Rom.6.6"
∷ word (π ∷ α ∷ ∙λ ∷ α ∷ ι ∷ ὸ ∷ ς ∷ []) "Rom.6.6"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.6.6"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "Rom.6.6"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ώ ∷ θ ∷ η ∷ []) "Rom.6.6"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.6.6"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ η ∷ θ ∷ ῇ ∷ []) "Rom.6.6"
∷ word (τ ∷ ὸ ∷ []) "Rom.6.6"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "Rom.6.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.6.6"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.6.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.6.6"
∷ word (μ ∷ η ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Rom.6.6"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.6.6"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.6.6"
∷ word (τ ∷ ῇ ∷ []) "Rom.6.6"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Rom.6.6"
∷ word (ὁ ∷ []) "Rom.6.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.6.7"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ α ∷ ν ∷ ὼ ∷ ν ∷ []) "Rom.6.7"
∷ word (δ ∷ ε ∷ δ ∷ ι ∷ κ ∷ α ∷ ί ∷ ω ∷ τ ∷ α ∷ ι ∷ []) "Rom.6.7"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.6.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.6.7"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.6.7"
∷ word (ε ∷ ἰ ∷ []) "Rom.6.8"
∷ word (δ ∷ ὲ ∷ []) "Rom.6.8"
∷ word (ἀ ∷ π ∷ ε ∷ θ ∷ ά ∷ ν ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.6.8"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Rom.6.8"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.6.8"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.6.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.6.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.6.8"
∷ word (σ ∷ υ ∷ ζ ∷ ή ∷ σ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.6.8"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.6.8"
∷ word (ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "Rom.6.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.6.9"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.6.9"
∷ word (ἐ ∷ γ ∷ ε ∷ ρ ∷ θ ∷ ε ∷ ὶ ∷ ς ∷ []) "Rom.6.9"
∷ word (ἐ ∷ κ ∷ []) "Rom.6.9"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Rom.6.9"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Rom.6.9"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ ν ∷ ῄ ∷ σ ∷ κ ∷ ε ∷ ι ∷ []) "Rom.6.9"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.6.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.6.9"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Rom.6.9"
∷ word (κ ∷ υ ∷ ρ ∷ ι ∷ ε ∷ ύ ∷ ε ∷ ι ∷ []) "Rom.6.9"
∷ word (ὃ ∷ []) "Rom.6.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.6.10"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "Rom.6.10"
∷ word (τ ∷ ῇ ∷ []) "Rom.6.10"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Rom.6.10"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "Rom.6.10"
∷ word (ἐ ∷ φ ∷ ά ∷ π ∷ α ∷ ξ ∷ []) "Rom.6.10"
∷ word (ὃ ∷ []) "Rom.6.10"
∷ word (δ ∷ ὲ ∷ []) "Rom.6.10"
∷ word (ζ ∷ ῇ ∷ []) "Rom.6.10"
∷ word (ζ ∷ ῇ ∷ []) "Rom.6.10"
∷ word (τ ∷ ῷ ∷ []) "Rom.6.10"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.6.10"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.6.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.6.11"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.6.11"
∷ word (∙λ ∷ ο ∷ γ ∷ ί ∷ ζ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Rom.6.11"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.6.11"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.6.11"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.6.11"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.6.11"
∷ word (τ ∷ ῇ ∷ []) "Rom.6.11"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Rom.6.11"
∷ word (ζ ∷ ῶ ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.6.11"
∷ word (δ ∷ ὲ ∷ []) "Rom.6.11"
∷ word (τ ∷ ῷ ∷ []) "Rom.6.11"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.6.11"
∷ word (ἐ ∷ ν ∷ []) "Rom.6.11"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.6.11"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.6.11"
∷ word (Μ ∷ ὴ ∷ []) "Rom.6.12"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.6.12"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ υ ∷ έ ∷ τ ∷ ω ∷ []) "Rom.6.12"
∷ word (ἡ ∷ []) "Rom.6.12"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.6.12"
∷ word (ἐ ∷ ν ∷ []) "Rom.6.12"
∷ word (τ ∷ ῷ ∷ []) "Rom.6.12"
∷ word (θ ∷ ν ∷ η ∷ τ ∷ ῷ ∷ []) "Rom.6.12"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.6.12"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.6.12"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.6.12"
∷ word (τ ∷ ὸ ∷ []) "Rom.6.12"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ύ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.6.12"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Rom.6.12"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Rom.6.12"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.6.12"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "Rom.6.13"
∷ word (π ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ά ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "Rom.6.13"
∷ word (τ ∷ ὰ ∷ []) "Rom.6.13"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "Rom.6.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.6.13"
∷ word (ὅ ∷ π ∷ ∙λ ∷ α ∷ []) "Rom.6.13"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ί ∷ α ∷ ς ∷ []) "Rom.6.13"
∷ word (τ ∷ ῇ ∷ []) "Rom.6.13"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Rom.6.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.6.13"
∷ word (π ∷ α ∷ ρ ∷ α ∷ σ ∷ τ ∷ ή ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Rom.6.13"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.6.13"
∷ word (τ ∷ ῷ ∷ []) "Rom.6.13"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.6.13"
∷ word (ὡ ∷ σ ∷ ε ∷ ὶ ∷ []) "Rom.6.13"
∷ word (ἐ ∷ κ ∷ []) "Rom.6.13"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Rom.6.13"
∷ word (ζ ∷ ῶ ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.6.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.6.13"
∷ word (τ ∷ ὰ ∷ []) "Rom.6.13"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "Rom.6.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.6.13"
∷ word (ὅ ∷ π ∷ ∙λ ∷ α ∷ []) "Rom.6.13"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "Rom.6.13"
∷ word (τ ∷ ῷ ∷ []) "Rom.6.13"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.6.13"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.6.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.6.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.6.14"
∷ word (ο ∷ ὐ ∷ []) "Rom.6.14"
∷ word (κ ∷ υ ∷ ρ ∷ ι ∷ ε ∷ ύ ∷ σ ∷ ε ∷ ι ∷ []) "Rom.6.14"
∷ word (ο ∷ ὐ ∷ []) "Rom.6.14"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Rom.6.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "Rom.6.14"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Rom.6.14"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.6.14"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.6.14"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Rom.6.14"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Rom.6.14"
∷ word (Τ ∷ ί ∷ []) "Rom.6.15"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.6.15"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ή ∷ σ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Rom.6.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.6.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.6.15"
∷ word (ἐ ∷ σ ∷ μ ∷ ὲ ∷ ν ∷ []) "Rom.6.15"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Rom.6.15"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.6.15"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.6.15"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Rom.6.15"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Rom.6.15"
∷ word (μ ∷ ὴ ∷ []) "Rom.6.15"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "Rom.6.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.6.16"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "Rom.6.16"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.6.16"
∷ word (ᾧ ∷ []) "Rom.6.16"
∷ word (π ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ά ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "Rom.6.16"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.6.16"
∷ word (δ ∷ ο ∷ ύ ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.6.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.6.16"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ή ∷ ν ∷ []) "Rom.6.16"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ί ∷ []) "Rom.6.16"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "Rom.6.16"
∷ word (ᾧ ∷ []) "Rom.6.16"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ύ ∷ ε ∷ τ ∷ ε ∷ []) "Rom.6.16"
∷ word (ἤ ∷ τ ∷ ο ∷ ι ∷ []) "Rom.6.16"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.6.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.6.16"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "Rom.6.16"
∷ word (ἢ ∷ []) "Rom.6.16"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ῆ ∷ ς ∷ []) "Rom.6.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.6.16"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.6.16"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Rom.6.17"
∷ word (δ ∷ ὲ ∷ []) "Rom.6.17"
∷ word (τ ∷ ῷ ∷ []) "Rom.6.17"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.6.17"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.6.17"
∷ word (ἦ ∷ τ ∷ ε ∷ []) "Rom.6.17"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ι ∷ []) "Rom.6.17"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.6.17"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.6.17"
∷ word (ὑ ∷ π ∷ η ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Rom.6.17"
∷ word (δ ∷ ὲ ∷ []) "Rom.6.17"
∷ word (ἐ ∷ κ ∷ []) "Rom.6.17"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "Rom.6.17"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.6.17"
∷ word (ὃ ∷ ν ∷ []) "Rom.6.17"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ δ ∷ ό ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Rom.6.17"
∷ word (τ ∷ ύ ∷ π ∷ ο ∷ ν ∷ []) "Rom.6.17"
∷ word (δ ∷ ι ∷ δ ∷ α ∷ χ ∷ ῆ ∷ ς ∷ []) "Rom.6.17"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ ε ∷ ρ ∷ ω ∷ θ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.6.18"
∷ word (δ ∷ ὲ ∷ []) "Rom.6.18"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.6.18"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.6.18"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.6.18"
∷ word (ἐ ∷ δ ∷ ο ∷ υ ∷ ∙λ ∷ ώ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Rom.6.18"
∷ word (τ ∷ ῇ ∷ []) "Rom.6.18"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ ῃ ∷ []) "Rom.6.18"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ι ∷ ν ∷ ο ∷ ν ∷ []) "Rom.6.19"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Rom.6.19"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.6.19"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.6.19"
∷ word (ἀ ∷ σ ∷ θ ∷ έ ∷ ν ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Rom.6.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.6.19"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Rom.6.19"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.6.19"
∷ word (ὥ ∷ σ ∷ π ∷ ε ∷ ρ ∷ []) "Rom.6.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.6.19"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ σ ∷ τ ∷ ή ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Rom.6.19"
∷ word (τ ∷ ὰ ∷ []) "Rom.6.19"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "Rom.6.19"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.6.19"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ α ∷ []) "Rom.6.19"
∷ word (τ ∷ ῇ ∷ []) "Rom.6.19"
∷ word (ἀ ∷ κ ∷ α ∷ θ ∷ α ∷ ρ ∷ σ ∷ ί ∷ ᾳ ∷ []) "Rom.6.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.6.19"
∷ word (τ ∷ ῇ ∷ []) "Rom.6.19"
∷ word (ἀ ∷ ν ∷ ο ∷ μ ∷ ί ∷ ᾳ ∷ []) "Rom.6.19"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.6.19"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.6.19"
∷ word (ἀ ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Rom.6.19"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.6.19"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Rom.6.19"
∷ word (π ∷ α ∷ ρ ∷ α ∷ σ ∷ τ ∷ ή ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Rom.6.19"
∷ word (τ ∷ ὰ ∷ []) "Rom.6.19"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "Rom.6.19"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.6.19"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ α ∷ []) "Rom.6.19"
∷ word (τ ∷ ῇ ∷ []) "Rom.6.19"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ ῃ ∷ []) "Rom.6.19"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.6.19"
∷ word (ἁ ∷ γ ∷ ι ∷ α ∷ σ ∷ μ ∷ ό ∷ ν ∷ []) "Rom.6.19"
∷ word (Ὅ ∷ τ ∷ ε ∷ []) "Rom.6.20"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.6.20"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ι ∷ []) "Rom.6.20"
∷ word (ἦ ∷ τ ∷ ε ∷ []) "Rom.6.20"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.6.20"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.6.20"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ θ ∷ ε ∷ ρ ∷ ο ∷ ι ∷ []) "Rom.6.20"
∷ word (ἦ ∷ τ ∷ ε ∷ []) "Rom.6.20"
∷ word (τ ∷ ῇ ∷ []) "Rom.6.20"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ ῃ ∷ []) "Rom.6.20"
∷ word (τ ∷ ί ∷ ν ∷ α ∷ []) "Rom.6.21"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.6.21"
∷ word (κ ∷ α ∷ ρ ∷ π ∷ ὸ ∷ ν ∷ []) "Rom.6.21"
∷ word (ε ∷ ἴ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "Rom.6.21"
∷ word (τ ∷ ό ∷ τ ∷ ε ∷ []) "Rom.6.21"
∷ word (ἐ ∷ φ ∷ []) "Rom.6.21"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "Rom.6.21"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Rom.6.21"
∷ word (ἐ ∷ π ∷ α ∷ ι ∷ σ ∷ χ ∷ ύ ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Rom.6.21"
∷ word (τ ∷ ὸ ∷ []) "Rom.6.21"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.6.21"
∷ word (τ ∷ έ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Rom.6.21"
∷ word (ἐ ∷ κ ∷ ε ∷ ί ∷ ν ∷ ω ∷ ν ∷ []) "Rom.6.21"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.6.21"
∷ word (ν ∷ υ ∷ ν ∷ ὶ ∷ []) "Rom.6.22"
∷ word (δ ∷ έ ∷ []) "Rom.6.22"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ ε ∷ ρ ∷ ω ∷ θ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.6.22"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.6.22"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.6.22"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.6.22"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ω ∷ θ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.6.22"
∷ word (δ ∷ ὲ ∷ []) "Rom.6.22"
∷ word (τ ∷ ῷ ∷ []) "Rom.6.22"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.6.22"
∷ word (ἔ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "Rom.6.22"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.6.22"
∷ word (κ ∷ α ∷ ρ ∷ π ∷ ὸ ∷ ν ∷ []) "Rom.6.22"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.6.22"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.6.22"
∷ word (ἁ ∷ γ ∷ ι ∷ α ∷ σ ∷ μ ∷ ό ∷ ν ∷ []) "Rom.6.22"
∷ word (τ ∷ ὸ ∷ []) "Rom.6.22"
∷ word (δ ∷ ὲ ∷ []) "Rom.6.22"
∷ word (τ ∷ έ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Rom.6.22"
∷ word (ζ ∷ ω ∷ ὴ ∷ ν ∷ []) "Rom.6.22"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ν ∷ []) "Rom.6.22"
∷ word (τ ∷ ὰ ∷ []) "Rom.6.23"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.6.23"
∷ word (ὀ ∷ ψ ∷ ώ ∷ ν ∷ ι ∷ α ∷ []) "Rom.6.23"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.6.23"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.6.23"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.6.23"
∷ word (τ ∷ ὸ ∷ []) "Rom.6.23"
∷ word (δ ∷ ὲ ∷ []) "Rom.6.23"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ σ ∷ μ ∷ α ∷ []) "Rom.6.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.6.23"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.6.23"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "Rom.6.23"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ς ∷ []) "Rom.6.23"
∷ word (ἐ ∷ ν ∷ []) "Rom.6.23"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.6.23"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.6.23"
∷ word (τ ∷ ῷ ∷ []) "Rom.6.23"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.6.23"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.6.23"
∷ word (Ἢ ∷ []) "Rom.7.1"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Rom.7.1"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Rom.7.1"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.7.1"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.7.1"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.7.1"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "Rom.7.1"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.7.1"
∷ word (ὁ ∷ []) "Rom.7.1"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Rom.7.1"
∷ word (κ ∷ υ ∷ ρ ∷ ι ∷ ε ∷ ύ ∷ ε ∷ ι ∷ []) "Rom.7.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.1"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "Rom.7.1"
∷ word (ἐ ∷ φ ∷ []) "Rom.7.1"
∷ word (ὅ ∷ σ ∷ ο ∷ ν ∷ []) "Rom.7.1"
∷ word (χ ∷ ρ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Rom.7.1"
∷ word (ζ ∷ ῇ ∷ []) "Rom.7.1"
∷ word (ἡ ∷ []) "Rom.7.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.7.2"
∷ word (ὕ ∷ π ∷ α ∷ ν ∷ δ ∷ ρ ∷ ο ∷ ς ∷ []) "Rom.7.2"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "Rom.7.2"
∷ word (τ ∷ ῷ ∷ []) "Rom.7.2"
∷ word (ζ ∷ ῶ ∷ ν ∷ τ ∷ ι ∷ []) "Rom.7.2"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ὶ ∷ []) "Rom.7.2"
∷ word (δ ∷ έ ∷ δ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.7.2"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.7.2"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.7.2"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.2"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ ά ∷ ν ∷ ῃ ∷ []) "Rom.7.2"
∷ word (ὁ ∷ []) "Rom.7.2"
∷ word (ἀ ∷ ν ∷ ή ∷ ρ ∷ []) "Rom.7.2"
∷ word (κ ∷ α ∷ τ ∷ ή ∷ ρ ∷ γ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Rom.7.2"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.7.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.2"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.7.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.2"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ό ∷ ς ∷ []) "Rom.7.2"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Rom.7.3"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.7.3"
∷ word (ζ ∷ ῶ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.7.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.3"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.7.3"
∷ word (μ ∷ ο ∷ ι ∷ χ ∷ α ∷ ∙λ ∷ ὶ ∷ ς ∷ []) "Rom.7.3"
∷ word (χ ∷ ρ ∷ η ∷ μ ∷ α ∷ τ ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "Rom.7.3"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.7.3"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Rom.7.3"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ὶ ∷ []) "Rom.7.3"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ ῳ ∷ []) "Rom.7.3"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.7.3"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.3"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ ά ∷ ν ∷ ῃ ∷ []) "Rom.7.3"
∷ word (ὁ ∷ []) "Rom.7.3"
∷ word (ἀ ∷ ν ∷ ή ∷ ρ ∷ []) "Rom.7.3"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ έ ∷ ρ ∷ α ∷ []) "Rom.7.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "Rom.7.3"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.7.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.3"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.7.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.3"
∷ word (μ ∷ ὴ ∷ []) "Rom.7.3"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.7.3"
∷ word (α ∷ ὐ ∷ τ ∷ ὴ ∷ ν ∷ []) "Rom.7.3"
∷ word (μ ∷ ο ∷ ι ∷ χ ∷ α ∷ ∙λ ∷ ί ∷ δ ∷ α ∷ []) "Rom.7.3"
∷ word (γ ∷ ε ∷ ν ∷ ο ∷ μ ∷ έ ∷ ν ∷ η ∷ ν ∷ []) "Rom.7.3"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ὶ ∷ []) "Rom.7.3"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ ῳ ∷ []) "Rom.7.3"
∷ word (Ὥ ∷ σ ∷ τ ∷ ε ∷ []) "Rom.7.4"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Rom.7.4"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.7.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.7.4"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.7.4"
∷ word (ἐ ∷ θ ∷ α ∷ ν ∷ α ∷ τ ∷ ώ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Rom.7.4"
∷ word (τ ∷ ῷ ∷ []) "Rom.7.4"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.7.4"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.7.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.4"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.7.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.4"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.7.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.7.4"
∷ word (τ ∷ ὸ ∷ []) "Rom.7.4"
∷ word (γ ∷ ε ∷ ν ∷ έ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.7.4"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.7.4"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ ῳ ∷ []) "Rom.7.4"
∷ word (τ ∷ ῷ ∷ []) "Rom.7.4"
∷ word (ἐ ∷ κ ∷ []) "Rom.7.4"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Rom.7.4"
∷ word (ἐ ∷ γ ∷ ε ∷ ρ ∷ θ ∷ έ ∷ ν ∷ τ ∷ ι ∷ []) "Rom.7.4"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.7.4"
∷ word (κ ∷ α ∷ ρ ∷ π ∷ ο ∷ φ ∷ ο ∷ ρ ∷ ή ∷ σ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Rom.7.4"
∷ word (τ ∷ ῷ ∷ []) "Rom.7.4"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.7.4"
∷ word (ὅ ∷ τ ∷ ε ∷ []) "Rom.7.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.7.5"
∷ word (ἦ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.7.5"
∷ word (ἐ ∷ ν ∷ []) "Rom.7.5"
∷ word (τ ∷ ῇ ∷ []) "Rom.7.5"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ί ∷ []) "Rom.7.5"
∷ word (τ ∷ ὰ ∷ []) "Rom.7.5"
∷ word (π ∷ α ∷ θ ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Rom.7.5"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.7.5"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ι ∷ ῶ ∷ ν ∷ []) "Rom.7.5"
∷ word (τ ∷ ὰ ∷ []) "Rom.7.5"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.7.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.5"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.7.5"
∷ word (ἐ ∷ ν ∷ η ∷ ρ ∷ γ ∷ ε ∷ ῖ ∷ τ ∷ ο ∷ []) "Rom.7.5"
∷ word (ἐ ∷ ν ∷ []) "Rom.7.5"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.7.5"
∷ word (μ ∷ έ ∷ ∙λ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Rom.7.5"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.7.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.7.5"
∷ word (τ ∷ ὸ ∷ []) "Rom.7.5"
∷ word (κ ∷ α ∷ ρ ∷ π ∷ ο ∷ φ ∷ ο ∷ ρ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Rom.7.5"
∷ word (τ ∷ ῷ ∷ []) "Rom.7.5"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ῳ ∷ []) "Rom.7.5"
∷ word (ν ∷ υ ∷ ν ∷ ὶ ∷ []) "Rom.7.6"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.6"
∷ word (κ ∷ α ∷ τ ∷ η ∷ ρ ∷ γ ∷ ή ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "Rom.7.6"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.7.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.6"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.7.6"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ α ∷ ν ∷ ό ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.7.6"
∷ word (ἐ ∷ ν ∷ []) "Rom.7.6"
∷ word (ᾧ ∷ []) "Rom.7.6"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ι ∷ χ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.7.6"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "Rom.7.6"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.7.6"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.7.6"
∷ word (ἐ ∷ ν ∷ []) "Rom.7.6"
∷ word (κ ∷ α ∷ ι ∷ ν ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "Rom.7.6"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.7.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.7.6"
∷ word (ο ∷ ὐ ∷ []) "Rom.7.6"
∷ word (π ∷ α ∷ ∙λ ∷ α ∷ ι ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "Rom.7.6"
∷ word (γ ∷ ρ ∷ ά ∷ μ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.7.6"
∷ word (Τ ∷ ί ∷ []) "Rom.7.7"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.7.7"
∷ word (ἐ ∷ ρ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.7.7"
∷ word (ὁ ∷ []) "Rom.7.7"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Rom.7.7"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.7.7"
∷ word (μ ∷ ὴ ∷ []) "Rom.7.7"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "Rom.7.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.7.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.7.7"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "Rom.7.7"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.7.7"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ ν ∷ []) "Rom.7.7"
∷ word (ε ∷ ἰ ∷ []) "Rom.7.7"
∷ word (μ ∷ ὴ ∷ []) "Rom.7.7"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.7.7"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.7.7"
∷ word (τ ∷ ή ∷ ν ∷ []) "Rom.7.7"
∷ word (τ ∷ ε ∷ []) "Rom.7.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.7.7"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Rom.7.7"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.7.7"
∷ word (ᾔ ∷ δ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.7.7"
∷ word (ε ∷ ἰ ∷ []) "Rom.7.7"
∷ word (μ ∷ ὴ ∷ []) "Rom.7.7"
∷ word (ὁ ∷ []) "Rom.7.7"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Rom.7.7"
∷ word (ἔ ∷ ∙λ ∷ ε ∷ γ ∷ ε ∷ ν ∷ []) "Rom.7.7"
∷ word (Ο ∷ ὐ ∷ κ ∷ []) "Rom.7.7"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ή ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.7.7"
∷ word (ἀ ∷ φ ∷ ο ∷ ρ ∷ μ ∷ ὴ ∷ ν ∷ []) "Rom.7.8"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.8"
∷ word (∙λ ∷ α ∷ β ∷ ο ∷ ῦ ∷ σ ∷ α ∷ []) "Rom.7.8"
∷ word (ἡ ∷ []) "Rom.7.8"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.7.8"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.7.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.7.8"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ῆ ∷ ς ∷ []) "Rom.7.8"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ι ∷ ρ ∷ γ ∷ ά ∷ σ ∷ α ∷ τ ∷ ο ∷ []) "Rom.7.8"
∷ word (ἐ ∷ ν ∷ []) "Rom.7.8"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Rom.7.8"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "Rom.7.8"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Rom.7.8"
∷ word (χ ∷ ω ∷ ρ ∷ ὶ ∷ ς ∷ []) "Rom.7.8"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.7.8"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.7.8"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.7.8"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ά ∷ []) "Rom.7.8"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.7.9"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.9"
∷ word (ἔ ∷ ζ ∷ ω ∷ ν ∷ []) "Rom.7.9"
∷ word (χ ∷ ω ∷ ρ ∷ ὶ ∷ ς ∷ []) "Rom.7.9"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.7.9"
∷ word (π ∷ ο ∷ τ ∷ έ ∷ []) "Rom.7.9"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ο ∷ ύ ∷ σ ∷ η ∷ ς ∷ []) "Rom.7.9"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.9"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.7.9"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ῆ ∷ ς ∷ []) "Rom.7.9"
∷ word (ἡ ∷ []) "Rom.7.9"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.7.9"
∷ word (ἀ ∷ ν ∷ έ ∷ ζ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Rom.7.9"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.7.10"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.10"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ο ∷ ν ∷ []) "Rom.7.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.7.10"
∷ word (ε ∷ ὑ ∷ ρ ∷ έ ∷ θ ∷ η ∷ []) "Rom.7.10"
∷ word (μ ∷ ο ∷ ι ∷ []) "Rom.7.10"
∷ word (ἡ ∷ []) "Rom.7.10"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ []) "Rom.7.10"
∷ word (ἡ ∷ []) "Rom.7.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.7.10"
∷ word (ζ ∷ ω ∷ ὴ ∷ ν ∷ []) "Rom.7.10"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "Rom.7.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.7.10"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "Rom.7.10"
∷ word (ἡ ∷ []) "Rom.7.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.7.11"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.7.11"
∷ word (ἀ ∷ φ ∷ ο ∷ ρ ∷ μ ∷ ὴ ∷ ν ∷ []) "Rom.7.11"
∷ word (∙λ ∷ α ∷ β ∷ ο ∷ ῦ ∷ σ ∷ α ∷ []) "Rom.7.11"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.7.11"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.7.11"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ῆ ∷ ς ∷ []) "Rom.7.11"
∷ word (ἐ ∷ ξ ∷ η ∷ π ∷ ά ∷ τ ∷ η ∷ σ ∷ έ ∷ ν ∷ []) "Rom.7.11"
∷ word (μ ∷ ε ∷ []) "Rom.7.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.7.11"
∷ word (δ ∷ ι ∷ []) "Rom.7.11"
∷ word (α ∷ ὐ ∷ τ ∷ ῆ ∷ ς ∷ []) "Rom.7.11"
∷ word (ἀ ∷ π ∷ έ ∷ κ ∷ τ ∷ ε ∷ ι ∷ ν ∷ ε ∷ ν ∷ []) "Rom.7.11"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "Rom.7.12"
∷ word (ὁ ∷ []) "Rom.7.12"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.7.12"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Rom.7.12"
∷ word (ἅ ∷ γ ∷ ι ∷ ο ∷ ς ∷ []) "Rom.7.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.7.12"
∷ word (ἡ ∷ []) "Rom.7.12"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ []) "Rom.7.12"
∷ word (ἁ ∷ γ ∷ ί ∷ α ∷ []) "Rom.7.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.7.12"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ί ∷ α ∷ []) "Rom.7.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.7.12"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ή ∷ []) "Rom.7.12"
∷ word (Τ ∷ ὸ ∷ []) "Rom.7.13"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.7.13"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ὸ ∷ ν ∷ []) "Rom.7.13"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Rom.7.13"
∷ word (ἐ ∷ γ ∷ έ ∷ ν ∷ ε ∷ τ ∷ ο ∷ []) "Rom.7.13"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.7.13"
∷ word (μ ∷ ὴ ∷ []) "Rom.7.13"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "Rom.7.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.7.13"
∷ word (ἡ ∷ []) "Rom.7.13"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.7.13"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.7.13"
∷ word (φ ∷ α ∷ ν ∷ ῇ ∷ []) "Rom.7.13"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.7.13"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.7.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.13"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ο ∷ ῦ ∷ []) "Rom.7.13"
∷ word (μ ∷ ο ∷ ι ∷ []) "Rom.7.13"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ α ∷ ζ ∷ ο ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "Rom.7.13"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "Rom.7.13"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.7.13"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Rom.7.13"
∷ word (κ ∷ α ∷ θ ∷ []) "Rom.7.13"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "Rom.7.13"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ω ∷ ∙λ ∷ ὸ ∷ ς ∷ []) "Rom.7.13"
∷ word (ἡ ∷ []) "Rom.7.13"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.7.13"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.7.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.7.13"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ῆ ∷ ς ∷ []) "Rom.7.13"
∷ word (Ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Rom.7.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.7.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.7.14"
∷ word (ὁ ∷ []) "Rom.7.14"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Rom.7.14"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ό ∷ ς ∷ []) "Rom.7.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.7.14"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.7.14"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.14"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ ι ∷ ν ∷ ό ∷ ς ∷ []) "Rom.7.14"
∷ word (ε ∷ ἰ ∷ μ ∷ ι ∷ []) "Rom.7.14"
∷ word (π ∷ ε ∷ π ∷ ρ ∷ α ∷ μ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "Rom.7.14"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Rom.7.14"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.7.14"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "Rom.7.14"
∷ word (ὃ ∷ []) "Rom.7.15"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.7.15"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.7.15"
∷ word (ο ∷ ὐ ∷ []) "Rom.7.15"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ω ∷ []) "Rom.7.15"
∷ word (ο ∷ ὐ ∷ []) "Rom.7.15"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.7.15"
∷ word (ὃ ∷ []) "Rom.7.15"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "Rom.7.15"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.7.15"
∷ word (π ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ω ∷ []) "Rom.7.15"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Rom.7.15"
∷ word (ὃ ∷ []) "Rom.7.15"
∷ word (μ ∷ ι ∷ σ ∷ ῶ ∷ []) "Rom.7.15"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.7.15"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ []) "Rom.7.15"
∷ word (ε ∷ ἰ ∷ []) "Rom.7.16"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.16"
∷ word (ὃ ∷ []) "Rom.7.16"
∷ word (ο ∷ ὐ ∷ []) "Rom.7.16"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "Rom.7.16"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.7.16"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ []) "Rom.7.16"
∷ word (σ ∷ ύ ∷ μ ∷ φ ∷ η ∷ μ ∷ ι ∷ []) "Rom.7.16"
∷ word (τ ∷ ῷ ∷ []) "Rom.7.16"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.7.16"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.7.16"
∷ word (κ ∷ α ∷ ∙λ ∷ ό ∷ ς ∷ []) "Rom.7.16"
∷ word (ν ∷ υ ∷ ν ∷ ὶ ∷ []) "Rom.7.17"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.17"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Rom.7.17"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.7.17"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.7.17"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Rom.7.17"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.7.17"
∷ word (ἡ ∷ []) "Rom.7.17"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ ῦ ∷ σ ∷ α ∷ []) "Rom.7.17"
∷ word (ἐ ∷ ν ∷ []) "Rom.7.17"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Rom.7.17"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.7.17"
∷ word (ο ∷ ἶ ∷ δ ∷ α ∷ []) "Rom.7.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.7.18"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.7.18"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.7.18"
∷ word (ο ∷ ἰ ∷ κ ∷ ε ∷ ῖ ∷ []) "Rom.7.18"
∷ word (ἐ ∷ ν ∷ []) "Rom.7.18"
∷ word (ἐ ∷ μ ∷ ο ∷ ί ∷ []) "Rom.7.18"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ []) "Rom.7.18"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.7.18"
∷ word (ἐ ∷ ν ∷ []) "Rom.7.18"
∷ word (τ ∷ ῇ ∷ []) "Rom.7.18"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ί ∷ []) "Rom.7.18"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.7.18"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ό ∷ ν ∷ []) "Rom.7.18"
∷ word (τ ∷ ὸ ∷ []) "Rom.7.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.7.18"
∷ word (θ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.7.18"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ κ ∷ ε ∷ ι ∷ τ ∷ α ∷ ί ∷ []) "Rom.7.18"
∷ word (μ ∷ ο ∷ ι ∷ []) "Rom.7.18"
∷ word (τ ∷ ὸ ∷ []) "Rom.7.18"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.18"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.7.18"
∷ word (τ ∷ ὸ ∷ []) "Rom.7.18"
∷ word (κ ∷ α ∷ ∙λ ∷ ὸ ∷ ν ∷ []) "Rom.7.18"
∷ word (ο ∷ ὔ ∷ []) "Rom.7.18"
∷ word (ο ∷ ὐ ∷ []) "Rom.7.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.7.19"
∷ word (ὃ ∷ []) "Rom.7.19"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "Rom.7.19"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ []) "Rom.7.19"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ό ∷ ν ∷ []) "Rom.7.19"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.7.19"
∷ word (ὃ ∷ []) "Rom.7.19"
∷ word (ο ∷ ὐ ∷ []) "Rom.7.19"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "Rom.7.19"
∷ word (κ ∷ α ∷ κ ∷ ὸ ∷ ν ∷ []) "Rom.7.19"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.7.19"
∷ word (π ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ω ∷ []) "Rom.7.19"
∷ word (ε ∷ ἰ ∷ []) "Rom.7.20"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.20"
∷ word (ὃ ∷ []) "Rom.7.20"
∷ word (ο ∷ ὐ ∷ []) "Rom.7.20"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "Rom.7.20"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.7.20"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ []) "Rom.7.20"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Rom.7.20"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.7.20"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.7.20"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Rom.7.20"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.7.20"
∷ word (ἡ ∷ []) "Rom.7.20"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ ῦ ∷ σ ∷ α ∷ []) "Rom.7.20"
∷ word (ἐ ∷ ν ∷ []) "Rom.7.20"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Rom.7.20"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.7.20"
∷ word (Ε ∷ ὑ ∷ ρ ∷ ί ∷ σ ∷ κ ∷ ω ∷ []) "Rom.7.21"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Rom.7.21"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.7.21"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.7.21"
∷ word (τ ∷ ῷ ∷ []) "Rom.7.21"
∷ word (θ ∷ έ ∷ ∙λ ∷ ο ∷ ν ∷ τ ∷ ι ∷ []) "Rom.7.21"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Rom.7.21"
∷ word (π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.7.21"
∷ word (τ ∷ ὸ ∷ []) "Rom.7.21"
∷ word (κ ∷ α ∷ ∙λ ∷ ὸ ∷ ν ∷ []) "Rom.7.21"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.7.21"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Rom.7.21"
∷ word (τ ∷ ὸ ∷ []) "Rom.7.21"
∷ word (κ ∷ α ∷ κ ∷ ὸ ∷ ν ∷ []) "Rom.7.21"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ κ ∷ ε ∷ ι ∷ τ ∷ α ∷ ι ∷ []) "Rom.7.21"
∷ word (σ ∷ υ ∷ ν ∷ ή ∷ δ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.7.22"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.7.22"
∷ word (τ ∷ ῷ ∷ []) "Rom.7.22"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.7.22"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.22"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.7.22"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.7.22"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.7.22"
∷ word (ἔ ∷ σ ∷ ω ∷ []) "Rom.7.22"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Rom.7.22"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ω ∷ []) "Rom.7.23"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.23"
∷ word (ἕ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "Rom.7.23"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.7.23"
∷ word (ἐ ∷ ν ∷ []) "Rom.7.23"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.7.23"
∷ word (μ ∷ έ ∷ ∙λ ∷ ε ∷ σ ∷ ί ∷ ν ∷ []) "Rom.7.23"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.7.23"
∷ word (ἀ ∷ ν ∷ τ ∷ ι ∷ σ ∷ τ ∷ ρ ∷ α ∷ τ ∷ ε ∷ υ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "Rom.7.23"
∷ word (τ ∷ ῷ ∷ []) "Rom.7.23"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.7.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.23"
∷ word (ν ∷ ο ∷ ό ∷ ς ∷ []) "Rom.7.23"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.7.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.7.23"
∷ word (α ∷ ἰ ∷ χ ∷ μ ∷ α ∷ ∙λ ∷ ω ∷ τ ∷ ί ∷ ζ ∷ ο ∷ ν ∷ τ ∷ ά ∷ []) "Rom.7.23"
∷ word (μ ∷ ε ∷ []) "Rom.7.23"
∷ word (ἐ ∷ ν ∷ []) "Rom.7.23"
∷ word (τ ∷ ῷ ∷ []) "Rom.7.23"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.7.23"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.7.23"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.7.23"
∷ word (τ ∷ ῷ ∷ []) "Rom.7.23"
∷ word (ὄ ∷ ν ∷ τ ∷ ι ∷ []) "Rom.7.23"
∷ word (ἐ ∷ ν ∷ []) "Rom.7.23"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.7.23"
∷ word (μ ∷ έ ∷ ∙λ ∷ ε ∷ σ ∷ ί ∷ ν ∷ []) "Rom.7.23"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.7.23"
∷ word (τ ∷ α ∷ ∙λ ∷ α ∷ ί ∷ π ∷ ω ∷ ρ ∷ ο ∷ ς ∷ []) "Rom.7.24"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.7.24"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "Rom.7.24"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.7.24"
∷ word (μ ∷ ε ∷ []) "Rom.7.24"
∷ word (ῥ ∷ ύ ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.7.24"
∷ word (ἐ ∷ κ ∷ []) "Rom.7.24"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.24"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.7.24"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.24"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "Rom.7.24"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "Rom.7.24"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Rom.7.25"
∷ word (τ ∷ ῷ ∷ []) "Rom.7.25"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.7.25"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.7.25"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.7.25"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.7.25"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.7.25"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.7.25"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.7.25"
∷ word (Ἄ ∷ ρ ∷ α ∷ []) "Rom.7.25"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.7.25"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.7.25"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.7.25"
∷ word (τ ∷ ῷ ∷ []) "Rom.7.25"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.7.25"
∷ word (ν ∷ ο ∷ ῒ ∷ []) "Rom.7.25"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ ω ∷ []) "Rom.7.25"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.7.25"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.7.25"
∷ word (τ ∷ ῇ ∷ []) "Rom.7.25"
∷ word (δ ∷ ὲ ∷ []) "Rom.7.25"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "Rom.7.25"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.7.25"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.7.25"
∷ word (Ο ∷ ὐ ∷ δ ∷ ὲ ∷ ν ∷ []) "Rom.8.1"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Rom.8.1"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Rom.8.1"
∷ word (κ ∷ α ∷ τ ∷ ά ∷ κ ∷ ρ ∷ ι ∷ μ ∷ α ∷ []) "Rom.8.1"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.8.1"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.8.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.8.1"
∷ word (ὁ ∷ []) "Rom.8.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.2"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Rom.8.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.2"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.8.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.2"
∷ word (ζ ∷ ω ∷ ῆ ∷ ς ∷ []) "Rom.8.2"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.8.2"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.8.2"
∷ word (ἠ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ έ ∷ ρ ∷ ω ∷ σ ∷ έ ∷ ν ∷ []) "Rom.8.2"
∷ word (σ ∷ ε ∷ []) "Rom.8.2"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.8.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.2"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.8.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.2"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.8.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.2"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "Rom.8.2"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.3"
∷ word (ἀ ∷ δ ∷ ύ ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "Rom.8.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.3"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.8.3"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.3"
∷ word (ᾧ ∷ []) "Rom.8.3"
∷ word (ἠ ∷ σ ∷ θ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "Rom.8.3"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.8.3"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.3"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ό ∷ ς ∷ []) "Rom.8.3"
∷ word (ὁ ∷ []) "Rom.8.3"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.8.3"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.8.3"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.8.3"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ν ∷ []) "Rom.8.3"
∷ word (π ∷ έ ∷ μ ∷ ψ ∷ α ∷ ς ∷ []) "Rom.8.3"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.3"
∷ word (ὁ ∷ μ ∷ ο ∷ ι ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.8.3"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Rom.8.3"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.8.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.3"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "Rom.8.3"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.8.3"
∷ word (κ ∷ α ∷ τ ∷ έ ∷ κ ∷ ρ ∷ ι ∷ ν ∷ ε ∷ []) "Rom.8.3"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.8.3"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "Rom.8.3"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.3"
∷ word (τ ∷ ῇ ∷ []) "Rom.8.3"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ί ∷ []) "Rom.8.3"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.8.4"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.4"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ί ∷ ω ∷ μ ∷ α ∷ []) "Rom.8.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.4"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.8.4"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ω ∷ θ ∷ ῇ ∷ []) "Rom.8.4"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.4"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.8.4"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.8.4"
∷ word (μ ∷ ὴ ∷ []) "Rom.8.4"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.8.4"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Rom.8.4"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.8.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.8.4"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.8.4"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.8.4"
∷ word (ο ∷ ἱ ∷ []) "Rom.8.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.5"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.8.5"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Rom.8.5"
∷ word (ὄ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.8.5"
∷ word (τ ∷ ὰ ∷ []) "Rom.8.5"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.5"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Rom.8.5"
∷ word (φ ∷ ρ ∷ ο ∷ ν ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.8.5"
∷ word (ο ∷ ἱ ∷ []) "Rom.8.5"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.5"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.8.5"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.8.5"
∷ word (τ ∷ ὰ ∷ []) "Rom.8.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.5"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.8.5"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.6"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.6"
∷ word (φ ∷ ρ ∷ ό ∷ ν ∷ η ∷ μ ∷ α ∷ []) "Rom.8.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.6"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Rom.8.6"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.8.6"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.6"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.6"
∷ word (φ ∷ ρ ∷ ό ∷ ν ∷ η ∷ μ ∷ α ∷ []) "Rom.8.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.6"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.8.6"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "Rom.8.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.6"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "Rom.8.6"
∷ word (δ ∷ ι ∷ ό ∷ τ ∷ ι ∷ []) "Rom.8.7"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.7"
∷ word (φ ∷ ρ ∷ ό ∷ ν ∷ η ∷ μ ∷ α ∷ []) "Rom.8.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.7"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Rom.8.7"
∷ word (ἔ ∷ χ ∷ θ ∷ ρ ∷ α ∷ []) "Rom.8.7"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.8.7"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "Rom.8.7"
∷ word (τ ∷ ῷ ∷ []) "Rom.8.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.7"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Rom.8.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.8.7"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "Rom.8.7"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ ά ∷ σ ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.8.7"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Rom.8.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.7"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ τ ∷ α ∷ ι ∷ []) "Rom.8.7"
∷ word (ο ∷ ἱ ∷ []) "Rom.8.8"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.8"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.8"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "Rom.8.8"
∷ word (ὄ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.8.8"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.8.8"
∷ word (ἀ ∷ ρ ∷ έ ∷ σ ∷ α ∷ ι ∷ []) "Rom.8.8"
∷ word (ο ∷ ὐ ∷ []) "Rom.8.8"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.8.8"
∷ word (Ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.8.9"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.9"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.8.9"
∷ word (ἐ ∷ σ ∷ τ ∷ ὲ ∷ []) "Rom.8.9"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.9"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "Rom.8.9"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.8.9"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.9"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.8.9"
∷ word (ε ∷ ἴ ∷ π ∷ ε ∷ ρ ∷ []) "Rom.8.9"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.8.9"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.8.9"
∷ word (ο ∷ ἰ ∷ κ ∷ ε ∷ ῖ ∷ []) "Rom.8.9"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.9"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.8.9"
∷ word (ε ∷ ἰ ∷ []) "Rom.8.9"
∷ word (δ ∷ έ ∷ []) "Rom.8.9"
∷ word (τ ∷ ι ∷ ς ∷ []) "Rom.8.9"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.8.9"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.8.9"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.8.9"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "Rom.8.9"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.8.9"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.8.9"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.8.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.8.9"
∷ word (ε ∷ ἰ ∷ []) "Rom.8.10"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.10"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.8.10"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.10"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.8.10"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.10"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.8.10"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "Rom.8.10"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ὸ ∷ ν ∷ []) "Rom.8.10"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.8.10"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "Rom.8.10"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.10"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.10"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.8.10"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "Rom.8.10"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.8.10"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.8.10"
∷ word (ε ∷ ἰ ∷ []) "Rom.8.11"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.11"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.11"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.8.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.11"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ α ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.8.11"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.8.11"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "Rom.8.11"
∷ word (ἐ ∷ κ ∷ []) "Rom.8.11"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Rom.8.11"
∷ word (ο ∷ ἰ ∷ κ ∷ ε ∷ ῖ ∷ []) "Rom.8.11"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.11"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.8.11"
∷ word (ὁ ∷ []) "Rom.8.11"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ α ∷ ς ∷ []) "Rom.8.11"
∷ word (ἐ ∷ κ ∷ []) "Rom.8.11"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Rom.8.11"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.8.11"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "Rom.8.11"
∷ word (ζ ∷ ῳ ∷ ο ∷ π ∷ ο ∷ ι ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "Rom.8.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.11"
∷ word (τ ∷ ὰ ∷ []) "Rom.8.11"
∷ word (θ ∷ ν ∷ η ∷ τ ∷ ὰ ∷ []) "Rom.8.11"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Rom.8.11"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.8.11"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.8.11"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.11"
∷ word (ἐ ∷ ν ∷ ο ∷ ι ∷ κ ∷ ο ∷ ῦ ∷ ν ∷ []) "Rom.8.11"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.8.11"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.8.11"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.11"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.8.11"
∷ word (Ἄ ∷ ρ ∷ α ∷ []) "Rom.8.12"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.8.12"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Rom.8.12"
∷ word (ὀ ∷ φ ∷ ε ∷ ι ∷ ∙λ ∷ έ ∷ τ ∷ α ∷ ι ∷ []) "Rom.8.12"
∷ word (ἐ ∷ σ ∷ μ ∷ έ ∷ ν ∷ []) "Rom.8.12"
∷ word (ο ∷ ὐ ∷ []) "Rom.8.12"
∷ word (τ ∷ ῇ ∷ []) "Rom.8.12"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "Rom.8.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.12"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.8.12"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Rom.8.12"
∷ word (ζ ∷ ῆ ∷ ν ∷ []) "Rom.8.12"
∷ word (ε ∷ ἰ ∷ []) "Rom.8.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.13"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.8.13"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Rom.8.13"
∷ word (ζ ∷ ῆ ∷ τ ∷ ε ∷ []) "Rom.8.13"
∷ word (μ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ε ∷ τ ∷ ε ∷ []) "Rom.8.13"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ ν ∷ ῄ ∷ σ ∷ κ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.8.13"
∷ word (ε ∷ ἰ ∷ []) "Rom.8.13"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.13"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.8.13"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Rom.8.13"
∷ word (π ∷ ρ ∷ ά ∷ ξ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.8.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.13"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.8.13"
∷ word (θ ∷ α ∷ ν ∷ α ∷ τ ∷ ο ∷ ῦ ∷ τ ∷ ε ∷ []) "Rom.8.13"
∷ word (ζ ∷ ή ∷ σ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Rom.8.13"
∷ word (ὅ ∷ σ ∷ ο ∷ ι ∷ []) "Rom.8.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.14"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.8.14"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.8.14"
∷ word (ἄ ∷ γ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.8.14"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ι ∷ []) "Rom.8.14"
∷ word (υ ∷ ἱ ∷ ο ∷ ί ∷ []) "Rom.8.14"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.8.14"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.8.14"
∷ word (ο ∷ ὐ ∷ []) "Rom.8.15"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.15"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ β ∷ ε ∷ τ ∷ ε ∷ []) "Rom.8.15"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.8.15"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Rom.8.15"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Rom.8.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.8.15"
∷ word (φ ∷ ό ∷ β ∷ ο ∷ ν ∷ []) "Rom.8.15"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.8.15"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ β ∷ ε ∷ τ ∷ ε ∷ []) "Rom.8.15"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.8.15"
∷ word (υ ∷ ἱ ∷ ο ∷ θ ∷ ε ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Rom.8.15"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.15"
∷ word (ᾧ ∷ []) "Rom.8.15"
∷ word (κ ∷ ρ ∷ ά ∷ ζ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.8.15"
∷ word (Α ∷ β ∷ β ∷ α ∷ []) "Rom.8.15"
∷ word (ὁ ∷ []) "Rom.8.15"
∷ word (π ∷ α ∷ τ ∷ ή ∷ ρ ∷ []) "Rom.8.15"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Rom.8.16"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.16"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.8.16"
∷ word (σ ∷ υ ∷ μ ∷ μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ε ∷ ῖ ∷ []) "Rom.8.16"
∷ word (τ ∷ ῷ ∷ []) "Rom.8.16"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.8.16"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.8.16"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.8.16"
∷ word (ἐ ∷ σ ∷ μ ∷ ὲ ∷ ν ∷ []) "Rom.8.16"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Rom.8.16"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.8.16"
∷ word (ε ∷ ἰ ∷ []) "Rom.8.17"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.17"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Rom.8.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.17"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ό ∷ μ ∷ ο ∷ ι ∷ []) "Rom.8.17"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ό ∷ μ ∷ ο ∷ ι ∷ []) "Rom.8.17"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.8.17"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.8.17"
∷ word (σ ∷ υ ∷ γ ∷ κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ό ∷ μ ∷ ο ∷ ι ∷ []) "Rom.8.17"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.8.17"
∷ word (ε ∷ ἴ ∷ π ∷ ε ∷ ρ ∷ []) "Rom.8.17"
∷ word (σ ∷ υ ∷ μ ∷ π ∷ ά ∷ σ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.8.17"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.8.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.17"
∷ word (σ ∷ υ ∷ ν ∷ δ ∷ ο ∷ ξ ∷ α ∷ σ ∷ θ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.8.17"
∷ word (Λ ∷ ο ∷ γ ∷ ί ∷ ζ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.8.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.18"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.8.18"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.8.18"
∷ word (ἄ ∷ ξ ∷ ι ∷ α ∷ []) "Rom.8.18"
∷ word (τ ∷ ὰ ∷ []) "Rom.8.18"
∷ word (π ∷ α ∷ θ ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Rom.8.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.18"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Rom.8.18"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ο ∷ ῦ ∷ []) "Rom.8.18"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.8.18"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.8.18"
∷ word (μ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ο ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "Rom.8.18"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "Rom.8.18"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ υ ∷ φ ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "Rom.8.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.8.18"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.8.18"
∷ word (ἡ ∷ []) "Rom.8.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.19"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ρ ∷ α ∷ δ ∷ ο ∷ κ ∷ ί ∷ α ∷ []) "Rom.8.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.19"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.8.19"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.8.19"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ ά ∷ ∙λ ∷ υ ∷ ψ ∷ ι ∷ ν ∷ []) "Rom.8.19"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.8.19"
∷ word (υ ∷ ἱ ∷ ῶ ∷ ν ∷ []) "Rom.8.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.19"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.8.19"
∷ word (ἀ ∷ π ∷ ε ∷ κ ∷ δ ∷ έ ∷ χ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.8.19"
∷ word (τ ∷ ῇ ∷ []) "Rom.8.20"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.20"
∷ word (μ ∷ α ∷ τ ∷ α ∷ ι ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "Rom.8.20"
∷ word (ἡ ∷ []) "Rom.8.20"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ ι ∷ ς ∷ []) "Rom.8.20"
∷ word (ὑ ∷ π ∷ ε ∷ τ ∷ ά ∷ γ ∷ η ∷ []) "Rom.8.20"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "Rom.8.20"
∷ word (ἑ ∷ κ ∷ ο ∷ ῦ ∷ σ ∷ α ∷ []) "Rom.8.20"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.8.20"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.8.20"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.8.20"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ ά ∷ ξ ∷ α ∷ ν ∷ τ ∷ α ∷ []) "Rom.8.20"
∷ word (ἐ ∷ φ ∷ []) "Rom.8.20"
∷ word (ἑ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ ι ∷ []) "Rom.8.20"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.8.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.21"
∷ word (α ∷ ὐ ∷ τ ∷ ὴ ∷ []) "Rom.8.21"
∷ word (ἡ ∷ []) "Rom.8.21"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ ι ∷ ς ∷ []) "Rom.8.21"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ ε ∷ ρ ∷ ω ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.8.21"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.8.21"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.21"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Rom.8.21"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.21"
∷ word (φ ∷ θ ∷ ο ∷ ρ ∷ ᾶ ∷ ς ∷ []) "Rom.8.21"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.8.21"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.8.21"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ ε ∷ ρ ∷ ί ∷ α ∷ ν ∷ []) "Rom.8.21"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.21"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Rom.8.21"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.8.21"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ ω ∷ ν ∷ []) "Rom.8.21"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.21"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.8.21"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Rom.8.22"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.22"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.8.22"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ []) "Rom.8.22"
∷ word (ἡ ∷ []) "Rom.8.22"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ ι ∷ ς ∷ []) "Rom.8.22"
∷ word (σ ∷ υ ∷ σ ∷ τ ∷ ε ∷ ν ∷ ά ∷ ζ ∷ ε ∷ ι ∷ []) "Rom.8.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.22"
∷ word (σ ∷ υ ∷ ν ∷ ω ∷ δ ∷ ί ∷ ν ∷ ε ∷ ι ∷ []) "Rom.8.22"
∷ word (ἄ ∷ χ ∷ ρ ∷ ι ∷ []) "Rom.8.22"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.22"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Rom.8.22"
∷ word (ο ∷ ὐ ∷ []) "Rom.8.23"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Rom.8.23"
∷ word (δ ∷ έ ∷ []) "Rom.8.23"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.8.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.23"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὶ ∷ []) "Rom.8.23"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.8.23"
∷ word (ἀ ∷ π ∷ α ∷ ρ ∷ χ ∷ ὴ ∷ ν ∷ []) "Rom.8.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.23"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.8.23"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.8.23"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.8.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.23"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὶ ∷ []) "Rom.8.23"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.23"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.8.23"
∷ word (σ ∷ τ ∷ ε ∷ ν ∷ ά ∷ ζ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.8.23"
∷ word (υ ∷ ἱ ∷ ο ∷ θ ∷ ε ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Rom.8.23"
∷ word (ἀ ∷ π ∷ ε ∷ κ ∷ δ ∷ ε ∷ χ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Rom.8.23"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.8.23"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ύ ∷ τ ∷ ρ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Rom.8.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.23"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.8.23"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.8.23"
∷ word (τ ∷ ῇ ∷ []) "Rom.8.24"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.24"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ ι ∷ []) "Rom.8.24"
∷ word (ἐ ∷ σ ∷ ώ ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "Rom.8.24"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ὶ ∷ ς ∷ []) "Rom.8.24"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.24"
∷ word (β ∷ ∙λ ∷ ε ∷ π ∷ ο ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "Rom.8.24"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.8.24"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.8.24"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ ς ∷ []) "Rom.8.24"
∷ word (ὃ ∷ []) "Rom.8.24"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.24"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ε ∷ ι ∷ []) "Rom.8.24"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.8.24"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ ζ ∷ ε ∷ ι ∷ []) "Rom.8.24"
∷ word (ε ∷ ἰ ∷ []) "Rom.8.25"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.25"
∷ word (ὃ ∷ []) "Rom.8.25"
∷ word (ο ∷ ὐ ∷ []) "Rom.8.25"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.8.25"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ ζ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.8.25"
∷ word (δ ∷ ι ∷ []) "Rom.8.25"
∷ word (ὑ ∷ π ∷ ο ∷ μ ∷ ο ∷ ν ∷ ῆ ∷ ς ∷ []) "Rom.8.25"
∷ word (ἀ ∷ π ∷ ε ∷ κ ∷ δ ∷ ε ∷ χ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.8.25"
∷ word (Ὡ ∷ σ ∷ α ∷ ύ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.8.26"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.26"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.26"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.26"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.8.26"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ ν ∷ τ ∷ ι ∷ ∙λ ∷ α ∷ μ ∷ β ∷ ά ∷ ν ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.8.26"
∷ word (τ ∷ ῇ ∷ []) "Rom.8.26"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ί ∷ ᾳ ∷ []) "Rom.8.26"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.8.26"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.26"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.26"
∷ word (τ ∷ ί ∷ []) "Rom.8.26"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ ξ ∷ ώ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.8.26"
∷ word (κ ∷ α ∷ θ ∷ ὸ ∷ []) "Rom.8.26"
∷ word (δ ∷ ε ∷ ῖ ∷ []) "Rom.8.26"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.8.26"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Rom.8.26"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.8.26"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Rom.8.26"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.26"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.8.26"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ ε ∷ ν ∷ τ ∷ υ ∷ γ ∷ χ ∷ ά ∷ ν ∷ ε ∷ ι ∷ []) "Rom.8.26"
∷ word (σ ∷ τ ∷ ε ∷ ν ∷ α ∷ γ ∷ μ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.8.26"
∷ word (ἀ ∷ ∙λ ∷ α ∷ ∙λ ∷ ή ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "Rom.8.26"
∷ word (ὁ ∷ []) "Rom.8.27"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.27"
∷ word (ἐ ∷ ρ ∷ α ∷ υ ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.8.27"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Rom.8.27"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "Rom.8.27"
∷ word (ο ∷ ἶ ∷ δ ∷ ε ∷ ν ∷ []) "Rom.8.27"
∷ word (τ ∷ ί ∷ []) "Rom.8.27"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.27"
∷ word (φ ∷ ρ ∷ ό ∷ ν ∷ η ∷ μ ∷ α ∷ []) "Rom.8.27"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.27"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.8.27"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.8.27"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.8.27"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "Rom.8.27"
∷ word (ἐ ∷ ν ∷ τ ∷ υ ∷ γ ∷ χ ∷ ά ∷ ν ∷ ε ∷ ι ∷ []) "Rom.8.27"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.8.27"
∷ word (ἁ ∷ γ ∷ ί ∷ ω ∷ ν ∷ []) "Rom.8.27"
∷ word (Ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Rom.8.28"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.28"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.8.28"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.8.28"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ σ ∷ ι ∷ []) "Rom.8.28"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.8.28"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "Rom.8.28"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Rom.8.28"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ε ∷ ῖ ∷ []) "Rom.8.28"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.8.28"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ό ∷ ν ∷ []) "Rom.8.28"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.8.28"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.8.28"
∷ word (π ∷ ρ ∷ ό ∷ θ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Rom.8.28"
∷ word (κ ∷ ∙λ ∷ η ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.8.28"
∷ word (ο ∷ ὖ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.8.28"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.8.29"
∷ word (ο ∷ ὓ ∷ ς ∷ []) "Rom.8.29"
∷ word (π ∷ ρ ∷ ο ∷ έ ∷ γ ∷ ν ∷ ω ∷ []) "Rom.8.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.29"
∷ word (π ∷ ρ ∷ ο ∷ ώ ∷ ρ ∷ ι ∷ σ ∷ ε ∷ ν ∷ []) "Rom.8.29"
∷ word (σ ∷ υ ∷ μ ∷ μ ∷ ό ∷ ρ ∷ φ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.8.29"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.29"
∷ word (ε ∷ ἰ ∷ κ ∷ ό ∷ ν ∷ ο ∷ ς ∷ []) "Rom.8.29"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.29"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "Rom.8.29"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.8.29"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.8.29"
