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184359c9471fcd0acb0a4d81629e219a8ac7276b
| 3,248
|
agda
|
Agda
|
notes/FOT/FOTC/Program/Mirror/MirrorMutualSL.agda
|
asr/fotc
|
2fc9f2b81052a2e0822669f02036c5750371b72d
|
[
"MIT"
] | 11
|
2015-09-03T20:53:42.000Z
|
2021-09-12T16:09:54.000Z
|
notes/FOT/FOTC/Program/Mirror/MirrorMutualSL.agda
|
asr/fotc
|
2fc9f2b81052a2e0822669f02036c5750371b72d
|
[
"MIT"
] | 2
|
2016-10-12T17:28:16.000Z
|
2017-01-01T14:34:26.000Z
|
notes/FOT/FOTC/Program/Mirror/MirrorMutualSL.agda
|
asr/fotc
|
2fc9f2b81052a2e0822669f02036c5750371b72d
|
[
"MIT"
] | 3
|
2016-09-19T14:18:30.000Z
|
2018-03-14T08:50:00.000Z
|
------------------------------------------------------------------------------
-- Proving mirror (mirror t) = t using mutual data types
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module FOT.FOTC.Program.Mirror.MirrorMutualSL where
infixr 5 _∷_ _++_
open import Function
open import Relation.Binary.PropositionalEquality
open ≡-Reasoning
------------------------------------------------------------------------------
-- The mutual data types
data Tree (A : Set) : Set
data Forest (A : Set) : Set
data Tree A where
tree : A → Forest A → Tree A
data Forest A where
[] : Forest A
_∷_ : Tree A → Forest A → Forest A
------------------------------------------------------------------------------
-- Auxiliary functions
_++_ : {A : Set} → Forest A → Forest A → Forest A
[] ++ ys = ys
(a ∷ xs) ++ ys = a ∷ xs ++ ys
map : {A B : Set} → (Tree A → Tree B) → Forest A → Forest B
map f [] = []
map f (a ∷ ts) = f a ∷ map f ts
reverse : {A : Set} → Forest A → Forest A
reverse [] = []
reverse (a ∷ ts) = reverse ts ++ a ∷ []
postulate
map-++ : {A B : Set}(f : Tree A → Tree B)(xs ys : Forest A) →
map f (xs ++ ys) ≡ map f xs ++ map f ys
reverse-++ : {A : Set}(xs ys : Forest A) →
reverse (xs ++ ys) ≡ reverse ys ++ reverse xs
------------------------------------------------------------------------------
-- The mirror function.
{-# TERMINATING #-}
mirror : {A : Set} → Tree A → Tree A
mirror (tree a ts) = tree a (reverse (map mirror ts))
------------------------------------------------------------------------------
-- The proof of the property.
mirror-involutive : {A : Set} → (t : Tree A) → mirror (mirror t) ≡ t
helper : {A : Set} → (ts : Forest A) →
reverse (map mirror (reverse (map mirror ts))) ≡ ts
mirror-involutive (tree a []) = refl
mirror-involutive (tree a (t ∷ ts)) =
begin
tree a (reverse (map mirror (reverse (map mirror ts) ++ mirror t ∷ [])))
≡⟨ cong (tree a) (helper (t ∷ ts)) ⟩
tree a (t ∷ ts)
∎
helper [] = refl
helper (t ∷ ts) =
begin
reverse (map mirror (reverse (map mirror ts) ++ mirror t ∷ []))
≡⟨ cong reverse
(map-++ mirror (reverse (map mirror ts)) (mirror t ∷ []))
⟩
reverse (map mirror (reverse (map mirror ts)) ++
(map mirror (mirror t ∷ [])))
≡⟨ subst (λ x → (reverse (map mirror (reverse (map mirror ts)) ++
(map mirror (mirror t ∷ [])))) ≡ x)
(reverse-++ (map mirror (reverse (map mirror ts)))
(map mirror (mirror t ∷ [])))
refl
⟩
reverse (map mirror (mirror t ∷ [])) ++
reverse (map mirror (reverse (map mirror ts)))
≡⟨ refl ⟩
mirror (mirror t) ∷ reverse (map mirror (reverse (map mirror ts)))
≡⟨ cong (flip _∷_ (reverse (map mirror (reverse (map mirror ts)))))
(mirror-involutive t)
⟩
t ∷ reverse (map mirror (reverse (map mirror ts)))
≡⟨ cong (_∷_ t) (helper ts) ⟩
t ∷ ts
∎
| 32.808081
| 78
| 0.452586
|
2392d9c28bfb7ad8830e7a1fde1c1631686193aa
| 4,366
|
agda
|
Agda
|
src/FRP/LTL/ISet/Causal.agda
|
agda/agda-frp-ltl
|
e88107d7d192cbfefd0a94505e6a5793afe1a7a5
|
[
"MIT"
] | 21
|
2015-07-02T20:25:05.000Z
|
2020-06-15T02:51:13.000Z
|
src/FRP/LTL/ISet/Causal.agda
|
agda/agda-frp-ltl
|
e88107d7d192cbfefd0a94505e6a5793afe1a7a5
|
[
"MIT"
] | 2
|
2015-03-01T07:01:31.000Z
|
2015-03-02T15:23:53.000Z
|
src/FRP/LTL/ISet/Causal.agda
|
agda/agda-frp-ltl
|
e88107d7d192cbfefd0a94505e6a5793afe1a7a5
|
[
"MIT"
] | 3
|
2015-03-01T07:33:00.000Z
|
2022-03-12T11:39:04.000Z
|
open import Coinduction using ( ∞ ; ♯_ ; ♭ )
open import Data.Product using ( ∃ ; _×_ ; _,_ ; proj₂ )
open import Data.Nat using ( ℕ ; zero ; suc )
open import Data.Empty using ( ⊥ )
open import FRP.LTL.ISet.Core using ( ISet ; M⟦_⟧ ; ⟦_⟧ ; ⌈_⌉ ; _,_ ; splitM⟦_⟧ ) renaming ( [_] to ⟪_⟫ )
open import FRP.LTL.ISet.Globally using ( □ ; [_] )
open import FRP.LTL.ISet.Stateless using ( _⇒_ ; _$_ )
open import FRP.LTL.RSet.Core using ( RSet )
open import FRP.LTL.Time.Bound using
( Time∞ ; fin ; +∞ ; _≺_ ; _≼_ ; _⋠_ ; ≺-Indn ; _,_
; ≺-impl-≼ ; ≼-refl ; _≼-trans_ ; ≡-impl-≽ ; ≺-impl-≢ ; ≺-impl-⋡ ; t≺+∞ ; ∞≼-impl-≡∞ ; ≺-indn )
open import FRP.LTL.Time.Interval using ( [_⟩ ; Int ; ↑ )
open import FRP.LTL.Util using ( ⊥-elim )
open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl )
open import Relation.Unary using ( _∈_ )
module FRP.LTL.ISet.Causal where
infixr 2 _⊵_
infixr 3 _⋙_ _≫_ _⊨_≫_
-- A ⊵ B is the causal function space from A to B
data _∙_⊸_∙_ (A : ISet) (s : Time∞) (B : ISet) (u : Time∞) : Set where
inp : .(s ≼ u) → .(u ≺ +∞) →
(∀ {t} .(s≺t : s ≺ t) → M⟦ A ⟧ [ s≺t ⟩ → ∞ (A ∙ t ⊸ B ∙ u)) →
(A ∙ s ⊸ B ∙ u)
out : ∀ {v} .(u≺v : u ≺ v) →
M⟦ B ⟧ [ u≺v ⟩ → ∞ (A ∙ s ⊸ B ∙ v) →
(A ∙ s ⊸ B ∙ u)
done : .(u ≡ +∞) →
(A ∙ s ⊸ B ∙ u)
_⊵_ : ISet → ISet → ISet
A ⊵ B = ⌈ (λ t → A ∙ fin t ⊸ B ∙ fin t) ⌉
-- Categorical structure
ar : ∀ {A B} t → M⟦ A ⇒ B ⟧ (↑ t) → (A ∙ fin t ⊸ B ∙ fin t)
ar {A} {B} t f = inp ≼-refl t≺+∞ P where
P : ∀ {u} .(t≺u : fin t ≺ u) → M⟦ A ⟧ [ t≺u ⟩ → ∞ (A ∙ u ⊸ B ∙ fin t)
P {+∞} t≺u σ = ♯ out t≺u (f $ σ) (♯ done refl)
P {fin u} t≺u σ with splitM⟦ A ⇒ B ⟧ [ t≺u ⟩ (↑ u) refl f
P {fin u} t≺u σ | (f₁ , f₂) = ♯ out t≺u (f₁ $ σ) (♯ ar u f₂)
arr : ∀ {A B} → ⟦ □ (A ⇒ B) ⇒ (A ⊵ B) ⟧
arr ⟪ ⟪ f ⟫ ⟫ = ⟪ (λ t t∈i → ar t (f t t∈i) ) ⟫
-- We could define id in terms of arr, but we define it explictly for efficiency.
id : ∀ {A} t → (A ∙ t ⊸ A ∙ t)
id +∞ = done refl
id (fin t) = inp ≼-refl t≺+∞ (λ {u} t≺u σ → ♯ out t≺u σ (♯ id u))
identity : ∀ {A} → ⟦ A ⊵ A ⟧
identity = ⟪ (λ t t∈i → id (fin t)) ⟫
-- The following typechecks but does not pass the termination checker,
-- due to the possibility of infinite left-to-right chatter:
-- _≫_ : ∀ {A B C s t u} → (A ∙ s ⊸ B ∙ t) → (B ∙ t ⊸ C ∙ u) → (A ∙ s ⊸ C ∙ u)
-- P ≫ out u≺w τ Q = out u≺w τ (♯ (P ≫ ♭ Q))
-- P ≫ done u≡∞ = done u≡∞
-- inp s≼t t≺∞ P ≫ inp t≼u u≺∞ Q = inp (s≼t ≼-trans t≼u) u≺∞ (λ s≺v σ → ♯ (♭ (P s≺v σ) ≫ inp t≼u u≺∞ Q))
-- done t≡∞ ≫ inp t≼u u≺∞ Q = ⊥-elim (≺-impl-≢ u≺∞ (∞≼-impl-≡∞ (≡-impl-≽ t≡∞ ≼-trans t≼u)))
-- out t≺v σ P ≫ inp t≼u u≺∞ Q = out t≺v σ P ≫ inp t≼u u≺∞ Q
-- We have to be explicit about the induction scheme in the case of left-to-right
-- communication, which is because, for any t and u ≺ ∞, there is a bound
-- on the length of any ≺-chain between t and u.
mutual
_⊨_≫_ : ∀ {A B C s t u} → (≺-Indn t u) → (A ∙ s ⊸ B ∙ t) → (B ∙ t ⊸ C ∙ u) → (A ∙ s ⊸ C ∙ u)
n , t+n≻u ⊨ P ≫ out u≺w τ Q = out u≺w τ (♯ (P ≫ ♭ Q))
n , t+n≻u ⊨ P ≫ done u≡∞ = done u≡∞
n , t+n≻u ⊨ inp s≼t t≺∞ P ≫ inp t≼u u≺∞ Q = inp (s≼t ≼-trans t≼u) u≺∞ (λ s≺v σ → ♯ (♭ (P s≺v σ) ≫ inp t≼u u≺∞ Q))
n , t+n≻u ⊨ done t≡∞ ≫ inp t≼u u≺∞ Q = ⊥-elim (≺-impl-≢ u≺∞ (∞≼-impl-≡∞ (≡-impl-≽ t≡∞ ≼-trans t≼u)))
zero , u≺t ⊨ out t≺v σ P ≫ inp t≼u u≺∞ Q = ⊥-elim (≺-impl-⋡ u≺t t≼u)
suc n , t+1+n≻u ⊨ out t≺v σ P ≫ inp t≼u u≺∞ Q = n , t+1+n≻u t≺v ⊨ ♭ P ≫ ♭ (Q t≺v σ)
_≫_ : ∀ {A B C s t u} → (A ∙ s ⊸ B ∙ t) → (B ∙ t ⊸ C ∙ u) → (A ∙ s ⊸ C ∙ u)
P ≫ out u≺w τ Q = out u≺w τ (♯ (P ≫ ♭ Q))
P ≫ done u≡∞ = done u≡∞
inp s≼t t≺∞ P ≫ inp t≼u u≺∞ Q = inp (s≼t ≼-trans t≼u) u≺∞ (λ s≺v σ → ♯ (♭ (P s≺v σ) ≫ inp t≼u u≺∞ Q))
done t≡∞ ≫ inp t≼u u≺∞ Q = ⊥-elim (≺-impl-≢ u≺∞ (∞≼-impl-≡∞ (≡-impl-≽ t≡∞ ≼-trans t≼u)))
out t≺v σ P ≫ inp t≼u u≺∞ Q = ≺-indn u≺∞ ⊨ out t≺v σ P ≫ inp t≼u u≺∞ Q
_⋙_ : ∀ {A B C} → ⟦ (A ⊵ B) ⇒ (B ⊵ C) ⇒ (A ⊵ C) ⟧
(⟪ ⟪ f ⟫ ⟫ ⋙ ⟪ ⟪ g ⟫ ⟫) = ⟪ (λ t t∈i → f t t∈i ≫ g t t∈i) ⟫
-- Apply a process to some of its output
_/_/_ : ∀ {A B s t u} → (A ∙ s ⊸ B ∙ u) → .(s≺t : s ≺ t) → M⟦ A ⟧ [ s≺t ⟩ → (A ∙ t ⊸ B ∙ u)
inp s≼u u≺∞ P / s≺t / σ = ♭ (P s≺t σ)
out u≺v τ P / s≺t / σ = out u≺v τ (♯ (♭ P / s≺t / σ))
done u≡∞ / s≺t / σ = done u≡∞
| 44.10101
| 121
| 0.444114
|
18c195a32ca88347bbde7df34297c485009b812d
| 3,432
|
agda
|
Agda
|
Cubical/HITs/PropositionalTruncation/MagicTrick.agda
|
L-TChen/cubical
|
60226aacd7b386aef95d43a0c29c4eec996348a8
|
[
"MIT"
] | null | null | null |
Cubical/HITs/PropositionalTruncation/MagicTrick.agda
|
L-TChen/cubical
|
60226aacd7b386aef95d43a0c29c4eec996348a8
|
[
"MIT"
] | 1
|
2022-01-27T02:07:48.000Z
|
2022-01-27T02:07:48.000Z
|
Cubical/HITs/PropositionalTruncation/MagicTrick.agda
|
L-TChen/cubical
|
60226aacd7b386aef95d43a0c29c4eec996348a8
|
[
"MIT"
] | null | null | null |
{-
Based on Nicolai Kraus' blog post:
The Truncation Map |_| : ℕ -> ‖ℕ‖ is nearly Invertible
https://homotopytypetheory.org/2013/10/28/the-truncation-map-_-ℕ-‖ℕ‖-is-nearly-invertible/
Defines [recover], which definitionally satisfies `recover ∣ x ∣ ≡ x` ([recover∣∣]) for homogeneous types
Also see the follow-up post by Jason Gross:
Composition is not what you think it is! Why “nearly invertible” isn’t.
https://homotopytypetheory.org/2014/02/24/composition-is-not-what-you-think-it-is-why-nearly-invertible-isnt/
-}
{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.HITs.PropositionalTruncation.MagicTrick where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function
open import Cubical.Foundations.Path
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Pointed.Homogeneous
open import Cubical.HITs.PropositionalTruncation.Base
open import Cubical.HITs.PropositionalTruncation.Properties
module Recover {ℓ} (A∙ : Pointed ℓ) (h : isHomogeneous A∙) where
private
A = typ A∙
a = pt A∙
toEquivPtd : ∥ A ∥ → Σ[ B∙ ∈ Pointed ℓ ] (A , a) ≡ B∙
toEquivPtd = rec isPropSingl (λ x → (A , x) , h x)
private
B∙ : ∥ A ∥ → Pointed ℓ
B∙ tx = fst (toEquivPtd tx)
-- the key observation is that B∙ ∣ x ∣ is definitionally equal to (A , x)
private
obvs : ∀ x → B∙ ∣ x ∣ ≡ (A , x)
obvs x = refl -- try it: `C-c C-n B∙ ∣ x ∣` gives `(A , x)`
-- thus any truncated element (of a homogeneous type) can be recovered by agda's normalizer!
recover : ∀ (tx : ∥ A ∥) → typ (B∙ tx)
recover tx = pt (B∙ tx)
recover∣∣ : ∀ (x : A) → recover ∣ x ∣ ≡ x
recover∣∣ x = refl -- try it: `C-c C-n recover ∣ x ∣` gives `x`
private
-- notice that the following typechecks because typ (B∙ ∣ x ∣) is definitionally equal to to A, but
-- `recover : ∥ A ∥ → A` does not because typ (B∙ tx) is not definitionally equal to A (though it is
-- judegmentally equal to A by cong typ (snd (toEquivPtd tx)) : A ≡ typ (B∙ tx))
obvs2 : A → A
obvs2 = recover ∘ ∣_∣
-- one might wonder if (cong recover (squash ∣ x ∣ ∣ y ∣)) therefore has type x ≡ y, but thankfully
-- typ (B∙ (squash ∣ x ∣ ∣ y ∣ i)) is *not* A (it's a messy hcomp involving h x and h y)
recover-squash : ∀ x y → -- x ≡ y -- this raises an error
PathP (λ i → typ (B∙ (squash ∣ x ∣ ∣ y ∣ i))) x y
recover-squash x y = cong recover (squash ∣ x ∣ ∣ y ∣)
-- Demo, adapted from:
-- https://bitbucket.org/nicolaikraus/agda/src/e30d70c72c6af8e62b72eefabcc57623dd921f04/trunc-inverse.lagda
private
open import Cubical.Data.Nat
open Recover (ℕ , zero) (isHomogeneousDiscrete discreteℕ)
-- only `∣hidden∣` is exported, `hidden` is no longer in scope
module _ where
private
hidden : ℕ
hidden = 17
∣hidden∣ : ∥ ℕ ∥
∣hidden∣ = ∣ hidden ∣
-- we can still recover the value, even though agda can no longer see `hidden`!
test : recover ∣hidden∣ ≡ 17
test = refl -- try it: `C-c C-n recover ∣hidden∣` gives `17`
-- `C-c C-n hidden` gives an error
-- Finally, note that the definition of recover is independent of the proof that A is homogeneous. Thus we
-- still can definitionally recover information hidden by ∣_∣ as long as we permit holes. Try replacing
-- `isHomogeneousDiscrete discreteℕ` above with a hole (`?`) and notice that everything still works
| 38.561798
| 111
| 0.653263
|
1842f4d7047a05afc31e0a68ece5d47bc38653db
| 2,886
|
agda
|
Agda
|
test/asset/agda-stdlib-1.0/Data/Vec/Relation/Unary/Any.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | null | null | null |
test/asset/agda-stdlib-1.0/Data/Vec/Relation/Unary/Any.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | null | null | null |
test/asset/agda-stdlib-1.0/Data/Vec/Relation/Unary/Any.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | null | null | null |
------------------------------------------------------------------------
-- The Agda standard library
--
-- Vectors where at least one element satisfies a given property
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Vec.Relation.Unary.Any {a} {A : Set a} where
open import Data.Empty
open import Data.Fin
open import Data.Nat using (zero; suc)
open import Data.Sum using (_⊎_; inj₁; inj₂; [_,_]′)
open import Data.Vec as Vec using (Vec; []; [_]; _∷_)
open import Data.Product as Prod using (∃; _,_)
open import Level using (_⊔_)
open import Relation.Nullary using (¬_; yes; no)
open import Relation.Nullary.Negation using (contradiction)
import Relation.Nullary.Decidable as Dec
open import Relation.Unary
------------------------------------------------------------------------
-- Any P xs means that at least one element in xs satisfies P.
data Any {p} (P : A → Set p) : ∀ {n} → Vec A n → Set (a ⊔ p) where
here : ∀ {n x} {xs : Vec A n} (px : P x) → Any P (x ∷ xs)
there : ∀ {n x} {xs : Vec A n} (pxs : Any P xs) → Any P (x ∷ xs)
------------------------------------------------------------------------
-- Operations on Any
module _ {p} {P : A → Set p} {n x} {xs : Vec A n} where
-- If the tail does not satisfy the predicate, then the head will.
head : ¬ Any P xs → Any P (x ∷ xs) → P x
head ¬pxs (here px) = px
head ¬pxs (there pxs) = contradiction pxs ¬pxs
-- If the head does not satisfy the predicate, then the tail will.
tail : ¬ P x → Any P (x ∷ xs) → Any P xs
tail ¬px (here px) = ⊥-elim (¬px px)
tail ¬px (there pxs) = pxs
-- Convert back and forth with sum
toSum : Any P (x ∷ xs) → P x ⊎ Any P xs
toSum (here px) = inj₁ px
toSum (there pxs) = inj₂ pxs
fromSum : P x ⊎ Any P xs → Any P (x ∷ xs)
fromSum = [ here , there ]′
map : ∀ {p q} {P : A → Set p} {Q : A → Set q} →
P ⊆ Q → ∀ {n} → Any P {n} ⊆ Any Q {n}
map g (here px) = here (g px)
map g (there pxs) = there (map g pxs)
index : ∀ {p} {P : A → Set p} {n} {xs : Vec A n} → Any P xs → Fin n
index (here px) = zero
index (there pxs) = suc (index pxs)
-- If any element satisfies P, then P is satisfied.
