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{-
Some basic utilities for reflection
-}
{-# OPTIONS --cubical --no-exact-split --safe #-}
module Cubical.Reflection.Base where
open import Cubical.Foundations.Prelude
open import Cubical.Data.List.Base
open import Cubical.Data.Nat.Base
import Agda.Builtin.Reflection as R
open import Agda.Builtin.String
_>>=_ = R.bindTC
_<|>_ = R.catchTC
_$_ : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → (A → B) → A → B
f $ a = f a
_>>_ : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → R.TC A → R.TC B → R.TC B
f >> g = f >>= λ _ → g
infixl 4 _>>=_ _>>_ _<|>_
infixr 3 _$_
liftTC : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → (A → B) → R.TC A → R.TC B
liftTC f ta = ta >>= λ a → R.returnTC (f a)
v : ℕ → R.Term
v n = R.var n []
pattern varg t = R.arg (R.arg-info R.visible R.relevant) t
pattern harg t = R.arg (R.arg-info R.hidden R.relevant) t
pattern _v∷_ a l = varg a ∷ l
pattern _h∷_ a l = harg a ∷ l
infixr 5 _v∷_ _h∷_
vlam : String → R.Term → R.Term
vlam str t = R.lam R.visible (R.abs str t)
newMeta = R.checkType R.unknown
|
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|
-- Andreas, 2013-11-08
module Issue532 where
module M (A : Set) where
postulate
ax : A
P : A → Set
record R (A : Set) : Set where
open M A public
field
f : P ax
open R
-- Error WAS:
-- Not a valid let-declaration
-- when scope checking let open M A public in (f : P ax) → Set₀
S : {A : Set} → R A → Set
S r = P r (ax r)
-- Works now.
|
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|
{-# OPTIONS --type-in-type #-} -- yes, there will be some cheating in this lecture
module Lec4Done where
open import Lec1Done
open import Lec2Done
open import Lec3Done
-- the identity functor (the identity action on objects and arrows)
ID : {C : Category} -> C => C
ID = id~> where open Category CATEGORY
-- composition of functors (composition of actions on objects and arrows)
_>F>_ : {C D E : Category} -> (C => D) -> (D => E) -> (C => E)
F >F> G = F >~> G where open Category CATEGORY
-- EXAMPLES
data Maybe (X : Set) : Set where
yes : (x : X) -> Maybe X
no : Maybe X
maybe : {X Y : Set} -> (X -> Y) -> Maybe X -> Maybe Y
maybe f (yes x) = yes (f x)
maybe f no = no
MAYBE : SET => SET
MAYBE = record
{ F-Obj = Maybe
; F-map = maybe
; F-map-id~> = extensionality \ { (yes x) -> refl (yes x) ; no -> refl no }
; F-map->~> = \ f g -> extensionality \ { (yes x) -> refl (yes (g (f x))) ; no -> refl no }
}
module MAYBE-CAT where
open Category SET
open _=>_ MAYBE
joinMaybe : {X : Set} -> Maybe (Maybe X) -> Maybe X
joinMaybe no = no -- goes wrong sooner
joinMaybe (yes mx) = mx -- maybe goes wrong later
MAYBE-CAT : Category
MAYBE-CAT = record
{ Obj = Set
; _~>_ = \ S T -> S -> Maybe T
; id~> = yes
; _>~>_ = \ f g -> f >~> maybe g >~> joinMaybe
; law-id~>>~> = \ f -> extensionality \ x -> help1 (f x)
; law->~>id~> = \ f -> extensionality \ x -> help2 (f x)
; law->~>>~> = \ f g h -> extensionality \ x -> help3 g h (f x)
} where
help1 : {T : Set} (w : Maybe T) -> joinMaybe (yes w) == w
help1 mx = refl mx
help2 : {T : Set} (w : Maybe T) -> joinMaybe (maybe yes w) == w
help2 (yes x) = refl (yes x)
help2 no = refl no
help3 : {R S T : Set} (g : R -> Maybe S) (h : S -> Maybe T)
(w : Maybe R) ->
joinMaybe (maybe h (joinMaybe (maybe g w))) ==
joinMaybe (maybe (\ x -> joinMaybe (maybe h (g x))) w)
help3 g h (yes x) with g x
help3 g h (yes x) | yes y = refl (h y)
help3 g h (yes x) | no = refl no
help3 g h no = refl no
module NATURAL-TRANSFORMATION {C D : Category} where
open Category
open _=>_
record _~~>_ (F G : C => D) : Set where
field
-- one operation
xform : {X : Obj C} -> _~>_ D (F-Obj F X) (F-Obj G X)
-- one law
xform-natural : {X Y : Obj C}(f : _~>_ C X Y) ->
_>~>_ D (F-map F f) (xform {Y})
==
_>~>_ D (xform {X}) (F-map G f)
module MAYBE-GADGETS where
open NATURAL-TRANSFORMATION {SET}{SET}
open Category SET
open _=>_ MAYBE
open _~~>_
unitMaybe : ID ~~> MAYBE
unitMaybe = record
{ xform = yes
; xform-natural = \ f -> refl (f >~> yes)
}
multMaybe : (MAYBE >F> MAYBE) ~~> MAYBE
multMaybe = record
{ xform = MAYBE-CAT.joinMaybe
; xform-natural = \ f -> extensionality \
{ (yes (yes x)) -> refl (yes (f x))
; (yes no) -> refl no
; no -> refl no }
}
law1 : {X : Set} -> (xform unitMaybe >~> xform multMaybe) == id~> {Maybe X}
law1 = extensionality \ { (yes x) -> refl (yes x) ; no -> refl no }
law2 : {X : Set} -> (F-map (xform unitMaybe) >~> xform multMaybe) == id~> {Maybe X}
law2 = extensionality \ { (yes x) -> refl (yes x) ; no -> refl no }
law3 : {X : Set} -> (xform multMaybe >~> xform multMaybe)
== (F-map (xform multMaybe) >~> xform multMaybe {X})
law3 = extensionality \ { (yes (yes mx)) -> refl mx ; (yes no) -> refl no ; no -> refl no }
MAYBE-CAT2 : Category
MAYBE-CAT2 = record
{ Obj = Set
; _~>_ = \ S T -> S -> Maybe T
; id~> = xform unitMaybe
; _>~>_ = \ f g -> f >~> F-map g >~> xform multMaybe
; law-id~>>~> = \ f ->
xform unitMaybe >~> F-map f >~> xform multMaybe
=< law->~>>~> (xform unitMaybe) (F-map f) (xform multMaybe) ]=
(xform unitMaybe >~> F-map f) >~> xform multMaybe
=< refl (_>~> xform multMaybe) =$= xform-natural unitMaybe f ]=
(f >~> xform unitMaybe) >~> xform multMaybe
=[ law->~>>~> f (xform unitMaybe) (xform multMaybe) >=
f >~> (xform unitMaybe >~> xform multMaybe)
=[ refl (f >~>_) =$= law1 >=
f >~> id~>
=[ law->~>id~> f >=
f [QED]
; law->~>id~> = \ f ->
f >~> (F-map (xform unitMaybe) >~> xform multMaybe)
=[ refl (f >~>_) =$= law2 >=
f >~> id~>
=[ law->~>id~> f >=
f [QED]
; law->~>>~> = \ f g h ->
(f >~> (F-map g >~> xform multMaybe)) >~> (F-map h >~> xform multMaybe)
=[ law->~>>~> f (F-map g >~> xform multMaybe) (F-map h >~> xform multMaybe) >=
f >~> (F-map g >~> xform multMaybe) >~> (F-map h >~> xform multMaybe)
=[ refl (f >~>_) =$= (
(F-map g >~> xform multMaybe) >~> (F-map h >~> xform multMaybe)
=[ law->~>>~> (F-map g) (xform multMaybe) (F-map h >~> xform multMaybe) >=
F-map g >~> (xform multMaybe >~> (F-map h >~> xform multMaybe))
=[ refl (F-map g >~>_) =$= (
xform multMaybe >~> (F-map h >~> xform multMaybe)
=< law->~>>~> (xform multMaybe) (F-map h) (xform multMaybe) ]=
(xform multMaybe >~> F-map h) >~> xform multMaybe
=< refl (_>~> xform multMaybe) =$= xform-natural multMaybe h ]=
(F-map (F-map h) >~> xform multMaybe) >~> xform multMaybe
=[ law->~>>~> (F-map (F-map h)) (xform multMaybe) (xform multMaybe) >=
F-map (F-map h) >~> (xform multMaybe >~> xform multMaybe)
=[ refl (F-map (F-map h) >~>_) =$= law3 >=
F-map (F-map h) >~> (F-map (xform multMaybe) >~> xform multMaybe)
=< law->~>>~> (F-map (F-map h)) (F-map (xform multMaybe)) (xform multMaybe) ]=
(F-map (F-map h) >~> F-map (xform multMaybe)) >~> xform multMaybe
=< refl (_>~> xform multMaybe) =$= F-map->~> (F-map h) (xform multMaybe) ]=
(F-map (F-map h >~> xform multMaybe) >~> xform multMaybe) [QED]
) >=
F-map g >~> (F-map (F-map h >~> xform multMaybe) >~> xform multMaybe)
=< law->~>>~> (F-map g) (F-map (F-map h >~> xform multMaybe)) (xform multMaybe) ]=
(F-map g >~> F-map (F-map h >~> xform multMaybe)) >~> xform multMaybe
=< refl (_>~> xform multMaybe) =$= F-map->~> g (F-map h >~> xform multMaybe) ]=
(F-map (g >~> F-map h >~> xform multMaybe) >~> xform multMaybe) [QED]
) >=
(f >~> F-map (g >~> F-map h >~> xform multMaybe) >~> xform multMaybe) [QED]
}
module MONAD {C : Category}(M : C => C) where
open NATURAL-TRANSFORMATION {C}{C}
open Category C
open _=>_ M
open _~~>_
record Monad : Set where
field
unit : ID ~~> M
mult : (M >F> M) ~~> M
unitMult : {X : Obj} -> (xform unit >~> xform mult) == id~> {F-Obj X}
multUnit : {X : Obj} -> (F-map (xform unit) >~> xform mult) == id~> {F-Obj X}
multMult : {X : Obj} -> (xform mult >~> xform mult) == (F-map (xform mult) >~> xform mult {X})
KLEISLI : Category
KLEISLI = record
{ Obj = Obj
; _~>_ = \ S T -> S ~> F-Obj T
; id~> = xform unit
; _>~>_ = \ f g -> f >~> F-map g >~> xform mult
; law-id~>>~> = \ f ->
xform unit >~> F-map f >~> xform mult
=< law->~>>~> (xform unit) (F-map f) (xform mult) ]=
(xform unit >~> F-map f) >~> xform mult
=< refl (_>~> xform mult) =$= xform-natural unit f ]=
(f >~> xform unit) >~> xform mult
=[ law->~>>~> f (xform unit) (xform mult) >=
f >~> (xform unit >~> xform mult)
=[ refl (f >~>_) =$= unitMult >=
f >~> id~>
=[ law->~>id~> f >=
f [QED]
; law->~>id~> = \ f ->
f >~> (F-map (xform unit) >~> xform mult)
=[ refl (f >~>_) =$= multUnit >=
f >~> id~>
=[ law->~>id~> f >=
f [QED]
; law->~>>~> = \ f g h ->
(f >~> (F-map g >~> xform mult)) >~> (F-map h >~> xform mult)
=[ law->~>>~> f (F-map g >~> xform mult) (F-map h >~> xform mult) >=
f >~> (F-map g >~> xform mult) >~> (F-map h >~> xform mult)
=[ refl (f >~>_) =$= (
(F-map g >~> xform mult) >~> (F-map h >~> xform mult)
=[ law->~>>~> (F-map g) (xform mult) (F-map h >~> xform mult) >=
F-map g >~> (xform mult >~> (F-map h >~> xform mult))
=[ refl (F-map g >~>_) =$= (
xform mult >~> (F-map h >~> xform mult)
=< law->~>>~> (xform mult) (F-map h) (xform mult) ]=
(xform mult >~> F-map h) >~> xform mult
=< refl (_>~> xform mult) =$= xform-natural mult h ]=
(F-map (F-map h) >~> xform mult) >~> xform mult
=[ law->~>>~> (F-map (F-map h)) (xform mult) (xform mult) >=
F-map (F-map h) >~> (xform mult >~> xform mult)
=[ refl (F-map (F-map h) >~>_) =$= multMult >=
F-map (F-map h) >~> (F-map (xform mult) >~> xform mult)
=< law->~>>~> (F-map (F-map h)) (F-map (xform mult)) (xform mult) ]=
(F-map (F-map h) >~> F-map (xform mult)) >~> xform mult
=< refl (_>~> xform mult) =$= F-map->~> (F-map h) (xform mult) ]=
(F-map (F-map h >~> xform mult) >~> xform mult) [QED]
) >=
F-map g >~> (F-map (F-map h >~> xform mult) >~> xform mult)
=< law->~>>~> (F-map g) (F-map (F-map h >~> xform mult)) (xform mult) ]=
(F-map g >~> F-map (F-map h >~> xform mult)) >~> xform mult
=< refl (_>~> xform mult) =$= F-map->~> g (F-map h >~> xform mult) ]=
(F-map (g >~> F-map h >~> xform mult) >~> xform mult) [QED]
) >=
(f >~> F-map (g >~> F-map h >~> xform mult) >~> xform mult) [QED]
}
data List (X : Set) : Set where
[] : List X
_,-_ : (x : X)(xs : List X) -> List X
list : {X Y : Set} -> (X -> Y) -> List X -> List Y
list f [] = []
list f (x ,- xs) = f x ,- list f xs
LIST : SET => SET
LIST = record
{ F-Obj = List
; F-map = list
; F-map-id~> = extensionality listId
; F-map->~> = \ f g -> extensionality (listCp f g)
} where
open Category SET
listId : {T : Set}(xs : List T) -> list id xs == xs
listId [] = refl []
listId (x ,- xs) = refl (_,-_ x) =$= listId xs
listCp : {R S T : Set} (f : R -> S) (g : S -> T) (xs : List R) →
list (f >~> g) xs == (list f >~> list g) xs
listCp f g [] = refl []
listCp f g (x ,- xs) = refl (_,-_ (g (f x))) =$= listCp f g xs
data Two : Set where tt ff : Two
data BitProcess (X : Set) : Set where -- in what way is X used?
stop : (x : X) -> BitProcess X -- stop with value x
send : (b : Two)(k : BitProcess X) -> BitProcess X -- send b, continue as k
recv : (kt kf : BitProcess X) -> BitProcess X -- receive bit, continue as
-- kt if tt, kf if ff
bpRun : forall {X} -> BitProcess X -- a process to run
-> List Two -- a list of bits waiting to be input
-> List Two -- the list of bits output
* Maybe -- and, if we don't run out of input
( X -- the resulting value
* List Two -- and the unused input
)
bpRun (stop x) bs = [] , yes (x , bs)
bpRun (send b k) bs with bpRun k bs
bpRun (send b k) bs | os , yes (x , is) = (b ,- os) , yes (x , is)
bpRun (send b k) bs | os , no = os , no
bpRun (recv kt kf) [] = [] , no
bpRun (recv kt kf) (tt ,- bs) = bpRun kt bs
bpRun (recv kt kf) (ff ,- bs) = bpRun kf bs
bitProcess : {X Y : Set} -> (X -> Y) -> BitProcess X -> BitProcess Y
bitProcess f (stop x) = stop (f x)
bitProcess f (send b k) = send b (bitProcess f k)
bitProcess f (recv kt kf) = recv (bitProcess f kt) (bitProcess f kf)
BITPROCESS : SET => SET -- actions on *values* lift to processes
BITPROCESS = record
{ F-Obj = BitProcess
; F-map = bitProcess
; F-map-id~> = extensionality helpId
; F-map->~> = \ f g -> extensionality (helpCp f g)
} where
open Category SET
helpId : {T : Set} (p : BitProcess T) -> bitProcess id p == p
helpId (stop x) = refl (stop x)
helpId (send b k) rewrite helpId k = refl (send b k)
helpId (recv kt kf) rewrite helpId kt | helpId kf = refl (recv kt kf)
helpCp : {R S T : Set} (f : R -> S)(g : S -> T) (p : BitProcess R) →
bitProcess (f >~> g) p == (bitProcess f >~> bitProcess g) p
helpCp f g (stop x) = refl (stop (g (f x)))
helpCp f g (send b k) rewrite helpCp f g k = refl (send b (bitProcess g (bitProcess f k)))
helpCp f g (recv kt kf) rewrite helpCp f g kt | helpCp f g kf
= refl (recv (bitProcess g (bitProcess f kt)) (bitProcess g (bitProcess f kf)))
|
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|
data Bool : Set where
true false : Bool
{-# BUILTIN BOOL Bool #-}
{-# BUILTIN TRUE true #-}
{-# BUILTIN FALSE true #-} -- hmm, no
|
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-- WARNING: This file was generated automatically by Vehicle
-- and should not be modified manually!
-- Metadata
-- - Agda version: 2.6.2
-- - AISEC version: 0.1.0.1
-- - Time generated: ???
{-# OPTIONS --allow-exec #-}
open import Vehicle
open import Vehicle.Data.Tensor
open import Data.Product
open import Data.Sum
open import Data.Integer as ℤ using (ℤ)
open import Data.Rational as ℚ using (ℚ)
open import Data.Fin as Fin using (Fin; #_)
open import Data.List
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary
module acasXu-temp-output where
InputVector : Set
InputVector = Tensor ℚ (5 ∷ [])
distanceToIntruder : ∀ {_x0 : Set} {{_x1 : HasNatLits _x0}} → _x0
distanceToIntruder = 0
angleToIntruder : ∀ {_x4 : Set} {{_x5 : HasNatLits _x4}} → _x4
angleToIntruder = 1
intruderHeading : ∀ {_x8 : Set} {{_x9 : HasNatLits _x8}} → _x8
intruderHeading = 2
speed : ∀ {_x12 : Set} {{_x13 : HasNatLits _x12}} → _x12
speed = 3
intruderSpeed : ∀ {_x16 : Set} {{_x17 : HasNatLits _x16}} → _x16
intruderSpeed = 4
OutputVector : Set
OutputVector = Tensor ℚ (5 ∷ [])
clearOfConflict : ∀ {_x20 : Set} {{_x21 : HasNatLits _x20}} → _x20
clearOfConflict = 0
weakLeft : ∀ {_x24 : Set} {{_x25 : HasNatLits _x24}} → _x24
weakLeft = 1
weakRight : ∀ {_x28 : Set} {{_x29 : HasNatLits _x28}} → _x28
weakRight = 2
strongLeft : ∀ {_x32 : Set} {{_x33 : HasNatLits _x32}} → _x32
strongLeft = 3
strongRight : ∀ {_x36 : Set} {{_x37 : HasNatLits _x36}} → _x36
strongRight = 4
postulate acasXu : InputVector → OutputVector
pi : ℚ
pi = ℤ.+ 392699 ℚ./ 125000
Advises : Fin 5 → (InputVector → Set)
Advises i x = ∀ (j : Fin 5) → i ≢ j → acasXu x i ℚ.< acasXu x j
IntruderDistantAndSlower : InputVector → Set
IntruderDistantAndSlower x = x distanceToIntruder ℚ.≥ ℤ.+ 55947691 ℚ./ 1000 × (x speed ℚ.≥ ℤ.+ 1145 ℚ./ 1 × x intruderSpeed ℚ.≤ ℤ.+ 60 ℚ./ 1)
abstract
property1 : ∀ (x : Tensor ℚ (5 ∷ [])) → IntruderDistantAndSlower x → acasXu x clearOfConflict ℚ.≤ ℤ.+ 1500 ℚ./ 1
property1 = checkSpecification record
{ proofCache = "/home/matthew/Code/AISEC/vehicle/proofcache.vclp"
}
abstract
property2 : ∀ (x : Tensor ℚ (5 ∷ [])) → IntruderDistantAndSlower x → ∃ λ (j : Fin 5) → acasXu x j ℚ.> acasXu x clearOfConflict
property2 = checkSpecification record
{ proofCache = "/home/matthew/Code/AISEC/vehicle/proofcache.vclp"
}
DirectlyAhead : InputVector → Set
DirectlyAhead x = (ℤ.+ 1500 ℚ./ 1 ℚ.≤ x distanceToIntruder × x distanceToIntruder ℚ.≤ ℤ.+ 1800 ℚ./ 1) × (ℚ.- (ℤ.+ 3 ℚ./ 50) ℚ.≤ x angleToIntruder × x angleToIntruder ℚ.≤ ℤ.+ 3 ℚ./ 50)
MovingTowards : InputVector → Set
MovingTowards x = x intruderHeading ℚ.≥ ℤ.+ 31 ℚ./ 10 × (x speed ℚ.≥ ℤ.+ 980 ℚ./ 1 × x intruderSpeed ℚ.≥ ℤ.+ 960 ℚ./ 1)
abstract
property3 : ∀ (x : Tensor ℚ (5 ∷ [])) → DirectlyAhead x × MovingTowards x → ¬ Advises clearOfConflict x
property3 = checkSpecification record
{ proofCache = "/home/matthew/Code/AISEC/vehicle/proofcache.vclp"
}
MovingAway : InputVector → Set
MovingAway x = x intruderHeading ≡ ℤ.+ 0 ℚ./ 1 × (ℤ.+ 1000 ℚ./ 1 ℚ.≤ x speed × (ℤ.+ 700 ℚ./ 1 ℚ.≤ x intruderSpeed × x intruderSpeed ℚ.≤ ℤ.+ 800 ℚ./ 1))
abstract
property4 : ∀ (x : Tensor ℚ (5 ∷ [])) → DirectlyAhead x × MovingAway x → ¬ Advises clearOfConflict x
property4 = checkSpecification record
{ proofCache = "/home/matthew/Code/AISEC/vehicle/proofcache.vclp"
}
NearAndApproachingFromLeft : InputVector → Set
NearAndApproachingFromLeft x = (ℤ.+ 250 ℚ./ 1 ℚ.≤ x distanceToIntruder × x distanceToIntruder ℚ.≤ ℤ.+ 400 ℚ./ 1) × ((ℤ.+ 1 ℚ./ 5 ℚ.≤ x angleToIntruder × x angleToIntruder ℚ.≤ ℤ.+ 2 ℚ./ 5) × ((ℚ.- pi ℚ.≤ x intruderHeading × x intruderHeading ℚ.≤ ℚ.- pi ℚ.+ ℤ.+ 1 ℚ./ 200) × ((ℤ.+ 100 ℚ./ 1 ℚ.≤ x speed × x speed ℚ.≤ ℤ.+ 400 ℚ./ 1) × (ℤ.+ 0 ℚ./ 1 ℚ.≤ x intruderSpeed × x intruderSpeed ℚ.≤ ℤ.+ 400 ℚ./ 1))))
abstract
property5 : ∀ (x : Tensor ℚ (5 ∷ [])) → NearAndApproachingFromLeft x → Advises strongRight x
property5 = checkSpecification record
{ proofCache = "/home/matthew/Code/AISEC/vehicle/proofcache.vclp"
}
IntruderFarAway : InputVector → Set
IntruderFarAway x = (ℤ.+ 12000 ℚ./ 1 ℚ.≤ x distanceToIntruder × x distanceToIntruder ℚ.≤ ℤ.+ 62000 ℚ./ 1) × ((ℚ.- pi ℚ.≤ x angleToIntruder × x angleToIntruder ℚ.≤ ℚ.- (ℤ.+ 7 ℚ./ 10) ⊎ ℤ.+ 7 ℚ./ 10 ℚ.≤ x angleToIntruder × x angleToIntruder ℚ.≤ pi) × ((ℚ.- pi ℚ.≤ x intruderHeading × x intruderHeading ℚ.≤ ℚ.- pi ℚ.+ ℤ.+ 1 ℚ./ 200) × ((ℤ.+ 100 ℚ./ 1 ℚ.≤ x speed × x speed ℚ.≤ ℤ.+ 1200 ℚ./ 1) × (ℤ.+ 0 ℚ./ 1 ℚ.≤ x intruderSpeed × x intruderSpeed ℚ.≤ ℤ.+ 1200 ℚ./ 1))))
abstract
property6 : ∀ (x : Tensor ℚ (5 ∷ [])) → IntruderFarAway x → Advises clearOfConflict x
property6 = checkSpecification record
{ proofCache = "/home/matthew/Code/AISEC/vehicle/proofcache.vclp"
}
LargeVerticalSeparation : InputVector → Set
LargeVerticalSeparation x = (ℤ.+ 0 ℚ./ 1 ℚ.≤ x distanceToIntruder × x distanceToIntruder ℚ.≤ ℤ.+ 60760 ℚ./ 1) × ((ℚ.- pi ℚ.≤ x angleToIntruder × x angleToIntruder ℚ.≤ pi) × ((ℚ.- pi ℚ.≤ x intruderHeading × x intruderHeading ℚ.≤ pi) × ((ℤ.+ 100 ℚ./ 1 ℚ.≤ x speed × x speed ℚ.≤ ℤ.+ 1200 ℚ./ 1) × (ℤ.+ 0 ℚ./ 1 ℚ.≤ x intruderSpeed × x intruderSpeed ℚ.≤ ℤ.+ 1200 ℚ./ 1))))
abstract
property7 : ∀ (x : Tensor ℚ (5 ∷ [])) → LargeVerticalSeparation x → ¬ Advises strongLeft x × ¬ Advises strongRight x
property7 = checkSpecification record
{ proofCache = "/home/matthew/Code/AISEC/vehicle/proofcache.vclp"
}
LargeVerticalSeparationAndPreviousWeakLeft : InputVector → Set
LargeVerticalSeparationAndPreviousWeakLeft x = (ℤ.+ 0 ℚ./ 1 ℚ.≤ x distanceToIntruder × x distanceToIntruder ℚ.≤ ℤ.+ 60760 ℚ./ 1) × ((ℚ.- pi ℚ.≤ x angleToIntruder × x angleToIntruder ℚ.≤ ℚ.- (ℤ.+ 3 ℚ./ 4) ℚ.* pi) × ((ℚ.- (ℤ.+ 1 ℚ./ 10) ℚ.≤ x intruderHeading × x intruderHeading ℚ.≤ ℤ.+ 1 ℚ./ 10) × ((ℤ.+ 600 ℚ./ 1 ℚ.≤ x speed × x speed ℚ.≤ ℤ.+ 1200 ℚ./ 1) × (ℤ.+ 600 ℚ./ 1 ℚ.≤ x intruderSpeed × x intruderSpeed ℚ.≤ ℤ.+ 1200 ℚ./ 1))))
abstract
property8 : ∀ (x : Tensor ℚ (5 ∷ [])) → LargeVerticalSeparationAndPreviousWeakLeft x → Advises clearOfConflict x ⊎ Advises weakLeft x
property8 = checkSpecification record
{ proofCache = "/home/matthew/Code/AISEC/vehicle/proofcache.vclp"
}
PreviousWeakRightAndNearbyIntruder : InputVector → Set
PreviousWeakRightAndNearbyIntruder x = (ℤ.+ 2000 ℚ./ 1 ℚ.≤ x distanceToIntruder × x distanceToIntruder ℚ.≤ ℤ.+ 7000 ℚ./ 1) × ((ℚ.- (ℤ.+ 2 ℚ./ 5) ℚ.≤ x angleToIntruder × x angleToIntruder ℚ.≤ ℚ.- (ℤ.+ 7 ℚ./ 50)) × ((ℚ.- pi ℚ.≤ x intruderHeading × x intruderHeading ℚ.≤ ℚ.- pi ℚ.+ ℤ.+ 1 ℚ./ 100) × ((ℤ.+ 100 ℚ./ 1 ℚ.≤ x speed × x speed ℚ.≤ ℤ.+ 150 ℚ./ 1) × (ℤ.+ 0 ℚ./ 1 ℚ.≤ x intruderSpeed × x intruderSpeed ℚ.≤ ℤ.+ 150 ℚ./ 1))))
abstract
property9 : ∀ (x : Tensor ℚ (5 ∷ [])) → PreviousWeakRightAndNearbyIntruder x → Advises strongLeft x
property9 = checkSpecification record
{ proofCache = "/home/matthew/Code/AISEC/vehicle/proofcache.vclp"
}
IntruderFarAway2 : InputVector → Set
IntruderFarAway2 x = (ℤ.+ 36000 ℚ./ 1 ℚ.≤ x distanceToIntruder × x distanceToIntruder ℚ.≤ ℤ.+ 60760 ℚ./ 1) × ((ℤ.+ 7 ℚ./ 10 ℚ.≤ x angleToIntruder × x angleToIntruder ℚ.≤ pi) × ((ℚ.- pi ℚ.≤ x intruderHeading × x intruderHeading ℚ.≤ ℚ.- pi ℚ.+ ℤ.+ 1 ℚ./ 100) × ((ℤ.+ 900 ℚ./ 1 ℚ.≤ x speed × x speed ℚ.≤ ℤ.+ 1200 ℚ./ 1) × (ℤ.+ 600 ℚ./ 1 ℚ.≤ x intruderSpeed × x intruderSpeed ℚ.≤ ℤ.+ 1200 ℚ./ 1))))
abstract
property10 : ∀ (x : Tensor ℚ (5 ∷ [])) → IntruderFarAway2 x → Advises clearOfConflict x
property10 = checkSpecification record
{ proofCache = "/home/matthew/Code/AISEC/vehicle/proofcache.vclp"
}
|
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------------------------------------------------------------------------------
-- Colist properties
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module FOTC.Data.Colist.PropertiesI where
open import FOTC.Base
open import FOTC.Base.List
open import FOTC.Data.Colist
------------------------------------------------------------------------------
-- Because a greatest post-fixed point is a fixed-point, then the
-- Colist predicate is also a pre-fixed point of the functional
-- ColistF, i.e.
--
-- ColistF Colist ≤ Colist (see FOTC.Data.Colist.Type).
Colist-in :
∀ {xs} →
xs ≡ [] ∨ (∃[ x' ] ∃[ xs' ] xs ≡ x' ∷ xs' ∧ Colist xs') →
Colist xs
Colist-in h = Colist-coind A h' h
where
A : D → Set
A xs = xs ≡ [] ∨ (∃[ x' ] ∃[ xs' ] xs ≡ x' ∷ xs' ∧ Colist xs')
h' : ∀ {xs} → A xs → xs ≡ [] ∨ (∃[ x' ] ∃[ xs' ] xs ≡ x' ∷ xs' ∧ A xs')
h' (inj₁ xs≡[]) = inj₁ xs≡[]
h' (inj₂ (x' , xs' , prf , Clxs')) = inj₂ (x' , xs' , prf , Colist-out Clxs')
|
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module _ where
open import Agda.Builtin.Nat
open import Agda.Builtin.Equality
module M (n : Nat) (m : Nat) where
foo : n ≡ 0 → Nat
foo refl = 0
module M' where
bar : Nat
bar = m
bad : Nat
bad = M'.bar -- internal error Monad.Signature:882
|
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|
module OldBasicILP.Syntax.Translation where
open import Common.Context public
import OldBasicILP.Syntax.ClosedHilbertSequential as CHS
import OldBasicILP.Syntax.ClosedHilbert as CH
-- Translation from closed Hilbert-style sequential to closed Hilbert-style.
mutual
chsᵀ→chᵀ : CHS.Ty → CH.Ty
chsᵀ→chᵀ (CHS.α P) = CH.α P
chsᵀ→chᵀ (A CHS.▻ B) = chsᵀ→chᵀ A CH.▻ chsᵀ→chᵀ B
chsᵀ→chᵀ (p CHS.⦂ A) = chsᴾ→chᴾ p CH.⦂ chsᵀ→chᵀ A
chsᵀ→chᵀ (A CHS.∧ B) = chsᵀ→chᵀ A CH.∧ chsᵀ→chᵀ B
chsᵀ→chᵀ CHS.⊤ = CH.⊤
chsᴾ→chᴾ : ∀ {Ξ A} → CHS.Proof Ξ A → CH.Proof (chsᵀ→chᵀ A)
chsᴾ→chᴾ CHS.[ d ] = CH.[ chsᴰ→ch d top ]
chsᴰ→ch : ∀ {Ξ A} → CHS.⊢ᴰ Ξ → A ∈ Ξ → CH.⊢ (chsᵀ→chᵀ A)
chsᴰ→ch (CHS.mp i j d) top = CH.app (chsᴰ→ch d i) (chsᴰ→ch d j)
chsᴰ→ch (CHS.ci d) top = CH.ci
chsᴰ→ch (CHS.ck d) top = CH.ck
chsᴰ→ch (CHS.cs d) top = CH.cs
chsᴰ→ch (CHS.nec `d d) top = CH.box (chsᴰ→ch `d top)
chsᴰ→ch (CHS.cdist {Ξ} {A} {B} {`Ξ₁} {`Ξ₂} {`d₁} {`d₂} d) top = oops {A} {B} {`Ξ₁} {`Ξ₂} {`d₁} {`d₂}
chsᴰ→ch (CHS.cup d) top = CH.cup
chsᴰ→ch (CHS.cdown d) top = CH.cdown
chsᴰ→ch (CHS.cpair d) top = CH.cpair
chsᴰ→ch (CHS.cfst d) top = CH.cfst
chsᴰ→ch (CHS.csnd d) top = CH.csnd
chsᴰ→ch (CHS.unit d) top = CH.unit
chsᴰ→ch (CHS.mp i j d) (pop k) = chsᴰ→ch d k
chsᴰ→ch (CHS.ci d) (pop k) = chsᴰ→ch d k
chsᴰ→ch (CHS.ck d) (pop k) = chsᴰ→ch d k
chsᴰ→ch (CHS.cs d) (pop k) = chsᴰ→ch d k
chsᴰ→ch (CHS.nec `d d) (pop k) = chsᴰ→ch d k
chsᴰ→ch (CHS.cdist d) (pop k) = chsᴰ→ch d k
chsᴰ→ch (CHS.cup d) (pop k) = chsᴰ→ch d k
chsᴰ→ch (CHS.cdown d) (pop k) = chsᴰ→ch d k
chsᴰ→ch (CHS.cpair d) (pop k) = chsᴰ→ch d k
chsᴰ→ch (CHS.cfst d) (pop k) = chsᴰ→ch d k
chsᴰ→ch (CHS.csnd d) (pop k) = chsᴰ→ch d k
chsᴰ→ch (CHS.unit d) (pop k) = chsᴰ→ch d k
-- FIXME: I can’t even postulate this.
-- postulate
-- ᴬlem₁ : ∀ {Ξ₁ Ξ₂ A B} {d₁ : CHS.⊢ᴰ Ξ₁ , A CHS.▻ B} {d₂ : CHS.⊢ᴰ Ξ₂ , A}
-- → chsᴰ→ch (CHS.appᴰ d₁ d₂) ≡ CH.app (chsᴰ→ch d₁ top) (chsᴰ→ch d₂ top)
postulate
oops : ∀ {A B Ξ₁ Ξ₂} {d₁ : CHS.⊢ᴰ Ξ₁ , A CHS.▻ B} {d₂ : CHS.⊢ᴰ Ξ₂ , A}
→ CH.⊢ chsᴾ→chᴾ CHS.[ d₁ ] CH.⦂ (chsᵀ→chᵀ A CH.▻ chsᵀ→chᵀ B) CH.▻
chsᴾ→chᴾ CHS.[ d₂ ] CH.⦂ chsᵀ→chᵀ A CH.▻
chsᴾ→chᴾ CHS.[ CHS.appᴰ d₁ d₂ ] CH.⦂ chsᵀ→chᵀ B
chs→ch : ∀ {A} → CHS.⊢ A → CH.⊢ (chsᵀ→chᵀ A)
chs→ch (Ξ , d) = chsᴰ→ch d top
-- Translation from closed Hilbert-style to closed Hilbert-style sequential.
mutual
chᵀ→chsᵀ : CH.Ty → CHS.Ty
chᵀ→chsᵀ (CH.α P) = CHS.α P
chᵀ→chsᵀ (A CH.▻ B) = chᵀ→chsᵀ A CHS.▻ chᵀ→chsᵀ B
chᵀ→chsᵀ (p CH.⦂ A) with chᴾ→chsᴾ p
chᵀ→chsᵀ (p CH.⦂ A) | (Ξ , p′) = p′ CHS.⦂ chᵀ→chsᵀ A
chᵀ→chsᵀ (A CH.∧ B) = chᵀ→chsᵀ A CHS.∧ chᵀ→chsᵀ B
chᵀ→chsᵀ CH.⊤ = CHS.⊤
chᴾ→chsᴾ : ∀ {A} → CH.Proof A → ∃ (λ Ξ → CHS.Proof Ξ (chᵀ→chsᵀ A))
chᴾ→chsᴾ CH.[ d ] with ch→chs d
chᴾ→chsᴾ CH.[ d ] | (Ξ , d′) = Ξ , CHS.[ d′ ]
ch→chs : ∀ {A} → CH.⊢ A → CHS.⊢ (chᵀ→chsᵀ A)
ch→chs (CH.app d₁ d₂) = CHS.app (ch→chs d₁) (ch→chs d₂)
ch→chs CH.ci = ∅ , CHS.ci CHS.nil
ch→chs CH.ck = ∅ , CHS.ck CHS.nil
ch→chs CH.cs = ∅ , CHS.cs CHS.nil
ch→chs (CH.box d) = CHS.box (ch→chs d)
ch→chs CH.cdist = ∅ , CHS.cdist CHS.nil
ch→chs CH.cup = ∅ , CHS.cup CHS.nil
ch→chs CH.cdown = ∅ , CHS.cdown CHS.nil
ch→chs CH.cpair = ∅ , CHS.cpair CHS.nil
ch→chs CH.cfst = ∅ , CHS.cfst CHS.nil
ch→chs CH.csnd = ∅ , CHS.csnd CHS.nil
ch→chs CH.unit = ∅ , CHS.unit CHS.nil
|
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|
-- No top-level module
f : Set₁
f = Set
|
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|
------------------------------------------------------------------------
-- The Agda standard library
--
-- This module is DEPRECATED. Please use
-- Data.Vec.Relation.Binary.Equality.Setoid directly.
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Relation.Binary
module Data.Vec.Relation.Equality.Setoid
{a ℓ} (S : Setoid a ℓ) where
open import Data.Vec.Relation.Binary.Equality.Setoid S public
|
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useful-lemma : ∀ {a} {A : Set a} → A
useful-lemma = useful-lemma
|
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|
------------------------------------------------------------------------
-- Very stable booleans
------------------------------------------------------------------------
{-# OPTIONS --cubical --safe #-}
-- The module is parametrised by a notion of equality. The higher
-- constructor of the HIT defining the very stable booleans uses path
-- equality, but the supplied notion of equality is used for other
-- things.
import Equality.Path as P
module Bool.Very-stable
{e⁺} (eq : ∀ {a p} → P.Equality-with-paths a p e⁺) where
open P.Derived-definitions-and-properties eq hiding (elim)
import Equality.Path.Univalence as PU
open import Prelude as Bool hiding (true; false)
open import Bijection equality-with-J using (_↔_)
import Bijection P.equality-with-J as PB
open import Equality.Decision-procedures equality-with-J
open import Equality.Path.Isomorphisms eq
open import Equality.Path.Isomorphisms.Univalence eq
open import Equivalence equality-with-J as Eq using (_≃_)
open import Erased.Cubical eq as Erased
import Erased.Cubical P.equality-with-paths as PE
open import Function-universe equality-with-J hiding (_∘_)
open import Injection equality-with-J using (Injective)
private
variable
a p : Level
A : Type a
P : A → Type p
b f r r-[] t : A
------------------------------------------------------------------------
-- Very stable booleans
-- Very stable booleans.
--
-- This type uses a construction due to Thierry Coquand and Fabian
-- Ruch. I think this construction is a variant of a construction due
-- to Rijke, Shulman and Spitters (see "Modalities in Homotopy Type
-- Theory").
data B̃ool : Type where
true false : B̃ool
stable : Erased B̃ool → B̃ool
stable-[]ᴾ : (b : B̃ool) → stable [ b ] P.≡ b
-- The constructor stable is a left inverse of [_].
stable-[] : (b : B̃ool) → stable [ b ] ≡ b
stable-[] = _↔_.from ≡↔≡ ∘ stable-[]ᴾ
-- B̃ool is very stable.
Very-stable-B̃ool : Very-stable B̃ool
Very-stable-B̃ool = Stable→Left-inverse→Very-stable stable stable-[]
------------------------------------------------------------------------
-- Eliminators
-- A dependent eliminator, expressed using paths.
elimᴾ :
(P : B̃ool → Type p) →
P true →
P false →
(r : ∀ b → Erased (P (erased b)) → P (stable b)) →
(∀ b (p : P b) →
P.[ (λ i → P (stable-[]ᴾ b i)) ] r [ b ] [ p ] ≡ p) →
∀ b → P b
elimᴾ P t f r r-[] = λ where
true → t
false → f
(stable b) → r b [ elimᴾ P t f r r-[] (erased b) ]
(stable-[]ᴾ b i) → r-[] b (elimᴾ P t f r r-[] b) i
-- A non-dependent eliminator, expressed using paths.
recᴾ :
A → A →
(r : Erased B̃ool → Erased A → A) →
(∀ b p → r [ b ] [ p ] P.≡ p) →
B̃ool → A
recᴾ = elimᴾ _
-- A dependent eliminator, expressed using equality.
elim :
(P : B̃ool → Type p) →
P true →
P false →
(r : ∀ b → Erased (P (erased b)) → P (stable b)) →
(∀ b (p : P b) → subst P (stable-[] b) (r [ b ] [ p ]) ≡ p) →
∀ b → P b
elim P t f r r-[] = elimᴾ P t f r (λ b p → subst≡→[]≡ (r-[] b p))
-- A "computation" rule.
elim-stable-[] :
dcong (elim P t f r r-[]) (stable-[] b) ≡
r-[] b (elim P t f r r-[] b)
elim-stable-[] = dcong-subst≡→[]≡ (refl _)
-- A non-dependent eliminator, expressed using equality.
rec :
A → A →
(r : Erased B̃ool → Erased A → A) →
(∀ b p → r [ b ] [ p ] ≡ p) →
B̃ool → A
rec t f r r-[] = recᴾ t f r (λ b p → _↔_.to ≡↔≡ (r-[] b p))
-- A "computation" rule.
rec-stable-[] :
cong (rec t f r r-[]) (stable-[] b) ≡
r-[] b (rec t f r r-[] b)
rec-stable-[] = cong-≡↔≡ (refl _)
-- A dependent eliminator that can be used when eliminating into very
-- stable type families.
elim-Very-stable :
(P : B̃ool → Type p) →
P true →
P false →
(∀ b → Very-stable (P b)) →
∀ b → P b
elim-Very-stable P t f s =
elim P t f
(λ b →
Erased (P (erased b)) ↝⟨ Erased.map (subst P (
erased b ≡⟨ sym $ stable-[] (erased b) ⟩∎
stable b ∎)) ⟩
Erased (P (stable b)) ↝⟨ Very-stable→Stable 0 (s (stable b)) ⟩□
P (stable b) □)
(λ b p →
Very-stable→Stable 1 (Very-stable→Very-stable-≡ 0 (s b)) _ _
[ subst P (stable-[] b)
(_≃_.from Eq.⟨ _ , s (stable [ b ]) ⟩
[ subst P (sym $ stable-[] b) p ]) ≡⟨ cong (subst P (stable-[] b)) $ sym $
_≃_.left-inverse-unique Eq.⟨ _ , s (stable [ b ]) ⟩ erased refl _ ⟩
subst P (stable-[] b)
(erased [ subst P (sym $ stable-[] b) p ]) ≡⟨⟩
subst P (stable-[] b) (subst P (sym $ stable-[] b) p) ≡⟨ subst-subst-sym _ _ _ ⟩∎
p ∎
])
-- A non-dependent eliminator that can be used when eliminating into
-- very stable type families.
rec-Very-stable : A → A → Very-stable A → B̃ool → A
rec-Very-stable t f s = elim-Very-stable _ t f (λ _ → s)
------------------------------------------------------------------------
-- Some properties
private
module Dummy where
-- A function mapping very stable booleans to types: true is mapped
-- to the unit type and false to the empty type.
private
T̃′ : B̃ool → Σ Type Very-stable
T̃′ = rec-Very-stable
(⊤ , Very-stable-⊤)
(⊥ , Very-stable-⊥)
(Very-stable-∃-Very-stable ext univ)
T̃ : B̃ool → Type
T̃ = proj₁ ∘ T̃′
-- Some computation rules that hold by definition.
_ : T̃ true ≡ ⊤
_ = refl _
_ : T̃ false ≡ ⊥
_ = refl _
-- The output of T̃ is very stable.
Very-stable-T̃ : ∀ b → Very-stable (T̃ b)
Very-stable-T̃ = proj₂ ∘ T̃′
-- The values true and false are not equal.
true≢false : true ≢ false
true≢false =
true ≡ false ↝⟨ cong T̃ ⟩
T̃ true ≡ T̃ false ↔⟨⟩
⊤ ≡ ⊥ ↝⟨ (λ eq → ≡⇒↝ _ eq _) ⟩□
⊥ □
module Alternative-T̃ where
-- A direct implementation of T̃.
private
T̃′ : B̃ool → Σ Type PE.Very-stable
T̃ : B̃ool → Type
T̃ b = proj₁ (T̃′ b)
Very-stable-T̃ : ∀ b → PE.Very-stable (T̃ b)
Very-stable-T̃ b = proj₂ (T̃′ b)
T̃′ true = ⊤ , PE.Very-stable-⊤
T̃′ false = ⊥ , PE.Very-stable-⊥
T̃′ (stable [ b ]) = PE.Erased (T̃ b) , PE.Very-stable-Erased
T̃′ (stable-[]ᴾ b i) = lemma₁ i , lemma₂ i
where
lemma₁ : PE.Erased (T̃ b) P.≡ T̃ b
lemma₁ = PU.≃⇒≡ (PE.Very-stable→Stable 0 (Very-stable-T̃ b))
lemma₂ :
P.[ (λ i → PE.Very-stable (lemma₁ i)) ]
PE.Very-stable-Erased ≡ Very-stable-T̃ b
lemma₂ =
P.heterogeneous-irrelevance₀
(PE.Very-stable-propositional P.ext)
-- Some computation rules that hold by definition.
_ : T̃ true ≡ ⊤
_ = refl _
_ : T̃ false ≡ ⊥
_ = refl _
_ : T̃ (stable [ b ]) ≡ PE.Erased (T̃ b)
_ = refl _
open Dummy public
-- A function from booleans to very stable booleans.
from-Bool : Bool → B̃ool
from-Bool = if_then true else false
-- The function from-Bool is injective.
--
-- This lemma was suggested by Mike Shulman.
from-Bool-injective : Injective from-Bool
from-Bool-injective {x = Bool.true} {y = Bool.true} = λ _ → refl _
from-Bool-injective {x = Bool.true} {y = Bool.false} = ⊥-elim ∘ true≢false
from-Bool-injective {x = Bool.false} {y = Bool.true} = ⊥-elim ∘ true≢false ∘ sym
from-Bool-injective {x = Bool.false} {y = Bool.false} = λ _ → refl _
-- B̃ool is equivalent to Erased Bool.
--
-- This lemma was suggested by Mike Shulman.
B̃ool≃Erased-Bool : B̃ool ≃ Erased Bool
B̃ool≃Erased-Bool = Eq.↔⇒≃ (record
{ surjection = record
{ logical-equivalence = record
{ to = to
; from = from
}
; right-inverse-of = λ b →
Very-stable→Stable 0
(Very-stable→Very-stable-≡ 0 Very-stable-Erased _ _)
[ to (from b) ≡⟨⟩
to (stable [ from-Bool (erased b) ]) ≡⟨ cong to (stable-[] _) ⟩
to (from-Bool (erased b)) ≡⟨ lemma (erased b) ⟩
[ erased b ] ≡⟨⟩
b ∎
]
}
; left-inverse-of = elim-Very-stable
_
(stable-[] true)
(stable-[] false)
(λ _ → Very-stable→Very-stable-≡ 0 Very-stable-B̃ool _ _)
})
where
to = rec-Very-stable [ Bool.true ] [ Bool.false ] Very-stable-Erased
from = stable ∘ Erased.map from-Bool
lemma : ∀ b → to (from-Bool b) ≡ [ b ]
lemma Bool.true = refl _
lemma Bool.false = refl _
------------------------------------------------------------------------
-- An example
private
-- At the time of writing it is unclear if there is a reasonable way
-- to compile very stable booleans. The constructors true and false
-- are distinct, and it is reasonable to think that a compiler will
-- compile them in different ways. Now consider the following two
-- very stable booleans:
b₁ : B̃ool
b₁ = stable [ true ]
b₂ : B̃ool
b₂ = stable [ false ]
-- The first one is equal to true, and the second one is equal to
-- false:
_ : b₁ ≡ true
_ = stable-[] true
_ : b₂ ≡ false
_ = stable-[] false
-- However, unlike the constructors true and false, b₁ and b₂ should
-- be compiled in the same way, because the argument to [_] is
-- erased.
-- One question, raised by Andrea Vezzosi, is whether it is possible
-- to write a program that gives /observably/ different results
-- (after erasure) for true and false. If this is the case, then the
-- erasure mechanism is not well-behaved.
|
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|
import Nat
open Nat using (Nat)
module TopLevelImport (n : Nat) where
it : Nat
it = n
|
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|
{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Foundations.Equiv.Reasoning where
open import Cubical.Foundations.Prelude using (refl; sym)
open import Cubical.Relation.Binary
-- Properties of equivalence
≃-reflexive : Reflexive _≃_
≃-reflexive = ?
|
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------------------------------------------------------------------------
-- Context extensions with the leftmost element in the outermost
-- position
------------------------------------------------------------------------
open import Data.Universe.Indexed
module deBruijn.Context.Extension.Left
{i u e} (Uni : IndexedUniverse i u e) where
import deBruijn.Context.Basics as Basics
import deBruijn.Context.Extension.Right as Right
open import Function
open import Level using (_⊔_)
open import Relation.Binary.PropositionalEquality as P using (_≡_)
open Basics Uni
open Right Uni
open P.≡-Reasoning
------------------------------------------------------------------------
-- Context extensions along with various operations making use of them
-- Context extensions.
infixr 5 _◅_
data Ctxt₊ (Γ : Ctxt) : Set (i ⊔ u ⊔ e) where
ε : Ctxt₊ Γ
_◅_ : (σ : Type Γ) (Γ₊ : Ctxt₊ (Γ ▻ σ)) → Ctxt₊ Γ
-- A synonym.
ε₊[_] : (Γ : Ctxt) → Ctxt₊ Γ
ε₊[ _ ] = ε
-- Appends a context extension to a context.
infixl 5 _++₊_
_++₊_ : (Γ : Ctxt) → Ctxt₊ Γ → Ctxt
Γ ++₊ ε = Γ
Γ ++₊ (σ ◅ Γ₊) = Γ ▻ σ ++₊ Γ₊
-- The following operations append a context extension to a context
-- extension.
infixl 5 _₊++₊_ _⁺++₊_
_₊++₊_ : ∀ {Γ} Γ₊ → Ctxt₊ (Γ ++₊ Γ₊) → Ctxt₊ Γ
ε ₊++₊ Γ₊₊ = Γ₊₊
(σ ◅ Γ₊) ₊++₊ Γ₊₊ = σ ◅ (Γ₊ ₊++₊ Γ₊₊)
_⁺++₊_ : ∀ {Γ} Γ⁺ → Ctxt₊ (Γ ++⁺ Γ⁺) → Ctxt₊ Γ
ε ⁺++₊ Γ₊ = Γ₊
Γ⁺ ▻ σ ⁺++₊ Γ₊ = Γ⁺ ⁺++₊ (σ ◅ Γ₊)
-- Application of context morphisms to context extensions.
infixl 8 _/̂₊_
_/̂₊_ : ∀ {Γ Δ} → Ctxt₊ Γ → Γ ⇨̂ Δ → Ctxt₊ Δ
ε /̂₊ ρ̂ = ε
(σ ◅ Γ₊) /̂₊ ρ̂ = σ /̂ ρ̂ ◅ Γ₊ /̂₊ ρ̂ ↑̂
-- N-ary lifting of context morphisms.
infixl 10 _↑̂₊_
_↑̂₊_ : ∀ {Γ Δ} (ρ̂ : Γ ⇨̂ Δ) Γ₊ → Γ ++₊ Γ₊ ⇨̂ Δ ++₊ Γ₊ /̂₊ ρ̂
ρ̂ ↑̂₊ ε = ρ̂
ρ̂ ↑̂₊ (σ ◅ Γ₊) = ρ̂ ↑̂ ↑̂₊ Γ₊
-- N-ary weakening.
ŵk₊ : ∀ {Γ} Γ₊ → Γ ⇨̂ Γ ++₊ Γ₊
ŵk₊ ε = îd
ŵk₊ (σ ◅ Γ₊) = ŵk[ σ ] ∘̂ ŵk₊ Γ₊
------------------------------------------------------------------------
-- Equality
-- Equality of context extensions.
record [Ctxt₊] : Set (i ⊔ u ⊔ e) where
constructor [_]
field
{Γ} : Ctxt
Γ₊ : Ctxt₊ Γ
infix 4 _≅-Ctxt₊_
_≅-Ctxt₊_ : ∀ {Γ₁} (Γ₊₁ : Ctxt₊ Γ₁)
{Γ₂} (Γ₊₂ : Ctxt₊ Γ₂) → Set _
Γ₊₁ ≅-Ctxt₊ Γ₊₂ = _≡_ {A = [Ctxt₊]} [ Γ₊₁ ] [ Γ₊₂ ]
≅-Ctxt₊-⇒-≡ : ∀ {Γ} {Γ₊₁ Γ₊₂ : Ctxt₊ Γ} →
Γ₊₁ ≅-Ctxt₊ Γ₊₂ → Γ₊₁ ≡ Γ₊₂
≅-Ctxt₊-⇒-≡ P.refl = P.refl
-- Certain uses of substitutivity can be removed.
drop-subst-Ctxt₊ : ∀ {a} {A : Set a} {x₁ x₂}
(f : A → Ctxt) {Γ₊} (x₁≡x₂ : x₁ ≡ x₂) →
P.subst (λ x → Ctxt₊ (f x)) x₁≡x₂ Γ₊ ≅-Ctxt₊ Γ₊
drop-subst-Ctxt₊ f P.refl = P.refl
------------------------------------------------------------------------
-- Some congruences
ε₊-cong : ∀ {Γ₁ Γ₂} → Γ₁ ≅-Ctxt Γ₂ → ε₊[ Γ₁ ] ≅-Ctxt₊ ε₊[ Γ₂ ]
ε₊-cong P.refl = P.refl
◅-cong : ∀ {Γ₁ σ₁} {Γ₊₁ : Ctxt₊ (Γ₁ ▻ σ₁)}
{Γ₂ σ₂} {Γ₊₂ : Ctxt₊ (Γ₂ ▻ σ₂)} →
Γ₊₁ ≅-Ctxt₊ Γ₊₂ → σ₁ ◅ Γ₊₁ ≅-Ctxt₊ σ₂ ◅ Γ₊₂
◅-cong P.refl = P.refl
++₊-cong : ∀ {Γ₁} {Γ₊₁ : Ctxt₊ Γ₁} {Γ₂} {Γ₊₂ : Ctxt₊ Γ₂} →
Γ₊₁ ≅-Ctxt₊ Γ₊₂ → Γ₁ ++₊ Γ₊₁ ≅-Ctxt Γ₂ ++₊ Γ₊₂
++₊-cong P.refl = P.refl
₊++₊-cong : ∀ {Γ₁ Γ₊₁} {Γ₊₊₁ : Ctxt₊ (Γ₁ ++₊ Γ₊₁)}
{Γ₂ Γ₊₂} {Γ₊₊₂ : Ctxt₊ (Γ₂ ++₊ Γ₊₂)} →
Γ₊₁ ≅-Ctxt₊ Γ₊₂ → Γ₊₊₁ ≅-Ctxt₊ Γ₊₊₂ →
Γ₊₁ ₊++₊ Γ₊₊₁ ≅-Ctxt₊ Γ₊₂ ₊++₊ Γ₊₊₂
₊++₊-cong P.refl P.refl = P.refl
⁺++₊-cong : ∀ {Γ₁ Γ⁺₁} {Γ₊₁ : Ctxt₊ (Γ₁ ++⁺ Γ⁺₁)}
{Γ₂ Γ⁺₂} {Γ₊₂ : Ctxt₊ (Γ₂ ++⁺ Γ⁺₂)} →
Γ⁺₁ ≅-Ctxt⁺ Γ⁺₂ → Γ₊₁ ≅-Ctxt₊ Γ₊₂ →
Γ⁺₁ ⁺++₊ Γ₊₁ ≅-Ctxt₊ Γ⁺₂ ⁺++₊ Γ₊₂
⁺++₊-cong P.refl P.refl = P.refl
/̂₊-cong : ∀ {Γ₁ Δ₁} {Γ₊₁ : Ctxt₊ Γ₁} {ρ̂₁ : Γ₁ ⇨̂ Δ₁}
{Γ₂ Δ₂} {Γ₊₂ : Ctxt₊ Γ₂} {ρ̂₂ : Γ₂ ⇨̂ Δ₂} →
Γ₊₁ ≅-Ctxt₊ Γ₊₂ → ρ̂₁ ≅-⇨̂ ρ̂₂ → Γ₊₁ /̂₊ ρ̂₁ ≅-Ctxt₊ Γ₊₂ /̂₊ ρ̂₂
/̂₊-cong P.refl P.refl = P.refl
↑̂₊-cong : ∀ {Γ₁ Δ₁} {ρ̂₁ : Γ₁ ⇨̂ Δ₁} {Γ₊₁ : Ctxt₊ Γ₁}
{Γ₂ Δ₂} {ρ̂₂ : Γ₂ ⇨̂ Δ₂} {Γ₊₂ : Ctxt₊ Γ₂} →
ρ̂₁ ≅-⇨̂ ρ̂₂ → Γ₊₁ ≅-Ctxt₊ Γ₊₂ → ρ̂₁ ↑̂₊ Γ₊₁ ≅-⇨̂ ρ̂₂ ↑̂₊ Γ₊₂
↑̂₊-cong P.refl P.refl = P.refl
ŵk₊-cong : ∀ {Γ₁} {Γ₊₁ : Ctxt₊ Γ₁} {Γ₂} {Γ₊₂ : Ctxt₊ Γ₂} →
Γ₊₁ ≅-Ctxt₊ Γ₊₂ → ŵk₊ Γ₊₁ ≅-⇨̂ ŵk₊ Γ₊₂
ŵk₊-cong P.refl = P.refl
------------------------------------------------------------------------
-- Some properties
abstract
-- Γ ++₊_ has at most one right identity.
private
++₊-right-identity-unique′ :
∀ {Γ} Γ⁺ Γ₊ → Γ ≅-Ctxt Γ ++⁺ Γ⁺ ++₊ Γ₊ → Γ₊ ≅-Ctxt₊ ε₊[ Γ ++⁺ Γ⁺ ]
++₊-right-identity-unique′ Γ⁺ ε _ = P.refl
++₊-right-identity-unique′ Γ⁺ (σ ◅ Γ₊) eq
with ++₊-right-identity-unique′ (Γ⁺ ▻ σ) Γ₊ eq
++₊-right-identity-unique′ Γ⁺ (σ ◅ .ε) eq | P.refl
with ++⁺-right-identity-unique (Γ⁺ ▻ σ) eq
... | ()
++₊-right-identity-unique :
∀ {Γ} Γ₊ → Γ ≅-Ctxt Γ ++₊ Γ₊ → Γ₊ ≅-Ctxt₊ ε₊[ Γ ]
++₊-right-identity-unique Γ₊ = ++₊-right-identity-unique′ ε Γ₊
-- Γ ++₊_ is left cancellative.
private
cancel-++₊-left′ : ∀ {Γ} Γ⁺₁ Γ₊₁ Γ⁺₂ Γ₊₂ →
Γ ++⁺ Γ⁺₁ ++₊ Γ₊₁ ≅-Ctxt Γ ++⁺ Γ⁺₂ ++₊ Γ₊₂ →
Γ⁺₁ ⁺++₊ Γ₊₁ ≅-Ctxt₊ Γ⁺₂ ⁺++₊ Γ₊₂
cancel-++₊-left′ Γ⁺₁ ε Γ⁺₂ ε eq = ⁺++₊-cong (cancel-++⁺-left Γ⁺₁ Γ⁺₂ eq) (ε₊-cong eq)
cancel-++₊-left′ Γ⁺₁ (σ₁ ◅ Γ₊₁) Γ⁺₂ (σ₂ ◅ Γ₊₂) eq = cancel-++₊-left′ (Γ⁺₁ ▻ σ₁) Γ₊₁ (Γ⁺₂ ▻ σ₂) Γ₊₂ eq
cancel-++₊-left′ Γ⁺₁ ε Γ⁺₂ (σ₂ ◅ Γ₊₂) eq = cancel-++₊-left′ Γ⁺₁ ε (Γ⁺₂ ▻ σ₂) Γ₊₂ eq
cancel-++₊-left′ Γ⁺₁ (σ₁ ◅ Γ₊₁) Γ⁺₂ ε eq = cancel-++₊-left′ (Γ⁺₁ ▻ σ₁) Γ₊₁ Γ⁺₂ ε eq
cancel-++₊-left : ∀ {Γ} (Γ₊₁ Γ₊₂ : Ctxt₊ Γ) →
Γ ++₊ Γ₊₁ ≅-Ctxt Γ ++₊ Γ₊₂ → Γ₊₁ ≅-Ctxt₊ Γ₊₂
cancel-++₊-left Γ₊₁ Γ₊₂ = cancel-++₊-left′ ε Γ₊₁ ε Γ₊₂
-- _++₊_/_₊++₊_ are associative.
++₊-++₊ : ∀ {Γ} Γ₊ Γ₊₊ → Γ ++₊ Γ₊ ++₊ Γ₊₊ ≅-Ctxt Γ ++₊ (Γ₊ ₊++₊ Γ₊₊)
++₊-++₊ ε Γ₊₊ = P.refl
++₊-++₊ (σ ◅ Γ₊) Γ₊₊ = ++₊-++₊ Γ₊ Γ₊₊
-- The identity substitution has no effect.
/̂₊-îd : ∀ {Γ} (Γ₊ : Ctxt₊ Γ) → Γ₊ /̂₊ îd ≅-Ctxt₊ Γ₊
/̂₊-îd ε = P.refl
/̂₊-îd (σ ◅ Γ₊) = ◅-cong (/̂₊-îd Γ₊)
-- The n-ary lifting of the identity substitution is the identity
-- substitution.
îd-↑̂₊ : ∀ {Γ} (Γ₊ : Ctxt₊ Γ) → îd ↑̂₊ Γ₊ ≅-⇨̂ îd[ Γ ++₊ Γ₊ ]
îd-↑̂₊ ε = P.refl
îd-↑̂₊ (σ ◅ Γ₊) = begin
[ îd ↑̂ ↑̂₊ Γ₊ ] ≡⟨ P.refl ⟩
[ îd ↑̂₊ Γ₊ ] ≡⟨ îd-↑̂₊ Γ₊ ⟩
[ îd ] ∎
-- The identity substitution has no effect even if lifted.
/̂-îd-↑̂₊ : ∀ {Γ} Γ₊ (σ : Type (Γ ++₊ Γ₊)) → σ /̂ îd ↑̂₊ Γ₊ ≅-Type σ
/̂-îd-↑̂₊ Γ₊ σ = begin
[ σ /̂ îd ↑̂₊ Γ₊ ] ≡⟨ /̂-cong (P.refl {x = [ σ ]}) (îd-↑̂₊ Γ₊) ⟩
[ σ /̂ îd ] ≡⟨ P.refl ⟩
[ σ ] ∎
-- Applying two substitutions is equivalent to applying their
-- composition.
/̂₊-∘̂ : ∀ {Γ Δ Ε} Γ₊ (ρ̂₁ : Γ ⇨̂ Δ) (ρ̂₂ : Δ ⇨̂ Ε) →
Γ₊ /̂₊ ρ̂₁ ∘̂ ρ̂₂ ≅-Ctxt₊ Γ₊ /̂₊ ρ̂₁ /̂₊ ρ̂₂
/̂₊-∘̂ ε ρ̂₁ ρ̂₂ = P.refl
/̂₊-∘̂ (σ ◅ Γ₊) ρ̂₁ ρ̂₂ = ◅-cong (/̂₊-∘̂ Γ₊ (ρ̂₁ ↑̂) (ρ̂₂ ↑̂))
-- _↑̂₊_ distributes over _∘̂_.
∘̂-↑̂₊ : ∀ {Γ Δ Ε} (ρ̂₁ : Γ ⇨̂ Δ) (ρ̂₂ : Δ ⇨̂ Ε) Γ₊ →
(ρ̂₁ ∘̂ ρ̂₂) ↑̂₊ Γ₊ ≅-⇨̂ ρ̂₁ ↑̂₊ Γ₊ ∘̂ ρ̂₂ ↑̂₊ (Γ₊ /̂₊ ρ̂₁)
∘̂-↑̂₊ ρ̂₁ ρ̂₂ ε = P.refl
∘̂-↑̂₊ ρ̂₁ ρ̂₂ (σ ◅ Γ₊) = begin
[ (ρ̂₁ ∘̂ ρ̂₂) ↑̂ ↑̂₊ Γ₊ ] ≡⟨ P.refl ⟩
[ (ρ̂₁ ↑̂ ∘̂ ρ̂₂ ↑̂) ↑̂₊ Γ₊ ] ≡⟨ ∘̂-↑̂₊ (ρ̂₁ ↑̂) (ρ̂₂ ↑̂) Γ₊ ⟩
[ (ρ̂₁ ↑̂ ↑̂₊ Γ₊) ∘̂ (ρ̂₂ ↑̂ ↑̂₊ (Γ₊ /̂₊ ρ̂₁ ↑̂)) ] ∎
-- A corollary.