∷ word (τ ∷ ὸ ∷ []) "Rom.8.29"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.8.29"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.8.29"
∷ word (π ∷ ρ ∷ ω ∷ τ ∷ ό ∷ τ ∷ ο ∷ κ ∷ ο ∷ ν ∷ []) "Rom.8.29"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.29"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.8.29"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.8.29"
∷ word (ο ∷ ὓ ∷ ς ∷ []) "Rom.8.30"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.30"
∷ word (π ∷ ρ ∷ ο ∷ ώ ∷ ρ ∷ ι ∷ σ ∷ ε ∷ ν ∷ []) "Rom.8.30"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.8.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.30"
∷ word (ἐ ∷ κ ∷ ά ∷ ∙λ ∷ ε ∷ σ ∷ ε ∷ ν ∷ []) "Rom.8.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.30"
∷ word (ο ∷ ὓ ∷ ς ∷ []) "Rom.8.30"
∷ word (ἐ ∷ κ ∷ ά ∷ ∙λ ∷ ε ∷ σ ∷ ε ∷ ν ∷ []) "Rom.8.30"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.8.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.30"
∷ word (ἐ ∷ δ ∷ ι ∷ κ ∷ α ∷ ί ∷ ω ∷ σ ∷ ε ∷ ν ∷ []) "Rom.8.30"
∷ word (ο ∷ ὓ ∷ ς ∷ []) "Rom.8.30"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.30"
∷ word (ἐ ∷ δ ∷ ι ∷ κ ∷ α ∷ ί ∷ ω ∷ σ ∷ ε ∷ ν ∷ []) "Rom.8.30"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.8.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.30"
∷ word (ἐ ∷ δ ∷ ό ∷ ξ ∷ α ∷ σ ∷ ε ∷ ν ∷ []) "Rom.8.30"
∷ word (Τ ∷ ί ∷ []) "Rom.8.31"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.8.31"
∷ word (ἐ ∷ ρ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.8.31"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.8.31"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "Rom.8.31"
∷ word (ε ∷ ἰ ∷ []) "Rom.8.31"
∷ word (ὁ ∷ []) "Rom.8.31"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.8.31"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.8.31"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.8.31"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.8.31"
∷ word (κ ∷ α ∷ θ ∷ []) "Rom.8.31"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.8.31"
∷ word (ὅ ∷ ς ∷ []) "Rom.8.32"
∷ word (γ ∷ ε ∷ []) "Rom.8.32"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.32"
∷ word (ἰ ∷ δ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.8.32"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "Rom.8.32"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.8.32"
∷ word (ἐ ∷ φ ∷ ε ∷ ί ∷ σ ∷ α ∷ τ ∷ ο ∷ []) "Rom.8.32"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.8.32"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.8.32"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.8.32"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.8.32"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "Rom.8.32"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "Rom.8.32"
∷ word (π ∷ ῶ ∷ ς ∷ []) "Rom.8.32"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "Rom.8.32"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.32"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Rom.8.32"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.8.32"
∷ word (τ ∷ ὰ ∷ []) "Rom.8.32"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Rom.8.32"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.8.32"
∷ word (χ ∷ α ∷ ρ ∷ ί ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.8.32"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.8.33"
∷ word (ἐ ∷ γ ∷ κ ∷ α ∷ ∙λ ∷ έ ∷ σ ∷ ε ∷ ι ∷ []) "Rom.8.33"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.8.33"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ε ∷ κ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.8.33"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.8.33"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.8.33"
∷ word (ὁ ∷ []) "Rom.8.33"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ῶ ∷ ν ∷ []) "Rom.8.33"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.8.34"
∷ word (ὁ ∷ []) "Rom.8.34"
∷ word (κ ∷ α ∷ τ ∷ α ∷ κ ∷ ρ ∷ ι ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.8.34"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.8.34"
∷ word (ὁ ∷ []) "Rom.8.34"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ α ∷ ν ∷ ώ ∷ ν ∷ []) "Rom.8.34"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.8.34"
∷ word (δ ∷ ὲ ∷ []) "Rom.8.34"
∷ word (ἐ ∷ γ ∷ ε ∷ ρ ∷ θ ∷ ε ∷ ί ∷ ς ∷ []) "Rom.8.34"
∷ word (ὅ ∷ ς ∷ []) "Rom.8.34"
∷ word (κ ∷ α ∷ ί ∷ []) "Rom.8.34"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.8.34"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.34"
∷ word (δ ∷ ε ∷ ξ ∷ ι ∷ ᾷ ∷ []) "Rom.8.34"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.34"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.8.34"
∷ word (ὃ ∷ ς ∷ []) "Rom.8.34"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.8.34"
∷ word (ἐ ∷ ν ∷ τ ∷ υ ∷ γ ∷ χ ∷ ά ∷ ν ∷ ε ∷ ι ∷ []) "Rom.8.34"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.8.34"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.8.34"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.8.35"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.8.35"
∷ word (χ ∷ ω ∷ ρ ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "Rom.8.35"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.8.35"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.35"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ς ∷ []) "Rom.8.35"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.35"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.8.35"
∷ word (θ ∷ ∙λ ∷ ῖ ∷ ψ ∷ ι ∷ ς ∷ []) "Rom.8.35"
∷ word (ἢ ∷ []) "Rom.8.35"
∷ word (σ ∷ τ ∷ ε ∷ ν ∷ ο ∷ χ ∷ ω ∷ ρ ∷ ί ∷ α ∷ []) "Rom.8.35"
∷ word (ἢ ∷ []) "Rom.8.35"
∷ word (δ ∷ ι ∷ ω ∷ γ ∷ μ ∷ ὸ ∷ ς ∷ []) "Rom.8.35"
∷ word (ἢ ∷ []) "Rom.8.35"
∷ word (∙λ ∷ ι ∷ μ ∷ ὸ ∷ ς ∷ []) "Rom.8.35"
∷ word (ἢ ∷ []) "Rom.8.35"
∷ word (γ ∷ υ ∷ μ ∷ ν ∷ ό ∷ τ ∷ η ∷ ς ∷ []) "Rom.8.35"
∷ word (ἢ ∷ []) "Rom.8.35"
∷ word (κ ∷ ί ∷ ν ∷ δ ∷ υ ∷ ν ∷ ο ∷ ς ∷ []) "Rom.8.35"
∷ word (ἢ ∷ []) "Rom.8.35"
∷ word (μ ∷ ά ∷ χ ∷ α ∷ ι ∷ ρ ∷ α ∷ []) "Rom.8.35"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.8.36"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.8.36"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.8.36"
∷ word (Ἕ ∷ ν ∷ ε ∷ κ ∷ ε ∷ ν ∷ []) "Rom.8.36"
∷ word (σ ∷ ο ∷ ῦ ∷ []) "Rom.8.36"
∷ word (θ ∷ α ∷ ν ∷ α ∷ τ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.8.36"
∷ word (ὅ ∷ ∙λ ∷ η ∷ ν ∷ []) "Rom.8.36"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.8.36"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "Rom.8.36"
∷ word (ἐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ σ ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "Rom.8.36"
∷ word (ὡ ∷ ς ∷ []) "Rom.8.36"
∷ word (π ∷ ρ ∷ ό ∷ β ∷ α ∷ τ ∷ α ∷ []) "Rom.8.36"
∷ word (σ ∷ φ ∷ α ∷ γ ∷ ῆ ∷ ς ∷ []) "Rom.8.36"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Rom.8.37"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.37"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "Rom.8.37"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.8.37"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ ν ∷ ι ∷ κ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.8.37"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.8.37"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.37"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ή ∷ σ ∷ α ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.8.37"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.8.37"
∷ word (π ∷ έ ∷ π ∷ ε ∷ ι ∷ σ ∷ μ ∷ α ∷ ι ∷ []) "Rom.8.38"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.8.38"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.8.38"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Rom.8.38"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.8.38"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Rom.8.38"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "Rom.8.38"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Rom.8.38"
∷ word (ἄ ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ο ∷ ι ∷ []) "Rom.8.38"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Rom.8.38"
∷ word (ἀ ∷ ρ ∷ χ ∷ α ∷ ὶ ∷ []) "Rom.8.38"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Rom.8.38"
∷ word (ἐ ∷ ν ∷ ε ∷ σ ∷ τ ∷ ῶ ∷ τ ∷ α ∷ []) "Rom.8.38"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Rom.8.38"
∷ word (μ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "Rom.8.38"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Rom.8.38"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.8.38"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Rom.8.39"
∷ word (ὕ ∷ ψ ∷ ω ∷ μ ∷ α ∷ []) "Rom.8.39"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Rom.8.39"
∷ word (β ∷ ά ∷ θ ∷ ο ∷ ς ∷ []) "Rom.8.39"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Rom.8.39"
∷ word (τ ∷ ι ∷ ς ∷ []) "Rom.8.39"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ ι ∷ ς ∷ []) "Rom.8.39"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Rom.8.39"
∷ word (δ ∷ υ ∷ ν ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.8.39"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.8.39"
∷ word (χ ∷ ω ∷ ρ ∷ ί ∷ σ ∷ α ∷ ι ∷ []) "Rom.8.39"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.8.39"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.39"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ς ∷ []) "Rom.8.39"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.8.39"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.8.39"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.8.39"
∷ word (ἐ ∷ ν ∷ []) "Rom.8.39"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.8.39"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.8.39"
∷ word (τ ∷ ῷ ∷ []) "Rom.8.39"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.8.39"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.8.39"
∷ word (Ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Rom.9.1"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Rom.9.1"
∷ word (ἐ ∷ ν ∷ []) "Rom.9.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.9.1"
∷ word (ο ∷ ὐ ∷ []) "Rom.9.1"
∷ word (ψ ∷ ε ∷ ύ ∷ δ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.9.1"
∷ word (σ ∷ υ ∷ μ ∷ μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ο ∷ ύ ∷ σ ∷ η ∷ ς ∷ []) "Rom.9.1"
∷ word (μ ∷ ο ∷ ι ∷ []) "Rom.9.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.9.1"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ι ∷ δ ∷ ή ∷ σ ∷ ε ∷ ώ ∷ ς ∷ []) "Rom.9.1"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.9.1"
∷ word (ἐ ∷ ν ∷ []) "Rom.9.1"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.9.1"
∷ word (ἁ ∷ γ ∷ ί ∷ ῳ ∷ []) "Rom.9.1"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.9.2"
∷ word (∙λ ∷ ύ ∷ π ∷ η ∷ []) "Rom.9.2"
∷ word (μ ∷ ο ∷ ί ∷ []) "Rom.9.2"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.9.2"
∷ word (μ ∷ ε ∷ γ ∷ ά ∷ ∙λ ∷ η ∷ []) "Rom.9.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.2"
∷ word (ἀ ∷ δ ∷ ι ∷ ά ∷ ∙λ ∷ ε ∷ ι ∷ π ∷ τ ∷ ο ∷ ς ∷ []) "Rom.9.2"
∷ word (ὀ ∷ δ ∷ ύ ∷ ν ∷ η ∷ []) "Rom.9.2"
∷ word (τ ∷ ῇ ∷ []) "Rom.9.2"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ ᾳ ∷ []) "Rom.9.2"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.9.2"
∷ word (η ∷ ὐ ∷ χ ∷ ό ∷ μ ∷ η ∷ ν ∷ []) "Rom.9.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.9.3"
∷ word (ἀ ∷ ν ∷ ά ∷ θ ∷ ε ∷ μ ∷ α ∷ []) "Rom.9.3"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.9.3"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.9.3"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.9.3"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.9.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.9.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.9.3"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.9.3"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.9.3"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ῶ ∷ ν ∷ []) "Rom.9.3"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.9.3"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.9.3"
∷ word (σ ∷ υ ∷ γ ∷ γ ∷ ε ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.9.3"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.9.3"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.9.3"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Rom.9.3"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ έ ∷ ς ∷ []) "Rom.9.4"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.9.4"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ η ∷ ∙λ ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "Rom.9.4"
∷ word (ὧ ∷ ν ∷ []) "Rom.9.4"
∷ word (ἡ ∷ []) "Rom.9.4"
∷ word (υ ∷ ἱ ∷ ο ∷ θ ∷ ε ∷ σ ∷ ί ∷ α ∷ []) "Rom.9.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.4"
∷ word (ἡ ∷ []) "Rom.9.4"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "Rom.9.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.4"
∷ word (α ∷ ἱ ∷ []) "Rom.9.4"
∷ word (δ ∷ ι ∷ α ∷ θ ∷ ῆ ∷ κ ∷ α ∷ ι ∷ []) "Rom.9.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.4"
∷ word (ἡ ∷ []) "Rom.9.4"
∷ word (ν ∷ ο ∷ μ ∷ ο ∷ θ ∷ ε ∷ σ ∷ ί ∷ α ∷ []) "Rom.9.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.4"
∷ word (ἡ ∷ []) "Rom.9.4"
∷ word (∙λ ∷ α ∷ τ ∷ ρ ∷ ε ∷ ί ∷ α ∷ []) "Rom.9.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.4"
∷ word (α ∷ ἱ ∷ []) "Rom.9.4"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ι ∷ []) "Rom.9.4"
∷ word (ὧ ∷ ν ∷ []) "Rom.9.5"
∷ word (ο ∷ ἱ ∷ []) "Rom.9.5"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ ε ∷ ς ∷ []) "Rom.9.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.5"
∷ word (ἐ ∷ ξ ∷ []) "Rom.9.5"
∷ word (ὧ ∷ ν ∷ []) "Rom.9.5"
∷ word (ὁ ∷ []) "Rom.9.5"
∷ word (χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.9.5"
∷ word (τ ∷ ὸ ∷ []) "Rom.9.5"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.9.5"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Rom.9.5"
∷ word (ὁ ∷ []) "Rom.9.5"
∷ word (ὢ ∷ ν ∷ []) "Rom.9.5"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.9.5"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.9.5"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.9.5"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ η ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.9.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.9.5"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.9.5"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ α ∷ ς ∷ []) "Rom.9.5"
∷ word (ἀ ∷ μ ∷ ή ∷ ν ∷ []) "Rom.9.5"
∷ word (Ο ∷ ὐ ∷ χ ∷ []) "Rom.9.6"
∷ word (ο ∷ ἷ ∷ ο ∷ ν ∷ []) "Rom.9.6"
∷ word (δ ∷ ὲ ∷ []) "Rom.9.6"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.9.6"
∷ word (ἐ ∷ κ ∷ π ∷ έ ∷ π ∷ τ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "Rom.9.6"
∷ word (ὁ ∷ []) "Rom.9.6"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "Rom.9.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.9.6"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.9.6"
∷ word (ο ∷ ὐ ∷ []) "Rom.9.6"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.9.6"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.9.6"
∷ word (ο ∷ ἱ ∷ []) "Rom.9.6"
∷ word (ἐ ∷ ξ ∷ []) "Rom.9.6"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ή ∷ ∙λ ∷ []) "Rom.9.6"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ι ∷ []) "Rom.9.6"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ή ∷ ∙λ ∷ []) "Rom.9.6"
∷ word (ο ∷ ὐ ∷ δ ∷ []) "Rom.9.7"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.9.7"
∷ word (ε ∷ ἰ ∷ σ ∷ ὶ ∷ ν ∷ []) "Rom.9.7"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ []) "Rom.9.7"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ά ∷ μ ∷ []) "Rom.9.7"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.9.7"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Rom.9.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Rom.9.7"
∷ word (Ἐ ∷ ν ∷ []) "Rom.9.7"
∷ word (Ἰ ∷ σ ∷ α ∷ ὰ ∷ κ ∷ []) "Rom.9.7"
∷ word (κ ∷ ∙λ ∷ η ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ί ∷ []) "Rom.9.7"
∷ word (σ ∷ ο ∷ ι ∷ []) "Rom.9.7"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ []) "Rom.9.7"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ []) "Rom.9.8"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.9.8"
∷ word (ο ∷ ὐ ∷ []) "Rom.9.8"
∷ word (τ ∷ ὰ ∷ []) "Rom.9.8"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Rom.9.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.9.8"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Rom.9.8"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "Rom.9.8"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Rom.9.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.9.8"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.9.8"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.9.8"
∷ word (τ ∷ ὰ ∷ []) "Rom.9.8"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Rom.9.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.9.8"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "Rom.9.8"
∷ word (∙λ ∷ ο ∷ γ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.9.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.9.8"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ []) "Rom.9.8"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "Rom.9.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.9.9"
∷ word (ὁ ∷ []) "Rom.9.9"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "Rom.9.9"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.9.9"
∷ word (Κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.9.9"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.9.9"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ὸ ∷ ν ∷ []) "Rom.9.9"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.9.9"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.9.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.9"
∷ word (ἔ ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "Rom.9.9"
∷ word (τ ∷ ῇ ∷ []) "Rom.9.9"
∷ word (Σ ∷ ά ∷ ρ ∷ ρ ∷ ᾳ ∷ []) "Rom.9.9"
∷ word (υ ∷ ἱ ∷ ό ∷ ς ∷ []) "Rom.9.9"
∷ word (ο ∷ ὐ ∷ []) "Rom.9.10"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Rom.9.10"
∷ word (δ ∷ έ ∷ []) "Rom.9.10"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.9.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.10"
∷ word (Ῥ ∷ ε ∷ β ∷ έ ∷ κ ∷ κ ∷ α ∷ []) "Rom.9.10"
∷ word (ἐ ∷ ξ ∷ []) "Rom.9.10"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Rom.9.10"
∷ word (κ ∷ ο ∷ ί ∷ τ ∷ η ∷ ν ∷ []) "Rom.9.10"
∷ word (ἔ ∷ χ ∷ ο ∷ υ ∷ σ ∷ α ∷ []) "Rom.9.10"
∷ word (Ἰ ∷ σ ∷ α ∷ ὰ ∷ κ ∷ []) "Rom.9.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.9.10"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.9.10"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.9.10"
∷ word (μ ∷ ή ∷ π ∷ ω ∷ []) "Rom.9.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.9.11"
∷ word (γ ∷ ε ∷ ν ∷ ν ∷ η ∷ θ ∷ έ ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.9.11"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "Rom.9.11"
∷ word (π ∷ ρ ∷ α ∷ ξ ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.9.11"
∷ word (τ ∷ ι ∷ []) "Rom.9.11"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ὸ ∷ ν ∷ []) "Rom.9.11"
∷ word (ἢ ∷ []) "Rom.9.11"
∷ word (φ ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.9.11"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.9.11"
∷ word (ἡ ∷ []) "Rom.9.11"
∷ word (κ ∷ α ∷ τ ∷ []) "Rom.9.11"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ο ∷ γ ∷ ὴ ∷ ν ∷ []) "Rom.9.11"
∷ word (π ∷ ρ ∷ ό ∷ θ ∷ ε ∷ σ ∷ ι ∷ ς ∷ []) "Rom.9.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.9.11"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.9.11"
∷ word (μ ∷ έ ∷ ν ∷ ῃ ∷ []) "Rom.9.11"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.9.12"
∷ word (ἐ ∷ ξ ∷ []) "Rom.9.12"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Rom.9.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Rom.9.12"
∷ word (ἐ ∷ κ ∷ []) "Rom.9.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.9.12"
∷ word (κ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.9.12"
∷ word (ἐ ∷ ρ ∷ ρ ∷ έ ∷ θ ∷ η ∷ []) "Rom.9.12"
∷ word (α ∷ ὐ ∷ τ ∷ ῇ ∷ []) "Rom.9.12"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.9.12"
∷ word (Ὁ ∷ []) "Rom.9.12"
∷ word (μ ∷ ε ∷ ί ∷ ζ ∷ ω ∷ ν ∷ []) "Rom.9.12"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ ε ∷ ι ∷ []) "Rom.9.12"
∷ word (τ ∷ ῷ ∷ []) "Rom.9.12"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ σ ∷ σ ∷ ο ∷ ν ∷ ι ∷ []) "Rom.9.12"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.9.13"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.9.13"
∷ word (Τ ∷ ὸ ∷ ν ∷ []) "Rom.9.13"
∷ word (Ἰ ∷ α ∷ κ ∷ ὼ ∷ β ∷ []) "Rom.9.13"
∷ word (ἠ ∷ γ ∷ ά ∷ π ∷ η ∷ σ ∷ α ∷ []) "Rom.9.13"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.9.13"
∷ word (δ ∷ ὲ ∷ []) "Rom.9.13"
∷ word (Ἠ ∷ σ ∷ α ∷ ῦ ∷ []) "Rom.9.13"
∷ word (ἐ ∷ μ ∷ ί ∷ σ ∷ η ∷ σ ∷ α ∷ []) "Rom.9.13"
∷ word (Τ ∷ ί ∷ []) "Rom.9.14"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.9.14"
∷ word (ἐ ∷ ρ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.9.14"
∷ word (μ ∷ ὴ ∷ []) "Rom.9.14"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ί ∷ α ∷ []) "Rom.9.14"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "Rom.9.14"
∷ word (τ ∷ ῷ ∷ []) "Rom.9.14"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.9.14"
∷ word (μ ∷ ὴ ∷ []) "Rom.9.14"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "Rom.9.14"
∷ word (τ ∷ ῷ ∷ []) "Rom.9.15"
∷ word (Μ ∷ ω ∷ ϋ ∷ σ ∷ ε ∷ ῖ ∷ []) "Rom.9.15"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.9.15"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.9.15"
∷ word (Ἐ ∷ ∙λ ∷ ε ∷ ή ∷ σ ∷ ω ∷ []) "Rom.9.15"
∷ word (ὃ ∷ ν ∷ []) "Rom.9.15"
∷ word (ἂ ∷ ν ∷ []) "Rom.9.15"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ῶ ∷ []) "Rom.9.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.15"
∷ word (ο ∷ ἰ ∷ κ ∷ τ ∷ ι ∷ ρ ∷ ή ∷ σ ∷ ω ∷ []) "Rom.9.15"
∷ word (ὃ ∷ ν ∷ []) "Rom.9.15"
∷ word (ἂ ∷ ν ∷ []) "Rom.9.15"
∷ word (ο ∷ ἰ ∷ κ ∷ τ ∷ ί ∷ ρ ∷ ω ∷ []) "Rom.9.15"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Rom.9.16"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.9.16"
∷ word (ο ∷ ὐ ∷ []) "Rom.9.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.9.16"
∷ word (θ ∷ έ ∷ ∙λ ∷ ο ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.9.16"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Rom.9.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.9.16"
∷ word (τ ∷ ρ ∷ έ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.9.16"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.9.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.9.16"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ῶ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.9.16"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.9.16"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.9.17"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.9.17"
∷ word (ἡ ∷ []) "Rom.9.17"
∷ word (γ ∷ ρ ∷ α ∷ φ ∷ ὴ ∷ []) "Rom.9.17"
∷ word (τ ∷ ῷ ∷ []) "Rom.9.17"
∷ word (Φ ∷ α ∷ ρ ∷ α ∷ ὼ ∷ []) "Rom.9.17"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.9.17"
∷ word (Ε ∷ ἰ ∷ ς ∷ []) "Rom.9.17"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Rom.9.17"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.9.17"
∷ word (ἐ ∷ ξ ∷ ή ∷ γ ∷ ε ∷ ι ∷ ρ ∷ ά ∷ []) "Rom.9.17"
∷ word (σ ∷ ε ∷ []) "Rom.9.17"
∷ word (ὅ ∷ π ∷ ω ∷ ς ∷ []) "Rom.9.17"
∷ word (ἐ ∷ ν ∷ δ ∷ ε ∷ ί ∷ ξ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "Rom.9.17"
∷ word (ἐ ∷ ν ∷ []) "Rom.9.17"
∷ word (σ ∷ ο ∷ ὶ ∷ []) "Rom.9.17"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.9.17"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ί ∷ ν ∷ []) "Rom.9.17"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.9.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.17"
∷ word (ὅ ∷ π ∷ ω ∷ ς ∷ []) "Rom.9.17"
∷ word (δ ∷ ι ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ῇ ∷ []) "Rom.9.17"
∷ word (τ ∷ ὸ ∷ []) "Rom.9.17"
∷ word (ὄ ∷ ν ∷ ο ∷ μ ∷ ά ∷ []) "Rom.9.17"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.9.17"
∷ word (ἐ ∷ ν ∷ []) "Rom.9.17"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "Rom.9.17"
∷ word (τ ∷ ῇ ∷ []) "Rom.9.17"
∷ word (γ ∷ ῇ ∷ []) "Rom.9.17"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Rom.9.18"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.9.18"
∷ word (ὃ ∷ ν ∷ []) "Rom.9.18"
∷ word (θ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ []) "Rom.9.18"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ε ∷ ῖ ∷ []) "Rom.9.18"
∷ word (ὃ ∷ ν ∷ []) "Rom.9.18"
∷ word (δ ∷ ὲ ∷ []) "Rom.9.18"
∷ word (θ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ []) "Rom.9.18"
∷ word (σ ∷ κ ∷ ∙λ ∷ η ∷ ρ ∷ ύ ∷ ν ∷ ε ∷ ι ∷ []) "Rom.9.18"
∷ word (Ἐ ∷ ρ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.9.19"
∷ word (μ ∷ ο ∷ ι ∷ []) "Rom.9.19"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.9.19"
∷ word (Τ ∷ ί ∷ []) "Rom.9.19"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.9.19"
∷ word (ἔ ∷ τ ∷ ι ∷ []) "Rom.9.19"
∷ word (μ ∷ έ ∷ μ ∷ φ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.9.19"
∷ word (τ ∷ ῷ ∷ []) "Rom.9.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.9.19"
∷ word (β ∷ ο ∷ υ ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.9.19"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.9.19"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.9.19"
∷ word (ἀ ∷ ν ∷ θ ∷ έ ∷ σ ∷ τ ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "Rom.9.19"
∷ word (ὦ ∷ []) "Rom.9.20"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ε ∷ []) "Rom.9.20"
∷ word (μ ∷ ε ∷ ν ∷ ο ∷ ῦ ∷ ν ∷ γ ∷ ε ∷ []) "Rom.9.20"
∷ word (σ ∷ ὺ ∷ []) "Rom.9.20"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.9.20"
∷ word (ε ∷ ἶ ∷ []) "Rom.9.20"
∷ word (ὁ ∷ []) "Rom.9.20"
∷ word (ἀ ∷ ν ∷ τ ∷ α ∷ π ∷ ο ∷ κ ∷ ρ ∷ ι ∷ ν ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Rom.9.20"
∷ word (τ ∷ ῷ ∷ []) "Rom.9.20"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.9.20"
∷ word (μ ∷ ὴ ∷ []) "Rom.9.20"
∷ word (ἐ ∷ ρ ∷ ε ∷ ῖ ∷ []) "Rom.9.20"
∷ word (τ ∷ ὸ ∷ []) "Rom.9.20"
∷ word (π ∷ ∙λ ∷ ά ∷ σ ∷ μ ∷ α ∷ []) "Rom.9.20"
∷ word (τ ∷ ῷ ∷ []) "Rom.9.20"
∷ word (π ∷ ∙λ ∷ ά ∷ σ ∷ α ∷ ν ∷ τ ∷ ι ∷ []) "Rom.9.20"
∷ word (Τ ∷ ί ∷ []) "Rom.9.20"
∷ word (μ ∷ ε ∷ []) "Rom.9.20"
∷ word (ἐ ∷ π ∷ ο ∷ ί ∷ η ∷ σ ∷ α ∷ ς ∷ []) "Rom.9.20"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.9.20"
∷ word (ἢ ∷ []) "Rom.9.21"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.9.21"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "Rom.9.21"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Rom.9.21"
∷ word (ὁ ∷ []) "Rom.9.21"
∷ word (κ ∷ ε ∷ ρ ∷ α ∷ μ ∷ ε ∷ ὺ ∷ ς ∷ []) "Rom.9.21"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.9.21"
∷ word (π ∷ η ∷ ∙λ ∷ ο ∷ ῦ ∷ []) "Rom.9.21"
∷ word (ἐ ∷ κ ∷ []) "Rom.9.21"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.9.21"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.9.21"
∷ word (φ ∷ υ ∷ ρ ∷ ά ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.9.21"
∷ word (π ∷ ο ∷ ι ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Rom.9.21"
∷ word (ὃ ∷ []) "Rom.9.21"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.9.21"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.9.21"
∷ word (τ ∷ ι ∷ μ ∷ ὴ ∷ ν ∷ []) "Rom.9.21"
∷ word (σ ∷ κ ∷ ε ∷ ῦ ∷ ο ∷ ς ∷ []) "Rom.9.21"
∷ word (ὃ ∷ []) "Rom.9.21"
∷ word (δ ∷ ὲ ∷ []) "Rom.9.21"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.9.21"
∷ word (ἀ ∷ τ ∷ ι ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Rom.9.21"
∷ word (ε ∷ ἰ ∷ []) "Rom.9.22"
∷ word (δ ∷ ὲ ∷ []) "Rom.9.22"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ ν ∷ []) "Rom.9.22"
∷ word (ὁ ∷ []) "Rom.9.22"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.9.22"
∷ word (ἐ ∷ ν ∷ δ ∷ ε ∷ ί ∷ ξ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.9.22"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.9.22"
∷ word (ὀ ∷ ρ ∷ γ ∷ ὴ ∷ ν ∷ []) "Rom.9.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.22"
∷ word (γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ σ ∷ α ∷ ι ∷ []) "Rom.9.22"
∷ word (τ ∷ ὸ ∷ []) "Rom.9.22"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.9.22"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.9.22"
∷ word (ἤ ∷ ν ∷ ε ∷ γ ∷ κ ∷ ε ∷ ν ∷ []) "Rom.9.22"
∷ word (ἐ ∷ ν ∷ []) "Rom.9.22"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῇ ∷ []) "Rom.9.22"
∷ word (μ ∷ α ∷ κ ∷ ρ ∷ ο ∷ θ ∷ υ ∷ μ ∷ ί ∷ ᾳ ∷ []) "Rom.9.22"
∷ word (σ ∷ κ ∷ ε ∷ ύ ∷ η ∷ []) "Rom.9.22"
∷ word (ὀ ∷ ρ ∷ γ ∷ ῆ ∷ ς ∷ []) "Rom.9.22"
∷ word (κ ∷ α ∷ τ ∷ η ∷ ρ ∷ τ ∷ ι ∷ σ ∷ μ ∷ έ ∷ ν ∷ α ∷ []) "Rom.9.22"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.9.22"
∷ word (ἀ ∷ π ∷ ώ ∷ ∙λ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Rom.9.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.23"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.9.23"
∷ word (γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ σ ∷ ῃ ∷ []) "Rom.9.23"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.9.23"
∷ word (π ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.9.23"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.9.23"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Rom.9.23"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.9.23"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.9.23"
∷ word (σ ∷ κ ∷ ε ∷ ύ ∷ η ∷ []) "Rom.9.23"
∷ word (ἐ ∷ ∙λ ∷ έ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.9.23"
∷ word (ἃ ∷ []) "Rom.9.23"
∷ word (π ∷ ρ ∷ ο ∷ η ∷ τ ∷ ο ∷ ί ∷ μ ∷ α ∷ σ ∷ ε ∷ ν ∷ []) "Rom.9.23"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.9.23"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "Rom.9.23"
∷ word (ο ∷ ὓ ∷ ς ∷ []) "Rom.9.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.24"
∷ word (ἐ ∷ κ ∷ ά ∷ ∙λ ∷ ε ∷ σ ∷ ε ∷ ν ∷ []) "Rom.9.24"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.9.24"
∷ word (ο ∷ ὐ ∷ []) "Rom.9.24"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Rom.9.24"
∷ word (ἐ ∷ ξ ∷ []) "Rom.9.24"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ω ∷ ν ∷ []) "Rom.9.24"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.9.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.24"
∷ word (ἐ ∷ ξ ∷ []) "Rom.9.24"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.9.24"
∷ word (ὡ ∷ ς ∷ []) "Rom.9.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.25"
∷ word (ἐ ∷ ν ∷ []) "Rom.9.25"
∷ word (τ ∷ ῷ ∷ []) "Rom.9.25"
∷ word (Ὡ ∷ σ ∷ η ∷ ὲ ∷ []) "Rom.9.25"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.9.25"
∷ word (Κ ∷ α ∷ ∙λ ∷ έ ∷ σ ∷ ω ∷ []) "Rom.9.25"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.9.25"
∷ word (ο ∷ ὐ ∷ []) "Rom.9.25"
∷ word (∙λ ∷ α ∷ ό ∷ ν ∷ []) "Rom.9.25"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.9.25"
∷ word (∙λ ∷ α ∷ ό ∷ ν ∷ []) "Rom.9.25"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.9.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.25"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.9.25"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.9.25"
∷ word (ἠ ∷ γ ∷ α ∷ π ∷ η ∷ μ ∷ έ ∷ ν ∷ η ∷ ν ∷ []) "Rom.9.25"
∷ word (ἠ ∷ γ ∷ α ∷ π ∷ η ∷ μ ∷ έ ∷ ν ∷ η ∷ ν ∷ []) "Rom.9.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.26"
∷ word (ἔ ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "Rom.9.26"
∷ word (ἐ ∷ ν ∷ []) "Rom.9.26"
∷ word (τ ∷ ῷ ∷ []) "Rom.9.26"
∷ word (τ ∷ ό ∷ π ∷ ῳ ∷ []) "Rom.9.26"
∷ word (ο ∷ ὗ ∷ []) "Rom.9.26"
∷ word (ἐ ∷ ρ ∷ ρ ∷ έ ∷ θ ∷ η ∷ []) "Rom.9.26"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.9.26"
∷ word (Ο ∷ ὐ ∷ []) "Rom.9.26"
∷ word (∙λ ∷ α ∷ ό ∷ ς ∷ []) "Rom.9.26"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.9.26"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.9.26"
∷ word (ἐ ∷ κ ∷ ε ∷ ῖ ∷ []) "Rom.9.26"
∷ word (κ ∷ ∙λ ∷ η ∷ θ ∷ ή ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.9.26"
∷ word (υ ∷ ἱ ∷ ο ∷ ὶ ∷ []) "Rom.9.26"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.9.26"
∷ word (ζ ∷ ῶ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.9.26"
∷ word (Ἠ ∷ σ ∷ α ∷ ΐ ∷ α ∷ ς ∷ []) "Rom.9.27"
∷ word (δ ∷ ὲ ∷ []) "Rom.9.27"
∷ word (κ ∷ ρ ∷ ά ∷ ζ ∷ ε ∷ ι ∷ []) "Rom.9.27"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.9.27"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.9.27"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ή ∷ ∙λ ∷ []) "Rom.9.27"
∷ word (Ἐ ∷ ὰ ∷ ν ∷ []) "Rom.9.27"
∷ word (ᾖ ∷ []) "Rom.9.27"
∷ word (ὁ ∷ []) "Rom.9.27"
∷ word (ἀ ∷ ρ ∷ ι ∷ θ ∷ μ ∷ ὸ ∷ ς ∷ []) "Rom.9.27"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.9.27"
∷ word (υ ∷ ἱ ∷ ῶ ∷ ν ∷ []) "Rom.9.27"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ὴ ∷ ∙λ ∷ []) "Rom.9.27"
∷ word (ὡ ∷ ς ∷ []) "Rom.9.27"
∷ word (ἡ ∷ []) "Rom.9.27"
∷ word (ἄ ∷ μ ∷ μ ∷ ο ∷ ς ∷ []) "Rom.9.27"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.9.27"
∷ word (θ ∷ α ∷ ∙λ ∷ ά ∷ σ ∷ σ ∷ η ∷ ς ∷ []) "Rom.9.27"
∷ word (τ ∷ ὸ ∷ []) "Rom.9.27"
∷ word (ὑ ∷ π ∷ ό ∷ ∙λ ∷ ε ∷ ι ∷ μ ∷ μ ∷ α ∷ []) "Rom.9.27"
∷ word (σ ∷ ω ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.9.27"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ν ∷ []) "Rom.9.28"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.9.28"
∷ word (σ ∷ υ ∷ ν ∷ τ ∷ ε ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "Rom.9.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.28"
∷ word (σ ∷ υ ∷ ν ∷ τ ∷ έ ∷ μ ∷ ν ∷ ω ∷ ν ∷ []) "Rom.9.28"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "Rom.9.28"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "Rom.9.28"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.9.28"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.9.28"
∷ word (γ ∷ ῆ ∷ ς ∷ []) "Rom.9.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.29"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.9.29"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ ί ∷ ρ ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "Rom.9.29"
∷ word (Ἠ ∷ σ ∷ α ∷ ΐ ∷ α ∷ ς ∷ []) "Rom.9.29"
∷ word (Ε ∷ ἰ ∷ []) "Rom.9.29"
∷ word (μ ∷ ὴ ∷ []) "Rom.9.29"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "Rom.9.29"
∷ word (Σ ∷ α ∷ β ∷ α ∷ ὼ ∷ θ ∷ []) "Rom.9.29"
∷ word (ἐ ∷ γ ∷ κ ∷ α ∷ τ ∷ έ ∷ ∙λ ∷ ι ∷ π ∷ ε ∷ ν ∷ []) "Rom.9.29"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.9.29"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ []) "Rom.9.29"
∷ word (ὡ ∷ ς ∷ []) "Rom.9.29"
∷ word (Σ ∷ ό ∷ δ ∷ ο ∷ μ ∷ α ∷ []) "Rom.9.29"
∷ word (ἂ ∷ ν ∷ []) "Rom.9.29"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ή ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "Rom.9.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.29"
∷ word (ὡ ∷ ς ∷ []) "Rom.9.29"
∷ word (Γ ∷ ό ∷ μ ∷ ο ∷ ρ ∷ ρ ∷ α ∷ []) "Rom.9.29"
∷ word (ἂ ∷ ν ∷ []) "Rom.9.29"
∷ word (ὡ ∷ μ ∷ ο ∷ ι ∷ ώ ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "Rom.9.29"
∷ word (Τ ∷ ί ∷ []) "Rom.9.30"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.9.30"
∷ word (ἐ ∷ ρ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.9.30"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.9.30"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Rom.9.30"
∷ word (τ ∷ ὰ ∷ []) "Rom.9.30"
∷ word (μ ∷ ὴ ∷ []) "Rom.9.30"
∷ word (δ ∷ ι ∷ ώ ∷ κ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "Rom.9.30"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.9.30"
∷ word (κ ∷ α ∷ τ ∷ έ ∷ ∙λ ∷ α ∷ β ∷ ε ∷ ν ∷ []) "Rom.9.30"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.9.30"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.9.30"
∷ word (δ ∷ ὲ ∷ []) "Rom.9.30"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.9.30"
∷ word (ἐ ∷ κ ∷ []) "Rom.9.30"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.9.30"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ὴ ∷ ∙λ ∷ []) "Rom.9.31"
∷ word (δ ∷ ὲ ∷ []) "Rom.9.31"
∷ word (δ ∷ ι ∷ ώ ∷ κ ∷ ω ∷ ν ∷ []) "Rom.9.31"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.9.31"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "Rom.9.31"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.9.31"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.9.31"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.9.31"
∷ word (ἔ ∷ φ ∷ θ ∷ α ∷ σ ∷ ε ∷ ν ∷ []) "Rom.9.31"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.9.32"
∷ word (τ ∷ ί ∷ []) "Rom.9.32"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.9.32"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.9.32"
∷ word (ἐ ∷ κ ∷ []) "Rom.9.32"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.9.32"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Rom.9.32"
∷ word (ὡ ∷ ς ∷ []) "Rom.9.32"
∷ word (ἐ ∷ ξ ∷ []) "Rom.9.32"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Rom.9.32"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ έ ∷ κ ∷ ο ∷ ψ ∷ α ∷ ν ∷ []) "Rom.9.32"
∷ word (τ ∷ ῷ ∷ []) "Rom.9.32"
∷ word (∙λ ∷ ί ∷ θ ∷ ῳ ∷ []) "Rom.9.32"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.9.32"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ κ ∷ ό ∷ μ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.9.32"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.9.33"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.9.33"
∷ word (Ἰ ∷ δ ∷ ο ∷ ὺ ∷ []) "Rom.9.33"
∷ word (τ ∷ ί ∷ θ ∷ η ∷ μ ∷ ι ∷ []) "Rom.9.33"
∷ word (ἐ ∷ ν ∷ []) "Rom.9.33"
∷ word (Σ ∷ ι ∷ ὼ ∷ ν ∷ []) "Rom.9.33"
∷ word (∙λ ∷ ί ∷ θ ∷ ο ∷ ν ∷ []) "Rom.9.33"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ κ ∷ ό ∷ μ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.9.33"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.33"
∷ word (π ∷ έ ∷ τ ∷ ρ ∷ α ∷ ν ∷ []) "Rom.9.33"
∷ word (σ ∷ κ ∷ α ∷ ν ∷ δ ∷ ά ∷ ∙λ ∷ ο ∷ υ ∷ []) "Rom.9.33"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.9.33"
∷ word (ὁ ∷ []) "Rom.9.33"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "Rom.9.33"
∷ word (ἐ ∷ π ∷ []) "Rom.9.33"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.9.33"
∷ word (ο ∷ ὐ ∷ []) "Rom.9.33"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ι ∷ σ ∷ χ ∷ υ ∷ ν ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.9.33"
∷ word (Ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Rom.10.1"
∷ word (ἡ ∷ []) "Rom.10.1"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.10.1"
∷ word (ε ∷ ὐ ∷ δ ∷ ο ∷ κ ∷ ί ∷ α ∷ []) "Rom.10.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.10.1"
∷ word (ἐ ∷ μ ∷ ῆ ∷ ς ∷ []) "Rom.10.1"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "Rom.10.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.10.1"
∷ word (ἡ ∷ []) "Rom.10.1"
∷ word (δ ∷ έ ∷ η ∷ σ ∷ ι ∷ ς ∷ []) "Rom.10.1"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.10.1"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.10.1"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "Rom.10.1"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.10.1"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.10.1"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.10.1"
∷ word (σ ∷ ω ∷ τ ∷ η ∷ ρ ∷ ί ∷ α ∷ ν ∷ []) "Rom.10.1"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ῶ ∷ []) "Rom.10.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.10.2"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.10.2"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.10.2"
∷ word (ζ ∷ ῆ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.10.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.10.2"
∷ word (ἔ ∷ χ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.10.2"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Rom.10.2"
∷ word (ο ∷ ὐ ∷ []) "Rom.10.2"
∷ word (κ ∷ α ∷ τ ∷ []) "Rom.10.2"
∷ word (ἐ ∷ π ∷ ί ∷ γ ∷ ν ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Rom.10.2"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.10.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.10.3"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.10.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.10.3"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.10.3"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.10.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.10.3"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.10.3"
∷ word (ἰ ∷ δ ∷ ί ∷ α ∷ ν ∷ []) "Rom.10.3"
∷ word (ζ ∷ η ∷ τ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.10.3"
∷ word (σ ∷ τ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Rom.10.3"
∷ word (τ ∷ ῇ ∷ []) "Rom.10.3"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ ῃ ∷ []) "Rom.10.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.10.3"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.10.3"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "Rom.10.3"
∷ word (ὑ ∷ π ∷ ε ∷ τ ∷ ά ∷ γ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.10.3"
∷ word (τ ∷ έ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Rom.10.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.10.4"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.10.4"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.10.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.10.4"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.10.4"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "Rom.10.4"
∷ word (τ ∷ ῷ ∷ []) "Rom.10.4"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ ι ∷ []) "Rom.10.4"
∷ word (Μ ∷ ω ∷ ϋ ∷ σ ∷ ῆ ∷ ς ∷ []) "Rom.10.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.10.5"
∷ word (γ ∷ ρ ∷ ά ∷ φ ∷ ε ∷ ι ∷ []) "Rom.10.5"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.10.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.10.5"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.10.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.10.5"
∷ word (ἐ ∷ κ ∷ []) "Rom.10.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.10.5"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.10.5"
∷ word (ὁ ∷ []) "Rom.10.5"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ α ∷ ς ∷ []) "Rom.10.5"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "Rom.10.5"
∷ word (ζ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.10.5"
∷ word (ἐ ∷ ν ∷ []) "Rom.10.5"
∷ word (α ∷ ὐ ∷ τ ∷ ῇ ∷ []) "Rom.10.5"
∷ word (ἡ ∷ []) "Rom.10.6"
∷ word (δ ∷ ὲ ∷ []) "Rom.10.6"
∷ word (ἐ ∷ κ ∷ []) "Rom.10.6"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.10.6"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ []) "Rom.10.6"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.10.6"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.10.6"
∷ word (Μ ∷ ὴ ∷ []) "Rom.10.6"
∷ word (ε ∷ ἴ ∷ π ∷ ῃ ∷ ς ∷ []) "Rom.10.6"
∷ word (ἐ ∷ ν ∷ []) "Rom.10.6"
∷ word (τ ∷ ῇ ∷ []) "Rom.10.6"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ ᾳ ∷ []) "Rom.10.6"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.10.6"
∷ word (Τ ∷ ί ∷ ς ∷ []) "Rom.10.6"
∷ word (ἀ ∷ ν ∷ α ∷ β ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.10.6"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.10.6"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.10.6"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ό ∷ ν ∷ []) "Rom.10.6"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ []) "Rom.10.6"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.10.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.10.6"
∷ word (κ ∷ α ∷ τ ∷ α ∷ γ ∷ α ∷ γ ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.10.6"
∷ word (ἤ ∷ []) "Rom.10.7"
∷ word (Τ ∷ ί ∷ ς ∷ []) "Rom.10.7"
∷ word (κ ∷ α ∷ τ ∷ α ∷ β ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.10.7"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.10.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.10.7"
∷ word (ἄ ∷ β ∷ υ ∷ σ ∷ σ ∷ ο ∷ ν ∷ []) "Rom.10.7"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ []) "Rom.10.7"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.10.7"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.10.7"
∷ word (ἐ ∷ κ ∷ []) "Rom.10.7"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Rom.10.7"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ α ∷ γ ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.10.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.10.8"
∷ word (τ ∷ ί ∷ []) "Rom.10.8"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.10.8"
∷ word (Ἐ ∷ γ ∷ γ ∷ ύ ∷ ς ∷ []) "Rom.10.8"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.10.8"
∷ word (τ ∷ ὸ ∷ []) "Rom.10.8"
∷ word (ῥ ∷ ῆ ∷ μ ∷ ά ∷ []) "Rom.10.8"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.10.8"
∷ word (ἐ ∷ ν ∷ []) "Rom.10.8"
∷ word (τ ∷ ῷ ∷ []) "Rom.10.8"
∷ word (σ ∷ τ ∷ ό ∷ μ ∷ α ∷ τ ∷ ί ∷ []) "Rom.10.8"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.10.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.10.8"
∷ word (ἐ ∷ ν ∷ []) "Rom.10.8"
∷ word (τ ∷ ῇ ∷ []) "Rom.10.8"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ ᾳ ∷ []) "Rom.10.8"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.10.8"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ []) "Rom.10.8"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.10.8"
∷ word (τ ∷ ὸ ∷ []) "Rom.10.8"
∷ word (ῥ ∷ ῆ ∷ μ ∷ α ∷ []) "Rom.10.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.10.8"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.10.8"
∷ word (ὃ ∷ []) "Rom.10.8"
∷ word (κ ∷ η ∷ ρ ∷ ύ ∷ σ ∷ σ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.10.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.10.9"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.10.9"
∷ word (ὁ ∷ μ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ή ∷ σ ∷ ῃ ∷ ς ∷ []) "Rom.10.9"
∷ word (ἐ ∷ ν ∷ []) "Rom.10.9"
∷ word (τ ∷ ῷ ∷ []) "Rom.10.9"
∷ word (σ ∷ τ ∷ ό ∷ μ ∷ α ∷ τ ∷ ί ∷ []) "Rom.10.9"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.10.9"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.10.9"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "Rom.10.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.10.9"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ σ ∷ ῃ ∷ ς ∷ []) "Rom.10.9"
∷ word (ἐ ∷ ν ∷ []) "Rom.10.9"
∷ word (τ ∷ ῇ ∷ []) "Rom.10.9"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ ᾳ ∷ []) "Rom.10.9"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.10.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.10.9"
∷ word (ὁ ∷ []) "Rom.10.9"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.10.9"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.10.9"
∷ word (ἤ ∷ γ ∷ ε ∷ ι ∷ ρ ∷ ε ∷ ν ∷ []) "Rom.10.9"
∷ word (ἐ ∷ κ ∷ []) "Rom.10.9"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Rom.10.9"
∷ word (σ ∷ ω ∷ θ ∷ ή ∷ σ ∷ ῃ ∷ []) "Rom.10.9"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ ᾳ ∷ []) "Rom.10.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.10.10"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.10.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.10.10"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Rom.10.10"
∷ word (σ ∷ τ ∷ ό ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.10.10"
∷ word (δ ∷ ὲ ∷ []) "Rom.10.10"
∷ word (ὁ ∷ μ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "Rom.10.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.10.10"
∷ word (σ ∷ ω ∷ τ ∷ η ∷ ρ ∷ ί ∷ α ∷ ν ∷ []) "Rom.10.10"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.10.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.10.11"
∷ word (ἡ ∷ []) "Rom.10.11"
∷ word (γ ∷ ρ ∷ α ∷ φ ∷ ή ∷ []) "Rom.10.11"
∷ word (Π ∷ ᾶ ∷ ς ∷ []) "Rom.10.11"
∷ word (ὁ ∷ []) "Rom.10.11"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "Rom.10.11"
∷ word (ἐ ∷ π ∷ []) "Rom.10.11"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.10.11"
∷ word (ο ∷ ὐ ∷ []) "Rom.10.11"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ι ∷ σ ∷ χ ∷ υ ∷ ν ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.10.11"
∷ word (ο ∷ ὐ ∷ []) "Rom.10.12"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Rom.10.12"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.10.12"
∷ word (δ ∷ ι ∷ α ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ []) "Rom.10.12"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ο ∷ υ ∷ []) "Rom.10.12"
∷ word (τ ∷ ε ∷ []) "Rom.10.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.10.12"
∷ word (Ἕ ∷ ∙λ ∷ ∙λ ∷ η ∷ ν ∷ ο ∷ ς ∷ []) "Rom.10.12"
∷ word (ὁ ∷ []) "Rom.10.12"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.10.12"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.10.12"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "Rom.10.12"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.10.12"
∷ word (π ∷ ∙λ ∷ ο ∷ υ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.10.12"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.10.12"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.10.12"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.10.12"
∷ word (ἐ ∷ π ∷ ι ∷ κ ∷ α ∷ ∙λ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "Rom.10.12"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "Rom.10.12"
∷ word (Π ∷ ᾶ ∷ ς ∷ []) "Rom.10.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.10.13"
∷ word (ὃ ∷ ς ∷ []) "Rom.10.13"
∷ word (ἂ ∷ ν ∷ []) "Rom.10.13"
∷ word (ἐ ∷ π ∷ ι ∷ κ ∷ α ∷ ∙λ ∷ έ ∷ σ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Rom.10.13"
∷ word (τ ∷ ὸ ∷ []) "Rom.10.13"
∷ word (ὄ ∷ ν ∷ ο ∷ μ ∷ α ∷ []) "Rom.10.13"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.10.13"
∷ word (σ ∷ ω ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.10.13"
∷ word (Π ∷ ῶ ∷ ς ∷ []) "Rom.10.14"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.10.14"
∷ word (ἐ ∷ π ∷ ι ∷ κ ∷ α ∷ ∙λ ∷ έ ∷ σ ∷ ω ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.10.14"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.10.14"
∷ word (ὃ ∷ ν ∷ []) "Rom.10.14"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.10.14"
∷ word (ἐ ∷ π ∷ ί ∷ σ ∷ τ ∷ ε ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "Rom.10.14"
∷ word (π ∷ ῶ ∷ ς ∷ []) "Rom.10.14"
∷ word (δ ∷ ὲ ∷ []) "Rom.10.14"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ σ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Rom.10.14"
∷ word (ο ∷ ὗ ∷ []) "Rom.10.14"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.10.14"
∷ word (ἤ ∷ κ ∷ ο ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "Rom.10.14"
∷ word (π ∷ ῶ ∷ ς ∷ []) "Rom.10.14"
∷ word (δ ∷ ὲ ∷ []) "Rom.10.14"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Rom.10.14"
∷ word (χ ∷ ω ∷ ρ ∷ ὶ ∷ ς ∷ []) "Rom.10.14"
∷ word (κ ∷ η ∷ ρ ∷ ύ ∷ σ ∷ σ ∷ ο ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Rom.10.14"
∷ word (π ∷ ῶ ∷ ς ∷ []) "Rom.10.15"
∷ word (δ ∷ ὲ ∷ []) "Rom.10.15"
∷ word (κ ∷ η ∷ ρ ∷ ύ ∷ ξ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Rom.10.15"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.10.15"
∷ word (μ ∷ ὴ ∷ []) "Rom.10.15"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ α ∷ ∙λ ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.10.15"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.10.15"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.10.15"
∷ word (Ὡ ∷ ς ∷ []) "Rom.10.15"
∷ word (ὡ ∷ ρ ∷ α ∷ ῖ ∷ ο ∷ ι ∷ []) "Rom.10.15"
∷ word (ο ∷ ἱ ∷ []) "Rom.10.15"
∷ word (π ∷ ό ∷ δ ∷ ε ∷ ς ∷ []) "Rom.10.15"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.10.15"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ι ∷ ζ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ω ∷ ν ∷ []) "Rom.10.15"
∷ word (τ ∷ ὰ ∷ []) "Rom.10.15"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ά ∷ []) "Rom.10.15"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Rom.10.16"
∷ word (ο ∷ ὐ ∷ []) "Rom.10.16"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.10.16"
∷ word (ὑ ∷ π ∷ ή ∷ κ ∷ ο ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "Rom.10.16"
∷ word (τ ∷ ῷ ∷ []) "Rom.10.16"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ῳ ∷ []) "Rom.10.16"
∷ word (Ἠ ∷ σ ∷ α ∷ ΐ ∷ α ∷ ς ∷ []) "Rom.10.16"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.10.16"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.10.16"
∷ word (Κ ∷ ύ ∷ ρ ∷ ι ∷ ε ∷ []) "Rom.10.16"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.10.16"
∷ word (ἐ ∷ π ∷ ί ∷ σ ∷ τ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "Rom.10.16"
∷ word (τ ∷ ῇ ∷ []) "Rom.10.16"
∷ word (ἀ ∷ κ ∷ ο ∷ ῇ ∷ []) "Rom.10.16"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.10.16"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Rom.10.17"
∷ word (ἡ ∷ []) "Rom.10.17"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "Rom.10.17"
∷ word (ἐ ∷ ξ ∷ []) "Rom.10.17"
∷ word (ἀ ∷ κ ∷ ο ∷ ῆ ∷ ς ∷ []) "Rom.10.17"
∷ word (ἡ ∷ []) "Rom.10.17"
∷ word (δ ∷ ὲ ∷ []) "Rom.10.17"
∷ word (ἀ ∷ κ ∷ ο ∷ ὴ ∷ []) "Rom.10.17"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.10.17"
∷ word (ῥ ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.10.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.10.17"
∷ word (Ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.10.18"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Rom.10.18"
∷ word (μ ∷ ὴ ∷ []) "Rom.10.18"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.10.18"
∷ word (ἤ ∷ κ ∷ ο ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "Rom.10.18"
∷ word (μ ∷ ε ∷ ν ∷ ο ∷ ῦ ∷ ν ∷ γ ∷ ε ∷ []) "Rom.10.18"
∷ word (Ε ∷ ἰ ∷ ς ∷ []) "Rom.10.18"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "Rom.10.18"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.10.18"
∷ word (γ ∷ ῆ ∷ ν ∷ []) "Rom.10.18"
∷ word (ἐ ∷ ξ ∷ ῆ ∷ ∙λ ∷ θ ∷ ε ∷ ν ∷ []) "Rom.10.18"
∷ word (ὁ ∷ []) "Rom.10.18"
∷ word (φ ∷ θ ∷ ό ∷ γ ∷ γ ∷ ο ∷ ς ∷ []) "Rom.10.18"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.10.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.10.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.10.18"
∷ word (τ ∷ ὰ ∷ []) "Rom.10.18"
∷ word (π ∷ έ ∷ ρ ∷ α ∷ τ ∷ α ∷ []) "Rom.10.18"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.10.18"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ η ∷ ς ∷ []) "Rom.10.18"
∷ word (τ ∷ ὰ ∷ []) "Rom.10.18"
∷ word (ῥ ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Rom.10.18"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.10.18"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.10.19"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Rom.10.19"
∷ word (μ ∷ ὴ ∷ []) "Rom.10.19"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ὴ ∷ ∙λ ∷ []) "Rom.10.19"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.10.19"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ []) "Rom.10.19"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.10.19"
∷ word (Μ ∷ ω ∷ ϋ ∷ σ ∷ ῆ ∷ ς ∷ []) "Rom.10.19"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.10.19"
∷ word (Ἐ ∷ γ ∷ ὼ ∷ []) "Rom.10.19"
∷ word (π ∷ α ∷ ρ ∷ α ∷ ζ ∷ η ∷ ∙λ ∷ ώ ∷ σ ∷ ω ∷ []) "Rom.10.19"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.10.19"
∷ word (ἐ ∷ π ∷ []) "Rom.10.19"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.10.19"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ ι ∷ []) "Rom.10.19"
∷ word (ἐ ∷ π ∷ []) "Rom.10.19"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ ι ∷ []) "Rom.10.19"
∷ word (ἀ ∷ σ ∷ υ ∷ ν ∷ έ ∷ τ ∷ ῳ ∷ []) "Rom.10.19"
∷ word (π ∷ α ∷ ρ ∷ ο ∷ ρ ∷ γ ∷ ι ∷ ῶ ∷ []) "Rom.10.19"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.10.19"
∷ word (Ἠ ∷ σ ∷ α ∷ ΐ ∷ α ∷ ς ∷ []) "Rom.10.20"
∷ word (δ ∷ ὲ ∷ []) "Rom.10.20"
∷ word (ἀ ∷ π ∷ ο ∷ τ ∷ ο ∷ ∙λ ∷ μ ∷ ᾷ ∷ []) "Rom.10.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.10.20"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.10.20"
∷ word (Ε ∷ ὑ ∷ ρ ∷ έ ∷ θ ∷ η ∷ ν ∷ []) "Rom.10.20"
∷ word (ἐ ∷ ν ∷ []) "Rom.10.20"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.10.20"
∷ word (ἐ ∷ μ ∷ ὲ ∷ []) "Rom.10.20"
∷ word (μ ∷ ὴ ∷ []) "Rom.10.20"
∷ word (ζ ∷ η ∷ τ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.10.20"
∷ word (ἐ ∷ μ ∷ φ ∷ α ∷ ν ∷ ὴ ∷ ς ∷ []) "Rom.10.20"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ό ∷ μ ∷ η ∷ ν ∷ []) "Rom.10.20"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.10.20"
∷ word (ἐ ∷ μ ∷ ὲ ∷ []) "Rom.10.20"
∷ word (μ ∷ ὴ ∷ []) "Rom.10.20"
∷ word (ἐ ∷ π ∷ ε ∷ ρ ∷ ω ∷ τ ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.10.20"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.10.21"
∷ word (δ ∷ ὲ ∷ []) "Rom.10.21"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.10.21"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ὴ ∷ ∙λ ∷ []) "Rom.10.21"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.10.21"
∷ word (Ὅ ∷ ∙λ ∷ η ∷ ν ∷ []) "Rom.10.21"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.10.21"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "Rom.10.21"
∷ word (ἐ ∷ ξ ∷ ε ∷ π ∷ έ ∷ τ ∷ α ∷ σ ∷ α ∷ []) "Rom.10.21"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Rom.10.21"
∷ word (χ ∷ ε ∷ ῖ ∷ ρ ∷ ά ∷ ς ∷ []) "Rom.10.21"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.10.21"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.10.21"
∷ word (∙λ ∷ α ∷ ὸ ∷ ν ∷ []) "Rom.10.21"
∷ word (ἀ ∷ π ∷ ε ∷ ι ∷ θ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ α ∷ []) "Rom.10.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.10.21"
∷ word (ἀ ∷ ν ∷ τ ∷ ι ∷ ∙λ ∷ έ ∷ γ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "Rom.10.21"
∷ word (Λ ∷ έ ∷ γ ∷ ω ∷ []) "Rom.11.1"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.11.1"
∷ word (μ ∷ ὴ ∷ []) "Rom.11.1"
∷ word (ἀ ∷ π ∷ ώ ∷ σ ∷ α ∷ τ ∷ ο ∷ []) "Rom.11.1"
∷ word (ὁ ∷ []) "Rom.11.1"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.11.1"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.11.1"
∷ word (∙λ ∷ α ∷ ὸ ∷ ν ∷ []) "Rom.11.1"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.11.1"
∷ word (μ ∷ ὴ ∷ []) "Rom.11.1"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "Rom.11.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.1"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.11.1"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.11.1"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ η ∷ ∙λ ∷ ί ∷ τ ∷ η ∷ ς ∷ []) "Rom.11.1"
∷ word (ε ∷ ἰ ∷ μ ∷ ί ∷ []) "Rom.11.1"
∷ word (ἐ ∷ κ ∷ []) "Rom.11.1"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.11.1"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ά ∷ μ ∷ []) "Rom.11.1"
∷ word (φ ∷ υ ∷ ∙λ ∷ ῆ ∷ ς ∷ []) "Rom.11.1"
∷ word (Β ∷ ε ∷ ν ∷ ι ∷ α ∷ μ ∷ ί ∷ ν ∷ []) "Rom.11.1"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.11.2"
∷ word (ἀ ∷ π ∷ ώ ∷ σ ∷ α ∷ τ ∷ ο ∷ []) "Rom.11.2"
∷ word (ὁ ∷ []) "Rom.11.2"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.11.2"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.11.2"
∷ word (∙λ ∷ α ∷ ὸ ∷ ν ∷ []) "Rom.11.2"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.11.2"
∷ word (ὃ ∷ ν ∷ []) "Rom.11.2"
∷ word (π ∷ ρ ∷ ο ∷ έ ∷ γ ∷ ν ∷ ω ∷ []) "Rom.11.2"
∷ word (ἢ ∷ []) "Rom.11.2"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.11.2"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "Rom.11.2"
∷ word (ἐ ∷ ν ∷ []) "Rom.11.2"
∷ word (Ἠ ∷ ∙λ ∷ ί ∷ ᾳ ∷ []) "Rom.11.2"
∷ word (τ ∷ ί ∷ []) "Rom.11.2"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.11.2"
∷ word (ἡ ∷ []) "Rom.11.2"
∷ word (γ ∷ ρ ∷ α ∷ φ ∷ ή ∷ []) "Rom.11.2"
∷ word (ὡ ∷ ς ∷ []) "Rom.11.2"
∷ word (ἐ ∷ ν ∷ τ ∷ υ ∷ γ ∷ χ ∷ ά ∷ ν ∷ ε ∷ ι ∷ []) "Rom.11.2"
∷ word (τ ∷ ῷ ∷ []) "Rom.11.2"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.11.2"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.11.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.11.2"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ή ∷ ∙λ ∷ []) "Rom.11.2"
∷ word (Κ ∷ ύ ∷ ρ ∷ ι ∷ ε ∷ []) "Rom.11.3"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.11.3"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ ή ∷ τ ∷ α ∷ ς ∷ []) "Rom.11.3"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.11.3"
∷ word (ἀ ∷ π ∷ έ ∷ κ ∷ τ ∷ ε ∷ ι ∷ ν ∷ α ∷ ν ∷ []) "Rom.11.3"
∷ word (τ ∷ ὰ ∷ []) "Rom.11.3"
∷ word (θ ∷ υ ∷ σ ∷ ι ∷ α ∷ σ ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ά ∷ []) "Rom.11.3"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.11.3"
∷ word (κ ∷ α ∷ τ ∷ έ ∷ σ ∷ κ ∷ α ∷ ψ ∷ α ∷ ν ∷ []) "Rom.11.3"
∷ word (κ ∷ ἀ ∷ γ ∷ ὼ ∷ []) "Rom.11.3"
∷ word (ὑ ∷ π ∷ ε ∷ ∙λ ∷ ε ∷ ί ∷ φ ∷ θ ∷ η ∷ ν ∷ []) "Rom.11.3"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ς ∷ []) "Rom.11.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.3"
∷ word (ζ ∷ η ∷ τ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.11.3"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.11.3"
∷ word (ψ ∷ υ ∷ χ ∷ ή ∷ ν ∷ []) "Rom.11.3"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.11.3"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.11.4"
∷ word (τ ∷ ί ∷ []) "Rom.11.4"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.11.4"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.11.4"
∷ word (ὁ ∷ []) "Rom.11.4"
∷ word (χ ∷ ρ ∷ η ∷ μ ∷ α ∷ τ ∷ ι ∷ σ ∷ μ ∷ ό ∷ ς ∷ []) "Rom.11.4"
∷ word (Κ ∷ α ∷ τ ∷ έ ∷ ∙λ ∷ ι ∷ π ∷ ο ∷ ν ∷ []) "Rom.11.4"
∷ word (ἐ ∷ μ ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "Rom.11.4"
∷ word (ἑ ∷ π ∷ τ ∷ α ∷ κ ∷ ι ∷ σ ∷ χ ∷ ι ∷ ∙λ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "Rom.11.4"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ α ∷ ς ∷ []) "Rom.11.4"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "Rom.11.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.11.4"
∷ word (ἔ ∷ κ ∷ α ∷ μ ∷ ψ ∷ α ∷ ν ∷ []) "Rom.11.4"
∷ word (γ ∷ ό ∷ ν ∷ υ ∷ []) "Rom.11.4"
∷ word (τ ∷ ῇ ∷ []) "Rom.11.4"
∷ word (Β ∷ ά ∷ α ∷ ∙λ ∷ []) "Rom.11.4"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.11.5"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.11.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.5"
∷ word (ἐ ∷ ν ∷ []) "Rom.11.5"
∷ word (τ ∷ ῷ ∷ []) "Rom.11.5"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Rom.11.5"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ῷ ∷ []) "Rom.11.5"
∷ word (∙λ ∷ ε ∷ ῖ ∷ μ ∷ μ ∷ α ∷ []) "Rom.11.5"
∷ word (κ ∷ α ∷ τ ∷ []) "Rom.11.5"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ο ∷ γ ∷ ὴ ∷ ν ∷ []) "Rom.11.5"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ο ∷ ς ∷ []) "Rom.11.5"
∷ word (γ ∷ έ ∷ γ ∷ ο ∷ ν ∷ ε ∷ ν ∷ []) "Rom.11.5"
∷ word (ε ∷ ἰ ∷ []) "Rom.11.6"
∷ word (δ ∷ ὲ ∷ []) "Rom.11.6"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ι ∷ []) "Rom.11.6"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Rom.11.6"
∷ word (ἐ ∷ ξ ∷ []) "Rom.11.6"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Rom.11.6"
∷ word (ἐ ∷ π ∷ ε ∷ ὶ ∷ []) "Rom.11.6"
∷ word (ἡ ∷ []) "Rom.11.6"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Rom.11.6"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Rom.11.6"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.11.6"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Rom.11.6"
∷ word (τ ∷ ί ∷ []) "Rom.11.7"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.11.7"
∷ word (ὃ ∷ []) "Rom.11.7"
∷ word (ἐ ∷ π ∷ ι ∷ ζ ∷ η ∷ τ ∷ ε ∷ ῖ ∷ []) "Rom.11.7"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ή ∷ ∙λ ∷ []) "Rom.11.7"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.11.7"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.11.7"
∷ word (ἐ ∷ π ∷ έ ∷ τ ∷ υ ∷ χ ∷ ε ∷ ν ∷ []) "Rom.11.7"
∷ word (ἡ ∷ []) "Rom.11.7"
∷ word (δ ∷ ὲ ∷ []) "Rom.11.7"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ο ∷ γ ∷ ὴ ∷ []) "Rom.11.7"
∷ word (ἐ ∷ π ∷ έ ∷ τ ∷ υ ∷ χ ∷ ε ∷ ν ∷ []) "Rom.11.7"
∷ word (ο ∷ ἱ ∷ []) "Rom.11.7"
∷ word (δ ∷ ὲ ∷ []) "Rom.11.7"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ο ∷ ὶ ∷ []) "Rom.11.7"
∷ word (ἐ ∷ π ∷ ω ∷ ρ ∷ ώ ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.11.7"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.11.8"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.11.8"
∷ word (Ἔ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "Rom.11.8"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.11.8"
∷ word (ὁ ∷ []) "Rom.11.8"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.11.8"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Rom.11.8"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ν ∷ ύ ∷ ξ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.11.8"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.11.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.11.8"
∷ word (μ ∷ ὴ ∷ []) "Rom.11.8"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ε ∷ ι ∷ ν ∷ []) "Rom.11.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.8"
∷ word (ὦ ∷ τ ∷ α ∷ []) "Rom.11.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.11.8"
∷ word (μ ∷ ὴ ∷ []) "Rom.11.8"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.11.8"
∷ word (ἕ ∷ ω ∷ ς ∷ []) "Rom.11.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.11.8"
∷ word (σ ∷ ή ∷ μ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "Rom.11.8"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "Rom.11.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.9"
∷ word (Δ ∷ α ∷ υ ∷ ὶ ∷ δ ∷ []) "Rom.11.9"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.11.9"
∷ word (Γ ∷ ε ∷ ν ∷ η ∷ θ ∷ ή ∷ τ ∷ ω ∷ []) "Rom.11.9"
∷ word (ἡ ∷ []) "Rom.11.9"
∷ word (τ ∷ ρ ∷ ά ∷ π ∷ ε ∷ ζ ∷ α ∷ []) "Rom.11.9"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.11.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.11.9"
∷ word (π ∷ α ∷ γ ∷ ί ∷ δ ∷ α ∷ []) "Rom.11.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.11.9"
∷ word (θ ∷ ή ∷ ρ ∷ α ∷ ν ∷ []) "Rom.11.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.11.9"
∷ word (σ ∷ κ ∷ ά ∷ ν ∷ δ ∷ α ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.11.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.11.9"
∷ word (ἀ ∷ ν ∷ τ ∷ α ∷ π ∷ ό ∷ δ ∷ ο ∷ μ ∷ α ∷ []) "Rom.11.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.11.9"
∷ word (σ ∷ κ ∷ ο ∷ τ ∷ ι ∷ σ ∷ θ ∷ ή ∷ τ ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "Rom.11.10"
∷ word (ο ∷ ἱ ∷ []) "Rom.11.10"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ο ∷ ὶ ∷ []) "Rom.11.10"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.11.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.11.10"
∷ word (μ ∷ ὴ ∷ []) "Rom.11.10"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ε ∷ ι ∷ ν ∷ []) "Rom.11.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.11.10"
∷ word (ν ∷ ῶ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.11.10"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.11.10"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.11.10"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.11.10"
∷ word (σ ∷ ύ ∷ γ ∷ κ ∷ α ∷ μ ∷ ψ ∷ ο ∷ ν ∷ []) "Rom.11.10"
∷ word (Λ ∷ έ ∷ γ ∷ ω ∷ []) "Rom.11.11"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.11.11"
∷ word (μ ∷ ὴ ∷ []) "Rom.11.11"
∷ word (ἔ ∷ π ∷ τ ∷ α ∷ ι ∷ σ ∷ α ∷ ν ∷ []) "Rom.11.11"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.11.11"
∷ word (π ∷ έ ∷ σ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Rom.11.11"
∷ word (μ ∷ ὴ ∷ []) "Rom.11.11"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "Rom.11.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.11.11"
∷ word (τ ∷ ῷ ∷ []) "Rom.11.11"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.11.11"
∷ word (π ∷ α ∷ ρ ∷ α ∷ π ∷ τ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.11.11"
∷ word (ἡ ∷ []) "Rom.11.11"
∷ word (σ ∷ ω ∷ τ ∷ η ∷ ρ ∷ ί ∷ α ∷ []) "Rom.11.11"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.11.11"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Rom.11.11"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.11.11"
∷ word (τ ∷ ὸ ∷ []) "Rom.11.11"
∷ word (π ∷ α ∷ ρ ∷ α ∷ ζ ∷ η ∷ ∙λ ∷ ῶ ∷ σ ∷ α ∷ ι ∷ []) "Rom.11.11"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ύ ∷ ς ∷ []) "Rom.11.11"
∷ word (ε ∷ ἰ ∷ []) "Rom.11.12"
∷ word (δ ∷ ὲ ∷ []) "Rom.11.12"
∷ word (τ ∷ ὸ ∷ []) "Rom.11.12"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ π ∷ τ ∷ ω ∷ μ ∷ α ∷ []) "Rom.11.12"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.11.12"
∷ word (π ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.11.12"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "Rom.11.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.12"
∷ word (τ ∷ ὸ ∷ []) "Rom.11.12"
∷ word (ἥ ∷ τ ∷ τ ∷ η ∷ μ ∷ α ∷ []) "Rom.11.12"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.11.12"
∷ word (π ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.11.12"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.11.12"
∷ word (π ∷ ό ∷ σ ∷ ῳ ∷ []) "Rom.11.12"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.11.12"
∷ word (τ ∷ ὸ ∷ []) "Rom.11.12"
∷ word (π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ μ ∷ α ∷ []) "Rom.11.12"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.11.12"
∷ word (Ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.11.13"
∷ word (δ ∷ ὲ ∷ []) "Rom.11.13"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Rom.11.13"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.11.13"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Rom.11.13"
∷ word (ἐ ∷ φ ∷ []) "Rom.11.13"
∷ word (ὅ ∷ σ ∷ ο ∷ ν ∷ []) "Rom.11.13"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.11.13"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.11.13"
∷ word (ε ∷ ἰ ∷ μ ∷ ι ∷ []) "Rom.11.13"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.11.13"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.11.13"
∷ word (ἀ ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ς ∷ []) "Rom.11.13"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.11.13"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "Rom.11.13"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.11.13"
∷ word (δ ∷ ο ∷ ξ ∷ ά ∷ ζ ∷ ω ∷ []) "Rom.11.13"
∷ word (ε ∷ ἴ ∷ []) "Rom.11.14"
∷ word (π ∷ ω ∷ ς ∷ []) "Rom.11.14"
∷ word (π ∷ α ∷ ρ ∷ α ∷ ζ ∷ η ∷ ∙λ ∷ ώ ∷ σ ∷ ω ∷ []) "Rom.11.14"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.11.14"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.11.14"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Rom.11.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.14"
∷ word (σ ∷ ώ ∷ σ ∷ ω ∷ []) "Rom.11.14"
∷ word (τ ∷ ι ∷ ν ∷ ὰ ∷ ς ∷ []) "Rom.11.14"
∷ word (ἐ ∷ ξ ∷ []) "Rom.11.14"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.11.14"
∷ word (ε ∷ ἰ ∷ []) "Rom.11.15"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.11.15"
∷ word (ἡ ∷ []) "Rom.11.15"
∷ word (ἀ ∷ π ∷ ο ∷ β ∷ ο ∷ ∙λ ∷ ὴ ∷ []) "Rom.11.15"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.11.15"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ∙λ ∷ α ∷ γ ∷ ὴ ∷ []) "Rom.11.15"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "Rom.11.15"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.11.15"
∷ word (ἡ ∷ []) "Rom.11.15"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ ∙λ ∷ η ∷ μ ∷ ψ ∷ ι ∷ ς ∷ []) "Rom.11.15"
