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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Setoids.Setoids
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Groups.Definition
module Groups.Subgroups.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S _+_) where
open import Setoids.Subset S
open Group G
record Subgroup {c : _} (pred : A → Set c) : Set (a ⊔ b ⊔ c) where
field
isSubset : subset pred
closedUnderPlus : {g h : A} → (pred g) → (pred h) → pred (g + h)
containsIdentity : pred 0G
closedUnderInverse : ({g : A} → (pred g) → (pred (inverse g)))
subgroupOp : {c : _} {pred : A → Set c} → (s : Subgroup pred) → Sg A pred → Sg A pred → Sg A pred
subgroupOp {pred = pred} record { closedUnderPlus = one } (a , prA) (b , prB) = (a + b) , one prA prB
subgroupIsGroup : {c : _} {pred : A → Set c} → (s : Subgroup pred) → Group (subsetSetoid (Subgroup.isSubset s)) (subgroupOp s)
Group.+WellDefined (subgroupIsGroup s) {m , prM} {n , prN} {x , prX} {y , prY} m=x n=y = +WellDefined m=x n=y
Group.0G (subgroupIsGroup record { containsIdentity = two }) = 0G , two
Group.inverse (subgroupIsGroup record { closedUnderInverse = three }) (a , prA) = (inverse a) , three prA
Group.+Associative (subgroupIsGroup s) {a , prA} {b , prB} {c , prC} = +Associative
Group.identRight (subgroupIsGroup s) {a , prA} = identRight
Group.identLeft (subgroupIsGroup s) {a , prA} = identLeft
Group.invLeft (subgroupIsGroup s) {a , prA} = invLeft
Group.invRight (subgroupIsGroup s) {a , prA} = invRight
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------------------------------------------------------------------------------
-- Generic well-founded induction on trees
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
-- Adapted from FOTC.Data.Nat.Induction.Acc.WellFounded.
module FOT.FOTC.Program.Mirror.Induction.Acc.WellFounded where
open import FOTC.Base
open import FOTC.Program.Mirror.Type
------------------------------------------------------------------------------
-- The accessibility predicate: x is accessible if everything which is
-- smaller than x is also accessible (inductively).
data Acc (_<_ : D → D → Set)(t : D) : Set where
acc : (∀ {t'} → Tree t' → t' < t → Acc _<_ t') → Acc _<_ t
accFold : {P : D → Set}(_<_ : D → D → Set) →
(∀ {t} → Tree t → (∀ {t'} → Tree t' → t' < t → P t') → P t) →
∀ {t} → Tree t → Acc _<_ t → P t
accFold _<_ f Tt (acc h) = f Tt (λ Tt' t'<t → accFold _<_ f Tt' (h Tt' t'<t))
-- The accessibility predicate encodes what it means to be
-- well-founded; if all elements are accessible, then _<_ is
-- well-founded.
WellFounded : (D → D → Set) → Set
WellFounded _<_ = ∀ {t} → Tree t → Acc _<_ t
WellFoundedInduction :
{P : D → Set}{_<_ : D → D → Set} →
WellFounded _<_ →
(∀ {t} → Tree t → (∀ {t'} → Tree t' → t' < t → P t') → P t) →
∀ {t} → Tree t → P t
WellFoundedInduction {_<_ = _<_} wf f Tt = accFold _<_ f Tt (wf Tt)
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------------------------------------------------------------------------
-- Example: Right recursive expression grammar
------------------------------------------------------------------------
module TotalRecognisers.Simple.Expression where
open import Codata.Musical.Notation
open import Data.Bool
open import Data.Char as Char using (Char)
open import Data.String as String using (String)
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary.Decidable
import TotalRecognisers.Simple as P
private
open module PC = P Char Char._≟_ hiding (_·_)
open PC.P using (_·_)
------------------------------------------------------------------------
-- Recognisers for bits and binary numbers
-- Bits.
bit = tok '0' ∣ tok '1'
-- Numbers.
number = bit · ♯ (empty ∣ number)
------------------------------------------------------------------------
-- An expression grammar
-- t ∷= f '+' t ∣ f
-- f ∷= a '*' f ∣ a
-- a ∷= '(' t ')' ∣ n
mutual
term = factor · ♯ (tok '+' · ♯ term)
∣ factor
factor = atom · ♯ (tok '*' · ♯ factor)
∣ atom
atom = tok '(' · ♯ term · ♯ tok ')'
∣ number
------------------------------------------------------------------------
-- Unit tests
module Tests where
test : ∀ {n} → P n → String → Bool
test p s = ⌊ String.toList s ∈? p ⌋
ex₁ : test term "0*(0+0)" ≡ true
ex₁ = refl
ex₂ : test term "0*(0+0" ≡ false
ex₂ = refl
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Indexed universes
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Universe.Indexed where
open import Data.Product
open import Data.Universe
open import Function
open import Level
------------------------------------------------------------------------
-- Definitions
record IndexedUniverse i u e : Set (suc (i ⊔ u ⊔ e)) where
field
I : Set i -- Index set.
U : I → Set u -- Codes.
El : ∀ {i} → U i → Set e -- Decoding function.
-- An indexed universe can be turned into an unindexed one.
unindexed-universe : Universe (i ⊔ u) e
unindexed-universe = record
{ U = ∃ λ i → U i
; El = El ∘ proj₂
}
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module Integer.Signed where
open import Data.Empty as ⊥
open import Data.Product as Σ
open import Data.Product.Relation.Pointwise.NonDependent
open import Data.Sum as ⊎
open import Data.Unit as ⊤
open import Equality
open import Function
open import Natural as ℕ
open import Quotient as /
open import Relation.Binary
open import Syntax
open Equality.FunctionProperties
⟦ℤ⟧ = ℕ ⊎ ℕ
⟦ℤ²⟧ = ⟦ℤ⟧ × ⟦ℤ⟧
infix 10 +1*_ -1*_
pattern +1*_ x = inj₁ x
pattern -1*_ x = inj₂ x
infix 1 _≈_
_≈_ : ⟦ℤ⟧ → ⟦ℤ⟧ → Set
+1* x ≈ +1* y = x ≡ y
+1* x ≈ -1* y = x ≡ 0 × y ≡ 0
-1* x ≈ +1* y = x ≡ 0 × y ≡ 0
-1* x ≈ -1* y = x ≡ y
≈-refl : ∀ {x} → x ≈ x
≈-refl {+1* x} = refl
≈-refl { -1* x} = refl
≈-sym : ∀ {x y} → x ≈ y → y ≈ x
≈-sym {+1* x} {+1* y} = sym
≈-sym {+1* x} { -1* y} = Σ.swap
≈-sym { -1* x} {+1* y} = Σ.swap
≈-sym { -1* x} { -1* y} = sym
≈-trans : ∀ {x y z} → x ≈ y → y ≈ z → x ≈ z
≈-trans {+1* x} {+1* y} {+1* z} p q = compPath p q
≈-trans {+1* x} {+1* y} { -1* z} p (q , r) = p ≫ q , r
≈-trans {+1* x} { -1* y} {+1* z} (p , q) (r , s) = p ≫ sym s
≈-trans {+1* x} { -1* y} { -1* z} (p , q) r = p , sym r ≫ q
≈-trans { -1* x} {+1* y} {+1* z} (p , q) r = p , sym r ≫ q
≈-trans { -1* x} {+1* y} { -1* z} (p , q) (r , s) = p ≫ sym s
≈-trans { -1* x} { -1* y} {+1* z} p (q , r) = p ≫ q , r
≈-trans { -1* x} { -1* y} { -1* z} p q = compPath p q
ℤ = ⟦ℤ⟧ / _≈_
ℤ² = ⟦ℤ²⟧ / Pointwise _≈_ _≈_
instance
⟦ℤ⟧-Number : Number ⟦ℤ⟧
⟦ℤ⟧-Number = record { Constraint = λ _ → ⊤ ; fromNat = λ n → +1* n }
ℤ-Number : Number ℤ
ℤ-Number = record { Constraint = λ _ → ⊤ ; fromNat = λ n → ⟦ fromNat n ⟧ }
⟦ℤ⟧-Negative : Negative ⟦ℤ⟧
⟦ℤ⟧-Negative = record { Constraint = λ _ → ⊤ ; fromNeg = λ n → -1* n }
ℤ-Negative : Negative ℤ
ℤ-Negative = record { Constraint = λ _ → ⊤ ; fromNeg = λ n → ⟦ fromNeg n ⟧ }
⟦negate⟧ : ⟦ℤ⟧ → ⟦ℤ⟧
⟦negate⟧ (+1* n) = -1* n
⟦negate⟧ (-1* n) = +1* n
negate : ℤ → ℤ
negate =
⟦negate⟧ // λ where
(+1* x) (+1* y) → id
(+1* x) (-1* y) → id
(-1* x) (+1* y) → id
(-1* x) (-1* y) → id
ℕ-minus : ℕ → ℕ → ⟦ℤ⟧
ℕ-minus zero n = -1* n
ℕ-minus m zero = +1* m
ℕ-minus (suc m) (suc n) = ℕ-minus m n
instance
ℕ-minus-syntax : minus-syntax-simple ℕ ℕ ⟦ℤ⟧
ℕ-minus-syntax = λ where ._-_ → ℕ-minus
ℕ-minus-identityˡ : ∀ a → ℕ-minus 0 a ≈ -1* a
ℕ-minus-identityˡ a = refl
ℕ-minus-identityʳ : ∀ a → ℕ-minus a 0 ≈ +1* a
ℕ-minus-identityʳ zero = refl , refl
ℕ-minus-identityʳ (suc a) = refl
⟦plus⟧ : ⟦ℤ²⟧ → ⟦ℤ⟧
⟦plus⟧ (+1* a , +1* b) = +1* (a + b)
⟦plus⟧ (+1* a , -1* b) = a - b
⟦plus⟧ (-1* a , +1* b) = b - a
⟦plus⟧ (-1* a , -1* b) = -1* (a + b)
plus : ℤ² → ℤ
plus =
⟦plus⟧ // λ where
(+1* a , +1* b) (+1* c , +1* d) (p , q) → cong₂ _+_ p q
(+1* a , +1* b) (+1* c , -1* d) (p , (q , r)) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ ,′ ⟪ r ⟫ of λ where
(♥ , ♥ , ♥) →
≈-trans
(⟨ ℕ.+-identityʳ a ⟩)
(≈-sym (ℕ-minus-identityʳ a))
(+1* a , +1* b) (-1* c , +1* d) ((p , q) , r) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ ,′ ⟪ r ⟫ of λ where
(♥ , ♥ , ♥) →
≈-sym (ℕ-minus-identityʳ b)
(+1* a , +1* b) (-1* c , -1* d) ((p , q) , (r , s)) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ ,′ ⟪ r ⟫ ,′ ⟪ s ⟫ of λ where
(♥ , ♥ , ♥ , ♥) →
refl , refl
(+1* a , -1* b) (+1* c , +1* d) (p , (q , r)) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ ,′ ⟪ r ⟫ of λ where
(♥ , ♥ , ♥) →
≈-trans {ℕ-minus a 0} {+1* a} {+1* (a + 0)}
(ℕ-minus-identityʳ a)
(sym ⟨ ℕ.+-identityʳ a ⟩)
(+1* a , -1* b) (+1* c , -1* d) (p , q) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ of λ where
(♥ , ♥) →
≈-refl
(+1* a , -1* b) (-1* c , +1* d) ((p , q) , (r , s)) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ ,′ ⟪ r ⟫ ,′ ⟪ s ⟫ of λ where
(♥ , ♥ , ♥ , ♥) →
refl
(+1* a , -1* b) (-1* c , -1* d) ((p , q) , r) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ ,′ ⟪ r ⟫ of λ where
(♥ , ♥ , ♥) →
refl
(-1* a , +1* b) (+1* c , +1* d) ((p , q) , r) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ ,′ ⟪ r ⟫ of λ where
(♥ , ♥ , ♥) →
ℕ-minus-identityʳ b
(-1* a , +1* b) (+1* c , -1* d) ((p , q) , (r , s)) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ ,′ ⟪ r ⟫ ,′ ⟪ s ⟫ of λ where
(♥ , ♥ , ♥ , ♥) →
refl
(-1* a , +1* b) (-1* c , +1* d) (p , q) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ of λ where
(♥ , ♥) →
≈-refl
(-1* a , +1* b) (-1* c , -1* d) (p , (q , r)) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ ,′ ⟪ r ⟫ of λ where
(♥ , ♥ , ♥) →
sym ⟨ ℕ.+-identityʳ a ⟩
(-1* a , -1* b) (+1* c , +1* d) ((p , q) , (r , s)) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ ,′ ⟪ r ⟫ ,′ ⟪ s ⟫ of λ where
(♥ , ♥ , ♥ , ♥) →
refl , refl
(-1* a , -1* b) (+1* c , -1* d) ((p , q) , r) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ ,′ ⟪ r ⟫ of λ where
(♥ , ♥ , ♥) →
refl
(-1* a , -1* b) (-1* c , +1* d) (p , (q , r)) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ ,′ ⟪ r ⟫ of λ where
(♥ , ♥ , ♥) →
⟨ ℕ.+-identityʳ a ⟩
(-1* a , -1* b) (-1* c , -1* d) (p , q) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ of λ where
(♥ , ♥) →
refl
⟦times⟧ : ⟦ℤ²⟧ → ⟦ℤ⟧
⟦times⟧ (+1* a , +1* b) = +1* (a * b)
⟦times⟧ (+1* a , -1* b) = -1* (a * b)
⟦times⟧ (-1* a , +1* b) = -1* (a * b)
⟦times⟧ (-1* a , -1* b) = +1* (a * b)
times : ℤ² → ℤ
times =
⟦times⟧ // λ where
(+1* a , +1* b) (+1* c , +1* d) (p , q) → cong₂ _*_ p q
(+1* a , -1* b) (+1* c , -1* d) (p , q) → cong₂ _*_ p q
(-1* a , +1* b) (-1* c , +1* d) (p , q) → cong₂ _*_ p q
(-1* a , -1* b) (-1* c , -1* d) (p , q) → cong₂ _*_ p q
(+1* a , +1* b) (+1* c , -1* d) (p , q , r) →
case ⟪ q ⟫ ,′ ⟪ r ⟫ of λ where
(♥ , ♥) → ⟨ *-zeroʳ a ⟩ , ⟨ *-zeroʳ c ⟩
(+1* a , -1* b) (+1* c , +1* d) (p , (q , r)) →
case ⟪ q ⟫ ,′ ⟪ r ⟫ of λ where
(♥ , ♥) → ⟨ *-zeroʳ a ⟩ , ⟨ *-zeroʳ c ⟩
(-1* a , +1* b) (-1* c , -1* d) (p , (q , r)) →
case ⟪ q ⟫ ,′ ⟪ r ⟫ of λ where
(♥ , ♥) → ⟨ *-zeroʳ a ⟩ , ⟨ *-zeroʳ c ⟩
(-1* a , -1* b) (-1* c , +1* d) (p , (q , r)) →
case ⟪ q ⟫ ,′ ⟪ r ⟫ of λ where
(♥ , ♥) → ⟨ *-zeroʳ a ⟩ , ⟨ *-zeroʳ c ⟩
(+1* a , +1* b) (-1* c , +1* d) ((p , q) , r) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ of λ where
(♥ , ♥) → refl , refl
(+1* a , -1* b) (-1* c , -1* d) ((p , q) , r) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ of λ where
(♥ , ♥) → refl , refl
(-1* a , +1* b) (+1* c , +1* d) ((p , q) , r) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ of λ where
(♥ , ♥) → refl , refl
(-1* a , -1* b) (+1* c , -1* d) ((p , q) , r) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ of λ where
(♥ , ♥) → refl , refl
(+1* a , +1* b) (-1* c , -1* d) ((p , q) , r) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ of λ where
(♥ , ♥) → refl
(+1* a , -1* b) (-1* c , +1* d) ((p , q) , r) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ of λ where
(♥ , ♥) → refl
(-1* a , +1* b) (+1* c , -1* d) ((p , q) , r) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ of λ where
(♥ , ♥) → refl
(-1* a , -1* b) (+1* c , +1* d) ((p , q) , r) →
case ⟪ p ⟫ ,′ ⟪ q ⟫ of λ where
(♥ , ♥) → refl
instance
ℤ-plus-syntax : plus-syntax-simple ℤ ℤ ℤ
ℤ-plus-syntax = λ where ._+_ → /.uncurry ≈-refl ≈-refl plus
ℤ-minus-syntax : minus-syntax-simple ℤ ℤ ℤ
ℤ-minus-syntax = λ where ._-_ x y → x + negate y
ℤ-times-syntax : times-syntax-simple ℤ ℤ ℤ
ℤ-times-syntax = λ where ._*_ → /.uncurry ≈-refl ≈-refl times
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-- 2014-02-08 Andreas
-- Eta-equality for records that are recursive via some data type
module _ where
open import Common.Equality
module Nested where
data List (A : Set) : Set where
[] : List A
_∷_ : (x : A)(xs : List A) → List A
record Tree (A : Set) : Set where
constructor tree
field
elem : A
subtrees : List (Tree A)
open Tree
test : ∀ {A} (t : Tree A) → t ≡ tree (elem t) (subtrees t)
test t = refl
-- we should have eta for Tree!
module Mutual where
mutual
data TreeList (A : Set) : Set where
[] : TreeList A
_∷_ : (x : Tree A)(xs : TreeList A) → TreeList A
record Tree (A : Set) : Set where
constructor tree
field
elem : A
subtrees : TreeList A
open Tree
test : ∀ {A} (t : Tree A) → t ≡ tree (elem t) (subtrees t)
test t = refl
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module JVM.Defaults.Syntax.Instructions.Show where
open import Data.String as String hiding (show)
open import Data.Nat.Show as Nat hiding (show)
open import Relation.Binary.PropositionalEquality
open import Relation.Ternary.Structures.Syntax
open import Relation.Ternary.Monad.Weakening
open import Relation.Ternary.Data.ReflexiveTransitive
open import JVM.Types
open import JVM.Contexts
open import JVM.Defaults.Syntax.Values
open import JVM.Defaults.Syntax.Instructions
open import JVM.Model using (↑; ↓)
show-instr : ∀ {Φ} → ⟨ Γ ∣ ψ₁ ⇒ ψ₂ ⟩ Φ → String
show-instr noop = "noop"
show-instr pop = "pop"
show-instr (push x) = "push"
where
show-const : Const a → String
show-const null = "null"
show-const unit = "unit"
show-const (num x) = Nat.show x
show-instr dup = "dup"
show-instr swap = "swap"
show-instr (bop x) = "bop"
show-instr new = "new"
show-instr read = "read"
show-instr write = "write"
show-instr (load (refl ⇈ wk)) = "load " ++ Nat.show (indexOf wk)
show-instr (store (refl ⇈ wk)) = "store " ++ Nat.show (indexOf wk)
show-instr (goto x) = "goto"
show-instr (if c x₁) = "if" ++ show-comp c
where
show-comp : Comparator → String
show-comp eq = "eq"
show-comp ne = "ne"
show-comp lt = "lt"
show-comp ge = "ge"
show-comp gt = "gt"
show-comp le = "le"
show-instr ret = "ret"
show-code : ∀ {Γ Φ} → Code Γ ψ₁ ψ₂ Φ → String
show-code (labeled (_ ∙⟨ _ ⟩ ↓ i)) = "<>: " ++ show-instr i
show-code (instr (↓ i)) = " " ++ show-instr i
show : ∀ {Φ} → ⟪ Γ ∣ ψ₁ ⇒ ψ₂ ⟫ Φ → String
show nil = ""
show (c ▹⟨ σ ⟩ is) = show-code c ++ "\n" ++ show is
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{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9.
Copyright (c) 2020, 2021, Oracle and/or its affiliates.
Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl
-}
module Haskell.Modules.ToBool where
open import Data.Bool hiding (not)
import Function
import Relation.Nullary as RN
import Relation.Nullary.Decidable.Core as RNDC
record ToBool {a}(A : Set a) : Set a where
field
toBool : A → Bool
open ToBool {{ ... }} public
not : ∀ {b} {B : Set b} ⦃ _ : ToBool B ⦄ → B → Bool
not b = Data.Bool.not (toBool b)
instance
ToBool-Bool : ToBool Bool
ToBool-Bool = record { toBool = Function.id }
ToBool-Dec : ∀{a}{A : Set a} → ToBool (RN.Dec A)
ToBool-Dec = record { toBool = RNDC.⌊_⌋ }
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------------------------------------------------------------------------
-- INCREMENTAL λ-CALCULUS
--
-- Logical relation for erasure with the Nehemiah plugin.
------------------------------------------------------------------------
module Nehemiah.Change.Implementation where
open import Nehemiah.Syntax.Type
open import Nehemiah.Syntax.Term
open import Nehemiah.Denotation.Value
open import Nehemiah.Denotation.Evaluation
open import Nehemiah.Change.Validity
open import Nehemiah.Change.Type
open import Nehemiah.Change.Value
open import Nehemiah.Change.Derive
open import Relation.Binary.PropositionalEquality
open import Data.Unit
open import Data.Product
open import Data.Integer
open import Structure.Bag.Nehemiah
import Parametric.Change.Implementation
Const ⟦_⟧Base ⟦_⟧Const ΔBase
⟦apply-base⟧ ⟦diff-base⟧ ⟦nil-base⟧ derive-const as Implementation
private
implements-base : ∀ ι {v : ⟦ ι ⟧Base} → Δ₍ ι ₎ v → ⟦ ΔBase ι ⟧Type → Set
implements-base base-int {v} Δv Δv′ = Δv ≡ Δv′
implements-base base-bag {v} Δv Δv′ = Δv ≡ Δv′
u⊟v≈u⊝v-base : ∀ ι → {u v : ⟦ ι ⟧Base} →
implements-base ι {v} (u ⊟₍ ι ₎ v) (⟦diff-base⟧ ι u v)
u⊟v≈u⊝v-base base-int = refl
u⊟v≈u⊝v-base base-bag = refl
nil-v≈⟦nil⟧-v-base : ∀ ι {v : ⟦ ι ⟧Base} →
implements-base ι (nil₍ ι ₎ v) (⟦nil-base⟧ ι v)
nil-v≈⟦nil⟧-v-base base-int = refl
nil-v≈⟦nil⟧-v-base base-bag = refl
carry-over-base : ∀ {ι}
{v : ⟦ ι ⟧Base}
(Δv : Δ₍ ι ₎ v)
{Δv′ : ⟦ ΔBase ι ⟧Type} (Δv≈Δv′ : implements-base ι {v} Δv Δv′) →
v ⊞₍ base ι ₎ Δv ≡ v ⟦⊕₍ base ι ₎⟧ Δv′
carry-over-base {base-int} {v} Δv Δv≈Δv′ = cong (_+_ v) Δv≈Δv′
carry-over-base {base-bag} Δv Δv≈Δv′ = cong (_++_ (before₍ bag ₎ Δv)) Δv≈Δv′
implementation-structure : Implementation.Structure
implementation-structure = record
{ implements-base = implements-base
; u⊟v≈u⊝v-base = u⊟v≈u⊝v-base
; nil-v≈⟦nil⟧-v-base = nil-v≈⟦nil⟧-v-base
; carry-over-base = carry-over-base
}
open Implementation.Structure implementation-structure public
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------------------------------------------------------------------------
-- The Agda standard library
--
-- The max operator derived from an arbitrary total order
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Relation.Binary
module Algebra.Construct.NaturalChoice.Max
{a ℓ₁ ℓ₂} (totalOrder : TotalOrder a ℓ₁ ℓ₂) where
open import Algebra.FunctionProperties
open import Relation.Binary.Construct.Converse using ()
renaming (totalOrder to flip)
open TotalOrder totalOrder renaming (Carrier to A)
------------------------------------------------------------------------
-- Max is just min with a flipped order
import Algebra.Construct.NaturalChoice.Min (flip totalOrder) as Min
infixl 6 _⊔_
_⊔_ : Op₂ A
_⊔_ = Min._⊓_
open Min public using ()
renaming
( ⊓-cong to ⊔-cong
; ⊓-idem to ⊔-idem
; ⊓-sel to ⊔-sel
; ⊓-assoc to ⊔-assoc
; ⊓-comm to ⊔-comm
; ⊓-identityˡ to ⊔-identityˡ
; ⊓-identityʳ to ⊔-identityʳ
; ⊓-identity to ⊔-identity
; ⊓-zeroˡ to ⊔-zeroˡ
; ⊓-zeroʳ to ⊔-zeroʳ
; ⊓-zero to ⊔-zero
; ⊓-isMagma to ⊔-isMagma
; ⊓-isSemigroup to ⊔-isSemigroup
; ⊓-isBand to ⊔-isBand
; ⊓-isSemilattice to ⊔-isSemilattice
; ⊓-isMonoid to ⊔-isMonoid
; ⊓-magma to ⊔-magma
; ⊓-semigroup to ⊔-semigroup
; ⊓-band to ⊔-band
; ⊓-semilattice to ⊔-semilattice
; ⊓-monoid to ⊔-monoid
; x⊓y≈y⇒y≤x to x⊔y≈y⇒x≤y
; x⊓y≈x⇒x≤y to x⊔y≈x⇒y≤x
)
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module string where
open import bool
open import eq
open import char
open import list
open import nat
open import unit
open import maybe
open import product
----------------------------------------------------------------------
-- datatypes
----------------------------------------------------------------------
postulate
Pair : (A B : Set) → Set
pair : {A B : Set} → A → B → Pair A B
pair-fst : {A B : Set} → Pair A B → A
pair-snd : {A B : Set} → Pair A B → B
{-# COMPILE GHC Pair = type (,) #-}
{-# COMPILE GHC pair = \ _ _ a b -> (a, b) #-}
{-# COMPILE GHC pair-fst = \ _ _ p -> fst p #-}
{-# COMPILE GHC pair-snd = \ _ _ p -> snd p #-}
postulate
string : Set
stringUncons : string → maybe (Pair char string)
stringFoldl : ∀{A : Set} → (A → char → A) → A → string → A
stringFoldr : ∀{A : Set} → (char → A → A) → A → string → A
{-# BUILTIN STRING string #-}
{-# COMPILE GHC stringUncons = Data.Text.uncons #-}
{-# COMPILE GHC stringFoldl = \ x -> Data.Text.foldl #-}
{-# COMPILE GHC stringFoldr = \ x -> Data.Text.foldr #-}
private
primitive
primStringToList : string → 𝕃 char
primStringAppend : string → string → string
primStringFromList : 𝕃 char → string
primStringEquality : string → string → 𝔹
-- see string-thms.agda for some axioms about the above primitive functions
----------------------------------------------------------------------
-- syntax
----------------------------------------------------------------------
infixr 6 _^_
infix 8 _=string_
----------------------------------------------------------------------
-- operations
----------------------------------------------------------------------
_^_ : string → string → string
_^_ = primStringAppend
string-uncons : string → maybe (char × string)
string-uncons x with stringUncons x
string-uncons x | nothing = nothing
string-uncons x | just x₁ = just (pair-fst x₁ , pair-snd x₁)
string-to-𝕃char : string → 𝕃 char
string-to-𝕃char = primStringToList
𝕃char-to-string : 𝕃 char → string
𝕃char-to-string = primStringFromList
_=string_ : string → string → 𝔹
_=string_ = primStringEquality
char-to-string : char → string
char-to-string c = 𝕃char-to-string [ c ]
string-append-t : ℕ → Set
string-append-t 0 = string → string
string-append-t (suc n) = string → (string-append-t n)
string-append-h : (n : ℕ) → string → string-append-t n
string-append-h 0 ret = λ x → ret ^ x
string-append-h (suc n) ret = λ x → (string-append-h n (ret ^ x))
string-append : (n : ℕ) → string-append-t n
string-append n = string-append-h n ""
string-concat : 𝕃 string → string
string-concat [] = ""
string-concat (s :: ss) = s ^ (string-concat ss)
string-concat-sep : (separator : string) → 𝕃 string → string
string-concat-sep sep [] = ""
string-concat-sep sep (s1 :: ss) with ss
... | [] = s1
... | s2 :: ss' = s1 ^ sep ^ (string-concat-sep sep ss)
string-concat-sep-map : ∀{A : Set} → (separator : string) → (A → string) → 𝕃 A → string
string-concat-sep-map sep f l = string-concat-sep sep (map f l)
escape-string-h : 𝕃 char → 𝕃 char
escape-string-h ('\n' :: cs) = '\\' :: 'n' :: (escape-string-h cs)
escape-string-h ('"' :: cs) = '\\' :: '"' :: (escape-string-h cs)
escape-string-h (c :: cs) = c :: escape-string-h cs
escape-string-h [] = []
escape-string : string → string
escape-string s = 𝕃char-to-string( escape-string-h( string-to-𝕃char s ) )
string-length : string → ℕ
string-length s = length (string-to-𝕃char s)
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{-# OPTIONS --without-K --safe #-}
module Categories.Functor.Monoidal where
open import Level
open import Data.Product using (Σ; _,_)
open import Categories.Category
open import Categories.Category.Product
open import Categories.Category.Monoidal
open import Categories.Functor hiding (id)
open import Categories.NaturalTransformation hiding (id)
private
variable
o ℓ e : Level
C D : Category o ℓ e
module _ (MC : Monoidal C) (MD : Monoidal D) where
private
module C = Category C
module D = Category D
module MC = Monoidal MC
module MD = Monoidal MD
record MonoidalFunctor : Set (levelOfTerm MC ⊔ levelOfTerm MD) where
field
F : Functor C D
open Functor F public
field
ε : D [ MD.unit , F₀ MC.unit ]
⊗-homo : NaturalTransformation (MD.⊗ ∘F (F ⁂ F)) (F ∘F MC.⊗)
module ⊗-homo = NaturalTransformation ⊗-homo
-- coherence condition
open D
open MD
open Commutation
field
associativity : ∀ {X Y Z} →
[ (F₀ X ⊗₀ F₀ Y) ⊗₀ F₀ Z ⇒ F₀ (X MC.⊗₀ Y MC.⊗₀ Z) ]⟨
⊗-homo.η (X , Y) ⊗₁ id ⇒⟨ F₀ (X MC.⊗₀ Y) ⊗₀ F₀ Z ⟩
⊗-homo.η (X MC.⊗₀ Y , Z) ⇒⟨ F₀ ((X MC.⊗₀ Y) MC.⊗₀ Z) ⟩
F₁ MC.associator.from
≈ associator.from ⇒⟨ F₀ X ⊗₀ F₀ Y ⊗₀ F₀ Z ⟩
id ⊗₁ ⊗-homo.η (Y , Z) ⇒⟨ F₀ X ⊗₀ F₀ (Y MC.⊗₀ Z) ⟩
⊗-homo.η (X , Y MC.⊗₀ Z)
⟩
unitaryˡ : ∀ {X} →
[ unit ⊗₀ F₀ X ⇒ F₀ X ]⟨
ε ⊗₁ id ⇒⟨ F₀ MC.unit ⊗₀ F₀ X ⟩
⊗-homo.η (MC.unit , X) ⇒⟨ F₀ (MC.unit MC.⊗₀ X) ⟩
F₁ MC.unitorˡ.from
≈ unitorˡ.from
⟩
unitaryʳ : ∀ {X} →
[ F₀ X ⊗₀ unit ⇒ F₀ X ]⟨
id ⊗₁ ε ⇒⟨ F₀ X ⊗₀ F₀ MC.unit ⟩
⊗-homo.η (X , MC.unit) ⇒⟨ F₀ (X MC.⊗₀ MC.unit) ⟩
F₁ MC.unitorʳ.from
≈ unitorʳ.from
⟩
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-- Andreas, 2017-01-08
-- Error range when "The name of the top level module does not match" to big
{-# OPTIONS --allow-unsolved-metas #-}
{-# OPTIONS --type-in-type #-}
-- {-# OPTIONS -v scope.checkModuleName:100 #-}
module ThisIsTheWrongName where
postulate Something : Set
-- WAS: Error range included option pragmas.
-- Expected range: only the module name
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{-# OPTIONS -v tc.cover.splittree:50 #-}
open import Agda.Builtin.Nat
data Vec (A : Set) : Nat → Set where
nil : Vec A zero
cons : (n : Nat) → A → Vec A n → Vec A (suc n)
append : {A : Set} (m n : Nat) → Vec A m → Vec A n → Vec A (m + n)
append .zero n nil ys = ys
append (.suc m) n (cons .m x xs) ys = cons (m + n) x (append m n xs ys)
open import Agda.Builtin.Equality
data _×_ (A B : Set) : Set where
pair : A → B → A × B
test : (p q : Nat × Nat) → p ≡ q → Set₁
test (.(pair x) y) (pair x .y) refl = Set
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open import Agda.Primitive using (_⊔_)
import Categories.Category as Category
import Categories.Category.Cartesian as Cartesian
open import MultiSorted.AlgebraicTheory
-- Finite products indexed by contexts
module MultiSorted.Product
{o ℓ e}
(𝒞 : Category.Category o ℓ e)
{𝓈 ℴ}
{Σ : Signature {𝓈} {ℴ}}
(interp-sort : Signature.sort Σ → Category.Category.Obj 𝒞) where
open Signature Σ
open Category.Category 𝒞
open HomReasoning
interp-sort-var : {Γ : Context} → var Γ → Obj
interp-sort-var {Γ} x = interp-sort (sort-of Γ x)
record Producted : Set (o ⊔ ℓ ⊔ e ⊔ 𝓈) where
field
prod : Context → Obj
π : ∀ {Γ} (x : var Γ) → prod Γ ⇒ interp-sort-var x
tuple : ∀ Γ {B} → ((x : var Γ) → B ⇒ interp-sort-var x) → B ⇒ prod Γ
project : ∀ {Γ} {B} {x : var Γ} {fs : (y : var Γ) → B ⇒ interp-sort-var y} → π x ∘ tuple Γ fs ≈ fs x
unique : ∀ {Γ} {B} {fs : (x : var Γ) → B ⇒ interp-sort-var x} {g : B ⇒ prod Γ} → (∀ i → π i ∘ g ≈ fs i) → tuple Γ fs ≈ g
η : ∀ {Γ} → tuple Γ π ≈ id
η = unique (λ i → identityʳ)
! : ∀ {B : Obj} → B ⇒ prod ctx-empty
! {B} = tuple ctx-empty ctx-empty-absurd
!-unique : ∀ {B : Obj} {f : B ⇒ prod ctx-empty} → ! ≈ f
!-unique {f = f} = unique ctx-empty-absurd
!-unique₂ : ∀ {B : Obj} {f g : B ⇒ prod ctx-empty} → f ≈ g
!-unique₂ = (⟺ !-unique) ○ !-unique
tuple-cong : ∀ {B : Obj} {Γ} {fs gs : (x : var Γ) → B ⇒ interp-sort-var x} → (∀ i → fs i ≈ gs i) →
tuple Γ fs ≈ tuple Γ gs
tuple-cong ξ = unique (λ i → project ○ ⟺ (ξ i))
∘-distribʳ-tuple : ∀ {B C} {Γ} {fs : (x : var Γ) → B ⇒ interp-sort-var x} {g : C ⇒ B} →
tuple Γ (λ x → fs x ∘ g) ≈ tuple Γ fs ∘ g
∘-distribʳ-tuple = unique (λ i → ⟺ assoc ○ ∘-resp-≈ˡ project)
-- A cartesian category has a standard products structure (which we need not use)
module _ (cartesian-𝒞 : Cartesian.Cartesian 𝒞) where
open Cartesian.Cartesian cartesian-𝒞
standard-prod : Context → Obj
standard-prod ctx-empty = ⊤
standard-prod (ctx-slot A) = interp-sort A
standard-prod (ctx-concat Γ Δ) = standard-prod Γ × standard-prod Δ
standard-π : ∀ {Γ} (x : var Γ) → standard-prod Γ ⇒ interp-sort-var x
standard-π var-var = id
standard-π (var-inl i) = standard-π i ∘ π₁
standard-π (var-inr i) = standard-π i ∘ π₂
standard-tuple : ∀ Γ {B} → ((x : var Γ) → B ⇒ interp-sort-var x) → B ⇒ standard-prod Γ
standard-tuple ctx-empty fs = !
standard-tuple (ctx-slot _) fs = fs var-var
standard-tuple (ctx-concat Γ Δ) fs = ⟨ standard-tuple Γ (λ i → fs (var-inl i)) , standard-tuple Δ (λ j → fs (var-inr j)) ⟩
standard-project : ∀ {Γ} {B} {x : var Γ} {fs : (x : var Γ) → B ⇒ interp-sort-var x} →
standard-π x ∘ standard-tuple Γ fs ≈ fs x
standard-project {x = var-var} = identityˡ
standard-project {x = var-inl x} = assoc ○ ((∘-resp-≈ʳ project₁) ○ standard-project {x = x})
standard-project {x = var-inr x} = assoc ○ ((∘-resp-≈ʳ project₂) ○ standard-project {x = x})
standard-unique : ∀ {Γ} {B} {fs : (x : var Γ) → B ⇒ interp-sort-var x} {g : B ⇒ standard-prod Γ} → (∀ i → standard-π i ∘ g ≈ fs i) →
standard-tuple Γ fs ≈ g
standard-unique {ctx-empty} ξ = !-unique _
standard-unique {ctx-slot _} ξ = ⟺ (ξ var-var) ○ identityˡ
standard-unique {ctx-concat Γ Δ} {fs = fs} ξ =
unique
(⟺ (standard-unique (λ x → sym-assoc ○ (ξ (var-inl x)))))
(⟺ (standard-unique (λ y → sym-assoc ○ (ξ (var-inr y)))))
StandardProducted : Producted
StandardProducted =
record
{ prod = standard-prod
; π = standard-π
; tuple = standard-tuple
; project = λ {Γ} → standard-project {Γ}
; unique = standard-unique }
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-- In a module instantiation 'module A = e', 'e' should have the form 'm e1 ..
-- en' where 'm' is a module name.
module NotAModuleExpr where
module Bad = \x -> x
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{-# OPTIONS --without-K --rewriting #-}
open import HoTT
import homotopy.WedgeExtension as WedgeExt
import homotopy.SuspAdjointLoop as SAL
module homotopy.Freudenthal where
{- lemma (move this where?) -}
private
move1-left-on-left : ∀ {i} {A : Type i} {x y : A} (p : x == y) (q : x == y)
→ ((! q) ∙ p == idp → p == q)
move1-left-on-left p idp h = h
module FreudenthalEquiv
{i} (n k : ℕ₋₂) (kle : k ≤T S n +2+ S n)
(X : Ptd i) {{cX : is-connected (S (S n)) (de⊙ X)}} where
Q : Susp (de⊙ X) → Type i
Q x = Trunc k (north == x)
⊙up : X ⊙→ ⊙Ω (⊙Susp X)
⊙up = SAL.η _
up = fst ⊙up
Codes-mer-args : WedgeExt.args {a₀ = pt X} {b₀ = [_] {n = k} (pt X)}
Codes-mer-args = record {n = S n; m = S n;
P = λ _ _ → (Trunc k (de⊙ X) , raise-level-≤T kle Trunc-level);
f = [_]; g = idf _; p = idp}
Codes-mer : de⊙ X → Trunc k (de⊙ X) → Trunc k (de⊙ X)
Codes-mer = WedgeExt.ext Codes-mer-args
Codes-mer-β-l : (λ a → Codes-mer a [ pt X ]) == [_]
Codes-mer-β-l = λ= $ WedgeExt.β-l {r = Codes-mer-args}
Codes-mer-β-r : (λ b → Codes-mer (pt X) b) == idf _
Codes-mer-β-r = λ= $ WedgeExt.β-r {r = Codes-mer-args}
Codes-mer-coh : app= Codes-mer-β-l (pt X) == app= Codes-mer-β-r [ pt X ]
Codes-mer-coh =
app= Codes-mer-β-l (pt X)
=⟨ app=-β (WedgeExt.β-l {r = Codes-mer-args}) (pt X) ⟩
WedgeExt.β-l {r = Codes-mer-args} (pt X)
=⟨ ! (move1-left-on-left _ _ (WedgeExt.coh {r = Codes-mer-args})) ⟩
WedgeExt.β-r {r = Codes-mer-args} [ pt X ]
=⟨ ! (app=-β (WedgeExt.β-r {r = Codes-mer-args}) [ pt X ]) ⟩
app= Codes-mer-β-r [ pt X ] ∎
Codes-mer-is-equiv : (x : de⊙ X) → is-equiv (Codes-mer x)
Codes-mer-is-equiv =
conn-extend (pointed-conn-out {n = S n} (de⊙ X) (pt X))
(λ x' → (is-equiv (Codes-mer x') , is-equiv-level))
(λ tt → transport is-equiv (! (Codes-mer-β-r)) (idf-is-equiv _))
Codes-mer-equiv : (x : de⊙ X) → Trunc k (de⊙ X) ≃ Trunc k (de⊙ X)
Codes-mer-equiv x = (Codes-mer x , Codes-mer-is-equiv x)
Codes-mer-inv-x₀ : <– (Codes-mer-equiv (pt X)) == idf _
Codes-mer-inv-x₀ =
ap is-equiv.g (conn-extend-β
(pointed-conn-out (de⊙ X) (pt X))
(λ x' → (is-equiv (Codes-mer x') , is-equiv-level))
_ unit)
∙ lemma (! (Codes-mer-β-r)) (snd $ ide _)
where lemma : ∀ {i j} {A : Type i} {B : Type j} {f g : A → B}
(α : f == g) (e : is-equiv f)
→ is-equiv.g (transport is-equiv α e) == is-equiv.g e
lemma idp e = idp
Codes : Susp (de⊙ X) → Type i
Codes = SuspRec.f (Trunc k (de⊙ X)) (Trunc k (de⊙ X)) (ua ∘ Codes-mer-equiv)
Codes-has-level : (x : Susp (de⊙ X)) → has-level k (Codes x)
Codes-has-level = Susp-elim Trunc-level Trunc-level
(λ _ → prop-has-all-paths-↓)
{-
favonia:
This equation should be true: [⊙Trunc-fmap ⊙up = (decodeN , decodeN-pt)].
Maybe there is a way to refactor the following code so that
pointedness is handled more elegantly.
-}
decodeN : Codes north → Trunc k (north' (de⊙ X) == north)
decodeN = Trunc-fmap up
decodeN-pt : decodeN [ pt X ] == [ idp ]
decodeN-pt = snd (⊙Trunc-fmap ⊙up)
decodeS : Codes south → Q south
decodeS = Trunc-fmap merid
encode₀ : {x : Susp (de⊙ X)} → north == x → Codes x
encode₀ α = transport Codes α [ pt X ]
encode : {x : Susp (de⊙ X)} → Trunc k (north == x) → Codes x
encode {x} tα = Trunc-rec {{Codes-has-level x}} encode₀ tα
abstract
encode-decodeN : (c : Codes north) → encode (decodeN c) == c
encode-decodeN = Trunc-elim
{{λ _ → =-preserves-level Trunc-level}}
(λ x →
encode (decodeN [ x ])
=⟨ idp ⟩
coe (ap Codes (merid x ∙ ! (merid (pt X)))) [ pt X ]
=⟨ ap-∙ Codes (merid x) (! (merid (pt X))) |in-ctx (λ w → coe w [ pt X ]) ⟩
coe (ap Codes (merid x) ∙ ap Codes (! (merid (pt X)))) [ pt X ]
=⟨ coe-∙ (ap Codes (merid x)) (ap Codes (! (merid (pt X)))) [ pt X ] ⟩
coe (ap Codes (! (merid (pt X)))) (coe (ap Codes (merid x)) [ pt X ])
=⟨ SuspRec.merid-β _ _ (ua ∘ Codes-mer-equiv) x
|in-ctx (λ w → coe (ap Codes (! (merid (pt X)))) (coe w [ pt X ])) ⟩
coe (ap Codes (! (merid (pt X)))) (coe (ua (Codes-mer-equiv x)) [ pt X ])
=⟨ coe-β (Codes-mer-equiv x) [ pt X ]
|in-ctx (λ w → coe (ap Codes (! (merid (pt X)))) w) ⟩
coe (ap Codes (! (merid (pt X)))) (Codes-mer x [ pt X ])
=⟨ app= Codes-mer-β-l x
|in-ctx (λ w → coe (ap Codes (! (merid (pt X)))) w) ⟩
coe (ap Codes (! (merid (pt X)))) [ x ]
=⟨ coe-ap-! Codes (merid (pt X)) [ x ] ⟩
coe! (ap Codes (merid (pt X))) [ x ]
=⟨ SuspRec.merid-β _ _ (ua ∘ Codes-mer-equiv) (pt X)
|in-ctx (λ w → coe! w [ x ]) ⟩
coe! (ua (Codes-mer-equiv (pt X))) [ x ]
=⟨ coe!-β (Codes-mer-equiv (pt X)) [ x ] ⟩
<– (Codes-mer-equiv (pt X)) [ x ]
=⟨ app= Codes-mer-inv-x₀ [ x ] ⟩
[ x ] ∎)
decode : {x : Susp (de⊙ X)} → Codes x → Q x
decode {x} = Susp-elim
{P = λ y → Codes y → Q y}
decodeN decodeS
(λ x' → ↓-→-from-transp (λ= (STS x')))
x
where
abstract
coh : {s₁ s₂ : Susp (de⊙ X)} (p : s₁ == s₂)
→ (ap (λ s → s ∙ p) (!-inv-r p))
== ∙-assoc p (! p) p ∙ ap (λ s → p ∙ s) (!-inv-l p) ∙ ∙-unit-r p
coh idp = idp
P : de⊙ X → de⊙ X → (S (n +2+ (S n))) -Type (lmax i i)
P = λ x₁ x₂ →
((transport Q (merid x₁) (Trunc-fmap up [ x₂ ])
== Trunc-fmap merid (transport Codes (merid x₁) [ x₂ ])),
=-preserves-level (raise-level-≤T kle Trunc-level))
f : (a : de⊙ X) → fst (P a (pt X))
f a =
transport Q (merid a) [ up (pt X) ]
=⟨ transport-Trunc (north ==_) (merid a) (up (pt X)) ⟩
[ transport (north ==_) (merid a) (up (pt X)) ]
=⟨ ap [_] $ transp-cst=idf {A = Susp (de⊙ X)} (merid a) (up (pt X)) ⟩
[ (merid (pt X) ∙ ! (merid (pt X))) ∙ merid a ]
=⟨ ap [_] $ ap (λ s → s ∙ merid a) (!-inv-r (merid (pt X))) ⟩
[ merid a ]
=⟨ idp ⟩
Trunc-fmap merid [ a ]
=⟨ ap (Trunc-fmap merid) (! (app= Codes-mer-β-l a)) ⟩
Trunc-fmap merid (Codes-mer a [ pt X ])
=⟨ ap (Trunc-fmap merid) (! (coe-β (Codes-mer-equiv a) [ pt X ])) ⟩
Trunc-fmap merid (coe (ua (Codes-mer-equiv a)) [ pt X ])
=⟨ ! (SuspRec.merid-β _ _ (ua ∘ Codes-mer-equiv) a)
|in-ctx (λ w → Trunc-fmap merid (coe w [ pt X ])) ⟩
Trunc-fmap merid (transport Codes (merid a) [ pt X ]) ∎
g : (b : de⊙ X) → fst (P (pt X) b)
g b =
transport Q (merid (pt X)) [ up b ]
=⟨ transport-Trunc (north ==_) (merid (pt X)) (up b) ⟩
[ transport (north ==_) (merid (pt X)) (up b) ]
=⟨ ap [_] $ transp-cst=idf {A = Susp (de⊙ X)} (merid (pt X)) (up b) ⟩
[ (merid b ∙ ! (merid (pt X))) ∙ merid (pt X) ]
=⟨ ap [_] $ ∙-assoc (merid b) (! (merid (pt X))) (merid (pt X))
∙ ap (λ s → merid b ∙ s) (!-inv-l (merid (pt X)))
∙ ∙-unit-r (merid b) ⟩
[ merid b ]
=⟨ idp ⟩
Trunc-fmap merid [ b ]
=⟨ ap (Trunc-fmap merid) (! (app= Codes-mer-β-r [ b ])) ⟩
Trunc-fmap merid (Codes-mer (pt X) [ b ])
=⟨ ap (Trunc-fmap merid) (! (coe-β (Codes-mer-equiv (pt X)) [ b ])) ⟩
Trunc-fmap merid (coe (ua (Codes-mer-equiv (pt X))) [ b ])
=⟨ ! (SuspRec.merid-β _ _ (ua ∘ Codes-mer-equiv) (pt X))
|in-ctx (λ w → Trunc-fmap merid (coe w [ b ])) ⟩
Trunc-fmap merid (transport Codes (merid (pt X)) [ b ]) ∎
p : f (pt X) == g (pt X)
p = ap2
(λ p₁ p₂ →
transport Q (merid (pt X)) [ up (pt X) ]
=⟨ transport-Trunc (north ==_) (merid (pt X)) (up (pt X)) ⟩
[ transport (north ==_) (merid (pt X)) (up (pt X)) ]
=⟨ ap [_] $ transp-cst=idf {A = Susp (de⊙ X)} (merid (pt X)) (up (pt X)) ⟩
[ (merid (pt X) ∙ ! (merid (pt X))) ∙ merid (pt X) ]
=⟨ ap [_] p₁ ⟩
[ merid (pt X) ]
=⟨ idp ⟩
Trunc-fmap merid [ pt X ]
=⟨ ap (Trunc-fmap merid) (! p₂) ⟩
Trunc-fmap merid (Codes-mer (pt X) [ pt X ])
=⟨ ap (Trunc-fmap merid) (! (coe-β (Codes-mer-equiv (pt X)) [ pt X ])) ⟩
Trunc-fmap merid (coe (ua (Codes-mer-equiv (pt X))) [ pt X ])
=⟨ ! (SuspRec.merid-β _ _ (ua ∘ Codes-mer-equiv) (pt X))
|in-ctx (λ w → Trunc-fmap merid (coe w [ pt X ])) ⟩
Trunc-fmap merid (transport Codes (merid (pt X)) [ pt X ]) ∎)
(coh (merid (pt X))) Codes-mer-coh
STS-args : WedgeExt.args {a₀ = pt X} {b₀ = pt X}
STS-args =
record {n = S n; m = S n; P = P; f = f; g = g; p = p}
STS : (x' : de⊙ X) (c : Codes north) →
transport Q (merid x') (Trunc-fmap up c)
== Trunc-fmap merid (transport Codes (merid x') c)
STS x' = Trunc-elim {{λ _ → =-preserves-level Trunc-level}}
(WedgeExt.ext STS-args x')
abstract
decode-encode : {x : Susp (de⊙ X)} (tα : Q x)
→ decode {x} (encode {x} tα) == tα
decode-encode {x} = Trunc-elim
{P = λ tα → decode {x} (encode {x} tα) == tα}
{{λ _ → =-preserves-level Trunc-level}}
(J (λ y p → decode {y} (encode {y} [ p ]) == [ p ])
(ap [_] (!-inv-r (merid (pt X)))))
eq : Trunc k (de⊙ X) ≃ Trunc k (north' (de⊙ X) == north)
eq = equiv decodeN encode decode-encode encode-decodeN
⊙eq : ⊙Trunc k X ⊙≃ ⊙Trunc k (⊙Ω (⊙Susp X))
⊙eq = ≃-to-⊙≃ eq (ap [_] (!-inv-r (merid (pt X))))
path : Trunc k (de⊙ X) == Trunc k (north' (de⊙ X) == north)
path = ua eq
⊙path : ⊙Trunc k X == ⊙Trunc k (⊙Ω (⊙Susp X))
⊙path = ⊙ua ⊙eq
{- Used to prove stability in iterated suspensions -}
module FreudenthalIso
{i} (n : ℕ₋₂) (k : ℕ) (kle : ⟨ S k ⟩ ≤T S n +2+ S n)
(X : Ptd i) {{_ : is-connected (S (S n)) (de⊙ X)}} where
open FreudenthalEquiv n ⟨ S k ⟩ kle X public
hom : Ω^S-group k (⊙Trunc ⟨ S k ⟩ X)
→ᴳ Ω^S-group k (⊙Trunc ⟨ S k ⟩ (⊙Ω (⊙Susp X)))
hom = Ω^S-group-fmap k (decodeN , decodeN-pt)
iso : Ω^S-group k (⊙Trunc ⟨ S k ⟩ X)
≃ᴳ Ω^S-group k (⊙Trunc ⟨ S k ⟩ (⊙Ω (⊙Susp X)))
iso = Ω^S-group-emap k ⊙eq
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module Structure.Groupoid.Functor where
open import Functional using (_on₂_)
open import Lang.Instance
import Lvl
open import Logic.Predicate
open import Structure.Category
import Structure.Category.Functor as Category
open import Structure.Function
open import Structure.Groupoid
import Structure.Relator.Names as Names
open import Structure.Setoid
open import Syntax.Function
open import Type
private variable ℓ ℓₒ ℓₘ ℓₗₒ ℓₗₘ ℓᵣₒ ℓᵣₘ ℓₑ ℓₗₑ ℓᵣₑ : Lvl.Level
private variable Obj Objₗ Objᵣ : Type{ℓ}
private variable _⟶_ _⟶ₗ_ _⟶ᵣ_ : Objₗ → Objᵣ → Type{ℓ}
module _
⦃ morphism-equivₗ : ∀{x y : Objₗ} → Equiv{ℓₗₑ}(x ⟶ₗ y) ⦄
⦃ morphism-equivᵣ : ∀{x y : Objᵣ} → Equiv{ℓᵣₑ}(x ⟶ᵣ y) ⦄
(Groupoidₗ : Groupoid(_⟶ₗ_))
(Groupoidᵣ : Groupoid(_⟶ᵣ_))
where
-- A covariant functor.
-- A mapping which transforms objects and morphisms from one category to another while "preserving" the groupoid structure.
-- A homomorphism between groupoids.
record Functor (F : Objₗ → Objᵣ) : Type{Lvl.of(Type.of(Groupoidₗ)) Lvl.⊔ Lvl.of(Type.of(Groupoidᵣ))} where
constructor intro
open Groupoid ⦃ … ⦄
private instance _ = Groupoidₗ
private instance _ = Groupoidᵣ
field
map : Names.Subrelation(_⟶ₗ_) ((_⟶ᵣ_) on₂ F)
field
⦃ map-function ⦄ : ∀{x y} → Function(map{x}{y})
⦃ op-preserving ⦄ : ∀{x y z : Objₗ}{f : y ⟶ₗ z}{g : x ⟶ₗ y} → (map(f ∘ g) ≡ map(f) ∘ map(g))
⦃ inv-preserving ⦄ : ∀{x y : Objₗ}{f : x ⟶ₗ y} → (map(inv f) ≡ inv(map f))
⦃ id-preserving ⦄ : ∀{x : Objₗ} → (map(id {x = x}) ≡ id)
categoryFunctor : Category.Functor(category ⦃ r = Groupoidₗ ⦄)(category ⦃ r = Groupoidᵣ ⦄) (F)
Category.Functor.map categoryFunctor = map
Category.Functor.map-function categoryFunctor = map-function
Category.Functor.op-preserving categoryFunctor = op-preserving
Category.Functor.id-preserving categoryFunctor = id-preserving
open Category.Functor(categoryFunctor) public hiding (map ; map-function ; op-preserving ; id-preserving)
module _
⦃ morphism-equiv : ∀{x y : Obj} → Equiv{ℓₑ}(x ⟶ y) ⦄
(Groupoid : Groupoid(_⟶_))
where
Endofunctor = Functor(Groupoid)(Groupoid)
module Endofunctor = Functor{Groupoidₗ = Groupoid}{Groupoidᵣ = Groupoid}
_→ᶠᵘⁿᶜᵗᵒʳ_ : GroupoidObject{ℓₗₒ}{ℓₗₘ}{ℓₗₑ} → GroupoidObject{ℓᵣₒ}{ℓᵣₘ}{ℓᵣₑ} → Type
grpₗ →ᶠᵘⁿᶜᵗᵒʳ grpᵣ = ∃(Functor (GroupoidObject.groupoid(grpₗ)) ((GroupoidObject.groupoid(grpᵣ))))
⟲ᶠᵘⁿᶜᵗᵒʳ_ : GroupoidObject{ℓₒ}{ℓₘ}{ℓₑ} → Type
⟲ᶠᵘⁿᶜᵗᵒʳ grp = grp →ᶠᵘⁿᶜᵗᵒʳ grp
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module _ where
open import Agda.Builtin.Nat using (mod-helper)
open import Common.Prelude
open import Common.Equality
_mod_ : Nat → Nat → Nat
n mod zero = 0
n mod suc m = mod-helper 0 m n m
{-# INLINE _mod_ #-}
primitive
primForce : ∀ {a b} {A : Set a} {B : A → Set b} (x : A) → (∀ x → B x) → B x
primForceLemma : ∀ {a b} {A : Set a} {B : A → Set b} (x : A) (f : ∀ x → B x) → primForce x f ≡ f x
force = primForce
forceLemma = primForceLemma
infixr 0 _$!_
_$!_ : ∀ {a b} {A : Set a} {B : A → Set b} → (∀ x → B x) → ∀ x → B x
f $! x = force x f
-- Memory leak without proper seq --
pow′ : Nat → Nat → Nat
pow′ zero acc = acc
pow′ (suc n) acc = pow′ n $! (acc + acc) mod 234576373
pow : Nat → Nat
pow n = pow′ n 1
main : IO Unit
main = printNat (pow 5000000)
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module PLRTree.Complete {A : Set} where
open import PLRTree {A}
open import PLRTree.Equality {A}
data _⋗_ : PLRTree → PLRTree → Set where
⋗lf : (x : A)
→ node perfect x leaf leaf ⋗ leaf
⋗nd : {l r l' r' : PLRTree}
(x x' : A)
→ l ≃ r
→ l' ≃ r'
→ l ⋗ l'
→ node perfect x l r ⋗ node perfect x' l' r'
mutual
data _⋘_ : PLRTree → PLRTree → Set where
x⋘ : (x y z : A)
→ node right x (node perfect y leaf leaf) leaf ⋘ node perfect z leaf leaf
l⋘ : {l r l' r' : PLRTree}
→ (x x' : A)
→ l ⋘ r
→ l' ≃ r'
→ r ≃ l'
→ (node left x l r) ⋘ (node perfect x' l' r')
r⋘ : {l r l' r' : PLRTree}
→ (x x' : A)
→ l ⋙ r
→ l' ≃ r'
→ l ⋗ l'
→ (node right x l r) ⋘ (node perfect x' l' r')
data _⋙_ : PLRTree → PLRTree → Set where
⋙p : {l r : PLRTree}
→ l ⋗ r
→ l ⋙ r
⋙l : {l r l' r' : PLRTree}
→ (x x' : A)
→ l ≃ r
→ l' ⋘ r'
→ l ⋗ r'
→ (node perfect x l r) ⋙ (node left x' l' r')
⋙r : {l r l' r' : PLRTree}
→ (x x' : A)
→ l ≃ r
→ l' ⋙ r'
→ l ≃ l'
→ (node perfect x l r) ⋙ (node right x' l' r')
data Complete : PLRTree → Set where
leaf : Complete leaf
perfect : {l r : PLRTree}
(x : A)
→ Complete l
→ Complete r
→ l ≃ r
→ Complete (node perfect x l r)
left : {l r : PLRTree}
(x : A)
→ Complete l
→ Complete r
→ l ⋘ r
→ Complete (node left x l r)
right : {l r : PLRTree}
(x : A)
→ Complete l
→ Complete r
→ l ⋙ r
→ Complete (node right x l r)
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{-
-- an ≃ equivalence of types can be lifted to a ≃S equivalence
-- (over their ≡-Setoids)
-- NOT NEEDED
lift≃ : ∀ {ℓ} → {A B : Set ℓ} → A ≃ B → (≡-Setoid A) ≃S (≡-Setoid B)
lift≃ {_} {A} {B} (f , mkqinv g α β) = equiv AS BS α' β'
where
module AA = Setoid (≡-Setoid A)
module BB = Setoid (≡-Setoid B)
AS : ≡-Setoid A ⟶ ≡-Setoid B
AS = →to⟶ f
BS : ≡-Setoid B ⟶ ≡-Setoid A
BS = →to⟶ g
α' : {x y : B} → P._≡_ x y → P._≡_ (f (g x)) y
α' = λ {x} {y} p → BB.trans (α x) p
β' : {x y : A} → P._≡_ x y → P._≡_ (g (f x)) y
β' = λ {x} {y} p → AA.trans (β x) p
-}
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{-# OPTIONS --without-K --rewriting #-}
open import lib.Basics
open import lib.NType2
open import lib.types.Bool
open import lib.types.Empty
open import lib.types.Paths
open import lib.types.Pi
open import lib.types.Sigma
{-
This file contains various lemmas that rely on lib.types.Paths or
functional extensionality for pointed maps.
-}
module lib.types.Pointed where
{- Pointed maps -}
-- function extensionality for pointed maps
⊙λ= : ∀ {i j} {X : Ptd i} {Y : Ptd j} {f g : X ⊙→ Y}
(p : ∀ x → fst f x == fst g x)
(α : snd f == snd g [ (λ y → y == pt Y) ↓ p (pt X) ])
→ f == g
⊙λ= {g = g} p α =
pair= (λ= p) (↓-app=cst-in (↓-idf=cst-out α ∙ ap (_∙ snd g) (! (app=-β p _))))
{-
⊙λ=' : ∀ {i j} {X : Ptd i} {Y : Ptd j} {f g : X ⊙→ Y}
(p : ∀ x → fst f x == fst g x) (α : snd f == p (pt X) ∙ snd g)
→ f == g
⊙λ=' p α = ⊙λ= p (↓-idf=cst-in α)
-}
-- associativity of pointed maps
⊙∘-assoc-pt : ∀ {i j k} {A : Type i} {B : Type j} {C : Type k}
{a₁ a₂ : A} (f : A → B) {b : B} (g : B → C) {c : C}
(p : a₁ == a₂) (q : f a₂ == b) (r : g b == c)
→ ⊙∘-pt (g ∘ f) p (⊙∘-pt g q r) == ⊙∘-pt g (⊙∘-pt f p q) r
⊙∘-assoc-pt _ _ idp _ _ = idp
⊙∘-assoc : ∀ {i j k l} {X : Ptd i} {Y : Ptd j} {Z : Ptd k} {W : Ptd l}
(h : Z ⊙→ W) (g : Y ⊙→ Z) (f : X ⊙→ Y)
→ ((h ⊙∘ g) ⊙∘ f) == (h ⊙∘ (g ⊙∘ f))
⊙∘-assoc (h , hpt) (g , gpt) (f , fpt) = ⊙λ= (λ _ → idp) (⊙∘-assoc-pt g h fpt gpt hpt)
{- Pointed equivalences -}
-- Extracting data from an pointed equivalence
module _ {i j} {X : Ptd i} {Y : Ptd j} (⊙e : X ⊙≃ Y) where
private
e : de⊙ X ≃ de⊙ Y
e = (fst (fst ⊙e) , snd ⊙e)
⊙≃-to-≃ = e
⊙–>-pt : –> e (pt X) == pt Y
⊙–>-pt = snd (fst ⊙e)
private
p = ⊙–>-pt
⊙–> : X ⊙→ Y
⊙–> = fst ⊙e
⊙<–-pt : <– e (pt Y) == pt X
⊙<–-pt = ap (<– e) (! ⊙–>-pt) ∙ <–-inv-l e (pt X)
⊙<– : Y ⊙→ X
⊙<– = <– e , ⊙<–-pt
infix 120 _⊙⁻¹
_⊙⁻¹ : Y ⊙≃ X
_⊙⁻¹ = ⊙<– , is-equiv-inverse (snd ⊙e)
⊙<–-inv-l : ⊙<– ⊙∘ ⊙–> == ⊙idf _
⊙<–-inv-l = ⊙λ= (<–-inv-l e) $ ↓-idf=cst-in $
ap (<– e) p ∙ ap (<– e) (! p) ∙ <–-inv-l e (pt X)
=⟨ ! (∙-assoc (ap (<– e) p) (ap (<– e) (! p)) (<–-inv-l e (pt X))) ⟩
(ap (<– e) p ∙ ap (<– e) (! p)) ∙ <–-inv-l e (pt X)
=⟨ ∙-ap (<– e) p (! p) ∙ ap (ap (<– e)) (!-inv-r p)
|in-ctx (λ w → w ∙ <–-inv-l e (pt X)) ⟩
<–-inv-l e (pt X)
=⟨ ! (∙-unit-r _) ⟩
<–-inv-l e (pt X) ∙ idp =∎
⊙<–-inv-r : ⊙–> ⊙∘ ⊙<– == ⊙idf _
⊙<–-inv-r = ⊙λ= (<–-inv-r e) $ ↓-idf=cst-in $
ap (–> e) (ap (<– e) (! p) ∙ <–-inv-l e (pt X)) ∙ p
=⟨ ap-∙ (–> e) (ap (<– e) (! p)) (<–-inv-l e (pt X))
|in-ctx (λ w → w ∙ p) ⟩
(ap (–> e) (ap (<– e) (! p)) ∙ ap (–> e) (<–-inv-l e (pt X))) ∙ p
=⟨ <–-inv-adj e (pt X)
|in-ctx (λ w → (ap (–> e) (ap (<– e) (! p)) ∙ w) ∙ p) ⟩
(ap (–> e) (ap (<– e) (! p)) ∙ <–-inv-r e (–> e (pt X))) ∙ p
=⟨ ∘-ap (–> e) (<– e) (! p)
|in-ctx (λ w → (w ∙ <–-inv-r e (–> e (pt X))) ∙ p) ⟩
(ap (–> e ∘ <– e) (! p) ∙ <–-inv-r e (–> e (pt X))) ∙ p
=⟨ ap (_∙ p) (! (↓-app=idf-out (apd (<–-inv-r e) (! p)))) ⟩
(<–-inv-r e (pt Y) ∙' (! p)) ∙ p
=⟨ ∙'=∙ (<–-inv-r e (pt Y)) (! p) |in-ctx _∙ p ⟩
(<–-inv-r e (pt Y) ∙ (! p)) ∙ p
=⟨ ∙-assoc (<–-inv-r e (pt Y)) (! p) p ⟩
<–-inv-r e (pt Y) ∙ (! p ∙ p)
=⟨ !-inv-l p |in-ctx (<–-inv-r e (pt Y)) ∙_ ⟩
<–-inv-r e (pt Y) ∙ idp =∎
module _ {i j k} {X : Ptd i} {Y : Ptd j} {Z : Ptd k} (⊙e : X ⊙≃ Y) where
post⊙∘-is-equiv : is-equiv (λ (k : Z ⊙→ X) → ⊙–> ⊙e ⊙∘ k)
post⊙∘-is-equiv = is-eq f g f-g g-f
where f = ⊙–> ⊙e ⊙∘_
g = ⊙<– ⊙e ⊙∘_
abstract
f-g = λ k → ! (⊙∘-assoc (⊙–> ⊙e) (⊙<– ⊙e) k) ∙ ap (_⊙∘ k) (⊙<–-inv-r ⊙e) ∙ ⊙∘-unit-l k
g-f = λ k → ! (⊙∘-assoc (⊙<– ⊙e) (⊙–> ⊙e) k) ∙ ap (_⊙∘ k) (⊙<–-inv-l ⊙e) ∙ ⊙∘-unit-l k
pre⊙∘-is-equiv : is-equiv (λ (k : Y ⊙→ Z) → k ⊙∘ ⊙–> ⊙e)
pre⊙∘-is-equiv = is-eq f g f-g g-f
where f = _⊙∘ ⊙–> ⊙e
g = _⊙∘ ⊙<– ⊙e
abstract
f-g = λ k → ⊙∘-assoc k (⊙<– ⊙e) (⊙–> ⊙e) ∙ ap (k ⊙∘_) (⊙<–-inv-l ⊙e) ∙ ⊙∘-unit-r k
g-f = λ k → ⊙∘-assoc k (⊙–> ⊙e) (⊙<– ⊙e) ∙ ap (k ⊙∘_) (⊙<–-inv-r ⊙e) ∙ ⊙∘-unit-r k
pre⊙∘-equiv : (Y ⊙→ Z) ≃ (X ⊙→ Z)
pre⊙∘-equiv = _ , pre⊙∘-is-equiv
{- Pointed maps out of bool -}
-- intuition : [f true] is fixed and the only changable part is [f false].
⊙Bool→-to-idf : ∀ {i} {X : Ptd i}
→ ⊙Bool ⊙→ X → de⊙ X
⊙Bool→-to-idf (h , _) = h false
⊙Bool→-equiv-idf : ∀ {i} (X : Ptd i)
→ (⊙Bool ⊙→ X) ≃ de⊙ X
⊙Bool→-equiv-idf {i} X = equiv ⊙Bool→-to-idf g f-g g-f
where
g : de⊙ X → ⊙Bool ⊙→ X
g x = (if_then pt X else x) , idp
abstract
f-g : ∀ x → ⊙Bool→-to-idf (g x) == x
f-g x = idp
g-f : ∀ H → g (⊙Bool→-to-idf H) == H
g-f (h , hpt) = pair=
(λ= lemma)
(↓-app=cst-in $
idp
=⟨ ! (!-inv-l hpt) ⟩
! hpt ∙ hpt
=⟨ ! (app=-β lemma true) |in-ctx (λ w → w ∙ hpt) ⟩
app= (λ= lemma) true ∙ hpt
=∎)
where lemma : ∀ b → fst (g (h false)) b == h b
lemma true = ! hpt
lemma false = idp
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module AndOrNot where
open import Data.Bool
open import Data.Product
test : Bool → Bool → Bool × Bool × Bool
test x y = x ∧ y , x ∨ y , not x
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open import Prelude
module Implicits.Resolution.GenericFinite.Algorithm.Soundness where
open import Induction.WellFounded
open import Induction.Nat
open import Data.List
open import Data.List.Any
open Membership-≡
open import Data.Fin.Substitution
open import Data.Nat.Base using (_<′_)
open import Data.Maybe as Maybe
open import Data.Nat hiding (_<_)
open import Data.Nat.Properties
open import Relation.Binary using (module DecTotalOrder)
open DecTotalOrder decTotalOrder using () renaming (refl to ≤-refl; trans to ≤-trans)
open import Implicits.Syntax
open import Implicits.Syntax.Type.Unification
open import Implicits.Syntax.Type.Unification.Lemmas as mgu hiding (sound)
open import Implicits.Substitutions
open import Implicits.Substitutions.Lemmas
open import Implicits.Resolution.GenericFinite.Resolution
open import Implicits.Resolution.GenericFinite.Algorithm
open import Implicits.Resolution.GenericFinite.TerminationCondition
open import Implicits.Resolution.Termination
open import Data.Vec hiding (_∈_)
open import Extensions.Bool as Bl
private
module M = MetaTypeMetaSubst
module ResolutionSound (cond : TerminationCondition) where
open TerminationCondition cond
open ResolutionAlgorithm cond
open ResolutionRules cond
open Lemmas
postulate lem₄ : ∀ {m ν} (a : MetaType m (suc ν)) u us →
from-meta ((M.open-meta a) M./ (us M.↑) M./ (M.sub u))
≡ (from-meta (a M./ (us M.↑tp))) tp[/tp from-meta u ]
open-↓-∀ : ∀ {ν m} {Δ : ICtx ν} {Φ} (a : MetaType m (suc ν)) τ u us →
Δ , Φ ⊢ (from-meta ((open-meta a) M./ (u ∷ us))) ↓ τ →
Δ , Φ ⊢ (from-meta (∀' a M./ us)) ↓ τ
open-↓-∀ {Δ = Δ} {Φ = Φ} a τ u us p = (i-tabs (from-meta u) (subst (λ v → Δ , Φ ⊢ v ↓ τ) (begin
from-meta (M._/_ (open-meta a) (u ∷ us))
≡⟨ cong (λ v → from-meta (M._/_ (open-meta a) v)) (sym $ us↑-⊙-sub-u≡u∷us u us) ⟩
from-meta ((open-meta a) M./ (us M.↑ M.⊙ (M.sub u)))
≡⟨ cong from-meta (/-⊙ (open-meta a)) ⟩
from-meta ((open-meta a) M./ us M.↑ M./ (M.sub u))
≡⟨ lem₄ a u us ⟩
from-meta (M._/_ a (us M.↑tp)) tp[/tp from-meta u ] ∎) p))
where open MetaTypeMetaLemmas hiding (subst)
mutual
match'-sound : ∀ {ν m} (Δ : ICtx ν) (Φ : TCtx) τ (r : MetaType m ν) →
(Φ↓ : T-Acc Φ) → (m↓ : m<-Acc r) →
Maybe.All (λ u → Δ , Φ ⊢ from-meta (r M./ u) ↓ τ) (match' Δ Φ τ r Φ↓ m↓)
match'-sound Δ Φ τ (a ⇒ b) Φ↓ (acc m↓)
with match' Δ Φ τ b Φ↓ (m↓ _ (b-m<-a⇒b a b))
| match'-sound Δ Φ τ b Φ↓ (m↓ _ (b-m<-a⇒b a b))
match'-sound Δ Φ τ (a ⇒ b) Φ↓ (acc m↓) | just u | just pu
with (step Φ Δ (from-meta (a M./ u)) (from-meta (b M./ u)) τ) <? Φ
match'-sound Δ Φ τ (a ⇒ b) (acc Φ↓) (acc m↓) | just u | just pu | no Φ> = nothing
match'-sound Δ Φ τ (a ⇒ b) (acc Φ↓) (acc m↓) | just u | just pu | yes Φ<
with resolve' Δ (step Φ Δ (from-meta (a M./ u)) (from-meta (b M./ u)) τ)
(from-meta (a M./ u)) (Φ↓ _ Φ<)
| sound' Δ (step Φ Δ (from-meta (a M./ u)) (from-meta (b M./ u)) τ)
(from-meta (a M./ u)) (Φ↓ _ Φ<)
match'-sound Δ Φ τ (a ⇒ b) (acc Φ↓) (acc m↓) | just u | just pu | yes Φ< | true | true px =
just (i-iabs Φ< px pu)
match'-sound Δ Φ τ (a ⇒ b) (acc Φ↓) (acc m↓) | just u | just pu | yes Φ< | false | false = nothing
match'-sound Δ Φ τ (a ⇒ b) (acc Φ↓) (acc m↓) | nothing | _ = nothing
match'-sound Δ Φ τ (∀' r) Φ↓ (acc m↓)
with match' Δ Φ τ (open-meta r) Φ↓ (m↓ _ (open-meta-a-m<-∀'a r))
| match'-sound Δ Φ τ (open-meta r) Φ↓ (m↓ _ (open-meta-a-m<-∀'a r))
match'-sound Δ Φ τ (∀' r) Φ↓ (acc m↓) | just (u ∷ us) | just px = just (open-↓-∀ r τ u us px)
match'-sound Δ Φ τ (∀' r) Φ↓ (acc m↓) | nothing | nothing = nothing
match'-sound Δ Φ τ (simpl x) Φ↓ m↓ with mgu (simpl x) τ | mgu.sound (simpl x) τ
match'-sound Δ Φ τ (simpl x) Φ↓ m↓ | just _ | just px =
just (subst (λ z → Δ , Φ ⊢ z ↓ τ) (sym px) (i-simp τ))
match'-sound Δ Φ τ (simpl x) Φ↓ m↓ | nothing | nothing = nothing
match-sound : ∀ {ν} (Δ : ICtx ν) (Φ : TCtx) τ r → (Φ↓ : T-Acc Φ) →
Bl.All (Δ , Φ ⊢ r ↓ τ) (match Δ Φ τ r Φ↓)
match-sound Δ Φ τ r Φ↓ with
match' Δ Φ τ (to-meta {zero} r) Φ↓ (m<-well-founded _) |
match'-sound Δ Φ τ (to-meta {zero} r) Φ↓ (m<-well-founded _)
match-sound Δ Φ τ r Φ↓ | just x | just px = true (subst (λ z → Δ , Φ ⊢ z ↓ τ) eq px)
where
eq : ∀ {ν} {a : Type ν} {s} → from-meta (to-meta {zero} a M./ s) ≡ a
eq {a = a} {s = []} = begin
from-meta (M._/_ (to-meta {zero} a) [])
≡⟨ cong (λ q → from-meta q) (MetaTypeMetaLemmas.id-vanishes (to-meta {zero} a)) ⟩
from-meta (to-meta {zero} a)
≡⟨ to-meta-zero-vanishes ⟩
a ∎
match-sound Δ Φ τ r Φ↓ | nothing | q = false
match1st-sound : ∀ {ν} (Δ : ICtx ν) (Φ : TCtx) (ρs : ICtx ν) → (τ : SimpleType ν) → (Φ↓ : T-Acc Φ) →
Bl.All (∃ λ r → (r ∈ ρs) × (Δ , Φ ⊢ r ↓ τ)) (match1st Δ Φ ρs τ Φ↓)
match1st-sound Δ Φ [] τ Φ↓ = false
match1st-sound Δ Φ (x ∷ ρs) τ Φ↓ with match Δ Φ τ x Φ↓ | match-sound Δ Φ τ x Φ↓
match1st-sound Δ Φ (x ∷ ρs) τ Φ↓ | true | true px = true (x , (here refl , px))
match1st-sound Δ Φ (x ∷ ρs) τ Φ↓ | false | false =
all-map (match1st-sound Δ Φ ρs τ Φ↓) (λ{ (r , r∈ρs , r↓τ) → r , ((there r∈ρs) , r↓τ) })
sound' : ∀ {ν} (Δ : ICtx ν) Φ r → (Φ↓ : T-Acc Φ) → Bl.All (Δ , Φ ⊢ᵣ r) (resolve' Δ Φ r Φ↓)
sound' Δ Φ (simpl x) Φ↓ =
all-map (match1st-sound Δ Φ Δ x Φ↓) (λ{ (r , r∈Δ , r↓τ) → r-simp r∈Δ r↓τ })
sound' Δ Φ (a ⇒ b) Φ↓ = all-map (sound' (a ∷ Δ) Φ b Φ↓) (λ x → r-iabs a x)
sound' Δ Φ (∀' r) Φ↓ = all-map (sound' (ictx-weaken Δ) Φ r Φ↓) r-tabs
sound : ∀ {ν} (Δ : ICtx ν) r {Φ} → (Φ↓ : T-Acc Φ) → Bl.All (Δ , Φ ⊢ᵣ r) (resolve Δ Φ r)
sound Δ r Φ↓ = sound' Δ _ r (wf-< _)
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{-# OPTIONS --without-K --safe #-}
module Categories.Adjoint.Instance.PathsOf where
-- PathsOf is adjoint to Underlying Quiver, i.e.
-- The "Underlying Graph" (of a Category) <-> "Path Category on a Quiver" Adjunction.
-- Lots of surprises here, of various level of surprisingness
-- 1. The PathCategory rises universe levels of arrows (and equivalences); the Underlying Quiver does not
-- (due to the Transitive Closure "Star" construction)
-- 2. As a consequence of (1), the eventual Adjunction will be for all Levels being the same
-- 3. To build a Functor from Categories to Quivers, then the object/hom levels are straightforward,
-- but "Categories" has NaturalIsomorphism as the notion of equivalence of Functors. This means
-- that, for example, the 2-point Groupoid is contractible
-- (thus equivalent to the 1-point Category), and yet those two "Graphs" shouldn't be considered to
-- be the same! This is because there isn't an equivalently natural notion of equivalence of
-- Graph (Quiver) Homomorphism.
-- Interestingly, this only shows up when trying to prove that the Underlying Functor respects
-- _≈_, which is not traditionally a requirement of a Functor! This requirement is usually left
-- implicit. (See Categories.Function.Construction.FreeCategory for the details)
--
-- In other words, the adjunction doesn't involve Cats, but StrictCats as one of the endpoints.
open import Level
open import Function using (_$_; flip) renaming (id to id→; _∘_ to _⊚_)
import Relation.Binary.PropositionalEquality as ≡
open import Relation.Binary.Construct.Closure.ReflexiveTransitive
open import Data.Quiver using (Quiver)
open import Data.Quiver.Paths
import Data.Quiver.Morphism as QM
open QM using (Morphism; _≃_)
open import Categories.Adjoint using (Adjoint)
open import Categories.Category.Core using (Category)
import Categories.Category.Construction.PathCategory as PC
open import Categories.Category.Instance.Quivers
open import Categories.Functor using (Functor)
open import Categories.Functor.Construction.PathsOf using (PathsOf)
open import Categories.Functor.Instance.UnderlyingQuiver using (Underlying₀; Underlying₁; Underlying)
open import Categories.NaturalTransformation using (ntHelper)
import Categories.Morphism.Reasoning as MR
module _ (o ℓ e : Level) where
-- the unit morphism from a Quiver X to U∘Free X.
unit : (X : Quiver o (o ⊔ ℓ) (o ⊔ ℓ ⊔ e)) → Morphism X (Underlying₀ (PC.PathCategory X))
unit X = let open Paths X in record { F₀ = id→ ; F₁ = return ; F-resp-≈ = _◅ ε }
module _ (X : Category o (o ⊔ ℓ) (o ⊔ ℓ ⊔ e)) where
open Category X
open HomReasoning
open Paths (Underlying₀ X)
-- "unwind" a path by using repeated composition
unwind : {A B : Obj} → Star _⇒_ A B → A ⇒ B
unwind = fold _⇒_ (flip _∘_) id
unwind-◅◅ : {A B C : Obj} {f : Star _⇒_ A B} {g : Star _⇒_ B C} →
unwind (f ◅◅ g) ≈ (unwind g) ∘ (unwind f)
unwind-◅◅ {f = ε} {g} = Equiv.sym identityʳ
unwind-◅◅ {f = x ◅ f} {g} = ∘-resp-≈ˡ (unwind-◅◅ {f = f} {g}) ○ assoc
unwind-resp-≈ : {A B : Obj} {f g : Star _⇒_ A B} → f ≈* g → unwind f ≈ unwind g
unwind-resp-≈ ε = Equiv.refl
unwind-resp-≈ (x ◅ eq) = ∘-resp-≈ (unwind-resp-≈ eq) x
unwindF : Functor (PC.PathCategory (Underlying₀ X)) X
unwindF = record
{ F₀ = id→
; F₁ = unwind
; identity = Category.Equiv.refl X
; homomorphism = λ { {f = f} {g} → unwind-◅◅ {f = f} {g} }
; F-resp-≈ = unwind-resp-≈
}
module _ (X : Quiver o (o ⊔ ℓ) (o ⊔ ℓ ⊔ e)) where
open Paths X
zig′ : {A B : Quiver.Obj X} → (f : Star (Quiver._⇒_ X) A B) →
unwind (PC.PathCategory X) (QM.qmap (unit X) f) ≈* f
zig′ ε = ε
zig′ (fs ◅ f) = Quiver.Equiv.refl X ◅ zig′ f
module _ {X Y : Category o (o ⊔ ℓ) (o ⊔ ℓ ⊔ e)} (F : Functor X Y) where
module X = Category X
module Y = Category Y
open Category.HomReasoning Y
open Functor F
unwind-natural : {A B : X.Obj} (f : Star X._⇒_ A B) → unwind Y (QM.qmap (Underlying₁ F) f) Y.≈ F₁ (unwind X f)
unwind-natural ε = Y.Equiv.sym identity
unwind-natural (x ◅ f) = Y.Equiv.sym (homomorphism ○ Category.∘-resp-≈ˡ Y (Y.Equiv.sym (unwind-natural f)))
Free⊣Underlying : Adjoint (PathsOf {o} {o ⊔ ℓ} {o ⊔ ℓ ⊔ e}) Underlying
Free⊣Underlying = record
{ unit = ntHelper record
{ η = unit
; commute = λ {X} {Y} f → let open Paths Y in record { F₀≡ = ≡.refl ; F₁≡ = Quiver.Equiv.refl Y ◅ ε }
}
; counit = ntHelper record
{ η = unwindF
; commute = λ {_} {Y} F → record
{ eq₀ = λ _ → ≡.refl
; eq₁ = λ f → toSquare Y (unwind-natural F f)
}
}
; zig = λ {G} → record
{ eq₀ = λ _ → ≡.refl
; eq₁ = λ f → toSquare (PC.PathCategory G) (zig′ G f)
}
; zag = λ {B} → record { F₀≡ = ≡.refl ; F₁≡ = Category.identityˡ B }
}
where
open MR using (toSquare)
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Type(s) used (only) when calling out to Haskell via the FFI
------------------------------------------------------------------------
{-# OPTIONS --without-K #-}
module Foreign.Haskell where
open import Level
-- A unit type.
data Unit : Set where
unit : Unit
{-# COMPILE GHC Unit = data () (()) #-}
{-# COMPILE UHC Unit = data __UNIT__ (__UNIT__) #-}
-- A pair type
record Pair {ℓ ℓ′ : Level} (A : Set ℓ) (B : Set ℓ′) : Set (ℓ ⊔ ℓ′) where
constructor _,_
field fst : A
snd : B
open Pair public
{-# FOREIGN GHC type AgdaPair l1 l2 a b = (a , b) #-}
{-# COMPILE GHC Pair = data MAlonzo.Code.Foreign.Haskell.AgdaPair ((,)) #-}
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Some unit types
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Unit where
open import Data.Sum
open import Relation.Nullary
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as PropEq
using (_≡_; refl)
-- Some types are defined in Data.Unit.Base.
open import Data.Unit.Base public
------------------------------------------------------------------------
-- Operations
infix 4 _≟_ _≤?_
_≟_ : Decidable {A = ⊤} _≡_
_ ≟ _ = yes refl
_≤?_ : Decidable _≤_
_ ≤? _ = yes _
total : Total _≤_
total _ _ = inj₁ _
------------------------------------------------------------------------
-- Properties
preorder : Preorder _ _ _
preorder = PropEq.preorder ⊤
setoid : Setoid _ _
setoid = PropEq.setoid ⊤
decTotalOrder : DecTotalOrder _ _ _
decTotalOrder = record
{ Carrier = ⊤
; _≈_ = _≡_
; _≤_ = _≤_
; isDecTotalOrder = record
{ isTotalOrder = record
{ isPartialOrder = record
{ isPreorder = record
{ isEquivalence = PropEq.isEquivalence
; reflexive = λ _ → _
; trans = λ _ _ → _
}
; antisym = antisym
}
; total = total
}
; _≟_ = _≟_
; _≤?_ = _≤?_
}
}
where
antisym : Antisymmetric _≡_ _≤_
antisym _ _ = refl
decSetoid : DecSetoid _ _
decSetoid = DecTotalOrder.Eq.decSetoid decTotalOrder
poset : Poset _ _ _
poset = DecTotalOrder.poset decTotalOrder
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Universe-sensitive functor and monad instances for the Product type.
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Algebra
module Data.Product.Categorical.Examples
{a e b} {A : Monoid a e} {B : Set b} where
open import Level using (Lift; lift; _⊔_)
open import Category.Functor using (RawFunctor)
open import Category.Monad using (RawMonad)
open import Data.Product
open import Data.Product.Relation.Binary.Pointwise.NonDependent
open import Function
import Function.Identity.Categorical as Id
open import Relation.Binary using (Rel)
open import Relation.Binary.PropositionalEquality using (_≡_; refl)
------------------------------------------------------------------------
-- Examples
-- Note that these examples are simple unit tests, because the type
-- checker verifies them.
private
module A = Monoid A
open import Data.Product.Categorical.Left A.rawMonoid b
_≈_ : Rel (A.Carrier × Lift a B) (e ⊔ a ⊔ b)
_≈_ = Pointwise A._≈_ _≡_
open RawFunctor functor
-- This type to the right of × needs to be a "lifted" version of (B : Set b)
-- that lives in the universe (Set (a ⊔ b)).
fmapIdₗ : (x : A.Carrier × Lift a B) → (id <$> x) ≈ x
fmapIdₗ x = A.refl , refl
open RawMonad monad
-- Now, let's show that "return" is a unit for >>=. We use Lift in exactly
-- the same way as above. The data (x : B) then needs to be "lifted" to
-- this new type (Lift B).
returnUnitL : ∀ {x : B} {f : Lift a B → A.Carrier × Lift a B} →
((return (lift x)) >>= f) ≈ f (lift x)
returnUnitL = A.identityˡ _ , refl
returnUnitR : {x : A.Carrier × Lift a B} → (x >>= return) ≈ x
returnUnitR = A.identityʳ _ , refl
-- And another (limited version of a) monad law...
bindCompose : ∀ {f g : Lift a B → A.Carrier × Lift a B} →
{x : A.Carrier × Lift a B} →
((x >>= f) >>= g) ≈ (x >>= (λ y → (f y >>= g)))
bindCompose = A.assoc _ _ _ , refl
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module InstanceArgumentsNotFound where
postulate A B : Set
f : {{a : A}} → B
test : B
test = f
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{-# OPTIONS --sized-types #-}
module FormalLanguage {ℓ} where
import Lvl
open import Sized.Data.List renaming (∅ to [])
open import Lang.Size
open import Data.Boolean
open import Data.Boolean.Operators
open Data.Boolean.Operators.Programming
open import Data.Boolean.Stmt
open import Functional
open import Relator.Equals
open import Type
-- Definitions:
-- A language is a set of words.
-- A word in a language is a list of valid symbols in the language.
-- A valid symbol in the language is an element in the alphabet of the language.
-- An alphabet of a language is a set.
-- A string is a word.
-- Standard conventions for variable naming in languages:
-- L is a language
-- Σ is an alphabet
Alphabet = Type{ℓ}
Word = List
-- Language is defined as a trie: (LHS is using the definition of Language, RHS is using the usual "semantics" of languages as sets):
-- For a language L
-- accepts-ε:
-- (accepts-ε(L) = 𝑇) ↔ (ε ∈ L)
-- accepts-ε(L) returns a boolean determining whether the empty word is accepted in the language.
-- suffix-lang:
-- ∀word∀c. (word ∈ suffix-lang(L)(c)) ↔ ((c 𝁼 word) ∈ L)
-- suffix-lang(L)(c) is the language that consists of the rest of the words when a word is starting with c in L.
-- Copied (with modifications) from: http://agda.readthedocs.io/en/v2.5.2/language/sized-types.html (2017-05-13)
-- which links the following paper: "Formal Languages, Formally and Coinductively, Dmitriy Traytel, FSCD (2016)" [https://www21.in.tum.de/~traytel/papers/fscd16-coind_lang/paper.pdf]
-- Example:
-- A language 𝔏 consists of 6 words:
-- 𝔏 = {"" , "aa" , "aaa" , "aab" , "aba" , "aaab"}
-- accepts-ε (𝔏) = 𝑇
-- suffix-lang(𝔏)(a) = {"a" , "aa" , "ab" , "ba" , "aab"}
-- accepts-ε (suffix-lang(𝔏)(a)) = 𝐹
-- suffix-lang(suffix-lang(𝔏)(a))(a) = {"" , "a" , "b" , "ab"}
-- suffix-lang(suffix-lang(𝔏)(a))(b) = {"a"}
record Language (Σ : Alphabet) {s : Size} : Type{ℓ} where
constructor intro
coinductive
field
accepts-ε : Bool
suffix-lang : ∀{sₛ : <ˢⁱᶻᵉ s} → Σ → Language(Σ){sₛ}
module Oper {Σ} where
infixl 1003 _∪_
infixl 1002 _∩_
infixl 1001 _𝁼_
infixl 1000 _*
-- The empty language.
-- The language that does not include any word at all.
∅ : ∀{s} → Language(Σ){s}
Language.accepts-ε ∅ = 𝐹
Language.suffix-lang ∅ _ = ∅
-- The empty word language.
-- The language with only the empty word.
ε : ∀{s} → Language(Σ){s}
Language.accepts-ε ε = 𝑇
Language.suffix-lang ε _ = ∅
-- The language of length 1 words that only accepts some symbols of its alphabet
alphabet-filter : ∀{s} → (Σ → Bool) → Language(Σ){s}
Language.accepts-ε (alphabet-filter f) = 𝐹
Language.suffix-lang (alphabet-filter f) c = if f(c) then ε else ∅
-- The single symbol language.
-- The language consisting of a single word with a single letter
-- Note: This is only possible when Alphabet has a computably decidable equality relation
single : ⦃ _ : ComputablyDecidable(_≡_) ⦄ → ∀{s} → Σ → Language(Σ){s}
single(a) = alphabet-filter(ComputablyDecidable.decide(_≡_) a)
-- The sublanguage filtered by a decidable predicate.
filter : ∀{s} → (Word(Σ) → Bool) → Language(Σ){s} → Language(Σ){s}
Language.accepts-ε (filter P(𝔏)) = P(List.∅)
Language.suffix-lang (filter P(𝔏)) c = filter (P ∘ tail) (Language.suffix-lang(𝔏)(c))
-- The language where every letter in the alphabet is applied to a function.
unmap : ∀{Σ₂}{s} → (Σ → Σ₂) → Language(Σ₂){s} → Language(Σ){s}
Language.accepts-ε (unmap f(𝔏)) = Language.accepts-ε (𝔏)
Language.suffix-lang (unmap f(𝔏)) c = unmap f(Language.suffix-lang(𝔏)(f(c)))
-- Union.