satisfied : ∀ {p} {P : A → Set p} {n} {xs : Vec A n} → Any P xs → ∃ P
satisfied (here px) = _ , px
satisfied (there pxs) = satisfied pxs
------------------------------------------------------------------------
-- Properties of predicates preserved by Any
module _ {p} {P : A → Set p} where
any : Decidable P → ∀ {n} → Decidable (Any P {n})
any P? [] = no λ()
any P? (x ∷ xs) with P? x
... | yes px = yes (here px)
... | no ¬px = Dec.map′ there (tail ¬px) (any P? xs)
satisfiable : Satisfiable P → ∀ {n} → Satisfiable (Any P {suc n})
satisfiable (x , p) {zero} = x ∷ [] , here p
satisfiable (x , p) {suc n} = Prod.map (x ∷_) there (satisfiable (x , p))
| 35.195122
| 75
| 0.517672
|
50b9d3de1d020995ceaf5e4f297b309dbb7be4f6
| 2,946
|
agda
|
Agda
|
Fields/FieldOfFractions/Addition.agda
|
Smaug123/agdaproofs
|
0f4230011039092f58f673abcad8fb0652e6b562
|
[
"MIT"
] | 4
|
2019-08-08T12:44:19.000Z
|
2022-01-28T06:04:15.000Z
|
Fields/FieldOfFractions/Addition.agda
|
Smaug123/agdaproofs
|
0f4230011039092f58f673abcad8fb0652e6b562
|
[
"MIT"
] | 14
|
2019-01-06T21:11:59.000Z
|
2020-04-11T11:03:39.000Z
|
Fields/FieldOfFractions/Addition.agda
|
Smaug123/agdaproofs
|
0f4230011039092f58f673abcad8fb0652e6b562
|
[
"MIT"
] | 1
|
2021-11-29T13:23:07.000Z
|
2021-11-29T13:23:07.000Z
|
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Definition
open import Rings.Definition
open import Rings.IntegralDomains.Definition
open import Setoids.Setoids
open import Sets.EquivalenceRelations
module Fields.FieldOfFractions.Addition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) where
open import Fields.FieldOfFractions.Setoid I
fieldOfFractionsPlus : fieldOfFractionsSet → fieldOfFractionsSet → fieldOfFractionsSet
fieldOfFractionsSet.num (fieldOfFractionsPlus (record { num = a ; denom = b ; denomNonzero = b!=0 }) (record { num = c ; denom = d ; denomNonzero = d!=0 })) = (a * d) + (b * c)
fieldOfFractionsSet.denom (fieldOfFractionsPlus (record { num = a ; denom = b ; denomNonzero = b!=0 }) (record { num = c ; denom = d ; denomNonzero = d!=0 })) = b * d
fieldOfFractionsSet.denomNonzero (fieldOfFractionsPlus (record { num = a ; denom = b ; denomNonzero = b!=0 }) (record { num = c ; denom = d ; denomNonzero = d!=0 })) = λ pr → exFalso (d!=0 (IntegralDomain.intDom I pr b!=0))
--record { num = ((a * d) + (b * c)) ; denom = b * d ; denomNonzero = λ pr → exFalso (d!=0 (IntegralDomain.intDom I pr b!=0)) }
plusWellDefined : {a b c d : fieldOfFractionsSet} → (Setoid._∼_ fieldOfFractionsSetoid a c) → (Setoid._∼_ fieldOfFractionsSetoid b d) → Setoid._∼_ fieldOfFractionsSetoid (fieldOfFractionsPlus a b) (fieldOfFractionsPlus c d)
plusWellDefined {record { num = a ; denom = b ; denomNonzero = b!=0 }} {record { num = c ; denom = d ; denomNonzero = d!=0 }} {record { num = e ; denom = f ; denomNonzero = f!=0 }} {record { num = g ; denom = h ; denomNonzero = h!=0 }} af=be ch=dg = need
where
open Setoid S
open Ring R
open Equivalence eq
have1 : (c * h) ∼ (d * g)
have1 = ch=dg
have2 : (a * f) ∼ (b * e)
have2 = af=be
need : (((a * d) + (b * c)) * (f * h)) ∼ ((b * d) * (((e * h) + (f * g))))
need = transitive (transitive (Ring.*Commutative R) (transitive (Ring.*DistributesOver+ R) (Group.+WellDefined (Ring.additiveGroup R) (transitive *Associative (transitive (*WellDefined (*Commutative) reflexive) (transitive (*WellDefined *Associative reflexive) (transitive (*WellDefined (*WellDefined have2 reflexive) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined (transitive (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative)) *Associative) reflexive) (symmetric *Associative))))))))) (transitive *Commutative (transitive (transitive (symmetric *Associative) (*WellDefined reflexive (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined have1 reflexive) (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))))))) *Associative))))) (symmetric (Ring.*DistributesOver+ R))
| 86.647059
| 970
| 0.688052
|
39e5a155d1e2fb5b22dd7fafe35f155097e50b9f
| 439
|
agda
|
Agda
|
Cubical/Data/Maybe/Base.agda
|
limemloh/cubical
|
df4ef7edffd1c1deb3d4ff342c7178e9901c44f1
|
[
"MIT"
] | 1
|
2020-03-23T23:52:11.000Z
|
2020-03-23T23:52:11.000Z
|
Cubical/Data/Maybe/Base.agda
|
limemloh/cubical
|
df4ef7edffd1c1deb3d4ff342c7178e9901c44f1
|
[
"MIT"
] | null | null | null |
Cubical/Data/Maybe/Base.agda
|
limemloh/cubical
|
df4ef7edffd1c1deb3d4ff342c7178e9901c44f1
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --cubical --safe #-}
module Cubical.Data.Maybe.Base where
open import Cubical.Core.Everything
private
variable
ℓ : Level
A B : Type ℓ
data Maybe (A : Type ℓ) : Type ℓ where
nothing : Maybe A
just : A → Maybe A
caseMaybe : (n j : B) → Maybe A → B
caseMaybe n _ nothing = n
caseMaybe _ j (just _) = j
map-Maybe : (A → B) → Maybe A → Maybe B
map-Maybe _ nothing = nothing
map-Maybe f (just x) = just (f x)
| 19.954545
| 39
| 0.630979
|
cb3e02a40b161e51d6e19925600d45949ecd6c3f
| 141
|
agda
|
Agda
|
src/Quasigroup/Properties.agda
|
Akshobhya1234/agda-NonAssociativeAlgebra
|
443e831e536b756acbd1afd0d6bae7bc0d288048
|
[
"MIT"
] | 2
|
2021-08-15T06:16:13.000Z
|
2021-08-17T09:14:03.000Z
|
src/Quasigroup/Properties.agda
|
Akshobhya1234/agda-NonAssociativeAlgebra
|
443e831e536b756acbd1afd0d6bae7bc0d288048
|
[
"MIT"
] | 2
|
2021-10-04T05:30:30.000Z
|
2021-10-09T08:24:56.000Z
|
src/Quasigroup/Properties.agda
|
Akshobhya1234/agda-NonAssociativeAlgebra
|
443e831e536b756acbd1afd0d6bae7bc0d288048
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --without-K --safe #-}
open import Algebra
module Quasigroup.Properties
{a ℓ} (Q : Quasigroup a ℓ) where
open Quasigroup Q
| 14.1
| 34
| 0.687943
|
397336dc5131c2bee347e796ccf1d95838bb042d
| 622
|
agda
|
Agda
|
test/Fail/Issue2621.agda
|
mdimjasevic/agda
|
8fb548356b275c7a1e79b768b64511ae937c738b
|
[
"BSD-3-Clause"
] | 1,989
|
2015-01-09T23:51:16.000Z
|
2022-03-30T18:20:48.000Z
|
test/Fail/Issue2621.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 4,066
|
2015-01-10T11:24:51.000Z
|
2022-03-31T21:14:49.000Z
|
test/Fail/Issue2621.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 371
|
2015-01-03T14:04:08.000Z
|
2022-03-30T19:00:30.000Z
|
open import Agda.Builtin.Nat renaming (Nat to ℕ)
open import Agda.Builtin.Equality
data Vec (A : Set) : ℕ → Set where
[] : Vec A zero
_∷_ : ∀{n} (x : A) (xs : Vec A n) → Vec A (suc n)
data All₂ {A : Set} {B : Set} (R : A → B → Set) : ∀ {k} → Vec A k → Vec B k → Set where
[] : All₂ R [] []
_∷_ : ∀ {k x y} {xs : Vec A k} {ys : Vec B k}
(r : R x y) (rs : All₂ R xs ys) → All₂ R (x ∷ xs) (y ∷ ys)
Det : ∀ {A : Set} {B : Set} (R : A → B → Set) → Set
Det R = ∀{a b c} → R a b → R a c → b ≡ c
detAll₂ : ∀ {A : Set} {B : Set} (R : A → B → Set) (h : Det R) → Det (All₂ R)
detAll₂ R h [] [] = refl
| 34.555556
| 87
| 0.463023
|
18c620be0697eecfd4ad30e457ee4c91a2d93ec3
| 2,470
|
agda
|
Agda
|
agda-stdlib-0.9/src/Induction/Lexicographic.agda
|
qwe2/try-agda
|
9d4c43b1609d3f085636376fdca73093481ab882
|
[
"Apache-2.0"
] | 1
|
2016-10-20T15:52:05.000Z
|
2016-10-20T15:52:05.000Z
|
agda-stdlib-0.9/src/Induction/Lexicographic.agda
|
qwe2/try-agda
|
9d4c43b1609d3f085636376fdca73093481ab882
|
[
"Apache-2.0"
] | null | null | null |
agda-stdlib-0.9/src/Induction/Lexicographic.agda
|
qwe2/try-agda
|
9d4c43b1609d3f085636376fdca73093481ab882
|
[
"Apache-2.0"
] | null | null | null |
------------------------------------------------------------------------
-- The Agda standard library
--
-- Lexicographic induction
------------------------------------------------------------------------
module Induction.Lexicographic where
open import Data.Product
open import Induction
open import Level
-- The structure of lexicographic induction.
Σ-Rec : ∀ {a b ℓ₁ ℓ₂ ℓ₃} {A : Set a} {B : A → Set b} →
RecStruct A (ℓ₁ ⊔ b) ℓ₂ → (∀ x → RecStruct (B x) ℓ₁ ℓ₃) →
RecStruct (Σ A B) _ _
Σ-Rec RecA RecB P (x , y) =
-- Either x is constant and y is "smaller", ...
RecB x (λ y' → P (x , y')) y
×
-- ...or x is "smaller" and y is arbitrary.
RecA (λ x' → ∀ y' → P (x' , y')) x
_⊗_ : ∀ {a b ℓ₁ ℓ₂ ℓ₃} {A : Set a} {B : Set b} →
RecStruct A (ℓ₁ ⊔ b) ℓ₂ → RecStruct B ℓ₁ ℓ₃ →
RecStruct (A × B) _ _
RecA ⊗ RecB = Σ-Rec RecA (λ _ → RecB)
-- Constructs a recursor builder for lexicographic induction.
Σ-rec-builder :
∀ {a b ℓ₁ ℓ₂ ℓ₃} {A : Set a} {B : A → Set b}
{RecA : RecStruct A (ℓ₁ ⊔ b) ℓ₂}
{RecB : ∀ x → RecStruct (B x) ℓ₁ ℓ₃} →
RecursorBuilder RecA → (∀ x → RecursorBuilder (RecB x)) →
RecursorBuilder (Σ-Rec RecA RecB)
Σ-rec-builder {RecA = RecA} {RecB = RecB} recA recB P f (x , y) =
(p₁ x y p₂x , p₂x)
where
p₁ : ∀ x y →
RecA (λ x' → ∀ y' → P (x' , y')) x →
RecB x (λ y' → P (x , y')) y
p₁ x y x-rec = recB x
(λ y' → P (x , y'))
(λ y y-rec → f (x , y) (y-rec , x-rec))
y
p₂ : ∀ x → RecA (λ x' → ∀ y' → P (x' , y')) x
p₂ = recA (λ x → ∀ y → P (x , y))
(λ x x-rec y → f (x , y) (p₁ x y x-rec , x-rec))
p₂x = p₂ x
[_⊗_] : ∀ {a b ℓ₁ ℓ₂ ℓ₃} {A : Set a} {B : Set b}
{RecA : RecStruct A (ℓ₁ ⊔ b) ℓ₂} {RecB : RecStruct B ℓ₁ ℓ₃} →
RecursorBuilder RecA → RecursorBuilder RecB →
RecursorBuilder (RecA ⊗ RecB)
[ recA ⊗ recB ] = Σ-rec-builder recA (λ _ → recB)
------------------------------------------------------------------------
-- Example
private
open import Data.Nat
open import Induction.Nat as N
-- The Ackermann function à la Rózsa Péter.
ackermann : ℕ → ℕ → ℕ
ackermann m n =
build [ N.rec-builder ⊗ N.rec-builder ]
(λ _ → ℕ)
(λ { (zero , n) _ → 1 + n
; (suc m , zero) (_ , ackm•) → ackm• 1
; (suc m , suc n) (ack[1+m]n , ackm•) → ackm• ack[1+m]n
})
(m , n)
| 30.875
| 72
| 0.452227
|
181e6c8d4453fc9b881090cd410cc1c358842807
| 1,693
|
agda
|
Agda
|
_assets/agda/Berardi.agda
|
ionathanch/ionathanch.github.io
|
d54cdaf24391b2726e491a18cba2d2d8ae3ac20b
|
[
"MIT"
] | null | null | null |
_assets/agda/Berardi.agda
|
ionathanch/ionathanch.github.io
|
d54cdaf24391b2726e491a18cba2d2d8ae3ac20b
|
[
"MIT"
] | null | null | null |
_assets/agda/Berardi.agda
|
ionathanch/ionathanch.github.io
|
d54cdaf24391b2726e491a18cba2d2d8ae3ac20b
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --type-in-type #-}
open import Data.Empty
open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
open import Function.Base using (_∘_; id)
open import Relation.Nullary using (¬_)
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; subst; cong; sym)
record _◁_ {ℓ} (A B : Set ℓ) : Set ℓ where
constructor _,_,_
field
ϕ : A → B
ψ : B → A
retract : ψ ∘ ϕ ≡ id
open _◁_
record _◁′_ {ℓ} (A B : Set ℓ) : Set ℓ where
constructor _,_,_
field
ϕ : A → B
ψ : B → A
retract : A ◁ B → ψ ∘ ϕ ≡ id
open _◁′_
postulate
EM : ∀ {ℓ} (A : Set ℓ) → A ⊎ (¬ A)
℘ : ∀ {ℓ} → Set ℓ → Set _
℘ X = X → Set
t : ∀ {ℓ} (A B : Set ℓ) → ℘ A ◁′ ℘ B
t A B with EM (℘ A ◁ ℘ B)
... | inj₁ ℘A◁℘B =
let ϕ , ψ , retract = ℘A◁℘B
in ϕ , ψ , λ _ → retract
... | inj₂ ¬℘A◁℘B =
(λ _ _ → ⊥) , (λ _ _ → ⊥) , λ ℘A◁℘B → ⊥-elim (¬℘A◁℘B ℘A◁℘B)
-- type-in-type allows U to be put into Set...
U : Set
U = ∀ X → ℘ X
-- ...so that u: U can be applied to U itself
projᵤ : U → ℘ U
projᵤ u = u U
injᵤ : ℘ U → U
injᵤ f X =
let _ , ψ , _ = t X U
ϕ , _ , _ = t U U
in ψ (ϕ f)
projᵤ∘injᵤ : projᵤ ∘ injᵤ ≡ id
projᵤ∘injᵤ = retract (t U U) (id , id , refl)
_∈_ : U → U → Set
u ∈ v = projᵤ u v
Russell : ℘ U
Russell u = ¬ u ∈ u
r : U
r = injᵤ Russell
-- up to here EM + impredicativity is enough,
-- as long as _≡_ itself is in the impredicative universe,
-- but to go further, large elimination (via subst) is required
r∈r≡r∉r : r ∈ r ≡ (¬ r ∈ r)
r∈r≡r∉r = cong (λ f → f Russell r) projᵤ∘injᵤ
r∉r : ¬ r ∈ r
r∉r r∈r =
let r∈r→r∉r = subst id r∈r≡r∉r
in r∈r→r∉r r∈r r∈r
false : ⊥
false =
let r∉r→r∈r = subst id (sym r∈r≡r∉r)
in r∉r (r∉r→r∈r r∉r)
| 21.1625
| 90
| 0.529829
|
cb3b80d4ad61c060046c8bc301109651d467d5b3
| 1,895
|
agda
|
Agda
|
test/succeed/UniversePolymorphism.agda
|
larrytheliquid/agda
|
477c8c37f948e6038b773409358fd8f38395f827
|
[
"MIT"
] | 1
|
2018-10-10T17:08:44.000Z
|
2018-10-10T17:08:44.000Z
|
test/succeed/UniversePolymorphism.agda
|
masondesu/agda
|
70c8a575c46f6a568c7518150a1a64fcd03aa437
|
[
"MIT"
] | null | null | null |
test/succeed/UniversePolymorphism.agda
|
masondesu/agda
|
70c8a575c46f6a568c7518150a1a64fcd03aa437
|
[
"MIT"
] | 1
|
2022-03-12T11:35:18.000Z
|
2022-03-12T11:35:18.000Z
|
-- {-# OPTIONS -v tc.conv:30 -v tc.conv.level:60 -v tc.meta.assign:15 #-}
module UniversePolymorphism where
open import Common.Level renaming (_⊔_ to max)
data Nat : Set where
zero : Nat
suc : Nat → Nat
infixr 40 _∷_
data Vec {i}(A : Set i) : Nat → Set i where
[] : Vec {i} A zero
_∷_ : ∀ {n} → A → Vec {i} A n → Vec {i} A (suc n)
map : ∀ {n a b}{A : Set a}{B : Set b} → (A → B) → Vec A n → Vec B n
map f [] = []
map f (x ∷ xs) = f x ∷ map f xs
vec : ∀ {n a}{A : Set a} → A → Vec A n
vec {zero} _ = []
vec {suc n} x = x ∷ vec x
_<*>_ : ∀ {n a b}{A : Set a}{B : Set b} → Vec (A → B) n → Vec A n → Vec B n
[] <*> [] = []
(f ∷ fs) <*> (x ∷ xs) = f x ∷ (fs <*> xs)
flip : ∀ {a b c}{A : Set a}{B : Set b}{C : Set c} →
(A → B → C) → B → A → C
flip f x y = f y x
module Zip where
Fun : ∀ {n a} → Vec (Set a) n → Set a → Set a
Fun [] B = B
Fun (A ∷ As) B = A → Fun As B
app : ∀ {n m a}(As : Vec (Set a) n)(B : Set a) →
Vec (Fun As B) m → Fun (map (flip Vec m) As) (Vec B m)
app [] B bs = bs
app (A ∷ As) B fs = λ as → app As B (fs <*> as)
zipWith : ∀ {n m a}(As : Vec (Set a) n)(B : Set a) →
Fun As B → Fun (map (flip Vec m) As) (Vec B m)
zipWith As B f = app As B (vec f)
zipWith₃ : ∀ {n a}{A B C D : Set a} → (A → B → C → D) → Vec A n → Vec B n → Vec C n → Vec D n
zipWith₃ = zipWith (_ ∷ _ ∷ _ ∷ []) _
data Σ {a b}(A : Set a)(B : A → Set b) : Set (max a b) where
_,_ : (x : A)(y : B x) → Σ A B
fst : ∀ {a b}{A : Set a}{B : A → Set b} → Σ A B → A
fst (x , y) = x
snd : ∀ {a b}{A : Set a}{B : A → Set b}(p : Σ A B) → B (fst p)
snd (x , y) = y
-- Normal Σ
List : ∀ {a} → Set a → Set a
List A = Σ _ (Vec A)
nil : ∀ {a}{A : Set a} → List A
nil = _ , []
cons : ∀ {a}{A : Set a} → A → List A → List A
cons x (_ , xs) = _ , x ∷ xs
AnyList : ∀ {i} → Set (lsuc i)
AnyList {i} = Σ (Set i) (List {i})
| 26.319444
| 95
| 0.451187
|
1c73be8d1ba82f4e72a434a5948dbe817fb98452
| 2,873
|
agda
|
Agda
|
SOAS/ContextMaps/Combinators.agda
|
JoeyEremondi/agda-soas
|
ff1a985a6be9b780d3ba2beff68e902394f0a9d8
|
[
"MIT"
] | 39
|
2021-11-09T20:39:55.000Z
|
2022-03-19T17:33:12.000Z
|
SOAS/ContextMaps/Combinators.agda
|
JoeyEremondi/agda-soas
|
ff1a985a6be9b780d3ba2beff68e902394f0a9d8
|
[
"MIT"
] | 1
|
2021-11-21T12:19:32.000Z
|
2021-11-21T12:19:32.000Z
|
SOAS/ContextMaps/Combinators.agda
|
JoeyEremondi/agda-soas
|
ff1a985a6be9b780d3ba2beff68e902394f0a9d8
|
[
"MIT"
] | 4
|
2021-11-09T20:39:59.000Z
|
2022-01-24T12:49:17.000Z
|
import SOAS.Families.Core
-- Combinators for context maps
module SOAS.ContextMaps.Combinators {T : Set}
(open SOAS.Families.Core {T})
(𝒳 : Familyₛ) where
open import SOAS.Common
open import SOAS.Context {T}
open import SOAS.Sorting
open import SOAS.Variable
open import SOAS.Families.Isomorphism
open import SOAS.Families.BCCC
private
variable
Γ Γ′ Δ Δ′ Θ : Ctx
α β τ : T
-- Sub from the empty context
empty : ∅ ~[ 𝒳 ]↝ Δ
empty ()
-- Combine two maps into the same context by concatenating the domain
copair : Γ ~[ 𝒳 ]↝ Θ → Δ ~[ 𝒳 ]↝ Θ → (Γ ∔ Δ) ~[ 𝒳 ]↝ Θ
copair {∅} σ ς v = ς v
copair {α ∙ Γ} σ ς new = σ new
copair {α ∙ Γ} σ ς (old v) = copair {Γ} (σ ∘ old) ς v
copair≈₁ : {σ₁ σ₂ : Γ ~[ 𝒳 ]↝ Θ}(ς : Δ ~[ 𝒳 ]↝ Θ){v : ℐ α (Γ ∔ Δ)}
→ ({τ : T}(v : ℐ τ Γ) → σ₁ v ≡ σ₂ v)
→ copair σ₁ ς v ≡ copair σ₂ ς v
copair≈₁ ς {v} p = cong (λ - → copair (λ {τ} → - {τ}) ς v) (dext (λ y → p y))
copair≈₂ : (σ : Γ ~[ 𝒳 ]↝ Θ){ς₁ ς₂ : Δ ~[ 𝒳 ]↝ Θ}{v : ℐ α (Γ ∔ Δ)}
→ ({τ : T}(v : ℐ τ Δ) → ς₁ v ≡ ς₂ v)
→ copair σ ς₁ v ≡ copair σ ς₂ v
copair≈₂ σ {v = v} p = cong (λ - → copair σ (λ {τ} → - {τ}) v) (dext (λ y → p y))
-- Split a map from a combined context into two maps
split : (Γ ∔ Δ) ~[ 𝒳 ]↝ Θ → Γ ~[ 𝒳 ]↝ Θ × Δ ~[ 𝒳 ]↝ Θ
split {∅} σ = (λ ()) , σ
split {α ∙ Γ} σ with split {Γ} (σ ∘ old)
... | ς₁ , ς₂ = (λ{ new → σ new ; (old v) → ς₁ v}) , ς₂
-- Expand the codomain of a renaming
expandʳ : ({Γ} Δ : Ctx) → Γ ↝ Γ ∔ Δ
expandʳ {α ∙ Γ} Δ new = new
expandʳ {α ∙ Γ} Δ (old v) = old (expandʳ Δ v)
expandˡ : (Γ {Δ} : Ctx) → Δ ↝ Γ ∔ Δ
expandˡ ∅ v = v
expandˡ (α ∙ Γ) v = old (expandˡ Γ v)