∷ word (ε ∷ ἰ ∷ []) "Rom.11.15"
∷ word (μ ∷ ὴ ∷ []) "Rom.11.15"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "Rom.11.15"
∷ word (ἐ ∷ κ ∷ []) "Rom.11.15"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Rom.11.15"
∷ word (ε ∷ ἰ ∷ []) "Rom.11.16"
∷ word (δ ∷ ὲ ∷ []) "Rom.11.16"
∷ word (ἡ ∷ []) "Rom.11.16"
∷ word (ἀ ∷ π ∷ α ∷ ρ ∷ χ ∷ ὴ ∷ []) "Rom.11.16"
∷ word (ἁ ∷ γ ∷ ί ∷ α ∷ []) "Rom.11.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.16"
∷ word (τ ∷ ὸ ∷ []) "Rom.11.16"
∷ word (φ ∷ ύ ∷ ρ ∷ α ∷ μ ∷ α ∷ []) "Rom.11.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.16"
∷ word (ε ∷ ἰ ∷ []) "Rom.11.16"
∷ word (ἡ ∷ []) "Rom.11.16"
∷ word (ῥ ∷ ί ∷ ζ ∷ α ∷ []) "Rom.11.16"
∷ word (ἁ ∷ γ ∷ ί ∷ α ∷ []) "Rom.11.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.16"
∷ word (ο ∷ ἱ ∷ []) "Rom.11.16"
∷ word (κ ∷ ∙λ ∷ ά ∷ δ ∷ ο ∷ ι ∷ []) "Rom.11.16"
∷ word (Ε ∷ ἰ ∷ []) "Rom.11.17"
∷ word (δ ∷ έ ∷ []) "Rom.11.17"
∷ word (τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "Rom.11.17"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.11.17"
∷ word (κ ∷ ∙λ ∷ ά ∷ δ ∷ ω ∷ ν ∷ []) "Rom.11.17"
∷ word (ἐ ∷ ξ ∷ ε ∷ κ ∷ ∙λ ∷ ά ∷ σ ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.11.17"
∷ word (σ ∷ ὺ ∷ []) "Rom.11.17"
∷ word (δ ∷ ὲ ∷ []) "Rom.11.17"
∷ word (ἀ ∷ γ ∷ ρ ∷ ι ∷ έ ∷ ∙λ ∷ α ∷ ι ∷ ο ∷ ς ∷ []) "Rom.11.17"
∷ word (ὢ ∷ ν ∷ []) "Rom.11.17"
∷ word (ἐ ∷ ν ∷ ε ∷ κ ∷ ε ∷ ν ∷ τ ∷ ρ ∷ ί ∷ σ ∷ θ ∷ η ∷ ς ∷ []) "Rom.11.17"
∷ word (ἐ ∷ ν ∷ []) "Rom.11.17"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.11.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.17"
∷ word (σ ∷ υ ∷ γ ∷ κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ὸ ∷ ς ∷ []) "Rom.11.17"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.11.17"
∷ word (ῥ ∷ ί ∷ ζ ∷ η ∷ ς ∷ []) "Rom.11.17"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.11.17"
∷ word (π ∷ ι ∷ ό ∷ τ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "Rom.11.17"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.11.17"
∷ word (ἐ ∷ ∙λ ∷ α ∷ ί ∷ α ∷ ς ∷ []) "Rom.11.17"
∷ word (ἐ ∷ γ ∷ έ ∷ ν ∷ ο ∷ υ ∷ []) "Rom.11.17"
∷ word (μ ∷ ὴ ∷ []) "Rom.11.18"
∷ word (κ ∷ α ∷ τ ∷ α ∷ κ ∷ α ∷ υ ∷ χ ∷ ῶ ∷ []) "Rom.11.18"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.11.18"
∷ word (κ ∷ ∙λ ∷ ά ∷ δ ∷ ω ∷ ν ∷ []) "Rom.11.18"
∷ word (ε ∷ ἰ ∷ []) "Rom.11.18"
∷ word (δ ∷ ὲ ∷ []) "Rom.11.18"
∷ word (κ ∷ α ∷ τ ∷ α ∷ κ ∷ α ∷ υ ∷ χ ∷ ᾶ ∷ σ ∷ α ∷ ι ∷ []) "Rom.11.18"
∷ word (ο ∷ ὐ ∷ []) "Rom.11.18"
∷ word (σ ∷ ὺ ∷ []) "Rom.11.18"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.11.18"
∷ word (ῥ ∷ ί ∷ ζ ∷ α ∷ ν ∷ []) "Rom.11.18"
∷ word (β ∷ α ∷ σ ∷ τ ∷ ά ∷ ζ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.11.18"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.11.18"
∷ word (ἡ ∷ []) "Rom.11.18"
∷ word (ῥ ∷ ί ∷ ζ ∷ α ∷ []) "Rom.11.18"
∷ word (σ ∷ έ ∷ []) "Rom.11.18"
∷ word (ἐ ∷ ρ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.11.19"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.11.19"
∷ word (Ἐ ∷ ξ ∷ ε ∷ κ ∷ ∙λ ∷ ά ∷ σ ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.11.19"
∷ word (κ ∷ ∙λ ∷ ά ∷ δ ∷ ο ∷ ι ∷ []) "Rom.11.19"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.11.19"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.11.19"
∷ word (ἐ ∷ γ ∷ κ ∷ ε ∷ ν ∷ τ ∷ ρ ∷ ι ∷ σ ∷ θ ∷ ῶ ∷ []) "Rom.11.19"
∷ word (κ ∷ α ∷ ∙λ ∷ ῶ ∷ ς ∷ []) "Rom.11.20"
∷ word (τ ∷ ῇ ∷ []) "Rom.11.20"
∷ word (ἀ ∷ π ∷ ι ∷ σ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Rom.11.20"
∷ word (ἐ ∷ ξ ∷ ε ∷ κ ∷ ∙λ ∷ ά ∷ σ ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.11.20"
∷ word (σ ∷ ὺ ∷ []) "Rom.11.20"
∷ word (δ ∷ ὲ ∷ []) "Rom.11.20"
∷ word (τ ∷ ῇ ∷ []) "Rom.11.20"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ι ∷ []) "Rom.11.20"
∷ word (ἕ ∷ σ ∷ τ ∷ η ∷ κ ∷ α ∷ ς ∷ []) "Rom.11.20"
∷ word (μ ∷ ὴ ∷ []) "Rom.11.20"
∷ word (ὑ ∷ ψ ∷ η ∷ ∙λ ∷ ὰ ∷ []) "Rom.11.20"
∷ word (φ ∷ ρ ∷ ό ∷ ν ∷ ε ∷ ι ∷ []) "Rom.11.20"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.11.20"
∷ word (φ ∷ ο ∷ β ∷ ο ∷ ῦ ∷ []) "Rom.11.20"
∷ word (ε ∷ ἰ ∷ []) "Rom.11.21"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.11.21"
∷ word (ὁ ∷ []) "Rom.11.21"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.11.21"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.11.21"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.11.21"
∷ word (φ ∷ ύ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.11.21"
∷ word (κ ∷ ∙λ ∷ ά ∷ δ ∷ ω ∷ ν ∷ []) "Rom.11.21"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.11.21"
∷ word (ἐ ∷ φ ∷ ε ∷ ί ∷ σ ∷ α ∷ τ ∷ ο ∷ []) "Rom.11.21"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Rom.11.21"
∷ word (σ ∷ ο ∷ ῦ ∷ []) "Rom.11.21"
∷ word (φ ∷ ε ∷ ί ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.11.21"
∷ word (ἴ ∷ δ ∷ ε ∷ []) "Rom.11.22"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.11.22"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ό ∷ τ ∷ η ∷ τ ∷ α ∷ []) "Rom.11.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.22"
∷ word (ἀ ∷ π ∷ ο ∷ τ ∷ ο ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Rom.11.22"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.11.22"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.11.22"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.11.22"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.11.22"
∷ word (π ∷ ε ∷ σ ∷ ό ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.11.22"
∷ word (ἀ ∷ π ∷ ο ∷ τ ∷ ο ∷ μ ∷ ί ∷ α ∷ []) "Rom.11.22"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.11.22"
∷ word (δ ∷ ὲ ∷ []) "Rom.11.22"
∷ word (σ ∷ ὲ ∷ []) "Rom.11.22"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ό ∷ τ ∷ η ∷ ς ∷ []) "Rom.11.22"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.11.22"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.11.22"
∷ word (ἐ ∷ π ∷ ι ∷ μ ∷ έ ∷ ν ∷ ῃ ∷ ς ∷ []) "Rom.11.22"
∷ word (τ ∷ ῇ ∷ []) "Rom.11.22"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "Rom.11.22"
∷ word (ἐ ∷ π ∷ ε ∷ ὶ ∷ []) "Rom.11.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.22"
∷ word (σ ∷ ὺ ∷ []) "Rom.11.22"
∷ word (ἐ ∷ κ ∷ κ ∷ ο ∷ π ∷ ή ∷ σ ∷ ῃ ∷ []) "Rom.11.22"
∷ word (κ ∷ ἀ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ ο ∷ ι ∷ []) "Rom.11.23"
∷ word (δ ∷ έ ∷ []) "Rom.11.23"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.11.23"
∷ word (μ ∷ ὴ ∷ []) "Rom.11.23"
∷ word (ἐ ∷ π ∷ ι ∷ μ ∷ έ ∷ ν ∷ ω ∷ σ ∷ ι ∷ []) "Rom.11.23"
∷ word (τ ∷ ῇ ∷ []) "Rom.11.23"
∷ word (ἀ ∷ π ∷ ι ∷ σ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Rom.11.23"
∷ word (ἐ ∷ γ ∷ κ ∷ ε ∷ ν ∷ τ ∷ ρ ∷ ι ∷ σ ∷ θ ∷ ή ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.11.23"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.11.23"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Rom.11.23"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.11.23"
∷ word (ὁ ∷ []) "Rom.11.23"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.11.23"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Rom.11.23"
∷ word (ἐ ∷ γ ∷ κ ∷ ε ∷ ν ∷ τ ∷ ρ ∷ ί ∷ σ ∷ α ∷ ι ∷ []) "Rom.11.23"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ύ ∷ ς ∷ []) "Rom.11.23"
∷ word (ε ∷ ἰ ∷ []) "Rom.11.24"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.11.24"
∷ word (σ ∷ ὺ ∷ []) "Rom.11.24"
∷ word (ἐ ∷ κ ∷ []) "Rom.11.24"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.11.24"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.11.24"
∷ word (φ ∷ ύ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.11.24"
∷ word (ἐ ∷ ξ ∷ ε ∷ κ ∷ ό ∷ π ∷ η ∷ ς ∷ []) "Rom.11.24"
∷ word (ἀ ∷ γ ∷ ρ ∷ ι ∷ ε ∷ ∙λ ∷ α ∷ ί ∷ ο ∷ υ ∷ []) "Rom.11.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.24"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "Rom.11.24"
∷ word (φ ∷ ύ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.11.24"
∷ word (ἐ ∷ ν ∷ ε ∷ κ ∷ ε ∷ ν ∷ τ ∷ ρ ∷ ί ∷ σ ∷ θ ∷ η ∷ ς ∷ []) "Rom.11.24"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.11.24"
∷ word (κ ∷ α ∷ ∙λ ∷ ∙λ ∷ ι ∷ έ ∷ ∙λ ∷ α ∷ ι ∷ ο ∷ ν ∷ []) "Rom.11.24"
∷ word (π ∷ ό ∷ σ ∷ ῳ ∷ []) "Rom.11.24"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.11.24"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ι ∷ []) "Rom.11.24"
∷ word (ο ∷ ἱ ∷ []) "Rom.11.24"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.11.24"
∷ word (φ ∷ ύ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.11.24"
∷ word (ἐ ∷ γ ∷ κ ∷ ε ∷ ν ∷ τ ∷ ρ ∷ ι ∷ σ ∷ θ ∷ ή ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.11.24"
∷ word (τ ∷ ῇ ∷ []) "Rom.11.24"
∷ word (ἰ ∷ δ ∷ ί ∷ ᾳ ∷ []) "Rom.11.24"
∷ word (ἐ ∷ ∙λ ∷ α ∷ ί ∷ ᾳ ∷ []) "Rom.11.24"
∷ word (Ο ∷ ὐ ∷ []) "Rom.11.25"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.11.25"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "Rom.11.25"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.11.25"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.11.25"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Rom.11.25"
∷ word (τ ∷ ὸ ∷ []) "Rom.11.25"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.11.25"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.11.25"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.11.25"
∷ word (μ ∷ ὴ ∷ []) "Rom.11.25"
∷ word (ἦ ∷ τ ∷ ε ∷ []) "Rom.11.25"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.11.25"
∷ word (φ ∷ ρ ∷ ό ∷ ν ∷ ι ∷ μ ∷ ο ∷ ι ∷ []) "Rom.11.25"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.11.25"
∷ word (π ∷ ώ ∷ ρ ∷ ω ∷ σ ∷ ι ∷ ς ∷ []) "Rom.11.25"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.11.25"
∷ word (μ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.11.25"
∷ word (τ ∷ ῷ ∷ []) "Rom.11.25"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ὴ ∷ ∙λ ∷ []) "Rom.11.25"
∷ word (γ ∷ έ ∷ γ ∷ ο ∷ ν ∷ ε ∷ ν ∷ []) "Rom.11.25"
∷ word (ἄ ∷ χ ∷ ρ ∷ ι ∷ []) "Rom.11.25"
∷ word (ο ∷ ὗ ∷ []) "Rom.11.25"
∷ word (τ ∷ ὸ ∷ []) "Rom.11.25"
∷ word (π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ μ ∷ α ∷ []) "Rom.11.25"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.11.25"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.11.25"
∷ word (ε ∷ ἰ ∷ σ ∷ έ ∷ ∙λ ∷ θ ∷ ῃ ∷ []) "Rom.11.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.26"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.11.26"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "Rom.11.26"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ὴ ∷ ∙λ ∷ []) "Rom.11.26"
∷ word (σ ∷ ω ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.11.26"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.11.26"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.11.26"
∷ word (Ἥ ∷ ξ ∷ ε ∷ ι ∷ []) "Rom.11.26"
∷ word (ἐ ∷ κ ∷ []) "Rom.11.26"
∷ word (Σ ∷ ι ∷ ὼ ∷ ν ∷ []) "Rom.11.26"
∷ word (ὁ ∷ []) "Rom.11.26"
∷ word (ῥ ∷ υ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Rom.11.26"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ρ ∷ έ ∷ ψ ∷ ε ∷ ι ∷ []) "Rom.11.26"
∷ word (ἀ ∷ σ ∷ ε ∷ β ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Rom.11.26"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.11.26"
∷ word (Ἰ ∷ α ∷ κ ∷ ώ ∷ β ∷ []) "Rom.11.26"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.27"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "Rom.11.27"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.11.27"
∷ word (ἡ ∷ []) "Rom.11.27"
∷ word (π ∷ α ∷ ρ ∷ []) "Rom.11.27"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "Rom.11.27"
∷ word (δ ∷ ι ∷ α ∷ θ ∷ ή ∷ κ ∷ η ∷ []) "Rom.11.27"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "Rom.11.27"
∷ word (ἀ ∷ φ ∷ έ ∷ ∙λ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "Rom.11.27"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Rom.11.27"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Rom.11.27"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.11.27"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.11.28"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.11.28"
∷ word (τ ∷ ὸ ∷ []) "Rom.11.28"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.11.28"
∷ word (ἐ ∷ χ ∷ θ ∷ ρ ∷ ο ∷ ὶ ∷ []) "Rom.11.28"
∷ word (δ ∷ ι ∷ []) "Rom.11.28"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.11.28"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.11.28"
∷ word (δ ∷ ὲ ∷ []) "Rom.11.28"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.11.28"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ο ∷ γ ∷ ὴ ∷ ν ∷ []) "Rom.11.28"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ὶ ∷ []) "Rom.11.28"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.11.28"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.11.28"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "Rom.11.28"
∷ word (ἀ ∷ μ ∷ ε ∷ τ ∷ α ∷ μ ∷ έ ∷ ∙λ ∷ η ∷ τ ∷ α ∷ []) "Rom.11.29"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.11.29"
∷ word (τ ∷ ὰ ∷ []) "Rom.11.29"
∷ word (χ ∷ α ∷ ρ ∷ ί ∷ σ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Rom.11.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.29"
∷ word (ἡ ∷ []) "Rom.11.29"
∷ word (κ ∷ ∙λ ∷ ῆ ∷ σ ∷ ι ∷ ς ∷ []) "Rom.11.29"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.11.29"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.11.29"
∷ word (ὥ ∷ σ ∷ π ∷ ε ∷ ρ ∷ []) "Rom.11.30"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.11.30"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.11.30"
∷ word (π ∷ ο ∷ τ ∷ ε ∷ []) "Rom.11.30"
∷ word (ἠ ∷ π ∷ ε ∷ ι ∷ θ ∷ ή ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Rom.11.30"
∷ word (τ ∷ ῷ ∷ []) "Rom.11.30"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.11.30"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Rom.11.30"
∷ word (δ ∷ ὲ ∷ []) "Rom.11.30"
∷ word (ἠ ∷ ∙λ ∷ ε ∷ ή ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Rom.11.30"
∷ word (τ ∷ ῇ ∷ []) "Rom.11.30"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ω ∷ ν ∷ []) "Rom.11.30"
∷ word (ἀ ∷ π ∷ ε ∷ ι ∷ θ ∷ ε ∷ ί ∷ ᾳ ∷ []) "Rom.11.30"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.11.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.31"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ι ∷ []) "Rom.11.31"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Rom.11.31"
∷ word (ἠ ∷ π ∷ ε ∷ ί ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.11.31"
∷ word (τ ∷ ῷ ∷ []) "Rom.11.31"
∷ word (ὑ ∷ μ ∷ ε ∷ τ ∷ έ ∷ ρ ∷ ῳ ∷ []) "Rom.11.31"
∷ word (ἐ ∷ ∙λ ∷ έ ∷ ε ∷ ι ∷ []) "Rom.11.31"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.11.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.31"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὶ ∷ []) "Rom.11.31"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Rom.11.31"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ η ∷ θ ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.11.31"
∷ word (σ ∷ υ ∷ ν ∷ έ ∷ κ ∷ ∙λ ∷ ε ∷ ι ∷ σ ∷ ε ∷ ν ∷ []) "Rom.11.32"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.11.32"
∷ word (ὁ ∷ []) "Rom.11.32"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.11.32"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.11.32"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.11.32"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.11.32"
∷ word (ἀ ∷ π ∷ ε ∷ ί ∷ θ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Rom.11.32"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.11.32"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.11.32"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.11.32"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ή ∷ σ ∷ ῃ ∷ []) "Rom.11.32"
∷ word (Ὦ ∷ []) "Rom.11.33"
∷ word (β ∷ ά ∷ θ ∷ ο ∷ ς ∷ []) "Rom.11.33"
∷ word (π ∷ ∙λ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "Rom.11.33"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.33"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ς ∷ []) "Rom.11.33"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.33"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.11.33"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.11.33"
∷ word (ὡ ∷ ς ∷ []) "Rom.11.33"
∷ word (ἀ ∷ ν ∷ ε ∷ ξ ∷ ε ∷ ρ ∷ α ∷ ύ ∷ ν ∷ η ∷ τ ∷ α ∷ []) "Rom.11.33"
∷ word (τ ∷ ὰ ∷ []) "Rom.11.33"
∷ word (κ ∷ ρ ∷ ί ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Rom.11.33"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.11.33"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.33"
∷ word (ἀ ∷ ν ∷ ε ∷ ξ ∷ ι ∷ χ ∷ ν ∷ ί ∷ α ∷ σ ∷ τ ∷ ο ∷ ι ∷ []) "Rom.11.33"
∷ word (α ∷ ἱ ∷ []) "Rom.11.33"
∷ word (ὁ ∷ δ ∷ ο ∷ ὶ ∷ []) "Rom.11.33"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.11.33"
∷ word (Τ ∷ ί ∷ ς ∷ []) "Rom.11.34"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.11.34"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ []) "Rom.11.34"
∷ word (ν ∷ ο ∷ ῦ ∷ ν ∷ []) "Rom.11.34"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.11.34"
∷ word (ἢ ∷ []) "Rom.11.34"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.11.34"
∷ word (σ ∷ ύ ∷ μ ∷ β ∷ ο ∷ υ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Rom.11.34"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.11.34"
∷ word (ἐ ∷ γ ∷ έ ∷ ν ∷ ε ∷ τ ∷ ο ∷ []) "Rom.11.34"
∷ word (ἢ ∷ []) "Rom.11.35"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.11.35"
∷ word (π ∷ ρ ∷ ο ∷ έ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "Rom.11.35"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.11.35"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.35"
∷ word (ἀ ∷ ν ∷ τ ∷ α ∷ π ∷ ο ∷ δ ∷ ο ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.11.35"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.11.35"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.11.36"
∷ word (ἐ ∷ ξ ∷ []) "Rom.11.36"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.11.36"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.36"
∷ word (δ ∷ ι ∷ []) "Rom.11.36"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.11.36"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.11.36"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.11.36"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.11.36"
∷ word (τ ∷ ὰ ∷ []) "Rom.11.36"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Rom.11.36"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.11.36"
∷ word (ἡ ∷ []) "Rom.11.36"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "Rom.11.36"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.11.36"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.11.36"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ α ∷ ς ∷ []) "Rom.11.36"
∷ word (ἀ ∷ μ ∷ ή ∷ ν ∷ []) "Rom.11.36"
∷ word (Π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "Rom.12.1"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.12.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.12.1"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Rom.12.1"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.12.1"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.12.1"
∷ word (ο ∷ ἰ ∷ κ ∷ τ ∷ ι ∷ ρ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.12.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.12.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.12.1"
∷ word (π ∷ α ∷ ρ ∷ α ∷ σ ∷ τ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Rom.12.1"
∷ word (τ ∷ ὰ ∷ []) "Rom.12.1"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Rom.12.1"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.12.1"
∷ word (θ ∷ υ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Rom.12.1"
∷ word (ζ ∷ ῶ ∷ σ ∷ α ∷ ν ∷ []) "Rom.12.1"
∷ word (ἁ ∷ γ ∷ ί ∷ α ∷ ν ∷ []) "Rom.12.1"
∷ word (ε ∷ ὐ ∷ ά ∷ ρ ∷ ε ∷ σ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.12.1"
∷ word (τ ∷ ῷ ∷ []) "Rom.12.1"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.12.1"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.12.1"
∷ word (∙λ ∷ ο ∷ γ ∷ ι ∷ κ ∷ ὴ ∷ ν ∷ []) "Rom.12.1"
∷ word (∙λ ∷ α ∷ τ ∷ ρ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "Rom.12.1"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.12.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.12.2"
∷ word (μ ∷ ὴ ∷ []) "Rom.12.2"
∷ word (σ ∷ υ ∷ σ ∷ χ ∷ η ∷ μ ∷ α ∷ τ ∷ ί ∷ ζ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Rom.12.2"
∷ word (τ ∷ ῷ ∷ []) "Rom.12.2"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ ι ∷ []) "Rom.12.2"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "Rom.12.2"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.12.2"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ μ ∷ ο ∷ ρ ∷ φ ∷ ο ∷ ῦ ∷ σ ∷ θ ∷ ε ∷ []) "Rom.12.2"
∷ word (τ ∷ ῇ ∷ []) "Rom.12.2"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ α ∷ ι ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "Rom.12.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.12.2"
∷ word (ν ∷ ο ∷ ό ∷ ς ∷ []) "Rom.12.2"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.12.2"
∷ word (τ ∷ ὸ ∷ []) "Rom.12.2"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ά ∷ ζ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.12.2"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.12.2"
∷ word (τ ∷ ί ∷ []) "Rom.12.2"
∷ word (τ ∷ ὸ ∷ []) "Rom.12.2"
∷ word (θ ∷ έ ∷ ∙λ ∷ η ∷ μ ∷ α ∷ []) "Rom.12.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.12.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.12.2"
∷ word (τ ∷ ὸ ∷ []) "Rom.12.2"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ὸ ∷ ν ∷ []) "Rom.12.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.12.2"
∷ word (ε ∷ ὐ ∷ ά ∷ ρ ∷ ε ∷ σ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.12.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.12.2"
∷ word (τ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ ο ∷ ν ∷ []) "Rom.12.2"
∷ word (Λ ∷ έ ∷ γ ∷ ω ∷ []) "Rom.12.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.12.3"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.12.3"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.12.3"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ο ∷ ς ∷ []) "Rom.12.3"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.12.3"
∷ word (δ ∷ ο ∷ θ ∷ ε ∷ ί ∷ σ ∷ η ∷ ς ∷ []) "Rom.12.3"
∷ word (μ ∷ ο ∷ ι ∷ []) "Rom.12.3"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "Rom.12.3"
∷ word (τ ∷ ῷ ∷ []) "Rom.12.3"
∷ word (ὄ ∷ ν ∷ τ ∷ ι ∷ []) "Rom.12.3"
∷ word (ἐ ∷ ν ∷ []) "Rom.12.3"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.12.3"
∷ word (μ ∷ ὴ ∷ []) "Rom.12.3"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ φ ∷ ρ ∷ ο ∷ ν ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.12.3"
∷ word (π ∷ α ∷ ρ ∷ []) "Rom.12.3"
∷ word (ὃ ∷ []) "Rom.12.3"
∷ word (δ ∷ ε ∷ ῖ ∷ []) "Rom.12.3"
∷ word (φ ∷ ρ ∷ ο ∷ ν ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.12.3"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.12.3"
∷ word (φ ∷ ρ ∷ ο ∷ ν ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.12.3"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.12.3"
∷ word (τ ∷ ὸ ∷ []) "Rom.12.3"
∷ word (σ ∷ ω ∷ φ ∷ ρ ∷ ο ∷ ν ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.12.3"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ ῳ ∷ []) "Rom.12.3"
∷ word (ὡ ∷ ς ∷ []) "Rom.12.3"
∷ word (ὁ ∷ []) "Rom.12.3"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.12.3"
∷ word (ἐ ∷ μ ∷ έ ∷ ρ ∷ ι ∷ σ ∷ ε ∷ ν ∷ []) "Rom.12.3"
∷ word (μ ∷ έ ∷ τ ∷ ρ ∷ ο ∷ ν ∷ []) "Rom.12.3"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.12.3"
∷ word (κ ∷ α ∷ θ ∷ ά ∷ π ∷ ε ∷ ρ ∷ []) "Rom.12.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.12.4"
∷ word (ἐ ∷ ν ∷ []) "Rom.12.4"
∷ word (ἑ ∷ ν ∷ ὶ ∷ []) "Rom.12.4"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.12.4"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.12.4"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "Rom.12.4"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.12.4"
∷ word (τ ∷ ὰ ∷ []) "Rom.12.4"
∷ word (δ ∷ ὲ ∷ []) "Rom.12.4"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "Rom.12.4"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Rom.12.4"
∷ word (ο ∷ ὐ ∷ []) "Rom.12.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.12.4"
∷ word (α ∷ ὐ ∷ τ ∷ ὴ ∷ ν ∷ []) "Rom.12.4"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "Rom.12.4"
∷ word (π ∷ ρ ∷ ᾶ ∷ ξ ∷ ι ∷ ν ∷ []) "Rom.12.4"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.12.5"
∷ word (ο ∷ ἱ ∷ []) "Rom.12.5"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὶ ∷ []) "Rom.12.5"
∷ word (ἓ ∷ ν ∷ []) "Rom.12.5"
∷ word (σ ∷ ῶ ∷ μ ∷ ά ∷ []) "Rom.12.5"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.12.5"
∷ word (ἐ ∷ ν ∷ []) "Rom.12.5"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.12.5"
∷ word (τ ∷ ὸ ∷ []) "Rom.12.5"
∷ word (δ ∷ ὲ ∷ []) "Rom.12.5"
∷ word (κ ∷ α ∷ θ ∷ []) "Rom.12.5"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "Rom.12.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ω ∷ ν ∷ []) "Rom.12.5"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "Rom.12.5"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.6"
∷ word (δ ∷ ὲ ∷ []) "Rom.12.6"
∷ word (χ ∷ α ∷ ρ ∷ ί ∷ σ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Rom.12.6"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.12.6"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.12.6"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Rom.12.6"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.12.6"
∷ word (δ ∷ ο ∷ θ ∷ ε ∷ ῖ ∷ σ ∷ α ∷ ν ∷ []) "Rom.12.6"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.12.6"
∷ word (δ ∷ ι ∷ ά ∷ φ ∷ ο ∷ ρ ∷ α ∷ []) "Rom.12.6"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "Rom.12.6"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "Rom.12.6"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.12.6"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.12.6"
∷ word (ἀ ∷ ν ∷ α ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ ν ∷ []) "Rom.12.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.12.6"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.12.6"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "Rom.12.7"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "Rom.12.7"
∷ word (ἐ ∷ ν ∷ []) "Rom.12.7"
∷ word (τ ∷ ῇ ∷ []) "Rom.12.7"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ ᾳ ∷ []) "Rom.12.7"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "Rom.12.7"
∷ word (ὁ ∷ []) "Rom.12.7"
∷ word (δ ∷ ι ∷ δ ∷ ά ∷ σ ∷ κ ∷ ω ∷ ν ∷ []) "Rom.12.7"
∷ word (ἐ ∷ ν ∷ []) "Rom.12.7"
∷ word (τ ∷ ῇ ∷ []) "Rom.12.7"
∷ word (δ ∷ ι ∷ δ ∷ α ∷ σ ∷ κ ∷ α ∷ ∙λ ∷ ί ∷ ᾳ ∷ []) "Rom.12.7"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "Rom.12.8"
∷ word (ὁ ∷ []) "Rom.12.8"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "Rom.12.8"
∷ word (ἐ ∷ ν ∷ []) "Rom.12.8"
∷ word (τ ∷ ῇ ∷ []) "Rom.12.8"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "Rom.12.8"
∷ word (ὁ ∷ []) "Rom.12.8"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ δ ∷ ι ∷ δ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.12.8"
∷ word (ἐ ∷ ν ∷ []) "Rom.12.8"
∷ word (ἁ ∷ π ∷ ∙λ ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "Rom.12.8"
∷ word (ὁ ∷ []) "Rom.12.8"
∷ word (π ∷ ρ ∷ ο ∷ ϊ ∷ σ ∷ τ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Rom.12.8"
∷ word (ἐ ∷ ν ∷ []) "Rom.12.8"
∷ word (σ ∷ π ∷ ο ∷ υ ∷ δ ∷ ῇ ∷ []) "Rom.12.8"
∷ word (ὁ ∷ []) "Rom.12.8"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ῶ ∷ ν ∷ []) "Rom.12.8"
∷ word (ἐ ∷ ν ∷ []) "Rom.12.8"
∷ word (ἱ ∷ ∙λ ∷ α ∷ ρ ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "Rom.12.8"
∷ word (Ἡ ∷ []) "Rom.12.9"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "Rom.12.9"
∷ word (ἀ ∷ ν ∷ υ ∷ π ∷ ό ∷ κ ∷ ρ ∷ ι ∷ τ ∷ ο ∷ ς ∷ []) "Rom.12.9"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ υ ∷ γ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.9"
∷ word (τ ∷ ὸ ∷ []) "Rom.12.9"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ό ∷ ν ∷ []) "Rom.12.9"
∷ word (κ ∷ ο ∷ ∙λ ∷ ∙λ ∷ ώ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Rom.12.9"
∷ word (τ ∷ ῷ ∷ []) "Rom.12.9"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ῷ ∷ []) "Rom.12.9"
∷ word (τ ∷ ῇ ∷ []) "Rom.12.10"
∷ word (φ ∷ ι ∷ ∙λ ∷ α ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ί ∷ ᾳ ∷ []) "Rom.12.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.12.10"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.12.10"
∷ word (φ ∷ ι ∷ ∙λ ∷ ό ∷ σ ∷ τ ∷ ο ∷ ρ ∷ γ ∷ ο ∷ ι ∷ []) "Rom.12.10"
∷ word (τ ∷ ῇ ∷ []) "Rom.12.10"
∷ word (τ ∷ ι ∷ μ ∷ ῇ ∷ []) "Rom.12.10"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.12.10"
∷ word (π ∷ ρ ∷ ο ∷ η ∷ γ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Rom.12.10"
∷ word (τ ∷ ῇ ∷ []) "Rom.12.11"
∷ word (σ ∷ π ∷ ο ∷ υ ∷ δ ∷ ῇ ∷ []) "Rom.12.11"
∷ word (μ ∷ ὴ ∷ []) "Rom.12.11"
∷ word (ὀ ∷ κ ∷ ν ∷ η ∷ ρ ∷ ο ∷ ί ∷ []) "Rom.12.11"
∷ word (τ ∷ ῷ ∷ []) "Rom.12.11"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.12.11"
∷ word (ζ ∷ έ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.11"
∷ word (τ ∷ ῷ ∷ []) "Rom.12.11"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.12.11"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.11"
∷ word (τ ∷ ῇ ∷ []) "Rom.12.12"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ ι ∷ []) "Rom.12.12"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.12"
∷ word (τ ∷ ῇ ∷ []) "Rom.12.12"
∷ word (θ ∷ ∙λ ∷ ί ∷ ψ ∷ ε ∷ ι ∷ []) "Rom.12.12"
∷ word (ὑ ∷ π ∷ ο ∷ μ ∷ έ ∷ ν ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.12"
∷ word (τ ∷ ῇ ∷ []) "Rom.12.12"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ ῇ ∷ []) "Rom.12.12"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ κ ∷ α ∷ ρ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.12"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Rom.12.13"
∷ word (χ ∷ ρ ∷ ε ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Rom.12.13"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.12.13"
∷ word (ἁ ∷ γ ∷ ί ∷ ω ∷ ν ∷ []) "Rom.12.13"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.13"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.12.13"
∷ word (φ ∷ ι ∷ ∙λ ∷ ο ∷ ξ ∷ ε ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "Rom.12.13"
∷ word (δ ∷ ι ∷ ώ ∷ κ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.13"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Rom.12.14"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.12.14"
∷ word (δ ∷ ι ∷ ώ ∷ κ ∷ ο ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.12.14"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Rom.12.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.12.14"
∷ word (μ ∷ ὴ ∷ []) "Rom.12.14"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ ᾶ ∷ σ ∷ θ ∷ ε ∷ []) "Rom.12.14"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.12.15"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Rom.12.15"
∷ word (χ ∷ α ∷ ι ∷ ρ ∷ ό ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.12.15"
∷ word (κ ∷ ∙λ ∷ α ∷ ί ∷ ε ∷ ι ∷ ν ∷ []) "Rom.12.15"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Rom.12.15"
∷ word (κ ∷ ∙λ ∷ α ∷ ι ∷ ό ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.12.15"
∷ word (τ ∷ ὸ ∷ []) "Rom.12.16"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Rom.12.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.12.16"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.12.16"
∷ word (φ ∷ ρ ∷ ο ∷ ν ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.16"
∷ word (μ ∷ ὴ ∷ []) "Rom.12.16"
∷ word (τ ∷ ὰ ∷ []) "Rom.12.16"
∷ word (ὑ ∷ ψ ∷ η ∷ ∙λ ∷ ὰ ∷ []) "Rom.12.16"
∷ word (φ ∷ ρ ∷ ο ∷ ν ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.16"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.12.16"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.12.16"
∷ word (τ ∷ α ∷ π ∷ ε ∷ ι ∷ ν ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.12.16"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ π ∷ α ∷ γ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Rom.12.16"
∷ word (μ ∷ ὴ ∷ []) "Rom.12.16"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Rom.12.16"
∷ word (φ ∷ ρ ∷ ό ∷ ν ∷ ι ∷ μ ∷ ο ∷ ι ∷ []) "Rom.12.16"
∷ word (π ∷ α ∷ ρ ∷ []) "Rom.12.16"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.12.16"
∷ word (μ ∷ η ∷ δ ∷ ε ∷ ν ∷ ὶ ∷ []) "Rom.12.17"
∷ word (κ ∷ α ∷ κ ∷ ὸ ∷ ν ∷ []) "Rom.12.17"
∷ word (ἀ ∷ ν ∷ τ ∷ ὶ ∷ []) "Rom.12.17"
∷ word (κ ∷ α ∷ κ ∷ ο ∷ ῦ ∷ []) "Rom.12.17"
∷ word (ἀ ∷ π ∷ ο ∷ δ ∷ ι ∷ δ ∷ ό ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.17"
∷ word (π ∷ ρ ∷ ο ∷ ν ∷ ο ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Rom.12.17"
∷ word (κ ∷ α ∷ ∙λ ∷ ὰ ∷ []) "Rom.12.17"
∷ word (ἐ ∷ ν ∷ ώ ∷ π ∷ ι ∷ ο ∷ ν ∷ []) "Rom.12.17"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.12.17"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "Rom.12.17"
∷ word (ε ∷ ἰ ∷ []) "Rom.12.18"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ τ ∷ ό ∷ ν ∷ []) "Rom.12.18"
∷ word (τ ∷ ὸ ∷ []) "Rom.12.18"
∷ word (ἐ ∷ ξ ∷ []) "Rom.12.18"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.12.18"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Rom.12.18"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.12.18"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "Rom.12.18"
∷ word (ε ∷ ἰ ∷ ρ ∷ η ∷ ν ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.18"
∷ word (μ ∷ ὴ ∷ []) "Rom.12.19"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.12.19"
∷ word (ἐ ∷ κ ∷ δ ∷ ι ∷ κ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.12.19"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ί ∷ []) "Rom.12.19"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.12.19"
∷ word (δ ∷ ό ∷ τ ∷ ε ∷ []) "Rom.12.19"
∷ word (τ ∷ ό ∷ π ∷ ο ∷ ν ∷ []) "Rom.12.19"
∷ word (τ ∷ ῇ ∷ []) "Rom.12.19"
∷ word (ὀ ∷ ρ ∷ γ ∷ ῇ ∷ []) "Rom.12.19"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.12.19"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Rom.12.19"
∷ word (Ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Rom.12.19"
∷ word (ἐ ∷ κ ∷ δ ∷ ί ∷ κ ∷ η ∷ σ ∷ ι ∷ ς ∷ []) "Rom.12.19"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.12.19"
∷ word (ἀ ∷ ν ∷ τ ∷ α ∷ π ∷ ο ∷ δ ∷ ώ ∷ σ ∷ ω ∷ []) "Rom.12.19"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.12.19"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "Rom.12.19"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.12.20"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.12.20"
∷ word (π ∷ ε ∷ ι ∷ ν ∷ ᾷ ∷ []) "Rom.12.20"
∷ word (ὁ ∷ []) "Rom.12.20"
∷ word (ἐ ∷ χ ∷ θ ∷ ρ ∷ ό ∷ ς ∷ []) "Rom.12.20"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.12.20"
∷ word (ψ ∷ ώ ∷ μ ∷ ι ∷ ζ ∷ ε ∷ []) "Rom.12.20"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "Rom.12.20"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.12.20"
∷ word (δ ∷ ι ∷ ψ ∷ ᾷ ∷ []) "Rom.12.20"
∷ word (π ∷ ό ∷ τ ∷ ι ∷ ζ ∷ ε ∷ []) "Rom.12.20"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "Rom.12.20"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.12.20"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.12.20"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ ν ∷ []) "Rom.12.20"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ α ∷ κ ∷ α ∷ ς ∷ []) "Rom.12.20"
∷ word (π ∷ υ ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.12.20"
∷ word (σ ∷ ω ∷ ρ ∷ ε ∷ ύ ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.12.20"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Rom.12.20"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.12.20"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "Rom.12.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.12.20"
∷ word (μ ∷ ὴ ∷ []) "Rom.12.21"
∷ word (ν ∷ ι ∷ κ ∷ ῶ ∷ []) "Rom.12.21"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Rom.12.21"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.12.21"
∷ word (κ ∷ α ∷ κ ∷ ο ∷ ῦ ∷ []) "Rom.12.21"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.12.21"
∷ word (ν ∷ ί ∷ κ ∷ α ∷ []) "Rom.12.21"
∷ word (ἐ ∷ ν ∷ []) "Rom.12.21"
∷ word (τ ∷ ῷ ∷ []) "Rom.12.21"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ῷ ∷ []) "Rom.12.21"
∷ word (τ ∷ ὸ ∷ []) "Rom.12.21"
∷ word (κ ∷ α ∷ κ ∷ ό ∷ ν ∷ []) "Rom.12.21"
∷ word (Π ∷ ᾶ ∷ σ ∷ α ∷ []) "Rom.13.1"
∷ word (ψ ∷ υ ∷ χ ∷ ὴ ∷ []) "Rom.13.1"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Rom.13.1"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ ε ∷ χ ∷ ο ∷ ύ ∷ σ ∷ α ∷ ι ∷ ς ∷ []) "Rom.13.1"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ α ∷ σ ∷ σ ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "Rom.13.1"
∷ word (ο ∷ ὐ ∷ []) "Rom.13.1"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.13.1"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.13.1"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ []) "Rom.13.1"
∷ word (ε ∷ ἰ ∷ []) "Rom.13.1"
∷ word (μ ∷ ὴ ∷ []) "Rom.13.1"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Rom.13.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.13.1"
∷ word (α ∷ ἱ ∷ []) "Rom.13.1"
∷ word (δ ∷ ὲ ∷ []) "Rom.13.1"
∷ word (ο ∷ ὖ ∷ σ ∷ α ∷ ι ∷ []) "Rom.13.1"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Rom.13.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.13.1"
∷ word (τ ∷ ε ∷ τ ∷ α ∷ γ ∷ μ ∷ έ ∷ ν ∷ α ∷ ι ∷ []) "Rom.13.1"
∷ word (ε ∷ ἰ ∷ σ ∷ ί ∷ ν ∷ []) "Rom.13.1"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "Rom.13.2"
∷ word (ὁ ∷ []) "Rom.13.2"
∷ word (ἀ ∷ ν ∷ τ ∷ ι ∷ τ ∷ α ∷ σ ∷ σ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Rom.13.2"
∷ word (τ ∷ ῇ ∷ []) "Rom.13.2"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ ᾳ ∷ []) "Rom.13.2"
∷ word (τ ∷ ῇ ∷ []) "Rom.13.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.13.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.13.2"
∷ word (δ ∷ ι ∷ α ∷ τ ∷ α ∷ γ ∷ ῇ ∷ []) "Rom.13.2"
∷ word (ἀ ∷ ν ∷ θ ∷ έ ∷ σ ∷ τ ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "Rom.13.2"
∷ word (ο ∷ ἱ ∷ []) "Rom.13.2"
∷ word (δ ∷ ὲ ∷ []) "Rom.13.2"
∷ word (ἀ ∷ ν ∷ θ ∷ ε ∷ σ ∷ τ ∷ η ∷ κ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "Rom.13.2"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.13.2"
∷ word (κ ∷ ρ ∷ ί ∷ μ ∷ α ∷ []) "Rom.13.2"
∷ word (∙λ ∷ ή ∷ μ ∷ ψ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.13.2"
∷ word (ο ∷ ἱ ∷ []) "Rom.13.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.13.3"
∷ word (ἄ ∷ ρ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.13.3"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.13.3"
∷ word (ε ∷ ἰ ∷ σ ∷ ὶ ∷ ν ∷ []) "Rom.13.3"
∷ word (φ ∷ ό ∷ β ∷ ο ∷ ς ∷ []) "Rom.13.3"
∷ word (τ ∷ ῷ ∷ []) "Rom.13.3"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ῷ ∷ []) "Rom.13.3"
∷ word (ἔ ∷ ρ ∷ γ ∷ ῳ ∷ []) "Rom.13.3"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.13.3"
∷ word (τ ∷ ῷ ∷ []) "Rom.13.3"
∷ word (κ ∷ α ∷ κ ∷ ῷ ∷ []) "Rom.13.3"
∷ word (θ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.13.3"
∷ word (δ ∷ ὲ ∷ []) "Rom.13.3"
∷ word (μ ∷ ὴ ∷ []) "Rom.13.3"
∷ word (φ ∷ ο ∷ β ∷ ε ∷ ῖ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.13.3"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.13.3"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Rom.13.3"
∷ word (τ ∷ ὸ ∷ []) "Rom.13.3"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ὸ ∷ ν ∷ []) "Rom.13.3"
∷ word (π ∷ ο ∷ ί ∷ ε ∷ ι ∷ []) "Rom.13.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.13.3"
∷ word (ἕ ∷ ξ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.13.3"
∷ word (ἔ ∷ π ∷ α ∷ ι ∷ ν ∷ ο ∷ ν ∷ []) "Rom.13.3"
∷ word (ἐ ∷ ξ ∷ []) "Rom.13.3"
∷ word (α ∷ ὐ ∷ τ ∷ ῆ ∷ ς ∷ []) "Rom.13.3"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.13.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.13.4"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ό ∷ ς ∷ []) "Rom.13.4"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.13.4"
∷ word (σ ∷ ο ∷ ὶ ∷ []) "Rom.13.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.13.4"
∷ word (τ ∷ ὸ ∷ []) "Rom.13.4"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ό ∷ ν ∷ []) "Rom.13.4"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.13.4"
∷ word (δ ∷ ὲ ∷ []) "Rom.13.4"
∷ word (τ ∷ ὸ ∷ []) "Rom.13.4"
∷ word (κ ∷ α ∷ κ ∷ ὸ ∷ ν ∷ []) "Rom.13.4"
∷ word (π ∷ ο ∷ ι ∷ ῇ ∷ ς ∷ []) "Rom.13.4"
∷ word (φ ∷ ο ∷ β ∷ ο ∷ ῦ ∷ []) "Rom.13.4"
∷ word (ο ∷ ὐ ∷ []) "Rom.13.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.13.4"
∷ word (ε ∷ ἰ ∷ κ ∷ ῇ ∷ []) "Rom.13.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.13.4"
∷ word (μ ∷ ά ∷ χ ∷ α ∷ ι ∷ ρ ∷ α ∷ ν ∷ []) "Rom.13.4"
∷ word (φ ∷ ο ∷ ρ ∷ ε ∷ ῖ ∷ []) "Rom.13.4"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.13.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.13.4"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ό ∷ ς ∷ []) "Rom.13.4"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.13.4"
∷ word (ἔ ∷ κ ∷ δ ∷ ι ∷ κ ∷ ο ∷ ς ∷ []) "Rom.13.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.13.4"
∷ word (ὀ ∷ ρ ∷ γ ∷ ὴ ∷ ν ∷ []) "Rom.13.4"
∷ word (τ ∷ ῷ ∷ []) "Rom.13.4"
∷ word (τ ∷ ὸ ∷ []) "Rom.13.4"
∷ word (κ ∷ α ∷ κ ∷ ὸ ∷ ν ∷ []) "Rom.13.4"
∷ word (π ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ο ∷ ν ∷ τ ∷ ι ∷ []) "Rom.13.4"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "Rom.13.5"
∷ word (ἀ ∷ ν ∷ ά ∷ γ ∷ κ ∷ η ∷ []) "Rom.13.5"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ ά ∷ σ ∷ σ ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.13.5"
∷ word (ο ∷ ὐ ∷ []) "Rom.13.5"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Rom.13.5"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.13.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.13.5"
∷ word (ὀ ∷ ρ ∷ γ ∷ ὴ ∷ ν ∷ []) "Rom.13.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.13.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.13.5"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.13.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.13.5"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ί ∷ δ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "Rom.13.5"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.13.6"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.13.6"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.13.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.13.6"
∷ word (φ ∷ ό ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.13.6"
∷ word (τ ∷ ε ∷ ∙λ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Rom.13.6"
∷ word (∙λ ∷ ε ∷ ι ∷ τ ∷ ο ∷ υ ∷ ρ ∷ γ ∷ ο ∷ ὶ ∷ []) "Rom.13.6"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.13.6"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.13.6"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.13.6"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.13.6"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Rom.13.6"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.13.6"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ κ ∷ α ∷ ρ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.13.6"
∷ word (ἀ ∷ π ∷ ό ∷ δ ∷ ο ∷ τ ∷ ε ∷ []) "Rom.13.7"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ []) "Rom.13.7"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Rom.13.7"
∷ word (ὀ ∷ φ ∷ ε ∷ ι ∷ ∙λ ∷ ά ∷ ς ∷ []) "Rom.13.7"
∷ word (τ ∷ ῷ ∷ []) "Rom.13.7"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.13.7"
∷ word (φ ∷ ό ∷ ρ ∷ ο ∷ ν ∷ []) "Rom.13.7"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.13.7"
∷ word (φ ∷ ό ∷ ρ ∷ ο ∷ ν ∷ []) "Rom.13.7"
∷ word (τ ∷ ῷ ∷ []) "Rom.13.7"
∷ word (τ ∷ ὸ ∷ []) "Rom.13.7"
∷ word (τ ∷ έ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Rom.13.7"
∷ word (τ ∷ ὸ ∷ []) "Rom.13.7"
∷ word (τ ∷ έ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Rom.13.7"
∷ word (τ ∷ ῷ ∷ []) "Rom.13.7"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.13.7"
∷ word (φ ∷ ό ∷ β ∷ ο ∷ ν ∷ []) "Rom.13.7"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.13.7"
∷ word (φ ∷ ό ∷ β ∷ ο ∷ ν ∷ []) "Rom.13.7"
∷ word (τ ∷ ῷ ∷ []) "Rom.13.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.13.7"
∷ word (τ ∷ ι ∷ μ ∷ ὴ ∷ ν ∷ []) "Rom.13.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.13.7"
∷ word (τ ∷ ι ∷ μ ∷ ή ∷ ν ∷ []) "Rom.13.7"
∷ word (Μ ∷ η ∷ δ ∷ ε ∷ ν ∷ ὶ ∷ []) "Rom.13.8"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ ν ∷ []) "Rom.13.8"
∷ word (ὀ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ ε ∷ τ ∷ ε ∷ []) "Rom.13.8"
∷ word (ε ∷ ἰ ∷ []) "Rom.13.8"
∷ word (μ ∷ ὴ ∷ []) "Rom.13.8"
∷ word (τ ∷ ὸ ∷ []) "Rom.13.8"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.13.8"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ᾶ ∷ ν ∷ []) "Rom.13.8"
∷ word (ὁ ∷ []) "Rom.13.8"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.13.8"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ ν ∷ []) "Rom.13.8"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.13.8"
∷ word (ἕ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "Rom.13.8"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Rom.13.8"
∷ word (π ∷ ε ∷ π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "Rom.13.8"
∷ word (τ ∷ ὸ ∷ []) "Rom.13.9"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Rom.13.9"
∷ word (Ο ∷ ὐ ∷ []) "Rom.13.9"
∷ word (μ ∷ ο ∷ ι ∷ χ ∷ ε ∷ ύ ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.13.9"
∷ word (Ο ∷ ὐ ∷ []) "Rom.13.9"
∷ word (φ ∷ ο ∷ ν ∷ ε ∷ ύ ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.13.9"
∷ word (Ο ∷ ὐ ∷ []) "Rom.13.9"
∷ word (κ ∷ ∙λ ∷ έ ∷ ψ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.13.9"
∷ word (Ο ∷ ὐ ∷ κ ∷ []) "Rom.13.9"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ή ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.13.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.13.9"
∷ word (ε ∷ ἴ ∷ []) "Rom.13.9"
∷ word (τ ∷ ι ∷ ς ∷ []) "Rom.13.9"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Rom.13.9"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ή ∷ []) "Rom.13.9"
∷ word (ἐ ∷ ν ∷ []) "Rom.13.9"
∷ word (τ ∷ ῷ ∷ []) "Rom.13.9"
∷ word (∙λ ∷ ό ∷ γ ∷ ῳ ∷ []) "Rom.13.9"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "Rom.13.9"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ α ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ α ∷ ι ∷ []) "Rom.13.9"
∷ word (ἐ ∷ ν ∷ []) "Rom.13.9"
∷ word (τ ∷ ῷ ∷ []) "Rom.13.9"
∷ word (Ἀ ∷ γ ∷ α ∷ π ∷ ή ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.13.9"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.13.9"
∷ word (π ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ο ∷ ν ∷ []) "Rom.13.9"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.13.9"
∷ word (ὡ ∷ ς ∷ []) "Rom.13.9"
∷ word (σ ∷ ε ∷ α ∷ υ ∷ τ ∷ ό ∷ ν ∷ []) "Rom.13.9"
∷ word (ἡ ∷ []) "Rom.13.10"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "Rom.13.10"
∷ word (τ ∷ ῷ ∷ []) "Rom.13.10"
∷ word (π ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ο ∷ ν ∷ []) "Rom.13.10"
∷ word (κ ∷ α ∷ κ ∷ ὸ ∷ ν ∷ []) "Rom.13.10"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.13.10"
∷ word (ἐ ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.13.10"
∷ word (π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ μ ∷ α ∷ []) "Rom.13.10"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.13.10"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Rom.13.10"
∷ word (ἡ ∷ []) "Rom.13.10"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "Rom.13.10"
∷ word (Κ ∷ α ∷ ὶ ∷ []) "Rom.13.11"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.13.11"
∷ word (ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "Rom.13.11"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.13.11"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ό ∷ ν ∷ []) "Rom.13.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.13.11"
∷ word (ὥ ∷ ρ ∷ α ∷ []) "Rom.13.11"
∷ word (ἤ ∷ δ ∷ η ∷ []) "Rom.13.11"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.13.11"
∷ word (ἐ ∷ ξ ∷ []) "Rom.13.11"
∷ word (ὕ ∷ π ∷ ν ∷ ο ∷ υ ∷ []) "Rom.13.11"
∷ word (ἐ ∷ γ ∷ ε ∷ ρ ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "Rom.13.11"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Rom.13.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.13.11"
∷ word (ἐ ∷ γ ∷ γ ∷ ύ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "Rom.13.11"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.13.11"
∷ word (ἡ ∷ []) "Rom.13.11"
∷ word (σ ∷ ω ∷ τ ∷ η ∷ ρ ∷ ί ∷ α ∷ []) "Rom.13.11"
∷ word (ἢ ∷ []) "Rom.13.11"
∷ word (ὅ ∷ τ ∷ ε ∷ []) "Rom.13.11"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ σ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Rom.13.11"
∷ word (ἡ ∷ []) "Rom.13.12"
∷ word (ν ∷ ὺ ∷ ξ ∷ []) "Rom.13.12"
∷ word (π ∷ ρ ∷ ο ∷ έ ∷ κ ∷ ο ∷ ψ ∷ ε ∷ ν ∷ []) "Rom.13.12"
∷ word (ἡ ∷ []) "Rom.13.12"
∷ word (δ ∷ ὲ ∷ []) "Rom.13.12"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ []) "Rom.13.12"
∷ word (ἤ ∷ γ ∷ γ ∷ ι ∷ κ ∷ ε ∷ ν ∷ []) "Rom.13.12"
∷ word (ἀ ∷ π ∷ ο ∷ β ∷ α ∷ ∙λ ∷ ώ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.13.12"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.13.12"
∷ word (τ ∷ ὰ ∷ []) "Rom.13.12"
∷ word (ἔ ∷ ρ ∷ γ ∷ α ∷ []) "Rom.13.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.13.12"
∷ word (σ ∷ κ ∷ ό ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.13.12"
∷ word (ἐ ∷ ν ∷ δ ∷ υ ∷ σ ∷ ώ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.13.12"
∷ word (δ ∷ ὲ ∷ []) "Rom.13.12"
∷ word (τ ∷ ὰ ∷ []) "Rom.13.12"
∷ word (ὅ ∷ π ∷ ∙λ ∷ α ∷ []) "Rom.13.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.13.12"
∷ word (φ ∷ ω ∷ τ ∷ ό ∷ ς ∷ []) "Rom.13.12"
∷ word (ὡ ∷ ς ∷ []) "Rom.13.13"
∷ word (ἐ ∷ ν ∷ []) "Rom.13.13"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "Rom.13.13"
∷ word (ε ∷ ὐ ∷ σ ∷ χ ∷ η ∷ μ ∷ ό ∷ ν ∷ ω ∷ ς ∷ []) "Rom.13.13"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ή ∷ σ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Rom.13.13"
∷ word (μ ∷ ὴ ∷ []) "Rom.13.13"
∷ word (κ ∷ ώ ∷ μ ∷ ο ∷ ι ∷ ς ∷ []) "Rom.13.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.13.13"
∷ word (μ ∷ έ ∷ θ ∷ α ∷ ι ∷ ς ∷ []) "Rom.13.13"
∷ word (μ ∷ ὴ ∷ []) "Rom.13.13"
∷ word (κ ∷ ο ∷ ί ∷ τ ∷ α ∷ ι ∷ ς ∷ []) "Rom.13.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.13.13"
∷ word (ἀ ∷ σ ∷ ε ∷ ∙λ ∷ γ ∷ ε ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Rom.13.13"
∷ word (μ ∷ ὴ ∷ []) "Rom.13.13"
∷ word (ἔ ∷ ρ ∷ ι ∷ δ ∷ ι ∷ []) "Rom.13.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.13.13"
∷ word (ζ ∷ ή ∷ ∙λ ∷ ῳ ∷ []) "Rom.13.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.13.14"
∷ word (ἐ ∷ ν ∷ δ ∷ ύ ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.13.14"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.13.14"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.13.14"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "Rom.13.14"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ν ∷ []) "Rom.13.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.13.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.13.14"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Rom.13.14"
∷ word (π ∷ ρ ∷ ό ∷ ν ∷ ο ∷ ι ∷ α ∷ ν ∷ []) "Rom.13.14"
∷ word (μ ∷ ὴ ∷ []) "Rom.13.14"
∷ word (π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ σ ∷ θ ∷ ε ∷ []) "Rom.13.14"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.13.14"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ς ∷ []) "Rom.13.14"
∷ word (Τ ∷ ὸ ∷ ν ∷ []) "Rom.14.1"
∷ word (δ ∷ ὲ ∷ []) "Rom.14.1"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ α ∷ []) "Rom.14.1"
∷ word (τ ∷ ῇ ∷ []) "Rom.14.1"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ι ∷ []) "Rom.14.1"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ∙λ ∷ α ∷ μ ∷ β ∷ ά ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Rom.14.1"
∷ word (μ ∷ ὴ ∷ []) "Rom.14.1"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.14.1"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ρ ∷ ί ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.14.1"
∷ word (δ ∷ ι ∷ α ∷ ∙λ ∷ ο ∷ γ ∷ ι ∷ σ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.14.1"
∷ word (ὃ ∷ ς ∷ []) "Rom.14.2"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.14.2"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ []) "Rom.14.2"
∷ word (φ ∷ α ∷ γ ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.14.2"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Rom.14.2"
∷ word (ὁ ∷ []) "Rom.14.2"
∷ word (δ ∷ ὲ ∷ []) "Rom.14.2"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.14.2"
∷ word (∙λ ∷ ά ∷ χ ∷ α ∷ ν ∷ α ∷ []) "Rom.14.2"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ε ∷ ι ∷ []) "Rom.14.2"
∷ word (ὁ ∷ []) "Rom.14.3"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ω ∷ ν ∷ []) "Rom.14.3"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.14.3"
∷ word (μ ∷ ὴ ∷ []) "Rom.14.3"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "Rom.14.3"
∷ word (μ ∷ ὴ ∷ []) "Rom.14.3"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ί ∷ τ ∷ ω ∷ []) "Rom.14.3"
∷ word (ὁ ∷ []) "Rom.14.3"
∷ word (δ ∷ ὲ ∷ []) "Rom.14.3"
∷ word (μ ∷ ὴ ∷ []) "Rom.14.3"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ω ∷ ν ∷ []) "Rom.14.3"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.14.3"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "Rom.14.3"
∷ word (μ ∷ ὴ ∷ []) "Rom.14.3"
∷ word (κ ∷ ρ ∷ ι ∷ ν ∷ έ ∷ τ ∷ ω ∷ []) "Rom.14.3"
∷ word (ὁ ∷ []) "Rom.14.3"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.14.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.14.3"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.14.3"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ ∙λ ∷ ά ∷ β ∷ ε ∷ τ ∷ ο ∷ []) "Rom.14.3"
∷ word (σ ∷ ὺ ∷ []) "Rom.14.4"
∷ word (τ ∷ ί ∷ ς ∷ []) "Rom.14.4"
∷ word (ε ∷ ἶ ∷ []) "Rom.14.4"
∷ word (ὁ ∷ []) "Rom.14.4"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ω ∷ ν ∷ []) "Rom.14.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ό ∷ τ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.14.4"
∷ word (ο ∷ ἰ ∷ κ ∷ έ ∷ τ ∷ η ∷ ν ∷ []) "Rom.14.4"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.4"
∷ word (ἰ ∷ δ ∷ ί ∷ ῳ ∷ []) "Rom.14.4"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.14.4"
∷ word (σ ∷ τ ∷ ή ∷ κ ∷ ε ∷ ι ∷ []) "Rom.14.4"
∷ word (ἢ ∷ []) "Rom.14.4"
∷ word (π ∷ ί ∷ π ∷ τ ∷ ε ∷ ι ∷ []) "Rom.14.4"
∷ word (σ ∷ τ ∷ α ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.14.4"
∷ word (δ ∷ έ ∷ []) "Rom.14.4"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ τ ∷ ε ∷ ῖ ∷ []) "Rom.14.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.14.4"
∷ word (ὁ ∷ []) "Rom.14.4"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "Rom.14.4"
∷ word (σ ∷ τ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Rom.14.4"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "Rom.14.4"
∷ word (Ὃ ∷ ς ∷ []) "Rom.14.5"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.14.5"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ ι ∷ []) "Rom.14.5"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "Rom.14.5"
∷ word (π ∷ α ∷ ρ ∷ []) "Rom.14.5"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "Rom.14.5"
∷ word (ὃ ∷ ς ∷ []) "Rom.14.5"
∷ word (δ ∷ ὲ ∷ []) "Rom.14.5"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ ι ∷ []) "Rom.14.5"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "Rom.14.5"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "Rom.14.5"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.14.5"
∷ word (ἐ ∷ ν ∷ []) "Rom.14.5"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.5"
∷ word (ἰ ∷ δ ∷ ί ∷ ῳ ∷ []) "Rom.14.5"
∷ word (ν ∷ ο ∷ ῒ ∷ []) "Rom.14.5"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ φ ∷ ο ∷ ρ ∷ ε ∷ ί ∷ σ ∷ θ ∷ ω ∷ []) "Rom.14.5"
∷ word (ὁ ∷ []) "Rom.14.6"
∷ word (φ ∷ ρ ∷ ο ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.14.6"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.14.6"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "Rom.14.6"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.14.6"
∷ word (φ ∷ ρ ∷ ο ∷ ν ∷ ε ∷ ῖ ∷ []) "Rom.14.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.6"
∷ word (ὁ ∷ []) "Rom.14.6"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ω ∷ ν ∷ []) "Rom.14.6"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.14.6"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ε ∷ ι ∷ []) "Rom.14.6"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ε ∷ ῖ ∷ []) "Rom.14.6"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.14.6"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.6"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.14.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.6"
∷ word (ὁ ∷ []) "Rom.14.6"
∷ word (μ ∷ ὴ ∷ []) "Rom.14.6"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ω ∷ ν ∷ []) "Rom.14.6"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.14.6"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.14.6"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ε ∷ ι ∷ []) "Rom.14.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.6"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ε ∷ ῖ ∷ []) "Rom.14.6"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.6"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.14.6"
∷ word (Ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "Rom.14.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.14.7"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.14.7"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "Rom.14.7"
∷ word (ζ ∷ ῇ ∷ []) "Rom.14.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.7"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "Rom.14.7"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "Rom.14.7"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ ν ∷ ῄ ∷ σ ∷ κ ∷ ε ∷ ι ∷ []) "Rom.14.7"
∷ word (ἐ ∷ ά ∷ ν ∷ []) "Rom.14.8"
∷ word (τ ∷ ε ∷ []) "Rom.14.8"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.14.8"
∷ word (ζ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.14.8"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.8"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.14.8"
∷ word (ζ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.14.8"
∷ word (ἐ ∷ ά ∷ ν ∷ []) "Rom.14.8"
∷ word (τ ∷ ε ∷ []) "Rom.14.8"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ ν ∷ ῄ ∷ σ ∷ κ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Rom.14.8"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.8"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.14.8"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ ν ∷ ῄ ∷ σ ∷ κ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.14.8"
∷ word (ἐ ∷ ά ∷ ν ∷ []) "Rom.14.8"
∷ word (τ ∷ ε ∷ []) "Rom.14.8"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.14.8"
∷ word (ζ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "Rom.14.8"
∷ word (ἐ ∷ ά ∷ ν ∷ []) "Rom.14.8"
∷ word (τ ∷ ε ∷ []) "Rom.14.8"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ ν ∷ ῄ ∷ σ ∷ κ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Rom.14.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.14.8"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.14.8"
∷ word (ἐ ∷ σ ∷ μ ∷ έ ∷ ν ∷ []) "Rom.14.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.14.9"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.14.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.14.9"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.14.9"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "Rom.14.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.9"
∷ word (ἔ ∷ ζ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Rom.14.9"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.14.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.9"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Rom.14.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.9"
∷ word (ζ ∷ ώ ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.14.9"
∷ word (κ ∷ υ ∷ ρ ∷ ι ∷ ε ∷ ύ ∷ σ ∷ ῃ ∷ []) "Rom.14.9"
∷ word (Σ ∷ ὺ ∷ []) "Rom.14.10"
∷ word (δ ∷ ὲ ∷ []) "Rom.14.10"
∷ word (τ ∷ ί ∷ []) "Rom.14.10"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ ι ∷ ς ∷ []) "Rom.14.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.14.10"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ό ∷ ν ∷ []) "Rom.14.10"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.14.10"
∷ word (ἢ ∷ []) "Rom.14.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.10"
∷ word (σ ∷ ὺ ∷ []) "Rom.14.10"
∷ word (τ ∷ ί ∷ []) "Rom.14.10"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.14.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.14.10"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ό ∷ ν ∷ []) "Rom.14.10"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.14.10"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.14.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.14.10"
∷ word (π ∷ α ∷ ρ ∷ α ∷ σ ∷ τ ∷ η ∷ σ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Rom.14.10"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.10"
∷ word (β ∷ ή ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.14.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.14.10"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.14.10"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.14.11"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Rom.14.11"
∷ word (Ζ ∷ ῶ ∷ []) "Rom.14.11"
∷ word (ἐ ∷ γ ∷ ώ ∷ []) "Rom.14.11"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.14.11"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "Rom.14.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.14.11"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Rom.14.11"
∷ word (κ ∷ ά ∷ μ ∷ ψ ∷ ε ∷ ι ∷ []) "Rom.14.11"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "Rom.14.11"
∷ word (γ ∷ ό ∷ ν ∷ υ ∷ []) "Rom.14.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.11"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ []) "Rom.14.11"
∷ word (γ ∷ ∙λ ∷ ῶ ∷ σ ∷ σ ∷ α ∷ []) "Rom.14.11"
∷ word (ἐ ∷ ξ ∷ ο ∷ μ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.14.11"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.11"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.14.11"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Rom.14.12"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.14.12"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.14.12"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "Rom.14.12"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.14.12"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ν ∷ []) "Rom.14.12"
∷ word (δ ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "Rom.14.12"
∷ word (Μ ∷ η ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Rom.14.13"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.14.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.14.13"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Rom.14.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.14.13"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.14.13"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ α ∷ τ ∷ ε ∷ []) "Rom.14.13"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.14.13"
∷ word (τ ∷ ὸ ∷ []) "Rom.14.13"
∷ word (μ ∷ ὴ ∷ []) "Rom.14.13"
∷ word (τ ∷ ι ∷ θ ∷ έ ∷ ν ∷ α ∷ ι ∷ []) "Rom.14.13"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ κ ∷ ο ∷ μ ∷ μ ∷ α ∷ []) "Rom.14.13"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.13"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ῷ ∷ []) "Rom.14.13"
∷ word (ἢ ∷ []) "Rom.14.13"
∷ word (σ ∷ κ ∷ ά ∷ ν ∷ δ ∷ α ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.14.13"
∷ word (ο ∷ ἶ ∷ δ ∷ α ∷ []) "Rom.14.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.14"
∷ word (π ∷ έ ∷ π ∷ ε ∷ ι ∷ σ ∷ μ ∷ α ∷ ι ∷ []) "Rom.14.14"
∷ word (ἐ ∷ ν ∷ []) "Rom.14.14"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.14.14"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.14.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.14.14"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ ν ∷ []) "Rom.14.14"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ὸ ∷ ν ∷ []) "Rom.14.14"
∷ word (δ ∷ ι ∷ []) "Rom.14.14"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.14.14"
∷ word (ε ∷ ἰ ∷ []) "Rom.14.14"
∷ word (μ ∷ ὴ ∷ []) "Rom.14.14"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.14"
∷ word (∙λ ∷ ο ∷ γ ∷ ι ∷ ζ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ῳ ∷ []) "Rom.14.14"
∷ word (τ ∷ ι ∷ []) "Rom.14.14"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ὸ ∷ ν ∷ []) "Rom.14.14"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.14.14"
∷ word (ἐ ∷ κ ∷ ε ∷ ί ∷ ν ∷ ῳ ∷ []) "Rom.14.14"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ό ∷ ν ∷ []) "Rom.14.14"
∷ word (ε ∷ ἰ ∷ []) "Rom.14.15"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.14.15"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.14.15"
∷ word (β ∷ ρ ∷ ῶ ∷ μ ∷ α ∷ []) "Rom.14.15"
∷ word (ὁ ∷ []) "Rom.14.15"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ό ∷ ς ∷ []) "Rom.14.15"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.14.15"
∷ word (∙λ ∷ υ ∷ π ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "Rom.14.15"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Rom.14.15"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.14.15"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "Rom.14.15"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.14.15"
∷ word (μ ∷ ὴ ∷ []) "Rom.14.15"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.15"
∷ word (β ∷ ρ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ί ∷ []) "Rom.14.15"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.14.15"
∷ word (ἐ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ ο ∷ ν ∷ []) "Rom.14.15"
∷ word (ἀ ∷ π ∷ ό ∷ ∙λ ∷ ∙λ ∷ υ ∷ ε ∷ []) "Rom.14.15"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.14.15"
∷ word (ο ∷ ὗ ∷ []) "Rom.14.15"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.14.15"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "Rom.14.15"
∷ word (μ ∷ ὴ ∷ []) "Rom.14.16"
∷ word (β ∷ ∙λ ∷ α ∷ σ ∷ φ ∷ η ∷ μ ∷ ε ∷ ί ∷ σ ∷ θ ∷ ω ∷ []) "Rom.14.16"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.14.16"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.14.16"
∷ word (τ ∷ ὸ ∷ []) "Rom.14.16"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ό ∷ ν ∷ []) "Rom.14.16"
∷ word (ο ∷ ὐ ∷ []) "Rom.14.17"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Rom.14.17"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.14.17"
∷ word (ἡ ∷ []) "Rom.14.17"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ []) "Rom.14.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.14.17"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.14.17"
∷ word (β ∷ ρ ∷ ῶ ∷ σ ∷ ι ∷ ς ∷ []) "Rom.14.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.17"
∷ word (π ∷ ό ∷ σ ∷ ι ∷ ς ∷ []) "Rom.14.17"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.14.17"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ []) "Rom.14.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.17"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "Rom.14.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.17"
∷ word (χ ∷ α ∷ ρ ∷ ὰ ∷ []) "Rom.14.17"
∷ word (ἐ ∷ ν ∷ []) "Rom.14.17"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.14.17"
∷ word (ἁ ∷ γ ∷ ί ∷ ῳ ∷ []) "Rom.14.17"
∷ word (ὁ ∷ []) "Rom.14.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.14.18"
∷ word (ἐ ∷ ν ∷ []) "Rom.14.18"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "Rom.14.18"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "Rom.14.18"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.18"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.14.18"
∷ word (ε ∷ ὐ ∷ ά ∷ ρ ∷ ε ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.14.18"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.18"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Rom.14.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.18"
∷ word (δ ∷ ό ∷ κ ∷ ι ∷ μ ∷ ο ∷ ς ∷ []) "Rom.14.18"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.14.18"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "Rom.14.18"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Rom.14.19"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.14.19"
∷ word (τ ∷ ὰ ∷ []) "Rom.14.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.14.19"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ ς ∷ []) "Rom.14.19"
∷ word (δ ∷ ι ∷ ώ ∷ κ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Rom.14.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.14.19"
∷ word (τ ∷ ὰ ∷ []) "Rom.14.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.14.19"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ῆ ∷ ς ∷ []) "Rom.14.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.14.19"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.14.19"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.14.19"
∷ word (μ ∷ ὴ ∷ []) "Rom.14.20"
∷ word (ἕ ∷ ν ∷ ε ∷ κ ∷ ε ∷ ν ∷ []) "Rom.14.20"
∷ word (β ∷ ρ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.14.20"
∷ word (κ ∷ α ∷ τ ∷ ά ∷ ∙λ ∷ υ ∷ ε ∷ []) "Rom.14.20"
∷ word (τ ∷ ὸ ∷ []) "Rom.14.20"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ν ∷ []) "Rom.14.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.14.20"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.14.20"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Rom.14.20"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Rom.14.20"
∷ word (κ ∷ α ∷ θ ∷ α ∷ ρ ∷ ά ∷ []) "Rom.14.20"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.14.20"
∷ word (κ ∷ α ∷ κ ∷ ὸ ∷ ν ∷ []) "Rom.14.20"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.20"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ῳ ∷ []) "Rom.14.20"
∷ word (τ ∷ ῷ ∷ []) "Rom.14.20"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.14.20"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ κ ∷ ό ∷ μ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.14.20"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ο ∷ ν ∷ τ ∷ ι ∷ []) "Rom.14.20"
∷ word (κ ∷ α ∷ ∙λ ∷ ὸ ∷ ν ∷ []) "Rom.14.21"
∷ word (τ ∷ ὸ ∷ []) "Rom.14.21"
∷ word (μ ∷ ὴ ∷ []) "Rom.14.21"
∷ word (φ ∷ α ∷ γ ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.14.21"
∷ word (κ ∷ ρ ∷ έ ∷ α ∷ []) "Rom.14.21"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "Rom.14.21"
∷ word (π ∷ ι ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.14.21"
∷ word (ο ∷ ἶ ∷ ν ∷ ο ∷ ν ∷ []) "Rom.14.21"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "Rom.14.21"
∷ word (ἐ ∷ ν ∷ []) "Rom.14.21"
∷ word (ᾧ ∷ []) "Rom.14.21"
∷ word (ὁ ∷ []) "Rom.14.21"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ό ∷ ς ∷ []) "Rom.14.21"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.14.21"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ κ ∷ ό ∷ π ∷ τ ∷ ε ∷ ι ∷ []) "Rom.14.21"