-- The language that includes any words that the two languages have.
_∪_ : ∀{s} → Language(Σ){s} → Language(Σ){s} → Language(Σ){s}
Language.accepts-ε (L₁ ∪ L₂) = Language.accepts-ε(L₁) || Language.accepts-ε(L₂)
Language.suffix-lang (L₁ ∪ L₂) c = Language.suffix-lang(L₁)(c) ∪ Language.suffix-lang(L₂)(c)
-- Intersection.
-- The language that only includes the words that both languages have in common.
_∩_ : ∀{s} → Language(Σ){s} → Language(Σ){s} → Language(Σ){s}
Language.accepts-ε (L₁ ∩ L₂) = Language.accepts-ε(L₁) && Language.accepts-ε(L₂)
Language.suffix-lang (L₁ ∩ L₂) c = Language.suffix-lang(L₁)(c) ∩ Language.suffix-lang(L₂)(c)
-- Concatenation.
-- The language that includes words that start with a word the first language and end in a word from the second language.
_𝁼_ : ∀{s} → Language(Σ){s} → Language(Σ){s} → Language(Σ){s}
Language.accepts-ε (L₁ 𝁼 L₂) = Language.accepts-ε(L₁) && Language.accepts-ε(L₂)
Language.suffix-lang (L₁ 𝁼 L₂) c =
if Language.accepts-ε(L₁)
then (Language.suffix-lang(L₁)(c) 𝁼 L₂) ∪ Language.suffix-lang(L₂)(c)
else (Language.suffix-lang(L₁)(c) 𝁼 L₂)
-- Star/Closure
-- The language that includes words in any number of concatenations with itself.
_* : ∀{s} → Language(Σ){s} → Language(Σ){s}
Language.accepts-ε (L *) = 𝑇
Language.suffix-lang (L *) c = Language.suffix-lang(L)(c) 𝁼 (L *)
-- Complement
-- The language that includes all words that a language does not have.
∁_ : ∀{s} → Language(Σ){s} → Language(Σ){s}
Language.accepts-ε (∁ L) = !(Language.accepts-ε(L))
Language.suffix-lang (∁ L) c = ∁(Language.suffix-lang(L)(c))
-- The universal language.
-- The language that includes all words in any combination of the alphabet.
-- The largest language (with most words) with a certain alphabet.
𝐔 : ∀{s} → Language(Σ){s}
𝐔 = ∁(∅)
-- Containment check
-- Checks whether a word is in the language.
_∈?_ : ∀{s} → Word{s = s}(Σ) → Language(Σ) → Bool
_∈?_ [] L = Language.accepts-ε(L)
_∈?_ (_⊰_ {sₗ} c w) L = _∈?_ {s = sₗ} w (Language.suffix-lang L c)
-- Containment
-- The relation of whether a word is in the language or not.
_∈_ : ∀{s} → Word{s = s}(Σ) → Language(Σ) → Type
_∈_ {s} a b = IsTrue(_∈?_ {s} a b)
[_]_∈_ : ∀(s) → Word{s = s}(Σ) → Language(Σ) → Type
[ s ] a ∈ b = _∈_ {s} a b
-- Uncontainment
-- The relation of whether a word is not in the language or not.
_∉_ : ∀{s} → Word{s = s}(Σ) → Language(Σ) → Type
_∉_ {s} a b = IsFalse(_∈?_ {s} a b)
[_]_∉_ : ∀(s) → Word{s = s}(Σ) → Language(Σ) → Type
[ s ] a ∉ b = _∉_ {s} a b
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module Data.Bin.Addition where
open import Data.List using ([]; _∷_)
open import Data.Bin using(addBits; addBitLists; addCarryToBitList)
open import Relation.Binary.PropositionalEquality
private
module PropEq = Relation.Binary.PropositionalEquality
open import Data.Fin using (zero; suc) renaming (toℕ to bitToℕ; _+_ to _+F_)
open import Data.Fin.Properties using (_+′_)
open import Data.Digit using (fromDigits)
open import Data.Product
import Data.Nat.Properties
open import Data.Bin.Bijection using (fromℕ-bijection)
module Solving where
open Relation.Binary.PropositionalEquality
open ≡-Reasoning
open import Algebra
-- open Algebra.Structures
open CommutativeSemiring Data.Nat.Properties.commutativeSemiring hiding (sym; refl)
open Data.Product
open Data.Nat.Properties.SemiringSolver
lem : ∀ {a b c as bs b' c' r}
→ c' * 2 + b' ≡ c + a + b
→ r ≡ c' + (as + bs)
→ b' + r * 2 ≡ c + (a + as * 2 + (b + bs * 2))
lem {a} {b} {c} {as} {bs} {b'} {c'} {r} eq1 eq2 =
begin
b' + r * 2
≡⟨ cong (λ x → b' + x * 2) eq2 ⟩
b' + (c' + (as + bs)) * 2
≡⟨ cong (λ x → b' + x) (proj₂ distrib 2 c' (as + bs)) ⟩
b' + (c' * 2 + (as + bs) * 2)
≡⟨ sym (+-assoc b' (c' * 2) ((as + bs) * 2)) ⟩
b' + c' * 2 + (as + bs) * 2
≡⟨ +-cong (+-comm b' (c' * 2)) refl ⟩
c' * 2 + b' + (as + bs) * 2
≡⟨ +-cong eq1 refl ⟩
c + a + b + (as + bs) * 2
≡⟨ solve 5 (λ a b c as bs →
c :+ a :+ b :+ (as :+ bs) :* con 2
:= c :+ (a :+ as :* con 2 :+ (b :+ bs :* con 2))
) refl a b c as bs ⟩
c + (a + as * 2 + (b + bs * 2))
∎
open import Data.Nat using () renaming (_+_ to _+_; _*_ to _*_)
open Data.Bin using (toℕ; toBits)
addBits-is-addition : ∀ {c a b} → bitToℕ (proj₁ (addBits c a b)) * 2 + bitToℕ (proj₂ (addBits c a b)) ≡ bitToℕ c + bitToℕ a + bitToℕ b
-- Brute force!!! (LOL)
addBits-is-addition {zero} {zero} {zero} = refl
addBits-is-addition {zero} {zero} {suc zero} = refl
addBits-is-addition {zero} {suc zero} {zero} = refl
addBits-is-addition {zero} {suc zero} {suc zero} = refl
addBits-is-addition {suc zero} {zero} {zero} = refl
addBits-is-addition {suc zero} {zero} {suc zero} = refl
addBits-is-addition {suc zero} {suc zero} {zero} = refl
addBits-is-addition {suc zero} {suc zero} {suc zero} = refl
addBits-is-addition {suc (suc ())}
addBits-is-addition {_} {suc (suc ())}
addBits-is-addition {_} {_} {suc (suc ())}
addCarryToBitLists-is-addition : ∀ c b → fromDigits (addCarryToBitList c b) ≡ bitToℕ c + fromDigits b
addCarryToBitLists-is-addition zero _ = refl
addCarryToBitLists-is-addition (suc zero) [] = refl
addCarryToBitLists-is-addition (suc zero) (zero ∷ t) = refl
addCarryToBitLists-is-addition (suc zero) (suc zero ∷ t) = cong (λ x → x * 2) (addCarryToBitLists-is-addition (suc zero) t)
addCarryToBitLists-is-addition (suc (suc ())) _
addCarryToBitLists-is-addition _ ((suc (suc ())) ∷ _)
open import Data.Nat.Properties using (isCommutativeSemiring)
open import Algebra.Structures
using (module IsCommutativeSemiring;
module IsCommutativeMonoid)
open IsCommutativeSemiring isCommutativeSemiring
using (+-isCommutativeMonoid)
open IsCommutativeMonoid +-isCommutativeMonoid using (identity; comm) renaming (∙-cong to +-cong)
addBitLists-is-addition : ∀ c a b → fromDigits (addBitLists c a b) ≡ bitToℕ c + (fromDigits a + fromDigits b)
addBitLists-is-addition c [] b = addCarryToBitLists-is-addition c b
addBitLists-is-addition c (a ∷ as) [] = trans (addCarryToBitLists-is-addition c (a ∷ as)) (+-cong {bitToℕ c} refl (sym (proj₂ identity (fromDigits (a ∷ as)))))
addBitLists-is-addition c (a ∷ as) (b ∷ bs) with addBits c a b | addBits-is-addition {c} {a} {b}
... | (c' , b') | abia with addBitLists c' as bs | addBitLists-is-addition c' as bs
... | r | ria = Solving.lem {bitToℕ a} {bitToℕ b} {bitToℕ c} {fromDigits as} {fromDigits bs} {bitToℕ b'} {bitToℕ c'} abia ria
open import Function.Inverse using (Inverse)
open import Function.Equality
open import Data.Bin.BitListBijection using () renaming (toℕ⟶ to bits-to-ℕ)
open import Data.Bin.Bijection using (Bits-inverse-Bin)
simplify-fromBits-to-ℕ : ∀ a → toℕ (Data.Bin.fromBits a) ≡ bits-to-ℕ ⟨$⟩ a
simplify-fromBits-to-ℕ a = Π.cong bits-to-ℕ (Inverse.left-inverse-of Bits-inverse-Bin a )
+-is-addition : ∀ a b → toℕ (Data.Bin._+_ a b) ≡ toℕ a + toℕ b
+-is-addition a b =
trans
(simplify-fromBits-to-ℕ (addBitLists zero (toBits a) (toBits b)))
(addBitLists-is-addition zero as bs)
where
as = toBits a
bs = toBits b
import Algebra.Lifting
open import Data.Nat using (ℕ)
open import Data.Bin using (Bin; 0#)
open import Algebra.Structures using (IsCommutativeMonoid)
private module Lifting = Algebra.Lifting _ _ fromℕ-bijection
is-commutativeMonoid : IsCommutativeMonoid _≡_ Data.Bin._+_ 0#
is-commutativeMonoid = lift-isCommutativeMonoid 0 +-isCommutativeMonoid
where
open Lifting.WithOp₂ _+_ Data.Bin._+_ +-is-addition
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Data.DiffInt.Properties where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Univalence
open import Cubical.Data.DiffInt.Base
open import Cubical.Data.Nat as ℕ using (suc; zero; isSetℕ; discreteℕ; ℕ) renaming (_+_ to _+ⁿ_; _·_ to _·ⁿ_)
open import Cubical.Data.Sigma
open import Cubical.Data.Bool
open import Cubical.Data.Int as Int using (Int; sucInt)
open import Cubical.Foundations.Path
open import Cubical.Foundations.Isomorphism
open import Cubical.Relation.Binary.Base
open import Cubical.Relation.Nullary
open import Cubical.HITs.SetQuotients
open BinaryRelation
relIsEquiv : isEquivRel rel
relIsEquiv = equivRel {A = ℕ × ℕ} relIsRefl relIsSym relIsTrans
where
open import Cubical.Data.Nat
relIsRefl : isRefl rel
relIsRefl (a0 , a1) = refl
relIsSym : isSym rel
relIsSym (a0 , a1) (b0 , b1) p = sym p
relIsTrans : isTrans rel
relIsTrans (a0 , a1) (b0 , b1) (c0 , c1) p0 p1 =
inj-m+ {m = (b0 + b1)} ((b0 + b1) + (a0 + c1) ≡⟨ +-assoc (b0 + b1) a0 c1 ⟩
((b0 + b1) + a0) + c1 ≡[ i ]⟨ +-comm b0 b1 i + a0 + c1 ⟩
((b1 + b0) + a0) + c1 ≡[ i ]⟨ +-comm (b1 + b0) a0 i + c1 ⟩
(a0 + (b1 + b0)) + c1 ≡[ i ]⟨ +-assoc a0 b1 b0 i + c1 ⟩
(a0 + b1) + b0 + c1 ≡⟨ sym (+-assoc (a0 + b1) b0 c1) ⟩
(a0 + b1) + (b0 + c1) ≡⟨ cong (λ p → p . fst + p .snd) (transport ΣPath≡PathΣ (p0 , p1))⟩
(b0 + a1) + (c0 + b1) ≡⟨ sym (+-assoc b0 a1 (c0 + b1))⟩
b0 + (a1 + (c0 + b1)) ≡[ i ]⟨ b0 + (a1 + +-comm c0 b1 i) ⟩
b0 + (a1 + (b1 + c0)) ≡[ i ]⟨ b0 + +-comm a1 (b1 + c0) i ⟩
b0 + ((b1 + c0) + a1) ≡[ i ]⟨ b0 + +-assoc b1 c0 a1 (~ i) ⟩
b0 + (b1 + (c0 + a1)) ≡⟨ +-assoc b0 b1 (c0 + a1)⟩
(b0 + b1) + (c0 + a1) ∎ )
relIsProp : BinaryRelation.isPropValued rel
relIsProp a b x y = isSetℕ _ _ _ _
discreteℤ : Discrete ℤ
discreteℤ = discreteSetQuotients (discreteΣ discreteℕ λ _ → discreteℕ) relIsProp relIsEquiv (λ _ _ → discreteℕ _ _)
isSetℤ : isSet ℤ
isSetℤ = Discrete→isSet discreteℤ
sucℤ' : ℕ × ℕ -> ℤ
sucℤ' (a⁺ , a⁻) = [ suc a⁺ , a⁻ ]
sucℤ'-respects-rel : (a b : ℕ × ℕ) → rel a b → sucℤ' a ≡ sucℤ' b
sucℤ'-respects-rel a@(a⁺ , a⁻) b@(b⁺ , b⁻) a~b = eq/ (suc a⁺ , a⁻) (suc b⁺ , b⁻) λ i → suc (a~b i)
sucℤ : ℤ -> ℤ
sucℤ = elim {R = rel} {B = λ _ → ℤ} (λ _ → isSetℤ) sucℤ' sucℤ'-respects-rel
predℤ' : ℕ × ℕ -> ℤ
predℤ' (a⁺ , a⁻) = [ a⁺ , suc a⁻ ]
⟦_⟧ : Int -> ℤ
⟦_⟧ (Int.pos n) = [ n , 0 ]
⟦_⟧ (Int.negsuc n) = [ 0 , suc n ]
fwd = ⟦_⟧
bwd' : ℕ × ℕ -> Int
bwd' (zero , a⁻) = Int.neg a⁻
bwd' (suc a⁺ , a⁻) = sucInt (bwd' (a⁺ , a⁻))
rel-suc : ∀ a⁺ a⁻ → rel (a⁺ , a⁻) (suc a⁺ , suc a⁻)
rel-suc a⁺ a⁻ = ℕ.+-suc a⁺ a⁻
bwd'-suc : ∀ a⁺ a⁻ → bwd' (a⁺ , a⁻) ≡ bwd' (suc a⁺ , suc a⁻)
bwd'-suc zero zero = refl
bwd'-suc zero (suc a⁻) = refl
bwd'-suc (suc a⁺) a⁻ i = sucInt (bwd'-suc a⁺ a⁻ i)
bwd'+ : ∀ m n → bwd' (m , m +ⁿ n) ≡ bwd' (0 , n)
bwd'+ zero n = refl
bwd'+ (suc m) n = sym (bwd'-suc m (m +ⁿ n)) ∙ bwd'+ m n
bwd'-respects-rel : (a b : ℕ × ℕ) → rel a b → bwd' a ≡ bwd' b
bwd'-respects-rel (zero , a⁻) ( b⁺ , b⁻) a~b = sym (bwd'+ b⁺ a⁻) ∙ (λ i → bwd' (b⁺ , a~b (~ i)))
bwd'-respects-rel (suc a⁺ , a⁻) (zero , b⁻) a~b = (λ i → bwd' (suc a⁺ , a~b (~ i))) ∙ sym (bwd'-suc a⁺ (a⁺ +ⁿ b⁻)) ∙ bwd'+ a⁺ b⁻
bwd'-respects-rel (suc a⁺ , a⁻) (suc b⁺ , b⁻) a~b i = sucInt (bwd'-respects-rel (a⁺ , a⁻) (b⁺ , b⁻) (ℕ.inj-m+ {1} {a⁺ +ⁿ b⁻} {b⁺ +ⁿ a⁻} a~b) i)
bwd : ℤ -> Int
bwd = elim {R = rel} {B = λ _ → Int} (λ _ → Int.isSetInt) bwd' bwd'-respects-rel
bwd-fwd : ∀ (x : Int) -> bwd (fwd x) ≡ x
bwd-fwd (Int.pos zero) = refl
bwd-fwd (Int.pos (suc n)) i = sucInt (bwd-fwd (Int.pos n) i)
bwd-fwd (Int.negsuc n) = refl
suc-⟦⟧ : ∀ x → sucℤ ⟦ x ⟧ ≡ ⟦ sucInt x ⟧
suc-⟦⟧ (Int.pos n) = refl
suc-⟦⟧ (Int.negsuc zero) = eq/ {R = rel} (1 , 1) (0 , 0) refl
suc-⟦⟧ (Int.negsuc (suc n)) = eq/ {R = rel} (1 , 2 +ⁿ n) (0 , 1 +ⁿ n) refl
fwd-bwd' : (a : ℕ × ℕ) → fwd (bwd [ a ]) ≡ [ a ]
fwd-bwd' a@(zero , zero) = refl
fwd-bwd' a@(zero , suc a⁻) = refl
fwd-bwd' a@(suc a⁺ , a⁻) = sym (suc-⟦⟧ (bwd [ a⁺ , a⁻ ])) ∙ (λ i → sucℤ (fwd-bwd' (a⁺ , a⁻) i))
fwd-bwd : ∀ (z : ℤ) -> fwd (bwd z) ≡ z
fwd-bwd = elimProp {R = rel} (λ _ → isSetℤ _ _) fwd-bwd'
Int≡ℤ : Int ≡ ℤ
Int≡ℤ = isoToPath (iso fwd bwd fwd-bwd bwd-fwd)
infix 8 -_
infixl 7 _·_
infixl 6 _+_
_+'_ : (a b : ℕ × ℕ) → ℤ
(a⁺ , a⁻) +' (b⁺ , b⁻) = [ a⁺ +ⁿ b⁺ , a⁻ +ⁿ b⁻ ]
private
commˡⁿ : ∀ a b c → a +ⁿ b +ⁿ c ≡ a +ⁿ c +ⁿ b
commˡⁿ a b c = sym (ℕ.+-assoc a b c) ∙ (λ i → a +ⁿ ℕ.+-comm b c i) ∙ ℕ.+-assoc a c b
lem0 : ∀ a b c d → (a +ⁿ b) +ⁿ (c +ⁿ d) ≡ (a +ⁿ c) +ⁿ (b +ⁿ d)
lem0 a b c d = ℕ.+-assoc (a +ⁿ b) c d ∙ (λ i → commˡⁿ a b c i +ⁿ d) ∙ sym (ℕ.+-assoc (a +ⁿ c) b d)
+ⁿ-creates-rel-≡ : ∀ a⁺ a⁻ x → _≡_ {A = ℤ} [ a⁺ , a⁻ ] [ a⁺ +ⁿ x , a⁻ +ⁿ x ]
+ⁿ-creates-rel-≡ a⁺ a⁻ x = eq/ (a⁺ , a⁻) (a⁺ +ⁿ x , a⁻ +ⁿ x) ((λ i → a⁺ +ⁿ ℕ.+-comm a⁻ x i) ∙ ℕ.+-assoc a⁺ x a⁻)
+-respects-relʳ : (a b c : ℕ × ℕ) → rel a b → (a +' c) ≡ (b +' c)
+-respects-relʳ a@(a⁺ , a⁻) b@(b⁺ , b⁻) c@(c⁺ , c⁻) p = eq/ {R = rel} (a⁺ +ⁿ c⁺ , a⁻ +ⁿ c⁻) (b⁺ +ⁿ c⁺ , b⁻ +ⁿ c⁻) (
(a⁺ +ⁿ c⁺) +ⁿ (b⁻ +ⁿ c⁻) ≡⟨ lem0 a⁺ c⁺ b⁻ c⁻ ⟩
(a⁺ +ⁿ b⁻) +ⁿ (c⁺ +ⁿ c⁻) ≡[ i ]⟨ p i +ⁿ (c⁺ +ⁿ c⁻) ⟩
(b⁺ +ⁿ a⁻) +ⁿ (c⁺ +ⁿ c⁻) ≡⟨ sym (lem0 b⁺ c⁺ a⁻ c⁻) ⟩
(b⁺ +ⁿ c⁺) +ⁿ (a⁻ +ⁿ c⁻) ∎)
+-respects-relˡ : (a b c : ℕ × ℕ) → rel b c → (a +' b) ≡ (a +' c)
+-respects-relˡ a@(a⁺ , a⁻) b@(b⁺ , b⁻) c@(c⁺ , c⁻) p = eq/ {R = rel} (a⁺ +ⁿ b⁺ , a⁻ +ⁿ b⁻) (a⁺ +ⁿ c⁺ , a⁻ +ⁿ c⁻) (
(a⁺ +ⁿ b⁺) +ⁿ (a⁻ +ⁿ c⁻) ≡⟨ lem0 a⁺ b⁺ a⁻ c⁻ ⟩
(a⁺ +ⁿ a⁻) +ⁿ (b⁺ +ⁿ c⁻) ≡[ i ]⟨ (a⁺ +ⁿ a⁻) +ⁿ p i ⟩
(a⁺ +ⁿ a⁻) +ⁿ (c⁺ +ⁿ b⁻) ≡⟨ sym (lem0 a⁺ c⁺ a⁻ b⁻) ⟩
(a⁺ +ⁿ c⁺) +ⁿ (a⁻ +ⁿ b⁻) ∎)
_+''_ : ℤ → ℤ → ℤ
_+''_ = rec2 {R = rel} {B = ℤ} φ _+'_ +-respects-relʳ +-respects-relˡ
where abstract φ = isSetℤ
-- normalization of isSetℤ explodes. Therefore we wrap this with expanded cases
_+_ : ℤ → ℤ → ℤ
x@([ _ ]) + y@([ _ ]) = x +'' y
x@([ _ ]) + y@(eq/ _ _ _ _) = x +'' y
x@(eq/ _ _ _ _) + y@([ _ ]) = x +'' y
x@(eq/ _ _ _ _) + y@(eq/ _ _ _ _) = x +'' y
x@(eq/ _ _ _ _) + y@(squash/ a b p q i j) = isSetℤ _ _ (cong (x +_) p) (cong (x +_) q) i j
x@([ _ ]) + y@(squash/ a b p q i j) = isSetℤ _ _ (cong (x +_) p) (cong (x +_) q) i j
x@(squash/ a b p q i j) + y = isSetℤ _ _ (cong (_+ y) p) (cong (_+ y) q) i j
-'_ : ℕ × ℕ → ℤ
-' (a⁺ , a⁻) = [ a⁻ , a⁺ ]
neg-respects-rel'-≡ : (a b : ℕ × ℕ) → rel a b → (-' a) ≡ (-' b)
neg-respects-rel'-≡ a@(a⁺ , a⁻) b@(b⁺ , b⁻) p = eq/ {R = rel} (a⁻ , a⁺) (b⁻ , b⁺) (ℕ.+-comm a⁻ b⁺ ∙ sym p ∙ ℕ.+-comm a⁺ b⁻)
-_ : ℤ → ℤ
-_ = rec {R = rel} {B = ℤ} isSetℤ -'_ neg-respects-rel'-≡
_·'_ : (a b : ℕ × ℕ) → ℤ
(a⁺ , a⁻) ·' (b⁺ , b⁻) = [ a⁺ ·ⁿ b⁺ +ⁿ a⁻ ·ⁿ b⁻ , a⁺ ·ⁿ b⁻ +ⁿ a⁻ ·ⁿ b⁺ ]
private
lem1 : ∀ a b c d → (a +ⁿ b) +ⁿ (c +ⁿ d) ≡ (a +ⁿ d) +ⁿ (b +ⁿ c)
lem1 a b c d = (λ i → (a +ⁿ b) +ⁿ ℕ.+-comm c d i) ∙ ℕ.+-assoc (a +ⁿ b) d c ∙ (λ i → commˡⁿ a b d i +ⁿ c) ∙ sym (ℕ.+-assoc (a +ⁿ d) b c)
·-respects-relʳ : (a b c : ℕ × ℕ) → rel a b → (a ·' c) ≡ (b ·' c)
·-respects-relʳ a@(a⁺ , a⁻) b@(b⁺ , b⁻) c@(c⁺ , c⁻) p = eq/ {R = rel} (a⁺ ·ⁿ c⁺ +ⁿ a⁻ ·ⁿ c⁻ , a⁺ ·ⁿ c⁻ +ⁿ a⁻ ·ⁿ c⁺) (b⁺ ·ⁿ c⁺ +ⁿ b⁻ ·ⁿ c⁻ , b⁺ ·ⁿ c⁻ +ⁿ b⁻ ·ⁿ c⁺) (
(a⁺ ·ⁿ c⁺ +ⁿ a⁻ ·ⁿ c⁻) +ⁿ (b⁺ ·ⁿ c⁻ +ⁿ b⁻ ·ⁿ c⁺) ≡⟨ lem1 (a⁺ ·ⁿ c⁺) (a⁻ ·ⁿ c⁻) (b⁺ ·ⁿ c⁻) (b⁻ ·ⁿ c⁺) ⟩
(a⁺ ·ⁿ c⁺ +ⁿ b⁻ ·ⁿ c⁺) +ⁿ (a⁻ ·ⁿ c⁻ +ⁿ b⁺ ·ⁿ c⁻) ≡[ i ]⟨ ℕ.·-distribʳ a⁺ b⁻ c⁺ i +ⁿ ℕ.·-distribʳ a⁻ b⁺ c⁻ i ⟩
((a⁺ +ⁿ b⁻) ·ⁿ c⁺) +ⁿ ((a⁻ +ⁿ b⁺) ·ⁿ c⁻) ≡[ i ]⟨ p i ·ⁿ c⁺ +ⁿ (ℕ.+-comm a⁻ b⁺ ∙ sym p ∙ ℕ.+-comm a⁺ b⁻) i ·ⁿ c⁻ ⟩
((b⁺ +ⁿ a⁻) ·ⁿ c⁺) +ⁿ ((b⁻ +ⁿ a⁺) ·ⁿ c⁻) ≡[ i ]⟨ ℕ.·-distribʳ b⁺ a⁻ c⁺ (~ i) +ⁿ ℕ.·-distribʳ b⁻ a⁺ c⁻ (~ i) ⟩
(b⁺ ·ⁿ c⁺ +ⁿ a⁻ ·ⁿ c⁺) +ⁿ (b⁻ ·ⁿ c⁻ +ⁿ a⁺ ·ⁿ c⁻) ≡⟨ sym (lem1 (b⁺ ·ⁿ c⁺) (b⁻ ·ⁿ c⁻) (a⁺ ·ⁿ c⁻) (a⁻ ·ⁿ c⁺)) ⟩
(b⁺ ·ⁿ c⁺ +ⁿ b⁻ ·ⁿ c⁻) +ⁿ (a⁺ ·ⁿ c⁻ +ⁿ a⁻ ·ⁿ c⁺) ∎)
·-respects-relˡ : (a b c : ℕ × ℕ) → rel b c → (a ·' b) ≡ (a ·' c)
·-respects-relˡ a@(a⁺ , a⁻) b@(b⁺ , b⁻) c@(c⁺ , c⁻) p = eq/ {R = rel} (a⁺ ·ⁿ b⁺ +ⁿ a⁻ ·ⁿ b⁻ , a⁺ ·ⁿ b⁻ +ⁿ a⁻ ·ⁿ b⁺) (a⁺ ·ⁿ c⁺ +ⁿ a⁻ ·ⁿ c⁻ , a⁺ ·ⁿ c⁻ +ⁿ a⁻ ·ⁿ c⁺) (
(a⁺ ·ⁿ b⁺ +ⁿ a⁻ ·ⁿ b⁻) +ⁿ (a⁺ ·ⁿ c⁻ +ⁿ a⁻ ·ⁿ c⁺) ≡⟨ lem0 (a⁺ ·ⁿ b⁺) (a⁻ ·ⁿ b⁻) (a⁺ ·ⁿ c⁻) (a⁻ ·ⁿ c⁺) ⟩
(a⁺ ·ⁿ b⁺ +ⁿ a⁺ ·ⁿ c⁻) +ⁿ (a⁻ ·ⁿ b⁻ +ⁿ a⁻ ·ⁿ c⁺) ≡[ i ]⟨ ℕ.·-distribˡ a⁺ b⁺ c⁻ i +ⁿ ℕ.·-distribˡ a⁻ b⁻ c⁺ i ⟩
(a⁺ ·ⁿ (b⁺ +ⁿ c⁻)) +ⁿ (a⁻ ·ⁿ (b⁻ +ⁿ c⁺)) ≡[ i ]⟨ a⁺ ·ⁿ p i +ⁿ a⁻ ·ⁿ (ℕ.+-comm b⁻ c⁺ ∙ sym p ∙ ℕ.+-comm b⁺ c⁻) i ⟩
(a⁺ ·ⁿ (c⁺ +ⁿ b⁻)) +ⁿ (a⁻ ·ⁿ (c⁻ +ⁿ b⁺)) ≡[ i ]⟨ ℕ.·-distribˡ a⁺ c⁺ b⁻ (~ i) +ⁿ ℕ.·-distribˡ a⁻ c⁻ b⁺ (~ i) ⟩
(a⁺ ·ⁿ c⁺ +ⁿ a⁺ ·ⁿ b⁻) +ⁿ (a⁻ ·ⁿ c⁻ +ⁿ a⁻ ·ⁿ b⁺) ≡⟨ sym (lem0 (a⁺ ·ⁿ c⁺) (a⁻ ·ⁿ c⁻) (a⁺ ·ⁿ b⁻) (a⁻ ·ⁿ b⁺)) ⟩
(a⁺ ·ⁿ c⁺ +ⁿ a⁻ ·ⁿ c⁻) +ⁿ (a⁺ ·ⁿ b⁻ +ⁿ a⁻ ·ⁿ b⁺) ∎)
_·''_ : ℤ → ℤ → ℤ
_·''_ = rec2 {R = rel} {B = ℤ} isSetℤ _·'_ ·-respects-relʳ ·-respects-relˡ
-- normalization of isSetℤ explodes. Therefore we wrap this with expanded cases
_·_ : ℤ → ℤ → ℤ
x@([ _ ]) · y@([ _ ]) = x ·'' y
x@([ _ ]) · y@(eq/ _ _ _ _) = x ·'' y
x@(eq/ _ _ _ _) · y@([ _ ]) = x ·'' y
x@(eq/ _ _ _ _) · y@(eq/ _ _ _ _) = x ·'' y
x@(eq/ _ _ _ _) · y@(squash/ a b p q i j) = isSetℤ _ _ (cong (x ·_) p) (cong (x ·_) q) i j
x@([ _ ]) · y@(squash/ a b p q i j) = isSetℤ _ _ (cong (x ·_) p) (cong (x ·_) q) i j
x@(squash/ a b p q i j) · y = isSetℤ _ _ (cong (_· y) p) (cong (_· y) q) i j
+-identityʳ : (x : ℤ) → x + 0 ≡ x
+-identityʳ = elimProp {R = rel} (λ _ → isSetℤ _ _)
λ{ (a⁺ , a⁻) i → [ ℕ.+-comm a⁺ 0 i , ℕ.+-comm a⁻ 0 i ] }
+-comm : (x y : ℤ) → x + y ≡ y + x
+-comm = elimProp2 {R = rel} (λ _ _ → isSetℤ _ _)
λ{ (a⁺ , a⁻) (b⁺ , b⁻) i → [ ℕ.+-comm a⁺ b⁺ i , ℕ.+-comm a⁻ b⁻ i ] }
+-inverseʳ : (x : ℤ) → x + (- x) ≡ 0
+-inverseʳ = elimProp {R = rel} (λ _ → isSetℤ _ _)
λ{ (a⁺ , a⁻) → eq/ {R = rel} (a⁺ +ⁿ a⁻ , a⁻ +ⁿ a⁺) (0 , 0) (ℕ.+-zero (a⁺ +ⁿ a⁻) ∙ ℕ.+-comm a⁺ a⁻) }
+-assoc : (x y z : ℤ) → x + (y + z) ≡ x + y + z
+-assoc = elimProp3 {R = rel} (λ _ _ _ → isSetℤ _ _)
λ{ (a⁺ , a⁻) (b⁺ , b⁻) (c⁺ , c⁻) i → [ ℕ.+-assoc a⁺ b⁺ c⁺ i , ℕ.+-assoc a⁻ b⁻ c⁻ i ] }
·-identityʳ : (x : ℤ) → x · 1 ≡ x
·-identityʳ = elimProp {R = rel} (λ _ → isSetℤ _ _) γ
where
γ : (a : ℕ × ℕ) → _
γ (a⁺ , a⁻) i = [ p i , q i ]
where
p : a⁺ ·ⁿ 1 +ⁿ a⁻ ·ⁿ 0 ≡ a⁺
p i = ℕ.+-comm (ℕ.·-identityʳ a⁺ i) (ℕ.0≡m·0 a⁻ (~ i)) i
q : a⁺ ·ⁿ 0 +ⁿ a⁻ ·ⁿ 1 ≡ a⁻
q i = ℕ.0≡m·0 a⁺ (~ i) +ⁿ ℕ.·-identityʳ a⁻ i
·-comm : (x y : ℤ) → x · y ≡ y · x
·-comm = elimProp2 {R = rel} (λ _ _ → isSetℤ _ _)
λ{ (a⁺ , a⁻) (b⁺ , b⁻) i → [ ℕ.·-comm a⁺ b⁺ i +ⁿ ℕ.·-comm a⁻ b⁻ i , ℕ.+-comm (ℕ.·-comm a⁺ b⁻ i) (ℕ.·-comm a⁻ b⁺ i) i ] }
·-distribˡ : (x y z : ℤ) → x · (y + z) ≡ x · y + x · z
·-distribˡ = elimProp3 {R = rel} (λ _ _ _ → isSetℤ _ _)
λ{ (a⁺ , a⁻) (b⁺ , b⁻) (c⁺ , c⁻) →
[ a⁺ ·ⁿ (b⁺ +ⁿ c⁺) +ⁿ a⁻ ·ⁿ (b⁻ +ⁿ c⁻)
, a⁺ ·ⁿ (b⁻ +ⁿ c⁻) +ⁿ a⁻ ·ⁿ (b⁺ +ⁿ c⁺)
] ≡[ i ]⟨ [ ℕ.·-distribˡ a⁺ b⁺ c⁺ (~ i) +ⁿ ℕ.·-distribˡ a⁻ b⁻ c⁻ (~ i) , ℕ.·-distribˡ a⁺ b⁻ c⁻ (~ i) +ⁿ ℕ.·-distribˡ a⁻ b⁺ c⁺ (~ i) ] ⟩
[ (a⁺ ·ⁿ b⁺ +ⁿ a⁺ ·ⁿ c⁺) +ⁿ (a⁻ ·ⁿ b⁻ +ⁿ a⁻ ·ⁿ c⁻)
, (a⁺ ·ⁿ b⁻ +ⁿ a⁺ ·ⁿ c⁻) +ⁿ (a⁻ ·ⁿ b⁺ +ⁿ a⁻ ·ⁿ c⁺)
] ≡[ i ]⟨ [ lem0 (a⁺ ·ⁿ b⁺) (a⁻ ·ⁿ b⁻) (a⁺ ·ⁿ c⁺) (a⁻ ·ⁿ c⁻) (~ i), lem0 (a⁺ ·ⁿ b⁻) (a⁺ ·ⁿ c⁻) (a⁻ ·ⁿ b⁺) (a⁻ ·ⁿ c⁺) i ] ⟩
[ a⁺ ·ⁿ b⁺ +ⁿ a⁻ ·ⁿ b⁻ +ⁿ (a⁺ ·ⁿ c⁺ +ⁿ a⁻ ·ⁿ c⁻)
, a⁺ ·ⁿ b⁻ +ⁿ a⁻ ·ⁿ b⁺ +ⁿ (a⁺ ·ⁿ c⁻ +ⁿ a⁻ ·ⁿ c⁺)
] ∎
}
·-assoc : (x y z : ℤ) → x · (y · z) ≡ x · y · z
·-assoc = elimProp3 {R = rel} (λ _ _ _ → isSetℤ _ _)
λ{ (a⁺ , a⁻) (b⁺ , b⁻) (c⁺ , c⁻) →
[ a⁺ ·ⁿ (b⁺ ·ⁿ c⁺ +ⁿ b⁻ ·ⁿ c⁻) +ⁿ a⁻ ·ⁿ (b⁺ ·ⁿ c⁻ +ⁿ b⁻ ·ⁿ c⁺)
, a⁺ ·ⁿ (b⁺ ·ⁿ c⁻ +ⁿ b⁻ ·ⁿ c⁺) +ⁿ a⁻ ·ⁿ (b⁺ ·ⁿ c⁺ +ⁿ b⁻ ·ⁿ c⁻)
] ≡[ i ]⟨ [ ℕ.·-distribˡ a⁺ (b⁺ ·ⁿ c⁺) (b⁻ ·ⁿ c⁻) (~ i) +ⁿ ℕ.·-distribˡ a⁻ (b⁺ ·ⁿ c⁻) (b⁻ ·ⁿ c⁺) (~ i)
, ℕ.·-distribˡ a⁺ (b⁺ ·ⁿ c⁻) (b⁻ ·ⁿ c⁺) (~ i) +ⁿ ℕ.·-distribˡ a⁻ (b⁺ ·ⁿ c⁺) (b⁻ ·ⁿ c⁻) (~ i) ] ⟩
[ (a⁺ ·ⁿ (b⁺ ·ⁿ c⁺) +ⁿ a⁺ ·ⁿ (b⁻ ·ⁿ c⁻)) +ⁿ (a⁻ ·ⁿ (b⁺ ·ⁿ c⁻) +ⁿ a⁻ ·ⁿ (b⁻ ·ⁿ c⁺))
, (a⁺ ·ⁿ (b⁺ ·ⁿ c⁻) +ⁿ a⁺ ·ⁿ (b⁻ ·ⁿ c⁺)) +ⁿ (a⁻ ·ⁿ (b⁺ ·ⁿ c⁺) +ⁿ a⁻ ·ⁿ (b⁻ ·ⁿ c⁻))
] ≡[ i ]⟨ [ (ℕ.·-assoc a⁺ b⁺ c⁺ i +ⁿ ℕ.·-assoc a⁺ b⁻ c⁻ i) +ⁿ (ℕ.·-assoc a⁻ b⁺ c⁻ i +ⁿ ℕ.·-assoc a⁻ b⁻ c⁺ i)
, (ℕ.·-assoc a⁺ b⁺ c⁻ i +ⁿ ℕ.·-assoc a⁺ b⁻ c⁺ i) +ⁿ (ℕ.·-assoc a⁻ b⁺ c⁺ i +ⁿ ℕ.·-assoc a⁻ b⁻ c⁻ i) ] ⟩
[ (a⁺ ·ⁿ b⁺ ·ⁿ c⁺ +ⁿ a⁺ ·ⁿ b⁻ ·ⁿ c⁻) +ⁿ (a⁻ ·ⁿ b⁺ ·ⁿ c⁻ +ⁿ a⁻ ·ⁿ b⁻ ·ⁿ c⁺)
, (a⁺ ·ⁿ b⁺ ·ⁿ c⁻ +ⁿ a⁺ ·ⁿ b⁻ ·ⁿ c⁺) +ⁿ (a⁻ ·ⁿ b⁺ ·ⁿ c⁺ +ⁿ a⁻ ·ⁿ b⁻ ·ⁿ c⁻)
] ≡[ i ]⟨ [ lem1 (a⁺ ·ⁿ b⁺ ·ⁿ c⁺) (a⁺ ·ⁿ b⁻ ·ⁿ c⁻) (a⁻ ·ⁿ b⁺ ·ⁿ c⁻) (a⁻ ·ⁿ b⁻ ·ⁿ c⁺) i
, lem1 (a⁺ ·ⁿ b⁺ ·ⁿ c⁻) (a⁺ ·ⁿ b⁻ ·ⁿ c⁺) (a⁻ ·ⁿ b⁺ ·ⁿ c⁺) (a⁻ ·ⁿ b⁻ ·ⁿ c⁻) i ] ⟩
[ (a⁺ ·ⁿ b⁺ ·ⁿ c⁺ +ⁿ a⁻ ·ⁿ b⁻ ·ⁿ c⁺) +ⁿ (a⁺ ·ⁿ b⁻ ·ⁿ c⁻ +ⁿ a⁻ ·ⁿ b⁺ ·ⁿ c⁻)
, (a⁺ ·ⁿ b⁺ ·ⁿ c⁻ +ⁿ a⁻ ·ⁿ b⁻ ·ⁿ c⁻) +ⁿ (a⁺ ·ⁿ b⁻ ·ⁿ c⁺ +ⁿ a⁻ ·ⁿ b⁺ ·ⁿ c⁺)
] ≡[ i ]⟨ [ ℕ.·-distribʳ (a⁺ ·ⁿ b⁺) (a⁻ ·ⁿ b⁻) c⁺ i +ⁿ ℕ.·-distribʳ (a⁺ ·ⁿ b⁻) (a⁻ ·ⁿ b⁺) c⁻ i
, ℕ.·-distribʳ (a⁺ ·ⁿ b⁺) (a⁻ ·ⁿ b⁻) c⁻ i +ⁿ ℕ.·-distribʳ (a⁺ ·ⁿ b⁻) (a⁻ ·ⁿ b⁺) c⁺ i ] ⟩
[ (a⁺ ·ⁿ b⁺ +ⁿ a⁻ ·ⁿ b⁻) ·ⁿ c⁺ +ⁿ (a⁺ ·ⁿ b⁻ +ⁿ a⁻ ·ⁿ b⁺) ·ⁿ c⁻
, (a⁺ ·ⁿ b⁺ +ⁿ a⁻ ·ⁿ b⁻) ·ⁿ c⁻ +ⁿ (a⁺ ·ⁿ b⁻ +ⁿ a⁻ ·ⁿ b⁺) ·ⁿ c⁺
] ∎
}
private
_ : Dec→Bool (discreteℤ [ (3 , 5) ] [ (4 , 6) ]) ≡ true
_ = refl
_ : Dec→Bool (discreteℤ [ (3 , 5) ] [ (4 , 7) ]) ≡ false
_ = refl
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open import Class.Listable
open import Data.List.Relation.Unary.All
open import Data.List.Relation.Unary.AllPairs
open import Data.List.Relation.Unary.Any
import Data.Vec.Recursive
import Data.Vec.Recursive.Categorical
open import Prelude
module Theory.PrimMeta where
private
variable
A B C : Set
M : Set → Set
data PrimMeta : Set where
EvalStmt : PrimMeta
ShellCmd : PrimMeta
CheckTerm : PrimMeta
Parse : PrimMeta
Normalize : PrimMeta
HeadNormalize : PrimMeta
InferType : PrimMeta
private
variable
m : PrimMeta
instance
PrimMeta-Eq : Eq PrimMeta
PrimMeta-Eq = Listable.Listable→Eq record
{ listing = EvalStmt ∷ ShellCmd ∷ CheckTerm ∷ Parse ∷ Normalize ∷ HeadNormalize ∷ InferType ∷ []
; unique = ((λ ()) ∷ (λ ()) ∷ (λ ()) ∷ (λ ()) ∷ (λ ()) ∷ (λ ()) ∷ []) ∷
((λ ()) ∷ (λ ()) ∷ (λ ()) ∷ (λ ()) ∷ (λ ()) ∷ []) ∷
((λ ()) ∷ (λ ()) ∷ (λ ()) ∷ (λ ()) ∷ []) ∷
((λ ()) ∷ (λ ()) ∷ (λ ()) ∷ []) ∷
((λ ()) ∷ (λ ()) ∷ []) ∷ ((λ ()) ∷ []) ∷ [] ∷ []
; complete = λ where
EvalStmt → here refl
ShellCmd → there (here refl)
CheckTerm → there (there (here refl))
Parse → there (there (there (here refl)))
Normalize → there (there (there (there (here refl))))
HeadNormalize → there (there (there (there (there (here refl)))))
InferType → there (there (there (there (there (there (here refl)))))) }
PrimMeta-EqB : EqB PrimMeta
PrimMeta-EqB = Eq→EqB
PrimMeta-Show : Show PrimMeta
PrimMeta-Show = record { show = helper }
where
helper : PrimMeta → String
helper EvalStmt = "EvalStmt"
helper ShellCmd = "ShellCmd"
helper CheckTerm = "CheckTerm"
helper Parse = "Parse"
helper Normalize = "Normalize"
helper HeadNormalize = "HeadNormalize"
helper InferType = "InferType"
primMetaArity : PrimMeta → ℕ
primMetaArity EvalStmt = 1
primMetaArity ShellCmd = 2
primMetaArity CheckTerm = 2
primMetaArity Parse = 3
primMetaArity Normalize = 1
primMetaArity HeadNormalize = 1
primMetaArity InferType = 1
primMetaArgs : Set → PrimMeta → Set
primMetaArgs A m = A Data.Vec.Recursive.^ (primMetaArity m)
mapPrimMetaArgs : (A → B) → primMetaArgs A m → primMetaArgs B m
mapPrimMetaArgs f = Data.Vec.Recursive.map f _
traversePrimMetaArgs : {{Monad M}} → (A → M B) → primMetaArgs A m → M (primMetaArgs B m)
traversePrimMetaArgs {{mon}} = Data.Vec.Recursive.Categorical.mapM mon
primMetaArgs-Show : (A → String) → primMetaArgs A m → String
primMetaArgs-Show showA = Data.Vec.Recursive.foldr "" showA (λ _ a s → showA a + s) _
primMetaArgsZipWith : (A → B → C) → primMetaArgs A m → primMetaArgs B m → primMetaArgs C m
primMetaArgsZipWith f x y = Data.Vec.Recursive.zipWith f _ x y
primMetaArgsSequence : {{Monad M}} → primMetaArgs (M A) m → M (primMetaArgs A m)
primMetaArgsSequence {{mon}} = Data.Vec.Recursive.Categorical.sequenceM mon
primMetaArgsAnd : primMetaArgs Bool m → Bool
primMetaArgsAnd = Data.Vec.Recursive.foldr {P = const Bool} true id (const _∧_) _
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module UniDB.Spec where
open import UniDB.Core public
record Vr (T : STX) : Set where
field
vr : {γ : Dom} → Ix γ → T γ
vr-inj : {γ : Dom} → Inj (vr {γ})
open Vr {{...}} public
record Wk (X : STX) : Set where
field
wk₁ : {γ : Dom} (x : X γ) → X (suc γ)
wk : {γ : Dom} (δ : Dom) (x : X γ) → X (γ ∪ δ)
wk-zero : {γ : Dom} (x : X γ) → wk 0 x ≡ x
wk-suc : {γ : Dom} (δ : Dom) (x : X γ) → wk (suc δ) x ≡ wk₁ (wk δ x)
wk1-wk₁ : {γ : Dom} (x : X γ) →
wk 1 x ≡ wk₁ x
wk1-wk₁ x = trans (wk-suc 0 x) (cong wk₁ (wk-zero x))
open Wk {{...}} public
instance
iVrIx : Vr Ix
vr {{iVrIx}} i = i
vr-inj {{iVrIx}} p = p
iWkIx : Wk Ix
wk₁ {{iWkIx}} = suc
wk {{iWkIx}} zero i = i
wk {{iWkIx}} (suc δ) i = suc (wk δ i)
wk-zero {{iWkIx}} x = refl
wk-suc {{iWkIx}} δ x = refl
--------------------------------------------------------------------------------
record WkVr (T : STX) {{vrT : Vr T}} {{wkT : Wk T}} : Set where
field
wk₁-vr : {γ : Dom} (i : Ix γ) →
wk₁ (vr {T} i) ≡ vr {T} (wk₁ i)
wk-vr : {γ : Dom} (δ : Dom) (i : Ix γ) →
wk δ (vr {T} i) ≡ vr {T} (wk δ i)
open WkVr {{...}} public
instance
iWkVrIx : WkVr Ix
wk₁-vr {{iWkVrIx}} i = refl
wk-vr {{iWkVrIx}} δ i = refl
--------------------------------------------------------------------------------
record Lk (T : STX) (Ξ : MOR) : Set where
field
lk : {γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) (i : Ix γ₁) → T γ₂
open Lk {{...}} public
record Up (Ξ : MOR) : Set where
infixl 9 _↑₁ _↑_
field
_↑₁ : {γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) → Ξ (suc γ₁) (suc γ₂)
_↑_ : {γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) (δ : Dom) → Ξ (γ₁ ∪ δ) (γ₂ ∪ δ)
↑-zero : {γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) → ξ ↑ zero ≡ ξ
↑-suc : {γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) (δ : Dom) → ξ ↑ suc δ ≡ ξ ↑ δ ↑₁
record Comp (Ξ : MOR) : Set where
infixl 8 _⊙_
field
_⊙_ : {γ₁ γ₂ γ₃ : Dom} (ξ₁ : Ξ γ₁ γ₂) (ξ₂ : Ξ γ₂ γ₃) → Ξ γ₁ γ₃
record Idm (Ξ : MOR) : Set where
field
idm : (γ : Dom) → Ξ γ γ
record Wkm (Ξ : MOR) : Set where
field
wkm : {γ : Dom} (δ : Dom) → Ξ γ (γ ∪ δ)
record Snoc (T : STX) (Ξ : MOR) : Set where
field
snoc : {γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) (t : T γ₂) → Ξ (suc γ₁) γ₂
record Beta (T : STX) (Ξ : MOR) : Set where
field
beta : {γ : Dom} (t : T γ) → Ξ (suc γ) γ
open Idm {{...}} public
open Wkm {{...}} public
open Up {{...}} public
open Comp {{...}} public
open Snoc {{...}} public
open Beta {{...}} public
--------------------------------------------------------------------------------
record LkRen
(T : STX) {{vrT : Vr T}}
(Ξ : MOR) {{lkIxΞ : Lk Ix Ξ}} {{lkTΞ : Lk T Ξ}} : Set where
field
lk-ren : {γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) (i : Ix γ₁) →
lk {T} ξ i ≡ vr (lk ξ i)
open LkRen {{...}} public
record LkUp
(T : STX) {{vrT : Vr T}} {{wkT : Wk T}}
(Ξ : MOR) {{lkTΞ : Lk T Ξ}} {{upΞ : Up Ξ}} : Set where
field
lk-↑₁-zero : {γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) →
lk {T} (ξ ↑₁) zero ≡ vr zero
lk-↑₁-suc : {γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) (i : Ix γ₁) →
lk {T} (ξ ↑₁) (suc i) ≡ wk₁ (lk ξ i)
lk-↑-∪ : {γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) (i : Ix γ₁) (δ : Dom) →
lk {T} (ξ ↑ δ) (wk δ i) ≡ wk δ (lk ξ i)
lk-↑-∪ ξ i zero = begin
lk (ξ ↑ 0) i ≡⟨ cong₂ lk ((↑-zero ξ)) refl ⟩
lk ξ i ≡⟨ sym (wk-zero (lk ξ i)) ⟩
wk 0 (lk ξ i) ∎
lk-↑-∪ ξ i (suc δ) = begin
lk (ξ ↑ suc δ) (suc (wk δ i)) ≡⟨ cong₂ lk (↑-suc ξ δ) refl ⟩
lk (ξ ↑ δ ↑₁) (suc (wk δ i)) ≡⟨ lk-↑₁-suc (ξ ↑ δ) (wk δ i) ⟩
wk₁ (lk (ξ ↑ δ) (wk δ i)) ≡⟨ cong wk₁ (lk-↑-∪ ξ i δ) ⟩
wk₁ (wk δ (lk ξ i)) ≡⟨ sym (wk-suc δ (lk ξ i)) ⟩
wk (suc δ) (lk ξ i) ∎
open LkUp {{...}} public
record LkRenComp
(T : STX) {{vrT : Vr T}}
(Ξ : MOR) {{lkIxΞ : Lk Ix Ξ}} {{lkTΞ : Lk T Ξ}} {{compΞ : Comp Ξ}}
{{lkRenTΞ : LkRen T Ξ}} : Set where
field
lk-ren-comp : {γ₁ γ₂ γ₃ : Dom} (ξ₁ : Ξ γ₁ γ₂) (ξ₂ : Ξ γ₂ γ₃)
(i : Ix γ₁) → lk {T} (ξ₁ ⊙ ξ₂) i ≡ lk {T} ξ₂ (lk {Ix} ξ₁ i)
open LkRenComp {{...}} public
record LkIdm
(T : STX) {{vrT : Vr T}}
(Ξ : MOR) {{lkTΞ : Lk T Ξ}} {{idmΞ : Idm Ξ}} : Set where
field
lk-idm : {γ : Dom} (i : Ix γ) → lk {T} (idm {Ξ} γ) i ≡ vr i
open LkIdm {{...}} public
record LkWkm
(T : STX) {{vrT : Vr T}}
(Ξ : MOR) {{lkTΞ : Lk T Ξ}} {{wkmΞ : Wkm Ξ}} : Set where
field
lk-wkm : {γ : Dom} (δ : Dom) (i : Ix γ) →
lk {T} (wkm {Ξ} δ) i ≡ vr (wk δ i)
open LkWkm {{...}} public
--------------------------------------------------------------------------------
record UpIdm (Ξ : MOR) {{upΞ : Up Ξ}} {{idmΞ : Idm Ξ}} : Set where
field
idm-↑₁ : {γ : Dom} → idm {Ξ} γ ↑₁ ≡ idm {Ξ} (suc γ)
idm-↑ : {γ : Dom} (δ : Dom) → idm {Ξ} γ ↑ δ ≡ idm {Ξ} (γ ∪ δ)
idm-↑ zero = ↑-zero (idm {Ξ} _)
idm-↑ (suc δ) = begin
idm _ ↑ suc δ ≡⟨ ↑-suc (idm _) δ ⟩
idm _ ↑ δ ↑₁ ≡⟨ cong _↑₁ (idm-↑ δ) ⟩
idm _ ↑₁ ≡⟨ idm-↑₁ ⟩
idm (suc (_ ∪ δ)) ∎
open UpIdm {{...}} public
record UpComp (Ξ : MOR) {{upΞ : Up Ξ}} {{compΞ : Comp Ξ}} : Set where
field
⊙-↑₁ : {γ₁ γ₂ γ₃ : Dom} (ξ₁ : Ξ γ₁ γ₂) (ξ₂ : Ξ γ₂ γ₃) →
(ξ₁ ⊙ ξ₂) ↑₁ ≡ (ξ₁ ↑₁) ⊙ (ξ₂ ↑₁)
⊙-↑ : {γ₁ γ₂ γ₃ : Dom} (ξ₁ : Ξ γ₁ γ₂) (ξ₂ : Ξ γ₂ γ₃) (δ : Dom) →
(ξ₁ ⊙ ξ₂) ↑ δ ≡ (ξ₁ ↑ δ) ⊙ (ξ₂ ↑ δ)
⊙-↑ ξ₁ ξ₂ zero = begin
(ξ₁ ⊙ ξ₂) ↑ 0 ≡⟨ ↑-zero (ξ₁ ⊙ ξ₂) ⟩
ξ₁ ⊙ ξ₂ ≡⟨ sym (cong₂ _⊙_ (↑-zero ξ₁) (↑-zero ξ₂)) ⟩
(ξ₁ ↑ 0) ⊙ (ξ₂ ↑ 0) ∎
⊙-↑ ξ₁ ξ₂ (suc δ) = begin
(ξ₁ ⊙ ξ₂) ↑ suc δ ≡⟨ ↑-suc (ξ₁ ⊙ ξ₂) δ ⟩
(ξ₁ ⊙ ξ₂) ↑ δ ↑₁ ≡⟨ cong _↑₁ (⊙-↑ ξ₁ ξ₂ δ) ⟩
((ξ₁ ↑ δ) ⊙ (ξ₂ ↑ δ)) ↑₁ ≡⟨ ⊙-↑₁ (ξ₁ ↑ δ) (ξ₂ ↑ δ) ⟩
(ξ₁ ↑ δ ↑₁) ⊙ (ξ₂ ↑ δ ↑₁) ≡⟨ sym (cong₂ _⊙_ (↑-suc ξ₁ δ) (↑-suc ξ₂ δ)) ⟩
(ξ₁ ↑ suc δ) ⊙ (ξ₂ ↑ suc δ) ∎
open UpComp {{...}} public
record CompIdm (Ξ : MOR) {{idmΞ : Idm Ξ}} {{compΞ : Comp Ξ}} : Set where
field
⊙-idm : {γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) → ξ ⊙ idm {Ξ} γ₂ ≡ ξ
idm-⊙ : {γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) → idm {Ξ} γ₁ ⊙ ξ ≡ ξ
open CompIdm {{...}} public
record CompAssoc (Ξ : MOR) {{compΞ : Comp Ξ}} : Set where
field
⊙-assoc : {γ₁ γ₂ γ₃ γ₄ : Dom} (ξ₁ : Ξ γ₁ γ₂) (ξ₂ : Ξ γ₂ γ₃) (ξ₃ : Ξ γ₃ γ₄) →
ξ₁ ⊙ (ξ₂ ⊙ ξ₃) ≡ (ξ₁ ⊙ ξ₂) ⊙ ξ₃
open CompAssoc {{...}} public
--------------------------------------------------------------------------------
record WkmBeta
(T : STX) (Ξ : MOR) {{idmΞ : Idm Ξ}} {{wkmΞ : Wkm Ξ}}
{{compΞ : Comp Ξ}} {{betaTΞ : Beta T Ξ}} : Set where
field
wkm-beta : {γ : Dom} (t : T γ) → wkm {Ξ} 1 ⊙ beta {T} {Ξ} t ≡ idm {Ξ} γ
open WkmBeta {{...}} public
record WkmHom
(Ξ : MOR) {{idmΞ : Idm Ξ}} {{wkmΞ : Wkm Ξ}} {{compΞ : Comp Ξ}} : Set where
field
wkm-zero : {γ : Dom} → wkm {Ξ} 0 ≡ idm {Ξ} γ
wkm-suc : {γ : Dom} (δ : Dom) → wkm {Ξ} {γ} (suc δ) ≡ wkm {Ξ} δ ⊙ wkm {Ξ} 1
open WkmHom {{...}} public
record WkmComm
(Ξ : MOR) {{upΞ : Up Ξ}} {{wkmΞ : Wkm Ξ}} {{compΞ : Comp Ξ}} : Set where
field
wkm₁-comm : {γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) →
ξ ⊙ wkm {Ξ} 1 ≡ wkm {Ξ} 1 ⊙ ξ ↑₁
wkm-comm : {{idmΞ : Idm Ξ}} {{wkmHomΞ : WkmHom Ξ}}
{{compIdmΞ : CompIdm Ξ}} {{compAssocΞ : CompAssoc Ξ}}
{γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) (δ : Dom) →
ξ ⊙ wkm {Ξ} δ ≡ wkm {Ξ} δ ⊙ ξ ↑ δ
wkm-comm {{iWkmCommReg}} ξ zero = begin
ξ ⊙ wkm 0 ≡⟨ cong (_⊙_ ξ) wkm-zero ⟩
ξ ⊙ idm _ ≡⟨ ⊙-idm ξ ⟩
ξ ≡⟨ sym (idm-⊙ ξ) ⟩
idm _ ⊙ ξ ≡⟨ sym (cong₂ _⊙_ wkm-zero (↑-zero ξ)) ⟩
wkm 0 ⊙ (ξ ↑ 0) ∎
wkm-comm {{iWkmCommReg}} ξ (suc δ) = begin
ξ ⊙ wkm (suc δ) ≡⟨ cong (_⊙_ ξ) (wkm-suc δ) ⟩
ξ ⊙ (wkm δ ⊙ wkm 1) ≡⟨ ⊙-assoc ξ (wkm δ) (wkm 1) ⟩
(ξ ⊙ wkm δ) ⊙ wkm 1 ≡⟨ cong (λ ξ → ξ ⊙ wkm 1) (wkm-comm ξ δ) ⟩
(wkm δ ⊙ (ξ ↑ δ)) ⊙ wkm 1 ≡⟨ sym (⊙-assoc (wkm δ) (ξ ↑ δ) (wkm 1)) ⟩
wkm δ ⊙ ((ξ ↑ δ) ⊙ wkm 1) ≡⟨ cong (_⊙_ (wkm δ)) (wkm₁-comm (ξ ↑ δ)) ⟩
wkm δ ⊙ (wkm 1 ⊙ (ξ ↑ δ ↑₁)) ≡⟨ ⊙-assoc (wkm δ) (wkm 1) (ξ ↑ δ ↑₁) ⟩
(wkm δ ⊙ wkm 1) ⊙ (ξ ↑ δ ↑₁) ≡⟨ cong (λ ρ → (wkm δ ⊙ wkm 1) ⊙ ρ) (sym (↑-suc ξ δ)) ⟩
(wkm δ ⊙ wkm 1) ⊙ (ξ ↑ suc δ) ≡⟨ cong (λ ρ → ρ ⊙ (ξ ↑ suc δ)) (sym (wkm-suc δ)) ⟩
wkm (suc δ) ⊙ (ξ ↑ suc δ) ∎
open WkmComm {{...}} public
--------------------------------------------------------------------------------
infix 1 [_]_≃_
record [_]_≃_
(T : STX)
{Ξ : MOR} {{lkTΞ : Lk T Ξ}}
{Ζ : MOR} {{lkTΖ : Lk T Ζ}}
{γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) (ζ : Ζ γ₁ γ₂) : Set where
field
lk≃ : (i : Ix γ₁) → lk {T} {Ξ} ξ i ≡ lk {T} {Ζ} ζ i
open [_]_≃_ public
module _
{T : STX} {{vrT : Vr T}} {{wkT : Wk T}}
{Ξ : MOR} {{lkTΞ : Lk T Ξ}} {{upΞ : Up Ξ}} {{lkUpTΞ : LkUp T Ξ}}
{Ζ : MOR} {{lkTΖ : Lk T Ζ}} {{upΖ : Up Ζ}} {{lkUpTΖ : LkUp T Ζ}}
where
≃-↑₁ : {γ₁ γ₂ : Dom} {ξ : Ξ γ₁ γ₂} {ζ : Ζ γ₁ γ₂}
(hyp : [ T ] ξ ≃ ζ) → [ T ] ξ ↑₁ ≃ ζ ↑₁
lk≃ (≃-↑₁ {γ₁} {γ₂} {ξ} {ζ} hyp) zero = begin
lk (ξ ↑₁) zero ≡⟨ lk-↑₁-zero ξ ⟩
vr zero ≡⟨ sym (lk-↑₁-zero ζ) ⟩
lk (ζ ↑₁) zero ∎
lk≃ (≃-↑₁ {γ₁} {γ₂} {ξ} {ζ} hyp) (suc i) = begin
lk (ξ ↑₁) (suc i) ≡⟨ lk-↑₁-suc ξ i ⟩
wk₁ (lk ξ i) ≡⟨ cong wk₁ (lk≃ hyp i) ⟩
wk₁ (lk ζ i) ≡⟨ sym (lk-↑₁-suc ζ i) ⟩
lk (ζ ↑₁) (suc i) ∎
≃-↑ : {γ₁ γ₂ : Dom} {ξ : Ξ γ₁ γ₂} {ζ : Ζ γ₁ γ₂}
(hyp : [ T ] ξ ≃ ζ) (δ : Dom) → [ T ] ξ ↑ δ ≃ ζ ↑ δ
≃-↑ {ξ = ξ} {ζ} hyp zero
rewrite ↑-zero {Ξ} ξ | ↑-zero {Ζ} ζ = hyp
≃-↑ {ξ = ξ} {ζ} hyp (suc δ)
rewrite ↑-suc {Ξ} ξ δ | ↑-suc {Ζ} ζ δ = ≃-↑₁ (≃-↑ hyp δ)
--------------------------------------------------------------------------------
infix 1 [_]_≅_
data [_]_≅_ (T : STX)
{Ξ : MOR} {{lkTΞ : Lk T Ξ}} {{upΞ : Up Ξ}}
{Ζ : MOR} {{lkTΖ : Lk T Ζ}} {{upΖ : Up Ζ}} :
{γ₁ γ₂ : Dom} (ξ : Ξ γ₁ γ₂) (ζ : Ζ γ₁ γ₂) → Set where
≃-to-≅ : {γ₁ γ₂ : Dom} {ξ : Ξ γ₁ γ₂} {ζ : Ζ γ₁ γ₂}
(hyp : (δ' : Dom) → [ T ] ξ ↑ δ' ≃ ζ ↑ δ') (δ : Dom) →
[ T ] ξ ↑ δ ≅ ζ ↑ δ
module _
{T : STX}
{Ξ : MOR} {{lkTΞ : Lk T Ξ}} {{upΞ : Up Ξ}}
{Ζ : MOR} {{lkTΖ : Lk T Ζ}} {{upΖ : Up Ζ}} where
≃-to-≅` : {γ₁ γ₂ : Dom} {ξ : Ξ γ₁ γ₂} {ζ : Ζ γ₁ γ₂}
(hyp : (δ' : Dom) → [ T ] ξ ↑ δ' ≃ ζ ↑ δ') → [ T ] ξ ≅ ζ
≃-to-≅` {ξ = ξ} {ζ} hyp = lem₂
where
lem : [ T ] ξ ↑ 0 ≅ ζ ↑ 0
lem = ≃-to-≅ hyp 0
lem₂ : [ T ] ξ ≅ ζ
lem₂ rewrite sym (↑-zero {Ξ} ξ) | sym (↑-zero {Ζ} ζ) = lem
≅-to-≃ : {γ₁ γ₂ : Dom} {ξ : Ξ γ₁ γ₂} {ζ : Ζ γ₁ γ₂}
(hyp : [ T ] ξ ≅ ζ) → [ T ] ξ ≃ ζ
≅-to-≃ (≃-to-≅ hyp δ) = hyp δ
≅-↑₁ : {γ₁ γ₂ : Dom} {ξ : Ξ γ₁ γ₂} {ζ : Ζ γ₁ γ₂}
(hyp : [ T ] ξ ≅ ζ) → [ T ] ξ ↑₁ ≅ ζ ↑₁
≅-↑₁ (≃-to-≅ {ξ = ξ} {ζ} hyp δ)
rewrite sym (↑-suc {Ξ} ξ δ) | sym (↑-suc {Ζ} ζ δ)
= ≃-to-≅ hyp (suc δ)
≅-↑ : {γ₁ γ₂ : Dom} {ξ : Ξ γ₁ γ₂} {ζ : Ζ γ₁ γ₂}
(hyp : [ T ] ξ ≅ ζ) (δ : Dom) → [ T ] ξ ↑ δ ≅ ζ ↑ δ
≅-↑ {ξ = ξ} {ζ} hyp zero
rewrite ↑-zero ξ | ↑-zero ζ
= hyp
≅-↑ {ξ = ξ} {ζ} hyp (suc δ)
rewrite ↑-suc ξ δ | ↑-suc ζ δ
= ≅-↑₁ (≅-↑ hyp δ)
module _
{T : STX} {{vrT : Vr T}}
{Ξ : MOR} {{lkIxΞ : Lk Ix Ξ}} {{lkTΞ : Lk T Ξ}} {{lkRenTΞ : LkRen T Ξ}} {{upΞ : Up Ξ}}
{Ζ : MOR} {{lkIxΖ : Lk Ix Ζ}} {{lkTΖ : Lk T Ζ}} {{lkRenTΖ : LkRen T Ζ}} {{upΖ : Up Ζ}} where
Ix≅-to-≅ : {γ₁ γ₂ : Dom} {ξ : Ξ γ₁ γ₂} {ζ : Ζ γ₁ γ₂} → [ Ix ] ξ ≅ ζ → [ T ] ξ ≅ ζ
Ix≅-to-≅ (≃-to-≅ {ξ = ξ} {ζ} hyp δ) = ≃-to-≅ (λ δ' → record { lk≃ = λ i → begin
lk {T} (ξ ↑ δ') i ≡⟨ lk-ren {T} {Ξ} (ξ ↑ δ') i ⟩
vr (lk {Ix} (ξ ↑ δ') i) ≡⟨ cong vr (lk≃ (hyp δ') i) ⟩
vr (lk {Ix} (ζ ↑ δ') i) ≡⟨ sym (lk-ren {T} {Ζ} (ζ ↑ δ') i) ⟩
lk {T} (ζ ↑ δ') i ∎}) δ
--------------------------------------------------------------------------------
record HComp (Ξ Ζ Θ : MOR) : Set where
infixl 8 _⊡_
field
_⊡_ : {γ₁ γ₂ γ₃ : Dom} (ξ : Ξ γ₁ γ₂) (ζ : Ζ γ₂ γ₃) → Θ γ₁ γ₃
open HComp {{...}} public
record UpHComp
(Ξ : MOR) {{upΞ : Up Ξ}}
(Ζ : MOR) {{upΖ : Up Ζ}}
(Θ : MOR) {{upΘ : Up Θ}}
{{hcompΞΖΘ : HComp Ξ Ζ Θ}} : Set where
field
⊡-↑₁ : {γ₁ γ₂ γ₃ : Dom} (ξ : Ξ γ₁ γ₂) (ζ : Ζ γ₂ γ₃) →
(_⊡_ {Θ = Θ} ξ ζ) ↑₁ ≡ (ξ ↑₁) ⊡ (ζ ↑₁)
⊡-↑ : {γ₁ γ₂ γ₃ : Dom} (ξ : Ξ γ₁ γ₂) (ζ : Ζ γ₂ γ₃) (δ : Dom) →
_↑_ {Θ} (ξ ⊡ ζ) δ ≡ (ξ ↑ δ) ⊡ (ζ ↑ δ)
⊡-↑ ξ ζ zero = begin
(ξ ⊡ ζ) ↑ 0 ≡⟨ ↑-zero {Θ} (_⊡_ {Ξ} {Ζ} {Θ} ξ ζ) ⟩
(ξ ⊡ ζ) ≡⟨ sym (cong₂ (_⊡_ {Ξ} {Ζ} {Θ}) (↑-zero ξ) (↑-zero ζ)) ⟩
(ξ ↑ 0) ⊡ (ζ ↑ 0) ∎
⊡-↑ ξ ζ (suc δ) = begin
(ξ ⊡ ζ) ↑ suc δ ≡⟨ ↑-suc (ξ ⊡ ζ) δ ⟩
(ξ ⊡ ζ) ↑ δ ↑₁ ≡⟨ cong _↑₁ ((⊡-↑ ξ ζ δ)) ⟩
(ξ ↑ δ ⊡ ζ ↑ δ) ↑₁ ≡⟨ ⊡-↑₁ (ξ ↑ δ) (ζ ↑ δ) ⟩
ξ ↑ δ ↑₁ ⊡ ζ ↑ δ ↑₁ ≡⟨ sym (cong₂ (_⊡_ {Ξ} {Ζ} {Θ}) (↑-suc ξ δ) (↑-suc ζ δ)) ⟩
ξ ↑ suc δ ⊡ ζ ↑ suc δ ∎
open UpHComp {{...}} public
record HCompIdmLeft
(Ξ : MOR) {{idmΞ : Idm Ξ}}
(Ζ : MOR) {{hcompΞΖΖ : HComp Ξ Ζ Ζ}} : Set where
field
idm-⊡ : {γ₁ γ₂ : Dom} (ζ : Ζ γ₁ γ₂) → idm {Ξ} γ₁ ⊡ ζ ≡ ζ
open HCompIdmLeft {{...}} public
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{-# OPTIONS --omega-in-omega --no-termination-check --overlapping-instances #-}
module Light.Implementation.Data.These where
open import Light.Library.Data.These using (Library ; Dependencies)
open import Light.Variable.Levels
open import Light.Level using (_⊔_)
dependencies : Dependencies
dependencies = record {}
instance library : Library dependencies
library = record { Implementation }
where
module Implementation where
data These (𝕒 : Set aℓ) (𝕓 : Set bℓ) : Set (aℓ ⊔ bℓ) where
this : 𝕒 → These 𝕒 𝕓
that : 𝕓 → These 𝕒 𝕓
these : 𝕒 → 𝕓 → These 𝕒 𝕓
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------------------------------------------------------------------------
-- 1-categories
------------------------------------------------------------------------
-- The code is based on the presentation in the HoTT book (but might
-- not follow it exactly).
{-# OPTIONS --without-K --safe #-}
open import Equality
module Category
{reflexive} (eq : ∀ {a p} → Equality-with-J a p reflexive) where
open import Bijection eq as Bijection using (_↔_)
open Derived-definitions-and-properties eq
open import Equivalence eq as Eq
using (_≃_; ⟨_,_⟩; module _≃_; Is-equivalence)
open import Function-universe eq as F hiding (id) renaming (_∘_ to _⊚_)
open import H-level eq
open import H-level.Closure eq
open import Logical-equivalence using (module _⇔_)
import Nat eq as Nat
open import Prelude as P hiding (id; Unit)
open import Univalence-axiom eq
------------------------------------------------------------------------
-- Precategories
-- This definition of precategories takes the type of objects as a
-- parameter.
Precategory-with-Obj :
∀ {ℓ₁} → Type ℓ₁ → (ℓ₂ : Level) → Type (ℓ₁ ⊔ lsuc ℓ₂)
Precategory-with-Obj Obj ℓ₂ =
-- Morphisms (a /set/).
∃ λ (HOM : Obj → Obj → Set ℓ₂) →
let Hom = λ X Y → proj₁ (HOM X Y) in
-- Identity.
∃ λ (id : ∀ {X} → Hom X X) →
-- Composition.
∃ λ (_∙_ : ∀ {X Y Z} → Hom Y Z → Hom X Y → Hom X Z) →
-- Identity laws.
(∀ {X Y} {f : Hom X Y} → (id ∙ f) ≡ f) ×
(∀ {X Y} {f : Hom X Y} → (f ∙ id) ≡ f) ×
-- Associativity.
(∀ {X Y Z U} {f : Hom X Y} {g : Hom Y Z} {h : Hom Z U} →
(h ∙ (g ∙ f)) ≡ ((h ∙ g) ∙ f))
-- Precategories.
Precategory′ : (ℓ₁ ℓ₂ : Level) → Type (lsuc (ℓ₁ ⊔ ℓ₂))
Precategory′ ℓ₁ ℓ₂ =
-- Objects.
∃ λ (Obj : Type ℓ₁) →
Precategory-with-Obj Obj ℓ₂
-- A wrapper.
record Precategory (ℓ₁ ℓ₂ : Level) : Type (lsuc (ℓ₁ ⊔ ℓ₂)) where
field
precategory : Precategory′ ℓ₁ ℓ₂
-- Objects.
Obj : Type ℓ₁
Obj = proj₁ precategory
-- Morphisms.
HOM : Obj → Obj → Set ℓ₂
HOM = proj₁ (proj₂ precategory)
-- The morphism type family.
Hom : Obj → Obj → Type ℓ₂
Hom X Y = proj₁ (HOM X Y)
-- The morphism types are sets.
Hom-is-set : ∀ {X Y} → Is-set (Hom X Y)
Hom-is-set = proj₂ (HOM _ _)
-- Identity.
id : ∀ {X} → Hom X X
id = proj₁ (proj₂ (proj₂ precategory))
-- Composition.
infixr 10 _∙_
_∙_ : ∀ {X Y Z} → Hom Y Z → Hom X Y → Hom X Z
_∙_ = proj₁ (proj₂ (proj₂ (proj₂ precategory)))
-- The left identity law.
left-identity : ∀ {X Y} {f : Hom X Y} → id ∙ f ≡ f
left-identity = proj₁ (proj₂ (proj₂ (proj₂ (proj₂ precategory))))
-- The right identity law.
right-identity : ∀ {X Y} {f : Hom X Y} → f ∙ id ≡ f
right-identity =
proj₁ (proj₂ (proj₂ (proj₂ (proj₂ (proj₂ precategory)))))
-- The associativity law.
assoc : ∀ {X Y Z U} {f : Hom X Y} {g : Hom Y Z} {h : Hom Z U} →
h ∙ (g ∙ f) ≡ (h ∙ g) ∙ f
assoc =
proj₂ (proj₂ (proj₂ (proj₂ (proj₂ (proj₂ precategory)))))
-- Isomorphisms.
Is-isomorphism : ∀ {X Y} → Hom X Y → Type ℓ₂
Is-isomorphism f = ∃ λ g → (f ∙ g ≡ id) × (g ∙ f ≡ id)
infix 4 _≅_
_≅_ : Obj → Obj → Type ℓ₂
X ≅ Y = ∃ λ (f : Hom X Y) → Is-isomorphism f
-- Some projections.
infix 15 _¹ _⁻¹ _¹⁻¹ _⁻¹¹
_¹ : ∀ {X Y} → X ≅ Y → Hom X Y
f ¹ = proj₁ f
_⁻¹ : ∀ {X Y} → X ≅ Y → Hom Y X
f ⁻¹ = proj₁ (proj₂ f)
_¹⁻¹ : ∀ {X Y} (f : X ≅ Y) → f ¹ ∙ f ⁻¹ ≡ id
f ¹⁻¹ = proj₁ (proj₂ (proj₂ f))
_⁻¹¹ : ∀ {X Y} (f : X ≅ Y) → f ⁻¹ ∙ f ¹ ≡ id
f ⁻¹¹ = proj₂ (proj₂ (proj₂ f))
abstract
-- "Is-isomorphism f" is a proposition.
Is-isomorphism-propositional :
∀ {X Y} (f : Hom X Y) →
Is-proposition (Is-isomorphism f)
Is-isomorphism-propositional f (g , fg , gf) (g′ , fg′ , g′f) =
Σ-≡,≡→≡ (g ≡⟨ sym left-identity ⟩
id ∙ g ≡⟨ cong (λ h → h ∙ g) $ sym g′f ⟩
(g′ ∙ f) ∙ g ≡⟨ sym assoc ⟩
g′ ∙ (f ∙ g) ≡⟨ cong (_∙_ g′) fg ⟩
g′ ∙ id ≡⟨ right-identity ⟩∎
g′ ∎)
(Σ-≡,≡→≡ (Hom-is-set _ _) (Hom-is-set _ _))
-- Isomorphism equality is equivalent to "forward morphism"
-- equality.
≡≃≡¹ : ∀ {X Y} {f g : X ≅ Y} → (f ≡ g) ≃ (f ¹ ≡ g ¹)
≡≃≡¹ {f = f} {g} =
(f ≡ g) ↔⟨ inverse $ ignore-propositional-component $ Is-isomorphism-propositional _ ⟩□
(f ¹ ≡ g ¹) □
-- The type of isomorphisms (between two objects) is a set.
≅-set : ∀ {X Y} → Is-set (X ≅ Y)
≅-set = Σ-closure 2 Hom-is-set
(λ _ → mono₁ 1 $ Is-isomorphism-propositional _)
-- Identity isomorphism.
id≅ : ∀ {X} → X ≅ X
id≅ = id , id , left-identity , right-identity
-- Composition of isomorphisms.
infixr 10 _∙≅_
_∙≅_ : ∀ {X Y Z} → Y ≅ Z → X ≅ Y → X ≅ Z
f ∙≅ g = (f ¹ ∙ g ¹) , (g ⁻¹ ∙ f ⁻¹) , fg f g , gf f g
where
abstract
fg : ∀ {X Y Z} (f : Y ≅ Z) (g : X ≅ Y) →
(f ¹ ∙ g ¹) ∙ (g ⁻¹ ∙ f ⁻¹) ≡ id
fg f g =
(f ¹ ∙ g ¹) ∙ (g ⁻¹ ∙ f ⁻¹) ≡⟨ sym assoc ⟩
f ¹ ∙ (g ¹ ∙ (g ⁻¹ ∙ f ⁻¹)) ≡⟨ cong (_∙_ (f ¹)) assoc ⟩
f ¹ ∙ ((g ¹ ∙ g ⁻¹) ∙ f ⁻¹) ≡⟨ cong (λ h → f ¹ ∙ (h ∙ f ⁻¹)) $ g ¹⁻¹ ⟩
f ¹ ∙ (id ∙ f ⁻¹) ≡⟨ cong (_∙_ (f ¹)) left-identity ⟩
f ¹ ∙ f ⁻¹ ≡⟨ f ¹⁻¹ ⟩∎
id ∎
gf : ∀ {X Y Z} (f : Y ≅ Z) (g : X ≅ Y) →
(g ⁻¹ ∙ f ⁻¹) ∙ (f ¹ ∙ g ¹) ≡ id
gf f g =
(g ⁻¹ ∙ f ⁻¹) ∙ (f ¹ ∙ g ¹) ≡⟨ sym assoc ⟩
g ⁻¹ ∙ (f ⁻¹ ∙ (f ¹ ∙ g ¹)) ≡⟨ cong (_∙_ (g ⁻¹)) assoc ⟩
g ⁻¹ ∙ ((f ⁻¹ ∙ f ¹) ∙ g ¹) ≡⟨ cong (λ h → g ⁻¹ ∙ (h ∙ g ¹)) $ f ⁻¹¹ ⟩
g ⁻¹ ∙ (id ∙ g ¹) ≡⟨ cong (_∙_ (g ⁻¹)) left-identity ⟩
g ⁻¹ ∙ g ¹ ≡⟨ g ⁻¹¹ ⟩∎
id ∎
-- The inverse of an isomorphism.
infix 15 _⁻¹≅
_⁻¹≅ : ∀ {X Y} → X ≅ Y → Y ≅ X
f ⁻¹≅ = f ⁻¹ , f ¹ , f ⁻¹¹ , f ¹⁻¹
-- Isomorphisms form a precategory.
precategory-≅ : Precategory ℓ₁ ℓ₂
precategory-≅ = record { precategory =
Obj ,
(λ X Y → (X ≅ Y) , ≅-set) ,
id≅ , _∙≅_ ,
_≃_.from ≡≃≡¹ left-identity ,
_≃_.from ≡≃≡¹ right-identity ,
_≃_.from ≡≃≡¹ assoc }
-- Equal objects are isomorphic.
≡→≅ : ∀ {X Y} → X ≡ Y → X ≅ Y
≡→≅ = elim (λ {X Y} _ → X ≅ Y) (λ _ → id≅)
-- "Computation rule" for ≡→≅.
≡→≅-refl : ∀ {X} → ≡→≅ (refl X) ≡ id≅
≡→≅-refl = elim-refl (λ {X Y} _ → X ≅ Y) _
-- Rearrangement lemma for ≡→≅.
≡→≅-¹ :
∀ {X Y} (X≡Y : X ≡ Y) →
≡→≅ X≡Y ¹ ≡ elim (λ {X Y} _ → Hom X Y) (λ _ → id) X≡Y
≡→≅-¹ {X} = elim¹
(λ X≡Y → ≡→≅ X≡Y ¹ ≡
elim (λ {X Y} _ → Hom X Y) (λ _ → id) X≡Y)
(≡→≅ (refl X) ¹ ≡⟨ cong _¹ ≡→≅-refl ⟩
id≅ ¹ ≡⟨⟩
id ≡⟨ sym $ elim-refl (λ {X Y} _ → Hom X Y) _ ⟩∎
elim (λ {X Y} _ → Hom X Y) (λ _ → id) (refl X) ∎)
-- A lemma that can be used to prove that ≡→≅ is an equivalence.
≡→≅-equivalence-lemma :
∀ {X} →
(≡≃≅ : ∀ {Y} → (X ≡ Y) ≃ (X ≅ Y)) →
_≃_.to ≡≃≅ (refl X) ¹ ≡ id →
∀ {Y} → Is-equivalence (≡→≅ {X = X} {Y = Y})
≡→≅-equivalence-lemma {X} ≡≃≅ ≡≃≅-refl {Y} =
Eq.respects-extensional-equality
(elim¹ (λ X≡Y → _≃_.to ≡≃≅ X≡Y ≡ ≡→≅ X≡Y)
(_≃_.to ≡≃≅ (refl X) ≡⟨ _≃_.from ≡≃≡¹ ≡≃≅-refl ⟩
id≅ ≡⟨ sym ≡→≅-refl ⟩∎
≡→≅ (refl X) ∎))
(_≃_.is-equivalence ≡≃≅)
-- An example: sets and functions. (Defined using extensionality.)
precategory-Set :
(ℓ : Level) →
Extensionality ℓ ℓ →
Precategory (lsuc ℓ) ℓ
precategory-Set ℓ ext = record { precategory =
-- Objects: sets.
Set ℓ ,
-- Morphisms: functions.
(λ { (A , A-set) (B , B-set) →
(A → B) , Π-closure ext 2 (λ _ → B-set) }) ,
-- Identity.
P.id ,
-- Composition.
(λ f g → f ∘ g) ,
-- Laws.
refl _ , refl _ , refl _ }
-- Isomorphisms in this category are equivalent to equivalences
-- (assuming extensionality).