-- Special cases of the above, when one of the contexts is a singleton
-- and the map from the singleton context is isomorphic to a term
-- Add a term to a context map
add : 𝒳 α Δ → Γ ~[ 𝒳 ]↝ Δ → (α ∙ Γ) ~[ 𝒳 ]↝ Δ
add t σ new = t
add t σ (old v) = σ v
-- Consider a term as a context map from the singleton context
asMap : 𝒳 α Γ → ⌊ α ⌋ ~[ 𝒳 ]↝ Γ
asMap t new = t
-- Separate a compound context map into a term and a residual map
detach : (τ ∙ Γ) ~[ 𝒳 ]↝ Δ → 𝒳 τ Δ × (Γ ~[ 𝒳 ]↝ Δ)
detach {_}{∅} σ = σ new , (λ ())
detach {_}{(α ∙ Γ)} σ = σ new , σ ∘ old
add[new][old] : (σ : τ ∙ Γ ~[ 𝒳 ]↝ Δ)(v : ℐ α (τ ∙ Γ))
→ add (σ new) (σ ∘ old) v ≡ σ v
add[new][old] σ new = refl
add[new][old] σ (old v) = refl
-- Remove a term from a compound context map
remove : (τ ∙ Γ) ~[ 𝒳 ]↝ Δ → Γ ~[ 𝒳 ]↝ Δ
remove {_} {∅} σ = λ ()
remove {_} {α ∙ Γ} σ = σ ∘ old
-- Add and remove are inverses
add∘remove : (σ : (τ ∙ Γ) ~[ 𝒳 ]↝ Δ) (v : ℐ α (τ ∙ Γ))
→ add (σ new) (remove σ) v
≡ σ v
add∘remove σ new = refl
add∘remove σ (old new) = refl
add∘remove σ (old (old v)) = refl
remove∘add : (σ : Γ ~[ 𝒳 ]↝ Δ) (t : 𝒳 τ Δ)(v : ℐ α Γ)
→ remove (add t σ) v
≡ σ v
remove∘add σ t new = refl
remove∘add σ t (old v) = refl
| 29.316327
| 81
| 0.528716
|
0b2698ae1e9a7143f683ad6ee7e202af3aab8e04
| 900
|
agda
|
Agda
|
agda/SList/Concatenation.agda
|
bgbianchi/sorting
|
b8d428bccbdd1b13613e8f6ead6c81a8f9298399
|
[
"MIT"
] | 6
|
2015-05-21T12:50:35.000Z
|
2021-08-24T22:11:15.000Z
|
agda/SList/Concatenation.agda
|
bgbianchi/sorting
|
b8d428bccbdd1b13613e8f6ead6c81a8f9298399
|
[
"MIT"
] | null | null | null |
agda/SList/Concatenation.agda
|
bgbianchi/sorting
|
b8d428bccbdd1b13613e8f6ead6c81a8f9298399
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --sized-types #-}
module SList.Concatenation (A : Set) where
open import Data.List
open import List.Permutation.Base A
open import Size
open import SList
lemma-⊕-/ : {xs ys : List A}{x y : A} → xs / x ⟶ ys → unsize A (_⊕_ A (size A xs) (y ∙ snil)) / x ⟶ unsize A (_⊕_ A (size A ys) (y ∙ snil))
lemma-⊕-/ /head = /head
lemma-⊕-/ (/tail xs/x⟶xs') = /tail (lemma-⊕-/ xs/x⟶xs')
lemma-⊕∼ : {xs ys : List A}(x : A) → xs ∼ ys → (x ∷ xs) ∼ unsize A (_⊕_ A (size A ys) (x ∙ snil))
lemma-⊕∼ x ∼[] = ∼x /head /head ∼[]
lemma-⊕∼ x (∼x xs/x⟶xs' ys/x⟶ys' xs'∼ys') = ∼x (/tail xs/x⟶xs') (lemma-⊕-/ ys/x⟶ys') (lemma-⊕∼ x xs'∼ys')
lemma-size-unsize : {ι : Size}(x : A) → (xs : SList A {ι}) → (unsize A (_⊕_ A (size A (unsize A xs)) (x ∙ snil))) ∼ unsize A (_⊕_ A xs (x ∙ snil))
lemma-size-unsize x snil = ∼x /head /head ∼[]
lemma-size-unsize x (y ∙ ys) = ∼x /head /head (lemma-size-unsize x ys)
| 37.5
| 146
| 0.553333
|
dccfa37a9b3252958a7ed0dc7f142c7a791e33a4
| 3,054
|
agda
|
Agda
|
src/Prelude/Show.agda
|
t-more/agda-prelude
|
da4fca7744d317b8843f2bc80a923972f65548d3
|
[
"MIT"
] | 111
|
2015-01-05T11:28:15.000Z
|
2022-02-12T23:29:26.000Z
|
src/Prelude/Show.agda
|
t-more/agda-prelude
|
da4fca7744d317b8843f2bc80a923972f65548d3
|
[
"MIT"
] | 59
|
2016-02-09T05:36:44.000Z
|
2022-01-14T07:32:36.000Z
|
src/Prelude/Show.agda
|
t-more/agda-prelude
|
da4fca7744d317b8843f2bc80a923972f65548d3
|
[
"MIT"
] | 24
|
2015-03-12T18:03:45.000Z
|
2021-04-22T06:10:41.000Z
|
module Prelude.Show where
open import Prelude.Unit
open import Prelude.String
open import Prelude.Char
open import Prelude.Nat
open import Prelude.Int
open import Prelude.Word
open import Prelude.Function
open import Prelude.List
open import Prelude.Fin
open import Prelude.Vec
open import Prelude.Maybe
open import Prelude.Sum
open import Prelude.Product
open import Prelude.Bool
open import Prelude.Ord
open import Prelude.Semiring
--- Class ---
ShowS = String → String
showString : String → ShowS
showString s r = s & r
showParen : Bool → ShowS → ShowS
showParen false s = s
showParen true s = showString "(" ∘ s ∘ showString ")"
record Show {a} (A : Set a) : Set a where
field
showsPrec : Nat → A → ShowS
show : A → String
show x = showsPrec 0 x ""
shows : A → ShowS
shows = showsPrec 0
open Show {{...}} public
simpleShowInstance : ∀ {a} {A : Set a} → (A → String) → Show A
showsPrec {{simpleShowInstance s}} _ x = showString (s x)
ShowBy : ∀ {a b} {A : Set a} {B : Set b} {{ShowB : Show B}} → (A → B) → Show A
showsPrec {{ShowBy f}} p = showsPrec p ∘ f
--- Instances ---
-- Bool --
instance
ShowBool : Show Bool
ShowBool = simpleShowInstance λ b → if b then "true" else "false"
-- Int --
instance
ShowInt : Show Int
ShowInt = simpleShowInstance primShowInteger
-- Nat --
instance
ShowNat : Show Nat
ShowNat = simpleShowInstance (primShowInteger ∘ pos)
-- Word64 --
instance
ShowWord64 : Show Word64
ShowWord64 = simpleShowInstance (show ∘ word64ToNat)
-- Char --
instance
ShowChar : Show Char
ShowChar = simpleShowInstance primShowChar
-- String --
instance
ShowString : Show String
ShowString = simpleShowInstance primShowString
-- List --
instance
ShowList : ∀ {a} {A : Set a} {{ShowA : Show A}} → Show (List A)
showsPrec {{ShowList}} _ [] = showString "[]"
showsPrec {{ShowList}} _ (x ∷ xs) =
showString "["
∘ foldl (λ r x → r ∘ showString ", " ∘ showsPrec 2 x) (showsPrec 2 x) xs
∘ showString "]"
-- Fin --
instance
ShowFin : ∀ {n} → Show (Fin n)
ShowFin = simpleShowInstance (show ∘ finToNat)
-- Vec --
instance
ShowVec : ∀ {a} {A : Set a} {n} {{ShowA : Show A}} → Show (Vec A n)
ShowVec = ShowBy vecToList
-- Maybe --
instance
ShowMaybe : ∀ {a} {A : Set a} {{ShowA : Show A}} → Show (Maybe A)
showsPrec {{ShowMaybe}} p nothing = showString "nothing"
showsPrec {{ShowMaybe}} p (just x) = showParen (p >? 9) (showString "just " ∘ showsPrec 10 x)
-- Either --
instance
ShowEither : ∀ {a b} {A : Set a} {B : Set b} {{ShowA : Show A}} {{ShowB : Show B}} → Show (Either A B)
showsPrec {{ShowEither}} p (left x) = showParen (p >? 9) $ showString "left " ∘ showsPrec 10 x
showsPrec {{ShowEither}} p (right x) = showParen (p >? 9) $ showString "right " ∘ showsPrec 10 x
-- Sigma --
instance
ShowSigma : ∀ {a b} {A : Set a} {B : A → Set b} {{ShowA : Show A}} {{ShowB : ∀ {x} → Show (B x)}} →
Show (Σ A B)
showsPrec {{ShowSigma}} p (x , y) = showParen (p >? 1) $ showsPrec 2 x ∘ showString ", " ∘ showsPrec 1 y
| 23.492308
| 106
| 0.638834
|
107a72458be8734ebfa4e893eda79078315b6ab0
| 1,439
|
agda
|
Agda
|
code/InftyConat.agda
|
ionathanch/msc-thesis
|
8fe15af8f9b5021dc50bcf96665e0988abf28f3c
|
[
"CC-BY-4.0"
] | null | null | null |
code/InftyConat.agda
|
ionathanch/msc-thesis
|
8fe15af8f9b5021dc50bcf96665e0988abf28f3c
|
[
"CC-BY-4.0"
] | null | null | null |
code/InftyConat.agda
|
ionathanch/msc-thesis
|
8fe15af8f9b5021dc50bcf96665e0988abf28f3c
|
[
"CC-BY-4.0"
] | null | null | null |
{-# OPTIONS --guardedness #-}
open import Agda.Builtin.Equality
open import Data.Empty
-- base : Size
-- next : Delay → Size
-- later : Size → Delay
record Delay : Set
data Size : Set
record Delay where
coinductive
constructor later
field now : Size
open Delay
data Size where
base : Size
next : Delay → Size
-- ω ≡ next ω' ≡ next (later (next ω')) ≡ ...
-- ≡ next (later ω) ≡ ...
ω : Size
ω = next ω' -- next (later (next ω))
where
ω' : Delay
-- ω' = later (next ω')
now ω' = next ω'
next' : Size → Size
next' s = next (later s)
{-# ETA Delay #-}
lim : ω ≡ next' ω
lim = refl
data FSize : Size → Set where
fbase : FSize base
fnext : {d : Delay} → FSize (now d) → FSize (next d)
inf : FSize ω → ⊥
inf (fnext s) = inf s
data Nat : Size → Set where
zero : (s : Size) → Nat (next' s)
succ : (s : Size) → Nat s → Nat (next' s)
shift : ∀ s → Nat s → Nat (next' s)
shift _ (zero s) = zero (next' s)
shift _ (succ s n) = succ (next' s) (shift s n)
postulate blocker : ∀ s → Nat s → Nat s
-- The guard condition passes due to recursion on n,
-- not due to recusion on the size, since it doesn't know that
-- `now s` is in fact smaller than s without using Agda's sized types
-- (or the old version of coinductive types with musical notation).
zeroify : ∀ s → Nat s → Nat s
zeroify base ()
zeroify (next s) (zero _) = zero _
zeroify (next s) (succ _ n) = shift (now s) (zeroify (now s) (blocker (now s) n))
| 22.84127
| 81
| 0.614315
|
50d5e2721bb2d67cb8635764fe98f326ceb27cc3
| 7,472
|
agda
|
Agda
|
src/Tactic/Nat/Subtract/By.agda
|
UlfNorell/agda-prelude
|
d704381936db6bd393e63aa2740345e7364f9732
|
[
"MIT"
] | 111
|
2015-01-05T11:28:15.000Z
|
2022-02-12T23:29:26.000Z
|
src/Tactic/Nat/Subtract/By.agda
|
UlfNorell/agda-prelude
|
d704381936db6bd393e63aa2740345e7364f9732
|
[
"MIT"
] | 59
|
2016-02-09T05:36:44.000Z
|
2022-01-14T07:32:36.000Z
|
src/Tactic/Nat/Subtract/By.agda
|
UlfNorell/agda-prelude
|
d704381936db6bd393e63aa2740345e7364f9732
|
[
"MIT"
] | 24
|
2015-03-12T18:03:45.000Z
|
2021-04-22T06:10:41.000Z
|
module Tactic.Nat.Subtract.By where
open import Prelude hiding (abs)
open import Builtin.Reflection
open import Tactic.Reflection.Quote
open import Tactic.Reflection.DeBruijn
open import Tactic.Reflection.Substitute
open import Tactic.Reflection
open import Control.Monad.State
open import Tactic.Nat.Reflect
open import Tactic.Nat.NF
open import Tactic.Nat.Exp
open import Tactic.Nat.Auto
open import Tactic.Nat.Auto.Lemmas
open import Tactic.Nat.Simpl.Lemmas
open import Tactic.Nat.Simpl
open import Tactic.Nat.Refute
open import Tactic.Nat.Reflect
open import Tactic.Nat.Subtract.Exp
open import Tactic.Nat.Subtract.Reflect
open import Tactic.Nat.Subtract.Lemmas
open import Tactic.Nat.Less.Lemmas
private
NFGoal : (R₁ R₂ : Nat → Nat → Set) (a b c d : SubNF) → Env Var → Set
NFGoal _R₁_ _R₂_ a b c d ρ = ⟦ a ⟧ns (atomEnvS ρ) R₁ ⟦ b ⟧ns (atomEnvS ρ) → ⟦ c ⟧ns (atomEnvS ρ) R₂ ⟦ d ⟧ns (atomEnvS ρ)
follows-diff-prf : {a b c d : Nat} → a ≤ b → b < c → c ≤ d → d ≡ suc (d - suc a) + a
follows-diff-prf {a} (diff! i) (diff! j) (diff! k) =
sym $ (λ z → suc z + a) $≡ lem-sub-zero (k + suc (j + (i + a))) (suc a) (i + j + k) auto ʳ⟨≡⟩
auto
decide-leq : ∀ u v ρ → Maybe (⟦ u ⟧ns (atomEnvS ρ) ≤ ⟦ v ⟧ns (atomEnvS ρ))
decide-leq u v ρ with cancel u v | (λ a b → cancel-sound-s′ a b u v (atomEnvS ρ))
... | [] , d | sound =
let eval x = ⟦ x ⟧ns (atomEnvS ρ) in
just (diff (eval d) $ sym (sound (suc (eval d)) 1 auto))
... | _ , _ | _ = nothing
by-proof-less-nf : ∀ u u₁ v v₁ ρ → Maybe (NFGoal _<_ _<_ u u₁ v v₁ ρ)
by-proof-less-nf u u₁ v v₁ ρ = do
v≤u ← decide-leq v u ρ
u₁≤v₁ ← decide-leq u₁ v₁ ρ
pure λ u<u₁ →
diff (⟦ v₁ ⟧ns (atomEnvS ρ) - suc (⟦ v ⟧ns (atomEnvS ρ)))
(follows-diff-prf v≤u u<u₁ u₁≤v₁)
by-proof-less : ∀ a a₁ b b₁ ρ → Maybe (SubExpLess a a₁ ρ → SubExpLess b b₁ ρ)
by-proof-less a a₁ b b₁ ρ with cancel (normSub a) (normSub a₁)
| cancel (normSub b) (normSub b₁)
| complicateSubLess a a₁ ρ
| simplifySubLess b b₁ ρ
... | u , u₁ | v , v₁ | compl | simpl = do
prf ← by-proof-less-nf u u₁ v v₁ ρ
pure (simpl ∘ prf ∘ compl)
lem-plus-zero-r : (a b : Nat) → a + b ≡ 0 → b ≡ 0
lem-plus-zero-r zero b eq = eq
lem-plus-zero-r (suc a) b ()
lem-leq-zero : {a b : Nat} → a ≤ b → b ≡ 0 → a ≡ 0
lem-leq-zero (diff k eq) refl = lem-plus-zero-r k _ (follows-from (sym eq))
⟨+⟩-sound-ns : ∀ {Atom} {{_ : Ord Atom}} u v (ρ : Env Atom) → ⟦ u +nf v ⟧ns ρ ≡ ⟦ u ⟧ns ρ + ⟦ v ⟧ns ρ
⟨+⟩-sound-ns u v ρ =
ns-sound (u +nf v) ρ ⟨≡⟩
⟨+⟩-sound u v ρ ⟨≡⟩ʳ
_+_ $≡ ns-sound u ρ *≡ ns-sound v ρ
by-proof-eq-nf : Nat → ∀ u u₁ v v₁ ρ → Maybe (NFGoal _≡_ _≡_ u u₁ v v₁ ρ)
by-proof-eq-sub : Nat → ∀ u u₁ v v₁ v₂ ρ → Maybe (NFGoal _≡_ _≡_ u u₁ [ 1 , [ v ⟨-⟩ v₁ ] ] v₂ ρ)
by-proof-eq-sub n u u₁ v v₁ v₂ ρ = do
let eval x = ⟦ x ⟧ns (atomEnvS ρ)
evals x = ⟦ x ⟧sns ρ
prf ← by-proof-eq-nf n u u₁ v (v₁ +nf v₂) ρ
pure (λ u=u₁ →
sym $ lem-sub-zero (evals v) (evals v₁) (eval v₂) $ sym $
lem-eval-sns-ns v ρ ⟨≡⟩
prf u=u₁ ⟨≡⟩
⟨+⟩-sound-ns v₁ v₂ (atomEnvS ρ) ⟨≡⟩ʳ
(_+ eval v₂) $≡ (lem-eval-sns-ns v₁ ρ))
by-proof-eq-sub₂ : Nat → ∀ u u₁ v v₁ v₂ v₃ ρ → Maybe (NFGoal _≡_ _≡_ u u₁ [ 1 , [ v ⟨-⟩ v₁ ] ] [ 1 , [ v₂ ⟨-⟩ v₃ ] ] ρ)
by-proof-eq-sub₂ n u u₁ v v₁ v₂ v₃ ρ = do
let eval x = ⟦ x ⟧ns (atomEnvS ρ)
evals x = ⟦ x ⟧sns ρ
prf ← by-proof-eq-nf n u u₁ (v₃ +nf v) (v₂ +nf v₁) ρ
pure λ u=u₁ →
lem-sub (evals v₂) (evals v₃) (evals v) (evals v₁) $
_+_ $≡ lem-eval-sns-ns v₃ ρ *≡ lem-eval-sns-ns v ρ ⟨≡⟩
⟨+⟩-sound-ns v₃ v (atomEnvS ρ) ʳ⟨≡⟩
prf u=u₁ ⟨≡⟩ ⟨+⟩-sound-ns v₂ v₁ (atomEnvS ρ) ⟨≡⟩ʳ
_+_ $≡ lem-eval-sns-ns v₂ ρ *≡ lem-eval-sns-ns v₁ ρ
-- More advanced tactics for equalities
-- a + b ≡ 0 → a ≡ 0
by-proof-eq-adv : Nat → ∀ u u₁ v v₁ ρ → Maybe (NFGoal _≡_ _≡_ u u₁ v v₁ ρ)
by-proof-eq-adv _ u [] v [] ρ = do leq ← decide-leq v u ρ; pure (lem-leq-zero leq)
by-proof-eq-adv _ [] u₁ v [] ρ = do leq ← decide-leq v u₁ ρ; pure (lem-leq-zero leq ∘ sym)
by-proof-eq-adv _ u [] [] v₁ ρ = do leq ← decide-leq v₁ u ρ; pure (sym ∘ lem-leq-zero leq)
by-proof-eq-adv _ [] u₁ [] v₁ ρ = do leq ← decide-leq v₁ u₁ ρ; pure (sym ∘ lem-leq-zero leq ∘ sym)
by-proof-eq-adv (suc n) u u₁ [ 1 , [ v ⟨-⟩ v₁ ] ] [ 1 , [ v₂ ⟨-⟩ v₃ ] ] ρ = by-proof-eq-sub₂ n u u₁ v v₁ v₂ v₃ ρ
by-proof-eq-adv n u u₁ [ 1 , [ v ⟨-⟩ v₁ ] ] v₂ ρ = by-proof-eq-sub n u u₁ v v₁ v₂ ρ
by-proof-eq-adv (suc n) u u₁ v₂ [ 1 , [ v ⟨-⟩ v₁ ] ] ρ = do
prf ← by-proof-eq-sub n u u₁ v v₁ v₂ ρ
pure (sym ∘ prf)
by-proof-eq-adv _ u u₁ v v₁ ρ = nothing
by-proof-eq-nf n u u₁ v v₁ ρ with u == v | u₁ == v₁
by-proof-eq-nf n u u₁ .u .u₁ ρ | yes refl | yes refl = just id
... | _ | _ with u == v₁ | u₁ == v -- try sym
by-proof-eq-nf n u u₁ .u₁ .u ρ | _ | _ | yes refl | yes refl = just sym
... | _ | _ = by-proof-eq-adv n u u₁ v v₁ ρ -- try advanced stuff
by-proof-eq : ∀ a a₁ b b₁ ρ → Maybe (SubExpEq a a₁ ρ → SubExpEq b b₁ ρ)
by-proof-eq a a₁ b b₁ ρ with cancel (normSub a) (normSub a₁)
| cancel (normSub b) (normSub b₁)
| complicateSubEq a a₁ ρ
| simplifySubEq b b₁ ρ
... | u , u₁ | v , v₁ | compl | simpl = do
prf ← by-proof-eq-nf 10 u u₁ v v₁ ρ
pure (simpl ∘ prf ∘ compl)
not-less-zero′ : {n : Nat} → n < 0 → ⊥
not-less-zero′ (diff _ ())
not-less-zero : {A : Set} {n : Nat} → n < 0 → A
not-less-zero n<0 = ⊥-elim (erase-⊥ (not-less-zero′ n<0))
less-one-is-zero : {n : Nat} → n < 1 → n ≡ 0
less-one-is-zero {zero} _ = refl
less-one-is-zero {suc n} (diff k eq) = refute eq
by-proof-less-eq-nf : ∀ u u₁ v v₁ ρ → Maybe (NFGoal _<_ _≡_ u u₁ v v₁ ρ)
by-proof-less-eq-nf u [] v v₁ ρ = just not-less-zero -- could've used refute, but we'll take it
by-proof-less-eq-nf u [ 1 , [] ] v v₁ ρ = do
prf ← by-proof-eq-nf 10 u [] v v₁ ρ
pure (prf ∘ less-one-is-zero)
by-proof-less-eq-nf u u₁ v v₁ ρ = nothing
by-proof-less-eq : ∀ a a₁ b b₁ ρ → Maybe (SubExpLess a a₁ ρ → SubExpEq b b₁ ρ)
by-proof-less-eq a a₁ b b₁ ρ with cancel (normSub a) (normSub a₁)
| cancel (normSub b) (normSub b₁)
| complicateSubLess a a₁ ρ
| simplifySubEq b b₁ ρ
... | u , u₁ | v , v₁ | compl | simpl = do
prf ← by-proof-less-eq-nf u u₁ v v₁ ρ
pure (simpl ∘ prf ∘ compl)
by-proof : ∀ hyp goal ρ → Maybe (⟦ hyp ⟧eqn ρ → ⟦ goal ⟧eqn ρ)
by-proof (a :≡ a₁) (b :≡ b₁) ρ = by-proof-eq a a₁ b b₁ ρ
by-proof (a :< a₁) (b :≡ b₁) ρ = by-proof-less-eq a a₁ b b₁ ρ
by-proof (a :< a₁) (b :< b₁) ρ = by-proof-less a a₁ b b₁ ρ
by-proof (a :≡ a₁) (b :< b₁) ρ = do
prf ← by-proof-less a (lit 1 ⟨+⟩ a₁) b b₁ ρ
pure λ eq → prf (diff 0 (cong suc (sym eq)))
by-tactic : Term → Type → TC Term
by-tactic prf g = do
ensureNoMetas prf
h ← inferNormalisedType prf
let t = pi (vArg h) (abs "_" (weaken 1 g))
just (hyp ∷ goal ∷ [] , Γ) ← termToSubHyps t
where _ → typeError $ strErr "Invalid goal:" ∷ termErr t ∷ []
pure $
applyTerm (safe
(getProof (quote cantProve) t $
def (quote by-proof)
( vArg (` hyp)
∷ vArg (` goal)
∷ vArg (quotedEnv Γ)
∷ [])) _) (vArg prf ∷ [])
| 41.977528
| 122
| 0.54296
|
dcc1e1fd5be07a2e1c1c96840e0eb5f632fd81ce
| 17,290
|
agda
|
Agda
|
Algebra/Monus.agda
|
oisdk/agda-playground
|
97a3aab1282b2337c5f43e2cfa3fa969a94c11b7
|
[
"MIT"
] | 6
|
2020-09-11T17:45:41.000Z
|
2021-11-16T08:11:34.000Z
|
Algebra/Monus.agda
|
oisdk/agda-playground
|
97a3aab1282b2337c5f43e2cfa3fa969a94c11b7
|
[
"MIT"
] | null | null | null |
Algebra/Monus.agda
|
oisdk/agda-playground
|
97a3aab1282b2337c5f43e2cfa3fa969a94c11b7
|
[
"MIT"
] | 1
|
2021-11-11T12:30:21.000Z
|
2021-11-11T12:30:21.000Z
|
{-# OPTIONS --safe #-}
-- This is a file for dealing with Monuses: these are monoids that are like the
-- positive half of a group. Much of my info on them comes from these papers:
--
-- * Wehrung, Friedrich. ‘Injective Positively Ordered Monoids I’. Journal of
-- Pure and Applied Algebra 83, no. 1 (11 November 1992): 43–82.