/̂-↑̂₊-/̂-ŵk-↑̂₊ : ∀ {Γ Δ} σ (ρ̂ : Γ ⇨̂ Δ) (Γ₊ : Ctxt₊ Γ) τ →
τ /̂ ρ̂ ↑̂₊ Γ₊ /̂ ŵk[ σ /̂ ρ̂ ] ↑̂₊ (Γ₊ /̂₊ ρ̂) ≅-Type
τ /̂ ŵk[ σ ] ↑̂₊ Γ₊ /̂ ρ̂ ↑̂ ↑̂₊ (Γ₊ /̂₊ ŵk)
/̂-↑̂₊-/̂-ŵk-↑̂₊ σ ρ̂ Γ₊ τ = /̂-cong (P.refl {x = [ τ ]}) (begin
[ ρ̂ ↑̂₊ Γ₊ ∘̂ ŵk ↑̂₊ (Γ₊ /̂₊ ρ̂) ] ≡⟨ P.sym $ ∘̂-↑̂₊ ρ̂ ŵk Γ₊ ⟩
[ (ρ̂ ∘̂ ŵk) ↑̂₊ Γ₊ ] ≡⟨ P.refl ⟩
[ (ŵk[ σ ] ∘̂ ρ̂ ↑̂) ↑̂₊ Γ₊ ] ≡⟨ ∘̂-↑̂₊ ŵk[ σ ] (ρ̂ ↑̂) Γ₊ ⟩
[ ŵk[ σ ] ↑̂₊ Γ₊ ∘̂ ρ̂ ↑̂ ↑̂₊ (Γ₊ /̂₊ ŵk) ] ∎)
-- ŵk₊ commutes (modulo lifting) with arbitrary context morphisms.
∘̂-ŵk₊ : ∀ {Γ Δ} (ρ̂ : Γ ⇨̂ Δ) Γ₊ →
ρ̂ ∘̂ ŵk₊ (Γ₊ /̂₊ ρ̂) ≅-⇨̂ ŵk₊ Γ₊ ∘̂ ρ̂ ↑̂₊ Γ₊
∘̂-ŵk₊ ρ̂ ε = P.refl
∘̂-ŵk₊ ρ̂ (σ ◅ Γ₊) = begin
[ ρ̂ ∘̂ ŵk ∘̂ ŵk₊ (Γ₊ /̂₊ ρ̂ ↑̂) ] ≡⟨ P.refl ⟩
[ ŵk[ σ ] ∘̂ ρ̂ ↑̂ ∘̂ ŵk₊ (Γ₊ /̂₊ ρ̂ ↑̂) ] ≡⟨ ∘̂-cong (P.refl {x = [ ŵk ]}) (∘̂-ŵk₊ (ρ̂ ↑̂) Γ₊) ⟩
[ ŵk ∘̂ ŵk₊ Γ₊ ∘̂ ρ̂ ↑̂ ↑̂₊ Γ₊ ] ∎
-- ŵk₊ is homomorphic with respect to _₊++₊_/_∘̂_.
ŵk₊-₊++₊ : ∀ {Γ} (Γ₊ : Ctxt₊ Γ) (Γ₊₊ : Ctxt₊ (Γ ++₊ Γ₊)) →
ŵk₊ (Γ₊ ₊++₊ Γ₊₊) ≅-⇨̂ ŵk₊ Γ₊ ∘̂ ŵk₊ Γ₊₊
ŵk₊-₊++₊ ε Γ₊₊ = P.refl
ŵk₊-₊++₊ (σ ◅ Γ₊) Γ₊₊ = ∘̂-cong (P.refl {x = [ ŵk ]}) (ŵk₊-₊++₊ Γ₊ Γ₊₊)
-- Two n-ary liftings can be merged into one.
↑̂₊-₊++₊ : ∀ {Γ Δ} (ρ̂ : Γ ⇨̂ Δ) Γ₊ Γ₊₊ →
ρ̂ ↑̂₊ (Γ₊ ₊++₊ Γ₊₊) ≅-⇨̂ ρ̂ ↑̂₊ Γ₊ ↑̂₊ Γ₊₊
↑̂₊-₊++₊ ρ̂ ε Γ₊₊ = P.refl
↑̂₊-₊++₊ ρ̂ (σ ◅ Γ₊) Γ₊₊ = ↑̂₊-₊++₊ (ρ̂ ↑̂) Γ₊ Γ₊₊
-- _/̂₊_ distributes over _₊++₊_ (sort of).
++₊-₊++₊-/̂₊ :
∀ {Γ Δ} (ρ̂ : Γ ⇨̂ Δ) Γ₊ Γ₊₊ →
Δ ++₊ (Γ₊ ₊++₊ Γ₊₊) /̂₊ ρ̂ ≅-Ctxt Δ ++₊ Γ₊ /̂₊ ρ̂ ++₊ Γ₊₊ /̂₊ ρ̂ ↑̂₊ Γ₊
++₊-₊++₊-/̂₊ ρ̂ ε Γ₊₊ = P.refl
++₊-₊++₊-/̂₊ {Δ = Δ} ρ̂ (σ ◅ Γ₊) Γ₊₊ = ++₊-₊++₊-/̂₊ (ρ̂ ↑̂) Γ₊ Γ₊₊
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{-
Macros (autoDesc, AutoStructure, AutoEquivStr, autoUnivalentStr) for automatically generating structure definitions.
For example:
autoDesc (λ (X : Type₀) → X → X × ℕ) ↦ recvar (var , constant ℕ)
We prefer to use the constant structure whenever possible, e.g., [autoDesc (λ (X : Type₀) → ℕ → ℕ)]
is [constant (ℕ → ℕ)] rather than [param ℕ (constant ℕ)].
Writing [auto* (λ X → ⋯)] doesn't seem to work, but [auto* (λ (X : Type ℓ) → ⋯)] does.
-}
{-# OPTIONS --cubical --no-exact-split --safe #-}
module Cubical.Structures.Auto where
open import Cubical.Foundations.Prelude
open import Cubical.Data.Sigma
open import Cubical.Data.Nat
open import Cubical.Data.List
open import Cubical.Data.Bool
open import Cubical.Data.Maybe
open import Cubical.Structures.Macro as Macro
import Agda.Builtin.Reflection as R
-- Magic number
private
FUEL = 10000
-- Mark part of a structure definition as functorial
abstract
Funct[_] : ∀ {ℓ} → Type ℓ → Type ℓ
Funct[ A ] = A
-- Some reflection utilities
private
_>>=_ = R.bindTC
_<|>_ = R.catchTC
_>>_ : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → R.TC A → R.TC B → R.TC B
f >> g = f >>= λ _ → g
infixl 4 _>>=_ _>>_ _<|>_
varg : ∀ {ℓ} {A : Type ℓ} → A → R.Arg A
varg = R.arg (R.arg-info R.visible R.relevant)
tLevel = R.def (quote Level) []
tType : R.Term → R.Term
tType ℓ = R.def (quote Type) [ varg ℓ ]
tFuncDesc : R.Term → R.Term
tFuncDesc ℓ = R.def (quote FuncDesc) [ varg ℓ ]
tDesc : R.Term → R.Term
tDesc ℓ = R.def (quote Desc) [ varg ℓ ]
func : (ℓ ℓ' : Level) → Type (ℓ-suc (ℓ-max ℓ ℓ'))
func ℓ ℓ' = Type ℓ → Type ℓ'
tStruct : R.Term → R.Term → R.Term
tStruct ℓ ℓ' = R.def (quote func) (varg ℓ ∷ varg ℓ' ∷ [])
newMeta = R.checkType R.unknown
-- We try to build a descriptor by unifying the provided structure with combinators we're aware of. We
-- redefine the structure combinators as the *Shape terms below so that we don't depend on the specific way
-- these are defined in other files (order of implicit arguments and so on); the syntactic analysis that goes
-- on here means that we would get mysterious errors if those changed.
private
constantShape : ∀ {ℓ'} (ℓ : Level) (A : Type ℓ') → (Type ℓ → Type ℓ')
constantShape _ A _ = A
pointedShape : (ℓ : Level) → Type ℓ → Type ℓ
pointedShape _ X = X
joinShape : ∀ {ℓ₀ ℓ₁} (ℓ : Level)
→ (Type ℓ → Type ℓ₀) → (Type ℓ → Type ℓ₁) → Type ℓ → Type (ℓ-max ℓ₀ ℓ₁)
joinShape _ A₀ A₁ X = A₀ X × A₁ X
paramShape : ∀ {ℓ₀ ℓ'} (ℓ : Level)
→ Type ℓ' → (Type ℓ → Type ℓ₀) → Type ℓ → Type (ℓ-max ℓ' ℓ₀)
paramShape _ A A₀ X = A → A₀ X
recvarShape : ∀ {ℓ₀} (ℓ : Level)
→ (Type ℓ → Type ℓ₀) → Type ℓ → Type (ℓ-max ℓ ℓ₀)
recvarShape _ A₀ X = X → A₀ X
maybeShape : ∀ {ℓ₀} (ℓ : Level)
→ (Type ℓ → Type ℓ₀) → Type ℓ → Type ℓ₀
maybeShape _ A₀ X = Maybe (A₀ X)
functorialShape : ∀ {ℓ₀} (ℓ : Level)
→ (Type ℓ → Type ℓ₀) → Type ℓ → Type ℓ₀
functorialShape _ A₀ X = Funct[ A₀ X ]
private
-- Build functorial structure descriptor from a function [t : Type ℓ → Type ℓ']
buildFuncDesc : ℕ → R.Term → R.Term → R.Term → R.TC R.Term
buildFuncDesc zero ℓ ℓ' t = R.typeError (R.strErr "Ran out of fuel! at \n" ∷ R.termErr t ∷ [])
buildFuncDesc (suc fuel) ℓ ℓ' t =
tryConstant t <|> tryPointed t <|> tryJoin t <|> tryParam t <|> tryMaybe t <|>
R.typeError (R.strErr "Can't automatically generate a functorial structure for\n" ∷ R.termErr t ∷ [])
where
tryConstant : R.Term → R.TC R.Term
tryConstant t =
newMeta (tType ℓ') >>= λ A →
R.unify t (R.def (quote constantShape) (varg ℓ ∷ varg A ∷ [])) >>
R.returnTC (R.con (quote FuncDesc.constant) (varg A ∷ []))
tryPointed : R.Term → R.TC R.Term
tryPointed t =
R.unify t (R.def (quote pointedShape) (varg ℓ ∷ [])) >>
R.returnTC (R.con (quote FuncDesc.var) [])
tryParam : R.Term → R.TC R.Term
tryParam t =
newMeta (tType R.unknown) >>= λ A →
newMeta tLevel >>= λ ℓ₀ →
newMeta (tStruct ℓ ℓ₀) >>= λ A₀ →
R.unify t (R.def (quote paramShape) (varg ℓ ∷ varg A ∷ varg A₀ ∷ [])) >>
buildFuncDesc fuel ℓ ℓ₀ A₀ >>= λ d₀ →
R.returnTC (R.con (quote FuncDesc.param) (varg A ∷ varg d₀ ∷ []))
tryJoin : R.Term → R.TC R.Term
tryJoin t =
newMeta tLevel >>= λ ℓ₀ →
newMeta tLevel >>= λ ℓ₁ →
newMeta (tStruct ℓ ℓ₀) >>= λ A₀ →
newMeta (tStruct ℓ ℓ₁) >>= λ A₁ →
R.unify t (R.def (quote joinShape) (varg ℓ ∷ varg A₀ ∷ varg A₁ ∷ [])) >>
buildFuncDesc fuel ℓ ℓ₀ A₀ >>= λ d₀ →
buildFuncDesc fuel ℓ ℓ₁ A₁ >>= λ d₁ →
R.returnTC (R.con (quote FuncDesc._,_) (varg d₀ ∷ varg d₁ ∷ []))
tryMaybe : R.Term → R.TC R.Term
tryMaybe t =
newMeta tLevel >>= λ ℓ₀ →
newMeta (tStruct ℓ ℓ₀) >>= λ A₀ →
R.unify t (R.def (quote maybeShape) (varg ℓ ∷ varg A₀ ∷ [])) >>
buildFuncDesc fuel ℓ ℓ₀ A₀ >>= λ d₀ →
R.returnTC (R.con (quote FuncDesc.maybe) (varg d₀ ∷ []))
autoFuncDesc' : R.Term → R.Term → R.TC Unit
autoFuncDesc' t hole =
R.inferType hole >>= λ H →
newMeta tLevel >>= λ ℓ →
newMeta tLevel >>= λ ℓ' →
R.unify (tFuncDesc ℓ) H >>
R.checkType t (tStruct ℓ ℓ') >>
buildFuncDesc FUEL ℓ ℓ' t >>= R.unify hole
-- Build structure descriptor from a function [t : Type ℓ → Type ℓ']
buildDesc : ℕ → R.Term → R.Term → R.Term → R.TC R.Term
buildDesc zero ℓ ℓ' t = R.typeError (R.strErr "Ran out of fuel! at \n" ∷ R.termErr t ∷ [])
buildDesc (suc fuel) ℓ ℓ' t =
tryConstant t <|> tryPointed t <|> tryJoin t <|> tryParam t <|> tryRecvar t <|> tryMaybe t <|>
tryFunct t <|>
R.typeError (R.strErr "Can't automatically generate a structure for\n" ∷ R.termErr t ∷ [])
where
tryConstant : R.Term → R.TC R.Term
tryConstant t =
newMeta (tType ℓ') >>= λ A →
R.unify t (R.def (quote constantShape) (varg ℓ ∷ varg A ∷ [])) >>
R.returnTC (R.con (quote Desc.constant) (varg A ∷ []))
tryPointed : R.Term → R.TC R.Term
tryPointed t =
R.unify t (R.def (quote pointedShape) (varg ℓ ∷ [])) >>
R.returnTC (R.con (quote Desc.var) [])
tryJoin : R.Term → R.TC R.Term
tryJoin t =
newMeta tLevel >>= λ ℓ₀ →
newMeta tLevel >>= λ ℓ₁ →
newMeta (tStruct ℓ ℓ₀) >>= λ A₀ →
newMeta (tStruct ℓ ℓ₁) >>= λ A₁ →
R.unify t (R.def (quote joinShape) (varg ℓ ∷ varg A₀ ∷ varg A₁ ∷ [])) >>
buildDesc fuel ℓ ℓ₀ A₀ >>= λ d₀ →
buildDesc fuel ℓ ℓ₁ A₁ >>= λ d₁ →
R.returnTC (R.con (quote Desc._,_) (varg d₀ ∷ varg d₁ ∷ []))
tryParam : R.Term → R.TC R.Term
tryParam t =
newMeta (tType R.unknown) >>= λ A →
newMeta tLevel >>= λ ℓ₀ →
newMeta (tStruct ℓ ℓ₀) >>= λ A₀ →
R.unify t (R.def (quote paramShape) (varg ℓ ∷ varg A ∷ varg A₀ ∷ [])) >>
buildDesc fuel ℓ ℓ₀ A₀ >>= λ d₀ →
R.returnTC (R.con (quote Desc.param) (varg A ∷ varg d₀ ∷ []))
tryRecvar : R.Term → R.TC R.Term
tryRecvar t =
newMeta tLevel >>= λ ℓ₀ →
newMeta (tStruct ℓ ℓ₀) >>= λ A₀ →
R.unify t (R.def (quote recvarShape) (varg ℓ ∷ varg A₀ ∷ [])) >>
buildDesc fuel ℓ ℓ₀ A₀ >>= λ d₀ →
R.returnTC (R.con (quote Desc.recvar) (varg d₀ ∷ []))
tryMaybe : R.Term → R.TC R.Term
tryMaybe t =
newMeta tLevel >>= λ ℓ₀ →
newMeta (tStruct ℓ ℓ₀) >>= λ A₀ →
R.unify t (R.def (quote maybeShape) (varg ℓ ∷ varg A₀ ∷ [])) >>
buildDesc fuel ℓ ℓ₀ A₀ >>= λ d₀ →
R.returnTC (R.con (quote Desc.maybe) (varg d₀ ∷ []))
tryFunct : R.Term → R.TC R.Term
tryFunct t =
newMeta (tStruct ℓ ℓ') >>= λ A₀ →
R.unify t (R.def (quote functorialShape) (varg ℓ ∷ varg A₀ ∷ [])) >>
buildFuncDesc fuel ℓ ℓ' A₀ >>= λ d₀ →
R.returnTC (R.con (quote Desc.functorial) (varg d₀ ∷ []))
autoDesc' : R.Term → R.Term → R.TC Unit
autoDesc' t hole =
R.inferType hole >>= λ H →
newMeta tLevel >>= λ ℓ →
newMeta tLevel >>= λ ℓ' →
R.unify (tDesc ℓ) H >>
R.checkType t (tStruct ℓ ℓ') >>
buildDesc FUEL ℓ ℓ' t >>= R.unify hole
macro
-- (Type ℓ → Type ℓ₁) → FuncDesc ℓ
autoFuncDesc : R.Term → R.Term → R.TC Unit
autoFuncDesc = autoFuncDesc'
-- (S : Type ℓ → Type ℓ₁) → ∀ {X Y} → (X → Y) → (S X → S Y)
autoFuncAction : R.Term → R.Term → R.TC Unit
autoFuncAction t hole =
newMeta (tFuncDesc R.unknown) >>= λ d →
R.unify hole (R.def (quote funcMacroAction) [ varg d ]) >>
autoFuncDesc' t d
-- (S : Type ℓ → Type ℓ₁) → ∀ {X} s → autoFuncAction S (idfun X) s ≡ s
autoFuncId : R.Term → R.Term → R.TC Unit
autoFuncId t hole =
newMeta (tFuncDesc R.unknown) >>= λ d →
R.unify hole (R.def (quote funcMacroId) [ varg d ]) >>
autoFuncDesc' t d
-- (S : Type ℓ → Type ℓ₁) → Desc ℓ
autoDesc : R.Term → R.Term → R.TC Unit
autoDesc = autoDesc'
-- (S : Type ℓ → Type ℓ₁) → (Type ℓ → Type ℓ₁)
-- Removes Funct[_] annotations
AutoStructure : R.Term → R.Term → R.TC Unit
AutoStructure t hole =
newMeta (tDesc R.unknown) >>= λ d →
R.unify hole (R.def (quote MacroStructure) [ varg d ]) >>
autoDesc' t d
-- (S : Type ℓ → Type ℓ₁) → StrIso (AutoStructure S) _
AutoEquivStr : R.Term → R.Term → R.TC Unit
AutoEquivStr t hole =
newMeta (tDesc R.unknown) >>= λ d →
R.unify hole (R.def (quote MacroEquivStr) [ varg d ]) >>
autoDesc' t d
-- (S : Type ℓ → Type ℓ₁) → SNS (AutoStructure S) (AutoEquivStr S)
autoUnivalentStr : R.Term → R.Term → R.TC Unit
autoUnivalentStr t hole =
newMeta (tDesc R.unknown) >>= λ d →
R.unify hole (R.def (quote MacroUnivalentStr) [ varg d ]) >>
autoDesc' t d
|
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module Preliminaries where
open import Agda.Primitive using (Level) renaming (lzero to lZ; lsuc to lS; _⊔_ to lmax)
-- ----------------------------------------------------------------------
-- functions
_o_ : {A B C : Set} → (B → C) → (A → B) → A → C
g o f = \ x → g (f x)
infixr 10 _o_
-- ----------------------------------------------------------------------
-- identity type
data _==_ {l : Level} {A : Set l} (M : A) : A → Set l where
Refl : M == M
Id : {l : Level} {A : Set l} (M : A) → A → Set l
Id M N = M == N
{-# BUILTIN EQUALITY _==_ #-}
{-# BUILTIN REFL Refl #-}
transport : {l1 : Level} {l2 : Level} {A : Set l1} (B : A → Set l2)
{a1 a2 : A} → a1 == a2 → (B a1 → B a2)
transport B Refl = λ x → x
! : {l : Level} {A : Set l} {M N : A} → M == N → N == M
! Refl = Refl
_∘_ : {l : Level} {A : Set l} {M N P : A}
→ N == P → M == N → M == P
β ∘ Refl = β
ap : {l1 l2 : Level} {A : Set l1} {B : Set l2} {M N : A}
(f : A → B) → M == N → (f M) == (f N)
ap f Refl = Refl
ap2 : {l1 l2 l3 : Level} {A : Set l1} {B : Set l2} {C : Set l3} {M N : A} {M' N' : B} (f : A -> B -> C) -> M == N -> M' == N' -> (f M M') == (f N N')
ap2 f Refl Refl = Refl
ap3 : {l1 l2 l3 l4 : Level} {A : Set l1} {B : Set l2} {C : Set l3} {D : Set l4} {M N : A} {M' N' : B} {M'' N'' : C}
(f : A → B → C → D) → M == N → M' == N' → M'' == N'' → (f M M' M'') == (f N N' N'')
ap3 f Refl Refl Refl = Refl
postulate
-- function extensionality
λ= : {l1 l2 : Level} {A : Set l1} {B : A -> Set l2} {f g : (x : A) -> B x} -> ((x : A) -> (f x) == (g x)) -> f == g
-- function extensionality for implicit functions
λ=i : {l1 l2 : Level} {A : Set l1} {B : A -> Set l2} {f g : {x : A} -> B x} -> ((x : A) -> (f {x}) == (g {x})) -> _==_ {_}{ {x : A} → B x } f g
private primitive primTrustMe : {l : Level} {A : Set l} {x y : A} -> x == y
infixr 9 _==_
infix 2 _∎
infixr 2 _=⟨_⟩_
_=⟨_⟩_ : {l : Level} {A : Set l} (x : A) {y z : A} → x == y → y == z → x == z
_ =⟨ p1 ⟩ p2 = (p2 ∘ p1)
_∎ : {l : Level} {A : Set l} (x : A) → x == x
_∎ _ = Refl
-- ----------------------------------------------------------------------
-- product types
record Unit : Set where
constructor <>
record Σ {l1 l2 : Level} {A : Set l1} (B : A -> Set l2) : Set (lmax l1 l2) where
constructor _,_
field
fst : A
snd : B fst
open Σ public
infixr 0 _,_
_×_ : {l1 l2 : Level} → Set l1 -> Set l2 -> Set (lmax l1 l2)
A × B = Σ (\ (_ : A) -> B)
infixr 10 _×_
-- ----------------------------------------------------------------------
-- booleans
data Bool : Set where
True : Bool
False : Bool
{-# COMPILED_DATA Bool Bool True False #-}
{-# BUILTIN BOOL Bool #-}
{-# BUILTIN TRUE True #-}
{-# BUILTIN FALSE False #-}
-- ----------------------------------------------------------------------
-- order
data Order : Set where
Less : Order
Equal : Order
Greater : Order
-- ----------------------------------------------------------------------
-- sums
data Void : Set where
abort : {A : Set} → Void → A
abort ()
data Either (A B : Set) : Set where
Inl : A → Either A B
Inr : B → Either A B
DecEq : Set → Set
DecEq A = (x y : A) → Either (x == y) (x == y → Void)
-- ----------------------------------------------------------------------
-- natural numbers
module Nat where
data Nat : Set where
Z : Nat
S : Nat -> Nat
-- let's you use numerals for Nat
{-# BUILTIN NATURAL Nat #-}
_+_ : Nat → Nat → Nat
Z + n = n
(S m) + n = S (m + n)
max : Nat → Nat → Nat
max Z n = n
max m Z = m
max (S m) (S n) = S (max m n)
equal : Nat → Nat → Bool
equal Z Z = True
equal Z (S _) = False
equal (S _) Z = False
equal (S m) (S n) = equal m n
compare : Nat → Nat → Order
compare Z Z = Equal
compare Z (S m) = Less
compare (S n) Z = Greater
compare (S n) (S m) = compare n m
open Nat public using (Nat ; Z ; S)
-- ----------------------------------------------------------------------
-- monad
module Monad where
record Monad : Set1 where
field
T : Set → Set
return : ∀ {A} → A → T A
_>>=_ : ∀ {A B} → T A → (A → T B) -> T B
-- ----------------------------------------------------------------------
-- options
module Maybe where
data Maybe {l : Level} (A : Set l) : Set l where
Some : A → Maybe A
None : Maybe A
Monad : Monad.Monad
Monad = record { T = Maybe; return = Some; _>>=_ = (λ {None _ → None; (Some v) f → f v}) }
open Maybe public using (Maybe;Some;None)
-- ----------------------------------------------------------------------
-- lists
module List where
data List {l : Level} (A : Set l) : Set l where
[] : List A
_::_ : A -> List A -> List A
{-# COMPILED_DATA List [] [] (:) #-}
{-# BUILTIN LIST List #-}
{-# BUILTIN NIL [] #-}
{-# BUILTIN CONS _::_ #-}
infixr 99 _::_
_++_ : {A : Set} → List A → List A → List A
[] ++ ys = ys
(x :: xs) ++ ys = x :: (xs ++ ys)
infixr 10 _++_
map : {l1 l2 : Level} {A : Set l1} {B : Set l2} → (A → B) → List A → List B
map f [] = []
map f (x :: xs) = f x :: map f xs
map-id : {l : Level} {A : Set l} (l : List A) → map (\ (x : A) → x) l == l
map-id [] = Refl
map-id (x :: l) with map (\ x -> x) l | map-id l
... | ._ | Refl = Refl
++-assoc : ∀ {A} (l1 l2 l3 : List A) → (l1 ++ l2) ++ l3 == l1 ++ (l2 ++ l3)
++-assoc [] l2 l3 = Refl
++-assoc (x :: xs) l2 l3 = ap (_::_ x) (++-assoc xs l2 l3)
open List public using (List ; [] ; _::_)
-- ----------------------------------------------------------------------
-- characters
module Char where
postulate {- Agda Primitive -}
Char : Set
{-# BUILTIN CHAR Char #-}
{-# COMPILED_TYPE Char Char #-}
private
primitive
primCharToNat : Char → Nat
primCharEquality : Char → Char → Bool
toNat : Char → Nat
toNat = primCharToNat
equalb : Char -> Char -> Bool
equalb = primCharEquality
-- need to go outside the real language a little to give the primitives good types,
-- but from the outside this should be safe
equal : DecEq Char
equal x y with equalb x y
... | True = Inl primTrustMe
... | False = Inr canthappen where
postulate canthappen : _
open Char public using (Char)
-- ----------------------------------------------------------------------
-- vectors
module Vector where
data Vec (A : Set) : Nat → Set where
[] : Vec A 0
_::_ : ∀ {n} → A → Vec A n → Vec A (S n)
infixr 99 _::_
Vec-elim : {A : Set} (P : {n : Nat} → Vec A n → Set)
→ (P [])
→ ({n : Nat} (x : A) (xs : Vec A n) → P xs → P (x :: xs))
→ {n : Nat} (v : Vec A n) → P v
Vec-elim P n c [] = n
Vec-elim P n c (y :: ys) = c y ys (Vec-elim P n c ys)
fromList : {A : Set} → List A → Σ \n → Vec A n
fromList [] = _ , []
fromList (x :: xs) = _ , x :: snd (fromList xs)
toList : {A : Set} {n : Nat} → Vec A n → List A
toList [] = []
toList (x :: xs) = x :: (toList xs)
toList' : {A : Set} → (Σ \ n → Vec A n) → List A
toList' (._ , []) = []
toList' (._ , (x :: xs)) = x :: (toList' (_ , xs))
-- ----------------------------------------------------------------------
-- strings
module String where
postulate {- Agda Primitive -}
String : Set
{-# BUILTIN STRING String #-}
{-# COMPILED_TYPE String String #-}
private
primitive
primStringToList : String -> List Char
primStringFromList : List Char -> String
primStringAppend : String -> String -> String
primStringEquality : String -> String -> Bool
equal : String -> String -> Bool
equal = primStringEquality
toList = primStringToList
fromList = primStringFromList
append = primStringAppend
toVec : String -> Σ \ m → Vector.Vec Char m
toVec = Vector.fromList o toList
|
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------------------------------------------------------------------------
-- Operators
------------------------------------------------------------------------
module Mixfix.Operator where
open import Data.Nat using (ℕ; zero; suc; _+_)
open import Data.Vec using (Vec)
open import Data.Product using (∃; ∃₂; _,_)
open import Data.Maybe using (Maybe; just; nothing)
open import Data.String using (String)
open import Relation.Nullary using (Dec; yes; no)
open import Relation.Binary.PropositionalEquality using (_≡_; refl)
open import Mixfix.Fixity
-- Name parts.
NamePart : Set
NamePart = String
-- Operators. The parameter arity is the internal arity of the
-- operator, i.e. the number of arguments taken between the first and
-- last name parts.
record Operator (fix : Fixity) (arity : ℕ) : Set where
field nameParts : Vec NamePart (1 + arity)
open Operator public
-- Predicate filtering out operators of the given fixity and
-- associativity.
hasFixity : ∀ fix → ∃₂ Operator → Maybe (∃ (Operator fix))
hasFixity fix (fix' , op) with fix ≟ fix'
hasFixity fix (.fix , op) | yes refl = just op
hasFixity fix (fix' , op) | _ = nothing
|
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Decision procedures for finite sets and subsets of finite sets
------------------------------------------------------------------------
module Data.Fin.Dec where
open import Function
import Data.Bool as Bool
open import Data.Nat hiding (_<_)
open import Data.Vec hiding (_∈_)
open import Data.Vec.Equality as VecEq
using () renaming (module PropositionalEquality to PropVecEq)
open import Data.Fin
open import Data.Fin.Subset
open import Data.Fin.Subset.Properties
open import Data.Product as Prod
open import Data.Empty
open import Function
import Function.Equivalence as Eq
open import Relation.Binary as B
import Relation.Binary.HeterogeneousEquality as H
open import Relation.Nullary
import Relation.Nullary.Decidable as Dec
open import Relation.Unary as U using (Pred)
infix 4 _∈?_
_∈?_ : ∀ {n} x (p : Subset n) → Dec (x ∈ p)
zero ∈? inside ∷ p = yes here
zero ∈? outside ∷ p = no λ()
suc n ∈? s ∷ p with n ∈? p
... | yes n∈p = yes (there n∈p)
... | no n∉p = no (n∉p ∘ drop-there)
private
restrictP : ∀ {p n} → (Fin (suc n) → Set p) → (Fin n → Set p)
restrictP P f = P (suc f)
restrict : ∀ {p n} {P : Fin (suc n) → Set p} →
U.Decidable P → U.Decidable (restrictP P)
restrict dec f = dec (suc f)
any? : ∀ {n} {P : Fin n → Set} →
U.Decidable P → Dec (∃ P)
any? {zero} dec = no λ { (() , _) }
any? {suc n} {P} dec with dec zero | any? (restrict dec)
... | yes p | _ = yes (_ , p)
... | _ | yes (_ , p') = yes (_ , p')
... | no ¬p | no ¬p' = no helper
where
helper : ∄ P
helper (zero , p) = ¬p p
helper (suc f , p') = ¬p' (_ , p')
nonempty? : ∀ {n} (p : Subset n) → Dec (Nonempty p)
nonempty? p = any? (λ x → x ∈? p)
private
restrict∈ : ∀ {p q n}
(P : Fin (suc n) → Set p) {Q : Fin (suc n) → Set q} →
(∀ {f} → Q f → Dec (P f)) →
(∀ {f} → restrictP Q f → Dec (restrictP P f))
restrict∈ _ dec {f} Qf = dec {suc f} Qf
decFinSubset : ∀ {p q n} {P : Fin n → Set p} {Q : Fin n → Set q} →
U.Decidable Q →
(∀ {f} → Q f → Dec (P f)) →
Dec (∀ {f} → Q f → P f)
decFinSubset {n = zero} _ _ = yes λ{}
decFinSubset {n = suc n} {P} {Q} decQ decP = helper
where
helper : Dec (∀ {f} → Q f → P f)
helper with decFinSubset (restrict decQ) (restrict∈ P decP)
helper | no ¬q⟶p = no (λ q⟶p → ¬q⟶p (λ {f} q → q⟶p {suc f} q))
helper | yes q⟶p with decQ zero
helper | yes q⟶p | yes q₀ with decP q₀
helper | yes q⟶p | yes q₀ | no ¬p₀ = no (λ q⟶p → ¬p₀ (q⟶p {zero} q₀))
helper | yes q⟶p | yes q₀ | yes p₀ = yes (λ {_} → hlpr _)
where
hlpr : ∀ f → Q f → P f
hlpr zero _ = p₀
hlpr (suc f) qf = q⟶p qf
helper | yes q⟶p | no ¬q₀ = yes (λ {_} → hlpr _)
where
hlpr : ∀ f → Q f → P f
hlpr zero q₀ = ⊥-elim (¬q₀ q₀)
hlpr (suc f) qf = q⟶p qf
all∈? : ∀ {n p} {P : Fin n → Set p} {q} →
(∀ {f} → f ∈ q → Dec (P f)) →
Dec (∀ {f} → f ∈ q → P f)
all∈? {q = q} dec = decFinSubset (λ f → f ∈? q) dec
all? : ∀ {n p} {P : Fin n → Set p} →
U.Decidable P → Dec (∀ f → P f)
all? dec with all∈? {q = ⊤} (λ {f} _ → dec f)
... | yes ∀p = yes (λ f → ∀p ∈⊤)
... | no ¬∀p = no (λ ∀p → ¬∀p (λ {f} _ → ∀p f))
decLift : ∀ {n} {P : Fin n → Set} →
U.Decidable P → U.Decidable (Lift P)
decLift dec p = all∈? (λ {x} _ → dec x)
private
restrictSP : ∀ {n} → Side → (Subset (suc n) → Set) → (Subset n → Set)
restrictSP s P p = P (s ∷ p)
restrictS : ∀ {n} {P : Subset (suc n) → Set} →
(s : Side) → U.Decidable P → U.Decidable (restrictSP s P)
restrictS s dec p = dec (s ∷ p)
anySubset? : ∀ {n} {P : Subset n → Set} →
U.Decidable P → Dec (∃ P)
anySubset? {zero} {P} dec with dec []
... | yes P[] = yes (_ , P[])
... | no ¬P[] = no helper
where
helper : ∄ P
helper ([] , P[]) = ¬P[] P[]
anySubset? {suc n} {P} dec with anySubset? (restrictS inside dec)
| anySubset? (restrictS outside dec)
... | yes (_ , Pp) | _ = yes (_ , Pp)
... | _ | yes (_ , Pp) = yes (_ , Pp)
... | no ¬Pp | no ¬Pp' = no helper
where
helper : ∄ P
helper (inside ∷ p , Pp) = ¬Pp (_ , Pp)
helper (outside ∷ p , Pp') = ¬Pp' (_ , Pp')
-- If a decidable predicate P over a finite set is sometimes false,
-- then we can find the smallest value for which this is the case.
¬∀⟶∃¬-smallest :
∀ n {p} (P : Fin n → Set p) → U.Decidable P →
¬ (∀ i → P i) → ∃ λ i → ¬ P i × ((j : Fin′ i) → P (inject j))
¬∀⟶∃¬-smallest zero P dec ¬∀iPi = ⊥-elim (¬∀iPi (λ()))
¬∀⟶∃¬-smallest (suc n) P dec ¬∀iPi with dec zero
¬∀⟶∃¬-smallest (suc n) P dec ¬∀iPi | no ¬P0 = (zero , ¬P0 , λ ())
¬∀⟶∃¬-smallest (suc n) P dec ¬∀iPi | yes P0 =
Prod.map suc (Prod.map id extend′) $
¬∀⟶∃¬-smallest n (λ n → P (suc n)) (dec ∘ suc) (¬∀iPi ∘ extend)
where
extend : (∀ i → P (suc i)) → (∀ i → P i)
extend ∀iP[1+i] zero = P0
extend ∀iP[1+i] (suc i) = ∀iP[1+i] i
extend′ : ∀ {i : Fin n} →
((j : Fin′ i) → P (suc (inject j))) →
((j : Fin′ (suc i)) → P (inject j))
extend′ g zero = P0
extend′ g (suc j) = g j
-- Decision procedure for _⊆_ (obtained via the natural lattice
-- order).
infix 4 _⊆?_
_⊆?_ : ∀ {n} → B.Decidable (_⊆_ {n = n})
p₁ ⊆? p₂ =
Dec.map (Eq.sym NaturalPoset.orders-equivalent) $
Dec.map′ PropVecEq.to-≡ PropVecEq.from-≡ $
VecEq.DecidableEquality._≟_ Bool.decSetoid p₁ (p₁ ∩ p₂)
|
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open import Silica
open import HeapProperties
open import Data.List.Membership.DecSetoid ≡-decSetoid
open import Data.List.Relation.Unary.Any
open TypeEnvContext
------------ Lemmas --------------
-- If an expression is well-typed in Δ, then all locations in the expression are in Δ.
locationsInExprAreInContext : ∀ {Δ Δ' e T fl}
→ ∀ {l : IndirectRef}
→ Δ ⊢ e ⦂ T ⊣ Δ'
→ FreeLocations e fl
→ l ∈ fl
----------------
→ ∃[ T' ] ((StaticEnv.locEnv Δ) ∋ l ⦂ T')
-- fl is empty, so l is in fl leads to a contradiction.
locationsInExprAreInContext (varTy x spl) varFL ()
-- l is related to e, so therefore we can point to where l is in Δ.
locationsInExprAreInContext (locTy {Δ = Δ''} {T₁ = T₁} l spl) (locFL l) (here refl) = ⟨ T₁ , Z ⟩
locationsInExprAreInContext (locTy {Δ = Δ''} {T₁} l spl) (locFL l) (there ())
locationsInExprAreInContext (objTy o spl) objValFL ()
locationsInExprAreInContext (boolTy b) boolFL ()
locationsInExprAreInContext ty voidFL ()
-- TODO: relax progress a little per statement of Theorem 5.1.
data Progress : Expr → Set where
step : ∀ (Σ Σ' : RuntimeEnv)
→ ∀ (e e' : Expr)
→ (Σ , e ⟶ Σ' , e')
-------------
→ Progress e
done : ∀ {e : Value}
---------
→ Progress (valExpr e)
progress : ∀ {e T Δ Δ'}
→ ∀ (Σ : RuntimeEnv)
→ Closed e
→ Σ & Δ ok
→ Δ ⊢ e ⦂ T ⊣ Δ'
---------------
→ Progress e
progress Σ (closed (simpleExpr (var x)) ()) consis (varTy x split) -- Contradiction: var x has free variables, but we assumed e was closed.
-- TODO: Refactor these cases to avoid duplication!
progress Σ cl consis@(ok {Σ} _ _ _ _ _) ty@(locTy l voidSplit) =
let
locationExistsInContext = locationsInExprAreInContext ty (locFL l) (here refl)
lInDelta = proj₂ locationExistsInContext
heapLookupResult = voidLookup consis lInDelta
in
step Σ Σ (simpleExpr (loc l)) (valExpr voidVal) (SElookup ty heapLookupResult)
progress Σ cl consis@(ok {Σ} _ _ _ _ _) ty@(locTy l booleanSplit) =
let
locationExistsInContext = locationsInExprAreInContext ty (locFL l) (here refl)
lInDelta = proj₂ locationExistsInContext
heapLookupResult = boolLookup consis lInDelta
in
step Σ Σ (simpleExpr (loc l)) (valExpr (boolVal (proj₁ heapLookupResult))) (SElookup ty (proj₂ heapLookupResult))
progress Σ cl consis@(ok {Σ} _ _ _ _ _) ty@(locTy l (unownedSplit _ _ _ _)) =
let
locationExistsInContext = locationsInExprAreInContext ty (locFL l) (here refl)
lInDelta = proj₂ locationExistsInContext
heapLookupResult = locLookup consis lInDelta
heapLookupFound = proj₁ (proj₂ heapLookupResult)
o = proj₁ heapLookupResult
in
step Σ Σ (simpleExpr (loc l)) (valExpr (objVal o)) (SElookup ty heapLookupFound)
progress Σ cl consis@(ok {Σ} _ _ _ _ _) ty@(locTy l (shared-shared-shared _)) =
let
locationExistsInContext = locationsInExprAreInContext ty (locFL l) (here refl)
lInDelta = proj₂ locationExistsInContext
heapLookupResult = locLookup consis lInDelta
heapLookupFound = proj₁ (proj₂ heapLookupResult)
o = proj₁ heapLookupResult
in
step Σ Σ (simpleExpr (loc l)) (valExpr (objVal o)) (SElookup ty heapLookupFound)
progress Σ cl consis@(ok {Σ} _ _ _ _ _) ty@(locTy l (owned-shared _)) =
let
locationExistsInContext = locationsInExprAreInContext ty (locFL l) (here refl)
lInDelta = proj₂ locationExistsInContext
heapLookupResult = locLookup consis lInDelta
heapLookupFound = proj₁ (proj₂ heapLookupResult)
o = proj₁ heapLookupResult
in
step Σ Σ (simpleExpr (loc l)) (valExpr (objVal o)) (SElookup ty heapLookupFound)
progress Σ cl consis@(ok {Σ} _ _ _ _ _) ty@(locTy l (states-shared _)) =
let
locationExistsInContext = locationsInExprAreInContext ty (locFL l) (here refl)
lInDelta = proj₂ locationExistsInContext
heapLookupResult = locLookup consis lInDelta
heapLookupFound = proj₁ (proj₂ heapLookupResult)
o = proj₁ heapLookupResult
in
step Σ Σ (simpleExpr (loc l)) (valExpr (objVal o)) (SElookup ty heapLookupFound)
progress Σ cl consis (objTy o split) = done
progress Σ cl consis (boolTy b) = done
progress Σ cl consis (voidTy) = done
progress Σ cl consis (assertTyₗ {s₁ = s} {l = l} tcEq subset) = step Σ Σ (assertₗ l s) (valExpr voidVal) (SEassertₗ {Σ} l s)
progress Σ cl consis (assertTyₓ {s₁ = s} {x = x} tcEq subset) = step Σ Σ (assertₓ x s) (valExpr voidVal) (SEassertₓ {Σ} x s)
progress Σ cl consis (newTy {Γ} {Δ} {Δ'} {Δ''} {states} {C} {st} {x₁} {x₂} stOK x₁Ty x₂Ty CInΓ refl) =
let
oFresh = ObjectRefContext.fresh (RuntimeEnv.μ Σ)
o = proj₁ oFresh
μ' = (RuntimeEnv.μ Σ) ObjectRefContext., o ⦂ record { contractName = C ; stateName = st ; x₁ = x₁ ; x₂ = x₂ }
Σ' = record Σ {μ = μ'}
in
step Σ Σ' (new C st x₁ x₂) (valExpr (objVal o)) (SEnew (proj₁ (proj₂ oFresh)) refl refl)
|
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Indexed binary relations
------------------------------------------------------------------------
-- This file contains some core definitions which are reexported by
-- Relation.Binary.Indexed.
module Relation.Binary.Indexed.Core where
open import Function
open import Level
import Relation.Binary.Core as B
import Relation.Binary.Core as P
------------------------------------------------------------------------
-- Indexed binary relations
-- Heterogeneous.
REL : ∀ {i₁ i₂ a₁ a₂} {I₁ : Set i₁} {I₂ : Set i₂} →
(I₁ → Set a₁) → (I₂ → Set a₂) → (ℓ : Level) → Set _
REL A₁ A₂ ℓ = ∀ {i₁ i₂} → A₁ i₁ → A₂ i₂ → Set ℓ
-- Homogeneous.
Rel : ∀ {i a} {I : Set i} → (I → Set a) → (ℓ : Level) → Set _
Rel A ℓ = REL A A ℓ
------------------------------------------------------------------------
-- Simple properties of indexed binary relations
-- Reflexivity.
Reflexive : ∀ {i a ℓ} {I : Set i} (A : I → Set a) → Rel A ℓ → Set _
Reflexive _ _∼_ = ∀ {i} → B.Reflexive (_∼_ {i})
-- Symmetry.
Symmetric : ∀ {i a ℓ} {I : Set i} (A : I → Set a) → Rel A ℓ → Set _
Symmetric _ _∼_ = ∀ {i j} → B.Sym (_∼_ {i} {j}) _∼_
-- Transitivity.
Transitive : ∀ {i a ℓ} {I : Set i} (A : I → Set a) → Rel A ℓ → Set _
Transitive _ _∼_ = ∀ {i j k} → B.Trans _∼_ (_∼_ {j}) (_∼_ {i} {k})
------------------------------------------------------------------------
-- Setoids
record IsEquivalence {i a ℓ} {I : Set i} (A : I → Set a)
(_≈_ : Rel A ℓ) : Set (i ⊔ a ⊔ ℓ) where
field
refl : Reflexive A _≈_
sym : Symmetric A _≈_
trans : Transitive A _≈_
reflexive : ∀ {i} → P._≡_ ⟨ B._⇒_ ⟩ _≈_ {i}
reflexive P.refl = refl
record Setoid {i} (I : Set i) c ℓ : Set (suc (i ⊔ c ⊔ ℓ)) where
infix 4 _≈_
field
Carrier : I → Set c
_≈_ : Rel Carrier ℓ
isEquivalence : IsEquivalence Carrier _≈_
open IsEquivalence isEquivalence public
|
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{-
Product of structures S and T: X ↦ S X × T X
-}
{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Structures.Relational.Product where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Function
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Structure
open import Cubical.Foundations.RelationalStructure
open import Cubical.Foundations.SIP
open import Cubical.Foundations.Univalence
open import Cubical.Data.Sigma
open import Cubical.HITs.PropositionalTruncation as Trunc
open import Cubical.HITs.SetQuotients
open import Cubical.Structures.Product
private
variable
ℓ ℓ₁ ℓ₁' ℓ₁'' ℓ₂ ℓ₂' ℓ₂'' : Level
-- Structured relations
ProductRelStr :
{S₁ : Type ℓ → Type ℓ₁} (ρ₁ : StrRel S₁ ℓ₁')
{S₂ : Type ℓ → Type ℓ₂} (ρ₂ : StrRel S₂ ℓ₂')
→ StrRel (ProductStructure S₁ S₂) (ℓ-max ℓ₁' ℓ₂')
ProductRelStr ρ₁ ρ₂ R (s₁ , s₂) (t₁ , t₂) =
ρ₁ R s₁ t₁ × ρ₂ R s₂ t₂
productSuitableRel :
{S₁ : Type ℓ → Type ℓ₁} {ρ₁ : StrRel S₁ ℓ₁'}
{S₂ : Type ℓ → Type ℓ₂} {ρ₂ : StrRel S₂ ℓ₂'}
→ SuitableStrRel S₁ ρ₁ → SuitableStrRel S₂ ρ₂
→ SuitableStrRel (ProductStructure S₁ S₂) (ProductRelStr ρ₁ ρ₂)
productSuitableRel θ₁ θ₂ .quo (X , s₁ , s₂) R (r₁ , r₂) .fst .fst =
θ₁ .quo (X , s₁) R r₁ .fst .fst , θ₂ .quo (X , s₂) R r₂ .fst .fst
productSuitableRel θ₁ θ₂ .quo (X , s₁ , s₂) R (r₁ , r₂) .fst .snd =
θ₁ .quo (X , s₁) R r₁ .fst .snd , θ₂ .quo (X , s₂) R r₂ .fst .snd
productSuitableRel θ₁ θ₂ .quo (X , s₁ , s₂) R (r₁ , r₂) .snd ((q₁ , q₂) , (c₁ , c₂)) i .fst =
θ₁ .quo (X , s₁) R r₁ .snd (q₁ , c₁) i .fst , θ₂ .quo (X , s₂) R r₂ .snd (q₂ , c₂) i .fst
productSuitableRel θ₁ θ₂ .quo (X , s₁ , s₂) R (r₁ , r₂) .snd ((q₁ , q₂) , (c₁ , c₂)) i .snd =
θ₁ .quo (X , s₁) R r₁ .snd (q₁ , c₁) i .snd , θ₂ .quo (X , s₂) R r₂ .snd (q₂ , c₂) i .snd
productSuitableRel θ₁ θ₂ .symmetric R (r₁ , r₂) =
θ₁ .symmetric R r₁ , θ₂ .symmetric R r₂
productSuitableRel θ₁ θ₂ .transitive R R' (r₁ , r₂) (r₁' , r₂') =
θ₁ .transitive R R' r₁ r₁' , θ₂ .transitive R R' r₂ r₂'
productSuitableRel θ₁ θ₂ .set setX =
isSet× (θ₁ .set setX) (θ₂ .set setX)
productSuitableRel θ₁ θ₂ .prop propR (s₁ , s₂) (t₁ , t₂) =
isProp× (θ₁ .prop propR s₁ t₁) (θ₂ .prop propR s₂ t₂)
productRelMatchesEquiv :
{S₁ : Type ℓ → Type ℓ₁} (ρ₁ : StrRel S₁ ℓ₁') {ι₁ : StrEquiv S₁ ℓ₁''}
{S₂ : Type ℓ → Type ℓ₂} (ρ₂ : StrRel S₂ ℓ₂') {ι₂ : StrEquiv S₂ ℓ₂''}
→ StrRelMatchesEquiv ρ₁ ι₁ → StrRelMatchesEquiv ρ₂ ι₂
→ StrRelMatchesEquiv (ProductRelStr ρ₁ ρ₂) (ProductEquivStr ι₁ ι₂)
productRelMatchesEquiv ρ₁ ρ₂ μ₁ μ₂ A B e =
Σ-cong-equiv (μ₁ _ _ e) (λ _ → μ₂ _ _ e)
productRelAction :
{S₁ : Type ℓ → Type ℓ₁} {ρ₁ : StrRel S₁ ℓ₁'} (α₁ : StrRelAction ρ₁)
{S₂ : Type ℓ → Type ℓ₂} {ρ₂ : StrRel S₂ ℓ₂'} (α₂ : StrRelAction ρ₂)
→ StrRelAction (ProductRelStr ρ₁ ρ₂)
productRelAction α₁ α₂ .actStr f (s₁ , s₂) = α₁ .actStr f s₁ , α₂ .actStr f s₂
productRelAction α₁ α₂ .actStrId (s₁ , s₂) = ΣPathP (α₁ .actStrId s₁ , α₂ .actStrId s₂)
productRelAction α₁ α₂ .actRel h _ _ (r₁ , r₂) = α₁ .actRel h _ _ r₁ , α₂ .actRel h _ _ r₂
productPositiveRel :
{S₁ : Type ℓ → Type ℓ₁} {ρ₁ : StrRel S₁ ℓ₁'} {θ₁ : SuitableStrRel S₁ ρ₁}
{S₂ : Type ℓ → Type ℓ₂} {ρ₂ : StrRel S₂ ℓ₂'} {θ₂ : SuitableStrRel S₂ ρ₂}
→ PositiveStrRel θ₁
→ PositiveStrRel θ₂
→ PositiveStrRel (productSuitableRel θ₁ θ₂)
productPositiveRel σ₁ σ₂ .act = productRelAction (σ₁ .act) (σ₂ .act)
productPositiveRel σ₁ σ₂ .reflexive (s₁ , s₂) = σ₁ .reflexive s₁ , σ₂ .reflexive s₂
productPositiveRel σ₁ σ₂ .detransitive R R' (rr'₁ , rr'₂) =
Trunc.rec squash
(λ {(s₁ , r₁ , r₁') →
Trunc.rec squash
(λ {(s₂ , r₂ , r₂') → ∣ (s₁ , s₂) , (r₁ , r₂) , (r₁' , r₂') ∣})
(σ₂ .detransitive R R' rr'₂)})
(σ₁ .detransitive R R' rr'₁)
productPositiveRel {S₁ = S₁} {ρ₁} {θ₁} {S₂} {ρ₂} {θ₂} σ₁ σ₂ .quo {X} R =
subst isEquiv
(funExt (elimProp (λ _ → productSuitableRel θ₁ θ₂ .set squash/ _ _) (λ _ → refl)))
(compEquiv
(isoToEquiv isom)
(Σ-cong-equiv (_ , σ₁ .quo R) (λ _ → _ , σ₂ .quo R)) .snd)
where
fwd :
ProductStructure S₁ S₂ X / ProductRelStr ρ₁ ρ₂ (R .fst .fst)
→ (S₁ X / ρ₁ (R .fst .fst)) × (S₂ X / ρ₂ (R .fst .fst))
fwd [ s₁ , s₂ ] = [ s₁ ] , [ s₂ ]
fwd (eq/ (s₁ , s₂) (t₁ , t₂) (r₁ , r₂) i) = eq/ s₁ t₁ r₁ i , eq/ s₂ t₂ r₂ i
fwd (squash/ _ _ p q i j) =
isSet× squash/ squash/ _ _ (cong fwd p) (cong fwd q) i j
bwd[] : S₁ X → S₂ X / ρ₂ (R .fst .fst)
→ ProductStructure S₁ S₂ X / ProductRelStr ρ₁ ρ₂ (R .fst .fst)
bwd[] s₁ [ s₂ ] = [ s₁ , s₂ ]
bwd[] s₁ (eq/ s₂ t₂ r₂ i) =
eq/ (s₁ , s₂) (s₁ , t₂) (posRelReflexive σ₁ R s₁ , r₂) i
bwd[] s₁ (squash/ _ _ p q i j) =
squash/ _ _ (λ j → bwd[] s₁ (p j)) (λ j → bwd[] s₁ (q j)) i j
bwd : S₁ X / ρ₁ (R .fst .fst) → S₂ X / ρ₂ (R .fst .fst)
→ ProductStructure S₁ S₂ X / ProductRelStr ρ₁ ρ₂ (R .fst .fst)
bwd [ s₁ ] u = bwd[] s₁ u
bwd (eq/ s₁ t₁ r₁ i) u = path u i
where
path : ∀ u → bwd [ s₁ ] u ≡ bwd [ t₁ ] u
path = elimProp (λ _ → squash/ _ _) (λ s₂ → eq/ (s₁ , s₂) (t₁ , s₂) (r₁ , posRelReflexive σ₂ R s₂))
bwd (squash/ _ _ p q i j) =
isSetΠ (λ _ → squash/) _ _ (cong bwd p) (cong bwd q) i j
open Iso
isom : Iso _ _
isom .fun = fwd
isom .inv = uncurry bwd
isom .rightInv =
uncurry
(elimProp (λ _ → isPropΠ λ _ → isSet× squash/ squash/ _ _)
(λ _ → elimProp (λ _ → isSet× squash/ squash/ _ _) (λ _ → refl)))
isom .leftInv = elimProp (λ _ → squash/ _ _) (λ _ → refl)
productRelMatchesTransp :
{S₁ : Type ℓ → Type ℓ₁} (ρ₁ : StrRel S₁ ℓ₁') (α₁ : EquivAction S₁)
{S₂ : Type ℓ → Type ℓ₂} (ρ₂ : StrRel S₂ ℓ₂') (α₂ : EquivAction S₂)
→ StrRelMatchesEquiv ρ₁ (EquivAction→StrEquiv α₁)
→ StrRelMatchesEquiv ρ₂ (EquivAction→StrEquiv α₂)
→ StrRelMatchesEquiv (ProductRelStr ρ₁ ρ₂) (EquivAction→StrEquiv (productEquivAction α₁ α₂))
productRelMatchesTransp _ _ _ _ μ₁ μ₂ _ _ e =
compEquiv (Σ-cong-equiv (μ₁ _ _ e) (λ _ → μ₂ _ _ e)) ΣPath≃PathΣ
|
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module List.Permutation.Pair (A : Set) where
open import Data.List
open import Data.Product
data _≈_ : List A → List A × List A → Set where
≈[]r : (xs : List A)
→ xs ≈ (xs , [])
≈[]l : (xs : List A)
→ xs ≈ ([] , xs)
≈xr : {x : A}{xs ys zs : List A}
→ xs ≈ (ys , zs)
→ (x ∷ xs) ≈ (ys , x ∷ zs)
≈xl : {x : A}{xs ys zs : List A}
→ xs ≈ (ys , zs)
→ (x ∷ xs) ≈ (x ∷ ys , zs)
|
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module Semantics where
open import Data.Nat hiding (_⊔_; _⊓_)
open import Data.Product
open import Data.Sum
open import Data.String using (String)
open import Data.Unit hiding (_≟_)
open import Data.Empty
open import Relation.Nullary
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_;_≢_; refl)
open Eq.≡-Reasoning
open import Level hiding (_⊔_) renaming (zero to lzero; suc to lsuc)
{- TODO:
* subtyping of refinement types
* union types
* intersection types
-}
Id = String
variable
x y : Id
ℓ : Level
data Expr : Set where
Nat : ℕ → Expr
Var : Id → Expr
Lam : Id → Expr → Expr
App : Expr → Expr → Expr
Pair : Expr → Expr → Expr
Fst Snd : Expr → Expr
Inl Inr : Expr → Expr
Case : Expr → Id → Expr → Id → Expr → Expr
data RawType : Set where
Nat : RawType
_⇒_ _⋆_ _⊹_ : RawType → RawType → RawType
ss⇒tt : ∀ {S S₁ T T₁ : RawType} → (S ⇒ S₁) ≡ (T ⇒ T₁) → (S ≡ T × S₁ ≡ T₁)
ss⇒tt refl = refl , refl
ss⋆tt : ∀ {S S₁ T T₁ : RawType} → (S ⋆ S₁) ≡ (T ⋆ T₁) → (S ≡ T × S₁ ≡ T₁)
ss⋆tt refl = refl , refl
ss⊹tt : ∀ {S S₁ T T₁ : RawType} → (S ⊹ S₁) ≡ (T ⊹ T₁) → (S ≡ T × S₁ ≡ T₁)
ss⊹tt refl = refl , refl
data Type : Set₁ where
Base : (P : ℕ → Set) → Type -- refinement
Nat : Type
_⇒_ : Type → Type → Type
_⋆_ : Type → Type → Type
_⊹_ : Type → Type → Type
T-Nat = Base (λ n → ⊤) -- all natural numbers
data ne : Type → Set where
ne-base : ∀ {P} → (∃P : Σ ℕ P) → ne (Base P)
ne-nat : ne Nat
ne-⇒ : ∀ {S T} → ne S → ne T → ne (S ⇒ T)
ne-⋆ : ∀ {S T} → ne S → ne T → ne (S ⋆ T)
ne-⊹L : ∀ {S T} → ne S → ne (S ⊹ T)
ne-⊹R : ∀ {S T} → ne T → ne (S ⊹ T)
data Env (A : Set ℓ) : Set ℓ where
· : Env A
_,_⦂_ : Env A → (x : Id) → (a : A) → Env A
∥_∥ : Type → RawType
∥ Base P ∥ = Nat
∥ Nat ∥ = Nat
∥ S ⇒ S₁ ∥ = ∥ S ∥ ⇒ ∥ S₁ ∥
∥ S ⋆ S₁ ∥ = ∥ S ∥ ⋆ ∥ S₁ ∥
∥ S ⊹ S₁ ∥ = ∥ S ∥ ⊹ ∥ S₁ ∥
_∨_ : (P Q : ℕ → Set) → ℕ → Set
P ∨ Q = λ n → P n ⊎ Q n
_∧_ : (P Q : ℕ → Set) → ℕ → Set
P ∧ Q = λ n → P n × Q n
implies : ∀ {P Q : ℕ → Set} → (n : ℕ) → P n → (P n ⊎ Q n)
implies n Pn = inj₁ Pn
p*q->p : ∀ {P Q : ℕ → Set} → (n : ℕ) → (P n × Q n) → P n
p*q->p n (Pn , Qn) = Pn
_⊔_ _⊓_ : (S T : Type) {r : ∥ S ∥ ≡ ∥ T ∥} → Type
(Base P ⊔ Base P₁) {refl} = Base (P ∨ P₁)
(Base P ⊔ Nat) = Nat
(Nat ⊔ Base P) = Nat
(Nat ⊔ Nat) = Nat
((S ⇒ S₁) ⊔ (T ⇒ T₁)) {r} with ss⇒tt r
... | sss , ttt = (S ⊓ T){sss} ⇒ (S₁ ⊔ T₁){ttt}
((S ⋆ S₁) ⊔ (T ⋆ T₁)) {r} with ss⋆tt r
... | sss , ttt = (S ⊔ T){sss} ⋆ (S₁ ⊔ T₁){ttt}
((S ⊹ S₁) ⊔ (T ⊹ T₁)) {r} with ss⊹tt r
... | sss , ttt = (S ⊔ T){sss} ⊹ (S₁ ⊔ T₁){ttt}
Base P ⊓ Base P₁ = Base (P ∧ P₁)
Base P ⊓ Nat = Base P
Nat ⊓ Base P = Base P
Nat ⊓ Nat = Nat
((S ⇒ S₁) ⊓ (T ⇒ T₁)){r} with ss⇒tt r
... | sss , ttt = (S ⊔ T){sss} ⇒ (S₁ ⊓ T₁){ttt}
((S ⋆ S₁) ⊓ (T ⋆ T₁)){r} with ss⋆tt r
... | sss , ttt = (S ⊓ T){sss} ⋆ (S₁ ⊓ T₁){ttt}
((S ⊹ S₁) ⊓ (T ⊹ T₁)){r} with ss⊹tt r
... | sss , ttt = (S ⊓ T){sss} ⊹ (S₁ ⊓ T₁){ttt}
variable
S T U S′ T′ U′ U″ : Type
Γ Γ₁ Γ₂ : Env Type
L M N : Expr
n : ℕ
P : ℕ → Set
data Split {A : Set ℓ} : Env A → Env A → Env A → Set ℓ where
nil : Split · · ·
lft : ∀ {a : A}{Γ Γ₁ Γ₂ : Env A} → Split Γ Γ₁ Γ₂ → Split (Γ , x ⦂ a) (Γ₁ , x ⦂ a) Γ₂
rgt : ∀ {a : A}{Γ Γ₁ Γ₂ : Env A} → Split Γ Γ₁ Γ₂ → Split (Γ , x ⦂ a) Γ₁ (Γ₂ , x ⦂ a)
data _⦂_∈_ {A : Set ℓ} : Id → A → Env A → Set ℓ where
found : ∀ {a : A}{E : Env A} →
x ⦂ a ∈ (E , x ⦂ a)
there : ∀ {a a' : A}{E : Env A} →
x ⦂ a ∈ E →
-- x ≢ y →
x ⦂ a ∈ (E , y ⦂ a')
data _<:_ : Type → Type → Set where
<:-refl :
T <: T
<:-base :
(P Q : ℕ → Set) →
(p→q : ∀ n → P n → Q n) →
Base P <: Base Q
<:-base-nat :
Base P <: Nat
<:-⇒ :
S′ <: S →
T <: T′ →
(S ⇒ T) <: (S′ ⇒ T′)
<:-⋆ :
S <: S′ →
T <: T′ →
(S ⋆ T) <: (S′ ⋆ T′)
<:-⊹ :
S <: S′ →
T <: T′ →
(S ⊹ T) <: (S′ ⊹ T′)
-- subtyping is compatible with raw types
<:-raw : S <: T → ∥ S ∥ ≡ ∥ T ∥
<:-raw <:-refl = refl
<:-raw (<:-base P Q p→q) = refl
<:-raw <:-base-nat = refl
<:-raw (<:-⇒ s<:t s<:t₁) = Eq.cong₂ _⇒_ (Eq.sym (<:-raw s<:t)) (<:-raw s<:t₁)
<:-raw (<:-⋆ s<:t s<:t₁) = Eq.cong₂ _⋆_ (<:-raw s<:t) (<:-raw s<:t₁)
<:-raw (<:-⊹ s<:t s<:t₁) = Eq.cong₂ _⊹_ (<:-raw s<:t) (<:-raw s<:t₁)
<:-⊔ : ∀ S T → {c : ∥ S ∥ ≡ ∥ T ∥} → S <: (S ⊔ T){c}
<:-⊓ : ∀ S T → {c : ∥ S ∥ ≡ ∥ T ∥} → (S ⊓ T){c} <: S
<:-⊔ (Base P) (Base P₁) {refl} = <:-base P (P ∨ P₁) implies
<:-⊔ (Base P) Nat = <:-base-nat
<:-⊔ Nat (Base P) = <:-refl
<:-⊔ Nat Nat = <:-refl
<:-⊔ (S ⇒ S₁) (T ⇒ T₁) {c} with ss⇒tt c
... | c1 , c2 = <:-⇒ (<:-⊓ S T) (<:-⊔ S₁ T₁)
<:-⊔ (S ⋆ S₁) (T ⋆ T₁) {c} with ss⋆tt c
... | c1 , c2 = <:-⋆ (<:-⊔ S T) (<:-⊔ S₁ T₁)
<:-⊔ (S ⊹ S₁) (T ⊹ T₁) {c} with ss⊹tt c
... | c1 , c2 = <:-⊹ (<:-⊔ S T) (<:-⊔ S₁ T₁)
<:-⊓ (Base P) (Base P₁) {refl} = <:-base (P ∧ P₁) P p*q->p
<:-⊓ (Base P) Nat = <:-refl
<:-⊓ Nat (Base P) = <:-base-nat
<:-⊓ Nat Nat = <:-refl
<:-⊓ (S ⇒ S₁) (T ⇒ T₁) {c} with ss⇒tt c
... | c1 , c2 = <:-⇒ (<:-⊔ S T) (<:-⊓ S₁ T₁)
<:-⊓ (S ⋆ S₁) (T ⋆ T₁) {c} with ss⋆tt c
... | c1 , c2 = <:-⋆ (<:-⊓ S T) (<:-⊓ S₁ T₁)
<:-⊓ (S ⊹ S₁) (T ⊹ T₁) {c} with ss⊹tt c
... | c1 , c2 = <:-⊹ (<:-⊓ S T) (<:-⊓ S₁ T₁)
-- should be in terms of RawType for evaluation
data _⊢_⦂_ : Env Type → Expr → Type → Set₁ where
nat' :
Γ ⊢ Nat n ⦂ Base (_≡_ n)
var :
(x∈ : x ⦂ T ∈ Γ) →
--------------------
Γ ⊢ Var x ⦂ T
lam :
(Γ , x ⦂ S) ⊢ M ⦂ T →
--------------------
Γ ⊢ Lam x M ⦂ (S ⇒ T)
app :
Γ ⊢ M ⦂ (S ⇒ T) →
Γ ⊢ N ⦂ S →
--------------------
Γ ⊢ App M N ⦂ T
pair :
Γ ⊢ M ⦂ S →
Γ ⊢ N ⦂ T →
--------------------
Γ ⊢ Pair M N ⦂ (S ⋆ T)
pair-E1 :
Γ ⊢ M ⦂ (S ⋆ T) →
--------------------
Γ ⊢ Fst M ⦂ S
pair-E2 :
Γ ⊢ M ⦂ (S ⋆ T) →
--------------------
Γ ⊢ Snd M ⦂ T
sum-I1 :
Γ ⊢ M ⦂ S →
--------------------
Γ ⊢ Inl M ⦂ (S ⊹ T)
sum-I2 :
Γ ⊢ N ⦂ T →
--------------------
Γ ⊢ Inl N ⦂ (S ⊹ T)
sum-E :
Γ ⊢ L ⦂ (S ⊹ T) →
(Γ , x ⦂ S) ⊢ M ⦂ U →
(Γ , y ⦂ T) ⊢ N ⦂ U →
--------------------
Γ ⊢ Case L x M y N ⦂ U
split-sym : Split Γ Γ₁ Γ₂ → Split Γ Γ₂ Γ₁
split-sym nil = nil
split-sym (lft sp) = rgt (split-sym sp)
split-sym (rgt sp) = lft (split-sym sp)
weaken-∈ : Split Γ Γ₁ Γ₂ → x ⦂ T ∈ Γ₁ → x ⦂ T ∈ Γ
weaken-∈ (lft sp) found = found
weaken-∈ (rgt sp) found = there (weaken-∈ sp found)
weaken-∈ (lft sp) (there x∈) = there (weaken-∈ sp x∈)
weaken-∈ (rgt sp) (there x∈) = there (weaken-∈ sp (there x∈))
weaken : Split Γ Γ₁ Γ₂ → Γ₁ ⊢ M ⦂ T → Γ ⊢ M ⦂ T
weaken sp (nat') = nat'
weaken sp (var x∈) = var (weaken-∈ sp x∈)
weaken sp (lam ⊢M) = lam (weaken (lft sp) ⊢M)
weaken sp (app ⊢M ⊢N) = app (weaken sp ⊢M) (weaken sp ⊢N)
weaken sp (pair ⊢M ⊢N) = pair (weaken sp ⊢M) (weaken sp ⊢N)
weaken sp (pair-E1 ⊢M) = pair-E1 (weaken sp ⊢M)
weaken sp (pair-E2 ⊢M) = pair-E2 (weaken sp ⊢M)
weaken sp (sum-I1 ⊢M) = sum-I1 (weaken sp ⊢M)
weaken sp (sum-I2 ⊢N) = sum-I2 (weaken sp ⊢N)
weaken sp (sum-E ⊢L ⊢M ⊢N) = sum-E (weaken sp ⊢L) (weaken (lft sp) ⊢M) (weaken (lft sp) ⊢N)
-- incorrectness typing
P=n : ℕ → ℕ → Set
P=n = λ n x → n ≡ x
data _⊢_÷_ : Env Type → Expr → Type → Set₁ where
nat' :
--------------------
· ⊢ Nat n ÷ Base (_≡_ n)
var1 :
( · , x ⦂ T) ⊢ Var x ÷ T
{-
var :
x ⦂ T ∈ Γ →
--------------------
Γ ⊢ Var x ÷ T
-}
lam :
(· , x ⦂ S) ⊢ M ÷ T →
--------------------
· ⊢ Lam x M ÷ (S ⇒ T)
pair :
Split Γ Γ₁ Γ₂ →
Γ₁ ⊢ M ÷ S →
Γ₂ ⊢ N ÷ T →
--------------------
Γ ⊢ Pair M N ÷ (S ⋆ T)
pair-E1 :
Γ ⊢ M ÷ (S ⋆ T) →
--------------------
Γ ⊢ Fst M ÷ S
pair-E2 :
Γ ⊢ M ÷ (S ⋆ T) →
--------------------
Γ ⊢ Snd M ÷ T
sum-E :
Split Γ Γ₁ Γ₂ →
Γ₁ ⊢ L ÷ (S ⊹ T) →
(Γ₂ , x ⦂ S) ⊢ M ÷ U →
(Γ₂ , y ⦂ T) ⊢ N ÷ U →
--------------------
Γ ⊢ Case L x M y N ÷ U
sum-E′ : ∀ {ru′=ru″} →
Split Γ Γ₁ Γ₂ →
Γ₁ ⊢ L ÷ (S ⊹ T) →
(Γ₂ , x ⦂ S) ⊢ M ÷ U′ →
(Γ₂ , y ⦂ T) ⊢ N ÷ U″ →
U ≡ (U′ ⊔ U″){ru′=ru″} →
--------------------
Γ ⊢ Case L x M y N ÷ U
{-
`sub` :
Γ ⊢ M ÷ S →
T <: S →
--------------------
Γ ⊢ M ÷ T
-}
record _←_ (A B : Set) : Set where
field
func : A → B
back : ∀ (b : B) → ∃ λ (a : A) → func a ≡ b
open _←_
T⟦_⟧ : Type → Set
T⟦ Base P ⟧ = Σ ℕ P
T⟦ Nat ⟧ = ℕ
T⟦ S ⇒ T ⟧ = T⟦ S ⟧ → T⟦ T ⟧
T⟦ S ⋆ T ⟧ = T⟦ S ⟧ × T⟦ T ⟧
T⟦ S ⊹ T ⟧ = T⟦ S ⟧ ⊎ T⟦ T ⟧
T'⟦_⟧ : Type → Set
T'⟦ Base P ⟧ = Σ ℕ P
T'⟦ Nat ⟧ = ℕ
T'⟦ S ⇒ T ⟧ = T'⟦ S ⟧ ← T'⟦ T ⟧
T'⟦ S ⋆ T ⟧ = T'⟦ S ⟧ × T'⟦ T ⟧
T'⟦ S ⊹ T ⟧ = T'⟦ S ⟧ ⊎ T'⟦ T ⟧
E⟦_⟧ : Env Type → Env Set
E⟦ · ⟧ = ·
E⟦ Γ , x ⦂ T ⟧ = E⟦ Γ ⟧ , x ⦂ T⟦ T ⟧
data iEnv : Env Set → Set where
· : iEnv ·
_,_⦂_ : ∀ {E}{A} → iEnv E → (x : Id) → (a : A) → iEnv (E , x ⦂ A)
lookup : (x ⦂ T ∈ Γ) → iEnv E⟦ Γ ⟧ → T⟦ T ⟧
lookup found (γ , _ ⦂ a) = a
lookup (there x∈) (γ , _ ⦂ a) = lookup x∈ γ
eval : Γ ⊢ M ⦂ T → iEnv E⟦ Γ ⟧ → T⟦ T ⟧
eval (nat'{n = n}) γ = n , refl
eval (var x∈) γ = lookup x∈ γ
eval (lam ⊢M) γ = λ s → eval ⊢M (γ , _ ⦂ s)
eval (app ⊢M ⊢N) γ = eval ⊢M γ (eval ⊢N γ)
eval (pair ⊢M ⊢N) γ = (eval ⊢M γ) , (eval ⊢N γ)
eval (pair-E1 ⊢M) γ = proj₁ (eval ⊢M γ)
eval (pair-E2 ⊢M) γ = proj₂ (eval ⊢M γ)
eval (sum-I1 ⊢M) γ = inj₁ (eval ⊢M γ)
eval (sum-I2 ⊢N) γ = inj₂ (eval ⊢N γ)
eval (sum-E{S = S}{T = T}{U = U} ⊢L ⊢M ⊢N) γ =
[ (λ s → eval ⊢M (γ , _ ⦂ s)) , (λ t → eval ⊢N (γ , _ ⦂ t)) ] (eval ⊢L γ)
corr : Γ ⊢ M ÷ T → Γ ⊢ M ⦂ T
corr (nat') = nat'
corr var1 = var found
-- corr (var x) = var x
corr (lam ⊢M) = lam (corr ⊢M)
corr (pair-E1 ÷M) = pair-E1 (corr ÷M)
corr (pair-E2 ÷M) = pair-E2 (corr ÷M)
corr (pair sp ÷M ÷N) = pair (weaken sp (corr ÷M)) (weaken (split-sym sp) (corr ÷N))
corr (sum-E sp ÷L ÷M ÷N) = sum-E (weaken sp (corr ÷L)) (weaken (lft (split-sym sp)) (corr ÷M)) (weaken (lft (split-sym sp)) (corr ÷N))
corr (sum-E′ sp ÷L ÷M ÷N U≡U′⊔U″) =
sum-E (weaken sp (corr ÷L)) (weaken (lft (split-sym sp)) (corr {!!})) (weaken (lft (split-sym sp)) (corr {!!}))
{-
corr (`sub` ÷M T<S) = {!÷M!}
-}
-- pick one element of a type to demonstrate non-emptiness
one : ∀ (T : Type) {net : ne T} → T⟦ T ⟧
one (Base P) {ne-base ∃P} = ∃P
one Nat = zero
one (T ⇒ T₁) {ne-⇒ ne-T ne-T₁} = λ x → one T₁ {ne-T₁}
one (T ⋆ T₁) {ne-⋆ ne-T ne-T₁} = (one T {ne-T}) , (one T₁ {ne-T₁})
one (T ⊹ T₁) {ne-⊹L ne-T} = inj₁ (one T {ne-T})
one (T ⊹ T₁) {ne-⊹R ne-T₁} = inj₂ (one T₁ {ne-T₁})
{- not needed
many : iEnv E⟦ Γ ⟧
many {·} = ·
many {Γ , x ⦂ T} = many , x ⦂ one T
gen : (x∈ : x ⦂ T ∈ Γ) (t : T⟦ T ⟧) → iEnv E⟦ Γ ⟧
gen found t = many , _ ⦂ t
gen (there x∈) t = (gen x∈ t) , _ ⦂ one {!!}
lookup-gen : (x∈ : x ⦂ T ∈ Γ) (t : T⟦ T ⟧) → lookup x∈ (gen x∈ t) ≡ t
lookup-gen found t = refl
lookup-gen (there x∈) t = lookup-gen x∈ t
-}
open Eq.≡-Reasoning
postulate
ext : ∀ {A B : Set}{f g : A → B} → (∀ x → f x ≡ g x) → f ≡ g
unsplit-env : Split Γ Γ₁ Γ₂ → iEnv E⟦ Γ₁ ⟧ → iEnv E⟦ Γ₂ ⟧ → iEnv E⟦ Γ ⟧
unsplit-env nil γ₁ γ₂ = ·
unsplit-env (lft sp) (γ₁ , _ ⦂ a) γ₂ = (unsplit-env sp γ₁ γ₂) , _ ⦂ a
unsplit-env (rgt sp) γ₁ (γ₂ , _ ⦂ a) = (unsplit-env sp γ₁ γ₂) , _ ⦂ a
unsplit-split : (sp : Split Γ Γ₁ Γ₂) (γ₁ : iEnv E⟦ Γ₁ ⟧) (γ₂ : iEnv E⟦ Γ₂ ⟧) →
unsplit-env sp γ₁ γ₂ ≡ unsplit-env (split-sym sp) γ₂ γ₁
unsplit-split nil γ₁ γ₂ = refl
unsplit-split (lft sp) (γ₁ , _ ⦂ a) γ₂ rewrite unsplit-split sp γ₁ γ₂ = refl
unsplit-split (rgt sp) γ₁ (γ₂ , _ ⦂ a) rewrite unsplit-split sp γ₁ γ₂ = refl
lookup-unsplit : (sp : Split Γ Γ₁ Γ₂) (γ₁ : iEnv E⟦ Γ₁ ⟧) (γ₂ : iEnv E⟦ Γ₂ ⟧) →
(x∈ : x ⦂ T ∈ Γ₁) →
lookup (weaken-∈ sp x∈) (unsplit-env sp γ₁ γ₂) ≡ lookup x∈ γ₁
lookup-unsplit (lft sp) (γ₁ , _ ⦂ a) γ₂ found = refl
lookup-unsplit (rgt sp) γ₁ (γ₂ , _ ⦂ a) found = lookup-unsplit sp γ₁ γ₂ found
lookup-unsplit (lft sp) (γ₁ , _ ⦂ a) γ₂ (there x∈) = lookup-unsplit sp γ₁ γ₂ x∈
lookup-unsplit (rgt sp) γ₁ (γ₂ , _ ⦂ a) (there x∈) = lookup-unsplit sp γ₁ γ₂ (there x∈)
eval-unsplit : (sp : Split Γ Γ₁ Γ₂) (γ₁ : iEnv E⟦ Γ₁ ⟧) (γ₂ : iEnv E⟦ Γ₂ ⟧) →
(⊢M : Γ₁ ⊢ M ⦂ T) →
eval (weaken sp ⊢M) (unsplit-env sp γ₁ γ₂) ≡ eval ⊢M γ₁
eval-unsplit sp γ₁ γ₂ (nat')= refl
eval-unsplit sp γ₁ γ₂ (var x∈) = lookup-unsplit sp γ₁ γ₂ x∈
eval-unsplit sp γ₁ γ₂ (lam ⊢M) = ext (λ s → eval-unsplit (lft sp) (γ₁ , _ ⦂ s) γ₂ ⊢M)
eval-unsplit sp γ₁ γ₂ (app ⊢M ⊢M₁)
rewrite eval-unsplit sp γ₁ γ₂ ⊢M | eval-unsplit sp γ₁ γ₂ ⊢M₁ = refl
eval-unsplit sp γ₁ γ₂ (pair ⊢M ⊢M₁)
rewrite eval-unsplit sp γ₁ γ₂ ⊢M | eval-unsplit sp γ₁ γ₂ ⊢M₁ = refl
eval-unsplit sp γ₁ γ₂ (pair-E1 ⊢M)
rewrite eval-unsplit sp γ₁ γ₂ ⊢M = refl
eval-unsplit sp γ₁ γ₂ (pair-E2 ⊢M)
rewrite eval-unsplit sp γ₁ γ₂ ⊢M = refl
eval-unsplit sp γ₁ γ₂ (sum-I1 ⊢M)
rewrite eval-unsplit sp γ₁ γ₂ ⊢M = refl
eval-unsplit sp γ₁ γ₂ (sum-I2 ⊢N)
rewrite eval-unsplit sp γ₁ γ₂ ⊢N = refl
eval-unsplit sp γ₁ γ₂ (sum-E ⊢L ⊢M ⊢N)
rewrite eval-unsplit sp γ₁ γ₂ ⊢L
| ext (λ s → eval-unsplit (lft sp) (γ₁ , _ ⦂ s) γ₂ ⊢M)
| ext (λ t → eval-unsplit (lft sp) (γ₁ , _ ⦂ t) γ₂ ⊢N)
= refl
-- soundness of the incorrectness rules
lave :
(÷M : Γ ⊢ M ÷ T) →
∀ (t : T⟦ T ⟧) →
∃ λ (γ : iEnv E⟦ Γ ⟧) →
eval (corr ÷M) γ ≡ t
lave nat' (n , refl) = · , refl
lave var1 t = (· , _ ⦂ t) , refl
-- lave (var x∈) t = (gen x∈ t) , lookup-gen x∈ t
lave (lam{x = x}{S = S} ÷M) t = · , ext aux
where
aux : (s : T⟦ S ⟧) → eval (corr ÷M) (· , x ⦂ s) ≡ t s
aux s with lave ÷M (t s)
... | (· , .x ⦂ a) , snd = {!!} -- impossible to complete!