≃≃≅-Set :
(ℓ : Level) (ext : Extensionality ℓ ℓ) →
let open Precategory (precategory-Set ℓ ext) in
(X Y : Obj) → (⌞ X ⌟ ≃ ⌞ Y ⌟) ≃ (X ≅ Y)
≃≃≅-Set ℓ ext X Y = Eq.↔⇒≃ record
{ surjection = record
{ logical-equivalence = record
{ to = λ X≃Y → _≃_.to X≃Y , _≃_.from X≃Y ,
apply-ext ext (_≃_.right-inverse-of X≃Y) ,
apply-ext ext (_≃_.left-inverse-of X≃Y)
; from = λ X≅Y → Eq.↔⇒≃ record
{ surjection = record
{ logical-equivalence = record
{ to = proj₁ X≅Y
; from = proj₁ (proj₂ X≅Y)
}
; right-inverse-of = λ x →
cong (_$ x) $ proj₁ (proj₂ (proj₂ X≅Y))
}
; left-inverse-of = λ x →
cong (_$ x) $ proj₂ (proj₂ (proj₂ X≅Y))
}
}
; right-inverse-of = λ X≅Y →
_≃_.from (≡≃≡¹ {X = X} {Y = Y}) (refl (proj₁ X≅Y))
}
; left-inverse-of = λ X≃Y →
Eq.lift-equality ext (refl (_≃_.to X≃Y))
}
where open Precategory (precategory-Set ℓ ext) using (≡≃≡¹)
-- Equality characterisation lemma for Precategory′.
equality-characterisation-Precategory′ :
∀ {ℓ₁ ℓ₂} {C D : Precategory′ ℓ₁ ℓ₂} →
Extensionality (ℓ₁ ⊔ ℓ₂) (ℓ₁ ⊔ lsuc ℓ₂) →
Univalence ℓ₁ →
Univalence ℓ₂ →
let module C = Precategory (record { precategory = C })
module D = Precategory (record { precategory = D })
in
(∃ λ (eqO : C.Obj ≃ D.Obj) →
∃ λ (eqH : ∀ X Y → C.Hom (_≃_.from eqO X) (_≃_.from eqO Y) ≃
D.Hom X Y) →
(∀ X → _≃_.to (eqH X X) C.id ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
_≃_.to (eqH X Z) (C._∙_ (_≃_.from (eqH Y Z) f)
(_≃_.from (eqH X Y) g)) ≡
f D.∙ g))
↔
C ≡ D
equality-characterisation-Precategory′ {ℓ₁} {ℓ₂} {C} {D}
ext univ₁ univ₂ =
(∃ λ (eqO : C.Obj ≃ D.Obj) →
∃ λ (eqH : ∀ X Y → C.Hom (_≃_.from eqO X) (_≃_.from eqO Y) ≃
D.Hom X Y) →
(∀ X → _≃_.to (eqH X X) C.id ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
_≃_.to (eqH X Z) (C._∙_ (_≃_.from (eqH Y Z) f)
(_≃_.from (eqH X Y) g)) ≡
f D.∙ g)) ↝⟨ ∃-cong (λ _ → inverse $
Σ-cong (∀-cong ext₁₁₂₊ λ _ →
∀-cong ext₁₂₊ λ _ →
≡≃≃ univ₂)
(λ _ → F.id)) ⟩
(∃ λ (eqO : C.Obj ≃ D.Obj) →
∃ λ (eqH : ∀ X Y → C.Hom (_≃_.from eqO X) (_≃_.from eqO Y) ≡
D.Hom X Y) →
(∀ X → ≡⇒→ (eqH X X) C.id ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
≡⇒→ (eqH X Z) (C._∙_ (≡⇒← (eqH Y Z) f) (≡⇒← (eqH X Y) g)) ≡
f D.∙ g)) ↝⟨ inverse $ Σ-cong (≡≃≃ univ₁) (λ _ → F.id) ⟩
(∃ λ (eqO : C.Obj ≡ D.Obj) →
∃ λ (eqH : ∀ X Y → C.Hom (≡⇒← eqO X) (≡⇒← eqO Y) ≡ D.Hom X Y) →
(∀ X → ≡⇒→ (eqH X X) C.id ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
≡⇒→ (eqH X Z) (C._∙_ (≡⇒← (eqH Y Z) f) (≡⇒← (eqH X Y) g)) ≡
f D.∙ g)) ↝⟨ ∃-cong (λ _ → inverse $
Σ-cong (∀-cong ext₁₁₂₊ λ _ →
∀-cong ext₁₂₊ λ _ →
inverse $
ignore-propositional-component $
H-level-propositional ext₂₂ 2)
(λ _ → F.id)) ⟩
(∃ λ (eqO : C.Obj ≡ D.Obj) →
∃ λ (eqH : ∀ X Y → C.HOM (≡⇒← eqO X) (≡⇒← eqO Y) ≡ D.HOM X Y) →
let eqH′ = λ X Y → proj₁ (Σ-≡,≡←≡ (eqH X Y))
in
(∀ X → ≡⇒→ (eqH′ X X) C.id ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
≡⇒→ (eqH′ X Z) (C._∙_ (≡⇒← (eqH′ Y Z) f) (≡⇒← (eqH′ X Y) g)) ≡
f D.∙ g)) ↝⟨ ∃-cong (λ _ → ∃-cong λ _ → ≡⇒↝ _ $
cong (λ (eqH′ : ∀ _ _ → _) →
(∀ X → ≡⇒→ (eqH′ X X) C.id ≡ D.id) ×
(∀ X Y Z f g →
≡⇒→ (eqH′ X Z)
(C._∙_ (≡⇒← (eqH′ Y Z) f) (≡⇒← (eqH′ X Y) g)) ≡ f D.∙ g))
(apply-ext ext₁₁₂₊ λ _ → apply-ext ext₁₂₊ λ _ →
proj₁-Σ-≡,≡←≡ _)) ⟩
(∃ λ (eqO : C.Obj ≡ D.Obj) →
∃ λ (eqH : ∀ X Y → C.HOM (≡⇒← eqO X) (≡⇒← eqO Y) ≡ D.HOM X Y) →
let eqH′ = λ X Y → cong proj₁ (eqH X Y)
in
(∀ X → ≡⇒→ (eqH′ X X) C.id ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
≡⇒→ (eqH′ X Z) (C._∙_ (≡⇒← (eqH′ Y Z) f) (≡⇒← (eqH′ X Y) g)) ≡
f D.∙ g)) ↝⟨ ∃-cong (λ _ → inverse $
Σ-cong (∀-cong ext₁₁₂₊ λ _ →
inverse $ Eq.extensionality-isomorphism ext₁₂₊)
(λ _ → F.id)) ⟩
(∃ λ (eqO : C.Obj ≡ D.Obj) →
∃ λ (eqH : ∀ X → (λ Y → C.HOM (≡⇒← eqO X) (≡⇒← eqO Y)) ≡ D.HOM X) →
let eqH′ = λ X Y → cong proj₁ (ext⁻¹ (eqH X) Y)
in
(∀ X → ≡⇒→ (eqH′ X X) C.id ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
≡⇒→ (eqH′ X Z) (C._∙_ (≡⇒← (eqH′ Y Z) f) (≡⇒← (eqH′ X Y) g)) ≡
f D.∙ g)) ↝⟨ ∃-cong (λ _ → inverse $
Σ-cong (inverse $ Eq.extensionality-isomorphism ext₁₁₂₊)
(λ _ → F.id)) ⟩
(∃ λ (eqO : C.Obj ≡ D.Obj) →
∃ λ (eqH : (λ X Y → C.HOM (≡⇒← eqO X) (≡⇒← eqO Y)) ≡ D.HOM) →
let eqH′ = λ X Y → cong proj₁ (ext⁻¹ (ext⁻¹ eqH X) Y)
in
(∀ X → ≡⇒→ (eqH′ X X) C.id ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
≡⇒→ (eqH′ X Z) (C._∙_ (≡⇒← (eqH′ Y Z) f) (≡⇒← (eqH′ X Y) g)) ≡
f D.∙ g)) ↝⟨ ∃-cong (λ eqO → inverse $
Σ-cong (inverse $ ≡⇒↝ equivalence (HOM-lemma eqO))
(λ _ → F.id)) ⟩
(∃ λ (eqO : C.Obj ≡ D.Obj) →
∃ λ (eqH : subst (λ Obj → Obj → Obj → Set _) eqO C.HOM ≡ D.HOM) →
let eqH′ = λ X Y →
cong proj₁
(ext⁻¹ (ext⁻¹ (≡⇒← (HOM-lemma eqO) eqH) X) Y)
in
(∀ X → ≡⇒→ (eqH′ X X) C.id ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
≡⇒→ (eqH′ X Z) (C._∙_ (≡⇒← (eqH′ Y Z) f) (≡⇒← (eqH′ X Y) g)) ≡
f D.∙ g)) ↝⟨ ∃-cong (λ eqO → ∃-cong λ eqH → ≡⇒↝ _ $
cong (λ (eqH′ : ∀ _ _ → _) →
(∀ X → ≡⇒→ (eqH′ X X) C.id ≡ D.id) ×
(∀ X Y Z f g →
≡⇒→ (eqH′ X Z)
(C._∙_ (≡⇒← (eqH′ Y Z) f) (≡⇒← (eqH′ X Y) g)) ≡ f D.∙ g))
(apply-ext ext₁₁₂₊ λ X → apply-ext ext₁₂₊ λ Y →
cong proj₁ (ext⁻¹ (ext⁻¹ (≡⇒← (HOM-lemma eqO) eqH) X) Y) ≡⟨⟩
cong proj₁ (cong (_$ Y) (cong (_$ X) (≡⇒← (HOM-lemma eqO) eqH))) ≡⟨ cong (cong _) $ cong-∘ _ _ _ ⟩
cong proj₁ (cong (λ f → f X Y) (≡⇒← (HOM-lemma eqO) eqH)) ≡⟨ cong-∘ _ _ _ ⟩∎
cong (λ F → ⌞ F X Y ⌟) (≡⇒← (HOM-lemma eqO) eqH) ∎)) ⟩
(∃ λ (eqO : C.Obj ≡ D.Obj) →
∃ λ (eqH : subst (λ Obj → Obj → Obj → Set _) eqO C.HOM ≡ D.HOM) →
let eqH′ = λ X Y → cong (λ F → ⌞ F X Y ⌟) (≡⇒← (HOM-lemma eqO) eqH)
in
(∀ X → ≡⇒→ (eqH′ X X) C.id ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
≡⇒→ (eqH′ X Z) (C._∙_ (≡⇒← (eqH′ Y Z) f) (≡⇒← (eqH′ X Y) g)) ≡
f D.∙ g)) ↝⟨ ∃-cong (λ eqO → ∃-cong λ eqH →
(∀-cong ext₁₂ λ _ →
≡⇒↝ _ $ cong (_≡ _) P-lemma)
×-cong
(∀-cong ext₁₁₂ λ X →
∀-cong ext₁₁₂ λ Y →
∀-cong ext₁₂ λ Z →
∀-cong ext₂₂ λ f →
∀-cong ext₂₂ λ g →
≡⇒↝ _ $ cong (_≡ _) Q-lemma)) ⟩
(∃ λ (eqO : C.Obj ≡ D.Obj) →
∃ λ (eqH : subst (λ Obj → Obj → Obj → Set _) eqO C.HOM ≡ D.HOM) →
(∀ X → subst₂ (uncurry P) eqO eqH C.id {X = X} ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
subst₂ (uncurry Q) eqO eqH C._∙_ f g ≡ f D.∙ g)) ↝⟨ Σ-assoc ⟩
(∃ λ (eq : ∃ λ (eqO : C.Obj ≡ D.Obj) →
subst (λ Obj → Obj → Obj → Set _) eqO C.HOM ≡ D.HOM) →
(∀ X → subst (uncurry P) (uncurry Σ-≡,≡→≡ eq) C.id {X = X} ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
subst (uncurry Q) (uncurry Σ-≡,≡→≡ eq) C._∙_ f g ≡ f D.∙ g)) ↝⟨ Σ-cong Bijection.Σ-≡,≡↔≡ (λ _ → F.id) ⟩
(∃ λ (eq : (C.Obj , C.HOM) ≡ (D.Obj , D.HOM)) →
(∀ X → subst (uncurry P) eq C.id {X = X} ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
subst (uncurry Q) eq C._∙_ f g ≡ f D.∙ g)) ↔⟨ ∃-cong (λ _ → ∃-cong λ _ → ∀-cong ext₁₁₂ λ _ →
∀-cong ext₁₁₂ λ _ → ∀-cong ext₁₂ λ _ →
∀-cong ext₂₂ λ _ →
Eq.extensionality-isomorphism ext₂₂) ⟩
(∃ λ (eq : (C.Obj , C.HOM) ≡ (D.Obj , D.HOM)) →
(∀ X → subst (uncurry P) eq C.id {X = X} ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) →
subst (uncurry Q) eq C._∙_ {X = X} f ≡ D._∙_ f)) ↔⟨ ∃-cong (λ _ → ∃-cong λ _ → ∀-cong ext₁₁₂ λ _ →
∀-cong ext₁₁₂ λ _ → ∀-cong ext₁₂ λ _ →
Eq.extensionality-isomorphism ext₂₂) ⟩
(∃ λ (eq : (C.Obj , C.HOM) ≡ (D.Obj , D.HOM)) →
(∀ X → subst (uncurry P) eq C.id {X = X} ≡ D.id)
×
(∀ X Y Z →
subst (uncurry Q) eq C._∙_ {X = X} {Y = Y} {Z = Z} ≡ D._∙_)) ↝⟨ ∃-cong (λ _ → ∃-cong λ _ →
∀-cong ext₁₁₂ λ _ → ∀-cong ext₁₁₂ λ _ →
implicit-extensionality-isomorphism ext₁₂) ⟩
(∃ λ (eq : (C.Obj , C.HOM) ≡ (D.Obj , D.HOM)) →
(∀ X → subst (uncurry P) eq C.id {X = X} ≡ D.id)
×
(∀ X Y →
(λ {_} → subst (uncurry Q) eq C._∙_ {X = X} {Y = Y}) ≡ D._∙_)) ↝⟨ ∃-cong (λ _ → ∃-cong λ _ → ∀-cong ext₁₁₂ λ _ →
implicit-extensionality-isomorphism ext₁₁₂) ⟩
(∃ λ (eq : (C.Obj , C.HOM) ≡ (D.Obj , D.HOM)) →
(∀ X → subst (uncurry P) eq C.id {X = X} ≡ D.id)
×
(∀ X → (λ {_ _} → subst (uncurry Q) eq C._∙_ {X = X}) ≡ D._∙_)) ↝⟨ ∃-cong (λ _ →
implicit-extensionality-isomorphism ext₁₂
×-cong
implicit-extensionality-isomorphism ext₁₁₂) ⟩
(∃ λ (eq : (C.Obj , C.HOM) ≡ (D.Obj , D.HOM)) →
(λ {_} → subst (uncurry P) eq (λ {_} → C.id)) ≡
(λ {_} → D.id)
×
(λ {_ _ _} → subst (uncurry Q) eq (λ {_ _ _} → C._∙_)) ≡
(λ {_ _ _} → D._∙_)) ↝⟨ ∃-cong (λ _ → ≡×≡↔≡) ⟩
(∃ λ (eq : (C.Obj , C.HOM) ≡ (D.Obj , D.HOM)) →
( (λ {_} → subst (uncurry P) eq (λ {_} → C.id))
, (λ {_ _ _} → subst (uncurry Q) eq (λ {_ _ _} → C._∙_))
) ≡
((λ {_} → D.id) , λ {_ _ _} → D._∙_)) ↝⟨ ∃-cong (λ _ → ≡⇒↝ _ $ cong (_≡ _) $ sym $ push-subst-, _ _) ⟩
(∃ λ (eq : (C.Obj , C.HOM) ≡ (D.Obj , D.HOM)) →
subst _ eq ((λ {_} → C.id) , λ {_ _ _} → C._∙_) ≡
((λ {_} → D.id) , λ {_ _ _} → D._∙_)) ↝⟨ Bijection.Σ-≡,≡↔≡ ⟩
((C.Obj , C.HOM) , (λ {_} → C.id) , λ {_ _ _} → C._∙_) ≡
((D.Obj , D.HOM) , (λ {_} → D.id) , λ {_ _ _} → D._∙_) ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ Σ-assoc) ⟩
(C.Obj , C.HOM , (λ {_} → C.id) , λ {_ _ _} → C._∙_) ≡
(D.Obj , D.HOM , (λ {_} → D.id) , λ {_ _ _} → D._∙_) ↝⟨ ignore-propositional-component (
×-closure 1 (implicit-Π-closure ext₁₁₂ 1 λ _ →
implicit-Π-closure ext₁₂ 1 λ _ →
implicit-Π-closure ext₂₂ 1 λ _ →
D.Hom-is-set) $
×-closure 1 (implicit-Π-closure ext₁₁₂ 1 λ _ →
implicit-Π-closure ext₁₂ 1 λ _ →
implicit-Π-closure ext₂₂ 1 λ _ →
D.Hom-is-set)
(implicit-Π-closure ext₁₁₂ 1 λ _ →
implicit-Π-closure ext₁₁₂ 1 λ _ →
implicit-Π-closure ext₁₁₂ 1 λ _ →
implicit-Π-closure ext₁₂ 1 λ _ →
implicit-Π-closure ext₂₂ 1 λ _ →
implicit-Π-closure ext₂₂ 1 λ _ →
implicit-Π-closure ext₂₂ 1 λ _ →
D.Hom-is-set)) ⟩
((C.Obj , C.HOM , (λ {_} → C.id) , λ {_ _ _} → C._∙_) , _) ≡
((D.Obj , D.HOM , (λ {_} → D.id) , λ {_ _ _} → D._∙_) , _) ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ rearrange) ⟩□
C ≡ D □
where
module C = Precategory (record { precategory = C })
module D = Precategory (record { precategory = D })
ext₁₁₂₊ : Extensionality ℓ₁ (ℓ₁ ⊔ lsuc ℓ₂)
ext₁₁₂₊ = lower-extensionality ℓ₂ lzero ext
ext₁₁₂ : Extensionality ℓ₁ (ℓ₁ ⊔ ℓ₂)
ext₁₁₂ = lower-extensionality ℓ₂ (lsuc ℓ₂) ext
ext₁₂₊ : Extensionality ℓ₁ (lsuc ℓ₂)
ext₁₂₊ = lower-extensionality ℓ₂ ℓ₁ ext
ext₁₂ : Extensionality ℓ₁ ℓ₂
ext₁₂ = lower-extensionality ℓ₂ _ ext
ext₂₂ : Extensionality ℓ₂ ℓ₂
ext₂₂ = lower-extensionality ℓ₁ _ ext
rearrange :
∀ {a b c d e}
{A : Type a} {B : A → Type b} {C : (a : A) → B a → Type c}
{D : (a : A) (b : B a) → C a b → Type d}
{E : (a : A) (b : B a) (c : C a b) → D a b c → Type e} →
(∃ λ (a : A) → ∃ λ (b : B a) → ∃ λ (c : C a b) → ∃ λ (d : D a b c) →
E a b c d)
↔
(∃ λ (p : ∃ λ (a : A) → ∃ λ (b : B a) → ∃ λ (c : C a b) → D a b c) →
E (proj₁ p) (proj₁ (proj₂ p)) (proj₁ (proj₂ (proj₂ p)))
(proj₂ (proj₂ (proj₂ p))))
rearrange {A = A} {B} {C} {D} {E} =
(∃ λ (a : A) → ∃ λ (b : B a) → ∃ λ (c : C a b) → ∃ λ (d : D a b c) →
E a b c d) ↝⟨ ∃-cong (λ _ → ∃-cong λ _ → Σ-assoc) ⟩
(∃ λ (a : A) → ∃ λ (b : B a) → ∃ λ (p : ∃ λ (c : C a b) → D a b c) →
E a b (proj₁ p) (proj₂ p)) ↝⟨ ∃-cong (λ _ → Σ-assoc) ⟩
(∃ λ (a : A) → ∃ λ (p : ∃ λ (b : B a) → ∃ λ (c : C a b) → D a b c) →
E a (proj₁ p) (proj₁ (proj₂ p)) (proj₂ (proj₂ p))) ↝⟨ Σ-assoc ⟩□
(∃ λ (p : ∃ λ (a : A) → ∃ λ (b : B a) → ∃ λ (c : C a b) → D a b c) →
E (proj₁ p) (proj₁ (proj₂ p)) (proj₁ (proj₂ (proj₂ p)))
(proj₂ (proj₂ (proj₂ p)))) □
≡⇒←-subst :
{C D : Type ℓ₁} {H : C → C → Set ℓ₂}
(eqO : C ≡ D) →
(λ X Y → H (≡⇒← eqO X) (≡⇒← eqO Y))
≡
subst (λ Obj → Obj → Obj → Set _) eqO H
≡⇒←-subst {C} {H = H} eqO =
elim¹ (λ eqO → (λ X Y → H (≡⇒← eqO X) (≡⇒← eqO Y)) ≡
subst (λ Obj → Obj → Obj → Set _) eqO H)
((λ X Y → H (≡⇒← (refl C) X) (≡⇒← (refl C) Y)) ≡⟨ cong (λ f X Y → H (f X) (f Y)) ≡⇒←-refl ⟩
H ≡⟨ sym $ subst-refl _ _ ⟩∎
subst (λ Obj → Obj → Obj → Set _) (refl C) H ∎)
eqO
≡⇒←-subst-refl : {C : Type ℓ₁} {H : C → C → Set ℓ₂} → _
≡⇒←-subst-refl {C} {H} =
≡⇒←-subst {H = H} (refl C) ≡⟨ elim¹-refl _ _ ⟩∎
trans (cong (λ f X Y → H (f X) (f Y)) ≡⇒←-refl)
(sym $ subst-refl _ _) ∎
HOM-lemma :
(eqO : C.Obj ≡ D.Obj) →
((λ X Y → C.HOM (≡⇒← eqO X) (≡⇒← eqO Y)) ≡ D.HOM)
≡
(subst (λ Obj → Obj → Obj → Set _) eqO C.HOM ≡ D.HOM)
HOM-lemma eqO = cong (_≡ _) (≡⇒←-subst eqO)
≡⇒→-lemma :
∀ {eqO eqH X Y} {f : C.Hom (≡⇒← eqO X) (≡⇒← eqO Y)} → _
≡⇒→-lemma {eqO} {eqH} {X} {Y} {f} =
≡⇒→ (cong (λ H → ⌞ H X Y ⌟) (≡⇒← (HOM-lemma eqO) eqH)) f ≡⟨ sym $ subst-in-terms-of-≡⇒↝ equivalence
(≡⇒← (HOM-lemma eqO) eqH) (λ H → ⌞ H X Y ⌟) _ ⟩
subst (λ H → ⌞ H X Y ⌟) (≡⇒← (HOM-lemma eqO) eqH) f ≡⟨ cong (λ eq → subst (λ H → ⌞ H X Y ⌟) eq _) $ sym $
subst-in-terms-of-inverse∘≡⇒↝ equivalence (≡⇒←-subst eqO) (_≡ _) _ ⟩
subst (λ H → ⌞ H X Y ⌟)
(subst (_≡ _) (sym $ ≡⇒←-subst eqO) eqH) f ≡⟨ cong (λ eq → subst (λ H → ⌞ H X Y ⌟) eq _) $
subst-trans (≡⇒←-subst eqO) ⟩
subst (λ H → ⌞ H X Y ⌟) (trans (≡⇒←-subst eqO) eqH) f ≡⟨ sym $ subst-subst _ _ _ _ ⟩∎
subst (λ H → ⌞ H X Y ⌟) eqH
(subst (λ H → ⌞ H X Y ⌟) (≡⇒←-subst eqO) f) ∎
≡⇒←-lemma : ∀ {eqO eqH X Y} {f : D.Hom X Y} → _
≡⇒←-lemma {eqO} {eqH} {X} {Y} {f} =
≡⇒← (cong (λ H → ⌞ H X Y ⌟) (≡⇒← (HOM-lemma eqO) eqH)) f ≡⟨ sym $ subst-in-terms-of-inverse∘≡⇒↝ equivalence
(≡⇒← (HOM-lemma eqO) eqH) (λ H → ⌞ H X Y ⌟) _ ⟩
subst (λ H → ⌞ H X Y ⌟) (sym $ ≡⇒← (HOM-lemma eqO) eqH) f ≡⟨ cong (λ eq → subst (λ H → ⌞ H X Y ⌟) (sym eq) _) $ sym $
subst-in-terms-of-inverse∘≡⇒↝ equivalence (≡⇒←-subst eqO) (_≡ _) _ ⟩
subst (λ H → ⌞ H X Y ⌟)
(sym $ subst (_≡ _) (sym $ ≡⇒←-subst eqO) eqH) f ≡⟨ cong (λ eq → subst (λ H → ⌞ H X Y ⌟) (sym eq) _) $
subst-trans (≡⇒←-subst eqO) ⟩
subst (λ H → ⌞ H X Y ⌟) (sym $ trans (≡⇒←-subst eqO) eqH) f ≡⟨ cong (λ eq → subst (λ H → ⌞ H X Y ⌟) eq _) $
sym-trans (≡⇒←-subst eqO) eqH ⟩
subst (λ H → ⌞ H X Y ⌟) (trans (sym eqH) (sym $ ≡⇒←-subst eqO)) f ≡⟨ sym $ subst-subst _ _ _ _ ⟩∎
subst (λ H → ⌞ H X Y ⌟) (sym $ ≡⇒←-subst eqO)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) f) ∎
expand-≡⇒←-subst :
∀ {C : Type ℓ₁} {X Y}
{F G : C → C → Set ℓ₂}
{eqH : subst (λ Obj → Obj → Obj → Set ℓ₂) (refl C) F ≡ G}
{f : ⌞ F (≡⇒← (refl C) X) (≡⇒← (refl C) Y) ⌟} →
_
expand-≡⇒←-subst {C} {X} {Y} {F} {eqH = eqH} {f} =
subst (λ H → ⌞ H X Y ⌟) eqH
(subst (λ H → ⌞ H X Y ⌟) (≡⇒←-subst (refl C)) f) ≡⟨ cong (λ eq → subst (λ H → ⌞ H X Y ⌟) eqH $ subst (λ H → ⌞ H X Y ⌟) eq f)
≡⇒←-subst-refl ⟩
subst (λ H → ⌞ H X Y ⌟) eqH
(subst (λ H → ⌞ H X Y ⌟)
(trans (cong (λ f X Y → F (f X) (f Y)) ≡⇒←-refl)
(sym $ subst-refl _ _))
f) ≡⟨ cong (subst (λ H → ⌞ H X Y ⌟) eqH) $ sym $
subst-subst _ _ _ _ ⟩
subst (λ H → ⌞ H X Y ⌟) eqH
(subst (λ H → ⌞ H X Y ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H X Y ⌟)
(cong (λ f X Y → F (f X) (f Y)) ≡⇒←-refl)
f)) ≡⟨ cong (λ f → subst (λ H → ⌞ H X Y ⌟) eqH $
subst (λ H → ⌞ H X Y ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _) f) $ sym $
subst-∘ (λ H → ⌞ H X Y ⌟) (λ f X Y → F (f X) (f Y)) ≡⇒←-refl ⟩∎
subst (λ H → ⌞ H X Y ⌟) eqH
(subst (λ H → ⌞ H X Y ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ f → ⌞ F (f X) (f Y) ⌟) ≡⇒←-refl f)) ∎
expand-sym-≡⇒←-subst :
∀ {C : Type ℓ₁} {X Y}
{F G : C → C → Set ℓ₂}
{eqH : subst (λ Obj → Obj → Obj → Set ℓ₂) (refl C) F ≡ G}
{f : ⌞ G X Y ⌟} →
_
expand-sym-≡⇒←-subst {C} {X} {Y} {F} {eqH = eqH} {f} =
subst (λ H → ⌞ H X Y ⌟) (sym $ ≡⇒←-subst (refl C))
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) f) ≡⟨ cong (λ eq → subst (λ H → ⌞ H X Y ⌟) (sym eq) $
subst (λ H → ⌞ H X Y ⌟) (sym eqH) f)
≡⇒←-subst-refl ⟩
subst (λ H → ⌞ H X Y ⌟)
(sym $ trans (cong (λ f X Y → F (f X) (f Y)) ≡⇒←-refl)
(sym $ subst-refl _ _))
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) f) ≡⟨ cong (λ eq → subst (λ H → ⌞ H X Y ⌟) eq $
subst (λ H → ⌞ H X Y ⌟) (sym eqH) f) $
sym-trans (cong (λ f X Y → F (f X) (f Y)) ≡⇒←-refl) _ ⟩
subst (λ H → ⌞ H X Y ⌟)
(trans (sym $ sym $
subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(sym $ cong (λ f X Y → F (f X) (f Y))
≡⇒←-refl))
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) f) ≡⟨ cong (λ eq → subst (λ H → ⌞ H X Y ⌟)
(trans eq (sym $ cong (λ f X Y → F (f X) (f Y)) ≡⇒←-refl)) $
subst (λ H → ⌞ H X Y ⌟) (sym eqH) f) $
sym-sym _ ⟩
subst (λ H → ⌞ H X Y ⌟)
(trans (subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(sym $ cong (λ f X Y → F (f X) (f Y))
≡⇒←-refl))
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) f) ≡⟨ sym $ subst-subst _ _ _ _ ⟩
subst (λ H → ⌞ H X Y ⌟)
(sym $ cong (λ f X Y → F (f X) (f Y)) ≡⇒←-refl)
(subst (λ H → ⌞ H X Y ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) f)) ≡⟨ cong (λ eq → subst (λ H → ⌞ H X Y ⌟) eq $
subst (λ H → ⌞ H X Y ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _) $
subst (λ H → ⌞ H X Y ⌟) (sym eqH) f) $ sym $
cong-sym (λ f X Y → F (f X) (f Y)) ≡⇒←-refl ⟩
subst (λ H → ⌞ H X Y ⌟)
(cong (λ f X Y → F (f X) (f Y)) $ sym ≡⇒←-refl)
(subst (λ H → ⌞ H X Y ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) f)) ≡⟨ sym $ subst-∘ _ _ _ ⟩∎
subst (λ f → ⌞ F (f X) (f Y) ⌟) (sym ≡⇒←-refl)
(subst (λ H → ⌞ H X Y ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) f)) ∎
subst-Σ-≡,≡→≡ :
∀ {C : Type ℓ₁}
{F G : C → C → Set ℓ₂}
{eqH : subst (λ Obj → Obj → Obj → Set ℓ₂) (refl C) F ≡ G}
{P : (Obj : Type ℓ₁) (HOM : Obj → Obj → Set ℓ₂) → Type (ℓ₁ ⊔ ℓ₂)} →
_
subst-Σ-≡,≡→≡ {C} {F} {eqH = eqH} {P} =
subst (P C) eqH ∘
subst (P C) (sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) F) ≡⟨ apply-ext (lower-extensionality lzero (lsuc ℓ₂) ext) (λ _ →
subst-subst (P C) _ _ _) ⟩
subst (P C) (trans (sym $ subst-refl _ _) eqH) ≡⟨ apply-ext (lower-extensionality lzero (lsuc ℓ₂) ext) (λ _ →
subst-∘ (uncurry P) (C ,_) _) ⟩
subst (uncurry P) (cong (C ,_) (trans (sym $ subst-refl _ _) eqH)) ≡⟨ cong (subst (uncurry P)) $ sym $ Σ-≡,≡→≡-reflˡ eqH ⟩∎
subst (uncurry P) (Σ-≡,≡→≡ (refl C) eqH) ∎
P = λ Obj (HOM : Obj → Obj → Set _) →
∀ {X} → ⌞ HOM X X ⌟
abstract
P-lemma :
∀ {eqO eqH X} →
≡⇒→ (cong (λ H → ⌞ H X X ⌟) (≡⇒← (HOM-lemma eqO) eqH)) C.id ≡
subst₂ (uncurry P) eqO eqH C.id {X = X}
P-lemma {eqO} {eqH} {X} =
≡⇒→ (cong (λ H → ⌞ H X X ⌟) (≡⇒← (HOM-lemma eqO) eqH)) C.id ≡⟨ ≡⇒→-lemma ⟩
subst (λ H → ⌞ H X X ⌟) eqH
(subst (λ H → ⌞ H X X ⌟) (≡⇒←-subst eqO)
(C.id {X = ≡⇒← eqO X})) ≡⟨ elim
(λ eqO →
∀ {X F G}
(eqH : subst (λ Obj → Obj → Obj → Set ℓ₂) eqO F ≡ G)
(id : ∀ X → ⌞ F X X ⌟) →
subst (λ H → ⌞ H X X ⌟) eqH
(subst (λ H → ⌞ H X X ⌟) (≡⇒←-subst eqO) (id (≡⇒← eqO X)))
≡
subst (uncurry P) (Σ-≡,≡→≡ eqO eqH) (λ {X} → id X))
(λ C {X F G} eqH id →
subst (λ H → ⌞ H X X ⌟) eqH
(subst (λ H → ⌞ H X X ⌟) (≡⇒←-subst (refl C))
(id (≡⇒← (refl C) X))) ≡⟨ expand-≡⇒←-subst ⟩
subst (λ H → ⌞ H X X ⌟) eqH
(subst
(λ H → ⌞ H X X ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ f → ⌞ F (f X) (f X) ⌟)
≡⇒←-refl
(id (≡⇒← (refl C) X)))) ≡⟨ cong (λ f → subst (λ H → ⌞ H X X ⌟) eqH
(subst
(λ H → ⌞ H X X ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
f)) $
dcong (λ f → id (f X)) ≡⇒←-refl ⟩
subst (λ H → ⌞ H X X ⌟) eqH
(subst (λ H → ⌞ H X X ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) F)
(id X)) ≡⟨ cong (subst (λ H → ⌞ H X X ⌟) eqH) $
push-subst-implicit-application _ _ ⟩
subst (λ H → ⌞ H X X ⌟) eqH
(subst (P C)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) F)
(λ {X} → id X) {X = X}) ≡⟨ push-subst-implicit-application _ _ ⟩
subst (P C) eqH
(subst (P C)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) F)
(λ {X} → id X)) {X = X} ≡⟨ cong (λ (f : P C F → P C G) → f _)
subst-Σ-≡,≡→≡ ⟩∎
subst (uncurry P) (Σ-≡,≡→≡ (refl C) eqH) (λ {X} → id X) ∎)
eqO eqH (λ _ → C.id) ⟩
subst (uncurry P) (Σ-≡,≡→≡ eqO eqH)
(λ {X} → C.id {X = X}) {X = X} ≡⟨⟩
subst₂ (uncurry P) eqO eqH C.id ∎
Q = λ Obj (HOM : Obj → Obj → Set _) →
∀ {X Y Z} → ⌞ HOM Y Z ⌟ → ⌞ HOM X Y ⌟ → ⌞ HOM X Z ⌟
push-Q :
{C : Type ℓ₁} {X Y Z : C} {F G : C → C → Set ℓ₂}
{c : (X Y Z : C) → ⌞ F Y Z ⌟ → ⌞ F X Y ⌟ → ⌞ F X Z ⌟}
{F≡G : F ≡ G} {f : ⌞ G Y Z ⌟} {g : ⌞ G X Y ⌟} →
subst (λ H → ⌞ H X Z ⌟) F≡G
(c X Y Z
(subst (λ H → ⌞ H Y Z ⌟) (sym F≡G) f)
(subst (λ H → ⌞ H X Y ⌟) (sym F≡G) g)) ≡
subst (Q C) F≡G (c _ _ _) f g
push-Q {C} {X} {Y} {Z} {c = c} {F≡G} {f} {g} =
subst (λ H → ⌞ H X Z ⌟) F≡G
(c X Y Z (subst (λ H → ⌞ H Y Z ⌟) (sym F≡G) f)
(subst (λ H → ⌞ H X Y ⌟) (sym F≡G) g)) ≡⟨ sym subst-→ ⟩
subst (λ H → ⌞ H X Y ⌟ → ⌞ H X Z ⌟) F≡G
(c X Y Z (subst (λ H → ⌞ H Y Z ⌟) (sym F≡G) f)) g ≡⟨ cong (_$ g) $ sym subst-→ ⟩
subst (λ H → ⌞ H Y Z ⌟ → ⌞ H X Y ⌟ → ⌞ H X Z ⌟) F≡G
(c X Y Z) f g ≡⟨ cong (λ h → h f g) $
push-subst-implicit-application _
(λ H Z → ⌞ H Y Z ⌟ → ⌞ H X Y ⌟ → ⌞ H X Z ⌟) ⟩
subst (λ H → ∀ {Z} → ⌞ H Y Z ⌟ → ⌞ H X Y ⌟ → ⌞ H X Z ⌟)
F≡G (c X Y _) f g ≡⟨ cong (λ h → h {Z = Z} f g) $
push-subst-implicit-application F≡G
(λ H Y → ∀ {Z} → ⌞ H Y Z ⌟ → ⌞ H X Y ⌟ → ⌞ H X Z ⌟) ⟩
subst (λ H → ∀ {Y Z} → ⌞ H Y Z ⌟ → ⌞ H X Y ⌟ → ⌞ H X Z ⌟)
F≡G (c X _ _) f g ≡⟨ cong (λ h → h {Y = Y} {Z = Z} f g) $
push-subst-implicit-application F≡G
(λ H X → ∀ {Y Z} → ⌞ H Y Z ⌟ → ⌞ H X Y ⌟ → ⌞ H X Z ⌟) ⟩∎
subst (Q C) F≡G (c _ _ _) f g ∎
abstract
Q-lemma :
∀ {eqO eqH X Y Z f g} →
let eqH′ = λ X Y →
cong (λ H → ⌞ H X Y ⌟) (≡⇒← (HOM-lemma eqO) eqH)
in
≡⇒→ (eqH′ X Z) (≡⇒← (eqH′ Y Z) f C.∙ ≡⇒← (eqH′ X Y) g) ≡
subst₂ (uncurry Q) eqO eqH C._∙_ f g
Q-lemma {eqO} {eqH} {X} {Y} {Z} {f} {g} =
let eqH′ = λ X Y →
cong (λ F → ⌞ F X Y ⌟) (≡⇒← (HOM-lemma eqO) eqH)
in
≡⇒→ (eqH′ X Z) (≡⇒← (eqH′ Y Z) f C.∙ ≡⇒← (eqH′ X Y) g) ≡⟨ cong₂ (λ f g → ≡⇒→ (eqH′ X Z) (f C.∙ g))
≡⇒←-lemma
≡⇒←-lemma ⟩
≡⇒→ (eqH′ X Z)
(subst (λ H → ⌞ H Y Z ⌟) (sym $ ≡⇒←-subst eqO)
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f)
C.∙
subst (λ H → ⌞ H X Y ⌟) (sym $ ≡⇒←-subst eqO)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g)) ≡⟨ ≡⇒→-lemma ⟩
subst (λ H → ⌞ H X Z ⌟) eqH
(subst (λ H → ⌞ H X Z ⌟) (≡⇒←-subst eqO)
(subst (λ H → ⌞ H Y Z ⌟) (sym $ ≡⇒←-subst eqO)
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f)
C.∙
subst (λ H → ⌞ H X Y ⌟) (sym $ ≡⇒←-subst eqO)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g))) ≡⟨ elim
(λ eqO → ∀ {X Y Z F G}
(eqH : subst (λ Obj → Obj → Obj → Set ℓ₂) eqO F ≡ G)
(comp : ∀ X Y Z →
⌞ F Y Z ⌟ → ⌞ F X Y ⌟ → ⌞ F X Z ⌟)
(f : ⌞ G Y Z ⌟) (g : ⌞ G X Y ⌟) →
subst (λ H → ⌞ H X Z ⌟) eqH
(subst (λ H → ⌞ H X Z ⌟) (≡⇒←-subst eqO)
(comp (≡⇒← eqO X) (≡⇒← eqO Y) (≡⇒← eqO Z)
(subst (λ H → ⌞ H Y Z ⌟) (sym $ ≡⇒←-subst eqO)
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f))
(subst (λ H → ⌞ H X Y ⌟) (sym $ ≡⇒←-subst eqO)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g))))
≡
subst (uncurry Q) (Σ-≡,≡→≡ eqO eqH) (λ {X Y Z} → comp X Y Z) f g)
(λ C {X Y Z F G} eqH comp f g →
subst (λ H → ⌞ H X Z ⌟) eqH
(subst (λ H → ⌞ H X Z ⌟) (≡⇒←-subst (refl C))
(comp (≡⇒← (refl C) X) (≡⇒← (refl C) Y) (≡⇒← (refl C) Z)
(subst (λ H → ⌞ H Y Z ⌟) (sym $ ≡⇒←-subst (refl C))
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f))
(subst (λ H → ⌞ H X Y ⌟) (sym $ ≡⇒←-subst (refl C))
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g)))) ≡⟨ cong₂ (λ f g →
subst (λ H → ⌞ H X Z ⌟) eqH $
subst (λ H → ⌞ H X Z ⌟) (≡⇒←-subst (refl C)) $
comp (≡⇒← (refl C) X) (≡⇒← (refl C) Y)
(≡⇒← (refl C) Z) f g)
expand-sym-≡⇒←-subst
expand-sym-≡⇒←-subst ⟩
subst (λ H → ⌞ H X Z ⌟) eqH
(subst (λ H → ⌞ H X Z ⌟) (≡⇒←-subst (refl C))
(comp (≡⇒← (refl C) X) (≡⇒← (refl C) Y) (≡⇒← (refl C) Z)
(subst (λ f → ⌞ F (f Y) (f Z) ⌟) (sym ≡⇒←-refl)
(subst (λ H → ⌞ H Y Z ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f)))
(subst (λ f → ⌞ F (f X) (f Y) ⌟) (sym ≡⇒←-refl)
(subst (λ H → ⌞ H X Y ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g))))) ≡⟨ expand-≡⇒←-subst ⟩
subst (λ H → ⌞ H X Z ⌟) eqH
(subst (λ H → ⌞ H X Z ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ f → ⌞ F (f X) (f Z) ⌟) ≡⇒←-refl
(comp (≡⇒← (refl C) X) (≡⇒← (refl C) Y)
(≡⇒← (refl C) Z)
(subst (λ f → ⌞ F (f Y) (f Z) ⌟) (sym ≡⇒←-refl)
(subst (λ H → ⌞ H Y Z ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f)))
(subst (λ f → ⌞ F (f X) (f Y) ⌟) (sym ≡⇒←-refl)
(subst (λ H → ⌞ H X Y ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g)))))) ≡⟨ cong (subst (λ H → ⌞ H X Z ⌟) eqH ∘
subst (λ H → ⌞ H X Z ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)) $
dcong′
(λ h eq →
comp (h X) (h Y) (h Z)
(subst (λ f → ⌞ F (f Y) (f Z) ⌟) (sym eq)
(subst (λ H → ⌞ H Y Z ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f)))
(subst (λ f → ⌞ F (f X) (f Y) ⌟) (sym eq)
(subst (λ H → ⌞ H X Y ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g))))
_ ⟩
subst (λ H → ⌞ H X Z ⌟) eqH
(subst (λ H → ⌞ H X Z ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(comp X Y Z
(subst (λ f → ⌞ F (f Y) (f Z) ⌟) (sym (refl P.id))
(subst (λ H → ⌞ H Y Z ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f)))
(subst (λ f → ⌞ F (f X) (f Y) ⌟) (sym (refl P.id))
(subst (λ H → ⌞ H X Y ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g))))) ≡⟨ cong₂ (λ p q →
subst (λ H → ⌞ H X Z ⌟) eqH
(subst (λ H → ⌞ H X Z ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(comp X Y Z
(subst (λ f → ⌞ F (f Y) (f Z) ⌟) p
(subst (λ H → ⌞ H Y Z ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f)))
(subst (λ f → ⌞ F (f X) (f Y) ⌟) q
(subst (λ H → ⌞ H X Y ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g))))))
(sym-refl {x = P.id})
(sym-refl {x = P.id}) ⟩
subst (λ H → ⌞ H X Z ⌟) eqH
(subst (λ H → ⌞ H X Z ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(comp X Y Z
(subst (λ f → ⌞ F (f Y) (f Z) ⌟) (refl P.id)
(subst (λ H → ⌞ H Y Z ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f)))
(subst (λ f → ⌞ F (f X) (f Y) ⌟) (refl P.id)
(subst (λ H → ⌞ H X Y ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g))))) ≡⟨ cong₂ (λ f g →
subst (λ H → ⌞ H X Z ⌟) eqH
(subst (λ H → ⌞ H X Z ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(comp X Y Z f g)))
(subst-refl _ _)
(subst-refl _ _) ⟩
subst (λ H → ⌞ H X Z ⌟) eqH
(subst (λ H → ⌞ H X Z ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(comp X Y Z
(subst (λ H → ⌞ H Y Z ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f))
(subst (λ H → ⌞ H X Y ⌟)
(subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g)))) ≡⟨ sym $ cong₂ (λ p q →
subst (λ H → ⌞ H X Z ⌟) eqH
(subst (λ H → ⌞ H X Z ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(comp X Y Z
(subst (λ H → ⌞ H Y Z ⌟) p
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f))
(subst (λ H → ⌞ H X Y ⌟) q
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g)))))
(sym-sym (subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _))
(sym-sym (subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)) ⟩
subst (λ H → ⌞ H X Z ⌟) eqH
(subst (λ H → ⌞ H X Z ⌟)
(sym $ subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(comp X Y Z
(subst (λ H → ⌞ H Y Z ⌟)
(sym $ sym $
subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f))
(subst (λ H → ⌞ H X Y ⌟)
(sym $ sym $
subst-refl (λ Obj → Obj → Obj → Set ℓ₂) _)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g)))) ≡⟨ cong (subst (λ H → ⌞ H X Z ⌟) eqH) push-Q ⟩
subst (λ H → ⌞ H X Z ⌟) eqH
(subst (Q C)
(sym $ subst-refl _ _)
(λ {X Y Z} → comp X Y Z)
(subst (λ H → ⌞ H Y Z ⌟) (sym eqH) f)
(subst (λ H → ⌞ H X Y ⌟) (sym eqH) g)) ≡⟨ push-Q ⟩
subst (Q C) eqH
(subst (Q C)
(sym $ subst-refl _ _)
(λ {X Y Z} → comp X Y Z)) f g ≡⟨ cong (λ (h : Q C F → Q C G) → h _ _ _)
subst-Σ-≡,≡→≡ ⟩∎
subst (uncurry Q)
(Σ-≡,≡→≡ (refl C) eqH)
(λ {X Y Z} → comp X Y Z) f g ∎)
eqO eqH (λ _ _ _ → C._∙_) f g ⟩
subst (uncurry Q) (Σ-≡,≡→≡ eqO eqH) C._∙_ f g ≡⟨⟩
subst₂ (uncurry Q) eqO eqH C._∙_ f g ∎
-- Equality characterisation lemma for Precategory.
equality-characterisation-Precategory :
∀ {ℓ₁ ℓ₂} {C D : Precategory ℓ₁ ℓ₂} →
Extensionality (ℓ₁ ⊔ ℓ₂) (ℓ₁ ⊔ lsuc ℓ₂) →
Univalence ℓ₁ →
Univalence ℓ₂ →
let module C = Precategory C
module D = Precategory D
in
(∃ λ (eqO : C.Obj ≃ D.Obj) →
∃ λ (eqH : ∀ X Y → C.Hom (_≃_.from eqO X) (_≃_.from eqO Y) ≃
D.Hom X Y) →
(∀ X → _≃_.to (eqH X X) C.id ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
_≃_.to (eqH X Z) (C._∙_ (_≃_.from (eqH Y Z) f)
(_≃_.from (eqH X Y) g)) ≡
f D.∙ g))
↔
C ≡ D
equality-characterisation-Precategory {ℓ₁} {ℓ₂} {C} {D}
ext univ₁ univ₂ =
_ ↝⟨ equality-characterisation-Precategory′ ext univ₁ univ₂ ⟩
C.precategory ≡ D.precategory ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ rearrange) ⟩□
C ≡ D □
where
module C = Precategory C
module D = Precategory D
rearrange : Precategory ℓ₁ ℓ₂ ↔ Precategory′ ℓ₁ ℓ₂
rearrange = record
{ surjection = record
{ logical-equivalence = record
{ to = Precategory.precategory
; from = λ C → record { precategory = C }
}
; right-inverse-of = λ _ → refl _
}
; left-inverse-of = λ _ → refl _
}
-- Lifts a precategory's object type.
lift-precategory-Obj :
∀ {ℓ₁} ℓ₁′ {ℓ₂} →
Precategory ℓ₁ ℓ₂ → Precategory (ℓ₁ ⊔ ℓ₁′) ℓ₂
lift-precategory-Obj ℓ₁′ C .Precategory.precategory =
↑ ℓ₁′ C.Obj
, (λ (lift A) (lift B) → C.HOM A B)
, C.id
, C._∙_
, C.left-identity
, C.right-identity
, C.assoc
where
module C = Precategory C
-- Lifts a precategory's morphism type family.
lift-precategory-Hom :
∀ {ℓ₁ ℓ₂} ℓ₂′ →
Precategory ℓ₁ ℓ₂ → Precategory ℓ₁ (ℓ₂ ⊔ ℓ₂′)
lift-precategory-Hom ℓ₂′ C .Precategory.precategory =
C.Obj
, (λ A B → ↑ ℓ₂′ (C.Hom A B)
, ↑-closure 2 C.Hom-is-set)
, lift C.id
, (λ (lift f) (lift g) → lift (f C.∙ g))
, cong lift C.left-identity
, cong lift C.right-identity
, cong lift C.assoc
where
module C = Precategory C
------------------------------------------------------------------------
-- Categories
Category′ : (ℓ₁ ℓ₂ : Level) → Type (lsuc (ℓ₁ ⊔ ℓ₂))
Category′ ℓ₁ ℓ₂ =
-- A precategory.
∃ λ (C : Precategory ℓ₁ ℓ₂) →
-- The function ≡→≅ is an equivalence (for each pair of objects).
∀ {X Y} → Is-equivalence (Precategory.≡→≅ C {X = X} {Y = Y})
-- A wrapper.
record Category (ℓ₁ ℓ₂ : Level) : Type (lsuc (ℓ₁ ⊔ ℓ₂)) where
field
category : Category′ ℓ₁ ℓ₂
-- Precategory.
precategory : Precategory ℓ₁ ℓ₂
precategory = proj₁ category
open Precategory precategory public hiding (precategory)
-- The function ≡→≅ is an equivalence (for each pair of objects).
≡→≅-equivalence : ∀ {X Y} → Is-equivalence (≡→≅ {X = X} {Y = Y})
≡→≅-equivalence = proj₂ category
≡≃≅ : ∀ {X Y} → (X ≡ Y) ≃ (X ≅ Y)
≡≃≅ = ⟨ _ , ≡→≅-equivalence ⟩
≅→≡ : ∀ {X Y} → X ≅ Y → X ≡ Y
≅→≡ = _≃_.from ≡≃≅
-- "Computation rule" for ≅→≡.
≅→≡-refl : ∀ {X} → ≅→≡ id≅ ≡ refl X
≅→≡-refl {X} =
≅→≡ id≅ ≡⟨ cong ≅→≡ $ sym ≡→≅-refl ⟩
≅→≡ (≡→≅ (refl X)) ≡⟨ _≃_.left-inverse-of ≡≃≅ _ ⟩∎
refl X ∎
-- Obj has h-level 3.
Obj-3 : H-level 3 Obj
Obj-3 =
respects-surjection
(_≃_.surjection (Eq.inverse ≡≃≅))
2 ≅-set
-- Isomorphisms form a category.
category-≅ : Category ℓ₁ ℓ₂
category-≅ = record { category = precategory-≅ , is-equiv }
where
module P≅ = Precategory precategory-≅
abstract
is-equiv : ∀ {X Y} → Is-equivalence (P≅.≡→≅ {X = X} {Y = Y})
is-equiv =
_⇔_.from (Is-equivalence≃Is-equivalence-CP _)
λ (X≅Y , X≅Y-iso) →
Σ-map (Σ-map
P.id
(λ {X≡Y} ≡→≅[X≡Y]≡X≅Y →
elim (λ {X Y} X≡Y →
(X≅Y : X ≅ Y) (X≅Y-iso : P≅.Is-isomorphism X≅Y) →
≡→≅ X≡Y ≡ X≅Y →
P≅.≡→≅ X≡Y ≡ (X≅Y , X≅Y-iso))
(λ X X≅X X≅X-iso ≡→≅[refl]≡X≅X →
P≅.≡→≅ (refl X) ≡⟨ P≅.≡→≅-refl ⟩
P≅.id≅ ≡⟨ Σ-≡,≡→≡ (id≅ ≡⟨ sym ≡→≅-refl ⟩
≡→≅ (refl X) ≡⟨ ≡→≅[refl]≡X≅X ⟩∎
X≅X ∎)
(P≅.Is-isomorphism-propositional _ _ _) ⟩∎
(X≅X , X≅X-iso) ∎)
X≡Y X≅Y X≅Y-iso
≡→≅[X≡Y]≡X≅Y))
(λ { {X≡Y , _} ∀y→≡y → λ { (X≡Y′ , ≡→≅[X≡Y′]≡X≅Y) →
let lemma =
≡→≅ X≡Y′ ≡⟨ elim (λ X≡Y′ → ≡→≅ X≡Y′ ≡ proj₁ (P≅.≡→≅ X≡Y′))
(λ X → ≡→≅ (refl X) ≡⟨ ≡→≅-refl ⟩
id≅ ≡⟨ cong proj₁ $ sym P≅.≡→≅-refl ⟩∎
proj₁ (P≅.≡→≅ (refl X)) ∎)
X≡Y′ ⟩
proj₁ (P≅.≡→≅ X≡Y′) ≡⟨ cong proj₁ ≡→≅[X≡Y′]≡X≅Y ⟩∎
X≅Y ∎ in
(X≡Y , _) ≡⟨ Σ-≡,≡→≡ (cong proj₁ (∀y→≡y (X≡Y′ , lemma))) (P≅.≅-set _ _) ⟩∎
(X≡Y′ , _) ∎ } }) $
_⇔_.to (Is-equivalence≃Is-equivalence-CP _)
≡→≅-equivalence X≅Y
-- Some equality rearrangement lemmas.
Hom-, : ∀ {X X′ Y Y′} {f : Hom X Y}
(p : X ≡ X′) (q : Y ≡ Y′) →
subst (uncurry Hom) (cong₂ _,_ p q) f ≡ ≡→≅ q ¹ ∙ f ∙ ≡→≅ p ⁻¹
Hom-, p q = elim
(λ p → ∀ q → ∀ {f} → subst (uncurry Hom) (cong₂ _,_ p q) f ≡
≡→≅ q ¹ ∙ f ∙ ≡→≅ p ⁻¹)
(λ X q → elim
(λ q → ∀ {f} → subst (uncurry Hom) (cong₂ _,_ (refl X) q) f ≡
≡→≅ q ¹ ∙ f ∙ ≡→≅ (refl X) ⁻¹)
(λ Y {f} →
subst (uncurry Hom) (cong₂ _,_ (refl X) (refl Y)) f ≡⟨ cong (λ eq → subst (uncurry Hom) eq f) $ cong₂-refl _,_ ⟩
subst (uncurry Hom) (refl (X , Y)) f ≡⟨ subst-refl (uncurry Hom) _ ⟩
f ≡⟨ sym left-identity ⟩
id ∙ f ≡⟨ cong (λ g → g ¹ ∙ f) $ sym ≡→≅-refl ⟩
≡→≅ (refl Y) ¹ ∙ f ≡⟨ sym right-identity ⟩
(≡→≅ (refl Y) ¹ ∙ f) ∙ id ≡⟨ sym assoc ⟩
≡→≅ (refl Y) ¹ ∙ f ∙ id ≡⟨ cong (λ g → ≡→≅ (refl Y) ¹ ∙ f ∙ g ⁻¹) $ sym ≡→≅-refl ⟩∎
≡→≅ (refl Y) ¹ ∙ f ∙ ≡→≅ (refl X) ⁻¹ ∎)
q)
p q
≡→≅-trans : ∀ {X Y Z} (p : X ≡ Y) (q : Y ≡ Z) →
≡→≅ (trans p q) ≡ ≡→≅ q ∙≅ ≡→≅ p
≡→≅-trans {X} = elim¹
(λ p → ∀ q → ≡→≅ (trans p q) ≡ ≡→≅ q ∙≅ ≡→≅ p)
(elim¹
(λ q → ≡→≅ (trans (refl X) q) ≡ ≡→≅ q ∙≅ ≡→≅ (refl X))
(≡→≅ (trans (refl X) (refl X)) ≡⟨ cong ≡→≅ trans-refl-refl ⟩
≡→≅ (refl X) ≡⟨ ≡→≅-refl ⟩
id≅ ≡⟨ sym $ Precategory.left-identity precategory-≅ ⟩
id≅ ∙≅ id≅ ≡⟨ sym $ cong₂ _∙≅_ ≡→≅-refl ≡→≅-refl ⟩∎
≡→≅ (refl X) ∙≅ ≡→≅ (refl X) ∎))
-- Equality of categories is isomorphic to equality of the underlying
-- precategories (assuming extensionality).
≡↔precategory≡precategory′ :
∀ {ℓ₁ ℓ₂} {C D : Category′ ℓ₁ ℓ₂} →
Extensionality (ℓ₁ ⊔ ℓ₂) (ℓ₁ ⊔ ℓ₂) →
C ≡ D ↔ proj₁ C ≡ proj₁ D
≡↔precategory≡precategory′ {ℓ₂ = ℓ₂} ext =
inverse $
ignore-propositional-component
(implicit-Π-closure (lower-extensionality ℓ₂ lzero ext) 1 λ _ →
implicit-Π-closure (lower-extensionality ℓ₂ lzero ext) 1 λ _ →
Eq.propositional ext _)
-- Equality of categories is isomorphic to equality of the underlying
-- precategories (assuming extensionality).
≡↔precategory≡precategory :
∀ {ℓ₁ ℓ₂} {C D : Category ℓ₁ ℓ₂} →
Extensionality (ℓ₁ ⊔ ℓ₂) (ℓ₁ ⊔ ℓ₂) →
C ≡ D ↔ Category.precategory C ≡ Category.precategory D
≡↔precategory≡precategory {C = C} {D = D} ext =
C ≡ D ↔⟨ Eq.≃-≡ (Eq.↔⇒≃ rearrange) ⟩
C.category ≡ D.category ↝⟨ ≡↔precategory≡precategory′ ext ⟩□
C.precategory ≡ D.precategory □
where
module C = Category C
module D = Category D
rearrange : Category′ _ _ ↔ Category _ _
rearrange = record
{ surjection = record
{ logical-equivalence = record
{ to = λ C → record { category = C }
; from = Category.category
}
; right-inverse-of = λ _ → refl _
}
; left-inverse-of = λ _ → refl _
}
-- Equality characterisation lemma for Category′.
equality-characterisation-Category′ :
∀ {ℓ₁ ℓ₂} {C D : Category′ ℓ₁ ℓ₂} →
Extensionality (ℓ₁ ⊔ ℓ₂) (ℓ₁ ⊔ lsuc ℓ₂) →
Univalence ℓ₁ →
Univalence ℓ₂ →
let module C = Category (record { category = C })
module D = Category (record { category = D })
in
(∃ λ (eqO : C.Obj ≃ D.Obj) →
∃ λ (eqH : ∀ X Y → C.Hom (_≃_.from eqO X) (_≃_.from eqO Y) ≃
D.Hom X Y) →
(∀ X → _≃_.to (eqH X X) C.id ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
_≃_.to (eqH X Z) (C._∙_ (_≃_.from (eqH Y Z) f)
(_≃_.from (eqH X Y) g)) ≡
f D.∙ g))
↔
C ≡ D
equality-characterisation-Category′ {ℓ₂ = ℓ₂} {C} {D} ext univ₁ univ₂ =
_ ↝⟨ equality-characterisation-Precategory ext univ₁ univ₂ ⟩
C.precategory ≡ D.precategory ↝⟨ inverse $ ≡↔precategory≡precategory′ (lower-extensionality lzero (lsuc ℓ₂) ext) ⟩□
C ≡ D □
where
module C = Category (record { category = C })
module D = Category (record { category = D })
-- Equality characterisation lemma for Category.
equality-characterisation-Category :
∀ {ℓ₁ ℓ₂} {C D : Category ℓ₁ ℓ₂} →
Extensionality (ℓ₁ ⊔ ℓ₂) (ℓ₁ ⊔ lsuc ℓ₂) →
Univalence ℓ₁ →
Univalence ℓ₂ →
let module C = Category C
module D = Category D
in
(∃ λ (eqO : C.Obj ≃ D.Obj) →
∃ λ (eqH : ∀ X Y → C.Hom (_≃_.from eqO X) (_≃_.from eqO Y) ≃
D.Hom X Y) →
(∀ X → _≃_.to (eqH X X) C.id ≡ D.id)
×
(∀ X Y Z (f : D.Hom Y Z) (g : D.Hom X Y) →
_≃_.to (eqH X Z) (C._∙_ (_≃_.from (eqH Y Z) f)
(_≃_.from (eqH X Y) g)) ≡
f D.∙ g))
↔
C ≡ D
equality-characterisation-Category {ℓ₂ = ℓ₂} {C} {D} ext univ₁ univ₂ =
_ ↝⟨ equality-characterisation-Precategory ext univ₁ univ₂ ⟩
C.precategory ≡ D.precategory ↝⟨ inverse $ ≡↔precategory≡precategory (lower-extensionality lzero (lsuc ℓ₂) ext) ⟩□
C ≡ D □
where
module C = Category C
module D = Category D
-- A lemma that can be used to turn a precategory into a category.
precategory-to-category :
∀ {c₁ c₂}
(C : Precategory c₁ c₂) →
let open Precategory C in
(≡≃≅ : ∀ {X Y} → (X ≡ Y) ≃ (X ≅ Y)) →
(∀ {X} → _≃_.to ≡≃≅ (refl X) ¹ ≡ id) →
Category c₁ c₂
precategory-to-category C ≡≃≅ ≡≃≅-refl = record
{ category = C , Precategory.≡→≅-equivalence-lemma C ≡≃≅ ≡≃≅-refl
}
-- A variant of the previous lemma for precategories with Set c₁ as
-- the type of objects. (The lemma is defined using extensionality and
-- univalence for sets.)
precategory-with-Set-to-category :
∀ {c₁ c₂} →
Extensionality c₁ c₁ →
((A B : Set c₁) → Univalence′ ⌞ A ⌟ ⌞ B ⌟) →
(C : Precategory-with-Obj (Set c₁) c₂) →
let open Precategory (record { precategory = _ , C }) in
(≃≃≅ : ∀ X Y → (⌞ X ⌟ ≃ ⌞ Y ⌟) ≃ (X ≅ Y)) →
(∀ X → _≃_.to (≃≃≅ X X) Eq.id ¹ ≡ id) →
Category (lsuc c₁) c₂
precategory-with-Set-to-category ext univ C ≃≃≅ ≃≃≅-id =
precategory-to-category C′ ≡≃≅ ≡≃≅-refl
where
C′ = record { precategory = _ , C }
open Precategory C′
-- _≡_ and _≅_ are pointwise equivalent…
cong-⌞⌟ : {X Y : Obj} → (X ≡ Y) ≃ (⌞ X ⌟ ≡ ⌞ Y ⌟)
cong-⌞⌟ = Eq.↔⇒≃ $ inverse $
ignore-propositional-component (H-level-propositional ext 2)
≡≃≅ : ∀ {X Y} → (X ≡ Y) ≃ (X ≅ Y)
≡≃≅ {X} {Y} = ≃≃≅ X Y ⊚ ≡≃≃ (univ X Y) ⊚ cong-⌞⌟
-- …and the proof maps reflexivity to the identity isomorphism.
≡≃≅-refl :
∀ {X} → _¹ {X = X} {Y = X} (_≃_.to ≡≃≅ (refl X)) ≡ id
≡≃≅-refl {X} = cong (_¹ {X = X} {Y = X}) (
_≃_.to (≃≃≅ X X) (≡⇒≃ (proj₁ (Σ-≡,≡←≡ (refl X)))) ≡⟨ cong (_≃_.to (≃≃≅ X X) ∘ ≡⇒≃ ∘ proj₁) Σ-≡,≡←≡-refl ⟩
_≃_.to (≃≃≅ X X) (≡⇒≃ (refl ⌞ X ⌟)) ≡⟨ cong (_≃_.to (≃≃≅ X X)) ≡⇒≃-refl ⟩
_≃_.to (≃≃≅ X X) Eq.id ≡⟨ _≃_.from (≡≃≡¹ {X = X} {Y = X}) $ ≃≃≅-id X ⟩∎
id≅ ∎)
-- An example: sets and functions. (Defined using extensionality and
-- univalence for sets.)
category-Set :
(ℓ : Level) →
Extensionality ℓ ℓ →
((A B : Set ℓ) → Univalence′ ⌞ A ⌟ ⌞ B ⌟) →
Category (lsuc ℓ) ℓ
category-Set ℓ ext univ =
precategory-with-Set-to-category
ext
univ
(proj₂ precategory)
(≃≃≅-Set ℓ ext)
(λ _ → refl P.id)
where
C = precategory-Set ℓ ext
open Precategory C
-- An example: sets and bijections. (Defined using extensionality and
-- univalence for sets.)
category-Set-≅ :
(ℓ : Level) →
Extensionality ℓ ℓ →
((A B : Set ℓ) → Univalence′ ⌞ A ⌟ ⌞ B ⌟) →
Category (lsuc ℓ) ℓ
category-Set-≅ ℓ ext univ =
Category.category-≅ (category-Set ℓ ext univ)
private
-- The objects are sets.
Obj-category-Set-≅ :
∀ ℓ (ext : Extensionality ℓ ℓ)
(univ : (A B : Set ℓ) → Univalence′ ⌞ A ⌟ ⌞ B ⌟) →
Category.Obj (category-Set-≅ ℓ ext univ) ≡ Set ℓ
Obj-category-Set-≅ _ _ _ = refl _
-- The morphisms are bijections.
Hom-category-Set-≅ :
∀ ℓ (ext : Extensionality ℓ ℓ)
(univ : (A B : Set ℓ) → Univalence′ ⌞ A ⌟ ⌞ B ⌟) →
Category.Hom (category-Set-≅ ℓ ext univ) ≡
Category._≅_ (category-Set ℓ ext univ)
Hom-category-Set-≅ _ _ _ = refl _
-- A trivial category (with a singleton type of objects and singleton
-- homsets).
Unit : ∀ ℓ₁ ℓ₂ → Category ℓ₁ ℓ₂
Unit ℓ₁ ℓ₂ =
precategory-to-category record
{ precategory =
↑ ℓ₁ ⊤
, (λ _ _ → ↑ ℓ₂ ⊤ , ↑⊤-set)
, _
, _
, refl _
, refl _
, refl _
}
(λ {x y} →
x ≡ y ↔⟨ ≡↔⊤ ⟩
⊤ ↔⟨ inverse ≡↔⊤ ⟩
lift tt ≡ lift tt ↔⟨ inverse $ drop-⊤-left-Σ ≡↔⊤ ⟩
lift tt ≡ lift tt × lift tt ≡ lift tt ↔⟨ inverse $ drop-⊤-left-Σ Bijection.↑↔ ⟩
↑ ℓ₂ ⊤ × lift tt ≡ lift tt × lift tt ≡ lift tt ↔⟨ inverse $ drop-⊤-left-Σ Bijection.↑↔ ⟩□
↑ ℓ₂ ⊤ × ↑ ℓ₂ ⊤ × lift tt ≡ lift tt × lift tt ≡ lift tt □)
(refl _)
where
↑⊤-set : ∀ {ℓ} → Is-set (↑ ℓ ⊤)
↑⊤-set = mono (Nat.zero≤ 2) (↑-closure 0 ⊤-contractible)
≡↔⊤ : ∀ {ℓ} {x y : ↑ ℓ ⊤} → (x ≡ y) ↔ ⊤
≡↔⊤ = _⇔_.to contractible⇔↔⊤ $
propositional⇒inhabited⇒contractible ↑⊤-set (refl _)
-- An "empty" category, without objects.
Empty : ∀ ℓ₁ ℓ₂ → Category ℓ₁ ℓ₂
Empty ℓ₁ ℓ₂ =
precategory-to-category record
{ precategory =
⊥
, ⊥-elim
, (λ {x} → ⊥-elim x)
, (λ {x} → ⊥-elim x)
, (λ {x} → ⊥-elim x)
, (λ {x} → ⊥-elim x)
, (λ {x} → ⊥-elim x)
}
(λ {x} → ⊥-elim x)
(λ {x} → ⊥-elim x)
-- Lifts a category's object type.
lift-category-Obj :
∀ {ℓ₁} ℓ₁′ {ℓ₂} →
Category ℓ₁ ℓ₂ → Category (ℓ₁ ⊔ ℓ₁′) ℓ₂
lift-category-Obj ℓ₁′ C .Category.category =
C′
, ≡→≅-equivalence
where
C′ = lift-precategory-Obj ℓ₁′ (Category.precategory C)
module C = Category C
module C′ = Precategory C′
≡→≅-equivalence :
{X Y : Precategory.Obj C′} →
Is-equivalence (C′.≡→≅ {X = X} {Y = Y})
≡→≅-equivalence {X = X} {Y = Y} =
_≃_.is-equivalence $
Eq.with-other-function
(X ≡ Y ↝⟨ inverse $ Eq.≃-≡ $ Eq.↔⇒≃ Bijection.↑↔ ⟩
lower X ≡ lower Y ↝⟨ Eq.⟨ _ , C.≡→≅-equivalence ⟩ ⟩
lower X C.≅ lower Y ↔⟨⟩
X C′.≅ Y □)
C′.≡→≅
(elim (λ X≡Y → C.≡→≅ (cong lower X≡Y) ≡ C′.≡→≅ X≡Y)
(λ X →
C.≡→≅ (cong lower (refl X)) ≡⟨ cong C.≡→≅ $ cong-refl lower ⟩
C.≡→≅ (refl (lower X)) ≡⟨ C.≡→≅-refl ⟩
C.id≅ ≡⟨⟩
C′.id≅ ≡⟨ sym C′.≡→≅-refl ⟩∎
C′.≡→≅ (refl X) ∎))
-- Lifts a category's morphism type family.
lift-category-Hom :
∀ {ℓ₁ ℓ₂} ℓ₂′ →
Category ℓ₁ ℓ₂ → Category ℓ₁ (ℓ₂ ⊔ ℓ₂′)
lift-category-Hom ℓ₂′ C .Category.category =
C′
, ≡→≅-equivalence
where
C′ = lift-precategory-Hom ℓ₂′ (Category.precategory C)
module C = Category C
module C′ = Precategory C′
≡→≅-equivalence :
{X Y : Precategory.Obj C′} →
Is-equivalence (C′.≡→≅ {X = X} {Y = Y})
≡→≅-equivalence {X = X} {Y = Y} =
_≃_.is-equivalence $
Eq.with-other-function
(X ≡ Y ↝⟨ Eq.⟨ _ , C.≡→≅-equivalence ⟩ ⟩
X C.≅ Y ↝⟨ equiv ⟩□
X C′.≅ Y □)
C′.≡→≅
(elim (λ X≡Y → _≃_.to equiv (C.≡→≅ X≡Y) ≡ C′.≡→≅ X≡Y)
(λ X →
_≃_.to equiv (C.≡→≅ (refl X)) ≡⟨ cong (_≃_.to equiv) C.≡→≅-refl ⟩
_≃_.to equiv C.id≅ ≡⟨ _≃_.from C′.≡≃≡¹ (refl _) ⟩
C′.id≅ ≡⟨ sym C′.≡→≅-refl ⟩∎
C′.≡→≅ (refl X) ∎))
where
equiv : ∀ {X Y} → (X C.≅ Y) ≃ (X C′.≅ Y)
equiv =
Σ-cong (inverse Bijection.↑↔) λ _ →
Σ-cong (inverse Bijection.↑↔) λ _ →
(Eq.≃-≡ $ Eq.↔⇒≃ Bijection.↑↔)
×-cong
(Eq.≃-≡ $ Eq.↔⇒≃ Bijection.↑↔)
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module Parse where
open import Common.Unit
open import Common.Char
open import Common.String
open import Common.List
open import Common.IO
parse : List String → List Char → String
parse (e ∷ []) [] = "ha"
parse (e ∷ []) (')' ∷ xs) = "ho"
parse (e ∷ es) (a ∷ xs) = parse (e ∷ es) xs
parse _ _ = "hi"
parseRegExp : String
parseRegExp = parse ("ff" ∷ []) ('a' ∷ [])
main : _
main = do
let w = parseRegExp
putStrLn w
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module Structure.Setoid.Category where
open import Data
import Data.Tuple as Tuple
open import Functional
open import Function.Equals
open import Function.Equals.Proofs
open import Function.Proofs
open import Logic.Predicate
import Lvl
open import Structure.Category
open import Structure.Categorical.Properties
open import Structure.Function
open import Structure.Operator
open import Structure.Relator.Equivalence
open import Structure.Relator.Properties
open import Structure.Setoid
open import Type
open import Type.Properties.Singleton
private variable ℓ ℓₑ ℓₒ : Lvl.Level
-- TODO: Maybe move this?
FunctionObject : Setoid{ℓₑ}{ℓₒ} → Setoid{ℓₑ}{ℓₒ} → Type
FunctionObject ([∃]-intro A) ([∃]-intro B) = ∃{Obj = (A → B)} Function
instance
FunctionObject-equiv : ∀{A B : Setoid{ℓₑ}{ℓₒ}} → Equiv(FunctionObject A B)
Equiv._≡_ FunctionObject-equiv = (_⊜_) on₂ [∃]-witness
Reflexivity.proof (Equivalence.reflexivity (Equiv.equivalence FunctionObject-equiv)) = reflexivity(_⊜_)
Symmetry.proof (Equivalence.symmetry (Equiv.equivalence FunctionObject-equiv)) = symmetry(_⊜_)
Transitivity.proof (Equivalence.transitivity (Equiv.equivalence FunctionObject-equiv)) = transitivity(_⊜_)
-- The setoid category contains setoids and functions respecting the congruence property in the setoid.
setoidCategory : Category{Obj = Setoid{ℓₑ}{ℓₒ}} FunctionObject
Category._∘_ setoidCategory ([∃]-intro f) ([∃]-intro g) = [∃]-intro (f ∘ g) ⦃ [∘]-function {f = f}{g = g} ⦄
Category.id setoidCategory = [∃]-intro id
BinaryOperator.congruence (Category.binaryOperator setoidCategory) f₁f₂ g₁g₂ = [⊜][∘]-binaryOperator-raw f₁f₂ g₁g₂
Morphism.Associativity.proof (Category.associativity setoidCategory) {x = _} {y = _} {z = _} {x = [∃]-intro f} {y = [∃]-intro g} {z = [∃]-intro h} = [∘]-associativity {f = f} {g = g} {h = h}
Morphism.Identityₗ.proof (Tuple.left (Category.identity setoidCategory)) = [∘]-identityₗ
Morphism.Identityᵣ.proof (Tuple.right (Category.identity setoidCategory)) = [∘]-identityᵣ
setoidCategoryObject : ∀{ℓₑ}{ℓₒ} → CategoryObject
setoidCategoryObject{ℓₑ}{ℓₒ} = intro(setoidCategory{ℓₑ}{ℓₒ})
module _ where
open import Data.Proofs
open import Relator.Equals
Empty-initialObject : Object.Initial(FunctionObject{ℓₑ}) ([∃]-intro Empty)
IsUnit.unit Empty-initialObject = [∃]-intro empty
_⊜_.proof (IsUnit.uniqueness Empty-initialObject) {}
Unit-terminalObject : Object.Terminal(FunctionObject{ℓₑ}) ([∃]-intro Unit)
IsUnit.unit Unit-terminalObject = [∃]-intro (const <>)
_⊜_.proof (IsUnit.uniqueness Unit-terminalObject) = [≡]-intro
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties for Conats
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe --sized-types #-}
module Codata.Conat.Properties where
open import Size
open import Data.Nat.Base using (ℕ; zero; suc)
open import Codata.Thunk
open import Codata.Conat
open import Codata.Conat.Bisimilarity
open import Function
open import Relation.Nullary
open import Relation.Nullary.Decidable using (map′)
open import Relation.Binary
private
variable
i : Size
0∸m≈0 : ∀ m → i ⊢ zero ∸ m ≈ zero
0∸m≈0 zero = refl
0∸m≈0 (suc m) = 0∸m≈0 m
sℕ≤s⁻¹ : ∀ {m n} → suc m ℕ≤ suc n → m ℕ≤ n .force
sℕ≤s⁻¹ (sℕ≤s p) = p
_ℕ≤?_ : Decidable _ℕ≤_
zero ℕ≤? n = yes zℕ≤n
suc m ℕ≤? zero = no (λ ())
suc m ℕ≤? suc n = map′ sℕ≤s sℕ≤s⁻¹ (m ℕ≤? n .force)
0ℕ+-identity : ∀ {n} → i ⊢ 0 ℕ+ n ≈ n
0ℕ+-identity = refl
+ℕ0-identity : ∀ {n} → i ⊢ n +ℕ 0 ≈ n
+ℕ0-identity {n = zero} = zero
+ℕ0-identity {n = suc n} = suc λ where .force → +ℕ0-identity
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------------------------------------------------------------------------
-- The semantics is deterministic
------------------------------------------------------------------------
open import Atom
module Deterministic (atoms : χ-atoms) where
open import Equality.Propositional
open import Prelude hiding (const)
open import Tactic.By.Propositional
open import Chi atoms
open χ-atoms atoms
Lookup-deterministic :
∀ {c₁ c₂ bs xs₁ xs₂ e₁ e₂} →
Lookup c₁ bs xs₁ e₁ → Lookup c₂ bs xs₂ e₂ →
c₁ ≡ c₂ → xs₁ ≡ xs₂ × e₁ ≡ e₂
Lookup-deterministic here here _ = refl , refl
Lookup-deterministic here (there q _) refl = ⊥-elim (q refl)
Lookup-deterministic (there p _) here refl = ⊥-elim (p refl)
Lookup-deterministic (there p₁ p₂) (there q₁ q₂) refl =
Lookup-deterministic p₂ q₂ refl
↦-deterministic :
∀ {e xs es e₁ e₂} →
e [ xs ← es ]↦ e₁ → e [ xs ← es ]↦ e₂ → e₁ ≡ e₂
↦-deterministic [] [] = refl
↦-deterministic (∷ p) (∷ q) = by (↦-deterministic p q)
mutual
⇓-deterministic : ∀ {e v₁ v₂} → e ⇓ v₁ → e ⇓ v₂ → v₁ ≡ v₂
⇓-deterministic (apply p₁ p₂ p₃) (apply q₁ q₂ q₃)
with ⇓-deterministic p₁ q₁ | ⇓-deterministic p₂ q₂
... | refl | refl = ⇓-deterministic p₃ q₃
⇓-deterministic (case p₁ p₂ p₃ p₄) (case q₁ q₂ q₃ q₄)
with ⇓-deterministic p₁ q₁
... | refl with Lookup-deterministic p₂ q₂ refl
... | refl , refl rewrite ↦-deterministic p₃ q₃ =
⇓-deterministic p₄ q₄
⇓-deterministic (rec p) (rec q) = ⇓-deterministic p q
⇓-deterministic lambda lambda = refl
⇓-deterministic (const ps) (const qs) = by (⇓⋆-deterministic ps qs)
⇓⋆-deterministic : ∀ {es vs₁ vs₂} → es ⇓⋆ vs₁ → es ⇓⋆ vs₂ → vs₁ ≡ vs₂
⇓⋆-deterministic [] [] = refl
⇓⋆-deterministic (p ∷ ps) (q ∷ qs) =
cong₂ _∷_ (⇓-deterministic p q) (⇓⋆-deterministic ps qs)
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{-# OPTIONS --allow-unsolved-metas #-}
module _ where
postulate
Functor : (Set → Set) → Set₁
fmap : {F : Set → Set} {{_ : Functor F}} {A B : Set} → (A → B) → F A → F B
postulate
Id : Set → Set
bla : {A : Set} → Id A → Id A
bla = fmap {{?}} (λ x → x) -- should not fail!
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{-
Finitely presented algebras.
An R-algebra A is finitely presented, if there merely is an exact sequence
of R-modules:
(f₁,⋯,fₘ) → R[X₁,⋯,Xₙ] → A → 0
(where f₁,⋯,fₘ ∈ R[X₁,⋯,Xₙ])
-}
{-# OPTIONS --safe #-}
module Cubical.Algebra.CommAlgebra.FPAlgebra where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Powerset
open import Cubical.Foundations.Function
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Structure
open import Cubical.Data.FinData
open import Cubical.Data.Nat
open import Cubical.Data.Vec
open import Cubical.Data.Sigma
open import Cubical.Data.Empty
open import Cubical.HITs.PropositionalTruncation
open import Cubical.Algebra.CommRing
open import Cubical.Algebra.CommRing.FGIdeal using (inclOfFGIdeal)
open import Cubical.Algebra.CommAlgebra
open import Cubical.Algebra.CommAlgebra.FreeCommAlgebra
renaming (inducedHom to freeInducedHom)
open import Cubical.Algebra.CommAlgebra.QuotientAlgebra
renaming (inducedHom to quotientInducedHom)
open import Cubical.Algebra.CommAlgebra.Ideal
open import Cubical.Algebra.CommAlgebra.FGIdeal
open import Cubical.Algebra.CommAlgebra.Instances.Initial
open import Cubical.Algebra.CommAlgebra.Instances.Unit
renaming (UnitCommAlgebra to TerminalCAlg)
open import Cubical.Algebra.CommAlgebra.Kernel
open import Cubical.Algebra.Algebra.Properties
open import Cubical.Algebra.Algebra
private
variable
ℓ : Level
module _ {R : CommRing ℓ} where
open Construction using (var)
Polynomials : (n : ℕ) → CommAlgebra R ℓ
Polynomials n = R [ Fin n ]
evPoly : {n : ℕ} (A : CommAlgebra R ℓ) → ⟨ Polynomials n ⟩ → FinVec ⟨ A ⟩ n → ⟨ A ⟩
evPoly A P values = fst (freeInducedHom A values) P
evPolyPoly : {n : ℕ} (P : ⟨ Polynomials n ⟩) → evPoly (Polynomials n) P var ≡ P
evPolyPoly {n = n} P = cong (λ u → fst u P) (inducedHomVar R (Fin n))
evPolyHomomorphic : {n : ℕ} (A B : CommAlgebra R ℓ) (f : CommAlgebraHom A B)
→ (P : ⟨ Polynomials n ⟩) → (values : FinVec ⟨ A ⟩ n)
→ (fst f) (evPoly A P values) ≡ evPoly B P (fst f ∘ values)
evPolyHomomorphic A B f P values =
(fst f) (evPoly A P values) ≡⟨ refl ⟩
(fst f) (fst (freeInducedHom A values) P) ≡⟨ refl ⟩
fst (f ∘a freeInducedHom A values) P ≡⟨ cong (λ u → fst u P) (natIndHomR f values) ⟩
fst (freeInducedHom B (fst f ∘ values)) P ≡⟨ refl ⟩
evPoly B P (fst f ∘ values) ∎
where open AlgebraHoms
module _ {m : ℕ} (n : ℕ) (relation : FinVec ⟨ Polynomials n ⟩ m) where
open CommAlgebraStr using (0a)
open Cubical.Algebra.Algebra.Properties.AlgebraHoms
relationsIdeal = generatedIdeal (Polynomials n) relation
abstract
{-
The following definitions are abstract because of type checking speed
problems - complete unfolding of FPAlgebra is triggered otherwise.
This also means, the where blocks contain more type declarations than usual.
-}
FPAlgebra : CommAlgebra R ℓ
FPAlgebra = Polynomials n / relationsIdeal
modRelations : CommAlgebraHom (Polynomials n) (Polynomials n / relationsIdeal)
modRelations = quotientMap (Polynomials n) relationsIdeal
generator : (i : Fin n) → ⟨ FPAlgebra ⟩
generator = fst modRelations ∘ var
relationsHold : (i : Fin m) → evPoly FPAlgebra (relation i) generator ≡ 0a (snd FPAlgebra)
relationsHold i =
evPoly FPAlgebra (relation i) generator
≡⟨ sym (evPolyHomomorphic (Polynomials n) FPAlgebra modRelations (relation i) var) ⟩
fst modRelations (evPoly (Polynomials n) (relation i) var)
≡⟨ cong (λ u → fst modRelations u) (evPolyPoly (relation i)) ⟩
fst modRelations (relation i)
≡⟨ isZeroFromIdeal {R = R}
{A = (Polynomials n)}
{I = relationsIdeal}
(relation i)
(incInIdeal (Polynomials n) relation i ) ⟩
0a (snd FPAlgebra) ∎
inducedHom :
(A : CommAlgebra R ℓ)
(values : FinVec ⟨ A ⟩ n)
(relationsHold : (i : Fin m) → evPoly A (relation i) values ≡ 0a (snd A))
→ CommAlgebraHom FPAlgebra A
inducedHom A values relationsHold =
quotientInducedHom
(Polynomials n)
relationsIdeal
A
freeHom
isInKernel
where
freeHom : CommAlgebraHom (Polynomials n) A
freeHom = freeInducedHom A values
isInKernel : fst (generatedIdeal (Polynomials n) relation)
⊆ fst (kernel (Polynomials n) A freeHom)
isInKernel = inclOfFGIdeal
(CommAlgebra→CommRing (Polynomials n))
relation
(kernel (Polynomials n) A freeHom)
relationsHold
inducedHomOnGenerators :
(A : CommAlgebra R ℓ)
(values : FinVec ⟨ A ⟩ n)
(relationsHold : (i : Fin m) → evPoly A (relation i) values ≡ 0a (snd A))
(i : Fin n)
→ fst (inducedHom A values relationsHold) (generator i) ≡ values i
inducedHomOnGenerators _ _ _ _ = refl
unique :
{A : CommAlgebra R ℓ}
(values : FinVec ⟨ A ⟩ n)
(relationsHold : (i : Fin m) → evPoly A (relation i) values ≡ 0a (snd A))
(f : CommAlgebraHom FPAlgebra A)
→ ((i : Fin n) → fst f (generator i) ≡ values i)
→ inducedHom A values relationsHold ≡ f
unique {A = A} values relationsHold f hasCorrectValues =
injectivePrecomp
(Polynomials n)
relationsIdeal
A
(inducedHom A values relationsHold)
f
(sym (
f' ≡⟨ sym (inv f') ⟩
freeInducedHom A (evaluateAt A f') ≡⟨ cong (freeInducedHom A) (funExt hasCorrectValues) ⟩
freeInducedHom A values ≡⟨ cong (freeInducedHom A) refl ⟩
freeInducedHom A (evaluateAt A iHom') ≡⟨ inv iHom' ⟩
iHom' ∎))
where
{-
Poly n
| \
modRelations f'
↓ ↘
FPAlgebra ─f→ A
-}
f' iHom' : CommAlgebraHom (Polynomials n) A
f' = compAlgebraHom modRelations f
iHom' = compAlgebraHom modRelations (inducedHom A values relationsHold)
inv : retract (Iso.fun (homMapIso {I = Fin n} A)) (Iso.inv (homMapIso A))
inv = Iso.leftInv (homMapIso {R = R} {I = Fin n} A)
{- ∀ A : Comm-R-Algebra,
∀ J : Finitely-generated-Ideal,
Hom(R[I]/J,A) is isomorphic to the Set of roots of the generators of J
-}
zeroLocus : (A : CommAlgebra R ℓ) → Type ℓ
zeroLocus A = Σ[ v ∈ FinVec ⟨ A ⟩ n ] ((i : Fin m) → evPoly A (relation i) v ≡ 0a (snd A))
inducedHomFP : (A : CommAlgebra R ℓ) →
zeroLocus A → CommAlgebraHom FPAlgebra A
inducedHomFP A d = inducedHom A (fst d) (snd d)
evaluateAtFP : {A : CommAlgebra R ℓ} →
CommAlgebraHom FPAlgebra A → zeroLocus A
evaluateAtFP {A} f = value ,
λ i → evPoly A (relation i) value ≡⟨ step1 (relation i) ⟩
fst compHom (evPoly (Polynomials n) (relation i) var) ≡⟨ refl ⟩
(fst f) ((fst modRelations)
(evPoly (Polynomials n) (relation i) var)) ≡⟨ cong (fst f)
(evPolyHomomorphic
(Polynomials n)
FPAlgebra
modRelations
(relation i) var) ⟩
(fst f) (evPoly FPAlgebra (relation i) generator) ≡⟨ cong (fst f) (relationsHold i) ⟩
(fst f) (0a (snd FPAlgebra)) ≡⟨ IsAlgebraHom.pres0 (snd f) ⟩
0a (snd A) ∎
where
compHom : CommAlgebraHom (Polynomials n) A
compHom = CommAlgebraHoms.compCommAlgebraHom (Polynomials n) FPAlgebra A modRelations f
value : FinVec ⟨ A ⟩ n
value = (Iso.fun (homMapIso A)) compHom
step1 : (x : ⟨ Polynomials n ⟩) → evPoly A x value ≡ fst compHom (evPoly (Polynomials n) x var)
step1 x = sym (evPolyHomomorphic (Polynomials n) A compHom x var)
FPHomIso : {A : CommAlgebra R ℓ} →
Iso (CommAlgebraHom FPAlgebra A) (zeroLocus A)
Iso.fun FPHomIso = evaluateAtFP
Iso.inv FPHomIso = inducedHomFP _
Iso.rightInv (FPHomIso {A}) =
λ b → Σ≡Prop
(λ x → isPropΠ
(λ i → isSetCommAlgebra A
(evPoly A (relation i) x)
(0a (snd A))))
refl
Iso.leftInv (FPHomIso {A}) =
λ a → Σ≡Prop (λ f → isPropIsCommAlgebraHom {ℓ} {R} {ℓ} {ℓ} {FPAlgebra} {A} f)
λ i → fst (unique {A}
(fst (evaluateAtFP {A} a))
(snd (evaluateAtFP a))
a
(λ j → refl)
i)
homMapPathFP : (A : CommAlgebra R ℓ)→ CommAlgebraHom FPAlgebra A ≡ zeroLocus A
homMapPathFP A = isoToPath (FPHomIso {A})
isSetZeroLocus : (A : CommAlgebra R ℓ) → isSet (zeroLocus A)
isSetZeroLocus A = J (λ y _ → isSet y)
(isSetAlgebraHom (CommAlgebra→Algebra FPAlgebra) (CommAlgebra→Algebra A))
(homMapPathFP A)
record FinitePresentation (A : CommAlgebra R ℓ) : Type ℓ where
field
n : ℕ
m : ℕ
relations : FinVec ⟨ Polynomials n ⟩ m
equiv : CommAlgebraEquiv (FPAlgebra n relations) A
isFPAlgebra : (A : CommAlgebra R ℓ) → Type _
isFPAlgebra A = ∥ FinitePresentation A ∥₁
isFPAlgebraIsProp : {A : CommAlgebra R ℓ} → isProp (isFPAlgebra A)
isFPAlgebraIsProp = isPropPropTrunc
module Instances {R : CommRing ℓ} where
open FinitePresentation
{- Every (multivariate) polynomial algebra is finitely presented -}
module _ (n : ℕ) where
private
A : CommAlgebra R ℓ
A = Polynomials n
emptyGen : FinVec (fst A) 0
emptyGen = λ ()
B : CommAlgebra R ℓ
B = FPAlgebra n emptyGen
polynomialAlgFP : FinitePresentation A
FinitePresentation.n polynomialAlgFP = n
m polynomialAlgFP = 0
relations polynomialAlgFP = emptyGen
equiv polynomialAlgFP =
-- Idea: A and B enjoy the same universal property.
toAAsEquiv , snd toA
where
toA : CommAlgebraHom B A
toA = inducedHom n emptyGen A Construction.var (λ ())
fromA : CommAlgebraHom A B
fromA = freeInducedHom B (generator _ _)
open AlgebraHoms
inverse1 : fromA ∘a toA ≡ idAlgebraHom _
inverse1 =
fromA ∘a toA
≡⟨ sym (unique _ _ _ _ _ (λ i → cong (fst fromA) (
fst toA (generator n emptyGen i)
≡⟨ inducedHomOnGenerators _ _ _ _ _ _ ⟩
Construction.var i
∎))) ⟩
inducedHom n emptyGen B (generator _ _) (relationsHold _ _)
≡⟨ unique _ _ _ _ _ (λ i → refl) ⟩
idAlgebraHom _
∎
inverse2 : toA ∘a fromA ≡ idAlgebraHom _
inverse2 = isoFunInjective (homMapIso A) _ _ (
evaluateAt A (toA ∘a fromA) ≡⟨ sym (naturalEvR {A = B} {B = A} toA fromA) ⟩
fst toA ∘ evaluateAt B fromA ≡⟨ refl ⟩
fst toA ∘ generator _ _ ≡⟨ funExt (inducedHomOnGenerators _ _ _ _ _)⟩
Construction.var ∎)
toAAsEquiv : ⟨ B ⟩ ≃ ⟨ A ⟩
toAAsEquiv = isoToEquiv (iso (fst toA)
(fst fromA)
(λ a i → fst (inverse2 i) a)
(λ b i → fst (inverse1 i) b))
{- The initial R-algebra is finitely presented -}
private
R[⊥] : CommAlgebra R ℓ
R[⊥] = Polynomials 0
emptyGen : FinVec (fst R[⊥]) 0
emptyGen = λ ()
R[⊥]/⟨0⟩ : CommAlgebra R ℓ
R[⊥]/⟨0⟩ = FPAlgebra 0 emptyGen
R[⊥]/⟨0⟩IsInitial : (B : CommAlgebra R ℓ)
→ isContr (CommAlgebraHom R[⊥]/⟨0⟩ B)
R[⊥]/⟨0⟩IsInitial B = iHom , uniqueness
where
iHom : CommAlgebraHom R[⊥]/⟨0⟩ B
iHom = inducedHom 0 emptyGen B (λ ()) (λ ())
uniqueness : (f : CommAlgebraHom R[⊥]/⟨0⟩ B) →
iHom ≡ f
uniqueness f = unique 0 emptyGen {A = B} (λ ()) (λ ()) f (λ ())
initialCAlgFP : FinitePresentation (initialCAlg R)
n initialCAlgFP = 0
m initialCAlgFP = 0
relations initialCAlgFP = emptyGen
equiv initialCAlgFP =
equivByInitiality R R[⊥]/⟨0⟩ R[⊥]/⟨0⟩IsInitial
{- The terminal R-algebra is finitely presented -}
private
unitGen : FinVec (fst R[⊥]) 1
unitGen zero = 1a
where open CommAlgebraStr (snd R[⊥])
R[⊥]/⟨1⟩ : CommAlgebra R ℓ
R[⊥]/⟨1⟩ = FPAlgebra 0 unitGen
terminalCAlgFP : FinitePresentation (TerminalCAlg R)
n terminalCAlgFP = 0
m terminalCAlgFP = 1
relations terminalCAlgFP = unitGen
equiv terminalCAlgFP = equivFrom1≡0 R R[⊥]/⟨1⟩ (sym (⋆-lid 1a) ∙ relationsHold 0 unitGen zero)
where open CommAlgebraStr (snd R[⊥]/⟨1⟩)
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{-# OPTIONS --without-K #-}
open import HoTT.Base
open import HoTT.Equivalence
open import HoTT.Equivalence.Lift
open import HoTT.Identity.Sigma
open import HoTT.Identity.Universe
open import HoTT.HLevel
open import HoTT.Logic
module HoTT.Exercises.Chapter3.Exercise10 {i} where
postulate
lem : LEM {lsuc i}
_ : Prop𝒰 i ≃ Prop𝒰 (lsuc i)
_ = f , qinv→isequiv (g , η , ε)
where
f : _
f P = LiftProp P
g : _
g P with lem P
... | inl _ = ⊤
... | inr _ = ⊥
η : g ∘ f ~ id
η P with lem (f P)
... | inl t = hlevel⁼ (ua (prop-equiv (const (lower t)) (const ★)))
... | inr f = hlevel⁼ (ua (prop-equiv 𝟎-rec (𝟎-rec ∘ f ∘ lift)))
ε : f ∘ g ~ id
ε P with lem P
... | inl t = hlevel⁼ (ua (Lift-equiv ∙ₑ prop-equiv (const t) (const ★)))
... | inr f = hlevel⁼ (ua (Lift-equiv ∙ₑ prop-equiv 𝟎-rec (𝟎-rec ∘ f)))
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{-# OPTIONS --without-K #-}
module horner where
open import Type
open import Type.Identities
open import Function.NP
open import Function.Extensionality
open import Data.Fin.NP using (Fin; Fin▹ℕ)
open import Data.Product renaming (proj₁ to fst; proj₂ to snd) hiding (map)
open import Data.Zero
open import Data.One
open import Data.Sum hiding (map)
open import Data.Nat.NP
open import Data.Nat.Properties
import Data.List as L
import Data.List.Properties as LP
open L using (List; []; _∷_)
open import Relation.Binary.PropositionalEquality.NP
open import HoTT
--open import Explore.Fin
ℕ< : ℕ → Set
ℕ< n = Σ ℕ λ x → x < n
sumFin : (n : ℕ) (f : Fin n → ℕ) → ℕ
sumFin n f = {!!}
sum< : (n : ℕ) (f : ℕ< n → ℕ) → ℕ
sum< n f = {!!}
prod< : (n : ℕ) (f : ℕ< n → ℕ) → ℕ
prod< n f = {!!}
{-
foo : ∀ n a x → sumFin n λ i → a i * x ^ i
foo = ?
bar : ∀ n a x → sumFin n λ i → a i * x ^ i
bar = ?