∷ word (ἢ ∷ []) "Rom.14.21"
∷ word (σ ∷ κ ∷ α ∷ ν ∷ δ ∷ α ∷ ∙λ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.14.21"
∷ word (ἢ ∷ []) "Rom.14.21"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ῖ ∷ []) "Rom.14.21"
∷ word (σ ∷ ὺ ∷ []) "Rom.14.22"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.14.22"
∷ word (ἣ ∷ ν ∷ []) "Rom.14.22"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ ς ∷ []) "Rom.14.22"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.14.22"
∷ word (σ ∷ ε ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.14.22"
∷ word (ἔ ∷ χ ∷ ε ∷ []) "Rom.14.22"
∷ word (ἐ ∷ ν ∷ ώ ∷ π ∷ ι ∷ ο ∷ ν ∷ []) "Rom.14.22"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.14.22"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.14.22"
∷ word (μ ∷ α ∷ κ ∷ ά ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "Rom.14.22"
∷ word (ὁ ∷ []) "Rom.14.22"
∷ word (μ ∷ ὴ ∷ []) "Rom.14.22"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ω ∷ ν ∷ []) "Rom.14.22"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.14.22"
∷ word (ἐ ∷ ν ∷ []) "Rom.14.22"
∷ word (ᾧ ∷ []) "Rom.14.22"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ά ∷ ζ ∷ ε ∷ ι ∷ []) "Rom.14.22"
∷ word (ὁ ∷ []) "Rom.14.23"
∷ word (δ ∷ ὲ ∷ []) "Rom.14.23"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ρ ∷ ι ∷ ν ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Rom.14.23"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.14.23"
∷ word (φ ∷ ά ∷ γ ∷ ῃ ∷ []) "Rom.14.23"
∷ word (κ ∷ α ∷ τ ∷ α ∷ κ ∷ έ ∷ κ ∷ ρ ∷ ι ∷ τ ∷ α ∷ ι ∷ []) "Rom.14.23"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.14.23"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.14.23"
∷ word (ἐ ∷ κ ∷ []) "Rom.14.23"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.14.23"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "Rom.14.23"
∷ word (δ ∷ ὲ ∷ []) "Rom.14.23"
∷ word (ὃ ∷ []) "Rom.14.23"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.14.23"
∷ word (ἐ ∷ κ ∷ []) "Rom.14.23"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.14.23"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "Rom.14.23"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "Rom.14.23"
∷ word (Ὀ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Rom.15.1"
∷ word (δ ∷ ὲ ∷ []) "Rom.15.1"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.15.1"
∷ word (ο ∷ ἱ ∷ []) "Rom.15.1"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ τ ∷ ο ∷ ὶ ∷ []) "Rom.15.1"
∷ word (τ ∷ ὰ ∷ []) "Rom.15.1"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Rom.15.1"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.15.1"
∷ word (ἀ ∷ δ ∷ υ ∷ ν ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "Rom.15.1"
∷ word (β ∷ α ∷ σ ∷ τ ∷ ά ∷ ζ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.15.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.1"
∷ word (μ ∷ ὴ ∷ []) "Rom.15.1"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.15.1"
∷ word (ἀ ∷ ρ ∷ έ ∷ σ ∷ κ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.15.1"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.15.2"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.15.2"
∷ word (τ ∷ ῷ ∷ []) "Rom.15.2"
∷ word (π ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ο ∷ ν ∷ []) "Rom.15.2"
∷ word (ἀ ∷ ρ ∷ ε ∷ σ ∷ κ ∷ έ ∷ τ ∷ ω ∷ []) "Rom.15.2"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.15.2"
∷ word (τ ∷ ὸ ∷ []) "Rom.15.2"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ὸ ∷ ν ∷ []) "Rom.15.2"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.15.2"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ή ∷ ν ∷ []) "Rom.15.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.15.3"
∷ word (ὁ ∷ []) "Rom.15.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.15.3"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "Rom.15.3"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "Rom.15.3"
∷ word (ἤ ∷ ρ ∷ ε ∷ σ ∷ ε ∷ ν ∷ []) "Rom.15.3"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.15.3"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.15.3"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.15.3"
∷ word (Ο ∷ ἱ ∷ []) "Rom.15.3"
∷ word (ὀ ∷ ν ∷ ε ∷ ι ∷ δ ∷ ι ∷ σ ∷ μ ∷ ο ∷ ὶ ∷ []) "Rom.15.3"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.15.3"
∷ word (ὀ ∷ ν ∷ ε ∷ ι ∷ δ ∷ ι ∷ ζ ∷ ό ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.15.3"
∷ word (σ ∷ ε ∷ []) "Rom.15.3"
∷ word (ἐ ∷ π ∷ έ ∷ π ∷ ε ∷ σ ∷ α ∷ ν ∷ []) "Rom.15.3"
∷ word (ἐ ∷ π ∷ []) "Rom.15.3"
∷ word (ἐ ∷ μ ∷ έ ∷ []) "Rom.15.3"
∷ word (ὅ ∷ σ ∷ α ∷ []) "Rom.15.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.15.4"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ γ ∷ ρ ∷ ά ∷ φ ∷ η ∷ []) "Rom.15.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.15.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.15.4"
∷ word (ἡ ∷ μ ∷ ε ∷ τ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "Rom.15.4"
∷ word (δ ∷ ι ∷ δ ∷ α ∷ σ ∷ κ ∷ α ∷ ∙λ ∷ ί ∷ α ∷ ν ∷ []) "Rom.15.4"
∷ word (ἐ ∷ γ ∷ ρ ∷ ά ∷ φ ∷ η ∷ []) "Rom.15.4"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.15.4"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.15.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.15.4"
∷ word (ὑ ∷ π ∷ ο ∷ μ ∷ ο ∷ ν ∷ ῆ ∷ ς ∷ []) "Rom.15.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.4"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.15.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.15.4"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.15.4"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.15.4"
∷ word (γ ∷ ρ ∷ α ∷ φ ∷ ῶ ∷ ν ∷ []) "Rom.15.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.15.4"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ α ∷ []) "Rom.15.4"
∷ word (ἔ ∷ χ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Rom.15.4"
∷ word (ὁ ∷ []) "Rom.15.5"
∷ word (δ ∷ ὲ ∷ []) "Rom.15.5"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.15.5"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.15.5"
∷ word (ὑ ∷ π ∷ ο ∷ μ ∷ ο ∷ ν ∷ ῆ ∷ ς ∷ []) "Rom.15.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.5"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.15.5"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.15.5"
∷ word (δ ∷ ῴ ∷ η ∷ []) "Rom.15.5"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.15.5"
∷ word (τ ∷ ὸ ∷ []) "Rom.15.5"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Rom.15.5"
∷ word (φ ∷ ρ ∷ ο ∷ ν ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.15.5"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "Rom.15.5"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Rom.15.5"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.15.5"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "Rom.15.5"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.15.6"
∷ word (ὁ ∷ μ ∷ ο ∷ θ ∷ υ ∷ μ ∷ α ∷ δ ∷ ὸ ∷ ν ∷ []) "Rom.15.6"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.6"
∷ word (ἑ ∷ ν ∷ ὶ ∷ []) "Rom.15.6"
∷ word (σ ∷ τ ∷ ό ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.15.6"
∷ word (δ ∷ ο ∷ ξ ∷ ά ∷ ζ ∷ η ∷ τ ∷ ε ∷ []) "Rom.15.6"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.15.6"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "Rom.15.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.6"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Rom.15.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.15.6"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.15.6"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.15.6"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.15.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.15.6"
∷ word (Δ ∷ ι ∷ ὸ ∷ []) "Rom.15.7"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ∙λ ∷ α ∷ μ ∷ β ∷ ά ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Rom.15.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.15.7"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.15.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.7"
∷ word (ὁ ∷ []) "Rom.15.7"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.15.7"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ ∙λ ∷ ά ∷ β ∷ ε ∷ τ ∷ ο ∷ []) "Rom.15.7"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.15.7"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.15.7"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "Rom.15.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.15.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.15.7"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Rom.15.8"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.15.8"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.15.8"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ν ∷ []) "Rom.15.8"
∷ word (γ ∷ ε ∷ γ ∷ ε ∷ ν ∷ ῆ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.15.8"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῆ ∷ ς ∷ []) "Rom.15.8"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.15.8"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Rom.15.8"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.15.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.15.8"
∷ word (τ ∷ ὸ ∷ []) "Rom.15.8"
∷ word (β ∷ ε ∷ β ∷ α ∷ ι ∷ ῶ ∷ σ ∷ α ∷ ι ∷ []) "Rom.15.8"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Rom.15.8"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "Rom.15.8"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.15.8"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ ω ∷ ν ∷ []) "Rom.15.8"
∷ word (τ ∷ ὰ ∷ []) "Rom.15.9"
∷ word (δ ∷ ὲ ∷ []) "Rom.15.9"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Rom.15.9"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.15.9"
∷ word (ἐ ∷ ∙λ ∷ έ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.15.9"
∷ word (δ ∷ ο ∷ ξ ∷ ά ∷ σ ∷ α ∷ ι ∷ []) "Rom.15.9"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.15.9"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "Rom.15.9"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.15.9"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.15.9"
∷ word (Δ ∷ ι ∷ ὰ ∷ []) "Rom.15.9"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.15.9"
∷ word (ἐ ∷ ξ ∷ ο ∷ μ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ή ∷ σ ∷ ο ∷ μ ∷ α ∷ ί ∷ []) "Rom.15.9"
∷ word (σ ∷ ο ∷ ι ∷ []) "Rom.15.9"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.9"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ σ ∷ ι ∷ []) "Rom.15.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.9"
∷ word (τ ∷ ῷ ∷ []) "Rom.15.9"
∷ word (ὀ ∷ ν ∷ ο ∷ μ ∷ α ∷ τ ∷ ί ∷ []) "Rom.15.9"
∷ word (σ ∷ ο ∷ υ ∷ []) "Rom.15.9"
∷ word (ψ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "Rom.15.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.10"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Rom.15.10"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.15.10"
∷ word (Ε ∷ ὐ ∷ φ ∷ ρ ∷ ά ∷ ν ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Rom.15.10"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Rom.15.10"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Rom.15.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.15.10"
∷ word (∙λ ∷ α ∷ ο ∷ ῦ ∷ []) "Rom.15.10"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.15.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.11"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Rom.15.11"
∷ word (Α ∷ ἰ ∷ ν ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Rom.15.11"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Rom.15.11"
∷ word (τ ∷ ὰ ∷ []) "Rom.15.11"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Rom.15.11"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.15.11"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.15.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.11"
∷ word (ἐ ∷ π ∷ α ∷ ι ∷ ν ∷ ε ∷ σ ∷ ά ∷ τ ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "Rom.15.11"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.15.11"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Rom.15.11"
∷ word (ο ∷ ἱ ∷ []) "Rom.15.11"
∷ word (∙λ ∷ α ∷ ο ∷ ί ∷ []) "Rom.15.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.12"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Rom.15.12"
∷ word (Ἠ ∷ σ ∷ α ∷ ΐ ∷ α ∷ ς ∷ []) "Rom.15.12"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Rom.15.12"
∷ word (Ἔ ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "Rom.15.12"
∷ word (ἡ ∷ []) "Rom.15.12"
∷ word (ῥ ∷ ί ∷ ζ ∷ α ∷ []) "Rom.15.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.15.12"
∷ word (Ἰ ∷ ε ∷ σ ∷ σ ∷ α ∷ ί ∷ []) "Rom.15.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.12"
∷ word (ὁ ∷ []) "Rom.15.12"
∷ word (ἀ ∷ ν ∷ ι ∷ σ ∷ τ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Rom.15.12"
∷ word (ἄ ∷ ρ ∷ χ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.15.12"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.15.12"
∷ word (ἐ ∷ π ∷ []) "Rom.15.12"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Rom.15.12"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Rom.15.12"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ι ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.15.12"
∷ word (ὁ ∷ []) "Rom.15.13"
∷ word (δ ∷ ὲ ∷ []) "Rom.15.13"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.15.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.15.13"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ ο ∷ ς ∷ []) "Rom.15.13"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ώ ∷ σ ∷ α ∷ ι ∷ []) "Rom.15.13"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.15.13"
∷ word (π ∷ ά ∷ σ ∷ η ∷ ς ∷ []) "Rom.15.13"
∷ word (χ ∷ α ∷ ρ ∷ ᾶ ∷ ς ∷ []) "Rom.15.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.13"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ ς ∷ []) "Rom.15.13"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.13"
∷ word (τ ∷ ῷ ∷ []) "Rom.15.13"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.15.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.15.13"
∷ word (τ ∷ ὸ ∷ []) "Rom.15.13"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ ν ∷ []) "Rom.15.13"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.15.13"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.13"
∷ word (τ ∷ ῇ ∷ []) "Rom.15.13"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ ι ∷ []) "Rom.15.13"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.13"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ []) "Rom.15.13"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.15.13"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.15.13"
∷ word (Π ∷ έ ∷ π ∷ ε ∷ ι ∷ σ ∷ μ ∷ α ∷ ι ∷ []) "Rom.15.14"
∷ word (δ ∷ έ ∷ []) "Rom.15.14"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Rom.15.14"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.15.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.14"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.15.14"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.15.14"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "Rom.15.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.15.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.15.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.14"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὶ ∷ []) "Rom.15.14"
∷ word (μ ∷ ε ∷ σ ∷ τ ∷ ο ∷ ί ∷ []) "Rom.15.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "Rom.15.14"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ω ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "Rom.15.14"
∷ word (π ∷ ε ∷ π ∷ ∙λ ∷ η ∷ ρ ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Rom.15.14"
∷ word (π ∷ ά ∷ σ ∷ η ∷ ς ∷ []) "Rom.15.14"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.15.14"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Rom.15.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.14"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.15.14"
∷ word (ν ∷ ο ∷ υ ∷ θ ∷ ε ∷ τ ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.15.14"
∷ word (τ ∷ ο ∷ ∙λ ∷ μ ∷ η ∷ ρ ∷ ό ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "Rom.15.15"
∷ word (δ ∷ ὲ ∷ []) "Rom.15.15"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "Rom.15.15"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.15.15"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.15.15"
∷ word (μ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.15.15"
∷ word (ὡ ∷ ς ∷ []) "Rom.15.15"
∷ word (ἐ ∷ π ∷ α ∷ ν ∷ α ∷ μ ∷ ι ∷ μ ∷ ν ∷ ῄ ∷ σ ∷ κ ∷ ω ∷ ν ∷ []) "Rom.15.15"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.15.15"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.15.15"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.15.15"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Rom.15.15"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.15.15"
∷ word (δ ∷ ο ∷ θ ∷ ε ∷ ῖ ∷ σ ∷ ά ∷ ν ∷ []) "Rom.15.15"
∷ word (μ ∷ ο ∷ ι ∷ []) "Rom.15.15"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Rom.15.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.15.15"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.15.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.15.16"
∷ word (τ ∷ ὸ ∷ []) "Rom.15.16"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ί ∷ []) "Rom.15.16"
∷ word (μ ∷ ε ∷ []) "Rom.15.16"
∷ word (∙λ ∷ ε ∷ ι ∷ τ ∷ ο ∷ υ ∷ ρ ∷ γ ∷ ὸ ∷ ν ∷ []) "Rom.15.16"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.15.16"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.15.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.15.16"
∷ word (τ ∷ ὰ ∷ []) "Rom.15.16"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Rom.15.16"
∷ word (ἱ ∷ ε ∷ ρ ∷ ο ∷ υ ∷ ρ ∷ γ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ α ∷ []) "Rom.15.16"
∷ word (τ ∷ ὸ ∷ []) "Rom.15.16"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.15.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.15.16"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.15.16"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.15.16"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Rom.15.16"
∷ word (ἡ ∷ []) "Rom.15.16"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ φ ∷ ο ∷ ρ ∷ ὰ ∷ []) "Rom.15.16"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.15.16"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.15.16"
∷ word (ε ∷ ὐ ∷ π ∷ ρ ∷ ό ∷ σ ∷ δ ∷ ε ∷ κ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.15.16"
∷ word (ἡ ∷ γ ∷ ι ∷ α ∷ σ ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "Rom.15.16"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.16"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.15.16"
∷ word (ἁ ∷ γ ∷ ί ∷ ῳ ∷ []) "Rom.15.16"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "Rom.15.17"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.15.17"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.15.17"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "Rom.15.17"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.15.17"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.15.17"
∷ word (τ ∷ ὰ ∷ []) "Rom.15.17"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.15.17"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.15.17"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "Rom.15.17"
∷ word (ο ∷ ὐ ∷ []) "Rom.15.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.15.18"
∷ word (τ ∷ ο ∷ ∙λ ∷ μ ∷ ή ∷ σ ∷ ω ∷ []) "Rom.15.18"
∷ word (τ ∷ ι ∷ []) "Rom.15.18"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.15.18"
∷ word (ὧ ∷ ν ∷ []) "Rom.15.18"
∷ word (ο ∷ ὐ ∷ []) "Rom.15.18"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ι ∷ ρ ∷ γ ∷ ά ∷ σ ∷ α ∷ τ ∷ ο ∷ []) "Rom.15.18"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Rom.15.18"
∷ word (δ ∷ ι ∷ []) "Rom.15.18"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "Rom.15.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.15.18"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ὴ ∷ ν ∷ []) "Rom.15.18"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.15.18"
∷ word (∙λ ∷ ό ∷ γ ∷ ῳ ∷ []) "Rom.15.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.18"
∷ word (ἔ ∷ ρ ∷ γ ∷ ῳ ∷ []) "Rom.15.18"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.19"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ []) "Rom.15.19"
∷ word (σ ∷ η ∷ μ ∷ ε ∷ ί ∷ ω ∷ ν ∷ []) "Rom.15.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.19"
∷ word (τ ∷ ε ∷ ρ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "Rom.15.19"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.19"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ []) "Rom.15.19"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.15.19"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "Rom.15.19"
∷ word (μ ∷ ε ∷ []) "Rom.15.19"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.15.19"
∷ word (Ἰ ∷ ε ∷ ρ ∷ ο ∷ υ ∷ σ ∷ α ∷ ∙λ ∷ ὴ ∷ μ ∷ []) "Rom.15.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.19"
∷ word (κ ∷ ύ ∷ κ ∷ ∙λ ∷ ῳ ∷ []) "Rom.15.19"
∷ word (μ ∷ έ ∷ χ ∷ ρ ∷ ι ∷ []) "Rom.15.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.15.19"
∷ word (Ἰ ∷ ∙λ ∷ ∙λ ∷ υ ∷ ρ ∷ ι ∷ κ ∷ ο ∷ ῦ ∷ []) "Rom.15.19"
∷ word (π ∷ ε ∷ π ∷ ∙λ ∷ η ∷ ρ ∷ ω ∷ κ ∷ έ ∷ ν ∷ α ∷ ι ∷ []) "Rom.15.19"
∷ word (τ ∷ ὸ ∷ []) "Rom.15.19"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.15.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.15.19"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.15.19"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Rom.15.20"
∷ word (δ ∷ ὲ ∷ []) "Rom.15.20"
∷ word (φ ∷ ι ∷ ∙λ ∷ ο ∷ τ ∷ ι ∷ μ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "Rom.15.20"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ζ ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.15.20"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "Rom.15.20"
∷ word (ὅ ∷ π ∷ ο ∷ υ ∷ []) "Rom.15.20"
∷ word (ὠ ∷ ν ∷ ο ∷ μ ∷ ά ∷ σ ∷ θ ∷ η ∷ []) "Rom.15.20"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "Rom.15.20"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.15.20"
∷ word (μ ∷ ὴ ∷ []) "Rom.15.20"
∷ word (ἐ ∷ π ∷ []) "Rom.15.20"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ό ∷ τ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.15.20"
∷ word (θ ∷ ε ∷ μ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "Rom.15.20"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ῶ ∷ []) "Rom.15.20"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.15.21"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Rom.15.21"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Rom.15.21"
∷ word (Ο ∷ ἷ ∷ ς ∷ []) "Rom.15.21"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.15.21"
∷ word (ἀ ∷ ν ∷ η ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ η ∷ []) "Rom.15.21"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "Rom.15.21"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.15.21"
∷ word (ὄ ∷ ψ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.15.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.21"
∷ word (ο ∷ ἳ ∷ []) "Rom.15.21"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.15.21"
∷ word (ἀ ∷ κ ∷ η ∷ κ ∷ ό ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "Rom.15.21"
∷ word (σ ∷ υ ∷ ν ∷ ή ∷ σ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.15.21"
∷ word (Δ ∷ ι ∷ ὸ ∷ []) "Rom.15.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.22"
∷ word (ἐ ∷ ν ∷ ε ∷ κ ∷ ο ∷ π ∷ τ ∷ ό ∷ μ ∷ η ∷ ν ∷ []) "Rom.15.22"
∷ word (τ ∷ ὰ ∷ []) "Rom.15.22"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.15.22"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.15.22"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.15.22"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.15.22"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.15.22"
∷ word (ν ∷ υ ∷ ν ∷ ὶ ∷ []) "Rom.15.23"
∷ word (δ ∷ ὲ ∷ []) "Rom.15.23"
∷ word (μ ∷ η ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Rom.15.23"
∷ word (τ ∷ ό ∷ π ∷ ο ∷ ν ∷ []) "Rom.15.23"
∷ word (ἔ ∷ χ ∷ ω ∷ ν ∷ []) "Rom.15.23"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.23"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.15.23"
∷ word (κ ∷ ∙λ ∷ ί ∷ μ ∷ α ∷ σ ∷ ι ∷ []) "Rom.15.23"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "Rom.15.23"
∷ word (ἐ ∷ π ∷ ι ∷ π ∷ ο ∷ θ ∷ ί ∷ α ∷ ν ∷ []) "Rom.15.23"
∷ word (δ ∷ ὲ ∷ []) "Rom.15.23"
∷ word (ἔ ∷ χ ∷ ω ∷ ν ∷ []) "Rom.15.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.15.23"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.15.23"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.15.23"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.15.23"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.15.23"
∷ word (ἱ ∷ κ ∷ α ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.15.23"
∷ word (ἐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.15.23"
∷ word (ὡ ∷ ς ∷ []) "Rom.15.24"
∷ word (ἂ ∷ ν ∷ []) "Rom.15.24"
∷ word (π ∷ ο ∷ ρ ∷ ε ∷ ύ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "Rom.15.24"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.15.24"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.15.24"
∷ word (Σ ∷ π ∷ α ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "Rom.15.24"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ ζ ∷ ω ∷ []) "Rom.15.24"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.15.24"
∷ word (δ ∷ ι ∷ α ∷ π ∷ ο ∷ ρ ∷ ε ∷ υ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Rom.15.24"
∷ word (θ ∷ ε ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.15.24"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.15.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.24"
∷ word (ὑ ∷ φ ∷ []) "Rom.15.24"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.15.24"
∷ word (π ∷ ρ ∷ ο ∷ π ∷ ε ∷ μ ∷ φ ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "Rom.15.24"
∷ word (ἐ ∷ κ ∷ ε ∷ ῖ ∷ []) "Rom.15.24"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Rom.15.24"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.15.24"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.15.24"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.15.24"
∷ word (μ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.15.24"
∷ word (ἐ ∷ μ ∷ π ∷ ∙λ ∷ η ∷ σ ∷ θ ∷ ῶ ∷ []) "Rom.15.24"
∷ word (ν ∷ υ ∷ ν ∷ ὶ ∷ []) "Rom.15.25"
∷ word (δ ∷ ὲ ∷ []) "Rom.15.25"
∷ word (π ∷ ο ∷ ρ ∷ ε ∷ ύ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.15.25"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.15.25"
∷ word (Ἰ ∷ ε ∷ ρ ∷ ο ∷ υ ∷ σ ∷ α ∷ ∙λ ∷ ὴ ∷ μ ∷ []) "Rom.15.25"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.15.25"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.15.25"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Rom.15.25"
∷ word (ε ∷ ὐ ∷ δ ∷ ό ∷ κ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.15.26"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.15.26"
∷ word (Μ ∷ α ∷ κ ∷ ε ∷ δ ∷ ο ∷ ν ∷ ί ∷ α ∷ []) "Rom.15.26"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.26"
∷ word (Ἀ ∷ χ ∷ α ∷ ΐ ∷ α ∷ []) "Rom.15.26"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "Rom.15.26"
∷ word (τ ∷ ι ∷ ν ∷ ὰ ∷ []) "Rom.15.26"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Rom.15.26"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.15.26"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.15.26"
∷ word (π ∷ τ ∷ ω ∷ χ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.15.26"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.15.26"
∷ word (ἁ ∷ γ ∷ ί ∷ ω ∷ ν ∷ []) "Rom.15.26"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.15.26"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.26"
∷ word (Ἰ ∷ ε ∷ ρ ∷ ο ∷ υ ∷ σ ∷ α ∷ ∙λ ∷ ή ∷ μ ∷ []) "Rom.15.26"
∷ word (ε ∷ ὐ ∷ δ ∷ ό ∷ κ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.15.27"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Rom.15.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.27"
∷ word (ὀ ∷ φ ∷ ε ∷ ι ∷ ∙λ ∷ έ ∷ τ ∷ α ∷ ι ∷ []) "Rom.15.27"
∷ word (ε ∷ ἰ ∷ σ ∷ ὶ ∷ ν ∷ []) "Rom.15.27"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.15.27"
∷ word (ε ∷ ἰ ∷ []) "Rom.15.27"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.15.27"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.15.27"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.15.27"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.15.27"
∷ word (ἐ ∷ κ ∷ ο ∷ ι ∷ ν ∷ ώ ∷ ν ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Rom.15.27"
∷ word (τ ∷ ὰ ∷ []) "Rom.15.27"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Rom.15.27"
∷ word (ὀ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.15.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.27"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.27"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.15.27"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ι ∷ κ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.15.27"
∷ word (∙λ ∷ ε ∷ ι ∷ τ ∷ ο ∷ υ ∷ ρ ∷ γ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Rom.15.27"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.15.27"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Rom.15.28"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.15.28"
∷ word (ἐ ∷ π ∷ ι ∷ τ ∷ ε ∷ ∙λ ∷ έ ∷ σ ∷ α ∷ ς ∷ []) "Rom.15.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.28"
∷ word (σ ∷ φ ∷ ρ ∷ α ∷ γ ∷ ι ∷ σ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Rom.15.28"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.15.28"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.15.28"
∷ word (κ ∷ α ∷ ρ ∷ π ∷ ὸ ∷ ν ∷ []) "Rom.15.28"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.15.28"
∷ word (ἀ ∷ π ∷ ε ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.15.28"
∷ word (δ ∷ ι ∷ []) "Rom.15.28"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.15.28"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.15.28"
∷ word (Σ ∷ π ∷ α ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "Rom.15.28"
∷ word (ο ∷ ἶ ∷ δ ∷ α ∷ []) "Rom.15.29"
∷ word (δ ∷ ὲ ∷ []) "Rom.15.29"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Rom.15.29"
∷ word (ἐ ∷ ρ ∷ χ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Rom.15.29"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.15.29"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.15.29"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.29"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.15.29"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ ς ∷ []) "Rom.15.29"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.15.29"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.15.29"
∷ word (Π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "Rom.15.30"
∷ word (δ ∷ ὲ ∷ []) "Rom.15.30"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.15.30"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Rom.15.30"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.15.30"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.15.30"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.15.30"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.15.30"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.15.30"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.15.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.30"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.15.30"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.15.30"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ς ∷ []) "Rom.15.30"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.15.30"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.15.30"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ γ ∷ ω ∷ ν ∷ ί ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ί ∷ []) "Rom.15.30"
∷ word (μ ∷ ο ∷ ι ∷ []) "Rom.15.30"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.30"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Rom.15.30"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ α ∷ ῖ ∷ ς ∷ []) "Rom.15.30"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.15.30"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "Rom.15.30"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.15.30"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.15.30"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "Rom.15.30"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.15.31"
∷ word (ῥ ∷ υ ∷ σ ∷ θ ∷ ῶ ∷ []) "Rom.15.31"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Rom.15.31"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.15.31"
∷ word (ἀ ∷ π ∷ ε ∷ ι ∷ θ ∷ ο ∷ ύ ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.15.31"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.31"
∷ word (τ ∷ ῇ ∷ []) "Rom.15.31"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ᾳ ∷ []) "Rom.15.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.15.31"
∷ word (ἡ ∷ []) "Rom.15.31"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ []) "Rom.15.31"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.15.31"
∷ word (ἡ ∷ []) "Rom.15.31"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.15.31"
∷ word (Ἰ ∷ ε ∷ ρ ∷ ο ∷ υ ∷ σ ∷ α ∷ ∙λ ∷ ὴ ∷ μ ∷ []) "Rom.15.31"
∷ word (ε ∷ ὐ ∷ π ∷ ρ ∷ ό ∷ σ ∷ δ ∷ ε ∷ κ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.15.31"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.15.31"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Rom.15.31"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Rom.15.31"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.15.32"
∷ word (ἐ ∷ ν ∷ []) "Rom.15.32"
∷ word (χ ∷ α ∷ ρ ∷ ᾷ ∷ []) "Rom.15.32"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ὼ ∷ ν ∷ []) "Rom.15.32"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Rom.15.32"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.15.32"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.15.32"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Rom.15.32"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Rom.15.32"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ ν ∷ α ∷ π ∷ α ∷ ύ ∷ σ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "Rom.15.32"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.15.32"
∷ word (ὁ ∷ []) "Rom.15.33"
∷ word (δ ∷ ὲ ∷ []) "Rom.15.33"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.15.33"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.15.33"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ ς ∷ []) "Rom.15.33"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Rom.15.33"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.15.33"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.15.33"
∷ word (ἀ ∷ μ ∷ ή ∷ ν ∷ []) "Rom.15.33"
∷ word (Σ ∷ υ ∷ ν ∷ ί ∷ σ ∷ τ ∷ η ∷ μ ∷ ι ∷ []) "Rom.16.1"
∷ word (δ ∷ ὲ ∷ []) "Rom.16.1"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.16.1"
∷ word (Φ ∷ ο ∷ ί ∷ β ∷ η ∷ ν ∷ []) "Rom.16.1"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.16.1"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὴ ∷ ν ∷ []) "Rom.16.1"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.16.1"
∷ word (ο ∷ ὖ ∷ σ ∷ α ∷ ν ∷ []) "Rom.16.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.1"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ν ∷ []) "Rom.16.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.16.1"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Rom.16.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.16.1"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.1"
∷ word (Κ ∷ ε ∷ γ ∷ χ ∷ ρ ∷ ε ∷ α ∷ ῖ ∷ ς ∷ []) "Rom.16.1"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Rom.16.2"
∷ word (α ∷ ὐ ∷ τ ∷ ὴ ∷ ν ∷ []) "Rom.16.2"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ δ ∷ έ ∷ ξ ∷ η ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.2"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.2"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.16.2"
∷ word (ἀ ∷ ξ ∷ ί ∷ ω ∷ ς ∷ []) "Rom.16.2"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.16.2"
∷ word (ἁ ∷ γ ∷ ί ∷ ω ∷ ν ∷ []) "Rom.16.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.2"
∷ word (π ∷ α ∷ ρ ∷ α ∷ σ ∷ τ ∷ ῆ ∷ τ ∷ ε ∷ []) "Rom.16.2"
∷ word (α ∷ ὐ ∷ τ ∷ ῇ ∷ []) "Rom.16.2"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.2"
∷ word (ᾧ ∷ []) "Rom.16.2"
∷ word (ἂ ∷ ν ∷ []) "Rom.16.2"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.16.2"
∷ word (χ ∷ ρ ∷ ῄ ∷ ζ ∷ ῃ ∷ []) "Rom.16.2"
∷ word (π ∷ ρ ∷ ά ∷ γ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.16.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.16.2"
∷ word (α ∷ ὐ ∷ τ ∷ ὴ ∷ []) "Rom.16.2"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ τ ∷ ά ∷ τ ∷ ι ∷ ς ∷ []) "Rom.16.2"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "Rom.16.2"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ή ∷ θ ∷ η ∷ []) "Rom.16.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.2"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "Rom.16.2"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.16.2"
∷ word (Ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.3"
∷ word (Π ∷ ρ ∷ ί ∷ σ ∷ κ ∷ α ∷ ν ∷ []) "Rom.16.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.3"
∷ word (Ἀ ∷ κ ∷ ύ ∷ ∙λ ∷ α ∷ ν ∷ []) "Rom.16.3"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.16.3"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ο ∷ ύ ∷ ς ∷ []) "Rom.16.3"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.16.3"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.16.3"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.16.3"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "Rom.16.4"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Rom.16.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.16.4"
∷ word (ψ ∷ υ ∷ χ ∷ ῆ ∷ ς ∷ []) "Rom.16.4"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.16.4"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.16.4"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.16.4"
∷ word (τ ∷ ρ ∷ ά ∷ χ ∷ η ∷ ∙λ ∷ ο ∷ ν ∷ []) "Rom.16.4"
∷ word (ὑ ∷ π ∷ έ ∷ θ ∷ η ∷ κ ∷ α ∷ ν ∷ []) "Rom.16.4"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "Rom.16.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Rom.16.4"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.16.4"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ς ∷ []) "Rom.16.4"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῶ ∷ []) "Rom.16.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.16.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.4"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ι ∷ []) "Rom.16.4"
∷ word (α ∷ ἱ ∷ []) "Rom.16.4"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ι ∷ []) "Rom.16.4"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.16.4"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Rom.16.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.16.5"
∷ word (κ ∷ α ∷ τ ∷ []) "Rom.16.5"
∷ word (ο ∷ ἶ ∷ κ ∷ ο ∷ ν ∷ []) "Rom.16.5"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.16.5"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Rom.16.5"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.5"
∷ word (Ἐ ∷ π ∷ α ∷ ί ∷ ν ∷ ε ∷ τ ∷ ο ∷ ν ∷ []) "Rom.16.5"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.16.5"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ό ∷ ν ∷ []) "Rom.16.5"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.16.5"
∷ word (ὅ ∷ ς ∷ []) "Rom.16.5"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Rom.16.5"
∷ word (ἀ ∷ π ∷ α ∷ ρ ∷ χ ∷ ὴ ∷ []) "Rom.16.5"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.16.5"
∷ word (Ἀ ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Rom.16.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.16.5"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ν ∷ []) "Rom.16.5"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.6"
∷ word (Μ ∷ α ∷ ρ ∷ ι ∷ ά ∷ μ ∷ []) "Rom.16.6"
∷ word (ἥ ∷ τ ∷ ι ∷ ς ∷ []) "Rom.16.6"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.16.6"
∷ word (ἐ ∷ κ ∷ ο ∷ π ∷ ί ∷ α ∷ σ ∷ ε ∷ ν ∷ []) "Rom.16.6"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.16.6"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.16.6"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.7"
∷ word (Ἀ ∷ ν ∷ δ ∷ ρ ∷ ό ∷ ν ∷ ι ∷ κ ∷ ο ∷ ν ∷ []) "Rom.16.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.7"
∷ word (Ἰ ∷ ο ∷ υ ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "Rom.16.7"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.16.7"
∷ word (σ ∷ υ ∷ γ ∷ γ ∷ ε ∷ ν ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.16.7"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.16.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.7"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ ι ∷ χ ∷ μ ∷ α ∷ ∙λ ∷ ώ ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.16.7"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.16.7"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ έ ∷ ς ∷ []) "Rom.16.7"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.16.7"
∷ word (ἐ ∷ π ∷ ί ∷ σ ∷ η ∷ μ ∷ ο ∷ ι ∷ []) "Rom.16.7"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.7"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.16.7"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "Rom.16.7"
∷ word (ο ∷ ἳ ∷ []) "Rom.16.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.7"
∷ word (π ∷ ρ ∷ ὸ ∷ []) "Rom.16.7"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "Rom.16.7"
∷ word (γ ∷ έ ∷ γ ∷ ο ∷ ν ∷ α ∷ ν ∷ []) "Rom.16.7"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.7"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.16.7"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.8"
∷ word (Ἀ ∷ μ ∷ π ∷ ∙λ ∷ ι ∷ ᾶ ∷ τ ∷ ο ∷ ν ∷ []) "Rom.16.8"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.16.8"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ό ∷ ν ∷ []) "Rom.16.8"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.16.8"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.8"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.16.8"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.9"
∷ word (Ο ∷ ὐ ∷ ρ ∷ β ∷ α ∷ ν ∷ ὸ ∷ ν ∷ []) "Rom.16.9"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.16.9"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ὸ ∷ ν ∷ []) "Rom.16.9"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.16.9"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.9"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.16.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.9"
∷ word (Σ ∷ τ ∷ ά ∷ χ ∷ υ ∷ ν ∷ []) "Rom.16.9"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.16.9"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ό ∷ ν ∷ []) "Rom.16.9"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.16.9"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.10"
∷ word (Ἀ ∷ π ∷ ε ∷ ∙λ ∷ ∙λ ∷ ῆ ∷ ν ∷ []) "Rom.16.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.16.10"
∷ word (δ ∷ ό ∷ κ ∷ ι ∷ μ ∷ ο ∷ ν ∷ []) "Rom.16.10"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.10"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.16.10"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.10"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.16.10"
∷ word (ἐ ∷ κ ∷ []) "Rom.16.10"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.16.10"
∷ word (Ἀ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ β ∷ ο ∷ ύ ∷ ∙λ ∷ ο ∷ υ ∷ []) "Rom.16.10"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.11"
∷ word (Ἡ ∷ ρ ∷ ῳ ∷ δ ∷ ί ∷ ω ∷ ν ∷ α ∷ []) "Rom.16.11"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.16.11"
∷ word (σ ∷ υ ∷ γ ∷ γ ∷ ε ∷ ν ∷ ῆ ∷ []) "Rom.16.11"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.16.11"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.11"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.16.11"
∷ word (ἐ ∷ κ ∷ []) "Rom.16.11"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.16.11"
∷ word (Ν ∷ α ∷ ρ ∷ κ ∷ ί ∷ σ ∷ σ ∷ ο ∷ υ ∷ []) "Rom.16.11"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.16.11"
∷ word (ὄ ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.16.11"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.11"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.16.11"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.12"
∷ word (Τ ∷ ρ ∷ ύ ∷ φ ∷ α ∷ ι ∷ ν ∷ α ∷ ν ∷ []) "Rom.16.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.12"
∷ word (Τ ∷ ρ ∷ υ ∷ φ ∷ ῶ ∷ σ ∷ α ∷ ν ∷ []) "Rom.16.12"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Rom.16.12"
∷ word (κ ∷ ο ∷ π ∷ ι ∷ ώ ∷ σ ∷ α ∷ ς ∷ []) "Rom.16.12"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.12"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.16.12"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.12"
∷ word (Π ∷ ε ∷ ρ ∷ σ ∷ ί ∷ δ ∷ α ∷ []) "Rom.16.12"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.16.12"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ή ∷ ν ∷ []) "Rom.16.12"
∷ word (ἥ ∷ τ ∷ ι ∷ ς ∷ []) "Rom.16.12"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.16.12"
∷ word (ἐ ∷ κ ∷ ο ∷ π ∷ ί ∷ α ∷ σ ∷ ε ∷ ν ∷ []) "Rom.16.12"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.12"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.16.12"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.13"
∷ word (Ῥ ∷ ο ∷ ῦ ∷ φ ∷ ο ∷ ν ∷ []) "Rom.16.13"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.16.13"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ε ∷ κ ∷ τ ∷ ὸ ∷ ν ∷ []) "Rom.16.13"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.13"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.16.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.13"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.16.13"
∷ word (μ ∷ η ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Rom.16.13"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.16.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.13"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "Rom.16.13"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.14"
∷ word (Ἀ ∷ σ ∷ ύ ∷ γ ∷ κ ∷ ρ ∷ ι ∷ τ ∷ ο ∷ ν ∷ []) "Rom.16.14"
∷ word (Φ ∷ ∙λ ∷ έ ∷ γ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "Rom.16.14"
∷ word (Ἑ ∷ ρ ∷ μ ∷ ῆ ∷ ν ∷ []) "Rom.16.14"
∷ word (Π ∷ α ∷ τ ∷ ρ ∷ ο ∷ β ∷ ᾶ ∷ ν ∷ []) "Rom.16.14"
∷ word (Ἑ ∷ ρ ∷ μ ∷ ᾶ ∷ ν ∷ []) "Rom.16.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.14"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.16.14"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Rom.16.14"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.16.14"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ύ ∷ ς ∷ []) "Rom.16.14"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.15"
∷ word (Φ ∷ ι ∷ ∙λ ∷ ό ∷ ∙λ ∷ ο ∷ γ ∷ ο ∷ ν ∷ []) "Rom.16.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.15"
∷ word (Ἰ ∷ ο ∷ υ ∷ ∙λ ∷ ί ∷ α ∷ ν ∷ []) "Rom.16.15"
∷ word (Ν ∷ η ∷ ρ ∷ έ ∷ α ∷ []) "Rom.16.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.15"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.16.15"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὴ ∷ ν ∷ []) "Rom.16.15"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.16.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.15"
∷ word (Ὀ ∷ ∙λ ∷ υ ∷ μ ∷ π ∷ ᾶ ∷ ν ∷ []) "Rom.16.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.15"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.16.15"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Rom.16.15"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Rom.16.15"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.16.15"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "Rom.16.15"
∷ word (Ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Rom.16.16"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Rom.16.16"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.16"
∷ word (φ ∷ ι ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Rom.16.16"
∷ word (ἁ ∷ γ ∷ ί ∷ ῳ ∷ []) "Rom.16.16"
∷ word (Ἀ ∷ σ ∷ π ∷ ά ∷ ζ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Rom.16.16"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.16.16"
∷ word (α ∷ ἱ ∷ []) "Rom.16.16"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ι ∷ []) "Rom.16.16"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ι ∷ []) "Rom.16.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.16.16"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.16.16"
∷ word (Π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "Rom.16.17"
∷ word (δ ∷ ὲ ∷ []) "Rom.16.17"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.16.17"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Rom.16.17"
∷ word (σ ∷ κ ∷ ο ∷ π ∷ ε ∷ ῖ ∷ ν ∷ []) "Rom.16.17"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.16.17"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Rom.16.17"
∷ word (δ ∷ ι ∷ χ ∷ ο ∷ σ ∷ τ ∷ α ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Rom.16.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.17"
∷ word (τ ∷ ὰ ∷ []) "Rom.16.17"
∷ word (σ ∷ κ ∷ ά ∷ ν ∷ δ ∷ α ∷ ∙λ ∷ α ∷ []) "Rom.16.17"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "Rom.16.17"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.16.17"
∷ word (δ ∷ ι ∷ δ ∷ α ∷ χ ∷ ὴ ∷ ν ∷ []) "Rom.16.17"
∷ word (ἣ ∷ ν ∷ []) "Rom.16.17"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.16.17"
∷ word (ἐ ∷ μ ∷ ά ∷ θ ∷ ε ∷ τ ∷ ε ∷ []) "Rom.16.17"
∷ word (π ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.16.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.17"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ ί ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "Rom.16.17"
∷ word (ἀ ∷ π ∷ []) "Rom.16.17"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.16.17"
∷ word (ο ∷ ἱ ∷ []) "Rom.16.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.16.18"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ι ∷ []) "Rom.16.18"
∷ word (τ ∷ ῷ ∷ []) "Rom.16.18"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.16.18"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.16.18"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Rom.16.18"
∷ word (ο ∷ ὐ ∷ []) "Rom.16.18"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Rom.16.18"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Rom.16.18"
∷ word (τ ∷ ῇ ∷ []) "Rom.16.18"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῶ ∷ ν ∷ []) "Rom.16.18"
∷ word (κ ∷ ο ∷ ι ∷ ∙λ ∷ ί ∷ ᾳ ∷ []) "Rom.16.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.18"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Rom.16.18"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.16.18"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ ς ∷ []) "Rom.16.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.18"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ ς ∷ []) "Rom.16.18"
∷ word (ἐ ∷ ξ ∷ α ∷ π ∷ α ∷ τ ∷ ῶ ∷ σ ∷ ι ∷ []) "Rom.16.18"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Rom.16.18"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "Rom.16.18"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Rom.16.18"
∷ word (ἀ ∷ κ ∷ ά ∷ κ ∷ ω ∷ ν ∷ []) "Rom.16.18"
∷ word (ἡ ∷ []) "Rom.16.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Rom.16.19"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.16.19"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ὴ ∷ []) "Rom.16.19"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.16.19"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Rom.16.19"
∷ word (ἀ ∷ φ ∷ ί ∷ κ ∷ ε ∷ τ ∷ ο ∷ []) "Rom.16.19"
∷ word (ἐ ∷ φ ∷ []) "Rom.16.19"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Rom.16.19"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Rom.16.19"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ω ∷ []) "Rom.16.19"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "Rom.16.19"
∷ word (δ ∷ ὲ ∷ []) "Rom.16.19"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.16.19"
∷ word (σ ∷ ο ∷ φ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.16.19"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Rom.16.19"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.16.19"
∷ word (τ ∷ ὸ ∷ []) "Rom.16.19"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ό ∷ ν ∷ []) "Rom.16.19"
∷ word (ἀ ∷ κ ∷ ε ∷ ρ ∷ α ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "Rom.16.19"
∷ word (δ ∷ ὲ ∷ []) "Rom.16.19"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Rom.16.19"
∷ word (τ ∷ ὸ ∷ []) "Rom.16.19"
∷ word (κ ∷ α ∷ κ ∷ ό ∷ ν ∷ []) "Rom.16.19"
∷ word (ὁ ∷ []) "Rom.16.20"
∷ word (δ ∷ ὲ ∷ []) "Rom.16.20"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Rom.16.20"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.16.20"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ ς ∷ []) "Rom.16.20"
∷ word (σ ∷ υ ∷ ν ∷ τ ∷ ρ ∷ ί ∷ ψ ∷ ε ∷ ι ∷ []) "Rom.16.20"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Rom.16.20"
∷ word (Σ ∷ α ∷ τ ∷ α ∷ ν ∷ ᾶ ∷ ν ∷ []) "Rom.16.20"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Rom.16.20"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Rom.16.20"
∷ word (π ∷ ό ∷ δ ∷ α ∷ ς ∷ []) "Rom.16.20"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.16.20"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.20"
∷ word (τ ∷ ά ∷ χ ∷ ε ∷ ι ∷ []) "Rom.16.20"
∷ word (ἡ ∷ []) "Rom.16.20"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Rom.16.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.16.20"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.16.20"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.16.20"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.16.20"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.16.20"
∷ word (μ ∷ ε ∷ θ ∷ []) "Rom.16.20"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.16.20"
∷ word (Ἀ ∷ σ ∷ π ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.16.21"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.16.21"
∷ word (Τ ∷ ι ∷ μ ∷ ό ∷ θ ∷ ε ∷ ο ∷ ς ∷ []) "Rom.16.21"
∷ word (ὁ ∷ []) "Rom.16.21"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ό ∷ ς ∷ []) "Rom.16.21"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.16.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.21"
∷ word (Λ ∷ ο ∷ ύ ∷ κ ∷ ι ∷ ο ∷ ς ∷ []) "Rom.16.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.21"
∷ word (Ἰ ∷ ά ∷ σ ∷ ω ∷ ν ∷ []) "Rom.16.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.21"
∷ word (Σ ∷ ω ∷ σ ∷ ί ∷ π ∷ α ∷ τ ∷ ρ ∷ ο ∷ ς ∷ []) "Rom.16.21"
∷ word (ο ∷ ἱ ∷ []) "Rom.16.21"
∷ word (σ ∷ υ ∷ γ ∷ γ ∷ ε ∷ ν ∷ ε ∷ ῖ ∷ ς ∷ []) "Rom.16.21"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.16.21"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ ζ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Rom.16.22"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.16.22"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Rom.16.22"
∷ word (Τ ∷ έ ∷ ρ ∷ τ ∷ ι ∷ ο ∷ ς ∷ []) "Rom.16.22"
∷ word (ὁ ∷ []) "Rom.16.22"
∷ word (γ ∷ ρ ∷ ά ∷ ψ ∷ α ∷ ς ∷ []) "Rom.16.22"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Rom.16.22"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "Rom.16.22"
∷ word (ἐ ∷ ν ∷ []) "Rom.16.22"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Rom.16.22"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.16.23"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.16.23"
∷ word (Γ ∷ ά ∷ ϊ ∷ ο ∷ ς ∷ []) "Rom.16.23"
∷ word (ὁ ∷ []) "Rom.16.23"
∷ word (ξ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "Rom.16.23"
∷ word (μ ∷ ο ∷ υ ∷ []) "Rom.16.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.23"
∷ word (ὅ ∷ ∙λ ∷ η ∷ ς ∷ []) "Rom.16.23"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.16.23"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Rom.16.23"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Rom.16.23"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Rom.16.23"
∷ word (Ἔ ∷ ρ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.16.23"
∷ word (ὁ ∷ []) "Rom.16.23"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Rom.16.23"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Rom.16.23"
∷ word (π ∷ ό ∷ ∙λ ∷ ε ∷ ω ∷ ς ∷ []) "Rom.16.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Rom.16.23"
∷ word (Κ ∷ ο ∷ ύ ∷ α ∷ ρ ∷ τ ∷ ο ∷ ς ∷ []) "Rom.16.23"
∷ word (ὁ ∷ []) "Rom.16.23"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ό ∷ ς ∷ []) "Rom.16.23"
∷ word (Ἡ ∷ []) "Rom.16.24"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Rom.16.24"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Rom.16.24"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Rom.16.24"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.16.24"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Rom.16.24"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Rom.16.24"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Rom.16.24"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Rom.16.24"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Rom.16.24"
∷ word (Ἀ ∷ μ ∷ ή ∷ ν ∷ []) "Rom.16.24"
∷ []
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Examples showing how the case expressions can be used
------------------------------------------------------------------------
module README.Case where
open import Data.Fin hiding (pred)
open import Data.Maybe hiding (from-just)
open import Data.Nat hiding (pred)
open import Function
-- Some simple examples.
empty : ∀ {a} {A : Set a} → Fin 0 → A
empty i = case i of λ()
pred : ℕ → ℕ
pred n = case n of λ
{ zero → zero
; (suc n) → n
}
from-just : ∀ {a} {A : Set a} (x : Maybe A) → From-just A x
from-just x = case x return From-just _ of λ
{ (just x) → x
; nothing → _
}
-- Note that some natural uses of case are rejected by the termination
-- checker:
--
-- plus : ℕ → ℕ → ℕ
-- plus m n = case m of λ
-- { zero → n
-- ; (suc m) → suc (plus m n)
-- }
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{-# OPTIONS --universe-polymorphism #-}
module Categories.FunctorCategory where
open import Data.Product
open import Categories.Category
import Categories.Functor as Cat
open import Categories.Functor hiding (equiv; id; _∘_; _≡_)
open import Categories.NaturalTransformation
open import Categories.Product
open import Categories.Square
Functors : ∀ {o ℓ e} {o′ ℓ′ e′} → Category o ℓ e → Category o′ ℓ′ e′ → Category _ _ _
Functors C D = record
{ Obj = Functor C D
; _⇒_ = NaturalTransformation
; _≡_ = _≡_
; _∘_ = _∘₁_
; id = id
; assoc = λ {_} {_} {_} {_} {f} {g} {h} → assoc₁ {X = f} {g} {h}
; identityˡ = λ {_} {_} {f} → identity₁ˡ {X = f}
; identityʳ = λ {_} {_} {f} → identity₁ʳ {X = f}
; equiv = λ {F} {G} → equiv {F = F} {G = G}
; ∘-resp-≡ = λ {_} {_} {_} {f} {h} {g} {i} → ∘₁-resp-≡ {f = f} {h} {g} {i}
}
eval : ∀ {o ℓ e} {o′ ℓ′ e′} {C : Category o ℓ e}{D : Category o′ ℓ′ e′} → Functor (Product (Functors C D) C) D
eval {C = C} {D = D} =
record {
F₀ = λ x → let F , c = x in F₀ F c;
F₁ = λ { {F , c₁} {G , c₂} (ε , f) → F₁ G f D.∘ η ε _};
identity = λ { {F , c} → begin
F₁ F C.id D.∘ D.id ↓⟨ D.identityʳ ⟩
F₁ F C.id ↓⟨ identity F ⟩
D.id ∎};
homomorphism = λ { {F , c₁} {G , c₂} {H , c₃} {ε₁ , f₁} {ε₂ , f₂} →
begin
F₁ H (f₂ C.∘ f₁) D.∘ η ε₂ c₁ D.∘ η ε₁ c₁ ↑⟨ D.assoc ⟩
(F₁ H (f₂ C.∘ f₁) D.∘ η ε₂ c₁) D.∘ η ε₁ c₁
↓⟨ D.∘-resp-≡ˡ (begin
F₁ H (C [ f₂ ∘ f₁ ]) D.∘ η ε₂ c₁ ↓⟨ D.∘-resp-≡ˡ (homomorphism H) ⟩
(F₁ H f₂ D.∘ F₁ H f₁) D.∘ η ε₂ c₁ ↓⟨ D.assoc ⟩
F₁ H f₂ D.∘ F₁ H f₁ D.∘ η ε₂ c₁ ↑⟨ D.∘-resp-≡ʳ (commute ε₂ f₁) ⟩
F₁ H f₂ D.∘ η ε₂ c₂ D.∘ F₁ G f₁ ↑⟨ D.assoc ⟩
(F₁ H f₂ D.∘ η ε₂ c₂) D.∘ F₁ G f₁ ∎)
⟩
((F₁ H f₂ D.∘ η ε₂ c₂) D.∘ F₁ G f₁) D.∘ η ε₁ c₁ ↓⟨ D.assoc ⟩
(F₁ H f₂ D.∘ η ε₂ c₂) D.∘ F₁ G f₁ D.∘ η ε₁ c₁ ∎ };
F-resp-≡ = λ { {F , c₁} {G , c₂} {ε₁ , f₁} {ε₂ , f₂} (ε₁≡ε₂ , f₁≡f₂)
→ D.∘-resp-≡ (F-resp-≡ G f₁≡f₂) ε₁≡ε₂} }
where
module C = Category C
module D = Category D
open Functor
open NaturalTransformation
open D.HomReasoning
Cat[-∘_] : ∀ {o₁ ℓ₁ e₁ o₂ ℓ₂ e₂ o₃ ℓ₃ e₃} {A : Category o₁ ℓ₁ e₁} {B : Category o₂ ℓ₂ e₂}
{C : Category o₃ ℓ₃ e₃} -> Functor A B -> Functor (Functors B C) (Functors A C)
Cat[-∘_] {C = C} r = record
{ F₀ = λ X → X Cat.∘ r
; F₁ = λ η → η ∘ʳ r
; identity = C.Equiv.refl
; homomorphism = C.Equiv.refl
; F-resp-≡ = λ x → x
}
where
module C = Category C
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{-# OPTIONS --without-K #-}
open import library.Basics hiding (Type ; Σ)
open import library.types.Sigma
open import Sec2preliminaries
module Sec3hedberg where
-- Lemma 3.2
discr→pathHasConst : {X : Type} → has-dec-eq X → pathHasConst X
discr→pathHasConst dec x₁ x₂ with (dec x₁ x₂)
discr→pathHasConst dec x₁ x₂ | inl p = (λ _ → p) , (λ _ _ → idp)
discr→pathHasConst dec x₁ x₂ | inr np = idf _ , λ p → Empty-elim (np p)
-- Lemma 3.3
pathHasConst→isSet : {X : Type} → pathHasConst X → is-set X
pathHasConst→isSet {X} pc x₁ x₂ = all-paths-is-prop paths-equal where
claim : (y₁ y₂ : X) → (p : y₁ == y₂) → p == ! (fst (pc _ _) idp) ∙ fst (pc _ _) p
claim x₁ .x₁ idp = ! (!-inv-l (fst (pc x₁ x₁) idp))
paths-equal : (p₁ p₂ : x₁ == x₂) → p₁ == p₂
paths-equal p₁ p₂ =
p₁ =⟨ claim _ _ p₁ ⟩
! (fst (pc _ _) idp) ∙ fst (pc _ _) p₁ =⟨ ap (λ q → (! (fst (pc x₁ x₁) idp)) ∙ q) (snd (pc _ _) p₁ p₂) ⟩ -- whiskering
! (fst (pc _ _) idp) ∙ fst (pc _ _) p₂ =⟨ ! (claim _ _ p₂) ⟩
p₂ ∎
-- Theorem 3.1
hedberg : {X : Type} → has-dec-eq X → is-set X
hedberg = pathHasConst→isSet ∘ discr→pathHasConst
-- Definition 3.4
stable : Type → Type
stable X = ¬ (¬ X) → X
separated : Type → Type
separated X = (x₁ x₂ : X) → stable (x₁ == x₂)
-- Lemma 3.5
sep→set : {X : Type} → (separated X) → ({x₁ x₂ : X} → Funext {¬ (x₁ == x₂)} {Empty}) → is-set X
sep→set {X} sep ⊥-funext = pathHasConst→isSet isPc where
isPc : pathHasConst X
isPc x₁ x₂ = f , c where
f : x₁ == x₂ → x₁ == x₂
f = (sep x₁ x₂) ∘ (λ p np → np p)
c : const f
c p₁ p₂ =
f p₁ =⟨ idp ⟩
(sep x₁ x₂) (λ np → np p₁) =⟨ ap (sep x₁ x₂)
(⊥-funext _ _ λ np → Empty-elim {A = λ _ → np p₁ == np p₂} (np p₁)) ⟩
(sep x₁ x₂) (λ np → np p₂) =⟨ idp ⟩
f p₂ ∎
-- Definition 3.6
postulate
Trunc : Type → Type
h-tr : (X : Type) → is-prop (Trunc X)
∣_∣ : {X : Type} → X → Trunc X
rec : {X : Type} → {P : Type} → (is-prop P) → (X → P) → Trunc X → P
-- the propositional β rule is derivable:
trunc-β : {X : Type} → {P : Type} → (pp : is-prop P) → (f : X → P) → (x : X) → rec pp f ∣ x ∣ == f x
trunc-β pp f x = prop-has-all-paths pp _ _
-- Lemma 3.7
module _ (X : Type) (P : Trunc X → Type) (h : (z : Trunc X) → is-prop (P z)) (k : (x : X) → P(∣ x ∣)) where
total : Type
total = Σ (Trunc X) P
j : X → total
j x = ∣ x ∣ , k x
total-prop : is-prop total
total-prop = Σ-level (h-tr X) h
total-map : Trunc X → total
total-map = rec total-prop j
induction : (z : Trunc X) → P z
induction z = transport P (prop-has-all-paths (h-tr _) _ _) (snd (total-map z))
-- comment: Trunc is functorial
trunc-functorial : {X Y : Type} → (X → Y) → (Trunc X → Trunc Y)
trunc-functorial {X} {Y} f = rec (h-tr Y) (∣_∣ ∘ f)
-- Theorem 3.8
impred : {X : Type} → (Trunc X ↔₀₁ ((P : Type) → (is-prop P) → (X → P) → P))
impred {X} = one , two where
one : Trunc X → (P : Type) → (is-prop P) → (X → P) → P
one z P p-prop f = rec p-prop f z
two : ((P : Type) → (is-prop P) → (X → P) → P) → Trunc X
two k = k (Trunc X) (h-tr _) ∣_∣
-- Definition 3.9
splitSup : Type → Type
splitSup X = Trunc X → X
hseparated : Type → Type
hseparated X = (x₁ x₂ : X) → splitSup (x₁ == x₂)
-- Theorem 3.10
set-characterizations : {X : Type} → (pathHasConst X → is-set X)
× ((is-set X → hseparated X)
× (hseparated X → pathHasConst X))
set-characterizations {X} = one , two , three where
one : pathHasConst X → is-set X
one = pathHasConst→isSet
two : is-set X → hseparated X
two h = λ x₁ x₂ → rec (h x₁ x₂) (idf _)
three : hseparated X → pathHasConst X
three hsep x₁ x₂ = f , c where
f = (hsep _ _) ∘ ∣_∣
c = λ p₁ p₂ → f p₁ =⟨ idp ⟩
hsep _ _ (∣ p₁ ∣) =⟨ ap (hsep _ _) (prop-has-all-paths (h-tr _) _ _) ⟩
hsep _ _ (∣ p₂ ∣) =⟨ idp ⟩
f p₂ ∎
-- The rest of this section is only a replication of the arguments that we have given so far (for that reason, the proofs are not given in the article).
-- They do not directly follow from the statements that we have proved before, but they directly imply them.
-- Of course, replication of arguments is not a good style for a formalization -
-- we chose this "disadvantageous" order purely as we believe it led to a better presentation in the article.
-- Lemma 3.11
pathHasConst→isSet-local : {X : Type} → {x₀ : X} → ((y : X) → hasConst (x₀ == y)) → (y : X) → is-prop (x₀ == y)
pathHasConst→isSet-local {X} {x₀} pc y = all-paths-is-prop paths-equal where
claim : (y : X) → (p : x₀ == y) → p == ! (fst (pc _) idp) ∙ fst (pc _) p
claim .x₀ idp = ! (!-inv-l (fst (pc _) idp))
paths-equal : (p₁ p₂ : x₀ == y) → p₁ == p₂
paths-equal p₁ p₂ =
p₁ =⟨ claim _ p₁ ⟩
! (fst (pc _) idp) ∙ fst (pc _) p₁ =⟨ ap (λ q → (! (fst (pc x₀) idp)) ∙ q) (snd (pc y) p₁ p₂) ⟩ -- whiskering
! (fst (pc _) idp) ∙ fst (pc _) p₂ =⟨ ! (claim _ p₂) ⟩
p₂ ∎
-- Theorem 3.12
hedberg-local : {X : Type} → {x₀ : X} → ((y : X) → (x₀ == y) + ¬(x₀ == y)) → (y : X) → is-prop (x₀ == y)
hedberg-local {X} {x₀} dec = pathHasConst→isSet-local local-pathHasConst where
local-pathHasConst : (y : X) → hasConst (x₀ == y)
local-pathHasConst y with (dec y)
local-pathHasConst y₁ | inl p = (λ _ → p) , (λ _ _ → idp)
local-pathHasConst y₁ | inr np = idf _ , (λ p → Empty-elim (np p))
-- Lemma 3.13
sep→set-local : {X : Type} → {x₀ : X} → ((y : X) → stable (x₀ == y)) → ({y : X} → Funext {¬ (x₀ == y)} {Empty}) → (y : X) → is-prop (x₀ == y)
sep→set-local {X} {x₀} sep ⊥-funext = pathHasConst→isSet-local is-pathHasConst where
is-pathHasConst : (y : X) → hasConst (x₀ == y)
is-pathHasConst y = f , c where
f : x₀ == y → x₀ == y
f = (sep y) ∘ (λ p np → np p)
c : const f
c p₁ p₂ =
f p₁ =⟨ idp ⟩
(sep _) (λ np → np p₁) =⟨ ap (sep _)
(⊥-funext _ _ λ np → Empty-elim {A = λ _ → np p₁ == np p₂} (np p₁)) ⟩
(sep _) (λ np → np p₂) =⟨ idp ⟩
f p₂ ∎
-- Theorem 3.14
set-characterizations-local : {X : Type} → {x₀ : X} →
(((y : X) → hasConst (x₀ == y)) → (y : X) → is-prop (x₀ == y))
× ((((y : X) → is-prop (x₀ == y)) → (y : X) → splitSup (x₀ == y))
× (((y : X) → splitSup (x₀ == y)) → (y : X) → hasConst (x₀ == y)))
set-characterizations-local {X} {x₀} = one , two , three where
one = pathHasConst→isSet-local
two : ((y : X) → is-prop (x₀ == y)) → (y : X) → splitSup (x₀ == y)
two h y = rec (h y) (idf _)
three : ((y : X) → splitSup (x₀ == y)) → (y : X) → hasConst (x₀ == y)
three hsep y = f , c where
f = (hsep _) ∘ ∣_∣
c = λ p₁ p₂ →
f p₁ =⟨ idp ⟩
(hsep _) ∣ p₁ ∣ =⟨ ap (hsep _) (prop-has-all-paths (h-tr _) _ _) ⟩
(hsep _) ∣ p₂ ∣ =⟨ idp ⟩
f p₂ ∎
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Relation.Binary.Raw where
open import Cubical.Relation.Binary.Base public
open import Cubical.Relation.Binary.Raw.Definitions public
open import Cubical.Relation.Binary.Raw.Structures public
open import Cubical.Relation.Binary.Raw.Bundles public
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{-# OPTIONS --allow-unsolved-metas #-}
postulate
Nat : Set
variable
A : _
F : _ → _
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties related to Fin, and operations making use of these
-- properties (or other properties not available in Data.Fin)
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Fin.Properties where
open import Category.Applicative using (RawApplicative)
open import Category.Functor using (RawFunctor)
open import Data.Bool.Base using (Bool; true; false; not; _∧_; _∨_)
open import Data.Empty using (⊥-elim)
open import Data.Fin.Base
open import Data.Fin.Patterns
open import Data.Nat.Base as ℕ using (ℕ; zero; suc; s≤s; z≤n; _∸_)
import Data.Nat.Properties as ℕₚ
open import Data.Unit using (tt)
open import Data.Product using (∃; ∃₂; ∄; _×_; _,_; map; proj₁; uncurry; <_,_>)
open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
open import Data.Sum.Properties using ([,]-map-commute; [,]-∘-distr)
open import Function.Base using (_∘_; id; _$_)
open import Function.Equivalence using (_⇔_; equivalence)
open import Function.Injection using (_↣_)
open import Relation.Binary as B hiding (Decidable)
open import Relation.Binary.PropositionalEquality as P
using (_≡_; _≢_; refl; sym; trans; cong; subst; module ≡-Reasoning)
open import Relation.Nullary.Decidable as Dec using (map′)
open import Relation.Nullary.Reflects
open import Relation.Nullary.Negation using (contradiction)
open import Relation.Nullary
using (Reflects; ofʸ; ofⁿ; Dec; _because_; does; proof; yes; no; ¬_)
open import Relation.Nullary.Product using (_×-dec_)
open import Relation.Nullary.Sum using (_⊎-dec_)
open import Relation.Unary as U
using (U; Pred; Decidable; _⊆_; Satisfiable; Universal)
open import Relation.Unary.Properties using (U?)
------------------------------------------------------------------------
-- Fin
------------------------------------------------------------------------
¬Fin0 : ¬ Fin 0
¬Fin0 ()
------------------------------------------------------------------------
-- Properties of _≡_
------------------------------------------------------------------------
suc-injective : ∀ {o} {m n : Fin o} → Fin.suc m ≡ suc n → m ≡ n
suc-injective refl = refl
infix 4 _≟_
_≟_ : ∀ {n} → B.Decidable {A = Fin n} _≡_
zero ≟ zero = yes refl
zero ≟ suc y = no λ()
suc x ≟ zero = no λ()
suc x ≟ suc y = map′ (cong suc) suc-injective (x ≟ y)
------------------------------------------------------------------------
-- Structures
≡-isDecEquivalence : ∀ {n} → IsDecEquivalence (_≡_ {A = Fin n})
≡-isDecEquivalence = record
{ isEquivalence = P.isEquivalence
; _≟_ = _≟_
}
------------------------------------------------------------------------
-- Bundles
≡-preorder : ℕ → Preorder _ _ _
≡-preorder n = P.preorder (Fin n)
≡-setoid : ℕ → Setoid _ _
≡-setoid n = P.setoid (Fin n)
≡-decSetoid : ℕ → DecSetoid _ _
≡-decSetoid n = record
{ isDecEquivalence = ≡-isDecEquivalence {n}
}
------------------------------------------------------------------------
-- toℕ
------------------------------------------------------------------------
toℕ-injective : ∀ {n} {i j : Fin n} → toℕ i ≡ toℕ j → i ≡ j
toℕ-injective {zero} {} {} _
toℕ-injective {suc n} {zero} {zero} eq = refl
toℕ-injective {suc n} {suc i} {suc j} eq =
cong suc (toℕ-injective (cong ℕ.pred eq))
toℕ-strengthen : ∀ {n} (i : Fin n) → toℕ (strengthen i) ≡ toℕ i
toℕ-strengthen zero = refl
toℕ-strengthen (suc i) = cong suc (toℕ-strengthen i)
toℕ-raise : ∀ {m} n (i : Fin m) → toℕ (raise n i) ≡ n ℕ.+ toℕ i
toℕ-raise zero i = refl
toℕ-raise (suc n) i = cong suc (toℕ-raise n i)
toℕ<n : ∀ {n} (i : Fin n) → toℕ i ℕ.< n
toℕ<n zero = s≤s z≤n
toℕ<n (suc i) = s≤s (toℕ<n i)
toℕ≤pred[n] : ∀ {n} (i : Fin n) → toℕ i ℕ.≤ ℕ.pred n
toℕ≤pred[n] zero = z≤n
toℕ≤pred[n] (suc {n = suc n} i) = s≤s (toℕ≤pred[n] i)
-- A simpler implementation of toℕ≤pred[n],
-- however, with a different reduction behavior.