lave (pair-E1 ÷M) t with lave ÷M (t , one {!!})
... | γ , ih = γ , Eq.cong proj₁ ih
lave (pair-E2 ÷M) t with lave ÷M (one {!!} , t)
... | γ , ih = γ , Eq.cong proj₂ ih
lave (pair sp ÷M ÷N) (s , t) with lave ÷M s | lave ÷N t
... | γ₁ , ih-M | γ₂ , ih-N =
unsplit-env sp γ₁ γ₂ ,
Eq.cong₂ _,_ (Eq.trans (eval-unsplit sp γ₁ γ₂ (corr ÷M)) ih-M)
(begin eval (weaken (split-sym sp) (corr ÷N)) (unsplit-env sp γ₁ γ₂)
≡⟨ Eq.cong (eval (weaken (split-sym sp) (corr ÷N))) (unsplit-split sp γ₁ γ₂) ⟩
eval (weaken (split-sym sp) (corr ÷N)) (unsplit-env (split-sym sp) γ₂ γ₁)
≡⟨ eval-unsplit (split-sym sp) γ₂ γ₁ (corr ÷N) ⟩
ih-N)
-- works, but unsatisfactory!
-- this proof uses only one branch of the case
-- this choice is possible because both branches ÷M and ÷N have the same type
-- in general, U could be the union of the types of ÷M and ÷N
lave (sum-E{S = S}{T = T}{U = U} sp ÷L ÷M ÷N) u
with lave ÷M u | lave ÷N u
... | (γ₁ , x ⦂ s) , ih-M | (γ₂ , y ⦂ t) , ih-N
with lave ÷L (inj₁ s)
... | γ₀ , ih-L
=
unsplit-env sp γ₀ γ₁ ,
(begin [
(λ s₁ →
eval (weaken (lft (split-sym sp)) (corr ÷M))
(unsplit-env sp γ₀ γ₁ , x ⦂ s₁))
,
(λ t₁ →
eval (weaken (lft (split-sym sp)) (corr ÷N))
(unsplit-env sp γ₀ γ₁ , y ⦂ t₁))
]
(eval (weaken sp (corr ÷L)) (unsplit-env sp γ₀ γ₁))
≡⟨ Eq.cong [
(λ s₁ →
eval (weaken (lft (split-sym sp)) (corr ÷M))
(unsplit-env sp γ₀ γ₁ , x ⦂ s₁))
,
(λ t₁ →
eval (weaken (lft (split-sym sp)) (corr ÷N))
(unsplit-env sp γ₀ γ₁ , y ⦂ t₁))
] (eval-unsplit sp γ₀ γ₁ (corr ÷L)) ⟩
[
(λ s₁ →
eval (weaken (lft (split-sym sp)) (corr ÷M))
(unsplit-env sp γ₀ γ₁ , x ⦂ s₁))
,
(λ t₁ →
eval (weaken (lft (split-sym sp)) (corr ÷N))
(unsplit-env sp γ₀ γ₁ , y ⦂ t₁))
]
(eval (corr ÷L) γ₀)
≡⟨ Eq.cong
[
(λ s₁ →
eval (weaken (lft (split-sym sp)) (corr ÷M))
(unsplit-env sp γ₀ γ₁ , x ⦂ s₁))
,
(λ t₁ →
eval (weaken (lft (split-sym sp)) (corr ÷N))
(unsplit-env sp γ₀ γ₁ , y ⦂ t₁))
]
ih-L ⟩
eval (weaken (lft (split-sym sp)) (corr ÷M)) (unsplit-env sp γ₀ γ₁ , x ⦂ s)
≡⟨ Eq.cong (λ γ → eval (weaken (lft (split-sym sp)) (corr ÷M)) (γ , x ⦂ s)) (unsplit-split sp γ₀ γ₁) ⟩
eval (weaken (lft (split-sym sp)) (corr ÷M)) (unsplit-env (split-sym sp) γ₁ γ₀ , x ⦂ s)
≡⟨⟩
eval (weaken (lft (split-sym sp)) (corr ÷M)) (unsplit-env (lft (split-sym sp)) (γ₁ , x ⦂ s) γ₀)
≡⟨ eval-unsplit (lft (split-sym sp)) (γ₁ , x ⦂ s) γ₀ (corr ÷M) ⟩
ih-M)
lave (sum-E′{S = S}{T = T}{U = U} sp ÷L ÷M ÷N uuu) u = {!!}
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module Oscar.Class.Associativity where
open import Oscar.Class.Preservativity
open import Oscar.Function
open import Oscar.Level
open import Oscar.Relation
record Associativity
{𝔞} {𝔄 : Set 𝔞} {𝔰} {_►_ : 𝔄 → 𝔄 → Set 𝔰}
(_◅_ : ∀ {m n} → m ► n → ∀ {l} → m ⟨ l ►_ ⟩→ n)
{ℓ}
(_≤_ : ∀ {m n} → m ► n → m ► n → Set ℓ)
: Set (𝔞 ⊔ 𝔰 ⊔ ℓ) where
field
associativity : ∀ {k l} (f : k ► l) {m} (g : l ► m) {n} (h : m ► n) → (h ◅ (g ◅ f)) ≤ ((h ◅ g) ◅ f)
instance `preservativity : ∀ {l} {m} {n} {w : m ► n} → Preservativity (λ ⋆ → _◅_ ⋆) (λ ⋆ → _◅_ ⋆) _≤_ (m ⟨ l ►_ ⟩→ n ∋ w ◅_) id (m ⟨ l ►_ ⟩→ n ∋ w ◅_)
Preservativity.preservativity `preservativity g f = associativity f g _
-- ⦃ `preservativity ⦄ : ∀ {l} {m} {n} {w : m ► n} → Preservativity (λ ⋆ → _◅_ ⋆) (λ ⋆ → _◅_ ⋆) _≤_ (m ⟨ l ►_ ⟩→ n ∋ w ◅_) id (m ⟨ l ►_ ⟩→ n ∋ w ◅_)
open Associativity ⦃ … ⦄ public
module _ where
private
postulate
A : Set
_⇒_ : A → A → Set
_∙_ : ∀ {m n} → m ⇒ n → ∀ {l} → m ⟨ l ⇒_ ⟩→ n
_≋_ : ∀ {m n} → m ⇒ n → m ⇒ n → Set
instance _ : Associativity _∙_ _≋_
test-associativity₁ : ∀ {k l} (f : k ⇒ l) {m} (g : l ⇒ m) {n} (h : m ⇒ n) → (h ∙ (g ∙ f)) ≋ ((h ∙ g) ∙ f)
test-associativity₁ = associativity {_◅_ = _∙_}
test-associativity₂ : ∀ {k l} (f : k ⇒ l) {m} (g : l ⇒ m) {n} (h : m ⇒ n) → (h ∙ (g ∙ f)) ≋ ((h ∙ g) ∙ f)
test-associativity₂ = associativity {_≤_ = _≋_}
-- Associativity : ∀
-- {𝔞} {𝔄 : Set 𝔞} {𝔰} {_►_ : 𝔄 → 𝔄 → Set 𝔰}
-- (_◅_ : ∀ {m n} → m ► n → ∀ {l} → m ⟨ l ►_ ⟩→ n)
-- {ℓ}
-- (_≤_ : ∀ {m n} → m ► n → m ► n → Set ℓ)
-- → Set (𝔞 ⊔ 𝔰 ⊔ ℓ)
-- Associativity {_►_ = _►_} _◅_ _≤_ =
-- ∀ {l} {m} {n} {w : m ► n} → Preservativity (λ ⋆ → _◅_ ⋆) (λ ⋆ → _◅_ ⋆) _≤_ (m ⟨ l ►_ ⟩→ n ∋ w ◅_) id (m ⟨ l ►_ ⟩→ n ∋ w ◅_)
-- -- record Associativity
-- -- {𝔞} {𝔄 : Set 𝔞} {𝔰} {_►_ : 𝔄 → 𝔄 → Set 𝔰}
-- -- (_◅_ : ∀ {m n} → m ► n → ∀ {l} → m ⟨ l ►_ ⟩→ n)
-- -- {𝔱} {▸ : 𝔄 → Set 𝔱}
-- -- (_◃_ : ∀ {m n} → m ► n → m ⟨ ▸ ⟩→ n)
-- -- {ℓ}
-- -- (_≤_ : ∀ {n} → ▸ n → ▸ n → Set ℓ)
-- -- : Set (𝔞 ⊔ 𝔰 ⊔ 𝔱 ⊔ ℓ) where
-- -- field
-- -- associativity : ∀ {l} (f : ▸ l) {m} (g : l ► m) {n} (h : m ► n) → (h ◃ (g ◃ f)) ≤ ((h ◅ g) ◃ f)
-- -- open Associativity ⦃ … ⦄ public
-- -- -- record Associativity
-- -- -- {𝔞} {𝔄 : Set 𝔞} {𝔰} {_►_ : 𝔄 → 𝔄 → Set 𝔰}
-- -- -- (_▻_ : ∀ {l m n} → l ► m → m ⟨ _► n ⟩→ l)
-- -- -- {ℓ}
-- -- -- (_≤_ : ∀ {m n} → m ► n → m ► n → Set ℓ)
-- -- -- : Set (𝔞 ⊔ 𝔰 ⊔ ℓ) where
-- -- -- field
-- -- -- ⦃ `preservativity ⦄ : ∀ l m n w → Preservativity (_▻_ {l = l} {m = m} {n = n}) (_▻_ {l = l}) _≤_ (w ▻_) id (w ▻_)
-- -- -- -- ⦃ `preservativity ⦄ : ∀ n l w → Preservativity (λ ⋆ → _◅_ {n = n} ⋆) (λ ⋆ → _◅_ {l = l} ⋆) _≤_ (_◅ w) id (w ◅_)
-- -- -- -- record Associativity
-- -- -- -- {𝔞} {𝔄 : Set 𝔞} {𝔰} {_►_ : 𝔄 → 𝔄 → Set 𝔰}
-- -- -- -- (_◅_ : ∀ {l m n} → m ► n → m ⟨ l ►_ ⟩→ n)
-- -- -- -- {ℓ}
-- -- -- -- (_≤_ : ∀ {m n} → m ► n → m ► n → Set ℓ)
-- -- -- -- : Set (𝔞 ⊔ 𝔰 ⊔ ℓ) where
-- -- -- -- field
-- -- -- -- ⦃ `preservativity ⦄ : ∀ l n w → Preservativity (flip (_◅_ {l = l} {n = n})) (flip _◅_) _≤_ (w ◅_) id (_◅ w)
-- -- -- -- -- ⦃ `preservativity ⦄ : ∀ n l w → Preservativity (λ ⋆ → _◅_ {n = n} ⋆) (λ ⋆ → _◅_ {l = l} ⋆) _≤_ (_◅ w) id (w ◅_)
-- -- -- -- -- record Associativity
-- -- -- -- -- {𝔞} {𝔄 : Set 𝔞} {𝔰} {_►_ : 𝔄 → 𝔄 → Set 𝔰}
-- -- -- -- -- (_◅_ : ∀ {m n} → m ► n → ∀ {l} → m ⟨ l ►_ ⟩→ n)
-- -- -- -- -- {ℓ}
-- -- -- -- -- (_≤_ : ∀ {m n} → m ► n → m ► n → Set ℓ)
-- -- -- -- -- : Set (𝔞 ⊔ 𝔰 ⊔ ℓ) where
-- -- -- -- -- field
-- -- -- -- -- ⦃ `preservativity ⦄ : ∀ n l w → Preservativity (λ ⋆ → _◅_ {n = n} ⋆) (λ ⋆ → _◅_ ⋆ {l = l}) _≤_ (_◅ w) id (w ◅_)
-- -- -- -- -- -- associativity : ∀ {k l} (f : k ► l) {m} (g : l ► m) {n} (h : m ► n) → (h ◅ (g ◅ f)) ≤ ((h ◅ g) ◅ f)
-- -- -- -- -- -- record Associativity
-- -- -- -- -- -- {a} {A : Set a} {b} {B : A → Set b} {c} {C : (x : A) → B x → Set c}
-- -- -- -- -- -- (_◅_ : ∀ {m n} → m ► n → ∀ {l} → m ⟨ l ►_ ⟩→ n)
-- -- -- -- -- -- {ℓ}
-- -- -- -- -- -- (_≤_ : ∀ {m n} → m ► n → m ► n → Set ℓ)
-- -- -- -- -- -- : Set (𝔞 ⊔ 𝔰 ⊔ ℓ) where
-- -- -- -- -- -- field
-- -- -- -- -- -- ⦃ `preservativity ⦄ : Preservativity
-- -- -- -- -- -- associativity : ∀ {k l} (f : k ► l) {m} (g : l ► m) {n} (h : m ► n) → (h ◅ (g ◅ f)) ≤ ((h ◅ g) ◅ f)
-- -- -- -- -- -- open Associativity ⦃ … ⦄ public
-- -- -- -- -- -- -- record Associativity
-- -- -- -- -- -- -- {𝔞} {𝔄 : Set 𝔞} {𝔰} {_►_ : 𝔄 → 𝔄 → Set 𝔰}
-- -- -- -- -- -- -- (_◅_ : ∀ {m n} → m ► n → ∀ {l} → m ⟨ l ►_ ⟩→ n)
-- -- -- -- -- -- -- {ℓ}
-- -- -- -- -- -- -- (_≤_ : ∀ {m n} → m ► n → m ► n → Set ℓ)
-- -- -- -- -- -- -- : Set (𝔞 ⊔ 𝔰 ⊔ ℓ) where
-- -- -- -- -- -- -- field
-- -- -- -- -- -- -- associativity : ∀ {k l} (f : k ► l) {m} (g : l ► m) {n} (h : m ► n) → (h ◅ (g ◅ f)) ≤ ((h ◅ g) ◅ f)
-- -- -- -- -- -- -- open Associativity ⦃ … ⦄ public
-- -- -- -- -- -- -- -- record Associativity
-- -- -- -- -- -- -- -- {𝔞} {𝔄 : Set 𝔞} {𝔰} {_►_ : 𝔄 → 𝔄 → Set 𝔰}
-- -- -- -- -- -- -- -- (_◅_ : ∀ {m n} → m ► n → ∀ {l} → m ⟨ l ►_ ⟩→ n)
-- -- -- -- -- -- -- -- {𝔱} {▸ : 𝔄 → Set 𝔱}
-- -- -- -- -- -- -- -- (_◃_ : ∀ {m n} → m ► n → m ⟨ ▸ ⟩→ n)
-- -- -- -- -- -- -- -- {ℓ}
-- -- -- -- -- -- -- -- (_≤_ : ∀ {n} → ▸ n → ▸ n → Set ℓ)
-- -- -- -- -- -- -- -- : Set (𝔞 ⊔ 𝔰 ⊔ 𝔱 ⊔ ℓ) where
-- -- -- -- -- -- -- -- field
-- -- -- -- -- -- -- -- associativity : ∀ {l} (f : ▸ l) {m} (g : l ► m) {n} (h : m ► n) → (h ◃ (g ◃ f)) ≤ ((h ◅ g) ◃ f)
-- -- -- -- -- -- -- -- open Associativity ⦃ … ⦄ public
-- -- -- -- -- -- -- -- association : ∀
-- -- -- -- -- -- -- -- {𝔞} {𝔄 : Set 𝔞} {𝔰} {_►_ : 𝔄 → 𝔄 → Set 𝔰}
-- -- -- -- -- -- -- -- (_◅_ : ∀ {m n} → m ► n → ∀ {l} → m ⟨ l ►_ ⟩→ n)
-- -- -- -- -- -- -- -- {𝔱} {▸ : 𝔄 → Set 𝔱}
-- -- -- -- -- -- -- -- {_◃_ : ∀ {m n} → m ► n → m ⟨ ▸ ⟩→ n}
-- -- -- -- -- -- -- -- {ℓ}
-- -- -- -- -- -- -- -- (_≤_ : ∀ {n} → ▸ n → ▸ n → Set ℓ)
-- -- -- -- -- -- -- -- ⦃ _ : Associativity _◅_ _◃_ _≤_ ⦄
-- -- -- -- -- -- -- -- → ∀ {l} (f : ▸ l) {m} (g : l ► m) {n} (h : m ► n) → (h ◃ (g ◃ f)) ≤ ((h ◅ g) ◃ f)
-- -- -- -- -- -- -- -- association _◅_ _≤_ = associativity {_◅_ = _◅_} {_≤_ = _≤_}
|
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{-# OPTIONS --safe #-}
module Cubical.ZCohomology.Groups.Connected where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.HLevels
open import Cubical.Data.Nat
open import Cubical.Data.Int renaming (_+_ to _+ℤ_; +Comm to +ℤ-comm ; +Assoc to +ℤ-assoc)
open import Cubical.Data.Sigma hiding (_×_)
open import Cubical.HITs.SetTruncation as ST
open import Cubical.HITs.PropositionalTruncation as PT
open import Cubical.HITs.Truncation as T
open import Cubical.HITs.Nullification
open import Cubical.Algebra.Group
open import Cubical.Algebra.Group.Instances.Int
open import Cubical.Algebra.Group.Morphisms
open import Cubical.Algebra.Group.MorphismProperties
open import Cubical.Homotopy.Connected
open import Cubical.ZCohomology.Base
open import Cubical.ZCohomology.GroupStructure
open import Cubical.ZCohomology.Groups.Unit
private
H⁰-connected-type : ∀ {ℓ} {A : Type ℓ} (a : A) → isConnected 2 A → Iso (coHom 0 A) ℤ
Iso.fun (H⁰-connected-type a con) = ST.rec isSetℤ λ f → f a
Iso.inv (H⁰-connected-type a con) b = ∣ (λ x → b) ∣₂
Iso.rightInv (H⁰-connected-type a con) b = refl
Iso.leftInv (H⁰-connected-type a con) =
ST.elim (λ _ → isOfHLevelPath 2 isSetSetTrunc _ _)
λ f → cong ∣_∣₂ (funExt λ x → T.rec₊ (isSetℤ _ _) (cong f) (isConnectedPath 1 con a x .fst))
open IsGroupHom
open Iso
H⁰-connected : ∀ {ℓ} {A : Type ℓ} (a : A) → ((x : A) → ∥ a ≡ x ∥₁) → GroupIso (coHomGr 0 A) ℤGroup
fun (fst (H⁰-connected a con)) = ST.rec isSetℤ (λ f → f a)
inv (fst (H⁰-connected a con)) b = ∣ (λ _ → b) ∣₂
rightInv (fst (H⁰-connected a con)) _ = refl
leftInv (fst (H⁰-connected a con)) =
ST.elim (λ _ → isProp→isSet (isSetSetTrunc _ _))
(λ f → cong ∣_∣₂ (funExt λ x → PT.rec (isSetℤ _ _) (cong f) (con x)))
snd (H⁰-connected a con) = makeIsGroupHom (ST.elim2 (λ _ _ → isProp→isSet (isSetℤ _ _)) λ x y → refl)
|
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module Primitive where
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|
-- Andreas, 2011-10-02
{-# OPTIONS --show-implicit #-}
module Issue483a where
data _≡_ {A : Set}(a : A) : A → Set where
refl : a ≡ a
data Empty : Set where
postulate A : Set
abort : .Empty → A
abort ()
test : let X : .Set1 → A
X = _
in (x : Empty) → X Set ≡ abort x
test x = refl
-- this should fail with message like
--
-- Cannot instantiate the metavariable _16 to abort x since it
-- contains the variable x which is not in scope of the metavariable
-- when checking that the expression refl has type _16 _ ≡ abort x
--
-- a solution like X = λ _ → abort x : Set1 → A
-- would be invalid even though x is irrelevant, because there is no
-- term of type Set1 → A
|
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postulate
A : Set
F : { x : A } → Set
G : ⦃ x : A ⦄ → Set
|
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Structures.NAryOp where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.SIP
open import Cubical.Functions.FunExtEquiv
open import Cubical.Data.Nat
open import Cubical.Data.Vec
module _ {ℓ₁ ℓ₂ : Level} where
NAryFunStructure : (n : ℕ) (S : Type ℓ₁ → Type ℓ₂)
→ Type ℓ₁ → Type (nAryLevel ℓ₁ ℓ₂ n)
NAryFunStructure n S X = nAryOp n X (S X)
-- iso for n-ary functions
NAryFunEquivStr : (n : ℕ) {S : Type ℓ₁ → Type ℓ₂} {ℓ₃ : Level} (ι : StrEquiv S ℓ₃)
→ StrEquiv (NAryFunStructure n S) (ℓ-max ℓ₁ ℓ₃)
NAryFunEquivStr n ι (X , fX) (Y , fY) e =
(xs : Vec X n) → ι (X , fX $ⁿ xs) (Y , fY $ⁿ map (equivFun e) xs) e
nAryFunUnivalentStr : {S : Type ℓ₁ → Type ℓ₂} (n : ℕ) {ℓ₃ : Level}
(ι : StrEquiv S ℓ₃) (θ : UnivalentStr S ι)
→ UnivalentStr (NAryFunStructure n S) (NAryFunEquivStr n ι)
nAryFunUnivalentStr n ι θ =
SNS→UnivalentStr (NAryFunEquivStr n ι) λ fX fY →
compEquiv
(equivPi λ xs → UnivalentStr→SNS _ ι θ _ _)
(nAryFunExtEquiv n fX fY)
module _ {ℓ₁ ℓ₂ : Level} where
-- unary
UnaryFunEquivStr : {S : Type ℓ₁ → Type ℓ₂} {ℓ₃ : Level} (ι : StrEquiv S ℓ₃)
→ StrEquiv (NAryFunStructure 1 S) (ℓ-max ℓ₁ ℓ₃)
UnaryFunEquivStr ι (A , f) (B , g) e =
(x : A) → ι (A , f x) (B , g (equivFun e x)) e
unaryFunUnivalentStr : {S : Type ℓ₁ → Type ℓ₂} {ℓ₃ : Level}
(ι : StrEquiv S ℓ₃) (θ : UnivalentStr S ι)
→ UnivalentStr (NAryFunStructure 1 S) (UnaryFunEquivStr ι)
unaryFunUnivalentStr ι θ =
SNS→UnivalentStr (UnaryFunEquivStr ι) λ fX fY →
compEquiv (equivPi λ _ → UnivalentStr→SNS _ ι θ _ _) funExtEquiv
-- binary
BinaryFunEquivStr : {S : Type ℓ₁ → Type ℓ₂} {ℓ₃ : Level} (ι : StrEquiv S ℓ₃)
→ StrEquiv (NAryFunStructure 2 S) (ℓ-max ℓ₁ ℓ₃)
BinaryFunEquivStr ι (A , f) (B , g) e =
(x y : A) → ι (A , f x y) (B , g (equivFun e x) (equivFun e y)) e
binaryFunUnivalentStr : {S : Type ℓ₁ → Type ℓ₂} {ℓ₃ : Level}
(ι : StrEquiv S ℓ₃) (θ : UnivalentStr S ι)
→ UnivalentStr (NAryFunStructure 2 S) (BinaryFunEquivStr ι)
binaryFunUnivalentStr ι θ =
SNS→UnivalentStr (BinaryFunEquivStr ι) λ fX fY →
compEquiv (equivPi λ _ → equivPi λ _ → UnivalentStr→SNS _ ι θ _ _) funExt₂Equiv
|
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{-# OPTIONS --safe #-}
open import Definition.Typed.EqualityRelation
module Definition.LogicalRelation.ShapeView {{eqrel : EqRelSet}} where
open EqRelSet {{...}}
open import Definition.Untyped as U
open import Definition.Typed
open import Definition.Typed.Weakening
open import Definition.Typed.Properties
open import Definition.LogicalRelation
open import Definition.LogicalRelation.Properties.Escape
open import Definition.LogicalRelation.Properties.Reflexivity
open import Tools.Product
open import Tools.Empty using (⊥; ⊥-elim)
import Tools.PropositionalEquality as PE
import Data.Fin as Fin
import Data.Nat as Nat
-- Type for maybe embeddings of reducible types
data MaybeEmb (l : TypeLevel) (⊩⟨_⟩ : TypeLevel → Set) : Set where
noemb : ⊩⟨ l ⟩ → MaybeEmb l ⊩⟨_⟩
emb : ∀ {l′} → l′ <∞ l → MaybeEmb l′ ⊩⟨_⟩ → MaybeEmb l ⊩⟨_⟩
-- Specific reducible types with possible embedding
_⊩⟨_⟩U_^_ : (Γ : Con Term) (l : TypeLevel) (A : Term) (ll : TypeLevel) → Set
Γ ⊩⟨ l ⟩U A ^ ll = MaybeEmb l (λ l′ → Γ ⊩′⟨ l′ ⟩U A ^ ll)
_⊩⟨_⟩ℕ_ : (Γ : Con Term) (l : TypeLevel) (A : Term) → Set
Γ ⊩⟨ l ⟩ℕ A = MaybeEmb l (λ l′ → Γ ⊩ℕ A)
_⊩⟨_⟩Empty_^_ : (Γ : Con Term) (l : TypeLevel) (A : Term) (ll : Level) → Set
Γ ⊩⟨ l ⟩Empty A ^ ll = MaybeEmb l (λ l′ → Γ ⊩Empty A ^ ll)
_⊩⟨_⟩ne_^[_,_] : (Γ : Con Term) (l : TypeLevel) (A : Term) (r : Relevance) (ll : Level) → Set
Γ ⊩⟨ l ⟩ne A ^[ r , ll ] = MaybeEmb l (λ l′ → Γ ⊩ne A ^[ r , ll ])
_⊩⟨_⟩Π_^[_,_] : (Γ : Con Term) (l : TypeLevel) (A : Term) → Relevance → Level → Set
Γ ⊩⟨ l ⟩Π A ^[ r , lΠ ] = MaybeEmb l (λ l′ → Γ ⊩′⟨ l′ ⟩Π A ^[ r , lΠ ])
_⊩⟨_⟩∃_^_ : (Γ : Con Term) (l : TypeLevel) (A : Term) (ll : Level) → Set
Γ ⊩⟨ l ⟩∃ A ^ ll = MaybeEmb l (λ l′ → Γ ⊩′⟨ l′ ⟩∃ A ^ ll)
-- Construct a general reducible type from a specific
U-intr : ∀ {l Γ A ll } → (UA : Γ ⊩⟨ l ⟩U A ^ ll) → Γ ⊩⟨ l ⟩ A ^ [ ! , ll ]
U-intr (noemb UA) = Uᵣ UA
U-intr {l = ι ¹} (emb emb< x) = emb emb< (U-intr x)
U-intr {l = ∞} (emb ∞< x) = emb ∞< (U-intr x)
ℕ-intr : ∀ {l A Γ} → Γ ⊩⟨ l ⟩ℕ A → Γ ⊩⟨ l ⟩ A ^ [ ! , ι ⁰ ]
ℕ-intr (noemb x) = ℕᵣ x
ℕ-intr {l = ι ¹} (emb emb< x) = emb emb< (ℕ-intr x)
ℕ-intr {l = ∞} (emb ∞< x) = emb ∞< (ℕ-intr x)
Empty-intr : ∀ {l A Γ ll} → Γ ⊩⟨ l ⟩Empty A ^ ll → Γ ⊩⟨ l ⟩ A ^ [ % , ι ll ]
Empty-intr (noemb x) = Emptyᵣ x
Empty-intr {l = ι ¹} (emb emb< x) = emb emb< (Empty-intr x)
Empty-intr {l = ∞} (emb ∞< x) = emb ∞< (Empty-intr x)
ne-intr : ∀ {l A Γ r ll} → Γ ⊩⟨ l ⟩ne A ^[ r , ll ] → Γ ⊩⟨ l ⟩ A ^ [ r , ι ll ]
ne-intr (noemb x) = ne x
ne-intr {l = ι ¹} (emb emb< x) = emb emb< (ne-intr x)
ne-intr {l = ∞} (emb ∞< x) = emb ∞< (ne-intr x)
Π-intr : ∀ {l A Γ r ll} → Γ ⊩⟨ l ⟩Π A ^[ r , ll ] → Γ ⊩⟨ l ⟩ A ^ [ r , ι ll ]
Π-intr (noemb x) = Πᵣ x
Π-intr {l = ι ¹} (emb emb< x) = emb emb< (Π-intr x)
Π-intr {l = ∞} (emb ∞< x) = emb ∞< (Π-intr x)
∃-intr : ∀ {l A Γ ll} → Γ ⊩⟨ l ⟩∃ A ^ ll → Γ ⊩⟨ l ⟩ A ^ [ % , ι ll ]
∃-intr (noemb x) = ∃ᵣ x
∃-intr {l = ι ¹} (emb emb< x) = emb emb< (∃-intr x)
∃-intr {l = ∞} (emb ∞< x) = emb ∞< (∃-intr x)
-- Construct a specific reducible type from a general with some criterion
U-elim′ : ∀ {l Γ A r l′ ll} → Γ ⊢ A ⇒* Univ r l′ ^ [ ! , ll ] → Γ ⊩⟨ l ⟩ A ^ [ ! , ll ] → Γ ⊩⟨ l ⟩U A ^ ll
U-elim′ D (Uᵣ′ A ll r l l< e D') = noemb (Uᵣ r l l< e D')
U-elim′ D (ℕᵣ D') = ⊥-elim (U≢ℕ (whrDet* (D , Uₙ) (red D' , ℕₙ)))
U-elim′ D (ne′ K D' neK K≡K) = ⊥-elim (U≢ne neK (whrDet* (D , Uₙ) (red D' , ne neK)))
U-elim′ D (Πᵣ′ rF lF lG _ _ F G D' ⊢F ⊢G A≡A [F] [G] G-ext) = ⊥-elim (U≢Π (whrDet* (D , Uₙ) (red D' , Πₙ)))
U-elim′ {ι ¹} D (emb emb< x) with U-elim′ D x
U-elim′ {ι ¹} D (emb emb< x) | noemb x₁ = emb emb< (noemb x₁)
U-elim′ {ι ¹} D (emb emb< x) | emb () x₁
U-elim′ {∞} D (emb ∞< x) with U-elim′ D x
U-elim′ {∞} D (emb ∞< x) | noemb x₁ = emb ∞< (noemb x₁)
U-elim′ {∞} D (emb ∞< x) | emb <l x₁ = emb {l′ = ι ¹} ∞< (emb <l x₁)
U-elim : ∀ {l Γ r l′ ll′} → Γ ⊩⟨ l ⟩ Univ r l′ ^ [ ! , ll′ ] → Γ ⊩⟨ l ⟩U Univ r l′ ^ ll′
U-elim [U] = U-elim′ (id (escape [U])) [U]
ℕ-elim′ : ∀ {l A Γ ll} → Γ ⊢ A ⇒* ℕ ^ [ ! , ll ] → Γ ⊩⟨ l ⟩ A ^ [ ! , ll ] → Γ ⊩⟨ l ⟩ℕ A
ℕ-elim′ D (Uᵣ′ _ _ _ _ l< PE.refl [[ _ , _ , d ]]) = ⊥-elim (U≢ℕ (whrDet* (d , Uₙ) (D , ℕₙ)))
ℕ-elim′ D (ℕᵣ D′) = noemb D′
ℕ-elim′ D (ne′ K D′ neK K≡K) =
⊥-elim (ℕ≢ne neK (whrDet* (D , ℕₙ) (red D′ , ne neK)))
ℕ-elim′ D (Πᵣ′ rF lF lG _ _ F G D′ ⊢F ⊢G A≡A [F] [G] G-ext) =
⊥-elim (ℕ≢Π (whrDet* (D , ℕₙ) (red D′ , Πₙ)))
ℕ-elim′ {ι ¹} D (emb emb< x) with ℕ-elim′ D x
ℕ-elim′ {ι ¹} D (emb emb< x) | noemb x₁ = emb emb< (noemb x₁)
ℕ-elim′ {ι ¹} D (emb emb< x) | emb () x₁
ℕ-elim′ {∞} D (emb ∞< x) with ℕ-elim′ D x
ℕ-elim′ {∞} D (emb ∞< x) | noemb x₁ = emb ∞< (noemb x₁)
ℕ-elim′ {∞} D (emb ∞< x) | emb <l x₁ = emb {l′ = ι ¹} ∞< (emb <l x₁)
ℕ-elim : ∀ {Γ l ll } → Γ ⊩⟨ l ⟩ ℕ ^ [ ! , ll ] → Γ ⊩⟨ l ⟩ℕ ℕ
ℕ-elim [ℕ] = ℕ-elim′ (id (escape [ℕ])) [ℕ]
Empty-elim′ : ∀ {l A ll Γ} → Γ ⊢ A ⇒* Empty ll ^ [ % , ι ll ] → Γ ⊩⟨ l ⟩ A ^ [ % , ι ll ] → Γ ⊩⟨ l ⟩Empty A ^ ll
Empty-elim′ D (Emptyᵣ D′) = noemb D′
Empty-elim′ D (ne′ K D′ neK K≡K) =
⊥-elim (Empty≢ne neK (whrDet* (D , Emptyₙ) (red D′ , ne neK)))
Empty-elim′ D (Πᵣ′ rF lF lG _ _ F G D′ ⊢F ⊢G A≡A [F] [G] G-ext) =
⊥-elim (Empty≢Π (whrDet* (D , Emptyₙ) (red D′ , Πₙ)))
Empty-elim′ D (∃ᵣ′ F G D′ ⊢F ⊢G A≡A [F] [G] G-ext) =
⊥-elim (Empty≢∃ (whrDet* (D , Emptyₙ) (red D′ , ∃ₙ)))
Empty-elim′ {ι ¹} D (emb emb< x) with Empty-elim′ D x
Empty-elim′ {ι ¹} D (emb emb< x) | noemb x₁ = emb emb< (noemb x₁)
Empty-elim′ {ι ¹} D (emb emb< x) | emb () x₁
Empty-elim′ {∞} D (emb ∞< x) with Empty-elim′ D x
Empty-elim′ {∞} D (emb ∞< x) | noemb x₁ = emb ∞< (noemb x₁)
Empty-elim′ {∞} D (emb ∞< x) | emb <l x₁ = emb {l′ = ι ¹} ∞< (emb <l x₁)
Empty-elim : ∀ {Γ l ll } → Γ ⊩⟨ l ⟩ Empty ll ^ [ % , ι ll ] → Γ ⊩⟨ l ⟩Empty Empty ll ^ ll
Empty-elim [Empty] = Empty-elim′ (id (escape [Empty])) [Empty]
ne-elim′ : ∀ {l A Γ K r ll ll'} → Γ ⊢ A ⇒* K ^ [ r , ι ll ] → Neutral K → Γ ⊩⟨ l ⟩ A ^ [ r , ll' ] → ι ll PE.≡ ll' → Γ ⊩⟨ l ⟩ne A ^[ r , ll ]
ne-elim′ D neK (Uᵣ′ _ _ _ _ l< PE.refl [[ _ , _ , d ]]) e = ⊥-elim (U≢ne neK (whrDet* (d , Uₙ) (D , ne neK)))
ne-elim′ D neK (ℕᵣ D′) e = ⊥-elim (ℕ≢ne neK (whrDet* (red D′ , ℕₙ) (D , ne neK)))
ne-elim′ D neK (ne (ne K D′ neK′ K≡K)) PE.refl = noemb (ne K D′ neK′ K≡K)
ne-elim′ D neK (Πᵣ′ rF lF lG _ _ F G D′ ⊢F ⊢G A≡A [F] [G] G-ext) e =
⊥-elim (Π≢ne neK (whrDet* (red D′ , Πₙ) (D , ne neK)))
ne-elim′ D neK (∃ᵣ′ F G D′ ⊢F ⊢G A≡A [F] [G] G-ext) e =
⊥-elim (∃≢ne neK (whrDet* (red D′ , ∃ₙ) (D , ne neK)))
ne-elim′ D neK (Emptyᵣ D′) e = ⊥-elim (Empty≢ne neK (whrDet* (red D′ , Emptyₙ) (D , ne neK)))
ne-elim′ {ι ¹} D neK (emb emb< x) e with ne-elim′ D neK x e
ne-elim′ {ι ¹} D neK (emb emb< x) e | noemb x₁ = emb emb< (noemb x₁)
ne-elim′ {ι ¹} D neK (emb emb< x) e | emb () x₁
ne-elim′ {∞} D neK (emb ∞< x) e with ne-elim′ D neK x e
ne-elim′ {∞} D _ (emb ∞< x) e | noemb x₁ = emb ∞< (noemb x₁)
ne-elim′ {∞} D _ (emb ∞< x) e | emb <l x₁ = emb {l′ = ι ¹} ∞< (emb <l x₁)
ne-elim : ∀ {Γ l K r ll} → Neutral K → Γ ⊩⟨ l ⟩ K ^ [ r , ι ll ] → Γ ⊩⟨ l ⟩ne K ^[ r , ll ]
ne-elim neK [K] = ne-elim′ (id (escape [K])) neK [K] PE.refl
Π-elim′ : ∀ {l A Γ F G rF lF lG r lΠ} → Γ ⊢ A ⇒* Π F ^ rF ° lF ▹ G ° lG ° lΠ ^ [ r , ι lΠ ] → Γ ⊩⟨ l ⟩ A ^ [ r , ι lΠ ] → Γ ⊩⟨ l ⟩Π A ^[ r , lΠ ]
Π-elim′ D (Uᵣ′ _ _ _ _ l< X [[ _ , _ , d ]]) = ⊥-elim (U≢Π (whrDet* (d , Uₙ) (D , Πₙ)))
Π-elim′ D (ℕᵣ D′) = ⊥-elim (ℕ≢Π (whrDet* (red D′ , ℕₙ) (D , Πₙ)))
Π-elim′ D (Emptyᵣ D′) = ⊥-elim (Empty≢Π (whrDet* (red D′ , Emptyₙ) (D , Πₙ)))
Π-elim′ D (ne′ K D′ neK K≡K) =
⊥-elim (Π≢ne neK (whrDet* (D , Πₙ) (red D′ , ne neK)))
Π-elim′ D (Πᵣ′ rF lF lG lF≤ lG≤ F G D′ ⊢F ⊢G A≡A [F] [G] G-ext) =
noemb (Πᵣ rF lF lG lF≤ lG≤ F G D′ ⊢F ⊢G A≡A [F] [G] G-ext)
Π-elim′ D (∃ᵣ′ F G D′ ⊢F ⊢G A≡A [F] [G] G-ext) =
⊥-elim (Π≢∃ (whrDet* (D , Πₙ) (red D′ , ∃ₙ)))
Π-elim′ {ι ¹} D (emb emb< x) with Π-elim′ D x
Π-elim′ {ι ¹} D (emb emb< x) | noemb x₁ = emb emb< (noemb x₁)
Π-elim′ {ι ¹} D (emb emb< x) | emb () x₁
Π-elim′ {∞} D (emb ∞< x) with Π-elim′ D x
Π-elim′ {∞} D (emb ∞< x) | noemb x₁ = emb ∞< (noemb x₁)
Π-elim′ {∞} D (emb ∞< x) | emb <l x₁ = emb {l′ = ι ¹} ∞< (emb <l x₁)
Π-elim : ∀ {Γ F G rF lF lG r lΠ l} → Γ ⊩⟨ l ⟩ Π F ^ rF ° lF ▹ G ° lG ° lΠ ^ [ r , ι lΠ ] → Γ ⊩⟨ l ⟩Π Π F ^ rF ° lF ▹ G ° lG ° lΠ ^[ r , lΠ ]
Π-elim [Π] = Π-elim′ (id (escape [Π])) [Π]
∃-elim′ : ∀ {l A Γ F G ll} → Γ ⊢ A ⇒* ∃ F ▹ G ^ [ % , ι ll ] → Γ ⊩⟨ l ⟩ A ^ [ % , ι ll ] → Γ ⊩⟨ l ⟩∃ A ^ ll
∃-elim′ D (Emptyᵣ D′) = ⊥-elim (Empty≢∃ (whrDet* (red D′ , Emptyₙ) (D , ∃ₙ)))
∃-elim′ D (ne′ K D′ neK K≡K) =
⊥-elim (∃≢ne neK (whrDet* (D , ∃ₙ) (red D′ , ne neK)))
∃-elim′ D (Πᵣ′ rF lF lG _ _ F G D′ ⊢F ⊢G A≡A [F] [G] G-ext) =
⊥-elim (Π≢∃ (whrDet* (red D′ , Πₙ) (D , ∃ₙ)))
∃-elim′ D (∃ᵣ′ F G D′ ⊢F ⊢G A≡A [F] [G] G-ext) =
noemb (∃ᵣ F G D′ ⊢F ⊢G A≡A [F] [G] G-ext)
∃-elim′ {ι ¹} D (emb emb< x) with ∃-elim′ D x
∃-elim′ {ι ¹} D (emb emb< x) | noemb x₁ = emb emb< (noemb x₁)
∃-elim′ {ι ¹} D (emb emb< x) | emb () x₁
∃-elim′ {∞} D (emb ∞< x) with ∃-elim′ D x
∃-elim′ {∞} D (emb ∞< x) | noemb x₁ = emb ∞< (noemb x₁)
∃-elim′ {∞} D (emb ∞< x) | emb <l x₁ = emb {l′ = ι ¹} ∞< (emb <l x₁)
∃-elim : ∀ {Γ F G l ll} → Γ ⊩⟨ l ⟩ ∃ F ▹ G ^ [ % , ι ll ] → Γ ⊩⟨ l ⟩∃ (∃ F ▹ G) ^ ll
∃-elim [∃] = ∃-elim′ (id (escape [∃])) [∃]
-- Extract a type and a level from a maybe embedding
extractMaybeEmb : ∀ {l ⊩⟨_⟩} → MaybeEmb l ⊩⟨_⟩ → ∃ λ l′ → ⊩⟨ l′ ⟩
extractMaybeEmb (noemb x) = _ , x
extractMaybeEmb (emb <l x) = extractMaybeEmb x
-- A view for constructor equality of types where embeddings are ignored
data ShapeView Γ : ∀ l l′ A B r r' (p : Γ ⊩⟨ l ⟩ A ^ r) (q : Γ ⊩⟨ l′ ⟩ B ^ r') → Set where
Uᵥ : ∀ {A B l l′ ll ll′} UA UB → ShapeView Γ l l′ A B [ ! , ll ] [ ! , ll′ ] (Uᵣ UA) (Uᵣ UB)
ℕᵥ : ∀ {A B l l′} ℕA ℕB → ShapeView Γ l l′ A B [ ! , ι ⁰ ] [ ! , ι ⁰ ] (ℕᵣ ℕA) (ℕᵣ ℕB)
Emptyᵥ : ∀ {A B l l′ ll ll'} EmptyA EmptyB → ShapeView Γ l l′ A B [ % , ι ll ] [ % , ι ll' ] (Emptyᵣ EmptyA) (Emptyᵣ EmptyB)
ne : ∀ {A B l l′ r lr r' lr'} neA neB
→ ShapeView Γ l l′ A B [ r , ι lr ] [ r' , ι lr' ] (ne neA) (ne neB)
Πᵥ : ∀ {A B l l′ r r' lΠ lΠ' } ΠA ΠB
→ ShapeView Γ l l′ A B [ r , ι lΠ ] [ r' , ι lΠ' ] (Πᵣ ΠA) (Πᵣ ΠB)
∃ᵥ : ∀ {A B l l′ ll ll'} ∃A ∃B
→ ShapeView Γ l l′ A B [ % , ι ll ] [ % , ι ll' ] (∃ᵣ ∃A) (∃ᵣ ∃B)
emb⁰¹ : ∀ {A B r r' l p q}
→ ShapeView Γ (ι ⁰) l A B r r' p q
→ ShapeView Γ (ι ¹) l A B r r' (emb emb< p) q
emb¹⁰ : ∀ {A B r r' l p q}
→ ShapeView Γ l (ι ⁰) A B r r' p q
→ ShapeView Γ l (ι ¹) A B r r' p (emb emb< q)
emb¹∞ : ∀ {A B r r' l p q}
→ ShapeView Γ (ι ¹) l A B r r' p q
→ ShapeView Γ ∞ l A B r r' (emb ∞< p) q
emb∞¹ : ∀ {A B r r' l p q}
→ ShapeView Γ l (ι ¹) A B r r' p q
→ ShapeView Γ l ∞ A B r r' p (emb ∞< q)
-- Construct a shape view from an equality
goodCases : ∀ {l l′ Γ A B r r'} ([A] : Γ ⊩⟨ l ⟩ A ^ r) ([B] : Γ ⊩⟨ l′ ⟩ B ^ r')
→ Γ ⊩⟨ l ⟩ A ≡ B ^ r / [A] → ShapeView Γ l l′ A B r r' [A] [B]
goodCases (Uᵣ UA) (Uᵣ UB) A≡B = Uᵥ UA UB
goodCases (Uᵣ′ _ _ _ _ _ _ ⊢Γ) (ℕᵣ D) D' = ⊥-elim (U≢ℕ (whrDet* (D' , Uₙ) (red D , ℕₙ)))
goodCases (Uᵣ′ _ _ _ _ _ _ ⊢Γ) (Emptyᵣ D) D' = ⊥-elim (U≢Empty (whrDet* (D' , Uₙ) (red D , Emptyₙ)))
goodCases (Uᵣ′ _ _ _ _ _ _ ⊢Γ) (ne′ K D neK K≡K) D' = ⊥-elim (U≢ne neK (whrDet* (D' , Uₙ) (red D , ne neK)))
goodCases (Uᵣ′ _ _ _ _ _ _ ⊢Γ) (Πᵣ′ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext) D' =
⊥-elim (U≢Π (whrDet* (D' , Uₙ) (red D , Πₙ)))
goodCases (Uᵣ′ _ _ _ _ _ _ ⊢Γ) (∃ᵣ′ F G D ⊢F ⊢G A≡A [F] [G] G-ext) D' =
⊥-elim (U≢∃ (whrDet* (D' , Uₙ) (red D , ∃ₙ)))
goodCases (ℕᵣ D) (Uᵣ′ _ _ _ _ _ _ D') A≡B = ⊥-elim (U≢ℕ (whrDet* (red D' , Uₙ) (A≡B , ℕₙ)))
goodCases (ℕᵣ _) (Emptyᵣ D') D =
⊥-elim (ℕ≢Empty (whrDet* (D , ℕₙ) (red D' , Emptyₙ)))
goodCases (ℕᵣ ℕA) (ℕᵣ ℕB) A≡B = ℕᵥ ℕA ℕB
goodCases (ℕᵣ D) (ne′ K D₁ neK K≡K) A≡B =
⊥-elim (ℕ≢ne neK (whrDet* (A≡B , ℕₙ) (red D₁ , ne neK)))
goodCases (ℕᵣ D) (Πᵣ′ rF lF lG _ _ F G D₁ ⊢F ⊢G A≡A [F] [G] G-ext) A≡B =
⊥-elim (ℕ≢Π (whrDet* (A≡B , ℕₙ) (red D₁ , Πₙ)))
goodCases (ℕᵣ D) (∃ᵣ′ F G D₁ ⊢F ⊢G A≡A [F] [G] G-ext) A≡B =
⊥-elim (ℕ≢∃ (whrDet* (A≡B , ℕₙ) (red D₁ , ∃ₙ)))
goodCases (Emptyᵣ D) (Uᵣ′ _ _ _ _ _ _ D') A≡B = ⊥-elim (U≢Empty (whrDet* (red D' , Uₙ) (A≡B , Emptyₙ)))
goodCases (Emptyᵣ _) (ℕᵣ D') D =
⊥-elim (ℕ≢Empty (whrDet* (red D' , ℕₙ) (D , Emptyₙ)))
goodCases (Emptyᵣ EmptyA) (Emptyᵣ EmptyB) A≡B = Emptyᵥ EmptyA EmptyB
goodCases (Emptyᵣ D) (ne′ K D₁ neK K≡K) A≡B =
⊥-elim (Empty≢ne neK (whrDet* (A≡B , Emptyₙ) (red D₁ , ne neK)))
goodCases (Emptyᵣ D) (Πᵣ′ rF lF lG _ _ F G D₁ ⊢F ⊢G A≡A [F] [G] G-ext) A≡B =
⊥-elim (Empty≢Π (whrDet* (A≡B , Emptyₙ) (red D₁ , Πₙ)))
goodCases (Emptyᵣ D) (∃ᵣ′ F G D₁ ⊢F ⊢G A≡A [F] [G] G-ext) A≡B =
⊥-elim (Empty≢∃ (whrDet* (A≡B , Emptyₙ) (red D₁ , ∃ₙ)))
goodCases (ne′ K D neK K≡K) (Uᵣ′ _ _ _ _ _ _ D') (ne₌ M D'' neM K≡M) =
⊥-elim (U≢ne neM (whrDet* (red D' , Uₙ) (red D'' , ne neM)))
goodCases (ne′ K D neK K≡K) (ℕᵣ D₁) (ne₌ M D′ neM K≡M) =
⊥-elim (ℕ≢ne neM (whrDet* (red D₁ , ℕₙ) (red D′ , ne neM)))
goodCases (ne′ K D neK K≡K) (Emptyᵣ D₁) (ne₌ M D′ neM K≡M) =
⊥-elim (Empty≢ne neM (whrDet* (red D₁ , Emptyₙ) (red D′ , ne neM)))
goodCases (ne neA) (ne neB) A≡B = ne neA neB
goodCases (ne′ K D neK K≡K) (Πᵣ′ rF lF lG _ _ F G D₁ ⊢F ⊢G A≡A [F] [G] G-ext) (ne₌ M D′ neM K≡M) =
⊥-elim (Π≢ne neM (whrDet* (red D₁ , Πₙ) (red D′ , ne neM)))
goodCases (ne′ K D neK K≡K) (∃ᵣ′ F G D₁ ⊢F ⊢G A≡A [F] [G] G-ext) (ne₌ M D′ neM K≡M) =
⊥-elim (∃≢ne neM (whrDet* (red D₁ , ∃ₙ) (red D′ , ne neM)))
goodCases (Πᵣ′ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext) (Uᵣ′ _ _ _ _ _ _ D')
(Π₌ F′ G′ D′ A≡B [F≡F′] [G≡G′]) =
⊥-elim (U≢Π (whrDet* (red D' , Uₙ) (D′ , Πₙ)))
goodCases (Πᵣ′ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext) (ℕᵣ D₁)
(Π₌ F′ G′ D′ A≡B [F≡F′] [G≡G′]) =
⊥-elim (ℕ≢Π (whrDet* (red D₁ , ℕₙ) (D′ , Πₙ)))
goodCases (Πᵣ′ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext) (Emptyᵣ D₁)
(Π₌ F′ G′ D′ A≡B [F≡F′] [G≡G′]) =
⊥-elim (Empty≢Π (whrDet* (red D₁ , Emptyₙ) (D′ , Πₙ)))
goodCases (Πᵣ′ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext) (ne′ K D₁ neK K≡K)
(Π₌ F′ G′ D′ A≡B [F≡F′] [G≡G′]) =
⊥-elim (Π≢ne neK (whrDet* (D′ , Πₙ) (red D₁ , ne neK)))
goodCases (Πᵣ ΠA) (Πᵣ ΠB) A≡B = Πᵥ ΠA ΠB
goodCases (Πᵣ′ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext)
(∃ᵣ′ F₁ G₁ D₁ ⊢F₁ ⊢G₁ A≡A₁ [F]₁ [G]₁ G-ext₁)
(Π₌ F′ G′ D′ A≡B [F≡F′] [G≡G′]) =
⊥-elim (Π≢∃ (whrDet* (D′ , Πₙ) (red D₁ , ∃ₙ)))
goodCases (∃ᵣ′ F G D ⊢F ⊢G A≡A [F] [G] G-ext) (Uᵣ′ _ _ _ _ _ _ D')
(∃₌ F′ G′ D′ A≡B [F≡F′] [G≡G′]) =
⊥-elim (U≢∃ (whrDet* (red D' , Uₙ) (D′ , ∃ₙ)))
goodCases (∃ᵣ′ F G D ⊢F ⊢G A≡A [F] [G] G-ext) (ℕᵣ D₁)
(∃₌ F′ G′ D′ A≡B [F≡F′] [G≡G′]) =
⊥-elim (ℕ≢∃ (whrDet* (red D₁ , ℕₙ) (D′ , ∃ₙ)))
goodCases (∃ᵣ′ F G D ⊢F ⊢G A≡A [F] [G] G-ext) (Emptyᵣ D₁)
(∃₌ F′ G′ D′ A≡B [F≡F′] [G≡G′]) =
⊥-elim (Empty≢∃ (whrDet* (red D₁ , Emptyₙ) (D′ , ∃ₙ)))
goodCases (∃ᵣ′ F G D ⊢F ⊢G A≡A [F] [G] G-ext) (ne′ K D₁ neK K≡K)
(∃₌ F′ G′ D′ A≡B [F≡F′] [G≡G′]) =
⊥-elim (∃≢ne neK (whrDet* (D′ , ∃ₙ) (red D₁ , ne neK)))
goodCases (∃ᵣ′ F₁ G₁ D₁ ⊢F₁ ⊢G₁ A≡A₁ [F]₁ [G]₁ G-ext₁)
(Πᵣ′ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext)
(∃₌ F′ G′ D′ A≡B [F≡F′] [G≡G′]) =
⊥-elim (Π≢∃ (whrDet* (red D , Πₙ) (D′ , ∃ₙ)))
goodCases (∃ᵣ ∃A) (∃ᵣ ∃B) A≡B = ∃ᵥ ∃A ∃B
goodCases {l} {ι ¹} [A] (emb emb< x) A≡B = emb¹⁰ (goodCases {l} {ι ⁰} [A] x A≡B)
goodCases {l} {∞} [A] (emb ∞< x) A≡B = emb∞¹ (goodCases {l} {ι ¹} [A] x A≡B)
goodCases {ι ¹} {l} (emb emb< x) [B] A≡B = emb⁰¹ (goodCases {ι ⁰} {l} x [B] A≡B)
goodCases {∞} {l} (emb ∞< x) [B] A≡B = emb¹∞ (goodCases {ι ¹} {l} x [B] A≡B)
-- Construct an shape view between two derivations of the same type
goodCasesRefl : ∀ {l l′ Γ A r r'} ([A] : Γ ⊩⟨ l ⟩ A ^ r) ([A′] : Γ ⊩⟨ l′ ⟩ A ^ r')
→ ShapeView Γ l l′ A A r r' [A] [A′]
goodCasesRefl [A] [A′] = goodCases [A] [A′] (reflEq [A])
-- A view for constructor equality between three types
data ShapeView₃ Γ : ∀ l l′ l″ A B C r1 r2 r3
(p : Γ ⊩⟨ l ⟩ A ^ r1)
(q : Γ ⊩⟨ l′ ⟩ B ^ r2)
(r : Γ ⊩⟨ l″ ⟩ C ^ r3) → Set where
Uᵥ : ∀ {A B C l l′ l″ ll ll′ ll″ } UA UB UC → ShapeView₃ Γ l l′ l″ A B C [ ! , ll ] [ ! , ll′ ] [ ! , ll″ ]
(Uᵣ UA) (Uᵣ UB) (Uᵣ UC)
ℕᵥ : ∀ {A B C l l′ l″} ℕA ℕB ℕC
→ ShapeView₃ Γ l l′ l″ A B C [ ! , ι ⁰ ] [ ! , ι ⁰ ] [ ! , ι ⁰ ] (ℕᵣ ℕA) (ℕᵣ ℕB) (ℕᵣ ℕC)
Emptyᵥ : ∀ {A B C l l′ l″ ll ll′ ll″} EmptyA EmptyB EmptyC
→ ShapeView₃ Γ l l′ l″ A B C [ % , ι ll ] [ % , ι ll′ ] [ % , ι ll″ ] (Emptyᵣ EmptyA) (Emptyᵣ EmptyB) (Emptyᵣ EmptyC)
ne : ∀ {A B C r1 r2 r3 l1 l2 l3 l l′ l″} neA neB neC
→ ShapeView₃ Γ l l′ l″ A B C [ r1 , ι l1 ] [ r2 , ι l2 ] [ r3 , ι l3 ] (ne neA) (ne neB) (ne neC)
Πᵥ : ∀ {A B C r1 r2 r3 lΠ1 lΠ2 lΠ3 l l′ l″} ΠA ΠB ΠC
→ ShapeView₃ Γ l l′ l″ A B C [ r1 , ι lΠ1 ] [ r2 , ι lΠ2 ] [ r3 , ι lΠ3 ] (Πᵣ ΠA) (Πᵣ ΠB) (Πᵣ ΠC)
∃ᵥ : ∀ {A B C l l′ l″ ll ll' ll''} ΠA ΠB ΠC
→ ShapeView₃ Γ l l′ l″ A B C [ % , ι ll ] [ % , ι ll' ] [ % , ι ll'' ] (∃ᵣ ΠA) (∃ᵣ ΠB) (∃ᵣ ΠC)
emb⁰¹¹ : ∀ {A B C l l′ r1 r2 r3 p q r}
→ ShapeView₃ Γ (ι ⁰) l l′ A B C r1 r2 r3 p q r
→ ShapeView₃ Γ (ι ¹) l l′ A B C r1 r2 r3 (emb emb< p) q r
emb¹⁰¹ : ∀ {A B C l l′ r1 r2 r3 p q r}
→ ShapeView₃ Γ l (ι ⁰) l′ A B C r1 r2 r3 p q r
→ ShapeView₃ Γ l (ι ¹) l′ A B C r1 r2 r3 p (emb emb< q) r
emb¹¹⁰ : ∀ {A B C l l′ r1 r2 r3 p q r}
→ ShapeView₃ Γ l l′ (ι ⁰) A B C r1 r2 r3 p q r
→ ShapeView₃ Γ l l′ (ι ¹) A B C r1 r2 r3 p q (emb emb< r)
emb¹∞∞ : ∀ {A B C l l′ r1 r2 r3 p q r}
→ ShapeView₃ Γ (ι ¹) l l′ A B C r1 r2 r3 p q r
→ ShapeView₃ Γ ∞ l l′ A B C r1 r2 r3 (emb ∞< p) q r
emb∞¹∞ : ∀ {A B C l l′ r1 r2 r3 p q r}
→ ShapeView₃ Γ l (ι ¹) l′ A B C r1 r2 r3 p q r
→ ShapeView₃ Γ l ∞ l′ A B C r1 r2 r3 p (emb ∞< q) r
emb∞∞¹ : ∀ {A B C l l′ r1 r2 r3 p q r}
→ ShapeView₃ Γ l l′ (ι ¹) A B C r1 r2 r3 p q r
→ ShapeView₃ Γ l l′ ∞ A B C r1 r2 r3 p q (emb ∞< r)
-- Combines two two-way views into a three-way view