-}
baz : ∀ n (u : ℕ< n → ℕ) → (sum< n λ { (i , p) → prod< i (λ { (j , q) → u (j , <-trans q p) }) }) ≡ {!!}
baz = {!!}
module _ n (u : ℕ< n → Set) {{_ : UA}} {{_ : FunExt}} where
open ≡-Reasoning
Baz : _ ≡ _
Baz = (Σ (ℕ< n) λ { (i , p) → Π (ℕ< i) λ { (j , q) → u (j , <-trans q p) } })
≡⟨ ! Σ-assoc ⟩
(Σ ℕ λ i → Σ (i < n) λ p → Π (ℕ< i) λ { (j , q) → u (j , <-trans q p) })
≡⟨ Σ=′ ℕ (λ i → Σ=′ (i < n) λ p → ΠΣ-curry) ⟩
(Σ ℕ λ i → Σ (i < n) λ p → Π ℕ λ j → Π (j < i) λ q → u (j , <-trans q p))
∎
module DataVersion (A : ★) where
open import Data.Tree.Binary
data T : BinTree A → ★ where
empty : T empty
_⊕_ : ∀ {t u} → (𝟙 ⊎ T t × T u) → T (fork t u)
module TypeVersion where
ε = 𝟙
_⊕_ : ★ → ★ → ★
_⊕_ = λ u z → ε ⊎ u × z
module ListVersion where
open L
open ≡-Reasoning
map-∘ = LP.map-compose
sum-lin : ∀ k xs → sum (map (_*_ k) xs) ≡ k * sum xs
sum-lin k [] = ℕ°.*-comm 0 k
sum-lin k (x ∷ xs) = k * x + sum (map (_*_ k) xs)
≡⟨ ap (_+_ (k * x)) (sum-lin k xs) ⟩
k * x + k * sum xs
≡⟨ ! fst ℕ°.distrib k x (sum xs) ⟩
k * (x + sum xs)
∎
lemma : ∀ x xss → sum (map product (map (_∷_ x) xss)) ≡ x * sum (map product xss)
lemma x xss = sum (map product (map (_∷_ x) xss))
≡⟨ ap sum (! map-∘ xss) ⟩
sum (map (product ∘ _∷_ x) xss)
≡⟨by-definition⟩
sum (map (_*_ x ∘ product) xss)
≡⟨ ap sum (map-∘ xss) ⟩
sum (map (_*_ x) (map product xss))
≡⟨ sum-lin x (map product xss) ⟩
x * sum (map product xss)
∎
ε = 1
_⊕_ = λ u z → ε + u * z
t3 = ∀ xs → sum (map product (inits xs)) ≡ foldr _⊕_ ε xs
t4 : t3
t4 [] = refl
t4 (x ∷ xs) = ap suc (lemma x (inits xs) ∙ ap (_*_ x) (t4 xs))
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{-# OPTIONS --without-K --safe #-}
module Data.Fin.Indexed.Properties where
open import Agda.Builtin.Nat using (_<_)
open import Data.Nat.Base
open import Data.Fin.Indexed.Base
open import Data.Bool
open import Data.Maybe.Base
private variable n m k : ℕ
weaken : ∀ {n} → Fin n → Fin (suc n)
weaken {suc n} f0 = f0
weaken {suc n} (fs x) = fs (weaken x)
-- x \\ y | x < y = just x
-- | x ≡ y = nothing
-- | x > y = just (x - 1)
_\\_ : Fin (suc n) → Fin (suc n) → Maybe (Fin n)
f0 \\ f0 = nothing
fs i \\ f0 = just i
_\\_ {suc n} (fs i) (fs j) = mapMaybe fs (i \\ j)
_\\_ {suc n} (f0 ) (fs j) = just f0
insert : Fin (suc n) → Fin n → Fin (suc n)
insert f0 j = fs j
insert (fs i) f0 = f0
insert (fs i) (fs j) = fs (insert i j)
weakens : ∀ n → Fin m → Fin (n + m)
weakens zero x = x
weakens (suc n) x = weaken (weakens n x)
_∔_ : Fin n → Fin m → Fin (n + m)
f0 ∔ m = weakens _ m
fs n ∔ m = fs (n ∔ m)
under : (Fin m → Fin k) → Fin (n + m) → Fin (n + k)
under {n = zero } f x = f x
under {n = suc n} f (fs x) = fs (under f x)
under {n = suc n} f f0 = f0
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-- {-# OPTIONS -v tc.pos:100 #-}
-- Records are allowed in mutual blocks.
module RecordInMutual where
import Common.Level
open import Common.Equality
mutual
record A : Set where
field p : D
record B : Set where
field q : A
data D : Set where
c : B -> D
open A
open B
-- A and B are guarded via D, so we have eta for A and for B:
etaA : {a : A} → a ≡ record { p = p a }
etaA = refl
etaB : {b : B} → b ≡ record { q = q b }
etaB = refl
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{-# OPTIONS --without-K --safe #-}
module Definition.Conversion.Weakening where
open import Definition.Untyped as U hiding (wk)
open import Definition.Untyped.Properties
open import Definition.Typed
open import Definition.Typed.Weakening
open import Definition.Typed.Consequences.Syntactic
open import Definition.Conversion
open import Definition.Conversion.Soundness
import Tools.PropositionalEquality as PE
open import Tools.Product
mutual
-- Weakening of algorithmic equality of neutrals.
wk~↑ : ∀ {ρ t u A Γ Δ} ([ρ] : ρ ∷ Δ ⊆ Γ) → ⊢ Δ
→ Γ ⊢ t ~ u ↑ A
→ Δ ⊢ U.wk ρ t ~ U.wk ρ u ↑ U.wk ρ A
wk~↑ {ρ} [ρ] ⊢Δ (var-refl x₁ x≡y) = var-refl (wkTerm [ρ] ⊢Δ x₁) (PE.cong (wkVar ρ) x≡y)
wk~↑ ρ ⊢Δ (app-cong {G = G} t~u x) =
PE.subst (λ x → _ ⊢ _ ~ _ ↑ x) (PE.sym (wk-β G))
(app-cong (wk~↓ ρ ⊢Δ t~u) (wkConv↑Term ρ ⊢Δ x))
wk~↑ ρ ⊢Δ (fst-cong p~r) =
fst-cong (wk~↓ ρ ⊢Δ p~r)
wk~↑ ρ ⊢Δ (snd-cong {G = G} p~r) =
PE.subst (λ x → _ ⊢ _ ~ _ ↑ x)
(PE.sym (wk-β G))
(snd-cong (wk~↓ ρ ⊢Δ p~r))
wk~↑ {ρ} {Δ = Δ} [ρ] ⊢Δ (natrec-cong {k} {l} {h} {g} {a₀} {b₀} {F} {G} x x₁ x₂ t~u) =
PE.subst (λ x → _ ⊢ U.wk ρ (natrec F a₀ h k) ~ _ ↑ x) (PE.sym (wk-β F))
(natrec-cong (wkConv↑ (lift [ρ]) (⊢Δ ∙ ℕⱼ ⊢Δ) x)
(PE.subst (λ x → _ ⊢ _ [conv↑] _ ∷ x) (wk-β F)
(wkConv↑Term [ρ] ⊢Δ x₁))
(PE.subst (λ x → Δ ⊢ U.wk ρ h [conv↑] U.wk ρ g ∷ x)
(wk-β-natrec _ F) (wkConv↑Term [ρ] ⊢Δ x₂))
(wk~↓ [ρ] ⊢Δ t~u))
wk~↑ {ρ} {Δ = Δ} [ρ] ⊢Δ (Emptyrec-cong {k} {l} {F} {G} x t~u) =
Emptyrec-cong (wkConv↑ [ρ] ⊢Δ x) (wk~↓ [ρ] ⊢Δ t~u)
-- Weakening of algorithmic equality of neutrals in WHNF.
wk~↓ : ∀ {ρ t u A Γ Δ} ([ρ] : ρ ∷ Δ ⊆ Γ) → ⊢ Δ
→ Γ ⊢ t ~ u ↓ A
→ Δ ⊢ U.wk ρ t ~ U.wk ρ u ↓ U.wk ρ A
wk~↓ {ρ} [ρ] ⊢Δ ([~] A₁ D whnfA k~l) =
[~] (U.wk ρ A₁) (wkRed* [ρ] ⊢Δ D) (wkWhnf ρ whnfA) (wk~↑ [ρ] ⊢Δ k~l)
-- Weakening of algorithmic equality of types.
wkConv↑ : ∀ {ρ A B Γ Δ} ([ρ] : ρ ∷ Δ ⊆ Γ) → ⊢ Δ
→ Γ ⊢ A [conv↑] B
→ Δ ⊢ U.wk ρ A [conv↑] U.wk ρ B
wkConv↑ {ρ} [ρ] ⊢Δ ([↑] A′ B′ D D′ whnfA′ whnfB′ A′<>B′) =
[↑] (U.wk ρ A′) (U.wk ρ B′) (wkRed* [ρ] ⊢Δ D) (wkRed* [ρ] ⊢Δ D′)
(wkWhnf ρ whnfA′) (wkWhnf ρ whnfB′) (wkConv↓ [ρ] ⊢Δ A′<>B′)
-- Weakening of algorithmic equality of types in WHNF.
wkConv↓ : ∀ {ρ A B Γ Δ} ([ρ] : ρ ∷ Δ ⊆ Γ) → ⊢ Δ
→ Γ ⊢ A [conv↓] B
→ Δ ⊢ U.wk ρ A [conv↓] U.wk ρ B
wkConv↓ ρ ⊢Δ (U-refl x) = U-refl ⊢Δ
wkConv↓ ρ ⊢Δ (ℕ-refl x) = ℕ-refl ⊢Δ
wkConv↓ ρ ⊢Δ (Empty-refl x) = Empty-refl ⊢Δ
wkConv↓ ρ ⊢Δ (Unit-refl x) = Unit-refl ⊢Δ
wkConv↓ ρ ⊢Δ (ne x) = ne (wk~↓ ρ ⊢Δ x)
wkConv↓ ρ ⊢Δ (Π-cong x A<>B A<>B₁) =
let ⊢ρF = wk ρ ⊢Δ x
in Π-cong ⊢ρF (wkConv↑ ρ ⊢Δ A<>B) (wkConv↑ (lift ρ) (⊢Δ ∙ ⊢ρF) A<>B₁)
wkConv↓ ρ ⊢Δ (Σ-cong x A<>B A<>B₁) =
let ⊢ρF = wk ρ ⊢Δ x
in Σ-cong ⊢ρF (wkConv↑ ρ ⊢Δ A<>B) (wkConv↑ (lift ρ) (⊢Δ ∙ ⊢ρF) A<>B₁)
-- Weakening of algorithmic equality of terms.
wkConv↑Term : ∀ {ρ t u A Γ Δ} ([ρ] : ρ ∷ Δ ⊆ Γ) → ⊢ Δ
→ Γ ⊢ t [conv↑] u ∷ A
→ Δ ⊢ U.wk ρ t [conv↑] U.wk ρ u ∷ U.wk ρ A
wkConv↑Term {ρ} [ρ] ⊢Δ ([↑]ₜ B t′ u′ D d d′ whnfB whnft′ whnfu′ t<>u) =
[↑]ₜ (U.wk ρ B) (U.wk ρ t′) (U.wk ρ u′)
(wkRed* [ρ] ⊢Δ D) (wkRed*Term [ρ] ⊢Δ d) (wkRed*Term [ρ] ⊢Δ d′)
(wkWhnf ρ whnfB) (wkWhnf ρ whnft′) (wkWhnf ρ whnfu′)
(wkConv↓Term [ρ] ⊢Δ t<>u)
-- Weakening of algorithmic equality of terms in WHNF.
wkConv↓Term : ∀ {ρ t u A Γ Δ} ([ρ] : ρ ∷ Δ ⊆ Γ) → ⊢ Δ
→ Γ ⊢ t [conv↓] u ∷ A
→ Δ ⊢ U.wk ρ t [conv↓] U.wk ρ u ∷ U.wk ρ A
wkConv↓Term ρ ⊢Δ (ℕ-ins x) =
ℕ-ins (wk~↓ ρ ⊢Δ x)
wkConv↓Term ρ ⊢Δ (Empty-ins x) =
Empty-ins (wk~↓ ρ ⊢Δ x)
wkConv↓Term ρ ⊢Δ (Unit-ins x) =
Unit-ins (wk~↓ ρ ⊢Δ x)
wkConv↓Term {ρ} [ρ] ⊢Δ (ne-ins t u x x₁) =
ne-ins (wkTerm [ρ] ⊢Δ t) (wkTerm [ρ] ⊢Δ u) (wkNeutral ρ x) (wk~↓ [ρ] ⊢Δ x₁)
wkConv↓Term ρ ⊢Δ (univ x x₁ x₂) =
univ (wkTerm ρ ⊢Δ x) (wkTerm ρ ⊢Δ x₁) (wkConv↓ ρ ⊢Δ x₂)
wkConv↓Term ρ ⊢Δ (zero-refl x) = zero-refl ⊢Δ
wkConv↓Term ρ ⊢Δ (suc-cong t<>u) = suc-cong (wkConv↑Term ρ ⊢Δ t<>u)
wkConv↓Term {ρ} {Δ = Δ} [ρ] ⊢Δ (η-eq {F = F} {G = G} x₁ x₂ y y₁ t<>u) =
let ⊢F , _ = syntacticΠ (syntacticTerm x₁)
⊢ρF = wk [ρ] ⊢Δ ⊢F
in η-eq (wkTerm [ρ] ⊢Δ x₁) (wkTerm [ρ] ⊢Δ x₂)
(wkFunction ρ y) (wkFunction ρ y₁)
(PE.subst₃ (λ x y z → Δ ∙ U.wk ρ F ⊢ x [conv↑] y ∷ z)
(PE.cong₂ _∘_ (PE.sym (wk1-wk≡lift-wk1 _ _)) PE.refl)
(PE.cong₂ _∘_ (PE.sym (wk1-wk≡lift-wk1 _ _)) PE.refl)
PE.refl
(wkConv↑Term (lift [ρ]) (⊢Δ ∙ ⊢ρF) t<>u))
wkConv↓Term {ρ} [ρ] ⊢Δ (Σ-η {G = G} ⊢p ⊢r pProd rProd fstConv sndConv) =
Σ-η (wkTerm [ρ] ⊢Δ ⊢p)
(wkTerm [ρ] ⊢Δ ⊢r)
(wkProduct ρ pProd)
(wkProduct ρ rProd)
(wkConv↑Term [ρ] ⊢Δ fstConv)
(PE.subst (λ x → _ ⊢ _ [conv↑] _ ∷ x)
(wk-β G)
(wkConv↑Term [ρ] ⊢Δ sndConv))
wkConv↓Term {ρ} [ρ] ⊢Δ (η-unit [t] [u] tWhnf uWhnf) =
η-unit (wkTerm [ρ] ⊢Δ [t]) (wkTerm [ρ] ⊢Δ [u])
(wkWhnf ρ tWhnf) (wkWhnf ρ uWhnf)
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-- Instances of Gaussian integers.
{-# OPTIONS --without-K --safe #-}
module GauInt.Instances where
open import Data.Integer using (+_ ; -[1+_] ; +[1+_])
open import Data.Nat using (suc )
open import Instances
open import GauInt.Base renaming (-_ to -𝔾_ ; _-_ to _-𝔾_ ; _+_ to _+𝔾_ ; _*_ to _*𝔾_ ; NonZero to NonZero𝔾 ; rank to rank𝔾)
-- Instances to overload operations.
instance
sr𝔾 : SemiRing 𝔾
_+_ {{sr𝔾}} = _+𝔾_
_*_ {{sr𝔾}} = _*𝔾_
0# {{sr𝔾}} = 0𝔾
1# {{sr𝔾}} = 1𝔾
instance
ring𝔾 : Ring 𝔾
ring𝔾 .sra = sr𝔾
ring𝔾 .-_ = -𝔾_
instance
Rank𝔾 : Rank 𝔾
Rank𝔾 .rank = rank𝔾
-- This depends on how the boolean equality on 𝔾 is defined. To be
-- precise, it depends on the order of comparing the components.
instance
nzp : ∀ {n} {y} -> NonZero𝔾 (+ suc n + y i)
nzp = _
nzn : ∀ {n} {y} -> NonZero𝔾 (-[1+ n ] + y i)
nzn = _
nzpi : ∀ {n} -> NonZero𝔾 (0# + (+ suc n) i)
nzpi = _
nzni : ∀ {n} -> NonZero𝔾 (0# + (-[1+ n ]) i)
nzni = _
instance
NZT𝔾 : NonZeroTypeclass 𝔾
NZT𝔾 .NonZero = NonZero𝔾
{-
-- Translation from NonZero predicate to non-equality.
test-t : ∀ (x : 𝔾) -> .{{NonZero x}} -> ¬ x ≡ 0#
test-t (+_ zero + +[1+ n ] i) = λ {()}
test-t (+_ zero + -[1+_] n i) = λ {()}
test-t (+[1+ n ] + x₁ i) = λ {()}
test-t (-[1+_] n + x₁ i) = λ {()}
open import Relation.Binary.Structures
open IsDecEquivalence {{...}}
open IsDecTotalOrder {{...}}
open import Data.Nat.Instances
open import Data.Integer.Instances
test : {!!}
test = 0ℤ ≤? 0ℤ
-}
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{-# OPTIONS --cubical --without-K #-}
open import Cubical.Core.Everything
open import Cubical.Foundations.Function
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Equiv
open import Cubical.Data.Sum
open import Cubical.Data.Unit
module Coequalizers where
record Graph {ℓ ℓ' : Level} (E : Type ℓ) (V : Type ℓ') : Type (ℓ-max ℓ ℓ') where
field
π₀ : E → V
π₁ : E → V
open Graph ⦃...⦄
data Coeq {ℓ ℓ' : Level} {V : Type ℓ} {E : Type ℓ'} (ev : Graph E V) :
Type (ℓ-max ℓ ℓ') where
c[_] : V → (Coeq ev)
quot : (e : E) → c[ (Graph.π₀ ev e) ] ≡ c[ (Graph.π₁ ev e) ]
_/_ : {ℓ ℓ' : Level} (V : Type ℓ) (E : Type ℓ') ⦃ ev : Graph E V ⦄ →
Type (ℓ-max ℓ ℓ')
_/_ V E ⦃ ev ⦄ = Coeq ev
infix 25 _/_
module CoprodCoeq {ℓ ℓ' ℓ'' : Level} (E₀ : Type ℓ) (E₁ : Type ℓ') (V : Type ℓ'')
⦃ ev : Graph (E₀ ⊎ E₁) V ⦄ where
instance
evE₀ : Graph E₀ V
evE₀ = record { π₀ = π₀ ∘ inl ; π₁ = π₁ ∘ inl }
ev-snd : Graph E₁ (V / E₀)
ev-snd = record { π₀ = c[_] ∘ π₀ ∘ inr ; π₁ = c[_] ∘ π₁ ∘ inr }
coeq-coprod-equiv : V / (E₀ ⊎ E₁) ≃ (V / E₀) / E₁
coeq-coprod-equiv = isoToEquiv (iso f g f-g g-f)
where
f : V / (E₀ ⊎ E₁) → (V / E₀) / E₁
f c[ x ] = c[ c[ x ] ]
f (quot (inl e) i) = c[ quot e i ]
f (quot (inr e) i) = quot e i
g : (V / E₀) / E₁ → V / (E₀ ⊎ E₁)
g c[ c[ x ] ] = c[ x ]
g c[ quot e i ] = quot (inl e) i
g (quot e i) = quot (inr e) i
f-g : (z : (V / E₀) / E₁) → (f (g z) ≡ z)
f-g c[ c[ x ] ] = refl
f-g c[ quot e i ] = refl
f-g (quot e i) = refl
g-f : (z : V / (E₀ ⊎ E₁)) → (g (f z) ≡ z)
g-f c[ x ] = refl
g-f (quot (inl e) i) = refl
g-f (quot (inr e) i) = refl
module TrivialExtension {ℓ ℓ' : Level} (V : Type ℓ) (v : V) where
instance
evt-l : Graph Unit (V ⊎ Unit)
evt-l = record { π₀ = λ x → inl v ; π₁ = inr }
evt-r : Graph Unit (Unit ⊎ V)
evt-r = record { π₀ = inl ; π₁ = λ _ → inr v }
te-equiv-l : V ≃ (V ⊎ Unit) / Unit
te-equiv-l = isoToEquiv (iso (c[_] ∘ inl) g f-g (λ _ → refl))
where
g : (V ⊎ Unit) / Unit → V
g c[ inl x ] = x
g c[ inr x ] = v
g (quot e i) = v
f-g : (z : (V ⊎ Unit) / Unit) → c[ inl (g z) ] ≡ z
f-g c[ inl x ] = refl
f-g c[ inr x ] = quot x
f-g (quot e i) j = quot e (i ∧ j)
te-equiv-r : V ≃ (Unit ⊎ V) / Unit
te-equiv-r = isoToEquiv (iso (c[_] ∘ inr) g f-g λ _ → refl)
where
g : (Unit ⊎ V) / Unit → V
g c[ inl x ] = v
g c[ inr x ] = x
g (quot e i) = v
f-g : (z : (Unit ⊎ V) / Unit) → c[ inr (g z) ] ≡ z
f-g c[ inl x ] i = quot x (~ i)
f-g c[ inr x ] = refl
f-g (quot e i) j = quot e (i ∨ ~ j)
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-- MIT License
-- Copyright (c) 2021 Luca Ciccone and Luca Padovani
-- Permission is hereby granted, free of charge, to any person
-- obtaining a copy of this software and associated documentation
-- files (the "Software"), to deal in the Software without
-- restriction, including without limitation the rights to use,
-- copy, modify, merge, publish, distribute, sublicense, and/or sell
-- copies of the Software, and to permit persons to whom the
-- Software is furnished to do so, subject to the following
-- conditions:
-- The above copyright notice and this permission notice shall be
-- included in all copies or substantial portions of the Software.
-- THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
-- EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
-- OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
-- NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
-- HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
-- WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
-- FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
-- OTHER DEALINGS IN THE SOFTWARE.
open import Level
open import Axiom.ExcludedMiddle
open import Axiom.Extensionality.Propositional
open import Relation.Nullary
open import Relation.Binary.PropositionalEquality using (_≡_; refl)
module Common where
postulate
excluded-middle : ExcludedMiddle Level.zero
extensionality : Extensionality Level.zero (Level.suc Level.zero)
record Message (ℙ : Set) : Set where
infix 4 _?=_
field
_?=_ : (x y : ℙ) -> Dec (x ≡ y)
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module _ where
module M where
data D : Set where
d : D
private
instance
x : D
x = d
! : ⦃ _ : M.D ⦄ → M.D
! ⦃ x ⦄ = x
y : M.D
y = !
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{-# OPTIONS --cubical #-}
open import Formalization.PredicateLogic.Signature
module Formalization.PredicateLogic.Constructive.SequentCalculus (𝔏 : Signature) where
open Signature(𝔏)
open import Data.Option
open import Data.List
open import Data.List.Functions using (map) renaming (singleton to · ; _++_ to _∪_)
open import Data.List.Relation.Permutation
import Data.ListSized as S
open import Formalization.PredicateLogic.Syntax(𝔏)
open import Formalization.PredicateLogic.Syntax.Substitution(𝔏)
open import Functional as Fn using ()
import Lvl
open import Numeral.Natural
open import Type
private variable ℓ : Lvl.Level
private variable args n vars vars₁ vars₂ : ℕ
private variable Γ Γ₁ Γ₂ Γ₃ : List(Formula(vars))
private variable Δ Δ₁ Δ₂ Δ₃ : Option(Formula(vars))
private variable φ φ₁ φ₂ ψ A B C : Formula(vars)
private variable p : Prop(n)
private variable f : Prop(args)
private variable x : S.List(Term(vars))(args)
_∪·_ : ∀{T : Type{ℓ}} → List(T) → T → List(T)
_∪·_ = Fn.swap(_⊰_)
infixl 1000 _∪·_
module Variant1 where
data _⇒_ : List(Formula(vars)) → Option(Formula(vars)) → Type{Lvl.𝐒(ℓₚ Lvl.⊔ ℓₒ)} where
axiom : ((· φ) ⇒ Some(φ))
weakenₗ : (Γ ⇒ Δ) → ((Γ ∪· A) ⇒ Δ)
permuteₗ : (Γ₁ permutes Γ₂) → (Γ₁ ⇒ Δ) → (Γ₂ ⇒ Δ)
contractₗ : ((Γ ∪· A ∪· A) ⇒ Δ) → ((Γ ∪· A) ⇒ Δ)
⊥ₗ : (Γ ∪· ⊥) ⇒ None
∧ₗₗ : ((Γ ∪· A) ⇒ Δ) → ((Γ ∪· (A ∧ B)) ⇒ Δ)
∧ₗᵣ : ((Γ ∪· B) ⇒ Δ) → ((Γ ∪· (A ∧ B)) ⇒ Δ)
∨ₗ : ((Γ ∪· A) ⇒ Δ) → ((Γ ∪· B) ⇒ Δ) → ((Γ ∪· (A ∨ B)) ⇒ Δ)
⟶ₗ : (Γ ⇒ Some(A)) → ((Γ ∪· B) ⇒ Δ) → ((Γ ∪· (A ⟶ B)) ⇒ Δ)
Ɐₗ : ∀{t} → ((Γ ∪· (substitute0 t A)) ⇒ Δ) → ((Γ ∪· (Ɐ A)) ⇒ Δ)
∃ₗ : ∀{v}{n} → ((Γ ∪· (substituteN n (var v) A)) ⇒ Δ) → ((Γ ∪· (∃ A)) ⇒ Δ)
weakenᵣ : (Γ ⇒ None) → (Γ ⇒ Some(A))
⊤ᵣ : ∅ ⇒ Some(⊤ {vars})
∧ᵣ : (Γ ⇒ Some(A)) → (Γ ⇒ Some(B)) → (Γ ⇒ Some(A ∧ B))
∨ᵣₗ : (Γ ⇒ Some(A)) → (Γ ⇒ Some(A ∨ B))
∨ᵣᵣ : (Γ ⇒ Some(B)) → (Γ ⇒ Some(A ∨ B))
⟶ᵣ : ((Γ ∪· A) ⇒ Some(B)) → (Γ ⇒ Some(A ⟶ B))
Ɐᵣ : ∀{v}{n} → (Γ ⇒ Some((substituteN n (var v) A))) → (Γ ⇒ Some(Ɐ A))
∃ᵣ : ∀{t} → (Γ ⇒ Some((substitute0 t A))) → (Γ ⇒ Some(∃ A))
import Logic.Propositional as Meta
no-empty-refl : Meta.¬(∅ ⇒ None{T = Formula vars})
no-empty-refl (permuteₗ perm p) rewrite Proofs.permutes-on-empty perm = no-empty-refl p
{-
no-callCC : Meta.¬(∅ ⇒ Some(((A ⟶ B) ⟶ A) ⟶ A))
no-callCC (permuteₗ perm p) rewrite permutes-on-empty perm = no-callCC p
no-callCC (weakenᵣ (permuteₗ perm p)) rewrite permutes-on-empty perm = no-empty-refl p
no-callCC (⟶ᵣ p) = no-callCC {!!}
{-no-callCC (⟶ᵣ (weakenₗ p)) = {!!}
no-callCC (⟶ᵣ (permuteₗ x p)) = {!!}
no-callCC (⟶ᵣ (contractₗ p)) = {!!}
no-callCC (⟶ᵣ (⟶ₗ p p₁)) = {!!}
no-callCC (⟶ᵣ (weakenᵣ p)) = {!!}
no-callCC (⟶ᵣ (∧ᵣ p p₁)) = {!!}
no-callCC (⟶ᵣ (∨ᵣₗ p)) = {!!}
no-callCC (⟶ᵣ (∨ᵣᵣ p)) = {!!}
no-callCC (⟶ᵣ (⟶ᵣ p)) = {!!}
no-callCC (⟶ᵣ (Ɐᵣ p)) = {!!}
no-callCC (⟶ᵣ (∃ᵣ p)) = {!!}
-}
-}
module Variant3 where
-- Note: Because this formalization is non-standard, a problem arises for Formula(𝟎): It it missing some of the quantification rules because it has no variables.
data _⇒_ : List(Formula(vars)) → Formula(vars) → Type{Lvl.𝐒(ℓₚ Lvl.⊔ ℓₒ)} where
axiom : ((Γ ∪· (f $ x)) ⇒ (f $ x))
permuteₗ : (Γ₁ permutes Γ₂) → (Γ₁ ⇒ C) → (Γ₂ ⇒ C)
⊥ₗ : (Γ ∪· ⊥) ⇒ A
∧ₗ : ((Γ ∪· A ∪· B) ⇒ C) → ((Γ ∪· (A ∧ B)) ⇒ C)
∨ₗ : ((Γ ∪· A) ⇒ C) → ((Γ ∪· B) ⇒ C) → ((Γ ∪· (A ∨ B)) ⇒ C)
⟶ₗ : ((Γ ∪· (A ⟶ B)) ⇒ A) → ((Γ ∪· B) ⇒ C) → ((Γ ∪· (A ⟶ B)) ⇒ C)
Ɐₗ : ∀{t} → ((Γ ∪· (substitute0 t A) ∪· (Ɐ A)) ⇒ C) → ((Γ ∪· (Ɐ A)) ⇒ C)
∃ₗ : ∀{n}{v} → ((Γ ∪· (substituteN n (var v) A)) ⇒ C) → ((Γ ∪· (∃ A)) ⇒ C)
⊤ᵣ : Γ ⇒ (⊤ {vars})
∧ᵣ : (Γ ⇒ A) → (Γ ⇒ B) → (Γ ⇒ (A ∧ B))
∨ᵣₗ : (Γ ⇒ A) → (Γ ⇒ (A ∨ B))
∨ᵣᵣ : (Γ ⇒ B) → (Γ ⇒ (A ∨ B))
⟶ᵣ : ((Γ ∪· A) ⇒ B) → (Γ ⇒ (A ⟶ B))
Ɐᵣ : ∀{n}{v} → (Γ ⇒ (substituteN n (var v) A)) → (Γ ⇒ (Ɐ A))
∃ᵣ : ∀{t} → (Γ ⇒ (substitute0 t A)) → (Γ ⇒ (∃ A))
import Logic.Propositional as Meta
open import Numeral.Finite
open import Type.Properties.Inhabited
weakenₗ : (Γ ⇒ ψ) → ((Γ ∪· φ) ⇒ ψ)
weakenₗ axiom = permuteₗ swap axiom
weakenₗ (permuteₗ x p) = permuteₗ (prepend x) (weakenₗ p)
weakenₗ ⊥ₗ = permuteₗ swap ⊥ₗ
weakenₗ (∧ₗ p) = permuteₗ swap (∧ₗ(permuteₗ (trans swap (prepend swap)) (weakenₗ p)))
weakenₗ (∨ₗ p q) = permuteₗ swap (∨ₗ (permuteₗ swap (weakenₗ p)) (permuteₗ swap (weakenₗ q)))
weakenₗ (⟶ₗ p q) = permuteₗ swap (⟶ₗ (permuteₗ swap (weakenₗ p)) (permuteₗ swap (weakenₗ q)))
weakenₗ (Ɐₗ p) = permuteₗ swap (Ɐₗ (permuteₗ (trans swap (prepend swap)) (weakenₗ p)))
weakenₗ (∃ₗ{n = n} p) = permuteₗ swap (∃ₗ{n = n} (permuteₗ swap (weakenₗ p)))
weakenₗ ⊤ᵣ = ⊤ᵣ
weakenₗ (∧ᵣ p q) = ∧ᵣ (weakenₗ p) (weakenₗ q)
weakenₗ (∨ᵣₗ p) = ∨ᵣₗ (weakenₗ p)
weakenₗ (∨ᵣᵣ p) = ∨ᵣᵣ (weakenₗ p)
weakenₗ (⟶ᵣ p) = ⟶ᵣ (permuteₗ swap (weakenₗ p))
weakenₗ (Ɐᵣ{n = n} p) = Ɐᵣ{n = n} (weakenₗ p)
weakenₗ (∃ᵣ p) = ∃ᵣ (weakenₗ p)
weaken-union : (Γ₂ ⇒ φ) → ((Γ₁ ∪ Γ₂) ⇒ φ)
weaken-union {Γ₁ = ∅} p = p
weaken-union {Γ₁ = φ ⊰ Γ₁} p = weakenₗ (weaken-union {Γ₁ = Γ₁} p)
open import Formalization.PredicateLogic.Syntax.Tree(𝔏)
open import Numeral.Natural.Relation.Order
open import Numeral.Natural.Relation.Order.Proofs
open import Relator.Equals
open import Relator.Equals.Proofs
open import Structure.Function
open import Structure.Relator.Properties
open import Syntax.Function
direct₊ : ∀{φ : Formula(𝐒(vars))} → ((Γ ∪· φ) ⇒ φ)
direct₊{Γ = Γ}{φ = φ} = induction-on-height(P) (\{vars}{φ} → proof{vars}{φ}) \() where
P = \{vars} (φ : Formula(vars)) → (vars ≢ 𝟎) → ∀{Γ} → (Γ ∪· φ) ⇒ φ
proof : ∀{φ : Formula(vars)} → (∀{vars}{ψ : Formula(vars)} → (height ψ < height φ) → P(ψ)) → P(φ)
proof {𝟎} _ nz with () ← nz [≡]-intro
proof {𝐒 _} {φ = f $ x} prev _ = axiom
proof {𝐒 _} {φ = ⊤} prev _ = weakenₗ ⊤ᵣ
proof {𝐒 _} {φ = ⊥} prev _ = ⊥ₗ
proof {𝐒 _} {φ = φ ∧ ψ} prev nz = ∧ᵣ (∧ₗ (permuteₗ swap (prev (∧-height-orderₗ{φ = φ}{ψ = ψ}) nz))) (∧ₗ (prev (∧-height-orderᵣ{ψ = ψ}{φ = φ}) nz))
proof {𝐒 _} {φ = φ ∨ ψ} prev nz = ∨ₗ (∨ᵣₗ (prev (∨-height-orderₗ{φ = φ}{ψ = ψ}) nz)) (∨ᵣᵣ (prev (∨-height-orderᵣ{ψ = ψ}{φ = φ}) nz))
proof {𝐒 _} {φ = φ ⟶ ψ} prev nz = ⟶ᵣ (permuteₗ swap (⟶ₗ (permuteₗ swap (prev (⟶-height-orderₗ{φ = φ}{ψ = ψ}) nz)) (prev (⟶-height-orderᵣ{ψ = ψ}{φ = φ}) nz)))
proof {𝐒 v} {φ = Ɐ φ} prev nz = Ɐᵣ{n = 𝟎}{v = 𝟎} (Ɐₗ{t = var 𝟎} (weakenₗ (prev (subtransitivityₗ(_≤_)(_≡_) (congruence₁(𝐒) (substitute-height{φ = φ})) (Ɐ-height-order{φ = φ})) nz)))
proof {𝐒 v} {φ = ∃ φ} prev nz = ∃ᵣ{t = var 𝟎} (∃ₗ{n = 𝟎}{v = 𝟎} (prev (subtransitivityₗ(_≤_)(_≡_) (congruence₁(𝐒) (substitute-height{φ = φ})) (Ɐ-height-order{φ = φ})) nz))
no-empty-refl : Meta.¬(∅ ⇒ (⊥ {vars}))
no-empty-refl (permuteₗ perm p) rewrite Proofs.permutes-on-empty perm = no-empty-refl p
no-empty-axiomₗ : Meta.¬(·(p $ x) ⇒ ⊥)
no-empty-axiomₗ (permuteₗ perm p) rewrite Proofs.permutes-on-singleton perm = no-empty-axiomₗ p
no-empty-axiomᵣ : Meta.¬(∅ ⇒ (p $ x))
no-empty-axiomᵣ (permuteₗ perm p) rewrite Proofs.permutes-on-empty perm = no-empty-axiomᵣ p
no-negated-axiomᵣ : Meta.¬(∅ ⇒ (¬(p $ x)))
no-negated-axiomᵣ (permuteₗ perm p) rewrite Proofs.permutes-on-empty perm = no-negated-axiomᵣ p
no-negated-axiomᵣ (⟶ᵣ p) = no-empty-axiomₗ p
-- 3.5.2
substitute-proof : ∀{t : 𝕟(vars₁) → Term(vars₂)} → (Γ ⇒ φ) → ((map(substitute t) Γ) ⇒ (substitute t φ))
substitute-proof p = {!!}
module _ ⦃ pos-prop : ◊(Prop(0)) ⦄ where
no-excludedMiddle : Meta.¬(∀{A : Formula(vars)} → (∅ ⇒ (A ∨ (¬ A))))
no-excludedMiddle as = proof(as{[◊]-existence $ S.∅}) where
proof : Meta.¬(∅ ⇒ ((p $ x) ∨ ¬(p $ x)))
proof (permuteₗ perm q) rewrite Proofs.permutes-on-empty perm = proof q
proof (∨ᵣₗ (permuteₗ perm q)) rewrite Proofs.permutes-on-empty perm = no-empty-axiomᵣ q
proof (∨ᵣᵣ (permuteₗ perm q)) rewrite Proofs.permutes-on-empty perm = no-negated-axiomᵣ q
proof (∨ᵣᵣ (permuteₗ perm (⟶ᵣ q))) rewrite Proofs.permutes-on-empty perm = no-empty-axiomₗ q
proof (∨ᵣᵣ (⟶ᵣ (permuteₗ perm q))) rewrite Proofs.permutes-on-singleton perm = no-empty-axiomₗ q
no-doubleNegation : Meta.¬(∀{A : Formula(vars)} → (∅ ⇒ ((¬ ¬ A) ⟶ A)))
no-doubleNegation as = proof(as{[◊]-existence $ S.∅}) where
proof : Meta.¬(∅ ⇒ ((¬ ¬(p $ x)) ⟶ (p $ x)))
proof (permuteₗ perm q) rewrite Proofs.permutes-on-empty perm = proof q
proof (⟶ᵣ (permuteₗ perm q)) = {!!}
proof (⟶ᵣ (⟶ₗ q (permuteₗ perm q₁))) = {!!}
proof (⟶ᵣ (⟶ₗ (permuteₗ perm q) ⊥ₗ)) = {!!}
proof (⟶ᵣ (⟶ₗ (⟶ₗ q q₁) ⊥ₗ)) = {!!}
proof (⟶ᵣ (⟶ₗ (⟶ᵣ q) ⊥ₗ)) = {!!}
test : ∀{T : Type{ℓ}} → (Γ₁ permutes Γ₂) → ((Γ₁ ⇒ φ) → T) → ((Γ₂ ⇒ φ) → T)
test perm p1 p2 = p1 (permuteₗ (symmetry(_permutes_) perm) p2)
{-# INLINE test #-}
no-callCC : Meta.¬(∀{A B : Formula(vars)} → (∅ ⇒ ((A ⟶ B) ⟶ A) ⟶ A))
no-callCC as = proof(as{[◊]-existence $ S.∅}{⊥}) where
proof2 : Meta.¬((∅ ∪· ((p $ x ⟶ ⊥) ⟶ p $ x) ∪· p $ x) ⇒ ⊥)
proof2 (permuteₗ x t) = {!t!}
proof3 : Meta.¬((∅ ∪· p $ x ∪· ((p $ x ⟶ ⊥) ⟶ p $ x)) ⇒ ⊥)
proof3 (permuteₗ perm p) = {!!}
proof3 (⟶ₗ p q) = {!!}
proof : Meta.¬(∅ ⇒ (((p $ x) ⟶ ⊥) ⟶ (p $ x)) ⟶ (p $ x))
proof (permuteₗ perm q) rewrite Proofs.permutes-on-empty perm = proof q
proof (⟶ᵣ (permuteₗ perm q)) rewrite Proofs.permutes-on-singleton perm = {!!}
proof (⟶ᵣ (⟶ₗ (permuteₗ perm q) q₁)) = {!!}
proof (⟶ᵣ (⟶ₗ (⟶ₗ q (permuteₗ x r)) s)) = {!!}
proof (⟶ᵣ (⟶ₗ (⟶ₗ q (⟶ᵣ r)) s)) = no-empty-axiomₗ (contract-axiom-bottom r) where
contract-axiom-bottom : ((·(p $ x) ∪· (p $ x)) ⇒ ⊥) → (·(p $ x) ⇒ ⊥)
contract-axiom-bottom (permuteₗ x p) = {!!}
proof (⟶ᵣ (⟶ₗ (⟶ᵣ (permuteₗ perm q)) r)) = proof2 (permuteₗ perm q)
-- proof (⟶ᵣ (⟶ₗ (⟶ᵣ (permuteₗ perm q)) r)) = test (trans swap (symmetry(_permutes_) perm)) (proof ↦ {!⟶ᵣ proof!}) q
{-proof (⟶ᵣ (⟶ₗ (⟶ᵣ (permuteₗ perm q)) r)) = test (symmetry(_permutes_) perm) proof2 q where
proof2 : Meta.¬((∅ ∪· ((p $ x ⟶ ⊥) ⟶ p $ x) ∪· p $ x) ⇒ ⊥)
proof2 (permuteₗ perm p) = test (symmetry(_permutes_) perm) proof2 p-}
{-proof (⟶ᵣ (⟶ₗ (⟶ᵣ (permuteₗ (prepend perm) q)) r)) rewrite permutes-on-singleton perm = {!!}
proof (⟶ᵣ (⟶ₗ (⟶ᵣ (permuteₗ swap (permuteₗ perm q))) r)) = {!!}
proof (⟶ᵣ (⟶ₗ (⟶ᵣ (permuteₗ swap (⟶ₗ q₁ (permuteₗ perm q₂)))) r)) = {!!}
proof (⟶ᵣ (⟶ₗ (⟶ᵣ (permuteₗ (trans perm₁ perm₂) q)) r)) = {!!}-}
--⟶ₗ : ((Γ ∪· (A ⟶ B)) ⇒ A) → ((Γ ∪· B) ⇒ A) → ((Γ ∪· (A ⟶ B)) ⇒ A)
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-- Andreas, 2020-09-26, issue #4946.
-- More liberal type signatures for constructors of sized types.
-- {-# OPTIONS -v tc.polarity:20 #-}
open import Agda.Builtin.Size
variable
i : Size
A : Set
data T : Size → Set → Set where
c : A → T i A → T (↑ i) A
-- The type of the constructor c is elaborated to
--
-- c : {A : Set} {i : Set} → A → T i A → T (↑ i) A
--
-- Thus, the size argument i is not the first.
-- Nevertheless, Agda recognize the first argument of T
-- as covariant.
test : T i A → T ∞ A
test x = x
-- Should pass.
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Relation.Nullary where
open import Cubical.Relation.Nullary.Base public
open import Cubical.Relation.Nullary.Properties public
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module Pullback where
open import Logic.Equivalence
open import Logic.Relations
open import Logic.Base
open import Category
open import Unique
module Pull (ℂ : Cat) where
private open module CC = Category.Category ℂ
private open module U = Uniq ℂ
record isPull {A B C D A' : Obj}(f : A ─→ B)(g : A ─→ C)(f' : C ─→ D)(g' : B ─→ D)(h₁ : A' ─→ C)(h₂ : A' ─→ B)(commut : f' ∘ h₁ == g' ∘ h₂) : Set1 where
field unique : ∃! \(h : A' ─→ A) -> (g ∘ h == h₁) /\ (f ∘ h == h₂)
record pullback {B C D : Obj}(g' : B ─→ D)(f' : C ─→ D) : Set1 where
field
A : Obj
f : A ─→ B
g : A ─→ C
comm : g' ∘ f == f' ∘ g
pull : (forall {A' : Obj}(h₁ : A' ─→ C)(h₂ : A' ─→ B)(commut : f' ∘ h₁ == g' ∘ h₂) -> isPull f g f' g' h₁ h₂ commut)
record PullCat : Set2 where
field pull : {B C D : Obj}(g' : B ─→ D)(f' : C ─→ D) -> pullback g' f'
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------------------------------------------------------------------------------
-- Co-inductive natural numbers
------------------------------------------------------------------------------
{-# OPTIONS --allow-unsolved-metas #-}
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module GFPs.Conat where
open import FOTC.Base
open import FOTC.Base.PropertiesI
------------------------------------------------------------------------------
-- Conat is a greatest fixed-point of a functor
-- The functor.
NatF : (D → Set) → D → Set
NatF A n = n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ A n')
-- The co-natural numbers are the greatest fixed-point of NatF.
postulate
Conat : D → Set
-- Conat is a post-fixed point of NatF, i.e.
--
-- Conat ≤ NatF Conat.
Conat-out : ∀ {n} → Conat n → n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ Conat n')
-- The higher-order version.
Conat-out-ho : ∀ {n} → Conat n → NatF Conat n
-- Conat is the greatest post-fixed point of NatF, i.e.
--
-- ∀ A. A ≤ NatF A ⇒ A ≤ Conat.
Conat-coind :
(A : D → Set) →
-- A is post-fixed point of ConatF.
(∀ {n} → A n → n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ A n')) →
-- Conat is greater than A.
∀ {n} → A n → Conat n
-- The higher-order version.
Conat-coind-ho :
(A : D → Set) → (∀ {n} → A n → NatF A n) → ∀ {n} → A n → Conat n
-- 22 December 2013. This is a stronger induction principle. If we
-- use it, we can use the trivial predicate A = λ x → x ≡ x in the
-- proofs. Unfortunately, we don't have a justification/proof for
-- this principle.
Conat-stronger-coind₁ :
∀ (A : D → Set) {n} →
(A n → n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ A n')) →
A n → Conat n
-- Other stronger co-induction principle
--
-- Adapted from (Paulson, 1997. p. 16).
Conat-stronger-coind₂ :
(A : D → Set) →
(∀ {n} → A n → (n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ A n')) ∨ Conat n) →
∀ {n} → A n → Conat n
------------------------------------------------------------------------------
-- Conat-out and Conat-out-ho are equivalents
Conat-out-fo : ∀ {n} → Conat n → n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ Conat n')
Conat-out-fo = Conat-out-ho
Conat-out-ho' : ∀ {n} → Conat n → NatF Conat n
Conat-out-ho' = Conat-out
------------------------------------------------------------------------------
-- Conat-coind and Conat-coind-ho are equivalents
Conat-coind-fo :
(A : D → Set) →
(∀ {n} → A n → n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ A n')) →
∀ {n} → A n → Conat n
Conat-coind-fo = Conat-coind-ho
Conat-coind-ho' :
(A : D → Set) → (∀ {n} → A n → NatF A n) → ∀ {n} → A n → Conat n
Conat-coind-ho' = Conat-coind
------------------------------------------------------------------------------
-- Because a greatest post-fixed point is a fixed-point, then the
-- Conat predicate is also a pre-fixed point of the functional NatF,
-- i.e.
--
-- NatF Conat ≤ Conat.
Conat-in : ∀ {n} →
n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ Conat n') →
Conat n
Conat-in h = Conat-coind A h' h
where
A : D → Set
A n = n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ Conat n')
h' : ∀ {n} → A n → n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ A n')
h' (inj₁ n≡0) = inj₁ n≡0
h' (inj₂ (n' , prf , Cn')) = inj₂ (n' , prf , Conat-out Cn')
-- The higher-order version.
Conat-in-ho : ∀ {n} → NatF Conat n → Conat n
Conat-in-ho = Conat-in
-- A different definition.
Conat-in' : (∀ {n} → n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ Conat n')) →
∀ {n} → Conat n
Conat-in' h = Conat-coind (λ m → m ≡ m) (h' h) refl
where
h' : (∀ {n} → n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ Conat n')) →
∀ {m} → m ≡ m → m ≡ zero ∨ (∃[ m' ] m ≡ succ₁ m' ∧ m' ≡ m')
h' h'' {m} _ with (h'' {m})
... | inj₁ m≡0 = inj₁ m≡0
... | inj₂ (m' , prf , _) = inj₂ (m' , prf , refl)
Conat-in-ho' : (∀ {n} → NatF Conat n) → ∀ {n} → Conat n
Conat-in-ho' = Conat-in'
------------------------------------------------------------------------------
-- From Conat-coind/Conat-stronger-coind₁ to Conat-stronger-coind₁/Conat-coind
Conat-coind'' :
(A : D → Set) →
(∀ {n} → A n → n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ A n')) →
∀ {n} → A n → Conat n
Conat-coind'' A h An = Conat-stronger-coind₁ A h An
-- 22 December 2013: We couln't prove Conat-stronger-coind₁ using
-- Conat-coind.
Conat-stronger-coind₁' :
∀ (A : D → Set) {n} →
(A n → n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ A n')) →
A n → Conat n
Conat-stronger-coind₁' A {n} h An = Conat-in (case prf₁ prf₂ (h An))
where
prf₁ : n ≡ zero → n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ Conat n')
prf₁ n≡0 = inj₁ n≡0
prf₂ : ∃[ n' ] n ≡ succ₁ n' ∧ A n' →
n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ Conat n')
prf₂ (n' , prf , An') = inj₂ (n' , prf , {!!})
------------------------------------------------------------------------------
-- From Conat-stronger-coind₂ to Conat-stronger-coind₁
-- 13 January 2014: We couln't prove Conat-stronger-coind₁ using
-- Conat-stronger-coind₂.
Conat-stronger-coind₁'' :
∀ (A : D → Set) {n} →
(A n → n ≡ zero ∨ (∃[ n' ] n ≡ succ₁ n' ∧ A n')) →
A n → Conat n
Conat-stronger-coind₁'' A h An = Conat-stronger-coind₂ A {!!} An
------------------------------------------------------------------------------
-- References
--
-- Paulson, L. C. (1997). Mechanizing Coinduction and Corecursion in
-- Higher-order Logic. Journal of Logic and Computation 7.2,
-- pp. 175–204.
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{-# OPTIONS --universe-polymorphism #-}
open import Categories.Category
module Categories.Object.Initial {o ℓ e} (C : Category o ℓ e) where
open Category C
open import Level
record Initial : Set (o ⊔ ℓ ⊔ e) where
field
⊥ : Obj
! : ∀ {A} → (⊥ ⇒ A)
.!-unique : ∀ {A} → (f : ⊥ ⇒ A) → ! ≡ f
.!-unique₂ : ∀ {A} → (f g : ⊥ ⇒ A) → f ≡ g
!-unique₂ f g =
begin
f
↑⟨ !-unique f ⟩
!
↓⟨ !-unique g ⟩
g
∎
where
open HomReasoning
.⊥-id : (f : ⊥ ⇒ ⊥) → f ≡ id
⊥-id f = !-unique₂ f id
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open import Data.Graph
module Data.Graph.Path.Cut {ℓᵥ ℓₑ} (g : FiniteGraph ℓᵥ ℓₑ) where
open import Data.Fin as Fin using (Fin; zero; suc)
open import Data.Fin.Properties as Fin-Props using (pigeonhole)
open import Data.List as List using (List; []; _∷_)
open import Data.List.Any as Any using (Any; here; there)
open import Data.List.Membership.Propositional as ∈L renaming (_∈_ to _∈L_)
open import Data.Nat as ℕ
open import Data.Nat.Properties as ℕ-Props
open import Data.Product as Σ
open import Data.Sum as ⊎
open import Finite
import Finite.Pigeonhole
open import Function
open import Induction.Nat
open import Induction.WellFounded
import Level as ℓ
open import Relation.Binary.PropositionalEquality
open import Relation.Binary.PreorderReasoning ≤-preorder
open import Relation.Nullary hiding (module Dec)
open import Relation.Nullary.Decidable as Dec
open import Relation.Nullary.Negation
open FiniteGraph g
open IsFinite
infix 3 _∈_
data _∈_ x : ∀ {a b n} → Path a b n → Set where
here : ∀ {b c n} {e : Edge x b} {p : Path b c n} → x ∈ e ∷ p
there : ∀ {a b c n} {e : Edge a b} {p : Path b c n} → x ∈ p → x ∈ e ∷ p
infix 3 _∈?_
_∈?_ : ∀ {a b n} x (p : Path a b n) → Dec (x ∈ p)
x ∈? [] = no λ ()
_∈?_ {a} x (e ∷ p) =
case decEqVertex a x of λ where
(yes refl) → yes here
(no a≢x) →
case x ∈? p of λ where
(yes i) → yes (there i)
(no ¬i) →
no λ where
here → contradiction refl a≢x
(there i) → contradiction i ¬i
index : ∀ {a b x n} {p : Path a b n} → x ∈ p → Fin n
index here = zero
index (there i) = suc (index i)
lookup : ∀ {a b n} → Path a b n → Fin n → Vertex
lookup {a} (e ∷ p) zero = a
lookup (e ∷ p) (suc i) = lookup p i
∈-lookup : ∀ {a b n} {p : Path a b n} (i : Fin n) → lookup p i ∈ p
∈-lookup {p = []} ()
∈-lookup {p = e ∷ p} zero = here
∈-lookup {p = e ∷ p} (suc i) = there (∈-lookup i)
finiteIndex : ∀ {a b n} (p : Path a b n) → Fin n → Fin (size vertexFinite)
finiteIndex p = Any.index ∘ membership vertexFinite ∘ lookup p
prefixLength : ∀ {a b x n} {p : Path a b n} → x ∈ p → ℕ
prefixLength here = zero
prefixLength (there i) = suc (prefixLength i)
suffixLength : ∀ {a b x n} {p : Path a b n} → x ∈ p → ℕ
suffixLength {n = n} here = n
suffixLength (there i) = suffixLength i
split : ∀ {a b x n} {p : Path a b n}
(i : x ∈ p) →
Path a x (prefixLength i) × Path x b (suffixLength i)
split {p = p} here = [] , p
split {p = e ∷ p} (there i) = Σ.map₁ (e ∷_) (split i)
prefix : ∀ {a b x n} {p : Path a b n} (i : x ∈ p) → Path a x (prefixLength i)
prefix = proj₁ ∘ split
suffix : ∀ {a b x n} {p : Path a b n} (i : x ∈ p) → Path x b (suffixLength i)
suffix = proj₂ ∘ split
splitLengthsAddUp : ∀ {a b x n} {p : Path a b n}
(i : x ∈ p) →
n ≡ prefixLength i + suffixLength i
splitLengthsAddUp here = refl
splitLengthsAddUp (there i) = cong suc (splitLengthsAddUp i)
data Repeats : ∀ {a b n} → Path a b n → Set where
here : ∀ {a b c n} {e : Edge a b} {p : Path b c n} → a ∈ p → Repeats (e ∷ p)
there : ∀ {a b c n} {e : Edge a b} {p : Path b c n} → Repeats p → Repeats (e ∷ p)
repeats? : ∀ {a b n} (p : Path a b n) → Dec (Repeats p)
repeats? [] = no λ ()
repeats? {a} (e ∷ p) =
case a ∈? p of λ where
(yes i) → yes (here i)
(no ¬i) →
case repeats? p of λ where
(yes r) → yes (there r)
(no ¬r) →
no λ where
(here i) → contradiction i ¬i
(there r) → contradiction r ¬r
Acyclic : ∀ {a b n} → Path a b n → Set
Acyclic p = ¬ Repeats p
acyclic? : ∀ {a b n} (p : Path a b n) → Dec (Acyclic p)
acyclic? = ¬? ∘ repeats?
data Segmented a b : ℕ → Set (ℓᵥ ℓ.⊔ ℓₑ) where
_◄_◄_ : ∀ {x m n l} →
Path a x m →
Path x x (suc n) →
Path x b l →
Segmented a b (m + suc n + l)
segment : ∀ {a b n} {p : Path a b n} → Repeats p → Segmented a b n
segment {p = []} ()
segment {p = e ∷ p} (here i) rewrite splitLengthsAddUp i = [] ◄ e ∷ prefix i ◄ suffix i
segment {p = e ∷ p} (there r) =
case segment r of λ where
(p₁ ◄ p₂ ◄ p₃) → (e ∷ p₁) ◄ p₂ ◄ p₃
cutLoop< : ∀ {a b n} {p : Path a b n} → Repeats p → Path< a b n
cutLoop< r = case segment r of λ where (_◄_◄_ {m = m} p₁ p₂ p₃) → -, lengthLem m , p₁ ++ p₃
where
lengthLem : ∀ x {y z} → suc (x + z) ≤ x + suc y + z
lengthLem zero = s≤s (n≤m+n _ _)
lengthLem (suc x) = s≤s (lengthLem x)
indicesLoop : ∀ {a b n i j} {p : Path a b n} → i ≢ j → lookup p i ≡ lookup p j → Repeats p
indicesLoop {i = zero} {zero} {e ∷ p} z≢z eq = contradiction refl z≢z
indicesLoop {i = zero} {suc j} {e ∷ p} _ refl = here (∈-lookup j)
indicesLoop {i = suc i} {zero} {e ∷ p} _ refl = here (∈-lookup i)
indicesLoop {i = suc i} {suc j} {e ∷ p} si≢sj eq = there (indicesLoop (si≢sj ∘ cong suc) eq)
findLoop : ∀ {a b n} (p : Path a b n) → n > size vertexFinite → Repeats p
findLoop p gt =
let i , j , i≢j , eq = pigeonhole gt (finiteIndex p) in
indicesLoop i≢j (indexOf-injective vertexFinite eq)
acyclic-length-≤ : ∀ {a b n} (p : Path a b n) → Acyclic p → n ≤ size vertexFinite
acyclic-length-≤ {n = n} p ¬r =
case n ≤? size vertexFinite of λ where
(yes le) → le
(no ¬le) → contradiction (findLoop p (≰⇒> ¬le)) ¬r
shortenPath : ∀ {a b n} → Path a b n → n > size vertexFinite → Path< a b n
shortenPath p = cutLoop< ∘ findLoop p
shortenPathEnough : ∀ {a b n}
(p : Path a b n) →
n > size vertexFinite →
Path≤ a b (size vertexFinite)
shortenPathEnough = <-rec _ wfRec _
where
wfRec =
λ n rec p gt →
let n′ , le , p′ = shortenPath p gt in
case size vertexFinite <? n′ of λ where
(yes n′>v) → rec _ le p′ n′>v
(no n′≯v) → -, ≮⇒≥ n′≯v , p′
shortEnoughPath : ∀ {a b n} (p : Path a b n) → Path≤ a b (size vertexFinite)
shortEnoughPath {n = n} p =
case size vertexFinite <? n of λ where
(yes n>v) → shortenPathEnough p n>v
(no n≯v) → -, ≮⇒≥ n≯v , p
cutAllLoops : ∀ {a b n} →
(p : Path a b n) →
Repeats p →
∃ λ (p : Path≤ a b n) → ¬ Repeats (proj₂ (proj₂ p))
cutAllLoops = <-rec _ wfRec _
where
wfRec = λ x rec p r →
case cutLoop< r of λ where
(n′ , lt , p′) →
case repeats? p′ of λ where
(yes r) →
case rec _ lt p′ r of λ where
((n′′ , le′′ , p′′) , ¬r′′) →
(n′′ , ≤-trans le′′ (<⇒≤ lt) , p′′) , ¬r′′
(no ¬r) → (n′ , <⇒≤ lt , p′) , ¬r
acyclicPath : ∀ {a b n} →
(p : Path a b n) →
∃ λ (p : Path≤ a b n) → ¬ Repeats (proj₂ (proj₂ p))
acyclicPath p =
case repeats? p of λ where
(yes r) → cutAllLoops p r
(no ¬r) → (-, ≤-refl , p) , ¬r
minimalPath : ∀ {a b n} →
Path a b n →
∃ λ (p : Path≤ a b (size vertexFinite)) → ¬ Repeats (proj₂ (proj₂ p))
minimalPath p =
let
x , x≤max , p′ = shortEnoughPath p
(y , y≤x , p′′) , ¬r = acyclicPath p′
in
(y , ≤-trans y≤x x≤max , p′′) , ¬r
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open import Type
open import Structure.Relator
open import Structure.Setoid renaming (_≡_ to _≡ₑ_)
-- TODO: Organize this module
module Structure.Sets.ZFC.Inductive {ℓₛ ℓₗ ℓₑ} {S : Type{ℓₛ}} ⦃ equiv : Equiv{ℓₑ}(S) ⦄ (_∈_ : S → S → Type{ℓₗ}) ⦃ [∈]-binaryRelator : BinaryRelator(_∈_) ⦄ where
open import Functional using (id)
open import Functional.Dependent
import Lvl
open import Logic.Predicate
open import Logic.Propositional
open import Logic.Propositional.Theorems
open import Structure.Relator.Proofs renaming ([≡]-binaryRelator to [≡ₑ]-binaryRelator)
open import Structure.Sets.ZFC(_∈_) ⦃ [∈]-binaryRelator ⦄
open import Structure.Sets.ZFC.Oper(_∈_)
import Structure.Sets.Names
open Structure.Sets.Names.From-[∈] (_∈_)
open import Structure.Sets.Quantifiers (_∈_)
open import Syntax.Function
private variable ℓ : Lvl.Level
private variable A B C D E N a b c d e x y z As : S
private variable P : S → Type{ℓ}
module _ ⦃ zfc : ZFC ⦄ where
open ZFC(zfc)
-- minSubset P(A) is the intersection of all subsets of A satisfying P.
-- Semantically, minSubset P(A) is the minimal subset of A satisfying P when A satisfies P and big intersection of a set containing A preserves P.
minSubset : (P : S → Type{ℓ}) → ⦃ rel : UnaryRelator(P) ⦄ → S → S
minSubset P(A) = ⋂(filter P(℘(A)))
-- The minimal subset is a subset of the given set when the given set satisfies the given property.
minSubset-subset : ⦃ rel : UnaryRelator(P) ⦄ → P(A) → (minSubset P(A) ⊆ A)
minSubset-subset {P = P}{A = A} pa xM = [∧]-elimᵣ([↔]-to-[→] intersection xM) filt where
filt : (A ∈ filter P(℘(A)))
filt = [↔]-to-[←] restricted-comprehension ([∧]-intro ℘-self pa)
-- A subset of the minimal subset is equal to the minimal subset if it and the given set satisfies the given property.
minSubset-subsets : ⦃ rel : UnaryRelator(P) ⦄ → P(A) → P(B) → (B ⊆ minSubset P(A)) → (B ⊇ minSubset P(A))
minSubset-subsets {P = P}{A = A} pa pb sub cont = [∧]-elimᵣ([↔]-to-[→] (restricted-comprehension ⦃ _ ⦄) cont) (filt-pow ([∧]-intro (minSubset-subset pa ∘ sub) pb)) where
filt-pow : ((B ⊆ A) ∧ P(B)) → (B ∈ filter P(℘(A)))
filt-pow ([∧]-intro sub pb) = [↔]-to-[←] restricted-comprehension ([∧]-intro ([↔]-to-[←] power sub) pb)
minSubset-satisfaction3 : ⦃ rel : UnaryRelator(P) ⦄ → (∀ₛ(℘(℘(A))) (As ↦ ((∀ₛ(As) P) → P(⋂ As)))) → P(A) → P(minSubset P(A))
minSubset-satisfaction3 preserv p = preserv ([↔]-to-[←] power ([∧]-elimₗ ∘ [↔]-to-[→] restricted-comprehension)) ([∧]-elimᵣ ∘ [↔]-to-[→] restricted-comprehension)
-- When the big intersection of a set containing A preserves P and A satisfies P, then the minimal subset satisfies P.
minSubset-satisfaction : ⦃ rel : UnaryRelator(P) ⦄ → (∀{As} → (A ∈ As) → (∀ₛ(As) P) → P(⋂ As)) → P(A) → P(minSubset P(A))
minSubset-satisfaction preserv p = preserv ([↔]-to-[←] restricted-comprehension ([∧]-intro ℘-self p)) ([∧]-elimᵣ ∘ [↔]-to-[→] restricted-comprehension)
-- The "smallest" inductive set is the set of natural numbers.
-- All elements which can be expressed using only 𝟎 and 𝐒.
ℕ : S
ℕ = ⋂(filter Inductive (℘(ω₀))) -- TODO: This pattern seems useful. See the module Inductive
-- The relation "lesser than" in this model of ℕ.
-- This works for all elements in ℕ by the definition of 𝟎 and 𝐒.
_<_ : S → S → Type
a < b = a ∈ b
_≤_ : S → S → Type
a ≤ b = (a < b) ∨ (a ≡ b)
_>_ : S → S → Type
a > b = b < a
_≥_ : S → S → Type
a ≥ b = b ≤ a
infixl 2000 _<_ _≤_ _>_ _≥_
𝕟 : S → S
𝕟(n) = filter(_< n) ⦃ binary-unaryRelatorᵣ ⦄ (ℕ)
-- The set ℕ contains zero and all successors.
ℕ-inductive : Inductive(ℕ)
ℕ-inductive = minSubset-satisfaction p infinity where
p : ∀{S} → (ω₀ ∈ S) → (∀ₛ(S) Inductive) → Inductive(⋂ S)
p {S} omega ind = [∧]-intro base step where
base : 𝟎 ∈ (⋂ S)
base = [↔]-to-[←] intersection ([∧]-intro ([∃]-intro ω₀ ⦃ [∧]-intro omega ([∧]-elimₗ infinity) ⦄) ([∧]-elimₗ ∘ ind))
step : (x ∈ (⋂ S)) → (𝐒(x) ∈ (⋂ S))
step xint = [↔]-to-[←] intersection ([∧]-intro ([∃]-intro ω₀ ⦃ [∧]-intro omega ([∧]-elimᵣ infinity ([∧]-elimᵣ([↔]-to-[→] intersection xint) omega)) ⦄) (\as → [∧]-elimᵣ(ind as) ([∧]-elimᵣ([↔]-to-[→] intersection xint) as)))
{-
ℕ-inclusionᵣ : (x ∈ ℕ) → ∃(A ↦ ((A ⊆ ω₀) ∧ Inductive(A)) ∧ (x ∈ A)) ∧ ∀ₗ(A ↦ (((A ⊆ ω₀) ∧ Inductive(A)) → (x ∈ A)))
ℕ-inclusionᵣ xℕ = [∧]-map ([∃]-map-proof ([∧]-map ([∧]-map ([↔]-to-[→] power) id ∘ [↔]-to-[→] restricted-comprehension) id) ∘ [↔]-to-[→] union) (\p q → p ([↔]-to-[←] restricted-comprehension ([∧]-map ([↔]-to-[←] power) id q))) ([↔]-to-[→] intersection xℕ)
-}
-- The natural numbers' set induction principle.
ℕ-set-induction : Inductive(N) → (N ⊆ ℕ) → (N ⊇ ℕ)
ℕ-set-induction = minSubset-subsets infinity
-- The induction principle of the natural numbers for the elements in the set ℕ.
ℕ-induction : ⦃ rel : UnaryRelator(P) ⦄ → P(𝟎) → (∀ₛ(ℕ) (n ↦ (P(n) → P(𝐒(n))))) → (∀ₛ(ℕ) P)
ℕ-induction {P = P} pz ps = [∧]-elimᵣ ∘ [↔]-to-[→] restricted-comprehension ∘ Pset-super where
Pset : S
Pset = filter P(ℕ)
Pset-𝟎 : (𝟎 ∈ Pset)
Pset-𝟎 = [↔]-to-[←] restricted-comprehension ([∧]-intro ([∧]-elimₗ ℕ-inductive) pz)
Pset-𝐒 : ∀ₛ(Pset) (n ↦ (𝐒(n) ∈ Pset))
Pset-𝐒 {n} nPset =
let Pn : P(n)
Pn = [∧]-elimᵣ([↔]-to-[→] restricted-comprehension nPset)
nℕ : (n ∈ ℕ)
nℕ = [∧]-elimₗ([↔]-to-[→] restricted-comprehension nPset)
in [↔]-to-[←] restricted-comprehension ([∧]-intro ([∧]-elimᵣ ℕ-inductive nℕ) (ps nℕ Pn))
Pset-super : ℕ ⊆ Pset
Pset-super = ℕ-set-induction ([∧]-intro Pset-𝟎 Pset-𝐒) ([∧]-elimₗ ∘ [↔]-to-[→] restricted-comprehension)
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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
open import LibraBFT.Impl.Consensus.BlockStorage.BlockStore
open import LibraBFT.Impl.Consensus.BlockStorage.BlockTree
open import LibraBFT.Impl.Consensus.ConsensusTypes.ExecutedBlock as ExecutedBlock
open import LibraBFT.Impl.Consensus.ConsensusTypes.Vote as Vote
open import LibraBFT.Impl.OBM.Prelude
open import LibraBFT.Impl.Properties.Util
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 Optics.All
open import Util.ByteString
open import Util.Hash
open import Util.KVMap as Map
open import Util.PKCS
open import Util.Prelude
open QCProps
open Invariants
module LibraBFT.Impl.Consensus.BlockStorage.Properties.BlockTree where
module insertBlockESpec (eb0 : ExecutedBlock) (bt : BlockTree) where
eb0Id = eb0 ^∙ ebId
open Reqs (eb0 ^∙ ebBlock) bt
-- This is not quite right. It does not yet account for the updating of the parent Block
-- Is it needed (see below)?
record Updated (hv : HashValue) (pre post : BlockTree) (eb : ExecutedBlock) : Set where
field
≢hv¬Upd : ∀ {hv'} → hv' ≢ hv → btGetBlock hv' post ≡ btGetBlock hv' pre
record ContractOk (bt“ : BlockTree) (eb : ExecutedBlock) : Set where
constructor mkContractOk
field
bt≡x : bt ≡ (bt“ & btIdToBlock ∙~ (bt ^∙ btIdToBlock))
-- The following two fields are not used, but something like this will be useful in proving
-- btiPres and may provide value in their own right
¬upd : ∀ {eb'} → btGetBlock eb0Id bt ≡ just eb' → bt ≡ bt“
upd : btGetBlock eb0Id bt ≡ nothing → Updated eb0Id bt bt“ eb
blocks≈ : NoHC1 → eb [ _≈Block_ ]L eb0 at ebBlock
btiPres : ∀ {eci} → Preserves BlockTreeInv (bt , eci) (bt“ , eci)
Contract : Either ErrLog (BlockTree × ExecutedBlock) → Set
Contract (Left _) = ⊤
Contract (Right (bt' , b)) = ContractOk bt' b
open insertBlockE
postulate -- TODO-1: prove; note that the contract is stronger than we need because insertBlockE
-- is called only when btGetBlock eb0Id bt ≡ nothing in LibraBFT
contract' : EitherD-weakestPre (step₀ eb0 bt) Contract
contract : Contract (insertBlockE.E eb0 bt)
contract = EitherD-contract (step₀ eb0 bt) Contract contract'
module insertQuorumCertESpec
(qc : QuorumCert) (bt0 : BlockTree) where
open insertQuorumCertE qc bt0
Ok : Set
Ok = ∃₂ λ bt1 il → insertQuorumCertE qc bt0 ≡ Right (bt1 , il)
private
Ok' : BlockTree → List InfoLog → Either ErrLog (BlockTree × List InfoLog) → Set
Ok' bt il m = m ≡ Right (bt , il)
record ContractOk (btPre btPost : BlockTree) (ilPre ilPost : List InfoLog) : Set where
constructor mkContractOk
field
noNewQCs : ∈Post⇒∈PreOrBT (_≡ qc) btPre btPost
ContractOk-trans : ∀ {btPre btInt btPost ilPre ilInt ilPost}
→ ContractOk btPre btInt ilPre ilInt
→ ContractOk btInt btPost ilInt ilPost
→ ContractOk btPre btPost ilPre ilPost
ContractOk-trans (mkContractOk noNewQCs) (mkContractOk noNewQCs₁) =
mkContractOk (∈Post⇒∈PreOr'-trans _∈BlockTree_ (_≡ qc) noNewQCs noNewQCs₁)
Contract : EitherD-Post ErrLog (BlockTree × List InfoLog)
Contract (Left _) = ⊤
Contract (Right (bt1 , il)) = ContractOk bt0 bt1 [] il
contract' : EitherD-weakestPre step₀ Contract
contract'
with safetyInvariant
...| Left e = tt
...| Right unit = contract-step₁'
where
contract-step₁' : EitherD-weakestPre (step₁ blockId) Contract
proj₁ contract-step₁' _ = tt
proj₂ contract-step₁' block _ = contract-step₂'
where
contract-step₂' : EitherD-weakestPre (step₂ blockId block) Contract
proj₁ contract-step₂' _ = tt
proj₂ contract-step₂' hcb _ =
contract-step₃'
where
contract-cont2' : ∀ (bt : BlockTree) (info : List InfoLog)
→ let (bt' , info') = continue2 bt info
in ContractOk bt bt' info info'
contract-cont2' bt info
with (bt ^∙ btHighestCommitCert ∙ qcCommitInfo ∙ biRound) <? (qc ^∙ qcCommitInfo ∙ biRound)
...| yes hqcR<qcR = mkContractOk (∈BlockTree-upd-hcc refl refl)
...| no hqcR≥qcR = mkContractOk (λ _ x → inj₁ x)
cont1-update-bt : BlockTree → BlockTree
cont1-update-bt bt = bt & btIdToQuorumCert ∙~ Map.insert blockId qc (bt ^∙ btIdToQuorumCert)
info' : List InfoLog → Bool → List InfoLog
info' il b = (fakeInfo ∷ il) ++ (if b then (fakeInfo ∷ []) else [])
contract-cont1' : ∀ (btPre : BlockTree) (infoPre : List InfoLog)
→ let (btPost , infoPost) = continue1 btPre blockId block infoPre
in ContractOk btPre btPost infoPre infoPost
contract-cont1' btPre infoPre
with Map.kvm-member blockId (btPre ^∙ btIdToQuorumCert)
...| true = mkContractOk (ContractOk.noNewQCs (contract-cont2' btPre (info' infoPre $ ExecutedBlock.isNilBlock block )))
...| false = ContractOk-trans {btInt = cont1-update-bt btPre} {ilInt = info' infoPre $ ExecutedBlock.isNilBlock block }
(mkContractOk (∈Post⇒∈PreOrBT-QCs≡ _ refl refl))
(mkContractOk (ContractOk.noNewQCs (contract-cont2'
(cont1-update-bt btPre)
(info' infoPre $ ExecutedBlock.isNilBlock block))))
bt' = bt0 & btHighestCertifiedBlockId ∙~ block ^∙ ebId
& btHighestQuorumCert ∙~ qc
contract-step₃' : EitherD-weakestPre (step₃ blockId block hcb) Contract
proj₁ contract-step₃' _ = ContractOk-trans
(mkContractOk (∈BlockTree-upd-hqc refl refl))
(contract-cont1' bt' (fakeInfo ∷ []))
proj₂ contract-step₃' _ = ContractOk-trans
(mkContractOk (∈Post⇒∈PreOr'-refl _∈BlockTree_ _))
(contract-cont1' bt0 [])
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open import Prelude
module Nat where
-- definitions
data Nat : Set where
Z : Nat
1+ : Nat → Nat
{-# BUILTIN NATURAL Nat #-}
_+_ : Nat → Nat → Nat
Z + m = m
1+ n + m = 1+ (n + m)
infixl 60 _+_
data _≤_ : Nat → Nat → Set where
≤refl : ∀{n} → n ≤ n
≤1+ : ∀{n m} → n ≤ m → n ≤ 1+ m
infix 40 _≤_
_<_ : Nat → Nat → Set
n < m = n ≤ m ∧ n ≠ m
infix 40 _<_
-- The `difference` operation accepts two numbers n and m and a proof that n ≤ m.
-- Alternative definitions could return an option, returning None if n > m,
-- or truncation, returning 0 if n > m. The latter may make some of the work easier,
-- if it allows us to avoid issues with proof relevance. Even so, many proofs would
-- still need to make use of the proof that n ≤ m to establish that the difference is
-- not arbitrarily 0. As such, we believe that refactoring to use the perhaps more
-- conventional truncation approach would not necessarily go far enough in cleaning up
-- proofs as to be worth the refactoring effort.
difference : ∀{n m} → n ≤ m → Nat
difference {n} {.n} ≤refl = Z
difference {n} {.(1+ _)} (≤1+ n≤m-1) = 1+ (difference n≤m-1)
-- basic theorems
-- the succ operation is injective
1+inj : ∀{n m} → 1+ n == 1+ m → n == m
1+inj refl = refl
1+ap : ∀{n m} → n == m → 1+ n == 1+ m
1+ap {n} {.n} refl = refl
1+ap-cp : ∀{n m} → 1+ n ≠ 1+ m → n ≠ m
1+ap-cp h1 h2 = h1 (1+ap h2)
1+inj-cp : ∀{n m} → n ≠ m → 1+ n ≠ 1+ m
1+inj-cp h1 h2 = h1 (1+inj h2)
-- equality of naturals is decidable. we represent this as computing a
-- choice of units, with inl <> meaning that the naturals are indeed the
-- same and inr <> that they are not.
natEQ : (x y : Nat) → ((x == y) ∨ ((x == y) → ⊥))
natEQ Z Z = Inl refl
natEQ Z (1+ y) = Inr (λ ())
natEQ (1+ x) Z = Inr (λ ())
natEQ (1+ x) (1+ y) with natEQ x y
natEQ (1+ x) (1+ .x) | Inl refl = Inl refl
... | Inr b = Inr (λ x₁ → b (1+inj x₁))
0≠1+n : ∀{n} → 0 ≠ 1+ n
0≠1+n = λ ()
-- _+_ theorems
n+Z==n : ∀{n} → n + Z == n
n+Z==n {Z} = refl
n+Z==n {1+ n} = 1+ap n+Z==n
n+1+m==1+n+m : ∀{n m} → n + 1+ m == 1+ (n + m)
n+1+m==1+n+m {Z} = refl
n+1+m==1+n+m {1+ n} = 1+ap n+1+m==1+n+m
n≠n+1+m : ∀{n m} → n ≠ n + 1+ m
n≠n+1+m {Z} {m} ()
n≠n+1+m {1+ n} {m} h = 1+inj-cp n≠n+1+m h
+comm : ∀{a b} → a + b == b + a
+comm {Z} {b} = ! n+Z==n
+comm {1+ a} {Z} = 1+ap n+Z==n
+comm {1+ a} {1+ b} with a + 1+ b | b + 1+ a | n+1+m==1+n+m {a} {b} | n+1+m==1+n+m {b} {a}
+comm {1+ a} {1+ b} | _ | _ | refl | refl = 1+ap (1+ap (+comm {a}))
+assc : ∀{a b c} → (a + b) + c == a + (b + c)
+assc {Z} = refl
+assc {1+ a} = 1+ap (+assc {a})
a+n==a+m→n==m : ∀{a n m} → a + n == a + m → n == m
a+n==a+m→n==m {Z} refl = refl
a+n==a+m→n==m {1+ a} a+n==a+m = a+n==a+m→n==m (1+inj a+n==a+m)
n+a==m+a→n==m : ∀{n m a} → n + a == m + a → n == m
n+a==m+a→n==m {n} {m} {a} n+a==m+a rewrite +comm {n} {a} | +comm {m} {a} = a+n==a+m→n==m n+a==m+a
+inj : ∀{a1 a2 b} → a1 + b == a2 + b → a1 == a2
+inj {a1} {a2} {Z} h
rewrite (n+Z==n {a1}) | (n+Z==n {a2}) = h
+inj {a1} {a2} {1+ b} h
rewrite n+1+m==1+n+m {a1} {b} | n+1+m==1+n+m {a2} {b} = +inj (1+inj h)
+inj-cp : ∀{a1 a2 b} → a1 ≠ a2 → a1 + b ≠ a2 + b
+inj-cp ne f = ne (+inj f)
-- even/odd theorems
even-inj : ∀{m n} → m + m == n + n → m == n
even-inj {Z} {Z} eq = refl
even-inj {1+ m} {1+ n} eq rewrite (n+1+m==1+n+m {n} {n}) | (n+1+m==1+n+m {m} {m})
= 1+ap (even-inj (1+inj (1+inj eq)))
even-not-odd : ∀{m n} → 1+ (m + m) ≠ (n + n)
even-not-odd {Z} {1+ n} eq rewrite (n+1+m==1+n+m {n} {n}) = 0≠1+n (1+inj eq)
even-not-odd {1+ m} {1+ n} eq rewrite (n+1+m==1+n+m {n} {n}) | (n+1+m==1+n+m {m} {m})
= even-not-odd {m} {n} (1+inj (1+inj eq))
-- _≤_ theorems
0≤n : ∀{n} → Z ≤ n
0≤n {Z} = ≤refl
0≤n {1+ n} = ≤1+ 0≤n
1+n≰0 : ∀{n} → 1+ n ≤ Z → ⊥
1+n≰0 = λ ()
n≤m→1+n≤1+m : ∀{n m} → n ≤ m → 1+ n ≤ 1+ m
n≤m→1+n≤1+m {n} {.n} ≤refl = ≤refl
n≤m→1+n≤1+m {n} {.(1+ _)} (≤1+ h) = ≤1+ (n≤m→1+n≤1+m h)
n≤m→s+n=m : ∀{n m} → n ≤ m → Σ[ s ∈ Nat ] (s + n == m)
n≤m→s+n=m ≤refl = Z , refl
n≤m→s+n=m (≤1+ n≤m)
with n≤m→s+n=m n≤m
... | _ , refl = _ , refl
1+n≤1+m→n≤m : ∀{n m} → 1+ n ≤ 1+ m → n ≤ m
1+n≤1+m→n≤m {n} {.n} ≤refl = ≤refl
1+n≤1+m→n≤m {n} {Z} (≤1+ h) = abort (1+n≰0 h)
1+n≤1+m→n≤m {n} {1+ m} (≤1+ h) = ≤1+ (1+n≤1+m→n≤m h)
n+s≤m+s→n≤m : ∀{n m s} → n + s ≤ m + s → n ≤ m
n+s≤m+s→n≤m {n} {m} {s = Z} n+s≤m+s
rewrite n+Z==n {n} | n+Z==n {m} = n+s≤m+s
n+s≤m+s→n≤m {n} {m} {s = 1+ s} n+s≤m+s
rewrite n+1+m==1+n+m {n} {s} | n+1+m==1+n+m {m} {s}
= n+s≤m+s→n≤m (1+n≤1+m→n≤m n+s≤m+s)
1+n≰n : ∀{n} → 1+ n ≤ n → ⊥
1+n≰n {Z} h = abort (1+n≰0 h)
1+n≰n {1+ n} h = 1+n≰n (1+n≤1+m→n≤m h)
≤total : ∀{n m} → n ≤ m ∨ m ≤ n
≤total {Z} {m} = Inl 0≤n
≤total {1+ n} {Z} = Inr (≤1+ 0≤n)
≤total {1+ n} {1+ m} with ≤total {n} {m}
≤total {1+ n} {1+ m} | Inl h = Inl (n≤m→1+n≤1+m h)
≤total {1+ n} {1+ m} | Inr h = Inr (n≤m→1+n≤1+m h)
≤trans : ∀{a b c} → a ≤ b → b ≤ c → a ≤ c
≤trans ≤refl b≤c = b≤c
≤trans (≤1+ a≤b) ≤refl = ≤1+ a≤b
≤trans (≤1+ a≤b) (≤1+ b≤c) = ≤1+ (≤trans (≤1+ a≤b) b≤c)
≤antisym : ∀{n m} → n ≤ m → m ≤ n → n == m
≤antisym {n} {.n} ≤refl m≤n = refl
≤antisym {n} {.(1+ _)} (≤1+ h1) h2 = abort (1+n≰n (≤trans h2 h1))
n≤n+m : ∀{n m} → n ≤ n + m
n≤n+m {n} {Z} with n + Z | n+Z==n {n}
n≤n+m {_} {Z} | _ | refl = ≤refl
n≤n+m {n} {1+ m} with n + 1+ m | ! (n+1+m==1+n+m {n} {m})
n≤n+m {n} {1+ m} | _ | refl = ≤trans n≤n+m (≤1+ ≤refl)
n≤m+n : ∀{n m} → n ≤ m + n
n≤m+n {n} {m} rewrite +comm {m} {n} = n≤n+m
-- _<_ theorems
n≮0 : ∀{n} → n < Z → ⊥
n≮0 {Z} (π3 , π4) = π4 refl
n≮0 {1+ n} (π3 , π4) = 1+n≰0 π3
n<1+n : ∀{n} → n < 1+ n
π1 n<1+n = ≤1+ ≤refl
π2 n<1+n ()
1+n<1+m→n<m : ∀{n m} → 1+ n < 1+ m → n < m
π1 (1+n<1+m→n<m (π3 , π4)) = 1+n≤1+m→n≤m π3
π2 (1+n<1+m→n<m (π3 , π4)) = 1+ap-cp π4
<trans : ∀{a b c} → a < b → b < c → a < c
π1 (<trans (π3 , π4) (π5 , π6)) = ≤trans π3 π5
π2 (<trans (π3 , π4) (≤refl , π6)) = abort (π6 refl)
π2 (<trans (π3 , π4) (≤1+ π5 , π6)) refl = 1+n≰n (≤trans π3 π5)
<antisym : ∀{n m} → n < m → m < n → ⊥
<antisym (n≤m , n≠m) (m≤n , _) = n≠m (≤antisym n≤m m≤n)
<antirefl : ∀{n} → n < n → ⊥
<antirefl (_ , ne) = abort (ne refl)
n<m→n<s+m : ∀{n m s} → n < m → n < s + m
n<m→n<s+m {s = Z} n<m = n<m
n<m→n<s+m {s = 1+ s} n<m =
<trans (n<m→n<s+m {s = s} n<m) n<1+n
n<m→1+n<1+m : ∀{n m} → n < m → 1+ n < 1+ m
n<m→1+n<1+m (π3 , π4) = n≤m→1+n≤1+m π3 , 1+inj-cp π4
n<m→s+1+n=m : ∀{n m} → n < m → Σ[ s ∈ Nat ] (s + 1+ n == m)
n<m→s+1+n=m (≤refl , ne) = abort (ne refl)
n<m→s+1+n=m {n = n} (≤1+ n≤m , _)
with n≤m→s+n=m n≤m
... | _ , refl = _ , (n+1+m==1+n+m {m = n})
0<1+n : ∀{n} → 0 < 1+ n
0<1+n {Z} = ≤1+ ≤refl , (λ ())
0<1+n {1+ n} = 0≤n , (λ ())
1+n≤m→n<m : ∀{n m} → 1+ n ≤ m → n < m
1+n≤m→n<m ≤refl = n<1+n
1+n≤m→n<m (≤1+ 1+n≤m) = <trans (1+n≤m→n<m 1+n≤m) n<1+n
n≤m→n<1+m : ∀{n m} → n ≤ m → n < 1+ m
n≤m→n<1+m {Z} n≤m = 0<1+n
n≤m→n<1+m {1+ n} n≤m = n<m→1+n<1+m (1+n≤m→n<m n≤m)
n<m→1+n≤m : ∀{n m} → n < m → 1+ n ≤ m
n<m→1+n≤m (≤refl , ne) = abort (ne refl)
n<m→1+n≤m (≤1+ n≤m , _) = n≤m→1+n≤1+m n≤m
n<m→n≤m : ∀{n m} → n < m → n ≤ m
n<m→n≤m n<m = 1+n≤1+m→n≤m (≤1+ (n<m→1+n≤m n<m))
<dec : (n m : Nat) → n < m ∨ n == m ∨ m < n
<dec n m with natEQ n m
... | Inl refl = Inr (Inl refl)
... | Inr ne with ≤total {n} {m}
... | Inl ≤refl = abort (ne refl)
... | Inl (≤1+ n≤m) = Inl (n≤m→n<1+m n≤m)
... | Inr ≤refl = abort (ne refl)
... | Inr (≤1+ m≤n) = Inr (Inr (n≤m→n<1+m m≤n))
<dec-refl : (n : Nat) → <dec n n == Inr (Inl refl)
<dec-refl n with <dec n n
<dec-refl n | Inl (_ , ne) = abort (ne refl)
<dec-refl n | Inr (Inl refl) = refl
<dec-refl n | Inr (Inr (_ , ne)) = abort (ne refl)
-- difference theorems
m-n+n==m : ∀{n m} → (n≤m : n ≤ m) → difference n≤m + n == m
m-n+n==m ≤refl = refl
m-n+n==m (≤1+ n≤m) = 1+ap (m-n+n==m n≤m)
n+m-n==m : ∀{n m} → (n≤n+m : n ≤ n + m) → difference n≤n+m == m
n+m-n==m {n} n≤n+m =
n+a==m+a→n==m (m-n+n==m n≤n+m · +comm {n})
a+b==c→a==c-b : ∀{a b c} → a + b == c → (b≤c : b ≤ c) → a == difference b≤c
a+b==c→a==c-b a+b==c b≤c
= n+a==m+a→n==m (a+b==c · ! (m-n+n==m b≤c))
diff-proof-irrelevance : ∀{n m} →
(n≤m1 n≤m2 : n ≤ m) →
difference n≤m1 == difference n≤m2
diff-proof-irrelevance ≤refl ≤refl = refl
diff-proof-irrelevance ≤refl (≤1+ n≤m2) = abort (1+n≰n n≤m2)
diff-proof-irrelevance (≤1+ n≤m1) ≤refl = abort (1+n≰n n≤m1)
diff-proof-irrelevance (≤1+ n≤m1) (≤1+ n≤m2) = 1+ap (diff-proof-irrelevance n≤m1 n≤m2)
m-n==1+m-1+n : ∀{n m} →
(n≤m : n ≤ m) →
(1+n≤1+m : 1+ n ≤ 1+ m) →
difference n≤m == difference 1+n≤1+m
m-n==1+m-1+n {n} {.n} ≤refl ≤refl = refl
m-n==1+m-1+n {n} {.n} ≤refl (≤1+ 1+n≤n) = abort (1+n≰n 1+n≤n)
m-n==1+m-1+n {.(1+ _)} {.(1+ _)} (≤1+ 1+m≤m) ≤refl = abort (1+n≰n 1+m≤m)
m-n==1+m-1+n {n} {.(1+ _)} (≤1+ n≤m) (≤1+ 1+n≤1+m) = 1+ap (m-n==1+m-1+n n≤m 1+n≤1+m)
m-n==m+s-n+s : ∀{n m s} →
(n≤m : n ≤ m) →
(n+s≤m+s : n + s ≤ m + s) →
difference n≤m == difference n+s≤m+s
m-n==m+s-n+s {n} {m} {s = Z} n≤m n+s≤m+s
rewrite n+Z==n {n} | n+Z==n {m}
= diff-proof-irrelevance n≤m n+s≤m+s
m-n==m+s-n+s {n} {m} {s = 1+ s} n≤m n+s≤m+s
rewrite n+1+m==1+n+m {n} {s} | n+1+m==1+n+m {m} {s}
= (m-n==m+s-n+s n≤m (1+n≤1+m→n≤m n+s≤m+s)) · (m-n==1+m-1+n (1+n≤1+m→n≤m n+s≤m+s) n+s≤m+s)
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module Prelude where
open import Agda.Primitive using (Level; lzero; lsuc) renaming (_⊔_ to lmax)
-- empty type
data ⊥ : Set where
-- from false, derive whatever
abort : ∀ {C : Set} → ⊥ → C
abort ()
-- unit
data ⊤ : Set where
<> : ⊤
-- sums
data _+_ (A B : Set) : Set where
Inl : A → A + B
Inr : B → A + B
-- pairs
infixr 1 _,_
record Σ {l1 l2 : Level} (A : Set l1) (B : A → Set l2) : Set (lmax l1 l2) where
constructor _,_
field
π1 : A
π2 : B π1
open Σ public
-- Sigma types, or dependent pairs, with nice notation.
syntax Σ A (\ x -> B) = Σ[ x ∈ A ] B
_×_ : {l1 : Level} {l2 : Level} → (Set l1) → (Set l2) → Set (lmax l1 l2)
A × B = Σ A λ _ → B
infixr 1 _×_
infixr 1 _+_
-- equality
data _==_ {l : Level} {A : Set l} (M : A) : A → Set l where
refl : M == M
infixr 9 _==_
-- disequality
_≠_ : {l : Level} {A : Set l} → (a b : A) → Set l
a ≠ b = (a == b) → ⊥
{-# BUILTIN EQUALITY _==_ #-}
-- transitivity of equality
_·_ : {l : Level} {α : Set l} {x y z : α} → x == y → y == z → x == z
refl · refl = refl
-- symmetry of equality
! : {l : Level} {α : Set l} {x y : α} → x == y → y == x
! refl = refl
-- ap, in the sense of HoTT, that all functions respect equality in their
-- arguments. named in a slightly non-standard way to avoid naming
-- clashes with hazelnut constructors.
ap1 : {l1 l2 : Level} {α : Set l1} {β : Set l2} {x y : α} (F : α → β)
→ x == y → F x == F y
ap1 F refl = refl
-- transport, in the sense of HoTT, that fibrations respect equality
tr : {l1 l2 : Level} {α : Set l1} {x y : α}
(B : α → Set l2)
→ x == y
→ B x
→ B y
tr B refl x₁ = x₁
-- options
data Maybe (A : Set) : Set where
Some : A → Maybe A
None : Maybe A
-- the some constructor is injective. perhaps unsurprisingly.
someinj : {A : Set} {x y : A} → Some x == Some y → x == y
someinj refl = refl
-- some isn't none.
somenotnone : {A : Set} {x : A} → Some x == None → ⊥
somenotnone ()
-- function extensionality, used to reason about contexts as finite
-- functions.
postulate
funext : {A : Set} {B : A → Set} {f g : (x : A) → (B x)} →
((x : A) → f x == g x) → f == g
-- non-equality is commutative
flip : {A : Set} {x y : A} → (x == y → ⊥) → (y == x → ⊥)
flip neq eq = neq (! eq)
-- two types are said to be equivalent, or isomorphic, if there is a pair
-- of functions between them where both round-trips are stable up to ==
_≃_ : Set → Set → Set
_≃_ A B = Σ[ f ∈ (A → B) ] Σ[ g ∈ (B → A) ]
(((a : A) → g (f a) == a) × (((b : B) → f (g b) == b)))
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----------------------------------------------------------------------
-- Functional big-step evaluation of terms in the partiality monad
----------------------------------------------------------------------
module SystemF.Eval where
open import Codata.Musical.Notation
open import Category.Monad
open import Category.Monad.Partiality.All as All using (All; now; later)
open import Data.Fin using (Fin; zero; suc)
open import Data.Maybe as Maybe using (just; nothing)
open import Data.Maybe.Relation.Unary.Any as MaybeAny using (just)
open import Data.Nat using (_+_)
open import Data.Vec using ([])
open import Function
open import Relation.Binary.PropositionalEquality as P using (_≡_)
open import Relation.Nullary
open import PartialityAndFailure as PF hiding (fail)
open PF.Equality hiding (fail)
open PF.Equivalence
private module M {f} = RawMonad (PF.monad {f})
open import SystemF.Type
open import SystemF.Term
open import SystemF.WtTerm
open TypeSubst using () renaming (_[/_] to _[/tp_])
open TermTypeSubst using () renaming (_[/_] to _[/tmTp_])
open TermTermSubst using () renaming (_[/_] to _[/tmTm_])
open WtTermTypeSubst using () renaming (_[/_]′ to _[/⊢tmTp_])
open WtTermTermSubst using () renaming (_[/_] to _[/⊢tmTm_])
----------------------------------------------------------------------
-- Functional call-by-value big-step semantics
-- The functional presentation of the big-step semantics below is
-- heavily inspired by Danielsson's ICFP'12 paper "Operational
-- Semantics Using the Partiality Monad". While the paper describes a
-- closure-based semantics, the semantics given below is
-- substitution-based. This simplifies the evaluation of recursive
-- terms, i.e those involving fixpoint combinators, which would
-- otherwise have to be evaluated in a cyclic evironment, i.e. one
-- already containing the value of the recursive term being evaluated.
--
-- NB: while it is not described in detail in the paper, Danielsson
-- actually provides an alternative, substitution-based implementation
-- in the accompanying code, which can be found at
--
-- http://www.cse.chalmers.se/~nad/publications/danielsson-semantics-partiality-monad.tgz
--
-- As pointed out in Danielsson's paper, the functional presentation
-- of the big-step semantics feels rather natural in that it follows
-- the form of an interpreter, and it has the added advantage of
-- proving that the semantics are deterministic and computable "for
-- free".
--
-- For more information about Danielson's paper see
--
-- http://www.cse.chalmers.se/~nad/publications/danielsson-semantics-partiality-monad.html
----------------------------------------------------------------------
-- Semantic domain and evaluation
-- Computations with potential failure and partiality effects.