-- If no one needs the reduction behavior of toℕ≤pred[n],
-- it can be removed in favor of toℕ≤pred[n]′.
toℕ≤pred[n]′ : ∀ {n} (i : Fin n) → toℕ i ℕ.≤ ℕ.pred n
toℕ≤pred[n]′ i = ℕₚ.<⇒≤pred (toℕ<n i)
------------------------------------------------------------------------
-- fromℕ
------------------------------------------------------------------------
toℕ-fromℕ : ∀ n → toℕ (fromℕ n) ≡ n
toℕ-fromℕ zero = refl
toℕ-fromℕ (suc n) = cong suc (toℕ-fromℕ n)
fromℕ-toℕ : ∀ {n} (i : Fin n) → fromℕ (toℕ i) ≡ strengthen i
fromℕ-toℕ zero = refl
fromℕ-toℕ (suc i) = cong suc (fromℕ-toℕ i)
------------------------------------------------------------------------
-- fromℕ<
------------------------------------------------------------------------
fromℕ<-toℕ : ∀ {m} (i : Fin m) (i<m : toℕ i ℕ.< m) → fromℕ< i<m ≡ i
fromℕ<-toℕ zero (s≤s z≤n) = refl
fromℕ<-toℕ (suc i) (s≤s (s≤s m≤n)) = cong suc (fromℕ<-toℕ i (s≤s m≤n))
toℕ-fromℕ< : ∀ {m n} (m<n : m ℕ.< n) → toℕ (fromℕ< m<n) ≡ m
toℕ-fromℕ< (s≤s z≤n) = refl
toℕ-fromℕ< (s≤s (s≤s m<n)) = cong suc (toℕ-fromℕ< (s≤s m<n))
-- fromℕ is a special case of fromℕ<.
fromℕ-def : ∀ n → fromℕ n ≡ fromℕ< ℕₚ.≤-refl
fromℕ-def zero = refl
fromℕ-def (suc n) = cong suc (fromℕ-def n)
------------------------------------------------------------------------
-- fromℕ<″
------------------------------------------------------------------------
fromℕ<≡fromℕ<″ : ∀ {m n} (m<n : m ℕ.< n) (m<″n : m ℕ.<″ n) →
fromℕ< m<n ≡ fromℕ<″ m m<″n
fromℕ<≡fromℕ<″ (s≤s z≤n) (ℕ.less-than-or-equal refl) = refl
fromℕ<≡fromℕ<″ (s≤s (s≤s m<n)) (ℕ.less-than-or-equal refl) =
cong suc (fromℕ<≡fromℕ<″ (s≤s m<n) (ℕ.less-than-or-equal refl))
toℕ-fromℕ<″ : ∀ {m n} (m<n : m ℕ.<″ n) → toℕ (fromℕ<″ m m<n) ≡ m
toℕ-fromℕ<″ {m} {n} m<n = begin
toℕ (fromℕ<″ m m<n) ≡⟨ cong toℕ (sym (fromℕ<≡fromℕ<″ (ℕₚ.≤″⇒≤ m<n) m<n)) ⟩
toℕ (fromℕ< _) ≡⟨ toℕ-fromℕ< (ℕₚ.≤″⇒≤ m<n) ⟩
m ∎
where open ≡-Reasoning
------------------------------------------------------------------------
-- cast
------------------------------------------------------------------------
toℕ-cast : ∀ {m n} .(eq : m ≡ n) (k : Fin m) → toℕ (cast eq k) ≡ toℕ k
toℕ-cast {n = suc n} eq zero = refl
toℕ-cast {n = suc n} eq (suc k) = cong suc (toℕ-cast (cong ℕ.pred eq) k)
------------------------------------------------------------------------
-- Properties of _≤_
------------------------------------------------------------------------
-- Relational properties
≤-reflexive : ∀ {n} → _≡_ ⇒ (_≤_ {n})
≤-reflexive refl = ℕₚ.≤-refl
≤-refl : ∀ {n} → Reflexive (_≤_ {n})
≤-refl = ≤-reflexive refl
≤-trans : ∀ {n} → Transitive (_≤_ {n})
≤-trans = ℕₚ.≤-trans
≤-antisym : ∀ {n} → Antisymmetric _≡_ (_≤_ {n})
≤-antisym x≤y y≤x = toℕ-injective (ℕₚ.≤-antisym x≤y y≤x)
≤-total : ∀ {n} → Total (_≤_ {n})
≤-total x y = ℕₚ.≤-total (toℕ x) (toℕ y)
≤-irrelevant : ∀ {n} → Irrelevant (_≤_ {n})
≤-irrelevant = ℕₚ.≤-irrelevant
infix 4 _≤?_ _<?_
_≤?_ : ∀ {n} → B.Decidable (_≤_ {n})
a ≤? b = toℕ a ℕₚ.≤? toℕ b
_<?_ : ∀ {n} → B.Decidable (_<_ {n})
m <? n = suc (toℕ m) ℕₚ.≤? toℕ n
------------------------------------------------------------------------
-- Structures
≤-isPreorder : ∀ {n} → IsPreorder _≡_ (_≤_ {n})
≤-isPreorder = record
{ isEquivalence = P.isEquivalence
; reflexive = ≤-reflexive
; trans = ≤-trans
}
≤-isPartialOrder : ∀ {n} → IsPartialOrder _≡_ (_≤_ {n})
≤-isPartialOrder = record
{ isPreorder = ≤-isPreorder
; antisym = ≤-antisym
}
≤-isTotalOrder : ∀ {n} → IsTotalOrder _≡_ (_≤_ {n})
≤-isTotalOrder = record
{ isPartialOrder = ≤-isPartialOrder
; total = ≤-total
}
≤-isDecTotalOrder : ∀ {n} → IsDecTotalOrder _≡_ (_≤_ {n})
≤-isDecTotalOrder = record
{ isTotalOrder = ≤-isTotalOrder
; _≟_ = _≟_
; _≤?_ = _≤?_
}
------------------------------------------------------------------------
-- Bundles
≤-preorder : ℕ → Preorder _ _ _
≤-preorder n = record
{ isPreorder = ≤-isPreorder {n}
}
≤-poset : ℕ → Poset _ _ _
≤-poset n = record
{ isPartialOrder = ≤-isPartialOrder {n}
}
≤-totalOrder : ℕ → TotalOrder _ _ _
≤-totalOrder n = record
{ isTotalOrder = ≤-isTotalOrder {n}
}
≤-decTotalOrder : ℕ → DecTotalOrder _ _ _
≤-decTotalOrder n = record
{ isDecTotalOrder = ≤-isDecTotalOrder {n}
}
------------------------------------------------------------------------
-- Properties of _<_
------------------------------------------------------------------------
-- Relational properties
<-irrefl : ∀ {n} → Irreflexive _≡_ (_<_ {n})
<-irrefl refl = ℕₚ.<-irrefl refl
<-asym : ∀ {n} → Asymmetric (_<_ {n})
<-asym = ℕₚ.<-asym
<-trans : ∀ {n} → Transitive (_<_ {n})
<-trans = ℕₚ.<-trans
<-cmp : ∀ {n} → Trichotomous _≡_ (_<_ {n})
<-cmp zero zero = tri≈ (λ()) refl (λ())
<-cmp zero (suc j) = tri< (s≤s z≤n) (λ()) (λ())
<-cmp (suc i) zero = tri> (λ()) (λ()) (s≤s z≤n)
<-cmp (suc i) (suc j) with <-cmp i j
... | tri< i<j i≢j j≮i = tri< (s≤s i<j) (i≢j ∘ suc-injective) (j≮i ∘ ℕₚ.≤-pred)
... | tri> i≮j i≢j j<i = tri> (i≮j ∘ ℕₚ.≤-pred) (i≢j ∘ suc-injective) (s≤s j<i)
... | tri≈ i≮j i≡j j≮i = tri≈ (i≮j ∘ ℕₚ.≤-pred) (cong suc i≡j) (j≮i ∘ ℕₚ.≤-pred)
<-respˡ-≡ : ∀ {n} → (_<_ {n}) Respectsˡ _≡_
<-respˡ-≡ refl x≤y = x≤y
<-respʳ-≡ : ∀ {n} → (_<_ {n}) Respectsʳ _≡_
<-respʳ-≡ refl x≤y = x≤y
<-resp₂-≡ : ∀ {n} → (_<_ {n}) Respects₂ _≡_
<-resp₂-≡ = <-respʳ-≡ , <-respˡ-≡
<-irrelevant : ∀ {n} → Irrelevant (_<_ {n})
<-irrelevant = ℕₚ.<-irrelevant
------------------------------------------------------------------------
-- Structures
<-isStrictPartialOrder : ∀ {n} → IsStrictPartialOrder _≡_ (_<_ {n})
<-isStrictPartialOrder = record
{ isEquivalence = P.isEquivalence
; irrefl = <-irrefl
; trans = <-trans
; <-resp-≈ = <-resp₂-≡
}
<-isStrictTotalOrder : ∀ {n} → IsStrictTotalOrder _≡_ (_<_ {n})
<-isStrictTotalOrder = record
{ isEquivalence = P.isEquivalence
; trans = <-trans
; compare = <-cmp
}
------------------------------------------------------------------------
-- Bundles
<-strictPartialOrder : ℕ → StrictPartialOrder _ _ _
<-strictPartialOrder n = record
{ isStrictPartialOrder = <-isStrictPartialOrder {n}
}
<-strictTotalOrder : ℕ → StrictTotalOrder _ _ _
<-strictTotalOrder n = record
{ isStrictTotalOrder = <-isStrictTotalOrder {n}
}
------------------------------------------------------------------------
-- Other properties
<⇒≢ : ∀ {n} {i j : Fin n} → i < j → i ≢ j
<⇒≢ i<i refl = ℕₚ.n≮n _ i<i
≤∧≢⇒< : ∀ {n} {i j : Fin n} → i ≤ j → i ≢ j → i < j
≤∧≢⇒< {i = zero} {zero} _ 0≢0 = contradiction refl 0≢0
≤∧≢⇒< {i = zero} {suc j} _ _ = s≤s z≤n
≤∧≢⇒< {i = suc i} {suc j} (s≤s i≤j) 1+i≢1+j =
s≤s (≤∧≢⇒< i≤j (1+i≢1+j ∘ (cong suc)))
------------------------------------------------------------------------
-- inject
------------------------------------------------------------------------
toℕ-inject : ∀ {n} {i : Fin n} (j : Fin′ i) →
toℕ (inject j) ≡ toℕ j
toℕ-inject {i = suc i} zero = refl
toℕ-inject {i = suc i} (suc j) = cong suc (toℕ-inject j)
------------------------------------------------------------------------
-- inject+
------------------------------------------------------------------------
toℕ-inject+ : ∀ {m} n (i : Fin m) → toℕ i ≡ toℕ (inject+ n i)
toℕ-inject+ n zero = refl
toℕ-inject+ n (suc i) = cong suc (toℕ-inject+ n i)
------------------------------------------------------------------------
-- inject₁
------------------------------------------------------------------------
inject₁-injective : ∀ {n} {i j : Fin n} → inject₁ i ≡ inject₁ j → i ≡ j
inject₁-injective {i = zero} {zero} i≡j = refl
inject₁-injective {i = suc i} {suc j} i≡j =
cong suc (inject₁-injective (suc-injective i≡j))
toℕ-inject₁ : ∀ {n} (i : Fin n) → toℕ (inject₁ i) ≡ toℕ i
toℕ-inject₁ zero = refl
toℕ-inject₁ (suc i) = cong suc (toℕ-inject₁ i)
toℕ-inject₁-≢ : ∀ {n}(i : Fin n) → n ≢ toℕ (inject₁ i)
toℕ-inject₁-≢ (suc i) = toℕ-inject₁-≢ i ∘ ℕₚ.suc-injective
------------------------------------------------------------------------
-- inject₁ and lower₁
inject₁-lower₁ : ∀ {n} (i : Fin (suc n)) (n≢i : n ≢ toℕ i) →
inject₁ (lower₁ i n≢i) ≡ i
inject₁-lower₁ {zero} zero 0≢0 = contradiction refl 0≢0
inject₁-lower₁ {suc n} zero _ = refl
inject₁-lower₁ {suc n} (suc i) n+1≢i+1 =
cong suc (inject₁-lower₁ i (n+1≢i+1 ∘ cong suc))
lower₁-inject₁′ : ∀ {n} (i : Fin n) (n≢i : n ≢ toℕ (inject₁ i)) →
lower₁ (inject₁ i) n≢i ≡ i
lower₁-inject₁′ zero _ = refl
lower₁-inject₁′ (suc i) n+1≢i+1 =
cong suc (lower₁-inject₁′ i (n+1≢i+1 ∘ cong suc))
lower₁-inject₁ : ∀ {n} (i : Fin n) →
lower₁ (inject₁ i) (toℕ-inject₁-≢ i) ≡ i
lower₁-inject₁ i = lower₁-inject₁′ i (toℕ-inject₁-≢ i)
lower₁-irrelevant : ∀ {n} (i : Fin (suc n)) n≢i₁ n≢i₂ →
lower₁ {n} i n≢i₁ ≡ lower₁ {n} i n≢i₂
lower₁-irrelevant {zero} zero 0≢0 _ = contradiction refl 0≢0
lower₁-irrelevant {suc n} zero _ _ = refl
lower₁-irrelevant {suc n} (suc i) _ _ =
cong suc (lower₁-irrelevant i _ _)
------------------------------------------------------------------------
-- inject≤
------------------------------------------------------------------------
toℕ-inject≤ : ∀ {m n} (i : Fin m) (le : m ℕ.≤ n) →
toℕ (inject≤ i le) ≡ toℕ i
toℕ-inject≤ {_} {suc n} zero _ = refl
toℕ-inject≤ {_} {suc n} (suc i) le = cong suc (toℕ-inject≤ i (ℕₚ.≤-pred le))
inject≤-refl : ∀ {n} (i : Fin n) (n≤n : n ℕ.≤ n) → inject≤ i n≤n ≡ i
inject≤-refl {suc n} zero _ = refl
inject≤-refl {suc n} (suc i) n≤n = cong suc (inject≤-refl i (ℕₚ.≤-pred n≤n))
inject≤-idempotent : ∀ {m n k} (i : Fin m)
(m≤n : m ℕ.≤ n) (n≤k : n ℕ.≤ k) (m≤k : m ℕ.≤ k) →
inject≤ (inject≤ i m≤n) n≤k ≡ inject≤ i m≤k
inject≤-idempotent {_} {suc n} {suc k} zero _ _ _ = refl
inject≤-idempotent {_} {suc n} {suc k} (suc i) m≤n n≤k _ =
cong suc (inject≤-idempotent i (ℕₚ.≤-pred m≤n) (ℕₚ.≤-pred n≤k) _)
------------------------------------------------------------------------
-- splitAt
------------------------------------------------------------------------
-- Fin (m + n) ≃ Fin m ⊎ Fin n
splitAt-inject+ : ∀ m n i → splitAt m (inject+ n i) ≡ inj₁ i
splitAt-inject+ (suc m) n zero = refl
splitAt-inject+ (suc m) n (suc i) rewrite splitAt-inject+ m n i = refl
splitAt-raise : ∀ m n i → splitAt m (raise {n} m i) ≡ inj₂ i
splitAt-raise zero n i = refl
splitAt-raise (suc m) n i rewrite splitAt-raise m n i = refl
inject+-raise-splitAt : ∀ m n i → [ inject+ n , raise {n} m ] (splitAt m i) ≡ i
inject+-raise-splitAt zero n i = refl
inject+-raise-splitAt (suc m) n zero = refl
inject+-raise-splitAt (suc m) n (suc i) = begin
[ inject+ n , raise {n} (suc m) ] (splitAt (suc m) (suc i)) ≡⟨ [,]-map-commute (splitAt m i) ⟩
[ suc ∘ (inject+ n) , suc ∘ (raise {n} m) ] (splitAt m i) ≡˘⟨ [,]-∘-distr {f = suc} (splitAt m i) ⟩
suc ([ inject+ n , raise {n} m ] (splitAt m i)) ≡⟨ cong suc (inject+-raise-splitAt m n i) ⟩
suc i ∎
where open ≡-Reasoning
------------------------------------------------------------------------
-- lift
------------------------------------------------------------------------
lift-injective : ∀ {m n} (f : Fin m → Fin n) →
(∀ {x y} → f x ≡ f y → x ≡ y) →
∀ k {x y} → lift k f x ≡ lift k f y → x ≡ y
lift-injective f inj zero eq = inj eq
lift-injective f inj (suc k) {0F} {0F} eq = refl
lift-injective f inj (suc k) {suc i} {suc y} eq = cong suc (lift-injective f inj k (suc-injective eq))
------------------------------------------------------------------------
-- _≺_
------------------------------------------------------------------------
≺⇒<′ : _≺_ ⇒ ℕ._<′_
≺⇒<′ (n ≻toℕ i) = ℕₚ.≤⇒≤′ (toℕ<n i)
<′⇒≺ : ℕ._<′_ ⇒ _≺_
<′⇒≺ {n} ℕ.≤′-refl = subst (_≺ suc n) (toℕ-fromℕ n)
(suc n ≻toℕ fromℕ n)
<′⇒≺ (ℕ.≤′-step m≤′n) with <′⇒≺ m≤′n
... | n ≻toℕ i = subst (_≺ suc n) (toℕ-inject₁ i) (suc n ≻toℕ _)
------------------------------------------------------------------------
-- pred
------------------------------------------------------------------------
<⇒≤pred : ∀ {n} {i j : Fin n} → j < i → j ≤ pred i
<⇒≤pred {i = suc i} {zero} j<i = z≤n
<⇒≤pred {i = suc i} {suc j} (s≤s j<i) =
subst (_ ℕ.≤_) (sym (toℕ-inject₁ i)) j<i
------------------------------------------------------------------------
-- _ℕ-_
------------------------------------------------------------------------
toℕ‿ℕ- : ∀ n i → toℕ (n ℕ- i) ≡ n ∸ toℕ i
toℕ‿ℕ- n zero = toℕ-fromℕ n
toℕ‿ℕ- (suc n) (suc i) = toℕ‿ℕ- n i
------------------------------------------------------------------------
-- _ℕ-ℕ_
------------------------------------------------------------------------
nℕ-ℕi≤n : ∀ n i → n ℕ-ℕ i ℕ.≤ n
nℕ-ℕi≤n n zero = ℕₚ.≤-refl
nℕ-ℕi≤n (suc n) (suc i) = begin
n ℕ-ℕ i ≤⟨ nℕ-ℕi≤n n i ⟩
n ≤⟨ ℕₚ.n≤1+n n ⟩
suc n ∎
where open ℕₚ.≤-Reasoning
------------------------------------------------------------------------
-- punchIn
------------------------------------------------------------------------
punchIn-injective : ∀ {m} i (j k : Fin m) →
punchIn i j ≡ punchIn i k → j ≡ k
punchIn-injective zero _ _ refl = refl
punchIn-injective (suc i) zero zero _ = refl
punchIn-injective (suc i) (suc j) (suc k) ↑j+1≡↑k+1 =
cong suc (punchIn-injective i j k (suc-injective ↑j+1≡↑k+1))
punchInᵢ≢i : ∀ {m} i (j : Fin m) → punchIn i j ≢ i
punchInᵢ≢i (suc i) (suc j) = punchInᵢ≢i i j ∘ suc-injective
------------------------------------------------------------------------
-- punchOut
------------------------------------------------------------------------
-- A version of 'cong' for 'punchOut' in which the inequality argument can be
-- changed out arbitrarily (reflecting the proof-irrelevance of that argument).
punchOut-cong : ∀ {n} (i : Fin (suc n)) {j k} {i≢j : i ≢ j} {i≢k : i ≢ k} → j ≡ k → punchOut i≢j ≡ punchOut i≢k
punchOut-cong zero {zero} {i≢j = 0≢0} = contradiction refl 0≢0
punchOut-cong zero {suc j} {zero} {i≢k = 0≢0} = contradiction refl 0≢0
punchOut-cong zero {suc j} {suc k} = suc-injective
punchOut-cong {suc n} (suc i) {zero} {zero} _ = refl
punchOut-cong {suc n} (suc i) {suc j} {suc k} = cong suc ∘ punchOut-cong i ∘ suc-injective
-- An alternative to 'punchOut-cong' in the which the new inequality argument is
-- specific. Useful for enabling the omission of that argument during equational
-- reasoning.
punchOut-cong′ : ∀ {n} (i : Fin (suc n)) {j k} {p : i ≢ j} (q : j ≡ k) → punchOut p ≡ punchOut (p ∘ sym ∘ trans q ∘ sym)
punchOut-cong′ i q = punchOut-cong i q
punchOut-injective : ∀ {m} {i j k : Fin (suc m)}
(i≢j : i ≢ j) (i≢k : i ≢ k) →
punchOut i≢j ≡ punchOut i≢k → j ≡ k
punchOut-injective {_} {zero} {zero} {_} 0≢0 _ _ = contradiction refl 0≢0
punchOut-injective {_} {zero} {_} {zero} _ 0≢0 _ = contradiction refl 0≢0
punchOut-injective {_} {zero} {suc j} {suc k} _ _ pⱼ≡pₖ = cong suc pⱼ≡pₖ
punchOut-injective {suc n} {suc i} {zero} {zero} _ _ _ = refl
punchOut-injective {suc n} {suc i} {suc j} {suc k} i≢j i≢k pⱼ≡pₖ =
cong suc (punchOut-injective (i≢j ∘ cong suc) (i≢k ∘ cong suc) (suc-injective pⱼ≡pₖ))
punchIn-punchOut : ∀ {m} {i j : Fin (suc m)} (i≢j : i ≢ j) →
punchIn i (punchOut i≢j) ≡ j
punchIn-punchOut {_} {zero} {zero} 0≢0 = contradiction refl 0≢0
punchIn-punchOut {_} {zero} {suc j} _ = refl
punchIn-punchOut {suc m} {suc i} {zero} i≢j = refl
punchIn-punchOut {suc m} {suc i} {suc j} i≢j =
cong suc (punchIn-punchOut (i≢j ∘ cong suc))
punchOut-punchIn : ∀ {n} i {j : Fin n} → punchOut {i = i} {j = punchIn i j} (punchInᵢ≢i i j ∘ sym) ≡ j
punchOut-punchIn zero {j} = refl
punchOut-punchIn (suc i) {zero} = refl
punchOut-punchIn (suc i) {suc j} = cong suc (begin
punchOut (punchInᵢ≢i i j ∘ suc-injective ∘ sym ∘ cong suc) ≡⟨ punchOut-cong i refl ⟩
punchOut (punchInᵢ≢i i j ∘ sym) ≡⟨ punchOut-punchIn i ⟩
j ∎)
where open ≡-Reasoning
------------------------------------------------------------------------
-- Quantification
------------------------------------------------------------------------
module _ {n p} {P : Pred (Fin (suc n)) p} where
∀-cons : P zero → Π[ P ∘ suc ] → Π[ P ]
∀-cons z s zero = z
∀-cons z s (suc i) = s i
∀-cons-⇔ : (P zero × Π[ P ∘ suc ]) ⇔ Π[ P ]
∀-cons-⇔ = equivalence (uncurry ∀-cons) < _$ zero , _∘ suc >
∃-here : P zero → ∃⟨ P ⟩
∃-here = zero ,_
∃-there : ∃⟨ P ∘ suc ⟩ → ∃⟨ P ⟩
∃-there = map suc id
∃-toSum : ∃⟨ P ⟩ → P zero ⊎ ∃⟨ P ∘ suc ⟩
∃-toSum ( zero , P₀ ) = inj₁ P₀
∃-toSum (suc f , P₁₊) = inj₂ (f , P₁₊)
⊎⇔∃ : (P zero ⊎ ∃⟨ P ∘ suc ⟩) ⇔ ∃⟨ P ⟩
⊎⇔∃ = equivalence [ ∃-here , ∃-there ] ∃-toSum
decFinSubset : ∀ {n p q} {P : Pred (Fin n) p} {Q : Pred (Fin n) q} →
Decidable Q → (∀ {f} → Q f → Dec (P f)) → Dec (Q ⊆ P)
decFinSubset {zero} Q? P? = yes λ {}
decFinSubset {suc n} {P = P} {Q} Q? P?
with Q? zero | ∀-cons {P = λ x → Q x → P x}
... | false because [¬Q0] | cons =
map′ (λ f {x} → cons (⊥-elim ∘ invert [¬Q0]) (λ x → f {x}) x)
(λ f {x} → f {suc x})
(decFinSubset (Q? ∘ suc) P?)
... | true because [Q0] | cons =
map′ (uncurry λ P0 rec {x} → cons (λ _ → P0) (λ x → rec {x}) x)
< _$ invert [Q0] , (λ f {x} → f {suc x}) >
(P? (invert [Q0]) ×-dec decFinSubset (Q? ∘ suc) P?)
any? : ∀ {n p} {P : Fin n → Set p} → Decidable P → Dec (∃ P)
any? {zero} {P = _} P? = no λ { (() , _) }
any? {suc n} {P = P} P? = Dec.map ⊎⇔∃ (P? zero ⊎-dec any? (P? ∘ suc))
all? : ∀ {n p} {P : Pred (Fin n) p} →
Decidable P → Dec (∀ f → P f)
all? P? = map′ (λ ∀p f → ∀p tt) (λ ∀p {x} _ → ∀p x)
(decFinSubset U? (λ {f} _ → P? f))
private
-- A nice computational property of `all?`:
-- The boolean component of the result is exactly the
-- obvious fold of boolean tests (`foldr _∧_ true`).
note : ∀ {p} {P : Pred (Fin 3) p} (P? : Decidable P) →
∃ λ z → does (all? P?) ≡ z
note P? = does (P? 0F) ∧ does (P? 1F) ∧ does (P? 2F) ∧ true
, refl
-- If a decidable predicate P over a finite set is sometimes false,
-- then we can find the smallest value for which this is the case.
¬∀⟶∃¬-smallest : ∀ n {p} (P : Pred (Fin n) p) → Decidable P →
¬ (∀ i → P i) → ∃ λ i → ¬ P i × ((j : Fin′ i) → P (inject j))
¬∀⟶∃¬-smallest zero P P? ¬∀P = contradiction (λ()) ¬∀P
¬∀⟶∃¬-smallest (suc n) P P? ¬∀P with P? zero
... | false because [¬P₀] = (zero , invert [¬P₀] , λ ())
... | true because [P₀] = map suc (map id (∀-cons (invert [P₀])))
(¬∀⟶∃¬-smallest n (P ∘ suc) (P? ∘ suc) (¬∀P ∘ (∀-cons (invert [P₀]))))
-- When P is a decidable predicate over a finite set the following
-- lemma can be proved.
¬∀⟶∃¬ : ∀ n {p} (P : Pred (Fin n) p) → Decidable P →
¬ (∀ i → P i) → (∃ λ i → ¬ P i)
¬∀⟶∃¬ n P P? ¬P = map id proj₁ (¬∀⟶∃¬-smallest n P P? ¬P)
-- The pigeonhole principle.
pigeonhole : ∀ {m n} → m ℕ.< n → (f : Fin n → Fin m) →
∃₂ λ i j → i ≢ j × f i ≡ f j
pigeonhole (s≤s z≤n) f = contradiction (f zero) λ()
pigeonhole (s≤s (s≤s m≤n)) f with any? (λ k → f zero ≟ f (suc k))
... | yes (j , f₀≡fⱼ) = zero , suc j , (λ()) , f₀≡fⱼ
... | no f₀≢fₖ with pigeonhole (s≤s m≤n) (λ j → punchOut (f₀≢fₖ ∘ (j ,_ )))
... | (i , j , i≢j , fᵢ≡fⱼ) =
suc i , suc j , i≢j ∘ suc-injective ,
punchOut-injective (f₀≢fₖ ∘ (i ,_)) _ fᵢ≡fⱼ
------------------------------------------------------------------------
-- Categorical
------------------------------------------------------------------------
module _ {f} {F : Set f → Set f} (RA : RawApplicative F) where
open RawApplicative RA
sequence : ∀ {n} {P : Pred (Fin n) f} →
(∀ i → F (P i)) → F (∀ i → P i)
sequence {zero} ∀iPi = pure λ()
sequence {suc n} ∀iPi = ∀-cons <$> ∀iPi zero ⊛ sequence (∀iPi ∘ suc)
module _ {f} {F : Set f → Set f} (RF : RawFunctor F) where
open RawFunctor RF
sequence⁻¹ : ∀ {A : Set f} {P : Pred A f} →
F (∀ i → P i) → (∀ i → F (P i))
sequence⁻¹ F∀iPi i = (λ f → f i) <$> F∀iPi
------------------------------------------------------------------------
-- If there is an injection from a type to a finite set, then the type
-- has decidable equality.
module _ {a} {A : Set a} where
eq? : ∀ {n} → A ↣ Fin n → B.Decidable {A = A} _≡_
eq? inj = Dec.via-injection inj _≟_
------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.
-- Version 0.15
cmp = <-cmp
{-# WARNING_ON_USAGE cmp
"Warning: cmp was deprecated in v0.15.
Please use <-cmp instead."
#-}
strictTotalOrder = <-strictTotalOrder
{-# WARNING_ON_USAGE strictTotalOrder
"Warning: strictTotalOrder was deprecated in v0.15.
Please use <-strictTotalOrder instead."
#-}
-- Version 0.16
to-from = toℕ-fromℕ
{-# WARNING_ON_USAGE to-from
"Warning: to-from was deprecated in v0.16.
Please use toℕ-fromℕ instead."
#-}
from-to = fromℕ-toℕ
{-# WARNING_ON_USAGE from-to
"Warning: from-to was deprecated in v0.16.
Please use fromℕ-toℕ instead."
#-}
bounded = toℕ<n
{-# WARNING_ON_USAGE bounded
"Warning: bounded was deprecated in v0.16.
Please use toℕ<n instead."
#-}
prop-toℕ-≤ = toℕ≤pred[n]
{-# WARNING_ON_USAGE prop-toℕ-≤
"Warning: prop-toℕ-≤ was deprecated in v0.16.
Please use toℕ≤pred[n] instead."
#-}
prop-toℕ-≤′ = toℕ≤pred[n]′
{-# WARNING_ON_USAGE prop-toℕ-≤′
"Warning: prop-toℕ-≤′ was deprecated in v0.16.
Please use toℕ≤pred[n]′ instead."
#-}
inject-lemma = toℕ-inject
{-# WARNING_ON_USAGE inject-lemma
"Warning: inject-lemma was deprecated in v0.16.
Please use toℕ-inject instead."
#-}
inject+-lemma = toℕ-inject+
{-# WARNING_ON_USAGE inject+-lemma
"Warning: inject+-lemma was deprecated in v0.16.
Please use toℕ-inject+ instead."
#-}
inject₁-lemma = toℕ-inject₁
{-# WARNING_ON_USAGE inject₁-lemma
"Warning: inject₁-lemma was deprecated in v0.16.
Please use toℕ-inject₁ instead."
#-}
inject≤-lemma = toℕ-inject≤
{-# WARNING_ON_USAGE inject≤-lemma
"Warning: inject≤-lemma was deprecated in v0.16.
Please use toℕ-inject≤ instead."
#-}
-- Version 0.17
≤+≢⇒< = ≤∧≢⇒<
{-# WARNING_ON_USAGE ≤+≢⇒<
"Warning: ≤+≢⇒< was deprecated in v0.17.
Please use ≤∧≢⇒< instead."
#-}
-- Version 1.0
≤-irrelevance = ≤-irrelevant
{-# WARNING_ON_USAGE ≤-irrelevance
"Warning: ≤-irrelevance was deprecated in v1.0.
Please use ≤-irrelevant instead."
#-}
<-irrelevance = <-irrelevant
{-# WARNING_ON_USAGE <-irrelevance
"Warning: <-irrelevance was deprecated in v1.0.
Please use <-irrelevant instead."
#-}
-- Version 1.1
infixl 6 _+′_
_+′_ : ∀ {m n} (i : Fin m) (j : Fin n) → Fin (ℕ.pred m ℕ.+ n)
i +′ j = inject≤ (i + j) (ℕₚ.+-monoˡ-≤ _ (toℕ≤pred[n] i))
{-# WARNING_ON_USAGE _+′_
"Warning: _+′_ was deprecated in v1.1.
Please use `raise` or `inject+` from `Data.Fin` instead."
#-}
-- Version 1.2
fromℕ≤-toℕ = fromℕ<-toℕ
{-# WARNING_ON_USAGE fromℕ≤-toℕ
"Warning: fromℕ≤-toℕ was deprecated in v1.2.
Please use fromℕ<-toℕ instead."
#-}
toℕ-fromℕ≤ = toℕ-fromℕ<
{-# WARNING_ON_USAGE toℕ-fromℕ≤
"Warning: toℕ-fromℕ≤ was deprecated in v1.2.
Please use toℕ-fromℕ< instead."
#-}
fromℕ≤≡fromℕ≤″ = fromℕ<≡fromℕ<″
{-# WARNING_ON_USAGE fromℕ≤≡fromℕ≤″
"Warning: fromℕ≤≡fromℕ≤″ was deprecated in v1.2.
Please use fromℕ<≡fromℕ<″ instead."
#-}
toℕ-fromℕ≤″ = toℕ-fromℕ<″
{-# WARNING_ON_USAGE toℕ-fromℕ≤″
"Warning: toℕ-fromℕ≤″ was deprecated in v1.2.
Please use toℕ-fromℕ<″ instead."
#-}
isDecEquivalence = ≡-isDecEquivalence
{-# WARNING_ON_USAGE isDecEquivalence
"Warning: isDecEquivalence was deprecated in v1.2.
Please use ≡-isDecEquivalence instead."
#-}
preorder = ≡-preorder
{-# WARNING_ON_USAGE preorder
"Warning: preorder was deprecated in v1.2.
Please use ≡-preorder instead."
#-}
setoid = ≡-setoid
{-# WARNING_ON_USAGE setoid
"Warning: setoid was deprecated in v1.2.
Please use ≡-setoid instead."
#-}
decSetoid = ≡-decSetoid
{-# WARNING_ON_USAGE decSetoid
"Warning: decSetoid was deprecated in v1.2.
Please use ≡-decSetoid instead."
#-}
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{-# OPTIONS --without-K --rewriting #-}
open import HoTT
open import homotopy.HSpace renaming (HSpaceStructure to HSS)
open import homotopy.Freudenthal
open import homotopy.IterSuspensionStable
open import homotopy.Pi2HSusp
open import homotopy.EM1HSpace
open import homotopy.EilenbergMacLane1
module homotopy.EilenbergMacLane where
private
-- helper
kle : (n : ℕ) → ⟨ S (S n) ⟩ ≤T ⟨ n ⟩ +2+ ⟨ n ⟩
kle O = inl idp
kle (S n) = ≤T-trans (≤T-ap-S (kle n))
(≤T-trans (inl (! (+2+-βr ⟨ n ⟩ ⟨ n ⟩)))
(inr ltS))
-- EM(G,n) when G is π₁(A,a₀)
module EMImplicit {i} {X : Ptd i} {{_ : is-connected 0 (de⊙ X)}}
{{X-level : has-level 1 (de⊙ X)}} (H-X : HSS X) where
private
A = de⊙ X
a₀ = pt X
⊙EM : (n : ℕ) → Ptd i
⊙EM O = ⊙Ω X
⊙EM (S n) = ⊙Trunc ⟨ S n ⟩ (⊙Susp^ n X)
module _ (n : ℕ) where
EM = de⊙ (⊙EM n)
EM-level : (n : ℕ) → has-level ⟨ n ⟩ (EM n)
EM-level O = has-level-apply X-level _ _
EM-level (S n) = Trunc-level
instance
EM-conn : (n : ℕ) → is-connected ⟨ n ⟩ (EM (S n))
EM-conn n = Trunc-preserves-conn (⊙Susp^-conn' n)
{-
π (S k) (EM (S n)) (embase (S n)) == π k (EM n) (embase n)
where k > 0 and n = S (S n')
-}
module Stable (k n : ℕ) (indexing : S k ≤ S (S n))
where
private
SSn : ℕ
SSn = S (S n)
lte : ⟨ S k ⟩ ≤T ⟨ SSn ⟩
lte = ⟨⟩-monotone-≤ $ indexing
Skle : S k ≤ (S n) *2
Skle = ≤-trans indexing (lemma n)
where lemma : (n' : ℕ) → S (S n') ≤ (S n') *2
lemma O = inl idp
lemma (S n') = ≤-trans (≤-ap-S (lemma n')) (inr ltS)
private
module SS = Susp^StableSucc k (S n) Skle (⊙Susp^ (S n) X) {{⊙Susp^-conn' (S n)}}
abstract
stable : πS (S k) (⊙EM (S SSn)) ≃ᴳ πS k (⊙EM SSn)
stable =
πS (S k) (⊙EM (S SSn))
≃ᴳ⟨ πS-Trunc-fuse-≤-iso _ _ _ (≤T-ap-S lte) ⟩
πS (S k) (⊙Susp^ SSn X)
≃ᴳ⟨ SS.stable ⟩
πS k (⊙Susp^ (S n) X)
≃ᴳ⟨ πS-Trunc-fuse-≤-iso _ _ _ lte ⁻¹ᴳ ⟩
πS k (⊙EM SSn)
≃ᴳ∎
module BelowDiagonal where
π₁ : (n : ℕ) → πS 0 (⊙EM (S (S n))) ≃ᴳ 0ᴳ
π₁ n =
contr-iso-0ᴳ (πS 0 (⊙EM (S (S n))))
(connected-at-level-is-contr
{{raise-level-≤T (≤T-ap-S (≤T-ap-S (-2≤T ⟨ n ⟩₋₂)))
(Trunc-level {n = 0})}})
-- some clutter here arises from the definition of <;
-- any simple way to avoid this?
πS-below : (k n : ℕ) → (S k < n)
→ πS k (⊙EM n) ≃ᴳ 0ᴳ
πS-below 0 .2 ltS = π₁ 0
πS-below 0 .3 (ltSR ltS) = π₁ 1
πS-below 0 (S (S n)) (ltSR (ltSR _)) = π₁ n
πS-below (S k) ._ ltS =
πS-below k _ ltS
∘eᴳ Stable.stable k k (inr ltS)
πS-below (S k) ._ (ltSR ltS) =
πS-below k _ (ltSR ltS)
∘eᴳ Stable.stable k (S k) (inr (ltSR ltS))
πS-below (S k) ._ (ltSR (ltSR ltS)) =
πS-below k _ (ltSR (ltSR ltS))
∘eᴳ Stable.stable k (S (S k)) (inr (ltSR (ltSR ltS)))
πS-below (S k) (S (S (S n))) (ltSR (ltSR (ltSR lt))) =
πS-below k _ (<-cancel-S (ltSR (ltSR (ltSR lt))))
∘eᴳ Stable.stable k n (inr (<-cancel-S (ltSR (ltSR (ltSR lt)))))
module OnDiagonal where
π₁ : πS 0 (⊙EM 1) ≃ᴳ πS 0 X
π₁ = πS-Trunc-fuse-≤-iso 0 1 X ≤T-refl
private
module Π₂ = Pi2HSusp H-X
π₂ : πS 1 (⊙EM 2) ≃ᴳ πS 0 X
π₂ = Π₂.π₂-Susp
∘eᴳ πS-Trunc-fuse-≤-iso 1 2 (⊙Susp (de⊙ X)) ≤T-refl
πS-diag : (n : ℕ) → πS n (⊙EM (S n)) ≃ᴳ πS 0 X
πS-diag 0 = π₁
πS-diag 1 = π₂
πS-diag (S (S n)) = πS-diag (S n)
∘eᴳ Stable.stable (S n) n ≤-refl
module AboveDiagonal where
πS-above : ∀ (k n : ℕ) → (n < S k)
→ πS k (⊙EM n) ≃ᴳ 0ᴳ
πS-above k n lt =
contr-iso-0ᴳ (πS k (⊙EM n))
(inhab-prop-is-contr
[ idp^ (S k) ]
{{Trunc-preserves-level 0 (Ω^-level -1 (S k) _
(raise-level-≤T (lemma lt) (EM-level n)))}})
where lemma : {k n : ℕ} → n < k → ⟨ n ⟩ ≤T (⟨ k ⟩₋₂ +2+ -1)
lemma ltS = inl (! (+2+-comm _ -1))
lemma (ltSR lt) = ≤T-trans (lemma lt) (inr ltS)
module Spectrum where
private
module Π₂ = Pi2HSusp H-X
spectrum0 : ⊙Ω (⊙EM 1) ⊙≃ ⊙EM 0
spectrum0 =
⊙Ω (⊙EM 1)
⊙≃⟨ ⊙Ω-⊙Trunc-comm 0 X ⟩
⊙Trunc 0 (⊙Ω X)
⊙≃⟨ ⊙unTrunc-equiv (⊙Ω X) ⟩
⊙Ω X ⊙≃∎
spectrum1 : ⊙Ω (⊙EM 2) ⊙≃ ⊙EM 1
spectrum1 =
⊙Ω (⊙EM 2)
⊙≃⟨ ⊙Ω-⊙Trunc-comm 1 (⊙Susp (de⊙ X)) ⟩
⊙Trunc 1 (⊙Ω (⊙Susp (de⊙ X)))
⊙≃⟨ Π₂.⊙eq ⟩
⊙EM 1 ⊙≃∎
sconn : (n : ℕ) → is-connected ⟨ S n ⟩ (de⊙ (⊙Susp^ (S n) X))
sconn n = ⊙Susp^-conn' (S n)
module FS (n : ℕ) =
FreudenthalEquiv ⟨ n ⟩₋₁ ⟨ S (S n) ⟩ (kle n)
(⊙Susp^ (S n) X) {{sconn n}}
Trunc-fmap-σloop-is-equiv : ∀ (n : ℕ)
→ is-equiv (Trunc-fmap {n = ⟨ S n ⟩} (σloop (⊙Susp^ n X)))
Trunc-fmap-σloop-is-equiv O = snd (Π₂.eq ⁻¹)
Trunc-fmap-σloop-is-equiv (S n) = snd (FS.eq n)
spectrumSS : (n : ℕ)
→ ⊙Ω (⊙EM (S (S (S n)))) ⊙≃ ⊙EM (S (S n))
spectrumSS n =
⊙Ω (⊙EM (S (S (S n))))
⊙≃⟨ ⊙Ω-⊙Trunc-comm ⟨ S (S n) ⟩ (⊙Susp^ (S (S n)) X) ⟩
⊙Trunc ⟨ S (S n) ⟩ (⊙Ω (⊙Susp^ (S (S n)) X))
⊙≃⟨ FS.⊙eq n ⊙⁻¹ ⟩
⊙EM (S (S n)) ⊙≃∎
⊙–>-spectrumSS : ∀ (n : ℕ)
→ ⊙–> (spectrumSS n) ◃⊙idf
=⊙∘
FS.⊙encodeN n ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm ⟨ S (S n) ⟩ (⊙Susp^ (S (S n)) X)) ◃⊙idf
⊙–>-spectrumSS n =
⊙–> (spectrumSS n) ◃⊙idf
=⊙∘⟨ =⊙∘-in {gs = ⊙<– (FS.⊙eq n) ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm ⟨ S (S n) ⟩ (⊙Susp^ (S (S n)) X)) ◃⊙idf} $
⊙λ= (⊙∘-unit-l _) ⟩
⊙<– (FS.⊙eq n) ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm ⟨ S (S n) ⟩ (⊙Susp^ (S (S n)) X)) ◃⊙idf
=⊙∘₁⟨ 0 & 1 & FS.⊙<–-⊙eq n ⟩
FS.⊙encodeN n ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm ⟨ S (S n) ⟩ (⊙Susp^ (S (S n)) X)) ◃⊙idf ∎⊙∘
spectrum : (n : ℕ) → ⊙Ω (⊙EM (S n)) ⊙≃ ⊙EM n
spectrum 0 = spectrum0
spectrum 1 = spectrum1
spectrum (S (S n)) = spectrumSS n
module EMImplicitMap {i} {j} {X : Ptd i} {Y : Ptd j} (f : X ⊙→ Y)
{{_ : is-connected 0 (de⊙ X)}} {{_ : is-connected 0 (de⊙ Y)}}
{{X-level : has-level 1 (de⊙ X)}} {{Y-level : has-level 1 (de⊙ Y)}}
(H-X : HSS X) (H-Y : HSS Y) where
⊙EM-fmap : ∀ n → EMImplicit.⊙EM H-X n ⊙→ EMImplicit.⊙EM H-Y n
⊙EM-fmap O = ⊙Ω-fmap f
⊙EM-fmap (S n) = ⊙Trunc-fmap (⊙Susp^-fmap n f)
module SpectrumNatural {i} {X Y : Ptd i} (f : X ⊙→ Y)
{{_ : is-connected 0 (de⊙ X)}} {{_ : is-connected 0 (de⊙ Y)}}
{{X-level : has-level 1 (de⊙ X)}} {{Y-level : has-level 1 (de⊙ Y)}}
(H-X : HSS X) (H-Y : HSS Y) where
open EMImplicitMap f H-X H-Y
open EMImplicit.Spectrum
⊙–>-spectrum0-natural :
⊙–> (spectrum0 H-Y) ◃⊙∘
⊙Ω-fmap (⊙Trunc-fmap f) ◃⊙idf
=⊙∘
⊙Ω-fmap f ◃⊙∘
⊙–> (spectrum0 H-X) ◃⊙idf
⊙–>-spectrum0-natural =
⊙–> (spectrum0 H-Y) ◃⊙∘
⊙Ω-fmap (⊙Trunc-fmap f) ◃⊙idf
=⊙∘⟨ 0 & 1 & ⊙expand (⊙–> (⊙unTrunc-equiv (⊙Ω Y)) ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm 0 Y) ◃⊙idf) ⟩
⊙–> (⊙unTrunc-equiv (⊙Ω Y)) ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm 0 Y) ◃⊙∘
⊙Ω-fmap (⊙Trunc-fmap f) ◃⊙idf
=⊙∘⟨ 1 & 2 & ⊙–>-⊙Ω-⊙Trunc-comm-natural-=⊙∘ 0 f ⟩
⊙–> (⊙unTrunc-equiv (⊙Ω Y)) ◃⊙∘
⊙Trunc-fmap (⊙Ω-fmap f) ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm (S (S ⟨-2⟩)) X) ◃⊙idf
=⊙∘⟨ 0 & 2 & ⊙–>-⊙unTrunc-equiv-natural-=⊙∘ (⊙Ω-fmap f) ⟩
⊙Ω-fmap f ◃⊙∘
⊙–> (⊙unTrunc-equiv (⊙Ω X)) ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm (S (S ⟨-2⟩)) X) ◃⊙idf
=⊙∘⟨ 1 & 2 & ⊙contract ⟩
⊙Ω-fmap f ◃⊙∘
⊙–> (spectrum0 H-X) ◃⊙idf ∎⊙∘
⊙–>-spectrum1-natural :
⊙–> (spectrum1 H-Y) ◃⊙∘
⊙Ω-fmap (⊙Trunc-fmap (⊙Susp-fmap (fst f))) ◃⊙idf
=⊙∘
⊙Trunc-fmap f ◃⊙∘
⊙–> (spectrum1 H-X) ◃⊙idf
⊙–>-spectrum1-natural =
⊙–> (spectrum1 H-Y) ◃⊙∘
⊙Ω-fmap (⊙Trunc-fmap (⊙Susp-fmap (fst f))) ◃⊙idf
=⊙∘⟨ 0 & 1 & ⊙expand (⊙–> (Pi2HSusp.⊙eq H-Y) ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm 1 (⊙Susp (de⊙ Y))) ◃⊙idf) ⟩
⊙–> (Pi2HSusp.⊙eq H-Y) ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm 1 (⊙Susp (de⊙ Y))) ◃⊙∘
⊙Ω-fmap (⊙Trunc-fmap (⊙Susp-fmap (fst f))) ◃⊙idf
=⊙∘⟨ 1 & 2 & ⊙–>-⊙Ω-⊙Trunc-comm-natural-=⊙∘ 1 (⊙Susp-fmap (fst f)) ⟩
⊙–> (Pi2HSusp.⊙eq H-Y) ◃⊙∘
⊙Trunc-fmap (⊙Ω-fmap (⊙Susp-fmap (fst f))) ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm 1 (⊙Susp (de⊙ X))) ◃⊙idf
=⊙∘⟨ 0 & 2 & Pi2HSuspNaturality.⊙encodeN-natural f H-X H-Y ⟩
⊙Trunc-fmap f ◃⊙∘
⊙–> (Pi2HSusp.⊙eq H-X) ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm (S (S (S ⟨-2⟩))) (⊙Susp (de⊙ X))) ◃⊙idf
=⊙∘⟨ 1 & 2 & ⊙contract ⟩
⊙Trunc-fmap f ◃⊙∘
⊙–> (spectrum1 H-X) ◃⊙idf ∎⊙∘
⊙–>-spectrumSS-natural : ∀ (n : ℕ)
→ ⊙–> (spectrumSS H-Y n) ◃⊙∘
⊙Ω-fmap (⊙Trunc-fmap (⊙Susp^-fmap (S (S n)) f)) ◃⊙idf
=⊙∘
⊙Trunc-fmap (⊙Susp^-fmap (S n) f) ◃⊙∘
⊙–> (spectrumSS H-X n) ◃⊙idf
⊙–>-spectrumSS-natural n =
⊙–> (spectrumSS H-Y n) ◃⊙∘
⊙Ω-fmap (⊙Trunc-fmap (⊙Susp^-fmap (S (S n)) f)) ◃⊙idf
=⊙∘⟨ 0 & 1 & ⊙–>-spectrumSS H-Y n ⟩
FS.⊙encodeN H-Y n ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm ⟨ S (S n) ⟩ (⊙Susp^ (S (S n)) Y)) ◃⊙∘
⊙Ω-fmap (⊙Trunc-fmap (⊙Susp^-fmap (S (S n)) f)) ◃⊙idf
=⊙∘⟨ 1 & 2 & ⊙–>-⊙Ω-⊙Trunc-comm-natural-=⊙∘ ⟨ S (S n) ⟩ (⊙Susp^-fmap (S (S n)) f) ⟩
FS.⊙encodeN H-Y n ◃⊙∘
⊙Trunc-fmap (⊙Ω-fmap (⊙Susp^-fmap (S (S n)) f)) ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm ⟨ S (S n) ⟩ (⊙Susp^ (S (S n)) X)) ◃⊙idf
=⊙∘⟨ 0 & 2 & FreudenthalEquivNatural.⊙encodeN-natural
⟨ n ⟩₋₁ ⟨ S (S n) ⟩ (kle n)
{X = ⊙Susp^ (S n) X} {Y = ⊙Susp^ (S n) Y} (⊙Susp^-fmap (S n) f)
{{sconn H-X n}} {{sconn H-Y n}} ⟩
⊙Trunc-fmap (⊙Susp^-fmap (S n) f) ◃⊙∘
FS.⊙encodeN H-X n ◃⊙∘
⊙–> (⊙Ω-⊙Trunc-comm ⟨ S (S n) ⟩ (⊙Susp^ (S (S n)) X)) ◃⊙idf
=⊙∘⟨ 1 & 2 & !⊙∘ $ ⊙–>-spectrumSS H-X n ⟩
⊙Trunc-fmap (⊙Susp^-fmap (S n) f) ◃⊙∘
⊙–> (spectrumSS H-X n) ◃⊙idf ∎⊙∘
abstract
⊙–>-spectrum-natural : ∀ (n : ℕ)
→ ⊙–> (spectrum H-Y n) ◃⊙∘
⊙Ω-fmap (⊙EM-fmap (S n)) ◃⊙idf
=⊙∘
⊙EM-fmap n ◃⊙∘
⊙–> (spectrum H-X n) ◃⊙idf
⊙–>-spectrum-natural 0 = ⊙–>-spectrum0-natural
⊙–>-spectrum-natural 1 = ⊙–>-spectrum1-natural
⊙–>-spectrum-natural (S (S n)) = ⊙–>-spectrumSS-natural n
⊙<–-spectrum-natural : ∀ (n : ℕ)
→ ⊙<– (spectrum H-Y n) ◃⊙∘
⊙EM-fmap n ◃⊙idf
=⊙∘
⊙Ω-fmap (⊙EM-fmap (S n)) ◃⊙∘
⊙<– (spectrum H-X n) ◃⊙idf
⊙<–-spectrum-natural n =
⊙<– (spectrum H-Y n) ◃⊙∘
⊙EM-fmap n ◃⊙idf
=⊙∘⟨ 2 & 0 & !⊙∘ $ ⊙<–-inv-r-=⊙∘ (spectrum H-X n) ⟩
⊙<– (spectrum H-Y n) ◃⊙∘
⊙EM-fmap n ◃⊙∘
⊙–> (spectrum H-X n) ◃⊙∘
⊙<– (spectrum H-X n) ◃⊙idf
=⊙∘⟨ 1 & 2 & !⊙∘ (⊙–>-spectrum-natural n) ⟩
⊙<– (spectrum H-Y n) ◃⊙∘
⊙–> (spectrum H-Y n) ◃⊙∘
⊙Ω-fmap (⊙Trunc-fmap (⊙Susp^-fmap n f)) ◃⊙∘
⊙<– (spectrum H-X n) ◃⊙idf
=⊙∘⟨ 0 & 2 & ⊙<–-inv-l-=⊙∘ (spectrum H-Y n) ⟩
⊙Ω-fmap (⊙Trunc-fmap (⊙Susp^-fmap n f)) ◃⊙∘
⊙<– (spectrum H-X n) ◃⊙idf ∎⊙∘
module EMExplicit {i} (G : AbGroup i) where
module HSpace = EM₁HSpace G
open EMImplicit
{X = ⊙EM₁ (AbGroup.grp G)}
{{EM₁-conn}}
{{EM₁-level₁ (AbGroup.grp G)}}
HSpace.H-⊙EM₁ public
open BelowDiagonal public using (πS-below)
πS-diag : (n : ℕ) → πS n (⊙EM (S n)) ≃ᴳ AbGroup.grp G
πS-diag n = π₁-EM₁ (AbGroup.grp G) ∘eᴳ OnDiagonal.πS-diag n
open AboveDiagonal public using (πS-above)
abstract
spectrum : (n : ℕ) → ⊙Ω (⊙EM (S n)) ⊙≃ ⊙EM n
spectrum = Spectrum.spectrum
spectrum-def : ∀ n → spectrum n == Spectrum.spectrum n
spectrum-def n = idp
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Core definitions for Characters
------------------------------------------------------------------------
module Data.Char.Core where
open import Data.Nat using (ℕ)
open import Data.Bool using (Bool; true; false)
------------------------------------------------------------------------
-- The type
postulate
Char : Set
{-# BUILTIN CHAR Char #-}
{-# COMPILED_TYPE Char Char #-}
------------------------------------------------------------------------
-- Primitive operations
primitive
primCharToNat : Char → ℕ
primCharEquality : Char → Char → Bool
-- primShowChar is in Data.String.Core for break an import cycle.