combine : ∀ {Γ l l′ l″ l‴ A B C r1 r2 r2' r3 [A] [B] [B]′ [C]}
→ ShapeView Γ l l′ A B r1 r2 [A] [B]
→ ShapeView Γ l″ l‴ B C r2' r3 [B]′ [C]
→ ShapeView₃ Γ l l′ l‴ A B C r1 r2 r3 [A] [B] [C]
combine (Uᵥ UA UB) (Uᵥ UB' UC) = Uᵥ UA UB UC
combine (Uᵥ UA (Uᵣ r l′ l< PE.refl D)) (ℕᵥ ℕA ℕB) =
⊥-elim (U≢ℕ (whrDet* (red D , Uₙ) (red ℕA , ℕₙ)))
combine (Uᵥ UA (Uᵣ r l′ l< PE.refl D)) (Emptyᵥ EmptyA EmptyB) =
⊥-elim (U≢Empty (whrDet* (red D , Uₙ) (red EmptyA , Emptyₙ)))
combine (Uᵥ UA (Uᵣ r l′ l< PE.refl D)) (ne (ne K D' neK K≡K) neB) =
⊥-elim (U≢ne neK (whrDet* (red D , Uₙ) (red D' , ne neK)))
combine (Uᵥ UA (Uᵣ r l′ l< PE.refl D)) (Πᵥ (Πᵣ rF lF lG _ _ F G D' ⊢F ⊢G A≡A [F] [G] G-ext) ΠB) =
⊥-elim (U≢Π (whrDet* (red D , Uₙ) (red D' , Πₙ)))
combine (Uᵥ UA (Uᵣ r l′ l< PE.refl D)) (∃ᵥ (∃ᵣ F G D' ⊢F ⊢G A≡A [F] [G] G-ext) ∃B) =
⊥-elim (U≢∃ (whrDet* (red D , Uₙ) (red D' , ∃ₙ)))
combine (ℕᵥ ℕA ℕB) (Uᵥ (Uᵣ r l′ l< PE.refl D) UB) =
⊥-elim (U≢ℕ (whrDet* (red D , Uₙ) (red ℕB , ℕₙ)))
combine (ℕᵥ ℕA ℕB) (Emptyᵥ EmptyA EmptyB) =
⊥-elim (ℕ≢Empty (whrDet* (red ℕB , ℕₙ) (red EmptyA , Emptyₙ)))
combine (ℕᵥ ℕA₁ ℕB₁) (ℕᵥ ℕA ℕB) = ℕᵥ ℕA₁ ℕB₁ ℕB
combine (ℕᵥ ℕA ℕB) (ne (ne K D neK K≡K) neB) =
⊥-elim (ℕ≢ne neK (whrDet* (red ℕB , ℕₙ) (red D , ne neK)))
combine (ℕᵥ ℕA ℕB) (Πᵥ (Πᵣ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext) ΠB) =
⊥-elim (ℕ≢Π (whrDet* (red ℕB , ℕₙ) (red D , Πₙ)))
combine (ℕᵥ ℕA ℕB) (∃ᵥ (∃ᵣ F G D ⊢F ⊢G A≡A [F] [G] G-ext) ∃B) =
⊥-elim (ℕ≢∃ (whrDet* (red ℕB , ℕₙ) (red D , ∃ₙ)))
combine (Emptyᵥ EmptyA EmptyB) (Uᵥ (Uᵣ r l′ l< PE.refl D) UB) =
⊥-elim (U≢Empty (whrDet* (red D , Uₙ) (red EmptyB , Emptyₙ)))
combine (Emptyᵥ EmptyA EmptyB) (ℕᵥ ℕA ℕB) =
⊥-elim (Empty≢ℕ (whrDet* (red EmptyB , Emptyₙ) (red ℕA , ℕₙ)))
combine (Emptyᵥ EmptyA₁ EmptyB₁) (Emptyᵥ EmptyA EmptyB) = Emptyᵥ EmptyA₁ EmptyB₁ EmptyB
combine (Emptyᵥ EmptyA EmptyB) (ne (ne K D neK K≡K) neB) =
⊥-elim (Empty≢ne neK (whrDet* (red EmptyB , Emptyₙ) (red D , ne neK)))
combine (Emptyᵥ EmptyA EmptyB) (Πᵥ (Πᵣ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext) ΠB) =
⊥-elim (Empty≢Π (whrDet* (red EmptyB , Emptyₙ) (red D , Πₙ)))
combine (Emptyᵥ EmptyA EmptyB) (∃ᵥ (∃ᵣ F G D ⊢F ⊢G A≡A [F] [G] G-ext) ∃B) =
⊥-elim (Empty≢∃ (whrDet* (red EmptyB , Emptyₙ) (red D , ∃ₙ)))
combine (ne neA (ne K D' neK K≡K)) (Uᵥ (Uᵣ r l′ l< PE.refl D) UB) =
⊥-elim (U≢ne neK (whrDet* (red D , Uₙ) (red D' , ne neK)))
combine (ne neA (ne K D neK K≡K)) (ℕᵥ ℕA ℕB) =
⊥-elim (ℕ≢ne neK (whrDet* (red ℕA , ℕₙ) (red D , ne neK)))
combine (ne neA (ne K D neK K≡K)) (Emptyᵥ EmptyA EmptyB) =
⊥-elim (Empty≢ne neK (whrDet* (red EmptyA , Emptyₙ) (red D , ne neK)))
combine (ne neA₁ neB₁) (ne neA neB) = ne neA₁ neB₁ neB
combine (ne neA (ne K D₁ neK K≡K)) (Πᵥ (Πᵣ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext) ΠB) =
⊥-elim (Π≢ne neK (whrDet* (red D , Πₙ) (red D₁ , ne neK)))
combine (ne neA (ne K D₁ neK K≡K)) (∃ᵥ (∃ᵣ F G D ⊢F ⊢G A≡A [F] [G] G-ext) ∃B) =
⊥-elim (∃≢ne neK (whrDet* (red D , ∃ₙ) (red D₁ , ne neK)))
combine (Πᵥ ΠA (Πᵣ rF lF lG _ _ F G D' ⊢F ⊢G A≡A [F] [G] G-ext)) (Uᵥ (Uᵣ r l′ l< PE.refl D) UB) =
⊥-elim (U≢Π (whrDet* (red D , Uₙ) (red D' , Πₙ)))
combine (Πᵥ ΠA (Πᵣ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext)) (ℕᵥ ℕA ℕB) =
⊥-elim (ℕ≢Π (whrDet* (red ℕA , ℕₙ) (red D , Πₙ)))
combine (Πᵥ ΠA (Πᵣ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext)) (Emptyᵥ EmptyA EmptyB) =
⊥-elim (Empty≢Π (whrDet* (red EmptyA , Emptyₙ) (red D , Πₙ)))
combine (Πᵥ ΠA (Πᵣ rF lF lG _ _ F G D₁ ⊢F ⊢G A≡A [F] [G] G-ext)) (ne (ne K D neK K≡K) neB) =
⊥-elim (Π≢ne neK (whrDet* (red D₁ , Πₙ) (red D , ne neK)))
combine (Πᵥ ΠA₁ ΠB₁) (Πᵥ ΠA ΠB) = Πᵥ ΠA₁ ΠB₁ ΠB
combine (Πᵥ ΠA (Πᵣ rF lF lG _ _ F G D ⊢F ⊢G A≡A [F] [G] G-ext))
(∃ᵥ (∃ᵣ F₁ G₁ D₁ ⊢F₁ ⊢G₁ A≡A₁ [F]₁ [G]₁ G-ext₁) ∃B) =
⊥-elim (Π≢∃ (whrDet* (red D , Πₙ) (red D₁ , ∃ₙ)))
combine (∃ᵥ ∃A (∃ᵣ F G D' ⊢F ⊢G A≡A [F] [G] G-ext)) (Uᵥ (Uᵣ r l′ l< PE.refl D) UB) =
⊥-elim (U≢∃ (whrDet* (red D , Uₙ) (red D' , ∃ₙ)))
combine (∃ᵥ ∃A (∃ᵣ F G D ⊢F ⊢G A≡A [F] [G] G-ext)) (ℕᵥ ℕA ℕB) =
⊥-elim (ℕ≢∃ (whrDet* (red ℕA , ℕₙ) (red D , ∃ₙ)))
combine (∃ᵥ ∃A (∃ᵣ F G D ⊢F ⊢G A≡A [F] [G] G-ext)) (Emptyᵥ EmptyA EmptyB) =
⊥-elim (Empty≢∃ (whrDet* (red EmptyA , Emptyₙ) (red D , ∃ₙ)))
combine (∃ᵥ ∃A (∃ᵣ F G D₁ ⊢F ⊢G A≡A [F] [G] G-ext)) (ne (ne K D neK K≡K) neB) =
⊥-elim (∃≢ne neK (whrDet* (red D₁ , ∃ₙ) (red D , ne neK)))
combine (∃ᵥ ΠA (∃ᵣ F G D ⊢F ⊢G A≡A [F] [G] G-ext))
(Πᵥ (Πᵣ rF₁ lF₁ lG₁ _ _ F₁ G₁ D₁ ⊢F₁ ⊢G₁ A≡A₁ [F]₁ [G]₁ G-ext₁) ∃B) =
⊥-elim (Π≢∃ (whrDet* (red D₁ , Πₙ) (red D , ∃ₙ)))
combine (∃ᵥ ∃A ∃B) (∃ᵥ ∃A₁ ∃B₁) = ∃ᵥ ∃A ∃B ∃B₁
combine (emb⁰¹ [AB]) [BC] = emb⁰¹¹ (combine [AB] [BC])
combine (emb¹⁰ [AB]) [BC] = emb¹⁰¹ (combine [AB] [BC])
combine [AB] (emb⁰¹ [BC]) = combine [AB] [BC]
combine [AB] (emb¹⁰ [BC]) = emb¹¹⁰ (combine [AB] [BC])
combine (emb¹∞ [AB]) [BC] = emb¹∞∞ (combine [AB] [BC])
combine (emb∞¹ [AB]) [BC] = emb∞¹∞ (combine [AB] [BC])
combine [AB] (emb¹∞ [BC]) = combine [AB] [BC]
combine [AB] (emb∞¹ [BC]) = emb∞∞¹ (combine [AB] [BC])
|
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{-
Smith Normal Form
Referrences:
Guillaume Cano, Cyril Cohen, Maxime Dénès, Anders Mörtberg, Vincent Siles,
"Formalized linear algebra over Elementary Divisor Rings in Coq"
(https://arxiv.org/abs/1601.07472)
-}
{-# OPTIONS --safe #-}
module Cubical.Algebra.IntegerMatrix.Smith where
open import Cubical.Algebra.IntegerMatrix.Smith.NormalForm public
open import Cubical.Algebra.IntegerMatrix.Smith.Normalization public using (smith)
|
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module API where
open import Algebra
open import Prelude
empty : ∀ {A} -> Graph A
empty = ε
vertex : ∀ {A} -> A -> Graph A
vertex = v
overlay : ∀ {A} -> Graph A -> Graph A -> Graph A
overlay = _+_
connect : ∀ {A} -> Graph A -> Graph A -> Graph A
connect = _*_
edge : ∀ {A} -> A -> A -> Graph A
edge x y = connect (vertex x) (vertex y)
overlays : ∀ {A} -> List (Graph A) -> Graph A
overlays = foldr overlay empty
connects : ∀ {A} -> List (Graph A) -> Graph A
connects = foldr connect empty
vertices : ∀ {A} -> List A -> Graph A
vertices = overlays ∘ map vertex
edges : ∀ {A} -> List (A × A) -> Graph A
edges = overlays ∘ map (uncurry edge)
graph : ∀ {A} -> List A -> List (A × A) -> Graph A
graph vs es = overlay (vertices vs) (edges es)
foldg : ∀ {A} {B : Set} -> B -> (A -> B) -> (B -> B -> B) -> (B -> B -> B) -> Graph A -> B
foldg {A} {B} e w o c = go
where
go : Graph A -> B
go ε = e
go (v x) = w x
go (x + y) = o (go x) (go y)
go (x * y) = c (go x) (go y)
path : ∀ {A} -> List A -> Graph A
path [] = empty
path (x :: []) = vertex x
path (x :: xs) = edges (zip (x :: xs) xs)
circuit : ∀ {A} -> List A -> Graph A
circuit [] = empty
circuit (x :: xs) = path ([ x ] ++ xs ++ [ x ])
clique : ∀ {A} -> List A -> Graph A
clique = connects ∘ map vertex
biclique : ∀ {A} -> List A -> List A -> Graph A
biclique xs ys = connect (vertices xs) (vertices ys)
star : ∀ {A} -> A -> List A -> Graph A
star x ys = connect (vertex x) (vertices ys)
|
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{-# OPTIONS --without-K #-}
module sets.list.properties where
open import level
open import equality.core
open import sets.list.core
module _ {i}{A : Set i} where
data all {j}(P : A → Set j) : List A → Set (i ⊔ j) where
mk-all : ∀ {x xs} → P x → all P xs → all P (x ∷ xs)
data any {j}(P : A → Set j) : List A → Set (i ⊔ j) where
hd-any : ∀ {x xs} → P x → any P (x ∷ xs)
tl-any : ∀ {x xs} → any P xs → any P (x ∷ xs)
elem : A → List A → Set i
elem x = any (λ x' → x ≡ x')
|
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import SOAS.Syntax.Signature as Sig
open import SOAS.Families.Core
-- Term syntax for a signature
module SOAS.Syntax.Term
{T : Set}(open Sig T)
{O : Set}(S : Signature O) where
open import SOAS.Syntax.Arguments {T}
open import SOAS.Metatheory.Syntax {T}
open import SOAS.Common
open import SOAS.Context {T}
open import SOAS.Variable
open import SOAS.Construction.Structure
open import SOAS.ContextMaps.Inductive
open import SOAS.Abstract.Hom
open import Categories.Object.Initial
open import Data.List.Base using (List ; [] ; [_] ; _∷_)
open import Data.Unit
open Signature S
private
variable
α β τ : T
Γ Δ Π : Ctx
module _ (𝔛 : Familyₛ) where
open import SOAS.Metatheory.MetaAlgebra ⅀F 𝔛
-- Grammar of terms for a (⅀,𝔛)-meta-algebra
data 𝕋 : Familyₛ where
con : ⅀ 𝕋 τ Γ → 𝕋 τ Γ
var : ℐ τ Γ → 𝕋 τ Γ
mvar : 𝔛 τ Π → Sub 𝕋 Π Γ → 𝕋 τ Γ
Tmᵃ : MetaAlg 𝕋
Tmᵃ = record { 𝑎𝑙𝑔 = con ; 𝑣𝑎𝑟 = var ; 𝑚𝑣𝑎𝑟 = λ 𝔪 ε → mvar 𝔪 (tabulate ε) }
-- 𝕋 is the initial meta-algebra
𝕋:Init : Initial 𝕄etaAlgebras
𝕋:Init = record
{ ⊥ = 𝕋 ⋉ Tmᵃ
; ⊥-is-initial = record
{ ! = λ{ {𝒜 ⋉ 𝒜ᵃ} → (𝕤𝕖𝕞 𝒜ᵃ) ⋉ 𝕤𝕖𝕞ᵃ⇒ 𝒜ᵃ }
; !-unique = λ { {𝒜 ⋉ 𝒜ᵃ}(g ⋉ gᵃ⇒) {x = t} → 𝕤𝕖𝕞! 𝒜ᵃ gᵃ⇒ t } } }
where
module _ {𝒜 : Familyₛ}(𝒜ᵃ : MetaAlg 𝒜) where
open MetaAlg 𝒜ᵃ
𝕤𝕖𝕞 : 𝕋 ⇾̣ 𝒜
𝔸 : (as : List (Ctx × T)) → Arg as 𝕋 Γ → Arg as 𝒜 Γ
𝔸 [] tt = tt
𝔸 (_ ∷ []) t = 𝕤𝕖𝕞 t
𝔸 (_ ∷ a ∷ as) (t , ts) = (𝕤𝕖𝕞 t , 𝔸 (a ∷ as) ts)
𝕊 : Sub 𝕋 Π Γ → Π ~[ 𝒜 ]↝ Γ
𝕊 (t ◂ σ) new = 𝕤𝕖𝕞 t
𝕊 (t ◂ σ) (old v) = 𝕊 σ v
𝕤𝕖𝕞 (con (o ⋮ a)) = 𝑎𝑙𝑔 (o ⋮ 𝔸 (Arity o) a)
𝕤𝕖𝕞 (var v) = 𝑣𝑎𝑟 v
𝕤𝕖𝕞 (mvar 𝔪 ε) = 𝑚𝑣𝑎𝑟 𝔪 (𝕊 ε)
𝕤𝕖𝕞ᵃ⇒ : MetaAlg⇒ Tmᵃ 𝒜ᵃ 𝕤𝕖𝕞
𝕤𝕖𝕞ᵃ⇒ = record
{ ⟨𝑎𝑙𝑔⟩ = λ{ {t = (o ⋮ a)} → cong (λ - → 𝑎𝑙𝑔 (o ⋮ -)) (𝔸-Arg₁ (Arity o) a) }
; ⟨𝑣𝑎𝑟⟩ = refl
; ⟨𝑚𝑣𝑎𝑟⟩ = λ{ {𝔪 = 𝔪}{ε} → cong (𝑚𝑣𝑎𝑟 𝔪) (dext (𝕊-tab ε)) }
}
where
𝔸-Arg₁ : (as : List (Ctx × T))(a : Arg as 𝕋 Γ)
→ 𝔸 as a ≡ Arg₁ as 𝕤𝕖𝕞 a
𝔸-Arg₁ [] tt = refl
𝔸-Arg₁ (_ ∷ []) t = refl
𝔸-Arg₁ (_ ∷ a ∷ as) (t , ap) = cong (_ ,_) (𝔸-Arg₁ (a ∷ as) ap)
𝕊-tab : (ε : Π ~[ 𝕋 ]↝ Γ)(v : ℐ α Π)
→ 𝕊 (tabulate ε) v ≡ 𝕤𝕖𝕞 (ε v)
𝕊-tab ε new = refl
𝕊-tab ε (old v) = 𝕊-tab (ε ∘ old) v
module _ {g : 𝕋 ⇾̣ 𝒜}(gᵃ⇒ : MetaAlg⇒ Tmᵃ 𝒜ᵃ g) where
open MetaAlg⇒ gᵃ⇒
𝕤𝕖𝕞! : (t : 𝕋 α Γ) → 𝕤𝕖𝕞 t ≡ g t
𝕊-ix : (ε : Sub 𝕋 Π Γ)(v : ℐ α Π) → 𝕊 ε v ≡ g (index ε v)
𝕊-ix (x ◂ ε) new = 𝕤𝕖𝕞! x
𝕊-ix (x ◂ ε) (old v) = 𝕊-ix ε v
𝔸-Arg₁ : (as : List (Ctx × T))(ar : Arg as 𝕋 Γ)
→ 𝔸 as ar ≡ Arg₁ as g ar
𝔸-Arg₁ [] tt = refl
𝔸-Arg₁ (_ ∷ []) t = 𝕤𝕖𝕞! t
𝔸-Arg₁ (_ ∷ a ∷ as) (t , ap) = cong₂ _,_ (𝕤𝕖𝕞! t) (𝔸-Arg₁ (a ∷ as) ap)
𝕤𝕖𝕞! (con (o ⋮ a)) rewrite 𝔸-Arg₁ (Arity o) a = sym ⟨𝑎𝑙𝑔⟩
𝕤𝕖𝕞! (var v) = sym ⟨𝑣𝑎𝑟⟩
𝕤𝕖𝕞! (mvar 𝔪 ε) rewrite cong (𝑚𝑣𝑎𝑟 𝔪) (dext (𝕊-ix ε)) =
trans (sym ⟨𝑚𝑣𝑎𝑟⟩) (cong (g ∘ mvar 𝔪) (tab∘ix≈id ε))
-- Syntax instance for a term grammar
𝕋:Syn : Syntax
𝕋:Syn = record
{ ⅀F = ⅀F
; ⅀:CS = ⅀:CompatStr
; 𝕋:Init = 𝕋:Init
; mvarᵢ = mvar }
|
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module SingleSorted.Example where
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl)
open import Data.Product using (_×_; proj₁; proj₂; <_,_>; ∃; ∃-syntax; _,_)
import Function using (_∘_)
open import SingleSorted.AlgebraicTheory
open import Categories.Category.Instance.Sets
open import Categories.Category.Cartesian
open import Agda.Primitive
open import Categories.Category
module Sets₀ where
𝒮 : Category (lsuc lzero) lzero lzero
𝒮 = Sets lzero
open Sets₀
open Category 𝒮
-- Function extensionality
postulate
funext : ∀ {X : Set} {Y : X → Set} {f g : ∀ (x : X) → (Y x)} → (∀ (x : X) → ((f x) ≡ (g x))) → (f ≡ g)
record singleton : Set where
constructor ⋆
η-singleton : ∀ (x : singleton) → ⋆ ≡ x
η-singleton ⋆ = refl
prod-elem-structure : ∀ {A B : Set} {x : A × B} → ∃[ a ] ∃[ b ] (a , b) ≡ x
prod-elem-structure {A} {B} {x} = proj₁ x Data.Product., (proj₂ x) , refl
first-factor : ∀ {X A B : Obj} {f : X ⇒ A} {g : X ⇒ B} → proj₁ ∘ < f , g > ≡ f
first-factor {X} {A} {B} {f} {g} = funext λ x → refl
second-factor : ∀ {X A B : Obj} {f : X ⇒ A} {g : X ⇒ B} → proj₂ ∘ < f , g > ≡ g
second-factor {X} {A} {B} {f} {g} = funext λ{ x → refl}
unique-map-into-product : ∀ {X A B : Set} {h : X → A × B} {f : X → A} {g : X → B} {x : X}
→ (proj₁ ∘ h) x ≡ f x
→ (proj₂ ∘ h) x ≡ g x
--------------------
→ < f , g > x ≡ h x
unique-map-into-product {X} {A} {B} {h} {f} {g} {x} eq1 eq2
rewrite first-factor {X} {A} {B} {f} {g} | second-factor {X} {A} {B} {f} {g}
= {!!}
cartesian-Sets : Cartesian 𝒮
cartesian-Sets =
record
{ terminal = record
{ ⊤ = singleton
; ⊤-is-terminal = record
{ ! = λ{ x → ⋆}
; !-unique = λ{ f {x} → η-singleton (f x)}
}
}
; products = record
{ product = λ {A} {B} →
record
{ A×B = A × B
; π₁ = proj₁
; π₂ = proj₂
; ⟨_,_⟩ = <_,_>
; project₁ = λ{ → refl}
; project₂ = λ{ → refl}
; unique = λ{ {X} {h} {f} {g} eq1 eq2 {x} → {!!}}
}
}
}
|
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------------------------------------------------------------------------
-- The Agda standard library
--
-- The Maybe type
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Maybe where
open import Data.Empty using (⊥)
open import Data.Unit using (⊤)
open import Data.Bool.Base using (T)
open import Data.Maybe.Relation.Unary.All
open import Data.Maybe.Relation.Unary.Any
open import Level using (Level)
private
variable
a : Level
A : Set a
------------------------------------------------------------------------
-- The base type and some operations
open import Data.Maybe.Base public
------------------------------------------------------------------------
-- Using Any and All to define Is-just and Is-nothing
Is-just : Maybe A → Set _
Is-just = Any (λ _ → ⊤)
Is-nothing : Maybe A → Set _
Is-nothing = All (λ _ → ⊥)
to-witness : ∀ {m : Maybe A} → Is-just m → A
to-witness (just {x = p} _) = p
to-witness-T : ∀ (m : Maybe A) → T (is-just m) → A
to-witness-T (just p) _ = p
|
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|
data Nat : Set where
zero : Nat
data _≡_ {A : Set} (x : A) : A → Set where
refl : x ≡ x
subst : ∀ {A : Set} (P : A → Set) {x y} → x ≡ y → P x → P y
subst P refl px = px
postulate
Eq : Set → Set
mkEq : {A : Set} (x y : A) → x ≡ y
_==_ : {A : Set} {{_ : Eq A}} (x y : A) → x ≡ y
A : Set
B : A → Set
C : ∀ x → B x → Set
case_of_ : ∀ {a b} {A : Set a} {B : Set b} → A → (A → B) → B
case x of f = f x
id : ∀ {a} (A : Set a) → A → A
id A x = x
eqTriple : {{_ : ∀ {x} {y : B x} → Eq (C x y)}}
(a : A) (b : B a) (c : C a b)
(a₁ : A) (b₁ : B a₁) (c : C a₁ b₁) → Nat
eqTriple a b c a₁ b₁ c₁ =
subst (λ a₂ → ∀ (b₂ : B a₂) (c₂ : _) → Nat) (mkEq a a₁)
(λ b₂ c₂ → subst (λ b₃ → ∀ c₃ → Nat) (mkEq b b₂)
(λ c₃ → case c == c₃ of λ eq → zero) c₂)
b₁ c₁
|
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------------------------------------------------------------------------
-- The actual maximum stack size of the compiled program matches the
-- maximum stack size of the instrumented source-level semantics
------------------------------------------------------------------------
open import Prelude
import Lambda.Syntax
module Lambda.Compiler-correctness.Sizes-match
{Name : Type}
(open Lambda.Syntax Name)
(def : Name → Tm 1)
where
open import Equality.Propositional as E using (refl)
open import Logical-equivalence using (_⇔_)
open import Tactic.By.Propositional using (by)
open import Prelude.Size
open import Colist E.equality-with-J hiding (_++_; length)
import Conat E.equality-with-J as Conat
open import Function-universe E.equality-with-J hiding (id; _∘_)
open import List E.equality-with-J using (_++_; length)
open import Monad E.equality-with-J
open import Nat E.equality-with-J
open import Vec.Data E.equality-with-J
open import Upper-bounds
open import Lambda.Compiler def
open import Lambda.Interpreter.Stack-sizes def as I
open import Lambda.Delay-crash-trace as DCT
using (Delay-crash-trace)
open import Lambda.Virtual-machine.Instructions Name
open import Lambda.Virtual-machine comp-name as VM
private
module C = Closure Code
module T = Closure Tm
open Delay-crash-trace using (tell)
------------------------------------------------------------------------
-- A lemma
-- A rearrangement lemma for ⟦_⟧⁺.
⟦⟧-· :
∀ {A n} (t₁ t₂ : Tm n)
{ρ} tc {k : T.Value → Delay-crash-trace (ℕ → ℕ) A ∞} →
DCT.[ ∞ ] ⟦ t₁ · t₂ ⟧ ρ tc >>= k ∼
do v₁ ← ⟦ t₁ ⟧ ρ false
v₂ ← ⟦ t₂ ⟧ ρ false
[ pred , pred ] v₁ ∙ v₂ >>= k
⟦⟧-· t₁ t₂ {ρ} tc {k} =
⟦ t₁ · t₂ ⟧ ρ tc >>= k DCT.∼⟨⟩
(do v₁ ← ⟦ t₁ ⟧ ρ false
v₂ ← ⟦ t₂ ⟧ ρ false
[ pred , pred ] v₁ ∙ v₂) >>= k DCT.∼⟨ DCT.symmetric (DCT.associativity (⟦ t₁ ⟧ _ _) _ _) ⟩
(do v₁ ← ⟦ t₁ ⟧ ρ false
(do v₂ ← ⟦ t₂ ⟧ ρ false
[ pred , pred ] v₁ ∙ v₂) >>= k) DCT.∼⟨ ((⟦ t₁ ⟧ _ _ DCT.∎) DCT.>>=-cong λ _ → DCT.symmetric (DCT.associativity (⟦ t₂ ⟧ _ _) _ _)) ⟩
(do v₁ ← ⟦ t₁ ⟧ ρ false
v₂ ← ⟦ t₂ ⟧ ρ false
[ pred , pred ] v₁ ∙ v₂ >>= k) DCT.∎
------------------------------------------------------------------------
-- Well-formed continuations and stacks
-- A continuation is OK with respect to a certain state if the
-- following property is satisfied.
Cont-OK :
Size → State → (T.Value → Delay-crash-trace (ℕ → ℕ) C.Value ∞) → Type
Cont-OK i ⟨ c , s , ρ ⟩ k =
∀ v → [ i ] VM.stack-sizes ⟨ c , val (comp-val v) ∷ s , ρ ⟩ ≂
numbers (k v) (1 + length s)
-- A workaround for what might be an Agda bug.
castC :
∀ {i} {j : Size< i} {s k} →
Cont-OK i s k → Cont-OK j s k
castC {s = ⟨ _ , _ , _ ⟩} c-ok = cast-≂ ∘ c-ok
-- If the In-tail-context parameter indicates that we are in a tail
-- context, then the stack must have a certain shape, and it must be
-- related to the continuation in a certain way.
data Stack-OK (i : Size)
(k : T.Value → Delay-crash-trace (ℕ → ℕ) C.Value ∞) :
In-tail-context → Stack → Type where
unrestricted : ∀ {s} → Stack-OK i k false s
restricted :
∀ {s n} {c : Code n} {ρ : C.Env n} →
(∀ v → [ i ] 2 + length s ∷′
VM.stack-sizes ⟨ c , val (comp-val v) ∷ s , ρ ⟩ ≂
numbers (k v) (2 + length s)) →
Stack-OK i k true (ret c ρ ∷ s)
-- Stacks that are OK in a tail context are OK in any context.
any-OK :
∀ {tc i k s} →
Stack-OK i k true s →
Stack-OK i k tc s
any-OK {false} = const unrestricted
any-OK {true} = id
------------------------------------------------------------------------
-- The stack sizes match
mutual
-- Some lemmas making up the main part of the correctness result.
⟦⟧-correct :
∀ {i n} (t : Tm n) (ρ : T.Env n) {tc c s}
{k : T.Value → Delay-crash-trace (ℕ → ℕ) C.Value ∞} →
Stack-OK i k tc s →
Cont-OK i ⟨ c , s , comp-env ρ ⟩ k →
[ i ] VM.stack-sizes ⟨ comp tc t c , s , comp-env ρ ⟩ ≂
numbers (⟦ t ⟧ ρ tc >>= k) (length s)
⟦⟧-correct (var x) ρ {tc} {c} {s} {k} _ c-ok =
VM.stack-sizes ⟨ var x ∷ c , s , comp-env ρ ⟩ ∼⟨ ∷∼∷′ ⟩≂
(length s ∷′
VM.stack-sizes ⟨ c , val (index (comp-env ρ) x) ∷ s , comp-env ρ ⟩) ≡⟨ by (comp-index ρ x) ⟩≂
(length s ∷′
VM.stack-sizes ⟨ c , val (comp-val (index ρ x)) ∷ s , comp-env ρ ⟩) ≂⟨ cons″-≂ (c-ok (index ρ x)) ⟩∼
(length s ∷′ numbers (k (index ρ x)) (1 + length s)) ∼⟨ symmetric-∼ ∷∼∷′ ⟩
numbers (⟦ var x ⟧ ρ tc >>= k) (length s) ∎
⟦⟧-correct (lam t) ρ {tc} {c} {s} {k} _ c-ok =
VM.stack-sizes ⟨ clo (comp-body t) ∷ c , s , comp-env ρ ⟩ ∼⟨ ∷∼∷′ ⟩≂
(length s ∷′
VM.stack-sizes ⟨ c , val (comp-val (T.lam t ρ)) ∷ s , comp-env ρ ⟩) ≂⟨ cons″-≂ (c-ok (T.lam t ρ)) ⟩∼
(length s ∷′ numbers (k (T.lam t ρ)) (1 + length s)) ∼⟨ symmetric-∼ ∷∼∷′ ⟩
numbers (⟦ lam t ⟧ ρ tc >>= k) (length s) ∎
⟦⟧-correct (t₁ · t₂) ρ {tc} {c} {s} {k} _ c-ok =
VM.stack-sizes ⟨ comp false t₁ (comp false t₂ (app ∷ c))
, s
, comp-env ρ
⟩ ≂⟨ (⟦⟧-correct t₁ _ unrestricted λ v₁ →
VM.stack-sizes ⟨ comp false t₂ (app ∷ c)
, val (comp-val v₁) ∷ s
, comp-env ρ
⟩ ≂⟨ (⟦⟧-correct t₂ _ unrestricted λ v₂ →
VM.stack-sizes ⟨ app ∷ c
, val (comp-val v₂) ∷ val (comp-val v₁) ∷ s
, comp-env ρ
⟩ ≂⟨ ∙-correct v₁ v₂ c-ok ⟩∼
numbers ([ pred , pred ] v₁ ∙ v₂ >>= k) (2 + length s) ∎) ⟩∼
numbers (do v₂ ← ⟦ t₂ ⟧ ρ false
[ pred , pred ] v₁ ∙ v₂ >>= k)
(1 + length s) ∎) ⟩∼
numbers (do v₁ ← ⟦ t₁ ⟧ ρ false
v₂ ← ⟦ t₂ ⟧ ρ false
[ pred , pred ] v₁ ∙ v₂ >>= k)
(length s) ∼⟨ symmetric-∼ (numbers-cong (⟦⟧-· t₁ t₂ tc)) ⟩
numbers (⟦ t₁ · t₂ ⟧ ρ tc >>= k) (length s) ∎
⟦⟧-correct (call f t) ρ {false} {c} {s} {k} unrestricted c-ok =
VM.stack-sizes ⟨ comp false t (cal f ∷ c) , s , comp-env ρ ⟩ ≂⟨ (⟦⟧-correct t _ unrestricted λ v →
VM.stack-sizes ⟨ cal f ∷ c , val (comp-val v) ∷ s , comp-env ρ ⟩ ∼⟨ ∷∼∷′ ⟩≂
1 + length s ∷′
VM.stack-sizes ⟨ comp-name f
, ret c (comp-env ρ) ∷ s
, comp-val v ∷ []
⟩ ≂⟨ cons′-≂ (_⇔_.from ≂′⇔≂″ λ { .force →
body-lemma (def f) [] (castC c-ok) }) ⟩∼
1 + length s ∷′
numbers (⟦ def f ⟧ (v ∷ []) true >>= tell pred ∘ return >>= k)
(1 + length s) ∼⟨ symmetric-∼ ∷∼∷′ ⟩
numbers ([ id , pred ] T.lam (def f) [] ∙ v >>= k)
(1 + length s) ∎) ⟩∼
numbers (⟦ t ⟧ ρ false >>= λ v →
[ id , pred ] T.lam (def f) [] ∙ v >>= k)
(length s) ∼⟨ numbers-cong (DCT.associativity (⟦ t ⟧ _ _) _ _) ⟩
numbers ((⟦ t ⟧ ρ false >>=
[ id , pred ] T.lam (def f) [] ∙_) >>= k)
(length s) ∼⟨⟩
numbers (⟦ call f t ⟧ ρ false >>= k) (length s) ∎
⟦⟧-correct {i} (call f t) ρ {true} {c} {ret c′ ρ′ ∷ s} {k}
s-ok@(restricted c-ok) _ =
VM.stack-sizes ⟨ comp false t (tcl f ∷ c)
, ret c′ ρ′ ∷ s
, comp-env ρ
⟩ ≂⟨ (⟦⟧-correct t _ unrestricted λ v →
VM.stack-sizes ⟨ tcl f ∷ c
, val (comp-val v) ∷ ret c′ ρ′ ∷ s
, comp-env ρ
⟩ ∼⟨ ∷∼∷′ ⟩≂
2 + length s ∷′
VM.stack-sizes ⟨ comp-name f
, ret c′ ρ′ ∷ s
, comp-val v ∷ []
⟩ ≡⟨⟩≂
2 + length s ∷′
VM.stack-sizes ⟨ comp-name f
, ret c′ ρ′ ∷ s
, comp-env (v ∷ [])
⟩ ≂⟨ cons′-≂ (_⇔_.from ≂′⇔≂″ λ { .force →
⟦⟧-correct (def f) (_ ∷ []) (restricted lemma) (λ v′ →
VM.stack-sizes ⟨ ret ∷ []
, val (comp-val v′) ∷ ret c′ ρ′ ∷ s
, comp-env (v ∷ [])
⟩ ∼⟨ ∷∼∷′ ⟩≂
2 + length s ∷′
VM.stack-sizes ⟨ c′ , val (comp-val v′) ∷ s , ρ′ ⟩ ≂⟨ lemma v′ ⟩∼
numbers (tell id (k v′)) (2 + length s) ∎) }) ⟩∼
2 + length s ∷′
numbers (⟦ def f ⟧ (v ∷ []) true >>= tell id ∘ k) (1 + length s) ∼⟨ (refl ∷ λ { .force →
numbers-cong (DCT.associativity (⟦ def f ⟧ _ _) _ _) }) ⟩
2 + length s ∷′
numbers (⟦ def f ⟧ (v ∷ []) true >>= tell id ∘ return >>= k)
(1 + length s) ∼⟨ symmetric-∼ ∷∼∷′ ⟩
numbers ([ pred , id ] T.lam (def f) [] ∙ v >>= k)
(2 + length s) ∎) ⟩∼
numbers (⟦ t ⟧ ρ false >>= λ v →
[ pred , id ] T.lam (def f) [] ∙ v >>= k)
(1 + length s) ∼⟨ numbers-cong (DCT.associativity (⟦ t ⟧ _ _) _ _) ⟩
numbers ((⟦ t ⟧ ρ false >>=
[ pred , id ] T.lam (def f) [] ∙_) >>= k)
(1 + length s) ∼⟨⟩
numbers (⟦ call f t ⟧ ρ true >>= k) (1 + length s) ∎
where
lemma : {j : Size< i} → _
lemma = λ v′ →
2 + length s ∷′
VM.stack-sizes ⟨ c′ , val (comp-val v′) ∷ s , ρ′ ⟩ ≂⟨ consʳ-≂ (inj₁ (here ≤-refl)) (cast-≂ (c-ok v′)) ⟩∼
2 + length s ∷′
numbers (k v′) (2 + length s) ∼⟨ symmetric-∼ ∷∼∷′ ⟩
numbers (tell id (k v′)) (2 + length s) ∎
⟦⟧-correct (con b) ρ {tc} {c} {s} {k} _ c-ok =
VM.stack-sizes ⟨ con b ∷ c , s , comp-env ρ ⟩ ∼⟨ ∷∼∷′ ⟩≂
(length s ∷′
VM.stack-sizes ⟨ c , val (comp-val (T.con b)) ∷ s , comp-env ρ ⟩) ≂⟨ cons″-≂ (c-ok (T.con b)) ⟩∼
(length s ∷′ numbers (k (T.con b)) (1 + length s)) ∼⟨ symmetric-∼ ∷∼∷′ ⟩
numbers (⟦ con b ⟧ ρ tc >>= k) (length s) ∎
⟦⟧-correct (if t₁ t₂ t₃) ρ {tc} {c} {s} {k} s-ok c-ok =
VM.stack-sizes ⟨ comp false t₁
(bra (comp tc t₂ []) (comp tc t₃ []) ∷ c)
, s
, comp-env ρ
⟩ ≂⟨ (⟦⟧-correct t₁ _ unrestricted λ v₁ → ⟦if⟧-correct v₁ t₂ t₃ s-ok c-ok) ⟩∼
numbers (⟦ t₁ ⟧ ρ false >>= λ v₁ → ⟦if⟧ v₁ t₂ t₃ ρ tc >>= k)
(length s) ∼⟨ numbers-cong (DCT.associativity (⟦ t₁ ⟧ _ _) _ _) ⟩
numbers ((⟦ t₁ ⟧ ρ false >>= λ v₁ → ⟦if⟧ v₁ t₂ t₃ ρ tc) >>= k)
(length s) ∎
body-lemma :
∀ {i n n′} (t : Tm (1 + n)) ρ {ρ′ : C.Env n′} {c s v}
{k : T.Value → Delay-crash-trace (ℕ → ℕ) C.Value ∞} →
Cont-OK i ⟨ c , s , ρ′ ⟩ k →
[ i ] VM.stack-sizes ⟨ comp-body t
, ret c ρ′ ∷ s
, comp-val v ∷ comp-env ρ
⟩ ≂
numbers (⟦ t ⟧ (v ∷ ρ) true >>= tell pred ∘ return >>= k)
(1 + length s)
body-lemma t ρ {ρ′} {c} {s} {v} {k} c-ok =
VM.stack-sizes ⟨ comp-body t
, ret c ρ′ ∷ s
, comp-val v ∷ comp-env ρ
⟩ ≡⟨⟩≂
VM.stack-sizes ⟨ comp-body t
, ret c ρ′ ∷ s
, comp-env (v ∷ ρ)
⟩ ≂⟨ (⟦⟧-correct t (_ ∷ _) (any-OK (restricted lemma)) λ v′ →
VM.stack-sizes ⟨ ret ∷ []
, val (comp-val v′) ∷ ret c ρ′ ∷ s
, comp-env (v ∷ ρ)
⟩ ∼⟨ ∷∼∷′ ⟩≂
2 + length s ∷′
VM.stack-sizes ⟨ c , val (comp-val v′) ∷ s , ρ′ ⟩ ≂⟨ lemma v′ ⟩∼
numbers (tell pred (k v′)) (2 + length s) ∎) ⟩∼
numbers (⟦ t ⟧ (v ∷ ρ) true >>= tell pred ∘ k) (1 + length s) ∼⟨ numbers-cong (DCT.associativity (⟦ t ⟧ _ _) _ _) ⟩
numbers (⟦ t ⟧ (v ∷ ρ) true >>= tell pred ∘ return >>= k)
(1 + length s) ∎
where
lemma = λ v′ →
2 + length s ∷′
VM.stack-sizes ⟨ c , val (comp-val v′) ∷ s , ρ′ ⟩ ≂⟨ cons″-≂ (c-ok v′) ⟩∼
2 + length s ∷′ numbers (k v′) (1 + length s) ∼⟨ symmetric-∼ ∷∼∷′ ⟩
numbers (tell pred (k v′)) (2 + length s) ∎
∙-correct :
∀ {i n} v₁ v₂ {ρ : C.Env n} {c s}
{k : T.Value → Delay-crash-trace (ℕ → ℕ) C.Value ∞} →
Cont-OK i ⟨ c , s , ρ ⟩ k →
[ i ] VM.stack-sizes ⟨ app ∷ c
, val (comp-val v₂) ∷ val (comp-val v₁) ∷ s
, ρ
⟩ ≂
numbers ([ pred , pred ] v₁ ∙ v₂ >>= k) (2 + length s)
∙-correct (T.lam t₁ ρ₁) v₂ {ρ} {c} {s} {k} c-ok =
VM.stack-sizes
⟨ app ∷ c
, val (comp-val v₂) ∷ val (comp-val (T.lam t₁ ρ₁)) ∷ s
, ρ
⟩ ∼⟨ ∷∼∷′ ⟩≂
2 + length s ∷′
VM.stack-sizes ⟨ comp-body t₁
, ret c ρ ∷ s
, comp-val v₂ ∷ comp-env ρ₁
⟩ ≂⟨ cons′-≂ (_⇔_.from ≂′⇔≂″ λ { .force → body-lemma t₁ _ (castC c-ok) }) ⟩∼
2 + length s ∷′
numbers (⟦ t₁ ⟧ (v₂ ∷ ρ₁) true >>= tell pred ∘ return >>= k)
(1 + length s) ∼⟨ symmetric-∼ ∷∼∷′ ⟩
numbers ([ pred , pred ] T.lam t₁ ρ₁ ∙ v₂ >>= k) (2 + length s) ∎
∙-correct (T.con b) v₂ {ρ} {c} {s} {k} _ =
VM.stack-sizes ⟨ app ∷ c
, val (comp-val v₂) ∷ val (C.con b) ∷ s
, ρ
⟩ ∼⟨ ∷∼∷′ ⟩≂
2 + length s ∷′ [] ∼⟨ symmetric-∼ ∷∼∷′ ⟩≂
numbers ([ pred , pred ] T.con b ∙ v₂ >>= k) (2 + length s) □≂
⟦if⟧-correct :
∀ {i n} v₁ t₂ t₃ {ρ : T.Env n} {tc c s}
{k : T.Value → Delay-crash-trace (ℕ → ℕ) C.Value ∞} →
Stack-OK i k tc s →
Cont-OK i ⟨ c , s , comp-env ρ ⟩ k →
[ i ] VM.stack-sizes ⟨ bra (comp tc t₂ []) (comp tc t₃ []) ∷ c
, val (comp-val v₁) ∷ s
, comp-env ρ
⟩ ≂
numbers (⟦if⟧ v₁ t₂ t₃ ρ tc >>= k) (1 + length s)
⟦if⟧-correct (T.lam t₁ ρ₁) t₂ t₃ {ρ} {tc} {c} {s} {k} _ _ =
VM.stack-sizes ⟨ bra (comp tc t₂ []) (comp tc t₃ []) ∷ c
, val (comp-val (T.lam t₁ ρ₁)) ∷ s
, comp-env ρ
⟩ ∼⟨ ∷∼∷′ ⟩≂
1 + length s ∷′ [] ∼⟨ symmetric-∼ ∷∼∷′ ⟩≂
numbers (⟦if⟧ (T.lam t₁ ρ₁) t₂ t₃ ρ tc >>= k) (1 + length s) □≂
⟦if⟧-correct (T.con true) t₂ t₃ {ρ} {tc} {c} {s} {k} s-ok c-ok =
VM.stack-sizes ⟨ bra (comp tc t₂ []) (comp tc t₃ []) ∷ c
, val (comp-val (T.con true)) ∷ s
, comp-env ρ
⟩ ∼⟨ ∷∼∷′ ⟩≂
1 + length s ∷′
VM.stack-sizes ⟨ comp tc t₂ [] ++ c , s , comp-env ρ ⟩ ≡⟨ by (comp-++ _ t₂) ⟩≂
1 + length s ∷′
VM.stack-sizes ⟨ comp tc t₂ c , s , comp-env ρ ⟩ ≂⟨ cons″-≂ (⟦⟧-correct t₂ _ s-ok c-ok) ⟩∼
1 + length s ∷′ numbers (⟦ t₂ ⟧ ρ tc >>= k) (length s) ∼⟨ symmetric-∼ ∷∼∷′ ⟩
numbers (⟦if⟧ (T.con true) t₂ t₃ ρ tc >>= k) (1 + length s) ∎
⟦if⟧-correct (T.con false) t₂ t₃ {ρ} {tc} {c} {s} {k} s-ok c-ok =
VM.stack-sizes ⟨ bra (comp tc t₂ []) (comp tc t₃ []) ∷ c
, val (comp-val (T.con false)) ∷ s
, comp-env ρ
⟩ ∼⟨ ∷∼∷′ ⟩≂
1 + length s ∷′
VM.stack-sizes ⟨ comp tc t₃ [] ++ c , s , comp-env ρ ⟩ ≡⟨ by (comp-++ _ t₃) ⟩≂
1 + length s ∷′
VM.stack-sizes ⟨ comp tc t₃ c , s , comp-env ρ ⟩ ≂⟨ cons″-≂ (⟦⟧-correct t₃ _ s-ok c-ok) ⟩∼
1 + length s ∷′ numbers (⟦ t₃ ⟧ ρ tc >>= k) (length s) ∼⟨ symmetric-∼ ∷∼∷′ ⟩
numbers (⟦if⟧ (T.con false) t₂ t₃ ρ tc >>= k) (1 + length s) ∎
-- The virtual machine and the interpreter produce related lists of
-- stack sizes.
--
-- (However, the traces are not necessarily bisimilar, see
-- Lambda.Interpreter.Stack-sizes.Counterexample.stack-sizes-not-bisimilar.)
stack-sizes-related :
(t : Tm 0) →
[ ∞ ] VM.stack-sizes ⟨ comp₀ t , [] , [] ⟩ ≂ I.stack-sizes t
stack-sizes-related t =
VM.stack-sizes ⟨ comp false t [] , [] , [] ⟩ ≂⟨ ⟦⟧-correct t [] unrestricted (λ _ → cons″-≂ (_ □≂)) ⟩∼
numbers (comp-val ⟨$⟩ ⟦ t ⟧ [] false) 0 ∼⟨ scanl-cong (DCT.trace-⟨$⟩ _) ⟩
numbers (⟦ t ⟧ [] false) 0 ∼⟨⟩
I.stack-sizes t ∎
-- The maximum stack sizes match.
maximum-stack-sizes-match :
∀ (t : Tm 0) {i v} →
LUB (I.stack-sizes t) i →
LUB (VM.stack-sizes ⟨ comp₀ t , [] , [] ⟩) v →
Conat.[ ∞ ] i ∼ v
maximum-stack-sizes-match t {i} {v} i-lub =
LUB (VM.stack-sizes ⟨ comp₀ t , [] , [] ⟩) v ↝⟨ LUB-≂ (stack-sizes-related t) ⟩
LUB (I.stack-sizes t) v ↝⟨ lub-unique i-lub ⟩□
Conat.[ ∞ ] i ∼ v □
|
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{-# OPTIONS --without-K #-}
open import Equivalence
module PathStructure.Id.Ap {a b} {A : Set a} {B : Set b}
(f : A → B) (qi : qinv f) where
open import Functoriality
open import GroupoidStructure
open import Homotopy
open import PathOperations
open import Types
private
g = π₁ qi
α = π₁ (π₂ qi)
β = π₂ (π₂ qi)
split-path : {a a′ : A} → a ≡ a′ → f a ≡ f a′
split-path = ap f
merge-path : {a a′ : A} → f a ≡ f a′ → a ≡ a′
merge-path p = β _ ⁻¹ · ap g p · β _
F : (b b′ : B) (p : b ≡ b′) → Set _
F b b′ p =
ap f (β (g b) ⁻¹ · ap g (ap (f ∘ g) p) · β (g b′))
≡ ap (f ∘ g) p
F-lemma : ∀ b → F b b refl
F-lemma b = ap (ap f)
( ap (λ y → β (g b) ⁻¹ · y)
(id·p _)
· p⁻¹·p (β (g b))
)
ap-β-lem : ∀ a → ap (g ∘ f) (β a) ≡ β (g (f a))
ap-β-lem a
= p·id _ ⁻¹
· ap (λ y → ap (g ∘ f) (β a) · y)
(p·p⁻¹ (β a) ⁻¹)
· p·q·r (ap (g ∘ f) (β a)) (β a) (β a ⁻¹)
· ap (λ y → y · β a ⁻¹)
(naturality _ _ β (β a) ⁻¹)
· ap (λ y → (β (g (f a)) · y) · β a ⁻¹)
(ap-id _)
· p·q·r (β (g (f a))) (β a) (β a ⁻¹) ⁻¹
· ap (λ y → β (g (f a)) · y)
(p·p⁻¹ (β a))
· p·id _
add-right : ∀ {a} {A : Set a} {a₁ a₂ a₃ : B} (p q : a₁ ≡ a₂) (r : a₂ ≡ a₃) →
(p ≡ q) ≡ (p · r ≡ q · r)
add-right {a₁ = a₁} p q r = J
(λ a₂ _ r → (p q : a₁ ≡ a₂) → (p ≡ q) ≡ (p · r ≡ q · r))
(λ _ p q →
tr (λ x → (p ≡ q) ≡ (p · refl ≡ x)) (p·id q ⁻¹)
(tr (λ x → (p ≡ q) ≡ (x ≡ q)) (p·id p ⁻¹) refl))
_ _ r p q
add-left : ∀ {a} {A : Set a} {a₁ a₂ a₃ : B} (p q : a₂ ≡ a₃) (r : a₁ ≡ a₂) →
(p ≡ q) ≡ (r · p ≡ r · q)
add-left {a₃ = a₃} p q r = J
(λ _ a₂ r → (p q : a₂ ≡ a₃) → (p ≡ q) ≡ (r · p ≡ r · q))
(λ _ p q →
tr (λ x → (p ≡ q) ≡ (refl · p ≡ x)) (id·p q ⁻¹)
(tr (λ x → (p ≡ q) ≡ (x ≡ q)) (id·p p ⁻¹) refl))
_ _ r p q
F-tr : ∀ a a′ (q : f a ≡ f a′) →
F (f a) (f a′) q ≡ (ap f (β a ⁻¹ · ap g q · β a′) ≡ q)
F-tr a a′ q =
( add-left {A = A}
(ap f (β a ⁻¹ · ap g q · β a′)) q (α (f a))
· add-right {A = A}
(α (f a) · ap f (β a ⁻¹ · ap g q · β a′)) (α (f a) · q) (α (f a′) ⁻¹)
· ap (λ x → (α (f a) · ap f (β a ⁻¹ · ap g q · β a′)) · α (f a′) ⁻¹ ≡ x)
( ap (λ y → (α (f a) · y) · α (f a′) ⁻¹)
(ap-id _ ⁻¹)
· ap (λ y → y · α (f a′) ⁻¹)
(naturality _ _ α q)
· p·q·r (ap (f ∘ g) q) (α (f a′)) (α (f a′) ⁻¹) ⁻¹
· ap (λ y → ap (f ∘ g) q · y)
(p·p⁻¹ (α (f a′)))
· p·id _
)
· ap (λ x → x ≡ ap (f ∘ g) q)
( ap (λ y → (α (f a) · y) · α (f a′) ⁻¹)
(ap-id _ ⁻¹)
· ap (λ y → y · α (f a′) ⁻¹)
(naturality _ _ α (ap f (β a ⁻¹ · ap g q · β a′)))
· p·q·r (ap (f ∘ g) (ap f (β a ⁻¹ · ap g q · β a′)))
(α (f a′)) (α (f a′) ⁻¹) ⁻¹
· ap (λ y → ap (f ∘ g) (ap f (β a ⁻¹ · ap g q · β a′)) · y)
(p·p⁻¹ (α (f a′)))
· p·id _
· ap-∘ f g (ap f (β a ⁻¹ · ap g q · β a′))
· ap (ap f)
( ap-∘ g f (β a ⁻¹ · ap g q · β a′) ⁻¹
· ap· (g ∘ f) (β a ⁻¹) (ap g q · β a′)
· ap (λ y → ap (g ∘ f) (β a ⁻¹) · y)
(ap· (g ∘ f) (ap g q) (β a′))
· ap (λ y → y · ap (g ∘ f) (ap g q) · ap (g ∘ f) (β a′))
(ap⁻¹ (g ∘ f) (β a))
· ap (λ y → y ⁻¹ · ap (g ∘ f) (ap g q) · ap (g ∘ f) (β a′))
(ap-β-lem a)
· ap (λ y → β (g (f a)) ⁻¹ · ap (g ∘ f) (ap g q) · y)
(ap-β-lem a′)
· ap (λ y → β (g (f a)) ⁻¹ · y · β (g (f a′)))
( ap-∘ g f (ap g q)
· ap (ap g)
(ap-∘ f g q ⁻¹)
)
)
)
) ⁻¹
proof-F : {a a′ : A} → split-path ∘ merge-path {a} {a′} ∼ id
proof-F p = tr id (F-tr _ _ p) (J F F-lemma _ _ p)
-- Copied from HoTT book, theorem 2.11.1.