-- https://doi.org/10.1016/0022-4049(92)90104-N.
-- * Wehrung, Friedrich. ‘Embedding Simple Commutative Monoids into Simple
-- Refinement Monoids’. Semigroup Forum 56, no. 1 (January 1998): 104–29.
-- https://doi.org/10.1007/s00233-002-7008-0.
-- * Amer, K. ‘Equationally Complete Classes of Commutative Monoids with Monus’.
-- Algebra Universalis 18, no. 1 (1 February 1984): 129–31.
-- https://doi.org/10.1007/BF01182254.
-- * Wehrung, Friedrich. ‘Metric Properties of Positively Ordered Monoids’.
-- Forum Mathematicum 5, no. 5 (1993).
-- https://doi.org/10.1515/form.1993.5.183.
-- * Wehrung, Friedrich. ‘Restricted Injectivity, Transfer Property and
-- Decompositions of Separative Positively Ordered Monoids.’ Communications in
-- Algebra 22, no. 5 (1 January 1994): 1747–81.
-- https://doi.org/10.1080/00927879408824934.
--
-- These monoids have a preorder defined on them, the algebraic preorder:
--
-- x ≤ y = ∃ z × (y ≡ x ∙ z)
--
-- The _∸_ operator extracts the z from above, if it exists.
module Algebra.Monus where
open import Prelude
open import Algebra
open import Relation.Binary
open import Path.Reasoning
open import Function.Reasoning
-- Positively ordered monoids.
--
-- These are monoids which have a preorder that respects the monoid operation
-- in a straightforward way.
record POM ℓ₁ ℓ₂ : Type (ℓsuc (ℓ₁ ℓ⊔ ℓ₂)) where
field commutativeMonoid : CommutativeMonoid ℓ₁
open CommutativeMonoid commutativeMonoid public
field preorder : Preorder 𝑆 ℓ₂
open Preorder preorder public renaming (refl to ≤-refl)
field
positive : ∀ x → ε ≤ x
≤-cong : ∀ x {y z} → y ≤ z → x ∙ y ≤ x ∙ z
x≤x∙y : ∀ {x y} → x ≤ x ∙ y
x≤x∙y {x} {y} = subst (_≤ x ∙ y) (∙ε x) (≤-cong x (positive y))
≤-congʳ : ∀ x {y z} → y ≤ z → y ∙ x ≤ z ∙ x
≤-congʳ x {y} {z} p = subst₂ _≤_ (comm x y) (comm x z) (≤-cong x p)
alg-≤-trans : ∀ {x y z k₁ k₂} → y ≡ x ∙ k₁ → z ≡ y ∙ k₂ → z ≡ x ∙ (k₁ ∙ k₂)
alg-≤-trans {x} {y} {z} {k₁} {k₂} y≡x∙k₁ z≡y∙k₂ =
z ≡⟨ z≡y∙k₂ ⟩
y ∙ k₂ ≡⟨ cong (_∙ k₂) y≡x∙k₁ ⟩
(x ∙ k₁) ∙ k₂ ≡⟨ assoc x k₁ k₂ ⟩
x ∙ (k₁ ∙ k₂) ∎
infix 4 _≺_
_≺_ : 𝑆 → 𝑆 → Type _
x ≺ y = ∃ k × (y ≡ x ∙ k) × (k ≢ ε)
-- Total Antisymmetric POM
record TAPOM ℓ₁ ℓ₂ : Type (ℓsuc (ℓ₁ ℓ⊔ ℓ₂)) where
field pom : POM ℓ₁ ℓ₂
open POM pom public using (preorder; _≤_; ≤-cong; ≤-congʳ; x≤x∙y; commutativeMonoid; positive)
open CommutativeMonoid commutativeMonoid public
field
_≤|≥_ : Total _≤_
antisym : Antisymmetric _≤_
totalOrder : TotalOrder 𝑆 ℓ₂ ℓ₂
totalOrder = fromPartialOrder (record { preorder = preorder ; antisym = antisym }) _≤|≥_
open TotalOrder totalOrder public hiding (_≤|≥_; antisym) renaming (refl to ≤-refl)
-- Every commutative monoid generates a positively ordered monoid
-- called the "algebraic" or "minimal" pom
module AlgebraicPOM {ℓ} (mon : CommutativeMonoid ℓ) where
commutativeMonoid = mon
open CommutativeMonoid mon
infix 4 _≤_
_≤_ : 𝑆 → 𝑆 → Type _
x ≤ y = ∃ z × (y ≡ x ∙ z)
-- The snd here is the same proof as alg-≤-trans, so could be refactored out.
≤-trans : Transitive _≤_
≤-trans (k₁ , _) (k₂ , _) .fst = k₁ ∙ k₂
≤-trans {x} {y} {z} (k₁ , y≡x∙k₁) (k₂ , z≡y∙k₂) .snd =
z ≡⟨ z≡y∙k₂ ⟩
y ∙ k₂ ≡⟨ cong (_∙ k₂) y≡x∙k₁ ⟩
(x ∙ k₁) ∙ k₂ ≡⟨ assoc x k₁ k₂ ⟩
x ∙ (k₁ ∙ k₂) ∎
preorder : Preorder 𝑆 ℓ
Preorder._≤_ preorder = _≤_
Preorder.refl preorder = ε , sym (∙ε _)
Preorder.trans preorder = ≤-trans
positive : ∀ x → ε ≤ x
positive x = x , sym (ε∙ x)
≤-cong : ∀ x {y z} → y ≤ z → x ∙ y ≤ x ∙ z
≤-cong x (k , z≡y∙k) = k , cong (x ∙_) z≡y∙k ; sym (assoc x _ k)
algebraic-pom : ∀ {ℓ} → CommutativeMonoid ℓ → POM ℓ ℓ
algebraic-pom mon = record { AlgebraicPOM mon }
-- Total Minimal POM
record TMPOM ℓ : Type (ℓsuc ℓ) where
field commutativeMonoid : CommutativeMonoid ℓ
pom : POM _ _
pom = algebraic-pom commutativeMonoid
open POM pom public
infix 4 _≤|≥_
field _≤|≥_ : Total _≤_
<⇒≺ : ∀ x y → y ≰ x → x ≺ y
<⇒≺ x y x<y with x ≤|≥ y
... | inr y≤x = ⊥-elim (x<y y≤x)
... | inl (k , y≡x∙k) = λ
where
.fst → k
.snd .fst → y≡x∙k
.snd .snd k≡ε → x<y (ε , sym (∙ε y ; y≡x∙k ; cong (x ∙_) k≡ε ; ∙ε x))
infixl 6 _∸_
_∸_ : 𝑆 → 𝑆 → 𝑆
x ∸ y = either′ (const ε) fst (x ≤|≥ y)
x∸y≤x : ∀ x y → x ∸ y ≤ x
x∸y≤x x y with x ≤|≥ y
... | inl (k , p) = positive x
... | inr (k , x≡y∙k) = y , x≡y∙k ; comm y k
-- Total Minimal Antisymmetric POM
record TMAPOM ℓ : Type (ℓsuc ℓ) where
field tmpom : TMPOM ℓ
open TMPOM tmpom public using (_≤_; _≤|≥_; positive; alg-≤-trans; _≺_; <⇒≺; _∸_; x∸y≤x)
field antisym : Antisymmetric _≤_
tapom : TAPOM _ _
TAPOM.pom tapom = TMPOM.pom tmpom
TAPOM._≤|≥_ tapom = _≤|≥_
TAPOM.antisym tapom = antisym
open TAPOM tapom public hiding (antisym; _≤|≥_; _≤_; positive)
zeroSumFree : ∀ x y → x ∙ y ≡ ε → x ≡ ε
zeroSumFree x y x∙y≡ε = antisym (y , sym x∙y≡ε) (positive x)
≤‿∸‿cancel : ∀ x y → x ≤ y → (y ∸ x) ∙ x ≡ y
≤‿∸‿cancel x y x≤y with y ≤|≥ x
... | inl y≤x = ε∙ x ; antisym x≤y y≤x
... | inr (k , y≡x∙k) = comm k x ; sym y≡x∙k
∸‿comm : ∀ x y → x ∙ (y ∸ x) ≡ y ∙ (x ∸ y)
∸‿comm x y with y ≤|≥ x | x ≤|≥ y
... | inl y≤x | inl x≤y = cong (_∙ ε) (antisym x≤y y≤x)
... | inr (k , y≡x∙k) | inl x≤y = sym y≡x∙k ; sym (∙ε y)
... | inl y≤x | inr (k , x≥y) = ∙ε x ; x≥y
... | inr (k₁ , y≡x∙k₁) | inr (k₂ , x≡y∙k₂) =
x ∙ k₁ ≡˘⟨ y≡x∙k₁ ⟩
y ≡⟨ antisym (k₂ , x≡y∙k₂) (k₁ , y≡x∙k₁) ⟩
x ≡⟨ x≡y∙k₂ ⟩
y ∙ k₂ ∎
∸‿≺ : ∀ x y → x ≢ ε → y ≢ ε → x ∸ y ≺ x
∸‿≺ x y x≢ε y≢ε with x ≤|≥ y
... | inl _ = x , sym (ε∙ x) , x≢ε
... | inr (k , x≡y∙k) = y , x≡y∙k ; comm y k , y≢ε
-- Commutative Monoids with Monus
record CMM ℓ : Type (ℓsuc ℓ) where
field commutativeMonoid : CommutativeMonoid ℓ
pom : POM _ _
pom = algebraic-pom commutativeMonoid
open POM pom public
field _∸_ : 𝑆 → 𝑆 → 𝑆
infixl 6 _∸_
field
∸‿comm : ∀ x y → x ∙ (y ∸ x) ≡ y ∙ (x ∸ y)
∸‿assoc : ∀ x y z → (x ∸ y) ∸ z ≡ x ∸ (y ∙ z)
∸‿inv : ∀ x → x ∸ x ≡ ε
ε∸ : ∀ x → ε ∸ x ≡ ε
∸ε : ∀ x → x ∸ ε ≡ x
∸ε x =
x ∸ ε ≡˘⟨ ε∙ (x ∸ ε) ⟩
ε ∙ (x ∸ ε) ≡⟨ ∸‿comm ε x ⟩
x ∙ (ε ∸ x) ≡⟨ cong (x ∙_) (ε∸ x) ⟩
x ∙ ε ≡⟨ ∙ε x ⟩
x ∎
∸≤ : ∀ x y → x ≤ y → x ∸ y ≡ ε
∸≤ x y (k , y≡x∙k) =
x ∸ y ≡⟨ cong (x ∸_) y≡x∙k ⟩
x ∸ (x ∙ k) ≡˘⟨ ∸‿assoc x x k ⟩
(x ∸ x) ∸ k ≡⟨ cong (_∸ k) (∸‿inv x) ⟩
ε ∸ k ≡⟨ ε∸ k ⟩
ε ∎
∣_-_∣ : 𝑆 → 𝑆 → 𝑆
∣ x - y ∣ = (x ∸ y) ∙ (y ∸ x)
_⊔₂_ : 𝑆 → 𝑆 → 𝑆
x ⊔₂ y = x ∙ y ∙ ∣ x - y ∣
_⊓₂_ : 𝑆 → 𝑆 → 𝑆
x ⊓₂ y = (x ∙ y) ∸ ∣ x - y ∣
-- Cancellative Commutative Monoids with Monus
record CCMM ℓ : Type (ℓsuc ℓ) where
field cmm : CMM ℓ
open CMM cmm public
field ∸‿cancel : ∀ x y → (x ∙ y) ∸ x ≡ y
cancelˡ : Cancellativeˡ _∙_
cancelˡ x y z x∙y≡x∙z =
y ≡˘⟨ ∸‿cancel x y ⟩
(x ∙ y) ∸ x ≡⟨ cong (_∸ x) x∙y≡x∙z ⟩
(x ∙ z) ∸ x ≡⟨ ∸‿cancel x z ⟩
z ∎
cancelʳ : Cancellativeʳ _∙_
cancelʳ x y z y∙x≡z∙x =
y ≡˘⟨ ∸‿cancel x y ⟩
(x ∙ y) ∸ x ≡⟨ cong (_∸ x) (comm x y) ⟩
(y ∙ x) ∸ x ≡⟨ cong (_∸ x) y∙x≡z∙x ⟩
(z ∙ x) ∸ x ≡⟨ cong (_∸ x) (comm z x) ⟩
(x ∙ z) ∸ x ≡⟨ ∸‿cancel x z ⟩
z ∎
zeroSumFree : ∀ x y → x ∙ y ≡ ε → x ≡ ε
zeroSumFree x y x∙y≡ε =
x ≡˘⟨ ∸‿cancel y x ⟩
(y ∙ x) ∸ y ≡⟨ cong (_∸ y) (comm y x) ⟩
(x ∙ y) ∸ y ≡⟨ cong (_∸ y) x∙y≡ε ⟩
ε ∸ y ≡⟨ ε∸ y ⟩
ε ∎
antisym : Antisymmetric _≤_
antisym {x} {y} (k₁ , y≡x∙k₁) (k₂ , x≡y∙k₂) =
x ≡⟨ x≡y∙k₂ ⟩
y ∙ k₂ ≡⟨ [ lemma ]⇒ y ∙ ε ≡ y ∙ (k₂ ∙ k₁)
⟨ cancelˡ y ε (k₂ ∙ k₁) ⟩⇒ ε ≡ k₂ ∙ k₁
⟨ sym ⟩⇒ k₂ ∙ k₁ ≡ ε
⟨ zeroSumFree k₂ k₁ ⟩⇒ k₂ ≡ ε
⟨ cong (y ∙_) ⟩⇒ y ∙ k₂ ≡ y ∙ ε ⇒∎ ⟩
y ∙ ε ≡⟨ ∙ε y ⟩
y ∎
where
lemma = ∙ε y ; alg-≤-trans x≡y∙k₂ y≡x∙k₁
partialOrder : PartialOrder _ _
PartialOrder.preorder partialOrder = preorder
PartialOrder.antisym partialOrder = antisym
≺⇒< : ∀ x y → x ≺ y → y ≰ x
≺⇒< x y (k₁ , y≡x∙k₁ , k₁≢ε) (k₂ , x≡y∙k₂) =
[ x ∙ ε ≡⟨ ∙ε x ⟩
x ≡⟨ x≡y∙k₂ ⟩
y ∙ k₂ ≡⟨ cong (_∙ k₂) y≡x∙k₁ ⟩
(x ∙ k₁) ∙ k₂ ≡⟨ assoc x k₁ k₂ ⟩
x ∙ (k₁ ∙ k₂) ∎ ]⇒ x ∙ ε ≡ x ∙ (k₁ ∙ k₂)
⟨ cancelˡ x ε (k₁ ∙ k₂) ⟩⇒ ε ≡ k₁ ∙ k₂
⟨ sym ⟩⇒ k₁ ∙ k₂ ≡ ε
⟨ zeroSumFree k₁ k₂ ⟩⇒ k₁ ≡ ε
⟨ k₁≢ε ⟩⇒ ⊥ ⇒∎
≤⇒<⇒≢ε : ∀ x y → (x≤y : x ≤ y) → y ≰ x → fst x≤y ≢ ε
≤⇒<⇒≢ε x y (k₁ , y≡x∙k₁) y≰x k₁≡ε = y≰x λ
where
.fst → ε
.snd → x ≡˘⟨ ∙ε x ⟩
x ∙ ε ≡˘⟨ cong (x ∙_) k₁≡ε ⟩
x ∙ k₁ ≡˘⟨ y≡x∙k₁ ⟩
y ≡˘⟨ ∙ε y ⟩
y ∙ ε ∎
≤-cancelʳ : ∀ x y z → y ∙ x ≤ z ∙ x → y ≤ z
≤-cancelʳ x y z (k , z∙x≡y∙x∙k) .fst = k
≤-cancelʳ x y z (k , z∙x≡y∙x∙k) .snd = cancelʳ x z (y ∙ k) $
z ∙ x ≡⟨ z∙x≡y∙x∙k ⟩
(y ∙ x) ∙ k ≡⟨ assoc y x k ⟩
y ∙ (x ∙ k) ≡⟨ cong (y ∙_) (comm x k) ⟩
y ∙ (k ∙ x) ≡˘⟨ assoc y k x ⟩
(y ∙ k) ∙ x ∎
≤-cancelˡ : ∀ x y z → x ∙ y ≤ x ∙ z → y ≤ z
≤-cancelˡ x y z (k , x∙z≡x∙y∙k) .fst = k
≤-cancelˡ x y z (k , x∙z≡x∙y∙k) .snd =
cancelˡ x z (y ∙ k) (x∙z≡x∙y∙k ; assoc x y k)
≺-irrefl : Irreflexive _≺_
≺-irrefl {x} (k , x≡x∙k , k≢ε) = k≢ε (sym (cancelˡ x ε k (∙ε x ; x≡x∙k)))
≤∸ : ∀ x y → (x≤y : x ≤ y) → y ∸ x ≡ fst x≤y
≤∸ x y (k , y≡x∙k) =
y ∸ x ≡⟨ cong (_∸ x) y≡x∙k ⟩
(x ∙ k) ∸ x ≡⟨ ∸‿cancel x k ⟩
k ∎
≤‿∸‿cancel : ∀ x y → x ≤ y → (y ∸ x) ∙ x ≡ y
≤‿∸‿cancel x y (k , y≡x∙k) =
(y ∸ x) ∙ x ≡⟨ cong (λ y → (y ∸ x) ∙ x) y≡x∙k ⟩
((x ∙ k) ∸ x) ∙ x ≡⟨ cong (_∙ x) (∸‿cancel x k) ⟩
k ∙ x ≡⟨ comm k x ⟩
x ∙ k ≡˘⟨ y≡x∙k ⟩
y ∎
-- Cancellative total minimal antisymmetric pom (has monus)
record CTMAPOM ℓ : Type (ℓsuc ℓ) where
field tmapom : TMAPOM ℓ
open TMAPOM tmapom public
field cancel : Cancellativeˡ _∙_