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module Imports.Bool where
data Bool : Set where
true false : Bool
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module AKS.Rational where
open import AKS.Rational.Base public
open import AKS.Rational.Properties public
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-- Andreas, 2017-12-03, issue #2826, reported by Snicksi, shrunk by sattlerc
data A : Set where
a : A
data B : Set where
b : (x : A) → B
foo : B → B
foo s with a
-- WAS: case splitting produces repeated variables in pattern
foo s | x = {!s!}
-- Expected: Something like the following
-- foo (b x₁) | x = {!!}
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{-# OPTIONS --without-K #-}
--open import HoTT
open import homotopy.3x3.PushoutPushout
open import homotopy.3x3.Transpose
import homotopy.3x3.To as To
import homotopy.3x3.From as From
open import homotopy.3x3.Common
module homotopy.3x3.FromTo2 {i} (d : Span^2 {i}) where
open Span^2 d
open M d hiding (Pushout^2)
open M (transpose d) using () renaming (module F₁∙ to F∙₁; f₁∙ to f∙₁;
module F₃∙ to F∙₃; f₃∙ to f∙₃;
v-h-span to h-v-span)
open M using (Pushout^2)
open To d
open From d
open import homotopy.3x3.FromToInit d
module M2 (c : A₂₂) where
coh : ∀ {i} {A : Type i} {a b c d : A}
{p q : a == b} (α : p == q)
{v w : b == d} {β β' : w == v} (eqβ : β == β')
{t u : a == c} {ε ε' : u == t} (eqε : ε == ε')
{r s : c == d} (ζ : r == s)
(γ : (q , v =□ t , s))
(δ : (p , w =□ u , r))
(eq : γ == δ ∙□-i/ ! β' / ε' / ∙□-o/ ! α / ζ /)
→ (α , β , γ =□□ δ , ε , ζ)
coh idp {β = idp} idp {ε = idp} idp idp _ _ x = x
end-lemma1 : ap (right ∘ f₃∙) (glue c) == ap (from ∘ i₄∙ ∘ f₃∙) (glue c) :> E∙₂Red.T-lhs c
end-lemma1 =
ap (right ∘ f₃∙) (glue c)
=⟨ ap-∘ right f₃∙ (glue c) ⟩
ap right (ap f₃∙ (glue c))
=⟨ ap (ap right) (F₃∙.glue-β c) ⟩
ap right (ap left (H₃₁ c) ∙ glue (f₃₂ c) ∙ ap right (H₃₃ c))
=⟨ ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c) ⟩
ap (right ∘ left) (H₃₁ c) ∙ ap right (glue (f₃₂ c)) ∙ ap (right ∘ right) (H₃₃ c)
=⟨ ! (from-to-r-g (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)) ⟩
ap (right ∘ left) (H₃₁ c) ∙ (ap (from ∘ i₄∙) (glue (f₃₂ c))) ∙ ap (right ∘ right) (H₃₃ c)
=⟨ ! (ap-∙∙`∘`∘ (from ∘ i₄∙) left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)) ⟩
ap (from ∘ i₄∙) (ap left (H₃₁ c) ∙ glue (f₃₂ c) ∙ ap right (H₃₃ c))
=⟨ ! (ap (ap (from ∘ i₄∙)) (F₃∙.glue-β c)) ⟩
ap (from ∘ i₄∙) (ap f₃∙ (glue c))
=⟨ ∘-ap (from ∘ i₄∙) f₃∙ (glue c) ⟩
ap (from ∘ i₄∙ ∘ f₃∙) (glue c) ∎
lemma1 : ↓-='-out (apd (from-to-r ∘ f₃∙) (glue c)) == end-lemma1
lemma1 =
↓-='-out (apd (from-to-r ∘ f₃∙) (glue c))
=⟨ apd-∘'' from-to-r f₃∙ (glue c) (F₃∙.glue-β c) |in-ctx ↓-='-out ⟩
↓-='-out (↓-ap-out= (λ b → from (to (right b)) == right b) f₃∙ (glue c) (F₃∙.glue-β c) (apd from-to-r (ap left (H₃₁ c) ∙ glue (f₃₂ c) ∙ ap right (H₃₃ c))))
=⟨ apd-∙∙`∘`∘ from-to-r left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c) |in-ctx (λ u → ↓-='-out (↓-ap-out= (λ b → from (to (right b)) == right b) f₃∙ (glue c) (F₃∙.glue-β c) u)) ⟩
↓-='-out (↓-ap-out= (λ b → from (to (right b)) == right b) f₃∙ (glue c) (F₃∙.glue-β c) (↓-ap-in _ _ (apd (λ _ → idp) (H₃₁ c)) ∙ᵈ apd from-to-r (glue (f₃₂ c)) ∙ᵈ ↓-ap-in _ _ (apd (λ _ → idp) (H₃₃ c))))
=⟨ FromToR.glue-β (f₃₂ c) |in-ctx (λ u → ↓-='-out (↓-ap-out= (λ b → from (to (right b)) == right b) f₃∙ (glue c) (F₃∙.glue-β c)
(↓-ap-in _ _ (apd (λ _ → idp) (H₃₁ c)) ∙ᵈ u ∙ᵈ ↓-ap-in _ _ (apd (λ _ → idp) (H₃₃ c))))) ⟩
↓-='-out (↓-ap-out= (λ b → from (to (right b)) == right b) f₃∙ (glue c) (F₃∙.glue-β c)
(↓-ap-in _ _ (apd (λ _ → idp) (H₃₁ c)) ∙ᵈ ↓-='-in (! (from-to-r-g (f₃₂ c))) ∙ᵈ ↓-ap-in _ _ (apd (λ _ → idp) (H₃₃ c))))
=⟨ lemma-a _ f₃∙ (glue c) (F₃∙.glue-β c) _ ⟩
↓-='-out (↓-ap-in _ _ (apd (λ x → idp {a = right (left x)}) (H₃₁ c)) ∙ᵈ ↓-='-in (! (from-to-r-g (f₃₂ c))) ∙ᵈ ↓-ap-in _ _ (apd (λ _ → idp) (H₃₃ c)))
∙□-i/ ap-∘ right f₃∙ (glue c) ∙ ap (ap right) (F₃∙.glue-β c)
/ ! (ap (ap (from ∘ i₄∙)) (F₃∙.glue-β c)) ∙ ∘-ap (from ∘ i₄∙) f₃∙ (glue c) /
=⟨ lemma-b (glue (f₃₂ c)) (apd (λ _ → idp) (H₃₁ c)) (! (from-to-r-g (f₃₂ c))) (apd (λ _ → idp) (H₃₃ c)) |in-ctx (λ u → u ∙□-i/ ap-∘ right f₃∙ (glue c) ∙ ap (ap right) (F₃∙.glue-β c)
/ ! (ap (ap (from ∘ i₄∙)) (F₃∙.glue-β c)) ∙ ∘-ap (from ∘ i₄∙) f₃∙ (glue c) /) ⟩
(↓-='-out (apd (λ x → idp {a = right (left x)}) (H₃₁ c))
∙□h ((! (from-to-r-g (f₃₂ c)))
∙□h (↓-='-out (apd (λ _ → idp) (H₃₃ c)))))
∙□-i/ ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)
/ ! (ap-∙∙`∘`∘ (from ∘ i₄∙) left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)) /
∙□-i/ ap-∘ right f₃∙ (glue c) ∙ ap (ap right) (F₃∙.glue-β c)
/ ! (ap (ap (from ∘ i₄∙)) (F₃∙.glue-β c)) ∙ ∘-ap (from ∘ i₄∙) f₃∙ (glue c) /
=⟨ coh3 |in-ctx (λ u → u
∙□-i/ ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)
/ ! (ap-∙∙`∘`∘ (from ∘ i₄∙) left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)) /
∙□-i/ ap-∘ right f₃∙ (glue c) ∙ ap (ap right) (F₃∙.glue-β c)
/ ! (ap (ap (from ∘ i₄∙)) (F₃∙.glue-β c)) ∙ ∘-ap (from ∘ i₄∙) f₃∙ (glue c) /) ⟩
((! (from-to-r-g (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙□-i/ ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)
/ ! (ap-∙∙`∘`∘ (from ∘ i₄∙) left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)) /
∙□-i/ ap-∘ right f₃∙ (glue c) ∙ ap (ap right) (F₃∙.glue-β c)
/ ! (ap (ap (from ∘ i₄∙)) (F₃∙.glue-β c)) ∙ ∘-ap (from ∘ i₄∙) f₃∙ (glue c) /
=⟨ coh' ⟩
end-lemma1 ∎ where
coh' : ∀ {i} {A : Type i} {x y : A} {a b c d e f g h : x == y} {p : a == b} {q : b == c}
{r : c == d} {s : d == e} {t : e == f} {u : f == g} {v : g == h}
→ (s ∙□-i/ r / t / ∙□-i/ p ∙ q / u ∙ v /) == (a =⟨ p ⟩ b =⟨ q ⟩ c =⟨ r ⟩ d =⟨ s ⟩ e =⟨ t ⟩ f =⟨ u ⟩ g =⟨ v ⟩ h ∎)
coh' {p = idp} {idp} {idp} {idp} {idp} {idp} {idp} = idp
coh2 : ∀ {i j} {A : Type i} {B : Type j} {f : A → B} {x y : A} {p : x == y}
→ ↓-='-out (apd (λ x → idp {a = f x}) p) == idp {a = ap f p}
coh2 {p = idp} = idp
coh3 : (↓-='-out (apd (λ x → idp {a = right (left x)}) (H₃₁ c))
∙□h ((! (from-to-r-g (f₃₂ c)))
∙□h (↓-='-out (apd (λ _ → idp) (H₃₃ c))))) == ((! (from-to-r-g (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
coh3 =
↓-='-out (apd (λ x → idp {a = right (left x)}) (H₃₁ c))
∙□h ((! (from-to-r-g (f₃₂ c)))
∙□h (↓-='-out (apd (λ _ → idp) (H₃₃ c))))
=⟨ coh2 {f = right ∘ left} {p = H₃₁ c} |in-ctx (λ u → u ∙□h ((! (from-to-r-g (f₃₂ c))) ∙□h (↓-='-out (apd (λ _ → idp) (H₃₃ c))))) ⟩
idp {a = ap (right ∘ left) (H₃₁ c)} ∙□h ((! (from-to-r-g (f₃₂ c)))
∙□h (↓-='-out (apd (λ _ → idp) (H₃₃ c))))
=⟨ coh2 {f = right ∘ right} {p = H₃₃ c} |in-ctx (λ u → idp {a = ap (right ∘ left) (H₃₁ c)} ∙□h ((! (from-to-r-g (f₃₂ c))) ∙□h u)) ⟩
idp {a = ap (right ∘ left) (H₃₁ c)} ∙□h ((! (from-to-r-g (f₃₂ c))) ∙□h idp {a = ap (right ∘ right) (H₃₃ c)})
=⟨ coh4 (ap (right ∘ left) (H₃₁ c)) (ap (right ∘ right) (H₃₃ c)) (! (from-to-r-g (f₃₂ c))) ⟩
((! (from-to-r-g (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))) ∎ where
coh4 : ∀ {i} {A : Type i} {x y z t : A} (p : x == y) (q : z == t) {s t : y == z} (r : s == t)
→ (idp {a = p} ∙□h (r ∙□h idp {a = q})) == (r |in-ctx (λ u → p ∙ u ∙ q))
coh4 idp idp idp = idp
end-lemma3 : ap (left ∘ f₁∙) (glue c) == ap (from ∘ i₀∙ ∘ f₁∙) (glue c)
end-lemma3 =
ap (left ∘ f₁∙) (glue c)
=⟨ ap-∘ left f₁∙ (glue c) ⟩
ap left (ap f₁∙ (glue c))
=⟨ ap (ap left) (F₁∙.glue-β c) ⟩
ap left (ap left (H₁₁ c) ∙ glue (f₁₂ c) ∙ ap right (H₁₃ c))
=⟨ ap-∙∙`∘`∘ left left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c) ⟩
ap (left ∘ left) (H₁₁ c) ∙ ap left (glue (f₁₂ c)) ∙ ap (left ∘ right) (H₁₃ c)
=⟨ ! (from-to-l-g (f₁₂ c)) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)) ⟩
ap (left ∘ left) (H₁₁ c) ∙ (ap (from ∘ i₀∙) (glue (f₁₂ c))) ∙ ap (left ∘ right) (H₁₃ c)
=⟨ ! (ap-∙∙`∘`∘ (from ∘ i₀∙) left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c)) ⟩
ap (from ∘ i₀∙) (ap left (H₁₁ c) ∙ glue (f₁₂ c) ∙ ap right (H₁₃ c))
=⟨ ! (ap (ap (from ∘ i₀∙)) (F₁∙.glue-β c)) ⟩
ap (from ∘ i₀∙) (ap f₁∙ (glue c))
=⟨ ∘-ap (from ∘ i₀∙) f₁∙ (glue c) ⟩
ap (from ∘ i₀∙ ∘ f₁∙) (glue c) ∎
lemma3 : ↓-='-out (apd (from-to-l ∘ f₁∙) (glue c)) == end-lemma3
lemma3 =
↓-='-out (apd (from-to-l ∘ f₁∙) (glue c))
=⟨ apd-∘'' from-to-l f₁∙ (glue c) (F₁∙.glue-β c) |in-ctx ↓-='-out ⟩
↓-='-out (↓-ap-out= (λ b → from (to (left b)) == left b) f₁∙ (glue c) (F₁∙.glue-β c) (apd from-to-l (ap left (H₁₁ c) ∙ glue (f₁₂ c) ∙ ap right (H₁₃ c))))
=⟨ lemma-a _ f₁∙ (glue c) (F₁∙.glue-β c) (apd from-to-l (ap left (H₁₁ c) ∙ glue (f₁₂ c) ∙ ap right (H₁₃ c))) ⟩
↓-='-out (apd from-to-l (ap left (H₁₁ c) ∙ glue (f₁₂ c) ∙ ap right (H₁₃ c)))
∙□-i/ ap-∘ left f₁∙ (glue c) ∙ ap (ap left) (F₁∙.glue-β c) / ! (ap (ap (from ∘ to ∘ left)) (F₁∙.glue-β c)) ∙ ∘-ap (from ∘ to ∘ left) f₁∙ (glue c) /
=⟨ apd-∙∙`∘`∘ from-to-l left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c) |in-ctx
(λ u → ↓-='-out u ∙□-i/ ap-∘ left f₁∙ (glue c) ∙ ap (ap left) (F₁∙.glue-β c) / ! (ap (ap (from ∘ to ∘ left)) (F₁∙.glue-β c)) ∙ ∘-ap (from ∘ to ∘ left) f₁∙ (glue c) /) ⟩
↓-='-out (↓-ap-in _ _ (apd (λ _ → idp) (H₁₁ c)) ∙ᵈ apd from-to-l (glue (f₁₂ c)) ∙ᵈ ↓-ap-in _ _ (apd (λ _ → idp) (H₁₃ c)))
∙□-i/ ap-∘ left f₁∙ (glue c) ∙ ap (ap left) (F₁∙.glue-β c) / ! (ap (ap (from ∘ to ∘ left)) (F₁∙.glue-β c)) ∙ ∘-ap (from ∘ to ∘ left) f₁∙ (glue c) /
=⟨ FromToL.glue-β (f₁₂ c) |in-ctx (λ u → ↓-='-out (↓-ap-in _ _ (apd (λ _ → idp) (H₁₁ c)) ∙ᵈ u ∙ᵈ ↓-ap-in _ _ (apd (λ _ → idp) (H₁₃ c)))
∙□-i/ ap-∘ left f₁∙ (glue c) ∙ ap (ap left) (F₁∙.glue-β c) / ! (ap (ap (from ∘ to ∘ left)) (F₁∙.glue-β c)) ∙ ∘-ap (from ∘ to ∘ left) f₁∙ (glue c) /) ⟩
↓-='-out (↓-ap-in _ _ (apd (λ _ → idp) (H₁₁ c)) ∙ᵈ ↓-='-in (! (from-to-l-g (f₁₂ c))) ∙ᵈ ↓-ap-in _ _ (apd (λ _ → idp) (H₁₃ c)))
∙□-i/ ap-∘ left f₁∙ (glue c) ∙ ap (ap left) (F₁∙.glue-β c) / ! (ap (ap (from ∘ to ∘ left)) (F₁∙.glue-β c)) ∙ ∘-ap (from ∘ to ∘ left) f₁∙ (glue c) /
=⟨ lemma-b (glue (f₁₂ c)) (apd (λ _ → idp) (H₁₁ c)) (! (from-to-l-g (f₁₂ c))) (apd (λ _ → idp) (H₁₃ c))
|in-ctx (λ u → u ∙□-i/ ap-∘ left f₁∙ (glue c) ∙ ap (ap left) (F₁∙.glue-β c) / ! (ap (ap (from ∘ to ∘ left)) (F₁∙.glue-β c)) ∙ ∘-ap (from ∘ to ∘ left) f₁∙ (glue c) /) ⟩
(↓-='-out (apd (λ _ → idp) (H₁₁ c)) ∙□h ((! (from-to-l-g (f₁₂ c))) ∙□h ↓-='-out (apd (λ _ → idp) (H₁₃ c))))
∙□-i/ ap-∙∙`∘`∘ left left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c) / ! (ap-∙∙`∘`∘ (from ∘ i₀∙) left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c)) /
∙□-i/ ap-∘ left f₁∙ (glue c) ∙ ap (ap left) (F₁∙.glue-β c) / ! (ap (ap (from ∘ to ∘ left)) (F₁∙.glue-β c)) ∙ ∘-ap (from ∘ to ∘ left) f₁∙ (glue c) /
=⟨ coh2 (left ∘ left) (left ∘ right) (H₁₁ c) (! (from-to-l-g (f₁₂ c))) (H₁₃ c) |in-ctx (λ u →
u ∙□-i/ ap-∙∙`∘`∘ left left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c) / ! (ap-∙∙`∘`∘ (from ∘ i₀∙) left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c)) /
∙□-i/ ap-∘ left f₁∙ (glue c) ∙ ap (ap left) (F₁∙.glue-β c) / ! (ap (ap (from ∘ to ∘ left)) (F₁∙.glue-β c)) ∙ ∘-ap (from ∘ to ∘ left) f₁∙ (glue c) /) ⟩
((! (from-to-l-g (f₁₂ c))) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)))
∙□-i/ ap-∙∙`∘`∘ left left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c) / ! (ap-∙∙`∘`∘ (from ∘ i₀∙) left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c)) /
∙□-i/ ap-∘ left f₁∙ (glue c) ∙ ap (ap left) (F₁∙.glue-β c) / ! (ap (ap (from ∘ to ∘ left)) (F₁∙.glue-β c)) ∙ ∘-ap (from ∘ to ∘ left) f₁∙ (glue c) /
=⟨ coh3 ⟩
end-lemma3 ∎ where
coh2 : ∀ {i i' j} {A : Type i} {A' : Type i'} {B : Type j} (f : A → B) (f' : A' → B)
{a b : A} {a' b' : A'} (p : a == b) {s s' : f b == f' a'} (q : s == s') (r : a' == b')
→ (↓-='-out (apd (λ x → idp) p) ∙□h (q ∙□h (↓-='-out (apd (λ _ → idp) r))))
== (q |in-ctx (λ u → ap f p ∙ u ∙ ap f' r))
coh2 f f' idp idp idp = idp
coh3 : ∀ {i} {A : Type i} {x y : A} {a b c d e f g h : x == y} {p : a == b} {q : b == c}
{r : c == d} {s : d == e} {t : e == f} {u : f == g} {v : g == h}
→ (s ∙□-i/ r / t / ∙□-i/ p ∙ q / u ∙ v /) == (a =⟨ p ⟩ b =⟨ q ⟩ c =⟨ r ⟩ d =⟨ s ⟩ e =⟨ t ⟩ f =⟨ u ⟩ g =⟨ v ⟩ h ∎)
coh3 {p = idp} {idp} {idp} {idp} {idp} {idp} {idp} = idp
{- Lemma 2 -}
lemma2-7 =
↓-='-out (ap↓ (ap from) (apd (glue {d = h-v-span}) (glue c)))
=⟨ from-glue-glue-β c |in-ctx ↓-='-out ⟩
↓-='-out (↓-='-in (E∙₂Red.coh! c (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /)
∙□-i/ E∙₂Red.lhs-o c / E∙₂Red.rhs-o c /)
◃/ From.glue-β (left (f₁₂ c)) / ! (From.glue-β (right (f₃₂ c))) /)
=⟨ thing _ (From.glue-β (left (f₁₂ c))) _ ⟩
E∙₂Red.coh! c (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /)
∙□-i/ E∙₂Red.lhs-o c / E∙₂Red.rhs-o c /
∙□-o/ From.glue-β (left (f₁₂ c)) / ! (From.glue-β (right (f₃₂ c))) / ∎
lemma2-6 =
ap□ from (↓-='-out (apd (glue {d = h-v-span}) (glue c)))
=⟨ ap□-↓-='-out-β _ _ from (apd (glue {d = h-v-span}) (glue c)) ⟩
↓-='-out (ap↓ (ap from) (apd (glue {d = h-v-span}) (glue c)))
∙□-i/ ∘-ap from (right ∘ f∙₃) (glue c) / ap-∘ from (left ∘ f∙₁) (glue c) /
=⟨ lemma2-7 |in-ctx (λ u → (u ∙□-i/ ∘-ap from (right ∘ f∙₃) (glue c) / ap-∘ from (left ∘ f∙₁) (glue c) /)) ⟩
E∙₂Red.coh! c (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /)
∙□-i/ E∙₂Red.lhs-o c / E∙₂Red.rhs-o c /
∙□-o/ From.glue-β (left (f₁₂ c)) / ! (From.glue-β (right (f₃₂ c))) /
∙□-i/ ∘-ap from (right ∘ f∙₃) (glue c) / ap-∘ from (left ∘ f∙₁) (glue c) / ∎
lemma2-5 =
ap□ from (↓-='-out (apd (glue {d = h-v-span}) (glue c))
∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /)
=⟨ ap□-∙□-i/ from _ (E₂∙Red.lhs-i c) _ ⟩
ap□ from (↓-='-out (apd (glue {d = h-v-span}) (glue c)))
∙□-i/ ap (ap from) (E₂∙Red.lhs-i c) / ap (ap from) (E₂∙Red.rhs-i c) /
=⟨ lemma2-6 |in-ctx (λ u → u ∙□-i/ ap (ap from) (E₂∙Red.lhs-i c) / ap (ap from) (E₂∙Red.rhs-i c) /) ⟩
E∙₂Red.coh! c (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /)
∙□-i/ E∙₂Red.lhs-o c / E∙₂Red.rhs-o c /
∙□-o/ From.glue-β (left (f₁₂ c)) / ! (From.glue-β (right (f₃₂ c))) /
∙□-i/ ∘-ap from (right ∘ f∙₃) (glue c) / ap-∘ from (left ∘ f∙₁) (glue c) /
∙□-i/ ap (ap from) (E₂∙Red.lhs-i c) / ap (ap from) (E₂∙Red.rhs-i c) / ∎
lemma2'-1 =
E₂∙Red.ap-ap-coh-lhs-i c from ∙ ap (ap from) (E₂∙Red.lhs-i c) ∙ ∘-ap from (right ∘ f∙₃) (glue c) ∙ E∙₂Red.lhs-o c
=⟨ eq1 {f = ap from}
{q = ! (ap-∙∙!`∘`∘ right left right (H₁₃ c) (glue (f₂₃ c)) (H₃₃ c))}
{ ! (F∙₃.glue-β c) |in-ctx (ap right)}
{∘-ap right f∙₃ (glue c)}
{p = E₂∙Red.ap-ap-coh-lhs-i c from}
{∘-ap from (right ∘ f∙₃) (glue c)}
{ap-∘ i∙₄ f∙₃ (glue c)}
{F∙₃.glue-β c |in-ctx (ap i∙₄)}
{ap-∙∙!`∘`∘ i∙₄ left right (H₁₃ c) (glue (f₂₃ c)) (H₃₃ c)}
{I∙₄.glue-β (f₂₃ c) |in-ctx (λ u → ap (left ∘ right) (H₁₃ c) ∙ u ∙ ! (ap (right ∘ right) (H₃₃ c)))}
⟩
(! (ap-∙∙!'`∘`∘ from (right ∘ left) (right ∘ right) (H₁₃ c) (ap right (glue (f₂₃ c))) (H₃₃ c))
∙ ap (ap from) (! (ap-∙∙!`∘`∘ right left right (H₁₃ c) (glue (f₂₃ c)) (H₃₃ c)))
∙ (ap (ap from) (! (F∙₃.glue-β c) |in-ctx (ap right))
∙ (ap (ap from) (∘-ap right f∙₃ (glue c))
∙ ∘-ap from (right ∘ f∙₃) (glue c)
∙ ap-∘ i∙₄ f∙₃ (glue c))
∙ (F∙₃.glue-β c |in-ctx (ap i∙₄)))
∙ ap-∙∙!`∘`∘ i∙₄ left right (H₁₃ c) (glue (f₂₃ c)) (H₃₃ c))
∙ (I∙₄.glue-β (f₂₃ c) |in-ctx (λ u → ap (left ∘ right) (H₁₃ c) ∙ u ∙ ! (ap (right ∘ right) (H₃₃ c))))
=⟨ ap-∘^3-coh from right f∙₃ (glue c) |in-ctx (λ u →
(! (ap-∙∙!'`∘`∘ from (right ∘ left) (right ∘ right) (H₁₃ c) (ap right (glue (f₂₃ c))) (H₃₃ c))
∙ ap (ap from) (! (ap-∙∙!`∘`∘ right left right (H₁₃ c) (glue (f₂₃ c)) (H₃₃ c)))
∙ (ap (ap from) (! (F∙₃.glue-β c) |in-ctx (ap right))
∙ u
∙ (F∙₃.glue-β c |in-ctx (ap i∙₄)))
∙ ap-∙∙!`∘`∘ i∙₄ left right (H₁₃ c) (glue (f₂₃ c)) (H₃₃ c))
∙ (I∙₄.glue-β (f₂₃ c) |in-ctx (λ u → ap (left ∘ right) (H₁₃ c) ∙ u ∙ ! (ap (right ∘ right) (H₃₃ c)))))
⟩
(! (ap-∙∙!'`∘`∘ from (right ∘ left) (right ∘ right) (H₁₃ c) (ap right (glue (f₂₃ c))) (H₃₃ c))
∙ ap (ap from) (! (ap-∙∙!`∘`∘ right left right (H₁₃ c) (glue (f₂₃ c)) (H₃₃ c)))
∙ (ap (ap from) (! (F∙₃.glue-β c) |in-ctx (ap right))
∙ ∘-ap from right (ap f∙₃ (glue c))
∙ (F∙₃.glue-β c |in-ctx (ap i∙₄)))
∙ ap-∙∙!`∘`∘ i∙₄ left right (H₁₃ c) (glue (f₂₃ c)) (H₃₃ c))
∙ (I∙₄.glue-β (f₂₃ c) |in-ctx (λ u → ap (left ∘ right) (H₁₃ c) ∙ u ∙ ! (ap (right ∘ right) (H₃₃ c))))
=⟨ ap-∘-ap-coh from right (F∙₃.glue-β c) |in-ctx (λ u →
(! (ap-∙∙!'`∘`∘ from (right ∘ left) (right ∘ right) (H₁₃ c) (ap right (glue (f₂₃ c))) (H₃₃ c))
∙ ap (ap from) (! (ap-∙∙!`∘`∘ right left right (H₁₃ c) (glue (f₂₃ c)) (H₃₃ c)))
∙ u
∙ ap-∙∙!`∘`∘ i∙₄ left right (H₁₃ c) (glue (f₂₃ c)) (H₃₃ c))
∙ (I∙₄.glue-β (f₂₃ c) |in-ctx (λ u → ap (left ∘ right) (H₁₃ c) ∙ u ∙ ! (ap (right ∘ right) (H₃₃ c))))) ⟩
(! (ap-∙∙!'`∘`∘ from (right ∘ left) (right ∘ right) (H₁₃ c) (ap right (glue (f₂₃ c))) (H₃₃ c))
∙ ap (ap from) (! (ap-∙∙!`∘`∘ right left right (H₁₃ c) (glue (f₂₃ c)) (H₃₃ c)))
∙ ∘-ap from right (ap left (H₁₃ c) ∙ glue (f₂₃ c) ∙ ap right (! (H₃₃ c)))
∙ ap-∙∙!`∘`∘ i∙₄ left right (H₁₃ c) (glue (f₂₃ c)) (H₃₃ c))
∙ (I∙₄.glue-β (f₂₃ c) |in-ctx (λ u → ap (left ∘ right) (H₁₃ c) ∙ u ∙ ! (ap (right ∘ right) (H₃₃ c))))
=⟨ ap-∘-ap-∙∙!`∘`∘-coh from right left right (H₁₃ c) (glue (f₂₃ c)) (H₃₃ c) |in-ctx (λ u →
u ∙ (I∙₄.glue-β (f₂₃ c) |in-ctx (λ u → ap (left ∘ right) (H₁₃ c) ∙ u ∙ ! (ap (right ∘ right) (H₃₃ c))))) ⟩
(∘-ap from right (glue (f₂₃ c)) |in-ctx (λ u → ap (left ∘ right) (H₁₃ c) ∙ u ∙ ! (ap (right ∘ right) (H₃₃ c))))
∙ (I∙₄.glue-β (f₂₃ c) |in-ctx (λ u → ap (left ∘ right) (H₁₃ c) ∙ u ∙ ! (ap (right ∘ right) (H₃₃ c))))
=⟨ ∙-|in-ctx (λ u → ap (left ∘ right) (H₁₃ c) ∙ u ∙ ! (ap (right ∘ right) (H₃₃ c))) (∘-ap from right (glue (f₂₃ c))) (I∙₄.glue-β (f₂₃ c)) ⟩
((∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) |in-ctx (λ u → ap (left ∘ right) (H₁₃ c) ∙ u ∙ ! (ap (right ∘ right) (H₃₃ c)))) ∎
where
eq1 : ∀ {i j} {A : Type i} {B : Type j} {f : A → B} {a b c d : A}
{q : a == b} {r : b == c} {s : c == d}
{g h k l m n : B} {p : g == f a} {t : f d == h} {u : h == k}
{v : k == l} {w : l == m} {x : m == n}
→ p ∙ ap f (_ =⟨ q ⟩ _ =⟨ r ⟩ _ =⟨ s ⟩ _ ∎) ∙ t
∙ (_ =⟨ u ⟩ _ =⟨ v ⟩ _ =⟨ w ⟩ _ =⟨ x ⟩ _ ∎)
== (p ∙ ap f q ∙ (ap f r ∙ (ap f s ∙ t ∙ u) ∙ v) ∙ w) ∙ x
eq1 {q = idp} {idp} {idp} {p = p} {idp} {idp} {idp} {idp} {idp} = ! (∙-unit-r (p ∙ idp))
lemma2'-2 :
E∙₂Red.rhs-o c
∙ ap-∘ from (left ∘ f∙₁) (glue c)
∙ ap (ap from) (E₂∙Red.rhs-i c)
∙ E₂∙Red.ap-ap-coh-rhs-i c from
== (! (∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c)) |in-ctx (λ u → ! (ap (left ∘ left) (H₁₁ c)) ∙ u ∙ ap (right ∘ left) (H₃₁ c)))
lemma2'-2 =
E∙₂Red.rhs-o c ∙ ap-∘ from (left ∘ f∙₁) (glue c) ∙ ap (ap from) (E₂∙Red.rhs-i c) ∙ E₂∙Red.ap-ap-coh-rhs-i c from
=⟨ eq1 {f = ap from}
{p = ! (I∙₀.glue-β (f₂₁ c)) |in-ctx (λ u → (! (ap (left ∘ left) (H₁₁ c)) ∙ u ∙ ap (right ∘ left) (H₃₁ c)))}
{ ! (ap-!∙∙`∘`∘ i∙₀ left right (H₁₁ c) (glue (f₂₁ c)) (H₃₁ c))}
{ ! (F∙₁.glue-β c) |in-ctx (λ u → ap i∙₀ u)}
{∘-ap i∙₀ f∙₁ (glue c)}
{ap-∘ from (left ∘ f∙₁) (glue c)}
{ap-∘ left f∙₁ (glue c)}
{F∙₁.glue-β c |in-ctx (ap left)}
{ap-!∙∙`∘`∘ left left right (H₁₁ c) (glue (f₂₁ c)) (H₃₁ c)}
{ap-!'∙∙`∘`∘ from (left ∘ left) (left ∘ right) (H₁₁ c) (ap left (glue (f₂₁ c))) (H₃₁ c)}
⟩
(! (I∙₀.glue-β (f₂₁ c)) |in-ctx (λ u → (! (ap (left ∘ left) (H₁₁ c)) ∙ u ∙ ap (right ∘ left) (H₃₁ c))))
∙ ((! (ap-!∙∙`∘`∘ i∙₀ left right (H₁₁ c) (glue (f₂₁ c)) (H₃₁ c)))
∙ ((! (F∙₁.glue-β c) |in-ctx (λ u → ap i∙₀ u))
∙ (∘-ap i∙₀ f∙₁ (glue c)
∙ ap-∘ from (left ∘ f∙₁) (glue c)
∙ ap (ap from) (ap-∘ left f∙₁ (glue c)))
∙ ap (ap from) (F∙₁.glue-β c |in-ctx (ap left)))
∙ ap (ap from) (ap-!∙∙`∘`∘ left left right (H₁₁ c) (glue (f₂₁ c)) (H₃₁ c))
∙ ap-!'∙∙`∘`∘ from (left ∘ left) (left ∘ right) (H₁₁ c) (ap left (glue (f₂₁ c))) (H₃₁ c))
=⟨ ap-∘^3-coh' from left f∙₁ (glue c)
|in-ctx (λ u → (! (I∙₀.glue-β (f₂₁ c)) |in-ctx (λ u → (! (ap (left ∘ left) (H₁₁ c)) ∙ u ∙ ap (right ∘ left) (H₃₁ c))))
∙ ((! (ap-!∙∙`∘`∘ i∙₀ left right (H₁₁ c) (glue (f₂₁ c)) (H₃₁ c)))
∙ ((! (F∙₁.glue-β c) |in-ctx (λ u → ap i∙₀ u))
∙ u
∙ ap (ap from) (F∙₁.glue-β c |in-ctx (ap left)))
∙ ap (ap from) (ap-!∙∙`∘`∘ left left right (H₁₁ c) (glue (f₂₁ c)) (H₃₁ c))
∙ ap-!'∙∙`∘`∘ from (left ∘ left) (left ∘ right) (H₁₁ c) (ap left (glue (f₂₁ c))) (H₃₁ c))) ⟩
(! (I∙₀.glue-β (f₂₁ c)) |in-ctx (λ u → (! (ap (left ∘ left) (H₁₁ c)) ∙ u ∙ ap (right ∘ left) (H₃₁ c))))
∙ ((! (ap-!∙∙`∘`∘ i∙₀ left right (H₁₁ c) (glue (f₂₁ c)) (H₃₁ c)))
∙ ((! (F∙₁.glue-β c) |in-ctx (ap i∙₀))
∙ ap-∘ from left (ap f∙₁ (glue c))
∙ ap (ap from) (F∙₁.glue-β c |in-ctx (ap left)))
∙ ap (ap from) (ap-!∙∙`∘`∘ left left right (H₁₁ c) (glue (f₂₁ c)) (H₃₁ c))
∙ ap-!'∙∙`∘`∘ from (left ∘ left) (left ∘ right) (H₁₁ c) (ap left (glue (f₂₁ c))) (H₃₁ c))
=⟨ ap-∘-ap-coh' from left (F∙₁.glue-β c)
|in-ctx (λ u → (! (I∙₀.glue-β (f₂₁ c)) |in-ctx (λ u → (! (ap (left ∘ left) (H₁₁ c)) ∙ u ∙ ap (right ∘ left) (H₃₁ c))))
∙ ((! (ap-!∙∙`∘`∘ i∙₀ left right (H₁₁ c) (glue (f₂₁ c)) (H₃₁ c)))
∙ u
∙ ap (ap from) (ap-!∙∙`∘`∘ left left right (H₁₁ c) (glue (f₂₁ c)) (H₃₁ c))
∙ ap-!'∙∙`∘`∘ from (left ∘ left) (left ∘ right) (H₁₁ c) (ap left (glue (f₂₁ c))) (H₃₁ c))) ⟩
(! (I∙₀.glue-β (f₂₁ c)) |in-ctx (λ u → (! (ap (left ∘ left) (H₁₁ c)) ∙ u ∙ ap (right ∘ left) (H₃₁ c))))
∙ ((! (ap-!∙∙`∘`∘ i∙₀ left right (H₁₁ c) (glue (f₂₁ c)) (H₃₁ c)))
∙ ap-∘ from left (ap left (! (H₁₁ c)) ∙ glue (f₂₁ c) ∙ ap right (H₃₁ c))
∙ ap (ap from) (ap-!∙∙`∘`∘ left left right (H₁₁ c) (glue (f₂₁ c)) (H₃₁ c))
∙ ap-!'∙∙`∘`∘ from (left ∘ left) (left ∘ right) (H₁₁ c) (ap left (glue (f₂₁ c))) (H₃₁ c))
=⟨ ap-∘-ap-!∙∙`∘`∘-coh from left left right (H₁₁ c) (glue (f₂₁ c)) (H₃₁ c)
|in-ctx (λ u → (! (I∙₀.glue-β (f₂₁ c)) |in-ctx (λ u → (! (ap (left ∘ left) (H₁₁ c)) ∙ u ∙ ap (right ∘ left) (H₃₁ c)))) ∙ u) ⟩
(! (I∙₀.glue-β (f₂₁ c)) |in-ctx (λ u → (! (ap (left ∘ left) (H₁₁ c)) ∙ u ∙ ap (right ∘ left) (H₃₁ c))))
∙ (ap-∘ from left (glue (f₂₁ c)) |in-ctx (λ u → ! (ap (left ∘ left) (H₁₁ c)) ∙ u ∙ ap (right ∘ left) (H₃₁ c)))
=⟨ coh2 (glue (f₂₁ c)) (I∙₀.glue-β (f₂₁ c)) ⟩
(! (∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c)) |in-ctx (λ u → ! (ap (left ∘ left) (H₁₁ c)) ∙ u ∙ ap (right ∘ left) (H₃₁ c))) ∎
where
eq1 : ∀ {i j} {A : Type i} {B : Type j} {f : A → B} {a b c d e : B} {k l m n : A} {o : B}
{p : a == b} {q : b == c} {r : c == d} {s : d == e} {t : e == f k} {u : k == l}
{v : l == m} {w : m == n} {x : f n == o}
→ (_ =⟨ p ⟩ _ =⟨ q ⟩ _ =⟨ r ⟩ _ =⟨ s ⟩ _ ∎) ∙ t ∙ ap f (_ =⟨ u ⟩ _ =⟨ v ⟩ _ =⟨ w ⟩ _ ∎) ∙ x
== (p ∙ (q ∙ ((r ∙ (s ∙ t ∙ ap f u) ∙ ap f v) ∙ ap f w ∙ x)))
eq1 {p = idp} {idp} {idp} {idp} {t = t} {idp} {idp} {idp} {idp} = ! (∙-unit-r (t ∙ idp)) ∙ ! (∙-unit-r ((t ∙ idp) ∙ idp))
coh2 : ∀ {i j k l} {A : Type i} {B : Type j} {C : Type k} {D : Type l}
{a a' : A} (q : a == a') {h : A → B} {g : B → C} {f : (g (h a) == g (h a')) → D} {r : g (h a) == g (h a')} (p : ap (g ∘ h) q == r)
→ ((! p) |in-ctx f) ∙ (ap-∘ g h q |in-ctx f) == (! (∘-ap g h q ∙ p) |in-ctx f)
coh2 idp p = ∙-unit-r _
lemma2-4' =
ap□ from (↓-='-out (apd (glue {d = h-v-span}) (glue c))
∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /)
∙□-i/ E₂∙Red.ap-ap-coh-lhs-i c from / E₂∙Red.ap-ap-coh-rhs-i c from /
=⟨ lemma2-5 |in-ctx (λ u → u ∙□-i/ E₂∙Red.ap-ap-coh-lhs-i c from / E₂∙Red.ap-ap-coh-rhs-i c from /) ⟩
E∙₂Red.coh! c (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /)
∙□-i/ E∙₂Red.lhs-o c / E∙₂Red.rhs-o c /
∙□-o/ From.glue-β (left (f₁₂ c)) / ! (From.glue-β (right (f₃₂ c))) /
∙□-i/ ∘-ap from (right ∘ f∙₃) (glue c) / ap-∘ from (left ∘ f∙₁) (glue c) /
∙□-i/ ap (ap from) (E₂∙Red.lhs-i c) / ap (ap from) (E₂∙Red.rhs-i c) /
∙□-i/ E₂∙Red.ap-ap-coh-lhs-i c from / E₂∙Red.ap-ap-coh-rhs-i c from /
=⟨ assoc (E∙₂Red.coh! c (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /))
(E∙₂Red.lhs-o c) (∘-ap from (right ∘ f∙₃) (glue c)) (ap (ap from) (E₂∙Red.lhs-i c))
(E₂∙Red.ap-ap-coh-lhs-i c from) _ _ _ _ (From.glue-β (left (f₁₂ c))) _ ⟩
E∙₂Red.coh! c (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /)
∙□-o/ From.glue-β (left (f₁₂ c)) / ! (From.glue-β (right (f₃₂ c))) /
∙□-i/ E₂∙Red.ap-ap-coh-lhs-i c from ∙ ap (ap from) (E₂∙Red.lhs-i c) ∙ ∘-ap from (right ∘ f∙₃) (glue c) ∙ E∙₂Red.lhs-o c
/ E∙₂Red.rhs-o c ∙ ap-∘ from (left ∘ f∙₁) (glue c) ∙ ap (ap from) (E₂∙Red.rhs-i c) ∙ E₂∙Red.ap-ap-coh-rhs-i c from /
=⟨ ∙□-i/-rewrite (E∙₂Red.coh! c (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /)
∙□-o/ From.glue-β (left (f₁₂ c)) / ! (From.glue-β (right (f₃₂ c))) /) lemma2'-1 lemma2'-2 ⟩ -- rewrite
E∙₂Red.coh! c (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /)
∙□-o/ From.glue-β (left (f₁₂ c)) / ! (From.glue-β (right (f₃₂ c))) /
∙□-i/ (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) |in-ctx (λ u → ap (left ∘ right) (H₁₃ c) ∙ u ∙ ! (ap (right ∘ right) (H₃₃ c)))
/ (! (∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c))) |in-ctx (λ u → ! (ap (left ∘ left) (H₁₁ c)) ∙ u ∙ ap (right ∘ left) (H₃₁ c)) / ∎ where
assoc : ∀ {i} {A : Type i} {a b b' c : A} {u u' : a == b} {v v1 v2 v3 v4 : b == c}
{w w1 w2 w3 w4 : a == b'} {x x' : b' == c} (α : (u , v =□ w , x))
(p1 : v1 == v) (p2 : v2 == v1) (p3 : v3 == v2) (p4 : v4 == v3)
(q1 : w == w1) (q2 : w1 == w2) (q3 : w2 == w3) (q4 : w3 == w4)
(r : u' == u) (s : x == x')
→ α ∙□-i/ p1 / q1 / ∙□-o/ r / s / ∙□-i/ p2 / q2 / ∙□-i/ p3 / q3 / ∙□-i/ p4 / q4 /
== α ∙□-o/ r / s / ∙□-i/ p4 ∙ p3 ∙ p2 ∙ p1 / q1 ∙ q2 ∙ q3 ∙ q4 /
assoc α idp idp idp idp idp idp idp idp idp idp = idp
lemma2-4'' =
E₂∙Red.ap-ap-coh c from (ap□ from (↓-='-out (apd (glue {d = h-v-span}) (glue c))
∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /)
∙□-i/ E₂∙Red.ap-ap-coh-lhs-i c from / E₂∙Red.ap-ap-coh-rhs-i c from /)
=⟨ lemma2-4' |in-ctx (E₂∙Red.ap-ap-coh c from) ⟩
E₂∙Red.ap-ap-coh c from (E∙₂Red.coh! c (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /)
∙□-o/ From.glue-β (left (f₁₂ c)) / ! (From.glue-β (right (f₃₂ c))) /
∙□-i/ (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) |in-ctx (λ u → ap (left ∘ right) (H₁₃ c) ∙ u ∙ ! (ap (right ∘ right) (H₃₃ c)))
/ (! (∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c))) |in-ctx (λ u → ! (ap (left ∘ left) (H₁₁ c)) ∙ u ∙ ap (right ∘ left) (H₃₁ c))/)
=⟨ lemma (∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c)) (From.glue-β (right (f₃₂ c))) (From.glue-β (left (f₁₂ c))) (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c))
(↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /) ⟩
↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /
∙□-o/ ∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c) / ! (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) /
∙□-i/ (From.glue-β (right (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))
/ (! (From.glue-β (left (f₁₂ c)))) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)) / ∎
lemma2-4 =
ap□ from (E₂∙Red.coh c (↓-='-out (apd (glue {d = h-v-span}) (glue c))
∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /))
=⟨ E₂∙Red.ap-ap-coh-β c from (↓-='-out (apd (glue {d = h-v-span}) (glue c)) ∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /) ⟩
E₂∙Red.ap-ap-coh c from (ap□ from (↓-='-out (apd (glue {d = h-v-span}) (glue c))
∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /)
∙□-i/ E₂∙Red.ap-ap-coh-lhs-i c from / E₂∙Red.ap-ap-coh-rhs-i c from /)
∙□-i/ E₂∙Red.ap-ap-coh-lhs-o c from / E₂∙Red.ap-ap-coh-rhs-o c from /
=⟨ lemma2-4'' |in-ctx (λ u → u ∙□-i/ E₂∙Red.ap-ap-coh-lhs-o c from / E₂∙Red.ap-ap-coh-rhs-o c from /) ⟩
↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /
∙□-o/ ∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c) / ! (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) /
∙□-i/ (From.glue-β (right (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))
/ (! (From.glue-β (left (f₁₂ c)))) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)) /
∙□-i/ E₂∙Red.ap-ap-coh-lhs-o c from / E₂∙Red.ap-ap-coh-rhs-o c from / ∎
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module _ where
open import Common.Prelude
open import Common.Equality
record Functor (F : Set → Set) : Set₁ where
field
fmap : ∀ {A B} → (A → B) → F A → F B
open Functor {{...}} public
record Applicative (F : Set → Set) : Set₁ where
field
pure : ∀ {A} → A → F A
_<*>_ : ∀ {A B} → F (A → B) → F A → F B
instance Fun : Functor F
defaultApplicativeFunctor : Functor F
fmap {{defaultApplicativeFunctor}} f x = pure f <*> x
open Applicative {{...}} public hiding (Fun)
-- Concrete instances --
data Vec (A : Set) : Nat → Set where
[] : Vec A zero
_∷_ : ∀ {n} → A → Vec A n → Vec A (suc n)
vmap : ∀ {A B n} → (A → B) → Vec A n → Vec B n
vmap f [] = []
vmap f (x ∷ xs) = f x ∷ vmap f xs
it : ∀ {a} {A : Set a} {{_ : A}} → A
it {{x}} = x
pureV : ∀ {n A} → A → Vec A n
pureV {zero} _ = []
pureV {suc n} x = x ∷ pureV x
instance
FunctorVec : ∀ {n} → Functor (λ A → Vec A n)
fmap {{FunctorVec}} = vmap
ApplicativeVec : ∀ {n} → Applicative (λ A → Vec A n)
pure {{ApplicativeVec}} x = pureV x
_<*>_ {{ApplicativeVec}} [] [] = []
_<*>_ {{ApplicativeVec}} (f ∷ fs) (x ∷ xs) = f x ∷ (fs <*> xs)
Applicative.Fun ApplicativeVec = FunctorVec
-- In this case there are two candidates for Functor Vec:
-- FunctorVec and Applicative.Fun ApplicativeVec
-- but since they are equal everything works out.
testVec : ∀ {n} → Vec Nat n → Vec Nat n → Vec Nat n
testVec xs ys = fmap _+_ xs <*> ys
what : ∀ {n} → FunctorVec {n} ≡ it
what = refl
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module _ where
open import Agda.Builtin.Nat
open import Agda.Builtin.Equality
postulate
String : Set
{-# BUILTIN STRING String #-}
primitive
@0 ⦃ primShowNat ⦄ : Nat → String
-- Wrong modality for primitive primShowNat
-- Got: instance, erased
-- Expected: visible, unrestricted
-- when checking that the type of the primitive function primShowNat
-- is Nat → String
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{-# OPTIONS --without-K #-}
module PathStructure.Sigma {a b} {A : Set a} {B : A → Set b} where
open import Equivalence
open import PathOperations
open import Transport
open import Types
ap₂-dep : ∀ {a b c} {A : Set a} {B : A → Set b} {C : Set c}
{x x′ : A} {y : B x} {y′ : B x′}
(f : (a : A) (b : B a) → C) (p : x ≡ x′) (q : tr B p y ≡ y′) →
f x y ≡ f x′ y′
ap₂-dep {B = B} f p q = J
(λ x x′ p → (y : B x) (y′ : B x′) (q : tr B p y ≡ y′) → f x y ≡ f x′ y′)
(λ x _ _ q → ap (f x) q)
_ _ p _ _ q
ap₂-dep-eq : ∀ {a b c} {A : Set a} {B : A → Set b} {C : Set c}
(f : (x : A) → B x → C)
{x x′ : A} (p p′ : x ≡ x′) (pp : p ≡ p′)
{y : B x} {y′ : B x′} (q : tr B p y ≡ y′) (q′ : tr B p′ y ≡ y′)
(qq : tr (λ z → tr B z y ≡ y′) pp q ≡ q′) →
ap₂-dep f p q ≡ ap₂-dep f p′ q′
ap₂-dep-eq {B = B} f {x = x} {x′ = x′} p p′ pp q q′ qq = J
(λ p p′ pp → (y : B x) (y′ : B x′)
(q : tr B p y ≡ y′) (q′ : tr B p′ y ≡ y′)
(qq : tr (λ z → tr B z y ≡ y′) pp q ≡ q′) →
ap₂-dep f p q ≡ ap₂-dep f p′ q′)
(λ p _ _ q q′ qq → J
(λ q q′ qq → ap₂-dep f p q ≡ ap₂-dep f p q′)
(λ _ → refl)
_ _ qq)
_ _ pp _ _ _ _ qq
split-path : {x y : Σ A B} →
x ≡ y → Σ (π₁ x ≡ π₁ y) (λ p → tr B p (π₂ x) ≡ π₂ y)
split-path {x = x} p = ap π₁ p , tr-∘ π₁ p (π₂ x) ⁻¹ · apd π₂ p
merge-path : {x y : Σ A B} →
Σ (π₁ x ≡ π₁ y) (λ p → tr B p (π₂ x) ≡ π₂ y) → x ≡ y
merge-path pq = ap₂-dep _,_ (π₁ pq) (π₂ pq)
split-merge-eq : {x y : Σ A B} →
(x ≡ y) ≃ Σ (π₁ x ≡ π₁ y) (λ p → tr B p (π₂ x) ≡ π₂ y)
split-merge-eq
= split-path
, (merge-path , λ pq → J
(λ x x′ p → (y : B x) (y′ : B x′) (q : tr B p y ≡ y′) →
split-path (merge-path (p , q)) ≡ p , q)
(λ x y y′ q → J {A = B x}
(λ _ _ q → split-path (merge-path (refl , q)) ≡ refl , q)
(λ _ → refl) _ _ q)
_ _ (π₁ pq) _ _ (π₂ pq))
, (merge-path , J
(λ _ _ p → merge-path (split-path p) ≡ p)
(λ _ → refl) _ _)
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Primitive IO: simple bindings to Haskell types and functions
------------------------------------------------------------------------
{-# OPTIONS --without-K #-}
module IO.Primitive where
open import Codata.Musical.Costring
open import Data.Char.Base
open import Data.String.Base
open import Foreign.Haskell
------------------------------------------------------------------------
-- The IO monad
open import Agda.Builtin.IO public using (IO)
infixl 1 _>>=_
postulate
return : ∀ {a} {A : Set a} → A → IO A
_>>=_ : ∀ {a b} {A : Set a} {B : Set b} → IO A → (A → IO B) → IO B
{-# COMPILE GHC return = \_ _ -> return #-}
{-# COMPILE GHC _>>=_ = \_ _ _ _ -> (>>=) #-}
{-# COMPILE UHC return = \_ _ x -> UHC.Agda.Builtins.primReturn x #-}
{-# COMPILE UHC _>>=_ = \_ _ _ _ x y -> UHC.Agda.Builtins.primBind x y #-}
------------------------------------------------------------------------
-- Simple lazy IO
-- Note that the functions below produce commands which, when
-- executed, may raise exceptions.
-- Note also that the semantics of these functions depends on the
-- version of the Haskell base library. If the version is 4.2.0.0 (or
-- later?), then the functions use the character encoding specified by
-- the locale. For older versions of the library (going back to at
-- least version 3) the functions use ISO-8859-1.
{-# FOREIGN GHC import qualified Data.Text #-}
{-# FOREIGN GHC import qualified Data.Text.IO #-}
{-# FOREIGN GHC import qualified System.IO #-}
{-# FOREIGN GHC import qualified Control.Exception #-}
postulate
getContents : IO Costring
readFile : String → IO Costring
writeFile : String → Costring → IO Unit
appendFile : String → Costring → IO Unit
putStr : Costring → IO Unit
putStrLn : Costring → IO Unit
-- Reads a finite file. Raises an exception if the file path refers
-- to a non-physical file (like "/dev/zero").
readFiniteFile : String → IO String
{-# FOREIGN GHC
readFiniteFile :: Data.Text.Text -> IO Data.Text.Text
readFiniteFile f = do
h <- System.IO.openFile (Data.Text.unpack f) System.IO.ReadMode
Control.Exception.bracketOnError (return ()) (\_ -> System.IO.hClose h)
(\_ -> System.IO.hFileSize h)
Data.Text.IO.hGetContents h
fromColist :: MAlonzo.Code.Codata.Musical.Colist.AgdaColist a -> [a]
fromColist MAlonzo.Code.Codata.Musical.Colist.Nil = []
fromColist (MAlonzo.Code.Codata.Musical.Colist.Cons x xs) =
x : fromColist (MAlonzo.RTE.flat xs)
toColist :: [a] -> MAlonzo.Code.Codata.Musical.Colist.AgdaColist a
toColist [] = MAlonzo.Code.Codata.Musical.Colist.Nil
toColist (x : xs) =
MAlonzo.Code.Codata.Musical.Colist.Cons x (MAlonzo.RTE.Sharp (toColist xs))
#-}
{-# COMPILE GHC getContents = fmap toColist getContents #-}
{-# COMPILE GHC readFile = fmap toColist . readFile . Data.Text.unpack #-}
{-# COMPILE GHC writeFile = \x -> writeFile (Data.Text.unpack x) . fromColist #-}
{-# COMPILE GHC appendFile = \x -> appendFile (Data.Text.unpack x) . fromColist #-}
{-# COMPILE GHC putStr = putStr . fromColist #-}
{-# COMPILE GHC putStrLn = putStrLn . fromColist #-}
{-# COMPILE GHC readFiniteFile = readFiniteFile #-}
{-# COMPILE UHC getContents = UHC.Agda.Builtins.primGetContents #-}
{-# COMPILE UHC readFile = UHC.Agda.Builtins.primReadFile #-}
{-# COMPILE UHC writeFile = UHC.Agda.Builtins.primWriteFile #-}
{-# COMPILE UHC appendFile = UHC.Agda.Builtins.primAppendFile #-}
{-# COMPILE UHC putStr = UHC.Agda.Builtins.primPutStr #-}
{-# COMPILE UHC putStrLn = UHC.Agda.Builtins.primPutStrLn #-}
{-# COMPILE UHC readFiniteFile = UHC.Agda.Builtins.primReadFiniteFile #-}
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module product where
open import Data.Nat using (ℕ; _*_)
open import lists using (List; foldr)
-- リストの要素の積
product : (List ℕ) → ℕ
product xs = foldr _*_ 1 xs
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{-# OPTIONS --universe-polymorphism #-}
module Issue229 where
open import Common.Level
data Works a b : Set (lsuc a ⊔ lsuc b) where
w : (A : Set a)(B : Set b) → Works a b
record Doesn'tWork a b : Set (lsuc a ⊔ lsuc b) where
field
A : Set a
B : Set b
-- An internal error has occurred. Please report this as a bug.
-- Location of the error: src/full/Agda/Interaction/Highlighting/Generate.hs:469
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{-# OPTIONS --safe --cubical #-}
module Data.List.Kleene where
open import Prelude
open import Data.Fin
mutual
infixr 5 _&_ ∹_
infixl 5 _⁺ _⋆
record _⁺ {a} (A : Set a) : Set a where
inductive
constructor _&_
field
head : A
tail : A ⋆
data _⋆ {a} (A : Set a) : Set a where
[] : A ⋆
∹_ : A ⁺ → A ⋆
open _⁺ public
mutual
foldr⁺ : (A → B → B) → B → A ⁺ → B
foldr⁺ f b (x & xs) = f x (foldr⋆ f b xs)
foldr⋆ : (A → B → B) → B → A ⋆ → B
foldr⋆ f b [] = b
foldr⋆ f b (∹ xs) = foldr⁺ f b xs
length⋆ : A ⋆ → ℕ
length⋆ = foldr⋆ (const suc) zero
length⁺ : A ⁺ → ℕ
length⁺ = foldr⁺ (const suc) zero
mutual
_!⁺_ : (xs : A ⁺) → Fin (length⁺ xs) → A
xs !⁺ f0 = xs .head
xs !⁺ fs i = xs .tail !⋆ i
_!⋆_ : (xs : A ⋆) → Fin (length⋆ xs) → A
(∹ xs) !⋆ i = xs !⁺ i
map⋆ : (A → B) → A ⋆ → B ⋆
map⋆ f = foldr⋆ (λ x xs → ∹ f x & xs) []
map⁺ : (A → B) → A ⁺ → B ⁺
map⁺ f (x & xs) = f x & map⋆ f xs
mutual
_⋆++⋆_ : A ⋆ → A ⋆ → A ⋆
[] ⋆++⋆ ys = ys
(∹ xs) ⋆++⋆ ys = ∹ (xs ⁺++⋆ ys)
_⁺++⋆_ : A ⁺ → A ⋆ → A ⁺
head (xs ⁺++⋆ ys) = head xs
tail (xs ⁺++⋆ ys) = tail xs ⋆++⋆ ys
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-- Lemma for `Constructive.Axiom.Properties.Base`
{-# OPTIONS --without-K --safe #-}
module Constructive.Axiom.Properties.Base.Lemma where
-- agda-stdlib
open import Data.Empty
open import Data.Nat
import Data.Nat.Properties as ℕₚ
open import Data.Product
open import Data.Sum as Sum
open import Data.Sum.Properties
open import Function.Base
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary
-- agda-misc
open import Constructive.Combinators
open import Constructive.Common
-- lemma for `ℕ-llpo⇒mp∨`
ℕ≤-any-dec : ∀ {p} {P : ℕ → Set p} → DecU P → DecU (λ n → ∃ λ m → m ≤ n × P m)
ℕ≤-any-dec {P = P} P? zero with P? 0
... | inj₁ P0 = inj₁ (0 , ℕₚ.≤-refl , P0)
... | inj₂ ¬P0 = inj₂ λ {(m , m≤0 , Pm) → ¬P0 (subst P (ℕₚ.n≤0⇒n≡0 m≤0) Pm)}
ℕ≤-any-dec P? (suc n) with P? 0
... | inj₁ P0 = inj₁ (0 , (z≤n , P0))
... | inj₂ ¬P0 with ℕ≤-any-dec (P? ∘ suc) n
ℕ≤-any-dec {P = P} P? (suc n) | inj₂ ¬P0 | inj₁ (m , m≤n , Psm) =
inj₁ (suc m , s≤s m≤n , Psm)
ℕ≤-any-dec {P = P} P? (suc n) | inj₂ ¬P0 | inj₂ ¬∃m→m≤n×Psm = inj₂ f
where
f : (∃ λ m → m ≤ suc n × P m) → ⊥
f (zero , m≤sn , Pm) = ¬P0 Pm
f (suc m , sm≤sn , Psm) = ¬∃m→m≤n×Psm (m , (ℕₚ.≤-pred sm≤sn , Psm))
private
1+n≰0 : ∀ n → ¬ (suc n ≤ 0)
1+n≰0 n ()
module _ {p} {P : ℕ → Set p} (P? : DecU P) where
ℕ<-any-dec : DecU (λ n → ∃ λ m → m < n × P m)
ℕ<-any-dec zero = inj₂ λ {(m , m<0 , _) → 1+n≰0 m m<0}
ℕ<-any-dec (suc n) with ℕ≤-any-dec P? n
... | inj₁ (m , m≤n , Pm) = inj₁ (m , s≤s m≤n , Pm)
... | inj₂ ¬∃m→m≤n×Pm = inj₂ (contraposition f ¬∃m→m≤n×Pm)
where
f : (∃ λ m → suc m ≤ suc n × P m) → ∃ λ m → m ≤ n × P m
f (m , sm≤sn , Pm) = m , (ℕₚ.≤-pred sm≤sn , Pm)
ℕ≤-all-dec : DecU (λ n → ∀ m → m ≤ n → P m)
ℕ≤-all-dec n with ℕ≤-any-dec (¬-DecU P?) n
... | inj₁ (m , m≤n , ¬Pm) = inj₂ λ ∀i→i≤n→Pi → ¬Pm (∀i→i≤n→Pi m m≤n)
... | inj₂ ¬∃m→m≤n׬Pm =
inj₁ λ m m≤n → DecU⇒stable P? m λ ¬Pm → ¬∃m→m≤n׬Pm (m , m≤n , ¬Pm)
module _ {p} {P : ℕ → Set p} (P? : DecU P) where
ℕ<-all-dec : DecU (λ n → ∀ m → m < n → P m)
ℕ<-all-dec n with ℕ<-any-dec (¬-DecU P?) n
... | inj₁ (m , m<n , ¬Pm) = inj₂ λ ∀i→i<n→Pi → ¬Pm (∀i→i<n→Pi m m<n)
... | inj₂ ¬∃m→m<n׬Pm =
inj₁ λ m m<n → DecU⇒stable P? m λ ¬Pm → ¬∃m→m<n׬Pm (m , m<n , ¬Pm)
-- lemma for llpo-ℕ⇒llpo
lemma₁ : ∀ {p} {P : ℕ → Set p} →
(∀ m n → m ≢ n → P m ⊎ P n) →
∃ (λ n → ¬ P (2 * n)) → ∃ (λ n → ¬ P (suc (2 * n))) → ⊥
lemma₁ ∀mn→m≢n→Pm⊎Pn (m , ¬P2m) (n , ¬P1+2n) with
∀mn→m≢n→Pm⊎Pn (2 * m) (suc (2 * n)) (ℕₚ.even≢odd m n)
... | inj₁ P2m = ¬P2m P2m
... | inj₂ P1+2n = ¬P1+2n P1+2n
parity : ℕ → ℕ ⊎ ℕ
parity zero = inj₁ zero
parity (suc zero) = inj₂ zero
parity (suc (suc n)) = Sum.map suc suc (parity n)
parity-even : ∀ n → parity (2 * n) ≡ inj₁ n
parity-even zero = refl
parity-even (suc n) = begin
parity (2 * (suc n)) ≡⟨ cong parity (ℕₚ.*-distribˡ-+ 2 1 n) ⟩
parity (2 + 2 * n) ≡⟨⟩
Sum.map suc suc (parity (2 * n)) ≡⟨ cong (Sum.map suc suc) (parity-even n) ⟩
Sum.map suc suc (inj₁ n) ≡⟨⟩
inj₁ (suc n) ∎
where open ≡-Reasoning
parity-odd : ∀ n → parity (suc (2 * n)) ≡ inj₂ n
parity-odd zero = refl
parity-odd (suc n) = begin
parity (suc (2 * suc n)) ≡⟨ cong (parity ∘ suc) $ ℕₚ.*-distribˡ-+ 2 1 n ⟩
parity (1 + (2 * 1 + 2 * n)) ≡⟨⟩
parity (2 + (1 + 2 * n)) ≡⟨⟩
Sum.map suc suc (parity (1 + 2 * n)) ≡⟨ cong (Sum.map suc suc) (parity-odd n) ⟩
Sum.map suc suc (inj₂ n) ≡⟨⟩
inj₂ (suc n) ∎
where open ≡-Reasoning
Sum-map-injective : ∀ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d}
{f : A → C} {g : B → D} →
(∀ {u v} → f u ≡ f v → u ≡ v) →
(∀ {u v} → g u ≡ g v → u ≡ v) → ∀ {x y} →
Sum.map f g x ≡ Sum.map f g y → x ≡ y
Sum-map-injective f-inj g-inj {inj₁ x} {inj₁ x₁} eq = cong inj₁ (f-inj (inj₁-injective eq))
Sum-map-injective f-ing g-inj {inj₂ y} {inj₂ y₁} eq = cong inj₂ (g-inj (inj₂-injective eq))
parity-injective : ∀ {m n} → parity m ≡ parity n → m ≡ n
parity-injective {zero} {zero} pm≡pn = refl
parity-injective {zero} {suc (suc n)} pm≡pn with parity n
parity-injective {zero} {suc (suc n)} () | inj₁ _
parity-injective {zero} {suc (suc n)} () | inj₂ _
parity-injective {suc (suc m)} {zero} pm≡pn with parity m
parity-injective {suc (suc m)} {zero} () | inj₁ _
parity-injective {suc (suc m)} {zero} () | inj₂ _
parity-injective {suc zero} {suc zero} pm≡pn = refl
parity-injective {suc zero} {suc (suc n)} pm≡pn with parity n
parity-injective {suc zero} {suc (suc n)} () | inj₁ _
parity-injective {suc zero} {suc (suc n)} () | inj₂ _
parity-injective {suc (suc m)} {suc zero} pm≡pn with parity m
parity-injective {suc (suc m)} {suc zero} () | inj₁ _
parity-injective {suc (suc m)} {suc zero} () | inj₂ _
parity-injective {suc (suc m)} {suc (suc n)} pm≡pn =
cong (suc ∘ suc)
(parity-injective
(Sum-map-injective ℕₚ.suc-injective ℕₚ.suc-injective pm≡pn))
parity-even′ : ∀ {m n} → parity m ≡ inj₁ n → m ≡ 2 * n
parity-even′ {m} {n} eq = parity-injective (begin
parity m ≡⟨ eq ⟩
inj₁ n ≡⟨ sym $ parity-even n ⟩
parity (2 * n) ∎)
where open ≡-Reasoning
parity-odd′ : ∀ {m n} → parity m ≡ inj₂ n → m ≡ 1 + 2 * n
parity-odd′ {m} {n} eq = parity-injective (begin
parity m ≡⟨ eq ⟩
inj₂ n ≡⟨ sym $ parity-odd n ⟩
parity (1 + 2 * n) ∎)
where open ≡-Reasoning
mix : ℕ ⊎ ℕ → ℕ
mix = Sum.[ (λ n → 2 * n) , (λ n → 1 + 2 * n) ]
parity-mix : ∀ s → parity (mix s) ≡ s
parity-mix (inj₁ n) = parity-even n
parity-mix (inj₂ n) = parity-odd n
mix-parity : ∀ n → mix (parity n) ≡ n
mix-parity n = parity-injective (parity-mix (parity n))
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module Dual where
open import Prelude using (flip)
open import Logic.Equivalence
open import Category
_op : Cat -> Cat
ℂ@(cat _ _ _ _ _ _ _ _ _) op = cat Obj
(\A B -> B ─→ A)
id
(\{_}{_}{_} -> flip _∘_)
(\{_}{_} -> Eq)
(\{_}{_}{_}{_}{_}{_}{_} -> flip cong)
(\{_}{_}{_} -> idR)
(\{_}{_}{_} -> idL)
(\{_}{_}{_}{_}{_}{_}{_} -> sym assoc)
where open module C = Cat ℂ
{-
open Poly-Cat
dualObj : {ℂ : Cat} -> Obj ℂ -> Obj (ℂ op)
dualObj {cat _ _ _ _ _ _ _ _ _}(obj A) = obj A
undualObj : {ℂ : Cat} -> Obj (ℂ op) -> Obj ℂ
undualObj {cat _ _ _ _ _ _ _ _ _}(obj A) = obj A
dualdualArr : {ℂ : Cat}{A B : Obj ℂ} -> A ─→ B -> dualObj B ─→ dualObj A
dualdualArr {cat _ _ _ _ _ _ _ _ _}{A = obj _}{B = obj _}(arr f) = arr f
dualundualArr : {ℂ : Cat}{A : Obj ℂ}{B : Obj (ℂ op)} ->
A ─→ undualObj B -> B ─→ dualObj A
dualundualArr {cat _ _ _ _ _ _ _ _ _}{A = obj _}{B = obj _}(arr f) = arr f
-}
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-- Andreas, 2013-03-20
-- Without termination checking disabled, the positivity checker
-- will throw an error.
module NoTerminationCheckPositivity where
open import Common.Level
module M {a}{A : Set a}(K : A → A → A) where
-- F fails the termination check
F : A → A
F X = K X (F X)
K : Set → Set → Set
K X Y = X
open M K
data E : Set where
e : F E → E
-- Since F is non-terminating and hence excluded from unfolding
-- in the positivity checker, it will complain unless termination
-- checking is off.
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{-# OPTIONS --cubical --safe --postfix-projections #-}
-- This file contains an implementation of the stack-based compiler for Hutton's
-- razor, as from:
--
-- P. Bahr and G. Hutton, “Calculating correct compilers,” Journal of
-- Functional Programming, vol. 25, no. e14, Sep. 2015,
-- doi: 10.1017/S0956796815000180.
--
-- The compiler is total and the evaluator is stack-safe, and furthermore we have
-- proven a full isomorphism between the code representation and the AST.
module Data.Dyck.Except where
open import Prelude
open import Data.Nat using (_+_)
open import Data.Vec.Iterated using (Vec; _∷_; []; foldlN; head)
private
variable
n : ℕ
--------------------------------------------------------------------------------
-- Language for Arithmetic Expressions
--------------------------------------------------------------------------------
data Expr (A : Type a) : Type a where
[_] : A → Expr A
_⊕_ : Expr A → Expr A → Expr A
throw : Expr A
catch : Expr A → Expr A → Expr A
--------------------------------------------------------------------------------
-- Code for the virtual stack machine.
--------------------------------------------------------------------------------
data Code (A : Type a) : ℕ → Type a where
HALT : Code A 1
PUSH : A → Code A (1 + n) → Code A n
ADD : Code A (1 + n) → Code A (2 + n)
RAISE : Code A (1 + n) → Code A n
CATCH : Code A (1 + n) → Code A (2 + n)
--------------------------------------------------------------------------------
-- Conversion from a Code to a Expr (evaluation / execution)
--------------------------------------------------------------------------------
code→expr⊙ : Code A n → Vec (Expr A) n → Expr A
code→expr⊙ HALT (v ∷ []) = v
code→expr⊙ (PUSH v is) st = code→expr⊙ is ([ v ] ∷ st)
code→expr⊙ (ADD is) (t₁ ∷ t₂ ∷ st) = code→expr⊙ is (t₂ ⊕ t₁ ∷ st)
code→expr⊙ (RAISE is) st = code→expr⊙ is (throw ∷ st)
code→expr⊙ (CATCH is) (t₁ ∷ t₂ ∷ st) = code→expr⊙ is (catch t₂ t₁ ∷ st)
code→expr : Code A zero → Expr A
code→expr ds = code→expr⊙ ds []
--------------------------------------------------------------------------------
-- Conversion from a Expr to a Code (compilation)
--------------------------------------------------------------------------------
expr→code⊙ : Expr A → Code A (1 + n) → Code A n
expr→code⊙ [ x ] = PUSH x
expr→code⊙ (xs ⊕ ys) = expr→code⊙ xs ∘ expr→code⊙ ys ∘ ADD
expr→code⊙ throw = RAISE
expr→code⊙ (catch xs ys) = expr→code⊙ xs ∘ expr→code⊙ ys ∘ CATCH
expr→code : Expr A → Code A 0
expr→code tr = expr→code⊙ tr HALT
--------------------------------------------------------------------------------
-- Execution
--------------------------------------------------------------------------------
Func : Type a → ℕ → Type _
Func A zero = Maybe A
Func A (suc n) = Maybe A → Func A n
open import Data.Maybe.Sugar
exec : Code ℕ n → Func ℕ n
exec HALT = λ x → x
exec (PUSH x xs) = exec xs (just x)
exec (ADD xs) = λ x₂ x₁ → exec xs ⦇ x₁ + x₂ ⦈
exec (RAISE xs) = exec xs nothing
exec (CATCH xs) = λ x₂ x₁ → exec xs (x₁ <|> x₂)
_ : exec (expr→code (catch ([ 3 ] ⊕ throw) ([ 10 ] ⊕ [ 2 ]))) ≡ just 12
_ = refl
--------------------------------------------------------------------------------
-- Proof of isomorphism
--------------------------------------------------------------------------------
expr→code→expr⊙ : {is : Code A (1 + n)} {st : Vec (Expr A) n} (e : Expr A) →
code→expr⊙ (expr→code⊙ e is) st ≡ code→expr⊙ is (e ∷ st)
expr→code→expr⊙ [ x ] = refl
expr→code→expr⊙ (xs ⊕ ys) = expr→code→expr⊙ xs ; expr→code→expr⊙ ys
expr→code→expr⊙ throw = refl
expr→code→expr⊙ (catch xs ys) = expr→code→expr⊙ xs ; expr→code→expr⊙ ys
code→expr→code⊙ : {st : Vec (Expr A) n} (is : Code A n) →
expr→code (code→expr⊙ is st) ≡ foldlN (Code A) expr→code⊙ is st
code→expr→code⊙ HALT = refl
code→expr→code⊙ (PUSH i is) = code→expr→code⊙ is
code→expr→code⊙ (ADD is) = code→expr→code⊙ is
code→expr→code⊙ (RAISE is) = code→expr→code⊙ is
code→expr→code⊙ (CATCH is) = code→expr→code⊙ is
prog-iso : Code A 0 ⇔ Expr A
prog-iso .fun = code→expr
prog-iso .inv = expr→code
prog-iso .rightInv = expr→code→expr⊙
prog-iso .leftInv = code→expr→code⊙
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-- {-# OPTIONS --show-implicit #-} -- in case of emergency
module ldlc-algo-nosingle where
open import Data.Nat renaming (_+_ to _+ᴺ_ ; _≤_ to _≤ᴺ_ ; _≥_ to _≥ᴺ_ ; _<_ to _<ᴺ_ ; _>_ to _>ᴺ_ ; _≟_ to _≟ᴺ_)
open import Data.Nat.Properties renaming (_<?_ to _<ᴺ?_)
open import Data.Integer renaming (_+_ to _+ᶻ_ ; _≤_ to _≤ᶻ_ ; _≥_ to _≥ᶻ_ ; _<_ to _<ᶻ_ ; _>_ to _>ᶻ_)
open import Data.Integer.Properties using (⊖-≥ ; 0≤n⇒+∣n∣≡n ; +-monoˡ-≤)
open import Data.Fin
open import Data.Fin.Subset renaming (∣_∣ to ∣_∣ˢ)
open import Data.Vec hiding (_++_ ; length)
open import Relation.Nullary
open import Relation.Nullary.Negation
open import Relation.Binary.PropositionalEquality renaming (trans to ≡-trans)
open import Data.Product
open import Data.Sum
open import aux
----------------------------------------------------------------------
module syntx where
data Exp {n : ℕ} : Set
data Val {n : ℕ} : Exp {n} → Set
data Ty {n : ℕ} : Set
data Exp {n} where
Var : ℕ → Exp {n}
UnitE : Exp {n}
Abs : Exp {n} → Exp {n}
App : {e : Exp {n}} → Exp {n} → Val e → Exp {n}
LabI : Fin n → Exp {n}
CaseE : {s : Subset n} {e : Exp {n}} → Val e → (f : ∀ l → l ∈ s → Exp {n}) → Exp {n}
Prod : Exp {n} → Exp {n} → Exp {n}
ProdV : {e : Exp {n}} → Val e → Exp {n} → Exp {n}
LetP : Exp {n} → Exp {n} → Exp {n}
LetE : Exp {n} → Exp {n} → Exp {n}
data Val {n} where
VUnit : Val UnitE
VVar : {i : ℕ} → Val (Var i)
VLab : {x : Fin n} → Val (LabI x)
VFun : {N : Exp} → Val (Abs N)
VProd : {e e' : Exp} → (v : Val e) → Val e' → Val (ProdV v e')
data Ty {n} where
UnitT : Ty
Label : Subset n → Ty
Pi : Ty {n} → Ty {n} → Ty
Sigma : Ty {n} → Ty {n} → Ty
CaseT : {s : Subset n} {e : Exp {n}} → Val e → (f : ∀ l → l ∈ s → Ty {n}) → Ty
-- variable in expression
data _∈`_ {n : ℕ} : ℕ → Exp {n} → Set where
in-var : {x : ℕ} → x ∈` (Var x)
in-abs : {x : ℕ} {e : Exp {n}} → x ∈` e → (ℕ.suc x) ∈` (Abs e)
in-app : {x : ℕ} {e e' : Exp {n}} {v : Val e'} → x ∈` e ⊎ x ∈` e' → x ∈` App e v
in-casee : {x : ℕ} {s : Subset n} {f : (∀ l → l ∈ s → Exp {n})} {e : Exp {n}} {v : Val e} → (∃₂ λ l i → x ∈` (f l i)) ⊎ x ∈` e → x ∈` CaseE v f
in-prod : {x : ℕ} {e e' : Exp {n}} → x ∈` e ⊎ (ℕ.suc x) ∈` e' → x ∈` Prod e e'
in-prodv : {x : ℕ} {e e' : Exp {n}} {v : Val e} → x ∈` e ⊎ x ∈` e' → x ∈` ProdV v e' -- (Pair-A-I => e' has 0 substituted away => just x, not suc x)
in-letp : {x : ℕ} {e e' : Exp {n}} → x ∈` e ⊎ (ℕ.suc (ℕ.suc x)) ∈` e' → x ∈` LetP e e'
in-lete : {x : ℕ} {e e' : Exp {n}} → x ∈` e ⊎ (ℕ.suc x) ∈` e' → x ∈` LetE e e'
-- variable in type
data _∈`ᵀ_ {n : ℕ} : ℕ → Ty {n} → Set where
in-pi : {x : ℕ} {A B : Ty {n}} → n ∈`ᵀ A ⊎ n ∈`ᵀ B → n ∈`ᵀ Pi A B
in-pigma : {x : ℕ} {A B : Ty {n}} → n ∈`ᵀ A ⊎ n ∈`ᵀ B → n ∈`ᵀ Sigma A B
in-case : {x : ℕ} {s : Subset n} {f : ∀ l → l ∈ s → Ty {n}} {e : Exp {n}} {v : Val e} → (∃₂ λ l i → x ∈`ᵀ (f l i)) ⊎ x ∈` e → x ∈`ᵀ CaseT v f
----------------------------------------------------------------------
----------------------------------------------------------------------
module substitution where
---- Substitution and Shifting
open syntx
↑ᴺ_,_[_] : ℤ → ℕ → ℕ → ℕ
↑ᴺ d , c [ x ]
with (x <ᴺ? c)
... | yes p = x
... | no ¬p = ∣ ℤ.pos x +ᶻ d ∣
↑_,_[_] : ∀ {n} → ℤ → ℕ → Exp {n} → Exp {n}
shift-val : ∀ {n d c} {e : Exp {n}} → Val e → Val (↑ d , c [ e ])
↑ d , c [ UnitE ] = UnitE
↑ d , c [ Var x ] = Var (↑ᴺ d , c [ x ])
↑ d , c [ Abs t ] = Abs (↑ d , (ℕ.suc c) [ t ])
↑ d , c [ App t v ] = App (↑ d , c [ t ]) (shift-val{d = d}{c = c} v)
↑ d , c [ LabI x ] = LabI x
↑ d , c [ CaseE{e = e} V f ] = CaseE (shift-val{d = d}{c = c} V) (λ l x → ↑ d , c [ f l x ])
↑ d , c [ Prod e e' ] = Prod (↑ d , c [ e ]) (↑ d , (ℕ.suc c) [ e' ])
↑ d , c [ ProdV e e' ] = ProdV (shift-val{d = d}{c = c} e) (↑ d , (ℕ.suc c) [ e' ])
↑ d , c [ LetP e e' ] = LetP (↑ d , c [ e ]) (↑ d , (ℕ.suc (ℕ.suc c)) [ e' ])
↑ d , c [ LetE e e' ] = LetE (↑ d , c [ e ]) (↑ d , (ℕ.suc c) [ e' ])
shift-val {n} {d} {c} {.UnitE} VUnit = VUnit
shift-val {n} {d} {c} {.(Var _)} VVar = VVar
shift-val {n} {d} {c} {.(LabI _)} VLab = VLab
shift-val {n} {d} {c} {.(Abs _)} VFun = VFun
shift-val {n} {d} {c} {.(ProdV V _)} (VProd V V₁) = VProd (shift-val V) (shift-val V₁)
-- shorthands
↑¹[_] : ∀ {n} → Exp {n} → Exp
↑¹[ e ] = ↑ (ℤ.pos 1) , 0 [ e ]
↑ⱽ¹[_] : ∀ {n} {e : Exp {n}} → Val e → Val (↑ (ℤ.pos 1) , 0 [ e ])
↑ⱽ¹[_] {n} {e} v = shift-val v
↑⁻¹[_] : ∀ {n} → Exp {n} → Exp
↑⁻¹[ e ] = ↑ (ℤ.negsuc 0) , 0 [ e ]
-- substitution
[_↦_]_ : ∀ {n} {e : Exp {n}} → ℕ → Val e → Exp {n} → Exp {n}
sub-val : ∀ {n k} {e e' : Exp {n}} {v : Val e'} → Val e → Val ([ k ↦ v ] e)
[_↦_]_ {n} {e} k v (Var x)
with (_≟ᴺ_ x k)
... | yes p = e
... | no ¬p = Var x
[ k ↦ v ] UnitE = UnitE
[ k ↦ v ] Abs e = Abs (([ ℕ.suc k ↦ shift-val{d = ℤ.pos 1}{c = 0} v ] e))
[_↦_]_ {n} {e'} k v (App{e = e₁} e v') = App ([ k ↦ v ] e) (sub-val{n}{k}{e₁}{e'}{v} v') -- ([ k ↦ v ] e₁)
[ k ↦ v ] LabI x = LabI x
[_↦_]_ {n} {e} k v (CaseE v' f) = CaseE (sub-val{n}{k}{e' = e}{v = v} v') (λ l x₁ → [ k ↦ v ] (f l x₁))
[ k ↦ v ] Prod e e₁ = Prod ([ k ↦ v ] e) ([ ℕ.suc k ↦ shift-val{d = ℤ.pos 1}{c = 0} v ] e₁)
[_↦_]_ {n} {e} k v (ProdV v' e') = ProdV (sub-val{n}{k}{e' = e}{v = v} v') ([ ℕ.suc k ↦ shift-val{d = ℤ.pos 1}{c = 0} v ] e')
[ k ↦ v ] LetP e e₁ = LetE ([ k ↦ v ] e) ([ (ℕ.suc (ℕ.suc k)) ↦ shift-val{d = ℤ.pos 2}{c = 0} v ] e₁)
[ k ↦ v ] LetE e e₁ = LetE ([ k ↦ v ] e) ([ (ℕ.suc k) ↦ shift-val{d = ℤ.pos 1}{c = 0} v ] e₁)
sub-val {n} {k} {.UnitE} {e'} {v} VUnit = VUnit
sub-val {n} {k} {(Var i)} {e'} {v} VVar
with (_≟ᴺ_ i k)
... | yes p = v
... | no ¬p = VVar
sub-val {n} {k} {.(LabI _)} {e'} {v} VLab = VLab
sub-val {n} {k} {.(Abs _)} {e'} {v} VFun = VFun
sub-val {n} {k} {.(ProdV v' _)} {e'} {v} (VProd v' v'') = VProd (sub-val v') (sub-val v'')
-- type substitution
[_↦_]ᵀ_ : ∀ {n} {e : Exp {n}} → ℕ → Val e → Ty {n} → Ty {n}
[ k ↦ s ]ᵀ UnitT = UnitT
[ k ↦ s ]ᵀ Label x = Label x
[ k ↦ s ]ᵀ Pi T T₁ = Pi ([ k ↦ s ]ᵀ T) ([ k ↦ s ]ᵀ T₁)
[ k ↦ s ]ᵀ Sigma T T₁ = Sigma ([ k ↦ s ]ᵀ T) ([ k ↦ s ]ᵀ T₁)
[_↦_]ᵀ_ {n} {e} k v (CaseT x f) = CaseT (sub-val{n}{k}{e' = e}{v = v} x) λ l x₁ → [ k ↦ v ]ᵀ (f l x₁)
----------------------------------------------------------------------
----------------------------------------------------------------------
module typing where
open syntx
open substitution
-- Type environment
data TEnv {n : ℕ} : Set where
[] : TEnv
⟨_,_⟩ : (T : Ty {n}) (Γ : TEnv {n}) → TEnv
_++_ : {n : ℕ} → TEnv {n} → TEnv {n} → TEnv {n}
[] ++ Γ' = Γ'
⟨ T , Γ ⟩ ++ Γ' = ⟨ T , Γ ++ Γ' ⟩
++-assoc : {n : ℕ} {Γ Γ' Γ'' : TEnv {n}} → Γ ++ (Γ' ++ Γ'') ≡ (Γ ++ Γ') ++ Γ''
++-assoc {n} {[]} {Γ'} {Γ''} = refl
++-assoc {n} {⟨ T , Γ ⟩} {Γ'} {Γ''} = cong (λ x → ⟨ T , x ⟩) (++-assoc{n}{Γ}{Γ'}{Γ''})
length : {n : ℕ} → TEnv {n} → ℕ
length {n} [] = zero
length {n} ⟨ T , Γ ⟩ = ℕ.suc (length Γ)
data _∶_∈_ {n : ℕ} : ℕ → Ty {n} → TEnv {n} → Set where
here : {T : Ty} {Γ : TEnv} → 0 ∶ T ∈ ⟨ T , Γ ⟩
there : {n : ℕ} {T₁ T₂ : Ty} {Γ : TEnv} → n ∶ T₁ ∈ Γ → (ℕ.suc n) ∶ T₁ ∈ ⟨ T₂ , Γ ⟩
---- Algorithmic Typing
-- Type environment formation
data ⊢_ok {n : ℕ} : TEnv {n} → Set
-- Type formation
data _⊢_ {n : ℕ} : TEnv {n}→ Ty {n} → Set
-- Type synthesis
data _⊢_⇒_ {n : ℕ} : TEnv {n} → Exp {n} → Ty {n} → Set
-- Type check
data _⊢_⇐_ {n : ℕ} : TEnv {n} → Exp {n} → Ty {n} → Set
-- Check subtype (⇐ instead of ⇒?)