proof-direct : {a a′ : A} → split-path ∘ merge-path {a} {a′} ∼ id
proof-direct {a} {a′} p
= ap (ap f)
(p·q·r (β a ⁻¹) (ap g p) (β a′))
· id·p _ ⁻¹
· ap (λ y → y · ap f ((β a ⁻¹ · ap g p) · β a′))
(p⁻¹·p (α (f a)) ⁻¹)
· p·q·r (α (f a) ⁻¹) (α (f a)) (ap f ((β a ⁻¹ · ap g p) · β a′)) ⁻¹
· ap (λ y → α (f a) ⁻¹ · α (f a) · y)
(ap-id _ ⁻¹)
· ap (λ y → α (f a) ⁻¹ · y)
(naturality _ _ α (ap f ((β a ⁻¹ · ap g p) · β a′)))
· ap (λ y → α (f a) ⁻¹ · y · α (f a′))
( ap-∘ f g (ap f ((β a ⁻¹ · ap g p) · β a′))
· ap (ap f)
( ap-∘ g f ((β a ⁻¹ · ap g p) · β a′) ⁻¹
· p·id _ ⁻¹
· ap (λ y → ap (g ∘ f) ((β a ⁻¹ · ap g p) · β a′) · y)
(p·p⁻¹ (β a′) ⁻¹)
· p·q·r (ap (g ∘ f) ((β a ⁻¹ · ap g p) · β a′)) (β a′) (β a′ ⁻¹)
· ap (λ y → y · β a′ ⁻¹)
( naturality _ _ β ((β a ⁻¹ · ap g p) · β a′) ⁻¹
· ap (λ y → β a · y)
(ap-id _)
)
· ap (λ y → y · β a′ ⁻¹)
(p·q·r (β a) (β a ⁻¹ · ap g p) (β a′))
· p·q·r (β a · β a ⁻¹ · ap g p) (β a′) (β a′ ⁻¹) ⁻¹
· ap (λ y → (β a · β a ⁻¹ · ap g p) · y)
(p·p⁻¹ (β a′))
· p·id _
· p·q·r (β a) (β a ⁻¹) (ap g p)
· ap (λ y → y · ap g p)
(p·p⁻¹ (β a))
· id·p _
)
· ap-∘ f g p ⁻¹
)
· ap (λ y → α (f a) ⁻¹ · y)
(naturality _ _ α p ⁻¹)
· p·q·r (α (f a) ⁻¹) (α (f a)) (ap id p)
· ap (λ y → y · ap id p)
(p⁻¹·p (α (f a)))
· ap-id _
split-merge-eq : {a a′ : A} → (a ≡ a′) ≃ (f a ≡ f a′)
split-merge-eq
= split-path
, (merge-path , proof-F)
, (merge-path , J
(λ _ _ p → merge-path (split-path p) ≡ p)
(λ _ → p⁻¹·p (β _))
_ _)
|
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{-# OPTIONS --without-K #-}
open import Types
open import Paths
open import HLevel
open import Equivalences
open import Univalence
module Funext {i} {A : Set i} where
-- Naive non dependent function extensionality
module FunextNonDep {j} {B : Set j} {f g : A → B} (h : (x : A) → f x ≡ g x)
where
private
equiv-comp : {B C : Set j} (f : B ≃ C)
→ is-equiv (λ (g : A → B) → (λ x → f ☆ (g x)))
equiv-comp f =
equiv-induction (λ {B} f → is-equiv (λ (g : A → B) → (λ x → f ☆ (g x))))
(λ A' y → id-is-equiv (A → A') y) f
free-path-space-B : Set j
free-path-space-B = Σ B (λ x → Σ B (λ y → x ≡ y))
d : A → free-path-space-B
d x = (f x , (f x , refl))
e : A → free-path-space-B
e x = (f x , (g x , h x))
abstract
π₁-is-equiv : is-equiv (λ (y : free-path-space-B) → π₁ y)
π₁-is-equiv =
iso-is-eq π₁ (λ z → (z , (z , refl))) (λ _ → refl)
(λ x' → Σ-eq refl
(Σ-eq (π₂ (π₂ x'))
(trans-cst≡id (π₂ (π₂ x')) refl)))
comp-π₁-is-equiv : is-equiv (λ (f : A → free-path-space-B)
→ (λ x → π₁ (f x)))
comp-π₁-is-equiv = equiv-comp ((λ (y : free-path-space-B) → π₁ y),
π₁-is-equiv)
d≡e : d ≡ e
d≡e = equiv-is-inj (_ , comp-π₁-is-equiv) _ _ refl
funext-nondep : f ≡ g
funext-nondep = ap (λ f' x → π₁ (π₂ (f' x))) d≡e
open FunextNonDep
-- Weak function extensionality (a product of contractible types is
-- contractible)
module WeakFunext {j} {P : A → Set j} (e : (x : A) → is-contr (P x)) where
P-is-unit : P ≡ (λ x → unit)
P-is-unit = funext-nondep (λ x → eq-to-path (contr-equiv-unit (e x)))
abstract
weak-funext : is-contr (Π A P)
weak-funext = transport (λ Q → is-contr (Π A Q)) (! P-is-unit)
((λ x → tt) , (λ y → funext-nondep (λ x → refl)))
open WeakFunext
-- Naive dependent function extensionality
module FunextDep {j} {P : A → Set j} {f g : Π A P} (h : (x : A) → f x ≡ g x)
where
Q : A → Set j
Q x = Σ (P x) (λ y → y ≡ f x)
abstract
Q-is-contr : (x : A) → is-contr (Q x)
Q-is-contr x = pathto-is-contr (f x)
ΠAQ-is-contr : is-contr (Π A Q)
ΠAQ-is-contr = weak-funext Q-is-contr
Q-f : Π A Q
Q-f x = (f x , refl)
Q-g : Π A Q
Q-g x = (g x , ! (h x))
abstract
Q-f≡Q-g : Q-f ≡ Q-g
Q-f≡Q-g = contr-has-all-paths ΠAQ-is-contr Q-f Q-g
funext-p : f ≡ g
funext-p = ap (λ u x → π₁ (u x)) Q-f≡Q-g
open FunextDep
happly : ∀ {j} {P : A → Set j} {f g : Π A P} (p : f ≡ g) → ((x : A) → f x ≡ g x)
happly p x = ap (λ u → u x) p
-- Strong function extensionality
module StrongFunextDep {j} {P : A → Set j} where
abstract
funext-refl : (f : Π A P)
→ funext-p (λ x → refl {a = f x}) ≡ refl
funext-refl f = ap (ap (λ u x → π₁ (u x)))
(contr-has-all-paths (≡-is-truncated _
(ΠAQ-is-contr (λ x → refl)))
(Q-f≡Q-g (λ x → refl)) refl)
abstract
funext-happly-p : {f g : Π A P} (p : f ≡ g)
→ funext-p (happly p) ≡ p
funext-happly-p {f} refl = funext-refl f
abstract
happly-path : {f : Π A P} {u v : (x : A) → Q (λ x → refl {a = f x}) x}
(p : u ≡ v) (x : A)
→ happly (ap (λ u x → π₁ (u x)) p) x ≡ π₂ (u x) ∘ ! (π₂ (v x))
happly-path {u = u} refl x = ! (opposite-right-inverse (π₂ (u x)))
abstract
happly-funext-p : {f g : Π A P} (h : (x : A) → f x ≡ g x)
→ happly (funext-p h) ≡ h
happly-funext-p h = funext-p (λ x → happly-path (Q-f≡Q-g h) x
∘ opposite-opposite (h x))
happly-is-equiv : {f g : Π A P} → is-equiv (happly {f = f} {g = g})
happly-is-equiv = iso-is-eq _ funext-p happly-funext-p funext-happly-p
funext-is-equiv-p : {f g : Π A P}
→ is-equiv (funext-p {f = f} {g = g})
funext-is-equiv-p = iso-is-eq _ happly funext-happly-p happly-funext-p
open StrongFunextDep
-- We only export the following
module _ {j} {P : A → Set j} {f g : Π A P} where
abstract
funext : (h : (x : A) → f x ≡ g x) → f ≡ g
funext = FunextDep.funext-p
funext-happly : (p : f ≡ g) → funext (happly p) ≡ p
funext-happly p = funext-happly-p p
happly-funext : (h : (x : A) → f x ≡ g x)
→ happly (funext h) ≡ h
happly-funext h = happly-funext-p h
funext-is-equiv : is-equiv funext
funext-is-equiv = StrongFunextDep.funext-is-equiv-p
funext-equiv : ((x : A) → f x ≡ g x) ≃ (f ≡ g)
funext-equiv = (funext , funext-is-equiv)
happly-equiv : (f ≡ g) ≃ ((x : A) → f x ≡ g x)
happly-equiv = (happly , happly-is-equiv)
|
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{-# OPTIONS --without-K #-}
module sets.core where
open import equality.core
module _ {i}{A : Set i}
(_<_ : A → A → Set i) where
data Ordering (x y : A) : Set i where
lt : x < y → Ordering x y
eq : x ≡ y → Ordering x y
gt : y < x → Ordering x y
|
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{-# OPTIONS --cubical --no-import-sorts --postfix-projections --safe #-}
module Cubical.Categories.Presheaf where
open import Cubical.Categories.Presheaf.Base public
open import Cubical.Categories.Presheaf.Properties public
open import Cubical.Categories.Presheaf.KanExtension public
|
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open import Agda.Builtin.Reflection
open import Agda.Builtin.Unit
macro
@0 m : @0 Set → Term → TC ⊤
m A B =
bindTC (quoteTC A) λ A →
unify A B
F : @0 Set → Set
F A = m A
|
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-- Andreas, 2016-10-03, issue #2233
-- Positivity check should return the same result when change
-- all involved definitions from non-abstract to abstract.
abstract
data Functor : Set where
Id : Functor
_·_ : Functor → Set → Set
Id · A = A
data ν (F : Functor) : Set where
inn : F · ν F → ν F
-- Should positivity check ok.
|
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open import WarningOnImport.Impo
C = A
|
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|
module Data.List.Properties.Extra {a}{A : Set a} where
open import Data.Nat
open import Data.Fin hiding (_<_)
open import Data.List
open import Data.Product hiding (map)
open import Data.Fin using (fromℕ≤; zero; suc)
open import Data.List.Relation.Unary.All hiding (map; lookup; _[_]≔_)
open import Data.List.Relation.Unary.Any hiding (map; lookup)
open import Data.List.Membership.Propositional
open import Data.List.Relation.Binary.Pointwise hiding (refl; map)
open import Relation.Binary.PropositionalEquality
∈-∷ʳ : ∀ (l : List A)(x : A) → x ∈ (l ∷ʳ x)
∈-∷ʳ [] x = here refl
∈-∷ʳ (x ∷ l) y = there (∈-∷ʳ l y)
infixl 10 _[_]≔_
_[_]≔_ : (l : List A) → Fin (length l) → A → List A
[] [ () ]≔ x
(x ∷ xs) [ zero ]≔ x' = x' ∷ xs
(x ∷ xs) [ suc i ]≔ y = x ∷ xs [ i ]≔ y
infixl 10 _[_]≔'_
_[_]≔'_ : ∀ {x} → (l : List A) → x ∈ l → A → List A
[] [ () ]≔' y
(x ∷ l) [ here px ]≔' y = y ∷ l
(x ∷ l) [ there px ]≔' y = x ∷ (l [ px ]≔' y)
≔'-[]= : ∀ {x}{l : List A} (p : x ∈ l) → ∀ {y} → y ∈ (l [ p ]≔' y)
≔'-[]= (here px) = here refl
≔'-[]= (there p) = there (≔'-[]= p)
drop-prefix : ∀ (l m : List A) → drop (length l) (l ++ m) ≡ m
drop-prefix [] m = refl
drop-prefix (x ∷ l) m = drop-prefix l m
|
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{-# OPTIONS --without-K --rewriting #-}
open import lib.Basics
open import lib.types.Cofiber
open import lib.types.Sigma
open import lib.types.Wedge
module lib.types.Smash {i j} (X : Ptd i) (Y : Ptd j) where
module ∨In× = WedgeRec {X = X} {Y = Y}
(λ x → (x , pt Y)) (λ y → (pt X , y)) idp
∨-in-× = ∨In×.f
∨-⊙in-× : X ⊙∨ Y ⊙→ X ⊙× Y
∨-⊙in-× = (∨In×.f , idp)
⊙Smash : Ptd (lmax i j)
⊙Smash = ⊙Cofiber ∨-⊙in-×
Smash = de⊙ ⊙Smash
_∧_ = Smash
_⊙∧_ = ⊙Smash
|
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{-# OPTIONS --universe-polymorphism #-}
module Categories.GlobularSet where
open import Level
open import Data.Unit
open import Categories.Support.PropositionalEquality
open import Categories.Category
open import Categories.Globe
open import Categories.Functor
open import Categories.Presheaf
open import Categories.Agda
record GlobularSet (o : Level) : Set (suc o) where
field
presheaf : Presheaf Globe (Sets o)
Trivial : GlobularSet zero
Trivial = record
{ presheaf = record
{ F₀ = λ x → ⊤
; F₁ = λ x y → y
; identity = ≣-refl
; homomorphism = ≣-refl
; F-resp-≡ = λ x → ≣-refl
}
}
zeroCell : ∀ {ℓ} → GlobularSet ℓ → Set ℓ
zeroCell G = Functor.F₀ (GlobularSet.presheaf G) 0
{-
lift : ∀ {ℓ} → (G : GlobularSet ℓ) → (A : Set ℓ) → (f g : zeroCell G → A) → GlobularSet ℓ
lift {ℓ} G A f g = record
{ presheaf = record
{ F₀ = F₀′
; F₁ = F₁′
; identity = ≣-refl
; homomorphism = {!!}
; F-resp-≡ = F-resp-≡′
}
}
where
open GlobularSet G
open Functor presheaf
F₀′ : ℕ → Set ℓ
F₀′ zero = A
F₀′ (suc n) = F₀ n
F₁′ : ∀ {A B} → GlobeHom B A → F₀′ A → F₀′ B
F₁′ I a = a
F₁′ {suc zero} (σ Z<Sn) a = f a
F₁′ {suc (suc m)} (σ Z<Sn) a = F₁′ {suc m} (σ Z<Sn) {!!}
F₁′ (σ (raise< n<m)) a = {!!}
F₁′ (τ n<m) a = {!!}
F-resp-≡′ : ∀ {A B} {F G : GlobeHom B A} → F ≣ G → {x : F₀′ A} → F₁′ F x ≣ F₁′ G x
F-resp-≡′ ≣-refl = ≣-refl
{-
objs : ∀ {ℓ} → (A : Set ℓ) → ℕ → Set ℓ
objs A zero = A
objs A (suc n) = Σ (objs A n) (λ x → Σ (objs A n) (λ y → x ≣ y))
base : ∀ {n} {ℓ} {A : Set ℓ} → objs A n → A
base {zero} o = o
base {suc n} (s , t , s≡t) = base {n} s
homs : ∀ {A B} {ℓ} (Q : Set ℓ) → GlobeHom B A → objs Q A → objs Q B
homs Q I o = o
homs {suc n} Q (σ Z<Sn) (s , t , s≡t) = base {n} s
homs Q (σ (raise< n<m)) (s , .s , ≣-refl) = homs Q (σ n<m) s , homs Q (σ n<m) s , ≣-refl
homs {suc n} Q (τ Z<Sn) (s , t , s≡t) = base {n} t
homs Q (τ (raise< n<m)) (s , .s , ≣-refl) = homs Q (τ n<m) s , homs Q (τ n<m) s , ≣-refl
Equality : ∀ {ℓ} (A : Set ℓ) → GlobularSet ℓ
Equality A = record
{ presheaf = record
{ F₀ = objs A
; F₁ = homs A
; identity = ≣-refl
; homomorphism = λ {_} {_} {_} {f} {g} → homomorphism {f = f} {g}
; F-resp-≡ = F-resp-≡
}
}
where
homomorphism : {X Y Z : ℕ} {f : GlobeHom Y X} {g : GlobeHom Z Y} {x : objs A X} → homs A (f ⊚ g) x ≣ homs A g (homs A f x)
homomorphism {f = I} = ≣-refl
homomorphism {f = σ n<m} {I} = ≣-refl
homomorphism {f = σ n<m} {σ n<m'} = {!!}
homomorphism {f = σ n<m} {τ n<m'} = {!!}
homomorphism {f = τ n<m} {I} = ≣-refl
homomorphism {f = τ n<m} {σ n<m'} = {!!}
homomorphism {f = τ n<m} {τ n<m'} = {!!}
F-resp-≡ : {A' B : ℕ} {F G : GlobeHom B A'} → F ≣ G → {x : objs A A'} → _
F-resp-≡ ≣-refl = ≣-refl
-}
-}
|
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{-# OPTIONS --warning ShadowingInTelescope #-}
postulate
_ : (A A : Set) → Set
|
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module BasicIS4.Metatheory.Gentzen-BasicKripkeAlechina where
open import BasicIS4.Syntax.Gentzen public
open import BasicIS4.Semantics.BasicKripkeAlechina public
-- Soundness with respect to all models, or evaluation.
mutual
eval : ∀ {A Γ} → Γ ⊢ A → Γ ⊨ A
eval (var i) γ = lookup i γ
eval (lam t) γ = λ ξ a → eval t (mono⊩⋆ ξ γ , a)
eval (app {A} {B} t u) γ = _⟪$⟫_ {A} {B} (eval t γ) (eval u γ)
eval (multibox ts u) γ = λ ξ ζ → eval u (thing ts γ ξ ζ)
eval (down {A} t) γ = ⟪↓⟫ {A} (eval t γ)
eval (pair t u) γ = eval t γ , eval u γ
eval (fst t) γ = π₁ (eval t γ)
eval (snd t) γ = π₂ (eval t γ)
eval unit γ = ∙
-- TODO: What is this?
thing : ∀ {{_ : Model}} {Δ Γ} {w : World}
→ Γ ⊢⋆ □⋆ Δ → w ⊩⋆ Γ → ∀ {w′} → w ≤ w′ → ∀ {v′} → w′ R v′ → v′ ⊩⋆ □⋆ Δ
thing {∅} ∙ γ ξ ζ = ∙
thing {Δ , B} (ts , t) γ ξ ζ = thing ts γ ξ ζ , λ ξ′ ζ′ →
let _ , (ξ″ , ζ″) = R⨾≤→≤⨾R (_ , (ζ , ξ′))
in eval t γ (trans≤ ξ ξ″) (transR ζ″ ζ′)
eval⋆ : ∀ {Ξ Γ} → Γ ⊢⋆ Ξ → Γ ⊨⋆ Ξ
eval⋆ {∅} ∙ γ = ∙
eval⋆ {Ξ , A} (ts , t) γ = eval⋆ ts γ , eval t γ
-- TODO: Correctness of evaluation with respect to conversion.
|
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------------------------------------------------------------------------
-- Some defined parsers
------------------------------------------------------------------------
-- Note that the fixpoint combinator ensures that _⋆ can be defined
-- without any need for library grammars (c.f.
-- RecursiveDescent.Inductive.Lib). However, use of the fixpoint
-- combinator can lead to code which is hard to understand, and should
-- be kept at a minimum.
-- This module also provides an example of a parser for which the
-- Index cannot be inferred.
module RecursiveDescent.InductiveWithFix.Lib where
open import RecursiveDescent.Index
open import RecursiveDescent.InductiveWithFix
open import Utilities
open import Data.Nat hiding (_≟_)
import Data.Vec as V; open V using (Vec)
import Data.Vec1 as V₁; open V₁ using (Vec₁)
import Data.List as L ; open L using (List)
open import Relation.Nullary
open import Data.Product.Record
open import Data.Product renaming (_,_ to pair)
open import Data.Bool hiding (_≟_)
open import Data.Function
open import Data.Maybe
open import Data.Unit hiding (_≟_)
------------------------------------------------------------------------
-- Applicative functor parsers
-- We could get these for free with better library support.
infixl 50 _⊛_ _<⊛_ _⊛>_ _<$>_ _<$_
_⊛_ : forall {tok nt i₁ i₂ r₁ r₂} ->
Parser tok nt i₁ (r₁ -> r₂) ->
Parser tok nt i₂ r₁ ->
Parser tok nt _ r₂
p₁ ⊛ p₂ = p₁ >>= \f -> p₂ >>= \x -> return (f x)
_<$>_ : forall {tok nt i r₁ r₂} ->
(r₁ -> r₂) ->
Parser tok nt i r₁ ->
Parser tok nt _ r₂
f <$> x = return f ⊛ x
_<⊛_ : forall {tok nt i₁ i₂ r₁ r₂} ->
Parser tok nt i₁ r₁ ->
Parser tok nt i₂ r₂ ->
Parser tok nt _ r₁
x <⊛ y = const <$> x ⊛ y
_⊛>_ : forall {tok nt i₁ i₂ r₁ r₂} ->
Parser tok nt i₁ r₁ ->
Parser tok nt i₂ r₂ ->
Parser tok nt _ r₂
x ⊛> y = flip const <$> x ⊛ y
_<$_ : forall {tok nt i r₁ r₂} ->
r₁ ->
Parser tok nt i r₂ ->
Parser tok nt _ r₁
x <$ y = const x <$> y
------------------------------------------------------------------------
-- Parsing sequences
infix 55 _⋆ _+
_⋆ : forall {tok nt c r} ->
Parser tok nt (false , c) r ->
Parser tok nt _ (List r)
p ⋆ = fix (return L.[] ∣ L._∷_ <$> lift p ⊛ ! fresh)
_+ : forall {tok nt c r} ->
Parser tok nt (false , c) r ->
Parser tok nt _ (List r)
p + = L._∷_ <$> p ⊛ p ⋆
chain₁ : forall {tok nt c₁ i₂ r}
-> Assoc
-> Parser tok nt (false , c₁) r
-> Parser tok nt i₂ (r -> r -> r)
-> Parser tok nt _ r
chain₁ a p op = chain₁-combine a <$> (pair <$> p ⊛ op) ⋆ ⊛ p
chain : forall {tok nt c₁ i₂ r}
-> Assoc
-> Parser tok nt (false , c₁) r
-> Parser tok nt i₂ (r -> r -> r)
-> r
-> Parser tok nt _ r
chain a p op x = return x ∣ chain₁ a p op
-- Note that the resulting index here cannot be inferred...
private
exactly'-corners : Corners -> ℕ -> Corners
exactly'-corners c zero = _
exactly'-corners c (suc n) = _
exactly' : forall {tok nt c r} n ->
Parser tok nt (false , c) r ->
Parser tok nt (false , exactly'-corners c n)
(Vec r (suc n))
exactly' zero p = V.[_] <$> p
exactly' (suc n) p = V._∷_ <$> p ⊛ exactly' n p
-- ...so we might as well generalise the function a little.
-- exactly n p parses n occurrences of p.
exactly-index : Index -> ℕ -> Index
exactly-index i zero = _
exactly-index i (suc n) = _
exactly : forall {tok nt i r} n ->
Parser tok nt i r ->
Parser tok nt (exactly-index i n) (Vec r n)
exactly zero p = return V.[]
exactly (suc n) p = V._∷_ <$> p ⊛ exactly n p
-- A function with a similar type:
sequence : forall {tok nt i r n} ->
Vec₁ (Parser tok nt i r) n ->
Parser tok nt (exactly-index i n) (Vec r n)
sequence V₁.[] = return V.[]
sequence (V₁._∷_ p ps) = V._∷_ <$> p ⊛ sequence ps
------------------------------------------------------------------------
-- sat and friends
sat : forall {tok nt r} ->
(tok -> Maybe r) -> Parser tok nt (0I ·I 1I) r
sat {tok} {nt} {r} p = symbol !>>= \c -> ok (p c)
where
okIndex : Maybe r -> Index
okIndex nothing = _
okIndex (just _) = _
ok : (x : Maybe r) -> Parser tok nt (okIndex x) r
ok nothing = fail
ok (just x) = return x
sat' : forall {tok nt} -> (tok -> Bool) -> Parser tok nt _ ⊤
sat' p = sat (boolToMaybe ∘ p)
any : forall {tok nt} -> Parser tok nt _ tok
any = sat just
------------------------------------------------------------------------
-- Some parsers which require a decidable token equality
open import Relation.Binary
module Token (D : DecSetoid) where
open DecSetoid D using (_≟_) renaming (carrier to tok)
open import Data.Vec1
-- Parsing a given token (or, really, a given equivalence class of
-- tokens).
sym : forall {nt} -> tok -> Parser tok nt _ tok
sym c = sat p
where
p : tok -> Maybe tok
p x with c ≟ x
... | yes _ = just x
... | no _ = nothing
-- Parsing a sequence of tokens.
string : forall {nt n} -> Vec tok n -> Parser tok nt _ (Vec tok n)
string cs = sequence (map₀₁ sym cs)
------------------------------------------------------------------------
-- Character parsers
import Data.Char as C
open C using (Char; _==_)
open Token C.decSetoid
digit : forall {nt} -> Parser Char nt _ ℕ
digit = 0 <$ sym '0'
∣ 1 <$ sym '1'
∣ 2 <$ sym '2'
∣ 3 <$ sym '3'
∣ 4 <$ sym '4'
∣ 5 <$ sym '5'
∣ 6 <$ sym '6'
∣ 7 <$ sym '7'
∣ 8 <$ sym '8'
∣ 9 <$ sym '9'
number : forall {nt} -> Parser Char nt _ ℕ
number = toNum <$> digit +
where toNum = L.foldr (\n x -> 10 * x + n) 0 ∘ L.reverse
-- whitespace recognises an incomplete but useful list of whitespace
-- characters.
whitespace : forall {nt} -> Parser Char nt _ ⊤
whitespace = sat' isSpace
where
isSpace = \c -> (c == ' ') ∨ (c == '\t') ∨ (c == '\n') ∨ (c == '\r')
|
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module RecordConstructors (Parameter : Set) where
-- Note that the fixity declaration has to be given outside of the
-- record definition.
infix 6 _⟨_⟩_
record R (X : Set) (Y : Set) : Set₁ where
constructor _⟨_⟩_
field
{A} : Set
f : A → X
{B} D {E} : Set
g : B → Y → E
postulate A : Set
r : R A A
r = f ⟨ A ⟩ λ (_ : A) → f
where
f : A → A
f x = x
data _≡_ {A : Set₁} (x : A) : A → Set where
refl : x ≡ x
lemma : r ≡ record {}
lemma = refl
-- Record constructors can be overloaded.
record R′ : Set₁ where
constructor _⟨_⟩_
field
T₁ T₂ T₃ : Set
data D : Set where
_⟨_⟩_ : D
r′ : R′
r′ = A ⟨ A ⟩ A
d : D
d = _⟨_⟩_
|
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module Relator.Equals.Proofs.Equivalence where
open import Functional
import Lvl
open import Lang.Instance
open import Logic.Propositional
open import Logic
open import Relator.Equals
open import Structure.Setoid using (Equiv) renaming (_≡_ to _≡ₛ_)
open import Structure.Function
open import Structure.Operator
open import Structure.Relator
open import Structure.Relator.Equivalence
import Structure.Relator.Names as Names
open import Structure.Relator.Properties
open import Structure.Type.Identity
open import Structure.Type.Identity.Proofs
open import Type
-- TODO: Consider using Structure.Type.Identity instead of these proofs
module One {ℓ} {T : Type{ℓ}} where
instance
[≡]-reflexivity : Reflexivity (_≡_ {T = T})
[≡]-reflexivity = intro [≡]-intro
instance
[≡]-identity-eliminator : ∀{ℓₚ} → IdentityEliminator{ℓₚ = ℓₚ}(_≡_ {T = T})
IdentityEliminator.elim [≡]-identity-eliminator _ proof {x = x}{y = .x} [≡]-intro = proof{x}
instance
[≡]-symmetry : Symmetry (_≡_ {T = T})
[≡]-symmetry = identity-eliminator-to-symmetry
instance
[≡]-transitivity : Transitivity (_≡_ {T = T})
[≡]-transitivity = identity-eliminator-to-transitivity
instance
[≡]-equivalence : Equivalence (_≡_ {T = T})
[≡]-equivalence = intro
[≡]-to-equivalence : ∀{ℓₗ}{x y : T} → (x ≡ y) → ⦃ equiv-T : Equiv{ℓₗ}(T) ⦄ → (_≡ₛ_ ⦃ equiv-T ⦄ x y)
[≡]-to-equivalence([≡]-intro) = reflexivity(_≡ₛ_)
[≡]-equiv : Equiv{ℓ}(T)
Equiv._≡_ [≡]-equiv = _≡_
Equiv.equivalence [≡]-equiv = [≡]-equivalence
-- Equality is a subrelation to every reflexive relation.
-- One interpretation of this is that identity is the smallest reflexive relation when a relation is interpreted as a set of tuples and size is the cardinality of the set.
instance
[≡]-sub-of-reflexive : ∀{ℓₗ}{_▫_ : T → T → Stmt{ℓₗ}} → ⦃ _ : Reflexivity(_▫_) ⦄ → ((_≡_) ⊆₂ (_▫_))
_⊆₂_.proof [≡]-sub-of-reflexive [≡]-intro = reflexivity(_)
-- Replaces occurrences of an element in a function
[≡]-substitutionᵣ : ∀{ℓ₂}{x y} → (x ≡ y) → ∀{f : T → Type{ℓ₂}} → f(x) → f(y)
[≡]-substitutionᵣ [≡]-intro p = p -- TODO: Express in terms of sub-of-reflexive which is transport so that functors are automatically something
-- Replaces occurrences of an element in a function
[≡]-substitutionₗ : ∀{ℓ₂}{x y} → (x ≡ y) → ∀{f : T → Type{ℓ₂}} → f(y) → f(x)
[≡]-substitutionₗ [≡]-intro p = p
[≡]-substitution : ∀{ℓ₂}{x y} → (x ≡ y) → ∀{f : T → Type{ℓ₂}} → (f(x) ↔ f(y))
[≡]-substitution eq = [↔]-intro ([≡]-substitutionₗ eq) ([≡]-substitutionᵣ eq)
[≡]-unary-relator : ∀{ℓ₂}{P : T → Stmt{ℓ₂}} → UnaryRelator ⦃ [≡]-equiv ⦄ (P)
UnaryRelator.substitution([≡]-unary-relator {P = P}) xy = [≡]-substitutionᵣ xy {P}
[≡]-binary-relator : ∀{ℓ₂}{P : T → T → Stmt{ℓ₂}} → BinaryRelator ⦃ [≡]-equiv ⦄ ⦃ [≡]-equiv ⦄ (P)
BinaryRelator.substitution [≡]-binary-relator [≡]-intro [≡]-intro = id
open One public
module Two {ℓ₁}{A : Type{ℓ₁}} {ℓ₂}{B : Type{ℓ₂}} where
-- Applies a function to each side of the equality (TODO: Remove this and use Function everywhere instead)
[≡]-with : (f : A → B) → ∀{x y : A} → (x ≡ y) → (f(x) ≡ f(y))
[≡]-with f [≡]-intro = [≡]-intro
[≡]-function : ∀{f} → Function ⦃ [≡]-equiv ⦄ ⦃ [≡]-equiv ⦄ (f)
Function.congruence([≡]-function {f}) eq = [≡]-with(f) eq
[≡]-to-function : ∀{ℓₗ} → ⦃ equiv-B : Equiv{ℓₗ}(B) ⦄ → ∀{f : A → B} → Function ⦃ [≡]-equiv ⦄ ⦃ equiv-B ⦄ (f)
Function.congruence ([≡]-to-function) [≡]-intro = reflexivity(_≡ₛ_)
open Two public
module Three {ℓ₁}{A : Type{ℓ₁}} {ℓ₂}{B : Type{ℓ₂}} {ℓ₃}{C : Type{ℓ₃}} where
-- Applies an operation to each side of the equality (TODO: Make this an instance of BinaryOperator instead)
[≡]-with-op : (_▫_ : A → B → C) → {a₁ a₂ : A}{b₁ b₂ : B} → (a₁ ≡ a₂) → (b₁ ≡ b₂) → ((a₁ ▫ b₁) ≡ (a₂ ▫ b₂))
[≡]-with-op (_▫_) [≡]-intro [≡]-intro = [≡]-intro
-- [≡]-with-op-[_] (_▫_) {a₁}{a₂} {b₁}{b₂} (a₁≡a₂) (b₁≡b₂) =
-- [≡]-elimᵣ (b₁≡b₂) {\x → (a₁ ▫ b₁) ≡ (a₂ ▫ x)} ([≡]-with(x ↦ (x ▫ b₁)) (a₁≡a₂))
[≡]-binary-operator : ∀{_▫_} → BinaryOperator ⦃ [≡]-equiv ⦄ ⦃ [≡]-equiv ⦄ ⦃ [≡]-equiv ⦄ (_▫_)
BinaryOperator.congruence([≡]-binary-operator {_▫_}) aeq beq = [≡]-with-op(_▫_) aeq beq
open Three public
module Four {ℓ₁}{A : Type{ℓ₁}} {ℓ₂}{B : Type{ℓ₂}} {ℓ₃}{C : Type{ℓ₃}} {ℓ₄}{D : Type{ℓ₄}} where
[≡]-trinary-operator : ∀{_▫_▫_ : A → B → C → D} → TrinaryOperator ⦃ [≡]-equiv ⦄ ⦃ [≡]-equiv ⦄ ⦃ [≡]-equiv ⦄ ⦃ [≡]-equiv ⦄ (_▫_▫_)
TrinaryOperator.congruence([≡]-trinary-operator {_▫_▫_}) [≡]-intro [≡]-intro [≡]-intro = [≡]-intro
open Four public
|
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{-# OPTIONS -v tc.lhs:50 #-}
{-# OPTIONS -v tc.coverage:50 #-}
open import Agda.Builtin.String
test : String → String
test x@"foo" = "bar"
test x = x
|
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{-# OPTIONS --universe-polymorphism #-}
module Issue204 where
open import Issue204.Dependency
postulate
ℓ : Level
r : R ℓ
d : D ℓ
open R r
open M d
|
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-- Andreas, 2014-09-09
mutual
{-# NON_TERMINATING #-}
f : Set
f = g
{-# TERMINATING #-}
g : Set
g = f
-- Expected error:
-- In a mutual block, either all functions must have the same (or no)
-- termination checking pragma.
|
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import Either
open import Boolean
module Logic where
id : ∀ { A : Set } → A → A
id x = x
Rel : Set → Set₁
Rel X = X → X → Set
Decidable : ∀ { X } → Rel X → Set
Decidable R = ∀ x y → Either (R x y) (¬ (R x y))
where open Either
modusPonens : { P Q : Set } → ( P → Q ) → P → Q
modusPonens = id
|
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module Data.Tuple where
import Lvl
open import Type
open import Syntax.Function
infixr 200 _⨯_ _,_
private variable ℓ ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Lvl.Level
private variable A B C A₁ A₂ B₁ B₂ : Type{ℓ}
-- Definition of a 2-tuple
record _⨯_ (A : Type{ℓ₁}) (B : Type{ℓ₂}) : Type{ℓ₁ Lvl.⊔ ℓ₂} where
constructor _,_
field
left : A
right : B
open _⨯_ public
elim : ∀{P : (A ⨯ B) → Type{ℓ}} → ((a : A) → (b : B) → P(a , b)) → ((p : (A ⨯ B)) → P(p))
elim f(a , b) = f a b
map : (A₁ → A₂) → (B₁ → B₂) → (A₁ ⨯ B₁) → (A₂ ⨯ B₂)
map f g (x , y) = (f(x) , g(y))
-- Curries a function taking a 2-tuple, transforming it to a function returning a function instead
curry : ((A ⨯ B) → C) → (A → B → C)
curry f x y = f(x , y)
-- Uncurries a function taking a function, transforming it to a function taking a 2-tuple instead
uncurry : (A → B → C) → ((A ⨯ B) → C)
uncurry = elim
mapLeft : (A₁ → A₂) → (A₁ ⨯ B) → (A₂ ⨯ B)
mapLeft f = map f (x ↦ x)
mapRight : let _ = A in (B₁ → B₂) → (A ⨯ B₁) → (A ⨯ B₂)
mapRight f = map (x ↦ x) f
associateLeft : (A ⨯ (B ⨯ C)) → ((A ⨯ B) ⨯ C)
associateLeft (x , (y , z)) = ((x , y) , z)
associateRight : ((A ⨯ B) ⨯ C) → (A ⨯ (B ⨯ C))
associateRight ((x , y) , z) = (x , (y , z))
-- Swaps the left and right elements of a 2-tuple
swap : (A ⨯ B) → (B ⨯ A)
swap(x , y) = (y , x)
repeat : A → (A ⨯ A)
repeat x = (x , x)
|
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------------------------------------------------------------------------
-- Experiments with Pure Type Systems (PTS)
------------------------------------------------------------------------
-- Author: Sandro Stucki
-- Copyright (c) 2015 EPFL
-- The code in this directory contains a (partial) Agda formalization
-- of Pure Type Systems (PTS).
--
-- The code makes heavy use of the Agda standard library, which is
-- freely available from
--
-- https://github.com/agda/agda-stdlib/
--
-- The code has been tested using Agda 2.5.3 and version 0.14 of the
-- Agda standard library.
module README where
------------------------------------------------------------------------
-- Modules related to pure type systems (PTS)
-- Syntax of (untyped) terms along with support for substitutions.
open import Pts.Syntax
-- Variants of β-reduction/equivalence and properties thereof.
open import Pts.Reduction.Cbv
open import Pts.Reduction.Full
open import Pts.Reduction.Parallel
-- Typing of terms, substitution lemmas and a proofs of type soundess
-- (preservation/subject reduction and progress).
open import Pts.Core
open import Pts.Typing
open import Pts.Typing.Progress
------------------------------------------------------------------------
-- Modules containing generic functionality
-- Extra lemmas that are derivable in the substitution framework of
-- the Agda standard library, as well as support for substitutions
-- lifted to relations and typed substitutions.
open import Data.Fin.Substitution.ExtraLemmas
open import Data.Fin.Substitution.Relation
open import Data.Fin.Substitution.Typed
-- Symmetric and equivalence closures of binary relations.
open import Relation.Binary.SymmetricClosure
open import Relation.Binary.EquivalenceClosure
-- Support for generic reduction relations.
open import Relation.Binary.Reduction
|
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-- {-# OPTIONS -v tc.meta:50 #-}
-- Andreas 2012-03-27, record pattern unification
module Issue376 where
import Common.Level
open import Common.Equality
open import Common.Irrelevance
record Sigma (A : Set)(B : A -> Set) : Set where
constructor _,_
field
fst : A
snd : B fst
open Sigma public
record Unit : Set where
constructor unit
bla1 : (A : Set) (a : A) ->
let X : Unit -> A
X = _
in X unit ≡ a
bla1 A a = refl
bla2 : (A : Set)(B : A -> Set) ->
let X : Sigma A B -> Sigma A B
X = _
in (x : A)(y : B x) -> X (x , y) ≡ (x , y)
bla2 A B x y = refl
-- _55 A B (x , y) := (x , y)
-- irrelevant records
bla3 : (A : Set)(B : A -> Set) ->
let X : .(z : Sigma A B) -> (C : .(Sigma A B) -> Set) -> (.(z : Sigma A B) -> C z) -> C z
X = _
in (x : A)(y : B x)(C : .(Sigma A B) -> Set)(k : .(z : Sigma A B) -> C z) ->
X (x , y) C k ≡ k (x , y)
bla3 A B x y C k = refl
-- nested irrelevance
bla4 : (A : Set) ->
let A' = Squash (Squash A) in
let X : .(z : A') -> (C : .A' -> Set) -> (.(z : A') -> C z) -> C z
X = _
in (a : A)(C : .A' -> Set)(k : .(z : A') -> C z) ->
X (squash (squash a)) C k ≡ k (squash (squash a))
bla4 A a C k = refl
-- projected bound var
bla5 : (A : Set) (B : A -> Set) ->
let X : (x : A) (y : B x) -> Sigma A B
X = _
in (z : Sigma A B) -> X (fst z) (snd z) ≡ z
bla5 A B z = refl
-- projected bound var
bla6 : (A : Set) (B : A -> Set) ->
let X : A -> A
X = _
in (z : Sigma A B) -> X (fst z) ≡ fst z
bla6 A B z = refl
|
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{-# OPTIONS --cubical --safe #-}
module Cubical.Structures.Group where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.HLevels
open import Cubical.Data.Sigma
open import Cubical.Foundations.SIP renaming (SNS-PathP to SNS)
open import Cubical.Structures.NAryOp
open import Cubical.Structures.Semigroup hiding (⟨_⟩)
open import Cubical.Structures.Monoid hiding (⟨_⟩)
private
variable
ℓ ℓ' : Level
raw-group-structure : Type ℓ → Type ℓ
raw-group-structure = raw-semigroup-structure
raw-group-is-SNS : SNS {ℓ} raw-group-structure _
raw-group-is-SNS = raw-semigroup-is-SNS
-- The neutral element and the inverse function will be derived from the
-- axioms, instead of being defined in the raw-group-structure in order
-- to have that isomorphisms between groups are equivalences that preserves
-- multiplication (so we don't have to show that they also preserve inversion
-- and neutral element, although they will preserve them).
group-axioms : (G : Type ℓ) → raw-group-structure G → Type ℓ
group-axioms G _·_ = i × ii
where
i = semigroup-axioms G _·_
ii = Σ[ e ∈ G ] ((x : G) → (x · e ≡ x) × (e · x ≡ x)) ×
((x : G) → Σ[ x' ∈ G ] (x · x' ≡ e) × (x' · x ≡ e))
group-structure : Type ℓ → Type ℓ
group-structure = add-to-structure raw-group-structure group-axioms
Group : Type (ℓ-suc ℓ)
Group {ℓ} = TypeWithStr ℓ group-structure
-- Extracting components of a group
⟨_⟩ : Group {ℓ} → Type ℓ
⟨ G , _ ⟩ = G
group-operation : (G : Group {ℓ}) → ⟨ G ⟩ → ⟨ G ⟩ → ⟨ G ⟩
group-operation (_ , f , _) = f
module group-operation-syntax where
group-operation-syntax : (G : Group {ℓ}) → ⟨ G ⟩ → ⟨ G ⟩ → ⟨ G ⟩
group-operation-syntax = group-operation
infixr 20 group-operation-syntax
syntax group-operation-syntax G x y = x ·⟨ G ⟩ y
open group-operation-syntax
group-is-set : (G : Group {ℓ}) → isSet ⟨ G ⟩
group-is-set (_ , _ , (P , _) , _) = P
group-assoc : (G : Group {ℓ})
→ (x y z : ⟨ G ⟩) → (x ·⟨ G ⟩ (y ·⟨ G ⟩ z)) ≡ ((x ·⟨ G ⟩ y) ·⟨ G ⟩ z)
group-assoc (_ , _ , (_ , P) , _) = P
-- Defining identity
group-id : (G : Group {ℓ}) → ⟨ G ⟩
group-id (_ , _ , _ , P) = fst P
group-rid : (G : Group {ℓ})
→ (x : ⟨ G ⟩) → x ·⟨ G ⟩ (group-id G) ≡ x
group-rid (_ , _ , _ , P) x = fst ((fst (snd P)) x)
group-lid : (G : Group {ℓ})
→ (x : ⟨ G ⟩) → (group-id G) ·⟨ G ⟩ x ≡ x
group-lid (_ , _ , _ , P) x = snd ((fst (snd P)) x)
-- Defining the inverse function
group-inv : (G : Group {ℓ}) → ⟨ G ⟩ → ⟨ G ⟩
group-inv (_ , _ , _ , P) x = fst ((snd (snd P)) x)
group-rinv : (G : Group {ℓ})
→ (x : ⟨ G ⟩) → x ·⟨ G ⟩ (group-inv G x) ≡ group-id G
group-rinv (_ , _ , _ , P) x = fst (snd ((snd (snd P)) x))
group-linv : (G : Group {ℓ})
→ (x : ⟨ G ⟩) → (group-inv G x) ·⟨ G ⟩ x ≡ group-id G
group-linv (_ , _ , _ , P) x = snd (snd ((snd (snd P)) x))
-- Iso for groups are those for monoids
group-iso : StrIso group-structure ℓ
group-iso = add-to-iso (nAryFunIso 2) group-axioms
-- Group axioms isProp
group-axioms-isProp : (X : Type ℓ)
→ (s : raw-group-structure X)
→ isProp (group-axioms X s)
group-axioms-isProp X s t = η t
where
𝒢 : Group
𝒢 = X , s , t
is-identity : X → Type _
is-identity e = (x : X) → (x ·⟨ 𝒢 ⟩ e ≡ x) × (e ·⟨ 𝒢 ⟩ x ≡ x)
α : (e : X) → isProp (is-identity e)
α e = isPropΠ (λ _ → isPropΣ (group-is-set 𝒢 _ _) (λ _ → group-is-set 𝒢 _ _))
β : (e : X) → is-identity e → isProp ((x : X) → Σ[ x' ∈ X ] (x ·⟨ 𝒢 ⟩ x' ≡ e) × (x' ·⟨ 𝒢 ⟩ x ≡ e))
β e is-identity-e =
isPropΠ λ { x (x' , _ , P) (x'' , Q , _) → ΣProp≡ (λ _ → isPropΣ (group-is-set 𝒢 _ _) λ _ → group-is-set 𝒢 _ _)
(inv-lemma ℳ x x' x'' P Q) }
where
ℳ : Monoid
ℳ = ⟨ 𝒢 ⟩ , (e , group-operation 𝒢) ,
group-is-set 𝒢 ,
group-assoc 𝒢 ,
(λ x → fst (is-identity-e x)) ,
(λ x → snd (is-identity-e x))
γ : isProp (Σ[ e ∈ X ] ((x : X) → (x ·⟨ 𝒢 ⟩ e ≡ x) × (e ·⟨ 𝒢 ⟩ x ≡ x)) ×
((x : X) → Σ[ x' ∈ X ] (x ·⟨ 𝒢 ⟩ x' ≡ e) × (x' ·⟨ 𝒢 ⟩ x ≡ e)))
γ (e , P , _) (e' , Q , _) = ΣProp≡ (λ e → isPropΣ (α e) λ is-identity-e → β e is-identity-e)
(e ≡⟨ sym (fst (Q e)) ⟩
e ·⟨ 𝒢 ⟩ e' ≡⟨ snd (P e') ⟩
e' ∎)
η : isProp (group-axioms X s)
η = isPropΣ (semigroup-axiom-isProp X s) λ _ → γ
-- Group paths equivalent to equality
group-is-SNS : SNS {ℓ} group-structure group-iso
group-is-SNS = add-axioms-SNS _ group-axioms-isProp (nAryFunSNS 2)
GroupPath : (M N : Group {ℓ}) → (M ≃[ group-iso ] N) ≃ (M ≡ N)
GroupPath = SIP group-is-SNS
-- Group ·syntax
module group-·syntax (G : Group {ℓ}) where
infixr 18 _·_
_·_ : ⟨ G ⟩ → ⟨ G ⟩ → ⟨ G ⟩
_·_ = group-operation G
₁ : ⟨ G ⟩
₁ = group-id G
infix 19 _⁻¹
_⁻¹ : ⟨ G ⟩ → ⟨ G ⟩
_⁻¹ = group-inv G
|
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module Text.Greek.SBLGNT.Eph where
open import Data.List
open import Text.Greek.Bible
open import Text.Greek.Script
open import Text.Greek.Script.Unicode
ΠΡΟΣ-ΕΦΕΣΙΟΥΣ : List (Word)
ΠΡΟΣ-ΕΦΕΣΙΟΥΣ =
word (Π ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Eph.1.1"
∷ word (ἀ ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ς ∷ []) "Eph.1.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.1.1"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.1.1"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.1.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.1.1"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.1.1"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Eph.1.1"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.1.1"
∷ word (ο ∷ ὖ ∷ σ ∷ ι ∷ ν ∷ []) "Eph.1.1"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.1"
∷ word (Ἐ ∷ φ ∷ έ ∷ σ ∷ ῳ ∷ []) "Eph.1.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.1"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.1.1"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.1.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.1.1"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Eph.1.2"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Eph.1.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.2"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "Eph.1.2"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Eph.1.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.1.2"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.1.2"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.1.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.2"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Eph.1.2"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.1.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.2"
∷ word (Ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ η ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.1.3"
∷ word (ὁ ∷ []) "Eph.1.3"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Eph.1.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.3"
∷ word (π ∷ α ∷ τ ∷ ὴ ∷ ρ ∷ []) "Eph.1.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.1.3"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Eph.1.3"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.1.3"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.1.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.3"
∷ word (ὁ ∷ []) "Eph.1.3"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ή ∷ σ ∷ α ∷ ς ∷ []) "Eph.1.3"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.1.3"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.3"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "Eph.1.3"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ ᾳ ∷ []) "Eph.1.3"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ῇ ∷ []) "Eph.1.3"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.3"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.1.3"
∷ word (ἐ ∷ π ∷ ο ∷ υ ∷ ρ ∷ α ∷ ν ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Eph.1.3"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.1.3"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Eph.1.4"
∷ word (ἐ ∷ ξ ∷ ε ∷ ∙λ ∷ έ ∷ ξ ∷ α ∷ τ ∷ ο ∷ []) "Eph.1.4"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.1.4"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.4"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Eph.1.4"
∷ word (π ∷ ρ ∷ ὸ ∷ []) "Eph.1.4"
∷ word (κ ∷ α ∷ τ ∷ α ∷ β ∷ ο ∷ ∙λ ∷ ῆ ∷ ς ∷ []) "Eph.1.4"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "Eph.1.4"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Eph.1.4"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.1.4"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "Eph.1.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.4"
∷ word (ἀ ∷ μ ∷ ώ ∷ μ ∷ ο ∷ υ ∷ ς ∷ []) "Eph.1.4"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ν ∷ ώ ∷ π ∷ ι ∷ ο ∷ ν ∷ []) "Eph.1.4"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.4"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.4"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "Eph.1.4"
∷ word (π ∷ ρ ∷ ο ∷ ο ∷ ρ ∷ ί ∷ σ ∷ α ∷ ς ∷ []) "Eph.1.5"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.1.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.1.5"
∷ word (υ ∷ ἱ ∷ ο ∷ θ ∷ ε ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Eph.1.5"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.1.5"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.1.5"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.1.5"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "Eph.1.5"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.1.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.1.5"
∷ word (ε ∷ ὐ ∷ δ ∷ ο ∷ κ ∷ ί ∷ α ∷ ν ∷ []) "Eph.1.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.1.5"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.1.5"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.1.6"
∷ word (ἔ ∷ π ∷ α ∷ ι ∷ ν ∷ ο ∷ ν ∷ []) "Eph.1.6"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Eph.1.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.6"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ο ∷ ς ∷ []) "Eph.1.6"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.6"
∷ word (ἧ ∷ ς ∷ []) "Eph.1.6"
∷ word (ἐ ∷ χ ∷ α ∷ ρ ∷ ί ∷ τ ∷ ω ∷ σ ∷ ε ∷ ν ∷ []) "Eph.1.6"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.1.6"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.6"
∷ word (τ ∷ ῷ ∷ []) "Eph.1.6"
∷ word (ἠ ∷ γ ∷ α ∷ π ∷ η ∷ μ ∷ έ ∷ ν ∷ ῳ ∷ []) "Eph.1.6"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.7"
∷ word (ᾧ ∷ []) "Eph.1.7"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Eph.1.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.1.7"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ύ ∷ τ ∷ ρ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Eph.1.7"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.1.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.1.7"
∷ word (α ∷ ἵ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.1.7"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.1.7"
∷ word (ἄ ∷ φ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Eph.1.7"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.1.7"
∷ word (π ∷ α ∷ ρ ∷ α ∷ π ∷ τ ∷ ω ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "Eph.1.7"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.1.7"
∷ word (τ ∷ ὸ ∷ []) "Eph.1.7"
∷ word (π ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ς ∷ []) "Eph.1.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.7"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ο ∷ ς ∷ []) "Eph.1.7"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.7"
∷ word (ἧ ∷ ς ∷ []) "Eph.1.8"
∷ word (ἐ ∷ π ∷ ε ∷ ρ ∷ ί ∷ σ ∷ σ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "Eph.1.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.1.8"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.1.8"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.8"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "Eph.1.8"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ ᾳ ∷ []) "Eph.1.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.8"
∷ word (φ ∷ ρ ∷ ο ∷ ν ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "Eph.1.8"
∷ word (γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ σ ∷ α ∷ ς ∷ []) "Eph.1.9"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "Eph.1.9"
∷ word (τ ∷ ὸ ∷ []) "Eph.1.9"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Eph.1.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.1.9"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.1.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.9"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.1.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.1.9"
∷ word (ε ∷ ὐ ∷ δ ∷ ο ∷ κ ∷ ί ∷ α ∷ ν ∷ []) "Eph.1.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.9"
∷ word (ἣ ∷ ν ∷ []) "Eph.1.9"
∷ word (π ∷ ρ ∷ ο ∷ έ ∷ θ ∷ ε ∷ τ ∷ ο ∷ []) "Eph.1.9"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.9"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Eph.1.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.1.10"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Eph.1.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.1.10"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.1.10"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.1.10"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ῶ ∷ ν ∷ []) "Eph.1.10"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ α ∷ ι ∷ ώ ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Eph.1.10"
∷ word (τ ∷ ὰ ∷ []) "Eph.1.10"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Eph.1.10"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.10"
∷ word (τ ∷ ῷ ∷ []) "Eph.1.10"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.1.10"
∷ word (τ ∷ ὰ ∷ []) "Eph.1.10"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Eph.1.10"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.1.10"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.1.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.10"
∷ word (τ ∷ ὰ ∷ []) "Eph.1.10"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Eph.1.10"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.10"
∷ word (γ ∷ ῆ ∷ ς ∷ []) "Eph.1.10"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.10"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Eph.1.10"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.11"
∷ word (ᾧ ∷ []) "Eph.1.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.11"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ η ∷ ρ ∷ ώ ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "Eph.1.11"
∷ word (π ∷ ρ ∷ ο ∷ ο ∷ ρ ∷ ι ∷ σ ∷ θ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.1.11"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.1.11"
∷ word (π ∷ ρ ∷ ό ∷ θ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Eph.1.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.1.11"
∷ word (τ ∷ ὰ ∷ []) "Eph.1.11"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Eph.1.11"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Eph.1.11"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.1.11"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.1.11"
∷ word (β ∷ ο ∷ υ ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "Eph.1.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.1.11"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.1.11"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.11"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.1.12"
∷ word (τ ∷ ὸ ∷ []) "Eph.1.12"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Eph.1.12"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.1.12"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.1.12"
∷ word (ἔ ∷ π ∷ α ∷ ι ∷ ν ∷ ο ∷ ν ∷ []) "Eph.1.12"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Eph.1.12"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.12"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.1.12"
∷ word (π ∷ ρ ∷ ο ∷ η ∷ ∙λ ∷ π ∷ ι ∷ κ ∷ ό ∷ τ ∷ α ∷ ς ∷ []) "Eph.1.12"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.12"
∷ word (τ ∷ ῷ ∷ []) "Eph.1.12"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.1.12"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.13"
∷ word (ᾧ ∷ []) "Eph.1.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.13"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Eph.1.13"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.1.13"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.1.13"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ν ∷ []) "Eph.1.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.13"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Eph.1.13"
∷ word (τ ∷ ὸ ∷ []) "Eph.1.13"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "Eph.1.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.13"
∷ word (σ ∷ ω ∷ τ ∷ η ∷ ρ ∷ ί ∷ α ∷ ς ∷ []) "Eph.1.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.1.13"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.13"
∷ word (ᾧ ∷ []) "Eph.1.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.13"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ σ ∷ α ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.1.13"
∷ word (ἐ ∷ σ ∷ φ ∷ ρ ∷ α ∷ γ ∷ ί ∷ σ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Eph.1.13"
∷ word (τ ∷ ῷ ∷ []) "Eph.1.13"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Eph.1.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.13"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "Eph.1.13"
∷ word (τ ∷ ῷ ∷ []) "Eph.1.13"
∷ word (ἁ ∷ γ ∷ ί ∷ ῳ ∷ []) "Eph.1.13"
∷ word (ὅ ∷ []) "Eph.1.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.1.14"
∷ word (ἀ ∷ ρ ∷ ρ ∷ α ∷ β ∷ ὼ ∷ ν ∷ []) "Eph.1.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.14"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ ς ∷ []) "Eph.1.14"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.1.14"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.1.14"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ύ ∷ τ ∷ ρ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Eph.1.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.14"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ ο ∷ ι ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.1.14"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.1.14"
∷ word (ἔ ∷ π ∷ α ∷ ι ∷ ν ∷ ο ∷ ν ∷ []) "Eph.1.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.14"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Eph.1.14"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.14"
∷ word (Δ ∷ ι ∷ ὰ ∷ []) "Eph.1.15"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Eph.1.15"
∷ word (κ ∷ ἀ ∷ γ ∷ ώ ∷ []) "Eph.1.15"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ ς ∷ []) "Eph.1.15"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.1.15"
∷ word (κ ∷ α ∷ θ ∷ []) "Eph.1.15"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.1.15"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.1.15"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.15"
∷ word (τ ∷ ῷ ∷ []) "Eph.1.15"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.1.15"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.1.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.15"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.1.15"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "Eph.1.15"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.1.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.1.15"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Eph.1.15"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.1.15"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "Eph.1.15"
∷ word (ο ∷ ὐ ∷ []) "Eph.1.16"
∷ word (π ∷ α ∷ ύ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Eph.1.16"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῶ ∷ ν ∷ []) "Eph.1.16"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Eph.1.16"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.1.16"
∷ word (μ ∷ ν ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "Eph.1.16"
∷ word (π ∷ ο ∷ ι ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Eph.1.16"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Eph.1.16"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.1.16"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ ῶ ∷ ν ∷ []) "Eph.1.16"
∷ word (μ ∷ ο ∷ υ ∷ []) "Eph.1.16"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.1.17"
∷ word (ὁ ∷ []) "Eph.1.17"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Eph.1.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.1.17"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Eph.1.17"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.1.17"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.1.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.17"
∷ word (ὁ ∷ []) "Eph.1.17"
∷ word (π ∷ α ∷ τ ∷ ὴ ∷ ρ ∷ []) "Eph.1.17"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.17"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Eph.1.17"
∷ word (δ ∷ ώ ∷ ῃ ∷ []) "Eph.1.17"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Eph.1.17"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Eph.1.17"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ς ∷ []) "Eph.1.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.17"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ ψ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.1.17"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.17"
∷ word (ἐ ∷ π ∷ ι ∷ γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "Eph.1.17"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.17"
∷ word (π ∷ ε ∷ φ ∷ ω ∷ τ ∷ ι ∷ σ ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "Eph.1.18"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.1.18"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.1.18"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.18"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "Eph.1.18"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.1.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.1.18"
∷ word (τ ∷ ὸ ∷ []) "Eph.1.18"
∷ word (ε ∷ ἰ ∷ δ ∷ έ ∷ ν ∷ α ∷ ι ∷ []) "Eph.1.18"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.1.18"
∷ word (τ ∷ ί ∷ ς ∷ []) "Eph.1.18"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.1.18"
∷ word (ἡ ∷ []) "Eph.1.18"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ὶ ∷ ς ∷ []) "Eph.1.18"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.18"
∷ word (κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.1.18"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.18"
∷ word (τ ∷ ί ∷ ς ∷ []) "Eph.1.18"
∷ word (ὁ ∷ []) "Eph.1.18"
∷ word (π ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ς ∷ []) "Eph.1.18"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.18"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Eph.1.18"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.18"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ ς ∷ []) "Eph.1.18"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.18"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.18"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.1.18"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Eph.1.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.19"
∷ word (τ ∷ ί ∷ []) "Eph.1.19"
∷ word (τ ∷ ὸ ∷ []) "Eph.1.19"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ ά ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Eph.1.19"
∷ word (μ ∷ έ ∷ γ ∷ ε ∷ θ ∷ ο ∷ ς ∷ []) "Eph.1.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.19"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.1.19"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.19"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.1.19"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.1.19"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.1.19"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Eph.1.19"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.1.19"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.1.19"
∷ word (ἐ ∷ ν ∷ έ ∷ ρ ∷ γ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Eph.1.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.1.19"
∷ word (κ ∷ ρ ∷ ά ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Eph.1.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.1.19"
∷ word (ἰ ∷ σ ∷ χ ∷ ύ ∷ ο ∷ ς ∷ []) "Eph.1.19"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.19"
∷ word (ἣ ∷ ν ∷ []) "Eph.1.20"
∷ word (ἐ ∷ ν ∷ ή ∷ ρ ∷ γ ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "Eph.1.20"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.20"
∷ word (τ ∷ ῷ ∷ []) "Eph.1.20"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.1.20"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ α ∷ ς ∷ []) "Eph.1.20"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Eph.1.20"
∷ word (ἐ ∷ κ ∷ []) "Eph.1.20"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Eph.1.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.20"
∷ word (κ ∷ α ∷ θ ∷ ί ∷ σ ∷ α ∷ ς ∷ []) "Eph.1.20"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.20"
∷ word (δ ∷ ε ∷ ξ ∷ ι ∷ ᾷ ∷ []) "Eph.1.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.20"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.20"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.1.20"
∷ word (ἐ ∷ π ∷ ο ∷ υ ∷ ρ ∷ α ∷ ν ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Eph.1.20"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ ά ∷ ν ∷ ω ∷ []) "Eph.1.21"
∷ word (π ∷ ά ∷ σ ∷ η ∷ ς ∷ []) "Eph.1.21"
∷ word (ἀ ∷ ρ ∷ χ ∷ ῆ ∷ ς ∷ []) "Eph.1.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.21"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Eph.1.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.21"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.1.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.21"
∷ word (κ ∷ υ ∷ ρ ∷ ι ∷ ό ∷ τ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "Eph.1.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.21"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.1.21"
∷ word (ὀ ∷ ν ∷ ό ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.1.21"
∷ word (ὀ ∷ ν ∷ ο ∷ μ ∷ α ∷ ζ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ []) "Eph.1.21"
∷ word (ο ∷ ὐ ∷ []) "Eph.1.21"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Eph.1.21"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.21"
∷ word (τ ∷ ῷ ∷ []) "Eph.1.21"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ ι ∷ []) "Eph.1.21"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "Eph.1.21"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Eph.1.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.21"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.21"
∷ word (τ ∷ ῷ ∷ []) "Eph.1.21"
∷ word (μ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ τ ∷ ι ∷ []) "Eph.1.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.22"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Eph.1.22"
∷ word (ὑ ∷ π ∷ έ ∷ τ ∷ α ∷ ξ ∷ ε ∷ ν ∷ []) "Eph.1.22"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Eph.1.22"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.1.22"
∷ word (π ∷ ό ∷ δ ∷ α ∷ ς ∷ []) "Eph.1.22"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.1.22"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Eph.1.22"
∷ word (ἔ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "Eph.1.22"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "Eph.1.22"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Eph.1.22"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Eph.1.22"
∷ word (τ ∷ ῇ ∷ []) "Eph.1.22"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "Eph.1.22"
∷ word (ἥ ∷ τ ∷ ι ∷ ς ∷ []) "Eph.1.23"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "Eph.1.23"
∷ word (τ ∷ ὸ ∷ []) "Eph.1.23"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "Eph.1.23"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.1.23"
∷ word (τ ∷ ὸ ∷ []) "Eph.1.23"
∷ word (π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ μ ∷ α ∷ []) "Eph.1.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.1.23"
∷ word (τ ∷ ὰ ∷ []) "Eph.1.23"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Eph.1.23"
∷ word (ἐ ∷ ν ∷ []) "Eph.1.23"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "Eph.1.23"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ []) "Eph.1.23"
∷ word (Κ ∷ α ∷ ὶ ∷ []) "Eph.2.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.2.1"
∷ word (ὄ ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Eph.2.1"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.2.1"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.2.1"
∷ word (π ∷ α ∷ ρ ∷ α ∷ π ∷ τ ∷ ώ ∷ μ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "Eph.2.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.1"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Eph.2.1"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Eph.2.1"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.2.1"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.2"
∷ word (α ∷ ἷ ∷ ς ∷ []) "Eph.2.2"
∷ word (π ∷ ο ∷ τ ∷ ε ∷ []) "Eph.2.2"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ ε ∷ π ∷ α ∷ τ ∷ ή ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Eph.2.2"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.2.2"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.2.2"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ α ∷ []) "Eph.2.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.2.2"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "Eph.2.2"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "Eph.2.2"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.2.2"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.2.2"
∷ word (ἄ ∷ ρ ∷ χ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "Eph.2.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.2.2"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Eph.2.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.2.2"
∷ word (ἀ ∷ έ ∷ ρ ∷ ο ∷ ς ∷ []) "Eph.2.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.2.2"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.2.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.2.2"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Eph.2.2"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Eph.2.2"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.2"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.2.2"
∷ word (υ ∷ ἱ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.2.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.2.2"
∷ word (ἀ ∷ π ∷ ε ∷ ι ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Eph.2.2"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.3"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "Eph.2.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.3"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Eph.2.3"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.2.3"
∷ word (ἀ ∷ ν ∷ ε ∷ σ ∷ τ ∷ ρ ∷ ά ∷ φ ∷ η ∷ μ ∷ έ ∷ ν ∷ []) "Eph.2.3"
∷ word (π ∷ ο ∷ τ ∷ ε ∷ []) "Eph.2.3"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.3"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Eph.2.3"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Eph.2.3"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.2.3"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Eph.2.3"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.2.3"
∷ word (π ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.2.3"
∷ word (τ ∷ ὰ ∷ []) "Eph.2.3"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Eph.2.3"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.2.3"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Eph.2.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.3"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.2.3"
∷ word (δ ∷ ι ∷ α ∷ ν ∷ ο ∷ ι ∷ ῶ ∷ ν ∷ []) "Eph.2.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.3"
∷ word (ἤ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Eph.2.3"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Eph.2.3"
∷ word (φ ∷ ύ ∷ σ ∷ ε ∷ ι ∷ []) "Eph.2.3"
∷ word (ὀ ∷ ρ ∷ γ ∷ ῆ ∷ ς ∷ []) "Eph.2.3"
∷ word (ὡ ∷ ς ∷ []) "Eph.2.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.3"
∷ word (ο ∷ ἱ ∷ []) "Eph.2.3"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ο ∷ ί ∷ []) "Eph.2.3"
∷ word (ὁ ∷ []) "Eph.2.4"
∷ word (δ ∷ ὲ ∷ []) "Eph.2.4"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Eph.2.4"
∷ word (π ∷ ∙λ ∷ ο ∷ ύ ∷ σ ∷ ι ∷ ο ∷ ς ∷ []) "Eph.2.4"
∷ word (ὢ ∷ ν ∷ []) "Eph.2.4"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.4"
∷ word (ἐ ∷ ∙λ ∷ έ ∷ ε ∷ ι ∷ []) "Eph.2.4"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.2.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.2.4"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "Eph.2.4"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "Eph.2.4"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.2.4"
∷ word (ἣ ∷ ν ∷ []) "Eph.2.4"
∷ word (ἠ ∷ γ ∷ ά ∷ π ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Eph.2.4"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.2.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.5"
∷ word (ὄ ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Eph.2.5"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.2.5"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.2.5"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.2.5"
∷ word (π ∷ α ∷ ρ ∷ α ∷ π ∷ τ ∷ ώ ∷ μ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "Eph.2.5"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ζ ∷ ω ∷ ο ∷ π ∷ ο ∷ ί ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Eph.2.5"
∷ word (τ ∷ ῷ ∷ []) "Eph.2.5"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.2.5"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ί ∷ []) "Eph.2.5"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "Eph.2.5"
∷ word (σ ∷ ε ∷ σ ∷ ῳ ∷ σ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Eph.2.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.6"
∷ word (σ ∷ υ ∷ ν ∷ ή ∷ γ ∷ ε ∷ ι ∷ ρ ∷ ε ∷ ν ∷ []) "Eph.2.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.6"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ κ ∷ ά ∷ θ ∷ ι ∷ σ ∷ ε ∷ ν ∷ []) "Eph.2.6"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.6"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.2.6"
∷ word (ἐ ∷ π ∷ ο ∷ υ ∷ ρ ∷ α ∷ ν ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Eph.2.6"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.2.6"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.2.6"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.2.7"
∷ word (ἐ ∷ ν ∷ δ ∷ ε ∷ ί ∷ ξ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Eph.2.7"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.7"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.2.7"
∷ word (α ∷ ἰ ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "Eph.2.7"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.2.7"
∷ word (ἐ ∷ π ∷ ε ∷ ρ ∷ χ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "Eph.2.7"
∷ word (τ ∷ ὸ ∷ []) "Eph.2.7"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ ά ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Eph.2.7"
∷ word (π ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ς ∷ []) "Eph.2.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.2.7"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ο ∷ ς ∷ []) "Eph.2.7"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.2.7"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.7"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "Eph.2.7"
∷ word (ἐ ∷ φ ∷ []) "Eph.2.7"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.2.7"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.7"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.2.7"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.2.7"
∷ word (τ ∷ ῇ ∷ []) "Eph.2.8"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Eph.2.8"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ί ∷ []) "Eph.2.8"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "Eph.2.8"
∷ word (σ ∷ ε ∷ σ ∷ ῳ ∷ σ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Eph.2.8"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.2.8"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.2.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.8"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Eph.2.8"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Eph.2.8"
∷ word (ἐ ∷ ξ ∷ []) "Eph.2.8"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.2.8"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.2.8"
∷ word (τ ∷ ὸ ∷ []) "Eph.2.8"
∷ word (δ ∷ ῶ ∷ ρ ∷ ο ∷ ν ∷ []) "Eph.2.8"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Eph.2.9"
∷ word (ἐ ∷ ξ ∷ []) "Eph.2.9"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Eph.2.9"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.2.9"
∷ word (μ ∷ ή ∷ []) "Eph.2.9"
∷ word (τ ∷ ι ∷ ς ∷ []) "Eph.2.9"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Eph.2.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.2.10"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Eph.2.10"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "Eph.2.10"
∷ word (π ∷ ο ∷ ί ∷ η ∷ μ ∷ α ∷ []) "Eph.2.10"
∷ word (κ ∷ τ ∷ ι ∷ σ ∷ θ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.2.10"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.10"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.2.10"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.2.10"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Eph.2.10"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ι ∷ ς ∷ []) "Eph.2.10"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.2.10"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "Eph.2.10"
∷ word (π ∷ ρ ∷ ο ∷ η ∷ τ ∷ ο ∷ ί ∷ μ ∷ α ∷ σ ∷ ε ∷ ν ∷ []) "Eph.2.10"
∷ word (ὁ ∷ []) "Eph.2.10"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Eph.2.10"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.2.10"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.10"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.2.10"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ή ∷ σ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Eph.2.10"
∷ word (Δ ∷ ι ∷ ὸ ∷ []) "Eph.2.11"
∷ word (μ ∷ ν ∷ η ∷ μ ∷ ο ∷ ν ∷ ε ∷ ύ ∷ ε ∷ τ ∷ ε ∷ []) "Eph.2.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Eph.2.11"
∷ word (π ∷ ο ∷ τ ∷ ὲ ∷ []) "Eph.2.11"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Eph.2.11"
∷ word (τ ∷ ὰ ∷ []) "Eph.2.11"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Eph.2.11"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.11"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ί ∷ []) "Eph.2.11"
∷ word (ο ∷ ἱ ∷ []) "Eph.2.11"
∷ word (∙λ ∷ ε ∷ γ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Eph.2.11"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ α ∷ []) "Eph.2.11"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Eph.2.11"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.2.11"