Comp : ∀ m n → Set
Comp m n = (Val m n) ?⊥
-- Following Danielsson's approach, we formulate the evaluation
-- function _⇓′ in an "embedded language" to work around the
-- limitations of Agda's guarded coinduction. The function _⇓′
-- returns "programs", i.e. instances of the type _?⊥P which
-- internalizes the monadic bind operation as a constructor. In a
-- second step, these programs are interpreted in the
-- partiality-and-failure monad by the function _⇓.
--
-- For details about this technique, see e.g. Danielsson's PAR'10
-- paper "Beating the Productivity Checker Using Embedded Languages".
--
-- SystemF.Eval.NoWorkarounds contains an alternative version of the
-- semantics which does not use the above-mentioned workaround. The
-- alternative definition, while (provably) equivalent to the one
-- given below, is more verbose and arguably less readable. However
-- the associated type soundness proof is simpler in that it requires
-- no additional compositionality lemmas.
module _ where
open PF.Workaround
-- Computation "programs".
CompP : ∀ m n → Set₁
CompP m n = (Val m n) ?⊥P
mutual
infix 7 _⇓′ _[_]′ _·′_
-- Evaluation of untyped (open) terms in _?⊥P.
_⇓′ : ∀ {m n} → Term m n → CompP m n
var x ⇓′ = fail
Λ t ⇓′ = return (Λ t)
λ' a t ⇓′ = return (λ' a t)
μ a t ⇓′ = later (♯ (t [/tmTm μ a t ] ⇓′))
(t [ a ]) ⇓′ = t ⇓′ >>= λ v → v [ a ]′
(s · t) ⇓′ = s ⇓′ >>= λ f → t ⇓′ >>= λ v → f ·′ v
fold a t ⇓′ = t ⇓′ >>= λ v → return (fold a v)
unfold a t ⇓′ = t ⇓′ >>= λ v → unfold′ a v
-- Call-by-value evaluation of type application in _?⊥P.
_[_]′ : ∀ {m n} → Val m n → Type n → CompP m n
(Λ t) [ a ]′ = later (♯ (t [/tmTp a ] ⇓′))
_ [ _ ]′ = fail
-- Call-by-value Evaluation of term application in _?⊥P.
_·′_ : ∀ {m n} → Val m n → Val m n → CompP m n
(λ' _ t) ·′ v = later (♯ (t [/tmTm ⌜ v ⌝ ] ⇓′))
_ ·′ _ = fail
-- Evaluation of recursive type unfolding in _?⊥P.
unfold′ : ∀ {m n} → Type (1 + n) → Val m n → CompP m n
unfold′ _ (fold _ v) = return v
unfold′ a _ = fail
infix 7 _⇓
-- Evaluation of untyped (open) terms in the partiality monad.
_⇓ : ∀ {m n} → Term m n → Comp m n
t ⇓ = ⟦ t ⇓′ ⟧P
----------------------------------------------------------------------
-- The semantics _⇓ is compositional
module _ where
open M
open PF.Reasoning
open PF.Workaround using (⟦_⟧P) renaming (_>>=_ to _>>=P_)
open PF.Workaround.Correct
infix 7 _[_]⇓ _·⇓_
-- Short hands for relating the semantics of composite terms to the
-- semantics of their subterms.
_[_]⇓ : ∀ {m n} → Comp m n → Type n → Comp m n
c [ a ]⇓ = c >>= λ v → ⟦ v [ a ]′ ⟧P
_·⇓_ : ∀ {m n} → Comp m n → Comp m n → Comp m n
c ·⇓ d = c >>= λ f → d >>= λ v → ⟦ f ·′ v ⟧P
fold⇓ : ∀ {m n} → Type (1 + n) → Comp m n → Comp m n
fold⇓ a c = c >>= λ v → return (fold a v)
unfold⇓ : ∀ {m n} → Type (1 + n) → Comp m n → Comp m n
unfold⇓ a c = c >>= λ v → ⟦ unfold′ a v ⟧P
-- The semantics of type application is compositional.
[]-comp : ∀ {m n} (t : Term m n) (a : Type n) → t [ a ] ⇓ ≅ (t ⇓) [ a ]⇓
[]-comp t a = >>=-hom (t ⇓′) _
-- The semantics of term application is compositional.
·-comp : ∀ {m n} (s t : Term m n) → s · t ⇓ ≅ (s ⇓) ·⇓ (t ⇓)
·-comp s t =
s · t ⇓
≅⟨ >>=-hom (s ⇓′) _ ⟩
(s ⇓ >>= λ f → ⟦ t ⇓′ >>=P (λ v → f ·′ v) ⟧P)
≅⟨ (s ⇓ ∎ >>=-cong λ _ → >>=-hom (t ⇓′) _) ⟩
(s ⇓) ·⇓ (t ⇓)
∎
-- The semantics of recursive type folding is compositional.
fold-comp : ∀ {m n} (a : Type (1 + n)) (t : Term m n) →
fold a t ⇓ ≅ fold⇓ a (t ⇓)
fold-comp a t = >>=-hom (t ⇓′) _
-- The semantics of recursive type unfolding is compositional.
unfold-comp : ∀ {m n} (a : Type (1 + n)) (t : Term m n) →
unfold a t ⇓ ≅ unfold⇓ a (t ⇓)
unfold-comp a t = >>=-hom (t ⇓′) _
----------------------------------------------------------------------
-- Type soundness
open PF using (fail)
infix 4 _⊢comp_∈_
-- A computation is well-typed if it is a well-typed value or it takes
-- a step towards a well-typed computation. Note that we exclude the
-- case of failing well-typed computations through the use of
-- Maybe.Any.
_⊢comp_∈_ : ∀ {m n} → Ctx m n → Comp m n → Type n → Set
Γ ⊢comp c ∈ a = All (MaybeAny.Any (λ v → Γ ⊢val v ∈ a)) c
-- Well-typed computations do not fail.
does-not-fail : ∀ {m n} {Γ : Ctx m n} {c a} → Γ ⊢comp c ∈ a → ¬ c ≈ fail
does-not-fail (now (MaybeAny.just _)) (now ())
does-not-fail (later ⊢c) (laterˡ c-fails) = does-not-fail (♭ ⊢c) c-fails
-- It remains to prove that well-typed terms evaluate to well-typed
-- computations. The proof ⊢_⇓ follows the same structure as _⇓ and
-- uses an analogous workaround to deal with guarded coinduction. It
-- is formulated in the language AllP of of Partiality.All "programs"
-- defined in All.Alternative.
open All.Alternative
open PF.Workaround using (⟦_⟧P)
infix 4 ⊢compP_∈_ ⊢val_∈_
-- Closed well-typed computation "programs".
⊢compP_∈_ : Comp 0 0 → Type 0 → Set₁
⊢compP c ∈ a = AllP (MaybeAny.Any (λ v → [] ⊢val v ∈ a)) c
-- A short hand for closed well-typed values.
⊢val_∈_ : Val 0 0 → Type 0 → Set
⊢val v ∈ a = [] ⊢val v ∈ a
mutual
infix 7 ⊢_⇓′ ⊢_[_]′ ⊢_·′_
-- Evaluation of closed terms preserves well-typedness in AllP.
⊢_⇓′ : ∀ {t a} → [] ⊢ t ∈ a → ⊢compP t ⇓ ∈ a
⊢ var () ⇓′
⊢ Λ ⊢t ⇓′ = now (just (Λ ⊢t))
⊢ λ' a ⊢t ⇓′ = now (just (λ' a ⊢t))
⊢ μ a ⊢t ⇓′ = later (♯ ⊢ ⊢t [/⊢tmTm μ a ⊢t ] ⇓′)
⊢_⇓′ {t [ a ]} (⊢t [ .a ] ) =
t [ a ] ⇓ ≅⟨ []-comp t a ⟩P
(t ⇓) [ a ]⇓ ⟨ ⊢ ⊢t ⇓′ >>=-congP (λ { .{_} (just ⊢v) →
⊢ ⊢v [ a ]′ }) ⟩P
⊢_⇓′ {s · t} (⊢s · ⊢t) =
s · t ⇓ ≅⟨ ·-comp s t ⟩P
(s ⇓) ·⇓ (t ⇓) ⟨ (⊢ ⊢s ⇓′ >>=-congP λ { .{_} (just ⊢f) →
⊢ ⊢t ⇓′ >>=-congP λ { .{_} (just ⊢v) →
⊢ ⊢f ·′ ⊢v }}) ⟩P
⊢_⇓′ {fold a t} (fold .a ⊢t) =
fold a t ⇓ ≅⟨ fold-comp a t ⟩P
fold⇓ a (t ⇓) ⟨ (⊢ ⊢t ⇓′ >>=-congP λ { .{_} (just ⊢v) →
now (just (fold a ⊢v)) }) ⟩P
⊢_⇓′ {unfold a t} (unfold .a ⊢t) =
unfold a t ⇓ ≅⟨ unfold-comp a t ⟩P
unfold⇓ a (t ⇓) ⟨ (⊢ ⊢t ⇓′ >>=-congP λ { .{_} (just ⊢v) →
⊢unfold′ a ⊢v }) ⟩P
-- Evaluation of type application preserves well-typedness in AllP.
⊢_[_]′ : ∀ {v a} → ⊢val v ∈ ∀' a → ∀ b → ⊢compP ⟦ v [ b ]′ ⟧P ∈ a [/tp b ]
⊢ Λ ⊢t [ a ]′ = later (♯ ⊢ ⊢t [/⊢tmTp a ] ⇓′)
-- Evaluation of term application preserves well-typedness in AllP.
⊢_·′_ : ∀ {f v a b} → ⊢val f ∈ a →' b → ⊢val v ∈ a → ⊢compP ⟦ f ·′ v ⟧P ∈ b
⊢ λ' a ⊢t ·′ ⊢v = later (♯ ⊢ ⊢t [/⊢tmTm ⊢⌜ ⊢v ⌝ ] ⇓′)
-- Evaluation of recursive type unfolding preserves well-typedness in AllP.
⊢unfold′ : ∀ {v} a → ⊢val v ∈ μ a → ⊢compP ⟦ unfold′ a v ⟧P ∈ a [/tp μ a ]
⊢unfold′ _ (fold ._ ⊢v) = now (just ⊢v)
infix 7 ⊢_⇓
-- Evaluation of closed terms preserves well-typedness.
⊢_⇓ : ∀ {t a} → [] ⊢ t ∈ a → [] ⊢comp t ⇓ ∈ a
⊢_⇓ = sound ∘ ⊢_⇓′
-- Type soundness: evaluation of well-typed terms does not fail.
type-soundness : ∀ {t a} → [] ⊢ t ∈ a → ¬ t ⇓ ≈ fail
type-soundness ⊢t = does-not-fail ⊢ ⊢t ⇓
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------------------------------------------------------------------------------
-- Group theory properties
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module GroupTheory.PropertiesI where
open import GroupTheory.Base
open import Common.FOL.Relation.Binary.EqReasoning
------------------------------------------------------------------------------
-- Congruence properties
-- The propositional equality is compatible with the binary operation.
·-leftCong : ∀ {a b c} → a ≡ b → a · c ≡ b · c
·-leftCong refl = refl
·-rightCong : ∀ {a b c} → b ≡ c → a · b ≡ a · c
·-rightCong refl = refl
-- The propositional equality is compatible with the inverse function.
⁻¹-cong : ∀ {a b} → a ≡ b → a ⁻¹ ≡ b ⁻¹
⁻¹-cong refl = refl
------------------------------------------------------------------------------
leftCancellation : ∀ {a b c} → a · b ≡ a · c → b ≡ c
leftCancellation {a} {b} {c} h =
b ≡⟨ sym (leftIdentity b) ⟩
ε · b ≡⟨ ·-leftCong (sym (leftInverse a)) ⟩
a ⁻¹ · a · b ≡⟨ assoc (a ⁻¹) a b ⟩
a ⁻¹ · (a · b) ≡⟨ ·-rightCong h ⟩
a ⁻¹ · (a · c) ≡⟨ sym (assoc (a ⁻¹) a c) ⟩
a ⁻¹ · a · c ≡⟨ ·-leftCong (leftInverse a) ⟩
ε · c ≡⟨ leftIdentity c ⟩
c ∎
-- A different proof without using congruence.
leftCancellation' : ∀ {a b c} → a · b ≡ a · c → b ≡ c
-- Paper proof (Mac Lane and Garret Birkhoff 1999. p. 48):
--
-- 1. a⁻¹(ab) = a⁻¹(ac) (hypothesis ab = ac)
-- 2. a⁻¹a(b) = a⁻¹a(c) (associative axiom)
-- 3. εb = εc (left-inverse axiom for a⁻¹)
-- 4. b = c (left-identity axiom)
leftCancellation' {a} {b} {c} h =
b ≡⟨ sym (leftIdentity b) ⟩
ε · b ≡⟨ subst (λ t → ε · b ≡ t · b) (sym (leftInverse a)) refl ⟩
a ⁻¹ · a · b ≡⟨ assoc (a ⁻¹) a b ⟩
a ⁻¹ · (a · b) ≡⟨ subst (λ t → a ⁻¹ · (a · b) ≡ a ⁻¹ · t) h refl ⟩
a ⁻¹ · (a · c) ≡⟨ sym (assoc (a ⁻¹) a c) ⟩
a ⁻¹ · a · c ≡⟨ subst (λ t → a ⁻¹ · a · c ≡ t · c) (leftInverse a) refl ⟩
ε · c ≡⟨ leftIdentity c ⟩
c ∎
-- Mac Lane and Garret Birkhoff (1999) p. 50, exercise 6.
rightIdentity : ∀ a → a · ε ≡ a
rightIdentity a = leftCancellation prf
where
prf : a ⁻¹ · (a · ε) ≡ a ⁻¹ · a
prf = a ⁻¹ · (a · ε) ≡⟨ sym (assoc (a ⁻¹) a ε) ⟩
a ⁻¹ · a · ε ≡⟨ ·-leftCong (leftInverse a) ⟩
ε · ε ≡⟨ leftIdentity ε ⟩
ε ≡⟨ sym (leftInverse a) ⟩
a ⁻¹ · a ∎
-- Mac Lane and Garret Birkhoff (1999) p. 50, exercise 6.
rightInverse : ∀ a → a · a ⁻¹ ≡ ε
rightInverse a = leftCancellation prf
where
prf : a ⁻¹ · (a · a ⁻¹) ≡ a ⁻¹ · ε
prf = a ⁻¹ · (a · a ⁻¹) ≡⟨ sym (assoc (a ⁻¹) a (a ⁻¹)) ⟩
a ⁻¹ · a · a ⁻¹ ≡⟨ ·-leftCong (leftInverse a) ⟩
ε · a ⁻¹ ≡⟨ leftIdentity (a ⁻¹) ⟩
a ⁻¹ ≡⟨ sym (rightIdentity (a ⁻¹)) ⟩
a ⁻¹ · ε ∎
rightCancellation : ∀ {a b c} → b · a ≡ c · a → b ≡ c
rightCancellation {a} {b} {c} h =
-- Paper proof:
--
-- 1. (ba)a⁻¹ = (ca)a⁻¹ (hypothesis ab = ac)
-- 2. (b)aa⁻¹ = (c)aa⁻¹ (associative axiom)
-- 3. bε = cε (right-inverse axiom for a⁻¹)
-- 4. b = c (right-identity axiom)
b ≡⟨ sym (rightIdentity b) ⟩
b · ε ≡⟨ ·-rightCong (sym (rightInverse a)) ⟩
b · (a · a ⁻¹) ≡⟨ sym (assoc b a (a ⁻¹)) ⟩
b · a · a ⁻¹ ≡⟨ ·-leftCong h ⟩
c · a · a ⁻¹ ≡⟨ assoc c a (a ⁻¹) ⟩
c · (a · a ⁻¹) ≡⟨ ·-rightCong (rightInverse a) ⟩
c · ε ≡⟨ rightIdentity c ⟩
c ∎
-- Adapted from the Agda standard library 0.8.1 (see
-- Algebra.Properties.Group.right-helper).
y≡x⁻¹[xy] : ∀ a b → b ≡ a ⁻¹ · (a · b)
y≡x⁻¹[xy] a b = b ≡⟨ sym (leftIdentity b) ⟩
ε · b ≡⟨ ·-leftCong (sym (leftInverse a)) ⟩
a ⁻¹ · a · b ≡⟨ assoc (a ⁻¹) a b ⟩
a ⁻¹ · (a · b) ∎
-- Adapted from the Agda standard library 0.8.1 (see
-- Algebra.Properties.Group.left-helper).
x≡[xy]y⁻¹ : ∀ a b → a ≡ (a · b) · b ⁻¹
x≡[xy]y⁻¹ a b = a ≡⟨ sym (rightIdentity a) ⟩
a · ε ≡⟨ ·-rightCong (sym (rightInverse b)) ⟩
a · (b · b ⁻¹) ≡⟨ sym (assoc a b (b ⁻¹)) ⟩
a · b · b ⁻¹ ∎
rightIdentityUnique : ∀ r → (∀ a → a · r ≡ a) → r ≡ ε
-- Paper proof (Mac Lane and Garret 1999. p. 48):
--
-- 1. r = εr (ε is an identity)
-- 2. εr = r (hypothesis)
-- 3. r = ε (transitivity)
rightIdentityUnique r h = trans (sym (leftIdentity r)) (h ε)
-- A more appropiate version to be used in the proofs. Adapted from
-- the Agda standard library 0.8.1 (see
-- Algebra.Properties.Group.right-identity-unique).
rightIdentityUnique' : ∀ a r → a · r ≡ a → r ≡ ε
rightIdentityUnique' a r h = r ≡⟨ y≡x⁻¹[xy] a r ⟩
a ⁻¹ · (a · r) ≡⟨ ·-rightCong h ⟩
a ⁻¹ · a ≡⟨ leftInverse a ⟩
ε ∎
leftIdentityUnique : ∀ l → (∀ a → l · a ≡ a) → l ≡ ε
-- Paper proof:
-- 1. l = le (ε is an identity)
-- 2. le = e (hypothesis)
-- 3. l = e (transitivity)
leftIdentityUnique l h = trans (sym (rightIdentity l)) (h ε)
-- A more appropiate version to be used in the proofs. Adapted from
-- the Agda standard library 0.8.1 (see
-- Algebra.Properties.Group.left-identity-unique).
leftIdentityUnique' : ∀ a l → l · a ≡ a → l ≡ ε
leftIdentityUnique' a l h = l ≡⟨ x≡[xy]y⁻¹ l a ⟩
l · a · a ⁻¹ ≡⟨ ·-leftCong h ⟩
a · a ⁻¹ ≡⟨ rightInverse a ⟩
ε ∎
rightInverseUnique : ∀ {a} → ∃[ r ] (a · r ≡ ε) ∧
(∀ r' → a · r' ≡ ε → r ≡ r')
rightInverseUnique {a} =
-- Paper proof:
--
-- 1. We know that (a⁻¹) is a right inverse for a.
-- 2. Let's suppose there is other right inverse r for a, i.e. ar ≡ ε, then
-- 2.1. aa⁻¹ = ε (right-inverse axiom)
-- 2.2. ar = ε (hypothesis)
-- 2.3. aa⁻¹ = ar (transitivity)
-- 2.4 a⁻¹ = a (left-cancellation)
_ , rightInverse a , prf
where
prf : ∀ r' → a · r' ≡ ε → a ⁻¹ ≡ r'
prf r' ar'≡ε = leftCancellation aa⁻¹≡ar'
where
aa⁻¹≡ar' : a · a ⁻¹ ≡ a · r'
aa⁻¹≡ar' = a · a ⁻¹ ≡⟨ rightInverse a ⟩
ε ≡⟨ sym ar'≡ε ⟩
a · r' ∎
-- A more appropiate version to be used in the proofs.
rightInverseUnique' : ∀ {a r} → a · r ≡ ε → a ⁻¹ ≡ r
rightInverseUnique' {a} {r} ar≡ε = leftCancellation aa⁻¹≡ar
where
aa⁻¹≡ar : a · a ⁻¹ ≡ a · r
aa⁻¹≡ar = a · a ⁻¹ ≡⟨ rightInverse a ⟩
ε ≡⟨ sym ar≡ε ⟩
a · r ∎
leftInverseUnique : ∀ {a} → ∃[ l ] (l · a ≡ ε) ∧
(∀ l' → l' · a ≡ ε → l ≡ l')
leftInverseUnique {a} =
-- Paper proof:
--
-- 1. We know that (a⁻¹) is a left inverse for a.
-- 2. Let's suppose there is other right inverse l for a, i.e. la ≡ ε, then
-- 2.1. a⁻¹a = ε (left-inverse axiom)
-- 2.2. la = ε (hypothesis)
-- 2.3. a⁻¹a = la (transitivity)
-- 2.4 a⁻¹ = l (right-cancellation)
_ , leftInverse a , prf
where
prf : ∀ l' → l' · a ≡ ε → a ⁻¹ ≡ l'
prf l' l'a≡ε = rightCancellation a⁻¹a≡l'a
where
a⁻¹a≡l'a : a ⁻¹ · a ≡ l' · a
a⁻¹a≡l'a = a ⁻¹ · a ≡⟨ leftInverse a ⟩
ε ≡⟨ sym l'a≡ε ⟩
l' · a ∎
-- A more appropiate version to be used in the proofs.
leftInverseUnique' : ∀ {a l} → l · a ≡ ε → a ⁻¹ ≡ l
leftInverseUnique' {a} {l} la≡ε = rightCancellation a⁻¹a≡la
where
a⁻¹a≡la : a ⁻¹ · a ≡ l · a
a⁻¹a≡la = a ⁻¹ · a ≡⟨ leftInverse a ⟩
ε ≡⟨ sym la≡ε ⟩
l · a ∎
⁻¹-involutive : ∀ a → a ⁻¹ ⁻¹ ≡ a
-- Paper proof:
--
-- 1. a⁻¹a = ε (left-inverse axiom)
-- 2. The previous equation states that a is the unique right
-- inverse (a⁻¹)⁻¹ of a⁻¹.
⁻¹-involutive a = rightInverseUnique' (leftInverse a)
identityInverse : ε ⁻¹ ≡ ε
-- Paper proof:
--
-- 1. εε = ε (left/right-identity axiom)
-- 2. The previous equation states that ε is the unique left/right
-- inverse ε⁻¹ of ε.
identityInverse = rightInverseUnique' (leftIdentity ε)
inverseDistribution : ∀ a b → (a · b) ⁻¹ ≡ b ⁻¹ · a ⁻¹
-- Paper proof:
--
-- (b⁻¹a⁻¹)(ab) = b⁻¹(a⁻¹(ab)) (associative axiom)
-- = b⁻¹(a⁻¹a)b (associative axiom)
-- = b⁻¹(εb) (left-inverse axiom)
-- = b⁻¹b (left-identity axiom)
-- = ε (left-inverse axiom)
-- Therefore, b⁻¹a⁻¹ is the unique left inverse of ab.
inverseDistribution a b = leftInverseUnique' b⁻¹a⁻¹[ab]≡ε
where
b⁻¹a⁻¹[ab]≡ε : b ⁻¹ · a ⁻¹ · (a · b) ≡ ε
b⁻¹a⁻¹[ab]≡ε =
b ⁻¹ · a ⁻¹ · (a · b)
≡⟨ assoc (b ⁻¹) (a ⁻¹) (a · b) ⟩
b ⁻¹ · (a ⁻¹ · (a · b))
≡⟨ ·-rightCong (sym (assoc (a ⁻¹) a b)) ⟩
b ⁻¹ · (a ⁻¹ · a · b)
≡⟨ ·-rightCong (·-leftCong (leftInverse a)) ⟩
b ⁻¹ · (ε · b)
≡⟨ ·-rightCong (leftIdentity b) ⟩
b ⁻¹ · b
≡⟨ leftInverse b ⟩
ε ∎
-- If the square of every element is the identity, the system is
-- commutative. From: TPTP 6.4.0 problem GRP/GRP001-2.p.
x²≡ε→comm : (∀ a → a · a ≡ ε) → ∀ {b c d} → b · c ≡ d → c · b ≡ d
-- Paper proof:
--
-- 1. d(bc) = dd (hypothesis bc = d)
-- 2. d(bc) = ε (hypothesis dd = ε)
-- 3. d(bc)c = c (by 2)
-- 4. db(cc) = c (associativity axiom)
-- 5. db = c (hypothesis cc = ε)
-- 6. (db)b = cb (by 5)
-- 7. d(bb) = cb (associativity axiom)
-- 6. d = cb (hypothesis bb = ε)
x²≡ε→comm h {b} {c} {d} bc≡d = sym d≡cb
where
db≡c : d · b ≡ c
db≡c =
d · b
≡⟨ sym (rightIdentity (d · b)) ⟩
d · b · ε
≡⟨ ·-rightCong (sym (h c)) ⟩
d · b · (c · c)
≡⟨ assoc d b (c · c) ⟩
d · (b · (c · c))
≡⟨ ·-rightCong (sym (assoc b c c)) ⟩
d · ((b · c) · c)
≡⟨ ·-rightCong (·-leftCong bc≡d) ⟩
d · (d · c)
≡⟨ sym (assoc d d c) ⟩
d · d · c
≡⟨ ·-leftCong (h d) ⟩
ε · c
≡⟨ leftIdentity c ⟩
c ∎
d≡cb : d ≡ c · b
d≡cb = d ≡⟨ sym (rightIdentity d) ⟩
d · ε ≡⟨ ·-rightCong (sym (h b)) ⟩
d · (b · b) ≡⟨ sym (assoc d b b) ⟩
d · b · b ≡⟨ ·-leftCong db≡c ⟩
c · b ∎
------------------------------------------------------------------------------
-- References
--
-- Mac Lane, S. and Birkhof, G. (1999). Algebra. 3rd ed. AMS Chelsea
-- Publishing.
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-- intrinsically-typed λ-calculus
module IntrinsicallyTypedLC where
open import Data.List
open import Data.List.Relation.Unary.All
open import Data.Unit
open import Data.Nat
-- definitions
data Ty : Set where
Tunit : Ty
Tfun : Ty → Ty → Ty
TEnv = List Ty
data _∈_ : Ty → TEnv → Set where
here : ∀ {t Φ} → t ∈ (t ∷ Φ)
there : ∀ {t t' Φ} → t ∈ Φ → t ∈ (t' ∷ Φ)
data Exp : TEnv → Ty → Set where
Var : ∀ {Φ t} → (x : t ∈ Φ) → Exp Φ t
Abs : ∀ {Φ t t'} → Exp (t ∷ Φ) t' → Exp Φ (Tfun t t')
App : ∀ {Φ t t'} → Exp Φ (Tfun t t') → Exp Φ t → Exp Φ t'
-- big-step semantics
Val : Ty → Set
Val Tunit = ⊤
Val (Tfun t t₁) = Val t → Val t₁
access : ∀ {t Φ} → t ∈ Φ → All Val Φ → Val t
access here (px ∷ ρ) = px
access (there x) (px ∷ ρ) = access x ρ
eval : ∀ {Φ t} → Exp Φ t → All Val Φ → Val t
eval (Var x) ρ = access x ρ
eval (Abs e) ρ = λ x → eval e (x ∷ ρ)
eval (App e e₁) ρ = (eval e ρ) (eval e₁ ρ)
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Heterogeneous equality
------------------------------------------------------------------------
-- This file contains some core definitions which are reexported by
-- Relation.Binary.HeterogeneousEquality.
{-# OPTIONS --with-K --safe #-}
module Relation.Binary.HeterogeneousEquality.Core where
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
------------------------------------------------------------------------
-- Heterogeneous equality
infix 4 _≅_
data _≅_ {ℓ} {A : Set ℓ} (x : A) : {B : Set ℓ} → B → Set ℓ where
refl : x ≅ x
------------------------------------------------------------------------
-- Conversion
≅-to-≡ : ∀ {a} {A : Set a} {x y : A} → x ≅ y → x ≡ y
≅-to-≡ refl = refl
≡-to-≅ : ∀ {a} {A : Set a} {x y : A} → x ≡ y → x ≅ y
≡-to-≅ refl = refl
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module Problem4 where
infixr 40 _::_
data List (A : Set) : Set where
[] : List A
_::_ : A -> List A -> List A
-- 4.1
map : {A B : Set} -> (A -> B) -> List A -> List B
map f [] = []
map f (x :: xs) = f x :: map f xs
infixr 40 _++_
_++_ : {A : Set} -> List A -> List A -> List A
[] ++ ys = ys
(x :: xs) ++ ys = x :: (xs ++ ys)
-- 4.2
infixr 40 _▹_
data All {A : Set}(P : A -> Set) : List A -> Set where
∅ : All P []
_▹_ : {x : A} -> P x -> {xs : List A} -> All P xs -> All P (x :: xs)
-- 4.3
data Some {A : Set}(P : A -> Set) : List A -> Set where
hd : {x : A} -> P x -> {xs : List A} -> Some P (x :: xs)
tl : {x : A}{xs : List A} -> Some P xs -> Some P (x :: xs)
-- 4.4
-- We need composition at a higher universe here.
_∘¹_ : {A B : Set}{C : B -> Set1}(f : (x : B) -> C x)
(g : A -> B)(x : A) -> C (g x)
(f ∘¹ g) x = f (g x)
-- You might have to give f explictly when applying this theorem.
all-map : {A B : Set}{P : A -> Set}{Q : B -> Set}{f : A -> B}{xs : List A} ->
({x : A} -> P x -> Q (f x)) ->
All P xs -> All Q (map f xs)
all-map h ∅ = ∅
all-map h (p ▹ ps) = h p ▹ all-map h ps
all-++ : {A : Set}{P : A -> Set}{xs ys : List A} ->
All P xs -> All P ys -> All P (xs ++ ys)
all-++ ∅ qs = qs
all-++ (p ▹ ps) qs = p ▹ (all-++ ps qs)
some-map : {A B : Set}{P : A -> Set}{Q : B -> Set}{f : A -> B}{xs : List A} ->
({x : A} -> P x -> Q (f x)) ->
Some P xs -> Some Q (map f xs)
some-map h (hd p) = hd (h p)
some-map h (tl ps) = tl (some-map h ps)
some-++-left : {A : Set}{P : A -> Set}{xs ys : List A} ->
Some P xs -> Some P (xs ++ ys)
some-++-left (hd p) = hd p
some-++-left (tl ps) = tl (some-++-left ps)
-- Here we can't expect to infer xs, so we make it explicit
some-++-right : {A : Set}{P : A -> Set}(xs : List A){ys : List A} ->
Some P ys -> Some P (xs ++ ys)
some-++-right [] p = p
some-++-right (x :: xs) p = tl (some-++-right xs p)
-- 4.5
data _==_ {A : Set}(x : A) : A -> Set where
refl : x == x
_∈_ : {A : Set} -> A -> List A -> Set
x ∈ xs = Some (_==_ x) xs
-- 4.6
record True : Set where
tt : True
tt = record {}
Nat = List True
zero : Nat
zero = []
suc : Nat -> Nat
suc n = tt :: n
Vec : Set -> Nat -> Set
Vec A n = All (\_ -> A) n
Fin : Nat -> Set
Fin n = Some (\_ -> True) n
-- 4.7
infixr 5 _,_
data _×_ (A : Set)(B : A -> Set) : Set where
_,_ : (x : A) -> B x -> A × B
_∧_ : Set -> Set -> Set
A ∧ B = A × (\_ -> B)
_!_ : {A : Set}{P : A -> Set}{Q : A -> Set}{xs : List A} ->
All P xs -> Some Q xs -> A × (\z -> P z ∧ Q z)
∅ ! ()
(p ▹ ps) ! hd q = (_ , p , q)
(p ▹ ps) ! tl q = ps ! q
-- 4.8
data False : Set where
¬_ : Set -> Set
¬ A = A -> False
data _∨_ (A B : Set) : Set where
inl : A -> A ∨ B
inr : B -> A ∨ B
data Bool : Set where
true : Bool
false : Bool
data IsTrue : Bool -> Set where
isTrue : IsTrue true
Holds : {A : Set} -> (A -> Bool) -> A -> Set
Holds p x = IsTrue (p x)
false-isn't-true : ¬ IsTrue false
false-isn't-true ()
decide : {A : Set}(p : A -> Bool)(x : A) ->
Holds p x ∨ ¬ Holds p x
decide p x with p x
... | true = inl isTrue
... | false = inr false-isn't-true
all : {A : Set}(p : A -> Bool)(xs : List A) ->
All (Holds p) xs ∨ Some (\x -> ¬ Holds p x) xs
all p [] = inl ∅
all p (x :: xs) with decide p x
... | inr npx = inr (hd npx)
... | inl px with all p xs
... | inr npxs = inr (tl npxs)
... | inl pxs = inl (px ▹ pxs)
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{-# OPTIONS --safe --experimental-lossy-unification #-}
module Cubical.ZCohomology.RingStructure.CupProduct where
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Pointed
open import Cubical.Data.Nat
open import Cubical.Data.Int hiding (_+'_ ; +'≡+ ; _+_)
open import Cubical.HITs.SetTruncation as ST
open import Cubical.HITs.Truncation as T
open import Cubical.HITs.S1 hiding (_·_)
open import Cubical.HITs.Sn
open import Cubical.HITs.Susp
open import Cubical.ZCohomology.Base
open import Cubical.ZCohomology.GroupStructure
open import Cubical.ZCohomology.Properties
infixl 30 _·₀_
infixr 35 _⌣ₖ_
infixr 35 _⌣_
--- This definition of ℕ-addition removes some unnecessary transports.
open PlusBis
-- Cup product with one integer (K₀) argument
_·₀_ : {n : ℕ} (m : ℤ) → coHomK n → coHomK n
_·₀_ {n = n} (pos zero) x = 0ₖ _
_·₀_ {n = n} (pos (suc m)) x = x +ₖ (pos m ·₀ x)
_·₀_ {n = n} (negsuc zero) x = -ₖ x
_·₀_ {n = n} (negsuc (suc m)) x = (negsuc m ·₀ x) -ₖ x
·₀-0ₖ : {n : ℕ} (m : ℤ) → _·₀_ m (0ₖ n) ≡ 0ₖ n
·₀-0ₖ (pos zero) = refl
·₀-0ₖ (pos (suc n)) = cong (0ₖ _ +ₖ_) (·₀-0ₖ (pos n)) ∙ rUnitₖ _ (0ₖ _)
·₀-0ₖ (negsuc zero) = -0ₖ
·₀-0ₖ (negsuc (suc n)) = cong (λ x → x -ₖ (0ₖ _)) (·₀-0ₖ (negsuc n)) ∙ rCancelₖ _ (0ₖ _)
-- Pointed version first (enables truncation elimination)
⌣ₖ∙ : (n m : ℕ) → coHomK n → coHomK-ptd m →∙ coHomK-ptd (n +' m)
fst (⌣ₖ∙ zero m a) b = a ·₀ b
snd (⌣ₖ∙ zero m a) = ·₀-0ₖ a
fst (⌣ₖ∙ (suc n) zero a) b = b ·₀ a
snd (⌣ₖ∙ (suc n) zero a) = refl
⌣ₖ∙ (suc n) (suc m) = T.rec (isOfHLevel↑∙ (suc n) m) (cup n m)
where
cup : (n m : ℕ) → S₊ (suc n) → coHomK-ptd (suc m) →∙ coHomK-ptd (suc (suc (n + m)))
fst (cup zero m base) _ = 0ₖ _
fst (cup zero m (loop i)) x = Kn→ΩKn+1 _ x i
fst (cup (suc n) m north) _ = 0ₖ _
fst (cup (suc n) m south) _ = 0ₖ _
fst (cup (suc n) m (merid a i)) x = Kn→ΩKn+1 _ (fst (cup n m a) x) i
snd (cup zero m base) = refl
snd (cup zero m (loop i)) k = Kn→ΩKn+10ₖ _ k i
snd (cup (suc n) m north) = refl
snd (cup (suc n) m south) = refl
snd (cup (suc n) m (merid a i)) k = (cong (Kn→ΩKn+1 _) (snd (cup n m a)) ∙ Kn→ΩKn+10ₖ _) k i
-- Non pointed version
_⌣ₖ_ : {n m : ℕ} → coHomK n → coHomK m → coHomK (n +' m)
_⌣ₖ_ {n = n} {m = m} x y = fst (⌣ₖ∙ n m x) y
-- Doubly pointed version
⌣ₖ∙∙ : (n m : ℕ) → coHomK-ptd n →∙ (coHomK-ptd m →∙ coHomK-ptd (n +' m) ∙)
fst (⌣ₖ∙∙ n m) = ⌣ₖ∙ n m
fst (snd (⌣ₖ∙∙ zero zero) i) x = 0
fst (snd (⌣ₖ∙∙ zero (suc m)) i) x = 0ₖ _
fst (snd (⌣ₖ∙∙ (suc n) zero) i) x = ·₀-0ₖ x i
fst (snd (⌣ₖ∙∙ (suc zero) (suc m)) i) x = 0ₖ _
fst (snd (⌣ₖ∙∙ (suc (suc n)) (suc m)) i) x = 0ₖ _
snd (snd (⌣ₖ∙∙ zero zero) i) = refl
snd (snd (⌣ₖ∙∙ zero (suc m)) i) = refl
snd (snd (⌣ₖ∙∙ (suc n) zero) i) = refl
snd (snd (⌣ₖ∙∙ (suc zero) (suc m)) i) = refl
snd (snd (⌣ₖ∙∙ (suc (suc n)) (suc m)) i) = refl
-- Cup product
_⌣_ : ∀ {ℓ} {A : Type ℓ} {n m : ℕ} → coHom n A → coHom m A → coHom (n +' m) A
_⌣_ = ST.rec2 squash₂ λ f g → ∣ (λ x → f x ⌣ₖ g x) ∣₂
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-- https://stackoverflow.com/questions/61037572/how-to-define-the-range-function-on-a-relation-in-agda-set-theory
module range where
open import Data.Unit
open import Data.Product renaming (_,_ to ⟨_,_⟩)
open import Data.Sum
open import Function
Subset : Set → Set₁
Subset A = A → Set
_∈_ : ∀ {A} → A → Subset A → Set
a ∈ P = P a
Relation : ∀ A B → Set₁
Relation A B = Subset (A × B)
Range : ∀ {A B} → Relation A B → Subset B
Range R b = ∃ (R ∘ ⟨_, b ⟩) -- equivalent to ∃ \a → R ⟨ a , b ⟩
_⊆_ : ∀ {A} → Subset A → Subset A → Set
A ⊆ B = ∀ x → x ∈ A → x ∈ B
wholeSet : ∀ A → Subset A
wholeSet _ _ = ⊤
∀subset⊆set : ∀ {A sub} → sub ⊆ wholeSet A
∀subset⊆set _ _ = tt
_∩_ : ∀ {A} → Subset A → Subset A → Subset A
(A ∩ B) x = x ∈ A × x ∈ B
open import Data.Nat
x : Set₁
x = Subset ℕ
y : Set₁
y = Relation ℕ ℕ
z : Subset ℕ
z = Range {ℕ} λ { ⟨ n1 , n2 ⟩ → (x₁ : Σ ℕ (λ _ → ℕ)) → {!!} ∈ {!!}}
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------------------------------------------------------------------------------
-- Agda-Prop Library.
-- A compilation of theorems in Propositional Logic
------------------------------------------------------------------------------
open import Data.Nat using ( ℕ )
module Data.PropFormula.Theorems ( n : ℕ ) where
------------------------------------------------------------------------------
open import Data.PropFormula.Theorems.Biimplication n public
open import Data.PropFormula.Theorems.Classical n public
open import Data.PropFormula.Theorems.Conjunction n public
open import Data.PropFormula.Theorems.Disjunction n public
open import Data.PropFormula.Theorems.Implication n public
open import Data.PropFormula.Theorems.Mixies n public
open import Data.PropFormula.Theorems.Negation n public
open import Data.PropFormula.Theorems.Weakening n public
------------------------------------------------------------------------------
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------------------------------------------------------------------------------
-- Testing the translation of definitions
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module Definition08 where
open import Common.FOL
postulate
P : D → Set
op : D → D
-- In this case the proof term `Pb` is referenced in the types of the
-- definitions of `c` and `d` via the `where` clause. Therefore in the
-- translation of `c` and `d`, we need to erase this proof term.
-- TODO (2016-04-02): This test case is invalid after fixing #22.
-- foo : D → ∀ {b} → P b → D
-- foo a Pb = a
-- where
-- c : D
-- c = a
-- {-# ATP definition c #-}
-- d : D
-- d = op c
-- {-# ATP definition d #-}
-- postulate bar : d ≡ op a
-- {-# ATP prove bar #-}
-- We need to have at least one conjecture to generate a TPTP file.
postulate bar : ∀ d → d ≡ d
{-# ATP prove bar #-}
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Data.Unit.Polymorphic where
open import Cubical.Core.Everything
open import Cubical.Foundations.Prelude
open import Cubical.Relation.Nullary using (yes)
open import Cubical.Relation.Binary.Raw using (Decidable)
import Cubical.Data.Unit.Base as ⊤
⊤ : {ℓ : Level} → Type ℓ
⊤ = Lift ⊤.⊤
pattern tt = lift ⊤.tt
infix 4 _≟_
_≟_ : {ℓ : Level} → Decidable {A = ⊤ {ℓ}} _≡_
_ ≟ _ = yes refl
isContr⊤ : {ℓ : Level} → isContr (⊤ {ℓ})
isContr⊤ = tt , λ {tt → refl}
isProp⊤ : {ℓ : Level} → isProp (⊤ {ℓ})
isProp⊤ _ _ i = tt
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{-# OPTIONS --cubical --no-import-sorts --safe --guardedness #-}
module Cubical.Codata.Conat.Bounded where
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Function
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Transport
open import Cubical.Foundations.Univalence
open import Cubical.Codata.Conat.Base
renaming (zero to czero; suc to csuc)
open import Cubical.Codata.Conat.Properties
open import Cubical.Data.Empty as Empty
open import Cubical.Data.Sigma
open import Cubical.Data.Sum hiding (rec)
open import Cubical.Data.Unit
open import Cubical.Relation.Nullary
open import Cubical.Data.Nat as Nat
import Cubical.Data.Fin.Recursive as Fin
private variable ℓ : Level
_≺_ : ℕ → Conat → Type _
_≺′_ : ℕ → Conat′ → Type _
n ≺ c = n ≺′ force c
_ ≺′ czero = ⊥
zero ≺′ csuc _ = Unit
suc n ≺′ csuc c = n ≺ c
isProp≺ : ∀ n c → isProp (n ≺ c)
isProp≺′ : ∀ n c → isProp (n ≺′ c)
isProp≺ n c = isProp≺′ n (force c)
isProp≺′ n czero = isProp⊥
isProp≺′ zero (csuc _) = isPropUnit
isProp≺′ (suc n) (csuc c') = isProp≺ n c'
isPropDep≺ : ∀ c → isPropDep (_≺ c)
isPropDep≺ c = isOfHLevel→isOfHLevelDep 1 (λ n → isProp≺ n c) {_} {_}
isPropDep≺′ : ∀ c → isPropDep (_≺′ c)
isPropDep≺′ c = isOfHLevel→isOfHLevelDep 1 (λ n → isProp≺′ n c) {_} {_}
private
apart : ℕ → ℕ → Type
apart zero zero = ⊥
apart (suc m) (suc n) = apart m n
apart _ _ = Unit
≢→apart : (i j : ℕ) → ¬ i ≡ j → apart i j
≢→apart zero zero ¬p = ¬p refl
≢→apart (suc i) (suc j) ¬p = ≢→apart i j (¬p ∘ cong suc)
≢→apart zero (suc j) _ = _
≢→apart (suc i) zero _ = _
apart→≢ : (i j : ℕ) → apart i j → ¬ i ≡ j
apart→≢ (suc i) zero _ = snotz
apart→≢ zero (suc j) _ = znots
apart→≢ (suc i) (suc j) i#j = apart→≢ i j i#j ∘ cong predℕ
isPropApart : ∀ m n → isProp (apart m n)
isPropApart 0 0 = isProp⊥
isPropApart (suc m) (suc n) = isPropApart m n
isPropApart (suc _) 0 = isPropUnit
isPropApart 0 (suc _) = isPropUnit
_#_ : ∀{P : ℕ → Type ℓ} → (l r : Σ ℕ P) → Type
(m , _) # (n , _) = apart m n
#→≢ : ∀{P : ℕ → Type ℓ} → (l r : Σ ℕ P) → l # r → ¬ l ≡ r
#→≢ (i , _) (j , _) d = apart→≢ i j d ∘ cong fst
isProp# : ∀{P : ℕ → Type ℓ} (l r : Σ ℕ P) → isProp (l # r)
isProp# (m , _) (n , _) = isPropApart m n
isProp#Depᵣ : ∀{P : ℕ → Type ℓ} (r : Σ ℕ P) → isPropDep (_# r)
isProp#Depᵣ r = isOfHLevel→isOfHLevelDep 1 (λ l → isProp# l r) {_} {_}
Bounded : Conat → Type
Bounded m = Σ[ n ∈ ℕ ] n ≺ m
Bounded′ : Conat′ → Type
Bounded′ m = Σ[ n ∈ ℕ ] n ≺′ m
discreteB′ : ∀ m → (i j : Bounded′ m) → (i ≡ j) ⊎ (i # j)
discreteB′ m (i , i≺m) (j , j≺m) with discreteℕ i j
... | yes p = inl λ i → p i , isPropDep≺′ m i≺m j≺m p i
... | no ¬p = inr (≢→apart i j ¬p)
≺∞ : ∀ n → n ≺ ∞
≺∞ zero = _
≺∞ (suc n) = ≺∞ n
Σ≺∞≃ℕ : Bounded ∞ ≃ ℕ
Σ≺∞≃ℕ = isoToEquiv λ where
.fun → fst
.inv n → n , ≺∞ n
.rightInv _ → refl
.leftInv (n , p) i → λ where
.fst → n
.snd → isProp≺ n ∞ (≺∞ n) p i
where open Iso
Σ≺∞≡ℕ : Bounded ∞ ≡ ℕ
Σ≺∞≡ℕ = ua Σ≺∞≃ℕ
_≺?_ : ∀ n c → Dec (n ≺ c)
n ≺? c with force c
_ ≺? c | czero = no (idfun ⊥)
zero ≺? c | csuc d = yes _
suc n ≺? c | csuc d = n ≺? d
≺-pred : ∀ n c → suc n ≺ c → n ≺ c
≺-pred n c sn≺c with force c
≺-pred zero c sn≺c | csuc d = _
≺-pred (suc n) c sn≺c | csuc d = ≺-pred n d sn≺c
≺?-yes : ∀ n c → (p : n ≺ c) → n ≺? c ≡ yes p
≺?-yes n c p with force c
≺?-yes zero c p | csuc c' = refl
≺?-yes (suc n) c p | csuc c' = ≺?-yes n c' p
∀≺-same : ∀ m n → (∀ k → (k ≺ m) ≡ (k ≺ n)) → m ≡ n
∀≺-same m n ∀≺ i .force with force m | force n
... | czero | czero = czero
... | csuc o | csuc p = csuc (∀≺-same o p (∀≺ ∘ suc) i)
... | csuc o | czero
= Empty.rec {A = csuc o ≡ czero} (transport (∀≺ 0) _) i
... | czero | csuc p
= Empty.rec {A = czero ≡ csuc p} (transport⁻ (∀≺ 0) _) i
Bounded→Fin : ∀ m → Bounded (embed m) → Fin.Fin m
Bounded→Fin (suc m) (0 , 0≺m) = Fin.zero
Bounded→Fin (suc m) (suc n , n≺m) = Fin.suc (Bounded→Fin m (n , n≺m))
module Untangle
{m n}
(f : Bounded′ (csuc m) → Bounded′ (csuc n))
(g : Bounded′ (csuc n) → Bounded′ (csuc m))
(rinv : section f g)
(linv : retract f g)
where
bzro : ∀{k} → Bounded′ (csuc k)
bzro = (zero , _)
bsuc : ∀{k} → Bounded k → Bounded′ (csuc k)
bsuc (l , l≺k) = (suc l , l≺k)
#-f : ∀ v u → v # u → f v # f u
#-f v u v#u with discreteB′ (csuc n) (f v) (f u)
... | inr fv#fu = fv#fu
... | inl fv≡fu
= rec (#→≢ v u v#u (sym (linv v) ∙∙ cong g (fv≡fu) ∙∙ linv u))
#-g : ∀ v u → v # u → g v # g u
#-g v u v#u with discreteB′ (csuc m) (g v) (g u)
... | inr gv#gu = gv#gu
... | inl gv≡gu
= rec (#→≢ v u v#u (sym (rinv v) ∙∙ cong f (gv≡gu) ∙∙ rinv u))
#-fg : ∀ v u → v # u → f (g v) # f (g u)
#-fg v u = #-f (g v) (g u) ∘ #-g v u
#-gf : ∀ v u → v # u → g (f v) # g (f u)
#-gf v u = #-g (f v) (f u) ∘ #-f v u
default : ∀{k} → (v d : Bounded′ (csuc k)) → v # d → Bounded k
default (suc l , l≺n) d _ = (l , l≺n)
default (0 , _) (suc l , l≺n) _ = (l , l≺n)
f- : Bounded m → Bounded n
f- v = default (f (bsuc v)) (f bzro) (#-f (bsuc v) bzro _)
g- : Bounded n → Bounded m
g- v = default (g (bsuc v)) (g bzro) (#-g (bsuc v) bzro _)
g-f-z : ∀ v u → g bzro ≡ bsuc v → g (bsuc u) ≡ bzro → g- u ≡ v
g-f-z (l , l≺m) u p q with g (bsuc u) | g bzro | #-g (bsuc u) bzro _
... | zero , _ | suc k , k≺m | #gf = λ where
i .fst → predℕ (p i .fst)
i .snd → isPropDep≺ m k≺m l≺m (cong (predℕ ∘ fst) p) i
... | w@(suc k , k≺m) | dg | #gf = rec (snotz (cong fst q))
g-f-s : ∀ v u → g (bsuc u) ≡ bsuc v → g- u ≡ v
g-f-s (l , l≺m) u p with g (bsuc u) | #-g (bsuc u) bzro _
... | suc k , k≺m | #gf = λ where
i .fst → predℕ (p i .fst)
i .snd → isPropDep≺ m k≺m l≺m (cong (predℕ ∘ fst) p) i
... | zero , k≺m | #gf = rec (znots (cong fst p))
g-f- : ∀ v → g- (f- v) ≡ v
g-f- v@(i , i≺m)
with f (bsuc v) | linv (bsuc v) | #-f (bsuc v) bzro _
... | suc j , j≺m | p | #f = g-f-s v (j , j≺m) p
... | zero , _ | p | #f with f bzro | linv bzro
... | suc k , k≺n | q = g-f-z v (k , k≺n) p q
f-g-z : ∀ v u → f bzro ≡ bsuc v → f (bsuc u) ≡ bzro → f- u ≡ v
f-g-z (l , l≺n) u p q with f (bsuc u) | f bzro | #-f (bsuc u) bzro _
... | zero , _ | suc k , k≺n | #fg = λ where
i .fst → predℕ (p i .fst)
i .snd → isPropDep≺ n k≺n l≺n (cong (predℕ ∘ fst) p) i
... | w@(suc k , k≺m) | df | #fg = rec (snotz (cong fst q))
f-g-s : ∀ v u → f (bsuc u) ≡ bsuc v → f- u ≡ v
f-g-s (l , l≺n) u p with f (bsuc u) | #-f (bsuc u) bzro _
... | suc k , k≺n | _ = λ where
i .fst → predℕ (p i .fst)
i .snd → isPropDep≺ n k≺n l≺n (cong (predℕ ∘ fst) p) i
... | zero , k≺m | _ = rec (znots (cong fst p))
f-g- : ∀ v → f- (g- v) ≡ v
f-g- v@(i , i≺n)
with g (bsuc v) | rinv (bsuc v) | #-g (bsuc v) bzro _
... | suc j , j≺m | p | #g = f-g-s v (j , j≺m) p
... | zero , _ | p | #g with g bzro | rinv bzro
... | suc k , k≺m | q = f-g-z v (k , k≺m) p q
open Iso
iso- : Iso (Bounded m) (Bounded n)
iso- .fun = f-
iso- .inv = g-
iso- .rightInv = f-g-
iso- .leftInv = g-f-
untangled
: ∀{m n}
→ Iso (Bounded′ (csuc m)) (Bounded′ (csuc n))
→ Iso (Bounded m) (Bounded n)
untangled isom = Untangle.iso- fun inv rightInv leftInv
where open Iso isom
Bounded-inj-iso : ∀ m n → Iso (Bounded m) (Bounded n) → m ≡ n
Bounded-inj-iso m n theIso i .force with force m | force n
... | czero | czero = czero
... | csuc l | csuc r
= csuc (Bounded-inj-iso l r (untangled theIso) i)
... | czero | csuc r
= rec {A = czero ≡ csuc r} (Iso.inv theIso (zero , _) .snd) i
... | csuc l | czero
= rec {A = csuc l ≡ czero} (Iso.fun theIso (zero , _) .snd) i
Bounded-inj : ∀ m n → Bounded m ≡ Bounded n → m ≡ n
Bounded-inj m n = Bounded-inj-iso m n ∘ pathToIso
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{-# OPTIONS --without-K --safe #-}
open import Categories.Category.Core using (Category)
open import Categories.Functor.Bifunctor using (Bifunctor)
module Categories.Category.Construction.Wedges {o ℓ e o′ ℓ′ e′} {C : Category o ℓ e} {D : Category o′ ℓ′ e′}
(F : Bifunctor (Category.op C) C D) where
open import Level
open import Categories.Category.Core using (Category)
open import Categories.Diagram.Wedge F
Wedges : Category (o ⊔ ℓ ⊔ e ⊔ o′ ⊔ ℓ′ ⊔ e′) (o ⊔ ℓ ⊔ e ⊔ o′ ⊔ ℓ′ ⊔ e′) e′
Wedges = record
{ Obj = Wedge
; _⇒_ = Wedge-Morphism
; _≈_ = λ M N → u M ≈ u N
; id = Wedge-id
; _∘_ = Wedge-Morphism-∘
; assoc = assoc
; sym-assoc = sym-assoc
; identityˡ = identityˡ
; identityʳ = identityʳ
; identity² = identity²
; equiv = record { refl = Equiv.refl ; sym = Equiv.sym ; trans = Equiv.trans }
; ∘-resp-≈ = ∘-resp-≈
}
where
open Wedge-Morphism
open Category D
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module Text.Greek.SBLGNT.2Cor where
open import Data.List
open import Text.Greek.Bible
open import Text.Greek.Script
open import Text.Greek.Script.Unicode
ΠΡΟΣ-ΚΟΡΙΝΘΙΟΥΣ-Β : List (Word)
ΠΡΟΣ-ΚΟΡΙΝΘΙΟΥΣ-Β =
word (Π ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "2Cor.1.1"
∷ word (ἀ ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ς ∷ []) "2Cor.1.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "2Cor.1.1"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.1.1"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.1.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.1.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.1"
∷ word (Τ ∷ ι ∷ μ ∷ ό ∷ θ ∷ ε ∷ ο ∷ ς ∷ []) "2Cor.1.1"
∷ word (ὁ ∷ []) "2Cor.1.1"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ς ∷ []) "2Cor.1.1"
∷ word (τ ∷ ῇ ∷ []) "2Cor.1.1"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "2Cor.1.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.1.1"
∷ word (τ ∷ ῇ ∷ []) "2Cor.1.1"
∷ word (ο ∷ ὔ ∷ σ ∷ ῃ ∷ []) "2Cor.1.1"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.1"
∷ word (Κ ∷ ο ∷ ρ ∷ ί ∷ ν ∷ θ ∷ ῳ ∷ []) "2Cor.1.1"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "2Cor.1.1"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.1.1"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.1.1"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.1.1"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.1.1"
∷ word (ο ∷ ὖ ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.1.1"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.1"
∷ word (ὅ ∷ ∙λ ∷ ῃ ∷ []) "2Cor.1.1"
∷ word (τ ∷ ῇ ∷ []) "2Cor.1.1"
∷ word (Ἀ ∷ χ ∷ α ∷ ΐ ∷ ᾳ ∷ []) "2Cor.1.1"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "2Cor.1.2"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.1.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.2"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "2Cor.1.2"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.1.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.1.2"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.1.2"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.2"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.1.2"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "2Cor.1.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.2"
∷ word (Ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ η ∷ τ ∷ ὸ ∷ ς ∷ []) "2Cor.1.3"
∷ word (ὁ ∷ []) "2Cor.1.3"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.1.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.3"
∷ word (π ∷ α ∷ τ ∷ ὴ ∷ ρ ∷ []) "2Cor.1.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.3"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.1.3"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.3"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "2Cor.1.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.3"
∷ word (ὁ ∷ []) "2Cor.1.3"
∷ word (π ∷ α ∷ τ ∷ ὴ ∷ ρ ∷ []) "2Cor.1.3"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.1.3"
∷ word (ο ∷ ἰ ∷ κ ∷ τ ∷ ι ∷ ρ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.3"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.1.3"
∷ word (π ∷ ά ∷ σ ∷ η ∷ ς ∷ []) "2Cor.1.3"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.1.3"
∷ word (ὁ ∷ []) "2Cor.1.4"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "2Cor.1.4"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.4"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.1.4"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "2Cor.1.4"
∷ word (τ ∷ ῇ ∷ []) "2Cor.1.4"
∷ word (θ ∷ ∙λ ∷ ί ∷ ψ ∷ ε ∷ ι ∷ []) "2Cor.1.4"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.1.4"
∷ word (τ ∷ ὸ ∷ []) "2Cor.1.4"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.1.4"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.4"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ ν ∷ []) "2Cor.1.4"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.1.4"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.4"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "2Cor.1.4"
∷ word (θ ∷ ∙λ ∷ ί ∷ ψ ∷ ε ∷ ι ∷ []) "2Cor.1.4"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.1.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.1.4"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.1.4"
∷ word (ἧ ∷ ς ∷ []) "2Cor.1.4"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.1.4"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὶ ∷ []) "2Cor.1.4"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "2Cor.1.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.4"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.1.4"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.1.5"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "2Cor.1.5"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ []) "2Cor.1.5"
∷ word (τ ∷ ὰ ∷ []) "2Cor.1.5"
∷ word (π ∷ α ∷ θ ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "2Cor.1.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.5"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.1.5"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.5"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "2Cor.1.5"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.1.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.5"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.5"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ []) "2Cor.1.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.5"
∷ word (ἡ ∷ []) "2Cor.1.5"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ι ∷ ς ∷ []) "2Cor.1.5"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.5"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.1.6"
∷ word (δ ∷ ὲ ∷ []) "2Cor.1.6"
∷ word (θ ∷ ∙λ ∷ ι ∷ β ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.1.6"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.1.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.1.6"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.6"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.1.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.6"
∷ word (σ ∷ ω ∷ τ ∷ η ∷ ρ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.1.6"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.1.6"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.1.6"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.1.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.1.6"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.6"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.1.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.1.6"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ η ∷ ς ∷ []) "2Cor.1.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.6"
∷ word (ὑ ∷ π ∷ ο ∷ μ ∷ ο ∷ ν ∷ ῇ ∷ []) "2Cor.1.6"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.1.6"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.1.6"
∷ word (π ∷ α ∷ θ ∷ η ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.1.6"
∷ word (ὧ ∷ ν ∷ []) "2Cor.1.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.6"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.1.6"
∷ word (π ∷ ά ∷ σ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.1.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.7"
∷ word (ἡ ∷ []) "2Cor.1.7"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ὶ ∷ ς ∷ []) "2Cor.1.7"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.7"
∷ word (β ∷ ε ∷ β ∷ α ∷ ί ∷ α ∷ []) "2Cor.1.7"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.1.7"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.7"
∷ word (ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.1.7"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.1.7"
∷ word (ὡ ∷ ς ∷ []) "2Cor.1.7"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ο ∷ ί ∷ []) "2Cor.1.7"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "2Cor.1.7"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.1.7"
∷ word (π ∷ α ∷ θ ∷ η ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.1.7"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "2Cor.1.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.1.7"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.1.7"
∷ word (Ο ∷ ὐ ∷ []) "2Cor.1.8"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.1.8"
∷ word (θ ∷ έ ∷ ∙λ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.1.8"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.8"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ε ∷ ῖ ∷ ν ∷ []) "2Cor.1.8"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "2Cor.1.8"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.1.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.1.8"
∷ word (θ ∷ ∙λ ∷ ί ∷ ψ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.1.8"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.1.8"
∷ word (γ ∷ ε ∷ ν ∷ ο ∷ μ ∷ έ ∷ ν ∷ η ∷ ς ∷ []) "2Cor.1.8"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.8"
∷ word (τ ∷ ῇ ∷ []) "2Cor.1.8"
∷ word (Ἀ ∷ σ ∷ ί ∷ ᾳ ∷ []) "2Cor.1.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.1.8"
∷ word (κ ∷ α ∷ θ ∷ []) "2Cor.1.8"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "2Cor.1.8"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.1.8"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ν ∷ []) "2Cor.1.8"
∷ word (ἐ ∷ β ∷ α ∷ ρ ∷ ή ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.1.8"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "2Cor.1.8"
∷ word (ἐ ∷ ξ ∷ α ∷ π ∷ ο ∷ ρ ∷ η ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.1.8"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.8"
∷ word (ζ ∷ ῆ ∷ ν ∷ []) "2Cor.1.8"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.1.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὶ ∷ []) "2Cor.1.9"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.9"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.1.9"
∷ word (τ ∷ ὸ ∷ []) "2Cor.1.9"
∷ word (ἀ ∷ π ∷ ό ∷ κ ∷ ρ ∷ ι ∷ μ ∷ α ∷ []) "2Cor.1.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.9"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.1.9"
∷ word (ἐ ∷ σ ∷ χ ∷ ή ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.1.9"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.1.9"
∷ word (μ ∷ ὴ ∷ []) "2Cor.1.9"
∷ word (π ∷ ε ∷ π ∷ ο ∷ ι ∷ θ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.1.9"
∷ word (ὦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.1.9"
∷ word (ἐ ∷ φ ∷ []) "2Cor.1.9"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.1.9"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.1.9"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.1.9"
∷ word (τ ∷ ῷ ∷ []) "2Cor.1.9"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "2Cor.1.9"
∷ word (τ ∷ ῷ ∷ []) "2Cor.1.9"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ ο ∷ ν ∷ τ ∷ ι ∷ []) "2Cor.1.9"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.1.9"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ο ∷ ύ ∷ ς ∷ []) "2Cor.1.9"
∷ word (ὃ ∷ ς ∷ []) "2Cor.1.10"
∷ word (ἐ ∷ κ ∷ []) "2Cor.1.10"
∷ word (τ ∷ η ∷ ∙λ ∷ ι ∷ κ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.1.10"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.1.10"
∷ word (ἐ ∷ ρ ∷ ρ ∷ ύ ∷ σ ∷ α ∷ τ ∷ ο ∷ []) "2Cor.1.10"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.10"
∷ word (ῥ ∷ ύ ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "2Cor.1.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.1.10"
∷ word (ὃ ∷ ν ∷ []) "2Cor.1.10"
∷ word (ἠ ∷ ∙λ ∷ π ∷ ί ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.1.10"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.1.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.10"
∷ word (ἔ ∷ τ ∷ ι ∷ []) "2Cor.1.10"
∷ word (ῥ ∷ ύ ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "2Cor.1.10"
∷ word (σ ∷ υ ∷ ν ∷ υ ∷ π ∷ ο ∷ υ ∷ ρ ∷ γ ∷ ο ∷ ύ ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.1.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.11"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.11"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.1.11"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.11"
∷ word (τ ∷ ῇ ∷ []) "2Cor.1.11"
∷ word (δ ∷ ε ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.1.11"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.1.11"
∷ word (ἐ ∷ κ ∷ []) "2Cor.1.11"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "2Cor.1.11"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "2Cor.1.11"
∷ word (τ ∷ ὸ ∷ []) "2Cor.1.11"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.1.11"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.11"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ σ ∷ μ ∷ α ∷ []) "2Cor.1.11"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.1.11"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "2Cor.1.11"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ η ∷ θ ∷ ῇ ∷ []) "2Cor.1.11"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.1.11"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.11"
∷ word (Ἡ ∷ []) "2Cor.1.12"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.1.12"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ σ ∷ ι ∷ ς ∷ []) "2Cor.1.12"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.12"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "2Cor.1.12"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "2Cor.1.12"
∷ word (τ ∷ ὸ ∷ []) "2Cor.1.12"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.1.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.1.12"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ι ∷ δ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.1.12"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.12"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.1.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.12"
∷ word (ἁ ∷ γ ∷ ι ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "2Cor.1.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.12"
∷ word (ε ∷ ἰ ∷ ∙λ ∷ ι ∷ κ ∷ ρ ∷ ι ∷ ν ∷ ε ∷ ί ∷ ᾳ ∷ []) "2Cor.1.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.12"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.1.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.1.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.12"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ ᾳ ∷ []) "2Cor.1.12"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ι ∷ κ ∷ ῇ ∷ []) "2Cor.1.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.1.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.12"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ι ∷ []) "2Cor.1.12"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.1.12"
∷ word (ἀ ∷ ν ∷ ε ∷ σ ∷ τ ∷ ρ ∷ ά ∷ φ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.1.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.12"
∷ word (τ ∷ ῷ ∷ []) "2Cor.1.12"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "2Cor.1.12"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ ω ∷ ς ∷ []) "2Cor.1.12"
∷ word (δ ∷ ὲ ∷ []) "2Cor.1.12"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.1.12"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.12"
∷ word (ο ∷ ὐ ∷ []) "2Cor.1.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.1.13"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ α ∷ []) "2Cor.1.13"
∷ word (γ ∷ ρ ∷ ά ∷ φ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.1.13"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.1.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.1.13"
∷ word (ἢ ∷ []) "2Cor.1.13"
∷ word (ἃ ∷ []) "2Cor.1.13"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ε ∷ τ ∷ ε ∷ []) "2Cor.1.13"
∷ word (ἢ ∷ []) "2Cor.1.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.13"
∷ word (ἐ ∷ π ∷ ι ∷ γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ε ∷ τ ∷ ε ∷ []) "2Cor.1.13"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ ζ ∷ ω ∷ []) "2Cor.1.13"
∷ word (δ ∷ ὲ ∷ []) "2Cor.1.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.1.13"
∷ word (ἕ ∷ ω ∷ ς ∷ []) "2Cor.1.13"
∷ word (τ ∷ έ ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.1.13"
∷ word (ἐ ∷ π ∷ ι ∷ γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.1.13"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "2Cor.1.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.14"
∷ word (ἐ ∷ π ∷ έ ∷ γ ∷ ν ∷ ω ∷ τ ∷ ε ∷ []) "2Cor.1.14"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.14"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.1.14"
∷ word (μ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.1.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.1.14"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ μ ∷ α ∷ []) "2Cor.1.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.14"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.1.14"
∷ word (κ ∷ α ∷ θ ∷ ά ∷ π ∷ ε ∷ ρ ∷ []) "2Cor.1.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.14"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.1.14"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.14"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.14"
∷ word (τ ∷ ῇ ∷ []) "2Cor.1.14"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "2Cor.1.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.14"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.1.14"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.14"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "2Cor.1.14"
∷ word (Κ ∷ α ∷ ὶ ∷ []) "2Cor.1.15"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ ῃ ∷ []) "2Cor.1.15"
∷ word (τ ∷ ῇ ∷ []) "2Cor.1.15"
∷ word (π ∷ ε ∷ π ∷ ο ∷ ι ∷ θ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.1.15"
∷ word (ἐ ∷ β ∷ ο ∷ υ ∷ ∙λ ∷ ό ∷ μ ∷ η ∷ ν ∷ []) "2Cor.1.15"
∷ word (π ∷ ρ ∷ ό ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "2Cor.1.15"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.1.15"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.15"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "2Cor.1.15"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.1.15"
∷ word (δ ∷ ε ∷ υ ∷ τ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "2Cor.1.15"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "2Cor.1.15"
∷ word (σ ∷ χ ∷ ῆ ∷ τ ∷ ε ∷ []) "2Cor.1.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.16"
∷ word (δ ∷ ι ∷ []) "2Cor.1.16"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.16"
∷ word (δ ∷ ι ∷ ε ∷ ∙λ ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "2Cor.1.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.1.16"
∷ word (Μ ∷ α ∷ κ ∷ ε ∷ δ ∷ ο ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "2Cor.1.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.16"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "2Cor.1.16"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.1.16"
∷ word (Μ ∷ α ∷ κ ∷ ε ∷ δ ∷ ο ∷ ν ∷ ί ∷ α ∷ ς ∷ []) "2Cor.1.16"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "2Cor.1.16"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.1.16"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.16"
∷ word (ὑ ∷ φ ∷ []) "2Cor.1.16"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.16"
∷ word (π ∷ ρ ∷ ο ∷ π ∷ ε ∷ μ ∷ φ ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.1.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.1.16"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.1.16"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ α ∷ ν ∷ []) "2Cor.1.16"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.1.17"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "2Cor.1.17"
∷ word (β ∷ ο ∷ υ ∷ ∙λ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.1.17"
∷ word (μ ∷ ή ∷ τ ∷ ι ∷ []) "2Cor.1.17"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "2Cor.1.17"
∷ word (τ ∷ ῇ ∷ []) "2Cor.1.17"
∷ word (ἐ ∷ ∙λ ∷ α ∷ φ ∷ ρ ∷ ί ∷ ᾳ ∷ []) "2Cor.1.17"
∷ word (ἐ ∷ χ ∷ ρ ∷ η ∷ σ ∷ ά ∷ μ ∷ η ∷ ν ∷ []) "2Cor.1.17"
∷ word (ἢ ∷ []) "2Cor.1.17"
∷ word (ἃ ∷ []) "2Cor.1.17"
∷ word (β ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.1.17"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.1.17"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "2Cor.1.17"
∷ word (β ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.1.17"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.1.17"
∷ word (ᾖ ∷ []) "2Cor.1.17"
∷ word (π ∷ α ∷ ρ ∷ []) "2Cor.1.17"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "2Cor.1.17"
∷ word (τ ∷ ὸ ∷ []) "2Cor.1.17"
∷ word (Ν ∷ α ∷ ὶ ∷ []) "2Cor.1.17"
∷ word (ν ∷ α ∷ ὶ ∷ []) "2Cor.1.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.17"
∷ word (τ ∷ ὸ ∷ []) "2Cor.1.17"
∷ word (Ο ∷ ὒ ∷ []) "2Cor.1.17"
∷ word (ο ∷ ὔ ∷ []) "2Cor.1.17"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "2Cor.1.18"
∷ word (δ ∷ ὲ ∷ []) "2Cor.1.18"
∷ word (ὁ ∷ []) "2Cor.1.18"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.1.18"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.1.18"
∷ word (ὁ ∷ []) "2Cor.1.18"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "2Cor.1.18"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.18"
∷ word (ὁ ∷ []) "2Cor.1.18"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.1.18"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.18"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.1.18"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "2Cor.1.18"
∷ word (Ν ∷ α ∷ ὶ ∷ []) "2Cor.1.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.18"
∷ word (Ο ∷ ὔ ∷ []) "2Cor.1.18"
∷ word (ὁ ∷ []) "2Cor.1.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.19"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.1.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.1.19"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ς ∷ []) "2Cor.1.19"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "2Cor.1.19"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "2Cor.1.19"
∷ word (ὁ ∷ []) "2Cor.1.19"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.19"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.1.19"
∷ word (δ ∷ ι ∷ []) "2Cor.1.19"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.19"
∷ word (κ ∷ η ∷ ρ ∷ υ ∷ χ ∷ θ ∷ ε ∷ ί ∷ ς ∷ []) "2Cor.1.19"
∷ word (δ ∷ ι ∷ []) "2Cor.1.19"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "2Cor.1.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.19"
∷ word (Σ ∷ ι ∷ ∙λ ∷ ο ∷ υ ∷ α ∷ ν ∷ ο ∷ ῦ ∷ []) "2Cor.1.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.19"
∷ word (Τ ∷ ι ∷ μ ∷ ο ∷ θ ∷ έ ∷ ο ∷ υ ∷ []) "2Cor.1.19"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.1.19"
∷ word (ἐ ∷ γ ∷ έ ∷ ν ∷ ε ∷ τ ∷ ο ∷ []) "2Cor.1.19"
∷ word (Ν ∷ α ∷ ὶ ∷ []) "2Cor.1.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.19"
∷ word (Ο ∷ ὒ ∷ []) "2Cor.1.19"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.1.19"
∷ word (Ν ∷ α ∷ ὶ ∷ []) "2Cor.1.19"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.19"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "2Cor.1.19"
∷ word (γ ∷ έ ∷ γ ∷ ο ∷ ν ∷ ε ∷ ν ∷ []) "2Cor.1.19"
∷ word (ὅ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.1.20"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.1.20"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ι ∷ []) "2Cor.1.20"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.1.20"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.20"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "2Cor.1.20"
∷ word (τ ∷ ὸ ∷ []) "2Cor.1.20"
∷ word (Ν ∷ α ∷ ί ∷ []) "2Cor.1.20"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "2Cor.1.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.20"
∷ word (δ ∷ ι ∷ []) "2Cor.1.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.20"
∷ word (τ ∷ ὸ ∷ []) "2Cor.1.20"
∷ word (Ἀ ∷ μ ∷ ὴ ∷ ν ∷ []) "2Cor.1.20"
∷ word (τ ∷ ῷ ∷ []) "2Cor.1.20"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "2Cor.1.20"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.1.20"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "2Cor.1.20"
∷ word (δ ∷ ι ∷ []) "2Cor.1.20"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.20"
∷ word (ὁ ∷ []) "2Cor.1.21"
∷ word (δ ∷ ὲ ∷ []) "2Cor.1.21"
∷ word (β ∷ ε ∷ β ∷ α ∷ ι ∷ ῶ ∷ ν ∷ []) "2Cor.1.21"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.21"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "2Cor.1.21"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.1.21"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.1.21"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "2Cor.1.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.21"
∷ word (χ ∷ ρ ∷ ί ∷ σ ∷ α ∷ ς ∷ []) "2Cor.1.21"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.21"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "2Cor.1.21"
∷ word (ὁ ∷ []) "2Cor.1.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.22"
∷ word (σ ∷ φ ∷ ρ ∷ α ∷ γ ∷ ι ∷ σ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.1.22"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.1.22"
∷ word (δ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.1.22"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.1.22"
∷ word (ἀ ∷ ρ ∷ ρ ∷ α ∷ β ∷ ῶ ∷ ν ∷ α ∷ []) "2Cor.1.22"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.1.22"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.1.22"
∷ word (ἐ ∷ ν ∷ []) "2Cor.1.22"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "2Cor.1.22"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.1.22"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.22"
∷ word (Ἐ ∷ γ ∷ ὼ ∷ []) "2Cor.1.23"
∷ word (δ ∷ ὲ ∷ []) "2Cor.1.23"
∷ word (μ ∷ ά ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ α ∷ []) "2Cor.1.23"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.1.23"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "2Cor.1.23"
∷ word (ἐ ∷ π ∷ ι ∷ κ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ μ ∷ α ∷ ι ∷ []) "2Cor.1.23"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.1.23"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.1.23"
∷ word (ἐ ∷ μ ∷ ὴ ∷ ν ∷ []) "2Cor.1.23"
∷ word (ψ ∷ υ ∷ χ ∷ ή ∷ ν ∷ []) "2Cor.1.23"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.1.23"
∷ word (φ ∷ ε ∷ ι ∷ δ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.1.23"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.23"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "2Cor.1.23"
∷ word (ἦ ∷ ∙λ ∷ θ ∷ ο ∷ ν ∷ []) "2Cor.1.23"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.1.23"
∷ word (Κ ∷ ό ∷ ρ ∷ ι ∷ ν ∷ θ ∷ ο ∷ ν ∷ []) "2Cor.1.23"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "2Cor.1.24"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.1.24"
∷ word (κ ∷ υ ∷ ρ ∷ ι ∷ ε ∷ ύ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.1.24"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.24"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.1.24"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.1.24"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.1.24"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ο ∷ ί ∷ []) "2Cor.1.24"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.1.24"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.1.24"
∷ word (χ ∷ α ∷ ρ ∷ ᾶ ∷ ς ∷ []) "2Cor.1.24"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.1.24"
∷ word (τ ∷ ῇ ∷ []) "2Cor.1.24"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.1.24"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ι ∷ []) "2Cor.1.24"
∷ word (ἑ ∷ σ ∷ τ ∷ ή ∷ κ ∷ α ∷ τ ∷ ε ∷ []) "2Cor.1.24"
∷ word (ἔ ∷ κ ∷ ρ ∷ ι ∷ ν ∷ α ∷ []) "2Cor.2.1"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.2.1"
∷ word (ἐ ∷ μ ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "2Cor.2.1"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.2.1"
∷ word (τ ∷ ὸ ∷ []) "2Cor.2.1"
∷ word (μ ∷ ὴ ∷ []) "2Cor.2.1"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "2Cor.2.1"
∷ word (ἐ ∷ ν ∷ []) "2Cor.2.1"
∷ word (∙λ ∷ ύ ∷ π ∷ ῃ ∷ []) "2Cor.2.1"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.2.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.2.1"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "2Cor.2.1"
∷ word (ε ∷ ἰ ∷ []) "2Cor.2.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.2.2"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "2Cor.2.2"
∷ word (∙λ ∷ υ ∷ π ∷ ῶ ∷ []) "2Cor.2.2"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.2.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.2.2"
∷ word (τ ∷ ί ∷ ς ∷ []) "2Cor.2.2"
∷ word (ὁ ∷ []) "2Cor.2.2"
∷ word (ε ∷ ὐ ∷ φ ∷ ρ ∷ α ∷ ί ∷ ν ∷ ω ∷ ν ∷ []) "2Cor.2.2"
∷ word (μ ∷ ε ∷ []) "2Cor.2.2"
∷ word (ε ∷ ἰ ∷ []) "2Cor.2.2"
∷ word (μ ∷ ὴ ∷ []) "2Cor.2.2"
∷ word (ὁ ∷ []) "2Cor.2.2"
∷ word (∙λ ∷ υ ∷ π ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.2.2"
∷ word (ἐ ∷ ξ ∷ []) "2Cor.2.2"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "2Cor.2.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.2.3"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "2Cor.2.3"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.2.3"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "2Cor.2.3"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.2.3"
∷ word (μ ∷ ὴ ∷ []) "2Cor.2.3"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ὼ ∷ ν ∷ []) "2Cor.2.3"
∷ word (∙λ ∷ ύ ∷ π ∷ η ∷ ν ∷ []) "2Cor.2.3"
∷ word (σ ∷ χ ∷ ῶ ∷ []) "2Cor.2.3"
∷ word (ἀ ∷ φ ∷ []) "2Cor.2.3"
∷ word (ὧ ∷ ν ∷ []) "2Cor.2.3"
∷ word (ἔ ∷ δ ∷ ε ∷ ι ∷ []) "2Cor.2.3"
∷ word (μ ∷ ε ∷ []) "2Cor.2.3"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ε ∷ ι ∷ ν ∷ []) "2Cor.2.3"
∷ word (π ∷ ε ∷ π ∷ ο ∷ ι ∷ θ ∷ ὼ ∷ ς ∷ []) "2Cor.2.3"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.2.3"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "2Cor.2.3"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.2.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.2.3"
∷ word (ἡ ∷ []) "2Cor.2.3"
∷ word (ἐ ∷ μ ∷ ὴ ∷ []) "2Cor.2.3"
∷ word (χ ∷ α ∷ ρ ∷ ὰ ∷ []) "2Cor.2.3"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.2.3"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.2.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "2Cor.2.3"
∷ word (ἐ ∷ κ ∷ []) "2Cor.2.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.2.4"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῆ ∷ ς ∷ []) "2Cor.2.4"
∷ word (θ ∷ ∙λ ∷ ί ∷ ψ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.2.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.2.4"
∷ word (σ ∷ υ ∷ ν ∷ ο ∷ χ ∷ ῆ ∷ ς ∷ []) "2Cor.2.4"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.2.4"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "2Cor.2.4"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.2.4"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.2.4"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "2Cor.2.4"
∷ word (δ ∷ α ∷ κ ∷ ρ ∷ ύ ∷ ω ∷ ν ∷ []) "2Cor.2.4"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "2Cor.2.4"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.2.4"
∷ word (∙λ ∷ υ ∷ π ∷ η ∷ θ ∷ ῆ ∷ τ ∷ ε ∷ []) "2Cor.2.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.2.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.2.4"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "2Cor.2.4"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.2.4"
∷ word (γ ∷ ν ∷ ῶ ∷ τ ∷ ε ∷ []) "2Cor.2.4"
∷ word (ἣ ∷ ν ∷ []) "2Cor.2.4"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "2Cor.2.4"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ ω ∷ ς ∷ []) "2Cor.2.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.2.4"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.2.4"
∷ word (Ε ∷ ἰ ∷ []) "2Cor.2.5"
∷ word (δ ∷ έ ∷ []) "2Cor.2.5"
∷ word (τ ∷ ι ∷ ς ∷ []) "2Cor.2.5"
∷ word (∙λ ∷ ε ∷ ∙λ ∷ ύ ∷ π ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "2Cor.2.5"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.2.5"
∷ word (ἐ ∷ μ ∷ ὲ ∷ []) "2Cor.2.5"
∷ word (∙λ ∷ ε ∷ ∙λ ∷ ύ ∷ π ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "2Cor.2.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.2.5"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.2.5"
∷ word (μ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.2.5"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.2.5"
∷ word (μ ∷ ὴ ∷ []) "2Cor.2.5"
∷ word (ἐ ∷ π ∷ ι ∷ β ∷ α ∷ ρ ∷ ῶ ∷ []) "2Cor.2.5"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "2Cor.2.5"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.2.5"
∷ word (ἱ ∷ κ ∷ α ∷ ν ∷ ὸ ∷ ν ∷ []) "2Cor.2.6"
∷ word (τ ∷ ῷ ∷ []) "2Cor.2.6"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "2Cor.2.6"
∷ word (ἡ ∷ []) "2Cor.2.6"
∷ word (ἐ ∷ π ∷ ι ∷ τ ∷ ι ∷ μ ∷ ί ∷ α ∷ []) "2Cor.2.6"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "2Cor.2.6"
∷ word (ἡ ∷ []) "2Cor.2.6"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "2Cor.2.6"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.2.6"
∷ word (π ∷ ∙λ ∷ ε ∷ ι ∷ ό ∷ ν ∷ ω ∷ ν ∷ []) "2Cor.2.6"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "2Cor.2.7"
∷ word (τ ∷ ο ∷ ὐ ∷ ν ∷ α ∷ ν ∷ τ ∷ ί ∷ ο ∷ ν ∷ []) "2Cor.2.7"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.2.7"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.2.7"
∷ word (χ ∷ α ∷ ρ ∷ ί ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.2.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.2.7"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ έ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.2.7"
∷ word (μ ∷ ή ∷ []) "2Cor.2.7"
∷ word (π ∷ ω ∷ ς ∷ []) "2Cor.2.7"
∷ word (τ ∷ ῇ ∷ []) "2Cor.2.7"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "2Cor.2.7"
∷ word (∙λ ∷ ύ ∷ π ∷ ῃ ∷ []) "2Cor.2.7"
∷ word (κ ∷ α ∷ τ ∷ α ∷ π ∷ ο ∷ θ ∷ ῇ ∷ []) "2Cor.2.7"
∷ word (ὁ ∷ []) "2Cor.2.7"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.2.7"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "2Cor.2.8"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "2Cor.2.8"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.2.8"
∷ word (κ ∷ υ ∷ ρ ∷ ῶ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.2.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.2.8"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "2Cor.2.8"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "2Cor.2.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.2.9"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.2.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.2.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.2.9"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "2Cor.2.9"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.2.9"
∷ word (γ ∷ ν ∷ ῶ ∷ []) "2Cor.2.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.2.9"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ὴ ∷ ν ∷ []) "2Cor.2.9"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.2.9"
∷ word (ε ∷ ἰ ∷ []) "2Cor.2.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.2.9"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "2Cor.2.9"
∷ word (ὑ ∷ π ∷ ή ∷ κ ∷ ο ∷ ο ∷ ί ∷ []) "2Cor.2.9"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "2Cor.2.9"
∷ word (ᾧ ∷ []) "2Cor.2.10"
∷ word (δ ∷ έ ∷ []) "2Cor.2.10"
∷ word (τ ∷ ι ∷ []) "2Cor.2.10"
∷ word (χ ∷ α ∷ ρ ∷ ί ∷ ζ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.2.10"
∷ word (κ ∷ ἀ ∷ γ ∷ ώ ∷ []) "2Cor.2.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.2.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.2.10"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "2Cor.2.10"
∷ word (ὃ ∷ []) "2Cor.2.10"
∷ word (κ ∷ ε ∷ χ ∷ ά ∷ ρ ∷ ι ∷ σ ∷ μ ∷ α ∷ ι ∷ []) "2Cor.2.10"
∷ word (ε ∷ ἴ ∷ []) "2Cor.2.10"
∷ word (τ ∷ ι ∷ []) "2Cor.2.10"
∷ word (κ ∷ ε ∷ χ ∷ ά ∷ ρ ∷ ι ∷ σ ∷ μ ∷ α ∷ ι ∷ []) "2Cor.2.10"
∷ word (δ ∷ ι ∷ []) "2Cor.2.10"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.2.10"
∷ word (ἐ ∷ ν ∷ []) "2Cor.2.10"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ώ ∷ π ∷ ῳ ∷ []) "2Cor.2.10"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.2.10"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.2.11"
∷ word (μ ∷ ὴ ∷ []) "2Cor.2.11"
∷ word (π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ ε ∷ κ ∷ τ ∷ η ∷ θ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.2.11"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "2Cor.2.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.2.11"
∷ word (Σ ∷ α ∷ τ ∷ α ∷ ν ∷ ᾶ ∷ []) "2Cor.2.11"
∷ word (ο ∷ ὐ ∷ []) "2Cor.2.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.2.11"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.2.11"
∷ word (τ ∷ ὰ ∷ []) "2Cor.2.11"
∷ word (ν ∷ ο ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "2Cor.2.11"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.2.11"
∷ word (Ἐ ∷ ∙λ ∷ θ ∷ ὼ ∷ ν ∷ []) "2Cor.2.12"
∷ word (δ ∷ ὲ ∷ []) "2Cor.2.12"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.2.12"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.2.12"
∷ word (Τ ∷ ρ ∷ ῳ ∷ ά ∷ δ ∷ α ∷ []) "2Cor.2.12"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.2.12"
∷ word (τ ∷ ὸ ∷ []) "2Cor.2.12"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.2.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.2.12"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.2.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.2.12"
∷ word (θ ∷ ύ ∷ ρ ∷ α ∷ ς ∷ []) "2Cor.2.12"
∷ word (μ ∷ ο ∷ ι ∷ []) "2Cor.2.12"
∷ word (ἀ ∷ ν ∷ ε ∷ ῳ ∷ γ ∷ μ ∷ έ ∷ ν ∷ η ∷ ς ∷ []) "2Cor.2.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.2.12"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "2Cor.2.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.2.13"
∷ word (ἔ ∷ σ ∷ χ ∷ η ∷ κ ∷ α ∷ []) "2Cor.2.13"
∷ word (ἄ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.2.13"
∷ word (τ ∷ ῷ ∷ []) "2Cor.2.13"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ί ∷ []) "2Cor.2.13"
∷ word (μ ∷ ο ∷ υ ∷ []) "2Cor.2.13"
∷ word (τ ∷ ῷ ∷ []) "2Cor.2.13"
∷ word (μ ∷ ὴ ∷ []) "2Cor.2.13"
∷ word (ε ∷ ὑ ∷ ρ ∷ ε ∷ ῖ ∷ ν ∷ []) "2Cor.2.13"
∷ word (μ ∷ ε ∷ []) "2Cor.2.13"
∷ word (Τ ∷ ί ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.2.13"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.2.13"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ό ∷ ν ∷ []) "2Cor.2.13"
∷ word (μ ∷ ο ∷ υ ∷ []) "2Cor.2.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.2.13"
∷ word (ἀ ∷ π ∷ ο ∷ τ ∷ α ∷ ξ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.2.13"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.2.13"
∷ word (ἐ ∷ ξ ∷ ῆ ∷ ∙λ ∷ θ ∷ ο ∷ ν ∷ []) "2Cor.2.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.2.13"
∷ word (Μ ∷ α ∷ κ ∷ ε ∷ δ ∷ ο ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "2Cor.2.13"
∷ word (Τ ∷ ῷ ∷ []) "2Cor.2.14"
∷ word (δ ∷ ὲ ∷ []) "2Cor.2.14"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "2Cor.2.14"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "2Cor.2.14"
∷ word (τ ∷ ῷ ∷ []) "2Cor.2.14"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ο ∷ τ ∷ ε ∷ []) "2Cor.2.14"
∷ word (θ ∷ ρ ∷ ι ∷ α ∷ μ ∷ β ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ ι ∷ []) "2Cor.2.14"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.2.14"
∷ word (ἐ ∷ ν ∷ []) "2Cor.2.14"
∷ word (τ ∷ ῷ ∷ []) "2Cor.2.14"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "2Cor.2.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.2.14"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.2.14"
∷ word (ὀ ∷ σ ∷ μ ∷ ὴ ∷ ν ∷ []) "2Cor.2.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.2.14"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.2.14"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.2.14"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ι ∷ []) "2Cor.2.14"
∷ word (δ ∷ ι ∷ []) "2Cor.2.14"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.2.14"
∷ word (ἐ ∷ ν ∷ []) "2Cor.2.14"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "2Cor.2.14"
∷ word (τ ∷ ό ∷ π ∷ ῳ ∷ []) "2Cor.2.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.2.15"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.2.15"
∷ word (ε ∷ ὐ ∷ ω ∷ δ ∷ ί ∷ α ∷ []) "2Cor.2.15"
∷ word (ἐ ∷ σ ∷ μ ∷ ὲ ∷ ν ∷ []) "2Cor.2.15"
∷ word (τ ∷ ῷ ∷ []) "2Cor.2.15"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "2Cor.2.15"
∷ word (ἐ ∷ ν ∷ []) "2Cor.2.15"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.2.15"
∷ word (σ ∷ ῳ ∷ ζ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.2.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.2.15"
∷ word (ἐ ∷ ν ∷ []) "2Cor.2.15"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.2.15"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ∙λ ∷ υ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.2.15"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "2Cor.2.16"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "2Cor.2.16"
∷ word (ὀ ∷ σ ∷ μ ∷ ὴ ∷ []) "2Cor.2.16"
∷ word (ἐ ∷ κ ∷ []) "2Cor.2.16"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.2.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.2.16"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.2.16"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "2Cor.2.16"
∷ word (δ ∷ ὲ ∷ []) "2Cor.2.16"
∷ word (ὀ ∷ σ ∷ μ ∷ ὴ ∷ []) "2Cor.2.16"
∷ word (ἐ ∷ κ ∷ []) "2Cor.2.16"
∷ word (ζ ∷ ω ∷ ῆ ∷ ς ∷ []) "2Cor.2.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.2.16"
∷ word (ζ ∷ ω ∷ ή ∷ ν ∷ []) "2Cor.2.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.2.16"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.2.16"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "2Cor.2.16"
∷ word (τ ∷ ί ∷ ς ∷ []) "2Cor.2.16"
∷ word (ἱ ∷ κ ∷ α ∷ ν ∷ ό ∷ ς ∷ []) "2Cor.2.16"
∷ word (ο ∷ ὐ ∷ []) "2Cor.2.17"
∷ word (γ ∷ ά ∷ ρ ∷ []) "2Cor.2.17"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.2.17"
∷ word (ὡ ∷ ς ∷ []) "2Cor.2.17"
∷ word (ο ∷ ἱ ∷ []) "2Cor.2.17"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὶ ∷ []) "2Cor.2.17"
∷ word (κ ∷ α ∷ π ∷ η ∷ ∙λ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.2.17"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.2.17"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ν ∷ []) "2Cor.2.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.2.17"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.2.17"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.2.17"
∷ word (ὡ ∷ ς ∷ []) "2Cor.2.17"
∷ word (ἐ ∷ ξ ∷ []) "2Cor.2.17"
∷ word (ε ∷ ἰ ∷ ∙λ ∷ ι ∷ κ ∷ ρ ∷ ι ∷ ν ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "2Cor.2.17"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.2.17"
∷ word (ὡ ∷ ς ∷ []) "2Cor.2.17"
∷ word (ἐ ∷ κ ∷ []) "2Cor.2.17"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.2.17"
∷ word (κ ∷ α ∷ τ ∷ έ ∷ ν ∷ α ∷ ν ∷ τ ∷ ι ∷ []) "2Cor.2.17"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.2.17"
∷ word (ἐ ∷ ν ∷ []) "2Cor.2.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "2Cor.2.17"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.2.17"
∷ word (Ἀ ∷ ρ ∷ χ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.3.1"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "2Cor.3.1"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.3.1"
∷ word (σ ∷ υ ∷ ν ∷ ι ∷ σ ∷ τ ∷ ά ∷ ν ∷ ε ∷ ι ∷ ν ∷ []) "2Cor.3.1"
∷ word (ἢ ∷ []) "2Cor.3.1"
∷ word (μ ∷ ὴ ∷ []) "2Cor.3.1"
∷ word (χ ∷ ρ ∷ ῄ ∷ ζ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.3.1"
∷ word (ὥ ∷ ς ∷ []) "2Cor.3.1"
∷ word (τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "2Cor.3.1"
∷ word (σ ∷ υ ∷ σ ∷ τ ∷ α ∷ τ ∷ ι ∷ κ ∷ ῶ ∷ ν ∷ []) "2Cor.3.1"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "2Cor.3.1"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.3.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.3.1"
∷ word (ἢ ∷ []) "2Cor.3.1"
∷ word (ἐ ∷ ξ ∷ []) "2Cor.3.1"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.3.1"
∷ word (ἡ ∷ []) "2Cor.3.2"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ []) "2Cor.3.2"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.3.2"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.3.2"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "2Cor.3.2"
∷ word (ἐ ∷ γ ∷ γ ∷ ε ∷ γ ∷ ρ ∷ α ∷ μ ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "2Cor.3.2"
∷ word (ἐ ∷ ν ∷ []) "2Cor.3.2"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "2Cor.3.2"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.3.2"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.3.2"
∷ word (γ ∷ ι ∷ ν ∷ ω ∷ σ ∷ κ ∷ ο ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "2Cor.3.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.3.2"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ ι ∷ ν ∷ ω ∷ σ ∷ κ ∷ ο ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "2Cor.3.2"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "2Cor.3.2"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.3.2"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "2Cor.3.2"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.3.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.3.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ὲ ∷ []) "2Cor.3.3"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ []) "2Cor.3.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.3.3"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ η ∷ θ ∷ ε ∷ ῖ ∷ σ ∷ α ∷ []) "2Cor.3.3"
∷ word (ὑ ∷ φ ∷ []) "2Cor.3.3"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.3.3"
∷ word (ἐ ∷ γ ∷ γ ∷ ε ∷ γ ∷ ρ ∷ α ∷ μ ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "2Cor.3.3"
∷ word (ο ∷ ὐ ∷ []) "2Cor.3.3"
∷ word (μ ∷ έ ∷ ∙λ ∷ α ∷ ν ∷ ι ∷ []) "2Cor.3.3"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.3.3"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "2Cor.3.3"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.3.3"
∷ word (ζ ∷ ῶ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.3.3"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.3.3"
∷ word (ἐ ∷ ν ∷ []) "2Cor.3.3"
∷ word (π ∷ ∙λ ∷ α ∷ ξ ∷ ὶ ∷ ν ∷ []) "2Cor.3.3"
∷ word (∙λ ∷ ι ∷ θ ∷ ί ∷ ν ∷ α ∷ ι ∷ ς ∷ []) "2Cor.3.3"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.3.3"
∷ word (ἐ ∷ ν ∷ []) "2Cor.3.3"
∷ word (π ∷ ∙λ ∷ α ∷ ξ ∷ ὶ ∷ ν ∷ []) "2Cor.3.3"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.3.3"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ί ∷ ν ∷ α ∷ ι ∷ ς ∷ []) "2Cor.3.3"
∷ word (Π ∷ ε ∷ π ∷ ο ∷ ί ∷ θ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.3.4"
∷ word (δ ∷ ὲ ∷ []) "2Cor.3.4"
∷ word (τ ∷ ο ∷ ι ∷ α ∷ ύ ∷ τ ∷ η ∷ ν ∷ []) "2Cor.3.4"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.3.4"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.3.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.3.4"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.3.4"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.3.4"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.3.4"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "2Cor.3.4"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "2Cor.3.5"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.3.5"
∷ word (ἀ ∷ φ ∷ []) "2Cor.3.5"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.3.5"
∷ word (ἱ ∷ κ ∷ α ∷ ν ∷ ο ∷ ί ∷ []) "2Cor.3.5"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.3.5"
∷ word (∙λ ∷ ο ∷ γ ∷ ί ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ί ∷ []) "2Cor.3.5"
∷ word (τ ∷ ι ∷ []) "2Cor.3.5"
∷ word (ὡ ∷ ς ∷ []) "2Cor.3.5"
∷ word (ἐ ∷ ξ ∷ []) "2Cor.3.5"
∷ word (α ∷ ὑ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.3.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.3.5"
∷ word (ἡ ∷ []) "2Cor.3.5"
∷ word (ἱ ∷ κ ∷ α ∷ ν ∷ ό ∷ τ ∷ η ∷ ς ∷ []) "2Cor.3.5"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.3.5"
∷ word (ἐ ∷ κ ∷ []) "2Cor.3.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.3.5"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.3.5"
∷ word (ὃ ∷ ς ∷ []) "2Cor.3.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.3.6"
∷ word (ἱ ∷ κ ∷ ά ∷ ν ∷ ω ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.3.6"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.3.6"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ό ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.3.6"
∷ word (κ ∷ α ∷ ι ∷ ν ∷ ῆ ∷ ς ∷ []) "2Cor.3.6"
∷ word (δ ∷ ι ∷ α ∷ θ ∷ ή ∷ κ ∷ η ∷ ς ∷ []) "2Cor.3.6"
∷ word (ο ∷ ὐ ∷ []) "2Cor.3.6"
∷ word (γ ∷ ρ ∷ ά ∷ μ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.3.6"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.3.6"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.3.6"
∷ word (τ ∷ ὸ ∷ []) "2Cor.3.6"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.3.6"
∷ word (γ ∷ ρ ∷ ά ∷ μ ∷ μ ∷ α ∷ []) "2Cor.3.6"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ τ ∷ έ ∷ ν ∷ ν ∷ ε ∷ ι ∷ []) "2Cor.3.6"
∷ word (τ ∷ ὸ ∷ []) "2Cor.3.6"
∷ word (δ ∷ ὲ ∷ []) "2Cor.3.6"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "2Cor.3.6"
∷ word (ζ ∷ ῳ ∷ ο ∷ π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ []) "2Cor.3.6"
∷ word (Ε ∷ ἰ ∷ []) "2Cor.3.7"
∷ word (δ ∷ ὲ ∷ []) "2Cor.3.7"
∷ word (ἡ ∷ []) "2Cor.3.7"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ []) "2Cor.3.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.3.7"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.3.7"
∷ word (ἐ ∷ ν ∷ []) "2Cor.3.7"
∷ word (γ ∷ ρ ∷ ά ∷ μ ∷ μ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.3.7"
∷ word (ἐ ∷ ν ∷ τ ∷ ε ∷ τ ∷ υ ∷ π ∷ ω ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "2Cor.3.7"
∷ word (∙λ ∷ ί ∷ θ ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.3.7"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ή ∷ θ ∷ η ∷ []) "2Cor.3.7"
∷ word (ἐ ∷ ν ∷ []) "2Cor.3.7"
∷ word (δ ∷ ό ∷ ξ ∷ ῃ ∷ []) "2Cor.3.7"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "2Cor.3.7"
∷ word (μ ∷ ὴ ∷ []) "2Cor.3.7"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.3.7"
∷ word (ἀ ∷ τ ∷ ε ∷ ν ∷ ί ∷ σ ∷ α ∷ ι ∷ []) "2Cor.3.7"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.3.7"
∷ word (υ ∷ ἱ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.3.7"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ὴ ∷ ∙λ ∷ []) "2Cor.3.7"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.3.7"
∷ word (τ ∷ ὸ ∷ []) "2Cor.3.7"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "2Cor.3.7"
∷ word (Μ ∷ ω ∷ ϋ ∷ σ ∷ έ ∷ ω ∷ ς ∷ []) "2Cor.3.7"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.3.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.3.7"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "2Cor.3.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.3.7"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "2Cor.3.7"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.3.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.3.7"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ η ∷ ν ∷ []) "2Cor.3.7"
∷ word (π ∷ ῶ ∷ ς ∷ []) "2Cor.3.8"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "2Cor.3.8"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.3.8"
∷ word (ἡ ∷ []) "2Cor.3.8"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ []) "2Cor.3.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.3.8"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.3.8"
∷ word (ἔ ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "2Cor.3.8"
∷ word (ἐ ∷ ν ∷ []) "2Cor.3.8"
∷ word (δ ∷ ό ∷ ξ ∷ ῃ ∷ []) "2Cor.3.8"
∷ word (ε ∷ ἰ ∷ []) "2Cor.3.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.3.9"
∷ word (τ ∷ ῇ ∷ []) "2Cor.3.9"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ ᾳ ∷ []) "2Cor.3.9"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.3.9"
∷ word (κ ∷ α ∷ τ ∷ α ∷ κ ∷ ρ ∷ ί ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.3.9"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "2Cor.3.9"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῷ ∷ []) "2Cor.3.9"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.3.9"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ []) "2Cor.3.9"
∷ word (ἡ ∷ []) "2Cor.3.9"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ []) "2Cor.3.9"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.3.9"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "2Cor.3.9"
∷ word (δ ∷ ό ∷ ξ ∷ ῃ ∷ []) "2Cor.3.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.3.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.3.10"
∷ word (ο ∷ ὐ ∷ []) "2Cor.3.10"
∷ word (δ ∷ ε ∷ δ ∷ ό ∷ ξ ∷ α ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "2Cor.3.10"
∷ word (τ ∷ ὸ ∷ []) "2Cor.3.10"
∷ word (δ ∷ ε ∷ δ ∷ ο ∷ ξ ∷ α ∷ σ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.3.10"
∷ word (ἐ ∷ ν ∷ []) "2Cor.3.10"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "2Cor.3.10"
∷ word (τ ∷ ῷ ∷ []) "2Cor.3.10"
∷ word (μ ∷ έ ∷ ρ ∷ ε ∷ ι ∷ []) "2Cor.3.10"
∷ word (ε ∷ ἵ ∷ ν ∷ ε ∷ κ ∷ ε ∷ ν ∷ []) "2Cor.3.10"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.3.10"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ α ∷ ∙λ ∷ ∙λ ∷ ο ∷ ύ ∷ σ ∷ η ∷ ς ∷ []) "2Cor.3.10"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "2Cor.3.10"
∷ word (ε ∷ ἰ ∷ []) "2Cor.3.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.3.11"
∷ word (τ ∷ ὸ ∷ []) "2Cor.3.11"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.3.11"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.3.11"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "2Cor.3.11"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῷ ∷ []) "2Cor.3.11"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.3.11"
∷ word (τ ∷ ὸ ∷ []) "2Cor.3.11"
∷ word (μ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.3.11"
∷ word (ἐ ∷ ν ∷ []) "2Cor.3.11"
∷ word (δ ∷ ό ∷ ξ ∷ ῃ ∷ []) "2Cor.3.11"
∷ word (Ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.3.12"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "2Cor.3.12"
∷ word (τ ∷ ο ∷ ι ∷ α ∷ ύ ∷ τ ∷ η ∷ ν ∷ []) "2Cor.3.12"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ α ∷ []) "2Cor.3.12"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῇ ∷ []) "2Cor.3.12"
∷ word (π ∷ α ∷ ρ ∷ ρ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "2Cor.3.12"
∷ word (χ ∷ ρ ∷ ώ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.3.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.3.13"
∷ word (ο ∷ ὐ ∷ []) "2Cor.3.13"
∷ word (κ ∷ α ∷ θ ∷ ά ∷ π ∷ ε ∷ ρ ∷ []) "2Cor.3.13"
∷ word (Μ ∷ ω ∷ ϋ ∷ σ ∷ ῆ ∷ ς ∷ []) "2Cor.3.13"
∷ word (ἐ ∷ τ ∷ ί ∷ θ ∷ ε ∷ ι ∷ []) "2Cor.3.13"
∷ word (κ ∷ ά ∷ ∙λ ∷ υ ∷ μ ∷ μ ∷ α ∷ []) "2Cor.3.13"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.3.13"
∷ word (τ ∷ ὸ ∷ []) "2Cor.3.13"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "2Cor.3.13"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.3.13"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.3.13"
∷ word (τ ∷ ὸ ∷ []) "2Cor.3.13"
∷ word (μ ∷ ὴ ∷ []) "2Cor.3.13"
∷ word (ἀ ∷ τ ∷ ε ∷ ν ∷ ί ∷ σ ∷ α ∷ ι ∷ []) "2Cor.3.13"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.3.13"
∷ word (υ ∷ ἱ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.3.13"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ὴ ∷ ∙λ ∷ []) "2Cor.3.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.3.13"
∷ word (τ ∷ ὸ ∷ []) "2Cor.3.13"
∷ word (τ ∷ έ ∷ ∙λ ∷ ο ∷ ς ∷ []) "2Cor.3.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.3.13"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ []) "2Cor.3.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.3.14"
∷ word (ἐ ∷ π ∷ ω ∷ ρ ∷ ώ ∷ θ ∷ η ∷ []) "2Cor.3.14"
∷ word (τ ∷ ὰ ∷ []) "2Cor.3.14"
∷ word (ν ∷ ο ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "2Cor.3.14"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.3.14"
∷ word (ἄ ∷ χ ∷ ρ ∷ ι ∷ []) "2Cor.3.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.3.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.3.14"
∷ word (σ ∷ ή ∷ μ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "2Cor.3.14"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "2Cor.3.14"
∷ word (τ ∷ ὸ ∷ []) "2Cor.3.14"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "2Cor.3.14"
∷ word (κ ∷ ά ∷ ∙λ ∷ υ ∷ μ ∷ μ ∷ α ∷ []) "2Cor.3.14"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.3.14"
∷ word (τ ∷ ῇ ∷ []) "2Cor.3.14"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.3.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.3.14"
∷ word (π ∷ α ∷ ∙λ ∷ α ∷ ι ∷ ᾶ ∷ ς ∷ []) "2Cor.3.14"
∷ word (δ ∷ ι ∷ α ∷ θ ∷ ή ∷ κ ∷ η ∷ ς ∷ []) "2Cor.3.14"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "2Cor.3.14"
∷ word (μ ∷ ὴ ∷ []) "2Cor.3.14"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ α ∷ ∙λ ∷ υ ∷ π ∷ τ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.3.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.3.14"
∷ word (ἐ ∷ ν ∷ []) "2Cor.3.14"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "2Cor.3.14"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "2Cor.3.14"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.3.15"
∷ word (ἕ ∷ ω ∷ ς ∷ []) "2Cor.3.15"
∷ word (σ ∷ ή ∷ μ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "2Cor.3.15"
∷ word (ἡ ∷ ν ∷ ί ∷ κ ∷ α ∷ []) "2Cor.3.15"
∷ word (ἂ ∷ ν ∷ []) "2Cor.3.15"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "2Cor.3.15"
∷ word (Μ ∷ ω ∷ ϋ ∷ σ ∷ ῆ ∷ ς ∷ []) "2Cor.3.15"
∷ word (κ ∷ ά ∷ ∙λ ∷ υ ∷ μ ∷ μ ∷ α ∷ []) "2Cor.3.15"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.3.15"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.3.15"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.3.15"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.3.15"
∷ word (κ ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "2Cor.3.15"
∷ word (ἡ ∷ ν ∷ ί ∷ κ ∷ α ∷ []) "2Cor.3.16"
∷ word (δ ∷ ὲ ∷ []) "2Cor.3.16"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "2Cor.3.16"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ρ ∷ έ ∷ ψ ∷ ῃ ∷ []) "2Cor.3.16"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.3.16"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.3.16"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ α ∷ ι ∷ ρ ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "2Cor.3.16"
∷ word (τ ∷ ὸ ∷ []) "2Cor.3.16"
∷ word (κ ∷ ά ∷ ∙λ ∷ υ ∷ μ ∷ μ ∷ α ∷ []) "2Cor.3.16"
∷ word (ὁ ∷ []) "2Cor.3.17"
∷ word (δ ∷ ὲ ∷ []) "2Cor.3.17"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "2Cor.3.17"
∷ word (τ ∷ ὸ ∷ []) "2Cor.3.17"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ ά ∷ []) "2Cor.3.17"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "2Cor.3.17"
∷ word (ο ∷ ὗ ∷ []) "2Cor.3.17"
∷ word (δ ∷ ὲ ∷ []) "2Cor.3.17"
∷ word (τ ∷ ὸ ∷ []) "2Cor.3.17"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "2Cor.3.17"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.3.17"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ ε ∷ ρ ∷ ί ∷ α ∷ []) "2Cor.3.17"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.3.18"
∷ word (δ ∷ ὲ ∷ []) "2Cor.3.18"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.3.18"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ ε ∷ κ ∷ α ∷ ∙λ ∷ υ ∷ μ ∷ μ ∷ έ ∷ ν ∷ ῳ ∷ []) "2Cor.3.18"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ώ ∷ π ∷ ῳ ∷ []) "2Cor.3.18"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.3.18"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "2Cor.3.18"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.3.18"
∷ word (κ ∷ α ∷ τ ∷ ο ∷ π ∷ τ ∷ ρ ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.3.18"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.3.18"
∷ word (α ∷ ὐ ∷ τ ∷ ὴ ∷ ν ∷ []) "2Cor.3.18"
∷ word (ε ∷ ἰ ∷ κ ∷ ό ∷ ν ∷ α ∷ []) "2Cor.3.18"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ μ ∷ ο ∷ ρ ∷ φ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.3.18"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.3.18"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "2Cor.3.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.3.18"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "2Cor.3.18"
∷ word (κ ∷ α ∷ θ ∷ ά ∷ π ∷ ε ∷ ρ ∷ []) "2Cor.3.18"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.3.18"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.3.18"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.3.18"
∷ word (Δ ∷ ι ∷ ὰ ∷ []) "2Cor.4.1"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.4.1"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.4.1"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.4.1"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "2Cor.4.1"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ η ∷ ν ∷ []) "2Cor.4.1"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "2Cor.4.1"
∷ word (ἠ ∷ ∙λ ∷ ε ∷ ή ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.4.1"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.4.1"
∷ word (ἐ ∷ γ ∷ κ ∷ α ∷ κ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.4.1"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.4.2"
∷ word (ἀ ∷ π ∷ ε ∷ ι ∷ π ∷ ά ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.4.2"
∷ word (τ ∷ ὰ ∷ []) "2Cor.4.2"
∷ word (κ ∷ ρ ∷ υ ∷ π ∷ τ ∷ ὰ ∷ []) "2Cor.4.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.4.2"
∷ word (α ∷ ἰ ∷ σ ∷ χ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "2Cor.4.2"
∷ word (μ ∷ ὴ ∷ []) "2Cor.4.2"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.4.2"
∷ word (ἐ ∷ ν ∷ []) "2Cor.4.2"
∷ word (π ∷ α ∷ ν ∷ ο ∷ υ ∷ ρ ∷ γ ∷ ί ∷ ᾳ ∷ []) "2Cor.4.2"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "2Cor.4.2"
∷ word (δ ∷ ο ∷ ∙λ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.4.2"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.4.2"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ν ∷ []) "2Cor.4.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.4.2"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.4.2"
∷ word (τ ∷ ῇ ∷ []) "2Cor.4.2"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.4.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.4.2"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "2Cor.4.2"
∷ word (σ ∷ υ ∷ ν ∷ ι ∷ σ ∷ τ ∷ ά ∷ ν ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.4.2"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.4.2"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.4.2"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "2Cor.4.2"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ί ∷ δ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.4.2"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "2Cor.4.2"
∷ word (ἐ ∷ ν ∷ ώ ∷ π ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.4.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.4.2"
∷ word (ε ∷ ἰ ∷ []) "2Cor.4.3"
∷ word (δ ∷ ὲ ∷ []) "2Cor.4.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.4.3"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "2Cor.4.3"
∷ word (κ ∷ ε ∷ κ ∷ α ∷ ∙λ ∷ υ ∷ μ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.4.3"
∷ word (τ ∷ ὸ ∷ []) "2Cor.4.3"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.4.3"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.4.3"
∷ word (ἐ ∷ ν ∷ []) "2Cor.4.3"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.4.3"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ∙λ ∷ υ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.4.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "2Cor.4.3"
∷ word (κ ∷ ε ∷ κ ∷ α ∷ ∙λ ∷ υ ∷ μ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.4.3"