data _⇒_≤_ {n : ℕ} : TEnv {n} → Ty {n} → Ty {n} → Set
-- Unfolding (e.g. CaseT ... ⇓ T)
data _⊢_⇓_ {n : ℕ} : TEnv {n} → Ty {n} → Ty {n} → Set
-- Conversion
data _⇒_≡_ {n : ℕ} : TEnv {n} → Ty {n} → Ty {n} → Set
-- Implementations
data ⊢_ok {n} where
empty : ⊢ [] ok
entry : {Γ : TEnv {n}} {A : Ty {n}} → ⊢ Γ ok → Γ ⊢ A → ⊢ ⟨ A , Γ ⟩ ok
data _⊢_ {n} where
UnitF : {Γ : TEnv {n}} → ⊢ Γ ok → Γ ⊢ UnitT
LabF : {Γ : TEnv {n}} {L : Subset n} → ⊢ Γ ok → Γ ⊢ Label L
PiF : {Γ : TEnv {n}} {A B : Ty {n}} → Γ ⊢ A → ⟨ A , Γ ⟩ ⊢ B → Γ ⊢ Pi A B
SigmaF : {Γ : TEnv {n}} {A B : Ty {n}} → Γ ⊢ A → ⟨ A , Γ ⟩ ⊢ B → Γ ⊢ Sigma A B
CaseF : {Γ : TEnv {n}} {L : Subset n} {e : Exp {n}} {V : Val e} {f : ∀ l → l ∈ L → Ty {n}} {f-ok : ∀ l → (i : l ∈ L) → Γ ⊢ (f l i)} → Γ ⊢ e ⇐ Label L → Γ ⊢ CaseT V f
data _⊢_⇐_ {n} where
SubTypeA : {Γ : TEnv {n}} {A B : Ty {n}} {M : Exp {n}}
→ Γ ⊢ M ⇒ A
→ Γ ⇒ A ≤ B
→ Γ ⊢ M ⇐ B
data _⇒_≤_ {n} where
ASubUnit : {Γ : TEnv {n}} → Γ ⇒ UnitT ≤ UnitT
ASubLabel : {Γ : TEnv {n}} {L L' : Subset n} → L ⊆ L' → Γ ⇒ Label L ≤ Label L'
ASubPi : {Γ : TEnv {n}} {A A' B B' : Ty {n}}
→ Γ ⇒ A' ≤ A
→ ⟨ A' , Γ ⟩ ⇒ B ≤ B'
→ Γ ⇒ Pi A B ≤ Pi A' B'
ASubSigma : {Γ : TEnv {n}} {A A' B B' : Ty {n}}
→ Γ ⇒ A ≤ A'
→ ⟨ A , Γ ⟩ ⇒ B ≤ B'
→ Γ ⇒ Sigma A B ≤ Sigma A' B'
ASubCaseLL : {Γ : TEnv {n}} {B : Ty {n}} {e : Exp {n}} {V : Val e} {l : Fin n} {L L' : Subset n} {f : ∀ l → l ∈ L → Ty {n}} {ins : l ∈ L}
→ Γ ⊢ e ⇒ Label ⁅ l ⁆
→ L' ⊆ L
→ Γ ⇒ (f l ins) ≤ B
→ Γ ⇒ CaseT V f ≤ B
ASubCaseLR : {Γ : TEnv {n}} {A : Ty {n}} {e : Exp {n}} {V : Val e} {l : Fin n} {L L' : Subset n} {f : ∀ l → l ∈ L → Ty {n}} {ins : l ∈ L}
→ Γ ⊢ e ⇒ Label ⁅ l ⁆
→ L' ⊆ L
→ Γ ⇒ A ≤ (f l ins)
→ Γ ⇒ A ≤ CaseT V f
ASubCaseXL : {Γ Γ' : TEnv {n}} {B D : Ty {n}} {L : Subset n} {f : ∀ l → l ∈ L → Ty {n}}
→ Γ ⇒ D ≤ Label L
→ (∀ l → (i : l ∈ L) → (Γ' ++ ⟨ Label ⁅ l ⁆ , Γ ⟩) ⇒ (f l i) ≤ B)
→ (Γ' ++ ⟨ D , Γ ⟩) ⇒ CaseT (VVar{i = length Γ'}) f ≤ B
ASubCaseXR : {Γ Γ' : TEnv {n}} {A D : Ty {n}} {L : Subset n} {f : ∀ l → l ∈ L → Ty {n}}
→ Γ ⇒ D ≤ Label L
→ (∀ l → (i : l ∈ L) → (Γ' ++ ⟨ Label ⁅ l ⁆ , Γ ⟩) ⇒ A ≤ (f l i))
→ (Γ' ++ ⟨ D , Γ ⟩) ⇒ A ≤ CaseT (VVar{i = length Γ'}) f
data _⊢_⇒_ {n} where
VarA : {Γ : TEnv {n}} {A : Ty {n}} {x : ℕ} → ⊢ Γ ok → x ∶ A ∈ Γ → Γ ⊢ Var x ⇒ A
UnitAI : {Γ : TEnv {n}} → ⊢ Γ ok → Γ ⊢ UnitE ⇒ UnitT
LabAI : {Γ : TEnv {n}} {l : Fin n} → ⊢ Γ ok → Γ ⊢ LabI l ⇒ Label ⁅ l ⁆
LabAEl : {Γ : TEnv {n}} {B : Ty {n}} {L L' : Subset n} {l : Fin n} {ins : l ∈ L'} {f : ∀ l → l ∈ L → Exp {n}}
→ Γ ⊢ (LabI l) ⇒ Label L'
→ (s : L' ⊆ L)
→ Γ ⊢ (f l (s ins)) ⇒ B
→ Γ ⊢ CaseE (VLab{x = l}) f ⇒ B
-- unification has problems with arbitrary functions, hence θ
-- see https://lists.chalmers.se/pipermail/agda/2020/012293.html
LabAEx : {Γ Γ' Θ : TEnv {n}} {D : Ty {n}} {L : Subset n} {f : ∀ l → l ∈ L → Exp {n}} {f-t : ∀ l → l ∈ L → Ty {n}} {eq : Θ ≡ (Γ' ++ ⟨ D , Γ ⟩)}
→ Γ ⇒ D ≤ Label L
→ (∀ l → (i : l ∈ L) → (Γ' ++ ⟨ Label ⁅ l ⁆ , Γ ⟩) ⊢ (f l i) ⇒ (f-t l i))
→ Θ ⊢ CaseE (VVar{i = length Γ'}) f ⇒ CaseT (VVar{i = length Γ'}) f-t
PiAI : {Γ : TEnv {n}} {A B : Ty {n}} {M : Exp {n}} → ⟨ A , Γ ⟩ ⊢ M ⇒ B → Γ ⊢ Abs M ⇒ Pi A B
PiAE : {Γ : TEnv {n}} {A B D : Ty {n}} {M e : Exp {n}} {V : Val e}
→ Γ ⊢ M ⇒ D
→ Γ ⊢ D ⇓ Pi A B
→ Γ ⊢ e ⇐ A
→ Γ ⊢ ([ 0 ↦ V ]ᵀ B)
→ Γ ⊢ App M V ⇒ ([ 0 ↦ V ]ᵀ B)
SigmaAI : {Γ : TEnv {n}} {A A' B : Ty {n}} {M N : Exp {n}}
→ Γ ⊢ M ⇐ A
→ Γ ⇒ A' ≤ A
→ ⟨ A' , Γ ⟩ ⊢ N ⇒ B
→ Γ ⊢ Prod M N ⇒ Sigma A B
PairAI : {Γ : TEnv {n}} {A B : Ty {n}} {e N : Exp {n}} {V : Val e}
→ Γ ⊢ e ⇒ A
→ Γ ⊢ N ⇒ B
→ Γ ⊢ ProdV V N ⇒ Sigma A B
SigmaAE : {Γ : TEnv {n}} {A B C D : Ty {n}} {M N : Exp {n}}
→ Γ ⊢ M ⇒ D
→ Γ ⊢ D ⇓ Sigma A B
→ ⟨ B , ⟨ A , Γ ⟩ ⟩ ⊢ N ⇒ C
→ (¬ (0 ∈`ᵀ C)) × (¬ (1 ∈`ᵀ C))
→ Γ ⊢ LetP M N ⇒ C
Let : {Γ : TEnv {n}} {A B : Ty {n}} {M N : Exp {n}}
→ ¬(0 ∈`ᵀ B)
→ Γ ⊢ M ⇒ A
→ ⟨ A , Γ ⟩ ⊢ N ⇒ B
→ Γ ⊢ LetE M N ⇒ B
data _⊢_⇓_ {n} where
AURefl-U : {Γ : TEnv {n}} → Γ ⊢ UnitT ⇓ UnitT
AURefl-L : {Γ : TEnv {n}} {L : Subset n} → Γ ⊢ Label L ⇓ Label L
AURefl-P : {Γ : TEnv {n}} {A B : Ty {n}} → Γ ⊢ Pi A B ⇓ Pi A B
AURefl-S : {Γ : TEnv {n}} {A B : Ty {n}} → Γ ⊢ Sigma A B ⇓ Sigma A B
AUCaseL : {Γ : TEnv {n}} {D : Ty {n}} {l : Fin n} {L L' : Subset n} {ins : l ∈ L} {f : ∀ l → l ∈ L → Ty {n}} {e : Exp {n}} {V : Val e}
→ Γ ⊢ e ⇒ Label ⁅ l ⁆
→ L' ⊆ L
→ Γ ⊢ (f l ins) ⇓ D
→ Γ ⊢ CaseT V f ⇓ D
AUCaseX-P : {Γ Γ' : TEnv {n}} {A D : Ty {n}} {L : Subset n} {fᴬ fᴮ fᴰ : (∀ l → l ∈ L → Ty {n})}
→ Γ ⇒ D ≤ Label L
→ (∀ l → (i : l ∈ L) → (Γ' ++ ⟨ Label ⁅ l ⁆ , Γ ⟩) ⊢ (fᴮ l i) ⇓ Pi (fᴬ l i) (fᴰ l i))
→ (∀ l → (i : l ∈ L) → (Γ' ++ ⟨ Label ⁅ l ⁆ , Γ ⟩) ⇒ A ≡ (fᴬ l i))
→ (Γ' ++ ⟨ D , Γ ⟩) ⊢ CaseT (VVar{i = length Γ'}) fᴮ ⇓ Pi A (CaseT (VVar{i = length Γ'}) fᴰ)
AUCaseX-S : {Γ Γ' : TEnv {n}} {A D : Ty {n}} {L : Subset n} {fᴬ fᴮ fᴰ : (∀ l → l ∈ L → Ty {n})}
→ Γ ⇒ D ≤ Label L
→ (∀ l → (i : l ∈ L) → (Γ' ++ ⟨ Label ⁅ l ⁆ , Γ ⟩) ⊢ (fᴮ l i) ⇓ Sigma (fᴬ l i) (fᴰ l i))
→ (∀ l → (i : l ∈ L) → (Γ' ++ ⟨ Label ⁅ l ⁆ , Γ ⟩) ⇒ A ≡ (fᴬ l i))
→ (Γ' ++ ⟨ D , Γ ⟩) ⊢ CaseT (VVar{i = length Γ'}) fᴮ ⇓ Sigma A (CaseT (VVar{i = length Γ'}) fᴰ)
data _⇒_≡_ {n} where
AConvUnit : {Γ : TEnv {n}} → Γ ⇒ UnitT ≡ UnitT
AConvLabel : {Γ : TEnv {n}} {L : Subset n} → Γ ⇒ Label L ≡ Label L
AConvPi : {Γ : TEnv {n}} {A A' B B' : Ty} → Γ ⇒ A ≡ A' → ⟨ A' , Γ ⟩ ⇒ B ≡ B' → Γ ⇒ Pi A B ≡ Pi A' B'
AConvSigma : {Γ : TEnv {n}} {A A' B B' : Ty} → Γ ⇒ A ≡ A' → ⟨ A , Γ ⟩ ⇒ B ≡ B' → Γ ⇒ Sigma A B ≡ Sigma A' B'
AConvCaseLL : {Γ : TEnv {n}} {B : Ty {n}} {e : Exp {n}} {V : Val e} {L L' : Subset n} {f : (∀ l → l ∈ L → Ty)} {l : Fin n} {ins : l ∈ L}
→ Γ ⊢ e ⇒ Label ⁅ l ⁆
→ L ⊆ L'
→ Γ ⇒ (f l ins) ≡ B
→ Γ ⇒ CaseT V f ≡ B
AConvCaseLR : {Γ : TEnv {n}} {A : Ty {n}} {e : Exp {n}} {V : Val e} {L L' : Subset n} {f : (∀ l → l ∈ L → Ty)} {l : Fin n} {ins : l ∈ L}
→ Γ ⊢ e ⇒ Label ⁅ l ⁆
→ L ⊆ L'
→ Γ ⇒ A ≡ (f l ins)
→ Γ ⇒ A ≡ CaseT V f
AConvCaseXL : {Γ Γ' : TEnv {n}} {B D : Ty {n}} {L : Subset n} {f : ∀ l → l ∈ L → Ty {n}}
→ Γ ⇒ D ≤ Label L
→ (∀ l → (i : l ∈ L) → (Γ' ++ ⟨ Label ⁅ l ⁆ , Γ ⟩) ⇒ (f l i) ≡ B)
→ (Γ' ++ ⟨ D , Γ ⟩) ⇒ CaseT (VVar{i = length Γ'}) f ≡ B
AConvCaseXR : {Γ Γ' : TEnv {n}} {A D : Ty {n}} {L : Subset n} {f : ∀ l → l ∈ L → Ty {n}}
→ Γ ⇒ D ≤ Label L
→ (∀ l → (i : l ∈ L) → (Γ' ++ ⟨ Label ⁅ l ⁆ , Γ ⟩) ⇒ A ≡ (f l i))
→ (Γ' ++ ⟨ D , Γ ⟩) ⇒ A ≡ CaseT (VVar{i = length Γ'}) f
----------------------------------------------------------------------
----------------------------------------------------------------------
module semantics where
open syntx
open substitution
data _↠_ {n : ℕ} : Exp {n} → Exp {n} → Set where
ξ-App : {e₁ e₁' e : Exp {n}} {v : Val e} → e₁ ↠ e₁' → App e₁ v ↠ App e₁' v
ξ-LetE : {e₁ e₁' e : Exp {n}} → e₁ ↠ e₁' → LetE e₁ e ↠ LetE e₁' e
ξ-Prod : {e₁ e₁' e : Exp {n}} → e₁ ↠ e₁' → Prod e₁ e ↠ Prod e₁' e
ξ-ProdV : {e e₂ e₂' : Exp {n}} {v : Val e} → e₂ ↠ e₂' → ProdV v e₂ ↠ ProdV v e₂'
ξ-LetP : {e₁ e₁' e₂ : Exp {n}} → e₁ ↠ e₁' → LetP e₁ e₂ ↠ LetP e₁' e₂
β-App : {e e' : Exp {n}} → (v : Val e') → (App (Abs e) v) ↠ (↑⁻¹[ ([ 0 ↦ ↑ⱽ¹[ v ] ] e) ])
β-Prod : {e e' : Exp {n}} {v : Val e} → Prod e e' ↠ ProdV v (↑⁻¹[ ([ 0 ↦ ↑ⱽ¹[ v ] ] e') ])
β-LetE : {e e' : Exp {n}} → (v : Val e) → LetE e e' ↠ (↑⁻¹[ ([ 0 ↦ ↑ⱽ¹[ v ] ] e') ])
β-LetP : {e e' e'' : Exp {n}} → (v : Val e) → (v' : Val e')
→ LetP (ProdV v e') e''
↠
↑ (ℤ.negsuc 1) , 0 [ ([ 0 ↦ ↑ⱽ¹[ v ] ]
([ 0 ↦ shift-val {n} {ℤ.pos 1} {1} v' ] e'')) ]
β-LabE : {s : Subset n} {f : ∀ l → l ∈ s → Exp {n}} {x : Fin n} → (ins : x ∈ s)
→ CaseE (VLab{x = x}) f ↠ f x ins
----------------------------------------------------------------------
----------------------------------------------------------------------
module progress where
open syntx
open substitution
open typing
open semantics
-- To eliminate the possible typing judgement (LabAEx) for case expressions,
-- we need ([] ≢ Γ' ++ ⟨ D , Γ ⟩. Agda does not know that no possible constructor
-- for this equality exists, because _++_ is an arbitrary function and therefore
-- "green slime" (see the link @ (LabAEx) rule).
--
-- Workaround: Argue with length of environments
env-len-++ : {n : ℕ} {Γ Γ' : TEnv {n}} → length (Γ ++ Γ') ≡ length Γ +ᴺ length Γ'
env-len-++ {n} {[]} {Γ'} = refl
env-len-++ {n} {⟨ T , Γ ⟩} {Γ'} = cong ℕ.suc (env-len-++ {n} {Γ} {Γ'})
env-len-> : {n : ℕ} {Γ : TEnv {n}} {T : Ty {n}} → length ⟨ T , Γ ⟩ >ᴺ 0
env-len-> {n} {Γ} {T} = s≤s z≤n
env-len->-++ : {n : ℕ} {Γ Γ' : TEnv {n}} → length Γ' >ᴺ 0 → length (Γ ++ Γ') >ᴺ 0
env-len->-++ {n} {Γ} {⟨ T , Γ' ⟩} gt rewrite (env-len-++ {n} {Γ} {⟨ T , Γ' ⟩})= ≤-trans gt (m≤n+m (length ⟨ T , Γ' ⟩) (length Γ))
env-len-eq : {n : ℕ} {Γ : TEnv {n}} {Γ' : TEnv {n}} → Γ ≡ Γ' → length Γ ≡ length Γ'
env-len-eq {n} {Γ} {.Γ} refl = refl
env-empty-++ : {n : ℕ} {Γ' Γ : TEnv {n}} {D : Ty {n}} → ¬ ([] ≡ Γ' ++ ⟨ D , Γ ⟩)
env-empty-++ {n} {Γ} {Γ'} {D} eq = contradiction (env-len-eq eq) (<⇒≢ (env-len->-++ (env-len->{T = D})))
-- Canonical forms
canonical-forms-pi : {n : ℕ} {e : Exp {n}} {A B D : Ty {n}} → [] ⊢ e ⇒ D → [] ⊢ D ⇓ Pi A B → Val e → (∃[ e' ](e ≡ Abs e'))
canonical-forms-pi {n} {.(Abs _)} {A} {B} {.(Pi _ _)} (PiAI{M = M} j) u v = M , refl
canonical-forms-sigma : {n : ℕ} {e : Exp {n}} {A B D : Ty {n}} → [] ⊢ e ⇒ D → [] ⊢ D ⇓ Sigma A B → Val e → (∃{A = Exp {n}} λ e' → ∃{A = Val e'} λ v → ∃ λ e'' → e ≡ ProdV{e = e'} v e'')
canonical-forms-sigma {n} {.(ProdV _ _)} {A} {B} {.(Sigma _ _)} (PairAI{e = e}{N}{V} j j₁) u v = e , (V , (N , refl))
-- Main theorem
data Progress {n : ℕ} (e : Exp {n}) {T : Ty} {j : [] ⊢ e ⇒ T} : Set where
step : {e' : Exp{n}} → e ↠ e' → Progress e
value : Val e → Progress e
progress : {n : ℕ} {e : Exp {n}} {T : Ty} → (j : [] ⊢ e ⇒ T) → Progress e {T} {j}
progress {n} {Var x} {T} (VarA x₁ x₂) = value VVar
progress {n} {UnitE} {.UnitT} (UnitAI x) = value VUnit
progress {n} {Abs e} {.(Pi _ _)} (PiAI j) = value VFun
progress {n} {App e x} {.([ 0 ↦ x ]ᵀ _)} (PiAE{A = A}{B = B}{D = D} j x₁ x₂ x₃)
with progress {n} {e} {D} j
... | step x₄ = step (ξ-App x₄)
... | value x₄
with canonical-forms-pi {n} {e} {A} {B} {D} j x₁ x₄
... | fst , snd rewrite snd = step (β-App x)
progress {n} {LabI x} {.(Label ⁅ x ⁆)} (LabAI x₁) = value VLab
progress {n} {Prod e e₁} {.(Sigma A B)} (SigmaAI {A = A} {B = B} (SubTypeA{A = A₁} x x₂) x₁ j)
with progress {n} {e} {A₁} x
... | step x₃ = step (ξ-Prod x₃)
... | value x₃ = step (β-Prod{v = x₃})
progress {n} {ProdV x e} {.(Sigma _ _)} (PairAI{A = A} {B = B} j j₁)
with progress {n} {e} {B} j₁
... | step x₁ = step (ξ-ProdV x₁)
... | value x₁ = value (VProd x x₁)
progress {n} {LetP e e₁} {T} (SigmaAE{A = A}{B = B}{D = D} j x j₁ x₁)
with progress {n} {e} {D} j
... | step x₂ = step (ξ-LetP x₂)
... | value x₂
with canonical-forms-sigma {n} {e} {A} {B} {D} j x x₂
... | fst , fst₁ , fst₂ , snd rewrite snd
with x₂
... | VProd v v₁ = step (β-LetP v v₁)
progress {n} {LetE e e₁} {T} (Let{A = A} x j j₁)
with progress {n} {e} {A} j
... | step x₁ = step (ξ-LetE x₁)
... | value x₁ = step (β-LetE x₁)
progress {n} {CaseE {e = .(LabI _)} VLab f} {T} (LabAEl {l = _} {ins = ins} (LabAI x₁) x j₁) = step (β-LabE (x ins))
progress {n} {CaseE .VVar f} {.(CaseT VVar _)} (LabAEx{eq = eq} x x₁) = contradiction eq env-empty-++
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{-# OPTIONS --without-K --safe #-}
open import Relation.Binary.PropositionalEquality using
( _≡_ ; refl ; sym ; trans ; subst ; cong )
module Cats.Category.Discrete {li} {I : Set li}
(I-set : ∀ {i j : I} (p q : i ≡ j) → p ≡ q)
where
open import Data.Product using (Σ-syntax ; _,_ ; proj₁ ; proj₂)
open import Data.Unit using (⊤)
open import Level
open import Cats.Category.Base
open import Cats.Functor using (Functor)
Obj : Set li
Obj = I
data _⇒_ : Obj → Obj → Set li where
id : ∀ {A} → A ⇒ A
⇒-contr : ∀ {A B} (f : A ⇒ B)
→ Σ[ p ∈ A ≡ B ] (subst (_⇒ B) p f ≡ id)
⇒-contr id = refl , refl
⇒-contr′ : ∀ {A} (f : A ⇒ A)
→ f ≡ id
⇒-contr′ f with ⇒-contr f
... | A≡A , f≡id rewrite I-set A≡A refl = f≡id
⇒-prop : ∀ {A B} (f g : A ⇒ B) → f ≡ g
⇒-prop {A} {B} id g = sym (⇒-contr′ g)
_∘_ : ∀ {A B C} → B ⇒ C → A ⇒ B → A ⇒ C
id ∘ id = id
Discrete : Category li li zero
Discrete = record
{ Obj = Obj
; _⇒_ = _⇒_
; _≈_ = λ _ _ → ⊤
; id = id
; _∘_ = _∘_
; equiv = _
; ∘-resp = _
; id-r = _
; id-l = _
; assoc = _
}
functor : ∀ {lo la l≈} {C : Category lo la l≈}
→ (I → Category.Obj C)
→ Functor Discrete C
functor {C = C} f = record
{ fobj = f
; fmap = fmap
; fmap-resp = λ {A} {B} {g} {h} _ → C.≈.reflexive (cong fmap (⇒-prop g h))
; fmap-id = C.≈.refl
; fmap-∘ = λ { {f = id} {id} → C.id-l }
}
where
module C = Category C
fmap : ∀ {A B} (g : A ⇒ B) → f A C.⇒ f B
fmap id = C.id
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{-# OPTIONS --cubical --safe --guardedness #-}
module Cubical.Codata.M where
open import Cubical.Foundations.Prelude
-- TODO move
module Helpers where
module _ {ℓ ℓ'} {A : Type ℓ} {x : A} (P : ∀ y → x ≡ y → Type ℓ') (d : P x refl) where
-- A version with y as an explicit argument can be used to make Agda
-- infer the family P.
J' : ∀ y (p : x ≡ y) → P y p
J' y p = J P d p
lem-transp : ∀ {ℓ} {A : Type ℓ} (i : I) → (B : Type ℓ) (P : A ≡ B) → (p : P i) → PathP (\ j → P j) (transp (\ k → P (~ k ∧ i)) (~ i) p)
(transp (\ k → P (k ∨ i)) i p)
lem-transp {A = A} i = J' _ (\ p j → transp (\ _ → A) ((~ j ∧ ~ i) ∨ (j ∧ i)) p )
transp-over : ∀ {ℓ} (A : I → Type ℓ) (i j : I) → A i → A j
transp-over A i k p = transp (\ j → A ((~ j ∧ i) ∨ (j ∧ k))) (~ i ∧ ~ k) p
transp-over-1 : ∀ {ℓ} (A : I → Type ℓ) (i j : I) → A i → A j
transp-over-1 A i k p = transp (\ j → A ((j ∨ i) ∧ (~ j ∨ k))) (i ∧ k) p
compPathD : {ℓ ℓ' : _} {X : Type ℓ} (F : X → Type ℓ') {A B C : X} (P : A ≡ B) (Q : B ≡ C)
→ ∀ {x y z} → (\ i → F (P i)) [ x ≡ y ] → (\ i → F (Q i)) [ y ≡ z ] → (\ i → F ((P ∙ Q) i)) [ x ≡ z ]
compPathD F {A = A} P Q {x} p q i =
comp (\ j → F (hfill (λ j → \ { (i = i0) → A ; (i = i1) → Q j })
(inS (P i))
j))
(λ j → \ { (i = i0) → x; (i = i1) → q j })
(p i)
open Helpers
IxCont : Type₀ → Type₁
IxCont X = Σ (X → Type₀) \ S → ∀ x → S x → X → Type₀
⟦_⟧ : ∀ {X : Type₀} → IxCont X → (X → Type₀) → (X → Type₀)
⟦ (S , P) ⟧ X x = Σ (S x) \ s → ∀ y → P x s y → X y
record M {X : Type₀} (C : IxCont X) (x : X) : Type₀ where
coinductive
field
head : C .fst x
tails : ∀ y → C .snd x head y → M C y
open M public
module _ {X : Type₀} {C : IxCont X} where
private
F = ⟦ C ⟧
out : ∀ x → M C x → F (M C) x
out x a = (a .head) , (a .tails)
mapF : ∀ {A B} → (∀ x → A x → B x) → ∀ x → F A x → F B x
mapF f x (s , t) = s , \ y p → f _ (t y p)
unfold : ∀ {A} (α : ∀ x → A x → F A x) → ∀ x → A x → M C x
unfold α x a .head = α x a .fst
unfold α x a .tails y p = unfold α y (α x a .snd y p)
-- We generalize the type to avoid upsetting --guardedness by
-- transporting after the corecursive call.
-- Recognizing hcomp/transp as guardedness-preserving could be a better solution.
unfold-η' : ∀ {A} (α : ∀ x → A x → F A x) → (h : ∀ x → A x → M C x)
→ (∀ (x : X) (a : A x) → out x (h x a) ≡ mapF h x (α x a))
→ ∀ (x : X) (a : A x) m → h x a ≡ m → m ≡ unfold α x a
unfold-η' α h eq x a m eq' = let heq = cong head (sym eq') ∙ cong fst (eq x a)
in \ where
i .head → heq i
i .tails y p →
let p0 = (transp-over-1 (\ k → C .snd x (heq k) y) i i1 p)
p1 = (transp-over (\ k → C .snd x (heq k) y) i i0 p)
pe = lem-transp i _ (\ k → C .snd x (heq k) y) p
tl = compPathD (λ p → C .snd x p y → M C y)
(cong head (sym eq')) (cong fst (eq x a))
(cong (\ f → f .tails y) (sym eq'))
(cong (\ f → f .snd y) (eq x a))
in unfold-η' α h eq y (α x a .snd y p0)
(m .tails y p1)
(sym (\ k → tl k (pe k)))
i
unfold-η : ∀ {A} (α : ∀ x → A x → F A x) → (h : ∀ x → A x → M C x)
→ (∀ (x : X) (a : A x) → out x (h x a) ≡ mapF h x (α x a))
→ ∀ (x : X) (a : A x) → h x a ≡ unfold α x a
unfold-η α h eq x a = unfold-η' α h eq x a _ refl
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-- Andreas, 2019-04-13, issue #3692, reported by xekoukou
-- Feature #1086, omission of trivially absurd clauses,
-- caused wrong polarity computation.
-- {-# OPTIONS -v tc.polarity:20 #-}
-- {-# OPTIONS -v tc.cover.missing:100 #-}
open import Agda.Builtin.Equality
open import Agda.Builtin.Nat
data _≤_ : (m n : Nat) → Set where
z≤n : ∀ {n} → zero ≤ n
s≤s : ∀ {m n} (m≤n : m ≤ n) → suc m ≤ suc n
≤-pred : ∀ {m n} → suc m ≤ suc n → m ≤ n
≤-pred (s≤s m≤n) = m≤n
data Fin : Nat → Set where
fzero : (n : Nat) → Fin (suc n)
fsuc : (n : Nat) (i : Fin n) → Fin (suc n)
inject≤ : ∀ m n → Fin m → m ≤ n → Fin n
inject≤ m (suc n) (fzero _) le = fzero _
inject≤ m (suc n) (fsuc m' i) le = fsuc _ (inject≤ m' n i (≤-pred le))
-- Agda 2.6.0 accepts inject≤ without the following clauses,
-- leading to faulty acceptance of test below.
-- inject≤ (suc _) zero (fzero _) ()
-- inject≤ (suc _) zero (fsuc _ _) ()
test : ∀ {m n} (i : Fin m) (m≤n m≤n' : m ≤ n)
→ inject≤ m n i m≤n ≡ inject≤ m n i m≤n'
test i m≤n m≤n' = refl
-- WAS: succeeded because of wrong polarity for inject≤
-- Expected error:
-- m≤n != m≤n' of type m ≤ n
-- when checking that the expression refl has type
-- inject≤ m n i m≤n ≡ inject≤ m n i m≤n'
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{-# OPTIONS --safe #-}
module Cubical.Algebra.Group.Instances.IntMod where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.HLevels
open import Cubical.Data.Empty as ⊥
open import Cubical.Data.Unit
open import Cubical.Data.Bool hiding (isProp≤)
open import Cubical.Data.Nat renaming (_+_ to _+ℕ_)
open import Cubical.Data.Nat.Mod
open import Cubical.Data.Nat.Order
open import Cubical.Data.Int renaming (_+_ to _+ℤ_)
open import Cubical.Data.Fin
open import Cubical.Data.Fin.Arithmetic
open import Cubical.Data.Sigma
open import Cubical.Algebra.Monoid.Base
open import Cubical.Algebra.Semigroup.Base
open import Cubical.Algebra.Group.Base
open import Cubical.Algebra.Group.MorphismProperties
open import Cubical.Algebra.Group.Morphisms
open import Cubical.Algebra.Group.Properties
open import Cubical.Algebra.Group.Instances.Unit
open import Cubical.Algebra.Group.Instances.Bool
open import Cubical.Algebra.Group.Instances.Int
open GroupStr
open IsGroup
open IsMonoid
ℤGroup/_ : ℕ → Group₀
ℤGroup/ zero = ℤGroup
fst (ℤGroup/ suc n) = Fin (suc n)
1g (snd (ℤGroup/ suc n)) = 0
GroupStr._·_ (snd (ℤGroup/ suc n)) = _+ₘ_
inv (snd (ℤGroup/ suc n)) = -ₘ_
isGroup (snd (ℤGroup/ suc n)) = makeIsGroup
isSetFin
(λ x y z → sym (+ₘ-assoc x y z))
+ₘ-rUnit
+ₘ-lUnit
+ₘ-rCancel
+ₘ-lCancel
ℤGroup/1≅Unit : GroupIso (ℤGroup/ 1) UnitGroup₀
ℤGroup/1≅Unit = contrGroupIsoUnit isContrFin1
Bool≅ℤGroup/2 : GroupIso BoolGroup (ℤGroup/ 2)
Iso.fun (fst Bool≅ℤGroup/2) false = 1
Iso.fun (fst Bool≅ℤGroup/2) true = 0
Iso.inv (fst Bool≅ℤGroup/2) (zero , p) = true
Iso.inv (fst Bool≅ℤGroup/2) (suc zero , p) = false
Iso.inv (fst Bool≅ℤGroup/2) (suc (suc x) , p) =
⊥.rec (¬-<-zero (predℕ-≤-predℕ (predℕ-≤-predℕ p)))
Iso.rightInv (fst Bool≅ℤGroup/2) (zero , p) =
Σ≡Prop (λ _ → isProp≤) refl
Iso.rightInv (fst Bool≅ℤGroup/2) (suc zero , p) =
Σ≡Prop (λ _ → isProp≤) refl
Iso.rightInv (fst Bool≅ℤGroup/2) (suc (suc x) , p) =
⊥.rec (¬-<-zero (predℕ-≤-predℕ (predℕ-≤-predℕ p)))
Iso.leftInv (fst Bool≅ℤGroup/2) false = refl
Iso.leftInv (fst Bool≅ℤGroup/2) true = refl
snd Bool≅ℤGroup/2 =
makeIsGroupHom λ { false false → refl
; false true → refl
; true false → refl
; true true → refl}
ℤGroup/2≅Bool : GroupIso (ℤGroup/ 2) BoolGroup
ℤGroup/2≅Bool = invGroupIso Bool≅ℤGroup/2
-- Definition of the quotient map homomorphism ℤ → ℤGroup/ (suc n)
-- as a group homomorphism.
ℤ→Fin : (n : ℕ) → ℤ → Fin (suc n)
ℤ→Fin n (pos x) = x mod (suc n) , mod< n x
ℤ→Fin n (negsuc x) = -ₘ (suc x mod suc n , mod< n (suc x))
ℤ→Fin-presinv : (n : ℕ) (x : ℤ) → ℤ→Fin n (- x) ≡ -ₘ ℤ→Fin n x
ℤ→Fin-presinv n (pos zero) =
Σ≡Prop (λ _ → isProp≤) ((λ _ → zero) ∙ sym (cong fst help))
where
help : (-ₘ_ {n = n} 0) ≡ 0
help = GroupTheory.inv1g (ℤGroup/ (suc n))
ℤ→Fin-presinv n (pos (suc x)) = Σ≡Prop (λ _ → isProp≤) refl
ℤ→Fin-presinv n (negsuc x) =
sym (GroupTheory.invInv (ℤGroup/ (suc n)) _)
-ₘ1-id : (n : ℕ)
→ Path (Fin (suc n)) (-ₘ (1 mod (suc n) , mod< n 1))
(n mod (suc n) , mod< n n)
-ₘ1-id zero = refl
-ₘ1-id (suc n) =
cong -ₘ_ (FinPathℕ ((1 mod suc (suc n)) , mod< (suc n) 1) 1
(modIndBase (suc n) 1 (n , +-comm n 2)) .snd)
∙ Σ≡Prop (λ _ → isProp≤)
((+inductionBase (suc n) _
(λ x _ → ((suc (suc n)) ∸ x) mod (suc (suc n))) λ _ x → x) 1
(n , (+-comm n 2)))
suc-ₘ1 : (n y : ℕ)
→ ((suc y mod suc n) , mod< n (suc y)) -ₘ (1 mod (suc n) , mod< n 1)
≡ (y mod suc n , mod< n y)
suc-ₘ1 zero y =
isContr→isProp
(isOfHLevelRetractFromIso 0 (fst ℤGroup/1≅Unit) isContrUnit) _ _
suc-ₘ1 (suc n) y =
(λ i → ((suc y mod suc (suc n)) , mod< (suc n) (suc y))
+ₘ (-ₘ1-id (suc n) i))
∙ Σ≡Prop (λ _ → isProp≤)
(cong (_mod (2 +ℕ n))
(cong (_+ℕ (suc n) mod (2 +ℕ n))
(mod+mod≡mod (suc (suc n)) 1 y))
∙∙ sym (mod+mod≡mod (suc (suc n))
((1 mod suc (suc n))
+ℕ (y mod suc (suc n))) (suc n))
∙∙ (mod-rCancel (suc (suc n)) ((1 mod suc (suc n))
+ℕ (y mod suc (suc n))) (suc n)
∙ cong (_mod (suc (suc n)))
(cong (_+ℕ (suc n mod suc (suc n)))
(+-comm (1 mod suc (suc n)) (y mod suc (suc n)))
∙ sym (+-assoc (y mod suc (suc n))
(1 mod suc (suc n)) (suc n mod suc (suc n))))
∙∙ mod-rCancel (suc (suc n)) (y mod suc (suc n))
((1 mod suc (suc n)) +ℕ (suc n mod suc (suc n)))
∙∙ (cong (_mod (2 +ℕ n))
(cong ((y mod suc (suc n)) +ℕ_)
(sym (mod+mod≡mod (suc (suc n)) 1 (suc n))
∙ zero-charac (suc (suc n)))
∙ +-comm _ 0)
∙ mod-idempotent y)))
1-ₘsuc : (n y : ℕ)
→ ((1 mod (suc n) , mod< n 1)
+ₘ (-ₘ (((suc y mod suc n) , mod< n (suc y)))))
≡ -ₘ ((y mod suc n) , mod< n y)
1-ₘsuc n y =
sym (GroupTheory.invInv (ℤGroup/ (suc n)) _)
∙ cong -ₘ_
(GroupTheory.invDistr (ℤGroup/ (suc n))
(modInd n 1 , mod< n 1) (-ₘ (modInd n (suc y) , mod< n (suc y)))
∙ cong (_-ₘ (modInd n 1 , mod< n 1))
(GroupTheory.invInv (ℤGroup/ (suc n)) (modInd n (suc y) , mod< n (suc y)))
∙ suc-ₘ1 n y)
isHomℤ→Fin : (n : ℕ) → IsGroupHom (snd ℤGroup) (ℤ→Fin n) (snd (ℤGroup/ (suc n)))
isHomℤ→Fin n =
makeIsGroupHom
λ { (pos x) y → pos+case x y
; (negsuc x) (pos y) →
cong (ℤ→Fin n) (+Comm (negsuc x) (pos y))
∙∙ pos+case y (negsuc x)
∙∙ +ₘ-comm (ℤ→Fin n (pos y)) (ℤ→Fin n (negsuc x))
; (negsuc x) (negsuc y) →
sym (cong (ℤ→Fin n) (-Dist+ (pos (suc x)) (pos (suc y))))
∙∙ ℤ→Fin-presinv n (pos (suc x) +ℤ (pos (suc y)))
∙∙ cong -ₘ_ (pos+case (suc x) (pos (suc y)))
∙∙ GroupTheory.invDistr (ℤGroup/ (suc n))
(modInd n (suc x)
, mod< n (suc x)) (modInd n (suc y) , mod< n (suc y))
∙∙ +ₘ-comm (ℤ→Fin n (negsuc y)) (ℤ→Fin n (negsuc x))}
where
+1case : (y : ℤ) → ℤ→Fin n (1 +ℤ y) ≡ ℤ→Fin n 1 +ₘ ℤ→Fin n y
+1case (pos zero) = sym (GroupStr.·IdR (snd (ℤGroup/ (suc n))) _)
+1case (pos (suc y)) =
cong (ℤ→Fin n) (+Comm 1 (pos (suc y)))
∙ Σ≡Prop (λ _ → isProp≤) (mod+mod≡mod (suc n) 1 (suc y))
+1case (negsuc zero) =
Σ≡Prop (λ _ → isProp≤) refl
∙ sym (GroupStr.·InvR (snd (ℤGroup/ (suc n))) (modInd n 1 , mod< n 1))
+1case (negsuc (suc y)) =
Σ≡Prop (λ _ → isProp≤)
(cong fst (cong (ℤ→Fin n) (+Comm 1 (negsuc (suc y))))
∙∙ cong fst (cong -ₘ_ (refl {x = suc y mod suc n , mod< n (suc y)}))
∙∙ cong fst (sym (1-ₘsuc n (suc y)))
∙ λ i → fst ((1 mod (suc n) , mod< n 1)
+ₘ (-ₘ (((suc (suc y) mod suc n) , mod< n (suc (suc y)))))))
pos+case : (x : ℕ) (y : ℤ)
→ ℤ→Fin n (pos x +ℤ y) ≡ ℤ→Fin n (pos x) +ₘ ℤ→Fin n y
pos+case zero y =
cong (ℤ→Fin n) (+Comm 0 y)
∙ sym (GroupStr.·IdL (snd (ℤGroup/ (suc n))) (ℤ→Fin n y))
pos+case (suc zero) y = +1case y
pos+case (suc (suc x)) y =
cong (ℤ→Fin n) (cong (_+ℤ y) (+Comm (pos (suc x)) 1)
∙ sym (+Assoc 1 (pos (suc x)) y))
∙∙ +1case (pos (suc x) +ℤ y)
∙∙ (cong ((modInd n 1 , mod< n 1) +ₘ_) (pos+case (suc x) y)
∙∙ sym (+ₘ-assoc (modInd n 1 , mod< n 1)
(modInd n (suc x) , mod< n (suc x)) (ℤ→Fin n y))
∙∙ cong (_+ₘ ℤ→Fin n y) (lem x))
where
lem : (x : ℕ)
→ (modInd n 1 , mod< n 1) +ₘ (modInd n (suc x) , mod< n (suc x))
≡ ℤ→Fin n (pos (suc (suc x)))
lem x =
Σ≡Prop (λ _ → isProp≤) (sym (mod+mod≡mod (suc n) 1 (suc x)))
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{-# OPTIONS --without-K --safe #-}
open import Categories.Category
module Categories.Functor.Slice {o ℓ e} (C : Category o ℓ e) where
open import Data.Product using (_,_)
open import Categories.Adjoint
open import Categories.Category.CartesianClosed
open import Categories.Category.CartesianClosed.Locally
open import Categories.Functor
open import Categories.Functor.Properties
open import Categories.Morphism.Reasoning C
open import Categories.NaturalTransformation
import Categories.Category.Slice as S
import Categories.Diagram.Pullback as P
import Categories.Category.Construction.Pullbacks as Pbs
open Category C
open HomReasoning
module _ {A : Obj} where
open S.SliceObj
open S.Slice⇒
Base-F : ∀ {o′ ℓ′ e′} {D : Category o′ ℓ′ e′} (F : Functor C D) → Functor (S.Slice C A) (S.Slice D (Functor.F₀ F A))
Base-F {D = D} F = record
{ F₀ = λ { (S.sliceobj arr) → S.sliceobj (F₁ arr) }
; F₁ = λ { (S.slicearr △) → S.slicearr ([ F ]-resp-∘ △) }
; identity = identity
; homomorphism = homomorphism
; F-resp-≈ = F-resp-≈
}
where module D = Category D
open Functor F
open S C
Forgetful : Functor (Slice A) C
Forgetful = record
{ F₀ = λ X → Y X
; F₁ = λ f → h f
; identity = refl
; homomorphism = refl
; F-resp-≈ = λ eq → eq
}
BaseChange! : ∀ {B} (f : B ⇒ A) → Functor (Slice B) (Slice A)
BaseChange! f = record
{ F₀ = λ X → sliceobj (f ∘ arr X)
; F₁ = λ g → slicearr (pullʳ (△ g))
; identity = refl
; homomorphism = refl
; F-resp-≈ = λ eq → eq
}
module _ (pullbacks : ∀ {X Y Z} (h : X ⇒ Z) (i : Y ⇒ Z) → P.Pullback C h i) where
private
open P C
module pullbacks {X Y Z} h i = Pullback (pullbacks {X} {Y} {Z} h i)
open pullbacks
BaseChange* : ∀ {B} (f : B ⇒ A) → Functor (Slice A) (Slice B)
BaseChange* f = record
{ F₀ = λ X → sliceobj (p₂ (arr X) f)
; F₁ = λ {X Y} g → slicearr {h = Pullback.p₂ (unglue (pullbacks (arr Y) f)
(Pullback-resp-≈ (pullbacks (arr X) f) (△ g) refl))}
(p₂∘universal≈h₂ (arr Y) f)
; identity = λ {X} → ⟺ (unique (arr X) f id-comm identityʳ)
; homomorphism = λ {X Y Z} {h i} → unique-diagram (arr Z) f (p₁∘universal≈h₁ (arr Z) f ○ assoc ○ ⟺ (pullʳ (p₁∘universal≈h₁ (arr Y) f)) ○ ⟺ (pullˡ (p₁∘universal≈h₁ (arr Z) f)))
(p₂∘universal≈h₂ (arr Z) f ○ ⟺ (p₂∘universal≈h₂ (arr Y) f) ○ ⟺ (pullˡ (p₂∘universal≈h₂ (arr Z) f)))
; F-resp-≈ = λ {X Y} eq″ → unique (arr Y) f (p₁∘universal≈h₁ (arr Y) f ○ ∘-resp-≈ˡ eq″) (p₂∘universal≈h₂ (arr Y) f)
}
!⊣* : ∀ {B} (f : B ⇒ A) → BaseChange! f ⊣ BaseChange* f
!⊣* f = record
{ unit = ntHelper record
{ η = λ X → slicearr (p₂∘universal≈h₂ (f ∘ arr X) f {eq = identityʳ})
; commute = λ {X Y} g → unique-diagram (f ∘ arr Y) f
(cancelˡ (p₁∘universal≈h₁ (f ∘ arr Y) f) ○ ⟺ (cancelʳ (p₁∘universal≈h₁ (f ∘ arr X) f)) ○ pushˡ (⟺ (p₁∘universal≈h₁ (f ∘ arr Y) f)))
(pullˡ (p₂∘universal≈h₂ (f ∘ arr Y) f) ○ △ g ○ ⟺ (p₂∘universal≈h₂ (f ∘ arr X) f) ○ pushˡ (⟺ (p₂∘universal≈h₂ (f ∘ arr Y) f)))
}
; counit = ntHelper record
{ η = λ X → slicearr (pullbacks.commute (arr X) f)
; commute = λ {X Y} g → p₁∘universal≈h₁ (arr Y) f
}
; zig = λ {X} → p₁∘universal≈h₁ (f ∘ arr X) f
; zag = λ {Y} → unique-diagram (arr Y) f
(pullˡ (p₁∘universal≈h₁ (arr Y) f) ○ pullʳ (p₁∘universal≈h₁ (f ∘ pullbacks.p₂ (arr Y) f) f))
(pullˡ (p₂∘universal≈h₂ (arr Y) f) ○ p₂∘universal≈h₂ (f ∘ pullbacks.p₂ (arr Y) f) f ○ ⟺ identityʳ)
}
pullback-functorial : ∀ {B} (f : B ⇒ A) → Functor (Slice A) C
pullback-functorial f = record
{ F₀ = λ X → p.P X
; F₁ = λ f → p⇒ _ _ f
; identity = λ {X} → sym (p.unique X id-comm id-comm)
; homomorphism = λ {_ Y Z} →
p.unique-diagram Z (p.p₁∘universal≈h₁ Z ○ ⟺ identityˡ ○ ⟺ (pullʳ (p.p₁∘universal≈h₁ Y)) ○ ⟺ (pullˡ (p.p₁∘universal≈h₁ Z)))
(p.p₂∘universal≈h₂ Z ○ assoc ○ ⟺ (pullʳ (p.p₂∘universal≈h₂ Y)) ○ ⟺ (pullˡ (p.p₂∘universal≈h₂ Z)))
; F-resp-≈ = λ {_ B} {h i} eq →
p.unique-diagram B (p.p₁∘universal≈h₁ B ○ ⟺ (p.p₁∘universal≈h₁ B))
(p.p₂∘universal≈h₂ B ○ ∘-resp-≈ˡ eq ○ ⟺ (p.p₂∘universal≈h₂ B))
}
where p : ∀ X → Pullback f (arr X)
p X = pullbacks f (arr X)
module p X = Pullback (p X)
p⇒ : ∀ X Y (g : Slice⇒ X Y) → p.P X ⇒ p.P Y
p⇒ X Y g = Pbs.Pullback⇒.pbarr pX⇒pY
where pX : Pbs.PullbackObj C A
pX = record { pullback = p X }
pY : Pbs.PullbackObj C A
pY = record { pullback = p Y }
pX⇒pY : Pbs.Pullback⇒ C A pX pY
pX⇒pY = record
{ mor₁ = Category.id C
; mor₂ = h g
; commute₁ = identityʳ
; commute₂ = △ g
}
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module Data.Vec.Any.Properties where
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as P using (_≡_; _≢_; inspect; refl)
open import Relation.Unary renaming (_⊆_ to _⋐_) using ()
open import Data.Product as Prod hiding (map; swap)
open import Data.Vec as Vec
open import Data.Vec.Any as Any
open import Data.Vec.Any.Membership.Propositional
--open import Data.Vec.Membership
open import Data.Bool.Base using (Bool; false; true; T)
open import Data.Bool.Properties
open import Data.Fin using (Fin) renaming (zero to fzero; suc to fsuc)
open import Data.Nat
open import Data.Nat.Properties
open import Data.Empty
open import Data.Sum as Sum hiding (map)
open import Function using (_∘_; _$_; id; flip; const)
open import Function.Inverse as Inv using (_↔_)
open import Function.Equality using (_⟨$⟩_)
open import Function.Equivalence as Eq using (_⇔_ ; module Equivalence)
open import Function.Related as Related using (Related)
open Related.EquationalReasoning
open import Relation.Binary.Sum
map-id : ∀ {a p n} {A : Set a} {P : A → Set p} (f : P ⋐ P) {xs : Vec A n} →
(∀ {x} (p : P x) → f p ≡ p) →
(p : Any P xs) → Any.map f p ≡ p
map-id f hyp (here px) = P.cong here $ hyp px
map-id f hyp (there p) = P.cong there $ map-id f hyp p
map-∘ : ∀ {a p q r n}{A : Set a}{P : A → Set p}{Q : A → Set q}{R : A → Set r}
(f : Q ⋐ R)(g : P ⋐ Q) {xs : Vec A n}(p : Any P xs) →
Any.map (f ∘ g) p ≡ Any.map f (Any.map g p)
map-∘ f g (here px) = P.refl
map-∘ f g (there p) = P.cong there $ map-∘ f g p
map∘find′ : ∀ {a n p}{A : Set a} {P : A → Set p}{xs : Vec A n} (p : Any P xs) →
let (x , x∈xs , px) = find′ p in
{f : (x ≡_) ⋐ P } →
f P.refl ≡ px →
Any.map f x∈xs ≡ p
map∘find′ (here px) hyp = P.cong here hyp
map∘find′ (there pxs) hyp = P.cong there $ map∘find′ pxs hyp
find′∘map : ∀ {a n p q}{A : Set a}{P : A → Set p}{Q : A → Set q}{xs : Vec A n}(p : Any P xs)(f : P ⋐ Q) →
find′ (Any.map f p) ≡ Prod.map id (Prod.map id f) (find′ p)
find′∘map (here px) f = P.refl
find′∘map (there pxs) f rewrite find′∘map pxs f = P.refl
find′-∈′ : ∀ {a n}{A : Set a}{x}{xs : Vec A n}(x∈xs : x ∈′ xs) →
find′ x∈xs ≡ (x , x∈xs , P.refl)
find′-∈′ (here P.refl) = P.refl
find′-∈′ (there x∈xs) rewrite find′-∈′ x∈xs = P.refl
lose′∘find′ : ∀ {a n p}{A : Set a} {P : A → Set p}{xs : Vec A n} (p : Any P xs) →
uncurry′ lose′ (proj₂ (find′ p)) ≡ p
lose′∘find′ (here px) = P.refl
lose′∘find′ (there p) = P.cong there $ lose′∘find′ p
find′∘lose′ : ∀ {a n p}{A : Set a} (P : A → Set p){x}{xs : Vec A n} (x∈xs : x ∈′ xs)(px : P x) →
find′ {P = P} (lose′ x∈xs px) ≡ (x , x∈xs , px)
find′∘lose′ P x∈xs px rewrite find′∘map x∈xs (flip (P.subst P) px) | find′-∈′ x∈xs = P.refl
∃∈-Any : ∀ {a p n}{A : Set a} {P : A → Set p}{xs : Vec A n} →
(∃ λ x → x ∈′ xs × P x) → Any P xs
∃∈-Any = uncurry′ lose′ ∘ proj₂
Any↔ : ∀ {a p n}{A : Set a} {P : A → Set p}{xs : Vec A n} →
(∃ λ x → x ∈′ xs × P x) ↔ Any P xs
Any↔ {P = P}{xs} = record {
to = P.→-to-⟶ ∃∈-Any
; from = P.→-to-⟶ (find′ {P = P})
; inverse-of = record {
left-inverse-of = λ { (x , x∈xs , px) → find′∘lose′ P x∈xs px}
; right-inverse-of = lose′∘find′
}
}
Any-cong : ∀ {k a p m n} {A : Set a} {P₁ P₂ : A → Set p} {xs₁ : Vec A m} {xs₂ : Vec A n} →
(∀ x → Related k (P₁ x) (P₂ x)) → xs₁ ∼[ k ] xs₂ → Related k (Any P₁ xs₁) (Any P₂ xs₂)
Any-cong {P₁ = P₁}{P₂}{xs₁}{xs₂} P₁∼P₂ xs₁∼xs₂ =
Any P₁ xs₁ ↔⟨ sym $ Any↔ {P = P₁} ⟩ (∃ λ x → x ∈′ xs₁ × P₁ x)
∼⟨ Σ.cong Inv.id (xs₁∼xs₂ ×-cong P₁∼P₂ _) ⟩(∃ λ x → x ∈′ xs₂ × P₂ x)
↔⟨ Any↔ {P = P₂} ⟩ Any P₂ xs₂ ∎
where
open import Relation.Binary.Product.Pointwise
import Relation.Binary.Sigma.Pointwise as Σ
swap : ∀ {a b p}{A : Set a}{B : Set b}{P : A → B → Set p}
{m n}{xs : Vec A m}{ys : Vec B n} → Any (λ x → Any (P x) ys) xs → Any (λ y → (Any (flip P y) xs)) ys
swap (here pys) = Any.map here pys
swap (there pxys) = Any.map there $ swap pxys
swap-there : ∀ {a b p}{A : Set a}{B : Set b}{P : A → B → Set p}
{m n x}{xs : Vec A m}{ys : Vec B n}(any : Any (λ x → Any (P x) ys) xs) →
swap (Any.map (there {x = x}) any) ≡ there (swap any)
swap-there (here pys) = P.refl
swap-there (there pxys) = P.cong (Any.map there) $ swap-there pxys
swap-invol : ∀ {a b p}{A : Set a}{B : Set b}{P : A → B → Set p}
{m n}{xs : Vec A m}{ys : Vec B n}(any : Any (λ x → Any (P x) ys) xs) →
swap (swap any) ≡ any
swap-invol (here (here px)) = P.refl
swap-invol (here (there pys)) = P.cong (Any.map there) $ swap-invol (here pys)
swap-invol (there pxys) = P.trans (swap-there (swap pxys)) $ P.cong there $ swap-invol pxys
swap↔ : ∀ {a b p}{A : Set a}{B : Set b}{P : A → B → Set p}
{m n}{xs : Vec A m}{ys : Vec B n} →
Any (λ x → Any (P x) ys) xs ↔ Any (λ y → (Any (flip P y) xs)) ys
swap↔ = record {
to = P.→-to-⟶ swap
; from = P.→-to-⟶ swap
; inverse-of = record {
left-inverse-of = swap-invol
; right-inverse-of = swap-invol
}
}
⊥↔Any⊥ : ∀ {a n}{A : Set a}{xs : Vec A n} → ⊥ ↔ Any (const ⊥) xs
⊥↔Any⊥ {A = A} = record {
to = P.→-to-⟶ λ()
; from = P.→-to-⟶ (λ p → from p)
; inverse-of = record {
left-inverse-of = λ()
; right-inverse-of = λ p → from p
}
}
where
from : ∀ {n}{xs : Vec A n} → Any (const ⊥) xs → ∀ {b} {B : Set b} → B
from (here ())
from (there p) = from p
⊥↔Any[] : ∀ {a}{A : Set a}{P : A → Set} → ⊥ ↔ Any P []
⊥↔Any[] = record {
to = P.→-to-⟶ λ()
; from = P.→-to-⟶ λ()
; inverse-of = record {
left-inverse-of = λ()
; right-inverse-of = λ()
}
}
-- any⇔
module _ {a p q}{A : Set a}{P : A → Set p}{Q : A → Set q} where
Any-⊎⁺ : ∀{n}{xs : Vec A n} → Any P xs ⊎ Any Q xs → Any (λ x → P x ⊎ Q x) xs
Any-⊎⁺ = [ Any.map inj₁ , Any.map inj₂ ]′
Any-⊎⁻ : ∀{n}{xs : Vec A n} → Any (λ x → P x ⊎ Q x) xs → Any P xs ⊎ Any Q xs
Any-⊎⁻ (here (inj₁ px)) = inj₁ (here px)
Any-⊎⁻ (here (inj₂ qx)) = inj₂ (here qx)
Any-⊎⁻ (there p) = Sum.map there there (Any-⊎⁻ p)
⊎↔ : ∀ {n}{xs : Vec A n} → (Any P xs ⊎ Any Q xs) ↔ Any (λ x → P x ⊎ Q x) xs
⊎↔ = record
{ to = P.→-to-⟶ Any-⊎⁺
; from = P.→-to-⟶ Any-⊎⁻
; inverse-of = record
{ left-inverse-of = from∘to
; right-inverse-of = to∘from
}
}
where
from∘to : ∀ {n}{xs : Vec A n} (p : Any P xs ⊎ Any Q xs) → Any-⊎⁻ (Any-⊎⁺ p) ≡ p
from∘to (inj₁ (here p)) = P.refl
from∘to (inj₁ (there p)) rewrite from∘to (inj₁ p) = P.refl
from∘to (inj₂ (here q)) = P.refl
from∘to (inj₂ (there q)) rewrite from∘to (inj₂ q) = P.refl
to∘from : ∀ {n}{xs : Vec A n} (p : Any (λ x → P x ⊎ Q x) xs) →
Any-⊎⁺ (Any-⊎⁻ p) ≡ p
to∘from (here (inj₁ p)) = P.refl
to∘from (here (inj₂ q)) = P.refl
to∘from (there p) with Any-⊎⁻ p | to∘from p
to∘from (there .(Any.map inj₁ p)) | inj₁ p | P.refl = P.refl
to∘from (there .(Any.map inj₂ q)) | inj₂ q | P.refl = P.refl
module _ {a b p q} {A : Set a} {B : Set b}
{P : A → Set p} {Q : B → Set q} where
Any-×⁺ : ∀ {m n}{xs : Vec A m} {ys : Vec B n} → Any P xs × Any Q ys →
Any (λ x → Any (λ y → P x × Q y) ys) xs
Any-×⁺ (p , q) = Any.map (λ p → Any.map (λ q → (p , q)) q) p
Any-×⁻ : ∀ {m n} {xs : Vec A m}{ys : Vec B n} → Any (λ x → Any (λ y → P x × Q y) ys) xs →
Any P xs × Any Q ys
Any-×⁻ pq with Prod.map id (Prod.map id find′) (find′ pq)
... | (x , x∈xs , y , y∈ys , p , q) = (lose′ x∈xs p , lose′ y∈ys q)
×↔ : ∀ {m n} {xs : Vec A m}{ys : Vec B n} →
(Any P xs × Any Q ys) ↔ Any (λ x → Any (λ y → P x × Q y) ys) xs
×↔ {m}{n}{xs} {ys} = record
{ to = P.→-to-⟶ Any-×⁺
; from = P.→-to-⟶ Any-×⁻
; inverse-of = record
{ left-inverse-of = from∘to
; right-inverse-of = to∘from
}
}
where
from∘to : ∀ (pq : Any P xs × Any Q ys)
→ Any-×⁻ (Any-×⁺ pq) ≡ pq
from∘to (p , q)
rewrite find′∘map {Q = λ x → Any (λ y → P x × Q y) ys}
p (λ px → Any.map (λ q → (px , q)) q)
| find′∘map {Q = λ y → P (proj₁ (find′ p)) × Q y}
q (λ q → proj₂ (proj₂ (find′ p)) , q)
| lose′∘find′ p
| lose′∘find′ q
= P.refl
to∘from : ∀ (pq : Any (λ x → Any (λ y → P x × Q y) ys) xs) → Any-×⁺ (Any-×⁻ pq) ≡ pq
to∘from pq with find′ pq
| (λ (f : (proj₁ (find′ pq) ≡_) ⋐ _) → map∘find′ pq {f})
... | (x , x∈xs , pq′) | lem₁
with find′ pq′
| (λ (f : (proj₁ (find′ pq′) ≡_) ⋐ _) → map∘find′ pq′ {f})
... | (y , y∈ys , p , q) | lem₂
rewrite P.sym $ map-∘ {R = λ x → Any (λ y → P x × Q y) ys}
(λ p → Any.map (λ q → p , q) (lose′ y∈ys q))
(λ y → P.subst P y p)
x∈xs
= lem₁ _ helper
where
helper : Any.map (λ q → p , q) (lose′ y∈ys q) ≡ pq′
helper rewrite P.sym $ map-∘ {R = λ y → P x × Q y}
(λ q → p , q)
(λ y → P.subst Q y q)
y∈ys
= lem₂ _ P.refl
module _ {a b} {A : Set a} {B : Set b} where
map⁺ : ∀ {p} {P : B → Set p} {f : A → B}{n}{xs : Vec A n} →
Any (P ∘ f) xs → Any P (Vec.map f xs)
map⁺ (here pfx) = here pfx
map⁺ (there p) = there (map⁺ p)
map⁻ : ∀ {p} {P : B → Set p} {f : A → B} {n}{xs : Vec A n} →
Any P (Vec.map f xs) → Any (P ∘ f) xs
map⁻ {xs = []} ()
map⁻ {xs = x ∷ xs} (here p) = here p
map⁻ {xs = x ∷ xs} (there p) = there $ map⁻ p
map⁺∘map⁻ : ∀ {p} {P : B → Set p} {f : A → B}{n}{xs : Vec A n} →
(p : Any P (Vec.map f xs)) → map⁺ (map⁻ p) ≡ p
map⁺∘map⁻ {xs = []} ()
map⁺∘map⁻ {xs = x ∷ xs} (here p) = P.refl
map⁺∘map⁻ {xs = x ∷ xs} (there p) = P.cong there (map⁺∘map⁻ p)
map⁻∘map⁺ : ∀ {p} (P : B → Set p) {f : A → B}{n} {xs : Vec A n} →
(p : Any (P ∘ f) xs) → map⁻ {P = P} (map⁺ p) ≡ p
map⁻∘map⁺ P (here p) = P.refl
map⁻∘map⁺ P (there p) = P.cong there (map⁻∘map⁺ P p)
map↔ : ∀ {p} {P : B → Set p} {f : A → B} {n}{xs : Vec A n} →
Any (P ∘ f) xs ↔ Any P (Vec.map f xs)
map↔ {P = P} {f = f} = record
{ to = P.→-to-⟶ $ map⁺ {P = P} {f = f}
; from = P.→-to-⟶ $ map⁻ {P = P} {f = f}
; inverse-of = record
{ left-inverse-of = map⁻∘map⁺ P
; right-inverse-of = map⁺∘map⁻
}
}
------------------------------------------------------------------------
-- _++_
module _ {a p} {A : Set a} {P : A → Set p} where
++⁺ˡ : ∀ {m n}{xs : Vec A m} {ys : Vec A n} → Any P xs → Any P (xs ++ ys)
++⁺ˡ (here p) = here p
++⁺ˡ (there p) = there (++⁺ˡ p)
++⁺ʳ : ∀ {m n} (xs : Vec A m) {ys : Vec A n} → Any P ys → Any P (xs ++ ys)
++⁺ʳ [] p = p
++⁺ʳ (x ∷ xs) p = there (++⁺ʳ xs p)
++⁻ : ∀ {m n} (xs : Vec A m) {ys : Vec A n} → Any P (xs ++ ys) → Any P xs ⊎ Any P ys
++⁻ [] p = inj₂ p
++⁻ (x ∷ xs) (here p) = inj₁ (here p)
++⁻ (x ∷ xs) (there p) = Sum.map there id (++⁻ xs p)
++⁺∘++⁻ : ∀{m n} (xs : Vec A m) {ys : Vec A n} (p : Any P (xs ++ ys)) →
[ ++⁺ˡ , ++⁺ʳ xs ]′ (++⁻ xs p) ≡ p
++⁺∘++⁻ [] p = refl
++⁺∘++⁻ (x ∷ xs) (here p) = refl
++⁺∘++⁻ (x ∷ xs) (there p) with ++⁻ xs p | ++⁺∘++⁻ xs p
++⁺∘++⁻ (x ∷ xs) (there p) | inj₁ p′ | ih = P.cong there ih
++⁺∘++⁻ (x ∷ xs) (there p) | inj₂ p′ | ih = P.cong there ih
++⁻∘++⁺ : ∀{m n} (xs : Vec A m) {ys : Vec A n} (p : Any P xs ⊎ Any P ys) →
++⁻ xs ([ ++⁺ˡ , ++⁺ʳ xs ]′ p) ≡ p
++⁻∘++⁺ [] (inj₁ ())
++⁻∘++⁺ [] (inj₂ p) = refl
++⁻∘++⁺ (x ∷ xs) (inj₁ (here p)) = refl
++⁻∘++⁺ (x ∷ xs) {ys} (inj₁ (there p)) rewrite ++⁻∘++⁺ xs {ys} (inj₁ p) = refl
++⁻∘++⁺ (x ∷ xs) (inj₂ p) rewrite ++⁻∘++⁺ xs (inj₂ p) = refl
++↔ : ∀{m n} {xs : Vec A m}{ys : Vec A n} → (Any P xs ⊎ Any P ys) ↔ Any P (xs ++ ys)
++↔ {xs = xs} = record
{ to = P.→-to-⟶ [ ++⁺ˡ , ++⁺ʳ xs ]′
; from = P.→-to-⟶ $ ++⁻ xs
; inverse-of = record
{ left-inverse-of = ++⁻∘++⁺ xs
; right-inverse-of = ++⁺∘++⁻ xs
}
}
++-comm : ∀ {m n} (xs : Vec A m) (ys : Vec A n) → Any P (xs ++ ys) → Any P (ys ++ xs)
++-comm xs ys = [ ++⁺ʳ ys , ++⁺ˡ ]′ ∘ ++⁻ xs
++-comm∘++-comm : ∀{m n} (xs : Vec A m) {ys : Vec A n} (p : Any P (xs ++ ys)) →
++-comm ys xs (++-comm xs ys p) ≡ p
++-comm∘++-comm [] {ys} p
rewrite ++⁻∘++⁺ ys {ys = []} (inj₁ p) = P.refl
++-comm∘++-comm (x ∷ xs) {ys} (here p)
rewrite ++⁻∘++⁺ ys {ys = x ∷ xs} (inj₂ (here p)) = P.refl
++-comm∘++-comm (x ∷ xs) (there p) with ++⁻ xs p | ++-comm∘++-comm xs p
++-comm∘++-comm (x ∷ xs) {ys} (there .([ ++⁺ʳ xs , ++⁺ˡ ]′ (++⁻ ys (++⁺ʳ ys p))))
| inj₁ p | P.refl
rewrite ++⁻∘++⁺ ys (inj₂ p)
| ++⁻∘++⁺ ys (inj₂ $ there {x = x} p) = P.refl
++-comm∘++-comm (x ∷ xs) {ys} (there .([ ++⁺ʳ xs , ++⁺ˡ ]′ (++⁻ ys (++⁺ˡ p))))
| inj₂ p | P.refl
rewrite ++⁻∘++⁺ ys {ys = xs} (inj₁ p)
| ++⁻∘++⁺ ys {ys = x ∷ xs} (inj₁ p) = P.refl
++↔++ : ∀{m n} (xs : Vec A m) (ys : Vec A n) → Any P (xs ++ ys) ↔ Any P (ys ++ xs)
++↔++ xs ys = record
{ to = P.→-to-⟶ $ ++-comm xs ys
; from = P.→-to-⟶ $ ++-comm ys xs
; inverse-of = record
{ left-inverse-of = ++-comm∘++-comm xs
; right-inverse-of = ++-comm∘++-comm ys
}
}
------------------------------------------------------------------------
-- tabulate
module _ {a p} {A : Set a} {P : A → Set p} where
tabulate⁺ : ∀ {n} {f : Fin n → A} i → P (f i) → Any P (tabulate f)
tabulate⁺ fzero p = here p
tabulate⁺ (fsuc i) p = there (tabulate⁺ i p)
tabulate⁻ : ∀ {n} {f : Fin n → A} →
Any P (tabulate f) → ∃ λ i → P (f i)
tabulate⁻ {zero} ()
tabulate⁻ {suc n} (here p) = fzero , p
tabulate⁻ {suc n} (there p) = Prod.map fsuc id (tabulate⁻ p)
------------------------------------------------------------------------
-- map-with-∈.
module _ {a b p} {A : Set a} {B : Set b} {P : B → Set p} where
map-with-∈′⁺ : ∀ {n}{xs : Vec A n} (f : ∀ {x} → x ∈′ xs → B) →
(∃₂ λ x (x∈xs : x ∈′ xs) → P (f x∈xs)) →
Any P (map-with-∈′ xs f)
map-with-∈′⁺ f (_ , here refl , p) = here p
map-with-∈′⁺ f (_ , there x∈xs , p) =
there $ map-with-∈′⁺ (f ∘ there) (_ , x∈xs , p)
map-with-∈′⁻ : ∀ {n} (xs : Vec A n) (f : ∀ {x} → x ∈′ xs → B) →
Any P (map-with-∈′ xs f) →
∃₂ λ x (x∈xs : x ∈′ xs) → P (f x∈xs)
map-with-∈′⁻ [] f ()
map-with-∈′⁻ (y ∷ xs) f (here p) = (y , here refl , p)
map-with-∈′⁻ (y ∷ xs) f (there p) =
Prod.map id (Prod.map there id) $ map-with-∈′⁻ xs (f ∘ there) p
map-with-∈′↔ : ∀{n}{xs : Vec A n} {f : ∀ {x} → x ∈′ xs → B} →
(∃₂ λ x (x∈xs : x ∈′ xs) → P (f x∈xs)) ↔ Any P (map-with-∈′ xs f)
map-with-∈′↔ = record
{ to = P.→-to-⟶ (map-with-∈′⁺ _)
; from = P.→-to-⟶ (map-with-∈′⁻ _ _)
; inverse-of = record
{ left-inverse-of = from∘to _
; right-inverse-of = to∘from _ _
}
}
where
from∘to : ∀ {n} {xs : Vec A n} (f : ∀ {x} → x ∈′ xs → B)
(p : ∃₂ λ x (x∈xs : x ∈′ xs) → P (f x∈xs)) →
map-with-∈′⁻ xs f (map-with-∈′⁺ f p) ≡ p
from∘to f (_ , here refl , p) = refl
from∘to f (_ , there x∈xs , p)
rewrite from∘to (f ∘ there) (_ , x∈xs , p) = refl
to∘from : ∀ {n}(xs : Vec A n) (f : ∀ {x} → x ∈′ xs → B)
(p : Any P (map-with-∈′ xs f)) →
map-with-∈′⁺ f (map-with-∈′⁻ xs f p) ≡ p
to∘from [] f ()
to∘from (y ∷ xs) f (here p) = refl
to∘from (y ∷ xs) f (there p) =
P.cong there $ to∘from xs (f ∘ there) p
------------------------------------------------------------------------
module _ {a p} {A : Set a} {P : A → Set p} where
return⁺ : ∀ {x} → P x → Any P ([ x ])
return⁺ = here
return⁻ : ∀ {x} → Any P ([ x ]) → P x
return⁻ (here p) = p
return⁻ (there ())
return⁺∘return⁻ : ∀ {x} (p : Any P [ x ]) →
return⁺ (return⁻ p) ≡ p
return⁺∘return⁻ (here p) = refl
return⁺∘return⁻ (there ())
return⁻∘return⁺ : ∀ {x} (p : P x) → return⁻ (return⁺ p) ≡ p
return⁻∘return⁺ p = refl
return↔ : ∀ {x} → P x ↔ Any P ([ x ])
return↔ = record
{ to = P.→-to-⟶ $ return⁺
; from = P.→-to-⟶ $ return⁻
; inverse-of = record
{ left-inverse-of = return⁻∘return⁺
; right-inverse-of = return⁺∘return⁻
}
}
-- _∷_.