∷ word (∙λ ∷ ε ∷ γ ∷ ο ∷ μ ∷ έ ∷ ν ∷ η ∷ ς ∷ []) "Eph.2.11"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῆ ∷ ς ∷ []) "Eph.2.11"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.11"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "Eph.2.11"
∷ word (χ ∷ ε ∷ ι ∷ ρ ∷ ο ∷ π ∷ ο ∷ ι ∷ ή ∷ τ ∷ ο ∷ υ ∷ []) "Eph.2.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Eph.2.12"
∷ word (ἦ ∷ τ ∷ ε ∷ []) "Eph.2.12"
∷ word (τ ∷ ῷ ∷ []) "Eph.2.12"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ῷ ∷ []) "Eph.2.12"
∷ word (ἐ ∷ κ ∷ ε ∷ ί ∷ ν ∷ ῳ ∷ []) "Eph.2.12"
∷ word (χ ∷ ω ∷ ρ ∷ ὶ ∷ ς ∷ []) "Eph.2.12"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.2.12"
∷ word (ἀ ∷ π ∷ η ∷ ∙λ ∷ ∙λ ∷ ο ∷ τ ∷ ρ ∷ ι ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Eph.2.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.2.12"
∷ word (π ∷ ο ∷ ∙λ ∷ ι ∷ τ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Eph.2.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.2.12"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ὴ ∷ ∙λ ∷ []) "Eph.2.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.12"
∷ word (ξ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Eph.2.12"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.2.12"
∷ word (δ ∷ ι ∷ α ∷ θ ∷ η ∷ κ ∷ ῶ ∷ ν ∷ []) "Eph.2.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.2.12"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "Eph.2.12"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ α ∷ []) "Eph.2.12"
∷ word (μ ∷ ὴ ∷ []) "Eph.2.12"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.2.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.12"
∷ word (ἄ ∷ θ ∷ ε ∷ ο ∷ ι ∷ []) "Eph.2.12"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.12"
∷ word (τ ∷ ῷ ∷ []) "Eph.2.12"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "Eph.2.12"
∷ word (ν ∷ υ ∷ ν ∷ ὶ ∷ []) "Eph.2.13"
∷ word (δ ∷ ὲ ∷ []) "Eph.2.13"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.13"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.2.13"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.2.13"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Eph.2.13"
∷ word (ο ∷ ἵ ∷ []) "Eph.2.13"
∷ word (π ∷ ο ∷ τ ∷ ε ∷ []) "Eph.2.13"
∷ word (ὄ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.2.13"
∷ word (μ ∷ α ∷ κ ∷ ρ ∷ ὰ ∷ ν ∷ []) "Eph.2.13"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ή ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Eph.2.13"
∷ word (ἐ ∷ γ ∷ γ ∷ ὺ ∷ ς ∷ []) "Eph.2.13"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.13"
∷ word (τ ∷ ῷ ∷ []) "Eph.2.13"
∷ word (α ∷ ἵ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Eph.2.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.2.13"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.2.13"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.2.14"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Eph.2.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.2.14"
∷ word (ἡ ∷ []) "Eph.2.14"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "Eph.2.14"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.2.14"
∷ word (ὁ ∷ []) "Eph.2.14"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ α ∷ ς ∷ []) "Eph.2.14"
∷ word (τ ∷ ὰ ∷ []) "Eph.2.14"
∷ word (ἀ ∷ μ ∷ φ ∷ ό ∷ τ ∷ ε ∷ ρ ∷ α ∷ []) "Eph.2.14"
∷ word (ἓ ∷ ν ∷ []) "Eph.2.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.14"
∷ word (τ ∷ ὸ ∷ []) "Eph.2.14"
∷ word (μ ∷ ε ∷ σ ∷ ό ∷ τ ∷ ο ∷ ι ∷ χ ∷ ο ∷ ν ∷ []) "Eph.2.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.2.14"
∷ word (φ ∷ ρ ∷ α ∷ γ ∷ μ ∷ ο ∷ ῦ ∷ []) "Eph.2.14"
∷ word (∙λ ∷ ύ ∷ σ ∷ α ∷ ς ∷ []) "Eph.2.14"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.2.14"
∷ word (ἔ ∷ χ ∷ θ ∷ ρ ∷ α ∷ ν ∷ []) "Eph.2.14"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.14"
∷ word (τ ∷ ῇ ∷ []) "Eph.2.14"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "Eph.2.14"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.2.14"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.2.15"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Eph.2.15"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.2.15"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "Eph.2.15"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.15"
∷ word (δ ∷ ό ∷ γ ∷ μ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "Eph.2.15"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ ή ∷ σ ∷ α ∷ ς ∷ []) "Eph.2.15"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.2.15"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.2.15"
∷ word (δ ∷ ύ ∷ ο ∷ []) "Eph.2.15"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ ῃ ∷ []) "Eph.2.15"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.15"
∷ word (α ∷ ὑ ∷ τ ∷ ῷ ∷ []) "Eph.2.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.2.15"
∷ word (ἕ ∷ ν ∷ α ∷ []) "Eph.2.15"
∷ word (κ ∷ α ∷ ι ∷ ν ∷ ὸ ∷ ν ∷ []) "Eph.2.15"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Eph.2.15"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ ν ∷ []) "Eph.2.15"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ ν ∷ []) "Eph.2.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.16"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ∙λ ∷ ά ∷ ξ ∷ ῃ ∷ []) "Eph.2.16"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.2.16"
∷ word (ἀ ∷ μ ∷ φ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "Eph.2.16"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.16"
∷ word (ἑ ∷ ν ∷ ὶ ∷ []) "Eph.2.16"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Eph.2.16"
∷ word (τ ∷ ῷ ∷ []) "Eph.2.16"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Eph.2.16"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.2.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.2.16"
∷ word (σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ο ∷ ῦ ∷ []) "Eph.2.16"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ τ ∷ ε ∷ ί ∷ ν ∷ α ∷ ς ∷ []) "Eph.2.16"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.2.16"
∷ word (ἔ ∷ χ ∷ θ ∷ ρ ∷ α ∷ ν ∷ []) "Eph.2.16"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.16"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Eph.2.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.17"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ὼ ∷ ν ∷ []) "Eph.2.17"
∷ word (ε ∷ ὐ ∷ η ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ σ ∷ α ∷ τ ∷ ο ∷ []) "Eph.2.17"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ ν ∷ []) "Eph.2.17"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Eph.2.17"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.2.17"
∷ word (μ ∷ α ∷ κ ∷ ρ ∷ ὰ ∷ ν ∷ []) "Eph.2.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.17"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ ν ∷ []) "Eph.2.17"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.2.17"
∷ word (ἐ ∷ γ ∷ γ ∷ ύ ∷ ς ∷ []) "Eph.2.17"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Eph.2.18"
∷ word (δ ∷ ι ∷ []) "Eph.2.18"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.2.18"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Eph.2.18"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.2.18"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ α ∷ γ ∷ ω ∷ γ ∷ ὴ ∷ ν ∷ []) "Eph.2.18"
∷ word (ο ∷ ἱ ∷ []) "Eph.2.18"
∷ word (ἀ ∷ μ ∷ φ ∷ ό ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ι ∷ []) "Eph.2.18"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.18"
∷ word (ἑ ∷ ν ∷ ὶ ∷ []) "Eph.2.18"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Eph.2.18"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.2.18"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.2.18"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Eph.2.18"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Eph.2.19"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Eph.2.19"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Eph.2.19"
∷ word (ἐ ∷ σ ∷ τ ∷ ὲ ∷ []) "Eph.2.19"
∷ word (ξ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Eph.2.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.19"
∷ word (π ∷ ά ∷ ρ ∷ ο ∷ ι ∷ κ ∷ ο ∷ ι ∷ []) "Eph.2.19"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Eph.2.19"
∷ word (ἐ ∷ σ ∷ τ ∷ ὲ ∷ []) "Eph.2.19"
∷ word (σ ∷ υ ∷ μ ∷ π ∷ ο ∷ ∙λ ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "Eph.2.19"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.2.19"
∷ word (ἁ ∷ γ ∷ ί ∷ ω ∷ ν ∷ []) "Eph.2.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.19"
∷ word (ο ∷ ἰ ∷ κ ∷ ε ∷ ῖ ∷ ο ∷ ι ∷ []) "Eph.2.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.2.19"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.2.19"
∷ word (ἐ ∷ π ∷ ο ∷ ι ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ η ∷ θ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.2.20"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Eph.2.20"
∷ word (τ ∷ ῷ ∷ []) "Eph.2.20"
∷ word (θ ∷ ε ∷ μ ∷ ε ∷ ∙λ ∷ ί ∷ ῳ ∷ []) "Eph.2.20"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.2.20"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ω ∷ ν ∷ []) "Eph.2.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.20"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ῶ ∷ ν ∷ []) "Eph.2.20"
∷ word (ὄ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Eph.2.20"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ γ ∷ ω ∷ ν ∷ ι ∷ α ∷ ί ∷ ο ∷ υ ∷ []) "Eph.2.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.2.20"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.2.20"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.2.20"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.21"
∷ word (ᾧ ∷ []) "Eph.2.21"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ []) "Eph.2.21"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ὴ ∷ []) "Eph.2.21"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ ρ ∷ μ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "Eph.2.21"
∷ word (α ∷ ὔ ∷ ξ ∷ ε ∷ ι ∷ []) "Eph.2.21"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.2.21"
∷ word (ν ∷ α ∷ ὸ ∷ ν ∷ []) "Eph.2.21"
∷ word (ἅ ∷ γ ∷ ι ∷ ο ∷ ν ∷ []) "Eph.2.21"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.21"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.2.21"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.22"
∷ word (ᾧ ∷ []) "Eph.2.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.2.22"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Eph.2.22"
∷ word (σ ∷ υ ∷ ν ∷ ο ∷ ι ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ε ∷ ῖ ∷ σ ∷ θ ∷ ε ∷ []) "Eph.2.22"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.2.22"
∷ word (κ ∷ α ∷ τ ∷ ο ∷ ι ∷ κ ∷ η ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Eph.2.22"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.2.22"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.2.22"
∷ word (ἐ ∷ ν ∷ []) "Eph.2.22"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Eph.2.22"
∷ word (Τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "Eph.3.1"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Eph.3.1"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Eph.3.1"
∷ word (Π ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Eph.3.1"
∷ word (ὁ ∷ []) "Eph.3.1"
∷ word (δ ∷ έ ∷ σ ∷ μ ∷ ι ∷ ο ∷ ς ∷ []) "Eph.3.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.3.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.3.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.3.1"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Eph.3.1"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.3.1"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.3.1"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Eph.3.1"
∷ word (ε ∷ ἴ ∷ []) "Eph.3.2"
∷ word (γ ∷ ε ∷ []) "Eph.3.2"
∷ word (ἠ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Eph.3.2"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.3.2"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Eph.3.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.3.2"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ο ∷ ς ∷ []) "Eph.3.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.3.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.3.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.3.2"
∷ word (δ ∷ ο ∷ θ ∷ ε ∷ ί ∷ σ ∷ η ∷ ς ∷ []) "Eph.3.2"
∷ word (μ ∷ ο ∷ ι ∷ []) "Eph.3.2"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.3.2"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.3.2"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.3.3"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ ά ∷ ∙λ ∷ υ ∷ ψ ∷ ι ∷ ν ∷ []) "Eph.3.3"
∷ word (ἐ ∷ γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ σ ∷ θ ∷ η ∷ []) "Eph.3.3"
∷ word (μ ∷ ο ∷ ι ∷ []) "Eph.3.3"
∷ word (τ ∷ ὸ ∷ []) "Eph.3.3"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Eph.3.3"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Eph.3.3"
∷ word (π ∷ ρ ∷ ο ∷ έ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "Eph.3.3"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.3"
∷ word (ὀ ∷ ∙λ ∷ ί ∷ γ ∷ ῳ ∷ []) "Eph.3.3"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.3.4"
∷ word (ὃ ∷ []) "Eph.3.4"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Eph.3.4"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.3.4"
∷ word (ν ∷ ο ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Eph.3.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.3.4"
∷ word (σ ∷ ύ ∷ ν ∷ ε ∷ σ ∷ ί ∷ ν ∷ []) "Eph.3.4"
∷ word (μ ∷ ο ∷ υ ∷ []) "Eph.3.4"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.4"
∷ word (τ ∷ ῷ ∷ []) "Eph.3.4"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ η ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.3.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.3.4"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.3.4"
∷ word (ὃ ∷ []) "Eph.3.5"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ α ∷ ι ∷ ς ∷ []) "Eph.3.5"
∷ word (γ ∷ ε ∷ ν ∷ ε ∷ α ∷ ῖ ∷ ς ∷ []) "Eph.3.5"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Eph.3.5"
∷ word (ἐ ∷ γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ σ ∷ θ ∷ η ∷ []) "Eph.3.5"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.3.5"
∷ word (υ ∷ ἱ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.3.5"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.3.5"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "Eph.3.5"
∷ word (ὡ ∷ ς ∷ []) "Eph.3.5"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Eph.3.5"
∷ word (ἀ ∷ π ∷ ε ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ φ ∷ θ ∷ η ∷ []) "Eph.3.5"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.3.5"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Eph.3.5"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "Eph.3.5"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.3.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.3.5"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ ή ∷ τ ∷ α ∷ ι ∷ ς ∷ []) "Eph.3.5"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.5"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Eph.3.5"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Eph.3.6"
∷ word (τ ∷ ὰ ∷ []) "Eph.3.6"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Eph.3.6"
∷ word (σ ∷ υ ∷ γ ∷ κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ό ∷ μ ∷ α ∷ []) "Eph.3.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.3.6"
∷ word (σ ∷ ύ ∷ σ ∷ σ ∷ ω ∷ μ ∷ α ∷ []) "Eph.3.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.3.6"
∷ word (σ ∷ υ ∷ μ ∷ μ ∷ έ ∷ τ ∷ ο ∷ χ ∷ α ∷ []) "Eph.3.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.3.6"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "Eph.3.6"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.3.6"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.3.6"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.3.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.3.6"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ο ∷ υ ∷ []) "Eph.3.6"
∷ word (ο ∷ ὗ ∷ []) "Eph.3.7"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ή ∷ θ ∷ η ∷ ν ∷ []) "Eph.3.7"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ς ∷ []) "Eph.3.7"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.3.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.3.7"
∷ word (δ ∷ ω ∷ ρ ∷ ε ∷ ὰ ∷ ν ∷ []) "Eph.3.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.3.7"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ο ∷ ς ∷ []) "Eph.3.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.3.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.3.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.3.7"
∷ word (δ ∷ ο ∷ θ ∷ ε ∷ ί ∷ σ ∷ η ∷ ς ∷ []) "Eph.3.7"
∷ word (μ ∷ ο ∷ ι ∷ []) "Eph.3.7"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.3.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.3.7"
∷ word (ἐ ∷ ν ∷ έ ∷ ρ ∷ γ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Eph.3.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.3.7"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.3.7"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.3.7"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Eph.3.8"
∷ word (τ ∷ ῷ ∷ []) "Eph.3.8"
∷ word (ἐ ∷ ∙λ ∷ α ∷ χ ∷ ι ∷ σ ∷ τ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ ῳ ∷ []) "Eph.3.8"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Eph.3.8"
∷ word (ἁ ∷ γ ∷ ί ∷ ω ∷ ν ∷ []) "Eph.3.8"
∷ word (ἐ ∷ δ ∷ ό ∷ θ ∷ η ∷ []) "Eph.3.8"
∷ word (ἡ ∷ []) "Eph.3.8"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Eph.3.8"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "Eph.3.8"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.3.8"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Eph.3.8"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Eph.3.8"
∷ word (τ ∷ ὸ ∷ []) "Eph.3.8"
∷ word (ἀ ∷ ν ∷ ε ∷ ξ ∷ ι ∷ χ ∷ ν ∷ ί ∷ α ∷ σ ∷ τ ∷ ο ∷ ν ∷ []) "Eph.3.8"
∷ word (π ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ς ∷ []) "Eph.3.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.3.8"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.3.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.3.9"
∷ word (φ ∷ ω ∷ τ ∷ ί ∷ σ ∷ α ∷ ι ∷ []) "Eph.3.9"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Eph.3.9"
∷ word (τ ∷ ί ∷ ς ∷ []) "Eph.3.9"
∷ word (ἡ ∷ []) "Eph.3.9"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ []) "Eph.3.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.3.9"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ η ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Eph.3.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.3.9"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ ε ∷ κ ∷ ρ ∷ υ ∷ μ ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ []) "Eph.3.9"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Eph.3.9"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.3.9"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ω ∷ ν ∷ []) "Eph.3.9"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.9"
∷ word (τ ∷ ῷ ∷ []) "Eph.3.9"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Eph.3.9"
∷ word (τ ∷ ῷ ∷ []) "Eph.3.9"
∷ word (τ ∷ ὰ ∷ []) "Eph.3.9"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Eph.3.9"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ α ∷ ν ∷ τ ∷ ι ∷ []) "Eph.3.9"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.3.10"
∷ word (γ ∷ ν ∷ ω ∷ ρ ∷ ι ∷ σ ∷ θ ∷ ῇ ∷ []) "Eph.3.10"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Eph.3.10"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Eph.3.10"
∷ word (ἀ ∷ ρ ∷ χ ∷ α ∷ ῖ ∷ ς ∷ []) "Eph.3.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.3.10"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Eph.3.10"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Eph.3.10"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.10"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.3.10"
∷ word (ἐ ∷ π ∷ ο ∷ υ ∷ ρ ∷ α ∷ ν ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Eph.3.10"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.3.10"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.3.10"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Eph.3.10"
∷ word (ἡ ∷ []) "Eph.3.10"
∷ word (π ∷ ο ∷ ∙λ ∷ υ ∷ π ∷ ο ∷ ί ∷ κ ∷ ι ∷ ∙λ ∷ ο ∷ ς ∷ []) "Eph.3.10"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ []) "Eph.3.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.3.10"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.3.10"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.3.11"
∷ word (π ∷ ρ ∷ ό ∷ θ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Eph.3.11"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.3.11"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ω ∷ ν ∷ []) "Eph.3.11"
∷ word (ἣ ∷ ν ∷ []) "Eph.3.11"
∷ word (ἐ ∷ π ∷ ο ∷ ί ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Eph.3.11"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.11"
∷ word (τ ∷ ῷ ∷ []) "Eph.3.11"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.3.11"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.3.11"
∷ word (τ ∷ ῷ ∷ []) "Eph.3.11"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.3.11"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.3.11"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.12"
∷ word (ᾧ ∷ []) "Eph.3.12"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Eph.3.12"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.3.12"
∷ word (π ∷ α ∷ ρ ∷ ρ ∷ η ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Eph.3.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.3.12"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ α ∷ γ ∷ ω ∷ γ ∷ ὴ ∷ ν ∷ []) "Eph.3.12"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.12"
∷ word (π ∷ ε ∷ π ∷ ο ∷ ι ∷ θ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "Eph.3.12"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.3.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.3.12"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.3.12"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.3.12"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "Eph.3.13"
∷ word (α ∷ ἰ ∷ τ ∷ ο ∷ ῦ ∷ μ ∷ α ∷ ι ∷ []) "Eph.3.13"
∷ word (μ ∷ ὴ ∷ []) "Eph.3.13"
∷ word (ἐ ∷ γ ∷ κ ∷ α ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ []) "Eph.3.13"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.13"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Eph.3.13"
∷ word (θ ∷ ∙λ ∷ ί ∷ ψ ∷ ε ∷ σ ∷ ί ∷ ν ∷ []) "Eph.3.13"
∷ word (μ ∷ ο ∷ υ ∷ []) "Eph.3.13"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Eph.3.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.3.13"
∷ word (ἥ ∷ τ ∷ ι ∷ ς ∷ []) "Eph.3.13"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "Eph.3.13"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "Eph.3.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.3.13"
∷ word (Τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "Eph.3.14"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Eph.3.14"
∷ word (κ ∷ ά ∷ μ ∷ π ∷ τ ∷ ω ∷ []) "Eph.3.14"
∷ word (τ ∷ ὰ ∷ []) "Eph.3.14"
∷ word (γ ∷ ό ∷ ν ∷ α ∷ τ ∷ ά ∷ []) "Eph.3.14"
∷ word (μ ∷ ο ∷ υ ∷ []) "Eph.3.14"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.3.14"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.3.14"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Eph.3.14"
∷ word (ἐ ∷ ξ ∷ []) "Eph.3.15"
∷ word (ο ∷ ὗ ∷ []) "Eph.3.15"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ []) "Eph.3.15"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ι ∷ ὰ ∷ []) "Eph.3.15"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.15"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.3.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.3.15"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Eph.3.15"
∷ word (γ ∷ ῆ ∷ ς ∷ []) "Eph.3.15"
∷ word (ὀ ∷ ν ∷ ο ∷ μ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Eph.3.15"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.3.16"
∷ word (δ ∷ ῷ ∷ []) "Eph.3.16"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Eph.3.16"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.3.16"
∷ word (τ ∷ ὸ ∷ []) "Eph.3.16"
∷ word (π ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ς ∷ []) "Eph.3.16"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.3.16"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Eph.3.16"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.3.16"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ []) "Eph.3.16"
∷ word (κ ∷ ρ ∷ α ∷ τ ∷ α ∷ ι ∷ ω ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "Eph.3.16"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.3.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.3.16"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.3.16"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.3.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.3.16"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.3.16"
∷ word (ἔ ∷ σ ∷ ω ∷ []) "Eph.3.16"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Eph.3.16"
∷ word (κ ∷ α ∷ τ ∷ ο ∷ ι ∷ κ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Eph.3.17"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.3.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Eph.3.17"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.3.17"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.3.17"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.3.17"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.17"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Eph.3.17"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Eph.3.17"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.3.17"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.17"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "Eph.3.17"
∷ word (ἐ ∷ ρ ∷ ρ ∷ ι ∷ ζ ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Eph.3.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.3.17"
∷ word (τ ∷ ε ∷ θ ∷ ε ∷ μ ∷ ε ∷ ∙λ ∷ ι ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Eph.3.17"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.3.18"
∷ word (ἐ ∷ ξ ∷ ι ∷ σ ∷ χ ∷ ύ ∷ σ ∷ η ∷ τ ∷ ε ∷ []) "Eph.3.18"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ α ∷ β ∷ έ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Eph.3.18"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Eph.3.18"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "Eph.3.18"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.3.18"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Eph.3.18"
∷ word (τ ∷ ί ∷ []) "Eph.3.18"
∷ word (τ ∷ ὸ ∷ []) "Eph.3.18"
∷ word (π ∷ ∙λ ∷ ά ∷ τ ∷ ο ∷ ς ∷ []) "Eph.3.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.3.18"
∷ word (μ ∷ ῆ ∷ κ ∷ ο ∷ ς ∷ []) "Eph.3.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.3.18"
∷ word (ὕ ∷ ψ ∷ ο ∷ ς ∷ []) "Eph.3.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.3.18"
∷ word (β ∷ ά ∷ θ ∷ ο ∷ ς ∷ []) "Eph.3.18"
∷ word (γ ∷ ν ∷ ῶ ∷ ν ∷ α ∷ ί ∷ []) "Eph.3.19"
∷ word (τ ∷ ε ∷ []) "Eph.3.19"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.3.19"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ ά ∷ ∙λ ∷ ∙λ ∷ ο ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "Eph.3.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.3.19"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.3.19"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "Eph.3.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.3.19"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.3.19"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.3.19"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ω ∷ θ ∷ ῆ ∷ τ ∷ ε ∷ []) "Eph.3.19"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.3.19"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "Eph.3.19"
∷ word (τ ∷ ὸ ∷ []) "Eph.3.19"
∷ word (π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ μ ∷ α ∷ []) "Eph.3.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.3.19"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.3.19"
∷ word (Τ ∷ ῷ ∷ []) "Eph.3.20"
∷ word (δ ∷ ὲ ∷ []) "Eph.3.20"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ μ ∷ έ ∷ ν ∷ ῳ ∷ []) "Eph.3.20"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Eph.3.20"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Eph.3.20"
∷ word (π ∷ ο ∷ ι ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Eph.3.20"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ ε ∷ κ ∷ π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.3.20"
∷ word (ὧ ∷ ν ∷ []) "Eph.3.20"
∷ word (α ∷ ἰ ∷ τ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Eph.3.20"
∷ word (ἢ ∷ []) "Eph.3.20"
∷ word (ν ∷ ο ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "Eph.3.20"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.3.20"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.3.20"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ν ∷ []) "Eph.3.20"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.3.20"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ η ∷ ν ∷ []) "Eph.3.20"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.20"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "Eph.3.20"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Eph.3.21"
∷ word (ἡ ∷ []) "Eph.3.21"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "Eph.3.21"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.21"
∷ word (τ ∷ ῇ ∷ []) "Eph.3.21"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "Eph.3.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.3.21"
∷ word (ἐ ∷ ν ∷ []) "Eph.3.21"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.3.21"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.3.21"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.3.21"
∷ word (π ∷ ά ∷ σ ∷ α ∷ ς ∷ []) "Eph.3.21"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Eph.3.21"
∷ word (γ ∷ ε ∷ ν ∷ ε ∷ ὰ ∷ ς ∷ []) "Eph.3.21"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.3.21"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ ο ∷ ς ∷ []) "Eph.3.21"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.3.21"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ω ∷ ν ∷ []) "Eph.3.21"
∷ word (ἀ ∷ μ ∷ ή ∷ ν ∷ []) "Eph.3.21"
∷ word (Π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "Eph.4.1"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Eph.4.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.4.1"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Eph.4.1"
∷ word (ὁ ∷ []) "Eph.4.1"
∷ word (δ ∷ έ ∷ σ ∷ μ ∷ ι ∷ ο ∷ ς ∷ []) "Eph.4.1"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.1"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.4.1"
∷ word (ἀ ∷ ξ ∷ ί ∷ ω ∷ ς ∷ []) "Eph.4.1"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Eph.4.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.1"
∷ word (κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.4.1"
∷ word (ἧ ∷ ς ∷ []) "Eph.4.1"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ή ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Eph.4.1"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Eph.4.2"
∷ word (π ∷ ά ∷ σ ∷ η ∷ ς ∷ []) "Eph.4.2"
∷ word (τ ∷ α ∷ π ∷ ε ∷ ι ∷ ν ∷ ο ∷ φ ∷ ρ ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "Eph.4.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.2"
∷ word (π ∷ ρ ∷ α ∷ ΰ ∷ τ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "Eph.4.2"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Eph.4.2"
∷ word (μ ∷ α ∷ κ ∷ ρ ∷ ο ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ς ∷ []) "Eph.4.2"
∷ word (ἀ ∷ ν ∷ ε ∷ χ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Eph.4.2"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ω ∷ ν ∷ []) "Eph.4.2"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.2"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "Eph.4.2"
∷ word (σ ∷ π ∷ ο ∷ υ ∷ δ ∷ ά ∷ ζ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.4.3"
∷ word (τ ∷ η ∷ ρ ∷ ε ∷ ῖ ∷ ν ∷ []) "Eph.4.3"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.4.3"
∷ word (ἑ ∷ ν ∷ ό ∷ τ ∷ η ∷ τ ∷ α ∷ []) "Eph.4.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.3"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.4.3"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.3"
∷ word (τ ∷ ῷ ∷ []) "Eph.4.3"
∷ word (σ ∷ υ ∷ ν ∷ δ ∷ έ ∷ σ ∷ μ ∷ ῳ ∷ []) "Eph.4.3"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.3"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ ς ∷ []) "Eph.4.3"
∷ word (ἓ ∷ ν ∷ []) "Eph.4.4"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "Eph.4.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.4"
∷ word (ἓ ∷ ν ∷ []) "Eph.4.4"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Eph.4.4"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Eph.4.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.4"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ή ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Eph.4.4"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.4"
∷ word (μ ∷ ι ∷ ᾷ ∷ []) "Eph.4.4"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ ι ∷ []) "Eph.4.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.4"
∷ word (κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.4.4"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.4.4"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "Eph.4.5"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "Eph.4.5"
∷ word (μ ∷ ί ∷ α ∷ []) "Eph.4.5"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "Eph.4.5"
∷ word (ἓ ∷ ν ∷ []) "Eph.4.5"
∷ word (β ∷ ά ∷ π ∷ τ ∷ ι ∷ σ ∷ μ ∷ α ∷ []) "Eph.4.5"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "Eph.4.6"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Eph.4.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.6"
∷ word (π ∷ α ∷ τ ∷ ὴ ∷ ρ ∷ []) "Eph.4.6"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Eph.4.6"
∷ word (ὁ ∷ []) "Eph.4.6"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Eph.4.6"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Eph.4.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.6"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.4.6"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Eph.4.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.6"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.6"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "Eph.4.6"
∷ word (Ἑ ∷ ν ∷ ὶ ∷ []) "Eph.4.7"
∷ word (δ ∷ ὲ ∷ []) "Eph.4.7"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ ῳ ∷ []) "Eph.4.7"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.4.7"
∷ word (ἐ ∷ δ ∷ ό ∷ θ ∷ η ∷ []) "Eph.4.7"
∷ word (ἡ ∷ []) "Eph.4.7"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Eph.4.7"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.4.7"
∷ word (τ ∷ ὸ ∷ []) "Eph.4.7"
∷ word (μ ∷ έ ∷ τ ∷ ρ ∷ ο ∷ ν ∷ []) "Eph.4.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.7"
∷ word (δ ∷ ω ∷ ρ ∷ ε ∷ ᾶ ∷ ς ∷ []) "Eph.4.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.7"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.4.7"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "Eph.4.8"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Eph.4.8"
∷ word (Ἀ ∷ ν ∷ α ∷ β ∷ ὰ ∷ ς ∷ []) "Eph.4.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.4.8"
∷ word (ὕ ∷ ψ ∷ ο ∷ ς ∷ []) "Eph.4.8"
∷ word (ᾐ ∷ χ ∷ μ ∷ α ∷ ∙λ ∷ ώ ∷ τ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "Eph.4.8"
∷ word (α ∷ ἰ ∷ χ ∷ μ ∷ α ∷ ∙λ ∷ ω ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Eph.4.8"
∷ word (ἔ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "Eph.4.8"
∷ word (δ ∷ ό ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Eph.4.8"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.4.8"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "Eph.4.8"
∷ word (τ ∷ ὸ ∷ []) "Eph.4.9"
∷ word (δ ∷ ὲ ∷ []) "Eph.4.9"
∷ word (Ἀ ∷ ν ∷ έ ∷ β ∷ η ∷ []) "Eph.4.9"
∷ word (τ ∷ ί ∷ []) "Eph.4.9"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.4.9"
∷ word (ε ∷ ἰ ∷ []) "Eph.4.9"
∷ word (μ ∷ ὴ ∷ []) "Eph.4.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Eph.4.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.9"
∷ word (κ ∷ α ∷ τ ∷ έ ∷ β ∷ η ∷ []) "Eph.4.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.4.9"
∷ word (τ ∷ ὰ ∷ []) "Eph.4.9"
∷ word (κ ∷ α ∷ τ ∷ ώ ∷ τ ∷ ε ∷ ρ ∷ α ∷ []) "Eph.4.9"
∷ word (μ ∷ έ ∷ ρ ∷ η ∷ []) "Eph.4.9"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.9"
∷ word (γ ∷ ῆ ∷ ς ∷ []) "Eph.4.9"
∷ word (ὁ ∷ []) "Eph.4.10"
∷ word (κ ∷ α ∷ τ ∷ α ∷ β ∷ ὰ ∷ ς ∷ []) "Eph.4.10"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ς ∷ []) "Eph.4.10"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.4.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.10"
∷ word (ὁ ∷ []) "Eph.4.10"
∷ word (ἀ ∷ ν ∷ α ∷ β ∷ ὰ ∷ ς ∷ []) "Eph.4.10"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ ά ∷ ν ∷ ω ∷ []) "Eph.4.10"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Eph.4.10"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.4.10"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ῶ ∷ ν ∷ []) "Eph.4.10"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.4.10"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ώ ∷ σ ∷ ῃ ∷ []) "Eph.4.10"
∷ word (τ ∷ ὰ ∷ []) "Eph.4.10"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Eph.4.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.11"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.4.11"
∷ word (ἔ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "Eph.4.11"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.4.11"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Eph.4.11"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Eph.4.11"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.4.11"
∷ word (δ ∷ ὲ ∷ []) "Eph.4.11"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ ή ∷ τ ∷ α ∷ ς ∷ []) "Eph.4.11"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.4.11"
∷ word (δ ∷ ὲ ∷ []) "Eph.4.11"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ι ∷ σ ∷ τ ∷ ά ∷ ς ∷ []) "Eph.4.11"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.4.11"
∷ word (δ ∷ ὲ ∷ []) "Eph.4.11"
∷ word (π ∷ ο ∷ ι ∷ μ ∷ έ ∷ ν ∷ α ∷ ς ∷ []) "Eph.4.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.11"
∷ word (δ ∷ ι ∷ δ ∷ α ∷ σ ∷ κ ∷ ά ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Eph.4.11"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.4.12"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.4.12"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ τ ∷ ι ∷ σ ∷ μ ∷ ὸ ∷ ν ∷ []) "Eph.4.12"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.4.12"
∷ word (ἁ ∷ γ ∷ ί ∷ ω ∷ ν ∷ []) "Eph.4.12"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.4.12"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ν ∷ []) "Eph.4.12"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ ς ∷ []) "Eph.4.12"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.4.12"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "Eph.4.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.12"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.4.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.12"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.4.12"
∷ word (μ ∷ έ ∷ χ ∷ ρ ∷ ι ∷ []) "Eph.4.13"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ν ∷ τ ∷ ή ∷ σ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Eph.4.13"
∷ word (ο ∷ ἱ ∷ []) "Eph.4.13"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.4.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.4.13"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.4.13"
∷ word (ἑ ∷ ν ∷ ό ∷ τ ∷ η ∷ τ ∷ α ∷ []) "Eph.4.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.13"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.4.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.13"
∷ word (ἐ ∷ π ∷ ι ∷ γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.4.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.13"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "Eph.4.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.13"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.4.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.4.13"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ α ∷ []) "Eph.4.13"
∷ word (τ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ ο ∷ ν ∷ []) "Eph.4.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.4.13"
∷ word (μ ∷ έ ∷ τ ∷ ρ ∷ ο ∷ ν ∷ []) "Eph.4.13"
∷ word (ἡ ∷ ∙λ ∷ ι ∷ κ ∷ ί ∷ α ∷ ς ∷ []) "Eph.4.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.13"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.4.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.13"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.4.13"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.4.14"
∷ word (μ ∷ η ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Eph.4.14"
∷ word (ὦ ∷ μ ∷ ε ∷ ν ∷ []) "Eph.4.14"
∷ word (ν ∷ ή ∷ π ∷ ι ∷ ο ∷ ι ∷ []) "Eph.4.14"
∷ word (κ ∷ ∙λ ∷ υ ∷ δ ∷ ω ∷ ν ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Eph.4.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.14"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ φ ∷ ε ∷ ρ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Eph.4.14"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "Eph.4.14"
∷ word (ἀ ∷ ν ∷ έ ∷ μ ∷ ῳ ∷ []) "Eph.4.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.14"
∷ word (δ ∷ ι ∷ δ ∷ α ∷ σ ∷ κ ∷ α ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "Eph.4.14"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.14"
∷ word (τ ∷ ῇ ∷ []) "Eph.4.14"
∷ word (κ ∷ υ ∷ β ∷ ε ∷ ί ∷ ᾳ ∷ []) "Eph.4.14"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.4.14"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "Eph.4.14"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.14"
∷ word (π ∷ α ∷ ν ∷ ο ∷ υ ∷ ρ ∷ γ ∷ ί ∷ ᾳ ∷ []) "Eph.4.14"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.4.14"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.4.14"
∷ word (μ ∷ ε ∷ θ ∷ ο ∷ δ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "Eph.4.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.14"
∷ word (π ∷ ∙λ ∷ ά ∷ ν ∷ η ∷ ς ∷ []) "Eph.4.14"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.4.15"
∷ word (δ ∷ ὲ ∷ []) "Eph.4.15"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.15"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "Eph.4.15"
∷ word (α ∷ ὐ ∷ ξ ∷ ή ∷ σ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Eph.4.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.4.15"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Eph.4.15"
∷ word (τ ∷ ὰ ∷ []) "Eph.4.15"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Eph.4.15"
∷ word (ὅ ∷ ς ∷ []) "Eph.4.15"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.4.15"
∷ word (ἡ ∷ []) "Eph.4.15"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ή ∷ []) "Eph.4.15"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "Eph.4.15"
∷ word (ἐ ∷ ξ ∷ []) "Eph.4.16"
∷ word (ο ∷ ὗ ∷ []) "Eph.4.16"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "Eph.4.16"
∷ word (τ ∷ ὸ ∷ []) "Eph.4.16"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "Eph.4.16"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ ρ ∷ μ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "Eph.4.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.16"
∷ word (σ ∷ υ ∷ μ ∷ β ∷ ι ∷ β ∷ α ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "Eph.4.16"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.4.16"
∷ word (π ∷ ά ∷ σ ∷ η ∷ ς ∷ []) "Eph.4.16"
∷ word (ἁ ∷ φ ∷ ῆ ∷ ς ∷ []) "Eph.4.16"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.16"
∷ word (ἐ ∷ π ∷ ι ∷ χ ∷ ο ∷ ρ ∷ η ∷ γ ∷ ί ∷ α ∷ ς ∷ []) "Eph.4.16"
∷ word (κ ∷ α ∷ τ ∷ []) "Eph.4.16"
∷ word (ἐ ∷ ν ∷ έ ∷ ρ ∷ γ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Eph.4.16"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.16"
∷ word (μ ∷ έ ∷ τ ∷ ρ ∷ ῳ ∷ []) "Eph.4.16"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Eph.4.16"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ ο ∷ υ ∷ []) "Eph.4.16"
∷ word (μ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "Eph.4.16"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.4.16"
∷ word (α ∷ ὔ ∷ ξ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "Eph.4.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.16"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.4.16"
∷ word (π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "Eph.4.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.4.16"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "Eph.4.16"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.4.16"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.16"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "Eph.4.16"
∷ word (Τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Eph.4.17"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Eph.4.17"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Eph.4.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.17"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ ύ ∷ ρ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Eph.4.17"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.17"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.4.17"
∷ word (μ ∷ η ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Eph.4.17"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.4.17"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ε ∷ ῖ ∷ ν ∷ []) "Eph.4.17"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Eph.4.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.17"
∷ word (τ ∷ ὰ ∷ []) "Eph.4.17"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Eph.4.17"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ε ∷ ῖ ∷ []) "Eph.4.17"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.17"
∷ word (μ ∷ α ∷ τ ∷ α ∷ ι ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "Eph.4.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.17"
∷ word (ν ∷ ο ∷ ὸ ∷ ς ∷ []) "Eph.4.17"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Eph.4.17"
∷ word (ἐ ∷ σ ∷ κ ∷ ο ∷ τ ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Eph.4.18"
∷ word (τ ∷ ῇ ∷ []) "Eph.4.18"
∷ word (δ ∷ ι ∷ α ∷ ν ∷ ο ∷ ί ∷ ᾳ ∷ []) "Eph.4.18"
∷ word (ὄ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.4.18"
∷ word (ἀ ∷ π ∷ η ∷ ∙λ ∷ ∙λ ∷ ο ∷ τ ∷ ρ ∷ ι ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Eph.4.18"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.18"
∷ word (ζ ∷ ω ∷ ῆ ∷ ς ∷ []) "Eph.4.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.18"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.4.18"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.4.18"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.4.18"
∷ word (ἄ ∷ γ ∷ ν ∷ ο ∷ ι ∷ α ∷ ν ∷ []) "Eph.4.18"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.4.18"
∷ word (ο ∷ ὖ ∷ σ ∷ α ∷ ν ∷ []) "Eph.4.18"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.18"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.4.18"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.4.18"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.4.18"
∷ word (π ∷ ώ ∷ ρ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Eph.4.18"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.18"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "Eph.4.18"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Eph.4.18"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "Eph.4.19"
∷ word (ἀ ∷ π ∷ η ∷ ∙λ ∷ γ ∷ η ∷ κ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "Eph.4.19"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.4.19"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ δ ∷ ω ∷ κ ∷ α ∷ ν ∷ []) "Eph.4.19"
∷ word (τ ∷ ῇ ∷ []) "Eph.4.19"
∷ word (ἀ ∷ σ ∷ ε ∷ ∙λ ∷ γ ∷ ε ∷ ί ∷ ᾳ ∷ []) "Eph.4.19"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.4.19"
∷ word (ἐ ∷ ρ ∷ γ ∷ α ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Eph.4.19"
∷ word (ἀ ∷ κ ∷ α ∷ θ ∷ α ∷ ρ ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Eph.4.19"
∷ word (π ∷ ά ∷ σ ∷ η ∷ ς ∷ []) "Eph.4.19"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.19"
∷ word (π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ ε ∷ ξ ∷ ί ∷ ᾳ ∷ []) "Eph.4.19"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Eph.4.20"
∷ word (δ ∷ ὲ ∷ []) "Eph.4.20"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "Eph.4.20"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Eph.4.20"
∷ word (ἐ ∷ μ ∷ ά ∷ θ ∷ ε ∷ τ ∷ ε ∷ []) "Eph.4.20"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.4.20"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ν ∷ []) "Eph.4.20"
∷ word (ε ∷ ἴ ∷ []) "Eph.4.21"
∷ word (γ ∷ ε ∷ []) "Eph.4.21"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Eph.4.21"
∷ word (ἠ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Eph.4.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.21"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.21"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Eph.4.21"
∷ word (ἐ ∷ δ ∷ ι ∷ δ ∷ ά ∷ χ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Eph.4.21"
∷ word (κ ∷ α ∷ θ ∷ ώ ∷ ς ∷ []) "Eph.4.21"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.4.21"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ []) "Eph.4.21"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.21"
∷ word (τ ∷ ῷ ∷ []) "Eph.4.21"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.4.21"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ έ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Eph.4.22"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.4.22"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.4.22"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.4.22"
∷ word (π ∷ ρ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "Eph.4.22"
∷ word (ἀ ∷ ν ∷ α ∷ σ ∷ τ ∷ ρ ∷ ο ∷ φ ∷ ὴ ∷ ν ∷ []) "Eph.4.22"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.4.22"
∷ word (π ∷ α ∷ ∙λ ∷ α ∷ ι ∷ ὸ ∷ ν ∷ []) "Eph.4.22"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Eph.4.22"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.4.22"
∷ word (φ ∷ θ ∷ ε ∷ ι ∷ ρ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "Eph.4.22"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.4.22"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Eph.4.22"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ς ∷ []) "Eph.4.22"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.22"
∷ word (ἀ ∷ π ∷ ά ∷ τ ∷ η ∷ ς ∷ []) "Eph.4.22"
∷ word (ἀ ∷ ν ∷ α ∷ ν ∷ ε ∷ ο ∷ ῦ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Eph.4.23"
∷ word (δ ∷ ὲ ∷ []) "Eph.4.23"
∷ word (τ ∷ ῷ ∷ []) "Eph.4.23"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Eph.4.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.23"
∷ word (ν ∷ ο ∷ ὸ ∷ ς ∷ []) "Eph.4.23"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.4.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.24"
∷ word (ἐ ∷ ν ∷ δ ∷ ύ ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Eph.4.24"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.4.24"
∷ word (κ ∷ α ∷ ι ∷ ν ∷ ὸ ∷ ν ∷ []) "Eph.4.24"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Eph.4.24"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.4.24"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.4.24"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "Eph.4.24"
∷ word (κ ∷ τ ∷ ι ∷ σ ∷ θ ∷ έ ∷ ν ∷ τ ∷ α ∷ []) "Eph.4.24"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.24"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ ῃ ∷ []) "Eph.4.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.24"
∷ word (ὁ ∷ σ ∷ ι ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "Eph.4.24"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.24"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Eph.4.24"
∷ word (Δ ∷ ι ∷ ὸ ∷ []) "Eph.4.25"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ έ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Eph.4.25"
∷ word (τ ∷ ὸ ∷ []) "Eph.4.25"
∷ word (ψ ∷ ε ∷ ῦ ∷ δ ∷ ο ∷ ς ∷ []) "Eph.4.25"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Eph.4.25"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Eph.4.25"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "Eph.4.25"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Eph.4.25"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.25"
∷ word (π ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ο ∷ ν ∷ []) "Eph.4.25"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.4.25"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Eph.4.25"
∷ word (ἐ ∷ σ ∷ μ ∷ ὲ ∷ ν ∷ []) "Eph.4.25"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ω ∷ ν ∷ []) "Eph.4.25"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "Eph.4.25"
∷ word (ὀ ∷ ρ ∷ γ ∷ ί ∷ ζ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Eph.4.26"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.26"
∷ word (μ ∷ ὴ ∷ []) "Eph.4.26"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ά ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "Eph.4.26"
∷ word (ὁ ∷ []) "Eph.4.26"
∷ word (ἥ ∷ ∙λ ∷ ι ∷ ο ∷ ς ∷ []) "Eph.4.26"
∷ word (μ ∷ ὴ ∷ []) "Eph.4.26"
∷ word (ἐ ∷ π ∷ ι ∷ δ ∷ υ ∷ έ ∷ τ ∷ ω ∷ []) "Eph.4.26"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Eph.4.26"
∷ word (π ∷ α ∷ ρ ∷ ο ∷ ρ ∷ γ ∷ ι ∷ σ ∷ μ ∷ ῷ ∷ []) "Eph.4.26"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.4.26"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "Eph.4.27"
∷ word (δ ∷ ί ∷ δ ∷ ο ∷ τ ∷ ε ∷ []) "Eph.4.27"
∷ word (τ ∷ ό ∷ π ∷ ο ∷ ν ∷ []) "Eph.4.27"
∷ word (τ ∷ ῷ ∷ []) "Eph.4.27"
∷ word (δ ∷ ι ∷ α ∷ β ∷ ό ∷ ∙λ ∷ ῳ ∷ []) "Eph.4.27"
∷ word (ὁ ∷ []) "Eph.4.28"
∷ word (κ ∷ ∙λ ∷ έ ∷ π ∷ τ ∷ ω ∷ ν ∷ []) "Eph.4.28"
∷ word (μ ∷ η ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Eph.4.28"
∷ word (κ ∷ ∙λ ∷ ε ∷ π ∷ τ ∷ έ ∷ τ ∷ ω ∷ []) "Eph.4.28"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Eph.4.28"
∷ word (δ ∷ ὲ ∷ []) "Eph.4.28"
∷ word (κ ∷ ο ∷ π ∷ ι ∷ ά ∷ τ ∷ ω ∷ []) "Eph.4.28"
∷ word (ἐ ∷ ρ ∷ γ ∷ α ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Eph.4.28"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Eph.4.28"
∷ word (ἰ ∷ δ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Eph.4.28"
∷ word (χ ∷ ε ∷ ρ ∷ σ ∷ ὶ ∷ ν ∷ []) "Eph.4.28"
∷ word (τ ∷ ὸ ∷ []) "Eph.4.28"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ό ∷ ν ∷ []) "Eph.4.28"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.4.28"
∷ word (ἔ ∷ χ ∷ ῃ ∷ []) "Eph.4.28"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ δ ∷ ι ∷ δ ∷ ό ∷ ν ∷ α ∷ ι ∷ []) "Eph.4.28"
∷ word (τ ∷ ῷ ∷ []) "Eph.4.28"
∷ word (χ ∷ ρ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "Eph.4.28"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ι ∷ []) "Eph.4.28"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "Eph.4.29"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "Eph.4.29"
∷ word (σ ∷ α ∷ π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.4.29"
∷ word (ἐ ∷ κ ∷ []) "Eph.4.29"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.29"
∷ word (σ ∷ τ ∷ ό ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.4.29"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.4.29"
∷ word (μ ∷ ὴ ∷ []) "Eph.4.29"
∷ word (ἐ ∷ κ ∷ π ∷ ο ∷ ρ ∷ ε ∷ υ ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "Eph.4.29"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Eph.4.29"
∷ word (ε ∷ ἴ ∷ []) "Eph.4.29"
∷ word (τ ∷ ι ∷ ς ∷ []) "Eph.4.29"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ὸ ∷ ς ∷ []) "Eph.4.29"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.4.29"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "Eph.4.29"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.4.29"
∷ word (χ ∷ ρ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Eph.4.29"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.4.29"
∷ word (δ ∷ ῷ ∷ []) "Eph.4.29"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Eph.4.29"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.4.29"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Eph.4.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.30"
∷ word (μ ∷ ὴ ∷ []) "Eph.4.30"
∷ word (∙λ ∷ υ ∷ π ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Eph.4.30"
∷ word (τ ∷ ὸ ∷ []) "Eph.4.30"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Eph.4.30"
∷ word (τ ∷ ὸ ∷ []) "Eph.4.30"
∷ word (ἅ ∷ γ ∷ ι ∷ ο ∷ ν ∷ []) "Eph.4.30"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.4.30"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.4.30"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.30"
∷ word (ᾧ ∷ []) "Eph.4.30"
∷ word (ἐ ∷ σ ∷ φ ∷ ρ ∷ α ∷ γ ∷ ί ∷ σ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Eph.4.30"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.4.30"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "Eph.4.30"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ υ ∷ τ ∷ ρ ∷ ώ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.4.30"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ []) "Eph.4.31"
∷ word (π ∷ ι ∷ κ ∷ ρ ∷ ί ∷ α ∷ []) "Eph.4.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.31"
∷ word (θ ∷ υ ∷ μ ∷ ὸ ∷ ς ∷ []) "Eph.4.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.31"
∷ word (ὀ ∷ ρ ∷ γ ∷ ὴ ∷ []) "Eph.4.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.31"
∷ word (κ ∷ ρ ∷ α ∷ υ ∷ γ ∷ ὴ ∷ []) "Eph.4.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.31"
∷ word (β ∷ ∙λ ∷ α ∷ σ ∷ φ ∷ η ∷ μ ∷ ί ∷ α ∷ []) "Eph.4.31"
∷ word (ἀ ∷ ρ ∷ θ ∷ ή ∷ τ ∷ ω ∷ []) "Eph.4.31"
∷ word (ἀ ∷ φ ∷ []) "Eph.4.31"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.4.31"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Eph.4.31"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "Eph.4.31"
∷ word (κ ∷ α ∷ κ ∷ ί ∷ ᾳ ∷ []) "Eph.4.31"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Eph.4.32"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.4.32"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Eph.4.32"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ο ∷ ί ∷ []) "Eph.4.32"
∷ word (ε ∷ ὔ ∷ σ ∷ π ∷ ∙λ ∷ α ∷ γ ∷ χ ∷ ν ∷ ο ∷ ι ∷ []) "Eph.4.32"
∷ word (χ ∷ α ∷ ρ ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Eph.4.32"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.4.32"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Eph.4.32"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.4.32"
∷ word (ὁ ∷ []) "Eph.4.32"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Eph.4.32"
∷ word (ἐ ∷ ν ∷ []) "Eph.4.32"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.4.32"
∷ word (ἐ ∷ χ ∷ α ∷ ρ ∷ ί ∷ σ ∷ α ∷ τ ∷ ο ∷ []) "Eph.4.32"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Eph.4.32"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Eph.5.1"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Eph.5.1"
∷ word (μ ∷ ι ∷ μ ∷ η ∷ τ ∷ α ∷ ὶ ∷ []) "Eph.5.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.5.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.5.1"
∷ word (ὡ ∷ ς ∷ []) "Eph.5.1"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Eph.5.1"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ά ∷ []) "Eph.5.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.2"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Eph.5.2"
∷ word (ἐ ∷ ν ∷ []) "Eph.5.2"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "Eph.5.2"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Eph.5.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.2"
∷ word (ὁ ∷ []) "Eph.5.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.5.2"
∷ word (ἠ ∷ γ ∷ ά ∷ π ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Eph.5.2"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.5.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.2"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "Eph.5.2"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "Eph.5.2"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Eph.5.2"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.5.2"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ φ ∷ ο ∷ ρ ∷ ὰ ∷ ν ∷ []) "Eph.5.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.2"
∷ word (θ ∷ υ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Eph.5.2"
∷ word (τ ∷ ῷ ∷ []) "Eph.5.2"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Eph.5.2"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.5.2"
∷ word (ὀ ∷ σ ∷ μ ∷ ὴ ∷ ν ∷ []) "Eph.5.2"
∷ word (ε ∷ ὐ ∷ ω ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "Eph.5.2"
∷ word (Π ∷ ο ∷ ρ ∷ ν ∷ ε ∷ ί ∷ α ∷ []) "Eph.5.3"
∷ word (δ ∷ ὲ ∷ []) "Eph.5.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.3"
∷ word (ἀ ∷ κ ∷ α ∷ θ ∷ α ∷ ρ ∷ σ ∷ ί ∷ α ∷ []) "Eph.5.3"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ []) "Eph.5.3"
∷ word (ἢ ∷ []) "Eph.5.3"
∷ word (π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ ε ∷ ξ ∷ ί ∷ α ∷ []) "Eph.5.3"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "Eph.5.3"
∷ word (ὀ ∷ ν ∷ ο ∷ μ ∷ α ∷ ζ ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "Eph.5.3"
∷ word (ἐ ∷ ν ∷ []) "Eph.5.3"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Eph.5.3"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Eph.5.3"
∷ word (π ∷ ρ ∷ έ ∷ π ∷ ε ∷ ι ∷ []) "Eph.5.3"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Eph.5.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.4"
∷ word (α ∷ ἰ ∷ σ ∷ χ ∷ ρ ∷ ό ∷ τ ∷ η ∷ ς ∷ []) "Eph.5.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.4"
∷ word (μ ∷ ω ∷ ρ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ []) "Eph.5.4"
∷ word (ἢ ∷ []) "Eph.5.4"
∷ word (ε ∷ ὐ ∷ τ ∷ ρ ∷ α ∷ π ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ []) "Eph.5.4"
∷ word (ἃ ∷ []) "Eph.5.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Eph.5.4"
∷ word (ἀ ∷ ν ∷ ῆ ∷ κ ∷ ε ∷ ν ∷ []) "Eph.5.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Eph.5.4"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Eph.5.4"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ί ∷ α ∷ []) "Eph.5.4"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Eph.5.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Eph.5.5"
∷ word (ἴ ∷ σ ∷ τ ∷ ε ∷ []) "Eph.5.5"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.5.5"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Eph.5.5"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "Eph.5.5"
∷ word (π ∷ ό ∷ ρ ∷ ν ∷ ο ∷ ς ∷ []) "Eph.5.5"
∷ word (ἢ ∷ []) "Eph.5.5"
∷ word (ἀ ∷ κ ∷ ά ∷ θ ∷ α ∷ ρ ∷ τ ∷ ο ∷ ς ∷ []) "Eph.5.5"
∷ word (ἢ ∷ []) "Eph.5.5"
∷ word (π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ έ ∷ κ ∷ τ ∷ η ∷ ς ∷ []) "Eph.5.5"
∷ word (ὅ ∷ []) "Eph.5.5"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.5.5"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ο ∷ ∙λ ∷ ά ∷ τ ∷ ρ ∷ η ∷ ς ∷ []) "Eph.5.5"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Eph.5.5"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "Eph.5.5"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Eph.5.5"
∷ word (ἐ ∷ ν ∷ []) "Eph.5.5"
∷ word (τ ∷ ῇ ∷ []) "Eph.5.5"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ί ∷ ᾳ ∷ []) "Eph.5.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.5.5"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.5.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.5"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.5.5"
∷ word (Μ ∷ η ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "Eph.5.6"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.5.6"
∷ word (ἀ ∷ π ∷ α ∷ τ ∷ ά ∷ τ ∷ ω ∷ []) "Eph.5.6"
∷ word (κ ∷ ε ∷ ν ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.5.6"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ι ∷ ς ∷ []) "Eph.5.6"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.5.6"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "Eph.5.6"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Eph.5.6"
∷ word (ἔ ∷ ρ ∷ χ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Eph.5.6"
∷ word (ἡ ∷ []) "Eph.5.6"
∷ word (ὀ ∷ ρ ∷ γ ∷ ὴ ∷ []) "Eph.5.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.5.6"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.5.6"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Eph.5.6"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.5.6"
∷ word (υ ∷ ἱ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.5.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.5.6"
∷ word (ἀ ∷ π ∷ ε ∷ ι ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Eph.5.6"
∷ word (μ ∷ ὴ ∷ []) "Eph.5.7"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Eph.5.7"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Eph.5.7"
∷ word (σ ∷ υ ∷ μ ∷ μ ∷ έ ∷ τ ∷ ο ∷ χ ∷ ο ∷ ι ∷ []) "Eph.5.7"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Eph.5.7"
∷ word (ἦ ∷ τ ∷ ε ∷ []) "Eph.5.8"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Eph.5.8"
∷ word (π ∷ ο ∷ τ ∷ ε ∷ []) "Eph.5.8"
∷ word (σ ∷ κ ∷ ό ∷ τ ∷ ο ∷ ς ∷ []) "Eph.5.8"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Eph.5.8"
∷ word (δ ∷ ὲ ∷ []) "Eph.5.8"
∷ word (φ ∷ ῶ ∷ ς ∷ []) "Eph.5.8"
∷ word (ἐ ∷ ν ∷ []) "Eph.5.8"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.5.8"
∷ word (ὡ ∷ ς ∷ []) "Eph.5.8"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Eph.5.8"
∷ word (φ ∷ ω ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.5.8"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Eph.5.8"
∷ word (ὁ ∷ []) "Eph.5.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Eph.5.9"
∷ word (κ ∷ α ∷ ρ ∷ π ∷ ὸ ∷ ς ∷ []) "Eph.5.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.5.9"
∷ word (φ ∷ ω ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.5.9"
∷ word (ἐ ∷ ν ∷ []) "Eph.5.9"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "Eph.5.9"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ω ∷ σ ∷ ύ ∷ ν ∷ ῃ ∷ []) "Eph.5.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.9"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ ῃ ∷ []) "Eph.5.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.9"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ ᾳ ∷ []) "Eph.5.9"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ά ∷ ζ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.5.10"
∷ word (τ ∷ ί ∷ []) "Eph.5.10"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.5.10"
∷ word (ε ∷ ὐ ∷ ά ∷ ρ ∷ ε ∷ σ ∷ τ ∷ ο ∷ ν ∷ []) "Eph.5.10"
∷ word (τ ∷ ῷ ∷ []) "Eph.5.10"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.5.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.11"
∷ word (μ ∷ ὴ ∷ []) "Eph.5.11"
∷ word (σ ∷ υ ∷ γ ∷ κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Eph.5.11"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.5.11"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ι ∷ ς ∷ []) "Eph.5.11"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.5.11"
∷ word (ἀ ∷ κ ∷ ά ∷ ρ ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "Eph.5.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.5.11"
∷ word (σ ∷ κ ∷ ό ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Eph.5.11"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Eph.5.11"
∷ word (δ ∷ ὲ ∷ []) "Eph.5.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.11"
∷ word (ἐ ∷ ∙λ ∷ έ ∷ γ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "Eph.5.11"
∷ word (τ ∷ ὰ ∷ []) "Eph.5.12"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Eph.5.12"
∷ word (κ ∷ ρ ∷ υ ∷ φ ∷ ῇ ∷ []) "Eph.5.12"
∷ word (γ ∷ ι ∷ ν ∷ ό ∷ μ ∷ ε ∷ ν ∷ α ∷ []) "Eph.5.12"
∷ word (ὑ ∷ π ∷ []) "Eph.5.12"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Eph.5.12"
∷ word (α ∷ ἰ ∷ σ ∷ χ ∷ ρ ∷ ό ∷ ν ∷ []) "Eph.5.12"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.5.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.12"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ ν ∷ []) "Eph.5.12"
∷ word (τ ∷ ὰ ∷ []) "Eph.5.13"
∷ word (δ ∷ ὲ ∷ []) "Eph.5.13"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Eph.5.13"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ γ ∷ χ ∷ ό ∷ μ ∷ ε ∷ ν ∷ α ∷ []) "Eph.5.13"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Eph.5.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.5.13"
∷ word (φ ∷ ω ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.5.13"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ο ∷ ῦ ∷ τ ∷ α ∷ ι ∷ []) "Eph.5.13"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "Eph.5.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Eph.5.14"