∷ word (ἐ ∷ ν ∷ []) "2Cor.4.4"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "2Cor.4.4"
∷ word (ὁ ∷ []) "2Cor.4.4"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.4.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.4"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.4.4"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.4.4"
∷ word (ἐ ∷ τ ∷ ύ ∷ φ ∷ ∙λ ∷ ω ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.4.4"
∷ word (τ ∷ ὰ ∷ []) "2Cor.4.4"
∷ word (ν ∷ ο ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "2Cor.4.4"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.4.4"
∷ word (ἀ ∷ π ∷ ί ∷ σ ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.4.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.4.4"
∷ word (τ ∷ ὸ ∷ []) "2Cor.4.4"
∷ word (μ ∷ ὴ ∷ []) "2Cor.4.4"
∷ word (α ∷ ὐ ∷ γ ∷ ά ∷ σ ∷ α ∷ ι ∷ []) "2Cor.4.4"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.4.4"
∷ word (φ ∷ ω ∷ τ ∷ ι ∷ σ ∷ μ ∷ ὸ ∷ ν ∷ []) "2Cor.4.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.4"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.4.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.4.4"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "2Cor.4.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.4"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.4"
∷ word (ὅ ∷ ς ∷ []) "2Cor.4.4"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "2Cor.4.4"
∷ word (ε ∷ ἰ ∷ κ ∷ ὼ ∷ ν ∷ []) "2Cor.4.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.4"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.4.4"
∷ word (ο ∷ ὐ ∷ []) "2Cor.4.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.4.5"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.4.5"
∷ word (κ ∷ η ∷ ρ ∷ ύ ∷ σ ∷ σ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.4.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.4.5"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "2Cor.4.5"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "2Cor.4.5"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.4.5"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.4.5"
∷ word (δ ∷ ὲ ∷ []) "2Cor.4.5"
∷ word (δ ∷ ο ∷ ύ ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.4.5"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.4.5"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.4.5"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "2Cor.4.5"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.4.6"
∷ word (ὁ ∷ []) "2Cor.4.6"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.4.6"
∷ word (ὁ ∷ []) "2Cor.4.6"
∷ word (ε ∷ ἰ ∷ π ∷ ώ ∷ ν ∷ []) "2Cor.4.6"
∷ word (Ἐ ∷ κ ∷ []) "2Cor.4.6"
∷ word (σ ∷ κ ∷ ό ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.4.6"
∷ word (φ ∷ ῶ ∷ ς ∷ []) "2Cor.4.6"
∷ word (∙λ ∷ ά ∷ μ ∷ ψ ∷ ε ∷ ι ∷ []) "2Cor.4.6"
∷ word (ὃ ∷ ς ∷ []) "2Cor.4.6"
∷ word (ἔ ∷ ∙λ ∷ α ∷ μ ∷ ψ ∷ ε ∷ ν ∷ []) "2Cor.4.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.4.6"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "2Cor.4.6"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.4.6"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.4.6"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.4.6"
∷ word (φ ∷ ω ∷ τ ∷ ι ∷ σ ∷ μ ∷ ὸ ∷ ν ∷ []) "2Cor.4.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.4.6"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.4.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.4.6"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "2Cor.4.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.6"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.4.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.4.6"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ώ ∷ π ∷ ῳ ∷ []) "2Cor.4.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.6"
∷ word (Ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.4.7"
∷ word (δ ∷ ὲ ∷ []) "2Cor.4.7"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.4.7"
∷ word (θ ∷ η ∷ σ ∷ α ∷ υ ∷ ρ ∷ ὸ ∷ ν ∷ []) "2Cor.4.7"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.4.7"
∷ word (ἐ ∷ ν ∷ []) "2Cor.4.7"
∷ word (ὀ ∷ σ ∷ τ ∷ ρ ∷ α ∷ κ ∷ ί ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.4.7"
∷ word (σ ∷ κ ∷ ε ∷ ύ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.4.7"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.4.7"
∷ word (ἡ ∷ []) "2Cor.4.7"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ ο ∷ ∙λ ∷ ὴ ∷ []) "2Cor.4.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.4.7"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.4.7"
∷ word (ᾖ ∷ []) "2Cor.4.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.4.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.4.7"
∷ word (μ ∷ ὴ ∷ []) "2Cor.4.7"
∷ word (ἐ ∷ ξ ∷ []) "2Cor.4.7"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.4.7"
∷ word (ἐ ∷ ν ∷ []) "2Cor.4.8"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "2Cor.4.8"
∷ word (θ ∷ ∙λ ∷ ι ∷ β ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.4.8"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.4.8"
∷ word (ο ∷ ὐ ∷ []) "2Cor.4.8"
∷ word (σ ∷ τ ∷ ε ∷ ν ∷ ο ∷ χ ∷ ω ∷ ρ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.4.8"
∷ word (ἀ ∷ π ∷ ο ∷ ρ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.4.8"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.4.8"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.4.8"
∷ word (ἐ ∷ ξ ∷ α ∷ π ∷ ο ∷ ρ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.4.8"
∷ word (δ ∷ ι ∷ ω ∷ κ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.4.9"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.4.9"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.4.9"
∷ word (ἐ ∷ γ ∷ κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ε ∷ ι ∷ π ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.4.9"
∷ word (κ ∷ α ∷ τ ∷ α ∷ β ∷ α ∷ ∙λ ∷ ∙λ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.4.9"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.4.9"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.4.9"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.4.9"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ο ∷ τ ∷ ε ∷ []) "2Cor.4.10"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.4.10"
∷ word (ν ∷ έ ∷ κ ∷ ρ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.4.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.10"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "2Cor.4.10"
∷ word (ἐ ∷ ν ∷ []) "2Cor.4.10"
∷ word (τ ∷ ῷ ∷ []) "2Cor.4.10"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "2Cor.4.10"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ φ ∷ έ ∷ ρ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.4.10"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.4.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.4.10"
∷ word (ἡ ∷ []) "2Cor.4.10"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "2Cor.4.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.10"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "2Cor.4.10"
∷ word (ἐ ∷ ν ∷ []) "2Cor.4.10"
∷ word (τ ∷ ῷ ∷ []) "2Cor.4.10"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "2Cor.4.10"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.4.10"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ω ∷ θ ∷ ῇ ∷ []) "2Cor.4.10"
∷ word (ἀ ∷ ε ∷ ὶ ∷ []) "2Cor.4.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.4.11"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.4.11"
∷ word (ο ∷ ἱ ∷ []) "2Cor.4.11"
∷ word (ζ ∷ ῶ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.4.11"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.4.11"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.4.11"
∷ word (π ∷ α ∷ ρ ∷ α ∷ δ ∷ ι ∷ δ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.4.11"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.4.11"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "2Cor.4.11"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.4.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.4.11"
∷ word (ἡ ∷ []) "2Cor.4.11"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "2Cor.4.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.11"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "2Cor.4.11"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ω ∷ θ ∷ ῇ ∷ []) "2Cor.4.11"
∷ word (ἐ ∷ ν ∷ []) "2Cor.4.11"
∷ word (τ ∷ ῇ ∷ []) "2Cor.4.11"
∷ word (θ ∷ ν ∷ η ∷ τ ∷ ῇ ∷ []) "2Cor.4.11"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "2Cor.4.11"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.4.11"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "2Cor.4.12"
∷ word (ὁ ∷ []) "2Cor.4.12"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.4.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.4.12"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.4.12"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "2Cor.4.12"
∷ word (ἡ ∷ []) "2Cor.4.12"
∷ word (δ ∷ ὲ ∷ []) "2Cor.4.12"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "2Cor.4.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.4.12"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.4.12"
∷ word (Ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.4.13"
∷ word (δ ∷ ὲ ∷ []) "2Cor.4.13"
∷ word (τ ∷ ὸ ∷ []) "2Cor.4.13"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "2Cor.4.13"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "2Cor.4.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.4.13"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.4.13"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.4.13"
∷ word (τ ∷ ὸ ∷ []) "2Cor.4.13"
∷ word (γ ∷ ε ∷ γ ∷ ρ ∷ α ∷ μ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.4.13"
∷ word (Ἐ ∷ π ∷ ί ∷ σ ∷ τ ∷ ε ∷ υ ∷ σ ∷ α ∷ []) "2Cor.4.13"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "2Cor.4.13"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ ∙λ ∷ η ∷ σ ∷ α ∷ []) "2Cor.4.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.4.13"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.4.13"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.4.13"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "2Cor.4.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.4.13"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.4.13"
∷ word (ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.4.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.4.14"
∷ word (ὁ ∷ []) "2Cor.4.14"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ α ∷ ς ∷ []) "2Cor.4.14"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.4.14"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "2Cor.4.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.4.14"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.4.14"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "2Cor.4.14"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "2Cor.4.14"
∷ word (ἐ ∷ γ ∷ ε ∷ ρ ∷ ε ∷ ῖ ∷ []) "2Cor.4.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.4.14"
∷ word (π ∷ α ∷ ρ ∷ α ∷ σ ∷ τ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.4.14"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "2Cor.4.14"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.4.14"
∷ word (τ ∷ ὰ ∷ []) "2Cor.4.15"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.4.15"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "2Cor.4.15"
∷ word (δ ∷ ι ∷ []) "2Cor.4.15"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.4.15"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.4.15"
∷ word (ἡ ∷ []) "2Cor.4.15"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "2Cor.4.15"
∷ word (π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ ά ∷ σ ∷ α ∷ σ ∷ α ∷ []) "2Cor.4.15"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.4.15"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.4.15"
∷ word (π ∷ ∙λ ∷ ε ∷ ι ∷ ό ∷ ν ∷ ω ∷ ν ∷ []) "2Cor.4.15"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.4.15"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.4.15"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ σ ∷ ῃ ∷ []) "2Cor.4.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.4.15"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.4.15"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "2Cor.4.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.4.15"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.4.15"
∷ word (Δ ∷ ι ∷ ὸ ∷ []) "2Cor.4.16"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.4.16"
∷ word (ἐ ∷ γ ∷ κ ∷ α ∷ κ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.4.16"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.4.16"
∷ word (ε ∷ ἰ ∷ []) "2Cor.4.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.4.16"
∷ word (ὁ ∷ []) "2Cor.4.16"
∷ word (ἔ ∷ ξ ∷ ω ∷ []) "2Cor.4.16"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.4.16"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "2Cor.4.16"
∷ word (δ ∷ ι ∷ α ∷ φ ∷ θ ∷ ε ∷ ί ∷ ρ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "2Cor.4.16"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.4.16"
∷ word (ὁ ∷ []) "2Cor.4.16"
∷ word (ἔ ∷ σ ∷ ω ∷ []) "2Cor.4.16"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.4.16"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ α ∷ ι ∷ ν ∷ ο ∷ ῦ ∷ τ ∷ α ∷ ι ∷ []) "2Cor.4.16"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "2Cor.4.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.4.16"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "2Cor.4.16"
∷ word (τ ∷ ὸ ∷ []) "2Cor.4.17"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.4.17"
∷ word (π ∷ α ∷ ρ ∷ α ∷ υ ∷ τ ∷ ί ∷ κ ∷ α ∷ []) "2Cor.4.17"
∷ word (ἐ ∷ ∙λ ∷ α ∷ φ ∷ ρ ∷ ὸ ∷ ν ∷ []) "2Cor.4.17"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.4.17"
∷ word (θ ∷ ∙λ ∷ ί ∷ ψ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.4.17"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.4.17"
∷ word (κ ∷ α ∷ θ ∷ []) "2Cor.4.17"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "2Cor.4.17"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.4.17"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "2Cor.4.17"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.4.17"
∷ word (β ∷ ά ∷ ρ ∷ ο ∷ ς ∷ []) "2Cor.4.17"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "2Cor.4.17"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "2Cor.4.17"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.4.17"
∷ word (μ ∷ ὴ ∷ []) "2Cor.4.18"
∷ word (σ ∷ κ ∷ ο ∷ π ∷ ο ∷ ύ ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.4.18"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.4.18"
∷ word (τ ∷ ὰ ∷ []) "2Cor.4.18"
∷ word (β ∷ ∙λ ∷ ε ∷ π ∷ ό ∷ μ ∷ ε ∷ ν ∷ α ∷ []) "2Cor.4.18"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.4.18"
∷ word (τ ∷ ὰ ∷ []) "2Cor.4.18"
∷ word (μ ∷ ὴ ∷ []) "2Cor.4.18"
∷ word (β ∷ ∙λ ∷ ε ∷ π ∷ ό ∷ μ ∷ ε ∷ ν ∷ α ∷ []) "2Cor.4.18"
∷ word (τ ∷ ὰ ∷ []) "2Cor.4.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.4.18"
∷ word (β ∷ ∙λ ∷ ε ∷ π ∷ ό ∷ μ ∷ ε ∷ ν ∷ α ∷ []) "2Cor.4.18"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ κ ∷ α ∷ ι ∷ ρ ∷ α ∷ []) "2Cor.4.18"
∷ word (τ ∷ ὰ ∷ []) "2Cor.4.18"
∷ word (δ ∷ ὲ ∷ []) "2Cor.4.18"
∷ word (μ ∷ ὴ ∷ []) "2Cor.4.18"
∷ word (β ∷ ∙λ ∷ ε ∷ π ∷ ό ∷ μ ∷ ε ∷ ν ∷ α ∷ []) "2Cor.4.18"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ α ∷ []) "2Cor.4.18"
∷ word (Ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.1"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.5.1"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.5.1"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "2Cor.5.1"
∷ word (ἡ ∷ []) "2Cor.5.1"
∷ word (ἐ ∷ π ∷ ί ∷ γ ∷ ε ∷ ι ∷ ο ∷ ς ∷ []) "2Cor.5.1"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.5.1"
∷ word (ο ∷ ἰ ∷ κ ∷ ί ∷ α ∷ []) "2Cor.5.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.1"
∷ word (σ ∷ κ ∷ ή ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.5.1"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ υ ∷ θ ∷ ῇ ∷ []) "2Cor.5.1"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "2Cor.5.1"
∷ word (ἐ ∷ κ ∷ []) "2Cor.5.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.5.1"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.1"
∷ word (ο ∷ ἰ ∷ κ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.5.1"
∷ word (ἀ ∷ χ ∷ ε ∷ ι ∷ ρ ∷ ο ∷ π ∷ ο ∷ ί ∷ η ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.5.1"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.5.1"
∷ word (ἐ ∷ ν ∷ []) "2Cor.5.1"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.5.1"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.5.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.5.2"
∷ word (ἐ ∷ ν ∷ []) "2Cor.5.2"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "2Cor.5.2"
∷ word (σ ∷ τ ∷ ε ∷ ν ∷ ά ∷ ζ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.2"
∷ word (τ ∷ ὸ ∷ []) "2Cor.5.2"
∷ word (ο ∷ ἰ ∷ κ ∷ η ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.5.2"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.5.2"
∷ word (τ ∷ ὸ ∷ []) "2Cor.5.2"
∷ word (ἐ ∷ ξ ∷ []) "2Cor.5.2"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ο ∷ ῦ ∷ []) "2Cor.5.2"
∷ word (ἐ ∷ π ∷ ε ∷ ν ∷ δ ∷ ύ ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.5.2"
∷ word (ἐ ∷ π ∷ ι ∷ π ∷ ο ∷ θ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.5.2"
∷ word (ε ∷ ἴ ∷ []) "2Cor.5.3"
∷ word (γ ∷ ε ∷ []) "2Cor.5.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.3"
∷ word (ἐ ∷ ν ∷ δ ∷ υ ∷ σ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.5.3"
∷ word (ο ∷ ὐ ∷ []) "2Cor.5.3"
∷ word (γ ∷ υ ∷ μ ∷ ν ∷ ο ∷ ὶ ∷ []) "2Cor.5.3"
∷ word (ε ∷ ὑ ∷ ρ ∷ ε ∷ θ ∷ η ∷ σ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.5.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.5.4"
∷ word (ο ∷ ἱ ∷ []) "2Cor.5.4"
∷ word (ὄ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.5.4"
∷ word (ἐ ∷ ν ∷ []) "2Cor.5.4"
∷ word (τ ∷ ῷ ∷ []) "2Cor.5.4"
∷ word (σ ∷ κ ∷ ή ∷ ν ∷ ε ∷ ι ∷ []) "2Cor.5.4"
∷ word (σ ∷ τ ∷ ε ∷ ν ∷ ά ∷ ζ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.4"
∷ word (β ∷ α ∷ ρ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.5.4"
∷ word (ἐ ∷ φ ∷ []) "2Cor.5.4"
∷ word (ᾧ ∷ []) "2Cor.5.4"
∷ word (ο ∷ ὐ ∷ []) "2Cor.5.4"
∷ word (θ ∷ έ ∷ ∙λ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.4"
∷ word (ἐ ∷ κ ∷ δ ∷ ύ ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.5.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.5.4"
∷ word (ἐ ∷ π ∷ ε ∷ ν ∷ δ ∷ ύ ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.5.4"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.5.4"
∷ word (κ ∷ α ∷ τ ∷ α ∷ π ∷ ο ∷ θ ∷ ῇ ∷ []) "2Cor.5.4"
∷ word (τ ∷ ὸ ∷ []) "2Cor.5.4"
∷ word (θ ∷ ν ∷ η ∷ τ ∷ ὸ ∷ ν ∷ []) "2Cor.5.4"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "2Cor.5.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.5.4"
∷ word (ζ ∷ ω ∷ ῆ ∷ ς ∷ []) "2Cor.5.4"
∷ word (ὁ ∷ []) "2Cor.5.5"
∷ word (δ ∷ ὲ ∷ []) "2Cor.5.5"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ α ∷ σ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.5.5"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.5.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.5.5"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "2Cor.5.5"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.5.5"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "2Cor.5.5"
∷ word (ὁ ∷ []) "2Cor.5.5"
∷ word (δ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.5.5"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.5.5"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.5.5"
∷ word (ἀ ∷ ρ ∷ ρ ∷ α ∷ β ∷ ῶ ∷ ν ∷ α ∷ []) "2Cor.5.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.5"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.5.5"
∷ word (Θ ∷ α ∷ ρ ∷ ρ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.5.6"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "2Cor.5.6"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ο ∷ τ ∷ ε ∷ []) "2Cor.5.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.6"
∷ word (ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.5.6"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.5.6"
∷ word (ἐ ∷ ν ∷ δ ∷ η ∷ μ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.5.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.5.6"
∷ word (τ ∷ ῷ ∷ []) "2Cor.5.6"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "2Cor.5.6"
∷ word (ἐ ∷ κ ∷ δ ∷ η ∷ μ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.6"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.5.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.6"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.5.6"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.5.7"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.5.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.5.7"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.7"
∷ word (ο ∷ ὐ ∷ []) "2Cor.5.7"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.5.7"
∷ word (ε ∷ ἴ ∷ δ ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.5.7"
∷ word (θ ∷ α ∷ ρ ∷ ρ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.8"
∷ word (δ ∷ ὲ ∷ []) "2Cor.5.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.8"
∷ word (ε ∷ ὐ ∷ δ ∷ ο ∷ κ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.8"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.5.8"
∷ word (ἐ ∷ κ ∷ δ ∷ η ∷ μ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.5.8"
∷ word (ἐ ∷ κ ∷ []) "2Cor.5.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.8"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.5.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.8"
∷ word (ἐ ∷ ν ∷ δ ∷ η ∷ μ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.5.8"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.5.8"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.5.8"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.5.8"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "2Cor.5.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.9"
∷ word (φ ∷ ι ∷ ∙λ ∷ ο ∷ τ ∷ ι ∷ μ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.5.9"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.5.9"
∷ word (ἐ ∷ ν ∷ δ ∷ η ∷ μ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.5.9"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.5.9"
∷ word (ἐ ∷ κ ∷ δ ∷ η ∷ μ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.5.9"
∷ word (ε ∷ ὐ ∷ ά ∷ ρ ∷ ε ∷ σ ∷ τ ∷ ο ∷ ι ∷ []) "2Cor.5.9"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "2Cor.5.9"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.5.9"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.5.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.5.10"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "2Cor.5.10"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.5.10"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ω ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.5.10"
∷ word (δ ∷ ε ∷ ῖ ∷ []) "2Cor.5.10"
∷ word (ἔ ∷ μ ∷ π ∷ ρ ∷ ο ∷ σ ∷ θ ∷ ε ∷ ν ∷ []) "2Cor.5.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.10"
∷ word (β ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.5.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.10"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.10"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.5.10"
∷ word (κ ∷ ο ∷ μ ∷ ί ∷ σ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "2Cor.5.10"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.5.10"
∷ word (τ ∷ ὰ ∷ []) "2Cor.5.10"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.5.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.10"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.5.10"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.5.10"
∷ word (ἃ ∷ []) "2Cor.5.10"
∷ word (ἔ ∷ π ∷ ρ ∷ α ∷ ξ ∷ ε ∷ ν ∷ []) "2Cor.5.10"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.5.10"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ὸ ∷ ν ∷ []) "2Cor.5.10"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.5.10"
∷ word (φ ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.5.10"
∷ word (Ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.5.11"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "2Cor.5.11"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.5.11"
∷ word (φ ∷ ό ∷ β ∷ ο ∷ ν ∷ []) "2Cor.5.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.11"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.5.11"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.5.11"
∷ word (π ∷ ε ∷ ί ∷ θ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.11"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "2Cor.5.11"
∷ word (δ ∷ ὲ ∷ []) "2Cor.5.11"
∷ word (π ∷ ε ∷ φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ώ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.5.11"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ ζ ∷ ω ∷ []) "2Cor.5.11"
∷ word (δ ∷ ὲ ∷ []) "2Cor.5.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.11"
∷ word (ἐ ∷ ν ∷ []) "2Cor.5.11"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "2Cor.5.11"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ι ∷ δ ∷ ή ∷ σ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.5.11"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.5.11"
∷ word (π ∷ ε ∷ φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ῶ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.5.11"
∷ word (ο ∷ ὐ ∷ []) "2Cor.5.12"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "2Cor.5.12"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.5.12"
∷ word (σ ∷ υ ∷ ν ∷ ι ∷ σ ∷ τ ∷ ά ∷ ν ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.12"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.5.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.5.12"
∷ word (ἀ ∷ φ ∷ ο ∷ ρ ∷ μ ∷ ὴ ∷ ν ∷ []) "2Cor.5.12"
∷ word (δ ∷ ι ∷ δ ∷ ό ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.5.12"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.5.12"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.5.12"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.5.12"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.5.12"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.5.12"
∷ word (ἔ ∷ χ ∷ η ∷ τ ∷ ε ∷ []) "2Cor.5.12"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.5.12"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.5.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.5.12"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ώ ∷ π ∷ ῳ ∷ []) "2Cor.5.12"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.5.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.12"
∷ word (μ ∷ ὴ ∷ []) "2Cor.5.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.5.12"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ ᾳ ∷ []) "2Cor.5.12"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.5.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.5.13"
∷ word (ἐ ∷ ξ ∷ έ ∷ σ ∷ τ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.13"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "2Cor.5.13"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.5.13"
∷ word (σ ∷ ω ∷ φ ∷ ρ ∷ ο ∷ ν ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.13"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.5.13"
∷ word (ἡ ∷ []) "2Cor.5.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.5.14"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "2Cor.5.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.14"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.14"
∷ word (σ ∷ υ ∷ ν ∷ έ ∷ χ ∷ ε ∷ ι ∷ []) "2Cor.5.14"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.5.14"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ α ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "2Cor.5.14"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.5.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.5.14"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "2Cor.5.14"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.5.14"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.5.14"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "2Cor.5.14"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "2Cor.5.14"
∷ word (ο ∷ ἱ ∷ []) "2Cor.5.14"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.5.14"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.5.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.15"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.5.15"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.5.15"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "2Cor.5.15"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.5.15"
∷ word (ο ∷ ἱ ∷ []) "2Cor.5.15"
∷ word (ζ ∷ ῶ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.5.15"
∷ word (μ ∷ η ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "2Cor.5.15"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.5.15"
∷ word (ζ ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.5.15"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.5.15"
∷ word (τ ∷ ῷ ∷ []) "2Cor.5.15"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.5.15"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.5.15"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ α ∷ ν ∷ ό ∷ ν ∷ τ ∷ ι ∷ []) "2Cor.5.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.15"
∷ word (ἐ ∷ γ ∷ ε ∷ ρ ∷ θ ∷ έ ∷ ν ∷ τ ∷ ι ∷ []) "2Cor.5.15"
∷ word (Ὥ ∷ σ ∷ τ ∷ ε ∷ []) "2Cor.5.16"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.5.16"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.5.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.16"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "2Cor.5.16"
∷ word (ο ∷ ὐ ∷ δ ∷ έ ∷ ν ∷ α ∷ []) "2Cor.5.16"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.16"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.5.16"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "2Cor.5.16"
∷ word (ε ∷ ἰ ∷ []) "2Cor.5.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.16"
∷ word (ἐ ∷ γ ∷ ν ∷ ώ ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.16"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.5.16"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "2Cor.5.16"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ν ∷ []) "2Cor.5.16"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.5.16"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "2Cor.5.16"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "2Cor.5.16"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.16"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "2Cor.5.17"
∷ word (ε ∷ ἴ ∷ []) "2Cor.5.17"
∷ word (τ ∷ ι ∷ ς ∷ []) "2Cor.5.17"
∷ word (ἐ ∷ ν ∷ []) "2Cor.5.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "2Cor.5.17"
∷ word (κ ∷ α ∷ ι ∷ ν ∷ ὴ ∷ []) "2Cor.5.17"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ ι ∷ ς ∷ []) "2Cor.5.17"
∷ word (τ ∷ ὰ ∷ []) "2Cor.5.17"
∷ word (ἀ ∷ ρ ∷ χ ∷ α ∷ ῖ ∷ α ∷ []) "2Cor.5.17"
∷ word (π ∷ α ∷ ρ ∷ ῆ ∷ ∙λ ∷ θ ∷ ε ∷ ν ∷ []) "2Cor.5.17"
∷ word (ἰ ∷ δ ∷ ο ∷ ὺ ∷ []) "2Cor.5.17"
∷ word (γ ∷ έ ∷ γ ∷ ο ∷ ν ∷ ε ∷ ν ∷ []) "2Cor.5.17"
∷ word (κ ∷ α ∷ ι ∷ ν ∷ ά ∷ []) "2Cor.5.17"
∷ word (τ ∷ ὰ ∷ []) "2Cor.5.18"
∷ word (δ ∷ ὲ ∷ []) "2Cor.5.18"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "2Cor.5.18"
∷ word (ἐ ∷ κ ∷ []) "2Cor.5.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.18"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.5.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.18"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ∙λ ∷ ά ∷ ξ ∷ α ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.5.18"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.5.18"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "2Cor.5.18"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.5.18"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.18"
∷ word (δ ∷ ό ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.5.18"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.5.18"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.5.18"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "2Cor.5.18"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.5.18"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ∙λ ∷ α ∷ γ ∷ ῆ ∷ ς ∷ []) "2Cor.5.18"
∷ word (ὡ ∷ ς ∷ []) "2Cor.5.19"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.5.19"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.5.19"
∷ word (ἦ ∷ ν ∷ []) "2Cor.5.19"
∷ word (ἐ ∷ ν ∷ []) "2Cor.5.19"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "2Cor.5.19"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ν ∷ []) "2Cor.5.19"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ∙λ ∷ ά ∷ σ ∷ σ ∷ ω ∷ ν ∷ []) "2Cor.5.19"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "2Cor.5.19"
∷ word (μ ∷ ὴ ∷ []) "2Cor.5.19"
∷ word (∙λ ∷ ο ∷ γ ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.5.19"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.5.19"
∷ word (τ ∷ ὰ ∷ []) "2Cor.5.19"
∷ word (π ∷ α ∷ ρ ∷ α ∷ π ∷ τ ∷ ώ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "2Cor.5.19"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.5.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.5.19"
∷ word (θ ∷ έ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.5.19"
∷ word (ἐ ∷ ν ∷ []) "2Cor.5.19"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.5.19"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.5.19"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ν ∷ []) "2Cor.5.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.5.19"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ∙λ ∷ α ∷ γ ∷ ῆ ∷ ς ∷ []) "2Cor.5.19"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.5.20"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.20"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "2Cor.5.20"
∷ word (π ∷ ρ ∷ ε ∷ σ ∷ β ∷ ε ∷ ύ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.5.20"
∷ word (ὡ ∷ ς ∷ []) "2Cor.5.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.20"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.5.20"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.5.20"
∷ word (δ ∷ ι ∷ []) "2Cor.5.20"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.5.20"
∷ word (δ ∷ ε ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.5.20"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.5.20"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.5.20"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ∙λ ∷ ά ∷ γ ∷ η ∷ τ ∷ ε ∷ []) "2Cor.5.20"
∷ word (τ ∷ ῷ ∷ []) "2Cor.5.20"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "2Cor.5.20"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.5.21"
∷ word (μ ∷ ὴ ∷ []) "2Cor.5.21"
∷ word (γ ∷ ν ∷ ό ∷ ν ∷ τ ∷ α ∷ []) "2Cor.5.21"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.5.21"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.5.21"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.5.21"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.5.21"
∷ word (ἐ ∷ π ∷ ο ∷ ί ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.5.21"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.5.21"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.5.21"
∷ word (γ ∷ ε ∷ ν ∷ ώ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.5.21"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ []) "2Cor.5.21"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.5.21"
∷ word (ἐ ∷ ν ∷ []) "2Cor.5.21"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "2Cor.5.21"
∷ word (Σ ∷ υ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.6.1"
∷ word (δ ∷ ὲ ∷ []) "2Cor.6.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.1"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.6.1"
∷ word (μ ∷ ὴ ∷ []) "2Cor.6.1"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.6.1"
∷ word (κ ∷ ε ∷ ν ∷ ὸ ∷ ν ∷ []) "2Cor.6.1"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.6.1"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "2Cor.6.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.6.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.6.1"
∷ word (δ ∷ έ ∷ ξ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.6.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.6.1"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "2Cor.6.2"
∷ word (γ ∷ ά ∷ ρ ∷ []) "2Cor.6.2"
∷ word (Κ ∷ α ∷ ι ∷ ρ ∷ ῷ ∷ []) "2Cor.6.2"
∷ word (δ ∷ ε ∷ κ ∷ τ ∷ ῷ ∷ []) "2Cor.6.2"
∷ word (ἐ ∷ π ∷ ή ∷ κ ∷ ο ∷ υ ∷ σ ∷ ά ∷ []) "2Cor.6.2"
∷ word (σ ∷ ο ∷ υ ∷ []) "2Cor.6.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.2"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.2"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "2Cor.6.2"
∷ word (σ ∷ ω ∷ τ ∷ η ∷ ρ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.6.2"
∷ word (ἐ ∷ β ∷ ο ∷ ή ∷ θ ∷ η ∷ σ ∷ ά ∷ []) "2Cor.6.2"
∷ word (σ ∷ ο ∷ ι ∷ []) "2Cor.6.2"
∷ word (ἰ ∷ δ ∷ ο ∷ ὺ ∷ []) "2Cor.6.2"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "2Cor.6.2"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.6.2"
∷ word (ε ∷ ὐ ∷ π ∷ ρ ∷ ό ∷ σ ∷ δ ∷ ε ∷ κ ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.6.2"
∷ word (ἰ ∷ δ ∷ ο ∷ ὺ ∷ []) "2Cor.6.2"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "2Cor.6.2"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ []) "2Cor.6.2"
∷ word (σ ∷ ω ∷ τ ∷ η ∷ ρ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.6.2"
∷ word (μ ∷ η ∷ δ ∷ ε ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.6.3"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.3"
∷ word (μ ∷ η ∷ δ ∷ ε ∷ ν ∷ ὶ ∷ []) "2Cor.6.3"
∷ word (δ ∷ ι ∷ δ ∷ ό ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.6.3"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ κ ∷ ο ∷ π ∷ ή ∷ ν ∷ []) "2Cor.6.3"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.6.3"
∷ word (μ ∷ ὴ ∷ []) "2Cor.6.3"
∷ word (μ ∷ ω ∷ μ ∷ η ∷ θ ∷ ῇ ∷ []) "2Cor.6.3"
∷ word (ἡ ∷ []) "2Cor.6.3"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ []) "2Cor.6.3"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.6.4"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.4"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "2Cor.6.4"
∷ word (σ ∷ υ ∷ ν ∷ ι ∷ σ ∷ τ ∷ ά ∷ ν ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.6.4"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.6.4"
∷ word (ὡ ∷ ς ∷ []) "2Cor.6.4"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.6.4"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.6.4"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.4"
∷ word (ὑ ∷ π ∷ ο ∷ μ ∷ ο ∷ ν ∷ ῇ ∷ []) "2Cor.6.4"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῇ ∷ []) "2Cor.6.4"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.4"
∷ word (θ ∷ ∙λ ∷ ί ∷ ψ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.6.4"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.4"
∷ word (ἀ ∷ ν ∷ ά ∷ γ ∷ κ ∷ α ∷ ι ∷ ς ∷ []) "2Cor.6.4"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.4"
∷ word (σ ∷ τ ∷ ε ∷ ν ∷ ο ∷ χ ∷ ω ∷ ρ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.6.4"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.5"
∷ word (π ∷ ∙λ ∷ η ∷ γ ∷ α ∷ ῖ ∷ ς ∷ []) "2Cor.6.5"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.5"
∷ word (φ ∷ υ ∷ ∙λ ∷ α ∷ κ ∷ α ∷ ῖ ∷ ς ∷ []) "2Cor.6.5"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.5"
∷ word (ἀ ∷ κ ∷ α ∷ τ ∷ α ∷ σ ∷ τ ∷ α ∷ σ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.6.5"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.5"
∷ word (κ ∷ ό ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.6.5"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.5"
∷ word (ἀ ∷ γ ∷ ρ ∷ υ ∷ π ∷ ν ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.6.5"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.5"
∷ word (ν ∷ η ∷ σ ∷ τ ∷ ε ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.6.5"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.6"
∷ word (ἁ ∷ γ ∷ ν ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "2Cor.6.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.6"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.6.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.6"
∷ word (μ ∷ α ∷ κ ∷ ρ ∷ ο ∷ θ ∷ υ ∷ μ ∷ ί ∷ ᾳ ∷ []) "2Cor.6.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.6"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "2Cor.6.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.6"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "2Cor.6.6"
∷ word (ἁ ∷ γ ∷ ί ∷ ῳ ∷ []) "2Cor.6.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.6"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "2Cor.6.6"
∷ word (ἀ ∷ ν ∷ υ ∷ π ∷ ο ∷ κ ∷ ρ ∷ ί ∷ τ ∷ ῳ ∷ []) "2Cor.6.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.7"
∷ word (∙λ ∷ ό ∷ γ ∷ ῳ ∷ []) "2Cor.6.7"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "2Cor.6.7"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.7"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ []) "2Cor.6.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.6.7"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.6.7"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.6.7"
∷ word (ὅ ∷ π ∷ ∙λ ∷ ω ∷ ν ∷ []) "2Cor.6.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.6.7"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "2Cor.6.7"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.6.7"
∷ word (δ ∷ ε ∷ ξ ∷ ι ∷ ῶ ∷ ν ∷ []) "2Cor.6.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.7"
∷ word (ἀ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ε ∷ ρ ∷ ῶ ∷ ν ∷ []) "2Cor.6.7"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.6.8"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "2Cor.6.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.8"
∷ word (ἀ ∷ τ ∷ ι ∷ μ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.6.8"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.6.8"
∷ word (δ ∷ υ ∷ σ ∷ φ ∷ η ∷ μ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.6.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.8"
∷ word (ε ∷ ὐ ∷ φ ∷ η ∷ μ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.6.8"
∷ word (ὡ ∷ ς ∷ []) "2Cor.6.8"
∷ word (π ∷ ∙λ ∷ ά ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.6.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.8"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.6.8"
∷ word (ὡ ∷ ς ∷ []) "2Cor.6.9"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.6.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.9"
∷ word (ἐ ∷ π ∷ ι ∷ γ ∷ ι ∷ ν ∷ ω ∷ σ ∷ κ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.6.9"
∷ word (ὡ ∷ ς ∷ []) "2Cor.6.9"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ ν ∷ ῄ ∷ σ ∷ κ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.6.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.9"
∷ word (ἰ ∷ δ ∷ ο ∷ ὺ ∷ []) "2Cor.6.9"
∷ word (ζ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.6.9"
∷ word (ὡ ∷ ς ∷ []) "2Cor.6.9"
∷ word (π ∷ α ∷ ι ∷ δ ∷ ε ∷ υ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.6.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.9"
∷ word (μ ∷ ὴ ∷ []) "2Cor.6.9"
∷ word (θ ∷ α ∷ ν ∷ α ∷ τ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.6.9"
∷ word (ὡ ∷ ς ∷ []) "2Cor.6.10"
∷ word (∙λ ∷ υ ∷ π ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.6.10"
∷ word (ἀ ∷ ε ∷ ὶ ∷ []) "2Cor.6.10"
∷ word (δ ∷ ὲ ∷ []) "2Cor.6.10"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.6.10"
∷ word (ὡ ∷ ς ∷ []) "2Cor.6.10"
∷ word (π ∷ τ ∷ ω ∷ χ ∷ ο ∷ ὶ ∷ []) "2Cor.6.10"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.6.10"
∷ word (δ ∷ ὲ ∷ []) "2Cor.6.10"
∷ word (π ∷ ∙λ ∷ ο ∷ υ ∷ τ ∷ ί ∷ ζ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.6.10"
∷ word (ὡ ∷ ς ∷ []) "2Cor.6.10"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ ν ∷ []) "2Cor.6.10"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.6.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.10"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "2Cor.6.10"
∷ word (κ ∷ α ∷ τ ∷ έ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.6.10"
∷ word (Τ ∷ ὸ ∷ []) "2Cor.6.11"
∷ word (σ ∷ τ ∷ ό ∷ μ ∷ α ∷ []) "2Cor.6.11"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.6.11"
∷ word (ἀ ∷ ν ∷ έ ∷ ῳ ∷ γ ∷ ε ∷ ν ∷ []) "2Cor.6.11"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.6.11"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.6.11"
∷ word (Κ ∷ ο ∷ ρ ∷ ί ∷ ν ∷ θ ∷ ι ∷ ο ∷ ι ∷ []) "2Cor.6.11"
∷ word (ἡ ∷ []) "2Cor.6.11"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ []) "2Cor.6.11"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.6.11"
∷ word (π ∷ ε ∷ π ∷ ∙λ ∷ ά ∷ τ ∷ υ ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "2Cor.6.11"
∷ word (ο ∷ ὐ ∷ []) "2Cor.6.12"
∷ word (σ ∷ τ ∷ ε ∷ ν ∷ ο ∷ χ ∷ ω ∷ ρ ∷ ε ∷ ῖ ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.6.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.12"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.6.12"
∷ word (σ ∷ τ ∷ ε ∷ ν ∷ ο ∷ χ ∷ ω ∷ ρ ∷ ε ∷ ῖ ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.6.12"
∷ word (δ ∷ ὲ ∷ []) "2Cor.6.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.12"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.6.12"
∷ word (σ ∷ π ∷ ∙λ ∷ ά ∷ γ ∷ χ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.6.12"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.6.12"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.6.13"
∷ word (δ ∷ ὲ ∷ []) "2Cor.6.13"
∷ word (α ∷ ὐ ∷ τ ∷ ὴ ∷ ν ∷ []) "2Cor.6.13"
∷ word (ἀ ∷ ν ∷ τ ∷ ι ∷ μ ∷ ι ∷ σ ∷ θ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.6.13"
∷ word (ὡ ∷ ς ∷ []) "2Cor.6.13"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.6.13"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "2Cor.6.13"
∷ word (π ∷ ∙λ ∷ α ∷ τ ∷ ύ ∷ ν ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "2Cor.6.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.13"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.6.13"
∷ word (Μ ∷ ὴ ∷ []) "2Cor.6.14"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.6.14"
∷ word (ἑ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ζ ∷ υ ∷ γ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.6.14"
∷ word (ἀ ∷ π ∷ ί ∷ σ ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.6.14"
∷ word (τ ∷ ί ∷ ς ∷ []) "2Cor.6.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.6.14"
∷ word (μ ∷ ε ∷ τ ∷ ο ∷ χ ∷ ὴ ∷ []) "2Cor.6.14"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ ῃ ∷ []) "2Cor.6.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.14"
∷ word (ἀ ∷ ν ∷ ο ∷ μ ∷ ί ∷ ᾳ ∷ []) "2Cor.6.14"
∷ word (ἢ ∷ []) "2Cor.6.14"
∷ word (τ ∷ ί ∷ ς ∷ []) "2Cor.6.14"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ί ∷ α ∷ []) "2Cor.6.14"
∷ word (φ ∷ ω ∷ τ ∷ ὶ ∷ []) "2Cor.6.14"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.6.14"
∷ word (σ ∷ κ ∷ ό ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.6.14"
∷ word (τ ∷ ί ∷ ς ∷ []) "2Cor.6.15"
∷ word (δ ∷ ὲ ∷ []) "2Cor.6.15"
∷ word (σ ∷ υ ∷ μ ∷ φ ∷ ώ ∷ ν ∷ η ∷ σ ∷ ι ∷ ς ∷ []) "2Cor.6.15"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.6.15"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.6.15"
∷ word (Β ∷ ε ∷ ∙λ ∷ ι ∷ ά ∷ ρ ∷ []) "2Cor.6.15"
∷ word (ἢ ∷ []) "2Cor.6.15"
∷ word (τ ∷ ί ∷ ς ∷ []) "2Cor.6.15"
∷ word (μ ∷ ε ∷ ρ ∷ ὶ ∷ ς ∷ []) "2Cor.6.15"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "2Cor.6.15"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "2Cor.6.15"
∷ word (ἀ ∷ π ∷ ί ∷ σ ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.6.15"
∷ word (τ ∷ ί ∷ ς ∷ []) "2Cor.6.16"
∷ word (δ ∷ ὲ ∷ []) "2Cor.6.16"
∷ word (σ ∷ υ ∷ γ ∷ κ ∷ α ∷ τ ∷ ά ∷ θ ∷ ε ∷ σ ∷ ι ∷ ς ∷ []) "2Cor.6.16"
∷ word (ν ∷ α ∷ ῷ ∷ []) "2Cor.6.16"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.6.16"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "2Cor.6.16"
∷ word (ε ∷ ἰ ∷ δ ∷ ώ ∷ ∙λ ∷ ω ∷ ν ∷ []) "2Cor.6.16"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.6.16"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.6.16"
∷ word (ν ∷ α ∷ ὸ ∷ ς ∷ []) "2Cor.6.16"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.6.16"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.6.16"
∷ word (ζ ∷ ῶ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.6.16"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "2Cor.6.16"
∷ word (ε ∷ ἶ ∷ π ∷ ε ∷ ν ∷ []) "2Cor.6.16"
∷ word (ὁ ∷ []) "2Cor.6.16"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.6.16"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.6.16"
∷ word (Ἐ ∷ ν ∷ ο ∷ ι ∷ κ ∷ ή ∷ σ ∷ ω ∷ []) "2Cor.6.16"
∷ word (ἐ ∷ ν ∷ []) "2Cor.6.16"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.6.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.16"
∷ word (ἐ ∷ μ ∷ π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ή ∷ σ ∷ ω ∷ []) "2Cor.6.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.16"
∷ word (ἔ ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.6.16"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.6.16"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "2Cor.6.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.16"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὶ ∷ []) "2Cor.6.16"
∷ word (ἔ ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ί ∷ []) "2Cor.6.16"
∷ word (μ ∷ ο ∷ υ ∷ []) "2Cor.6.16"
∷ word (∙λ ∷ α ∷ ό ∷ ς ∷ []) "2Cor.6.16"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "2Cor.6.17"
∷ word (ἐ ∷ ξ ∷ έ ∷ ∙λ ∷ θ ∷ α ∷ τ ∷ ε ∷ []) "2Cor.6.17"
∷ word (ἐ ∷ κ ∷ []) "2Cor.6.17"
∷ word (μ ∷ έ ∷ σ ∷ ο ∷ υ ∷ []) "2Cor.6.17"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.6.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.17"
∷ word (ἀ ∷ φ ∷ ο ∷ ρ ∷ ί ∷ σ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "2Cor.6.17"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "2Cor.6.17"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "2Cor.6.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.17"
∷ word (ἀ ∷ κ ∷ α ∷ θ ∷ ά ∷ ρ ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.6.17"
∷ word (μ ∷ ὴ ∷ []) "2Cor.6.17"
∷ word (ἅ ∷ π ∷ τ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.6.17"
∷ word (κ ∷ ἀ ∷ γ ∷ ὼ ∷ []) "2Cor.6.17"
∷ word (ε ∷ ἰ ∷ σ ∷ δ ∷ έ ∷ ξ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.6.17"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.6.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.18"
∷ word (ἔ ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.6.18"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.6.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.6.18"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "2Cor.6.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.18"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.6.18"
∷ word (ἔ ∷ σ ∷ ε ∷ σ ∷ θ ∷ έ ∷ []) "2Cor.6.18"
∷ word (μ ∷ ο ∷ ι ∷ []) "2Cor.6.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.6.18"
∷ word (υ ∷ ἱ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.6.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.6.18"
∷ word (θ ∷ υ ∷ γ ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "2Cor.6.18"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "2Cor.6.18"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "2Cor.6.18"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ο ∷ κ ∷ ρ ∷ ά ∷ τ ∷ ω ∷ ρ ∷ []) "2Cor.6.18"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ α ∷ ς ∷ []) "2Cor.7.1"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "2Cor.7.1"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.7.1"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "2Cor.7.1"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.7.1"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ί ∷ []) "2Cor.7.1"
∷ word (κ ∷ α ∷ θ ∷ α ∷ ρ ∷ ί ∷ σ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.7.1"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.7.1"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.7.1"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὸ ∷ ς ∷ []) "2Cor.7.1"
∷ word (μ ∷ ο ∷ ∙λ ∷ υ ∷ σ ∷ μ ∷ ο ∷ ῦ ∷ []) "2Cor.7.1"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "2Cor.7.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.7.1"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.7.1"
∷ word (ἐ ∷ π ∷ ι ∷ τ ∷ ε ∷ ∙λ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.7.1"
∷ word (ἁ ∷ γ ∷ ι ∷ ω ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "2Cor.7.1"
∷ word (ἐ ∷ ν ∷ []) "2Cor.7.1"
∷ word (φ ∷ ό ∷ β ∷ ῳ ∷ []) "2Cor.7.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.7.1"
∷ word (Χ ∷ ω ∷ ρ ∷ ή ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "2Cor.7.2"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.7.2"
∷ word (ο ∷ ὐ ∷ δ ∷ έ ∷ ν ∷ α ∷ []) "2Cor.7.2"
∷ word (ἠ ∷ δ ∷ ι ∷ κ ∷ ή ∷ σ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.7.2"
∷ word (ο ∷ ὐ ∷ δ ∷ έ ∷ ν ∷ α ∷ []) "2Cor.7.2"
∷ word (ἐ ∷ φ ∷ θ ∷ ε ∷ ί ∷ ρ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.7.2"
∷ word (ο ∷ ὐ ∷ δ ∷ έ ∷ ν ∷ α ∷ []) "2Cor.7.2"
∷ word (ἐ ∷ π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ ε ∷ κ ∷ τ ∷ ή ∷ σ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.7.2"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.7.3"
∷ word (κ ∷ α ∷ τ ∷ ά ∷ κ ∷ ρ ∷ ι ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.7.3"
∷ word (ο ∷ ὐ ∷ []) "2Cor.7.3"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "2Cor.7.3"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ ί ∷ ρ ∷ η ∷ κ ∷ α ∷ []) "2Cor.7.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.7.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.7.3"
∷ word (ἐ ∷ ν ∷ []) "2Cor.7.3"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "2Cor.7.3"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.7.3"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "2Cor.7.3"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.7.3"
∷ word (τ ∷ ὸ ∷ []) "2Cor.7.3"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ π ∷ ο ∷ θ ∷ α ∷ ν ∷ ε ∷ ῖ ∷ ν ∷ []) "2Cor.7.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.7.3"
∷ word (σ ∷ υ ∷ ζ ∷ ῆ ∷ ν ∷ []) "2Cor.7.3"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ή ∷ []) "2Cor.7.4"
∷ word (μ ∷ ο ∷ ι ∷ []) "2Cor.7.4"
∷ word (π ∷ α ∷ ρ ∷ ρ ∷ η ∷ σ ∷ ί ∷ α ∷ []) "2Cor.7.4"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.7.4"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.7.4"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ή ∷ []) "2Cor.7.4"
∷ word (μ ∷ ο ∷ ι ∷ []) "2Cor.7.4"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ σ ∷ ι ∷ ς ∷ []) "2Cor.7.4"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.7.4"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.4"
∷ word (π ∷ ε ∷ π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "2Cor.7.4"
∷ word (τ ∷ ῇ ∷ []) "2Cor.7.4"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.7.4"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.7.4"
∷ word (τ ∷ ῇ ∷ []) "2Cor.7.4"
∷ word (χ ∷ α ∷ ρ ∷ ᾷ ∷ []) "2Cor.7.4"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.7.4"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "2Cor.7.4"
∷ word (τ ∷ ῇ ∷ []) "2Cor.7.4"
∷ word (θ ∷ ∙λ ∷ ί ∷ ψ ∷ ε ∷ ι ∷ []) "2Cor.7.4"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.4"
∷ word (Κ ∷ α ∷ ὶ ∷ []) "2Cor.7.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.7.5"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ό ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.7.5"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.7.5"
∷ word (Μ ∷ α ∷ κ ∷ ε ∷ δ ∷ ο ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "2Cor.7.5"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.7.5"
∷ word (ἔ ∷ σ ∷ χ ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "2Cor.7.5"
∷ word (ἄ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.7.5"
∷ word (ἡ ∷ []) "2Cor.7.5"
∷ word (σ ∷ ὰ ∷ ρ ∷ ξ ∷ []) "2Cor.7.5"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.7.5"
∷ word (ἐ ∷ ν ∷ []) "2Cor.7.5"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "2Cor.7.5"
∷ word (θ ∷ ∙λ ∷ ι ∷ β ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.7.5"
∷ word (ἔ ∷ ξ ∷ ω ∷ θ ∷ ε ∷ ν ∷ []) "2Cor.7.5"
∷ word (μ ∷ ά ∷ χ ∷ α ∷ ι ∷ []) "2Cor.7.5"
∷ word (ἔ ∷ σ ∷ ω ∷ θ ∷ ε ∷ ν ∷ []) "2Cor.7.5"
∷ word (φ ∷ ό ∷ β ∷ ο ∷ ι ∷ []) "2Cor.7.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.7.6"
∷ word (ὁ ∷ []) "2Cor.7.6"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "2Cor.7.6"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.7.6"
∷ word (τ ∷ α ∷ π ∷ ε ∷ ι ∷ ν ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.7.6"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ κ ∷ ά ∷ ∙λ ∷ ε ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.7.6"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.7.6"
∷ word (ὁ ∷ []) "2Cor.7.6"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.7.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.7.6"
∷ word (τ ∷ ῇ ∷ []) "2Cor.7.6"
∷ word (π ∷ α ∷ ρ ∷ ο ∷ υ ∷ σ ∷ ί ∷ ᾳ ∷ []) "2Cor.7.6"
∷ word (Τ ∷ ί ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.7.6"
∷ word (ο ∷ ὐ ∷ []) "2Cor.7.7"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.7.7"
∷ word (δ ∷ ὲ ∷ []) "2Cor.7.7"
∷ word (ἐ ∷ ν ∷ []) "2Cor.7.7"
∷ word (τ ∷ ῇ ∷ []) "2Cor.7.7"
∷ word (π ∷ α ∷ ρ ∷ ο ∷ υ ∷ σ ∷ ί ∷ ᾳ ∷ []) "2Cor.7.7"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.7.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.7.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.7.7"
∷ word (ἐ ∷ ν ∷ []) "2Cor.7.7"
∷ word (τ ∷ ῇ ∷ []) "2Cor.7.7"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.7.7"
∷ word (ᾗ ∷ []) "2Cor.7.7"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ κ ∷ ∙λ ∷ ή ∷ θ ∷ η ∷ []) "2Cor.7.7"
∷ word (ἐ ∷ φ ∷ []) "2Cor.7.7"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.7.7"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ω ∷ ν ∷ []) "2Cor.7.7"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.7.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.7.7"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.7"
∷ word (ἐ ∷ π ∷ ι ∷ π ∷ ό ∷ θ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.7.7"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.7.7"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.7"
∷ word (ὀ ∷ δ ∷ υ ∷ ρ ∷ μ ∷ ό ∷ ν ∷ []) "2Cor.7.7"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.7.7"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.7"
∷ word (ζ ∷ ῆ ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.7.7"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.7.7"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "2Cor.7.7"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "2Cor.7.7"
∷ word (μ ∷ ε ∷ []) "2Cor.7.7"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.7.7"
∷ word (χ ∷ α ∷ ρ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.7.7"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.7.8"
∷ word (ε ∷ ἰ ∷ []) "2Cor.7.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.7.8"
∷ word (ἐ ∷ ∙λ ∷ ύ ∷ π ∷ η ∷ σ ∷ α ∷ []) "2Cor.7.8"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.7.8"
∷ word (ἐ ∷ ν ∷ []) "2Cor.7.8"
∷ word (τ ∷ ῇ ∷ []) "2Cor.7.8"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ῇ ∷ []) "2Cor.7.8"
∷ word (ο ∷ ὐ ∷ []) "2Cor.7.8"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ μ ∷ έ ∷ ∙λ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.7.8"
∷ word (ε ∷ ἰ ∷ []) "2Cor.7.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.7.8"
∷ word (μ ∷ ε ∷ τ ∷ ε ∷ μ ∷ ε ∷ ∙λ ∷ ό ∷ μ ∷ η ∷ ν ∷ []) "2Cor.7.8"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ω ∷ []) "2Cor.7.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.7.8"
∷ word (ἡ ∷ []) "2Cor.7.8"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ []) "2Cor.7.8"
∷ word (ἐ ∷ κ ∷ ε ∷ ί ∷ ν ∷ η ∷ []) "2Cor.7.8"
∷ word (ε ∷ ἰ ∷ []) "2Cor.7.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.7.8"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.7.8"
∷ word (ὥ ∷ ρ ∷ α ∷ ν ∷ []) "2Cor.7.8"
∷ word (ἐ ∷ ∙λ ∷ ύ ∷ π ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.7.8"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.7.8"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "2Cor.7.9"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ω ∷ []) "2Cor.7.9"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "2Cor.7.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.7.9"
∷ word (ἐ ∷ ∙λ ∷ υ ∷ π ∷ ή ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "2Cor.7.9"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.7.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.7.9"
∷ word (ἐ ∷ ∙λ ∷ υ ∷ π ∷ ή ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "2Cor.7.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.7.9"
∷ word (μ ∷ ε ∷ τ ∷ ά ∷ ν ∷ ο ∷ ι ∷ α ∷ ν ∷ []) "2Cor.7.9"
∷ word (ἐ ∷ ∙λ ∷ υ ∷ π ∷ ή ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "2Cor.7.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.7.9"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.7.9"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "2Cor.7.9"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.7.9"
∷ word (ἐ ∷ ν ∷ []) "2Cor.7.9"
∷ word (μ ∷ η ∷ δ ∷ ε ∷ ν ∷ ὶ ∷ []) "2Cor.7.9"
∷ word (ζ ∷ η ∷ μ ∷ ι ∷ ω ∷ θ ∷ ῆ ∷ τ ∷ ε ∷ []) "2Cor.7.9"
∷ word (ἐ ∷ ξ ∷ []) "2Cor.7.9"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.9"
∷ word (ἡ ∷ []) "2Cor.7.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.7.10"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.7.10"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "2Cor.7.10"
∷ word (∙λ ∷ ύ ∷ π ∷ η ∷ []) "2Cor.7.10"
∷ word (μ ∷ ε ∷ τ ∷ ά ∷ ν ∷ ο ∷ ι ∷ α ∷ ν ∷ []) "2Cor.7.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.7.10"
∷ word (σ ∷ ω ∷ τ ∷ η ∷ ρ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.7.10"
∷ word (ἀ ∷ μ ∷ ε ∷ τ ∷ α ∷ μ ∷ έ ∷ ∙λ ∷ η ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.7.10"
∷ word (ἐ ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "2Cor.7.10"
∷ word (ἡ ∷ []) "2Cor.7.10"
∷ word (δ ∷ ὲ ∷ []) "2Cor.7.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.7.10"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "2Cor.7.10"
∷ word (∙λ ∷ ύ ∷ π ∷ η ∷ []) "2Cor.7.10"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.7.10"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "2Cor.7.10"
∷ word (ἰ ∷ δ ∷ ο ∷ ὺ ∷ []) "2Cor.7.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.7.11"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "2Cor.7.11"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.7.11"
∷ word (τ ∷ ὸ ∷ []) "2Cor.7.11"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.7.11"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "2Cor.7.11"
∷ word (∙λ ∷ υ ∷ π ∷ η ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.7.11"
∷ word (π ∷ ό ∷ σ ∷ η ∷ ν ∷ []) "2Cor.7.11"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ι ∷ ρ ∷ γ ∷ ά ∷ σ ∷ α ∷ τ ∷ ο ∷ []) "2Cor.7.11"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.7.11"
∷ word (σ ∷ π ∷ ο ∷ υ ∷ δ ∷ ή ∷ ν ∷ []) "2Cor.7.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.7.11"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.7.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.7.11"
∷ word (ἀ ∷ γ ∷ α ∷ ν ∷ ά ∷ κ ∷ τ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.7.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.7.11"
∷ word (φ ∷ ό ∷ β ∷ ο ∷ ν ∷ []) "2Cor.7.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.7.11"
∷ word (ἐ ∷ π ∷ ι ∷ π ∷ ό ∷ θ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.7.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.7.11"
∷ word (ζ ∷ ῆ ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.7.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.7.11"
∷ word (ἐ ∷ κ ∷ δ ∷ ί ∷ κ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.7.11"
∷ word (ἐ ∷ ν ∷ []) "2Cor.7.11"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "2Cor.7.11"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ σ ∷ τ ∷ ή ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "2Cor.7.11"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.7.11"
∷ word (ἁ ∷ γ ∷ ν ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.7.11"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.7.11"
∷ word (τ ∷ ῷ ∷ []) "2Cor.7.11"
∷ word (π ∷ ρ ∷ ά ∷ γ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "2Cor.7.11"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "2Cor.7.12"
∷ word (ε ∷ ἰ ∷ []) "2Cor.7.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.7.12"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "2Cor.7.12"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.7.12"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "2Cor.7.12"
∷ word (ἕ ∷ ν ∷ ε ∷ κ ∷ ε ∷ ν ∷ []) "2Cor.7.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.7.12"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ή ∷ σ ∷ α ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.7.12"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "2Cor.7.12"
∷ word (ἕ ∷ ν ∷ ε ∷ κ ∷ ε ∷ ν ∷ []) "2Cor.7.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.7.12"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ η ∷ θ ∷ έ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.7.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.7.12"
∷ word (ἕ ∷ ν ∷ ε ∷ κ ∷ ε ∷ ν ∷ []) "2Cor.7.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.7.12"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ω ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.7.12"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.7.12"
∷ word (σ ∷ π ∷ ο ∷ υ ∷ δ ∷ ὴ ∷ ν ∷ []) "2Cor.7.12"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.12"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.7.12"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.7.12"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.12"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.7.12"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.7.12"
∷ word (ἐ ∷ ν ∷ ώ ∷ π ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.7.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.7.12"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.7.12"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.7.13"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.7.13"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ε ∷ κ ∷ ∙λ ∷ ή ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.7.13"
∷ word (Ἐ ∷ π ∷ ὶ ∷ []) "2Cor.7.13"
∷ word (δ ∷ ὲ ∷ []) "2Cor.7.13"
∷ word (τ ∷ ῇ ∷ []) "2Cor.7.13"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.7.13"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.13"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ ω ∷ ς ∷ []) "2Cor.7.13"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.7.13"
∷ word (ἐ ∷ χ ∷ ά ∷ ρ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.7.13"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.7.13"
∷ word (τ ∷ ῇ ∷ []) "2Cor.7.13"
∷ word (χ ∷ α ∷ ρ ∷ ᾷ ∷ []) "2Cor.7.13"
∷ word (Τ ∷ ί ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.7.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.7.13"
∷ word (ἀ ∷ ν ∷ α ∷ π ∷ έ ∷ π ∷ α ∷ υ ∷ τ ∷ α ∷ ι ∷ []) "2Cor.7.13"
∷ word (τ ∷ ὸ ∷ []) "2Cor.7.13"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "2Cor.7.13"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.7.13"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.7.13"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.7.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.7.14"
∷ word (ε ∷ ἴ ∷ []) "2Cor.7.14"
∷ word (τ ∷ ι ∷ []) "2Cor.7.14"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "2Cor.7.14"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.7.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.14"
∷ word (κ ∷ ε ∷ κ ∷ α ∷ ύ ∷ χ ∷ η ∷ μ ∷ α ∷ ι ∷ []) "2Cor.7.14"
∷ word (ο ∷ ὐ ∷ []) "2Cor.7.14"
∷ word (κ ∷ α ∷ τ ∷ ῃ ∷ σ ∷ χ ∷ ύ ∷ ν ∷ θ ∷ η ∷ ν ∷ []) "2Cor.7.14"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.7.14"
∷ word (ὡ ∷ ς ∷ []) "2Cor.7.14"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "2Cor.7.14"
∷ word (ἐ ∷ ν ∷ []) "2Cor.7.14"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ ᾳ ∷ []) "2Cor.7.14"
∷ word (ἐ ∷ ∙λ ∷ α ∷ ∙λ ∷ ή ∷ σ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.7.14"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.7.14"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "2Cor.7.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.7.14"
∷ word (ἡ ∷ []) "2Cor.7.14"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ σ ∷ ι ∷ ς ∷ []) "2Cor.7.14"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.14"
∷ word (ἡ ∷ []) "2Cor.7.14"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.7.14"
∷ word (Τ ∷ ί ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.7.14"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ []) "2Cor.7.14"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ή ∷ θ ∷ η ∷ []) "2Cor.7.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.7.15"
∷ word (τ ∷ ὰ ∷ []) "2Cor.7.15"
∷ word (σ ∷ π ∷ ∙λ ∷ ά ∷ γ ∷ χ ∷ ν ∷ α ∷ []) "2Cor.7.15"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.7.15"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ ω ∷ ς ∷ []) "2Cor.7.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.7.15"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.7.15"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "2Cor.7.15"
∷ word (ἀ ∷ ν ∷ α ∷ μ ∷ ι ∷ μ ∷ ν ∷ ῃ ∷ σ ∷ κ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ []) "2Cor.7.15"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.7.15"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.7.15"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.7.15"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ή ∷ ν ∷ []) "2Cor.7.15"
∷ word (ὡ ∷ ς ∷ []) "2Cor.7.15"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "2Cor.7.15"
∷ word (φ ∷ ό ∷ β ∷ ο ∷ υ ∷ []) "2Cor.7.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.7.15"
∷ word (τ ∷ ρ ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "2Cor.7.15"
∷ word (ἐ ∷ δ ∷ έ ∷ ξ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.7.15"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "2Cor.7.15"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ω ∷ []) "2Cor.7.16"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.7.16"
∷ word (ἐ ∷ ν ∷ []) "2Cor.7.16"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "2Cor.7.16"
∷ word (θ ∷ α ∷ ρ ∷ ρ ∷ ῶ ∷ []) "2Cor.7.16"
∷ word (ἐ ∷ ν ∷ []) "2Cor.7.16"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.7.16"
∷ word (Γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ ζ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.8.1"
∷ word (δ ∷ ὲ ∷ []) "2Cor.8.1"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.8.1"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "2Cor.8.1"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.8.1"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "2Cor.8.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.8.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.8.1"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.8.1"
∷ word (δ ∷ ε ∷ δ ∷ ο ∷ μ ∷ έ ∷ ν ∷ η ∷ ν ∷ []) "2Cor.8.1"
∷ word (ἐ ∷ ν ∷ []) "2Cor.8.1"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "2Cor.8.1"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.8.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.8.1"
∷ word (Μ ∷ α ∷ κ ∷ ε ∷ δ ∷ ο ∷ ν ∷ ί ∷ α ∷ ς ∷ []) "2Cor.8.1"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.8.2"
∷ word (ἐ ∷ ν ∷ []) "2Cor.8.2"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῇ ∷ []) "2Cor.8.2"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ῇ ∷ []) "2Cor.8.2"
∷ word (θ ∷ ∙λ ∷ ί ∷ ψ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.8.2"
∷ word (ἡ ∷ []) "2Cor.8.2"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ί ∷ α ∷ []) "2Cor.8.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.8.2"
∷ word (χ ∷ α ∷ ρ ∷ ᾶ ∷ ς ∷ []) "2Cor.8.2"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.8.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.2"
∷ word (ἡ ∷ []) "2Cor.8.2"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.8.2"
∷ word (β ∷ ά ∷ θ ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.8.2"
∷ word (π ∷ τ ∷ ω ∷ χ ∷ ε ∷ ί ∷ α ∷ []) "2Cor.8.2"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.8.2"
∷ word (ἐ ∷ π ∷ ε ∷ ρ ∷ ί ∷ σ ∷ σ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.8.2"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.8.2"
∷ word (τ ∷ ὸ ∷ []) "2Cor.8.2"
∷ word (π ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.8.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.8.2"
∷ word (ἁ ∷ π ∷ ∙λ ∷ ό ∷ τ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.8.2"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.8.2"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.8.3"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.8.3"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ν ∷ []) "2Cor.8.3"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ῶ ∷ []) "2Cor.8.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.3"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "2Cor.8.3"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ν ∷ []) "2Cor.8.3"
∷ word (α ∷ ὐ ∷ θ ∷ α ∷ ί ∷ ρ ∷ ε ∷ τ ∷ ο ∷ ι ∷ []) "2Cor.8.3"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "2Cor.8.4"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῆ ∷ ς ∷ []) "2Cor.8.4"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.8.4"
∷ word (δ ∷ ε ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.8.4"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.8.4"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "2Cor.8.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.8.4"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "2Cor.8.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.8.4"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ ς ∷ []) "2Cor.8.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.8.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.8.4"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.8.4"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.8.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.5"
∷ word (ο ∷ ὐ ∷ []) "2Cor.8.5"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "2Cor.8.5"
∷ word (ἠ ∷ ∙λ ∷ π ∷ ί ∷ σ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.8.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.8.5"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.8.5"
∷ word (ἔ ∷ δ ∷ ω ∷ κ ∷ α ∷ ν ∷ []) "2Cor.8.5"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.8.5"
∷ word (τ ∷ ῷ ∷ []) "2Cor.8.5"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "2Cor.8.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.5"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.8.5"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.8.5"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.8.5"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.8.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.8.6"
∷ word (τ ∷ ὸ ∷ []) "2Cor.8.6"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ έ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.8.6"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.8.6"
∷ word (Τ ∷ ί ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.8.6"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.8.6"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "2Cor.8.6"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ ν ∷ ή ∷ ρ ∷ ξ ∷ α ∷ τ ∷ ο ∷ []) "2Cor.8.6"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "2Cor.8.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.6"
∷ word (ἐ ∷ π ∷ ι ∷ τ ∷ ε ∷ ∙λ ∷ έ ∷ σ ∷ ῃ ∷ []) "2Cor.8.6"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.8.6"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.8.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.6"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.8.6"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "2Cor.8.6"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ η ∷ ν ∷ []) "2Cor.8.6"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.8.7"
∷ word (ὥ ∷ σ ∷ π ∷ ε ∷ ρ ∷ []) "2Cor.8.7"
∷ word (ἐ ∷ ν ∷ []) "2Cor.8.7"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "2Cor.8.7"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ ε ∷ τ ∷ ε ∷ []) "2Cor.8.7"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ι ∷ []) "2Cor.8.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.7"
∷ word (∙λ ∷ ό ∷ γ ∷ ῳ ∷ []) "2Cor.8.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.7"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.8.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.7"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "2Cor.8.7"
∷ word (σ ∷ π ∷ ο ∷ υ ∷ δ ∷ ῇ ∷ []) "2Cor.8.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.7"
∷ word (τ ∷ ῇ ∷ []) "2Cor.8.7"
∷ word (ἐ ∷ ξ ∷ []) "2Cor.8.7"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.7"
∷ word (ἐ ∷ ν ∷ []) "2Cor.8.7"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.8.7"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "2Cor.8.7"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.8.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.7"
∷ word (ἐ ∷ ν ∷ []) "2Cor.8.7"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ ῃ ∷ []) "2Cor.8.7"
∷ word (τ ∷ ῇ ∷ []) "2Cor.8.7"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ι ∷ []) "2Cor.8.7"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ η ∷ τ ∷ ε ∷ []) "2Cor.8.7"
∷ word (Ο ∷ ὐ ∷ []) "2Cor.8.8"
∷ word (κ ∷ α ∷ τ ∷ []) "2Cor.8.8"
∷ word (ἐ ∷ π ∷ ι ∷ τ ∷ α ∷ γ ∷ ὴ ∷ ν ∷ []) "2Cor.8.8"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "2Cor.8.8"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.8.8"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.8.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.8.8"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ ω ∷ ν ∷ []) "2Cor.8.8"
∷ word (σ ∷ π ∷ ο ∷ υ ∷ δ ∷ ῆ ∷ ς ∷ []) "2Cor.8.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.8"
∷ word (τ ∷ ὸ ∷ []) "2Cor.8.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.8.8"
∷ word (ὑ ∷ μ ∷ ε ∷ τ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "2Cor.8.8"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ς ∷ []) "2Cor.8.8"
∷ word (γ ∷ ν ∷ ή ∷ σ ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.8.8"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ά ∷ ζ ∷ ω ∷ ν ∷ []) "2Cor.8.8"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ε ∷ τ ∷ ε ∷ []) "2Cor.8.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.8.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.8.9"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "2Cor.8.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.8.9"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.8.9"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.9"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "2Cor.8.9"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.8.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.8.9"
∷ word (δ ∷ ι ∷ []) "2Cor.8.9"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.8.9"
∷ word (ἐ ∷ π ∷ τ ∷ ώ ∷ χ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.8.9"
∷ word (π ∷ ∙λ ∷ ο ∷ ύ ∷ σ ∷ ι ∷ ο ∷ ς ∷ []) "2Cor.8.9"
∷ word (ὤ ∷ ν ∷ []) "2Cor.8.9"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.8.9"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.8.9"
∷ word (τ ∷ ῇ ∷ []) "2Cor.8.9"
∷ word (ἐ ∷ κ ∷ ε ∷ ί ∷ ν ∷ ο ∷ υ ∷ []) "2Cor.8.9"
∷ word (π ∷ τ ∷ ω ∷ χ ∷ ε ∷ ί ∷ ᾳ ∷ []) "2Cor.8.9"
∷ word (π ∷ ∙λ ∷ ο ∷ υ ∷ τ ∷ ή ∷ σ ∷ η ∷ τ ∷ ε ∷ []) "2Cor.8.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.10"
∷ word (γ ∷ ν ∷ ώ ∷ μ ∷ η ∷ ν ∷ []) "2Cor.8.10"
∷ word (ἐ ∷ ν ∷ []) "2Cor.8.10"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "2Cor.8.10"
∷ word (δ ∷ ί ∷ δ ∷ ω ∷ μ ∷ ι ∷ []) "2Cor.8.10"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.8.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.8.10"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.8.10"
∷ word (σ ∷ υ ∷ μ ∷ φ ∷ έ ∷ ρ ∷ ε ∷ ι ∷ []) "2Cor.8.10"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "2Cor.8.10"
∷ word (ο ∷ ὐ ∷ []) "2Cor.8.10"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.8.10"
∷ word (τ ∷ ὸ ∷ []) "2Cor.8.10"
∷ word (π ∷ ο ∷ ι ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.8.10"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.8.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.10"
∷ word (τ ∷ ὸ ∷ []) "2Cor.8.10"
∷ word (θ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ ν ∷ []) "2Cor.8.10"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ ν ∷ ή ∷ ρ ∷ ξ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.8.10"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.8.10"
∷ word (π ∷ έ ∷ ρ ∷ υ ∷ σ ∷ ι ∷ []) "2Cor.8.10"
∷ word (ν ∷ υ ∷ ν ∷ ὶ ∷ []) "2Cor.8.11"
∷ word (δ ∷ ὲ ∷ []) "2Cor.8.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.11"
∷ word (τ ∷ ὸ ∷ []) "2Cor.8.11"
∷ word (π ∷ ο ∷ ι ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.8.11"
∷ word (ἐ ∷ π ∷ ι ∷ τ ∷ ε ∷ ∙λ ∷ έ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "2Cor.8.11"
∷ word (ὅ ∷ π ∷ ω ∷ ς ∷ []) "2Cor.8.11"
∷ word (κ ∷ α ∷ θ ∷ ά ∷ π ∷ ε ∷ ρ ∷ []) "2Cor.8.11"
∷ word (ἡ ∷ []) "2Cor.8.11"
∷ word (π ∷ ρ ∷ ο ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ []) "2Cor.8.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.8.11"
∷ word (θ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ ν ∷ []) "2Cor.8.11"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "2Cor.8.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.11"
∷ word (τ ∷ ὸ ∷ []) "2Cor.8.11"
∷ word (ἐ ∷ π ∷ ι ∷ τ ∷ ε ∷ ∙λ ∷ έ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.8.11"
∷ word (ἐ ∷ κ ∷ []) "2Cor.8.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.8.11"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ ν ∷ []) "2Cor.8.11"
∷ word (ε ∷ ἰ ∷ []) "2Cor.8.12"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.8.12"
∷ word (ἡ ∷ []) "2Cor.8.12"
∷ word (π ∷ ρ ∷ ο ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ []) "2Cor.8.12"
∷ word (π ∷ ρ ∷ ό ∷ κ ∷ ε ∷ ι ∷ τ ∷ α ∷ ι ∷ []) "2Cor.8.12"
∷ word (κ ∷ α ∷ θ ∷ ὸ ∷ []) "2Cor.8.12"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "2Cor.8.12"
∷ word (ἔ ∷ χ ∷ ῃ ∷ []) "2Cor.8.12"
∷ word (ε ∷ ὐ ∷ π ∷ ρ ∷ ό ∷ σ ∷ δ ∷ ε ∷ κ ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.8.12"
∷ word (ο ∷ ὐ ∷ []) "2Cor.8.12"
∷ word (κ ∷ α ∷ θ ∷ ὸ ∷ []) "2Cor.8.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.8.12"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "2Cor.8.12"
∷ word (ο ∷ ὐ ∷ []) "2Cor.8.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.8.13"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.8.13"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.8.13"
∷ word (ἄ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ς ∷ []) "2Cor.8.13"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.8.13"
∷ word (θ ∷ ∙λ ∷ ῖ ∷ ψ ∷ ι ∷ ς ∷ []) "2Cor.8.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.8.13"
∷ word (ἐ ∷ ξ ∷ []) "2Cor.8.13"
∷ word (ἰ ∷ σ ∷ ό ∷ τ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.8.13"
∷ word (ἐ ∷ ν ∷ []) "2Cor.8.14"
∷ word (τ ∷ ῷ ∷ []) "2Cor.8.14"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "2Cor.8.14"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ῷ ∷ []) "2Cor.8.14"
∷ word (τ ∷ ὸ ∷ []) "2Cor.8.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.14"
∷ word (π ∷ ε ∷ ρ ∷ ί ∷ σ ∷ σ ∷ ε ∷ υ ∷ μ ∷ α ∷ []) "2Cor.8.14"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.8.14"
∷ word (τ ∷ ὸ ∷ []) "2Cor.8.14"
∷ word (ἐ ∷ κ ∷ ε ∷ ί ∷ ν ∷ ω ∷ ν ∷ []) "2Cor.8.14"
∷ word (ὑ ∷ σ ∷ τ ∷ έ ∷ ρ ∷ η ∷ μ ∷ α ∷ []) "2Cor.8.14"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.8.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.14"
∷ word (τ ∷ ὸ ∷ []) "2Cor.8.14"
∷ word (ἐ ∷ κ ∷ ε ∷ ί ∷ ν ∷ ω ∷ ν ∷ []) "2Cor.8.14"
∷ word (π ∷ ε ∷ ρ ∷ ί ∷ σ ∷ σ ∷ ε ∷ υ ∷ μ ∷ α ∷ []) "2Cor.8.14"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "2Cor.8.14"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.8.14"
∷ word (τ ∷ ὸ ∷ []) "2Cor.8.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.14"
∷ word (ὑ ∷ σ ∷ τ ∷ έ ∷ ρ ∷ η ∷ μ ∷ α ∷ []) "2Cor.8.14"
∷ word (ὅ ∷ π ∷ ω ∷ ς ∷ []) "2Cor.8.14"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "2Cor.8.14"
∷ word (ἰ ∷ σ ∷ ό ∷ τ ∷ η ∷ ς ∷ []) "2Cor.8.14"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "2Cor.8.15"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "2Cor.8.15"
∷ word (Ὁ ∷ []) "2Cor.8.15"
∷ word (τ ∷ ὸ ∷ []) "2Cor.8.15"
∷ word (π ∷ ο ∷ ∙λ ∷ ὺ ∷ []) "2Cor.8.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.8.15"
∷ word (ἐ ∷ π ∷ ∙λ ∷ ε ∷ ό ∷ ν ∷ α ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.8.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.15"
∷ word (ὁ ∷ []) "2Cor.8.15"
∷ word (τ ∷ ὸ ∷ []) "2Cor.8.15"
∷ word (ὀ ∷ ∙λ ∷ ί ∷ γ ∷ ο ∷ ν ∷ []) "2Cor.8.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.8.15"
∷ word (ἠ ∷ ∙λ ∷ α ∷ τ ∷ τ ∷ ό ∷ ν ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.8.15"
∷ word (Χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "2Cor.8.16"
∷ word (δ ∷ ὲ ∷ []) "2Cor.8.16"
∷ word (τ ∷ ῷ ∷ []) "2Cor.8.16"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "2Cor.8.16"
∷ word (τ ∷ ῷ ∷ []) "2Cor.8.16"
∷ word (δ ∷ ι ∷ δ ∷ ό ∷ ν ∷ τ ∷ ι ∷ []) "2Cor.8.16"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.8.16"
∷ word (α ∷ ὐ ∷ τ ∷ ὴ ∷ ν ∷ []) "2Cor.8.16"
∷ word (σ ∷ π ∷ ο ∷ υ ∷ δ ∷ ὴ ∷ ν ∷ []) "2Cor.8.16"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.8.16"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.16"
∷ word (ἐ ∷ ν ∷ []) "2Cor.8.16"
∷ word (τ ∷ ῇ ∷ []) "2Cor.8.16"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ ᾳ ∷ []) "2Cor.8.16"
∷ word (Τ ∷ ί ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.8.16"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.8.17"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.8.17"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "2Cor.8.17"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.8.17"
∷ word (ἐ ∷ δ ∷ έ ∷ ξ ∷ α ∷ τ ∷ ο ∷ []) "2Cor.8.17"
∷ word (σ ∷ π ∷ ο ∷ υ ∷ δ ∷ α ∷ ι ∷ ό ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ς ∷ []) "2Cor.8.17"
∷ word (δ ∷ ὲ ∷ []) "2Cor.8.17"
∷ word (ὑ ∷ π ∷ ά ∷ ρ ∷ χ ∷ ω ∷ ν ∷ []) "2Cor.8.17"
∷ word (α ∷ ὐ ∷ θ ∷ α ∷ ί ∷ ρ ∷ ε ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.8.17"
∷ word (ἐ ∷ ξ ∷ ῆ ∷ ∙λ ∷ θ ∷ ε ∷ ν ∷ []) "2Cor.8.17"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.8.17"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.8.17"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ π ∷ έ ∷ μ ∷ ψ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.8.18"
∷ word (δ ∷ ὲ ∷ []) "2Cor.8.18"
∷ word (μ ∷ ε ∷ τ ∷ []) "2Cor.8.18"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.8.18"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.8.18"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "2Cor.8.18"
∷ word (ο ∷ ὗ ∷ []) "2Cor.8.18"
∷ word (ὁ ∷ []) "2Cor.8.18"
∷ word (ἔ ∷ π ∷ α ∷ ι ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.8.18"
∷ word (ἐ ∷ ν ∷ []) "2Cor.8.18"
∷ word (τ ∷ ῷ ∷ []) "2Cor.8.18"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ῳ ∷ []) "2Cor.8.18"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.8.18"
∷ word (π ∷ α ∷ σ ∷ ῶ ∷ ν ∷ []) "2Cor.8.18"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.8.18"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ι ∷ ῶ ∷ ν ∷ []) "2Cor.8.18"
∷ word (ο ∷ ὐ ∷ []) "2Cor.8.19"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.8.19"
∷ word (δ ∷ ὲ ∷ []) "2Cor.8.19"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.8.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.19"
∷ word (χ ∷ ε ∷ ι ∷ ρ ∷ ο ∷ τ ∷ ο ∷ ν ∷ η ∷ θ ∷ ε ∷ ὶ ∷ ς ∷ []) "2Cor.8.19"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "2Cor.8.19"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.8.19"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ι ∷ ῶ ∷ ν ∷ []) "2Cor.8.19"
∷ word (σ ∷ υ ∷ ν ∷ έ ∷ κ ∷ δ ∷ η ∷ μ ∷ ο ∷ ς ∷ []) "2Cor.8.19"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.19"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "2Cor.8.19"
∷ word (τ ∷ ῇ ∷ []) "2Cor.8.19"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ι ∷ []) "2Cor.8.19"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ ῃ ∷ []) "2Cor.8.19"
∷ word (τ ∷ ῇ ∷ []) "2Cor.8.19"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ ῃ ∷ []) "2Cor.8.19"
∷ word (ὑ ∷ φ ∷ []) "2Cor.8.19"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.19"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.8.19"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.8.19"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.8.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.8.19"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.8.19"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "2Cor.8.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.19"
∷ word (π ∷ ρ ∷ ο ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.8.19"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.19"
∷ word (σ ∷ τ ∷ ε ∷ ∙λ ∷ ∙λ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.8.20"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.8.20"
∷ word (μ ∷ ή ∷ []) "2Cor.8.20"
∷ word (τ ∷ ι ∷ ς ∷ []) "2Cor.8.20"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.8.20"
∷ word (μ ∷ ω ∷ μ ∷ ή ∷ σ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "2Cor.8.20"
∷ word (ἐ ∷ ν ∷ []) "2Cor.8.20"
∷ word (τ ∷ ῇ ∷ []) "2Cor.8.20"
∷ word (ἁ ∷ δ ∷ ρ ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "2Cor.8.20"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ ῃ ∷ []) "2Cor.8.20"
∷ word (τ ∷ ῇ ∷ []) "2Cor.8.20"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ ῃ ∷ []) "2Cor.8.20"
∷ word (ὑ ∷ φ ∷ []) "2Cor.8.20"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.20"
∷ word (π ∷ ρ ∷ ο ∷ ν ∷ ο ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.8.21"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.8.21"
∷ word (κ ∷ α ∷ ∙λ ∷ ὰ ∷ []) "2Cor.8.21"
∷ word (ο ∷ ὐ ∷ []) "2Cor.8.21"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.8.21"
∷ word (ἐ ∷ ν ∷ ώ ∷ π ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.8.21"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.8.21"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.8.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.21"
∷ word (ἐ ∷ ν ∷ ώ ∷ π ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.8.21"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "2Cor.8.21"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ π ∷ έ ∷ μ ∷ ψ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.8.22"
∷ word (δ ∷ ὲ ∷ []) "2Cor.8.22"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.8.22"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.8.22"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "2Cor.8.22"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.22"
∷ word (ὃ ∷ ν ∷ []) "2Cor.8.22"
∷ word (ἐ ∷ δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ά ∷ σ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.8.22"
∷ word (ἐ ∷ ν ∷ []) "2Cor.8.22"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.8.22"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ά ∷ κ ∷ ι ∷ ς ∷ []) "2Cor.8.22"
∷ word (σ ∷ π ∷ ο ∷ υ ∷ δ ∷ α ∷ ῖ ∷ ο ∷ ν ∷ []) "2Cor.8.22"
∷ word (ὄ ∷ ν ∷ τ ∷ α ∷ []) "2Cor.8.22"
∷ word (ν ∷ υ ∷ ν ∷ ὶ ∷ []) "2Cor.8.22"
∷ word (δ ∷ ὲ ∷ []) "2Cor.8.22"
∷ word (π ∷ ο ∷ ∙λ ∷ ὺ ∷ []) "2Cor.8.22"
∷ word (σ ∷ π ∷ ο ∷ υ ∷ δ ∷ α ∷ ι ∷ ό ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "2Cor.8.22"
∷ word (π ∷ ε ∷ π ∷ ο ∷ ι ∷ θ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.8.22"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῇ ∷ []) "2Cor.8.22"
∷ word (τ ∷ ῇ ∷ []) "2Cor.8.22"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.8.22"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.8.22"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.8.23"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.8.23"
∷ word (Τ ∷ ί ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.8.23"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ὸ ∷ ς ∷ []) "2Cor.8.23"
∷ word (ἐ ∷ μ ∷ ὸ ∷ ς ∷ []) "2Cor.8.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.23"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.8.23"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.8.23"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ό ∷ ς ∷ []) "2Cor.8.23"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.8.23"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ὶ ∷ []) "2Cor.8.23"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.23"
∷ word (ἀ ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ι ∷ []) "2Cor.8.23"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ι ∷ ῶ ∷ ν ∷ []) "2Cor.8.23"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "2Cor.8.23"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.8.23"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.8.24"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "2Cor.8.24"
∷ word (ἔ ∷ ν ∷ δ ∷ ε ∷ ι ∷ ξ ∷ ι ∷ ν ∷ []) "2Cor.8.24"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.8.24"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ς ∷ []) "2Cor.8.24"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.8.24"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.24"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.8.24"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.8.24"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.8.24"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.8.24"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.8.24"
∷ word (ἐ ∷ ν ∷ δ ∷ ε ∷ ι ∷ κ ∷ ν ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.8.24"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.8.24"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "2Cor.8.24"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.8.24"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ι ∷ ῶ ∷ ν ∷ []) "2Cor.8.24"
∷ word (Π ∷ ε ∷ ρ ∷ ὶ ∷ []) "2Cor.9.1"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "2Cor.9.1"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.9.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.9.1"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ ς ∷ []) "2Cor.9.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.9.1"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.9.1"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.9.1"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.9.1"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ό ∷ ν ∷ []) "2Cor.9.1"
∷ word (μ ∷ ο ∷ ί ∷ []) "2Cor.9.1"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "2Cor.9.1"
∷ word (τ ∷ ὸ ∷ []) "2Cor.9.1"
∷ word (γ ∷ ρ ∷ ά ∷ φ ∷ ε ∷ ι ∷ ν ∷ []) "2Cor.9.1"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.9.1"
∷ word (ο ∷ ἶ ∷ δ ∷ α ∷ []) "2Cor.9.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.9.2"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.9.2"
∷ word (π ∷ ρ ∷ ο ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.9.2"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.9.2"
∷ word (ἣ ∷ ν ∷ []) "2Cor.9.2"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.9.2"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.9.2"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ῶ ∷ μ ∷ α ∷ ι ∷ []) "2Cor.9.2"
∷ word (Μ ∷ α ∷ κ ∷ ε ∷ δ ∷ ό ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.9.2"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.9.2"
∷ word (Ἀ ∷ χ ∷ α ∷ ΐ ∷ α ∷ []) "2Cor.9.2"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ σ ∷ κ ∷ ε ∷ ύ ∷ α ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "2Cor.9.2"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.9.2"
∷ word (π ∷ έ ∷ ρ ∷ υ ∷ σ ∷ ι ∷ []) "2Cor.9.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.2"
∷ word (τ ∷ ὸ ∷ []) "2Cor.9.2"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.9.2"
∷ word (ζ ∷ ῆ ∷ ∙λ ∷ ο ∷ ς ∷ []) "2Cor.9.2"
∷ word (ἠ ∷ ρ ∷ έ ∷ θ ∷ ι ∷ σ ∷ ε ∷ []) "2Cor.9.2"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.9.2"
∷ word (π ∷ ∙λ ∷ ε ∷ ί ∷ ο ∷ ν ∷ α ∷ ς ∷ []) "2Cor.9.2"
∷ word (ἔ ∷ π ∷ ε ∷ μ ∷ ψ ∷ α ∷ []) "2Cor.9.3"
∷ word (δ ∷ ὲ ∷ []) "2Cor.9.3"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.9.3"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ύ ∷ ς ∷ []) "2Cor.9.3"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.9.3"
∷ word (μ ∷ ὴ ∷ []) "2Cor.9.3"
∷ word (τ ∷ ὸ ∷ []) "2Cor.9.3"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ μ ∷ α ∷ []) "2Cor.9.3"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.9.3"
∷ word (τ ∷ ὸ ∷ []) "2Cor.9.3"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.9.3"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.9.3"
∷ word (κ ∷ ε ∷ ν ∷ ω ∷ θ ∷ ῇ ∷ []) "2Cor.9.3"
∷ word (ἐ ∷ ν ∷ []) "2Cor.9.3"
∷ word (τ ∷ ῷ ∷ []) "2Cor.9.3"
∷ word (μ ∷ έ ∷ ρ ∷ ε ∷ ι ∷ []) "2Cor.9.3"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "2Cor.9.3"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.9.3"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "2Cor.9.3"
∷ word (ἔ ∷ ∙λ ∷ ε ∷ γ ∷ ο ∷ ν ∷ []) "2Cor.9.3"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ σ ∷ κ ∷ ε ∷ υ ∷ α ∷ σ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.9.3"
∷ word (ἦ ∷ τ ∷ ε ∷ []) "2Cor.9.3"
∷ word (μ ∷ ή ∷ []) "2Cor.9.4"
∷ word (π ∷ ω ∷ ς ∷ []) "2Cor.9.4"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "2Cor.9.4"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.9.4"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "2Cor.9.4"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "2Cor.9.4"
∷ word (Μ ∷ α ∷ κ ∷ ε ∷ δ ∷ ό ∷ ν ∷ ε ∷ ς ∷ []) "2Cor.9.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.4"
∷ word (ε ∷ ὕ ∷ ρ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.9.4"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.9.4"
∷ word (ἀ ∷ π ∷ α ∷ ρ ∷ α ∷ σ ∷ κ ∷ ε ∷ υ ∷ ά ∷ σ ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.9.4"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ι ∷ σ ∷ χ ∷ υ ∷ ν ∷ θ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.9.4"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.9.4"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.9.4"
∷ word (μ ∷ ὴ ∷ []) "2Cor.9.4"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.9.4"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.9.4"
∷ word (ἐ ∷ ν ∷ []) "2Cor.9.4"
∷ word (τ ∷ ῇ ∷ []) "2Cor.9.4"