module cmm where
∸≤ : ∀ x y → x ≤ y → x ∸ y ≡ ε
∸≤ x y x≤y with x ≤|≥ y
∸≤ x y x≤y | inl _ = refl
∸≤ x y (k₁ , y≡x∙k₁) | inr (k₂ , x≡y∙k₂) =
[ lemma ]⇒ y ∙ ε ≡ y ∙ (k₂ ∙ k₁)
⟨ cancel y ε (k₂ ∙ k₁) ⟩⇒ ε ≡ k₂ ∙ k₁
⟨ sym ⟩⇒ k₂ ∙ k₁ ≡ ε
⟨ zeroSumFree k₂ k₁ ⟩⇒ k₂ ≡ ε ⇒∎
where
lemma = ∙ε y ; alg-≤-trans x≡y∙k₂ y≡x∙k₁
∸‿inv : ∀ x → x ∸ x ≡ ε
∸‿inv x = ∸≤ x x ≤-refl
ε∸ : ∀ x → ε ∸ x ≡ ε
ε∸ x = ∸≤ ε x (positive x)
∸‿assoc : ∀ x y z → (x ∸ y) ∸ z ≡ x ∸ (y ∙ z)
∸‿assoc x y z with x ≤|≥ y
∸‿assoc x y z | inl x≤y = ε∸ z ; sym (∸≤ x (y ∙ z) (≤-trans x≤y x≤x∙y))
∸‿assoc x y z | inr x≥y with x ≤|≥ y ∙ z
∸‿assoc x y z | inr (k₁ , x≡y∙k₁) | inl (k₂ , y∙z≡x∙k₂) = ∸≤ k₁ z (k₂ , lemma)
where
lemma : z ≡ k₁ ∙ k₂
lemma = cancel y z (k₁ ∙ k₂) (alg-≤-trans x≡y∙k₁ y∙z≡x∙k₂)
∸‿assoc x y z | inr (k , x≡y∙k) | inr x≥y∙z with k ≤|≥ z
∸‿assoc x y z | inr (k₁ , x≡y∙k₁) | inr (k₂ , x≡y∙z∙k₂) | inl (k₃ , z≡k₁∙k₃) =
[ lemma₁ ]⇒ ε ≡ k₂ ∙ k₃
⟨ sym ⟩⇒ k₂ ∙ k₃ ≡ ε
⟨ zeroSumFree k₂ k₃ ⟩⇒ k₂ ≡ ε
⟨ sym ⟩⇒ ε ≡ k₂ ⇒∎
where
lemma₃ =
y ∙ k₁ ≡˘⟨ x≡y∙k₁ ⟩
x ≡⟨ x≡y∙z∙k₂ ⟩
(y ∙ z) ∙ k₂ ≡⟨ assoc y z k₂ ⟩
y ∙ (z ∙ k₂) ∎
lemma₂ =
k₁ ∙ ε ≡⟨ ∙ε k₁ ⟩
k₁ ≡⟨ alg-≤-trans z≡k₁∙k₃ (cancel y k₁ (z ∙ k₂) lemma₃) ⟩
k₁ ∙ (k₃ ∙ k₂) ∎
lemma₁ =
ε ≡⟨ cancel k₁ ε (k₃ ∙ k₂) lemma₂ ⟩
k₃ ∙ k₂ ≡⟨ comm k₃ k₂ ⟩
k₂ ∙ k₃ ∎
∸‿assoc x y z | inr (k₁ , x≡y∙k₁) | inr (k₂ , x≡y∙z∙k₂) | inr (k₃ , k₁≡z∙k₃) =
cancel z k₃ k₂ lemma₂
where
lemma₁ =
y ∙ k₁ ≡˘⟨ x≡y∙k₁ ⟩
x ≡⟨ x≡y∙z∙k₂ ⟩
(y ∙ z) ∙ k₂ ≡⟨ assoc y z k₂ ⟩
y ∙ (z ∙ k₂) ∎
lemma₂ =
z ∙ k₃ ≡˘⟨ k₁≡z∙k₃ ⟩
k₁ ≡⟨ cancel y k₁ (z ∙ k₂) lemma₁ ⟩
z ∙ k₂ ∎
open cmm public
∸‿cancel : ∀ x y → (x ∙ y) ∸ x ≡ y
∸‿cancel x y with (x ∙ y) ≤|≥ x
... | inl x∙y≤x = sym (cancel x y ε (antisym x∙y≤x x≤x∙y ; sym (∙ε x)))
... | inr (k , x∙y≡x∙k) = sym (cancel x y k x∙y≡x∙k)
ccmm : CCMM _
ccmm = record { ∸‿cancel = ∸‿cancel
; cmm = record { cmm
; ∸‿comm = ∸‿comm
; commutativeMonoid = commutativeMonoid } }
open CCMM ccmm public
using (cancelʳ; cancelˡ; ∸ε; ≺⇒<; ≤⇒<⇒≢ε; _⊔₂_; _⊓₂_; ≺-irrefl; ≤∸)
∸‿< : ∀ x y → x ≢ ε → y ≢ ε → x ∸ y < x
∸‿< x y x≢ε y≢ε = ≺⇒< (x ∸ y) x (∸‿≺ x y x≢ε y≢ε)
∸‿<-< : ∀ x y → x < y → x ≢ ε → y ∸ x < y
∸‿<-< x y x<y x≢ε = ∸‿< y x (λ y≡ε → x<y (x , sym (cong (_∙ x) y≡ε ; ε∙ x))) x≢ε
2× : 𝑆 → 𝑆
2× x = x ∙ x
open import Relation.Binary.Lattice totalOrder
double-max : ∀ x y → 2× (x ⊔ y) ≡ x ⊔₂ y
double-max x y with x ≤|≥ y | y ≤|≥ x
double-max x y | inl x≤y | inl y≤x =
x ∙ x ≡⟨ cong (x ∙_) (antisym x≤y y≤x) ⟩
x ∙ y ≡˘⟨ ∙ε (x ∙ y) ⟩
(x ∙ y) ∙ ε ≡˘⟨ cong ((x ∙ y) ∙_) (ε∙ ε) ⟩
(x ∙ y) ∙ (ε ∙ ε) ∎
double-max x y | inl x≤y | inr (k , y≡x∙k) =
y ∙ y ≡⟨ cong (y ∙_) y≡x∙k ⟩
y ∙ (x ∙ k) ≡˘⟨ assoc y x k ⟩
(y ∙ x) ∙ k ≡⟨ cong (_∙ k) (comm y x) ⟩
(x ∙ y) ∙ k ≡˘⟨ cong ((x ∙ y) ∙_) (ε∙ k) ⟩
(x ∙ y) ∙ (ε ∙ k) ∎
double-max x y | inr (k , x≡y∙k) | inl y≤x =
x ∙ x ≡⟨ cong (x ∙_) x≡y∙k ⟩
x ∙ (y ∙ k) ≡˘⟨ assoc x y k ⟩
(x ∙ y) ∙ k ≡˘⟨ cong ((x ∙ y) ∙_) (∙ε k) ⟩
(x ∙ y) ∙ (k ∙ ε) ∎
double-max x y | inr (k₁ , x≡y∙k₁) | inr (k₂ , y≡x∙k₂) =
x ∙ x ≡⟨ cong (x ∙_) (antisym (k₂ , y≡x∙k₂) (k₁ , x≡y∙k₁)) ⟩
x ∙ y ≡⟨ cong₂ _∙_ x≡y∙k₁ y≡x∙k₂ ⟩
(y ∙ k₁) ∙ (x ∙ k₂) ≡˘⟨ assoc (y ∙ k₁) x k₂ ⟩
((y ∙ k₁) ∙ x) ∙ k₂ ≡⟨ cong (_∙ k₂) (comm (y ∙ k₁) x) ⟩
(x ∙ (y ∙ k₁)) ∙ k₂ ≡˘⟨ cong (_∙ k₂) (assoc x y k₁) ⟩
((x ∙ y) ∙ k₁) ∙ k₂ ≡⟨ assoc (x ∙ y) k₁ k₂ ⟩
(x ∙ y) ∙ (k₁ ∙ k₂) ∎
open import Data.Sigma.Properties
≤-prop : ∀ x y → isProp (x ≤ y)
≤-prop x y (k₁ , y≡x∙k₁) (k₂ , y≡x∙k₂) = Σ≡Prop (λ _ → total⇒isSet _ _) (cancelˡ x k₁ k₂ (sym y≡x∙k₁ ; y≡x∙k₂))
open import Cubical.Foundations.HLevels using (isProp×)
open import Data.Empty.Properties using (isProp¬)
≺-prop : ∀ x y → isProp (x ≺ y)
≺-prop x y (k₁ , y≡x∙k₁ , k₁≢ε) (k₂ , y≡x∙k₂ , k₂≢ε) = Σ≡Prop (λ k → isProp× (total⇒isSet _ _) (isProp¬ _)) (cancelˡ x k₁ k₂ (sym y≡x∙k₁ ; y≡x∙k₂))
-- We can construct the viterbi semiring by adjoining a top element to
-- a tapom
module Viterbi {ℓ₁} {ℓ₂} (tapom : TAPOM ℓ₁ ℓ₂) where
open TAPOM tapom
open import Relation.Binary.Construct.UpperBound totalOrder
open import Relation.Binary.Lattice totalOrder
⟨⊓⟩∙ : _∙_ Distributesˡ _⊓_
⟨⊓⟩∙ x y z with x <? y | (x ∙ z) <? (y ∙ z)
... | yes x<y | yes xz<yz = refl
... | no x≮y | no xz≮yz = refl
... | no x≮y | yes xz<yz = ⊥-elim (<⇒≱ xz<yz (≤-congʳ z (≮⇒≥ x≮y)))
... | yes x<y | no xz≮yz = antisym (≤-congʳ z (<⇒≤ x<y)) (≮⇒≥ xz≮yz)
∙⟨⊓⟩ : _∙_ Distributesʳ _⊓_
∙⟨⊓⟩ x y z = comm x (y ⊓ z) ; ⟨⊓⟩∙ y z x ; cong₂ _⊓_ (comm y x) (comm z x)
open UBSugar
module NS where
𝑅 = ⌈∙⌉
0# 1# : 𝑅
_*_ _+_ : 𝑅 → 𝑅 → 𝑅
1# = ⌈ ε ⌉
⌈⊤⌉ * y = ⌈⊤⌉
⌈ x ⌉ * ⌈⊤⌉ = ⌈⊤⌉
⌈ x ⌉ * ⌈ y ⌉ = ⌈ x ∙ y ⌉
0# = ⌈⊤⌉
⌈⊤⌉ + y = y
⌈ x ⌉ + ⌈⊤⌉ = ⌈ x ⌉
⌈ x ⌉ + ⌈ y ⌉ = ⌈ x ⊓ y ⌉
*-assoc : Associative _*_
*-assoc ⌈⊤⌉ _ _ = refl
*-assoc ⌈ _ ⌉ ⌈⊤⌉ _ = refl
*-assoc ⌈ _ ⌉ ⌈ _ ⌉ ⌈⊤⌉ = refl
*-assoc ⌈ x ⌉ ⌈ y ⌉ ⌈ z ⌉ = cong ⌈_⌉ (assoc x y z)
*-com : Commutative _*_
*-com ⌈⊤⌉ ⌈⊤⌉ = refl
*-com ⌈⊤⌉ ⌈ _ ⌉ = refl
*-com ⌈ _ ⌉ ⌈⊤⌉ = refl
*-com ⌈ x ⌉ ⌈ y ⌉ = cong ⌈_⌉ (comm x y)
⟨+⟩* : _*_ Distributesˡ _+_
⟨+⟩* ⌈⊤⌉ _ _ = refl
⟨+⟩* ⌈ _ ⌉ ⌈⊤⌉ ⌈⊤⌉ = refl
⟨+⟩* ⌈ _ ⌉ ⌈⊤⌉ ⌈ _ ⌉ = refl
⟨+⟩* ⌈ x ⌉ ⌈ y ⌉ ⌈⊤⌉ = refl
⟨+⟩* ⌈ x ⌉ ⌈ y ⌉ ⌈ z ⌉ = cong ⌈_⌉ (⟨⊓⟩∙ x y z)
+-assoc : Associative _+_
+-assoc ⌈⊤⌉ _ _ = refl
+-assoc ⌈ x ⌉ ⌈⊤⌉ _ = refl
+-assoc ⌈ x ⌉ ⌈ _ ⌉ ⌈⊤⌉ = refl
+-assoc ⌈ x ⌉ ⌈ y ⌉ ⌈ z ⌉ = cong ⌈_⌉ (⊓-assoc x y z)
0+ : ∀ x → 0# + x ≡ x
0+ ⌈⊤⌉ = refl
0+ ⌈ _ ⌉ = refl
+0 : ∀ x → x + 0# ≡ x
+0 ⌈ _ ⌉ = refl
+0 ⌈⊤⌉ = refl
1* : ∀ x → 1# * x ≡ x
1* ⌈⊤⌉ = refl
1* ⌈ x ⌉ = cong ⌈_⌉ (ε∙ x)
*1 : ∀ x → x * 1# ≡ x
*1 ⌈⊤⌉ = refl
*1 ⌈ x ⌉ = cong ⌈_⌉ (∙ε x)
0* : ∀ x → 0# * x ≡ 0#
0* _ = refl
open NS
nearSemiring : NearSemiring _
nearSemiring = record { NS }
+-comm : Commutative _+_
+-comm ⌈⊤⌉ ⌈⊤⌉ = refl
+-comm ⌈⊤⌉ ⌈ _ ⌉ = refl
+-comm ⌈ _ ⌉ ⌈⊤⌉ = refl
+-comm ⌈ x ⌉ ⌈ y ⌉ = cong ⌈_⌉ (⊓-comm x y)
*0 : ∀ x → x * ⌈⊤⌉ ≡ ⌈⊤⌉
*0 ⌈ _ ⌉ = refl
*0 ⌈⊤⌉ = refl
*⟨+⟩ : _*_ Distributesʳ _+_
*⟨+⟩ x y z = *-com x (y + z) ; ⟨+⟩* y z x ; cong₂ _+_ (*-com y x) (*-com z x)
viterbi : ∀ {ℓ₁ ℓ₂} → TAPOM ℓ₁ ℓ₂ → Semiring ℓ₁
viterbi tapom = record { Viterbi tapom }
| 30.875
| 149
| 0.452053
|
103beec4ae5a5a342889dd40a17f7b007d4a8f0c
| 2,619
|
agda
|
Agda
|
BasicIPC/Metatheory/Hilbert-KripkeConcreteGluedGentzen.agda
|
mietek/hilbert-gentzen
|
fcd187db70f0a39b894fe44fad0107f61849405c
|
[
"X11"
] | 29
|
2016-07-03T18:51:56.000Z
|
2022-01-01T10:29:18.000Z
|
BasicIPC/Metatheory/Hilbert-KripkeConcreteGluedGentzen.agda
|
mietek/hilbert-gentzen
|
fcd187db70f0a39b894fe44fad0107f61849405c
|
[
"X11"
] | 1
|
2018-06-10T09:11:22.000Z
|
2018-06-10T09:11:22.000Z
|
BasicIPC/Metatheory/Hilbert-KripkeConcreteGluedGentzen.agda
|
mietek/hilbert-gentzen
|
fcd187db70f0a39b894fe44fad0107f61849405c
|
[
"X11"
] | null | null | null |
module BasicIPC.Metatheory.Hilbert-KripkeConcreteGluedGentzen where
open import BasicIPC.Syntax.Hilbert public
open import BasicIPC.Semantics.KripkeConcreteGluedGentzen public
-- Internalisation of syntax as syntax representation in a particular model.
module _ {{_ : Model}} where
[_] : ∀ {A Γ} → Γ ⊢ A → Γ [⊢] A
[ var i ] = [var] i
[ app t u ] = [app] [ t ] [ u ]
[ ci ] = [ci]
[ ck ] = [ck]
[ cs ] = [cs]
[ cpair ] = [cpair]
[ cfst ] = [cfst]
[ csnd ] = [csnd]
[ unit ] = [unit]
-- Soundness with respect to all models, or evaluation.
eval : ∀ {A Γ} → Γ ⊢ A → Γ ⊨ A
eval (var i) γ = lookup i γ
eval (app t u) γ = eval t γ ⟪$⟫ eval u γ
eval ci γ = [ci] ⅋ K I
eval ck γ = [ck] ⅋ K ⟪K⟫
eval cs γ = [cs] ⅋ K ⟪S⟫′
eval cpair γ = [cpair] ⅋ K _⟪,⟫′_
eval cfst γ = [cfst] ⅋ K π₁
eval csnd γ = [csnd] ⅋ K π₂
eval unit γ = ∙
-- TODO: Correctness of evaluation with respect to conversion.
-- The canonical model.
private
instance
canon : Model
canon = record
{ _⊩ᵅ_ = λ w P → unwrap w ⊢ α P
; mono⊩ᵅ = λ ξ t → mono⊢ (unwrap≤ ξ) t
; _[⊢]_ = _⊢_
; mono[⊢] = mono⊢
; [var] = var
; [lam] = lam
; [app] = app
; [pair] = pair
; [fst] = fst
; [snd] = snd
; [unit] = unit
}
-- Soundness and completeness with respect to the canonical model.
mutual
reflectᶜ : ∀ {A w} → unwrap w ⊢ A → w ⊩ A
reflectᶜ {α P} t = t ⅋ t
reflectᶜ {A ▻ B} t = t ⅋ λ ξ a → reflectᶜ (app (mono⊢ (unwrap≤ ξ) t) (reifyᶜ a))
reflectᶜ {A ∧ B} t = reflectᶜ (fst t) , reflectᶜ (snd t)
reflectᶜ {⊤} t = ∙
reifyᶜ : ∀ {A w} → w ⊩ A → unwrap w ⊢ A
reifyᶜ {α P} s = syn s
reifyᶜ {A ▻ B} s = syn s
reifyᶜ {A ∧ B} s = pair (reifyᶜ (π₁ s)) (reifyᶜ (π₂ s))
reifyᶜ {⊤} s = unit
reflectᶜ⋆ : ∀ {Ξ w} → unwrap w ⊢⋆ Ξ → w ⊩⋆ Ξ
reflectᶜ⋆ {∅} ∙ = ∙
reflectᶜ⋆ {Ξ , A} (ts , t) = reflectᶜ⋆ ts , reflectᶜ t
reifyᶜ⋆ : ∀ {Ξ w} → w ⊩⋆ Ξ → unwrap w ⊢⋆ Ξ
reifyᶜ⋆ {∅} ∙ = ∙
reifyᶜ⋆ {Ξ , A} (ts , t) = reifyᶜ⋆ ts , reifyᶜ t
-- Reflexivity and transitivity.
refl⊩⋆ : ∀ {w} → w ⊩⋆ unwrap w
refl⊩⋆ = reflectᶜ⋆ refl⊢⋆
trans⊩⋆ : ∀ {w w′ w″} → w ⊩⋆ unwrap w′ → w′ ⊩⋆ unwrap w″ → w ⊩⋆ unwrap w″
trans⊩⋆ ts us = reflectᶜ⋆ (trans⊢⋆ (reifyᶜ⋆ ts) (reifyᶜ⋆ us))
-- Completeness with respect to all models, or quotation.
quot : ∀ {A Γ} → Γ ⊨ A → Γ ⊢ A
quot s = reifyᶜ (s refl⊩⋆)
-- Normalisation by evaluation.
norm : ∀ {A Γ} → Γ ⊢ A → Γ ⊢ A
norm = quot ∘ eval
-- TODO: Correctness of normalisation with respect to conversion.