∷↔ : ∀ {a p n} {A : Set a} (P : A → Set p) {x} {xs : Vec A n} →
(P x ⊎ Any P xs) ↔ Any P (x ∷ xs)
∷↔ P {x} {xs} =
(P x ⊎ Any P xs) ↔⟨ return↔ {P = P} ⊎-cong (Any P xs ∎) ⟩
(Any P [ x ] ⊎ Any P xs) ↔⟨ ++↔ {P = P} {xs = [ x ]} ⟩
Any P (x ∷ xs) ∎
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{-# OPTIONS -v tc.unquote.clause:30 #-}
open import Common.Reflection
open import Common.Prelude
data Box : Set → Set₁ where
box : (A : Set) → Box A
unquoteDecl test = define (vArg test) (funDef
(pi
(vArg (sort (lit 0)))
(abs "A" (pi
(vArg (def (quote Box) (vArg (var 0 []) ∷ [])))
(abs "x" (sort (lit 0))))))
(clause
(vArg dot ∷
vArg (con (quote box) (vArg (var "dot") ∷ [])) ∷ [])
(var 1 [])
∷ []))
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-- A generic proof of confluence for a rewriting system with the diamond property.
module Confluence where
open import Prelude
postulate
Tm : Set
_=>_ : Tm -> Tm -> Set
diamond : ∀{M N P : Tm} → (M => N) → (M => P) →
Σ[ Q :: Tm ] (N => Q × P => Q)
data _=>*_ : Tm → Tm → Set where
eval-refl : {e : Tm} → e =>* e
eval-cons : {e e' e'' : Tm} →
(S1 : e => e') → (D : e' =>* e'') → e =>* e''
strip : ∀{M N P : Tm} → (M => N) → (M =>* P) →
Σ[ Q :: Tm ] (N =>* Q × P => Q)
strip S1 eval-refl = , (eval-refl , S1)
strip S1 (eval-cons S2 D) with diamond S1 S2
... | Q' , S1' , S2' with strip S2' D
... | Q , D1 , S' = Q , ((eval-cons S1' D1) , S')
confluence : ∀{M N P : Tm} → (M =>* N) → (M =>* P) →
Σ[ Q :: Tm ] (N =>* Q × P =>* Q)
confluence eval-refl D2 = , (D2 , eval-refl)
confluence (eval-cons S1 D1) D2 with strip S1 D2
... | M' , D3 , S3 with confluence D1 D3
... | Q , D4 , D4' = Q , D4 , eval-cons S3 D4'
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{-# OPTIONS --without-K #-}
-- The core properties behind exploration functions
module Explore.Properties where
open import Level.NP using (_⊔_; ₀; ₁; ₛ; Lift)
open import Type using (★₀; ★₁; ★_)
open import Function.NP using (id; _∘′_; _∘_; flip; const; Π; Cmp)
open import Algebra using (Semigroup; module Semigroup; Monoid; module Monoid; CommutativeMonoid; module CommutativeMonoid)
import Algebra.FunctionProperties.NP
import Algebra.FunctionProperties.Derived as FPD
open import Algebra.FunctionProperties.Core using (Op₂)
import Algebra.FunctionProperties.Eq
open Algebra.FunctionProperties.Eq.Explicits using (Injective)
open import Data.Nat.NP using (_+_; _*_; _≤_; _+°_)
open import Data.Product using (Σ; _×_; _,_; proj₁; proj₂)
open import Data.Sum using (_⊎_)
open import Data.Zero using (𝟘)
open import Data.One using (𝟙)
open import Data.Two using (𝟚; ✓)
open import Data.Fin using (Fin)
open import Relation.Binary.NP using (module Setoid-Reasoning; _Preserves₂_⟶_⟶_)
open import Relation.Binary.PropositionalEquality using (_≡_; _≗_)
open import Explore.Core
module FP = Algebra.FunctionProperties.NP Π
module SgrpExtra {c ℓ} (sg : Semigroup c ℓ) where
open Semigroup sg
open Setoid-Reasoning (Semigroup.setoid sg) public
C : ★ _
C = Carrier
_≈°_ : ∀ {a} {A : ★ a} (f g : A → C) → ★ _
f ≈° g = ∀ x → f x ≈ g x
_∙°_ : ∀ {a} {A : ★ a} (f g : A → C) → A → C
(f ∙° g) x = f x ∙ g x
infixl 7 _-∙-_
_-∙-_ : _∙_ Preserves₂ _≈_ ⟶ _≈_ ⟶ _≈_
_-∙-_ = ∙-cong
!_ = sym
module Sgrp {c ℓ} (sg : Semigroup c ℓ) where
open Semigroup sg public
open SgrpExtra sg public
module RawMon {c} {C : ★ c} (rawMon : C × Op₂ C) where
ε = proj₁ rawMon
_∙_ = proj₂ rawMon
module Mon {c ℓ} (m : Monoid c ℓ) where
open Monoid m public
sg = semigroup
open SgrpExtra semigroup public
RawMon = C × Op₂ C
rawMon : RawMon
rawMon = ε , _∙_
module CMon {c ℓ} (cm : CommutativeMonoid c ℓ) where
open CommutativeMonoid cm public
sg = semigroup
m = monoid
open SgrpExtra sg public
open FP _≈_
∙-interchange : Interchange _∙_ _∙_
∙-interchange = FPD.FromAssocCommCong.interchange
_≈_ isEquivalence
_∙_ assoc comm (λ _ → flip ∙-cong refl)
module _ {ℓ a} {A : ★ a} where
ExploreInd : ∀ p → Explore ℓ A → ★ (a ⊔ ₛ (ℓ ⊔ p))
ExploreInd p exp =
∀ (P : Explore ℓ A → ★ p)
(Pε : P empty-explore)
(P∙ : ∀ {e₀ e₁ : Explore ℓ A} → P e₀ → P e₁ → P (merge-explore e₀ e₁))
(Pf : ∀ x → P (point-explore x))
→ P exp
module _ {p} where
point-explore-ind : (x : A) → ExploreInd p (point-explore x)
point-explore-ind x _ _ _ Pf = Pf x
empty-explore-ind : ExploreInd p empty-explore
empty-explore-ind _ Pε _ _ = Pε
merge-explore-ind : ∀ {e₀ e₁ : Explore ℓ A}
→ ExploreInd p e₀ → ExploreInd p e₁
→ ExploreInd p (merge-explore e₀ e₁)
merge-explore-ind Pe₀ Pe₁ P Pε _P∙_ Pf = (Pe₀ P Pε _P∙_ Pf) P∙ (Pe₁ P Pε _P∙_ Pf)
ExploreInd₀ : Explore ℓ A → ★ _
ExploreInd₀ = ExploreInd ₀
ExploreInd₁ : Explore ℓ A → ★ _
ExploreInd₁ = ExploreInd ₁
BigOpMonInd : ∀ p {c} (M : Monoid c ℓ) → BigOpMon M A → ★ _
BigOpMonInd p {c} M exp =
∀ (P : BigOpMon M A → ★ p)
(Pε : P (const ε))
(P∙ : ∀ {e₀ e₁ : BigOpMon M A} → P e₀ → P e₁ → P (λ f → e₀ f ∙ e₁ f))
(Pf : ∀ x → P (λ f → f x))
(P≈ : ∀ {e e'} → e ≈° e' → P e → P e')
→ P exp
where open Mon M
module _ (eᴬ : Explore {a} (ₛ ℓ) A) where
Πᵉ : (A → ★ ℓ) → ★ ℓ
Πᵉ = eᴬ (Lift 𝟙) _×_
Σᵉ : (A → ★ ℓ) → ★ ℓ
Σᵉ = eᴬ (Lift 𝟘) _⊎_
module _ {ℓ a} {A : ★ a} where
module _ (eᴬ : Explore (ₛ ℓ) A) where
Lookup : ★ (ₛ ℓ ⊔ a)
Lookup = ∀ {P : A → ★ ℓ} → Πᵉ eᴬ P → Π A P
-- alternative name suggestion: tabulate
Reify : ★ (a ⊔ ₛ ℓ)
Reify = ∀ {P : A → ★ ℓ} → Π A P → Πᵉ eᴬ P
Unfocus : ★ (a ⊔ ₛ ℓ)
Unfocus = ∀ {P : A → ★ ℓ} → Σᵉ eᴬ P → Σ A P
-- alternative name suggestion: inject
Focus : ★ (a ⊔ ₛ ℓ)
Focus = ∀ {P : A → ★ ℓ} → Σ A P → Σᵉ eᴬ P
Adequate-Σ : ((A → ★ ℓ) → ★ _) → ★ _
Adequate-Σ Σᴬ = ∀ F → Σᴬ F ≡ Σ A F
Adequate-Π : ((A → ★ ℓ) → ★ _) → ★ _
Adequate-Π Πᴬ = ∀ F → Πᴬ F ≡ Π A F
-- This module could be parameterised by the relation on types, here _≡_
module Universal-Adequacy {ℓu ℓe ℓr ℓa}
(U : ★_ ℓu)(El : U → ★_ ℓe)
(_≈_ : ★_ ℓe → ★_ (ℓa ⊔ ℓe) → ★_ ℓr){A : ★_ ℓa} where
Adequate-univ-sum : ((A → U) → U) → ★_ (ℓa ⊔ (ℓr ⊔ ℓu))
Adequate-univ-sum sumᴬ = ∀ f → El (sumᴬ f) ≈ Σ A (El ∘ f)
Adequate-univ-product : ((A → U) → U) → ★_ (ℓa ⊔ (ℓr ⊔ ℓu))
Adequate-univ-product productᴬ = ∀ f → El (productᴬ f) ≈ Π A (El ∘ f)
module Adequacy {ℓa ℓr}(_≈_ : ★₀ → ★_ ℓa → ★_ ℓr){A : ★_ ℓa} where
-- Universal-Adequacy.Adequate-univ-sum ℕ Fin _≡_
Adequate-sum : Sum A → ★_(ℓa ⊔ ℓr)
Adequate-sum sumᴬ = ∀ f → Fin (sumᴬ f) ≈ Σ A (Fin ∘ f)
-- Universal-Adequacy.Adequate-univ-product ℕ Fin _≡_
Adequate-product : Product A → ★_(ℓa ⊔ ℓr)
Adequate-product productᴬ = ∀ f → Fin (productᴬ f) ≈ Π A (Fin ∘ f)
-- Universal-Adequacy.Adequate-univ-product 𝟚 ✓ _≡_
Adequate-any : (any : BigOp 𝟚 A) → ★_(ℓa ⊔ ℓr)
Adequate-any anyᴬ = ∀ f → ✓ (anyᴬ f) ≈ Σ A (✓ ∘ f)
-- Universal-Adequacy.Adequate-univ-product 𝟚 ✓ _≡_
Adequate-all : (all : BigOp 𝟚 A) → ★_(ℓa ⊔ ℓr)
Adequate-all allᴬ = ∀ f → ✓ (allᴬ f) ≈ Π A (✓ ∘ f)
module _ {m a}{M : ★ m}{A : ★ a}([⊕] : BigOp M A) where
BigOpStableUnder : (p : A → A) → ★ _
BigOpStableUnder p = ∀ f → [⊕] f ≡ [⊕] (f ∘ p)
-- Extensionality of a big-operator
BigOp= : ★ _
BigOp= = {f g : A → M} → f ≗ g → [⊕] f ≡ [⊕] g
module _ {ℓ a} {A : ★ a} (eᴬ : Explore ℓ A) where
StableUnder : (A → A) → ★ _
StableUnder p = ∀ {M}(ε : M) op → BigOpStableUnder (eᴬ ε op) p
-- Extensionality of an exploration function
Explore= : ★ _
Explore= = ∀ {M}(ε : M) op → BigOp= (eᴬ ε op)
ExploreExt = Explore=
module _ {ℓ a} {A : ★ a} r (eᴬ : Explore ℓ A) where
ExploreMono : ★ _
ExploreMono = ∀ {C} (_⊆_ : C → C → ★ r)
{z₀ z₁} (z₀⊆z₁ : z₀ ⊆ z₁)
{_∙_} (∙-mono : _∙_ Preserves₂ _⊆_ ⟶ _⊆_ ⟶ _⊆_)
{f g} →
(∀ x → f x ⊆ g x) → eᴬ z₀ _∙_ f ⊆ eᴬ z₁ _∙_ g
ExploreMonExt : ★ _
ExploreMonExt =
∀ (m : Monoid ℓ r) {f g}
→ let open Mon m
bigop = eᴬ ε _∙_
in
f ≈° g → bigop f ≈ bigop g
Exploreε : ★ _
Exploreε = ∀ (m : Monoid ℓ r) →
let open Mon m in
eᴬ ε _∙_ (const ε) ≈ ε
ExploreLinˡ : ★ _
ExploreLinˡ =
∀ m _◎_ f k →
let open Mon {ℓ} {r} m
open FP _≈_ in
k ◎ ε ≈ ε →
_◎_ DistributesOverˡ _∙_ →
eᴬ ε _∙_ (λ x → k ◎ f x) ≈ k ◎ eᴬ ε _∙_ f
ExploreLinʳ : ★ _
ExploreLinʳ =
∀ m _◎_ f k →
let open Mon {ℓ} {r} m
open FP _≈_ in
ε ◎ k ≈ ε →
_◎_ DistributesOverʳ _∙_ →
eᴬ ε _∙_ (λ x → f x ◎ k) ≈ eᴬ ε _∙_ f ◎ k
ExploreHom : ★ _
ExploreHom =
∀ cm f g →
let open CMon {ℓ} {r} cm in
eᴬ ε _∙_ (f ∙° g) ≈ eᴬ ε _∙_ f ∙ eᴬ ε _∙_ g
{-
ExploreSwap'' : ∀ {b} → ★ _
ExploreSwap'' {b}
= ∀ (monM : Monoid _) (monN : Monoid _) →
let module M = Mon {_} {r} monM in
let module N = Mon {_} {r} monN in
∀ {h : M.C → N.C}
(h-ε : h M.ε ≈ N.ε)
(h-∙ : ∀ x y → h (x M.∙ y) ≈ h x N.∙ h y)
f
→ eᴬ ε _∙_ (h ∘ f) ≈ h (eᴬ ε _∙_ f)
-}
-- derived from lift-hom with the source monoid being (a → m)
ExploreSwap : ∀ {b} → ★ _
ExploreSwap {b}
= ∀ {B : ★ b} mon →
let open Mon {_} {r} mon in
∀ {opᴮ : (B → C) → C}
(opᴮ-ε : opᴮ (const ε) ≈ ε)
(hom : ∀ f g → opᴮ (f ∙° g) ≈ opᴮ f ∙ opᴮ g)
f
→ eᴬ ε _∙_ (opᴮ ∘ f) ≈ opᴮ (eᴬ ε _∙_ ∘ flip f)
module _ {a} {A : ★ a} (sumᴬ : Sum A) where
SumStableUnder : (A → A) → ★ _
SumStableUnder p = ∀ f → sumᴬ f ≡ sumᴬ (f ∘ p)
SumStableUnderInjection : ★ _
SumStableUnderInjection = ∀ p → Injective p → SumStableUnder p
SumInd : ★(₁ ⊔ a)
SumInd = ∀ (P : Sum A → ★₀)
(P0 : P (λ f → 0))
(P+ : ∀ {s₀ s₁ : Sum A} → P s₀ → P s₁ → P (λ f → s₀ f + s₁ f))
(Pf : ∀ x → P (λ f → f x))
→ P sumᴬ
SumExt : ★ _
SumExt = ∀ {f g} → f ≗ g → sumᴬ f ≡ sumᴬ g
SumZero : ★ _
SumZero = sumᴬ (λ _ → 0) ≡ 0
SumLin : ★ _
SumLin = ∀ f k → sumᴬ (λ x → k * f x) ≡ k * sumᴬ f
SumHom : ★ _
SumHom = ∀ f g → sumᴬ (λ x → f x + g x) ≡ sumᴬ f + sumᴬ g
SumMono : ★ _
SumMono = ∀ {f g} → (∀ x → f x ≤ g x) → sumᴬ f ≤ sumᴬ g
SumConst : ★ _
SumConst = ∀ x → sumᴬ (const x) ≡ sumᴬ (const 1) * x
SumSwap : ★ _
SumSwap = ∀ {B : ★₀}
{sumᴮ : Sum B}
(sumᴮ-0 : sumᴮ (const 0) ≡ 0)
(hom : ∀ f g → sumᴮ (f +° g) ≡ sumᴮ f + sumᴮ g)
f
→ sumᴬ (sumᴮ ∘ f) ≡ sumᴮ (sumᴬ ∘ flip f)
module _ {a} {A : ★ a} (countᴬ : Count A) where
CountStableUnder : (A → A) → ★ _
CountStableUnder p = ∀ f → countᴬ f ≡ countᴬ (f ∘ p)
CountExt : ★ _
CountExt = ∀ {f g} → f ≗ g → countᴬ f ≡ countᴬ g
Unique : Cmp A → ★ _
Unique cmp = ∀ x → countᴬ (cmp x) ≡ 1
-- -}
-- -}
-- -}
-- -}
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module Tactic.Reflection.Telescope where
open import Prelude hiding (abs)
open import Builtin.Reflection
open import Tactic.Reflection.DeBruijn
Telescope = List (Arg Type)
telView : Type → Telescope × Type
telView (pi a (abs _ b)) = first (_∷_ a) (telView b)
telView a = [] , a
visibleArity : Type → Nat
visibleArity = length ∘ filter isVisible ∘ fst ∘ telView
telPi : Telescope → Type → Type
telPi tel b = foldr (λ a b → pi a (abs "_" b)) b tel
arity : Name → TC Nat
arity f = length ∘ fst ∘ telView <$> getType f
argTel : Name → TC Telescope
argTel f = fst ∘ telView <$> getType f
telePat : Telescope → List (Arg Pattern)
telePat = map (var "_" <$_)
private
teleArgs′ : Nat → List (Arg Type) → List (Arg Term)
teleArgs′ (suc n) (a ∷ Γ) = (var n [] <$ a) ∷ teleArgs′ n Γ
teleArgs′ _ _ = []
teleArgs : Telescope → List (Arg Term)
teleArgs Γ = teleArgs′ (length Γ) Γ
conParams : Name → TC Nat
conParams c =
caseM getDefinition c of λ
{ (data-cons d) → getParameters d
; _ → pure 0 }
conPat : Name → TC Pattern
conPat c = do
np ← conParams c
con c ∘ telePat ∘ drop np <$> argTel c
conTerm : Name → TC Term
conTerm c = do
np ← conParams c
con c ∘ teleArgs ∘ drop np <$> argTel c
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module Common.Char where
postulate
Char : Set
{-# BUILTIN CHAR Char #-}
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------------------------------------------------------------------------
-- Extra lemmas about substitutions
------------------------------------------------------------------------
module Data.Fin.Substitution.ExtraLemmas where
open import Data.Fin as Fin using (Fin; zero; suc)
open import Data.Fin.Substitution
open import Data.Fin.Substitution.Lemmas
open import Data.Nat using (ℕ; zero; suc; _+_)
open import Data.Star using (Star; ε; _◅_; _▻_)
open import Data.Vec using (Vec; _∷_; lookup; map)
open import Data.Vec.All hiding (lookup; map)
open import Data.Vec.Properties using (lookup-map; map-cong; map-∘)
open import Function using (_∘_; _$_; flip)
open import Relation.Binary.PropositionalEquality as PropEq hiding (subst)
open PropEq.≡-Reasoning
-- Simple extension of substitutions.
--
-- FIXME: this should go into Data.Fin.Substitution.
record Extension (T : ℕ → Set) : Set where
infixr 5 _/∷_
field weaken : ∀ {n} → T n → T (suc n) -- Weakens Ts.
-- Extension.
_/∷_ : ∀ {m n} → T (suc n) → Sub T m n → Sub T (suc m) (suc n)
t /∷ ρ = t ∷ map weaken ρ
-- Helper module
module SimpleExt {T} (simple : Simple T) where
open Simple simple public
extension : Extension T
extension = record { weaken = weaken }
open Extension extension public hiding (weaken)
-- An generalized version of Data.Fin.Lemmas.Lemmas₀
--
-- FIXME: this should go into Data.Fin.Substitution.Lemmas.
module ExtLemmas₀ {T} (lemmas₀ : Lemmas₀ T) where
open Lemmas₀ lemmas₀ public hiding (lookup-map-weaken-↑⋆)
open SimpleExt simple
-- A generalized variant of Lemmas₀.lookup-map-weaken-↑⋆.
lookup-map-weaken-↑⋆ : ∀ {m n} k x {ρ : Sub T m n} {t} →
lookup x (map weaken ρ ↑⋆ k) ≡
lookup (Fin.lift k suc x) ((t /∷ ρ) ↑⋆ k)
lookup-map-weaken-↑⋆ zero x = refl
lookup-map-weaken-↑⋆ (suc k) zero = refl
lookup-map-weaken-↑⋆ (suc k) (suc x) {ρ} {t} = begin
lookup x (map weaken (map weaken ρ ↑⋆ k))
≡⟨ lookup-map x weaken _ ⟩
weaken (lookup x (map weaken ρ ↑⋆ k))
≡⟨ cong weaken (lookup-map-weaken-↑⋆ k x) ⟩
weaken (lookup (Fin.lift k suc x) ((t /∷ ρ) ↑⋆ k))
≡⟨ sym (lookup-map (Fin.lift k suc x) weaken _) ⟩
lookup (Fin.lift k suc x) (map weaken ((t /∷ ρ) ↑⋆ k))
∎
-- A generalized version of Data.Fin.Lemmas.Lemmas₄
--
-- FIXME: this should go into Data.Fin.Substitution.Lemmas.
module ExtLemmas₄ {T} (lemmas₄ : Lemmas₄ T) where
open Lemmas₄ lemmas₄ public hiding (⊙-wk; wk-commutes)
open Lemmas₃ lemmas₃ using (lookup-wk-↑⋆-⊙; /✶-↑✶′)
open SimpleExt simple using (_/∷_)
open ExtLemmas₀ lemmas₀ using (lookup-map-weaken-↑⋆)
private
⊙-wk′ : ∀ {m n} {ρ : Sub T m n} {t} k →
ρ ↑⋆ k ⊙ wk ↑⋆ k ≡ wk ↑⋆ k ⊙ (t /∷ ρ) ↑⋆ k
⊙-wk′ {ρ = ρ} {t} k = sym (begin
wk ↑⋆ k ⊙ (t /∷ ρ) ↑⋆ k ≡⟨ lemma ⟩
map weaken ρ ↑⋆ k ≡⟨ cong (λ ρ′ → ρ′ ↑⋆ k) map-weaken ⟩
(ρ ⊙ wk) ↑⋆ k ≡⟨ ↑⋆-distrib k ⟩
ρ ↑⋆ k ⊙ wk ↑⋆ k ∎)
where
lemma = extensionality λ x → begin
lookup x (wk ↑⋆ k ⊙ (t /∷ ρ) ↑⋆ k)
≡⟨ lookup-wk-↑⋆-⊙ k ⟩
lookup (Fin.lift k suc x) ((t /∷ ρ) ↑⋆ k)
≡⟨ sym (lookup-map-weaken-↑⋆ k x) ⟩
lookup x (map weaken ρ ↑⋆ k)
∎
⊙-wk : ∀ {m n} {ρ : Sub T m n} {t} → ρ ⊙ wk ≡ wk ⊙ (t /∷ ρ)
⊙-wk = ⊙-wk′ zero
wk-commutes : ∀ {m n} {ρ : Sub T m n} {t′} t →
t / ρ / wk ≡ t / wk / (t′ /∷ ρ)
wk-commutes {ρ = ρ} {t} = /✶-↑✶′ (ε ▻ ρ ▻ wk) (ε ▻ wk ▻ (t /∷ ρ)) ⊙-wk′ zero
-- A handy helper lemma: weakening commutes with substitution.
weaken-/ : ∀ {m n} {ρ : Sub T m n} {t′} t →
weaken (t / ρ) ≡ weaken t / (t′ /∷ ρ)
weaken-/ {ρ = ρ} {t′} t = begin
weaken (t / ρ) ≡⟨ sym /-wk ⟩
t / ρ / wk ≡⟨ wk-commutes t ⟩
t / wk / (t′ /∷ ρ) ≡⟨ cong₂ _/_ /-wk refl ⟩
weaken t / (t′ /∷ ρ) ∎
-- A generalize version of Data.Fin.Lemmas.AppLemmas
--
-- FIXME: this should go into Data.Fin.Substitution.Lemmas.
module ExtAppLemmas {T₁ T₂} (appLemmas : AppLemmas T₁ T₂) where
open AppLemmas appLemmas public hiding (wk-commutes)
open SimpleExt simple using (_/∷_)
private module L₄ = ExtLemmas₄ lemmas₄
wk-commutes : ∀ {m n} {ρ : Sub T₂ m n} {t′} t →
t / ρ / wk ≡ t / wk / (t′ /∷ ρ)
wk-commutes {ρ = ρ} {t} = ⨀→/✶ (ε ▻ ρ ▻ wk) (ε ▻ wk ▻ (t /∷ ρ)) L₄.⊙-wk
-- Lemmas relating T₃ substitutions in T₁ and T₂.
record LiftAppLemmas (T₁ T₂ T₃ : ℕ → Set) : Set where
field
lift : ∀ {n} → T₃ n → T₂ n
application₁₃ : Application T₁ T₃
application₂₃ : Application T₂ T₃
lemmas₂ : Lemmas₄ T₂
lemmas₃ : Lemmas₄ T₃
private
module L₂ = ExtLemmas₄ lemmas₂
module L₃ = ExtLemmas₄ lemmas₃
module A₁ = Application application₁₃
module A₂ = Application application₂₃
field
-- Lifting commutes with application of T₃ substitutions.
lift-/ : ∀ {m n} t {σ : Sub T₃ m n} → lift (t L₃./ σ) ≡ lift t A₂./ σ
-- Lifting preserves variables.
lift-var : ∀ {n} (x : Fin n) → lift (L₃.var x) ≡ L₂.var x
-- Sequences of T₃ substitutions are equivalent when applied to
-- T₁s if they are equivalent when applied to T₂ variables.
/✶-↑✶ :
∀ {m n} (σs₁ σs₂ : Subs T₃ m n) →
(∀ k x → L₂.var x A₂./✶ σs₁ L₃.↑✶ k ≡ L₂.var x A₂./✶ σs₂ L₃.↑✶ k) →
∀ k t → t A₁./✶ σs₁ L₃.↑✶ k ≡ t A₁./✶ σs₂ L₃.↑✶ k
lift-lookup-⊙ : ∀ {m n k} x {σ₁ : Sub T₃ m n} {σ₂ : Sub T₃ n k} →
lift (lookup x (σ₁ L₃.⊙ σ₂)) ≡ lift (lookup x σ₁) A₂./ σ₂
lift-lookup-⊙ x {σ₁} {σ₂} = begin
lift (lookup x (σ₁ L₃.⊙ σ₂)) ≡⟨ cong lift (L₃.lookup-⊙ x) ⟩
lift (lookup x σ₁ L₃./ σ₂) ≡⟨ lift-/ _ ⟩
lift (lookup x σ₁) A₂./ σ₂ ∎
lift-lookup-⨀ : ∀ {m n} x (σs : Subs T₃ m n) →
lift (lookup x (L₃.⨀ σs)) ≡ L₂.var x A₂./✶ σs
lift-lookup-⨀ x ε = begin
lift (lookup x L₃.id) ≡⟨ cong lift (L₃.lookup-id x) ⟩
lift (L₃.var x) ≡⟨ lift-var x ⟩
L₂.var x ∎
lift-lookup-⨀ x (σ ◅ ε) = begin
lift (lookup x σ) ≡⟨ cong lift (sym L₃.var-/) ⟩
lift (L₃.var x L₃./ σ) ≡⟨ lift-/ _ ⟩
lift (L₃.var x) A₂./ σ ≡⟨ cong₂ A₂._/_ (lift-var x) refl ⟩
L₂.var x A₂./ σ ∎
lift-lookup-⨀ x (σ ◅ (σ′ ◅ σs′)) = begin
lift (lookup x (L₃.⨀ σs L₃.⊙ σ))
≡⟨ lift-lookup-⊙ x ⟩
lift (lookup x (L₃.⨀ σs)) A₂./ σ
≡⟨ cong₂ A₂._/_ (lift-lookup-⨀ x (σ′ ◅ σs′)) refl ⟩
L₂.var x A₂./✶ σs A₂./ σ
∎
where σs = σ′ ◅ σs′
-- Sequences of T₃ substitutions are equivalent when applied to
-- T₁s if they are equivalent when applied as composites.
/✶-↑✶′ : ∀ {m n} (σs₁ σs₂ : Subs T₃ m n) →
(∀ k → L₃.⨀ (σs₁ L₃.↑✶ k) ≡ L₃.⨀ (σs₂ L₃.↑✶ k)) →
∀ k t → t A₁./✶ σs₁ L₃.↑✶ k ≡ t A₁./✶ σs₂ L₃.↑✶ k
/✶-↑✶′ σs₁ σs₂ hyp = /✶-↑✶ σs₁ σs₂ (λ k x → begin
L₂.var x A₂./✶ σs₁ L₃.↑✶ k
≡⟨ sym (lift-lookup-⨀ x (σs₁ L₃.↑✶ k)) ⟩
lift (lookup x (L₃.⨀ (σs₁ L₃.↑✶ k)))
≡⟨ cong (lift ∘ lookup x) (hyp k) ⟩
lift (lookup x (L₃.⨀ (σs₂ L₃.↑✶ k)))
≡⟨ lift-lookup-⨀ x (σs₂ L₃.↑✶ k) ⟩
L₂.var x A₂./✶ σs₂ L₃.↑✶ k
∎)
-- Derived lemmas about applications of T₃ substitutions to T₁s.
appLemmas : AppLemmas T₁ T₃
appLemmas = record
{ application = application₁₃
; lemmas₄ = lemmas₃
; id-vanishes = /✶-↑✶′ (ε ▻ L₃.id) ε L₃.id-↑⋆ 0
; /-⊙ = /✶-↑✶′ (ε ▻ _ L₃.⊙ _) (ε ▻ _ ▻ _) L₃.↑⋆-distrib 0
}
open ExtAppLemmas appLemmas public
hiding (application; lemmas₂; lemmas₃; var; weaken; subst; simple)
-- Lemmas relating T₂ and T₃ substitutions in T₁.
record LiftSubLemmas (T₁ T₂ T₃ : ℕ → Set) : Set where
field
application₁₂ : Application T₁ T₂
liftAppLemmas : LiftAppLemmas T₁ T₂ T₃
open LiftAppLemmas liftAppLemmas hiding (/✶-↑✶; /-wk)
private
module L₃ = ExtLemmas₄ lemmas₃
module L₂ = ExtLemmas₄ lemmas₂
module A₁₂ = Application application₁₂
module A₁₃ = Application (AppLemmas.application appLemmas)
module A₂₃ = Application application₂₃
field
-- Weakening commutes with lifting.
weaken-lift : ∀ {n} (t : T₃ n) → L₂.weaken (lift t) ≡ lift (L₃.weaken t)
-- Applying a composition of T₂ substitutions to T₁s
-- corresponds to two consecutive applications.
/-⊙₂ : ∀ {m n k} {σ₁ : Sub T₂ m n} {σ₂ : Sub T₂ n k} t →
t A₁₂./ σ₁ L₂.⊙ σ₂ ≡ t A₁₂./ σ₁ A₁₂./ σ₂
-- Sequences of T₃ substitutions are equivalent to T₂
-- substitutions when applied to T₁s if they are equivalent when
-- applied to variables.
/✶-↑✶₁ :
∀ {m n} (σs₁ : Subs T₃ m n) (σs₂ : Subs T₂ m n) →
(∀ k x → L₂.var x A₂₃./✶ σs₁ ↑✶ k ≡ L₂.var x L₂./✶ σs₂ L₂.↑✶ k) →
∀ k t → t A₁₃./✶ σs₁ ↑✶ k ≡ t A₁₂./✶ σs₂ L₂.↑✶ k
-- Sequences of T₃ substitutions are equivalent to T₂
-- substitutions when applied to T₂s if they are equivalent when
-- applied to variables.
/✶-↑✶₂ :
∀ {m n} (σs₁ : Subs T₃ m n) (σs₂ : Subs T₂ m n) →
(∀ k x → L₂.var x A₂₃./✶ σs₁ ↑✶ k ≡ L₂.var x L₂./✶ σs₂ L₂.↑✶ k) →
∀ k t → t A₂₃./✶ σs₁ ↑✶ k ≡ t L₂./✶ σs₂ L₂.↑✶ k
-- Lifting of T₃ substitutions to T₂ substitutions.
liftSub : ∀ {m n} → Sub T₃ m n → Sub T₂ m n
liftSub σ = map lift σ
-- The two types of lifting commute.
liftSub-↑⋆ : ∀ {m n} (σ : Sub T₃ m n) k →
liftSub σ L₂.↑⋆ k ≡ liftSub (σ ↑⋆ k)
liftSub-↑⋆ σ zero = refl
liftSub-↑⋆ σ (suc k) = cong₂ _∷_ (sym (lift-var _)) (begin
map L₂.weaken (liftSub σ L₂.↑⋆ k) ≡⟨ cong (map _) (liftSub-↑⋆ σ k) ⟩
map L₂.weaken (map lift (σ ↑⋆ k)) ≡⟨ sym (map-∘ _ _ _) ⟩
map (L₂.weaken ∘ lift) (σ ↑⋆ k) ≡⟨ map-cong weaken-lift _ ⟩
map (lift ∘ L₃.weaken) (σ ↑⋆ k) ≡⟨ map-∘ _ _ _ ⟩
map lift (map L₃.weaken (σ ↑⋆ k)) ∎)
-- The identity substitutions are equivalent up to lifting.
liftSub-id : ∀ {n} → liftSub (L₃.id {n}) ≡ L₂.id {n}
liftSub-id {zero} = refl
liftSub-id {suc n} = begin
liftSub (L₃.id L₃.↑) ≡⟨ sym (liftSub-↑⋆ L₃.id 1) ⟩
liftSub L₃.id L₂.↑ ≡⟨ cong L₂._↑ liftSub-id ⟩
L₂.id ∎
-- Weakening is equivalent up to lifting.
liftSub-wk⋆ : ∀ k {n} → liftSub (L₃.wk⋆ k {n}) ≡ L₂.wk⋆ k {n}
liftSub-wk⋆ zero = liftSub-id
liftSub-wk⋆ (suc k) = begin
liftSub (map L₃.weaken (L₃.wk⋆ k)) ≡⟨ sym (map-∘ _ _ _) ⟩
map (lift ∘ L₃.weaken) (L₃.wk⋆ k) ≡⟨ sym (map-cong weaken-lift _) ⟩
map (L₂.weaken ∘ lift) (L₃.wk⋆ k) ≡⟨ map-∘ _ _ _ ⟩
map L₂.weaken (liftSub (L₃.wk⋆ k)) ≡⟨ cong (map _) (liftSub-wk⋆ k) ⟩
map L₂.weaken (L₂.wk⋆ k) ∎
-- Weakening is equivalent up to lifting.
liftSub-wk : ∀ {n} → liftSub (L₃.wk {n}) ≡ L₂.wk {n}
liftSub-wk = liftSub-wk⋆ 1
-- Single variable substitution is equivalent up to lifting.
liftSub-sub : ∀ {n} (t : T₃ n) → liftSub (L₃.sub t) ≡ L₂.sub (lift t)
liftSub-sub t = cong₂ _∷_ refl liftSub-id
-- Lifting commutes with application to variables.
var-/-liftSub-↑⋆ : ∀ {m n} (σ : Sub T₃ m n) k x →
L₂.var x A₂₃./ σ ↑⋆ k ≡ L₂.var x L₂./ liftSub σ L₂.↑⋆ k
var-/-liftSub-↑⋆ σ k x = begin
L₂.var x A₂₃./ σ ↑⋆ k
≡⟨ cong₂ A₂₃._/_ (sym (lift-var x)) refl ⟩
lift (L₃.var x) A₂₃./ σ ↑⋆ k
≡⟨ sym (lift-/ _) ⟩
lift (L₃.var x L₃./ σ ↑⋆ k)
≡⟨ cong lift L₃.var-/ ⟩
lift (lookup x (σ ↑⋆ k))
≡⟨ sym (lookup-map x _ _) ⟩
lookup x (liftSub (σ ↑⋆ k))
≡⟨ sym L₂.var-/ ⟩
L₂.var x L₂./ liftSub (σ ↑⋆ k)
≡⟨ cong (L₂._/_ (L₂.var x)) (sym (liftSub-↑⋆ σ k)) ⟩
L₂.var x L₂./ liftSub σ L₂.↑⋆ k
∎
-- Lifting commutes with application.
/-liftSub : ∀ {m n} t {σ : Sub T₃ m n} →
t A₁₃./ σ ≡ t A₁₂./ liftSub σ
/-liftSub t {σ} = /✶-↑✶₁ (ε ▻ σ) (ε ▻ liftSub σ)
(var-/-liftSub-↑⋆ σ) zero t
-- Weakening is equivalent up to choice of application.
/-wk : ∀ {n} {t : T₁ n} → t A₁₃./ L₃.wk ≡ t A₁₂./ L₂.wk
/-wk {t = t} = begin
t A₁₃./ L₃.wk ≡⟨ /-liftSub t ⟩
t A₁₂./ liftSub L₃.wk ≡⟨ cong₂ A₁₂._/_ refl liftSub-wk ⟩
t A₁₂./ L₂.wk ∎
-- Single-variable substitution is equivalent up to choice of
-- application.
/-sub : ∀ {n} t (s : T₃ n) →
t A₁₃./ L₃.sub s ≡ t A₁₂./ L₂.sub (lift s)
/-sub t s = begin
t A₁₃./ L₃.sub s ≡⟨ /-liftSub t ⟩
t A₁₂./ liftSub (L₃.sub s) ≡⟨ cong₂ A₁₂._/_ refl (liftSub-sub s) ⟩
t A₁₂./ L₂.sub (lift s) ∎
-- Lifting commutes with application.
/-sub-↑ : ∀ {m n} t s (σ : Sub T₃ m n) →
t A₁₂./ L₂.sub s A₁₃./ σ ≡ (t A₁₃./ σ ↑) A₁₂./ L₂.sub (s A₂₃./ σ)
/-sub-↑ t s σ = begin
t A₁₂./ L₂.sub s A₁₃./ σ
≡⟨ /-liftSub _ ⟩
t A₁₂./ L₂.sub s A₁₂./ liftSub σ
≡⟨ sym (/-⊙₂ t) ⟩
t A₁₂./ (L₂.sub s L₂.⊙ liftSub σ)
≡⟨ cong₂ A₁₂._/_ refl (L₂.sub-⊙ s) ⟩
t A₁₂./ (liftSub σ L₂.↑ L₂.⊙ L₂.sub (s L₂./ liftSub σ))
≡⟨ /-⊙₂ t ⟩
t A₁₂./ liftSub σ L₂.↑ A₁₂./ L₂.sub (s L₂./ liftSub σ)
≡⟨ cong₂ (A₁₂._/_ ∘ A₁₂._/_ t) (liftSub-↑⋆ _ 1)
(cong L₂.sub (sym (/-liftSub₂ s))) ⟩
t A₁₂./ liftSub (σ ↑) A₁₂./ L₂.sub (s A₂₃./ σ)
≡⟨ cong₂ A₁₂._/_ (sym (/-liftSub t)) refl ⟩
t A₁₃./ σ ↑ A₁₂./ L₂.sub (s A₂₃./ σ)
∎
where
/-liftSub₂ : ∀ {m n} s {σ : Sub T₃ m n} →
s A₂₃./ σ ≡ s L₂./ liftSub σ
/-liftSub₂ s {σ} = /✶-↑✶₂ (ε ▻ σ) (ε ▻ liftSub σ)
(var-/-liftSub-↑⋆ σ) zero s
-- Lemmas for a term-like T₁ derived from term lemmas for T₂
record TermLikeLemmas (T₁ T₂ : ℕ → Set) : Set₁ where
field
app : ∀ {T₃} → Lift T₃ T₂ → ∀ {m n} → T₁ m → Sub T₃ m n → T₁ n
termLemmas : TermLemmas T₂
open TermLemmas termLemmas using (termSubst)
open TermSubst termSubst using (var; termLift; varLift; module Lifted)
field /✶-↑✶₁ : ∀ {T₃} {lift : Lift T₃ T₂} →
let open Application (record { _/_ = app lift })
using () renaming (_/✶_ to _/✶₁_)
open Lifted lift
using (_↑✶_) renaming (_/✶_ to _/✶₂_)
in
∀ {m n} (σs₁ : Subs T₃ m n) (σs₂ : Subs T₃ m n) →
(∀ k x → var x /✶₂ σs₁ ↑✶ k ≡ var x /✶₂ σs₂ ↑✶ k) →
∀ k t → t /✶₁ σs₁ ↑✶ k ≡ t /✶₁ σs₂ ↑✶ k
termApplication : Application T₁ T₂
termApplication = record { _/_ = app termLift }
varApplication : Application T₁ Fin
varApplication = record { _/_ = app varLift }
field /✶-↑✶₂ : let open Application varApplication
using () renaming (_/✶_ to _/✶₁₃_)
open Application termApplication
using () renaming (_/✶_ to _/✶₁₂_)
open Lifted varLift
using () renaming (_↑✶_ to _↑✶₃_; _/✶_ to _/✶₂₃_)
open TermSubst termSubst
using () renaming (_↑✶_ to _↑✶₂_; _/✶_ to _/✶₂₂_)
in
∀ {m n} (σs₁ : Subs Fin m n) (σs₂ : Subs T₂ m n) →
(∀ k x → var x /✶₂₃ σs₁ ↑✶₃ k ≡ var x /✶₂₂ σs₂ ↑✶₂ k) →
∀ k t → t /✶₁₃ σs₁ ↑✶₃ k ≡ t /✶₁₂ σs₂ ↑✶₂ k
-- An instantiation of the above lemmas for T₂ substitutions in T₁s.
termLiftAppLemmas : LiftAppLemmas T₁ T₂ T₂
termLiftAppLemmas = record
{ lift = Lift.lift termLift
; application₁₃ = termApplication
; application₂₃ = TermLemmas.application termLemmas
; lemmas₂ = TermLemmas.lemmas₄ termLemmas
; lemmas₃ = TermLemmas.lemmas₄ termLemmas
; lift-/ = λ _ → refl
; lift-var = λ _ → refl
; /✶-↑✶ = /✶-↑✶₁
}
open LiftAppLemmas termLiftAppLemmas public hiding (/-wk)
-- An instantiation of the above lemmas for variable substitutions
-- (renamings) in T₁s.
varLiftSubLemmas : LiftSubLemmas T₁ T₂ Fin
varLiftSubLemmas = record
{ application₁₂ = termApplication
; liftAppLemmas = record
{ lift = Lift.lift varLift
; application₁₃ = varApplication
; application₂₃ = Lifted.application varLift
; lemmas₂ = TermLemmas.lemmas₄ termLemmas
; lemmas₃ = VarLemmas.lemmas₄
; lift-/ = λ _ → sym (TermLemmas.app-var termLemmas)
; lift-var = λ _ → refl
; /✶-↑✶ = /✶-↑✶₁
}
; weaken-lift = λ _ → TermLemmas.weaken-var termLemmas
; /-⊙₂ = AppLemmas./-⊙ appLemmas
; /✶-↑✶₁ = /✶-↑✶₂
; /✶-↑✶₂ = TermLemmas./✶-↑✶ termLemmas
}
open Application varApplication public using () renaming (_/_ to _/Var_)
open LiftSubLemmas varLiftSubLemmas public hiding (/✶-↑✶₁; /✶-↑✶₂)
renaming (liftAppLemmas to varLiftAppLemmas)
-- Weakening of T₁s.
weaken : ∀ {n} → T₁ n → T₁ (suc n)
weaken t = t /Var VarSubst.wk
-- A version of wk-sub-vanishes for T₁s.
weaken-sub : ∀ {n t′} → (t : T₁ n) → weaken t / sub t′ ≡ t
weaken-sub t = begin
weaken t / sub _ ≡⟨ cong₂ _/_ /-wk refl ⟩
t / wk / sub _ ≡⟨ wk-sub-vanishes t ⟩
t ∎
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open import Relation.Binary.Core
module PLRTree.DropLast.Permutation {A : Set}
(_≤_ : A → A → Set)
(tot≤ : Total _≤_) where
open import Data.List
open import Data.Sum
open import List.Permutation.Base A
open import List.Permutation.Base.Concatenation A
open import List.Permutation.Base.Equivalence A
open import PLRTree {A}
open import PLRTree.Complete {A}
open import PLRTree.Compound {A}
open import PLRTree.Drop _≤_ tot≤
open import PLRTree.DropLast.Complete _≤_ tot≤
open import PLRTree.Equality {A}
lemma-last-dropLast : {t : Tag}{x : A}{l r : PLRTree} → Complete (node t x l r) → (last (node t x l r) compound ∷ flatten (dropLast (node t x l r))) ∼ flatten (node t x l r)
lemma-last-dropLast (perfect {l} {r} x cl cr l≃r)
with l | r | l≃r
... | leaf | leaf | ≃lf = ∼x /head /head ∼[]
... | node perfect x' l' r' | node perfect x'' l'' r'' | ≃nd .x' .x'' l'≃r' l''≃r'' l'≃l'' =
let _l = node perfect x' l' r' ;
_r = node perfect x'' l'' r'' ;
zxflfrd∼xflzfrd = ∼x /head (lemma++/ {last _r compound} {x ∷ flatten _l}) refl∼ ;
xflzfrd∼xflfr = lemma++∼l {x ∷ flatten _l} (lemma-last-dropLast cr)
in trans∼ zxflfrd∼xflzfrd xflzfrd∼xflfr
lemma-last-dropLast (left {l} {r} x cl cr l⋘r)
with l | r | l⋘r | lemma-dropLast-⋘ l⋘r
... | leaf | _ | () | _
... | node perfect x' l' r' | _ | () | _
... | node left x' l' r' | node perfect x'' l'' r'' | l⋘ .x' .x'' l'⋘r' l''≃r'' r'≃l'' | inj₁ ld⋘r
with dropLast (node left x' l' r') | ld⋘r | lemma-last-dropLast cl
... | leaf | () | _
... | node perfect _ _ _ | () | _
... | node left x''' l''' r''' | ld⋘r' | zfldfr∼flfr = ∼x (/tail /head) /head (lemma++∼r zfldfr∼flfr)
... | node right x''' l''' r''' | ld⋘r' | zfldfr∼flfr = ∼x (/tail /head) /head (lemma++∼r zfldfr∼flfr)
lemma-last-dropLast (left x cl cr l⋘r) | node left x' l' r' | node perfect x'' l'' r'' | l⋘ .x' .x'' l'⋘r' l''≃r'' r'≃l'' | inj₂ ld≃r
with dropLast (node left x' l' r') | ld≃r | lemma-last-dropLast cl
... | leaf | () | _
... | node perfect x''' l''' r''' | ld≃r' | zfldfr∼flfr = ∼x (/tail /head) /head (lemma++∼r zfldfr∼flfr)
... | node left _ _ _ | () | _
... | node right _ _ _ | () | _
lemma-last-dropLast (left x cl cr l⋘r) | node right x' l' r' | leaf | () | _
lemma-last-dropLast (left x cl cr l⋘r) | node right x' (node perfect x'' leaf leaf) leaf | node perfect x''' leaf leaf | x⋘ .x' .x'' .x''' | inj₁ ()
lemma-last-dropLast (left x cl cr l⋘r) | node right x' (node perfect x'' leaf leaf) leaf | node perfect x''' leaf leaf | x⋘ .x' .x'' .x''' | inj₂ x'≃x''' =
∼x (/tail /head) /head (lemma++∼r (lemma-last-dropLast cl))
lemma-last-dropLast (left x cl cr l⋘r) | node right x' l' r' | node perfect x'' l'' r'' | r⋘ .x' .x'' l⋙r l''≃r'' l'⋗l'' | inj₁ ld⋘r
with dropLast (node right x' l' r') | ld⋘r | lemma-last-dropLast cl
... | leaf | () | _
... | node perfect _ _ _ | () | _
... | node left x''' l''' r''' | ld⋘r' | zfldfr∼flfr = ∼x (/tail /head) /head (lemma++∼r zfldfr∼flfr)
... | node right x''' l''' r''' | ld⋘r' | zfldfr∼flfr = ∼x (/tail /head) /head (lemma++∼r zfldfr∼flfr)
lemma-last-dropLast (left x cl cr l⋘r) | node right x' l' r' | node perfect x'' l'' r'' | r⋘ .x' .x'' l⋙r l''≃r'' l'⋗l'' | inj₂ ld≃r
with dropLast (node right x' l' r') | ld≃r | lemma-last-dropLast cl
... | leaf | () | _
... | node perfect x''' l''' r''' | ld≃r' | zfldfr∼flfr = ∼x (/tail /head) /head (lemma++∼r zfldfr∼flfr)
... | node left _ _ _ | () | _
... | node right _ _ _ | () | _
lemma-last-dropLast (left x cl cr l⋘r) | node right x' l' r' | node left x'' l'' r'' | () | _
lemma-last-dropLast (left x cl cr l⋘r) | node right x' l' r' | node right x'' l'' r'' | () | _
lemma-last-dropLast (right {l} {r} x cl cr l⋙r)
with l | r | l⋙r
... | leaf | _ | ⋙p ()
... | node perfect x' leaf leaf | leaf | ⋙p (⋗lf .x') = ∼x (/tail /head) /head (∼x /head /head ∼[])
... | node perfect x' leaf leaf | node left _ _ _ | ⋙p ()
... | node perfect x' leaf leaf | node right _ _ _ | ⋙p ()
... | node perfect x' l' r' | node perfect x'' l'' r'' | ⋙p (⋗nd .x' .x'' l'≃r' l''≃r'' l'⋗l'') =
∼x (/tail /head) /head (lemma++∼r (lemma-last-dropLast cl))
... | node perfect x' l' r' | node left x'' l'' r'' | ⋙l .x' .x'' l'≃r' l''⋘r'' l'⋗r''
with lemma-dropLast-⋙ (⋙l x' x'' l'≃r' l''⋘r'' l'⋗r'')
... | inj₁ l⋙rd =
let _l = node perfect x' l' r' ;
_r = node left x'' l'' r'' ;
zxflfrd∼xflzfrd = ∼x /head (lemma++/ {last _r compound} {x ∷ flatten _l}) refl∼ ;
xflzfrd∼xflfr = lemma++∼l {x ∷ flatten _l} (lemma-last-dropLast cr)
in trans∼ zxflfrd∼xflzfrd xflzfrd∼xflfr
... | inj₂ ()
lemma-last-dropLast (right x cl cr l⋙r) | node perfect x' l' r' | node right x'' l'' r'' | ⋙r .x' .x'' l'≃r' l''⋙r'' l'≃l''
with lemma-dropLast-⋙ (⋙r x' x'' l'≃r' l''⋙r'' l'≃l'')
... | inj₁ l⋙rd =
let _l = node perfect x' l' r' ;
_r = node right x'' l'' r'' ;
zxflfrd∼xflzfrd = ∼x /head (lemma++/ {last _r compound} {x ∷ flatten _l}) refl∼ ;
xflzfrd∼xflfr = lemma++∼l {x ∷ flatten _l} (lemma-last-dropLast cr)
in trans∼ zxflfrd∼xflzfrd xflzfrd∼xflfr
... | inj₂ ()
lemma-last-dropLast (right x cl cr l⋙r) | node left _ _ _ | _ | ⋙p ()
lemma-last-dropLast (right x cl cr l⋙r) | node right _ _ _ | _ | ⋙p ()
lemma-dropLast-∼ : {t : Tag}{x : A}{l r : PLRTree} → Complete (node t x l r) → flatten (setRoot (last (node t x l r) compound) (dropLast (node t x l r))) ∼ (flatten l ++ flatten r)
lemma-dropLast-∼ (perfect {l} {r} x cl cr l≃r)
with l | r | l≃r
... | leaf | leaf | ≃lf = ∼[]
... | node perfect x' l' r' | node perfect x'' l'' r'' | ≃nd .x' .x'' l'≃r' l''≃r'' l'≃l'' =
let _l = node perfect x' l' r' ;
_r = node perfect x'' l'' r'' ;
zflfrd∼flzfrd = ∼x /head (lemma++/ {last _r compound} {flatten _l}) refl∼ ;
flzfrd∼flfr = lemma++∼l {flatten _l} (lemma-last-dropLast cr)
in trans∼ zflfrd∼flzfrd flzfrd∼flfr
lemma-dropLast-∼ (left {l} {r} x cl cr l⋘r)
with l | r | l⋘r | lemma-dropLast-⋘ l⋘r
... | leaf | _ | () | _
... | node perfect x' l' r' | _ | () | _
... | node left x' l' r' | node perfect x'' l'' r'' | l⋘ .x' .x'' l'⋘r' l''≃r'' r'≃l'' | inj₁ ld⋘r
with dropLast (node left x' l' r') | ld⋘r | lemma-last-dropLast cl
... | leaf | () | _
... | node perfect _ _ _ | () | _
... | node left x''' l''' r''' | ld⋘r' | zfld∼fl = lemma++∼r zfld∼fl
... | node right x''' l''' r''' | ld⋘r' | zfld∼fl = lemma++∼r zfld∼fl
lemma-dropLast-∼ (left x cl cr l⋘r) | node left x' l' r' | node perfect x'' l'' r'' | l⋘ .x' .x'' l'⋘r' l''≃r'' r'≃l'' | inj₂ ld≃r
with dropLast (node left x' l' r') | ld≃r | lemma-last-dropLast cl
... | leaf | () | _
... | node perfect x''' l''' r''' | ld≃r' | zfld∼fl = lemma++∼r zfld∼fl
... | node left _ _ _ | () | _
... | node right _ _ _ | () | _
lemma-dropLast-∼ (left x cl cr l⋘r) | node right x' l' r' | leaf | () | _
lemma-dropLast-∼ (left x cl cr l⋘r) | node right x' (node perfect x'' leaf leaf) leaf | node perfect x''' leaf leaf | x⋘ .x' .x'' .x''' | inj₁ ()
lemma-dropLast-∼ (left x cl cr l⋘r) | node right x' (node perfect x'' leaf leaf) leaf | node perfect x''' leaf leaf | x⋘ .x' .x'' .x''' | inj₂ x'≃x''' =
lemma++∼r (lemma-last-dropLast cl)
lemma-dropLast-∼ (left x cl cr l⋘r) | node right x' l' r' | node perfect x'' l'' r'' | r⋘ .x' .x'' l⋙r l''≃r'' l'⋗l'' | inj₁ ld⋘r
with dropLast (node right x' l' r') | ld⋘r | lemma-last-dropLast cl
... | leaf | () | _
... | node perfect _ _ _ | () | _
... | node left x''' l''' r''' | ld⋘r' | zfld∼fl = lemma++∼r zfld∼fl
... | node right x''' l''' r''' | ld⋘r' | zfld∼fl = lemma++∼r zfld∼fl
lemma-dropLast-∼ (left x cl cr l⋘r) | node right x' l' r' | node perfect x'' l'' r'' | r⋘ .x' .x'' l⋙r l''≃r'' l'⋗l'' | inj₂ ld≃r
with dropLast (node right x' l' r') | ld≃r | lemma-last-dropLast cl
... | leaf | () | _
... | node perfect x''' l''' r''' | ld≃r' | zfld∼fl = lemma++∼r zfld∼fl
... | node left _ _ _ | () | _
... | node right _ _ _ | () | _
lemma-dropLast-∼ (left x cl cr l⋘r) | node right x' l' r' | node left x'' l'' r'' | () | _
lemma-dropLast-∼ (left x cl cr l⋘r) | node right x' l' r' | node right x'' l'' r'' | () | _
lemma-dropLast-∼ (right {l} {r} x cl cr l⋙r)
with l | r | l⋙r
... | leaf | _ | ⋙p ()
... | node perfect x' leaf leaf | leaf | ⋙p (⋗lf .x') = ∼x /head /head ∼[]
... | node perfect x' leaf leaf | node left _ _ _ | ⋙p ()
... | node perfect x' leaf leaf | node right _ _ _ | ⋙p ()
... | node perfect x' l' r' | node perfect x'' l'' r'' | ⋙p (⋗nd .x' .x'' l'≃r' l''≃r'' l'⋗l'') = lemma++∼r (lemma-last-dropLast cl)
... | node perfect x' l' r' | node left x'' l'' r'' | ⋙l .x' .x'' l'≃r' l''⋘r'' l'⋗r''
with lemma-dropLast-⋙ (⋙l x' x'' l'≃r' l''⋘r'' l'⋗r'')
... | inj₁ l⋙rd =
let _l = node perfect x' l' r' ;
_r = node left x'' l'' r'' ;
zflfrd∼flzfrd = ∼x /head (lemma++/ {last _r compound} {flatten _l}) refl∼ ;
flzfrd∼flfr = lemma++∼l {flatten _l} (lemma-last-dropLast cr)
in trans∼ zflfrd∼flzfrd flzfrd∼flfr
... | inj₂ ()
lemma-dropLast-∼ (right x cl cr l⋙r) | node perfect x' l' r' | node right x'' l'' r'' | ⋙r .x' .x'' l'≃r' l''⋙r'' l'≃l''
with lemma-dropLast-⋙ (⋙r x' x'' l'≃r' l''⋙r'' l'≃l'')
... | inj₁ l⋙rd =
let _l = node perfect x' l' r' ;
_r = node right x'' l'' r'' ;
zflfrd∼flzfrd = ∼x /head (lemma++/ {last _r compound} {flatten _l}) refl∼ ;
flzfrd∼flfr = lemma++∼l {flatten _l} (lemma-last-dropLast cr)
in trans∼ zflfrd∼flzfrd flzfrd∼flfr
... | inj₂ ()
lemma-dropLast-∼ (right x cl cr l⋙r) | node left _ _ _ | _ | ⋙p ()
lemma-dropLast-∼ (right x cl cr l⋙r) | node right _ _ _ | _ | ⋙p ()
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module Structs where
open import Data.List
open import Data.String
open import Data.Product hiding (map)
open import Relation.Binary.PropositionalEquality as PE hiding ([_])
open import Data.List.Any hiding (map)
open module M {ℓ}{X : Set ℓ} = Membership (PE.setoid X )
import Level
data StructEnd {ℓ} : Set ℓ where
end : StructEnd
data StructField {ℓ}(s : String)(t : Set ℓ) : Set ℓ where
_∹_ : (x : String) → {_ : x ≡ s} → t → StructField s t
FieldTypes = λ ℓ → List (String × Set ℓ)
Struct : ∀{ℓ} → FieldTypes ℓ → Set ℓ
Struct [] = StructEnd
Struct {ℓ} ( (n , t) ∷ l) = StructField {ℓ} n t × Struct l
infixr 6 _∶_∷_
data StructLit {ℓ} : FieldTypes ℓ → Set (Level.suc ℓ) where
[] : StructLit []
_∶_∷_ : ∀{t : Set ℓ}{ls} → (s : String) → t → StructLit ls → StructLit ((s , t) ∷ ls)
⟨_⟩ : ∀{ℓ}{ls} → StructLit {ℓ} ls → Struct ls
⟨ [] ⟩ = end
⟨ s ∶ x ∷ xs ⟩ = _∹_ s {refl} x , ⟨ xs ⟩
instance
found-it : ∀{ℓ}{X : Set ℓ}{x : X}{xs : List X}
→ x ∈ (x ∷ xs)
found-it = here refl
keep-looking : ∀{ℓ}{X : Set ℓ}{x y : X}{xs : List X}
→ ⦃ t : y ∈ xs ⦄
→ y ∈ (x ∷ xs)
keep-looking ⦃ t ⦄ = there t
lookup : ∀{ℓ ℓ′}{K : Set ℓ}{V : Set ℓ′}{s}
→ (l : List (K × V)) → s ∈ (map proj₁ l) → V
lookup [] ()
lookup (x ∷ l) (here refl) = proj₂ x
lookup (x ∷ l) (there p) = lookup l p
update : ∀{ℓ ℓ′}{K : Set ℓ}{V : Set ℓ′}{s}
→ (l : List (K × V)) → s ∈ (map proj₁ l) → V → List (K × V)
update [] () _
update (x ∷ l) (here refl) n = (proj₁ x , n) ∷ l
update (x ∷ l) (there p) n = x ∷ update l p n
field-names = λ{ℓ}(ls : FieldTypes ℓ) → map proj₁ ls
infixl 6 _∙_
_∙_ : ∀{ℓ}{ls : FieldTypes ℓ}
→ Struct ls
→ (s : String) ⦃ prf : s ∈ field-names ls ⦄
→ lookup ls prf
_∙_ {ls = []} _ x ⦃ () ⦄
_∙_ {ls = x ∷ ls} (tg ∹ v , r) ._ ⦃ here refl ⦄ = v
_∙_ {ls = x ∷ ls} (k , r) x₁ ⦃ there prf ⦄ = _∙_ {ls = ls} r x₁ ⦃ prf ⦄
infixl 6 _∙_≔_
module TypeChangingAssignment where
infixl 6 _∙_≔′_
_∙_≔′_ : ∀{ℓ}{ls : FieldTypes ℓ}{X : Set ℓ}
→ Struct ls
→ (s : String) ⦃ prf : s ∈ field-names ls ⦄
→ X
→ Struct (update ls prf X)
_∙_≔′_ {ls = []} s t ⦃ () ⦄ n
_∙_≔′_ {ls = (q , _) ∷ ls} (_∹_ .q {refl} v , r) ._ ⦃ here refl ⦄ n = _∹_ q {refl} n , r
_∙_≔′_ {ls = x ∷ ls} (t , r) k ⦃ there prf ⦄ n = t , (r ∙ k ≔′ n)
memb-update : ∀{ℓ ℓ′}{K : Set ℓ}{V : Set ℓ′}{ls : List (K × V)}{k}{x : V} → ⦃ prf : k ∈ map proj₁ ls ⦄ → k ∈ map proj₁ (update ls prf x)
memb-update {ls = []} ⦃ () ⦄
memb-update {ls = x ∷ ls} {{prf = here refl}} = here refl
memb-update {ls = x ∷ ls} {{prf = there prf}} = there (memb-update {ls = ls} ⦃ prf ⦄)
lookup-update : ∀{ℓ ℓ′}{K : Set ℓ}{V : Set ℓ′}{ls : List (K × V)}{k}{x : V}
→ ⦃ prf : k ∈ map proj₁ ls ⦄
→ lookup (update ls prf x) (memb-update ⦃ prf ⦄) ≡ x
lookup-update {ls = []} ⦃ prf = () ⦄
lookup-update {ls = x ∷ ls} ⦃ prf = here refl ⦄ = refl
lookup-update {ls = x ∷ ls} ⦃ prf = there prf ⦄ = lookup-update {ls = ls} ⦃ prf ⦄
update-lookup : ∀{ℓ ℓ′}{K : Set ℓ}{V : Set ℓ′}{ls : List (K × V)}{k}
→ ⦃ prf : k ∈ map proj₁ ls ⦄
→ update ls prf (lookup ls prf) ≡ ls
update-lookup {ls = []} ⦃ () ⦄
update-lookup {ls = x ∷ ls} ⦃ here refl ⦄ = refl
update-lookup {ls = x ∷ ls} ⦃ there prf ⦄ rewrite update-lookup {ls = ls} ⦃ prf ⦄ = refl
_∙_≔_ : ∀{ℓ}{ls : FieldTypes ℓ}
→ Struct ls
→ (s : String) ⦃ prf : s ∈ field-names ls ⦄
→ lookup ls prf
→ Struct ls
_∙_≔_ {ls = []} s f ⦃ () ⦄ v
_∙_≔_ {ls = x ∷ ls} (_∹_ ._ {refl} _ , r) ._ ⦃ here refl ⦄ v = _∹_ (proj₁ x) {refl} v , r
_∙_≔_ {ls = x ∷ ls} (k , r) f ⦃ there prf ⦄ v = k , r ∙ f ≔ v
spurious-update : ∀{ℓ}{ls : FieldTypes ℓ}{s : Struct ls}{f}
→ ⦃ prf : f ∈ field-names ls ⦄
→ (s ∙ f ≔ (s ∙ f)) ≡ s
spurious-update {ls = []} ⦃ () ⦄
spurious-update {ls = _ ∷ _}{_∹_ ._ {refl} _ , _} ⦃ here refl ⦄ = refl
spurious-update {ls = _ ∷ _}{_∹_ ._ {refl} _ , _} ⦃ there prf ⦄ = cong (_,_ _) spurious-update
read-after-update : ∀{ℓ}{ls : FieldTypes ℓ}{s : Struct ls}{f}
→ ⦃ prf : f ∈ field-names ls ⦄
→ {v : lookup ls prf}
→ (s ∙ f ≔ v) ∙ f ≡ v
read-after-update {ls = []} ⦃ () ⦄
read-after-update {ls = _ ∷ _}{ _∹_ ._ {refl} _ , _ } ⦃ here refl ⦄ = refl
read-after-update {ls = _ ∷ _}{ _∹_ ._ {refl} _ , _ } ⦃ there prf ⦄ = read-after-update ⦃ prf ⦄
frame-inertia : ∀{ℓ}{ls : FieldTypes ℓ}{s : Struct ls}{f f′}
→ f ≢ f′
→ ⦃ prf : f ∈ field-names ls ⦄
→ ⦃ prf′ : f′ ∈ field-names ls ⦄
→ {v : lookup ls prf′}
→ (s ∙ f′ ≔ v) ∙ f ≡ (s ∙ f)
frame-inertia {ls = []} _ ⦃ () ⦄
frame-inertia {ls = _ ∷ ls}{ _∹_ ._ {refl} _ , _ } _ ⦃ here refl ⦄ ⦃ there prf′ ⦄ = refl
frame-inertia {ls = _ ∷ ls}{ _∹_ ._ {refl} _ , _ } _ ⦃ there prf ⦄ ⦃ here refl ⦄ = refl
frame-inertia {ls = _ ∷ ls}{ _∹_ ._ {refl} _ , _ } p ⦃ there prf ⦄ ⦃ there prf′ ⦄ = frame-inertia {ls = ls} p
frame-inertia {ls = _ ∷ ls}{ _∹_ ._ {refl} _ , _ } p ⦃ here refl ⦄ ⦃ here refl ⦄ with p refl
... | ()
append : ∀{ℓ}{ls : FieldTypes ℓ}{ms : FieldTypes ℓ} → Struct ls → Struct ms → Struct (Data.List._++_ ls ms)
append {ls = [] } s₁ s₂ = s₂
append {ls = x ∷ xs} (f , rest) s₂ = f , append rest s₂
drop-append : ∀{ℓ}{ls : FieldTypes ℓ}{ms : FieldTypes ℓ} → Struct (Data.List._++_ ls ms) → Struct ms
drop-append {ls = []} m = m
drop-append {ls = x ∷ xs} (f , rest) = drop-append {ls = xs} rest
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