∷ word (ὑ ∷ π ∷ ο ∷ σ ∷ τ ∷ ά ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.9.4"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ ῃ ∷ []) "2Cor.9.4"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ κ ∷ α ∷ ῖ ∷ ο ∷ ν ∷ []) "2Cor.9.5"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "2Cor.9.5"
∷ word (ἡ ∷ γ ∷ η ∷ σ ∷ ά ∷ μ ∷ η ∷ ν ∷ []) "2Cor.9.5"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ έ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.9.5"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.9.5"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.9.5"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.9.5"
∷ word (π ∷ ρ ∷ ο ∷ έ ∷ ∙λ ∷ θ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.9.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.9.5"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.9.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.5"
∷ word (π ∷ ρ ∷ ο ∷ κ ∷ α ∷ τ ∷ α ∷ ρ ∷ τ ∷ ί ∷ σ ∷ ω ∷ σ ∷ ι ∷ []) "2Cor.9.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.9.5"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ π ∷ η ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ μ ∷ έ ∷ ν ∷ η ∷ ν ∷ []) "2Cor.9.5"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.9.5"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.9.5"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ η ∷ ν ∷ []) "2Cor.9.5"
∷ word (ἑ ∷ τ ∷ ο ∷ ί ∷ μ ∷ η ∷ ν ∷ []) "2Cor.9.5"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.9.5"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "2Cor.9.5"
∷ word (ὡ ∷ ς ∷ []) "2Cor.9.5"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.9.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.5"
∷ word (μ ∷ ὴ ∷ []) "2Cor.9.5"
∷ word (ὡ ∷ ς ∷ []) "2Cor.9.5"
∷ word (π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ ε ∷ ξ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.9.5"
∷ word (Τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.9.6"
∷ word (δ ∷ έ ∷ []) "2Cor.9.6"
∷ word (ὁ ∷ []) "2Cor.9.6"
∷ word (σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ ω ∷ ν ∷ []) "2Cor.9.6"
∷ word (φ ∷ ε ∷ ι ∷ δ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ω ∷ ς ∷ []) "2Cor.9.6"
∷ word (φ ∷ ε ∷ ι ∷ δ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ω ∷ ς ∷ []) "2Cor.9.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.6"
∷ word (θ ∷ ε ∷ ρ ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.9.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.6"
∷ word (ὁ ∷ []) "2Cor.9.6"
∷ word (σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ ω ∷ ν ∷ []) "2Cor.9.6"
∷ word (ἐ ∷ π ∷ []) "2Cor.9.6"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.9.6"
∷ word (ἐ ∷ π ∷ []) "2Cor.9.6"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.9.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.6"
∷ word (θ ∷ ε ∷ ρ ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.9.6"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.9.7"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "2Cor.9.7"
∷ word (π ∷ ρ ∷ ο ∷ ῄ ∷ ρ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "2Cor.9.7"
∷ word (τ ∷ ῇ ∷ []) "2Cor.9.7"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ ᾳ ∷ []) "2Cor.9.7"
∷ word (μ ∷ ὴ ∷ []) "2Cor.9.7"
∷ word (ἐ ∷ κ ∷ []) "2Cor.9.7"
∷ word (∙λ ∷ ύ ∷ π ∷ η ∷ ς ∷ []) "2Cor.9.7"
∷ word (ἢ ∷ []) "2Cor.9.7"
∷ word (ἐ ∷ ξ ∷ []) "2Cor.9.7"
∷ word (ἀ ∷ ν ∷ ά ∷ γ ∷ κ ∷ η ∷ ς ∷ []) "2Cor.9.7"
∷ word (ἱ ∷ ∙λ ∷ α ∷ ρ ∷ ὸ ∷ ν ∷ []) "2Cor.9.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.9.7"
∷ word (δ ∷ ό ∷ τ ∷ η ∷ ν ∷ []) "2Cor.9.7"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ᾷ ∷ []) "2Cor.9.7"
∷ word (ὁ ∷ []) "2Cor.9.7"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "2Cor.9.7"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ τ ∷ ε ∷ ῖ ∷ []) "2Cor.9.8"
∷ word (δ ∷ ὲ ∷ []) "2Cor.9.8"
∷ word (ὁ ∷ []) "2Cor.9.8"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.9.8"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "2Cor.9.8"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "2Cor.9.8"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ῦ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.9.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.9.8"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.9.8"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.9.8"
∷ word (ἐ ∷ ν ∷ []) "2Cor.9.8"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "2Cor.9.8"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ο ∷ τ ∷ ε ∷ []) "2Cor.9.8"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "2Cor.9.8"
∷ word (α ∷ ὐ ∷ τ ∷ ά ∷ ρ ∷ κ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "2Cor.9.8"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.9.8"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ η ∷ τ ∷ ε ∷ []) "2Cor.9.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.9.8"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "2Cor.9.8"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ν ∷ []) "2Cor.9.8"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ό ∷ ν ∷ []) "2Cor.9.8"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "2Cor.9.9"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "2Cor.9.9"
∷ word (Ἐ ∷ σ ∷ κ ∷ ό ∷ ρ ∷ π ∷ ι ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.9.9"
∷ word (ἔ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "2Cor.9.9"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.9.9"
∷ word (π ∷ έ ∷ ν ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.9.9"
∷ word (ἡ ∷ []) "2Cor.9.9"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ []) "2Cor.9.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.9.9"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "2Cor.9.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.9.9"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.9.9"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ α ∷ []) "2Cor.9.9"
∷ word (ὁ ∷ []) "2Cor.9.10"
∷ word (δ ∷ ὲ ∷ []) "2Cor.9.10"
∷ word (ἐ ∷ π ∷ ι ∷ χ ∷ ο ∷ ρ ∷ η ∷ γ ∷ ῶ ∷ ν ∷ []) "2Cor.9.10"
∷ word (σ ∷ π ∷ ό ∷ ρ ∷ ο ∷ ν ∷ []) "2Cor.9.10"
∷ word (τ ∷ ῷ ∷ []) "2Cor.9.10"
∷ word (σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ ο ∷ ν ∷ τ ∷ ι ∷ []) "2Cor.9.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.10"
∷ word (ἄ ∷ ρ ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.9.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.9.10"
∷ word (β ∷ ρ ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.9.10"
∷ word (χ ∷ ο ∷ ρ ∷ η ∷ γ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.9.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.10"
∷ word (π ∷ ∙λ ∷ η ∷ θ ∷ υ ∷ ν ∷ ε ∷ ῖ ∷ []) "2Cor.9.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.9.10"
∷ word (σ ∷ π ∷ ό ∷ ρ ∷ ο ∷ ν ∷ []) "2Cor.9.10"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.9.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.10"
∷ word (α ∷ ὐ ∷ ξ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.9.10"
∷ word (τ ∷ ὰ ∷ []) "2Cor.9.10"
∷ word (γ ∷ ε ∷ ν ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "2Cor.9.10"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.9.10"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "2Cor.9.10"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.9.10"
∷ word (ἐ ∷ ν ∷ []) "2Cor.9.11"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "2Cor.9.11"
∷ word (π ∷ ∙λ ∷ ο ∷ υ ∷ τ ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.9.11"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.9.11"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "2Cor.9.11"
∷ word (ἁ ∷ π ∷ ∙λ ∷ ό ∷ τ ∷ η ∷ τ ∷ α ∷ []) "2Cor.9.11"
∷ word (ἥ ∷ τ ∷ ι ∷ ς ∷ []) "2Cor.9.11"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "2Cor.9.11"
∷ word (δ ∷ ι ∷ []) "2Cor.9.11"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.9.11"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.9.11"
∷ word (τ ∷ ῷ ∷ []) "2Cor.9.11"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "2Cor.9.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.9.12"
∷ word (ἡ ∷ []) "2Cor.9.12"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ []) "2Cor.9.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.9.12"
∷ word (∙λ ∷ ε ∷ ι ∷ τ ∷ ο ∷ υ ∷ ρ ∷ γ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.9.12"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ η ∷ ς ∷ []) "2Cor.9.12"
∷ word (ο ∷ ὐ ∷ []) "2Cor.9.12"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.9.12"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "2Cor.9.12"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ α ∷ ν ∷ α ∷ π ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ῦ ∷ σ ∷ α ∷ []) "2Cor.9.12"
∷ word (τ ∷ ὰ ∷ []) "2Cor.9.12"
∷ word (ὑ ∷ σ ∷ τ ∷ ε ∷ ρ ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "2Cor.9.12"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.9.12"
∷ word (ἁ ∷ γ ∷ ί ∷ ω ∷ ν ∷ []) "2Cor.9.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.9.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.12"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ ο ∷ υ ∷ σ ∷ α ∷ []) "2Cor.9.12"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.9.12"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "2Cor.9.12"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ι ∷ ῶ ∷ ν ∷ []) "2Cor.9.12"
∷ word (τ ∷ ῷ ∷ []) "2Cor.9.12"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "2Cor.9.12"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.9.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.9.13"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ῆ ∷ ς ∷ []) "2Cor.9.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.9.13"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ ς ∷ []) "2Cor.9.13"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ η ∷ ς ∷ []) "2Cor.9.13"
∷ word (δ ∷ ο ∷ ξ ∷ ά ∷ ζ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.9.13"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.9.13"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "2Cor.9.13"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.9.13"
∷ word (τ ∷ ῇ ∷ []) "2Cor.9.13"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ α ∷ γ ∷ ῇ ∷ []) "2Cor.9.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.9.13"
∷ word (ὁ ∷ μ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.9.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.9.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.9.13"
∷ word (τ ∷ ὸ ∷ []) "2Cor.9.13"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.9.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.9.13"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.9.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.13"
∷ word (ἁ ∷ π ∷ ∙λ ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "2Cor.9.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.9.13"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ί ∷ α ∷ ς ∷ []) "2Cor.9.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.9.13"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.9.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.9.13"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "2Cor.9.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.9.14"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.9.14"
∷ word (δ ∷ ε ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.9.14"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.9.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.9.14"
∷ word (ἐ ∷ π ∷ ι ∷ π ∷ ο ∷ θ ∷ ο ∷ ύ ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.9.14"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.9.14"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.9.14"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.9.14"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ ά ∷ ∙λ ∷ ∙λ ∷ ο ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "2Cor.9.14"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "2Cor.9.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.9.14"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.9.14"
∷ word (ἐ ∷ φ ∷ []) "2Cor.9.14"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.9.14"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "2Cor.9.15"
∷ word (τ ∷ ῷ ∷ []) "2Cor.9.15"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "2Cor.9.15"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.9.15"
∷ word (τ ∷ ῇ ∷ []) "2Cor.9.15"
∷ word (ἀ ∷ ν ∷ ε ∷ κ ∷ δ ∷ ι ∷ η ∷ γ ∷ ή ∷ τ ∷ ῳ ∷ []) "2Cor.9.15"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.9.15"
∷ word (δ ∷ ω ∷ ρ ∷ ε ∷ ᾷ ∷ []) "2Cor.9.15"
∷ word (Α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "2Cor.10.1"
∷ word (δ ∷ ὲ ∷ []) "2Cor.10.1"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "2Cor.10.1"
∷ word (Π ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "2Cor.10.1"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "2Cor.10.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.10.1"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.10.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.10.1"
∷ word (π ∷ ρ ∷ α ∷ ΰ ∷ τ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.10.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.10.1"
∷ word (ἐ ∷ π ∷ ι ∷ ε ∷ ι ∷ κ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "2Cor.10.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.10.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.10.1"
∷ word (ὃ ∷ ς ∷ []) "2Cor.10.1"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.10.1"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "2Cor.10.1"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "2Cor.10.1"
∷ word (τ ∷ α ∷ π ∷ ε ∷ ι ∷ ν ∷ ὸ ∷ ς ∷ []) "2Cor.10.1"
∷ word (ἐ ∷ ν ∷ []) "2Cor.10.1"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.10.1"
∷ word (ἀ ∷ π ∷ ὼ ∷ ν ∷ []) "2Cor.10.1"
∷ word (δ ∷ ὲ ∷ []) "2Cor.10.1"
∷ word (θ ∷ α ∷ ρ ∷ ρ ∷ ῶ ∷ []) "2Cor.10.1"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.10.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.10.1"
∷ word (δ ∷ έ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.10.2"
∷ word (δ ∷ ὲ ∷ []) "2Cor.10.2"
∷ word (τ ∷ ὸ ∷ []) "2Cor.10.2"
∷ word (μ ∷ ὴ ∷ []) "2Cor.10.2"
∷ word (π ∷ α ∷ ρ ∷ ὼ ∷ ν ∷ []) "2Cor.10.2"
∷ word (θ ∷ α ∷ ρ ∷ ρ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.10.2"
∷ word (τ ∷ ῇ ∷ []) "2Cor.10.2"
∷ word (π ∷ ε ∷ π ∷ ο ∷ ι ∷ θ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.10.2"
∷ word (ᾗ ∷ []) "2Cor.10.2"
∷ word (∙λ ∷ ο ∷ γ ∷ ί ∷ ζ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.10.2"
∷ word (τ ∷ ο ∷ ∙λ ∷ μ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.10.2"
∷ word (ἐ ∷ π ∷ ί ∷ []) "2Cor.10.2"
∷ word (τ ∷ ι ∷ ν ∷ α ∷ ς ∷ []) "2Cor.10.2"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.10.2"
∷ word (∙λ ∷ ο ∷ γ ∷ ι ∷ ζ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.10.2"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.10.2"
∷ word (ὡ ∷ ς ∷ []) "2Cor.10.2"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.10.2"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "2Cor.10.2"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "2Cor.10.2"
∷ word (ἐ ∷ ν ∷ []) "2Cor.10.3"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "2Cor.10.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.10.3"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.10.3"
∷ word (ο ∷ ὐ ∷ []) "2Cor.10.3"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.10.3"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "2Cor.10.3"
∷ word (σ ∷ τ ∷ ρ ∷ α ∷ τ ∷ ε ∷ υ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.10.3"
∷ word (τ ∷ ὰ ∷ []) "2Cor.10.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.10.4"
∷ word (ὅ ∷ π ∷ ∙λ ∷ α ∷ []) "2Cor.10.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.10.4"
∷ word (σ ∷ τ ∷ ρ ∷ α ∷ τ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "2Cor.10.4"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.10.4"
∷ word (ο ∷ ὐ ∷ []) "2Cor.10.4"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ι ∷ κ ∷ ὰ ∷ []) "2Cor.10.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.10.4"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.10.4"
∷ word (τ ∷ ῷ ∷ []) "2Cor.10.4"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "2Cor.10.4"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.10.4"
∷ word (κ ∷ α ∷ θ ∷ α ∷ ί ∷ ρ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.10.4"
∷ word (ὀ ∷ χ ∷ υ ∷ ρ ∷ ω ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.10.4"
∷ word (∙λ ∷ ο ∷ γ ∷ ι ∷ σ ∷ μ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.10.4"
∷ word (κ ∷ α ∷ θ ∷ α ∷ ι ∷ ρ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.10.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.10.5"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "2Cor.10.5"
∷ word (ὕ ∷ ψ ∷ ω ∷ μ ∷ α ∷ []) "2Cor.10.5"
∷ word (ἐ ∷ π ∷ α ∷ ι ∷ ρ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.10.5"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.10.5"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.10.5"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.10.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.10.5"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.10.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.10.5"
∷ word (α ∷ ἰ ∷ χ ∷ μ ∷ α ∷ ∙λ ∷ ω ∷ τ ∷ ί ∷ ζ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.10.5"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "2Cor.10.5"
∷ word (ν ∷ ό ∷ η ∷ μ ∷ α ∷ []) "2Cor.10.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.10.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.10.5"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ὴ ∷ ν ∷ []) "2Cor.10.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.10.5"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.10.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.10.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.10.6"
∷ word (ἑ ∷ τ ∷ ο ∷ ί ∷ μ ∷ ῳ ∷ []) "2Cor.10.6"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.10.6"
∷ word (ἐ ∷ κ ∷ δ ∷ ι ∷ κ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.10.6"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "2Cor.10.6"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ο ∷ ή ∷ ν ∷ []) "2Cor.10.6"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "2Cor.10.6"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ω ∷ θ ∷ ῇ ∷ []) "2Cor.10.6"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.10.6"
∷ word (ἡ ∷ []) "2Cor.10.6"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ή ∷ []) "2Cor.10.6"
∷ word (Τ ∷ ὰ ∷ []) "2Cor.10.7"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.10.7"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "2Cor.10.7"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ε ∷ τ ∷ ε ∷ []) "2Cor.10.7"
∷ word (ε ∷ ἴ ∷ []) "2Cor.10.7"
∷ word (τ ∷ ι ∷ ς ∷ []) "2Cor.10.7"
∷ word (π ∷ έ ∷ π ∷ ο ∷ ι ∷ θ ∷ ε ∷ ν ∷ []) "2Cor.10.7"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "2Cor.10.7"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.10.7"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.10.7"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.10.7"
∷ word (∙λ ∷ ο ∷ γ ∷ ι ∷ ζ ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "2Cor.10.7"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "2Cor.10.7"
∷ word (ἐ ∷ φ ∷ []) "2Cor.10.7"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.10.7"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.10.7"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "2Cor.10.7"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "2Cor.10.7"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.10.7"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "2Cor.10.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.10.7"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.10.7"
∷ word (ἐ ∷ ά ∷ ν ∷ []) "2Cor.10.8"
∷ word (τ ∷ ε ∷ []) "2Cor.10.8"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.10.8"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ό ∷ τ ∷ ε ∷ ρ ∷ ό ∷ ν ∷ []) "2Cor.10.8"
∷ word (τ ∷ ι ∷ []) "2Cor.10.8"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "2Cor.10.8"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "2Cor.10.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.10.8"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.10.8"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.10.8"
∷ word (ἧ ∷ ς ∷ []) "2Cor.10.8"
∷ word (ἔ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "2Cor.10.8"
∷ word (ὁ ∷ []) "2Cor.10.8"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "2Cor.10.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.10.8"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "2Cor.10.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.10.8"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.10.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.10.8"
∷ word (κ ∷ α ∷ θ ∷ α ∷ ί ∷ ρ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.10.8"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.10.8"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.10.8"
∷ word (α ∷ ἰ ∷ σ ∷ χ ∷ υ ∷ ν ∷ θ ∷ ή ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.10.8"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.10.9"
∷ word (μ ∷ ὴ ∷ []) "2Cor.10.9"
∷ word (δ ∷ ό ∷ ξ ∷ ω ∷ []) "2Cor.10.9"
∷ word (ὡ ∷ ς ∷ []) "2Cor.10.9"
∷ word (ἂ ∷ ν ∷ []) "2Cor.10.9"
∷ word (ἐ ∷ κ ∷ φ ∷ ο ∷ β ∷ ε ∷ ῖ ∷ ν ∷ []) "2Cor.10.9"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.10.9"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.10.9"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.10.9"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "2Cor.10.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.10.10"
∷ word (Α ∷ ἱ ∷ []) "2Cor.10.10"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ α ∷ ὶ ∷ []) "2Cor.10.10"
∷ word (μ ∷ έ ∷ ν ∷ []) "2Cor.10.10"
∷ word (φ ∷ η ∷ σ ∷ ί ∷ ν ∷ []) "2Cor.10.10"
∷ word (β ∷ α ∷ ρ ∷ ε ∷ ῖ ∷ α ∷ ι ∷ []) "2Cor.10.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.10.10"
∷ word (ἰ ∷ σ ∷ χ ∷ υ ∷ ρ ∷ α ∷ ί ∷ []) "2Cor.10.10"
∷ word (ἡ ∷ []) "2Cor.10.10"
∷ word (δ ∷ ὲ ∷ []) "2Cor.10.10"
∷ word (π ∷ α ∷ ρ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ []) "2Cor.10.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.10.10"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.10.10"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ὴ ∷ ς ∷ []) "2Cor.10.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.10.10"
∷ word (ὁ ∷ []) "2Cor.10.10"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "2Cor.10.10"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ θ ∷ ε ∷ ν ∷ η ∷ μ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.10.10"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.10.11"
∷ word (∙λ ∷ ο ∷ γ ∷ ι ∷ ζ ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "2Cor.10.11"
∷ word (ὁ ∷ []) "2Cor.10.11"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.10.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.10.11"
∷ word (ο ∷ ἷ ∷ ο ∷ ί ∷ []) "2Cor.10.11"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.10.11"
∷ word (τ ∷ ῷ ∷ []) "2Cor.10.11"
∷ word (∙λ ∷ ό ∷ γ ∷ ῳ ∷ []) "2Cor.10.11"
∷ word (δ ∷ ι ∷ []) "2Cor.10.11"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "2Cor.10.11"
∷ word (ἀ ∷ π ∷ ό ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.10.11"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ι ∷ []) "2Cor.10.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.10.11"
∷ word (π ∷ α ∷ ρ ∷ ό ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.10.11"
∷ word (τ ∷ ῷ ∷ []) "2Cor.10.11"
∷ word (ἔ ∷ ρ ∷ γ ∷ ῳ ∷ []) "2Cor.10.11"
∷ word (Ο ∷ ὐ ∷ []) "2Cor.10.12"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.10.12"
∷ word (τ ∷ ο ∷ ∙λ ∷ μ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.10.12"
∷ word (ἐ ∷ γ ∷ κ ∷ ρ ∷ ῖ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.10.12"
∷ word (ἢ ∷ []) "2Cor.10.12"
∷ word (σ ∷ υ ∷ γ ∷ κ ∷ ρ ∷ ῖ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.10.12"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ύ ∷ ς ∷ []) "2Cor.10.12"
∷ word (τ ∷ ι ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.10.12"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.10.12"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.10.12"
∷ word (σ ∷ υ ∷ ν ∷ ι ∷ σ ∷ τ ∷ α ∷ ν ∷ ό ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.10.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.10.12"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὶ ∷ []) "2Cor.10.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.10.12"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.10.12"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.10.12"
∷ word (μ ∷ ε ∷ τ ∷ ρ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.10.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.10.12"
∷ word (σ ∷ υ ∷ γ ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.10.12"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.10.12"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.10.12"
∷ word (ο ∷ ὐ ∷ []) "2Cor.10.12"
∷ word (σ ∷ υ ∷ ν ∷ ι ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.10.12"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.10.13"
∷ word (δ ∷ ὲ ∷ []) "2Cor.10.13"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.10.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.10.13"
∷ word (τ ∷ ὰ ∷ []) "2Cor.10.13"
∷ word (ἄ ∷ μ ∷ ε ∷ τ ∷ ρ ∷ α ∷ []) "2Cor.10.13"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ η ∷ σ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.10.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.10.13"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.10.13"
∷ word (τ ∷ ὸ ∷ []) "2Cor.10.13"
∷ word (μ ∷ έ ∷ τ ∷ ρ ∷ ο ∷ ν ∷ []) "2Cor.10.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.10.13"
∷ word (κ ∷ α ∷ ν ∷ ό ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.10.13"
∷ word (ο ∷ ὗ ∷ []) "2Cor.10.13"
∷ word (ἐ ∷ μ ∷ έ ∷ ρ ∷ ι ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.10.13"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.10.13"
∷ word (ὁ ∷ []) "2Cor.10.13"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.10.13"
∷ word (μ ∷ έ ∷ τ ∷ ρ ∷ ο ∷ υ ∷ []) "2Cor.10.13"
∷ word (ἐ ∷ φ ∷ ι ∷ κ ∷ έ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.10.13"
∷ word (ἄ ∷ χ ∷ ρ ∷ ι ∷ []) "2Cor.10.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.10.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.10.13"
∷ word (ο ∷ ὐ ∷ []) "2Cor.10.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.10.14"
∷ word (ὡ ∷ ς ∷ []) "2Cor.10.14"
∷ word (μ ∷ ὴ ∷ []) "2Cor.10.14"
∷ word (ἐ ∷ φ ∷ ι ∷ κ ∷ ν ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.10.14"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.10.14"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.10.14"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ ε ∷ κ ∷ τ ∷ ε ∷ ί ∷ ν ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.10.14"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ύ ∷ ς ∷ []) "2Cor.10.14"
∷ word (ἄ ∷ χ ∷ ρ ∷ ι ∷ []) "2Cor.10.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.10.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.10.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.10.14"
∷ word (ἐ ∷ φ ∷ θ ∷ ά ∷ σ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.10.14"
∷ word (ἐ ∷ ν ∷ []) "2Cor.10.14"
∷ word (τ ∷ ῷ ∷ []) "2Cor.10.14"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ῳ ∷ []) "2Cor.10.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.10.14"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.10.14"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.10.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.10.15"
∷ word (τ ∷ ὰ ∷ []) "2Cor.10.15"
∷ word (ἄ ∷ μ ∷ ε ∷ τ ∷ ρ ∷ α ∷ []) "2Cor.10.15"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ώ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.10.15"
∷ word (ἐ ∷ ν ∷ []) "2Cor.10.15"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ο ∷ τ ∷ ρ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.10.15"
∷ word (κ ∷ ό ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.10.15"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ α ∷ []) "2Cor.10.15"
∷ word (δ ∷ ὲ ∷ []) "2Cor.10.15"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.10.15"
∷ word (α ∷ ὐ ∷ ξ ∷ α ∷ ν ∷ ο ∷ μ ∷ έ ∷ ν ∷ η ∷ ς ∷ []) "2Cor.10.15"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.10.15"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.10.15"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.10.15"
∷ word (ἐ ∷ ν ∷ []) "2Cor.10.15"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.10.15"
∷ word (μ ∷ ε ∷ γ ∷ α ∷ ∙λ ∷ υ ∷ ν ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.10.15"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.10.15"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.10.15"
∷ word (κ ∷ α ∷ ν ∷ ό ∷ ν ∷ α ∷ []) "2Cor.10.15"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.10.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.10.15"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "2Cor.10.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.10.16"
∷ word (τ ∷ ὰ ∷ []) "2Cor.10.16"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ έ ∷ κ ∷ ε ∷ ι ∷ ν ∷ α ∷ []) "2Cor.10.16"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.10.16"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.10.16"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.10.16"
∷ word (ἐ ∷ ν ∷ []) "2Cor.10.16"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ο ∷ τ ∷ ρ ∷ ί ∷ ῳ ∷ []) "2Cor.10.16"
∷ word (κ ∷ α ∷ ν ∷ ό ∷ ν ∷ ι ∷ []) "2Cor.10.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.10.16"
∷ word (τ ∷ ὰ ∷ []) "2Cor.10.16"
∷ word (ἕ ∷ τ ∷ ο ∷ ι ∷ μ ∷ α ∷ []) "2Cor.10.16"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.10.16"
∷ word (Ὁ ∷ []) "2Cor.10.17"
∷ word (δ ∷ ὲ ∷ []) "2Cor.10.17"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ώ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.10.17"
∷ word (ἐ ∷ ν ∷ []) "2Cor.10.17"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "2Cor.10.17"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ά ∷ σ ∷ θ ∷ ω ∷ []) "2Cor.10.17"
∷ word (ο ∷ ὐ ∷ []) "2Cor.10.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.10.18"
∷ word (ὁ ∷ []) "2Cor.10.18"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "2Cor.10.18"
∷ word (σ ∷ υ ∷ ν ∷ ι ∷ σ ∷ τ ∷ ά ∷ ν ∷ ω ∷ ν ∷ []) "2Cor.10.18"
∷ word (ἐ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ ό ∷ ς ∷ []) "2Cor.10.18"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "2Cor.10.18"
∷ word (δ ∷ ό ∷ κ ∷ ι ∷ μ ∷ ο ∷ ς ∷ []) "2Cor.10.18"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.10.18"
∷ word (ὃ ∷ ν ∷ []) "2Cor.10.18"
∷ word (ὁ ∷ []) "2Cor.10.18"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "2Cor.10.18"
∷ word (σ ∷ υ ∷ ν ∷ ί ∷ σ ∷ τ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.10.18"
∷ word (Ὄ ∷ φ ∷ ε ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.11.1"
∷ word (ἀ ∷ ν ∷ ε ∷ ί ∷ χ ∷ ε ∷ σ ∷ θ ∷ έ ∷ []) "2Cor.11.1"
∷ word (μ ∷ ο ∷ υ ∷ []) "2Cor.11.1"
∷ word (μ ∷ ι ∷ κ ∷ ρ ∷ ό ∷ ν ∷ []) "2Cor.11.1"
∷ word (τ ∷ ι ∷ []) "2Cor.11.1"
∷ word (ἀ ∷ φ ∷ ρ ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "2Cor.11.1"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.11.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.1"
∷ word (ἀ ∷ ν ∷ έ ∷ χ ∷ ε ∷ σ ∷ θ ∷ έ ∷ []) "2Cor.11.1"
∷ word (μ ∷ ο ∷ υ ∷ []) "2Cor.11.1"
∷ word (ζ ∷ η ∷ ∙λ ∷ ῶ ∷ []) "2Cor.11.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.11.2"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.11.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.11.2"
∷ word (ζ ∷ ή ∷ ∙λ ∷ ῳ ∷ []) "2Cor.11.2"
∷ word (ἡ ∷ ρ ∷ μ ∷ ο ∷ σ ∷ ά ∷ μ ∷ η ∷ ν ∷ []) "2Cor.11.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.11.2"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.11.2"
∷ word (ἑ ∷ ν ∷ ὶ ∷ []) "2Cor.11.2"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ὶ ∷ []) "2Cor.11.2"
∷ word (π ∷ α ∷ ρ ∷ θ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "2Cor.11.2"
∷ word (ἁ ∷ γ ∷ ν ∷ ὴ ∷ ν ∷ []) "2Cor.11.2"
∷ word (π ∷ α ∷ ρ ∷ α ∷ σ ∷ τ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.11.2"
∷ word (τ ∷ ῷ ∷ []) "2Cor.11.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "2Cor.11.2"
∷ word (φ ∷ ο ∷ β ∷ ο ∷ ῦ ∷ μ ∷ α ∷ ι ∷ []) "2Cor.11.3"
∷ word (δ ∷ ὲ ∷ []) "2Cor.11.3"
∷ word (μ ∷ ή ∷ []) "2Cor.11.3"
∷ word (π ∷ ω ∷ ς ∷ []) "2Cor.11.3"
∷ word (ὡ ∷ ς ∷ []) "2Cor.11.3"
∷ word (ὁ ∷ []) "2Cor.11.3"
∷ word (ὄ ∷ φ ∷ ι ∷ ς ∷ []) "2Cor.11.3"
∷ word (ἐ ∷ ξ ∷ η ∷ π ∷ ά ∷ τ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.11.3"
∷ word (Ε ∷ ὕ ∷ α ∷ ν ∷ []) "2Cor.11.3"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.3"
∷ word (τ ∷ ῇ ∷ []) "2Cor.11.3"
∷ word (π ∷ α ∷ ν ∷ ο ∷ υ ∷ ρ ∷ γ ∷ ί ∷ ᾳ ∷ []) "2Cor.11.3"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.11.3"
∷ word (φ ∷ θ ∷ α ∷ ρ ∷ ῇ ∷ []) "2Cor.11.3"
∷ word (τ ∷ ὰ ∷ []) "2Cor.11.3"
∷ word (ν ∷ ο ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "2Cor.11.3"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.11.3"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.11.3"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.11.3"
∷ word (ἁ ∷ π ∷ ∙λ ∷ ό ∷ τ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.11.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.3"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.11.3"
∷ word (ἁ ∷ γ ∷ ν ∷ ό ∷ τ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.11.3"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.11.3"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.11.3"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.11.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ν ∷ []) "2Cor.11.3"
∷ word (ε ∷ ἰ ∷ []) "2Cor.11.4"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "2Cor.11.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.11.4"
∷ word (ὁ ∷ []) "2Cor.11.4"
∷ word (ἐ ∷ ρ ∷ χ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "2Cor.11.4"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.11.4"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "2Cor.11.4"
∷ word (κ ∷ η ∷ ρ ∷ ύ ∷ σ ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.11.4"
∷ word (ὃ ∷ ν ∷ []) "2Cor.11.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.11.4"
∷ word (ἐ ∷ κ ∷ η ∷ ρ ∷ ύ ∷ ξ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.11.4"
∷ word (ἢ ∷ []) "2Cor.11.4"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "2Cor.11.4"
∷ word (ἕ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "2Cor.11.4"
∷ word (∙λ ∷ α ∷ μ ∷ β ∷ ά ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "2Cor.11.4"
∷ word (ὃ ∷ []) "2Cor.11.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.11.4"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ β ∷ ε ∷ τ ∷ ε ∷ []) "2Cor.11.4"
∷ word (ἢ ∷ []) "2Cor.11.4"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.11.4"
∷ word (ἕ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "2Cor.11.4"
∷ word (ὃ ∷ []) "2Cor.11.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.11.4"
∷ word (ἐ ∷ δ ∷ έ ∷ ξ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.11.4"
∷ word (κ ∷ α ∷ ∙λ ∷ ῶ ∷ ς ∷ []) "2Cor.11.4"
∷ word (ἀ ∷ ν ∷ έ ∷ χ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.11.4"
∷ word (∙λ ∷ ο ∷ γ ∷ ί ∷ ζ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.11.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.11.5"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ ν ∷ []) "2Cor.11.5"
∷ word (ὑ ∷ σ ∷ τ ∷ ε ∷ ρ ∷ η ∷ κ ∷ έ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.11.5"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.11.5"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ ∙λ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.11.5"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ω ∷ ν ∷ []) "2Cor.11.5"
∷ word (ε ∷ ἰ ∷ []) "2Cor.11.6"
∷ word (δ ∷ ὲ ∷ []) "2Cor.11.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.6"
∷ word (ἰ ∷ δ ∷ ι ∷ ώ ∷ τ ∷ η ∷ ς ∷ []) "2Cor.11.6"
∷ word (τ ∷ ῷ ∷ []) "2Cor.11.6"
∷ word (∙λ ∷ ό ∷ γ ∷ ῳ ∷ []) "2Cor.11.6"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.11.6"
∷ word (ο ∷ ὐ ∷ []) "2Cor.11.6"
∷ word (τ ∷ ῇ ∷ []) "2Cor.11.6"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.11.6"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.11.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.6"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "2Cor.11.6"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ώ ∷ σ ∷ α ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.11.6"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.6"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.11.6"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.11.6"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.11.6"
∷ word (Ἢ ∷ []) "2Cor.11.7"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.11.7"
∷ word (ἐ ∷ π ∷ ο ∷ ί ∷ η ∷ σ ∷ α ∷ []) "2Cor.11.7"
∷ word (ἐ ∷ μ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "2Cor.11.7"
∷ word (τ ∷ α ∷ π ∷ ε ∷ ι ∷ ν ∷ ῶ ∷ ν ∷ []) "2Cor.11.7"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.11.7"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.11.7"
∷ word (ὑ ∷ ψ ∷ ω ∷ θ ∷ ῆ ∷ τ ∷ ε ∷ []) "2Cor.11.7"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.11.7"
∷ word (δ ∷ ω ∷ ρ ∷ ε ∷ ὰ ∷ ν ∷ []) "2Cor.11.7"
∷ word (τ ∷ ὸ ∷ []) "2Cor.11.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.11.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.11.7"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.11.7"
∷ word (ε ∷ ὐ ∷ η ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ι ∷ σ ∷ ά ∷ μ ∷ η ∷ ν ∷ []) "2Cor.11.7"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.11.7"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ α ∷ ς ∷ []) "2Cor.11.8"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.11.8"
∷ word (ἐ ∷ σ ∷ ύ ∷ ∙λ ∷ η ∷ σ ∷ α ∷ []) "2Cor.11.8"
∷ word (∙λ ∷ α ∷ β ∷ ὼ ∷ ν ∷ []) "2Cor.11.8"
∷ word (ὀ ∷ ψ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.11.8"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.11.8"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.11.8"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.11.8"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "2Cor.11.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.9"
∷ word (π ∷ α ∷ ρ ∷ ὼ ∷ ν ∷ []) "2Cor.11.9"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.11.9"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.11.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.9"
∷ word (ὑ ∷ σ ∷ τ ∷ ε ∷ ρ ∷ η ∷ θ ∷ ε ∷ ὶ ∷ ς ∷ []) "2Cor.11.9"
∷ word (ο ∷ ὐ ∷ []) "2Cor.11.9"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ν ∷ ά ∷ ρ ∷ κ ∷ η ∷ σ ∷ α ∷ []) "2Cor.11.9"
∷ word (ο ∷ ὐ ∷ θ ∷ ε ∷ ν ∷ ό ∷ ς ∷ []) "2Cor.11.9"
∷ word (τ ∷ ὸ ∷ []) "2Cor.11.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.11.9"
∷ word (ὑ ∷ σ ∷ τ ∷ έ ∷ ρ ∷ η ∷ μ ∷ ά ∷ []) "2Cor.11.9"
∷ word (μ ∷ ο ∷ υ ∷ []) "2Cor.11.9"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ α ∷ ν ∷ ε ∷ π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "2Cor.11.9"
∷ word (ο ∷ ἱ ∷ []) "2Cor.11.9"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ὶ ∷ []) "2Cor.11.9"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ό ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.11.9"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "2Cor.11.9"
∷ word (Μ ∷ α ∷ κ ∷ ε ∷ δ ∷ ο ∷ ν ∷ ί ∷ α ∷ ς ∷ []) "2Cor.11.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.9"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.9"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "2Cor.11.9"
∷ word (ἀ ∷ β ∷ α ∷ ρ ∷ ῆ ∷ []) "2Cor.11.9"
∷ word (ἐ ∷ μ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "2Cor.11.9"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.11.9"
∷ word (ἐ ∷ τ ∷ ή ∷ ρ ∷ η ∷ σ ∷ α ∷ []) "2Cor.11.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.9"
∷ word (τ ∷ η ∷ ρ ∷ ή ∷ σ ∷ ω ∷ []) "2Cor.11.9"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "2Cor.11.10"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ []) "2Cor.11.10"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.11.10"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.10"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "2Cor.11.10"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.11.10"
∷ word (ἡ ∷ []) "2Cor.11.10"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ σ ∷ ι ∷ ς ∷ []) "2Cor.11.10"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "2Cor.11.10"
∷ word (ο ∷ ὐ ∷ []) "2Cor.11.10"
∷ word (φ ∷ ρ ∷ α ∷ γ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "2Cor.11.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.11.10"
∷ word (ἐ ∷ μ ∷ ὲ ∷ []) "2Cor.11.10"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.10"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.11.10"
∷ word (κ ∷ ∙λ ∷ ί ∷ μ ∷ α ∷ σ ∷ ι ∷ []) "2Cor.11.10"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.11.10"
∷ word (Ἀ ∷ χ ∷ α ∷ ΐ ∷ α ∷ ς ∷ []) "2Cor.11.10"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.11.11"
∷ word (τ ∷ ί ∷ []) "2Cor.11.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.11.11"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.11.11"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ []) "2Cor.11.11"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.11.11"
∷ word (ὁ ∷ []) "2Cor.11.11"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.11.11"
∷ word (ο ∷ ἶ ∷ δ ∷ ε ∷ ν ∷ []) "2Cor.11.11"
∷ word (Ὃ ∷ []) "2Cor.11.12"
∷ word (δ ∷ ὲ ∷ []) "2Cor.11.12"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ []) "2Cor.11.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.12"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ ω ∷ []) "2Cor.11.12"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.11.12"
∷ word (ἐ ∷ κ ∷ κ ∷ ό ∷ ψ ∷ ω ∷ []) "2Cor.11.12"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.11.12"
∷ word (ἀ ∷ φ ∷ ο ∷ ρ ∷ μ ∷ ὴ ∷ ν ∷ []) "2Cor.11.12"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.11.12"
∷ word (θ ∷ ε ∷ ∙λ ∷ ό ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.11.12"
∷ word (ἀ ∷ φ ∷ ο ∷ ρ ∷ μ ∷ ή ∷ ν ∷ []) "2Cor.11.12"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.11.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.12"
∷ word (ᾧ ∷ []) "2Cor.11.12"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ῶ ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "2Cor.11.12"
∷ word (ε ∷ ὑ ∷ ρ ∷ ε ∷ θ ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.11.12"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "2Cor.11.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.12"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.11.12"
∷ word (ο ∷ ἱ ∷ []) "2Cor.11.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.11.13"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ι ∷ []) "2Cor.11.13"
∷ word (ψ ∷ ε ∷ υ ∷ δ ∷ α ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ι ∷ []) "2Cor.11.13"
∷ word (ἐ ∷ ρ ∷ γ ∷ ά ∷ τ ∷ α ∷ ι ∷ []) "2Cor.11.13"
∷ word (δ ∷ ό ∷ ∙λ ∷ ι ∷ ο ∷ ι ∷ []) "2Cor.11.13"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ σ ∷ χ ∷ η ∷ μ ∷ α ∷ τ ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.11.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.11.13"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.11.13"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.11.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.14"
∷ word (ο ∷ ὐ ∷ []) "2Cor.11.14"
∷ word (θ ∷ α ∷ ῦ ∷ μ ∷ α ∷ []) "2Cor.11.14"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "2Cor.11.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.11.14"
∷ word (ὁ ∷ []) "2Cor.11.14"
∷ word (Σ ∷ α ∷ τ ∷ α ∷ ν ∷ ᾶ ∷ ς ∷ []) "2Cor.11.14"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ σ ∷ χ ∷ η ∷ μ ∷ α ∷ τ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "2Cor.11.14"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.11.14"
∷ word (ἄ ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.11.14"
∷ word (φ ∷ ω ∷ τ ∷ ό ∷ ς ∷ []) "2Cor.11.14"
∷ word (ο ∷ ὐ ∷ []) "2Cor.11.15"
∷ word (μ ∷ έ ∷ γ ∷ α ∷ []) "2Cor.11.15"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "2Cor.11.15"
∷ word (ε ∷ ἰ ∷ []) "2Cor.11.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.15"
∷ word (ο ∷ ἱ ∷ []) "2Cor.11.15"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.11.15"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.11.15"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ σ ∷ χ ∷ η ∷ μ ∷ α ∷ τ ∷ ί ∷ ζ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "2Cor.11.15"
∷ word (ὡ ∷ ς ∷ []) "2Cor.11.15"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.11.15"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "2Cor.11.15"
∷ word (ὧ ∷ ν ∷ []) "2Cor.11.15"
∷ word (τ ∷ ὸ ∷ []) "2Cor.11.15"
∷ word (τ ∷ έ ∷ ∙λ ∷ ο ∷ ς ∷ []) "2Cor.11.15"
∷ word (ἔ ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "2Cor.11.15"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.11.15"
∷ word (τ ∷ ὰ ∷ []) "2Cor.11.15"
∷ word (ἔ ∷ ρ ∷ γ ∷ α ∷ []) "2Cor.11.15"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.11.15"
∷ word (Π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "2Cor.11.16"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "2Cor.11.16"
∷ word (μ ∷ ή ∷ []) "2Cor.11.16"
∷ word (τ ∷ ί ∷ ς ∷ []) "2Cor.11.16"
∷ word (μ ∷ ε ∷ []) "2Cor.11.16"
∷ word (δ ∷ ό ∷ ξ ∷ ῃ ∷ []) "2Cor.11.16"
∷ word (ἄ ∷ φ ∷ ρ ∷ ο ∷ ν ∷ α ∷ []) "2Cor.11.16"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "2Cor.11.16"
∷ word (ε ∷ ἰ ∷ []) "2Cor.11.16"
∷ word (δ ∷ ὲ ∷ []) "2Cor.11.16"
∷ word (μ ∷ ή ∷ []) "2Cor.11.16"
∷ word (γ ∷ ε ∷ []) "2Cor.11.16"
∷ word (κ ∷ ἂ ∷ ν ∷ []) "2Cor.11.16"
∷ word (ὡ ∷ ς ∷ []) "2Cor.11.16"
∷ word (ἄ ∷ φ ∷ ρ ∷ ο ∷ ν ∷ α ∷ []) "2Cor.11.16"
∷ word (δ ∷ έ ∷ ξ ∷ α ∷ σ ∷ θ ∷ έ ∷ []) "2Cor.11.16"
∷ word (μ ∷ ε ∷ []) "2Cor.11.16"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.11.16"
∷ word (κ ∷ ἀ ∷ γ ∷ ὼ ∷ []) "2Cor.11.16"
∷ word (μ ∷ ι ∷ κ ∷ ρ ∷ ό ∷ ν ∷ []) "2Cor.11.16"
∷ word (τ ∷ ι ∷ []) "2Cor.11.16"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "2Cor.11.16"
∷ word (ὃ ∷ []) "2Cor.11.17"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "2Cor.11.17"
∷ word (ο ∷ ὐ ∷ []) "2Cor.11.17"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.11.17"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.11.17"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "2Cor.11.17"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.11.17"
∷ word (ὡ ∷ ς ∷ []) "2Cor.11.17"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.17"
∷ word (ἀ ∷ φ ∷ ρ ∷ ο ∷ σ ∷ ύ ∷ ν ∷ ῃ ∷ []) "2Cor.11.17"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.17"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ ῃ ∷ []) "2Cor.11.17"
∷ word (τ ∷ ῇ ∷ []) "2Cor.11.17"
∷ word (ὑ ∷ π ∷ ο ∷ σ ∷ τ ∷ ά ∷ σ ∷ ε ∷ ι ∷ []) "2Cor.11.17"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.11.17"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.11.17"
∷ word (ἐ ∷ π ∷ ε ∷ ὶ ∷ []) "2Cor.11.18"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὶ ∷ []) "2Cor.11.18"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ῶ ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "2Cor.11.18"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.11.18"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "2Cor.11.18"
∷ word (κ ∷ ἀ ∷ γ ∷ ὼ ∷ []) "2Cor.11.18"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.11.18"
∷ word (ἡ ∷ δ ∷ έ ∷ ω ∷ ς ∷ []) "2Cor.11.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.11.19"
∷ word (ἀ ∷ ν ∷ έ ∷ χ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.11.19"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.11.19"
∷ word (ἀ ∷ φ ∷ ρ ∷ ό ∷ ν ∷ ω ∷ ν ∷ []) "2Cor.11.19"
∷ word (φ ∷ ρ ∷ ό ∷ ν ∷ ι ∷ μ ∷ ο ∷ ι ∷ []) "2Cor.11.19"
∷ word (ὄ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.11.19"
∷ word (ἀ ∷ ν ∷ έ ∷ χ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.11.20"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.11.20"
∷ word (ε ∷ ἴ ∷ []) "2Cor.11.20"
∷ word (τ ∷ ι ∷ ς ∷ []) "2Cor.11.20"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.11.20"
∷ word (κ ∷ α ∷ τ ∷ α ∷ δ ∷ ο ∷ υ ∷ ∙λ ∷ ο ∷ ῖ ∷ []) "2Cor.11.20"
∷ word (ε ∷ ἴ ∷ []) "2Cor.11.20"
∷ word (τ ∷ ι ∷ ς ∷ []) "2Cor.11.20"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ σ ∷ θ ∷ ί ∷ ε ∷ ι ∷ []) "2Cor.11.20"
∷ word (ε ∷ ἴ ∷ []) "2Cor.11.20"
∷ word (τ ∷ ι ∷ ς ∷ []) "2Cor.11.20"
∷ word (∙λ ∷ α ∷ μ ∷ β ∷ ά ∷ ν ∷ ε ∷ ι ∷ []) "2Cor.11.20"
∷ word (ε ∷ ἴ ∷ []) "2Cor.11.20"
∷ word (τ ∷ ι ∷ ς ∷ []) "2Cor.11.20"
∷ word (ἐ ∷ π ∷ α ∷ ί ∷ ρ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "2Cor.11.20"
∷ word (ε ∷ ἴ ∷ []) "2Cor.11.20"
∷ word (τ ∷ ι ∷ ς ∷ []) "2Cor.11.20"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.11.20"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "2Cor.11.20"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.11.20"
∷ word (δ ∷ έ ∷ ρ ∷ ε ∷ ι ∷ []) "2Cor.11.20"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.11.21"
∷ word (ἀ ∷ τ ∷ ι ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.11.21"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "2Cor.11.21"
∷ word (ὡ ∷ ς ∷ []) "2Cor.11.21"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.11.21"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.11.21"
∷ word (ἠ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ή ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.11.21"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.21"
∷ word (ᾧ ∷ []) "2Cor.11.21"
∷ word (δ ∷ []) "2Cor.11.21"
∷ word (ἄ ∷ ν ∷ []) "2Cor.11.21"
∷ word (τ ∷ ι ∷ ς ∷ []) "2Cor.11.21"
∷ word (τ ∷ ο ∷ ∙λ ∷ μ ∷ ᾷ ∷ []) "2Cor.11.21"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.21"
∷ word (ἀ ∷ φ ∷ ρ ∷ ο ∷ σ ∷ ύ ∷ ν ∷ ῃ ∷ []) "2Cor.11.21"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "2Cor.11.21"
∷ word (τ ∷ ο ∷ ∙λ ∷ μ ∷ ῶ ∷ []) "2Cor.11.21"
∷ word (κ ∷ ἀ ∷ γ ∷ ώ ∷ []) "2Cor.11.21"
∷ word (Ἑ ∷ β ∷ ρ ∷ α ∷ ῖ ∷ ο ∷ ί ∷ []) "2Cor.11.22"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.11.22"
∷ word (κ ∷ ἀ ∷ γ ∷ ώ ∷ []) "2Cor.11.22"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ η ∷ ∙λ ∷ ῖ ∷ τ ∷ α ∷ ί ∷ []) "2Cor.11.22"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.11.22"
∷ word (κ ∷ ἀ ∷ γ ∷ ώ ∷ []) "2Cor.11.22"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ []) "2Cor.11.22"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ά ∷ μ ∷ []) "2Cor.11.22"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.11.22"
∷ word (κ ∷ ἀ ∷ γ ∷ ώ ∷ []) "2Cor.11.22"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ι ∷ []) "2Cor.11.23"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.11.23"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.11.23"
∷ word (π ∷ α ∷ ρ ∷ α ∷ φ ∷ ρ ∷ ο ∷ ν ∷ ῶ ∷ ν ∷ []) "2Cor.11.23"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "2Cor.11.23"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.11.23"
∷ word (ἐ ∷ γ ∷ ώ ∷ []) "2Cor.11.23"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.23"
∷ word (κ ∷ ό ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.11.23"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ ω ∷ ς ∷ []) "2Cor.11.23"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.23"
∷ word (φ ∷ υ ∷ ∙λ ∷ α ∷ κ ∷ α ∷ ῖ ∷ ς ∷ []) "2Cor.11.23"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ ω ∷ ς ∷ []) "2Cor.11.23"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.23"
∷ word (π ∷ ∙λ ∷ η ∷ γ ∷ α ∷ ῖ ∷ ς ∷ []) "2Cor.11.23"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ α ∷ ∙λ ∷ ∙λ ∷ ό ∷ ν ∷ τ ∷ ω ∷ ς ∷ []) "2Cor.11.23"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.23"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.11.23"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ά ∷ κ ∷ ι ∷ ς ∷ []) "2Cor.11.23"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "2Cor.11.24"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ω ∷ ν ∷ []) "2Cor.11.24"
∷ word (π ∷ ε ∷ ν ∷ τ ∷ ά ∷ κ ∷ ι ∷ ς ∷ []) "2Cor.11.24"
∷ word (τ ∷ ε ∷ σ ∷ σ ∷ ε ∷ ρ ∷ ά ∷ κ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "2Cor.11.24"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "2Cor.11.24"
∷ word (μ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.11.24"
∷ word (ἔ ∷ ∙λ ∷ α ∷ β ∷ ο ∷ ν ∷ []) "2Cor.11.24"
∷ word (τ ∷ ρ ∷ ὶ ∷ ς ∷ []) "2Cor.11.25"
∷ word (ἐ ∷ ρ ∷ α ∷ β ∷ δ ∷ ί ∷ σ ∷ θ ∷ η ∷ ν ∷ []) "2Cor.11.25"
∷ word (ἅ ∷ π ∷ α ∷ ξ ∷ []) "2Cor.11.25"
∷ word (ἐ ∷ ∙λ ∷ ι ∷ θ ∷ ά ∷ σ ∷ θ ∷ η ∷ ν ∷ []) "2Cor.11.25"
∷ word (τ ∷ ρ ∷ ὶ ∷ ς ∷ []) "2Cor.11.25"
∷ word (ἐ ∷ ν ∷ α ∷ υ ∷ ά ∷ γ ∷ η ∷ σ ∷ α ∷ []) "2Cor.11.25"
∷ word (ν ∷ υ ∷ χ ∷ θ ∷ ή ∷ μ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "2Cor.11.25"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.25"
∷ word (τ ∷ ῷ ∷ []) "2Cor.11.25"
∷ word (β ∷ υ ∷ θ ∷ ῷ ∷ []) "2Cor.11.25"
∷ word (π ∷ ε ∷ π ∷ ο ∷ ί ∷ η ∷ κ ∷ α ∷ []) "2Cor.11.25"
∷ word (ὁ ∷ δ ∷ ο ∷ ι ∷ π ∷ ο ∷ ρ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.11.26"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ά ∷ κ ∷ ι ∷ ς ∷ []) "2Cor.11.26"
∷ word (κ ∷ ι ∷ ν ∷ δ ∷ ύ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.11.26"
∷ word (π ∷ ο ∷ τ ∷ α ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.11.26"
∷ word (κ ∷ ι ∷ ν ∷ δ ∷ ύ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.11.26"
∷ word (∙λ ∷ ῃ ∷ σ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.11.26"
∷ word (κ ∷ ι ∷ ν ∷ δ ∷ ύ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.11.26"
∷ word (ἐ ∷ κ ∷ []) "2Cor.11.26"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.11.26"
∷ word (κ ∷ ι ∷ ν ∷ δ ∷ ύ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.11.26"
∷ word (ἐ ∷ ξ ∷ []) "2Cor.11.26"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "2Cor.11.26"
∷ word (κ ∷ ι ∷ ν ∷ δ ∷ ύ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.11.26"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.26"
∷ word (π ∷ ό ∷ ∙λ ∷ ε ∷ ι ∷ []) "2Cor.11.26"
∷ word (κ ∷ ι ∷ ν ∷ δ ∷ ύ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.11.26"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.26"
∷ word (ἐ ∷ ρ ∷ η ∷ μ ∷ ί ∷ ᾳ ∷ []) "2Cor.11.26"
∷ word (κ ∷ ι ∷ ν ∷ δ ∷ ύ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.11.26"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.26"
∷ word (θ ∷ α ∷ ∙λ ∷ ά ∷ σ ∷ σ ∷ ῃ ∷ []) "2Cor.11.26"
∷ word (κ ∷ ι ∷ ν ∷ δ ∷ ύ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.11.26"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.26"
∷ word (ψ ∷ ε ∷ υ ∷ δ ∷ α ∷ δ ∷ έ ∷ ∙λ ∷ φ ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.11.26"
∷ word (κ ∷ ό ∷ π ∷ ῳ ∷ []) "2Cor.11.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.27"
∷ word (μ ∷ ό ∷ χ ∷ θ ∷ ῳ ∷ []) "2Cor.11.27"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.27"
∷ word (ἀ ∷ γ ∷ ρ ∷ υ ∷ π ∷ ν ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.11.27"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ά ∷ κ ∷ ι ∷ ς ∷ []) "2Cor.11.27"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.27"
∷ word (∙λ ∷ ι ∷ μ ∷ ῷ ∷ []) "2Cor.11.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.27"
∷ word (δ ∷ ί ∷ ψ ∷ ε ∷ ι ∷ []) "2Cor.11.27"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.27"
∷ word (ν ∷ η ∷ σ ∷ τ ∷ ε ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.11.27"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ά ∷ κ ∷ ι ∷ ς ∷ []) "2Cor.11.27"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.27"
∷ word (ψ ∷ ύ ∷ χ ∷ ε ∷ ι ∷ []) "2Cor.11.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.27"
∷ word (γ ∷ υ ∷ μ ∷ ν ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "2Cor.11.27"
∷ word (χ ∷ ω ∷ ρ ∷ ὶ ∷ ς ∷ []) "2Cor.11.28"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.11.28"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ κ ∷ τ ∷ ὸ ∷ ς ∷ []) "2Cor.11.28"
∷ word (ἡ ∷ []) "2Cor.11.28"
∷ word (ἐ ∷ π ∷ ί ∷ σ ∷ τ ∷ α ∷ σ ∷ ί ∷ ς ∷ []) "2Cor.11.28"
∷ word (μ ∷ ο ∷ ι ∷ []) "2Cor.11.28"
∷ word (ἡ ∷ []) "2Cor.11.28"
∷ word (κ ∷ α ∷ θ ∷ []) "2Cor.11.28"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "2Cor.11.28"
∷ word (ἡ ∷ []) "2Cor.11.28"
∷ word (μ ∷ έ ∷ ρ ∷ ι ∷ μ ∷ ν ∷ α ∷ []) "2Cor.11.28"
∷ word (π ∷ α ∷ σ ∷ ῶ ∷ ν ∷ []) "2Cor.11.28"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.11.28"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ι ∷ ῶ ∷ ν ∷ []) "2Cor.11.28"
∷ word (τ ∷ ί ∷ ς ∷ []) "2Cor.11.29"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ῖ ∷ []) "2Cor.11.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.29"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.11.29"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ῶ ∷ []) "2Cor.11.29"
∷ word (τ ∷ ί ∷ ς ∷ []) "2Cor.11.29"
∷ word (σ ∷ κ ∷ α ∷ ν ∷ δ ∷ α ∷ ∙λ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "2Cor.11.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.29"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.11.29"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "2Cor.11.29"
∷ word (π ∷ υ ∷ ρ ∷ ο ∷ ῦ ∷ μ ∷ α ∷ ι ∷ []) "2Cor.11.29"
∷ word (Ε ∷ ἰ ∷ []) "2Cor.11.30"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ᾶ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.11.30"
∷ word (δ ∷ ε ∷ ῖ ∷ []) "2Cor.11.30"
∷ word (τ ∷ ὰ ∷ []) "2Cor.11.30"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.11.30"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "2Cor.11.30"
∷ word (μ ∷ ο ∷ υ ∷ []) "2Cor.11.30"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.11.30"
∷ word (ὁ ∷ []) "2Cor.11.31"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.11.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.31"
∷ word (π ∷ α ∷ τ ∷ ὴ ∷ ρ ∷ []) "2Cor.11.31"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.11.31"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.11.31"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "2Cor.11.31"
∷ word (ο ∷ ἶ ∷ δ ∷ ε ∷ ν ∷ []) "2Cor.11.31"
∷ word (ὁ ∷ []) "2Cor.11.31"
∷ word (ὢ ∷ ν ∷ []) "2Cor.11.31"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ η ∷ τ ∷ ὸ ∷ ς ∷ []) "2Cor.11.31"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.11.31"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.11.31"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ α ∷ ς ∷ []) "2Cor.11.31"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.11.31"
∷ word (ο ∷ ὐ ∷ []) "2Cor.11.31"
∷ word (ψ ∷ ε ∷ ύ ∷ δ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.11.31"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.32"
∷ word (Δ ∷ α ∷ μ ∷ α ∷ σ ∷ κ ∷ ῷ ∷ []) "2Cor.11.32"
∷ word (ὁ ∷ []) "2Cor.11.32"
∷ word (ἐ ∷ θ ∷ ν ∷ ά ∷ ρ ∷ χ ∷ η ∷ ς ∷ []) "2Cor.11.32"
∷ word (Ἁ ∷ ρ ∷ έ ∷ τ ∷ α ∷ []) "2Cor.11.32"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.11.32"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ έ ∷ ω ∷ ς ∷ []) "2Cor.11.32"
∷ word (ἐ ∷ φ ∷ ρ ∷ ο ∷ ύ ∷ ρ ∷ ε ∷ ι ∷ []) "2Cor.11.32"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.11.32"
∷ word (π ∷ ό ∷ ∙λ ∷ ι ∷ ν ∷ []) "2Cor.11.32"
∷ word (Δ ∷ α ∷ μ ∷ α ∷ σ ∷ κ ∷ η ∷ ν ∷ ῶ ∷ ν ∷ []) "2Cor.11.32"
∷ word (π ∷ ι ∷ ά ∷ σ ∷ α ∷ ι ∷ []) "2Cor.11.32"
∷ word (μ ∷ ε ∷ []) "2Cor.11.32"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.33"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.11.33"
∷ word (θ ∷ υ ∷ ρ ∷ ί ∷ δ ∷ ο ∷ ς ∷ []) "2Cor.11.33"
∷ word (ἐ ∷ ν ∷ []) "2Cor.11.33"
∷ word (σ ∷ α ∷ ρ ∷ γ ∷ ά ∷ ν ∷ ῃ ∷ []) "2Cor.11.33"
∷ word (ἐ ∷ χ ∷ α ∷ ∙λ ∷ ά ∷ σ ∷ θ ∷ η ∷ ν ∷ []) "2Cor.11.33"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.11.33"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.11.33"
∷ word (τ ∷ ε ∷ ί ∷ χ ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.11.33"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.11.33"
∷ word (ἐ ∷ ξ ∷ έ ∷ φ ∷ υ ∷ γ ∷ ο ∷ ν ∷ []) "2Cor.11.33"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "2Cor.11.33"
∷ word (χ ∷ ε ∷ ῖ ∷ ρ ∷ α ∷ ς ∷ []) "2Cor.11.33"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.11.33"
∷ word (Κ ∷ α ∷ υ ∷ χ ∷ ᾶ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.12.1"
∷ word (δ ∷ ε ∷ ῖ ∷ []) "2Cor.12.1"
∷ word (ο ∷ ὐ ∷ []) "2Cor.12.1"
∷ word (σ ∷ υ ∷ μ ∷ φ ∷ έ ∷ ρ ∷ ο ∷ ν ∷ []) "2Cor.12.1"
∷ word (μ ∷ έ ∷ ν ∷ []) "2Cor.12.1"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.12.1"
∷ word (δ ∷ ὲ ∷ []) "2Cor.12.1"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.12.1"
∷ word (ὀ ∷ π ∷ τ ∷ α ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.12.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.1"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ ψ ∷ ε ∷ ι ∷ ς ∷ []) "2Cor.12.1"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.12.1"
∷ word (ο ∷ ἶ ∷ δ ∷ α ∷ []) "2Cor.12.2"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "2Cor.12.2"
∷ word (ἐ ∷ ν ∷ []) "2Cor.12.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "2Cor.12.2"
∷ word (π ∷ ρ ∷ ὸ ∷ []) "2Cor.12.2"
∷ word (ἐ ∷ τ ∷ ῶ ∷ ν ∷ []) "2Cor.12.2"
∷ word (δ ∷ ε ∷ κ ∷ α ∷ τ ∷ ε ∷ σ ∷ σ ∷ ά ∷ ρ ∷ ω ∷ ν ∷ []) "2Cor.12.2"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.12.2"
∷ word (ἐ ∷ ν ∷ []) "2Cor.12.2"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "2Cor.12.2"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.12.2"
∷ word (ο ∷ ἶ ∷ δ ∷ α ∷ []) "2Cor.12.2"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.12.2"
∷ word (ἐ ∷ κ ∷ τ ∷ ὸ ∷ ς ∷ []) "2Cor.12.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.12.2"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.12.2"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.12.2"
∷ word (ο ∷ ἶ ∷ δ ∷ α ∷ []) "2Cor.12.2"
∷ word (ὁ ∷ []) "2Cor.12.2"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.12.2"
∷ word (ο ∷ ἶ ∷ δ ∷ ε ∷ ν ∷ []) "2Cor.12.2"
∷ word (ἁ ∷ ρ ∷ π ∷ α ∷ γ ∷ έ ∷ ν ∷ τ ∷ α ∷ []) "2Cor.12.2"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.12.2"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.12.2"
∷ word (ἕ ∷ ω ∷ ς ∷ []) "2Cor.12.2"
∷ word (τ ∷ ρ ∷ ί ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.12.2"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ο ∷ ῦ ∷ []) "2Cor.12.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.3"
∷ word (ο ∷ ἶ ∷ δ ∷ α ∷ []) "2Cor.12.3"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.12.3"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.12.3"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "2Cor.12.3"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.12.3"
∷ word (ἐ ∷ ν ∷ []) "2Cor.12.3"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "2Cor.12.3"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "2Cor.12.3"
∷ word (χ ∷ ω ∷ ρ ∷ ὶ ∷ ς ∷ []) "2Cor.12.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.12.3"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.12.3"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.12.3"
∷ word (ο ∷ ἶ ∷ δ ∷ α ∷ []) "2Cor.12.3"
∷ word (ὁ ∷ []) "2Cor.12.3"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.12.3"
∷ word (ο ∷ ἶ ∷ δ ∷ ε ∷ ν ∷ []) "2Cor.12.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.12.4"
∷ word (ἡ ∷ ρ ∷ π ∷ ά ∷ γ ∷ η ∷ []) "2Cor.12.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.12.4"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.12.4"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ δ ∷ ε ∷ ι ∷ σ ∷ ο ∷ ν ∷ []) "2Cor.12.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.4"
∷ word (ἤ ∷ κ ∷ ο ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.12.4"
∷ word (ἄ ∷ ρ ∷ ρ ∷ η ∷ τ ∷ α ∷ []) "2Cor.12.4"
∷ word (ῥ ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "2Cor.12.4"
∷ word (ἃ ∷ []) "2Cor.12.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.12.4"
∷ word (ἐ ∷ ξ ∷ ὸ ∷ ν ∷ []) "2Cor.12.4"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ῳ ∷ []) "2Cor.12.4"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.12.4"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.12.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.12.5"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.12.5"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.12.5"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.12.5"
∷ word (δ ∷ ὲ ∷ []) "2Cor.12.5"
∷ word (ἐ ∷ μ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.12.5"
∷ word (ο ∷ ὐ ∷ []) "2Cor.12.5"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.12.5"
∷ word (ε ∷ ἰ ∷ []) "2Cor.12.5"
∷ word (μ ∷ ὴ ∷ []) "2Cor.12.5"
∷ word (ἐ ∷ ν ∷ []) "2Cor.12.5"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "2Cor.12.5"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.12.5"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "2Cor.12.6"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.12.6"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ σ ∷ ω ∷ []) "2Cor.12.6"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.12.6"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.12.6"
∷ word (ἔ ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.12.6"
∷ word (ἄ ∷ φ ∷ ρ ∷ ω ∷ ν ∷ []) "2Cor.12.6"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "2Cor.12.6"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.12.6"
∷ word (ἐ ∷ ρ ∷ ῶ ∷ []) "2Cor.12.6"
∷ word (φ ∷ ε ∷ ί ∷ δ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.12.6"
∷ word (δ ∷ έ ∷ []) "2Cor.12.6"
∷ word (μ ∷ ή ∷ []) "2Cor.12.6"
∷ word (τ ∷ ι ∷ ς ∷ []) "2Cor.12.6"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.12.6"
∷ word (ἐ ∷ μ ∷ ὲ ∷ []) "2Cor.12.6"
∷ word (∙λ ∷ ο ∷ γ ∷ ί ∷ σ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "2Cor.12.6"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.12.6"
∷ word (ὃ ∷ []) "2Cor.12.6"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ε ∷ ι ∷ []) "2Cor.12.6"
∷ word (μ ∷ ε ∷ []) "2Cor.12.6"
∷ word (ἢ ∷ []) "2Cor.12.6"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ ε ∷ ι ∷ []) "2Cor.12.6"
∷ word (τ ∷ ι ∷ []) "2Cor.12.6"
∷ word (ἐ ∷ ξ ∷ []) "2Cor.12.6"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "2Cor.12.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.7"
∷ word (τ ∷ ῇ ∷ []) "2Cor.12.7"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ ο ∷ ∙λ ∷ ῇ ∷ []) "2Cor.12.7"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.12.7"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ ψ ∷ ε ∷ ω ∷ ν ∷ []) "2Cor.12.7"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "2Cor.12.7"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.12.7"
∷ word (μ ∷ ὴ ∷ []) "2Cor.12.7"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ α ∷ ί ∷ ρ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "2Cor.12.7"
∷ word (ἐ ∷ δ ∷ ό ∷ θ ∷ η ∷ []) "2Cor.12.7"
∷ word (μ ∷ ο ∷ ι ∷ []) "2Cor.12.7"
∷ word (σ ∷ κ ∷ ό ∷ ∙λ ∷ ο ∷ ψ ∷ []) "2Cor.12.7"
∷ word (τ ∷ ῇ ∷ []) "2Cor.12.7"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ί ∷ []) "2Cor.12.7"
∷ word (ἄ ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ο ∷ ς ∷ []) "2Cor.12.7"
∷ word (Σ ∷ α ∷ τ ∷ α ∷ ν ∷ ᾶ ∷ []) "2Cor.12.7"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.12.7"
∷ word (μ ∷ ε ∷ []) "2Cor.12.7"
∷ word (κ ∷ ο ∷ ∙λ ∷ α ∷ φ ∷ ί ∷ ζ ∷ ῃ ∷ []) "2Cor.12.7"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.12.7"
∷ word (μ ∷ ὴ ∷ []) "2Cor.12.7"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ α ∷ ί ∷ ρ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "2Cor.12.7"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.12.8"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "2Cor.12.8"
∷ word (τ ∷ ρ ∷ ὶ ∷ ς ∷ []) "2Cor.12.8"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.12.8"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "2Cor.12.8"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ κ ∷ ά ∷ ∙λ ∷ ε ∷ σ ∷ α ∷ []) "2Cor.12.8"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.12.8"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ῇ ∷ []) "2Cor.12.8"
∷ word (ἀ ∷ π ∷ []) "2Cor.12.8"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "2Cor.12.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.9"
∷ word (ε ∷ ἴ ∷ ρ ∷ η ∷ κ ∷ έ ∷ ν ∷ []) "2Cor.12.9"
∷ word (μ ∷ ο ∷ ι ∷ []) "2Cor.12.9"
∷ word (Ἀ ∷ ρ ∷ κ ∷ ε ∷ ῖ ∷ []) "2Cor.12.9"
∷ word (σ ∷ ο ∷ ι ∷ []) "2Cor.12.9"
∷ word (ἡ ∷ []) "2Cor.12.9"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "2Cor.12.9"
∷ word (μ ∷ ο ∷ υ ∷ []) "2Cor.12.9"
∷ word (ἡ ∷ []) "2Cor.12.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.12.9"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ς ∷ []) "2Cor.12.9"
∷ word (ἐ ∷ ν ∷ []) "2Cor.12.9"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ί ∷ ᾳ ∷ []) "2Cor.12.9"
∷ word (τ ∷ ε ∷ ∙λ ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "2Cor.12.9"
∷ word (ἥ ∷ δ ∷ ι ∷ σ ∷ τ ∷ α ∷ []) "2Cor.12.9"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "2Cor.12.9"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.12.9"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.12.9"
∷ word (ἐ ∷ ν ∷ []) "2Cor.12.9"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "2Cor.12.9"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.12.9"
∷ word (μ ∷ ο ∷ υ ∷ []) "2Cor.12.9"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.12.9"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ κ ∷ η ∷ ν ∷ ώ ∷ σ ∷ ῃ ∷ []) "2Cor.12.9"
∷ word (ἐ ∷ π ∷ []) "2Cor.12.9"
∷ word (ἐ ∷ μ ∷ ὲ ∷ []) "2Cor.12.9"
∷ word (ἡ ∷ []) "2Cor.12.9"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ς ∷ []) "2Cor.12.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.12.9"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.12.9"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "2Cor.12.10"
∷ word (ε ∷ ὐ ∷ δ ∷ ο ∷ κ ∷ ῶ ∷ []) "2Cor.12.10"
∷ word (ἐ ∷ ν ∷ []) "2Cor.12.10"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.12.10"
∷ word (ἐ ∷ ν ∷ []) "2Cor.12.10"
∷ word (ὕ ∷ β ∷ ρ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.12.10"
∷ word (ἐ ∷ ν ∷ []) "2Cor.12.10"
∷ word (ἀ ∷ ν ∷ ά ∷ γ ∷ κ ∷ α ∷ ι ∷ ς ∷ []) "2Cor.12.10"
∷ word (ἐ ∷ ν ∷ []) "2Cor.12.10"
∷ word (δ ∷ ι ∷ ω ∷ γ ∷ μ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.12.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.10"
∷ word (σ ∷ τ ∷ ε ∷ ν ∷ ο ∷ χ ∷ ω ∷ ρ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "2Cor.12.10"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.12.10"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.12.10"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "2Cor.12.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.12.10"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ῶ ∷ []) "2Cor.12.10"
∷ word (τ ∷ ό ∷ τ ∷ ε ∷ []) "2Cor.12.10"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ τ ∷ ό ∷ ς ∷ []) "2Cor.12.10"
∷ word (ε ∷ ἰ ∷ μ ∷ ι ∷ []) "2Cor.12.10"
∷ word (Γ ∷ έ ∷ γ ∷ ο ∷ ν ∷ α ∷ []) "2Cor.12.11"
∷ word (ἄ ∷ φ ∷ ρ ∷ ω ∷ ν ∷ []) "2Cor.12.11"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.12.11"
∷ word (μ ∷ ε ∷ []) "2Cor.12.11"
∷ word (ἠ ∷ ν ∷ α ∷ γ ∷ κ ∷ ά ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "2Cor.12.11"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "2Cor.12.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.12.11"
∷ word (ὤ ∷ φ ∷ ε ∷ ι ∷ ∙λ ∷ ο ∷ ν ∷ []) "2Cor.12.11"
∷ word (ὑ ∷ φ ∷ []) "2Cor.12.11"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.12.11"
∷ word (σ ∷ υ ∷ ν ∷ ί ∷ σ ∷ τ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "2Cor.12.11"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ ν ∷ []) "2Cor.12.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.12.11"
∷ word (ὑ ∷ σ ∷ τ ∷ έ ∷ ρ ∷ η ∷ σ ∷ α ∷ []) "2Cor.12.11"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.12.11"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ ∙λ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.12.11"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ω ∷ ν ∷ []) "2Cor.12.11"
∷ word (ε ∷ ἰ ∷ []) "2Cor.12.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.11"
∷ word (ο ∷ ὐ ∷ δ ∷ έ ∷ ν ∷ []) "2Cor.12.11"
∷ word (ε ∷ ἰ ∷ μ ∷ ι ∷ []) "2Cor.12.11"
∷ word (τ ∷ ὰ ∷ []) "2Cor.12.12"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "2Cor.12.12"
∷ word (σ ∷ η ∷ μ ∷ ε ∷ ῖ ∷ α ∷ []) "2Cor.12.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.12.12"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ο ∷ υ ∷ []) "2Cor.12.12"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ι ∷ ρ ∷ γ ∷ ά ∷ σ ∷ θ ∷ η ∷ []) "2Cor.12.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.12.12"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.12.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.12.12"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "2Cor.12.12"
∷ word (ὑ ∷ π ∷ ο ∷ μ ∷ ο ∷ ν ∷ ῇ ∷ []) "2Cor.12.12"
∷ word (σ ∷ η ∷ μ ∷ ε ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.12.12"
∷ word (τ ∷ ε ∷ []) "2Cor.12.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.12"
∷ word (τ ∷ έ ∷ ρ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.12.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.12"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.12.12"
∷ word (τ ∷ ί ∷ []) "2Cor.12.13"
∷ word (γ ∷ ά ∷ ρ ∷ []) "2Cor.12.13"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "2Cor.12.13"
∷ word (ὃ ∷ []) "2Cor.12.13"
∷ word (ἡ ∷ σ ∷ σ ∷ ώ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "2Cor.12.13"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.12.13"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "2Cor.12.13"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ὰ ∷ ς ∷ []) "2Cor.12.13"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "2Cor.12.13"
∷ word (ε ∷ ἰ ∷ []) "2Cor.12.13"
∷ word (μ ∷ ὴ ∷ []) "2Cor.12.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.12.13"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "2Cor.12.13"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "2Cor.12.13"
∷ word (ο ∷ ὐ ∷ []) "2Cor.12.13"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ν ∷ ά ∷ ρ ∷ κ ∷ η ∷ σ ∷ α ∷ []) "2Cor.12.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.12.13"
∷ word (χ ∷ α ∷ ρ ∷ ί ∷ σ ∷ α ∷ σ ∷ θ ∷ έ ∷ []) "2Cor.12.13"
∷ word (μ ∷ ο ∷ ι ∷ []) "2Cor.12.13"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.12.13"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.12.13"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ η ∷ ν ∷ []) "2Cor.12.13"
∷ word (Ἰ ∷ δ ∷ ο ∷ ὺ ∷ []) "2Cor.12.14"
∷ word (τ ∷ ρ ∷ ί ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.12.14"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.12.14"
∷ word (ἑ ∷ τ ∷ ο ∷ ί ∷ μ ∷ ω ∷ ς ∷ []) "2Cor.12.14"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "2Cor.12.14"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "2Cor.12.14"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.12.14"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.12.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.14"
∷ word (ο ∷ ὐ ∷ []) "2Cor.12.14"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ν ∷ α ∷ ρ ∷ κ ∷ ή ∷ σ ∷ ω ∷ []) "2Cor.12.14"
∷ word (ο ∷ ὐ ∷ []) "2Cor.12.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.12.14"
∷ word (ζ ∷ η ∷ τ ∷ ῶ ∷ []) "2Cor.12.14"
∷ word (τ ∷ ὰ ∷ []) "2Cor.12.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.12.14"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.12.14"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.12.14"
∷ word (ο ∷ ὐ ∷ []) "2Cor.12.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.12.14"
∷ word (ὀ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ ε ∷ ι ∷ []) "2Cor.12.14"
∷ word (τ ∷ ὰ ∷ []) "2Cor.12.14"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "2Cor.12.14"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.12.14"
∷ word (γ ∷ ο ∷ ν ∷ ε ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.12.14"
∷ word (θ ∷ η ∷ σ ∷ α ∷ υ ∷ ρ ∷ ί ∷ ζ ∷ ε ∷ ι ∷ ν ∷ []) "2Cor.12.14"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.12.14"
∷ word (ο ∷ ἱ ∷ []) "2Cor.12.14"
∷ word (γ ∷ ο ∷ ν ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.12.14"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.12.14"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "2Cor.12.14"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "2Cor.12.15"
∷ word (δ ∷ ὲ ∷ []) "2Cor.12.15"
∷ word (ἥ ∷ δ ∷ ι ∷ σ ∷ τ ∷ α ∷ []) "2Cor.12.15"
∷ word (δ ∷ α ∷ π ∷ α ∷ ν ∷ ή ∷ σ ∷ ω ∷ []) "2Cor.12.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.15"
∷ word (ἐ ∷ κ ∷ δ ∷ α ∷ π ∷ α ∷ ν ∷ η ∷ θ ∷ ή ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.12.15"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.12.15"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.12.15"
∷ word (ψ ∷ υ ∷ χ ∷ ῶ ∷ ν ∷ []) "2Cor.12.15"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.12.15"
∷ word (ε ∷ ἰ ∷ []) "2Cor.12.15"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ ω ∷ ς ∷ []) "2Cor.12.15"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.12.15"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ ν ∷ []) "2Cor.12.15"
∷ word (ἧ ∷ σ ∷ σ ∷ ο ∷ ν ∷ []) "2Cor.12.15"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ μ ∷ α ∷ ι ∷ []) "2Cor.12.15"
∷ word (ἔ ∷ σ ∷ τ ∷ ω ∷ []) "2Cor.12.16"
∷ word (δ ∷ έ ∷ []) "2Cor.12.16"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "2Cor.12.16"
∷ word (ο ∷ ὐ ∷ []) "2Cor.12.16"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ β ∷ ά ∷ ρ ∷ η ∷ σ ∷ α ∷ []) "2Cor.12.16"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.12.16"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.12.16"
∷ word (ὑ ∷ π ∷ ά ∷ ρ ∷ χ ∷ ω ∷ ν ∷ []) "2Cor.12.16"
∷ word (π ∷ α ∷ ν ∷ ο ∷ ῦ ∷ ρ ∷ γ ∷ ο ∷ ς ∷ []) "2Cor.12.16"
∷ word (δ ∷ ό ∷ ∙λ ∷ ῳ ∷ []) "2Cor.12.16"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.12.16"
∷ word (ἔ ∷ ∙λ ∷ α ∷ β ∷ ο ∷ ν ∷ []) "2Cor.12.16"
∷ word (μ ∷ ή ∷ []) "2Cor.12.17"
∷ word (τ ∷ ι ∷ ν ∷ α ∷ []) "2Cor.12.17"
∷ word (ὧ ∷ ν ∷ []) "2Cor.12.17"
∷ word (ἀ ∷ π ∷ έ ∷ σ ∷ τ ∷ α ∷ ∙λ ∷ κ ∷ α ∷ []) "2Cor.12.17"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.12.17"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.12.17"
∷ word (δ ∷ ι ∷ []) "2Cor.12.17"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.12.17"
∷ word (ἐ ∷ π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ έ ∷ κ ∷ τ ∷ η ∷ σ ∷ α ∷ []) "2Cor.12.17"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.12.17"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ κ ∷ ά ∷ ∙λ ∷ ε ∷ σ ∷ α ∷ []) "2Cor.12.18"
∷ word (Τ ∷ ί ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.12.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.18"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ π ∷ έ ∷ σ ∷ τ ∷ ε ∷ ι ∷ ∙λ ∷ α ∷ []) "2Cor.12.18"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.12.18"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ό ∷ ν ∷ []) "2Cor.12.18"
∷ word (μ ∷ ή ∷ τ ∷ ι ∷ []) "2Cor.12.18"
∷ word (ἐ ∷ π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ έ ∷ κ ∷ τ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "2Cor.12.18"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.12.18"
∷ word (Τ ∷ ί ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.12.18"
∷ word (ο ∷ ὐ ∷ []) "2Cor.12.18"
∷ word (τ ∷ ῷ ∷ []) "2Cor.12.18"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "2Cor.12.18"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "2Cor.12.18"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ ε ∷ π ∷ α ∷ τ ∷ ή ∷ σ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.12.18"
∷ word (ο ∷ ὐ ∷ []) "2Cor.12.18"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.12.18"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.12.18"
∷ word (ἴ ∷ χ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.12.18"
∷ word (Π ∷ ά ∷ ∙λ ∷ α ∷ ι ∷ []) "2Cor.12.19"
∷ word (δ ∷ ο ∷ κ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "2Cor.12.19"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.12.19"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.12.19"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.12.19"
∷ word (κ ∷ α ∷ τ ∷ έ ∷ ν ∷ α ∷ ν ∷ τ ∷ ι ∷ []) "2Cor.12.19"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.12.19"
∷ word (ἐ ∷ ν ∷ []) "2Cor.12.19"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "2Cor.12.19"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.12.19"
∷ word (τ ∷ ὰ ∷ []) "2Cor.12.19"
∷ word (δ ∷ ὲ ∷ []) "2Cor.12.19"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "2Cor.12.19"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ί ∷ []) "2Cor.12.19"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.12.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.12.19"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.12.19"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ῆ ∷ ς ∷ []) "2Cor.12.19"
∷ word (φ ∷ ο ∷ β ∷ ο ∷ ῦ ∷ μ ∷ α ∷ ι ∷ []) "2Cor.12.20"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.12.20"
∷ word (μ ∷ ή ∷ []) "2Cor.12.20"
∷ word (π ∷ ω ∷ ς ∷ []) "2Cor.12.20"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ὼ ∷ ν ∷ []) "2Cor.12.20"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "2Cor.12.20"
∷ word (ο ∷ ἵ ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.12.20"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "2Cor.12.20"
∷ word (ε ∷ ὕ ∷ ρ ∷ ω ∷ []) "2Cor.12.20"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.12.20"
∷ word (κ ∷ ἀ ∷ γ ∷ ὼ ∷ []) "2Cor.12.20"
∷ word (ε ∷ ὑ ∷ ρ ∷ ε ∷ θ ∷ ῶ ∷ []) "2Cor.12.20"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.12.20"
∷ word (ο ∷ ἷ ∷ ο ∷ ν ∷ []) "2Cor.12.20"
∷ word (ο ∷ ὐ ∷ []) "2Cor.12.20"
∷ word (θ ∷ έ ∷ ∙λ ∷ ε ∷ τ ∷ ε ∷ []) "2Cor.12.20"
∷ word (μ ∷ ή ∷ []) "2Cor.12.20"
∷ word (π ∷ ω ∷ ς ∷ []) "2Cor.12.20"
∷ word (ἔ ∷ ρ ∷ ι ∷ ς ∷ []) "2Cor.12.20"
∷ word (ζ ∷ ῆ ∷ ∙λ ∷ ο ∷ ς ∷ []) "2Cor.12.20"
∷ word (θ ∷ υ ∷ μ ∷ ο ∷ ί ∷ []) "2Cor.12.20"
∷ word (ἐ ∷ ρ ∷ ι ∷ θ ∷ ε ∷ ῖ ∷ α ∷ ι ∷ []) "2Cor.12.20"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ α ∷ ∙λ ∷ ι ∷ α ∷ ί ∷ []) "2Cor.12.20"
∷ word (ψ ∷ ι ∷ θ ∷ υ ∷ ρ ∷ ι ∷ σ ∷ μ ∷ ο ∷ ί ∷ []) "2Cor.12.20"
∷ word (φ ∷ υ ∷ σ ∷ ι ∷ ώ ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "2Cor.12.20"
∷ word (ἀ ∷ κ ∷ α ∷ τ ∷ α ∷ σ ∷ τ ∷ α ∷ σ ∷ ί ∷ α ∷ ι ∷ []) "2Cor.12.20"
∷ word (μ ∷ ὴ ∷ []) "2Cor.12.21"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "2Cor.12.21"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ό ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.12.21"
∷ word (μ ∷ ο ∷ υ ∷ []) "2Cor.12.21"
∷ word (τ ∷ α ∷ π ∷ ε ∷ ι ∷ ν ∷ ώ ∷ σ ∷ ῃ ∷ []) "2Cor.12.21"
∷ word (μ ∷ ε ∷ []) "2Cor.12.21"
∷ word (ὁ ∷ []) "2Cor.12.21"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "2Cor.12.21"
∷ word (μ ∷ ο ∷ υ ∷ []) "2Cor.12.21"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.12.21"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.12.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.21"
∷ word (π ∷ ε ∷ ν ∷ θ ∷ ή ∷ σ ∷ ω ∷ []) "2Cor.12.21"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.12.21"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "2Cor.12.21"
∷ word (π ∷ ρ ∷ ο ∷ η ∷ μ ∷ α ∷ ρ ∷ τ ∷ η ∷ κ ∷ ό ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.12.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.21"
∷ word (μ ∷ ὴ ∷ []) "2Cor.12.21"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ ν ∷ ο ∷ η ∷ σ ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.12.21"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.12.21"
∷ word (τ ∷ ῇ ∷ []) "2Cor.12.21"
∷ word (ἀ ∷ κ ∷ α ∷ θ ∷ α ∷ ρ ∷ σ ∷ ί ∷ ᾳ ∷ []) "2Cor.12.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.21"
∷ word (π ∷ ο ∷ ρ ∷ ν ∷ ε ∷ ί ∷ ᾳ ∷ []) "2Cor.12.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.12.21"
∷ word (ἀ ∷ σ ∷ ε ∷ ∙λ ∷ γ ∷ ε ∷ ί ∷ ᾳ ∷ []) "2Cor.12.21"
∷ word (ᾗ ∷ []) "2Cor.12.21"
∷ word (ἔ ∷ π ∷ ρ ∷ α ∷ ξ ∷ α ∷ ν ∷ []) "2Cor.12.21"
∷ word (Τ ∷ ρ ∷ ί ∷ τ ∷ ο ∷ ν ∷ []) "2Cor.13.1"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.13.1"
∷ word (ἔ ∷ ρ ∷ χ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.13.1"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.13.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.13.1"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "2Cor.13.1"
∷ word (σ ∷ τ ∷ ό ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.13.1"
∷ word (δ ∷ ύ ∷ ο ∷ []) "2Cor.13.1"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ ύ ∷ ρ ∷ ω ∷ ν ∷ []) "2Cor.13.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.13.1"
∷ word (τ ∷ ρ ∷ ι ∷ ῶ ∷ ν ∷ []) "2Cor.13.1"
∷ word (σ ∷ τ ∷ α ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "2Cor.13.1"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "2Cor.13.1"
∷ word (ῥ ∷ ῆ ∷ μ ∷ α ∷ []) "2Cor.13.1"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ ί ∷ ρ ∷ η ∷ κ ∷ α ∷ []) "2Cor.13.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.13.2"
∷ word (π ∷ ρ ∷ ο ∷ ∙λ ∷ έ ∷ γ ∷ ω ∷ []) "2Cor.13.2"
∷ word (ὡ ∷ ς ∷ []) "2Cor.13.2"
∷ word (π ∷ α ∷ ρ ∷ ὼ ∷ ν ∷ []) "2Cor.13.2"
∷ word (τ ∷ ὸ ∷ []) "2Cor.13.2"
∷ word (δ ∷ ε ∷ ύ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "2Cor.13.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.13.2"
∷ word (ἀ ∷ π ∷ ὼ ∷ ν ∷ []) "2Cor.13.2"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "2Cor.13.2"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.13.2"
∷ word (π ∷ ρ ∷ ο ∷ η ∷ μ ∷ α ∷ ρ ∷ τ ∷ η ∷ κ ∷ ό ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.13.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.13.2"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.13.2"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ο ∷ ῖ ∷ ς ∷ []) "2Cor.13.2"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.13.2"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.13.2"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "2Cor.13.2"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ω ∷ []) "2Cor.13.2"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.13.2"
∷ word (τ ∷ ὸ ∷ []) "2Cor.13.2"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "2Cor.13.2"
∷ word (ο ∷ ὐ ∷ []) "2Cor.13.2"
∷ word (φ ∷ ε ∷ ί ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "2Cor.13.2"
∷ word (ἐ ∷ π ∷ ε ∷ ὶ ∷ []) "2Cor.13.3"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ὴ ∷ ν ∷ []) "2Cor.13.3"
∷ word (ζ ∷ η ∷ τ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "2Cor.13.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.13.3"
∷ word (ἐ ∷ ν ∷ []) "2Cor.13.3"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "2Cor.13.3"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.13.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.13.3"
∷ word (ὃ ∷ ς ∷ []) "2Cor.13.3"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.13.3"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.13.3"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.13.3"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ῖ ∷ []) "2Cor.13.3"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.13.3"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ τ ∷ ε ∷ ῖ ∷ []) "2Cor.13.3"
∷ word (ἐ ∷ ν ∷ []) "2Cor.13.3"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.13.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.13.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.13.4"
∷ word (ἐ ∷ σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ώ ∷ θ ∷ η ∷ []) "2Cor.13.4"
∷ word (ἐ ∷ ξ ∷ []) "2Cor.13.4"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "2Cor.13.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.13.4"
∷ word (ζ ∷ ῇ ∷ []) "2Cor.13.4"
∷ word (ἐ ∷ κ ∷ []) "2Cor.13.4"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.13.4"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.13.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.13.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.13.4"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.13.4"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.13.4"
∷ word (ἐ ∷ ν ∷ []) "2Cor.13.4"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "2Cor.13.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.13.4"
∷ word (ζ ∷ ή ∷ σ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.13.4"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "2Cor.13.4"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "2Cor.13.4"
∷ word (ἐ ∷ κ ∷ []) "2Cor.13.4"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ω ∷ ς ∷ []) "2Cor.13.4"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.13.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.13.4"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.13.4"
∷ word (Ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.13.5"
∷ word (π ∷ ε ∷ ι ∷ ρ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ ε ∷ []) "2Cor.13.5"
∷ word (ε ∷ ἰ ∷ []) "2Cor.13.5"
∷ word (ἐ ∷ σ ∷ τ ∷ ὲ ∷ []) "2Cor.13.5"
∷ word (ἐ ∷ ν ∷ []) "2Cor.13.5"
∷ word (τ ∷ ῇ ∷ []) "2Cor.13.5"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ι ∷ []) "2Cor.13.5"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.13.5"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ ε ∷ []) "2Cor.13.5"
∷ word (ἢ ∷ []) "2Cor.13.5"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.13.5"
∷ word (ἐ ∷ π ∷ ι ∷ γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ε ∷ τ ∷ ε ∷ []) "2Cor.13.5"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "2Cor.13.5"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.13.5"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "2Cor.13.5"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "2Cor.13.5"
∷ word (ἐ ∷ ν ∷ []) "2Cor.13.5"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "2Cor.13.5"
∷ word (ε ∷ ἰ ∷ []) "2Cor.13.5"
∷ word (μ ∷ ή ∷ τ ∷ ι ∷ []) "2Cor.13.5"
∷ word (ἀ ∷ δ ∷ ό ∷ κ ∷ ι ∷ μ ∷ ο ∷ ί ∷ []) "2Cor.13.5"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "2Cor.13.5"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ ζ ∷ ω ∷ []) "2Cor.13.6"
∷ word (δ ∷ ὲ ∷ []) "2Cor.13.6"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.13.6"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.13.6"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "2Cor.13.6"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.13.6"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.13.6"
∷ word (ἐ ∷ σ ∷ μ ∷ ὲ ∷ ν ∷ []) "2Cor.13.6"
∷ word (ἀ ∷ δ ∷ ό ∷ κ ∷ ι ∷ μ ∷ ο ∷ ι ∷ []) "2Cor.13.6"
∷ word (ε ∷ ὐ ∷ χ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.13.7"
∷ word (δ ∷ ὲ ∷ []) "2Cor.13.7"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "2Cor.13.7"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "2Cor.13.7"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "2Cor.13.7"
∷ word (μ ∷ ὴ ∷ []) "2Cor.13.7"
∷ word (π ∷ ο ∷ ι ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "2Cor.13.7"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.13.7"
∷ word (κ ∷ α ∷ κ ∷ ὸ ∷ ν ∷ []) "2Cor.13.7"
∷ word (μ ∷ η ∷ δ ∷ έ ∷ ν ∷ []) "2Cor.13.7"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "2Cor.13.7"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.13.7"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.13.7"
∷ word (δ ∷ ό ∷ κ ∷ ι ∷ μ ∷ ο ∷ ι ∷ []) "2Cor.13.7"
∷ word (φ ∷ α ∷ ν ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.13.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "2Cor.13.7"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.13.7"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.13.7"
∷ word (τ ∷ ὸ ∷ []) "2Cor.13.7"
∷ word (κ ∷ α ∷ ∙λ ∷ ὸ ∷ ν ∷ []) "2Cor.13.7"
∷ word (π ∷ ο ∷ ι ∷ ῆ ∷ τ ∷ ε ∷ []) "2Cor.13.7"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.13.7"
∷ word (δ ∷ ὲ ∷ []) "2Cor.13.7"
∷ word (ὡ ∷ ς ∷ []) "2Cor.13.7"
∷ word (ἀ ∷ δ ∷ ό ∷ κ ∷ ι ∷ μ ∷ ο ∷ ι ∷ []) "2Cor.13.7"
∷ word (ὦ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.13.7"
∷ word (ο ∷ ὐ ∷ []) "2Cor.13.8"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.13.8"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ θ ∷ ά ∷ []) "2Cor.13.8"
∷ word (τ ∷ ι ∷ []) "2Cor.13.8"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.13.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.13.8"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "2Cor.13.8"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "2Cor.13.8"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "2Cor.13.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.13.8"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "2Cor.13.8"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.13.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "2Cor.13.9"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "2Cor.13.9"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.13.9"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "2Cor.13.9"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "2Cor.13.9"
∷ word (δ ∷ ὲ ∷ []) "2Cor.13.9"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ τ ∷ ο ∷ ὶ ∷ []) "2Cor.13.9"
∷ word (ἦ ∷ τ ∷ ε ∷ []) "2Cor.13.9"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.13.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.13.9"
∷ word (ε ∷ ὐ ∷ χ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "2Cor.13.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.13.9"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.13.9"
∷ word (κ ∷ α ∷ τ ∷ ά ∷ ρ ∷ τ ∷ ι ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.13.9"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "2Cor.13.10"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "2Cor.13.10"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "2Cor.13.10"
∷ word (ἀ ∷ π ∷ ὼ ∷ ν ∷ []) "2Cor.13.10"
∷ word (γ ∷ ρ ∷ ά ∷ φ ∷ ω ∷ []) "2Cor.13.10"
∷ word (ἵ ∷ ν ∷ α ∷ []) "2Cor.13.10"
∷ word (π ∷ α ∷ ρ ∷ ὼ ∷ ν ∷ []) "2Cor.13.10"
∷ word (μ ∷ ὴ ∷ []) "2Cor.13.10"
∷ word (ἀ ∷ π ∷ ο ∷ τ ∷ ό ∷ μ ∷ ω ∷ ς ∷ []) "2Cor.13.10"
∷ word (χ ∷ ρ ∷ ή ∷ σ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "2Cor.13.10"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "2Cor.13.10"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "2Cor.13.10"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "2Cor.13.10"
∷ word (ἣ ∷ ν ∷ []) "2Cor.13.10"
∷ word (ὁ ∷ []) "2Cor.13.10"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "2Cor.13.10"
∷ word (ἔ ∷ δ ∷ ω ∷ κ ∷ έ ∷ ν ∷ []) "2Cor.13.10"
∷ word (μ ∷ ο ∷ ι ∷ []) "2Cor.13.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.13.10"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "2Cor.13.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.13.10"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "2Cor.13.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "2Cor.13.10"
∷ word (κ ∷ α ∷ θ ∷ α ∷ ί ∷ ρ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "2Cor.13.10"
∷ word (Λ ∷ ο ∷ ι ∷ π ∷ ό ∷ ν ∷ []) "2Cor.13.11"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "2Cor.13.11"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ε ∷ τ ∷ ε ∷ []) "2Cor.13.11"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ τ ∷ ί ∷ ζ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.13.11"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.13.11"
∷ word (τ ∷ ὸ ∷ []) "2Cor.13.11"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "2Cor.13.11"
∷ word (φ ∷ ρ ∷ ο ∷ ν ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "2Cor.13.11"
∷ word (ε ∷ ἰ ∷ ρ ∷ η ∷ ν ∷ ε ∷ ύ ∷ ε ∷ τ ∷ ε ∷ []) "2Cor.13.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.13.11"
∷ word (ὁ ∷ []) "2Cor.13.11"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "2Cor.13.11"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "2Cor.13.11"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ς ∷ []) "2Cor.13.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.13.11"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ ς ∷ []) "2Cor.13.11"
∷ word (ἔ ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "2Cor.13.11"
∷ word (μ ∷ ε ∷ θ ∷ []) "2Cor.13.11"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.13.11"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "2Cor.13.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "2Cor.13.12"
∷ word (ἐ ∷ ν ∷ []) "2Cor.13.12"
∷ word (ἁ ∷ γ ∷ ί ∷ ῳ ∷ []) "2Cor.13.12"
∷ word (φ ∷ ι ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "2Cor.13.12"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ ζ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "2Cor.13.12"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "2Cor.13.12"
∷ word (ο ∷ ἱ ∷ []) "2Cor.13.12"
∷ word (ἅ ∷ γ ∷ ι ∷ ο ∷ ι ∷ []) "2Cor.13.12"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "2Cor.13.12"
∷ word (ἡ ∷ []) "2Cor.13.13"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "2Cor.13.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.13.13"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.13.13"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "2Cor.13.13"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "2Cor.13.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.13.13"
∷ word (ἡ ∷ []) "2Cor.13.13"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "2Cor.13.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.13.13"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "2Cor.13.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "2Cor.13.13"
∷ word (ἡ ∷ []) "2Cor.13.13"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ί ∷ α ∷ []) "2Cor.13.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "2Cor.13.13"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ υ ∷ []) "2Cor.13.13"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "2Cor.13.13"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "2Cor.13.13"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "2Cor.13.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "2Cor.13.13"
∷ []
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{-# OPTIONS --without-K #-}
module Agda.Builtin.Unit where
record ⊤ : Set where
instance constructor tt
{-# BUILTIN UNIT ⊤ #-}
{-# COMPILED_DATA ⊤ () () #-}
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open import Agda.Primitive
data Foo : Setω where
foo : Foo
bar : Foo → Foo
bar foo = foo
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{-# OPTIONS --guarded #-}
postulate
test : (@tick _ : _) → Set
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-- ---------------------------------
-- this is the agda file implementing my own work on logic, viz. a relational
-- framework for logic.