| 24.942857
| 82
| 0.520809
|
4a9428b6926ffbce01225be45746be59e3ef8cd1
| 1,163
|
agda
|
Agda
|
expn-eval-holds.agda
|
JimFixGroupResearch/imper-ial
|
80d9411b2869614cae488cd4a6272894146c9f3c
|
[
"MIT"
] | null | null | null |
expn-eval-holds.agda
|
JimFixGroupResearch/imper-ial
|
80d9411b2869614cae488cd4a6272894146c9f3c
|
[
"MIT"
] | null | null | null |
expn-eval-holds.agda
|
JimFixGroupResearch/imper-ial
|
80d9411b2869614cae488cd4a6272894146c9f3c
|
[
"MIT"
] | null | null | null |
open import lib
open import eq-reas-nouni
equiv = _≡_
Val = nat
data Expn : Set where
val : Val -> Expn
plus : Expn -> Expn -> Expn
eval : Expn -> Val
eval (val v) = v
eval (plus e1 e2) = (eval e1) + (eval e2)
data evalsTo : Expn -> Val -> Set where
e-val : forall {v : Val}
------------------------
-> (evalsTo (val v) v)
e-add : forall {e1 e2 : Expn}{v1 v2 : Val}
-> (evalsTo e1 v1)
-> (evalsTo e2 v2)
-------------------------------------
-> (evalsTo (plus e1 e2) (v1 + v2))
e-thm-fwd : forall {e : Expn}{v : Val}
-> evalsTo e v -> equiv (eval e) v
e-thm-fwd (e-val{v}) =
begin
eval (val v)
equiv[ refl ]
v
qed
e-thm-fwd (e-add{e1}{e2}{v1}{v2} e1-evalsTo-v1 e2-evalsTo-v2) =
let
eval-e1-is-v1 = e-thm-fwd e1-evalsTo-v1
eval-e2-is-v2 = e-thm-fwd e2-evalsTo-v2
in begin
eval (plus e1 e2)
equiv[ refl ]
(eval e1) + (eval e2)
equiv[ cong2 _+_ eval-e1-is-v1 eval-e2-is-v2 ]
v1 + v2
qed
e-thm-alt : forall (e : Expn) -> evalsTo e (eval e)
e-thm-alt (val v) = e-val
e-thm-alt (plus e1 e2) = (e-add (e-thm-alt e1) (e-thm-alt e2))
| 20.767857
| 63
| 0.509888
|
c546e94dfde065392185a63d743ce078b33bbbdf
| 206
|
agda
|
Agda
|
test/interaction/SplitResult.agda
|
redfish64/autonomic-agda
|
c0ae7d20728b15d7da4efff6ffadae6fe4590016
|
[
"BSD-3-Clause"
] | null | null | null |
test/interaction/SplitResult.agda
|
redfish64/autonomic-agda
|
c0ae7d20728b15d7da4efff6ffadae6fe4590016
|
[
"BSD-3-Clause"
] | null | null | null |
test/interaction/SplitResult.agda
|
redfish64/autonomic-agda
|
c0ae7d20728b15d7da4efff6ffadae6fe4590016
|
[
"BSD-3-Clause"
] | null | null | null |
{-# OPTIONS --copatterns #-}
module SplitResult where
open import Common.Product
test : {A B : Set} (a : A) (b : B) → A × B
test a b = {!!}
testFun : {A B : Set} (a : A) (b : B) → A × B
testFun = {!!}
| 15.846154
| 45
| 0.524272
|
20a415fa9e4af58e38d1d5f0e76f9a478602ce67
| 8,619
|
agda
|
Agda
|
Cubical/Algebra/Ring/QuotientRing.agda
|
FernandoLarrain/cubical
|
9acdecfa6437ec455568be4e5ff04849cc2bc13b
|
[
"MIT"
] | 1
|
2022-03-05T00:29:41.000Z
|
2022-03-05T00:29:41.000Z
|
Cubical/Algebra/Ring/QuotientRing.agda
|
FernandoLarrain/cubical
|
9acdecfa6437ec455568be4e5ff04849cc2bc13b
|
[
"MIT"
] | null | null | null |
Cubical/Algebra/Ring/QuotientRing.agda
|
FernandoLarrain/cubical
|
9acdecfa6437ec455568be4e5ff04849cc2bc13b
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --safe #-}
module Cubical.Algebra.Ring.QuotientRing where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Structure
open import Cubical.Foundations.Powerset using (_∈_; _⊆_) -- \in, \sub=
open import Cubical.HITs.SetQuotients.Base renaming (_/_ to _/ₛ_)
open import Cubical.HITs.SetQuotients.Properties
open import Cubical.Algebra.Ring
open import Cubical.Algebra.Ring.Ideal
open import Cubical.Algebra.Ring.Kernel
private
variable
ℓ : Level
module _ (R' : Ring ℓ) (I : ⟨ R' ⟩ → hProp ℓ) (I-isIdeal : isIdeal R' I) where
open RingStr (snd R')
private R = ⟨ R' ⟩
open isIdeal I-isIdeal
open RingTheory R'
R/I : Type ℓ
R/I = R /ₛ (λ x y → x - y ∈ I)
private
homogeneity : ∀ (x a b : R)
→ (a - b ∈ I)
→ (x + a) - (x + b) ∈ I
homogeneity x a b p = subst (λ u → u ∈ I) (translatedDifference x a b) p
isSetR/I : isSet R/I
isSetR/I = squash/
[_]/I : (a : R) → R/I
[ a ]/I = [ a ]
lemma : (x y a : R)
→ x - y ∈ I
→ [ x + a ]/I ≡ [ y + a ]/I
lemma x y a x-y∈I = eq/ (x + a) (y + a) (subst (λ u → u ∈ I) calculate x-y∈I)
where calculate : x - y ≡ (x + a) - (y + a)
calculate =
x - y ≡⟨ translatedDifference a x y ⟩
((a + x) - (a + y)) ≡⟨ cong (λ u → u - (a + y)) (+Comm _ _) ⟩
((x + a) - (a + y)) ≡⟨ cong (λ u → (x + a) - u) (+Comm _ _) ⟩
((x + a) - (y + a)) ∎
pre-+/I : R → R/I → R/I
pre-+/I x = elim
(λ _ → squash/)
(λ y → [ x + y ])
λ y y' diffrenceInIdeal
→ eq/ (x + y) (x + y') (homogeneity x y y' diffrenceInIdeal)
pre-+/I-DescendsToQuotient : (x y : R) → (x - y ∈ I)
→ pre-+/I x ≡ pre-+/I y
pre-+/I-DescendsToQuotient x y x-y∈I i r = pointwise-equal r i
where
pointwise-equal : ∀ (u : R/I)
→ pre-+/I x u ≡ pre-+/I y u
pointwise-equal = elimProp (λ u → isSetR/I (pre-+/I x u) (pre-+/I y u))
(λ a → lemma x y a x-y∈I)
_+/I_ : R/I → R/I → R/I
x +/I y = (elim R/I→R/I-isSet pre-+/I pre-+/I-DescendsToQuotient x) y
where
R/I→R/I-isSet : R/I → isSet (R/I → R/I)
R/I→R/I-isSet _ = isSetΠ (λ _ → squash/)
+/I-comm : (x y : R/I) → x +/I y ≡ y +/I x
+/I-comm = elimProp2 (λ _ _ → squash/ _ _) eq
where eq : (x y : R) → [ x ] +/I [ y ] ≡ [ y ] +/I [ x ]
eq x y i = [ +Comm x y i ]
+/I-assoc : (x y z : R/I) → x +/I (y +/I z) ≡ (x +/I y) +/I z
+/I-assoc = elimProp3 (λ _ _ _ → squash/ _ _) eq
where eq : (x y z : R) → [ x ] +/I ([ y ] +/I [ z ]) ≡ ([ x ] +/I [ y ]) +/I [ z ]
eq x y z i = [ +Assoc x y z i ]
0/I : R/I
0/I = [ 0r ]
1/I : R/I
1/I = [ 1r ]
-/I : R/I → R/I
-/I = elim (λ _ → squash/) (λ x' → [ - x' ]) eq
where
eq : (x y : R) → (x - y ∈ I) → [ - x ] ≡ [ - y ]
eq x y x-y∈I = eq/ (- x) (- y) (subst (λ u → u ∈ I) eq' (isIdeal.-closed I-isIdeal x-y∈I))
where
eq' = - (x + (- y)) ≡⟨ sym (-Dist _ _) ⟩
(- x) - (- y) ∎
+/I-rinv : (x : R/I) → x +/I (-/I x) ≡ 0/I
+/I-rinv = elimProp (λ x → squash/ _ _) eq
where
eq : (x : R) → [ x ] +/I (-/I [ x ]) ≡ 0/I
eq x i = [ +Rinv x i ]
+/I-rid : (x : R/I) → x +/I 0/I ≡ x
+/I-rid = elimProp (λ x → squash/ _ _) eq
where
eq : (x : R) → [ x ] +/I 0/I ≡ [ x ]
eq x i = [ +Rid x i ]
_·/I_ : R/I → R/I → R/I
_·/I_ =
elim (λ _ → isSetΠ (λ _ → squash/))
(λ x → left· x)
eq'
where
eq : (x y y' : R) → (y - y' ∈ I) → [ x · y ] ≡ [ x · y' ]
eq x y y' y-y'∈I = eq/ _ _
(subst (λ u → u ∈ I)
(x · (y - y') ≡⟨ ·Rdist+ _ _ _ ⟩
((x · y) + x · (- y')) ≡⟨ cong (λ u → (x · y) + u)
(-DistR· x y') ⟩
(x · y) - (x · y') ∎)
(isIdeal.·-closedLeft I-isIdeal x y-y'∈I))
left· : (x : R) → R/I → R/I
left· x = elim (λ y → squash/)
(λ y → [ x · y ])
(eq x)
eq' : (x x' : R) → (x - x' ∈ I) → left· x ≡ left· x'
eq' x x' x-x'∈I i y = elimProp (λ y → squash/ (left· x y) (left· x' y))
(λ y → eq′ y)
y i
where
eq′ : (y : R) → left· x [ y ] ≡ left· x' [ y ]
eq′ y = eq/ (x · y) (x' · y)
(subst (λ u → u ∈ I)
((x - x') · y ≡⟨ ·Ldist+ x (- x') y ⟩
x · y + (- x') · y ≡⟨ cong
(λ u → x · y + u)
(-DistL· x' y) ⟩
x · y - x' · y ∎)
(isIdeal.·-closedRight I-isIdeal y x-x'∈I))
-- more or less copy paste from '+/I' - this is preliminary anyway
·/I-assoc : (x y z : R/I) → x ·/I (y ·/I z) ≡ (x ·/I y) ·/I z
·/I-assoc = elimProp3 (λ _ _ _ → squash/ _ _) eq
where eq : (x y z : R) → [ x ] ·/I ([ y ] ·/I [ z ]) ≡ ([ x ] ·/I [ y ]) ·/I [ z ]
eq x y z i = [ ·Assoc x y z i ]
·/I-lid : (x : R/I) → 1/I ·/I x ≡ x
·/I-lid = elimProp (λ x → squash/ _ _) eq
where
eq : (x : R) → 1/I ·/I [ x ] ≡ [ x ]
eq x i = [ ·Lid x i ]
·/I-rid : (x : R/I) → x ·/I 1/I ≡ x
·/I-rid = elimProp (λ x → squash/ _ _) eq
where
eq : (x : R) → [ x ] ·/I 1/I ≡ [ x ]
eq x i = [ ·Rid x i ]
/I-ldist : (x y z : R/I) → (x +/I y) ·/I z ≡ (x ·/I z) +/I (y ·/I z)
/I-ldist = elimProp3 (λ _ _ _ → squash/ _ _) eq
where
eq : (x y z : R) → ([ x ] +/I [ y ]) ·/I [ z ] ≡ ([ x ] ·/I [ z ]) +/I ([ y ] ·/I [ z ])
eq x y z i = [ ·Ldist+ x y z i ]
/I-rdist : (x y z : R/I) → x ·/I (y +/I z) ≡ (x ·/I y) +/I (x ·/I z)
/I-rdist = elimProp3 (λ _ _ _ → squash/ _ _) eq
where
eq : (x y z : R) → [ x ] ·/I ([ y ] +/I [ z ]) ≡ ([ x ] ·/I [ y ]) +/I ([ x ] ·/I [ z ])
eq x y z i = [ ·Rdist+ x y z i ]
asRing : Ring ℓ
asRing = makeRing 0/I 1/I _+/I_ _·/I_ -/I isSetR/I
+/I-assoc +/I-rid +/I-rinv +/I-comm
·/I-assoc ·/I-rid ·/I-lid /I-rdist /I-ldist
_/_ : (R : Ring ℓ) → (I : IdealsIn R) → Ring ℓ
R / (I , IisIdeal) = asRing R I IisIdeal
[_]/I : {R : Ring ℓ} {I : IdealsIn R} → (a : ⟨ R ⟩) → ⟨ R / I ⟩
[ a ]/I = [ a ]
module UniversalProperty (R : Ring ℓ) (I : IdealsIn R) where
open RingStr ⦃...⦄
open RingTheory ⦃...⦄
Iₛ = fst I
private
instance
_ = R
_ = snd R
module _ {S : Ring ℓ} (φ : RingHom R S) where
open IsRingHom
open RingHomTheory φ
private
instance
_ = S
_ = snd S
f = fst φ
module φ = IsRingHom (snd φ)
inducedHom : Iₛ ⊆ kernel φ → RingHom (R / I) S
fst (inducedHom Iₛ⊆kernel) =
elim
(λ _ → isSetRing S)
f
λ r₁ r₂ r₁-r₂∈I → equalByDifference (f r₁) (f r₂)
(f r₁ - f r₂ ≡⟨ cong (λ u → f r₁ + u) (sym (φ.pres- _)) ⟩
f r₁ + f (- r₂) ≡⟨ sym (φ.pres+ _ _) ⟩
f (r₁ - r₂) ≡⟨ Iₛ⊆kernel (r₁ - r₂) r₁-r₂∈I ⟩
0r ∎)
pres0 (snd (inducedHom Iₛ⊆kernel)) = φ.pres0
pres1 (snd (inducedHom Iₛ⊆kernel)) = φ.pres1
pres+ (snd (inducedHom Iₛ⊆kernel)) =
elimProp2 (λ _ _ → isSetRing S _ _) φ.pres+
pres· (snd (inducedHom Iₛ⊆kernel)) =
elimProp2 (λ _ _ → isSetRing S _ _) φ.pres·
pres- (snd (inducedHom Iₛ⊆kernel)) =
elimProp (λ _ → isSetRing S _ _) φ.pres-
solution : (p : Iₛ ⊆ kernel φ)
→ (x : ⟨ R ⟩) → inducedHom p $ [ x ] ≡ φ $ x
solution p x = refl
unique : (p : Iₛ ⊆ kernel φ)
→ (ψ : RingHom (R / I) S) → (ψIsSolution : (x : ⟨ R ⟩) → ψ $ [ x ] ≡ φ $ x)
→ (x : ⟨ R ⟩) → ψ $ [ x ] ≡ inducedHom p $ [ x ]
unique p ψ ψIsSolution x = ψIsSolution x
| 36.367089
| 98
| 0.37893
|
cba6530c118a0c35d3931962b11b4bb485d87717
| 1,922
|
agda
|
Agda
|
Cubical/Data/Fin/Base.agda
|
limemloh/cubical
|
df4ef7edffd1c1deb3d4ff342c7178e9901c44f1
|
[
"MIT"
] | null | null | null |
Cubical/Data/Fin/Base.agda
|
limemloh/cubical
|
df4ef7edffd1c1deb3d4ff342c7178e9901c44f1
|
[
"MIT"
] | null | null | null |
Cubical/Data/Fin/Base.agda
|
limemloh/cubical
|
df4ef7edffd1c1deb3d4ff342c7178e9901c44f1
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --cubical --safe #-}
module Cubical.Data.Fin.Base where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function
open import Cubical.Foundations.HLevels
open import Cubical.Data.Empty
open import Cubical.Data.Nat
open import Cubical.Data.Nat.Order
open import Cubical.Data.Sum
open import Cubical.Relation.Nullary
-- Finite types.
--
-- Currently it is most convenient to define these as a subtype of the
-- natural numbers, because indexed inductive definitions don't behave
-- well with cubical Agda. This definition also has some more general
-- attractive properties, of course, such as easy conversion back to
-- ℕ.
Fin : ℕ → Type₀
Fin n = Σ[ k ∈ ℕ ] k < n
private
variable
ℓ : Level
k : ℕ
fzero : Fin (suc k)
fzero = (0 , suc-≤-suc zero-≤)
-- It is easy, using this representation, to take the successor of a
-- number as a number in the next largest finite type.
fsuc : Fin k → Fin (suc k)
fsuc (k , l) = (suc k , suc-≤-suc l)
-- Conversion back to ℕ is trivial...
toℕ : Fin k → ℕ
toℕ = fst
-- ... and injective.
toℕ-injective : ∀{fj fk : Fin k} → toℕ fj ≡ toℕ fk → fj ≡ fk
toℕ-injective {fj = fj} {fk} = ΣProp≡ (λ _ → m≤n-isProp)
-- A case analysis helper for induction.
fsplit
: ∀(fj : Fin (suc k))
→ (fzero ≡ fj) ⊎ (Σ[ fk ∈ Fin k ] fsuc fk ≡ fj)
fsplit (0 , k<sn) = inl (toℕ-injective refl)
fsplit (suc k , k<sn) = inr ((k , pred-≤-pred k<sn) , toℕ-injective refl)
-- Fin 0 is empty
¬Fin0 : ¬ Fin 0
¬Fin0 (k , k<0) = ¬-<-zero k<0
-- The full inductive family eliminator for finite types.
finduction
: ∀(P : ∀{k} → Fin k → Type ℓ)
→ (∀{k} → P {suc k} fzero)
→ (∀{k} {fn : Fin k} → P fn → P (fsuc fn))
→ {k : ℕ} → (fn : Fin k) → P fn
finduction P fz fs {zero} = ⊥-elim ∘ ¬Fin0
finduction P fz fs {suc k} fj
= case fsplit fj return (λ _ → P fj) of λ
{ (inl p) → subst P p fz
; (inr (fk , p)) → subst P p (fs (finduction P fz fs fk))
}
| 27.070423
| 73
| 0.641519
|
107e07e2e57ff79db94a5a6328ba00e0b3744f21
| 517
|
agda
|
Agda
|
agda-stdlib/src/Data/Product/N-ary/Properties.agda
|
DreamLinuxer/popl21-artifact
|
fb380f2e67dcb4a94f353dbaec91624fcb5b8933
|
[
"MIT"
] | 5
|
2020-10-07T12:07:53.000Z
|
2020-10-10T21:41:32.000Z
|
agda-stdlib/src/Data/Product/N-ary/Properties.agda
|
DreamLinuxer/popl21-artifact
|
fb380f2e67dcb4a94f353dbaec91624fcb5b8933
|
[
"MIT"
] | null | null | null |
agda-stdlib/src/Data/Product/N-ary/Properties.agda
|
DreamLinuxer/popl21-artifact
|
fb380f2e67dcb4a94f353dbaec91624fcb5b8933
|
[
"MIT"
] | 1
|
2021-11-04T06:54:45.000Z
|
2021-11-04T06:54:45.000Z
|
------------------------------------------------------------------------
-- The Agda standard library
--
-- This module is DEPRECATED. Please use Data.Vec.Recursive.Properties
-- instead.
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Product.N-ary.Properties where
{-# WARNING_ON_IMPORT
"Data.Product.N-ary.Properties was deprecated in v1.1.
Use Data.Vec.Recursive.Properties instead."
#-}
open import Data.Vec.Recursive.Properties public
| 28.722222
| 72
| 0.539652
|
0bb1ccb7e6126acc97946fb3fb28fd9f0b2e715a
| 14,084
|
agda
|
Agda
|
JamesSecondComposite.agda
|
guillaumebrunerie/JamesConstruction
|
89fbc29473d2d1ed1a45c3c0e56288cdcf77050b
|
[
"MIT"
] | 5
|
2016-12-07T04:34:52.000Z
|
2018-11-16T22:10:16.000Z
|
JamesSecondComposite.agda
|
guillaumebrunerie/JamesConstruction
|
89fbc29473d2d1ed1a45c3c0e56288cdcf77050b
|
[
"MIT"
] | null | null | null |
JamesSecondComposite.agda
|
guillaumebrunerie/JamesConstruction
|
89fbc29473d2d1ed1a45c3c0e56288cdcf77050b
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --without-K --rewriting #-}
open import PathInduction
open import Pushout
module JamesSecondComposite {i} (A : Type i) (⋆A : A) where
open import JamesTwoMaps A ⋆A public
-- Second composite
to-ε∞ : to ε∞ == εJ
to-ε∞ = idp
to-α∞-in∞ : (a : A) (n : ℕ) (x : J n) → to (α∞ a (in∞ n x)) == αJ a (inJ n x)
to-α∞-in∞ a n x = inJS-α n a x
to-α∞-push∞ : (a : A) (n : ℕ) (x : J n) → Square (ap to (ap (α∞ a) (push∞ n x))) (inJS-α n a x) idp (ap (αJ a) (ap to (push∞ n x)))
to-α∞-push∞ = λ a n x → & coh (ap-shape-∙! to (push∞-β {in∞* = α∞-in∞ a} {push∞* = α∞-push∞ a} n x))
(to-push∞-S n (α n a x))
(ap-δJ (inJS-α n a x))
(∘-ap _ _ _)
(inJSS-γ n a x)
(ap-square (αJ a) (to-push∞ n x)) module Toα∞Push∞ where
coh : Coh ({A : Type i} {a c : A} {q : a == c} {d : A} {r : a == d}
{b : A} {s : b == d} {p : a == b} (p= : p == r ∙ ! s)
{r' : a == d} (y= : r == r') {g : A} {w : d == g}
{v : c == g} (sq : Square r' q w v) {e : A} {u : c == e}
{s' : b == d} (s= : s == s') {t : b == e} (u= : Square s' t w (! u ∙ v))
{x : c == b} (x= : Square x idp t u)
→ Square p q idp x)
coh = path-induction
to-α∞ : (a : A) (x : J∞A) → to (α∞ a x) == αJ a (to x)
to-α∞ a = J∞A-elim (to-α∞-in∞ a) (λ n x → ↓-='-from-square (ap-∘ to (α∞ a) (push∞ n x)) (ap-∘ (αJ a) to (push∞ n x)) (square-symmetry (to-α∞-push∞ a n x)))
to-δ∞-in∞ : (n : ℕ) (x : J n) → Square (ap to (δ∞ (in∞ n x))) idp (to-α∞ ⋆A (in∞ n x)) (δJ (to (in∞ n x)))
to-δ∞-in∞ = λ n x → & coh (∘-ap _ _ _) (ap-square to (δ∞-in∞-β n x)) (to-push∞ n x) (inJS-β n x) module Toδ∞In where
coh : Coh ({A : Type i} {a b : A} {p : a == b} {c : A} {r r' : b == c} (r= : r == r')
{s : a == c} (sq1 : Square p idp r s) {d : A} {t : c == d}
{u : a == d} (sq2 : Square s idp t u)
{rt : b == d} (rt= : Square r' rt t idp)
→ Square p idp rt u)
coh = path-induction
module _ (n : ℕ) (x : J n) where
side1 : Square (ap to (push∞ (S n) (α n ⋆A x))) (inJS-α n ⋆A x) (ap (αJ ⋆A) (inJS-α n ⋆A x)) (δJ (αJ ⋆A (inJ n x)))
side1 = to-push∞-S n (α n ⋆A x) |∙ idp |∙ ap-δJ (inJS-α n ⋆A x)
side2 : Square (ap to (ap (in∞ (S n)) (β n x))) (inJS-α n ⋆A x) (inJS-ι n x) idp
side2 = idp |∙ ∘-ap to (in∞ (S n)) (β n x) |∙ inJS-β n x
side3 : Square (ap to (ap (in∞ (S (S n))) (ap inr (β n x)))) (ap (αJ ⋆A) (inJS-α n ⋆A x)) (ap (αJ ⋆A) (inJS-ι n x)) (ap (αJ ⋆A) idp)
side3 = idp |∙ (∘-ap to (in∞ (S (S n))) _ ∙ ap-inJSS-ι n (β n x)) |∙ ap-square (αJ ⋆A) (inJS-β n x)
side4 : Square (ap to (push∞ (S n) (ι n x))) (inJS-ι n x) (ap (λ z → αJ ⋆A z) (inJS-ι n x)) (δJ (αJ ⋆A (inJ n x)))
side4 = to-push∞-S n (ι n x) |∙ idp |∙ ap-δJ (inJS-ι n x)
side5 : Square (ap to (ap (in∞ (S (S n))) (γ n ⋆A x))) (ap (αJ ⋆A) (inJS-ι n x)) (ap (αJ ⋆A) (inJS-α n ⋆A x)) (γJ ⋆A (inJ n x))
side5 = ∘-ap to (in∞ (S (S n))) (γ n ⋆A x) |∙ inJSS-γ n ⋆A x
side6 : Square (ap to (ap (in∞ (S (S n))) (β (S n) (ι n x)))) (ap (αJ ⋆A) (inJS-ι n x)) (ap (αJ ⋆A) (inJS-ι n x)) idp
side6 = ∘-ap to (in∞ (S (S n))) _ |∙ inJS-βS n (ι n x) |∙ vid-square
side7 : Square (ap to (ap (α∞ ⋆A) (push∞ n x))) (inJS-α n ⋆A x) (ap (αJ ⋆A) (inJS-ι n x)) (ap (αJ ⋆A) (δJ (inJ n x)))