∷ word (τ ∷ ὸ ∷ []) "Eph.5.14"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "Eph.5.14"
∷ word (φ ∷ ῶ ∷ ς ∷ []) "Eph.5.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.5.14"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "Eph.5.14"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Eph.5.14"
∷ word (Ἔ ∷ γ ∷ ε ∷ ι ∷ ρ ∷ ε ∷ []) "Eph.5.14"
∷ word (ὁ ∷ []) "Eph.5.14"
∷ word (κ ∷ α ∷ θ ∷ ε ∷ ύ ∷ δ ∷ ω ∷ ν ∷ []) "Eph.5.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.14"
∷ word (ἀ ∷ ν ∷ ά ∷ σ ∷ τ ∷ α ∷ []) "Eph.5.14"
∷ word (ἐ ∷ κ ∷ []) "Eph.5.14"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.5.14"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Eph.5.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.14"
∷ word (ἐ ∷ π ∷ ι ∷ φ ∷ α ∷ ύ ∷ σ ∷ ε ∷ ι ∷ []) "Eph.5.14"
∷ word (σ ∷ ο ∷ ι ∷ []) "Eph.5.14"
∷ word (ὁ ∷ []) "Eph.5.14"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "Eph.5.14"
∷ word (Β ∷ ∙λ ∷ έ ∷ π ∷ ε ∷ τ ∷ ε ∷ []) "Eph.5.15"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Eph.5.15"
∷ word (ἀ ∷ κ ∷ ρ ∷ ι ∷ β ∷ ῶ ∷ ς ∷ []) "Eph.5.15"
∷ word (π ∷ ῶ ∷ ς ∷ []) "Eph.5.15"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Eph.5.15"
∷ word (μ ∷ ὴ ∷ []) "Eph.5.15"
∷ word (ὡ ∷ ς ∷ []) "Eph.5.15"
∷ word (ἄ ∷ σ ∷ ο ∷ φ ∷ ο ∷ ι ∷ []) "Eph.5.15"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Eph.5.15"
∷ word (ὡ ∷ ς ∷ []) "Eph.5.15"
∷ word (σ ∷ ο ∷ φ ∷ ο ∷ ί ∷ []) "Eph.5.15"
∷ word (ἐ ∷ ξ ∷ α ∷ γ ∷ ο ∷ ρ ∷ α ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Eph.5.16"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.5.16"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ό ∷ ν ∷ []) "Eph.5.16"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Eph.5.16"
∷ word (α ∷ ἱ ∷ []) "Eph.5.16"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ι ∷ []) "Eph.5.16"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ α ∷ ί ∷ []) "Eph.5.16"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "Eph.5.16"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.5.17"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Eph.5.17"
∷ word (μ ∷ ὴ ∷ []) "Eph.5.17"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Eph.5.17"
∷ word (ἄ ∷ φ ∷ ρ ∷ ο ∷ ν ∷ ε ∷ ς ∷ []) "Eph.5.17"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Eph.5.17"
∷ word (σ ∷ υ ∷ ν ∷ ί ∷ ε ∷ τ ∷ ε ∷ []) "Eph.5.17"
∷ word (τ ∷ ί ∷ []) "Eph.5.17"
∷ word (τ ∷ ὸ ∷ []) "Eph.5.17"
∷ word (θ ∷ έ ∷ ∙λ ∷ η ∷ μ ∷ α ∷ []) "Eph.5.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.5.17"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Eph.5.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.18"
∷ word (μ ∷ ὴ ∷ []) "Eph.5.18"
∷ word (μ ∷ ε ∷ θ ∷ ύ ∷ σ ∷ κ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Eph.5.18"
∷ word (ο ∷ ἴ ∷ ν ∷ ῳ ∷ []) "Eph.5.18"
∷ word (ἐ ∷ ν ∷ []) "Eph.5.18"
∷ word (ᾧ ∷ []) "Eph.5.18"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.5.18"
∷ word (ἀ ∷ σ ∷ ω ∷ τ ∷ ί ∷ α ∷ []) "Eph.5.18"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Eph.5.18"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ῦ ∷ σ ∷ θ ∷ ε ∷ []) "Eph.5.18"
∷ word (ἐ ∷ ν ∷ []) "Eph.5.18"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Eph.5.18"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.5.19"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.5.19"
∷ word (ψ ∷ α ∷ ∙λ ∷ μ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.5.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.19"
∷ word (ὕ ∷ μ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "Eph.5.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.19"
∷ word (ᾠ ∷ δ ∷ α ∷ ῖ ∷ ς ∷ []) "Eph.5.19"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ α ∷ ῖ ∷ ς ∷ []) "Eph.5.19"
∷ word (ᾄ ∷ δ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.5.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.19"
∷ word (ψ ∷ ά ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.5.19"
∷ word (τ ∷ ῇ ∷ []) "Eph.5.19"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ ᾳ ∷ []) "Eph.5.19"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.5.19"
∷ word (τ ∷ ῷ ∷ []) "Eph.5.19"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.5.19"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.5.20"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ο ∷ τ ∷ ε ∷ []) "Eph.5.20"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Eph.5.20"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Eph.5.20"
∷ word (ἐ ∷ ν ∷ []) "Eph.5.20"
∷ word (ὀ ∷ ν ∷ ό ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Eph.5.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.5.20"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Eph.5.20"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.5.20"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.5.20"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.5.20"
∷ word (τ ∷ ῷ ∷ []) "Eph.5.20"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Eph.5.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.20"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ί ∷ []) "Eph.5.20"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ α ∷ σ ∷ σ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Eph.5.21"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "Eph.5.21"
∷ word (ἐ ∷ ν ∷ []) "Eph.5.21"
∷ word (φ ∷ ό ∷ β ∷ ῳ ∷ []) "Eph.5.21"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.5.21"
∷ word (Α ∷ ἱ ∷ []) "Eph.5.22"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ ε ∷ ς ∷ []) "Eph.5.22"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.5.22"
∷ word (ἰ ∷ δ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Eph.5.22"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ά ∷ σ ∷ ι ∷ ν ∷ []) "Eph.5.22"
∷ word (ὡ ∷ ς ∷ []) "Eph.5.22"
∷ word (τ ∷ ῷ ∷ []) "Eph.5.22"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.5.22"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Eph.5.23"
∷ word (ἀ ∷ ν ∷ ή ∷ ρ ∷ []) "Eph.5.23"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.5.23"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ὴ ∷ []) "Eph.5.23"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.5.23"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ὸ ∷ ς ∷ []) "Eph.5.23"
∷ word (ὡ ∷ ς ∷ []) "Eph.5.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.23"
∷ word (ὁ ∷ []) "Eph.5.23"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.5.23"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ὴ ∷ []) "Eph.5.23"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.5.23"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Eph.5.23"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.5.23"
∷ word (σ ∷ ω ∷ τ ∷ ὴ ∷ ρ ∷ []) "Eph.5.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.5.23"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.5.23"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Eph.5.24"
∷ word (ὡ ∷ ς ∷ []) "Eph.5.24"
∷ word (ἡ ∷ []) "Eph.5.24"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ []) "Eph.5.24"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ ά ∷ σ ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Eph.5.24"
∷ word (τ ∷ ῷ ∷ []) "Eph.5.24"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.5.24"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Eph.5.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.24"
∷ word (α ∷ ἱ ∷ []) "Eph.5.24"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ ε ∷ ς ∷ []) "Eph.5.24"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.5.24"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ά ∷ σ ∷ ι ∷ ν ∷ []) "Eph.5.24"
∷ word (ἐ ∷ ν ∷ []) "Eph.5.24"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ί ∷ []) "Eph.5.24"
∷ word (Ο ∷ ἱ ∷ []) "Eph.5.25"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ ε ∷ ς ∷ []) "Eph.5.25"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ᾶ ∷ τ ∷ ε ∷ []) "Eph.5.25"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Eph.5.25"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ ς ∷ []) "Eph.5.25"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Eph.5.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.25"
∷ word (ὁ ∷ []) "Eph.5.25"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.5.25"
∷ word (ἠ ∷ γ ∷ ά ∷ π ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Eph.5.25"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.5.25"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Eph.5.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.25"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "Eph.5.25"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "Eph.5.25"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Eph.5.25"
∷ word (α ∷ ὐ ∷ τ ∷ ῆ ∷ ς ∷ []) "Eph.5.25"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.5.26"
∷ word (α ∷ ὐ ∷ τ ∷ ὴ ∷ ν ∷ []) "Eph.5.26"
∷ word (ἁ ∷ γ ∷ ι ∷ ά ∷ σ ∷ ῃ ∷ []) "Eph.5.26"
∷ word (κ ∷ α ∷ θ ∷ α ∷ ρ ∷ ί ∷ σ ∷ α ∷ ς ∷ []) "Eph.5.26"
∷ word (τ ∷ ῷ ∷ []) "Eph.5.26"
∷ word (∙λ ∷ ο ∷ υ ∷ τ ∷ ρ ∷ ῷ ∷ []) "Eph.5.26"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.5.26"
∷ word (ὕ ∷ δ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.5.26"
∷ word (ἐ ∷ ν ∷ []) "Eph.5.26"
∷ word (ῥ ∷ ή ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Eph.5.26"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.5.27"
∷ word (π ∷ α ∷ ρ ∷ α ∷ σ ∷ τ ∷ ή ∷ σ ∷ ῃ ∷ []) "Eph.5.27"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.5.27"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "Eph.5.27"
∷ word (ἔ ∷ ν ∷ δ ∷ ο ∷ ξ ∷ ο ∷ ν ∷ []) "Eph.5.27"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.5.27"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Eph.5.27"
∷ word (μ ∷ ὴ ∷ []) "Eph.5.27"
∷ word (ἔ ∷ χ ∷ ο ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "Eph.5.27"
∷ word (σ ∷ π ∷ ί ∷ ∙λ ∷ ο ∷ ν ∷ []) "Eph.5.27"
∷ word (ἢ ∷ []) "Eph.5.27"
∷ word (ῥ ∷ υ ∷ τ ∷ ί ∷ δ ∷ α ∷ []) "Eph.5.27"
∷ word (ἤ ∷ []) "Eph.5.27"
∷ word (τ ∷ ι ∷ []) "Eph.5.27"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.5.27"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ύ ∷ τ ∷ ω ∷ ν ∷ []) "Eph.5.27"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Eph.5.27"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.5.27"
∷ word (ᾖ ∷ []) "Eph.5.27"
∷ word (ἁ ∷ γ ∷ ί ∷ α ∷ []) "Eph.5.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.27"
∷ word (ἄ ∷ μ ∷ ω ∷ μ ∷ ο ∷ ς ∷ []) "Eph.5.27"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Eph.5.28"
∷ word (ὀ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Eph.5.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.28"
∷ word (ο ∷ ἱ ∷ []) "Eph.5.28"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ ε ∷ ς ∷ []) "Eph.5.28"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ᾶ ∷ ν ∷ []) "Eph.5.28"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Eph.5.28"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῶ ∷ ν ∷ []) "Eph.5.28"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ ς ∷ []) "Eph.5.28"
∷ word (ὡ ∷ ς ∷ []) "Eph.5.28"
∷ word (τ ∷ ὰ ∷ []) "Eph.5.28"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῶ ∷ ν ∷ []) "Eph.5.28"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Eph.5.28"
∷ word (ὁ ∷ []) "Eph.5.28"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ ν ∷ []) "Eph.5.28"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.5.28"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.5.28"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ []) "Eph.5.28"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "Eph.5.28"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ᾷ ∷ []) "Eph.5.28"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "Eph.5.29"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Eph.5.29"
∷ word (π ∷ ο ∷ τ ∷ ε ∷ []) "Eph.5.29"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.5.29"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.5.29"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Eph.5.29"
∷ word (ἐ ∷ μ ∷ ί ∷ σ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Eph.5.29"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Eph.5.29"
∷ word (ἐ ∷ κ ∷ τ ∷ ρ ∷ έ ∷ φ ∷ ε ∷ ι ∷ []) "Eph.5.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.29"
∷ word (θ ∷ ά ∷ ∙λ ∷ π ∷ ε ∷ ι ∷ []) "Eph.5.29"
∷ word (α ∷ ὐ ∷ τ ∷ ή ∷ ν ∷ []) "Eph.5.29"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Eph.5.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.29"
∷ word (ὁ ∷ []) "Eph.5.29"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.5.29"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.5.29"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Eph.5.29"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Eph.5.30"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "Eph.5.30"
∷ word (ἐ ∷ σ ∷ μ ∷ ὲ ∷ ν ∷ []) "Eph.5.30"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.5.30"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.5.30"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.5.30"
∷ word (ἀ ∷ ν ∷ τ ∷ ὶ ∷ []) "Eph.5.31"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "Eph.5.31"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ε ∷ ί ∷ ψ ∷ ε ∷ ι ∷ []) "Eph.5.31"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "Eph.5.31"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.5.31"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Eph.5.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.31"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.5.31"
∷ word (μ ∷ η ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Eph.5.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.31"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ κ ∷ ο ∷ ∙λ ∷ ∙λ ∷ η ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Eph.5.31"
∷ word (τ ∷ ῇ ∷ []) "Eph.5.31"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ὶ ∷ []) "Eph.5.31"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.5.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.31"
∷ word (ἔ ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Eph.5.31"
∷ word (ο ∷ ἱ ∷ []) "Eph.5.31"
∷ word (δ ∷ ύ ∷ ο ∷ []) "Eph.5.31"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.5.31"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Eph.5.31"
∷ word (μ ∷ ί ∷ α ∷ ν ∷ []) "Eph.5.31"
∷ word (τ ∷ ὸ ∷ []) "Eph.5.32"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Eph.5.32"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Eph.5.32"
∷ word (μ ∷ έ ∷ γ ∷ α ∷ []) "Eph.5.32"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "Eph.5.32"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Eph.5.32"
∷ word (δ ∷ ὲ ∷ []) "Eph.5.32"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Eph.5.32"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.5.32"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Eph.5.32"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.32"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.5.32"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.5.32"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Eph.5.32"
∷ word (π ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "Eph.5.33"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.5.33"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Eph.5.33"
∷ word (ο ∷ ἱ ∷ []) "Eph.5.33"
∷ word (κ ∷ α ∷ θ ∷ []) "Eph.5.33"
∷ word (ἕ ∷ ν ∷ α ∷ []) "Eph.5.33"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "Eph.5.33"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.5.33"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.5.33"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ []) "Eph.5.33"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Eph.5.33"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ά ∷ τ ∷ ω ∷ []) "Eph.5.33"
∷ word (ὡ ∷ ς ∷ []) "Eph.5.33"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ό ∷ ν ∷ []) "Eph.5.33"
∷ word (ἡ ∷ []) "Eph.5.33"
∷ word (δ ∷ ὲ ∷ []) "Eph.5.33"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "Eph.5.33"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.5.33"
∷ word (φ ∷ ο ∷ β ∷ ῆ ∷ τ ∷ α ∷ ι ∷ []) "Eph.5.33"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.5.33"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ α ∷ []) "Eph.5.33"
∷ word (Τ ∷ ὰ ∷ []) "Eph.6.1"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Eph.6.1"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ύ ∷ ε ∷ τ ∷ ε ∷ []) "Eph.6.1"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.6.1"
∷ word (γ ∷ ο ∷ ν ∷ ε ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Eph.6.1"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.6.1"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.1"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.6.1"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Eph.6.1"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Eph.6.1"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.6.1"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ ο ∷ ν ∷ []) "Eph.6.1"
∷ word (τ ∷ ί ∷ μ ∷ α ∷ []) "Eph.6.2"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.6.2"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Eph.6.2"
∷ word (σ ∷ ο ∷ υ ∷ []) "Eph.6.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.2"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.6.2"
∷ word (μ ∷ η ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "Eph.6.2"
∷ word (ἥ ∷ τ ∷ ι ∷ ς ∷ []) "Eph.6.2"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "Eph.6.2"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ []) "Eph.6.2"
∷ word (π ∷ ρ ∷ ώ ∷ τ ∷ η ∷ []) "Eph.6.2"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.2"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ᾳ ∷ []) "Eph.6.2"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.6.3"
∷ word (ε ∷ ὖ ∷ []) "Eph.6.3"
∷ word (σ ∷ ο ∷ ι ∷ []) "Eph.6.3"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Eph.6.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.3"
∷ word (ἔ ∷ σ ∷ ῃ ∷ []) "Eph.6.3"
∷ word (μ ∷ α ∷ κ ∷ ρ ∷ ο ∷ χ ∷ ρ ∷ ό ∷ ν ∷ ι ∷ ο ∷ ς ∷ []) "Eph.6.3"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Eph.6.3"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.6.3"
∷ word (γ ∷ ῆ ∷ ς ∷ []) "Eph.6.3"
∷ word (Κ ∷ α ∷ ὶ ∷ []) "Eph.6.4"
∷ word (ο ∷ ἱ ∷ []) "Eph.6.4"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ ε ∷ ς ∷ []) "Eph.6.4"
∷ word (μ ∷ ὴ ∷ []) "Eph.6.4"
∷ word (π ∷ α ∷ ρ ∷ ο ∷ ρ ∷ γ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ ε ∷ []) "Eph.6.4"
∷ word (τ ∷ ὰ ∷ []) "Eph.6.4"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Eph.6.4"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.6.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Eph.6.4"
∷ word (ἐ ∷ κ ∷ τ ∷ ρ ∷ έ ∷ φ ∷ ε ∷ τ ∷ ε ∷ []) "Eph.6.4"
∷ word (α ∷ ὐ ∷ τ ∷ ὰ ∷ []) "Eph.6.4"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.4"
∷ word (π ∷ α ∷ ι ∷ δ ∷ ε ∷ ί ∷ ᾳ ∷ []) "Eph.6.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.4"
∷ word (ν ∷ ο ∷ υ ∷ θ ∷ ε ∷ σ ∷ ί ∷ ᾳ ∷ []) "Eph.6.4"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Eph.6.4"
∷ word (Ο ∷ ἱ ∷ []) "Eph.6.5"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ι ∷ []) "Eph.6.5"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ύ ∷ ε ∷ τ ∷ ε ∷ []) "Eph.6.5"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.6.5"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Eph.6.5"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Eph.6.5"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Eph.6.5"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Eph.6.5"
∷ word (φ ∷ ό ∷ β ∷ ο ∷ υ ∷ []) "Eph.6.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.5"
∷ word (τ ∷ ρ ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Eph.6.5"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.5"
∷ word (ἁ ∷ π ∷ ∙λ ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "Eph.6.5"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.6.5"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "Eph.6.5"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.6.5"
∷ word (ὡ ∷ ς ∷ []) "Eph.6.5"
∷ word (τ ∷ ῷ ∷ []) "Eph.6.5"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Eph.6.5"
∷ word (μ ∷ ὴ ∷ []) "Eph.6.6"
∷ word (κ ∷ α ∷ τ ∷ []) "Eph.6.6"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ο ∷ δ ∷ ο ∷ υ ∷ ∙λ ∷ ί ∷ α ∷ ν ∷ []) "Eph.6.6"
∷ word (ὡ ∷ ς ∷ []) "Eph.6.6"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ά ∷ ρ ∷ ε ∷ σ ∷ κ ∷ ο ∷ ι ∷ []) "Eph.6.6"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Eph.6.6"
∷ word (ὡ ∷ ς ∷ []) "Eph.6.6"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ι ∷ []) "Eph.6.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.6.6"
∷ word (π ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.6.6"
∷ word (τ ∷ ὸ ∷ []) "Eph.6.6"
∷ word (θ ∷ έ ∷ ∙λ ∷ η ∷ μ ∷ α ∷ []) "Eph.6.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.6.6"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.6.6"
∷ word (ἐ ∷ κ ∷ []) "Eph.6.6"
∷ word (ψ ∷ υ ∷ χ ∷ ῆ ∷ ς ∷ []) "Eph.6.6"
∷ word (μ ∷ ε ∷ τ ∷ []) "Eph.6.7"
∷ word (ε ∷ ὐ ∷ ν ∷ ο ∷ ί ∷ α ∷ ς ∷ []) "Eph.6.7"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.6.7"
∷ word (ὡ ∷ ς ∷ []) "Eph.6.7"
∷ word (τ ∷ ῷ ∷ []) "Eph.6.7"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.6.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.7"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Eph.6.7"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "Eph.6.7"
∷ word (ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "Eph.6.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Eph.6.8"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "Eph.6.8"
∷ word (ὃ ∷ []) "Eph.6.8"
∷ word (ἂ ∷ ν ∷ []) "Eph.6.8"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ ῃ ∷ []) "Eph.6.8"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ό ∷ ν ∷ []) "Eph.6.8"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Eph.6.8"
∷ word (κ ∷ ο ∷ μ ∷ ί ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Eph.6.8"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "Eph.6.8"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Eph.6.8"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "Eph.6.8"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Eph.6.8"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "Eph.6.8"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ θ ∷ ε ∷ ρ ∷ ο ∷ ς ∷ []) "Eph.6.8"
∷ word (Κ ∷ α ∷ ὶ ∷ []) "Eph.6.9"
∷ word (ο ∷ ἱ ∷ []) "Eph.6.9"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ι ∷ []) "Eph.6.9"
∷ word (τ ∷ ὰ ∷ []) "Eph.6.9"
∷ word (α ∷ ὐ ∷ τ ∷ ὰ ∷ []) "Eph.6.9"
∷ word (π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Eph.6.9"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.6.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ύ ∷ ς ∷ []) "Eph.6.9"
∷ word (ἀ ∷ ν ∷ ι ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.6.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.6.9"
∷ word (ἀ ∷ π ∷ ε ∷ ι ∷ ∙λ ∷ ή ∷ ν ∷ []) "Eph.6.9"
∷ word (ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "Eph.6.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Eph.6.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.9"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Eph.6.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.9"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.6.9"
∷ word (ὁ ∷ []) "Eph.6.9"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ό ∷ ς ∷ []) "Eph.6.9"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.6.9"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.9"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.6.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.9"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ω ∷ π ∷ ο ∷ ∙λ ∷ η ∷ μ ∷ ψ ∷ ί ∷ α ∷ []) "Eph.6.9"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Eph.6.9"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.6.9"
∷ word (π ∷ α ∷ ρ ∷ []) "Eph.6.9"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Eph.6.9"
∷ word (Τ ∷ ο ∷ ῦ ∷ []) "Eph.6.10"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ο ∷ ῦ ∷ []) "Eph.6.10"
∷ word (ἐ ∷ ν ∷ δ ∷ υ ∷ ν ∷ α ∷ μ ∷ ο ∷ ῦ ∷ σ ∷ θ ∷ ε ∷ []) "Eph.6.10"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.10"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.6.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.10"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.10"
∷ word (τ ∷ ῷ ∷ []) "Eph.6.10"
∷ word (κ ∷ ρ ∷ ά ∷ τ ∷ ε ∷ ι ∷ []) "Eph.6.10"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.6.10"
∷ word (ἰ ∷ σ ∷ χ ∷ ύ ∷ ο ∷ ς ∷ []) "Eph.6.10"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.6.10"
∷ word (ἐ ∷ ν ∷ δ ∷ ύ ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Eph.6.11"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.6.11"
∷ word (π ∷ α ∷ ν ∷ ο ∷ π ∷ ∙λ ∷ ί ∷ α ∷ ν ∷ []) "Eph.6.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.6.11"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.6.11"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.6.11"
∷ word (τ ∷ ὸ ∷ []) "Eph.6.11"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Eph.6.11"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.6.11"
∷ word (σ ∷ τ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "Eph.6.11"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.6.11"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Eph.6.11"
∷ word (μ ∷ ε ∷ θ ∷ ο ∷ δ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Eph.6.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.6.11"
∷ word (δ ∷ ι ∷ α ∷ β ∷ ό ∷ ∙λ ∷ ο ∷ υ ∷ []) "Eph.6.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Eph.6.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Eph.6.12"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.6.12"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "Eph.6.12"
∷ word (ἡ ∷ []) "Eph.6.12"
∷ word (π ∷ ά ∷ ∙λ ∷ η ∷ []) "Eph.6.12"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.6.12"
∷ word (α ∷ ἷ ∷ μ ∷ α ∷ []) "Eph.6.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.12"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Eph.6.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Eph.6.12"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.6.12"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Eph.6.12"
∷ word (ἀ ∷ ρ ∷ χ ∷ ά ∷ ς ∷ []) "Eph.6.12"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.6.12"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Eph.6.12"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Eph.6.12"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.6.12"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.6.12"
∷ word (κ ∷ ο ∷ σ ∷ μ ∷ ο ∷ κ ∷ ρ ∷ ά ∷ τ ∷ ο ∷ ρ ∷ α ∷ ς ∷ []) "Eph.6.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.6.12"
∷ word (σ ∷ κ ∷ ό ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Eph.6.12"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "Eph.6.12"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.6.12"
∷ word (τ ∷ ὰ ∷ []) "Eph.6.12"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ὰ ∷ []) "Eph.6.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.6.12"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ί ∷ α ∷ ς ∷ []) "Eph.6.12"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.12"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.6.12"
∷ word (ἐ ∷ π ∷ ο ∷ υ ∷ ρ ∷ α ∷ ν ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Eph.6.12"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.6.13"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Eph.6.13"
∷ word (ἀ ∷ ν ∷ α ∷ ∙λ ∷ ά ∷ β ∷ ε ∷ τ ∷ ε ∷ []) "Eph.6.13"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.6.13"
∷ word (π ∷ α ∷ ν ∷ ο ∷ π ∷ ∙λ ∷ ί ∷ α ∷ ν ∷ []) "Eph.6.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.6.13"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.6.13"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.6.13"
∷ word (δ ∷ υ ∷ ν ∷ η ∷ θ ∷ ῆ ∷ τ ∷ ε ∷ []) "Eph.6.13"
∷ word (ἀ ∷ ν ∷ τ ∷ ι ∷ σ ∷ τ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "Eph.6.13"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.13"
∷ word (τ ∷ ῇ ∷ []) "Eph.6.13"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "Eph.6.13"
∷ word (τ ∷ ῇ ∷ []) "Eph.6.13"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ᾷ ∷ []) "Eph.6.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.13"
∷ word (ἅ ∷ π ∷ α ∷ ν ∷ τ ∷ α ∷ []) "Eph.6.13"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ α ∷ σ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Eph.6.13"
∷ word (σ ∷ τ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "Eph.6.13"
∷ word (σ ∷ τ ∷ ῆ ∷ τ ∷ ε ∷ []) "Eph.6.14"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Eph.6.14"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ ζ ∷ ω ∷ σ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Eph.6.14"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.6.14"
∷ word (ὀ ∷ σ ∷ φ ∷ ὺ ∷ ν ∷ []) "Eph.6.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.6.14"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.14"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ ᾳ ∷ []) "Eph.6.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.14"
∷ word (ἐ ∷ ν ∷ δ ∷ υ ∷ σ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Eph.6.14"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.6.14"
∷ word (θ ∷ ώ ∷ ρ ∷ α ∷ κ ∷ α ∷ []) "Eph.6.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.6.14"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "Eph.6.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.15"
∷ word (ὑ ∷ π ∷ ο ∷ δ ∷ η ∷ σ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Eph.6.15"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Eph.6.15"
∷ word (π ∷ ό ∷ δ ∷ α ∷ ς ∷ []) "Eph.6.15"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.15"
∷ word (ἑ ∷ τ ∷ ο ∷ ι ∷ μ ∷ α ∷ σ ∷ ί ∷ ᾳ ∷ []) "Eph.6.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.6.15"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ο ∷ υ ∷ []) "Eph.6.15"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.6.15"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ ς ∷ []) "Eph.6.15"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.16"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "Eph.6.16"
∷ word (ἀ ∷ ν ∷ α ∷ ∙λ ∷ α ∷ β ∷ ό ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.6.16"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.6.16"
∷ word (θ ∷ υ ∷ ρ ∷ ε ∷ ὸ ∷ ν ∷ []) "Eph.6.16"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Eph.6.16"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.6.16"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.16"
∷ word (ᾧ ∷ []) "Eph.6.16"
∷ word (δ ∷ υ ∷ ν ∷ ή ∷ σ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Eph.6.16"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Eph.6.16"
∷ word (τ ∷ ὰ ∷ []) "Eph.6.16"
∷ word (β ∷ έ ∷ ∙λ ∷ η ∷ []) "Eph.6.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.6.16"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ο ∷ ῦ ∷ []) "Eph.6.16"
∷ word (π ∷ ε ∷ π ∷ υ ∷ ρ ∷ ω ∷ μ ∷ έ ∷ ν ∷ α ∷ []) "Eph.6.16"
∷ word (σ ∷ β ∷ έ ∷ σ ∷ α ∷ ι ∷ []) "Eph.6.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.17"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.6.17"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ α ∷ ί ∷ α ∷ ν ∷ []) "Eph.6.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.6.17"
∷ word (σ ∷ ω ∷ τ ∷ η ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Eph.6.17"
∷ word (δ ∷ έ ∷ ξ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Eph.6.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.17"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Eph.6.17"
∷ word (μ ∷ ά ∷ χ ∷ α ∷ ι ∷ ρ ∷ α ∷ ν ∷ []) "Eph.6.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.6.17"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Eph.6.17"
∷ word (ὅ ∷ []) "Eph.6.17"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Eph.6.17"
∷ word (ῥ ∷ ῆ ∷ μ ∷ α ∷ []) "Eph.6.17"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.6.17"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Eph.6.18"
∷ word (π ∷ ά ∷ σ ∷ η ∷ ς ∷ []) "Eph.6.18"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ ῆ ∷ ς ∷ []) "Eph.6.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.18"
∷ word (δ ∷ ε ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.6.18"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Eph.6.18"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.18"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "Eph.6.18"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ῷ ∷ []) "Eph.6.18"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.18"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Eph.6.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.6.18"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Eph.6.18"
∷ word (ἀ ∷ γ ∷ ρ ∷ υ ∷ π ∷ ν ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Eph.6.18"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.18"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "Eph.6.18"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ κ ∷ α ∷ ρ ∷ τ ∷ ε ∷ ρ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "Eph.6.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.18"
∷ word (δ ∷ ε ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "Eph.6.18"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "Eph.6.18"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Eph.6.18"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.6.18"
∷ word (ἁ ∷ γ ∷ ί ∷ ω ∷ ν ∷ []) "Eph.6.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.19"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Eph.6.19"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "Eph.6.19"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.6.19"
∷ word (μ ∷ ο ∷ ι ∷ []) "Eph.6.19"
∷ word (δ ∷ ο ∷ θ ∷ ῇ ∷ []) "Eph.6.19"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "Eph.6.19"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.19"
∷ word (ἀ ∷ ν ∷ ο ∷ ί ∷ ξ ∷ ε ∷ ι ∷ []) "Eph.6.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.6.19"
∷ word (σ ∷ τ ∷ ό ∷ μ ∷ α ∷ τ ∷ ό ∷ ς ∷ []) "Eph.6.19"
∷ word (μ ∷ ο ∷ υ ∷ []) "Eph.6.19"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.19"
∷ word (π ∷ α ∷ ρ ∷ ρ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "Eph.6.19"
∷ word (γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ σ ∷ α ∷ ι ∷ []) "Eph.6.19"
∷ word (τ ∷ ὸ ∷ []) "Eph.6.19"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Eph.6.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Eph.6.19"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ο ∷ υ ∷ []) "Eph.6.19"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Eph.6.20"
∷ word (ο ∷ ὗ ∷ []) "Eph.6.20"
∷ word (π ∷ ρ ∷ ε ∷ σ ∷ β ∷ ε ∷ ύ ∷ ω ∷ []) "Eph.6.20"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.20"
∷ word (ἁ ∷ ∙λ ∷ ύ ∷ σ ∷ ε ∷ ι ∷ []) "Eph.6.20"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.6.20"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.20"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Eph.6.20"
∷ word (π ∷ α ∷ ρ ∷ ρ ∷ η ∷ σ ∷ ι ∷ ά ∷ σ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "Eph.6.20"
∷ word (ὡ ∷ ς ∷ []) "Eph.6.20"
∷ word (δ ∷ ε ∷ ῖ ∷ []) "Eph.6.20"
∷ word (μ ∷ ε ∷ []) "Eph.6.20"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Eph.6.20"
∷ word (Ἵ ∷ ν ∷ α ∷ []) "Eph.6.21"
∷ word (δ ∷ ὲ ∷ []) "Eph.6.21"
∷ word (ε ∷ ἰ ∷ δ ∷ ῆ ∷ τ ∷ ε ∷ []) "Eph.6.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.21"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Eph.6.21"
∷ word (τ ∷ ὰ ∷ []) "Eph.6.21"
∷ word (κ ∷ α ∷ τ ∷ []) "Eph.6.21"
∷ word (ἐ ∷ μ ∷ έ ∷ []) "Eph.6.21"
∷ word (τ ∷ ί ∷ []) "Eph.6.21"
∷ word (π ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ω ∷ []) "Eph.6.21"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Eph.6.21"
∷ word (γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "Eph.6.21"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Eph.6.21"
∷ word (Τ ∷ υ ∷ χ ∷ ι ∷ κ ∷ ὸ ∷ ς ∷ []) "Eph.6.21"
∷ word (ὁ ∷ []) "Eph.6.21"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.6.21"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ς ∷ []) "Eph.6.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.21"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Eph.6.21"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ς ∷ []) "Eph.6.21"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.21"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Eph.6.21"
∷ word (ὃ ∷ ν ∷ []) "Eph.6.22"
∷ word (ἔ ∷ π ∷ ε ∷ μ ∷ ψ ∷ α ∷ []) "Eph.6.22"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.6.22"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Eph.6.22"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Eph.6.22"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Eph.6.22"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Eph.6.22"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Eph.6.22"
∷ word (γ ∷ ν ∷ ῶ ∷ τ ∷ ε ∷ []) "Eph.6.22"
∷ word (τ ∷ ὰ ∷ []) "Eph.6.22"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "Eph.6.22"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.6.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.22"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ έ ∷ σ ∷ ῃ ∷ []) "Eph.6.22"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Eph.6.22"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "Eph.6.22"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.6.22"
∷ word (Ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "Eph.6.23"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.6.23"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ῖ ∷ ς ∷ []) "Eph.6.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.23"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "Eph.6.23"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Eph.6.23"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Eph.6.23"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Eph.6.23"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Eph.6.23"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "Eph.6.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Eph.6.23"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Eph.6.23"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Eph.6.23"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Eph.6.23"
∷ word (ἡ ∷ []) "Eph.6.24"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Eph.6.24"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Eph.6.24"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Eph.6.24"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Eph.6.24"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ώ ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Eph.6.24"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Eph.6.24"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Eph.6.24"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Eph.6.24"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "Eph.6.24"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Eph.6.24"
∷ word (ἐ ∷ ν ∷ []) "Eph.6.24"
∷ word (ἀ ∷ φ ∷ θ ∷ α ∷ ρ ∷ σ ∷ ί ∷ ᾳ ∷ []) "Eph.6.24"
∷ []
|
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{-# OPTIONS --safe #-}
module Cubical.Algebra.Lattice.Properties where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Equiv.HalfAdjoint
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Univalence
open import Cubical.Foundations.Transport
open import Cubical.Foundations.SIP
open import Cubical.Data.Sigma
open import Cubical.Structures.Axioms
open import Cubical.Structures.Auto
open import Cubical.Structures.Macro
open import Cubical.Algebra.Semigroup
open import Cubical.Algebra.Monoid
open import Cubical.Algebra.CommMonoid
open import Cubical.Algebra.Semilattice
open import Cubical.Algebra.Lattice.Base
open import Cubical.Relation.Binary.Poset
private
variable
ℓ : Level
module LatticeTheory (L' : Lattice ℓ) where
private L = fst L'
open LatticeStr (snd L')
0lLeftAnnihilates∧l : ∀ (x : L) → 0l ∧l x ≡ 0l
0lLeftAnnihilates∧l x = 0l ∧l x ≡⟨ cong (0l ∧l_) (sym (∨lLid _)) ⟩
0l ∧l (0l ∨l x) ≡⟨ ∧lAbsorb∨l _ _ ⟩
0l ∎
0lRightAnnihilates∧l : ∀ (x : L) → x ∧l 0l ≡ 0l
0lRightAnnihilates∧l _ = ∧lComm _ _ ∙ 0lLeftAnnihilates∧l _
1lLeftAnnihilates∨l : ∀ (x : L) → 1l ∨l x ≡ 1l
1lLeftAnnihilates∨l x = 1l ∨l x ≡⟨ cong (1l ∨l_) (sym (∧lLid _)) ⟩
1l ∨l (1l ∧l x) ≡⟨ ∨lAbsorb∧l _ _ ⟩
1l ∎
1lRightAnnihilates∨l : ∀ (x : L) → x ∨l 1l ≡ 1l
1lRightAnnihilates∨l _ = ∨lComm _ _ ∙ 1lLeftAnnihilates∨l _
module Order (L' : Lattice ℓ) where
private L = fst L'
open LatticeStr (snd L')
open JoinSemilattice (Lattice→JoinSemilattice L') renaming (_≤_ to _≤j_ ; IndPoset to JoinPoset)
open MeetSemilattice (Lattice→MeetSemilattice L') renaming (_≤_ to _≤m_ ; IndPoset to MeetPoset)
≤j→≤m : ∀ x y → x ≤j y → x ≤m y
≤j→≤m x y x∨ly≡y = x ∧l y ≡⟨ cong (x ∧l_) (sym x∨ly≡y) ⟩
x ∧l (x ∨l y) ≡⟨ ∧lAbsorb∨l _ _ ⟩
x ∎
≤m→≤j : ∀ x y → x ≤m y → x ≤j y
≤m→≤j x y x∧ly≡x = x ∨l y ≡⟨ ∨lComm _ _ ⟩
y ∨l x ≡⟨ cong (y ∨l_) (sym x∧ly≡x) ⟩
y ∨l (x ∧l y) ≡⟨ cong (y ∨l_) (∧lComm _ _) ⟩
y ∨l (y ∧l x) ≡⟨ ∨lAbsorb∧l _ _ ⟩
y ∎
≤Equiv : ∀ (x y : L) → (x ≤j y) ≃ (x ≤m y)
≤Equiv x y = propBiimpl→Equiv (isSetLattice L' _ _) (isSetLattice L' _ _) (≤j→≤m x y) (≤m→≤j x y)
IndPosetPath : JoinPoset ≡ MeetPoset
IndPosetPath = PosetPath _ _ .fst ((idEquiv _) , isposetequiv ≤Equiv )
-- transport inequalities from ≤m to ≤j
∧lIsMinJoin : ∀ x y z → z ≤j x → z ≤j y → z ≤j x ∧l y
∧lIsMinJoin _ _ _ z≤jx z≤jy = ≤m→≤j _ _ (∧lIsMin _ _ _ (≤j→≤m _ _ z≤jx) (≤j→≤m _ _ z≤jy))
∧≤LCancelJoin : ∀ x y → x ∧l y ≤j y
∧≤LCancelJoin x y = ≤m→≤j _ _ (∧≤LCancel x y)
module _ {L M : Lattice ℓ} (φ ψ : LatticeHom L M) where
open LatticeStr ⦃...⦄
open IsLatticeHom
private
instance
_ = L
_ = M
_ = snd L
_ = snd M
LatticeHom≡f : fst φ ≡ fst ψ → φ ≡ ψ
LatticeHom≡f = Σ≡Prop λ f → isPropIsLatticeHom _ f _
|
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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Cosets
open import Groups.Homomorphisms.Definition
open import Rings.Homomorphisms.Definition
open import Groups.Lemmas
open import Groups.Definition
open import Setoids.Setoids
open import Setoids.Functions.Lemmas
open import Rings.Definition
open import Sets.EquivalenceRelations
open import Rings.Ideals.Definition
open import Fields.Fields
open import Fields.Lemmas
open import Rings.Cosets
open import Rings.Ideals.Maximal.Definition
open import Rings.Ideals.Lemmas
open import Rings.Ideals.Prime.Definition
open import Rings.IntegralDomains.Definition
open import Rings.Ideals.Prime.Lemmas
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Rings.Ideals.Maximal.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} {c : _} {pred : A → Set c} (i : Ideal R pred) where
open Ring R
open Group additiveGroup
open Setoid S
open Equivalence eq
idealMaximalImpliesQuotientField : ({d : Level} → MaximalIdeal i {d}) → Field (cosetRing R i)
Field.allInvertible (idealMaximalImpliesQuotientField max) cosetA cosetA!=0 = ans' , ans''
where
gen : Ideal (cosetRing R i) (generatedIdealPred (cosetRing R i) cosetA)
gen = generatedIdeal (cosetRing R i) cosetA
inv : Ideal R (inverseImagePred {S = S} {T = cosetSetoid additiveGroup (Ideal.isSubgroup i)} (GroupHom.wellDefined (RingHom.groupHom (cosetRingHom R i))) (Ideal.isSubset gen))
inv = inverseImageIsIdeal (cosetRing R i) (cosetRingHom R i) gen
containsOnce : {a : A} → (Ideal.predicate i a) → (Ideal.predicate inv a)
containsOnce {x} ix = x , ((x , Ideal.closedUnderPlus i (Ideal.closedUnderInverse i ix) (Ideal.isSubset i *Commutative (Ideal.accumulatesTimes i ix))) ,, Ideal.isSubset i (symmetric invLeft) (Ideal.containsIdentity i))
notInI : A
notInI = cosetA
notInIIsInInv : Ideal.predicate inv notInI
notInIIsInInv = cosetA , ((1R , Ideal.isSubset i {0R} (symmetric (transitive (+WellDefined reflexive (transitive *Commutative identIsIdent)) (invLeft {cosetA}))) (Ideal.containsIdentity i)) ,, Ideal.isSubset i (symmetric invLeft) (Ideal.containsIdentity i))
notInIPr : (Ideal.predicate i notInI) → False
notInIPr iInI = cosetA!=0 (Ideal.isSubset i (transitive (symmetric identLeft) (+WellDefined (symmetric (invIdent additiveGroup)) reflexive)) iInI)
ans : {a : A} → Ideal.predicate inv a
ans = MaximalIdeal.isMaximal max inv containsOnce (notInI , (notInIIsInInv ,, notInIPr))
ans' : A
ans' with ans {1R}
... | _ , ((b , _) ,, _) = b
ans'' : pred (inverse (Ring.1R (cosetRing R i)) + (ans' * cosetA))
ans'' with ans {1R}
ans'' | a , ((b , predCAb-a) ,, pred1-a) = Ideal.isSubset i (transitive (+WellDefined (invContravariant additiveGroup) reflexive) (transitive +Associative (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight)) *Commutative))) (Ideal.closedUnderPlus i (Ideal.closedUnderInverse i pred1-a) predCAb-a)
Field.nontrivial (idealMaximalImpliesQuotientField max) 1=0 = MaximalIdeal.notContainedIsNotContained max (Ideal.isSubset i (identIsIdent {MaximalIdeal.notContained (max {lzero})}) (Ideal.accumulatesTimes i p))
where
have : pred (inverse 1R)
have = Ideal.isSubset i identRight 1=0
p : pred 1R
p = Ideal.isSubset i (invTwice additiveGroup 1R) (Ideal.closedUnderInverse i have)
quotientFieldImpliesIdealMaximal : Field (cosetRing R i) → ({d : _} → MaximalIdeal i {d})
MaximalIdeal.notContained (quotientFieldImpliesIdealMaximal f) = Ring.1R (cosetRing R i)
MaximalIdeal.notContainedIsNotContained (quotientFieldImpliesIdealMaximal f) p1R = Field.nontrivial f (memberDividesImpliesMember R i p1R ((inverse 1R + 0R) , identIsIdent))
MaximalIdeal.isMaximal (quotientFieldImpliesIdealMaximal f) {bigger} biggerIdeal contained (a , (biggerA ,, notPredA)) = Ideal.isSubset biggerIdeal identIsIdent (Ideal.accumulatesTimes biggerIdeal v)
where
inv : Sg A (λ t → pred (inverse 1R + (t * a)))
inv = Field.allInvertible f a λ r → notPredA (translate' R i r)
r : A
r = underlying inv
s : pred (inverse 1R + (r * a))
s with inv
... | _ , p = p
t : bigger (inverse 1R + (r * a))
t = contained s
u : bigger (inverse (r * a))
u = Ideal.closedUnderInverse biggerIdeal (Ideal.isSubset biggerIdeal *Commutative (Ideal.accumulatesTimes biggerIdeal biggerA))
v : bigger 1R
v = Ideal.isSubset biggerIdeal (invTwice additiveGroup 1R) (Ideal.closedUnderInverse biggerIdeal (Ideal.isSubset biggerIdeal (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) (Ideal.closedUnderPlus biggerIdeal t u)))
idealMaximalImpliesIdealPrime : ({d : _} → MaximalIdeal i {d}) → PrimeIdeal i
idealMaximalImpliesIdealPrime max = quotientIntDomImpliesIdealPrime i (fieldIsIntDom (idealMaximalImpliesQuotientField max))
maximalIdealWellDefined : {d : _} {pred2 : A → Set d} (i2 : Ideal R pred2) → ({x : A} → pred x → pred2 x) → ({x : A} → pred2 x → pred x) → {e : _} → MaximalIdeal i {e} → MaximalIdeal i2 {e}
MaximalIdeal.notContained (maximalIdealWellDefined i2 pToP2 p2ToP record { notContained = notContained ; notContainedIsNotContained = notContainedIsNotContained ; isMaximal = isMaximal }) = notContained
MaximalIdeal.notContainedIsNotContained (maximalIdealWellDefined i2 pToP2 p2ToP record { notContained = notContained ; notContainedIsNotContained = notContainedIsNotContained ; isMaximal = isMaximal }) p2Not = notContainedIsNotContained (p2ToP p2Not)
MaximalIdeal.isMaximal (maximalIdealWellDefined i2 pToP2 p2ToP record { notContained = notContained ; notContainedIsNotContained = notContainedIsNotContained ; isMaximal = isMaximal }) {biggerPred} bigger pred2InBigger (r , (biggerPredR ,, notP2r)) {x} = isMaximal bigger (λ p → pred2InBigger (pToP2 p)) (r , (biggerPredR ,, λ p → notP2r (pToP2 p)))
|
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open import Agda.Builtin.Bool
open import Agda.Builtin.Char
open import Agda.Builtin.Nat
open import Agda.Builtin.Equality
char : Char → Set
char 'A' with 'O'
char _ | _ = Char
char _ = Char
lit : Nat → Set
lit 5 with 0
lit _ | _ = Nat
lit _ = Nat
con : Nat → Set
con zero with zero
con _ | _ = Nat
con (suc x) with zero
con _ | _ = Nat
record R : Set where
coinductive -- disallow matching
field f : Bool
n : Nat
data P (r : R) : Nat → Set where
fTrue : R.f r ≡ true → P r zero
nSuc : P r (suc (R.n r))
data Q : (b : Bool) (n : Nat) → Set where
true! : Q true zero
suc! : ∀{b n} → Q b (suc n)
test : (r : R) {n : Nat} (p : P r n) → Q (R.f r) n
test r nSuc = suc!
test r (fTrue p) with R.f r
test _ (fTrue ()) | false
test _ _ | true = true! -- underscore instead of (isTrue _)
-- -- Note that `with` first and then splitting on p does not work:
--
-- foo : (r : R) {n : Nat} (p : P r n) → Q (R.f r) n
-- foo r p with R.f r
-- foo r (fTrue x) | false = {!x!} -- cannot split on x
-- foo r (fTrue _) | true = true!
-- foo r nSuc | _ = suc!
|
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open import Agda.Builtin.List
open import Agda.Builtin.Reflection
macro
print : (Set → Set) → Term → TC _
print t _ = bindTC (quoteTC t) λ t →
typeError (termErr t ∷ [])
-- Prints λ { X → X }.
Test₁ : Set
Test₁ = {!print (λ { X → X })!}
module _ (A : Set) where
-- Prints λ { A₁ X → X }.
Test₂ : Set
Test₂ = {!print (λ { X → X })!}
-- Prints λ { B₁ X → X } B.
Test₃ : Set → Set
Test₃ B = {!print (λ { X → X })!}
module _ (A : Set) where
-- Prints λ { A₁ X → A₁ }. Note that A is not mentioned at all.
Test₄ : Set
Test₄ = {!print (λ { X → A })!}
-- Prints λ { B₁ X → A } B.
Test₅ : Set → Set
Test₅ B = {!print (λ { X → A })!}
|
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{-# OPTIONS --cubical --no-import-sorts --allow-unsolved-metas #-}
module Number.Instances.QuoInt where
open import Cubical.Foundations.Everything hiding (⋆) renaming (_⁻¹ to _⁻¹ᵖ; assoc to ∙-assoc)
open import Cubical.Foundations.Logic renaming (inl to inlᵖ; inr to inrᵖ)
open import Cubical.Relation.Nullary.Base renaming (¬_ to ¬ᵗ_)
open import Cubical.Relation.Binary.Base
open import Cubical.Data.Sum.Base renaming (_⊎_ to infixr 4 _⊎_)
open import Cubical.Data.Sigma.Base renaming (_×_ to infixr 4 _×_)
open import Cubical.Data.Sigma
open import Cubical.Data.Bool as Bool using (Bool; not; true; false)
open import Cubical.Data.Empty renaming (elim to ⊥-elim; ⊥ to ⊥⊥) -- `⊥` and `elim`
open import Cubical.Foundations.Logic renaming (¬_ to ¬ᵖ_; inl to inlᵖ; inr to inrᵖ)
open import Function.Base using (it; _∋_; _$_)
open import Cubical.Foundations.Isomorphism
open import Cubical.HITs.PropositionalTruncation --.Properties
open import Utils using (!_; !!_)
open import MoreLogic.Reasoning
open import MoreLogic.Definitions
open import MoreLogic.Properties
open import MorePropAlgebra.Definitions hiding (_≤''_)
open import MorePropAlgebra.Structures
open import MorePropAlgebra.Bundles
open import MorePropAlgebra.Consequences
open import Number.Structures2
open import Number.Bundles2
open import Cubical.Data.Nat.Literals
open import Cubical.Data.Nat using (suc; zero; ℕ; HasFromNat)
open import Number.Prelude.Nat using (¬-<ⁿ-zero; +ⁿ-comm; ¬suc<ⁿ0; _+ⁿ_; _·ⁿ_; ·ⁿ-reflects-≡ˡ')
open import Number.Instances.QuoIntFromInt public
open import Cubical.HITs.Ints.QuoInt as QuoInt using
( ℤ
; HasFromNat
; _+_
; Int≡ℤ
; signed
; posneg
; ℤ→Int
; sucℤ
; predℤ
; sign
; abs
; pos
; neg
; +-comm
; +-assoc
; sucℤ-+ʳ
; sucℤ-+ˡ
; spos
; sneg
; *-pos-suc
; negate-invol
) renaming
( isSetℤ to is-set
; _*_ to _·_
; -_ to infixl 6 -_
; *-comm to ·-comm
)
open IsLinearlyOrderedCommRing is-LinearlyOrderedCommRing using
( _-_
; <-irrefl
; <-trans
; +-<-ext
; +-rinv
; +-identity
; ·-identity
; _≤_
; ·-preserves-<
; <-tricho
; <-asym
; _#_
; +-inverse
; ·-assoc
)
0<1 : [ 0 < 1 ]
0<1 = 0 , refl
-- TODO: import these properties from somewhere else
+-reflects-< : ∀ x y z → [ (x + z < y + z) ⇒ ( x < y ) ]
+-preserves-< : ∀ x y z → [ ( x < y ) ⇒ (x + z < y + z) ]
+-creates-< : ∀ x y z → [ ( x < y ) ⇔ (x + z < y + z) ]
+-preserves-< a b x = snd (
( a < b ) ⇒ᵖ⟨ transport (λ i → [ sym (fst (+-identity a)) i < sym (fst (+-identity b)) i ]) ⟩
( a + 0 < b + 0 ) ⇒ᵖ⟨ transport (λ i → [ a + sym (+-rinv x) i < b + sym (+-rinv x) i ]) ⟩
( a + (x - x) < b + (x - x)) ⇒ᵖ⟨ transport (λ i → [ +-assoc a x (- x) i < +-assoc b x (- x) i ]) ⟩
((a + x) - x < (b + x) - x ) ⇒ᵖ⟨ +-<-ext (a + x) (- x) (b + x) (- x) ⟩
((a + x < b + x) ⊔ (- x < - x)) ⇒ᵖ⟨ (λ q → case q as (a + x < b + x) ⊔ (- x < - x) ⇒ a + x < b + x of λ
{ (inl a+x<b+x) → a+x<b+x -- somehow ⊥-elim needs a hint in the next line
; (inr -x<-x ) → ⊥-elim {A = λ _ → [ a + x < b + x ]} (<-irrefl (- x) -x<-x)
}) ⟩
a + x < b + x ◼ᵖ)
+-reflects-< x y z = snd
( x + z < y + z ⇒ᵖ⟨ +-preserves-< (x + z) (y + z) (- z) ⟩
(x + z) - z < (y + z) - z ⇒ᵖ⟨ transport (λ i → [ +-assoc x z (- z) (~ i) < +-assoc y z (- z) (~ i) ]) ⟩
x + (z - z) < y + (z - z) ⇒ᵖ⟨ transport (λ i → [ x + +-rinv z i < y + +-rinv z i ]) ⟩
x + 0 < y + 0 ⇒ᵖ⟨ transport (λ i → [ fst (+-identity x) i < fst (+-identity y) i ]) ⟩
x < y ◼ᵖ)
+-creates-< x y z .fst = +-preserves-< x y z
+-creates-< x y z .snd = +-reflects-< x y z
suc-creates-< : ∀ x y → [ (x < y) ⇔ (sucℤ x < sucℤ y) ]
suc-creates-< x y .fst p = substₚ (λ p → sucℤ x < p) (∣ +-comm y (pos 1) ∣) $ substₚ (λ p → p < y + pos 1) (∣ +-comm x (pos 1) ∣) (+-preserves-< x y (pos 1) p)
suc-creates-< x y .snd p = +-reflects-< x y (pos 1) $ substₚ (λ p → p < y + pos 1) (∣ +-comm (pos 1) x ∣) $ substₚ (λ p → sucℤ x < p) (∣ +-comm (pos 1) y ∣) p
·-creates-< : ∀ a b x → [ 0 < x ] → [ (a < b) ⇔ ((a · x) < (b · x)) ]
·-creates-< a b x p .fst q = ·-preserves-< a b x p q
·-creates-< a b x p .snd q = ·-reflects-< a b x p q
·-creates-<ˡ : ∀ a b x → [ 0 < x ] → [ (a < b) ⇔ ((x · a) < (x · b)) ]
·-creates-<ˡ a b x p .fst q = transport (λ i → [ ·-comm a x i < ·-comm b x i ]) $ ·-preserves-< a b x p q
·-creates-<ˡ a b x p .snd q = ·-reflects-< a b x p (transport (λ i → [ ·-comm x a i < ·-comm x b i ]) q)
·-creates-<-≡ : ∀ a b x → [ 0 < x ] → (a < b) ≡ ((a · x) < (b · x))
·-creates-<-≡ a b x p = ⇔toPath' (·-creates-< a b x p)
·-creates-<ˡ-≡ : ∀ a b x → [ 0 < x ] → (a < b) ≡ ((x · a) < (x · b))
·-creates-<ˡ-≡ a b x p = ⇔toPath' (·-creates-<ˡ a b x p)
+-creates-≤ : ∀ a b x → [ (a ≤ b) ⇔ ((a + x) ≤ (b + x)) ]
+-creates-≤ a b x = {! !}
·-creates-≤ : ∀ a b x → [ 0 ≤ x ] → [ (a ≤ b) ⇔ ((a · x) ≤ (b · x)) ]
·-creates-≤ a b x 0≤x .fst p = {! !}
·-creates-≤ a b x 0≤x .snd p = {! !}
·-creates-≤-≡ : ∀ a b x → [ 0 ≤ x ] → (a ≤ b) ≡ ((a · x) ≤ (b · x))
·-creates-≤-≡ a b x 0≤x = uncurry ⇔toPath $ ·-creates-≤ a b x 0≤x
≤-dicho : ∀ x y → [ (x ≤ y) ⊔ (y ≤ x) ]
≤-dicho x y with <-tricho x y
... | inl (inl x<y) = inlᵖ $ <-asym x y x<y
... | inl (inr y<x) = inrᵖ $ <-asym y x y<x
... | inr x≡y = inlᵖ $ subst (λ p → [ ¬(p < x) ]) x≡y (<-irrefl x)
ℤlattice : Lattice {ℓ-zero} {ℓ-zero}
ℤlattice = record { LinearlyOrderedCommRing bundle renaming (≤-Lattice to is-Lattice) }
open import MorePropAlgebra.Properties.Lattice ℤlattice
open OnSet is-set hiding (+-min-distribʳ; ·-min-distribʳ; +-max-distribʳ; ·-max-distribʳ)
≤-min-+ : ∀ a b c w → [ w ≤ (a + c) ] → [ w ≤ (b + c) ] → [ w ≤ (min a b + c) ]
≤-max-+ : ∀ a b c w → [ (a + c) ≤ w ] → [ (b + c) ≤ w ] → [ (max a b + c) ≤ w ]
≤-min-· : ∀ a b c w → [ w ≤ (a · c) ] → [ w ≤ (b · c) ] → [ w ≤ (min a b · c) ]
≤-max-· : ∀ a b c w → [ (a · c) ≤ w ] → [ (b · c) ≤ w ] → [ (max a b · c) ≤ w ]
≤-min-+ = OnType.≤-dicho⇒+.≤-min-+ _+_ ≤-dicho
≤-max-+ = OnType.≤-dicho⇒+.≤-max-+ _+_ ≤-dicho
≤-min-· = OnType.≤-dicho⇒·.≤-min-· _·_ ≤-dicho
≤-max-· = OnType.≤-dicho⇒·.≤-max-· _·_ ≤-dicho
+-min-distribʳ : ∀ x y z → (min x y + z) ≡ min (x + z) (y + z)
·-min-distribʳ : ∀ x y z → [ 0 ≤ z ] → (min x y · z) ≡ min (x · z) (y · z)
+-max-distribʳ : ∀ x y z → (max x y + z) ≡ max (x + z) (y + z)
·-max-distribʳ : ∀ x y z → [ 0 ≤ z ] → (max x y · z) ≡ max (x · z) (y · z)
+-min-distribˡ : ∀ x y z → (z + min x y) ≡ min (z + x) (z + y)
·-min-distribˡ : ∀ x y z → [ 0 ≤ z ] → (z · min x y) ≡ min (z · x) (z · y)
+-max-distribˡ : ∀ x y z → (z + max x y) ≡ max (z + x) (z + y)
·-max-distribˡ : ∀ x y z → [ 0 ≤ z ] → (z · max x y) ≡ max (z · x) (z · y)
+-min-distribʳ = OnSet.+-min-distribʳ is-set _+_ +-creates-≤ ≤-min-+
·-min-distribʳ = OnSet.·-min-distribʳ is-set 0 _·_ ·-creates-≤ ≤-min-·
+-max-distribʳ = OnSet.+-max-distribʳ is-set _+_ +-creates-≤ ≤-max-+
·-max-distribʳ = OnSet.·-max-distribʳ is-set 0 _·_ ·-creates-≤ ≤-max-·
+-min-distribˡ x y z = +-comm z (min x y) ∙ +-min-distribʳ x y z ∙ (λ i → min (+-comm x z i) (+-comm y z i))
·-min-distribˡ x y z p = ·-comm z (min x y) ∙ ·-min-distribʳ x y z p ∙ (λ i → min (·-comm x z i) (·-comm y z i))
+-max-distribˡ x y z = +-comm z (max x y) ∙ +-max-distribʳ x y z ∙ (λ i → max (+-comm x z i) (+-comm y z i))
·-max-distribˡ x y z p = ·-comm z (max x y) ∙ ·-max-distribʳ x y z p ∙ (λ i → max (·-comm x z i) (·-comm y z i))
pos<pos[suc] : ∀ x → [ pos x < pos (suc x) ]
pos<pos[suc] 0 = 0<1
pos<pos[suc] (suc x) = suc-creates-< (pos x) (pos (suc x)) .fst (pos<pos[suc] x)
0<ᶻpos[suc] : ∀ x → [ 0 < pos (suc x) ]
0<ᶻpos[suc] 0 = 0<1
0<ᶻpos[suc] (suc x) = <-trans 0 (pos (suc x)) (pos (suc (suc x))) (0<ᶻpos[suc] x) (suc-creates-< (pos x) (pos (suc x)) .fst (pos<pos[suc] x))
·-nullifiesˡ : ∀(x : ℤ) → 0 · x ≡ 0
·-nullifiesˡ (pos zero) = refl
·-nullifiesˡ (neg zero) = refl
·-nullifiesˡ (posneg i) = refl
·-nullifiesˡ (pos (suc n)) = refl
·-nullifiesˡ (neg (suc n)) = sym posneg
·-nullifiesʳ : ∀(x : ℤ) → x · 0 ≡ 0
·-nullifiesʳ x = ·-comm x 0 ∙ ·-nullifiesˡ x
·-preserves-0< : ∀ a b → [ 0 < a ] → [ 0 < b ] → [ 0 < a · b ]
·-preserves-0< a b p q = subst (λ p → [ p < a · b ]) (·-nullifiesˡ b) (·-preserves-< 0 a b q p)
private
term : ∀ b x → Type ℓ-zero
term b x = [ ((pos 0 < x) ⇒ (pos 0 < b)) ⊓ ((pos 0 < b) ⇒ (pos 0 < x)) ]
·-reflects-<ˡ : (x y z : ℤ) → [ pos 0 < z ] → [ z · x < z · y ] → [ x < y ]
·-reflects-<ˡ x y z p q = ·-reflects-< x y z p (transport (λ i → [ ·-comm z x i < ·-comm z y i ]) q)
-flips-<0 : ∀ x → [ (x < 0) ⇔ (0 < (- x)) ]
-flips-<0 (signed spos zero) = (λ x → x) , (λ x → x)
-flips-<0 (signed sneg zero) = (λ x → x) , (λ x → x)
-flips-<0 (ℤ.posneg i) = (λ x → x) , (λ x → x)
-flips-<0 (signed spos (suc n)) .fst p = ¬-<ⁿ-zero p
-flips-<0 (signed sneg (suc n)) .fst tt = n , +ⁿ-comm n 1
-flips-<0 (signed sneg (suc n)) .snd p = tt
-- NOTE: this could be a path, if we make `+-creates-<` into a path
-flips-< : ∀ x y → [ x < y ] → [ - y < - x ]
-flips-< x y p = (
( x < y ) ⇒ᵖ⟨ +-creates-< x y (- y) .fst ⟩
( x - y < y - y) ⇒ᵖ⟨ transport (λ i → [ +-comm x (- y) i < +-rinv y i ]) ⟩
( (- y) + x < 0 ) ⇒ᵖ⟨ +-creates-< ((- y) + x) 0 (- x) .fst ⟩
(((- y) + x) - x < 0 - x) ⇒ᵖ⟨ transport (λ i → [ +-assoc (- y) x (- x) (~ i) < +-identity (- x) .snd i ]) ⟩
( (- y) + (x - x) < - x ) ⇒ᵖ⟨ transport (λ i → [ (- y) + +-rinv x i < - x ]) ⟩
( (- y) + 0 < - x ) ⇒ᵖ⟨ transport (λ i → [ +-identity (- y) .fst i < - x ]) ⟩
( - y < - x ) ◼ᵖ) .snd p
-flips-<-⇔ : ∀ x y → [ (x < y) ⇔ (- y < - x) ]
-flips-<-⇔ x y .fst = -flips-< x y
-flips-<-⇔ x y .snd p = transport (λ i → [ negate-invol x i < negate-invol y i ]) $ -flips-< (- y) (- x) p
-flips-<-≡ : ∀ x y → (x < y) ≡ (- y < - x)
-flips-<-≡ x y = ⇔toPath' (-flips-<-⇔ x y)
-identity-· : ∀ a → (- 1) · a ≡ - a
-identity-· (pos zero) j = posneg (~ i0 ∨ ~ j)
-identity-· (neg zero) j = posneg (~ i1 ∨ ~ j)
-identity-· (posneg i) j = posneg (~ i ∨ ~ j)
-identity-· (pos (suc n)) i = neg (suc (+ⁿ-comm n 0 i))
-identity-· (neg (suc n)) i = pos (suc (+ⁿ-comm n 0 i))
-distˡ : ∀ a b → -(a · b) ≡ (- a) · b
-distˡ a b =
-(a · b) ≡⟨ sym $ -identity-· (a · b) ⟩
(- 1) · (a · b) ≡⟨ ·-assoc (- 1) a b ⟩
((- 1) · a) · b ≡⟨ (λ i → -identity-· a i · b) ⟩
(- a) · b ∎
private
lem : ∀ z → [ z < 0 ] → [ 0 < - z ]
lem z p = subst (λ p → [ p < - z ]) (sym posneg) $ -flips-< z 0 p
·-creates-<-flippedˡ-≡ : (x y z : ℤ) → [ z < 0 ] → (z · x < z · y) ≡ (y < x)
·-creates-<-flippedˡ-≡ x y z p =
z · x < z · y ≡⟨ -flips-<-≡ (z · x) (z · y) ⟩
- (z · y) < - (z · x) ≡⟨ (λ i → -distˡ z y i < -distˡ z x i) ⟩
(- z) · y < (- z) · x ≡⟨ sym $ ·-creates-<ˡ-≡ y x (- z) (lem z p) ⟩
y < x ∎
·-creates-<-flippedʳ-≡ : (x y z : ℤ) → [ z < 0 ] → (x · z < y · z) ≡ (y < x)
·-creates-<-flippedʳ-≡ x y z p = (λ i → ·-comm x z i < ·-comm y z i) ∙ ·-creates-<-flippedˡ-≡ x y z p
·-reflects-<-flippedˡ : (x y z : ℤ) → [ z < 0 ] → [ z · x < z · y ] → [ y < x ]
·-reflects-<-flippedˡ x y z p q = pathTo⇒ (·-creates-<-flippedˡ-≡ x y z p) q
-- (z · x < z · y ⇒ᵖ⟨ -flips-< (z · x) (z · y) ⟩
-- - (z · y) < - (z · x) ⇒ᵖ⟨ transport (λ i → [ -distˡ z y i < -distˡ z x i ]) ⟩
-- (- z) · y < (- z) · x ⇒ᵖ⟨ ·-creates-<ˡ y x (- z) (lem z p) .snd ⟩
-- y < x ◼ᵖ) .snd q
·-reflects-<-flippedʳ : (x y z : ℤ) → [ z < 0 ] → [ x · z < y · z ] → [ y < x ]
·-reflects-<-flippedʳ x y z p q = ·-reflects-<-flippedˡ x y z p (transport (λ i → [ ·-comm x z i < ·-comm y z i ]) q)
-- ·-preserves-<-flippedˡ : (x y z : ℤ) → [ z < 0 ] → [ x < y ] → [ z · y < z · x ]
-- ·-preserves-<-flippedˡ x y z p q = {! !}
·-reflects-0< : ∀ a b → [ 0 < a · b ] → [ ((0 < a) ⇔ (0 < b)) ⊓ ((a < 0) ⇔ (b < 0)) ]
·-reflects-0< a b p .fst .fst q = ·-reflects-<ˡ 0 b a q (transport (λ i → [ ·-nullifiesʳ a (~ i) < a · b ]) p)
·-reflects-0< a b p .fst .snd q = ·-reflects-< 0 a b q (transport (λ i → [ ·-nullifiesˡ b (~ i) < a · b ]) p)
·-reflects-0< a b p .snd .fst q = ·-reflects-<-flippedˡ 0 b a q (transport (λ i → [ ·-nullifiesʳ a (~ i) < a · b ]) p)
·-reflects-0< a b p .snd .snd q = ·-reflects-<-flippedʳ 0 a b q (transport (λ i → [ ·-nullifiesˡ b (~ i) < a · b ]) p)
#-dicho : ∀ x → [ x # 0 ] ⊎ (x ≡ 0)
#-dicho x = <-tricho x 0
⊕-identityʳ : ∀ s → (s Bool.⊕ spos) ≡ s
⊕-identityʳ spos = refl
⊕-identityʳ sneg = refl
·-preserves-signˡ : ∀ x y → [ 0 < y ] → sign (x · y) ≡ sign x
·-preserves-signˡ x (signed spos zero) p = ⊥-elim {A = λ _ → sign (x · ℤ.posneg i0) ≡ sign x} (¬-<ⁿ-zero p)
·-preserves-signˡ x (signed sneg zero) p = ⊥-elim {A = λ _ → sign (x · ℤ.posneg i1) ≡ sign x} (¬-<ⁿ-zero p)
·-preserves-signˡ x (ℤ.posneg i) p = ⊥-elim {A = λ _ → sign (x · ℤ.posneg i ) ≡ sign x} (¬-<ⁿ-zero p)
·-preserves-signˡ (signed spos zero) (signed spos (suc n)) p = refl
·-preserves-signˡ (signed sneg zero) (signed spos (suc n)) p = refl
·-preserves-signˡ (ℤ.posneg i) (signed spos (suc n)) p = refl
·-preserves-signˡ (signed s (suc n₁)) (signed spos (suc n)) p = ⊕-identityʳ s
#⇒≢ : ∀ x → [ x # 0 ] → ¬ᵗ(0 ≡ x)
#⇒≢ x (inl p) q = <-irrefl 0 $ subst (λ p → [ p < pos 0 ]) (sym q) p
#⇒≢ x (inr p) q = <-irrefl 0 $ subst (λ p → [ pos 0 < p ]) (sym q) p
<-split-pos : ∀ z → [ 0 < z ] → Σ[ n ∈ ℕ ] z ≡ pos (suc n)
<-split-pos (pos zero) p = ⊥-elim {A = λ _ → Σ[ n ∈ ℕ ] posneg i0 ≡ pos (suc n)} (<-irrefl 0 p)
<-split-pos (neg zero) p = ⊥-elim {A = λ _ → Σ[ n ∈ ℕ ] posneg i1 ≡ pos (suc n)} (<-irrefl 0 p)
<-split-pos (posneg i) p = ⊥-elim {A = λ _ → Σ[ n ∈ ℕ ] posneg i ≡ pos (suc n)} (<-irrefl 0 p)
<-split-pos (pos (suc n)) p = n , refl
<-split-neg : ∀ z → [ z < 0 ] → Σ[ n ∈ ℕ ] z ≡ neg (suc n)
<-split-neg (pos zero) p = ⊥-elim {A = λ _ → Σ[ n ∈ ℕ ] posneg i0 ≡ neg (suc n)} (<-irrefl 0 p)
<-split-neg (neg zero) p = ⊥-elim {A = λ _ → Σ[ n ∈ ℕ ] posneg i1 ≡ neg (suc n)} (<-irrefl 0 p)
<-split-neg (posneg i) p = ⊥-elim {A = λ _ → Σ[ n ∈ ℕ ] posneg i ≡ neg (suc n)} (<-irrefl 0 p)
<-split-neg (pos (suc m)) p = ⊥-elim {A = λ _ → Σ[ n ∈ ℕ ] pos (suc m) ≡ neg (suc n)} (¬suc<ⁿ0 m p)
<-split-neg (neg (suc m)) p = m , refl
#-split-abs : ∀ a → [ a # 0 ] → Σ[ x ∈ _ ] (abs a ≡ suc (abs x))
#-split-abs a (inl a<0) with <-split-neg a a<0; ... | (n , p) = neg n , λ i → abs (p i)
#-split-abs a (inr 0<a) with <-split-pos a 0<a; ... | (n , p) = pos n , λ i → abs (p i)
-- this is QuoInt.signed-zero
signed0≡0 : ∀ s → signed s 0 ≡ 0
signed0≡0 spos = refl
signed0≡0 sneg i = posneg (~ i)
·-sucIntʳ⁺ : ∀ m n → m · pos (suc n) ≡ m + m · pos n
·-sucIntʳ⁺ m n = ·-comm m (pos (suc n)) ∙ *-pos-suc n m ∙ (λ i → m + ·-comm (pos n) m i)
signed-respects-+ : ∀ s a b → signed s (a +ⁿ b) ≡ signed s a + signed s b
signed-respects-+ spos zero b = refl
signed-respects-+ sneg zero b = refl
signed-respects-+ spos (suc a) b i = sucℤ $ signed-respects-+ spos a b i
signed-respects-+ sneg (suc a) b i = predℤ $ signed-respects-+ sneg a b i
-- this is QuoInt.signed-inv
sign-abs-identity : ∀ a → signed (sign a) (abs a) ≡ a
sign-abs-identity (pos zero) j = posneg (i0 ∧ j)
sign-abs-identity (neg zero) j = posneg (i1 ∧ j)
sign-abs-identity (posneg i) j = posneg (i ∧ j)
sign-abs-identity (pos (suc n)) = refl
sign-abs-identity (neg (suc n)) = refl
signed-reflects-≡₁ : ∀ s₁ s₂ n → signed s₁ (suc n) ≡ signed s₂ (suc n) → s₁ ≡ s₂
signed-reflects-≡₁ s₁ s₂ n p i = sign (p i)
signed-reflects-≡₂ : ∀ s₁ s₂ a b → signed s₁ a ≡ signed s₂ b → a ≡ b
signed-reflects-≡₂ s₁ s₂ a b p i = abs (p i)
-abs : ∀ a → abs (- a) ≡ abs a
-abs (signed s n) = refl
-abs (posneg i) = refl
-reflects-≡ : ∀ a b → - a ≡ - b → a ≡ b
-reflects-≡ a b p = sym (negate-invol a) ∙ (λ i → - p i) ∙ negate-invol b
abs-preserves-· : ∀ a b → abs (a · b) ≡ abs a ·ⁿ abs b
abs-preserves-· a b = refl
sign-abs-≡ : ∀ a b → sign a ≡ sign b → abs a ≡ abs b → a ≡ b
sign-abs-≡ a b p q = transport (λ i → sign-abs-identity a i ≡ sign-abs-identity b i) λ i → signed (p i) (q i)
0<-sign : ∀ z → [ 0 < z ] → sign z ≡ spos
0<-sign z p i = sign $ <-split-pos z p .snd i
<0-sign : ∀ z → [ z < 0 ] → sign z ≡ sneg
<0-sign z p i = sign $ <-split-neg z p .snd i
sign-pos : ∀ n → sign (pos n) ≡ spos
sign-pos zero = refl
sign-pos (suc n) = refl
-- inj-*sm : l * suc m ≡ n * suc m → l ≡ n
-- inj-*sm {zero} {m} {n} p = 0≡n*sm→0≡n p
-- inj-*sm {l} {m} {zero} p = sym (0≡n*sm→0≡n (sym p))
-- inj-*sm {suc l} {m} {suc n} p = cong suc (inj-*sm (inj-m+ {m = suc m} p))
private
lem1 : ∀ a x → sign a ≡ sign (signed (sign a) (abs a +ⁿ x ·ⁿ abs a))
lem1 (pos zero) x = sym $ sign-pos (x ·ⁿ 0)
lem1 (neg zero) x = sym $ sign-pos (x ·ⁿ 0)
lem1 (posneg i) x = sym $ sign-pos (x ·ⁿ 0)
lem1 (pos (suc n)) x = refl
lem1 (neg (suc n)) x = refl
·-reflects-≡ˡ⁺ : ∀ a b x → (pos (suc x)) · a ≡ (pos (suc x)) · b → a ≡ b
·-reflects-≡ˡ⁺ a b x p = sym (sign-abs-identity a) ∙ (λ i → signed (κ i) (γ i)) ∙ sign-abs-identity b where
φ : suc x ·ⁿ abs a ≡ suc x ·ⁿ abs b
φ = signed-reflects-≡₂ _ _ _ _ p
γ : abs a ≡ abs b
γ = ·ⁿ-reflects-≡ˡ' {x} {abs a} {abs b} φ
κ = transport ( sign (signed (sign a) (suc x ·ⁿ abs a))
≡ sign (signed (sign b) (suc x ·ⁿ abs b)) ≡⟨ (λ i → lem1 a x (~ i) ≡ lem1 b x (~ i)) ⟩
sign a ≡ sign b ∎) (λ i → sign (p i))
·-reflects-≡ˡ⁻ : ∀ a b x → (neg (suc x)) · a ≡ (neg (suc x)) · b → a ≡ b
·-reflects-≡ˡ⁻ a b x p = sym (sign-abs-identity a) ∙ γ ∙ sign-abs-identity b where
φ : suc x ·ⁿ abs a ≡ suc x ·ⁿ abs b
φ = signed-reflects-≡₂ _ _ _ _ p
κ : abs a ≡ abs b
κ = ·ⁿ-reflects-≡ˡ' {x} {abs a} {abs b} φ
γ : signed (sign a) (abs a) ≡ signed (sign b) (abs b)
γ with #-dicho a
... | inl a#0 = -reflects-≡ _ _ (λ i → signed (θ i) (κ i)) where
abstract
c = #-split-abs a a#0 .fst
q₁ : abs a ≡ suc (abs c)
q₁ = #-split-abs a a#0 .snd
q₂ : abs b ≡ suc (abs c)
q₂ = sym κ ∙ q₁
θ : not (sign a) ≡ not (sign b)
θ = signed-reflects-≡₁ _ _ _ (transport (λ i → signed (not (sign a)) (suc x ·ⁿ q₁ i) ≡ signed (not (sign b)) (suc x ·ⁿ q₂ i)) p)
... | inr a≡0 = cong₂ signed refl (λ i → abs (a≡0 i)) ∙ signed0≡0 (sign a) ∙ sym (signed0≡0 (sign b)) ∙ cong₂ signed refl ((λ i → abs (a≡0 (~ i))) ∙ κ)
·-reflects-≡ˡ : ∀ a b x → [ x # 0 ] → x · a ≡ x · b → a ≡ b
·-reflects-≡ˡ a b x (inl x<0) q = let (y , r) = <-split-neg x x<0 in ·-reflects-≡ˡ⁻ a b y (transport (λ i → r i · a ≡ r i · b) q)
·-reflects-≡ˡ a b x (inr 0<x) q = let (y , r) = <-split-pos x 0<x in ·-reflects-≡ˡ⁺ a b y (transport (λ i → r i · a ≡ r i · b) q)
·-reflects-≡ʳ : ∀ a b x → [ x # 0 ] → a · x ≡ b · x → a ≡ b
·-reflects-≡ʳ a b x p q = ·-reflects-≡ˡ a b x p (·-comm x a ∙ q ∙ ·-comm b x)
|
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------------------------------------------------------------------------------
-- We do not know how erase a proof term in the translation
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module NotErasedProofTerm where
postulate
D : Set
_≡_ : D → D → Set
_≤_ : D → D → Set
zero : D
succ : D → D
data N : D → Set where
zN : N zero
sN : ∀ {n} → N n → N (succ n)
thm : ∀ n → N n → (∀ k → k ≤ k) → n ≡ n
thm n Nn h = prf
where
-- The internal type of prf is
-- ∀ (n : D) (Nn : N n) (h : ∀ k → k ≤ k) → ...