-- These codes have been checked by Agda 2.6.0
-- ---------------------------------
module RL where
-- this file use agda standard library
open import Agda.Builtin.Equality
open import Relation.Nullary
open import Data.Empty
open import Data.Product
-- ---------------------------------
-- Logical Frame
-- ---------------------------------
record Frame : Set₁ where
field
Φ : Set -- the set of language
bot : Φ -- contradiction
top : Φ -- validity
_⇌_ : Φ → Φ → Set -- ⇌ is interpreted as consistency relation
symm : ∀ (x y : Φ) → x ⇌ y → y ⇌ x -- ⇌ is symmetric
alre : ∀ (x : Φ) → ¬ (x ≡ bot) → x ⇌ x -- except for ⊥, ⇌ is reflexive
⊥-contra : ∀ (x : Φ) → ¬ (x ⇌ bot) -- ⊥ is in-⇌ with any x∈Φ
⊤-validi : ∀ (x : Φ) → ¬ (x ≡ bot) → (x ⇌ top) -- ⊤ ⇌ everything e.x. ⊥
substitution : {A : Frame} (x y z : Frame.Φ A) -- ∀x,y,z∈Φ.(x≡y ∧ x⇌z → y⇌z)
→ x ≡ y → Frame._⇌_ A x z → Frame._⇌_ A y z
substitution {A} x y z p q rewrite p = q
cons→non-contra : {A : Frame} (x : Frame.Φ A) -- ∀x∈Φ.(∃y∈Φ.(x⇌y) → x≠⊥)
→ ∃[ y ] Frame._⇌_ A x y → ¬ (x ≡ Frame.bot A)
cons→non-contra {A} x (y , f) q = Frame.⊥-contra A y w
where
s : Frame._⇌_ A (Frame.bot A) y
s = substitution {A} x (Frame.bot A) y q f
w : Frame._⇌_ A y (Frame.bot A)
w = Frame.symm A (Frame.bot A) y s
module Example₁ where
data tf : Set where
𝟘x 𝟙x : tf
data _↔_ : tf → tf → Set where
t-t : 𝟙x ↔ 𝟙x
symm-tf : ∀ (x y : tf) → x ↔ y → y ↔ x
symm-tf 𝟙x 𝟙x t-t = t-t
alre-tf : ∀ (x : tf) → ¬ (x ≡ 𝟘x) → x ↔ x
alre-tf 𝟙x _ = t-t
alre-tf 𝟘x p = ⊥-elim (p refl)
𝟙x-validi : ∀ (x : tf) → ¬ (x ≡ 𝟘x) → (x ↔ 𝟙x)
𝟙x-validi 𝟙x _ = t-t
𝟙x-validi 𝟘x ¬p = ⊥-elim (¬p refl)
𝟘x-contra : ∀ (x : tf) → ¬ (x ↔ 𝟘x)
𝟘x-contra x ()
tfFrame : Frame -- the smallest possible normal frame
tfFrame = record { Φ = tf
; bot = 𝟘x
; top = 𝟙x
; _⇌_ = _↔_
; symm = symm-tf
; alre = alre-tf
; ⊥-contra = 𝟘x-contra
; ⊤-validi = 𝟙x-validi }
-- ---------------------------------
-- Logical Consequence
-- ---------------------------------
record _⊢_ {A : Frame} (a b : Frame.Φ A) : Set where -- logical consequence
field -- a⊢b ⇔ ∀x∈Φ.(x⇌a → x⇌b)
fromCons : ∀ (x : Frame.Φ A) → Frame._⇌_ A x a → Frame._⇌_ A x b
-- ---------------------------------
-- properties of ⊢
refl-⊢ : {A : Frame} (a : Frame.Φ A) → _⊢_ {A} a a -- reflexive
refl-⊢ {A} a = record { fromCons = p }
where
p : ∀ (x : Frame.Φ A) → Frame._⇌_ A x a → Frame._⇌_ A x a
p x q = q
trans-⊢ : {A : Frame} (a b c : Frame.Φ A) -- transitive
→ _⊢_ {A} a b → _⊢_ {A} b c → _⊢_ {A} a c
trans-⊢ {A} a b c p q = record { fromCons = f }
where
f : ∀ (x : Frame.Φ A) → Frame._⇌_ A x a → Frame._⇌_ A x c
f x h = _⊢_.fromCons q x (_⊢_.fromCons p x h)
-- ---------------------------------
-- ---------------------------------
-- ∀x∈Φ.(⊥⊢x)
bot-cons : {A : Frame} (x : Frame.Φ A) -- ∀x∈Φ.∀y∈Φ.(y⇌⊥ → y⇌x)
→ ∀ (y : Frame.Φ A) → Frame._⇌_ A y (Frame.bot A) → Frame._⇌_ A y x
bot-cons {A} x y p = ⊥-elim (Frame.⊥-contra A y p)
bot-to-every : {A : Frame} (x : Frame.Φ A) → _⊢_ {A} (Frame.bot A) x
bot-to-every {A} x = record { fromCons = bot-cons {A} x }
-- ---------------------------------
-- ---------------------------------
-- ∀x∈Φ.(x⊢⊤)
top-cons : {A : Frame} (y x : Frame.Φ A) -- ∀y∈Φ.∀x∈Φ.(x⇌y → x⇌⊤)
→ Frame._⇌_ A x y → Frame._⇌_ A x (Frame.top A)
top-cons {A} y x p = Frame.⊤-validi A x (cons→non-contra {A} x (y , p))
top-from-every : {A : Frame} (x : Frame.Φ A) → _⊢_ {A} x (Frame.top A)
top-from-every {A} x = record { fromCons = top-cons {A} x }
-- ---------------------------------
-- ---------------------------------
-- the criteria for a Reasoning frame
-- ---------------------------------
record Reasoning (A : Frame) : Set₁ where
field -- basically, reas says every consistent pair is testified
reas : ∀ (x y : Frame.Φ A) → Frame._⇌_ A x y
→ ∃[ z ] ((¬ (z ≡ Frame.bot A)) × ((_⊢_ {A} z x) × (_⊢_ {A} z y)))
module Example₂ where
open Example₁
𝟘x⊢𝟘x : _⊢_ {tfFrame} 𝟘x 𝟘x
𝟘x⊢𝟘x = refl-⊢ {tfFrame} 𝟘x
𝟘x⊢𝟙x : _⊢_ {tfFrame} 𝟘x 𝟙x
𝟘x⊢𝟙x = bot-to-every {tfFrame} 𝟙x
𝟙x⊢𝟙x : _⊢_ {tfFrame} 𝟙x 𝟙x
𝟙x⊢𝟙x = refl-⊢ {tfFrame} 𝟙x
reas-tf : ∀ (x y : tf) → (x ↔ y)
→ ∃[ z ] ((¬ (z ≡ 𝟘x)) × ((_⊢_ {tfFrame} z x) × (_⊢_ {tfFrame} z y)))
reas-tf 𝟘x _ ()
reas-tf 𝟙x 𝟙x _ = (𝟙x , (p , (𝟙x⊢𝟙x , 𝟙x⊢𝟙x)))
where
p : ¬ (𝟙x ≡ 𝟘x)
p ()
Reasoning-tf : Reasoning tfFrame
Reasoning-tf = record { reas = reas-tf }
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module ICFPPrelude where
record ⊤ : Set where
constructor ⟨⟩
data ⊥ : Set where
¬_ : Set → Set
¬_ A = A → ⊥
data Nat : Set where
zero : Nat
suc : Nat → Nat
{-# BUILTIN NATURAL Nat #-}
_+_ : Nat → Nat → Nat
zero + n = n
suc m + n = suc (m + n)
infixr 2 _∪_
data _∪_ A B : Set where
inl : A → A ∪ B
inr : B → A ∪ B
data _≡_ {a}{A : Set a}(x : A) : A → Set a where
refl : x ≡ x
{-# BUILTIN EQUALITY _≡_ #-}
{-# BUILTIN REFL refl #-}
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data D (A : Set) : Set₁ where
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{-# OPTIONS --cubical --safe #-}
module Data.Lift where
open import Level
record Lift {a} ℓ (A : Type a) : Type (a ℓ⊔ ℓ) where
constructor lift
field lower : A
open Lift public
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module Oscar.Category.Semigroupoid where
open import Oscar.Data.Equality
open import Oscar.Level
open import Oscar.Relation
record Semigroupoid
{𝔬} {𝔒 : Ø 𝔬}
{𝔪} (_⊸_ : 𝔒 → 𝔒 → Ø 𝔪)
: Ø 𝔬 ∙̂ 𝔪 where
infixr 9 _∙_
field
_∙_ : ∀ {m n} → m ⊸ n → ∀ {l} → l ⊸ m → l ⊸ n
∙-associativity : ∀ {k l} (f : k ⊸ l) {m} (g : l ⊸ m) {n} (h : m ⊸ n) → (h ∙ g) ∙ f ≡ h ∙ g ∙ f
record Category
{𝔬} {𝔒 : Ø 𝔬}
{𝔪} (_⊸_ : 𝔒 → 𝔒 → Ø 𝔪)
: Ø 𝔬 ∙̂ 𝔪 where
field
semigroupoid : Semigroupoid _⊸_
open Semigroupoid semigroupoid public
field
ε : ∀ {n} → n ⊸ n
ε-left-identity : ∀ {m n} (σ : m ⊸ n) → ε ∙ σ ≡ σ
ε-right-identity : ∀ {m n} (σ : m ⊸ n) → σ ∙ ε ≡ σ
record RCategory
{𝔬} {𝔒 : Ø 𝔬}
{𝔪} {_⊸_ : 𝔒 → 𝔒 → Ø 𝔪}
(category : Category _⊸_)
: Ø 𝔬 ∙̂ 𝔪 where
open Category category public hiding (semigroupoid)
module MCategory
{𝔬} {𝔒 : Ø 𝔬}
{𝔪} {_⊸_ : 𝔒 → 𝔒 → Ø 𝔪}
(category : Category _⊸_)
where
open Category category public hiding (semigroupoid)
-- open import Oscar.Category.Morphism
-- open import Oscar.Category.Setoid
-- open import Oscar.Level
-- open import Oscar.Relation
-- module _ {𝔬} {𝔪} {𝔮} (𝔐 : Morphism 𝔬 𝔪 𝔮) (open Morphism 𝔐) where
-- record IsSemigroupoid (_∙_ : Transitive _↦_) : Set (lsuc (𝔬 ⊔ 𝔪 ⊔ 𝔮)) where
-- field
-- extensionality : Extensional _∙_ _≞_
-- associativity : Associative _∙_ _≞_
-- open IsSemigroupoid ⦃ … ⦄ public
-- infixr 4 _,_
-- record Semigroupoid 𝔬 𝔪 𝔮 : Set (lsuc (𝔬 ⊔ 𝔪 ⊔ 𝔮))
-- where
-- constructor _,_
-- field
-- 𝔐 : Morphism 𝔬 𝔪 𝔮
-- open Morphism 𝔐 public
-- infixl 7 _∙_
-- field _∙_ : Transitive _↦_
-- field ⦃ isSemigroupoid ⦄ : IsSemigroupoid 𝔐 _∙_
-- open IsSemigroupoid isSemigroupoid public
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open import Prelude hiding (id; Bool; _∷_; [])
module Examples.Infinite where
open import Implicits.Syntax
open import Implicits.WellTyped
open import Implicits.Substitutions
open import Implicits.Syntax.Type.Unification
open import Implicits.Resolution.Infinite.Resolution
open TypingRules _⊢ᵣ_
open import Data.Maybe
open import Data.List
tc-bool = 0
tc-int = 1
Bool : ∀ {n} → Type n
Bool = simpl (tc tc-bool)
Int : ∀ {n} → Type n
Int = simpl (tc tc-int)
module Ex₁ where
Δ : ICtx zero
Δ = (∀' (TVar zero ⇒ TVar zero)) ∷ []
p : Δ ⊢ᵣ (Int ⇒ Int)
p = r-iabs (r-simp (there (here refl))
(i-tabs
Int
(i-iabs (r-simp (here refl) (i-simp (tc tc-int))) (i-simp (tc tc-int)))))
module Ex₂ where
implicitly : ∀ {ν n} → Term ν n
implicitly = Λ (ρ α (var zero))
where
α = TVar zero
Implicitly : ∀ {ν n} {K : Ktx ν n} → K ⊢ implicitly ∈ (∀' (TVar zero ⇒ TVar zero))
Implicitly = Λ (ρ (ua-simp [] All.[]) (var zero))
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{-# OPTIONS --without-K --safe #-}
open import Categories.Category
module Categories.Category.CartesianClosed {o ℓ e} (𝒞 : Category o ℓ e) where
open import Level
open import Function using (_$_; flip)
open import Data.Product using (Σ; _,_; uncurry)
open import Categories.Functor renaming (id to idF)
open import Categories.Functor.Bifunctor
open import Categories.NaturalTransformation hiding (id)
open import Categories.NaturalTransformation.Properties
open import Categories.Category.Cartesian 𝒞
open import Categories.Category.Monoidal.Closed
open import Categories.Object.Product 𝒞
hiding (repack≡id; repack∘; repack-cancel; up-to-iso; transport-by-iso)
open import Categories.Object.Exponential 𝒞 hiding (repack)
open import Categories.Morphism 𝒞
open import Categories.Morphism.Reasoning 𝒞
private
module 𝒞 = Category 𝒞
open Category 𝒞
open HomReasoning
variable
A B C : Obj
f g h i : A ⇒ B
-- Cartesian closed category
-- is a category with all products and exponentials
record CartesianClosed : Set (levelOfTerm 𝒞) where
infixr 9 _^_
-- an alternative notation for exponential, which emphasizes its internal hom natural
infixr 5 _⇨_
field
cartesian : Cartesian
exp : Exponential A B
module exp {A B} = Exponential (exp {A} {B})
_^_ : Obj → Obj → Obj
B ^ A = exp.B^A {A} {B}
_⇨_ : Obj → Obj → Obj
_⇨_ = flip _^_
module cartesian = Cartesian cartesian
open cartesian public
B^A×A : ∀ B A → Product (B ^ A) A
B^A×A B A = exp.product {A} {B}
eval : Product.A×B (B^A×A B A) ⇒ B
eval = exp.eval
λg : C × A ⇒ B → C ⇒ B ^ A
λg f = exp.λg product f
λ-cong : f ≈ g → λg f ≈ λg g
λ-cong eq = exp.λ-cong product eq
_×id : (f : C ⇒ B ^ A) → C × A ⇒ [[ B^A×A B A ]]
f ×id = [ product ⇒ exp.product ] f ×id
β : eval ∘ λg f ×id ≈ f
β = exp.β product
η-exp : λg (eval ∘ f ×id) ≈ f
η-exp = exp.η product
λ-unique : eval ∘ f ×id ≈ g → f ≈ λg g
λ-unique = exp.λ-unique product
λ-unique₂ : eval ∘ f ×id ≈ eval ∘ g ×id → f ≈ g
λ-unique₂ = exp.λ-unique′ product
-- the annoying detail is that B^A×A is NOT the same as B ^ A × A, but they are isomorphic.
-- make some infra so that the latter (which is more intuitive) can be used.
B^A×A-iso : Product.A×B (B^A×A B A) ≅ B ^ A × A
B^A×A-iso {B = B} {A = A} = record
{ from = repack exp.product product
; to = repack product exp.product
; iso = record
{ isoˡ = begin
repack product exp.product ∘ repack exp.product product
≈⟨ [ exp.product ]⟨⟩∘ ⟩
[ exp.product ]⟨ π₁ ∘ repack exp.product product , π₂ ∘ repack exp.product product ⟩
≈⟨ Product.⟨⟩-cong₂ exp.product project₁ project₂ ⟩
[ exp.product ]⟨ [ exp.product ]π₁ , [ exp.product ]π₂ ⟩
≈⟨ Product.η exp.product ⟩
id
∎
; isoʳ = begin
repack exp.product product ∘ repack product exp.product
≈⟨ ⟨⟩∘ ⟩
⟨ [ exp.product ]π₁ ∘ repack product exp.product , [ exp.product ]π₂ ∘ repack product exp.product ⟩
≈⟨ ⟨⟩-cong₂ (Product.project₁ exp.product) (Product.project₂ exp.product) ⟩
⟨ π₁ , π₂ ⟩
≈⟨ η ⟩
id
∎
}
}
eval′ : B ^ A × A ⇒ B
eval′ = eval ∘ to
where open _≅_ B^A×A-iso
λ-unique′ : eval′ ∘ (f ⁂ id) ≈ g → f ≈ λg g
λ-unique′ eq = exp.λ-unique product (⟺ (pullʳ [ product ⇒ product ⇒ exp.product ]repack∘×) ○ eq)
λ-unique₂′ : eval′ ∘ (f ⁂ id) ≈ eval′ ∘ (g ⁂ id) → f ≈ g
λ-unique₂′ eq = (λ-unique′ eq) ○ ⟺ (λ-unique′ refl)
β′ : eval′ ∘ (λg f ⁂ id) ≈ f
β′ {f = f} = begin
eval′ ∘ (λg f ⁂ id) ≈⟨ pullʳ [ product ⇒ product ⇒ exp.product ]repack∘× ⟩
eval ∘ λg f ×id ≈⟨ β ⟩
f ∎
η-exp′ : λg (eval′ ∘ (f ⁂ id)) ≈ f
η-exp′ = sym (λ-unique′ refl)
η-id′ : λg (eval′ {B = B} {A = A}) ≈ id
η-id′ = sym (λ-unique′ (elimʳ (id×id product)))
⊤^A≅⊤ : ⊤ ^ A ≅ ⊤
⊤^A≅⊤ = record
{ from = !
; to = λg !
; iso = record
{ isoˡ = λ-unique₂ !-unique₂
; isoʳ = ⊤-id _
}
}
A^⊤≅A : A ^ ⊤ ≅ A
A^⊤≅A = record
{ from = let open _≅_ A×⊤≅A in eval′ ∘ to
; to = let open _≅_ A×⊤≅A in λg from
; iso = record
{ isoˡ = λ-unique₂′ $ begin
eval′ ∘ ((λg π₁ ∘ eval′ ∘ ⟨ id , ! ⟩) ⁂ id) ≈˘⟨ refl⟩∘⟨ first∘first ⟩
eval′ ∘ ((λg π₁ ⁂ id) ∘ ((eval′ ∘ ⟨ id , ! ⟩) ⁂ id)) ≈⟨ pullˡ β′ ⟩
π₁ ∘ ((eval′ ∘ ⟨ id , ! ⟩) ⁂ id) ≈⟨ helper ⟩
eval′ ∘ (id ⁂ id) ∎
; isoʳ = firstid ! $ begin
((eval′ ∘ ⟨ id , ! ⟩) ∘ λg π₁) ⁂ id ≈˘⟨ first∘first ⟩
(eval′ ∘ ⟨ id , ! ⟩ ⁂ id) ∘ (λg π₁ ⁂ id) ≈⟨ helper′ ⟩∘⟨refl ⟩
(⟨ id , ! ⟩ ∘ eval′) ∘ (λg π₁ ⁂ id) ≈⟨ pullʳ β′ ⟩
⟨ id , ! ⟩ ∘ π₁ ≈⟨ ⟨⟩∘ ⟩
⟨ id ∘ π₁ , ! ∘ π₁ ⟩ ≈⟨ ⟨⟩-cong₂ identityˡ !-unique₂ ⟩
⟨ π₁ , π₂ ⟩ ≈⟨ η ⟩
id ∎
}
}
where helper = begin
π₁ ∘ ((eval′ ∘ ⟨ id , ! ⟩) ⁂ id) ≈⟨ project₁ ⟩
(eval′ ∘ ⟨ id , ! ⟩) ∘ π₁ ≈⟨ pullʳ ⟨⟩∘ ⟩
eval′ ∘ ⟨ id ∘ π₁ , ! ∘ π₁ ⟩ ≈⟨ refl⟩∘⟨ ⟨⟩-congˡ !-unique₂ ⟩
eval′ ∘ (id ⁂ id) ∎
helper′ = let open _≅_ A×⊤≅A in begin
(eval′ ∘ ⟨ id , ! ⟩) ⁂ id ≈⟨ introˡ isoˡ ⟩
(⟨ id , ! ⟩ ∘ π₁) ∘ ((eval′ ∘ ⟨ id , ! ⟩) ⁂ id) ≈⟨ pullʳ helper ⟩
⟨ id , ! ⟩ ∘ (eval′ ∘ (id ⁂ id)) ≈⟨ refl⟩∘⟨ elimʳ (id×id product) ⟩
⟨ id , ! ⟩ ∘ eval′ ∎
-- we use -⇨- to represent the bifunctor.
-- -^- would generate a bifunctor of type Bifunctor 𝒞 𝒞.op 𝒞 which is not very typical.
-⇨- : Bifunctor 𝒞.op 𝒞 𝒞
-⇨- = record
{ F₀ = uncurry _⇨_
; F₁ = λ where
(f , g) → λg (g ∘ eval′ ∘ second f)
; identity = λ-cong (identityˡ ○ (elimʳ (id×id product))) ○ η-id′
; homomorphism = λ-unique₂′ helper
; F-resp-≈ = λ where
(eq₁ , eq₂) → λ-cong (∘-resp-≈ eq₂ (∘-resp-≈ʳ (⁂-cong₂ refl eq₁)))
}
where helper : eval′ ∘ first (λg ((g ∘ f) ∘ eval′ ∘ second (h ∘ i)))
≈ eval′ ∘ first (λg (g ∘ eval′ ∘ second i) ∘ λg (f ∘ eval′ ∘ second h))
helper {g = g} {f = f} {h = h} {i = i} = begin
eval′ ∘ first (λg ((g ∘ f) ∘ eval′ ∘ second (h ∘ i))) ≈⟨ β′ ⟩
(g ∘ f) ∘ eval′ ∘ second (h ∘ i) ≈˘⟨ refl⟩∘⟨ pullʳ second∘second ⟩
(g ∘ f) ∘ (eval′ ∘ second h) ∘ second i ≈˘⟨ pullˡ refl ⟩
g ∘ f ∘ (eval′ ∘ second h) ∘ second i ≈˘⟨ refl⟩∘⟨ assoc ⟩
g ∘ (f ∘ eval′ ∘ second h) ∘ second i ≈˘⟨ refl⟩∘⟨ pullˡ β′ ⟩
g ∘ eval′ ∘ first (λg (f ∘ eval′ ∘ second h)) ∘ second i ≈⟨ refl⟩∘⟨ pushʳ first↔second ⟩
g ∘ (eval′ ∘ second i) ∘ first (λg (f ∘ eval′ ∘ second h)) ≈˘⟨ assoc ⟩
(g ∘ eval′ ∘ second i) ∘ first (λg (f ∘ eval′ ∘ second h)) ≈˘⟨ pullˡ β′ ⟩
eval′ ∘ first (λg (g ∘ eval′ ∘ second i)) ∘ first (λg (f ∘ eval′ ∘ second h)) ≈⟨ refl⟩∘⟨ first∘first ⟩
eval′ ∘ first (λg (g ∘ eval′ ∘ second i) ∘ λg (f ∘ eval′ ∘ second h)) ∎
_⇨- : Obj → Endofunctor 𝒞
_⇨- = appˡ -⇨-
-⇨_ : Obj → Functor 𝒞.op 𝒞
-⇨_ = appʳ -⇨-
module _ where
private
A⇨[-×A] : Obj → Endofunctor 𝒞
A⇨[-×A] A = A ⇨- ∘F -× A
module A⇨[-×A] {A} = Functor (A⇨[-×A] A)
[A⇨-]×A : Obj → Endofunctor 𝒞
[A⇨-]×A A = -× A ∘F A ⇨-
module [A⇨-]×A {A} = Functor ([A⇨-]×A A)
closedMonoidal : Closed monoidal
closedMonoidal = record
{ [-,-] = -⇨-
; adjoint = λ {A} → record
{ unit = ntHelper record
{ η = λ _ → λg id
; commute = λ f → λ-unique₂′ $ begin
eval′ ∘ first (λg id ∘ f) ≈˘⟨ refl⟩∘⟨ first∘first ⟩
eval′ ∘ first (λg id) ∘ first f ≈⟨ cancelˡ β′ ⟩
first f ≈˘⟨ cancelʳ β′ ⟩
(first f ∘ eval′) ∘ first (λg id) ≈˘⟨ ∘-resp-≈ʳ (elimʳ (id×id product)) ⟩∘⟨refl ⟩
(first f ∘ eval′ ∘ first id) ∘ first (λg id) ≈˘⟨ pullˡ β′ ⟩
eval′ ∘ first (A⇨[-×A].F₁ f) ∘ first (λg id) ≈⟨ refl⟩∘⟨ first∘first ⟩
eval′ ∘ first (A⇨[-×A].F₁ f ∘ λg id) ∎
}
; counit = ntHelper record
{ η = λ _ → eval′
; commute = λ f → begin
eval′ ∘ [A⇨-]×A.F₁ f ≈⟨ β′ ⟩
f ∘ eval′ ∘ first id ≈⟨ refl⟩∘⟨ elimʳ (id×id product) ⟩
f ∘ eval′ ∎
}
; zig = β′
; zag = λ-unique₂′ $ begin
eval′ ∘ first (λg (eval′ ∘ eval′ ∘ second id) ∘ λg id)
≈˘⟨ refl⟩∘⟨ first∘first ⟩
eval′ ∘ first (λg (eval′ ∘ eval′ ∘ second id)) ∘ first (λg id)
≈⟨ pullˡ β′ ⟩
(eval′ ∘ eval′ ∘ second id) ∘ first (λg id)
≈⟨ ∘-resp-≈ʳ (elimʳ (id×id product)) ⟩∘⟨refl ⟩
(eval′ ∘ eval′) ∘ first (λg id) ≈⟨ cancelʳ β′ ⟩
eval′ ≈˘⟨ elimʳ (id×id product) ⟩
eval′ ∘ first id ∎
}
; mate = λ {X Y} f → record
{ commute₁ = λ-unique₂′ $ begin
eval′ ∘ first (λg (second f ∘ eval′ ∘ second id) ∘ λg id) ≈˘⟨ refl⟩∘⟨ first∘first ⟩
eval′ ∘ first (λg (second f ∘ eval′ ∘ second id)) ∘ first (λg id) ≈⟨ pullˡ β′ ⟩
(second f ∘ eval′ ∘ second id) ∘ first (λg id) ≈⟨ ∘-resp-≈ʳ (elimʳ (id×id product)) ⟩∘⟨refl ⟩
(second f ∘ eval′) ∘ first (λg id) ≈⟨ cancelʳ β′ ⟩
second f ≈˘⟨ cancelˡ β′ ⟩
eval′ ∘ first (λg id) ∘ second f ≈⟨ pushʳ first↔second ⟩
(eval′ ∘ second f) ∘ first (λg id) ≈˘⟨ identityˡ ⟩∘⟨refl ⟩
(id ∘ eval′ ∘ second f) ∘ first (λg id) ≈˘⟨ pullˡ β′ ⟩
eval′ ∘ first (λg (id ∘ eval′ ∘ second f)) ∘ first (λg id) ≈⟨ refl⟩∘⟨ first∘first ⟩
eval′ ∘ first (λg (id ∘ eval′ ∘ second f) ∘ λg id) ∎
; commute₂ = begin
eval′ ∘ first (λg (id ∘ eval′ ∘ second f)) ≈⟨ β′ ⟩
id ∘ eval′ ∘ second f ≈⟨ identityˡ ⟩
eval′ ∘ second f ∎
}
}
module closedMonoidal = Closed closedMonoidal
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{-
Agda Implementors' Meeting VI
Göteborg
May 24 - 30, 2007
Hello Agda!
Ulf Norell
-}
-- Now we're getting somewhere! Inductive families of datatypes.
module Families where
-- You can import modules defined in other files.
-- More details later...
--open import Naturals
data Nat : Set where
zero : Nat
suc : Nat -> Nat
infixl 60 _+_
infixl 80 _*_
_+_ : Nat -> Nat -> Nat
zero + m = m
suc n + m = suc (n + m)
_*_ : Nat -> Nat -> Nat
zero * m = zero
suc n * m = m + n * m
-- Think of an inductive family...
module Vec where
data Vec (A : Set) : Nat -> Set where
[] : Vec A zero
_::_ : {n : Nat} -> A -> Vec A n -> Vec A (suc n)
infixr 40 _::_
-- Some simple functions
head : {A : Set}{n : Nat} -> Vec A (suc n) -> A
head (x :: _) = x -- no need for a [] case
-- Does the definition look familiar?
map : {A B : Set}{n : Nat} -> (A -> B) -> Vec A n -> Vec B n
map f [] = []
map f (x :: xs) = f x :: map f xs
t1 : Vec Nat (suc (suc zero))
t1 = map (_+_ three) (zero :: suc three :: [])
where three = suc (suc (suc zero))
infixr 40 _++_
_++_ : {A : Set}{n m : Nat} -> Vec A n -> Vec A m -> Vec A (n + m)
[] ++ ys = ys
(x :: xs) ++ ys = x :: (xs ++ ys)
{-
Wait a second.. what's really going on here?
All the indices were conveniently implicit!
-}
-- Ok. Let's make the implicit stuff explicit.
{-
module WhatsGoingOnHere? where
open Vec using (Vec; []; _::_)
-}
-- Now what's this funny dot thing?
map' : {A B : Set}(n : Nat) -> (A -> B) -> Vec A n -> Vec B n
map' .zero f [] = []
map' .(suc _) f (x :: xs) = f x :: map' _ f xs
-- Basically the dot means: inside is not a pattern at all but a
-- term whose value is uniquely determined by type checking
-- the actual pattern.
-- In the cases above the types of the patterns
-- [] and (_::_ {n} x xs)
-- forces the first argument to be zero and suc n respectively.
-- So, that's what we write.
-- We could spend hours talking about this, but let's move on...
-- Let's do some other interesting families.
-- The identity type.
data _==_ {A : Set}(x : A) : A -> Set where
refl : x == x
infix 30 _==_
infix 20 ¬_
-- In the presence of families we get a lot more empty types.
data Bool : Set where
true : Bool
false : Bool
data False : Set where
¬_ : Set -> Set
¬ A = A -> False
_≠_ : {A : Set} -> A -> A -> Set
x ≠ y = ¬ x == y
true≠false : true == false -> False -- true ≠ false
true≠false ()
-- [The following example might have worked at AIM6, but it does not
-- work now, so I commented it out. /NAD]
-- lem : (n : Nat) -> n == suc n -> False
-- lem n ()
-- Why does this work: true == false is an empty type.
{-
What's next?
-}
-- Actually, inductive families are sufficiently fun that
-- you'll never get bored, but there's even more fun to be had.
-- Move on to: With.agda
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primitive
primLevelSuc : _
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{-# OPTIONS --without-K --safe #-}
module PiPointed where
open import Data.Empty
open import Data.Unit
open import Data.Sum
open import Data.Product renaming (map to map×)
open import Relation.Binary.PropositionalEquality
open import Singleton
open import PlainPi
infixr 90 _#_
------------------------------------------------------------------------------
-- Pointed types and singleton types
data ∙𝕌 : Set where
_#_ : (t : 𝕌) → (v : ⟦ t ⟧) → ∙𝕌
_∙×ᵤ_ : ∙𝕌 → ∙𝕌 → ∙𝕌
_∙+ᵤl_ : ∙𝕌 → ∙𝕌 → ∙𝕌
_∙+ᵤr_ : ∙𝕌 → ∙𝕌 → ∙𝕌
Singᵤ : ∙𝕌 → ∙𝕌
∙⟦_⟧ : ∙𝕌 → Σ[ A ∈ Set ] A
∙⟦ t # v ⟧ = ⟦ t ⟧ , v
∙⟦ T₁ ∙×ᵤ T₂ ⟧ = zip _×_ _,_ ∙⟦ T₁ ⟧ ∙⟦ T₂ ⟧
∙⟦ T₁ ∙+ᵤl T₂ ⟧ = zip _⊎_ (λ x _ → inj₁ x) ∙⟦ T₁ ⟧ ∙⟦ T₂ ⟧
∙⟦ T₁ ∙+ᵤr T₂ ⟧ = zip _⊎_ (λ _ y → inj₂ y) ∙⟦ T₁ ⟧ ∙⟦ T₂ ⟧
∙⟦ Singᵤ T ⟧ = < uncurry Singleton , (λ y → proj₂ y , refl) > ∙⟦ T ⟧
data _∙⟶_ : ∙𝕌 → ∙𝕌 → Set where
∙c : {t₁ t₂ : 𝕌} {v : ⟦ t₁ ⟧} → (c : t₁ ⟷ t₂) →
t₁ # v ∙⟶ t₂ # (eval c v)
∙id⟷ : {T : ∙𝕌} → T ∙⟶ T
_∙⊚_ : {T₁ T₂ T₃ : ∙𝕌} → (T₁ ∙⟶ T₂) → (T₂ ∙⟶ T₃) → (T₁ ∙⟶ T₃)
∙Singᵤ : (T₁ T₂ : ∙𝕌) → (T₁ ∙⟶ T₂) → (Singᵤ T₁ ∙⟶ Singᵤ T₂)
-- monad
return : (T : ∙𝕌) → T ∙⟶ Singᵤ T
join : (T : ∙𝕌) → Singᵤ (Singᵤ T) ∙⟶ Singᵤ T
tensorl : (T₁ T₂ : ∙𝕌) → (Singᵤ T₁ ∙×ᵤ T₂) ∙⟶ Singᵤ (T₁ ∙×ᵤ T₂)
tensorr : (T₁ T₂ : ∙𝕌) → (T₁ ∙×ᵤ Singᵤ T₂) ∙⟶ Singᵤ (T₁ ∙×ᵤ T₂)
tensor : (T₁ T₂ : ∙𝕌) → (Singᵤ T₁ ∙×ᵤ Singᵤ T₂) ∙⟶ Singᵤ (T₁ ∙×ᵤ T₂)
untensor : (T₁ T₂ : ∙𝕌) → Singᵤ (T₁ ∙×ᵤ T₂) ∙⟶ (Singᵤ T₁ ∙×ᵤ Singᵤ T₂)
plusl : (T₁ T₂ : ∙𝕌) → (Singᵤ T₁ ∙+ᵤl T₂) ∙⟶ Singᵤ (T₁ ∙+ᵤl T₂)
plusr : (T₁ T₂ : ∙𝕌) → (T₁ ∙+ᵤr Singᵤ T₂) ∙⟶ Singᵤ (T₁ ∙+ᵤr T₂)
-- plus : (T₁ T₂ : ∙𝕌) → (Singᵤ T₁ ∙+ᵤl Singᵤ T₂) ∙⟶ Singᵤ (T₁ ∙+ᵤl T₂) -- lobsided, feels wrong
-- comonad
extract : (T : ∙𝕌) → Singᵤ T ∙⟶ T
duplicate : (T : ∙𝕌) → Singᵤ T ∙⟶ Singᵤ (Singᵤ T)
cotensorl : (T₁ T₂ : ∙𝕌) → Singᵤ (T₁ ∙×ᵤ T₂) ∙⟶ (Singᵤ T₁ ∙×ᵤ T₂)
cotensorr : (T₁ T₂ : ∙𝕌) → Singᵤ (T₁ ∙×ᵤ T₂) ∙⟶ (T₁ ∙×ᵤ Singᵤ T₂)
coplusl : (T₁ T₂ : ∙𝕌) → Singᵤ (T₁ ∙+ᵤl T₂) ∙⟶ (Singᵤ T₁ ∙+ᵤl T₂)
coplusr : (T₁ T₂ : ∙𝕌) → Singᵤ (T₁ ∙+ᵤr T₂) ∙⟶ (T₁ ∙+ᵤr Singᵤ T₂)
∙eval : {T₁ T₂ : ∙𝕌} → (C : T₁ ∙⟶ T₂) →
let (t₁ , v₁) = ∙⟦ T₁ ⟧
(t₂ , v₂) = ∙⟦ T₂ ⟧
in Σ (t₁ → t₂) (λ f → f v₁ ≡ v₂)
∙eval ∙id⟷ = (λ x → x) , refl
∙eval (∙c c) = eval c , refl
∙eval (C₁ ∙⊚ C₂) with ∙eval C₁ | ∙eval C₂
... | (f , p) | (g , q) = (λ x → g (f x)) , trans (cong g p) q
∙eval (∙Singᵤ T₁ T₂ C) with ∙⟦ T₁ ⟧ | ∙⟦ T₂ ⟧ | ∙eval C
... | t₁ , v₁ | t₂ , .(f v₁) | f , refl = (λ {(x , refl) → f x , refl}) , refl
∙eval (return T) = (λ _ → proj₂ ∙⟦ T ⟧ , refl) , refl
∙eval (join T) = (λ { (._ , refl) → (proj₂ ∙⟦ T ⟧) , refl} ) , refl
∙eval (tensorl T₁ T₂) = (λ {_ → (proj₂ ∙⟦ T₁ ⟧ , proj₂ ∙⟦ T₂ ⟧) , refl}) , refl
∙eval (tensorr T₁ T₂) = (λ {_ → (proj₂ ∙⟦ T₁ ⟧ , proj₂ ∙⟦ T₂ ⟧) , refl}) , refl
∙eval (tensor T₁ T₂) = (λ {_ → (proj₂ ∙⟦ T₁ ⟧ , proj₂ ∙⟦ T₂ ⟧) , refl}) , refl
∙eval (untensor T₁ T₂) = (λ _ → ((proj₂ ∙⟦ T₁ ⟧ , refl) , (proj₂ ∙⟦ T₂ ⟧ , refl))) , refl
∙eval (plusl T₁ T₂) = (λ _ → inj₁ (proj₂ ∙⟦ T₁ ⟧) , refl) , refl
∙eval (plusr T₁ T₂) = (λ _ → inj₂ (proj₂ ∙⟦ T₂ ⟧) , refl) , refl
-- ∙eval (plus T₁ T₂) with ∙⟦ T₁ ⟧ | ∙⟦ T₂ ⟧
-- ... | t₁ , v₁ | t₂ , v₂ = (λ _ → inj₁ v₁ , refl) , refl
∙eval (extract T) = (λ {(w , refl) → w}) , refl
∙eval (duplicate T) = (λ {(w , refl) → (w , refl) , refl}) , refl
∙eval (cotensorl T₁ T₂) = (λ _ → ((proj₂ ∙⟦ T₁ ⟧ , refl) , proj₂ ∙⟦ T₂ ⟧)) , refl
∙eval (cotensorr T₁ T₂) = (λ _ → (proj₂ ∙⟦ T₁ ⟧ , (proj₂ ∙⟦ T₂ ⟧) , refl)) , refl
∙eval (coplusl T₁ T₂) = (λ _ → inj₁ (proj₂ ∙⟦ T₁ ⟧ , refl)) , refl
∙eval (coplusr T₁ T₂) = (λ _ → inj₂ (proj₂ ∙⟦ T₂ ⟧ , refl)) , refl
-----------------------------------------------------------------------------
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{-# OPTIONS --no-universe-polymorphism #-}
open import Induction.WellFounded as WF
open import Induction.Nat
open import Relation.Binary.Core hiding (Total)
open import Relation.Unary as U using (Decidable)
open import Relation.Nullary
open import Function using (_on_)
open import Data.Nat
import Level as L using (zero)
open import Data.List
open import BagEquality
module Lists where
infixr 8 _<l_
_<l_ : {A : Set} → Rel (List A) L.zero
_<l_ = Data.Nat._<′_ on length
<l-wellFounded : {A : Set} → Well-founded (_<l_ {A = A})
<l-wellFounded = newWf where
module InverseOfProj = WF.Inverse-image length
newWf = InverseOfProj.well-founded <′-well-founded
data Ordered : {A : Set} → Rel A _ → List A → Set where
NilIsOrd : {A : Set} {LEQ : Rel A _} → Ordered LEQ []
SingleIsOrd : {A : Set} {LEQ : Rel A _} {x : A} → Ordered LEQ [ x ]
HeadTailOrd : {A : Set} {LEQ : Rel A _} {x y : A} {zs : List A} → LEQ x y → Ordered LEQ (y ∷ zs) → Ordered LEQ (x ∷ (y ∷ zs))
cons-Order-cong : {A : Set} → {LEQ : Rel A L.zero} → {x : A} → {ys : List A} → (∀ z → z ∈ ys → LEQ x z) → Ordered LEQ ys → Ordered LEQ (x ∷ ys)
cons-Order-cong {A} {LEQ} {y} {[]} _ _ = SingleIsOrd
cons-Order-cong {A} {LEQ} {x} {y ∷ ys} xLEQ ord-ys = HeadTailOrd (xLEQ y (inj₁ refl)) ord-ys
++-Order-cong : {A : Set} {LEQ : Rel A L.zero} → {xs ys : List A} {y : A} → (∀ z → z ∈ xs → LEQ z y) → Ordered LEQ xs → Ordered LEQ (y ∷ ys) → Ordered LEQ (xs ++ (y ∷ ys))
++-Order-cong {A} {LEQ} {[]} {ys} {y} _ _ ord-y∷ys = ord-y∷ys
++-Order-cong {A} {LEQ} {x ∷ []} {ys} {y} yGEQ _ ord-y∷ys = HeadTailOrd (yGEQ x (inj₁ refl)) ord-y∷ys
++-Order-cong {A} {LEQ} {x₁ ∷ (x₂ ∷ xs)} {ys} {y} yGEQ (HeadTailOrd x₁LEQx₂ ord-x∷xs) ord-y∷ys = HeadTailOrd x₁LEQx₂ (++-Order-cong (λ z zIn → yGEQ z (inj₂ zIn)) ord-x∷xs ord-y∷ys)
Total : {A : Set} → Rel A L.zero → Set
Total {A} LEQ = (x y : A) → LEQ x y ⊕ LEQ y x
data ListPrimitive : {A : Set} → List A → Set where
NilIsPrim : ∀{A} → ListPrimitive {A = A} []
consIsNotPrim : {A : Set} → {x : A} → {xs : List A} → ¬ ListPrimitive (x ∷ xs)
consIsNotPrim ()
primDec : {A : Set} → U.Decidable (ListPrimitive {A = A})
primDec [] = yes NilIsPrim
primDec (x ∷ xs) = no consIsNotPrim
data ListPrimitive2 : {A : Set} → List A → Set where
NilIsPrim2 : ∀{A} → ListPrimitive2 {A = A} []
SingleIsPrim2 : ∀{A} {x : A} → ListPrimitive2 (x ∷ [])
consConsIsNotPrim2 : {A : Set} → {x y : A} → {zs : List A} → ¬ ListPrimitive2 (x ∷ (y ∷ zs))
consConsIsNotPrim2 ()
prim2Dec : {A : Set} → U.Decidable (ListPrimitive2 {A = A})
prim2Dec [] = yes NilIsPrim2
prim2Dec (x ∷ []) = yes SingleIsPrim2
prim2Dec (x ∷ (y ∷ zs)) = no consConsIsNotPrim2
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module Generic.Lib.Equality.Propositional where
open import Level
open import Relation.Binary
open import Data.Empty
infix 3 _≡_ _≢_ _≗_
data _≡_ {α} {A : Set α} (x : A) : A -> Set where
instance refl : x ≡ x
pattern lrefl = lift refl
_≢_ : ∀ {α} {A : Set α} -> A -> A -> Set
x ≢ y = x ≡ y -> ⊥
_≗_ : ∀ {α β} {A : Set α} {B : A -> Set β} -> (∀ x -> B x) -> (∀ x -> B x) -> Set α
f ≗ g = ∀ x -> f x ≡ g x
sym : ∀ {α} {A : Set α} {x y : A} -> x ≡ y -> y ≡ x
sym refl = refl
trans : ∀ {α} {A : Set α} {x y z : A} -> x ≡ y -> y ≡ z -> x ≡ z
trans refl refl = refl
left : ∀ {α} {A : Set α} {x y z : A} -> y ≡ x -> z ≡ x -> y ≡ z
left refl refl = refl
right : ∀ {α} {A : Set α} {x y z : A} -> x ≡ y -> x ≡ z -> y ≡ z
right refl refl = refl
subst : ∀ {α β} {A : Set α} {x y} -> (B : A -> Set β) -> x ≡ y -> B x -> B y
subst B refl z = z
cong : ∀ {α β} {A : Set α} {B : Set β} {x y} -> (f : A -> B) -> x ≡ y -> f x ≡ f y
cong f refl = refl
cong₂ : ∀ {α β γ} {A : Set α} {B : Set β} {C : Set γ} {x₁ x₂ y₁ y₂}
-> (g : A -> B -> C) -> x₁ ≡ x₂ -> y₁ ≡ y₂ -> g x₁ y₁ ≡ g x₂ y₂
cong₂ g refl refl = refl
≡-Setoid : ∀ {α} -> Set α -> Setoid α zero
≡-Setoid A = record
{ Carrier = A
; _≈_ = _≡_
; isEquivalence = record
{ refl = refl
; sym = sym
; trans = trans
}
}
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module agda where
open import IO
main = run (putStrLn "Happy New Year 1396")
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module Data.QuadTree.Implementation.PublicFunctions where
open import Haskell.Prelude renaming (zero to Z; suc to S)
open import Data.Lens.Lens
open import Data.Logic
open import Data.QuadTree.Implementation.PropDepthRelation
open import Data.QuadTree.Implementation.Definition
open import Data.QuadTree.Implementation.ValidTypes
open import Data.QuadTree.Implementation.QuadrantLenses
open import Data.QuadTree.Implementation.DataLenses
open import Data.QuadTree.Implementation.SafeFunctions
{-# FOREIGN AGDA2HS
{-# LANGUAGE Safe #-}
{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE Rank2Types #-}
import Data.Nat
import Data.Lens.Lens
import Data.Logic
import Data.QuadTree.Implementation.Definition
import Data.QuadTree.Implementation.ValidTypes
import Data.QuadTree.Implementation.QuadrantLenses
import Data.QuadTree.Implementation.DataLenses
import Data.QuadTree.Implementation.SafeFunctions
#-}
---- Unsafe functions (Original)
makeTree : {t : Set} {{eqT : Eq t}} -> (size : Nat × Nat) -> t -> QuadTree t
makeTree size v = qtFromSafe $ makeTreeSafe size v
{-# COMPILE AGDA2HS makeTree #-}
getLocation : {t : Set} {{eqT : Eq t}}
-> (loc : Nat × Nat) -> {dep : Nat}
-> (qt : QuadTree t)
-> {.(IsTrue (isInsideQuadTree loc qt))}
-> {.(IsTrue (isValid dep (treeToQuadrant qt)))} -> {.(IsTrue (dep == maxDepth qt))}
-> t
getLocation loc qt {inside} {p} {q} = getLocationSafe loc (maxDepth qt) (qtToSafe qt {p} {q}) {inside}
{-# COMPILE AGDA2HS getLocation #-}
setLocation : {t : Set} {{eqT : Eq t}}
-> (loc : Nat × Nat) -> t
-> {dep : Nat} -> (qt : QuadTree t)
-> {.(IsTrue (isInsideQuadTree loc qt))}
-> {.(IsTrue (isValid dep (treeToQuadrant qt)))} -> {.(IsTrue (dep == maxDepth qt))}
-> QuadTree t
setLocation loc v qt {inside} {p} {q} = qtFromSafe $ setLocationSafe loc (maxDepth qt) v (qtToSafe qt {p} {q}) {inside}
{-# COMPILE AGDA2HS setLocation #-}
mapLocation : {t : Set} {{eqT : Eq t}}
-> (loc : Nat × Nat) -> (t -> t)
-> {dep : Nat} -> (qt : QuadTree t)
-> {.(IsTrue (isInsideQuadTree loc qt))}
-> {.(IsTrue (isValid dep (treeToQuadrant qt)))} -> {.(IsTrue (dep == maxDepth qt))}
-> QuadTree t
mapLocation loc f qt {inside} {p} {q} = qtFromSafe $ mapLocationSafe loc (maxDepth qt) f (qtToSafe qt {p} {q}) {inside}
{-# COMPILE AGDA2HS mapLocation #-} | {
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-- Copyright: (c) 2016 Ertugrul Söylemez
-- License: BSD3
-- Maintainer: Ertugrul Söylemez <[email protected]>
module Algebra.Group.Semigroup where
open import Algebra.Category
open import Core
-- A semigroup is an associative binary function.
record SemigroupOver {a r} (A : Set a) (Eq : Equiv {r = r} A) : Set (a ⊔ r) where
open Equiv Eq
field
_⋄_ : A → A → A
⋄-cong : ∀ {x1 x2 y1 y2} → x1 ≈ x2 → y1 ≈ y2 → x1 ⋄ y1 ≈ x2 ⋄ y2
assoc : ∀ x y z → (x ⋄ y) ⋄ z ≈ x ⋄ (y ⋄ z)
semigroupoid : Semigroupoid
semigroupoid =
record {
Ob = ⊤;
Hom = λ _ _ → A;
Eq = Eq;
_∘_ = _⋄_;
∘-cong = ⋄-cong;
assoc = assoc
}
open Semigroupoid semigroupoid public
using (Epic; Monic)
record Semigroup {a r} : Set (lsuc (a ⊔ r)) where
field
A : Set a
Eq : Equiv {r = r} A
semigroupOver : SemigroupOver A Eq
open Equiv Eq public
open SemigroupOver semigroupOver public
-- Commutative semigroups.
record IsComm {a r} (S : Semigroup {a} {r}) : Set (a ⊔ r) where
open Semigroup S
field
comm : ∀ x y → x ⋄ y ≈ y ⋄ x
-- A semigroup morphism is a function that maps the elements of one
-- semigroup to another while preserving the compositional structure.
record SemigroupMorphism
{sa sr ta tr}
(S : Semigroup {sa} {sr})
(T : Semigroup {ta} {tr})
: Set (sa ⊔ ta ⊔ sr ⊔ tr)
where
private
module S = Semigroup S
module T = Semigroup T
field
map : S.A → T.A
map-cong : ∀ {x y} → x S.≈ y → map x T.≈ map y
⋄-preserving : ∀ x y → map (x S.⋄ y) T.≈ map x T.⋄ map y
semifunctor : Semifunctor S.semigroupoid T.semigroupoid
semifunctor =
record {
F = λ _ → tt;
map = map;
map-cong = map-cong;
∘-preserving = ⋄-preserving
}
-- An equivalence relation on semigroup morphisms that considers
-- semigroup morphisms to be equal iff they map every element in the
-- domain to the same element in the codomain.
SemigroupMorphismEq :
∀ {sa sr ta tr}
{S : Semigroup {sa} {sr}} {T : Semigroup {ta} {tr}}
→ Equiv (SemigroupMorphism S T)
SemigroupMorphismEq {T = T} =
record {
_≈_ = λ F G →
let module F = SemigroupMorphism F
module G = SemigroupMorphism G
in ∀ x → F.map x ≈ G.map x;
refl = λ _ → refl;
sym = λ x≈y x → sym (x≈y x);
trans = λ x≈y y≈z x → trans (x≈y x) (y≈z x)
}
where
module T = Semigroup T
open Equiv T.Eq
-- The category of semigroups and semigroup morphisms.
Semigroups : ∀ {a r} → Category
Semigroups {a} {r} =
record {
semigroupoid = record {
Ob = Semigroup {a} {r};
Hom = SemigroupMorphism;
Eq = SemigroupMorphismEq;
_∘_ = _∘_;
∘-cong = λ {_} {_} {_} {F1} {F2} {G1} {G2} → ∘-cong {F1 = F1} {F2} {G1} {G2};
assoc = assoc
};
id = id;
left-id = left-right-id;
right-id = left-right-id
}
where
open module MyEquiv {S T} = Equiv (SemigroupMorphismEq {S = S} {T = T})
_∘_ : ∀ {S T U} → SemigroupMorphism T U → SemigroupMorphism S T → SemigroupMorphism S U
_∘_ {S} {T} {U} F G =
record {
map = F.map Sets.∘ G.map;
map-cong = λ p → F.map-cong (G.map-cong p);
⋄-preserving = λ x y →
U.begin
F.map (G.map (x S.⋄ y)) U.≈[ F.map-cong (G.⋄-preserving _ _) ]
F.map (G.map x T.⋄ G.map y) U.≈[ F.⋄-preserving _ _ ]
F.map (G.map x) U.⋄ F.map (G.map y)
U.qed
}
where
module S = Semigroup S
module T = Semigroup T
module U = Semigroup U
module F = SemigroupMorphism F
module G = SemigroupMorphism G
∘-cong :
∀ {S T U}
{F1 F2 : SemigroupMorphism T U}
{G1 G2 : SemigroupMorphism S T}
→ F1 ≈ F2 → G1 ≈ G2 → F1 ∘ G1 ≈ F2 ∘ G2
∘-cong {U = U} {F1} {F2} {G1} {G2} F1≈F2 G1≈G2 x =
U.begin
F1.map (G1.map x) U.≈[ F1.map-cong (G1≈G2 _) ]
F1.map (G2.map x) U.≈[ F1≈F2 _ ]
F2.map (G2.map x)
U.qed
where
module U = Semigroup U
module F1 = SemigroupMorphism F1
module F2 = SemigroupMorphism F2
module G1 = SemigroupMorphism G1
module G2 = SemigroupMorphism G2
assoc :
∀ {S T U V}
(F : SemigroupMorphism U V)
(G : SemigroupMorphism T U)
(H : SemigroupMorphism S T)
→ (F ∘ G) ∘ H ≈ F ∘ (G ∘ H)
assoc {V = V} _ _ _ _ = V.refl
where
module V = Semigroup V
id : ∀ {S} → SemigroupMorphism S S
id {S} =
record {
map = λ x → x;
map-cong = λ x → x;
⋄-preserving = λ _ _ → Semigroup.refl S
}
left-right-id :
∀ {S T}
(F : SemigroupMorphism S T)
→ F ≈ F
left-right-id {T = T} _ _ = T.refl
where
module T = Semigroup T
module Semigroups {a r} = Category (Semigroups {a} {r})
-- Product semigroups
_×S_ : ∀ {sa sr ta tr} → Semigroup {sa} {sr} → Semigroup {ta} {tr} → Semigroup
S ×S T =
record {
A = S.A × T.A;
Eq = record {
_≈_ = λ { (x1 , x2) (y1 , y2) → (x1 S.≈ y1) × (x2 T.≈ y2) };
refl = S.refl , T.refl;
sym = λ { (p1 , p2) → S.sym p1 , T.sym p2 };
trans = λ { (p1 , p2) (q1 , q2) → S.trans p1 q1 , T.trans p2 q2 }
};
semigroupOver = record {
_⋄_ = λ { (x1 , x2) (y1 , y2) → (x1 S.⋄ y1) , (x2 T.⋄ y2) };
⋄-cong = λ { (p1 , p2) (q1 , q2) → S.⋄-cong p1 q1 , T.⋄-cong p2 q2 };
assoc = λ { (x1 , x2) (y1 , y2) (z1 , z2) → S.assoc x1 y1 z1 , T.assoc x2 y2 z2 }
}
}
where
module S = Semigroup S
module T = Semigroup T
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module Categories.Sets where
open import Library
open import Categories
Sets : ∀{l} → Cat
Sets {l} = record{
Obj = Set l;
Hom = λ X Y → X → Y;
iden = id;
comp = λ f g → f ∘ g;
idl = refl;
idr = refl;
ass = refl}
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{-# OPTIONS --type-in-type #-}
module Nested where
record Σ₁ (A : Set)(B : A → Set) : Set where
constructor _,_
field fst : A
snd : B fst
infixr 2 _,_
record Σ (A : Set)(B : A → Set) : Set where
field p : Σ₁ A B
open Σ₁ p public
open Σ
data ⊤ : Set where
tt : ⊤
∃ : {A : Set}(B : A → Set) → Set
∃ B = Σ _ B
infix 10 _≡_
data _≡_ {A : Set}(a : A) : {B : Set} → B → Set where
refl : a ≡ a
Cat : Set
Cat =
∃ λ (Obj : Set) →
∃ λ (Hom : Obj → Obj → Set) →
∃ λ (id : ∀ X → Hom X X) →
∃ λ (_○_ : ∀ {X Y Z} → Hom Y Z → Hom X Y → Hom X Z) →
∃ λ (idl : ∀ {X Y}{f : Hom X Y} → id Y ○ f ≡ f) →
∃ λ (idr : ∀ {X Y}{f : Hom X Y} → f ○ id X ≡ f) →
∃ λ (assoc : ∀ {W X Y Z}{f : Hom W X}{g : Hom X Y}{h : Hom Y Z} →
(h ○ g) ○ f ≡ h ○ (g ○ f)) →
⊤
Obj : (C : Cat) → Set
Obj C = fst C
Hom : (C : Cat) → Obj C → Obj C → Set
Hom C = fst (snd C)
id : (C : Cat) → ∀ X → Hom C X X
id C = fst (snd (snd C))
comp : (C : Cat) → ∀ {X Y Z} → Hom C Y Z → Hom C X Y → Hom C X Z
comp C = fst (snd (snd (snd C)))
idl : (C : Cat) → ∀ {X Y}{f : Hom C X Y} → comp C (id C Y) f ≡ f
idl C = fst (snd (snd (snd (snd C))))
idr : (C : Cat) → ∀ {X Y}{f : Hom C X Y} → comp C f (id C X) ≡ f
idr C = fst (snd (snd (snd (snd (snd C)))))
assoc : (C : Cat) → ∀ {W X Y Z}{f : Hom C W X}{g : Hom C X Y}{h : Hom C Y Z} →
comp C (comp C h g) f ≡ comp C h (comp C g f)
assoc C = fst (snd (snd (snd (snd (snd (snd C))))))
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-- Andreas, 2018-10-29, issue #3246
-- More things are now allowed in mutual blocks.
-- @mutual@ just pushes the definition parts to the bottom.
-- Definitions exist for data, record, functions, and pattern synonyms.
{-# BUILTIN FLOAT Float #-} -- not (yet) allowed in mutual block
mutual
import Agda.Builtin.Bool
open Agda.Builtin.Bool
f : Bool
f = g -- pushed to bottom
-- module M where -- not (yet) allowed in mutual block
module B = Agda.Builtin.Bool
primitive
primFloatEquality : Float → Float → Bool
{-# INJECTIVE primFloatEquality #-} -- certainly a lie
open import Agda.Builtin.Equality
{-# DISPLAY primFloatEquality x y = x ≡ y #-}
postulate A : Set
{-# COMPILE GHC A = type Integer #-}
variable x : Bool
g : Bool
g = true -- pushed to bottom
{-# STATIC g #-}
record R : Set where
coinductive
field foo : R
{-# ETA R #-}
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module Oscar.Property.Equivalence where
open import Oscar.Function
open import Oscar.Level
open import Oscar.Property.Reflexivity
open import Oscar.Property.Symmetry
open import Oscar.Property.Transitivity
record Equivalence {𝔬} {⋆ : Set 𝔬} {𝔮} (_≋_ : ⋆ → ⋆ → Set 𝔮) : Set (𝔬 ⊔ 𝔮) where
field
⦃ ′reflexivity ⦄ : Reflexivity _≋_
⦃ ′symmetry ⦄ : Symmetry _≋_
⦃ ′transitivity ⦄ : Transitivity _≋_
open Equivalence ⦃ … ⦄ public
instance
Equivalence⋆ : ∀
{𝔬} {⋆ : Set 𝔬} {𝔮} {_≋_ : ⋆ → ⋆ → Set 𝔮}
⦃ _ : Reflexivity _≋_ ⦄
⦃ _ : Symmetry _≋_ ⦄
⦃ _ : Transitivity _≋_ ⦄
→ Equivalence _≋_
Equivalence.′reflexivity Equivalence⋆ = it
Equivalence.′symmetry Equivalence⋆ = it
Equivalence.′transitivity Equivalence⋆ = it
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module Chunk where
open import Codata.Stream using (chunksOf; iterate; take)
open import Data.Nat using (ℕ; suc)
open import Data.Vec using (Vec; []; _∷_)
-- mylist 3 = [[1,2,3],[4,5,6],[7,8,9]]
myVec : (n : ℕ) → Vec (Vec ℕ n) n
myVec n = take n (chunksOf n (iterate suc 1))
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open import Prelude
module Implicits.Resolution.Finite.Decidable where
open import Coinduction
open import Data.Fin.Substitution
open import Data.List.Any
open Membership-≡
open import Implicits.Syntax
open import Implicits.Syntax.Type.Unification
open import Implicits.Substitutions
open import Implicits.Substitutions.Lemmas
open import Implicits.Resolution.Finite.Resolution
open import Function.Inverse as Inv using (_↔_; module Inverse)
open import Function.Equality hiding (cong)
open import Data.List.Any.Properties using (Any↔)
module M = MetaTypeMetaSubst
mutual
{-# NO_TERMINATION_CHECK #-}
match' : ∀ {m ν} (Δ : ICtx ν) s r∈Δ τ → (r : MetaType m ν) →
Dec (∃ λ u → Δ & s , r∈Δ ⊢ from-meta (r M./ u) ↓ τ)
match' Δ s r∈Δ τ (simpl x) with mgu (simpl x) τ
match' Δ s r∈Δ τ (simpl x) | just (u , p) =
yes (
(asub u) ,
subst (λ a → Δ & s , r∈Δ ⊢ a ↓ τ) (sym $ mgu-unifies (simpl x) τ (u , p)) (i-simp τ)
)
match' Δ s r∈Δ τ (simpl x) | nothing = no (λ{ (proj₁ , proj₂) → {!!} })
match' Δ s r∈Δ τ (a ⇒ b) with match' Δ s r∈Δ τ b
match' Δ s r∈Δ τ (a ⇒ b) | no ¬p = no {!!}
match' Δ s r∈Δ τ (a ⇒ b) | yes (u , b/u↓τ) with
all (λ x → (r∈Δ , (from-meta (a M./ u))) ?⊬dom x) s
match' Δ s r∈Δ τ (a ⇒ b) | yes (u , b/u↓τ) | yes p
-- The following with clause fails to satisfy termination checking
with let a' = from-meta (a M./ u) in resolve' Δ ((r∈Δ , a') List.∷ s) a'
match' Δ s r∈Δ τ (a ⇒ b) | yes (u , b/u↓τ) | yes p | yes ⊢ᵣa = yes (u , i-iabs p ⊢ᵣa b/u↓τ)
match' Δ s r∈Δ τ (a ⇒ b) | yes (u , b/u↓τ) | yes p | no ¬⊢ᵣa =
no (λ{ (u' , i-iabs x ⊢ᵣa a⇒b↓τ) → {!¬⊢ᵣa ⊢ᵣa!} })
match' Δ s r∈Δ τ (a ⇒ b) | yes (u , b/u↓τ) | no ¬p = no {!!}
-- The following with clause fails to satisfy termination checking
match' Δ s r∈Δ τ (∀' a) with match' Δ s r∈Δ τ (open-meta a)
match' Δ s r∈Δ τ (∀' a) | yes p = yes $ lem p
where
lem : (∃ λ u → Δ & s , r∈Δ ⊢ (from-meta ((open-meta a) M./ u)) ↓ τ) →
∃ λ u' → Δ & s , r∈Δ ⊢ (from-meta (∀' a M./ u')) ↓ τ
lem (u ∷ us , p) = us , (i-tabs (from-meta u) (subst (λ v → Δ & s , r∈Δ ⊢ v ↓ τ) (begin
from-meta (M._/_ (open-meta a) (u ∷ us))
≡⟨ cong (λ v → from-meta (M._/_ (open-meta a) v)) (sym $ us↑-⊙-sub-u≡u∷us u us) ⟩
from-meta ((open-meta a) M./ (us M.↑ M.⊙ (M.sub u)))
≡⟨ cong from-meta (/-⊙ (open-meta a)) ⟩
from-meta ((open-meta a) M./ us M.↑ M./ (M.sub u))
≡⟨ lem' a u us ⟩
from-meta (M._/_ a (us M.↑tp)) tp[/tp from-meta u ] ∎) p))
where open MetaTypeMetaLemmas hiding (subst)
match' Δ s r∈Δ τ (∀' r) | no ¬p = no {!!}
-- match defers to match', which concerns itself with MetaTypes.
-- If match' finds a match, we can use the fact that we have zero unification variables open here
-- to show that we found the right thing.
match : ∀ {ν} (Δ : ICtx ν) s r∈Δ → ∀ r τ → Dec (Δ & s , r∈Δ ⊢ r ↓ τ)
match Δ s r∈Δ a τ with match' Δ s r∈Δ τ (to-meta {zero} a)
match Δ s r∈Δ a τ | yes x = yes (lem x)
where
eq : ∀ {ν} {a : Type ν} {s} → from-meta (to-meta {zero} a M./ s) ≡ a
eq {a = a} {s = []} = begin
from-meta (M._/_ (to-meta {zero} a) [])
≡⟨ cong (λ q → from-meta q) (MetaTypeMetaLemmas.id-vanishes (to-meta {zero} a)) ⟩
from-meta (to-meta {zero} a)
≡⟨ to-meta-zero-vanishes ⟩
a ∎
lem : ∃ (λ u → Δ & s , r∈Δ ⊢ from-meta (to-meta {zero} a M./ u) ↓ τ) →
Δ & s , r∈Δ ⊢ a ↓ τ
lem ([] , proj₂) = subst (λ u → Δ & s , r∈Δ ⊢ u ↓ τ) eq proj₂
match Δ s r∈Δ a τ | no ¬p = no {!!}
match1st : ∀ {ν} (Δ : ICtx ν) s τ → Dec (Any (λ r → Δ & s , r ⊢ r ↓ τ) Δ)
match1st Δ s τ = match1st' Δ Δ s τ -- any (λ r → match Δ r τ)
where
match1st' : ∀ {ν} (Δ ρs : ICtx ν) s → (τ : SimpleType ν) →
Dec (Any (λ r → Δ & s , r ⊢ r ↓ τ) ρs)
match1st' Δ List.[] s τ = no {!!}
match1st' Δ (x List.∷ xs) s τ with match Δ s x x τ
match1st' Δ (x List.∷ xs) s τ | yes px = yes (here px)
match1st' Δ (x List.∷ xs) s τ | no ¬p with match1st' Δ xs s τ
match1st' Δ (x List.∷ xs) s τ | no ¬p | yes y = yes (there y)
match1st' Δ (x List.∷ xs) s τ | no ¬p | no ¬q = no {!!}
resolve' : ∀ {ν} (Δ : ICtx ν) s r → (Dec (Δ & s ⊢ᵣ r))
resolve' Δ s (simpl x) with match1st Δ s x
resolve' Δ s (simpl x) | yes p =
let r , r∈Δ , r↓x = (Inverse.from Any↔) Π.⟨$⟩ p in yes (r-simp r∈Δ r↓x)
resolve' Δ s (simpl x) | no ¬p =
no (λ{ (r-simp x₁ x₂) → ¬p (_⟨$⟩_ (Inverse.to Any↔) (_ , (x₁ , x₂))) })
resolve' Δ s (a ⇒ b) with resolve' (a List.∷ Δ) s b
resolve' Δ s (a ⇒ b) | yes p = yes (r-iabs p)
resolve' Δ s (a ⇒ b) | no ¬p = no (λ{ (r-iabs x) → ¬p x })
resolve' Δ s (∀' r) with resolve' (ictx-weaken Δ) (stack-weaken s) r
resolve' Δ s (∀' r) | yes p = yes (r-tabs p)
resolve' Δ s (∀' r) | no ¬p = no λ{ (r-tabs x) → ¬p x }
resolve : ∀ {ν} (Δ : ICtx ν) r → (Dec (Δ & List.[] ⊢ᵣ r))
resolve Δ r = resolve' Δ List.[] r
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{-# OPTIONS --rewriting #-}
-- {-# OPTIONS -v rewriting:100 #-}
postulate
_↦_ : ∀ {i} {A : Set i} → A → A → Set i
{-# BUILTIN REWRITE _↦_ #-}
postulate
Unit : Set
tt : Unit
module _ {i} (P : Unit → Set i) (tt* : P tt) where
postulate
Unit-elim : (x : Unit) → P x
Unit-β : Unit-elim tt ↦ tt*
{-# REWRITE Unit-β #-}
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