side7 = adapt-square (to-α∞-push∞ ⋆A n x ∙□ ap-square (αJ ⋆A) (to-push∞ n x)) ∙idp idp∙
to-push∞-βn : Cube (ap-square to (natural-square (push∞ (S n)) (β n x) idp (ap-∘ (in∞ (S (S n))) inr (β n x))))
hid-square
side1
side2
side3
side4
to-push∞-βn = & (ap-square-natural-square to) (β n x) (push∞ (S n)) ∙idp idp
-∙³ & natural-square-homotopy (to-push∞-S n) (β n x)
|∙³ & (natural-square-∘ (β n x) (inJS n)) δJ idp idp
-∙³ & natural-square= δJ idp (coh1 to (in∞ (S n)) (β n x)) (coh2 to (in∞ (S (S n))) inr (β n x) (ap-∘ (αJ ⋆A) (inJS n) (β n x)))
|∙³ & natural-cube2 δJ (inJS-β n x) (ap-square-idf (inJS-β n x)) (hid-flatcube _) where
coh1 : {A B C : Type i} (g : B → C) (f : A → B) {x y : A} (p : x == y) → Square idp (∘-ap (λ z → z) (g ∘ f) p ∙ ap-∘ g f p) (ap-idf _) (∘-ap g f p)
coh1 g f idp = ids
coh2 : {A B C D : Type i} (h : C → D) (g : B → C) (f : A → B) {x y : A} (p : x == y) {z : h (g (f x)) == h (g (f y))} (q : ap (h ∘ g ∘ f) p == z)
→ Square idp (! q ∙ ap-∘ h (g ∘ f) p ∙ ap (ap h) (ap-∘ g f p)) idp (∘-ap h g (ap f p) ∙ ∘-ap (h ∘ g) f p ∙ q)
coh2 h g f idp idp = ids
to-in∞-η : Cube (ap-square to (ap-square (in∞ (S (S n))) (η n x)))
(horiz-degen-square (ηJ (inJ n x)))
side5
vid-square
side3
side6
to-in∞-η = adapt-cube-idp (∘-ap-square to (in∞ (S (S n))) (η n x)
|∙³ inJSS-η n x)
idp (& coh1) (& coh2) idp where
coh1 : Coh ({A : Type i} {a b : A} {p : a == b} → idp |∙ vid-square {p = p} == vid-square)
coh1 = path-induction
coh2 : Coh ({A : Type i} {a b c d : A} {p : a == b} {q : a == c} {r : b == d} {s : c == d} {sq : Square p q r s}
{p' : a == b} {p= : p' == p} {p'' : a == b} {p=' : p'' == p'} → p=' |∙ p= |∙ sq == idp |∙ (p=' ∙ p=) |∙ sq)
coh2 = path-induction
coh-last-one : Coh ({A : Type i} {a b : A} {p : a == b} {c : A} {q : a == c} → p == q ∙ ! (! p ∙ q))
coh-last-one = path-induction
last-one : Cube (horiz-degen-square (ap-shape-∙! to (push∞-β {in∞* = α∞-in∞ ⋆A} {push∞* = α∞-push∞ ⋆A} n x)))
(horiz-degen-square (& coh-last-one))
side7
vid-square
vid-square
(vcomp! side1 side5)
last-one = & coh where
coh : Coh ({A : Type i} {a : A}
{c : A} {q : a == c} {d : A} {r : a == d} {b : A} {s : b == d}
{p : a == b} {p= : p == r ∙ ! s}
{r' : a == d} {y= : r == r'}
{g : A} {w : d == g} {v : c == g} {sq : Square r' q w v}
{e : A} {u : c == e} {s' : b == d} {s= : s == s'}
{t : b == e} {u= : Square s' t w (! u ∙ v)}
{x : c == b} {x= : Square x idp t u}
→ Cube (horiz-degen-square p=) (horiz-degen-square (& coh-last-one)) (adapt-square (& Toα∞Push∞.coh p= y= sq s= u= x= ∙□ x=) ∙idp idp∙) vid-square vid-square (vcomp! (y= |∙ idp |∙ sq) (s= |∙ u=)))
coh = path-induction
δ∞Push∞Coh-coh : Coh ({A : Type i} {a b : A} {p : a == b} {d : A} {s : d == b}
{c : A} {r : b == c} {f : A} {u : f == c} {e : A} {t : e == c}
{w : f == e} {eta : Square w idp t u}
{v : d == e} {nat : Square v s t r}
{vw : d == f} {vw-eq : vw == v ∙ ! w}
{a' : _} {a= : a == a'}
{b' : _} {b= : b == b'}
{p' : _} {p= : Square p a= b= p'}
{d' : _} {d= : d == d'}
{s' : _} {s= : Square s d= b= s'}
{c' : _} {c= : c == c'}
{r' : _} {r= : Square r b= c= r'}
{f' : _} {f= : f == f'}
{u' : _} {u= : Square u f= c= u'}
{e' : _} {e= : e == e'}
{t' : _} {t= : Square t e= c= t'}
{w' : _} {w= : Square w f= e= w'}
{eta' : _} (eta= : Cube eta eta' w= vid-square t= u=)
{v' : _} {v= : Square v d= e= v'}
{nat' : _} (nat= : Cube nat nat' v= s= t= r=)
{vw' : _} {vw= : Square vw d= f= vw'}
{vw-eq' : _} (vw-eq= : Cube (horiz-degen-square vw-eq) (horiz-degen-square vw-eq') vw= vid-square vid-square (vcomp! v= w=))
→ Cube (& δ∞Push∞.coh p eta nat vw-eq) (& δ∞Push∞.coh p' eta' nat' vw-eq') (vcomp! p= s=) p= vw= (vcomp! r= u=))
δ∞Push∞Coh-coh = path-induction
piece1 : FlatCube (ap-square to (& δ∞Push∞.coh (push∞ n x) (ap-square (in∞ (S (S n))) (η n x))
(natural-square (push∞ (S n)) (β n x) idp (ap-∘ _ _ _))
(push∞-β {in∞* = α∞-in∞ ⋆A} {push∞* = α∞-push∞ ⋆A} n x)))
(& δ∞Push∞.coh (ap to (push∞ n x))
(ap-square to (ap-square (in∞ (S (S n))) (η n x)))
(ap-square to
(natural-square (push∞ (S n)) (β n x) idp
(ap-∘ (in∞ (S (S n))) (ι (S n)) (β n x))))
(ap-shape-∙! to (push∞-β {in∞* = α∞-in∞ ⋆A} {push∞* = α∞-push∞ ⋆A} n x)))
(ap-∙! _ _ _) idp idp (ap-∙! _ _ _)
piece1 = & (δ∞Push∞.ap-coh to) {b = in∞ (S n) (ι n x)} {d = in∞ (S n) (α n ⋆A x)}
piece2 : Cube (& δ∞Push∞.coh (ap to (push∞ n x))
(ap-square to (ap-square (in∞ (S (S n))) (η n x)))
(ap-square to (natural-square (push∞ (S n)) (β n x) idp (ap-∘ (in∞ (S (S n))) inr (β n x))))
(ap-shape-∙! to (push∞-β {in∞* = α∞-in∞ ⋆A} {push∞* = α∞-push∞ ⋆A} n x)))
(& δ∞Push∞.coh (δJ (inJ n x)) (horiz-degen-square (ηJ (inJ n x))) hid-square (& coh-last-one))
(vcomp! (to-push∞ n x) side2)
(to-push∞ n x)
side7
(vcomp! side4 side6)
piece2 = & δ∞Push∞Coh-coh to-in∞-η to-push∞-βn last-one
piece3 : FlatCube (& δ∞Push∞.coh (δJ (inJ n x)) (horiz-degen-square (ηJ (inJ n x))) hid-square (& coh-last-one))
(ap-δJ (δJ (inJ n x)))
∙idp idp idp ∙idp
piece3 = & coh where
coh : Coh ({A : Type i} {a b : A} {p : a == b} {c : A} {q q' : b == c} {sq : Square p p q q'}
→ FlatCube (& δ∞Push∞.coh p (horiz-degen-square (& (ηIfy.coh αJ δJ) sq)) hid-square (& coh-last-one)) sq ∙idp idp idp ∙idp)
coh = path-induction
piece4 : Cube (ap-δJ (ap to (push∞ n x)))
(ap-δJ (δJ (inJ n x)))
hid-square (to-push∞ n x) (ap-square (αJ ⋆A) (to-push∞ n x)) (ap-δJ (inJS-ι n x))
piece4 = & natural-cube2 δJ (to-push∞ n x) (ap-square-idf _) (hid-flatcube _)
to-δ∞-push∞ : Cube (ap-square to (natural-square δ∞ (push∞ n x) (ap-idf _) idp))
(ap-δJ (ap to (push∞ n x)))
(to-δ∞-in∞ n x)
hid-square
(to-α∞-push∞ ⋆A n x)
(to-δ∞-in∞ (S n) (ι n x))
to-δ∞-push∞ = adapt-cube (ap (ap-square to) (natural-square-β δ∞ (push∞ n x) (push∞-βd n x))
-∙³ piece1
|∙³ piece2
∙³x piece3
|∙³ !³ piece4)
idp ∙idp !-inv-r !-inv-r
(& coh1 (& (coh-flat to) {b = in∞ (S n) (α n ⋆A x)})) (& coh2) (& coh3) (& coh4 (& (coh-flat to) {a = in∞ (S n) (ι n x)})) where
flat : Coh ({A : Type i} {a c : A} {s : a == c} {b : A} {r : b == c} {p : a == b} (sq : Square p idp r s) → p == s ∙ ! r)
flat = path-induction
coh-flat : {A B : Type i} (f : A → B) → Coh ({a c : A} {s : a == c} {b : A} {r : b == c} → & flat (ap-square f (& δ∞Inβ.coh)) == ap-∙! f s r)
coh-flat f = path-induction
coh1 : Coh ({A : Type i} {a b : A} {p : a == b} {c : A} {r r' : b == c} {r= : r == r'}
{s : a == c} {sq1 : Square p idp r s} {d : A} {t : c == d}
{u : a == d} {sq2 : Square s idp t u}
{rt : b == d} {sq3 : Square r' rt t idp}
{p= : p == s ∙ ! r} (p== : & flat sq1 == p=)
→ adapt-square (p= |∙ vcomp! sq2 (idp |∙ r= |∙ sq3) ∙□ ∙idp |∙ !² hid-square) idp ∙idp == & Toδ∞In.coh r= sq1 sq2 sq3)
coh1 = path-induction
coh2 : Coh ({A : Type i} {a b : A} {p : a == b} {c : A} {r : b == c} {s : a == c} {sq : Square p idp r s}
→ adapt-square (idp |∙ sq ∙□ idp |∙ !² sq) idp !-inv-r == hid-square)
coh2 = path-induction
coh3 : Coh ({A : Type i} {a b : A} {p : a == b} {c : A} {q : a == c} {s : c == b} {sq1 : Square p q idp s} {d : A} {t : b == d} {u : c == d} {sq2 : Square s idp t u}
→ adapt-square (idp |∙ adapt-square (sq1 ∙□ sq2) ∙idp idp∙ ∙□ idp |∙ !² sq2) ∙idp !-inv-r == sq1)
coh3 = path-induction
coh4 : Coh ({A : Type i} {a b c d : A} {p : a == b} {q : a == c} {r : b == d} {s : c == d} {sq : Square p q r s}
{p' : a == b} {p=□ : Square p' idp idp p} {t : b == b} {t=□ : t == idp} {t' : b == b} {t=' : t' == t}
{pt' : a == b} {pt=2 : Square pt' idp t' p'} {pt= : pt' == p' ∙ ! t'} (pt== : & flat pt=2 == pt=)
→ adapt-square (pt= |∙ vcomp! (horiz-degen-path p=□ |∙ idp |∙ sq) (t=' |∙ t=□ |∙ vid-square) ∙□ ∙idp |∙ !² sq) !-inv-r !-inv-r
== & Toδ∞In.coh t=' pt=2 p=□ (horiz-degen-square t=□))
coh4 = path-induction
natural-square-idp-symm : {A B : Type i} {f : A → B} {x y : A} {p : x == y} → natural-square (λ a → idp {a = f a}) p idp idp == square-symmetry hid-square
natural-square-idp-symm {p = idp} = idp
to-δ∞ : (x : J∞A) → Square (ap to (δ∞ x)) idp (to-α∞ ⋆A x) (δJ (to x))
to-δ∞ = J∞A-elim to-δ∞-in∞ (λ n x → cube-to-↓-path idp (ap-∘ _ _ _) idp (ap-∘ _ _ _)
(adapt-cube-idp (cube-rotate (to-δ∞-push∞ n x))
(& (ap-square-natural-square to) (push∞ n x) δ∞ coh ∙idp)
(! natural-square-idp-symm)
(! (natural-square-β (to-α∞ ⋆A) (push∞ n x) (push∞-βd n x)))
(! (& (natural-square-∘ (push∞ n x) to) δJ coh2 (!-inv-l))))) where
coh : {A B : Type i} {f : A → B} {x y : A} {p : x == y} → ap-∘ f (λ z → z) p ∙ ap (ap f) (ap-idf p) == idp
coh {p = idp} = idp
coh2 : {A B : Type i} {f : A → B} {x y : A} {p : x == y} → ∘-ap (λ z → z) f p ∙ idp == ap-idf (ap f p)
coh2 {p = idp} = idp
to-from : (x : JA) → to (from x) == x
to-from = JA-elim to-ε∞
(λ a x y → to-α∞ a (from x) ∙ ap (αJ a) y)
(λ x y → ↓-='-from-square (ap-∘ to from (δJ x) ∙ ap (ap to) (δJ-β x)) (ap-idf (δJ x))
(square-symmetry (adapt-square (to-δ∞ (from x) ∙□ ap-δJ y) idp∙ idp)))
| 54.378378
| 211
| 0.403295
|
0ebb5769b22605b000fb8613c7dd9932bb11c827
| 546
|
agda
|
Agda
|
Monads/MonadMorphs.agda
|
jmchapman/Relative-Monads
|
74707d3538bf494f4bd30263d2f5515a84733865
|
[
"MIT"
] | 21
|
2015-07-30T01:25:12.000Z
|
2021-02-13T18:02:18.000Z
|
Monads/MonadMorphs.agda
|
jmchapman/Relative-Monads
|
74707d3538bf494f4bd30263d2f5515a84733865
|
[
"MIT"
] | 3
|
2019-01-13T13:12:33.000Z
|
2019-05-29T09:50:26.000Z
|
Monads/MonadMorphs.agda
|
jmchapman/Relative-Monads
|
74707d3538bf494f4bd30263d2f5515a84733865
|
[
"MIT"
] | 1
|
2019-11-04T21:33:13.000Z
|
2019-11-04T21:33:13.000Z
|
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})
| 26
| 75
| 0.521978
|
0b941fcf7f494f2277f5c7621d8d6913cb9fd426
| 1,211
|
agda
|
Agda
|
archive/agda-2/Oscar/Data/Vec/Properties.agda
|
m0davis/oscar
|
52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb
|
[
"RSA-MD"
] | null | null | null |
archive/agda-2/Oscar/Data/Vec/Properties.agda
|
m0davis/oscar
|
52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb
|
[
"RSA-MD"
] | 1
|
2019-04-29T00:35:04.000Z
|
2019-05-11T23:33:04.000Z
|
archive/agda-2/Oscar/Data/Vec/Properties.agda
|
m0davis/oscar
|
52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb
|
[
"RSA-MD"
] | null | null | null |
module Oscar.Data.Vec.Properties where
open import Oscar.Data.Vec
open import Data.Vec.Properties public
open import Data.Nat
open import Data.Product renaming (map to mapP)
open import Relation.Binary.PropositionalEquality
open import Data.Fin
map-∈ : ∀ {a b} {A : Set a} {B : Set b} {y : B} {f : A → B} {n} {xs : Vec A n} → y ∈ map f xs → ∃ λ x → f x ≡ y
map-∈ {xs = []} ()
map-∈ {xs = x ∷ xs} here = x , refl
map-∈ {xs = x ∷ xs} (there y∈mapfxs) = map-∈ y∈mapfxs
∈-map₂ : ∀ {a b} {A : Set a} {B : Set b} {m n : ℕ}
→ ∀ {c} {F : Set c} (f : A → B → F)
{xs : Vec A m} {ys : Vec B n}
{x y} → x ∈ xs → y ∈ ys → (f x y) ∈ map₂ f xs ys
∈-map₂ f {xs = x ∷ xs} {ys} here y∈ys =
∈-++ₗ (∈-map (f x) y∈ys)
∈-map₂ f {xs = x ∷ xs} {ys} (there x∈xs) y∈ys =
∈-++ᵣ (map (f x) ys) (∈-map₂ f x∈xs y∈ys)
lookup-delete-thin : ∀ {a n} {A : Set a} (x : Fin (suc n)) (y : Fin n) (v : Vec A (suc n)) →
lookup y (delete x v) ≡ lookup (thin x y) v
lookup-delete-thin zero zero (_ ∷ _) = refl
lookup-delete-thin zero (suc _) (_ ∷ _) = refl
lookup-delete-thin (suc _) zero (_ ∷ _) = refl
lookup-delete-thin (suc x) (suc y) (_ ∷ v) = lookup-delete-thin x y v
| 36.69697
| 111
| 0.526837
|
2062ea9432d568e12b3e55ba10995a1d3e529eee
| 415
|
agda
|
Agda
|
test/Succeed/Issue3640.agda
|
mdimjasevic/agda
|
8fb548356b275c7a1e79b768b64511ae937c738b
|
[
"BSD-3-Clause"
] | 1,989
|
2015-01-09T23:51:16.000Z
|
2022-03-30T18:20:48.000Z
|
test/Succeed/Issue3640.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 4,066
|
2015-01-10T11:24:51.000Z
|
2022-03-31T21:14:49.000Z
|
test/Succeed/Issue3640.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 371
|
2015-01-03T14:04:08.000Z
|
2022-03-30T19:00:30.000Z
|
-- Andreas, 2019-03-25, issue #3640, reported by gallais
{-# OPTIONS --sized-types #-}
-- {-# OPTIONS -v tc.polarity:40 #-}
module _ where
open import Agda.Builtin.Size
module M (_ : Set) where
data U : Size → Set where
node : ∀ {i} → U (↑ i)
module L (A B : Set) where
open M A
-- WAS: crash because of number of parameters in size-index checki
-- of L.U was wrongly calculated.
-- Should succeed.
| 18.863636
| 66
| 0.643373
|
50d81505f942f8d033657f1f1f5fdc6aebd78c4e
| 366
|
agda
|
Agda
|
test/interaction/Issue535.agda
|
pthariensflame/agda
|
222c4c64b2ccf8e0fc2498492731c15e8fef32d4
|
[
"BSD-3-Clause"
] | 3
|
2015-03-28T14:51:03.000Z
|
2015-12-07T20:14:00.000Z
|
test/interaction/Issue535.agda
|
Blaisorblade/Agda
|
802a28aa8374f15fe9d011ceb80317fdb1ec0949
|
[
"BSD-3-Clause"
] | null | null | null |
test/interaction/Issue535.agda
|
Blaisorblade/Agda
|
802a28aa8374f15fe9d011ceb80317fdb1ec0949
|
[
"BSD-3-Clause"
] | 1
|
2019-03-05T20:02:38.000Z
|
2019-03-05T20:02:38.000Z
|
module Issue535 where
data Nat : Set where
zero : Nat
suc : Nat → Nat
data Vec A : Nat → Set where
[] : Vec A zero
_∷_ : ∀ {n} → A → Vec A n → Vec A (suc n)
replicate : ∀ {A n} → A → Vec A n
replicate {n = n} x = {!n!}
replicate′ : ∀ {n A} → A → Vec A n
replicate′ {n} x = {!n!}
extlam : Nat → {n m : Nat} → Vec Nat n
extlam = λ { x {m = m} → {!m!} }
| 18.3
| 43
| 0.505464
|
10377696692f89008752fa02c3aff712c836de87
| 7,992
|
agda
|
Agda
|
Cubical/Functions/FunExtEquiv.agda
|
Edlyr/cubical
|
5de11df25b79ee49d5c084fbbe6dfc66e4147a2e
|
[
"MIT"
] | null | null | null |
Cubical/Functions/FunExtEquiv.agda
|
Edlyr/cubical
|
5de11df25b79ee49d5c084fbbe6dfc66e4147a2e
|
[
"MIT"
] | null | null | null |
Cubical/Functions/FunExtEquiv.agda
|
Edlyr/cubical
|
5de11df25b79ee49d5c084fbbe6dfc66e4147a2e
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Functions.FunExtEquiv where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.CartesianKanOps
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Function
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Univalence
open import Cubical.Data.Vec
open import Cubical.Data.Nat
open import Cubical.Reflection.StrictEquiv
private
variable
ℓ ℓ₁ ℓ₂ ℓ₃ : Level
-- Function extensionality is an equivalence
module _ {A : Type ℓ} {B : A → I → Type ℓ₁}
{f : (x : A) → B x i0} {g : (x : A) → B x i1} where
funExtEquiv : (∀ x → PathP (B x) (f x) (g x)) ≃ PathP (λ i → ∀ x → B x i) f g
unquoteDef funExtEquiv = defStrictEquiv funExtEquiv funExt funExt⁻
funExtPath : (∀ x → PathP (B x) (f x) (g x)) ≡ PathP (λ i → ∀ x → B x i) f g
funExtPath = ua funExtEquiv
funExtIso : Iso (∀ x → PathP (B x) (f x) (g x)) (PathP (λ i → ∀ x → B x i) f g)
funExtIso = iso funExt funExt⁻ (λ x → refl {x = x}) (λ x → refl {x = x})
-- Function extensionality for binary functions
funExt₂ : {A : Type ℓ} {B : A → Type ℓ₁} {C : (x : A) → B x → I → Type ℓ₂}
{f : (x : A) → (y : B x) → C x y i0}
{g : (x : A) → (y : B x) → C x y i1}
→ ((x : A) (y : B x) → PathP (C x y) (f x y) (g x y))
→ PathP (λ i → ∀ x y → C x y i) f g
funExt₂ p i x y = p x y i
-- Function extensionality for binary functions is an equivalence
module _ {A : Type ℓ} {B : A → Type ℓ₁} {C : (x : A) → B x → I → Type ℓ₂}
{f : (x : A) → (y : B x) → C x y i0}
{g : (x : A) → (y : B x) → C x y i1} where
private
appl₂ : PathP (λ i → ∀ x y → C x y i) f g → ∀ x y → PathP (C x y) (f x y) (g x y)
appl₂ eq x y i = eq i x y
funExt₂Equiv : (∀ x y → PathP (C x y) (f x y) (g x y)) ≃ (PathP (λ i → ∀ x y → C x y i) f g)
unquoteDef funExt₂Equiv = defStrictEquiv funExt₂Equiv funExt₂ appl₂
funExt₂Path : (∀ x y → PathP (C x y) (f x y) (g x y)) ≡ (PathP (λ i → ∀ x y → C x y i) f g)
funExt₂Path = ua funExt₂Equiv
-- Function extensionality for ternary functions
funExt₃ : {A : Type ℓ} {B : A → Type ℓ₁} {C : (x : A) → B x → Type ℓ₂}
{D : (x : A) → (y : B x) → C x y → I → Type ℓ₃}
{f : (x : A) → (y : B x) → (z : C x y) → D x y z i0}
{g : (x : A) → (y : B x) → (z : C x y) → D x y z i1}
→ ((x : A) (y : B x) (z : C x y) → PathP (D x y z) (f x y z) (g x y z))
→ PathP (λ i → ∀ x y z → D x y z i) f g
funExt₃ p i x y z = p x y z i
-- Function extensionality for ternary functions is an equivalence
module _ {A : Type ℓ} {B : A → Type ℓ₁} {C : (x : A) → B x → Type ℓ₂}
{D : (x : A) → (y : B x) → C x y → I → Type ℓ₃}
{f : (x : A) → (y : B x) → (z : C x y) → D x y z i0}
{g : (x : A) → (y : B x) → (z : C x y) → D x y z i1} where
private
appl₃ : PathP (λ i → ∀ x y z → D x y z i) f g → ∀ x y z → PathP (D x y z) (f x y z) (g x y z)
appl₃ eq x y z i = eq i x y z
funExt₃Equiv : (∀ x y z → PathP (D x y z) (f x y z) (g x y z)) ≃ (PathP (λ i → ∀ x y z → D x y z i) f g)