-- Apia can erase the proof term Nn, but it cannot erase the proof
-- term h.
postulate prf : n ≡ n
{-# ATP prove prf #-}
|
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open import Nat
open import Prelude
open import core
open import contexts
open import lemmas-disjointness
open import exchange
open import lemmas-freshG
-- this module contains all the proofs of different weakening structural
-- properties that we use for the hypothetical judgements
module weakening where
mutual
weaken-subst-Δ : ∀{Δ1 Δ2 Γ σ Γ'} → Δ1 ## Δ2
→ Δ1 , Γ ⊢ σ :s: Γ'
→ (Δ1 ∪ Δ2) , Γ ⊢ σ :s: Γ'
weaken-subst-Δ disj (STAId x) = STAId x
weaken-subst-Δ disj (STASubst subst x) = STASubst (weaken-subst-Δ disj subst) (weaken-ta-Δ1 disj x)
weaken-ta-Δ1 : ∀{Δ1 Δ2 Γ d τ} → Δ1 ## Δ2
→ Δ1 , Γ ⊢ d :: τ
→ (Δ1 ∪ Δ2) , Γ ⊢ d :: τ
weaken-ta-Δ1 disj TAConst = TAConst
weaken-ta-Δ1 disj (TAVar x₁) = TAVar x₁
weaken-ta-Δ1 disj (TALam x₁ wt) = TALam x₁ (weaken-ta-Δ1 disj wt)
weaken-ta-Δ1 disj (TAAp wt wt₁) = TAAp (weaken-ta-Δ1 disj wt) (weaken-ta-Δ1 disj wt₁)
weaken-ta-Δ1 {Δ1} {Δ2} {Γ} disj (TAEHole {u = u} {Γ' = Γ'} x x₁) = TAEHole (x∈∪l Δ1 Δ2 u _ x ) (weaken-subst-Δ disj x₁)
weaken-ta-Δ1 {Δ1} {Δ2} {Γ} disj (TANEHole {Γ' = Γ'} {u = u} x wt x₁) = TANEHole (x∈∪l Δ1 Δ2 u _ x) (weaken-ta-Δ1 disj wt) (weaken-subst-Δ disj x₁)
weaken-ta-Δ1 disj (TACast wt x) = TACast (weaken-ta-Δ1 disj wt) x
weaken-ta-Δ1 disj (TAFailedCast wt x x₁ x₂) = TAFailedCast (weaken-ta-Δ1 disj wt) x x₁ x₂
weaken-ta-Δ1 disj (TAFst wt) = TAFst (weaken-ta-Δ1 disj wt)
weaken-ta-Δ1 disj (TASnd wt) = TASnd (weaken-ta-Δ1 disj wt)
weaken-ta-Δ1 disj (TAPair wt wt₁) = TAPair (weaken-ta-Δ1 disj wt) (weaken-ta-Δ1 disj wt₁)
-- this is a little bit of a time saver. since ∪ is commutative on
-- disjoint contexts, and we need that premise anyway in both positions,
-- there's no real reason to repeat the inductive argument above
weaken-ta-Δ2 : ∀{Δ1 Δ2 Γ d τ} → Δ1 ## Δ2
→ Δ2 , Γ ⊢ d :: τ
→ (Δ1 ∪ Δ2) , Γ ⊢ d :: τ
weaken-ta-Δ2 {Δ1} {Δ2} {Γ} {d} {τ} disj D = tr (λ q → q , Γ ⊢ d :: τ) (∪comm Δ2 Δ1 (##-comm disj)) (weaken-ta-Δ1 (##-comm disj) D)
-- note that these statements are somewhat stronger than usual. this is
-- because we don't have implcit α-conversion. this reifies the
-- often-silent on paper assumption that if you collide with a bound
-- variable you can just α-convert it away and not worry.
-- used in both cases below, so factored into a lemma to save repeated code
lem-reassoc : {A : Set} {Γ Γ' : A ctx} {x : Nat} {τ : A} → (x # Γ') → (Γ ,, (x , τ)) ∪ Γ' == (Γ ∪ Γ') ,, (x , τ)
lem-reassoc {A} {Γ} {Γ'} {x} {τ} apt with lem-apart-sing-disj apt
... | disj = (∪assoc Γ (■ (x , τ)) Γ' disj) ·
(ap1 (λ qq → Γ ∪ qq) (∪comm (■ (x , τ)) (Γ') disj) ·
! (∪assoc Γ Γ' (■ (x , τ)) (##-comm disj)))
-- first we prove a general form of weakening, that you can add any
-- context Γ as long as it's fresh with respect to e
mutual
weaken-synth-∪ : ∀{e τ Γ Γ'} → freshΓ Γ' e → Γ ⊢ e => τ → (Γ ∪ Γ') ⊢ e => τ
weaken-synth-∪ frsh SConst = SConst
weaken-synth-∪ frsh (SAsc x) = SAsc (weaken-ana-∪ (freshΓ-asc frsh) x)
weaken-synth-∪ {Γ = Γ} {Γ' = Γ'} frsh (SVar {x = x} x₁) = SVar (x∈∪l Γ Γ' x _ x₁)
weaken-synth-∪ frsh (SAp x wt x₁ x₂) = SAp x (weaken-synth-∪ (freshΓ-ap1 frsh) wt)
x₁
(weaken-ana-∪ (freshΓ-ap2 frsh) x₂)
weaken-synth-∪ frsh SEHole = SEHole
weaken-synth-∪ frsh (SNEHole x wt) = SNEHole x (weaken-synth-∪ (freshΓ-nehole frsh) wt)
weaken-synth-∪ {Γ = Γ} {Γ' = Γ'} frsh (SLam {e = e} {τ2 = τ2} {x = x} x₁ wt)
with ctxindirect Γ' x
... | Inl qq = abort (lem-fresh-lam2 (frsh x qq))
... | Inr qq = SLam (apart-parts Γ Γ' x x₁ qq)
(tr (λ qq → qq ⊢ e => τ2) (lem-reassoc {Γ = Γ} qq)
(weaken-synth-∪ (freshΓ-lam2 frsh) wt))
weaken-synth-∪ frsh (SFst wt x) = SFst (weaken-synth-∪ (freshΓ-fst frsh) wt) x
weaken-synth-∪ frsh (SSnd wt x) = SSnd (weaken-synth-∪ (freshΓ-snd frsh) wt) x
weaken-synth-∪ frsh (SPair hd wt wt₁) = SPair hd (weaken-synth-∪ (freshΓ-pair1 frsh) wt) (weaken-synth-∪ (freshΓ-pair2 frsh) wt₁)
weaken-ana-∪ : ∀{e τ Γ Γ'} → freshΓ Γ' e → Γ ⊢ e <= τ → (Γ ∪ Γ') ⊢ e <= τ
weaken-ana-∪ frsh (ASubsume x x₁) = ASubsume (weaken-synth-∪ frsh x) x₁
weaken-ana-∪ {Γ = Γ} {Γ' = Γ'} frsh (ALam {e = e} {τ2 = τ2} {x = x} x₁ x₂ wt)
with ctxindirect Γ' x
... | Inl qq = abort (lem-fresh-lam1 (frsh x qq))
... | Inr qq = ALam (apart-parts Γ Γ' x x₁ qq)
x₂
(tr (λ qq → qq ⊢ e <= τ2) (lem-reassoc {Γ = Γ} qq)
(weaken-ana-∪ (freshΓ-lam1 frsh) wt))
-- it follows from this that if the term is closed, you can weaken with
-- any context that's fresh in e
weaken-synth-closed : ∀{e τ Γ} → freshΓ Γ e → ∅ ⊢ e => τ → Γ ⊢ e => τ
weaken-synth-closed {e} {τ} {Γ} f wt = tr (λ qq → qq ⊢ e => τ) ∅∪1 (weaken-synth-∪ f wt)
weaken-ana-closed : ∀{e τ Γ} → freshΓ Γ e → ∅ ⊢ e <= τ → Γ ⊢ e <= τ
weaken-ana-closed {e} {τ} {Γ} f wt = tr (λ qq → qq ⊢ e <= τ) ∅∪1 (weaken-ana-∪ f wt)
-- the very structural forms also follow from the union weakening, since
-- ,, is defined by ∪
weaken-synth : ∀{ x Γ e τ τ'} → freshe x e
→ Γ ⊢ e => τ
→ (Γ ,, (x , τ')) ⊢ e => τ
weaken-synth f wt = weaken-synth-∪ (fresh-freshΓ f) wt
weaken-ana : ∀{x Γ e τ τ'} → freshe x e
→ Γ ⊢ e <= τ
→ (Γ ,, (x , τ')) ⊢ e <= τ
weaken-ana f wt = weaken-ana-∪ (fresh-freshΓ f) wt
mutual
weaken-subst-Γ : ∀{ x Γ Δ σ Γ' τ} →
envfresh x σ →
Δ , Γ ⊢ σ :s: Γ' →
Δ , (Γ ,, (x , τ)) ⊢ σ :s: Γ'
weaken-subst-Γ {Γ = Γ} (EFId x₁) (STAId x₂) = STAId (λ x τ x₃ → x∈∪l Γ _ x τ (x₂ x τ x₃) )
weaken-subst-Γ {x = x} {Γ = Γ} (EFSubst x₁ efrsh x₂) (STASubst {y = y} {τ = τ'} subst x₃) =
STASubst (exchange-subst-Γ {Γ = Γ} (flip x₂) (weaken-subst-Γ {Γ = Γ ,, (y , τ')} efrsh subst))
(weaken-ta x₁ x₃)
weaken-ta : ∀{x Γ Δ d τ τ'} →
fresh x d →
Δ , Γ ⊢ d :: τ →
Δ , Γ ,, (x , τ') ⊢ d :: τ
weaken-ta _ TAConst = TAConst
weaken-ta {x} {Γ} {_} {_} {τ} {τ'} (FVar x₂) (TAVar x₃) = TAVar (x∈∪l Γ (■ (x , τ')) _ _ x₃)
weaken-ta {x = x} frsh (TALam {x = y} x₂ wt) with natEQ x y
weaken-ta (FLam x₁ x₂) (TALam x₃ wt) | Inl refl = abort (x₁ refl)
weaken-ta {Γ = Γ} {τ' = τ'} (FLam x₁ x₃) (TALam {x = y} x₄ wt) | Inr x₂ = TALam (apart-extend1 Γ (flip x₁) x₄) (exchange-ta-Γ {Γ = Γ} (flip x₁) (weaken-ta x₃ wt))
weaken-ta (FAp frsh frsh₁) (TAAp wt wt₁) = TAAp (weaken-ta frsh wt) (weaken-ta frsh₁ wt₁)
weaken-ta (FHole x₁) (TAEHole x₂ x₃) = TAEHole x₂ (weaken-subst-Γ x₁ x₃)
weaken-ta (FNEHole x₁ frsh) (TANEHole x₂ wt x₃) = TANEHole x₂ (weaken-ta frsh wt) (weaken-subst-Γ x₁ x₃)
weaken-ta (FCast frsh) (TACast wt x₁) = TACast (weaken-ta frsh wt) x₁
weaken-ta (FFailedCast frsh) (TAFailedCast wt x₁ x₂ x₃) = TAFailedCast (weaken-ta frsh wt) x₁ x₂ x₃
weaken-ta (FFst frsh) (TAFst wt) = TAFst (weaken-ta frsh wt)
weaken-ta (FSnd frsh) (TASnd wt) = TASnd (weaken-ta frsh wt)
weaken-ta (FPair frsh frsh₁) (TAPair wt wt₁) = TAPair (weaken-ta frsh wt) (weaken-ta frsh₁ wt₁)
|
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|
-- Bug: With abstraction depended on bound variable names!
module Issue233 where
postulate
T : (Set → Set) → Set
mkT : (F : Set → Set) → T F
foo : T (λ A → A)
foo with λ (B : Set) → B
... | F = mkT F
|
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open import Data.List using ( List ; [] ; _∷_ )
open import Data.Nat using ( ℕ ; zero ; suc )
open import Data.Maybe using ( Maybe ; just ; nothing )
open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; subst )
open import Relation.Binary.PropositionalEquality.TrustMe using ( trustMe )
module AssocFree.Util where
≡-relevant : ∀ {α} {A : Set α} {a b : A} → .(a ≡ b) → (a ≡ b)
≡-relevant a≡b = trustMe
δsubst₂ : ∀ {a b p} {A : Set a} {B : A → Set b} (P : ∀ a → B a → Set p)
{x₁ x₂ y₁ y₂} → (x₁≡x₂ : x₁ ≡ x₂) → (subst B x₁≡x₂ y₁ ≡ y₂) → P x₁ y₁ → P x₂ y₂
δsubst₂ P refl refl p = p
subst-natural : ∀ {α β γ} {A : Set α} {B : A → Set β} {C : A → Set γ} {a₁ a₂ : A}
(p : a₁ ≡ a₂) (f : ∀ {a} → B a → C a) (b : B a₁) →
subst C p (f b) ≡ f (subst B p b)
subst-natural refl f b = refl
lookup : ∀ {α} {A : Set α} → List A → ℕ → Maybe A
lookup [] n = nothing
lookup (a ∷ as) zero = just a
lookup (a ∷ as) (suc n) = lookup as n
|
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{-# OPTIONS --without-K #-}
module Util.HoTT.Univalence.Axiom where
open import Util.HoTT.Equiv
open import Util.HoTT.Univalence.Statement
open import Util.Prelude
open import Util.Relation.Binary.PropositionalEquality using (Σ-≡⁻)
private
variable
α β γ : Level
A B C : Set α
postulate
univalence : ∀ {α} → Univalence α
≃→≡ : A ≃ B → A ≡ B
≃→≡ A≃B = univalence A≃B .proj₁ .proj₁
≡→≃∘≃→≡ : (p : A ≃ B) → ≡→≃ (≃→≡ p) ≡ p
≡→≃∘≃→≡ p = univalence p .proj₁ .proj₂
≃→≡∘≡→≃ : (p : A ≡ B) → ≃→≡ (≡→≃ p) ≡ p
≃→≡∘≡→≃ p = Σ-≡⁻ (univalence (≡→≃ p) .proj₂ (p , refl)) .proj₁
≃→≡-≡→≃-coh : (p : A ≡ B)
→ subst (λ q → ≡→≃ q ≡ ≡→≃ p) (≃→≡∘≡→≃ p) (≡→≃∘≃→≡ (≡→≃ p)) ≡ refl
≃→≡-≡→≃-coh p = Σ-≡⁻ (univalence (≡→≃ p) .proj₂ (p , refl)) .proj₂
≅→≡ : A ≅ B → A ≡ B
≅→≡ = ≃→≡ ∘ ≅→≃
|
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------------------------------------------------------------------------
-- The Agda standard library
--
-- IO
------------------------------------------------------------------------
module IO where
open import Coinduction
open import Data.Unit
open import Data.String
open import Data.Colist
open import Function
import IO.Primitive as Prim
open import Level
------------------------------------------------------------------------
-- The IO monad
-- One cannot write "infinitely large" computations with the
-- postulated IO monad in IO.Primitive without turning off the
-- termination checker (or going via the FFI, or perhaps abusing
-- something else). The following coinductive deep embedding is
-- introduced to avoid this problem. Possible non-termination is
-- isolated to the run function below.
infixl 1 _>>=_ _>>_
data IO {a} (A : Set a) : Set (suc a) where
lift : (m : Prim.IO A) → IO A
return : (x : A) → IO A
_>>=_ : {B : Set a} (m : ∞ (IO B)) (f : (x : B) → ∞ (IO A)) → IO A
_>>_ : {B : Set a} (m₁ : ∞ (IO B)) (m₂ : ∞ (IO A)) → IO A
{-# NON_TERMINATING #-}
run : ∀ {a} {A : Set a} → IO A → Prim.IO A
run (lift m) = m
run (return x) = Prim.return x
run (m >>= f) = Prim._>>=_ (run (♭ m )) λ x → run (♭ (f x))
run (m₁ >> m₂) = Prim._>>=_ (run (♭ m₁)) λ _ → run (♭ m₂)
------------------------------------------------------------------------
-- Utilities
sequence : ∀ {a} {A : Set a} → Colist (IO A) → IO (Colist A)
sequence [] = return []
sequence (c ∷ cs) = ♯ c >>= λ x →
♯ (♯ sequence (♭ cs) >>= λ xs →
♯ return (x ∷ ♯ xs))
-- The reason for not defining sequence′ in terms of sequence is
-- efficiency (the unused results could cause unnecessary memory use).
sequence′ : ∀ {a} {A : Set a} → Colist (IO A) → IO (Lift ⊤)
sequence′ [] = return _
sequence′ (c ∷ cs) = ♯ c >>
♯ (♯ sequence′ (♭ cs) >>
♯ return _)
mapM : ∀ {a b} {A : Set a} {B : Set b} →
(A → IO B) → Colist A → IO (Colist B)
mapM f = sequence ∘ map f
mapM′ : {A B : Set} → (A → IO B) → Colist A → IO (Lift ⊤)
mapM′ f = sequence′ ∘ map f
------------------------------------------------------------------------
-- Simple lazy IO
-- Note that the functions below produce commands which, when
-- executed, may raise exceptions.
-- Note also that the semantics of these functions depends on the
-- version of the Haskell base library. If the version is 4.2.0.0 (or
-- later?), then the functions use the character encoding specified by
-- the locale. For older versions of the library (going back to at
-- least version 3) the functions use ISO-8859-1.
getContents : IO Costring
getContents = lift Prim.getContents
readFile : String → IO Costring
readFile f = lift (Prim.readFile f)
-- Reads a finite file. Raises an exception if the file path refers to
-- a non-physical file (like "/dev/zero").
readFiniteFile : String → IO String
readFiniteFile f = lift (Prim.readFiniteFile f)
writeFile∞ : String → Costring → IO ⊤
writeFile∞ f s =
♯ lift (Prim.writeFile f s) >>
♯ return _
writeFile : String → String → IO ⊤
writeFile f s = writeFile∞ f (toCostring s)
appendFile∞ : String → Costring → IO ⊤
appendFile∞ f s =
♯ lift (Prim.appendFile f s) >>
♯ return _
appendFile : String → String → IO ⊤
appendFile f s = appendFile∞ f (toCostring s)
putStr∞ : Costring → IO ⊤
putStr∞ s =
♯ lift (Prim.putStr s) >>
♯ return _
putStr : String → IO ⊤
putStr s = putStr∞ (toCostring s)
putStrLn∞ : Costring → IO ⊤
putStrLn∞ s =
♯ lift (Prim.putStrLn s) >>
♯ return _
putStrLn : String → IO ⊤
putStrLn s = putStrLn∞ (toCostring s)
|
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-- {-# OPTIONS -v tc.cover.cover:10 -v tc.cover.splittree:100 -v tc.cover.strategy:100 -v tc.cc:100 #-}
module Issue365 where
{- Basic data types -}
data Nat : Set where
zero : Nat
succ : Nat -> Nat
data Fin : Nat -> Set where
fzero : {n : Nat} -> Fin (succ n)
fsucc : {n : Nat} -> Fin n -> Fin (succ n)
data Vec (A : Set) : Nat -> Set where
[] : Vec A zero
_::_ : {n : Nat} -> A -> Vec A n -> Vec A (succ n)
data _==_ {A : Set} (x : A) : A -> Set where
refl : x == x
{- Function composition -}
_◦_ : {A : Set} {B : A -> Set} {C : (x : A) -> B x -> Set}
(f : {x : A} (y : B x) -> C x y) (g : (x : A) -> B x)
(x : A) -> C x (g x)
(f ◦ g) x = f (g x)
{- Indexing and tabulating -}
_!_ : {n : Nat} {A : Set} -> Vec A n -> Fin n -> A
[] ! ()
(x :: xs) ! fzero = x
(x :: xs) ! (fsucc i) = xs ! i
tabulate : {n : Nat} {A : Set} -> (Fin n -> A) -> Vec A n
tabulate {zero} f = []
tabulate {succ n} f = f fzero :: tabulate (f ◦ fsucc)
lem-tab-! : forall {A n} (xs : Vec A n) -> tabulate (_!_ xs) == xs
lem-tab-! {A} {zero} [] = refl
lem-tab-! {A} {succ n} (x :: xs) with tabulate (_!_ xs) | lem-tab-! xs
lem-tab-! {A} {succ _} (x :: xs) | .xs | refl = refl
|
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module Operator.Equals {ℓ} where
import Lvl
open import Data.Boolean
open import Functional
open import Relator.Equals{ℓ}
open import Type{ℓ}
-- Type class for run-time checking of equality
record Equals(T : Type) : Type where
infixl 100 _==_
field
_==_ : T → T → Bool
field
⦃ completeness ⦄ : ∀{a b : T} → (a ≡ b) → (a == b ≡ 𝑇)
open Equals ⦃ ... ⦄ using (_==_) public
|
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open import Data.Product using ( ∃ ; _×_ ; _,_ ; proj₁ ; proj₂ )
open import Relation.Unary using ( _∈_ )
open import Web.Semantic.DL.TBox.Interp using ( Δ ; _⊨_≈_ ) renaming
( Interp to Interp′ ; emp to emp′ )
open import Web.Semantic.DL.Signature using ( Signature )
open import Web.Semantic.Util using ( False ; id )
module Web.Semantic.DL.ABox.Interp where
infixr 4 _,_
infixr 5 _*_
{-
An interpretation of a signature Σ (made of concept and role names)
over a set X of individuals consists of
- a Signature interpreation I
- a mapping from X do Δ I, the domain of interpretation of I
Note: In RDF the members of X are sets of IRIs, BNodes or Literals, but
IRIs can also refer to TBox elements.
-}
data Interp (Σ : Signature) (X : Set) : Set₁ where
-- I is a full Interpreation (Interp')
-- The function X → Δ {Σ} I interprets the variables in X
_,_ : ∀ I → (X → Δ {Σ} I) → (Interp Σ X)
-- extract the Signature Interpretation, forgetting the interpretation of variables
⌊_⌋ : ∀ {Σ X} → Interp Σ X → Interp′ Σ
⌊ I , i ⌋ = I
-- return the individuals function for an interpretation
ind : ∀ {Σ X} → (I : Interp Σ X) → X → Δ ⌊ I ⌋
ind (I , i) = i
-- paired individuals function for an interpretation, useful for relations/roles
ind² : ∀ {Σ X} → (I : Interp Σ X) → (X × X) → (Δ ⌊ I ⌋ × Δ ⌊ I ⌋)
ind² I (x , y) = (ind I x , ind I y)
-- why * ?
_*_ : ∀ {Σ X Y} → (Y → X) → Interp Σ X → Interp Σ Y
f * I = (⌊ I ⌋ , λ y → ind I (f y))
-- Empty interpretation
emp : ∀ {Σ} → Interp Σ False
emp = (emp′ , id)
data Surjective {Σ X} (I : Interp Σ X) : Set where
-- y is a variable i.e. y : X
-- (ind I y), x : Δ
-- all elements x of the domain Δ, have a variable y that it is an interpretation of
surj : (∀ x → ∃ λ y → ⌊ I ⌋ ⊨ x ≈ ind I y) → (I ∈ Surjective)
ind⁻¹ : ∀ {Σ X} {I : Interp Σ X} → (I ∈ Surjective) → (Δ ⌊ I ⌋ → X)
ind⁻¹ (surj i) x = proj₁ (i x)
surj✓ : ∀ {Σ X} {I : Interp Σ X} (I∈Surj : I ∈ Surjective) → ∀ x → (⌊ I ⌋ ⊨ x ≈ ind I (ind⁻¹ I∈Surj x))
surj✓ (surj i) x = proj₂ (i x)
|
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{-# OPTIONS --cubical #-}
module Cubical.Categories.Everything where
import Cubical.Categories.Category
import Cubical.Categories.Functor
import Cubical.Categories.NaturalTransformation
import Cubical.Categories.Presheaves
import Cubical.Categories.Sets
import Cubical.Categories.Type
|
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Vectors defined by recursion
------------------------------------------------------------------------
-- What is the point of this module? The n-ary products below are intended
-- to be used with a fixed n, in which case the nil constructor can be
-- avoided: pairs are represented as pairs (x , y), not as triples
-- (x , y , unit).
-- Additionally, vectors defined by recursion enjoy η-rules. That is to say
-- that two vectors of known length are definitionally equal whenever their
-- elements are.
{-# OPTIONS --without-K --safe #-}
module Data.Vec.Recursive where
open import Level using (Level; Lift; lift)
open import Data.Nat.Base as Nat using (ℕ; zero; suc)
open import Data.Empty
open import Data.Fin.Base as Fin using (Fin; zero; suc)
open import Data.Product as Prod using (_×_; _,_; proj₁; proj₂)
open import Data.Sum.Base as Sum using (_⊎_)
open import Data.Unit.Base
open import Data.Vec.Base as Vec using (Vec; _∷_)
open import Function
open import Relation.Unary
open import Agda.Builtin.Equality using (_≡_)
private
variable
a b c p : Level
A : Set a
B : Set b
C : Set c
-- Types and patterns
------------------------------------------------------------------------
pattern 2+_ n = suc (suc n)
infix 8 _^_
_^_ : Set a → ℕ → Set a
A ^ 0 = Lift _ ⊤
A ^ 1 = A
A ^ 2+ n = A × A ^ suc n
pattern [] = lift tt
infix 3 _∈[_]_
_∈[_]_ : {A : Set a} → A → ∀ n → A ^ n → Set a
a ∈[ 0 ] as = Lift _ ⊥
a ∈[ 1 ] a′ = a ≡ a′
a ∈[ 2+ n ] a′ , as = a ≡ a′ ⊎ a ∈[ suc n ] as
-- Basic operations
------------------------------------------------------------------------
cons : ∀ n → A → A ^ n → A ^ suc n
cons 0 a _ = a
cons (suc n) a as = a , as
uncons : ∀ n → A ^ suc n → A × A ^ n
uncons 0 a = a , lift tt
uncons (suc n) (a , as) = a , as
head : ∀ n → A ^ suc n → A
head n as = proj₁ (uncons n as)
tail : ∀ n → A ^ suc n → A ^ n
tail n as = proj₂ (uncons n as)
fromVec : ∀[ Vec A ⇒ (A ^_) ]
fromVec = Vec.foldr (_ ^_) (cons _) _
toVec : Π[ (A ^_) ⇒ Vec A ]
toVec 0 as = Vec.[]
toVec (suc n) as = head n as ∷ toVec n (tail n as)
lookup : ∀ {n} (k : Fin n) → A ^ n → A
lookup zero = head _
lookup (suc {n} k) = lookup k ∘′ tail n
replicate : ∀ n → A → A ^ n
replicate n a = fromVec (Vec.replicate a)
tabulate : ∀ n → (Fin n → A) → A ^ n
tabulate n f = fromVec (Vec.tabulate f)
append : ∀ m n → A ^ m → A ^ n → A ^ (m Nat.+ n)
append 0 n xs ys = ys
append 1 n x ys = cons n x ys
append (2+ m) n (x , xs) ys = x , append (suc m) n xs ys
splitAt : ∀ m n → A ^ (m Nat.+ n) → A ^ m × A ^ n
splitAt 0 n xs = [] , xs
splitAt (suc m) n xs =
let (ys , zs) = splitAt m n (tail (m Nat.+ n) xs) in
cons m (head (m Nat.+ n) xs) ys , zs
-- Manipulating N-ary products
------------------------------------------------------------------------
map : (A → B) → ∀ n → A ^ n → B ^ n
map f 0 as = lift tt
map f 1 a = f a
map f (2+ n) (a , as) = f a , map f (suc n) as
ap : ∀ n → (A → B) ^ n → A ^ n → B ^ n
ap 0 fs ts = []
ap 1 f t = f t
ap (2+ n) (f , fs) (t , ts) = f t , ap (suc n) fs ts
module _ {P : ℕ → Set p} where
foldr : P 0 → (A → P 1) → (∀ n → A → P (suc n) → P (2+ n)) →
∀ n → A ^ n → P n
foldr p0 p1 p2+ 0 as = p0
foldr p0 p1 p2+ 1 a = p1 a
foldr p0 p1 p2+ (2+ n) (a , as) = p2+ n a (foldr p0 p1 p2+ (suc n) as)
foldl : (P : ℕ → Set p) →
P 0 → (A → P 1) → (∀ n → A → P (suc n) → P (2+ n)) →
∀ n → A ^ n → P n
foldl P p0 p1 p2+ 0 as = p0
foldl P p0 p1 p2+ 1 a = p1 a
foldl P p0 p1 p2+ (2+ n) (a , as) = let p1′ = p1 a in
foldl (P ∘′ suc) p1′ (λ a → p2+ 0 a p1′) (p2+ ∘ suc) (suc n) as
reverse : ∀ n → A ^ n → A ^ n
reverse = foldl (_ ^_) [] id (λ n → _,_)
zipWith : (A → B → C) → ∀ n → A ^ n → B ^ n → C ^ n
zipWith f 0 as bs = []
zipWith f 1 a b = f a b
zipWith f (2+ n) (a , as) (b , bs) = f a b , zipWith f (suc n) as bs
unzipWith : (A → B × C) → ∀ n → A ^ n → B ^ n × C ^ n
unzipWith f 0 as = [] , []
unzipWith f 1 a = f a
unzipWith f (2+ n) (a , as) = Prod.zip _,_ _,_ (f a) (unzipWith f (suc n) as)
zip : ∀ n → A ^ n → B ^ n → (A × B) ^ n
zip = zipWith _,_
unzip : ∀ n → (A × B) ^ n → A ^ n × B ^ n
unzip = unzipWith id
|
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open import Prelude
open import RW.Utils.Monads
-- Some Error monad utilities, a là Haskell.
module RW.Utils.Error where
open import Data.String
open Monad {{...}}
-- Error Typeclass
record IsError {a}(A : Set a) : Set a where
field
showError : A → String
open IsError {{...}}
instance
IsError-String : IsError String
IsError-String = record { showError = λ s → s }
-- Error Monad
Err : ∀{a} → (E : Set a) ⦃ isErr : IsError E ⦄
→ Set a → Set a
Err e a = e ⊎ a
throwError : ∀{a}{E A : Set a} ⦃ isErr : IsError E ⦄
→ E → Err E A
throwError = i1
catchError : ∀{a}{E A : Set a} ⦃ isErr : IsError E ⦄
→ Err E A → (E → Err E A) → Err E A
catchError (i2 a) _ = i2 a
catchError (i1 e) f = f e
instance
MonadError : ∀{e}{E : Set e} ⦃ isErr : IsError E ⦄
→ Monad (Err E)
MonadError = record
{ return = i2
; _>>=_ = λ { (i1 err) _ → i1 err
; (i2 x ) f → f x
}
}
runErr : ∀{a}{E A : Set a} ⦃ isErr : IsError E ⦄
→ Err E A → String ⊎ A
runErr (i2 a) = i2 a
runErr ⦃ s ⦄ (i1 e) = i1 (IsError.showError s e)
|
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|
-- Andreas, 2018-04-10, issue #3581, reported by 3abc, test case by Andrea
-- Regression in the termination checker introduced together
-- with collecting function calls also in the type signatures
-- (fix of #1556).
open import Agda.Builtin.Bool
open import Agda.Builtin.Nat
I = Bool
i0 = true
i1 = false
record PathP {l} (A : I → Set l) (x : A i0) (y : A i1) : Set l where
field
apply : ∀ i → A i
open PathP
_[_≡_] = PathP
_≡_ : ∀ {l} {A : Set l} → A → A → Set l
_≡_ {A = A} = PathP (\ _ → A)
refl : ∀ {l} {A : Set l} {x : A} → x ≡ x
refl {x = x} .apply _ = x
cong' : ∀ {l ℓ'} {A : Set l}{B : A → Set ℓ'} (f : (a : A) → B a) {x y} (p : x ≡ y)
→ PathP (λ i → B (p .apply i)) (f (p .apply i0)) (f (p .apply i1))
cong' f p .apply = λ i → f (p .apply i)
foo : Nat → Nat
foo zero = zero
foo (suc n) = Z .apply true .apply true
where
postulate
Z : (\ _ → foo n ≡ foo n) [ (cong' foo (refl {x = n})) ≡ (\ { .apply i → cong' foo (refl {x = n}) .apply i }) ]
|
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------------------------------------------------------------------------------
-- Testing Agda internal terms: @Var Nat Args@ when @Args = []@
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module AgdaInternalTerms.VarEmptyArgumentsTerm where
postulate D : Set
postulate id : (P : D → Set)(x : D) → P x → P x
{-# ATP prove id #-}
|
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{-# OPTIONS --universe-polymorphism #-}
module Categories.Groupoid where
open import Level
open import Categories.Category
import Categories.Morphisms
record Groupoid {o ℓ e} (C : Category o ℓ e) : Set (o ⊔ ℓ ⊔ e) where
private module C = Category C
open C using (_⇒_)
open Categories.Morphisms C
field
_⁻¹ : ∀ {A B} → (A ⇒ B) → (B ⇒ A)
iso : ∀ {A B} {f : A ⇒ B} → Iso f (f ⁻¹)
|
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------------------------------------------------------------------------
-- Lemmas
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe --exact-split #-}
module Math.Combinatorics.ListFunction.Properties.Lemma where
open import Data.List hiding (_∷ʳ_)
import Data.List.Properties as Lₚ
open import Data.List.Relation.Binary.Sublist.Propositional using (_⊆_; []; _∷_; _∷ʳ_)
open import Data.Product as Prod using (proj₁; proj₂; _×_; _,_)
open import Function
open import Relation.Binary.PropositionalEquality
module _ {a} {A : Set a} where
[]⊆xs : ∀ (xs : List A) → [] ⊆ xs
[]⊆xs [] = []
[]⊆xs (x ∷ xs) = x ∷ʳ []⊆xs xs
module _ {a b} {A : Set a} {B : Set b} where
lemma₁ : ∀ (f : A → B) (x : A) (xss : List (List A)) →
map (λ ys → f x ∷ ys) (map (map f) xss) ≡ map (map f) (map (λ ys → x ∷ ys) xss)
lemma₁ f x xss = begin
map (λ ys → f x ∷ ys) (map (map f) xss) ≡⟨ sym $ Lₚ.map-compose xss ⟩
map (λ ys → f x ∷ map f ys) xss ≡⟨ Lₚ.map-compose xss ⟩
map (map f) (map (λ ys → x ∷ ys) xss) ∎
where open ≡-Reasoning
module _ {a b c} {A : Set a} {B : Set b} {C : Set c} where
proj₁-map₁ : ∀ (f : A → B) (t : A × C) → proj₁ (Prod.map₁ f t) ≡ f (Prod.proj₁ t)
proj₁-map₁ _ _ = refl
module _ {a b} {A : Set a} {B : Set b} where
proj₁-map₂ : ∀ (f : B → B) (t : A × B) → proj₁ (Prod.map₂ f t) ≡ proj₁ t
proj₁-map₂ _ _ = refl
proj₁′ : A × B → A
proj₁′ = proj₁
|
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module _ where
open import Agda.Builtin.Equality using (_≡_; refl)
-- First example --
module M (A : Set) where
record R : Set where
data D : Set where
open R (record {})
postulate
x : A
F : D → Set₁
F _ rewrite refl {x = x} = Set
-- Second example --
record ⊤ : Set where
no-eta-equality
constructor tt
data Box (A : Set) : Set where
[_] : A → Box A
Unit : Set
Unit = Box ⊤
F : Unit → Set → Set
F [ _ ] x = x
G : {P : Unit → Set} → ((x : ⊤) → P [ x ]) → ((x : Unit) → P x)
G f [ x ] = f x
record R : Set₁ where
no-eta-equality
field
f : (x : Unit) → Box (F x ⊤)
data ⊥ : Set where
r : R
r = record { f = G [_] }
open R r
H : ⊥ → Set₁
H _ rewrite refl {x = tt} = Set
|
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{-# OPTIONS --sized-types #-}
open import FRP.JS.Bool using ( Bool ; true ; false ) renaming ( _≟_ to _≟b_ )
open import FRP.JS.Nat using ( ℕ )
open import FRP.JS.Float using ( ℝ ) renaming ( _≟_ to _≟n_ )
open import FRP.JS.String using ( String ) renaming ( _≟_ to _≟s_ )
open import FRP.JS.Array using ( Array ) renaming ( lookup? to alookup? ; _≟[_]_ to _≟a[_]_ )
open import FRP.JS.Object using ( Object ) renaming ( lookup? to olookup? ; _≟[_]_ to _≟o[_]_ )
open import FRP.JS.Maybe using ( Maybe ; just ; nothing )
open import FRP.JS.Size using ( Size ; ↑_ )
module FRP.JS.JSON where
data JSON : {σ : Size} → Set where
null : ∀ {σ} → JSON {σ}
string : ∀ {σ} → String → JSON {σ}
float : ∀ {σ} → ℝ → JSON {σ}
bool : ∀ {σ} → Bool → JSON {σ}
array : ∀ {σ} → Array (JSON {σ}) → JSON {↑ σ}
object : ∀ {σ} → Object (JSON {σ}) → JSON {↑ σ}
{-# COMPILED_JS JSON function(x,v) {
if (x === null) { return v.null(null); }
else if (x.constructor === String) { return v.string(null,x); }
else if (x.constructor === Number) { return v.float(null,x); }
else if (x.constructor === Boolean) { return v.bool(null,x); }
else if (x.constructor === Array) { return v.array(null,x); }
else { return v.object(null,x); }
} #-}
{-# COMPILED_JS null function() { return null; } #-}
{-# COMPILED_JS string function() { return function(x) { return x; }; } #-}
{-# COMPILED_JS float function() { return function(x) { return x; }; } #-}
{-# COMPILED_JS bool function() { return function(x) { return x; }; } #-}
{-# COMPILED_JS array function() { return function(x) { return x; }; } #-}
{-# COMPILED_JS object function() { return function(x) { return x; }; } #-}
postulate
show : JSON → String
parse : String → Maybe JSON
{-# COMPILED_JS show JSON.stringify #-}
{-# COMPILED_JS parse require("agda.box").handle(JSON.parse) #-}
Key : Bool → Set
Key true = String
Key false = ℕ
lookup? : ∀ {σ} → Maybe (JSON {↑ σ}) → ∀ {b} → Key b → Maybe (JSON {σ})
lookup? (just (object js)) {true} k = olookup? js k
lookup? (just (array js)) {false} i = alookup? js i
lookup? _ _ = nothing
_≟_ : ∀ {σ τ} → JSON {σ} → JSON {τ} → Bool
null ≟ null = true
string s ≟ string t = s ≟s t
float m ≟ float n = m ≟n n
bool b ≟ bool c = b ≟b c
array js ≟ array ks = js ≟a[ _≟_ ] ks
object js ≟ object ks = js ≟o[ _≟_ ] ks
_ ≟ _ = false
|
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|
-- Testing the version option on a file with errors.
--
-- N.B. It is necessary to change the Issue1244a.out file when using
-- different versions of Agda.
foo : Set → Set
foo a = b
|
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module Luau.Addr where
open import Agda.Builtin.Bool using (true; false)
open import Agda.Builtin.Equality using (_≡_)
open import Agda.Builtin.Nat using (Nat; _==_)
open import Agda.Builtin.String using (String)
open import Agda.Builtin.TrustMe using (primTrustMe)
open import Properties.Dec using (Dec; yes; no)
open import Properties.Equality using (_≢_)
Addr : Set
Addr = Nat
_≡ᴬ_ : (a b : Addr) → Dec (a ≡ b)
a ≡ᴬ b with a == b
a ≡ᴬ b | false = no p where postulate p : (a ≢ b)
a ≡ᴬ b | true = yes primTrustMe
|
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{-# OPTIONS --without-K --safe #-}
module Categories.Category.Construction.Presheaves where
-- The Category of Presheaves over a Category C, i.e.
-- the Functor Category [ C.op , Setoids ]
-- Again, the levels are made explicit to show the generality and constraints.
-- CoPreasheaves are defined here as well, for convenience
-- The main reson to name them is that some things (like CoYoneda)
-- look more natural/symmetric then.
open import Level
open import Categories.Category
open import Categories.Category.Construction.Functors
open import Categories.Category.Instance.Setoids using (Setoids)
Presheaves′ : ∀ o′ ℓ′ {o ℓ e : Level} → Category o ℓ e →
Category (o ⊔ ℓ ⊔ e ⊔ suc (o′ ⊔ ℓ′)) (o ⊔ ℓ ⊔ o′ ⊔ ℓ′) (o ⊔ o′ ⊔ ℓ′)
Presheaves′ o′ ℓ′ C = Functors (Category.op C) (Setoids o′ ℓ′)
Presheaves : ∀ {o ℓ e o′ ℓ′ : Level} → Category o ℓ e →
Category (o ⊔ ℓ ⊔ e ⊔ suc (o′ ⊔ ℓ′)) (o ⊔ ℓ ⊔ o′ ⊔ ℓ′) (o ⊔ o′ ⊔ ℓ′)
Presheaves {o} {ℓ} {e} {o′} {ℓ′} C = Presheaves′ o′ ℓ′ C
CoPresheaves′ : ∀ o′ ℓ′ {o ℓ e : Level} → Category o ℓ e →
Category (o ⊔ ℓ ⊔ e ⊔ suc (o′ ⊔ ℓ′)) (o ⊔ ℓ ⊔ o′ ⊔ ℓ′) (o ⊔ o′ ⊔ ℓ′)
CoPresheaves′ o′ ℓ′ C = Functors C (Setoids o′ ℓ′)
CoPresheaves : ∀ {o ℓ e o′ ℓ′ : Level} → Category o ℓ e →
Category (o ⊔ ℓ ⊔ e ⊔ suc (o′ ⊔ ℓ′)) (o ⊔ ℓ ⊔ o′ ⊔ ℓ′) (o ⊔ o′ ⊔ ℓ′)
CoPresheaves {o} {ℓ} {e} {o′} {ℓ′} C = CoPresheaves′ o′ ℓ′ C
|
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module Monads.MonadMorphs where
open import Library
open import Functors
open import Categories
open import Monads
open Fun
open Monad
record MonadMorph {a b}{C : Cat {a}{b}}(M M' : Monad C) : Set (a ⊔ b) where
constructor monadmorph
open Cat C
field morph : ∀ {X} → Hom (T M X) (T M' X)
lawη : ∀ {X} → comp morph (η M {X}) ≅ η M' {X}
lawbind : ∀ {X Y}{k : Hom X (T M Y)} →
comp (morph {Y}) (bind M k)
≅
comp (bind M' (comp (morph {Y}) k)) (morph {X})
|
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-- Occurs check when unifying indices in patterns
module OccursCheck where
data Nat : Set where
zero : Nat
suc : Nat -> Nat
data _==_ {A : Set}(x : A) : A -> Set where
refl : x == x
f : {n : Nat} -> n == suc n -> Nat
f refl = zero
|
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{-# OPTIONS --cubical --no-import-sorts #-}
open import Cubical.Foundations.Everything renaming (_⁻¹ to _⁻¹ᵖ; assoc to ∙-assoc)
open import Function.Base using (_∋_)
import Cubical.Algebra.Group as Std
-- import Cubical.Structures.Group.Properties
open import MorePropAlgebra.Bundles
module MorePropAlgebra.Properties.Group {ℓ} (assumptions : Group {ℓ}) where
open Group assumptions renaming (Carrier to G)
import MorePropAlgebra.Properties.Monoid
module Monoid'Properties = MorePropAlgebra.Properties.Monoid (record { Group assumptions }) -- how does this even work without renaming?
module Monoid' = Monoid (record { Group assumptions })
( Monoid') = Monoid ∋ (record { Group assumptions })
stdIsGroup : Std.IsGroup 0g _+_ (-_)
stdIsGroup .Std.IsGroup.isMonoid = Monoid'Properties.stdIsMonoid
stdIsGroup .Std.IsGroup.inverse = is-inverse
stdGroup : Std.Group {ℓ}
stdGroup = record { Group assumptions ; isGroup = stdIsGroup }
module GroupLemmas' = Std.GroupLemmas stdGroup
-- module GroupLemmas (G : Group {ℓ}) where
-- open Group G public
-- abstract
-- simplL : (a : Carrier) {b c : Carrier} → a + b ≡ a + c → b ≡ c
-- simplR : {a b : Carrier} (c : Carrier) → a + c ≡ b + c → a ≡ b
-- invInvo : (a : Carrier) → - (- a) ≡ a
-- invId : - 0g ≡ 0g
-- idUniqueL : {e : Carrier} (x : Carrier) → e + x ≡ x → e ≡ 0g
-- idUniqueR : (x : Carrier) {e : Carrier} → x + e ≡ x → e ≡ 0g
-- invUniqueL : {g h : Carrier} → g + h ≡ 0g → g ≡ - h
-- invUniqueR : {g h : Carrier} → g + h ≡ 0g → h ≡ - g
-- invDistr : (a b : Carrier) → - (a + b) ≡ - b - a
-- private
-- simplR = GroupLemmas.simplR G
abstract
invUniqueL : {g h : G} → g + h ≡ 0g → g ≡ - h
invUniqueL {g} {h} p = GroupLemmas'.simplR h (p ∙ sym (is-invl h))
-- ported from `Algebra.Properties.Group`
private
right-helper : ∀ x y → y ≡ - x + (x + y)
right-helper x y = (
y ≡⟨ sym (snd (is-identity y)) ⟩
0g + y ≡⟨ cong₂ _+_ (sym (snd (is-inverse x))) refl ⟩
((- x) + x) + y ≡⟨ sym (is-assoc (- x) x y) ⟩
(- x) + (x + y) ∎)
-- alternative:
-- follows from uniqueness of -
-- (a + -a) ≡ 0
-- ∃! b . a + b = 0
-- show that a is an additive inverse of - a then it must be THE additive inverse of - a and has to be called - - a which is a by uniqueness
-involutive : ∀ x → - - x ≡ x
-involutive x = (
(- (- x)) ≡⟨ sym (fst (is-identity _)) ⟩
(- (- x)) + 0g ≡⟨ cong₂ _+_ refl (sym (snd (is-inverse _))) ⟩
(- (- x)) + (- x + x) ≡⟨ sym (right-helper (- x) x) ⟩
x ∎)
|
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module BSTree {A : Set}(_≤_ : A → A → Set) where
open import BTree {A}
data _⊴*_ : A → BTree → Set where
gelf : {x : A}
→ x ⊴* leaf
gend : {x y : A}{l r : BTree}
→ x ≤ y
→ x ⊴* l
→ x ⊴* (node y l r)
data _*⊴_ : BTree → A → Set where
lelf : {x : A}
→ leaf *⊴ x
lend : {x y : A}{l r : BTree}
→ y ≤ x
→ r *⊴ x
→ (node y l r) *⊴ x
data BSTree : BTree → Set where
slf : BSTree leaf
snd : {x : A}{l r : BTree}
→ BSTree l
→ BSTree r
→ l *⊴ x
→ x ⊴* r
→ BSTree (node x l r)
|
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module Issue1252 where
data Bool : Set where
true false : Bool
{-# COMPILED_DATA Bool Bool True False #-}
foo : Bool → Bool
foo true = false
foo false = true
{-# COMPILED_EXPORT foo foohs #-}
|
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module Hello where
open import IO using (run; putStrLn)
import IO.Primitive as Prim using (IO)
open import Data.Nat using (ℕ)
import Data.Nat.Show as Nat using (show)
open import Data.Unit using (⊤) -- This is no upper case 't'
open import Data.String using (_++_)
age : ℕ
age = 28
main : Prim.IO ⊤
main = run (putStrLn ("Hello World! I'm " ++ Nat.show age))
|
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-- Based on an example due to Thorsten Altenkirch. See "Recursion with
-- boxes", http://sneezy.cs.nott.ac.uk/fplunch/weblog/?p=104.
module SubjectReduction where
open import Common.Coinduction
Eq : {A : Set} → A → A → Set1
Eq {A} x y = (P : A → Set) → P x → P y
refl : ∀ {A} (x : A) → Eq x x
refl x P Px = Px
data Stream : Set where
tick : ∞ Stream → Stream
ticks : Stream
ticks = tick (♯ ticks)
l₁ : Eq ticks (tick _) -- Eq ticks (tick (♯ ticks))
l₁ = refl ticks
l₂ : ∀ {s t} → Eq s t → Eq (tick (♯ s)) (tick (♯ t))
l₂ eq = λ P Px → eq (λ s → P (tick (♯ s))) Px
Goal : Set1
Goal = Eq (tick (♯ ticks)) (tick (♯ tick (♯ ticks)))
_∶_ : (A : Set1) → A → A
_ ∶ x = x
l₃ : Goal
l₃ = ((_ → Eq (tick (♯ ticks)) (tick (♯ tick (♯ ticks)))) ∶ l₂) l₁
-- If l₃ is accepted, then it evaluates to λ P Px → Px, but the
-- following code is not accepted:
-- l₃′ : Goal
-- l₃′ = λ P Px → Px
|
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module Data.List.Decidable where
import Lvl
open import Data.Boolean
import Data.Boolean.Operators
open Data.Boolean.Operators.Programming
open import Data.Tuple
open import Data.List
open import Data.List.Functions
open import Data.List.Equiv
open import Functional
open import Logic.Propositional
open import Logic.Propositional.Theorems
open import Structure.Operator
open import Structure.Relator.Properties
open import Structure.Setoid
open import Type
open import Type.Properties.Decidable
open import Type.Properties.Decidable.Proofs
private variable ℓ ℓₑ ℓₑₗ : Lvl.Level
private variable T : Type{ℓ}
module _
⦃ equiv : Equiv{ℓₑ}(T) ⦄
⦃ equiv-list : Equiv{ℓₑₗ}(List(T)) ⦄
⦃ ext : Extensionality(equiv-list) ⦄
{_≡?_ : T → T → Bool}
where
instance
[≡]-decider : ⦃ dec : Decider(2)(_≡_ {T = T})(_≡?_) ⦄ → Decider(2)(_≡_ {T = List(T)})(satisfiesAll₂(_≡?_) (const 𝐹) (const 𝐹))
[≡]-decider {x = ∅} {∅} = true (reflexivity(_≡_))
[≡]-decider {x = ∅} {y ⊰ ys} = false [∅][⊰]-unequal
[≡]-decider {x = x ⊰ xs} {∅} = false ([∅][⊰]-unequal ∘ symmetry(_≡_))
[≡]-decider {x = x ⊰ xs} {y ⊰ ys} = step{f = id} (true ∘ uncurry (congruence₂(_⊰_))) (false ∘ contrapositiveᵣ [⊰]-generalized-cancellation) (tuple-decider ⦃ dec-Q = [≡]-decider {x = xs} {ys} ⦄)
|
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module Structure.Type.Identity.Proofs where
import Lvl
open import Functional using (_→ᶠ_ ; id ; _on₂_ ; swap ; _$_ ; apply)
open import Logic
open import Logic.Propositional
open import Logic.Propositional.Proofs.Structures
open import Structure.Function
open import Structure.Setoid using (Equiv ; intro) renaming (_≡_ to _≡ₛ_)
open import Structure.Relator.Equivalence
import Structure.Relator.Names as Names
open import Structure.Relator.Properties
open import Structure.Relator.Properties.Proofs
open import Structure.Relator
open import Structure.Type.Identity
open import Syntax.Function
open import Type
private variable ℓ ℓ₁ ℓ₂ ℓₑ₁ ℓₑ₂ ℓₑ ℓₘₑ ℓₚ ℓₒ : Lvl.Level
private variable T A B : Type{ℓ}
private variable x y : T
private variable Id _≡_ _▫_ : T → T → Stmt{ℓ}
module _ {_≡_ : Type{ℓ} → Type{ℓ} → Stmt{ℓₑ}} ⦃ minRefl : MinimalReflexiveRelation{ℓₚ = ℓ}(_≡_) ⦄ where
minimal-reflection-coercion : ((_≡_) ⊆₂ (_→ᶠ_))
minimal-reflection-coercion = minRefl{_→ᶠ_}
module _
{_≡_ : A → A → Stmt{ℓₑ₁}} ⦃ minRefl : MinimalReflexiveRelation{ℓₚ = ℓₑ₂}(_≡_) ⦄
{_▫_ : B → B → Stmt{ℓₑ₂}} ⦃ op-refl : Reflexivity(_▫_) ⦄
{f : A → B}
where
minimal-reflection-transport : (_≡_) ⊆₂ ((_▫_) on₂ f)
minimal-reflection-transport = intro (sub₂(_≡_)((_▫_) on₂ f) ⦃ minRefl ⦃ on₂-reflexivity ⦄ ⦄)
module _ ⦃ equiv-A : Equiv{ℓₑ₁}(A) ⦄ ⦃ minRefl : MinimalReflexiveRelation{ℓₚ = ℓₑ₂}(Equiv._≡_ equiv-A) ⦄ where
minimal-reflection-to-function : ⦃ equiv-B : Equiv{ℓₑ₂}(B) ⦄ {f : A → B} → Function ⦃ equiv-A ⦄ ⦃ equiv-B ⦄ (f)
minimal-reflection-to-function {f = f} = intro (sub₂(Equiv._≡_ equiv-A)((_≡ₛ_) on₂ f) ⦃ minimal-reflection-transport ⦄)
module _ ⦃ equiv-T : Equiv{ℓₑ}(T) ⦄ ⦃ minRefl : MinimalReflexiveRelation{ℓₚ = ℓ}(Equiv._≡_ equiv-T) ⦄ where
minimal-reflection-to-relator : ∀{P : T → Stmt{ℓ}} → UnaryRelator(P)
minimal-reflection-to-relator {P = P} = intro (sub₂(Equiv._≡_ equiv-T)((_→ᶠ_) on₂ P) ⦃ minimal-reflection-transport ⦄)
module _ ⦃ minRefl : MinimalReflexiveRelation(_≡_) ⦄ where
minimal-reflection-to-flipped-transitivityᵣ : Names.FlippedTransitivityᵣ(_≡_)
minimal-reflection-to-flipped-transitivityᵣ {z = z} = sub₂(_≡_)((_→ᶠ_) on₂ (_≡ z)) ⦃ minimal-reflection-transport ⦄
module _ ⦃ refl : Reflexivity(_≡_) ⦄ ⦃ minRefl : MinimalReflexiveRelation(_≡_) ⦄ where
minimal-reflection-to-symmetry : Symmetry(_≡_)
Symmetry.proof minimal-reflection-to-symmetry = sub₂(_≡_)(swap(_≡_)) ⦃ minRefl ⦃ swap-reflexivity ⦄ ⦄
minimal-reflection-to-transitivity : Transitivity(_≡_)
Transitivity.proof minimal-reflection-to-transitivity xy yz = minimal-reflection-to-flipped-transitivityᵣ (symmetry(_≡_) ⦃ minimal-reflection-to-symmetry ⦄ xy) yz
minimal-reflection-to-equivalence : Equivalence(_≡_)
Equivalence.reflexivity minimal-reflection-to-equivalence = refl
Equivalence.symmetry minimal-reflection-to-equivalence = minimal-reflection-to-symmetry
Equivalence.transitivity minimal-reflection-to-equivalence = minimal-reflection-to-transitivity
module _ ⦃ refl-Id : Reflexivity(Id) ⦄ ⦃ identElim : IdentityEliminator{ℓₚ = ℓₚ}(Id) ⦄ where
identity-eliminator-to-reflexive-subrelation : MinimalReflexiveRelation(Id)
identity-eliminator-to-reflexive-subrelation {_▫_ = _▫_} = intro(idElim(Id) (\{x y} _ → (x ▫ y)) (reflexivity(_▫_)))
module _ ⦃ refl : Reflexivity(Id) ⦄ ⦃ identElim : IdentityEliminator(Id) ⦄ where
identity-eliminator-to-symmetry : Symmetry(Id)
identity-eliminator-to-symmetry = minimal-reflection-to-symmetry ⦃ minRefl = identity-eliminator-to-reflexive-subrelation ⦄
identity-eliminator-to-flipped-transitivityᵣ : Names.FlippedTransitivityᵣ(Id)
identity-eliminator-to-flipped-transitivityᵣ = minimal-reflection-to-flipped-transitivityᵣ ⦃ minRefl = identity-eliminator-to-reflexive-subrelation ⦄
identity-eliminator-to-transitivity : Transitivity(Id)
identity-eliminator-to-transitivity = minimal-reflection-to-transitivity ⦃ minRefl = identity-eliminator-to-reflexive-subrelation ⦄
identity-eliminator-to-equivalence : Equivalence(Id)
Equivalence.reflexivity identity-eliminator-to-equivalence = refl
Equivalence.symmetry identity-eliminator-to-equivalence = identity-eliminator-to-symmetry
Equivalence.transitivity identity-eliminator-to-equivalence = identity-eliminator-to-transitivity
|
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-- Andreas, 2020-03-18, issue #4520, reported by Dylan Ede.
--
-- Make the error message concerning ambiguous names
-- in BUILTIN declarations more precise.
open import Agda.Primitive
open import Agda.Builtin.FromNat
open import Agda.Builtin.Nat renaming (Nat to ℕ)
private
variable
ℓ ℓ' : Level
record FromNat (A : Set ℓ) : Set (lsuc ℓ) where
field
Constraint : ℕ → Set ℓ
fromNat : (n : ℕ) ⦃ _ : Constraint n ⦄ → A
open FromNat ⦃ ... ⦄ public using (fromNat)
{-# BUILTIN FROMNAT fromNat #-}
-- ERROR WAS:
-- The argument to BUILTIN FROMNAT must be an unambiguous defined name
-- when scope checking the declaration
-- {-# BUILTIN FROMNAT fromNat #-}
-- NEW ERROR:
-- Name fromNat is ambiguous
-- when scope checking the declaration
-- {-# BUILTIN FROMNAT fromNat #-}
|
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|
{-# OPTIONS --experimental-irrelevance #-}
open import Agda.Builtin.Bool
open import Agda.Builtin.Equality
data Box (A : Set) : ..(x : A) → Set where
c : (x : A) → Box A x
unbox : {A : Set} → .(x : A) → Box A x → A
unbox a (c b) = b
.b : Bool
b = true
b' : Bool
b' = unbox b (c _)
|
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