unquoteDef funExt₃Equiv = defStrictEquiv funExt₃Equiv funExt₃ appl₃
funExt₃Path : (∀ x y z → PathP (D x y z) (f x y z) (g x y z)) ≡ (PathP (λ i → ∀ x y z → D x y z i) f g)
funExt₃Path = ua funExt₃Equiv
-- n-ary non-dependent funext
nAryFunExt : {X : Type ℓ} {Y : I → Type ℓ₁} (n : ℕ) (fX : nAryOp n X (Y i0)) (fY : nAryOp n X (Y i1))
→ ((xs : Vec X n) → PathP Y (fX $ⁿ xs) (fY $ⁿ map (λ x → x) xs))
→ PathP (λ i → nAryOp n X (Y i)) fX fY
nAryFunExt zero fX fY p = p []
nAryFunExt (suc n) fX fY p i x = nAryFunExt n (fX x) (fY x) (λ xs → p (x ∷ xs)) i
-- n-ary funext⁻
nAryFunExt⁻ : (n : ℕ) {X : Type ℓ} {Y : I → Type ℓ₁} (fX : nAryOp n X (Y i0)) (fY : nAryOp n X (Y i1))
→ PathP (λ i → nAryOp n X (Y i)) fX fY
→ ((xs : Vec X n) → PathP Y (fX $ⁿ xs) (fY $ⁿ map (λ x → x) xs))
nAryFunExt⁻ zero fX fY p [] = p
nAryFunExt⁻ (suc n) fX fY p (x ∷ xs) = nAryFunExt⁻ n (fX x) (fY x) (λ i → p i x) xs
nAryFunExtEquiv : (n : ℕ) {X : Type ℓ} {Y : I → Type ℓ₁} (fX : nAryOp n X (Y i0)) (fY : nAryOp n X (Y i1))
→ ((xs : Vec X n) → PathP Y (fX $ⁿ xs) (fY $ⁿ map (λ x → x) xs)) ≃ PathP (λ i → nAryOp n X (Y i)) fX fY
nAryFunExtEquiv n {X} {Y} fX fY = isoToEquiv (iso (nAryFunExt n fX fY) (nAryFunExt⁻ n fX fY)
(linv n fX fY) (rinv n fX fY))
where
linv : (n : ℕ) (fX : nAryOp n X (Y i0)) (fY : nAryOp n X (Y i1))
(p : PathP (λ i → nAryOp n X (Y i)) fX fY)
→ nAryFunExt n fX fY (nAryFunExt⁻ n fX fY p) ≡ p
linv zero fX fY p = refl
linv (suc n) fX fY p i j x = linv n (fX x) (fY x) (λ k → p k x) i j
rinv : (n : ℕ) (fX : nAryOp n X (Y i0)) (fY : nAryOp n X (Y i1))
(p : (xs : Vec X n) → PathP Y (fX $ⁿ xs) (fY $ⁿ map (λ x → x) xs))
→ nAryFunExt⁻ n fX fY (nAryFunExt n fX fY p) ≡ p
rinv zero fX fY p i [] = p []
rinv (suc n) fX fY p i (x ∷ xs) = rinv n (fX x) (fY x) (λ ys i → p (x ∷ ys) i) i xs
-- Funext when the domain also depends on the interval
funExtDep : {A : I → Type ℓ} {B : (i : I) → A i → Type ℓ₁}
{f : (x : A i0) → B i0 x} {g : (x : A i1) → B i1 x}
→ ({x₀ : A i0} {x₁ : A i1} (p : PathP A x₀ x₁) → PathP (λ i → B i (p i)) (f x₀) (g x₁))
→ PathP (λ i → (x : A i) → B i x) f g
funExtDep {A = A} {B} {f} {g} h i x =
comp
(λ k → B i (coei→i A i x k))
(λ k → λ
{ (i = i0) → f (coei→i A i0 x k)
; (i = i1) → g (coei→i A i1 x k)
})
(h (λ j → coei→j A i j x) i)
funExtDep⁻ : {A : I → Type ℓ} {B : (i : I) → A i → Type ℓ₁}
{f : (x : A i0) → B i0 x} {g : (x : A i1) → B i1 x}
→ PathP (λ i → (x : A i) → B i x) f g
→ ({x₀ : A i0} {x₁ : A i1} (p : PathP A x₀ x₁) → PathP (λ i → B i (p i)) (f x₀) (g x₁))
funExtDep⁻ q p i = q i (p i)
funExtDepEquiv : {A : I → Type ℓ} {B : (i : I) → A i → Type ℓ₁}
{f : (x : A i0) → B i0 x} {g : (x : A i1) → B i1 x}
→ ({x₀ : A i0} {x₁ : A i1} (p : PathP A x₀ x₁) → PathP (λ i → B i (p i)) (f x₀) (g x₁))
≃ PathP (λ i → (x : A i) → B i x) f g
funExtDepEquiv {A = A} {B} {f} {g} = isoToEquiv isom
where
open Iso
isom : Iso _ _
isom .fun = funExtDep
isom .inv = funExtDep⁻
isom .rightInv q m i x =
comp
(λ k → B i (coei→i A i x (k ∨ m)))
(λ k → λ
{ (i = i0) → f (coei→i A i0 x (k ∨ m))
; (i = i1) → g (coei→i A i1 x (k ∨ m))
; (m = i1) → q i x
})
(q i (coei→i A i x m))
isom .leftInv h m p i =
comp
(λ k → B i (lemi→i m k))
(λ k → λ
{ (i = i0) → f (lemi→i m k)
; (i = i1) → g (lemi→i m k)
; (m = i1) → h p i
})
(h (λ j → lemi→j j m) i)
where
lemi→j : ∀ j → coei→j A i j (p i) ≡ p j
lemi→j j =
coei→j (λ k → coei→j A i k (p i) ≡ p k) i j (coei→i A i (p i))
lemi→i : PathP (λ m → lemi→j i m ≡ p i) (coei→i A i (p i)) refl
lemi→i =
sym (coei→i (λ k → coei→j A i k (p i) ≡ p k) i (coei→i A i (p i)))
◁ λ m k → lemi→j i (m ∨ k)
heteroHomotopy≃Homotopy : {A : I → Type ℓ} {B : (i : I) → Type ℓ₁}
{f : A i0 → B i0} {g : A i1 → B i1}
→ ({x₀ : A i0} {x₁ : A i1} → PathP A x₀ x₁ → PathP B (f x₀) (g x₁))
≃ ((x₀ : A i0) → PathP B (f x₀) (g (transport (λ i → A i) x₀)))
heteroHomotopy≃Homotopy {A = A} {B} {f} {g} = isoToEquiv isom
where
open Iso
isom : Iso _ _
isom .fun h x₀ = h (isContrSinglP A x₀ .fst .snd)
isom .inv k {x₀} {x₁} p =
subst (λ fib → PathP B (f x₀) (g (fib .fst))) (isContrSinglP A x₀ .snd (x₁ , p)) (k x₀)
isom .rightInv k = funExt λ x₀ →
cong (λ α → subst (λ fib → PathP B (f x₀) (g (fib .fst))) α (k x₀))
(isProp→isSet isPropSinglP (isContrSinglP A x₀ .fst) _
(isContrSinglP A x₀ .snd (isContrSinglP A x₀ .fst))
refl)
∙ transportRefl (k x₀)
isom .leftInv h j {x₀} {x₁} p =
transp
(λ i → PathP B (f x₀) (g (isContrSinglP A x₀ .snd (x₁ , p) (i ∨ j) .fst)))
j
(h (isContrSinglP A x₀ .snd (x₁ , p) j .snd))
| 41.409326
| 106
| 0.49537
|
18508e4172c1b1c83a8e65f05c70c783c69c8d90
| 377
|
agda
|
Agda
|
test/succeed/Issue348.agda
|
asr/agda-kanso
|
aa10ae6a29dc79964fe9dec2de07b9df28b61ed5
|
[
"MIT"
] | 1
|
2018-10-10T17:08:44.000Z
|
2018-10-10T17:08:44.000Z
|
test/succeed/Issue348.agda
|
np/agda-git-experiment
|
20596e9dd9867166a64470dd24ea68925ff380ce
|
[
"MIT"
] | null | null | null |
test/succeed/Issue348.agda
|
np/agda-git-experiment
|
20596e9dd9867166a64470dd24ea68925ff380ce
|
[
"MIT"
] | null | null | null |
module Issue348 where
import Common.Irrelevance
data _==_ {A : Set1}(a : A) : A -> Set where
refl : a == a
record R : Set1 where
constructor mkR
field
.fromR : Set
reflR : (r : R) -> r == r
reflR r = refl {a = _}
-- issue: unsolved metavars resolved 2010-10-15 by making eta-expansion
-- more lazy (do not eta expand all meta variable listeners, see MetaVars.hs
| 23.5625
| 76
| 0.66313
|
fbc6248e11a68734ff420525f97f46d4dd1e9679
| 5,042
|
agda
|
Agda
|
homotopy/SuspAdjointLoop.agda
|
danbornside/HoTT-Agda
|
1695a7f3dc60177457855ae846bbd86fcd96983e
|
[
"MIT"
] | null | null | null |
homotopy/SuspAdjointLoop.agda
|
danbornside/HoTT-Agda
|
1695a7f3dc60177457855ae846bbd86fcd96983e
|
[
"MIT"
] | null | null | null |
homotopy/SuspAdjointLoop.agda
|
danbornside/HoTT-Agda
|
1695a7f3dc60177457855ae846bbd86fcd96983e
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --without-K #-}
open import HoTT
module homotopy.SuspAdjointLoop where
module SuspAdjointLoop {i j} (X : Ptd i) (Y : Ptd j) where
private
A = fst X; a₀ = snd X
B = fst Y; b₀ = snd Y
R : {b : B}
→ Σ (Suspension A → B) (λ h → h (north A) == b)
→ Σ (A → (b == b)) (λ k → k a₀ == idp)
R (h , idp) =
(λ a → ap h (merid A a ∙ ! (merid A a₀))) ,
ap (ap h) (!-inv-r (merid A a₀))
L : {b : B}
→ Σ (A → (b == b)) (λ k → k a₀ == idp)
→ Σ (Suspension A → B) (λ h → h (north A) == b)
L {b} (k , _) = (SuspensionRec.f A b b k) , idp
{- Show that R ∘ L ∼ idf -}
R-L : {b : B} → ∀ K → R {b} (L K) == K
R-L {b} (k , kpt) = ⊙λ= R-L-fst R-L-snd
where
R-L-fst : (a : A)
→ ap (SuspensionRec.f A b b k) (merid A a ∙ ! (merid A a₀)) == k a
R-L-fst a =
ap-∙ (SuspensionRec.f A b b k) (merid A a) (! (merid A a₀))
∙ ap2 _∙_ (SuspensionRec.glue-β A b b k a)
(ap-! (SuspensionRec.f A b b k) (merid A a₀)
∙ ap ! (SuspensionRec.glue-β A b b k a₀ ∙ kpt))
∙ ∙-unit-r (k a)
-- lemmas generalize to do some path induction for R-L-snd
lemma₁ : ∀ {i j} {A : Type i} {B : Type j} (f : A → B) {a₁ a₂ : A}
(p : a₁ == a₂) {q r : f a₁ == f a₂} {s : f a₁ == f a₁}
(α : ap f p == q) (β : q == r) (γ : q ∙ ! r == s) (δ : s == idp)
(σ : !-inv-r r == transport (λ t → t ∙ ! r == idp) β (γ ∙ δ))
→ ap (ap f) (!-inv-r p)
== (ap-∙ f p (! p) ∙ ap2 _∙_ α (ap-! f p ∙ ap ! (α ∙ β)) ∙ γ) ∙ δ
lemma₁ f idp idp idp γ idp σ = σ
lemma₂ : ∀ {i} {A : Type i} {x : A} {p : x == x} (α : p == idp)
→ idp == transport (λ t → t ∙ idp == idp) α (∙-unit-r p ∙ α)
lemma₂ idp = idp
R-L-snd : ap (ap (SuspensionRec.f A b b k)) (!-inv-r (merid A a₀))
== R-L-fst a₀ ∙ kpt
R-L-snd =
ap (ap (SuspensionRec.f A b b k)) (!-inv-r (merid A a₀))
=⟨ lemma₁ (SuspensionRec.f A b b k) (merid A a₀)
(SuspensionRec.glue-β A b b k a₀)
kpt (∙-unit-r (k a₀)) kpt (lemma₂ kpt) ⟩
R-L-fst a₀ ∙ kpt ∎
{- Show that L ∘ R ∼ idf -}
L-R : {b : B} → ∀ H → L {b} (R H) == H
L-R (h , idp) = ⊙λ= L-R-fst idp
where
fst-lemma : ∀ {i j} {A : Type i} {B : Type j} {x y z : A}
(f : A → B) (p : x == y) (q : z == y)
→ ap f p == ap f (p ∙ ! q) ∙' ap f q
fst-lemma _ idp idp = idp
L-R-fst : (σ : Suspension A) →
SuspensionRec.f A (h (north _)) (h (north _)) (fst (R (h , idp))) σ == h σ
L-R-fst = Suspension-elim A
idp
(ap h (merid A a₀))
(λ a → ↓-='-in $
ap h (merid A a)
=⟨ fst-lemma h (merid A a) (merid A a₀) ⟩
ap h (merid A a ∙ ! (merid A a₀)) ∙' ap h (merid A a₀)
=⟨ ! (SuspensionRec.glue-β A _ _ (fst (R (h , idp))) a)
|in-ctx (λ w → w ∙' (ap h (merid A a₀))) ⟩
ap (fst (L (R (h , idp)))) (merid A a) ∙' ap h (merid A a₀)
∎)
{- Show that R respects basepoint -}
pres-ident : {b : B}
→ R {b} ((λ _ → b) , idp) == ((λ _ → idp) , idp)
pres-ident {b} = ⊙λ=
(λ a → ap-cst b (merid A a ∙ ! (merid A a₀)))
(ap (ap (λ _ → b)) (!-inv-r (merid A a₀))
=⟨ lemma (merid A a₀) b ⟩
ap-cst b (merid A a₀ ∙ ! (merid A a₀))
=⟨ ! (∙-unit-r _) ⟩
ap-cst b (merid A a₀ ∙ ! (merid A a₀)) ∙ idp ∎)
where
lemma : ∀ {i j} {A : Type i} {B : Type j} {x y : A} (p : x == y) (b : B)
→ ap (ap (λ _ → b)) (!-inv-r p) == ap-cst b (p ∙ ! p)
lemma idp b = idp
{- Show that if there is a composition operation ⊙ on B, then R respects
that composition, that is R {b ⊙ c} (F ⊙ G) == R {b} F ∙ R {c} G -}
-- lift a composition operation on the codomain to the function space
comp-lift : ∀ {i j} {A : Type i} {B C D : Type j}
(a : A) (b : B) (c : C) (_⊙_ : B → C → D)
→ Σ (A → B) (λ f → f a == b)
→ Σ (A → C) (λ g → g a == c)
→ Σ (A → D) (λ h → h a == b ⊙ c)
comp-lift a b c _⊙_ (f , fpt) (g , gpt) =
(λ x → f x ⊙ g x) , ap2 _⊙_ fpt gpt
pres-comp-fst : ∀ {i j} {A : Type i} {B : Type j} (f g : A → B)
(_⊙_ : B → B → B) {a₁ a₂ : A} (p : a₁ == a₂)
→ ap (λ x → f x ⊙ g x) p == ap2 _⊙_ (ap f p) (ap g p)
pres-comp-fst f g _⊙_ idp = idp
pres-comp-snd : ∀ {i j} {A : Type i} {B : Type j} (f g : A → B)
(_⊙_ : B → B → B) {a₁ a₂ : A} (q : a₁ == a₂)
→ ap (ap (λ x → f x ⊙ g x)) (!-inv-r q)
== pres-comp-fst f g _⊙_ (q ∙ ! q)
∙ ap2 (ap2 _⊙_) (ap (ap f) (!-inv-r q)) (ap (ap g) (!-inv-r q))
pres-comp-snd f g _⊙_ idp = idp
pres-comp : {b c : B} (_⊙_ : B → B → B)
(F : Σ (Suspension A → B) (λ f → f (north A) == b))
(G : Σ (Suspension A → B) (λ f → f (north A) == c))
→ R (comp-lift (north A) b c _⊙_ F G)
== comp-lift a₀ idp idp (ap2 _⊙_) (R F) (R G)
pres-comp _⊙_ (f , idp) (g , idp) = ⊙λ=
(λ a → pres-comp-fst f g _⊙_ (merid A a ∙ ! (merid A a₀)))
(pres-comp-snd f g _⊙_ (merid A a₀))
eqv : fst (⊙Susp X ⊙→ Y) ≃ fst (X ⊙→ ⊙Ω Y)
eqv = equiv R L R-L L-R
⊙path : (⊙Susp X ⊙→ Y) == (X ⊙→ ⊙Ω Y)
⊙path = ⊙ua eqv pres-ident
| 36.273381
| 80
| 0.437723
|
5058d594d351d8d66b1321027d427dce4227c7e9
| 233
|
agda
|
Agda
|
prototyping/Properties/Remember.agda
|
JohnnyMorganz/luau
|
f2191b9e4da6a4bb2d9d344ebd7941ec2f00844b
|
[
"MIT"
] | 1
|
2021-11-06T08:03:00.000Z
|
2021-11-06T08:03:00.000Z
|
prototyping/Properties/Remember.agda
|
JohnnyMorganz/luau
|
f2191b9e4da6a4bb2d9d344ebd7941ec2f00844b
|
[
"MIT"
] | null | null | null |
prototyping/Properties/Remember.agda
|
JohnnyMorganz/luau
|
f2191b9e4da6a4bb2d9d344ebd7941ec2f00844b
|
[
"MIT"
] | null | null | null |
module Properties.Remember where
open import Agda.Builtin.Equality using (_≡_; refl)
data Remember {A : Set} (a : A) : Set where
_,_ : ∀ b → (a ≡ b) → Remember(a)
remember : ∀ {A} (a : A) → Remember(a)
remember a = (a , refl)
| 23.3
| 51
| 0.613734
|
50764d9c82c253a42ebe263f62af24b30116ee37
| 535
|
agda
|
Agda
|
test/Succeed/DoNotEtaExpandMVarsWhenComparingAgainstRecord.agda
|
mdimjasevic/agda
|
8fb548356b275c7a1e79b768b64511ae937c738b
|
[
"BSD-3-Clause"
] | 1,989
|
2015-01-09T23:51:16.000Z
|
2022-03-30T18:20:48.000Z
|
test/Succeed/DoNotEtaExpandMVarsWhenComparingAgainstRecord.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 4,066
|
2015-01-10T11:24:51.000Z
|
2022-03-31T21:14:49.000Z
|
test/Succeed/DoNotEtaExpandMVarsWhenComparingAgainstRecord.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 371
|
2015-01-03T14:04:08.000Z
|
2022-03-30T19:00:30.000Z
|
-- 2010-10-15
-- 2018-06-09
{-# OPTIONS --irrelevant-projections #-}
module DoNotEtaExpandMVarsWhenComparingAgainstRecord where
open import Common.Irrelevance
data _==_ {A : Set1}(a : A) : A -> Set where
refl : a == a
record IR : Set1 where
constructor mkIR
field
.fromIR : Set
open IR
reflIR2 : (r : IR) -> _ == mkIR (fromIR r)
reflIR2 r = refl {a = _}
-- this would fail if
-- ? = mkIR (fromIR r)
-- would be solved by
-- mkIR ?1 = mkIR (fromIR r)
-- because then no constraint is generated for ?1 due to triviality
| 19.814815
| 67
| 0.659813
|
4d2954d388d9a6dabcef626e1f778e99ebe6cee7
| 28,475
|
agda
|
Agda
|
homotopy/3x3/FromTo3.agda
|
danbornside/HoTT-Agda
|
1695a7f3dc60177457855ae846bbd86fcd96983e
|
[
"MIT"
] | 1
|
2021-06-30T00:17:55.000Z
|
2021-06-30T00:17:55.000Z
|
homotopy/3x3/FromTo3.agda
|
danbornside/HoTT-Agda
|
1695a7f3dc60177457855ae846bbd86fcd96983e
|
[
"MIT"
] | null | null | null |
homotopy/3x3/FromTo3.agda
|
danbornside/HoTT-Agda
|
1695a7f3dc60177457855ae846bbd86fcd96983e
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --without-K #-}
--open import HoTT
open import homotopy.3x3.PushoutPushout
open import homotopy.3x3.Transpose
import homotopy.3x3.To as To
import homotopy.3x3.From as From
open import homotopy.3x3.Common
module homotopy.3x3.FromTo3 {i} (d : Span^2 {i}) where
open Span^2 d
open M d hiding (Pushout^2)
open M (transpose d) using () renaming (module F₁∙ to F∙₁; f₁∙ to f∙₁;
module F₃∙ to F∙₃; f₃∙ to f∙₃;
v-h-span to h-v-span)
open M using (Pushout^2)
open To d
open From d
open import homotopy.3x3.FromToInit d
open import homotopy.3x3.FromTo2 d
module M3 (c : A₂₂) where
open M2 c
lemma2-3 =
ap□ from (E₂∙Red.coh c (↓-='-out (apd (glue {d = h-v-span}) (glue c))
∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /)
∙□-i/ E₂∙Red.lhs-o c / E₂∙Red.rhs-o c /)
=⟨ ap□-∙□-i/ from _ (E₂∙Red.lhs-o c) (E₂∙Red.rhs-o c) ⟩
ap□ from (E₂∙Red.coh c (↓-='-out (apd (glue {d = h-v-span}) (glue c))
∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /))
∙□-i/ ap (ap from) (E₂∙Red.lhs-o c) / ap (ap from) (E₂∙Red.rhs-o c) /
=⟨ lemma2-4 |in-ctx (λ u → u ∙□-i/ ap (ap from) (E₂∙Red.lhs-o c) / ap (ap from) (E₂∙Red.rhs-o c) /) ⟩
↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /
∙□-o/ ∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c) / ! (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) /
∙□-i/ (From.glue-β (right (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))
/ (! (From.glue-β (left (f₁₂ c)))) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)) /
∙□-i/ E₂∙Red.ap-ap-coh-lhs-o c from / E₂∙Red.ap-ap-coh-rhs-o c from /
∙□-i/ ap (ap from) (E₂∙Red.lhs-o c) / ap (ap from) (E₂∙Red.rhs-o c) / ∎
lemma2'-3 =
ap-∘ from (i₄∙ ∘ f₃∙) (glue c)
∙ ap (ap from) (E₂∙Red.lhs-o c)
∙ E₂∙Red.ap-ap-coh-lhs-o c from
∙ ((From.glue-β (right (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ E∙₂Red.lhs-i c
=⟨ coh2 (ap from)
(ap-∘ from (i₄∙ ∘ f₃∙) (glue c))
(ap-∘ i₄∙ f₃∙ (glue c))
_
_
_
(ap-∙∙`∘`∘ from (left ∘ right) (right ∘ right) (H₃₁ c) (glue (right (f₃₂ c))) (H₃₃ c))
((From.glue-β (right (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
(! (ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)))
_
_ ⟩
(ap-∘ from (i₄∙ ∘ f₃∙) (glue c)
∙ ap (ap from) (ap-∘ i₄∙ f₃∙ (glue c))
∙ ap (ap from) (F₃∙.glue-β c |in-ctx (ap i₄∙)))
-- ap from (ap i₄∙ (ap left (H₃₁ c) ∙ glue (f₃₂ c) ∙ ap right (H₃₃ c)))
∙ (ap (ap from) (ap-∙∙`∘`∘ i₄∙ left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c))
∙ ap (ap from) (I₄∙.glue-β (f₃₂ c) |in-ctx (λ u → ap (left ∘ right) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ ap-∙∙`∘`∘ from (left ∘ right) (right ∘ right) (H₃₁ c) (glue (right (f₃₂ c))) (H₃₃ c))
-- ap (right ∘ left) (H₃₁ c) ∙ ap from (glue (right (f₃₂ c))) ∙ ap (right ∘ right) (H₃₃ c)
∙ (((From.glue-β (right (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ ! (ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c))
∙ ! (ap (ap right) (F₃∙.glue-β c))
∙ ∘-ap right f₃∙ (glue c))
=⟨ ap-∘-ap-∙∙`∘`∘-coh from i₄∙ left right (H₃₁ c) (I₄∙.glue-β (f₃₂ c)) (H₃₃ c)
|in-ctx (λ u →
(ap-∘ from (i₄∙ ∘ f₃∙) (glue c)
∙ ap (ap from) (ap-∘ i₄∙ f₃∙ (glue c))
∙ ap (ap from) (F₃∙.glue-β c |in-ctx (ap i₄∙)))
∙ u
∙ (((From.glue-β (right (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ ! (ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c))
∙ ! (ap (ap right) (F₃∙.glue-β c))
∙ ∘-ap right f₃∙ (glue c))) ⟩
(ap-∘ from (i₄∙ ∘ f₃∙) (glue c)
∙ ap (ap from) (ap-∘ i₄∙ f₃∙ (glue c))
∙ ap (ap from) (F₃∙.glue-β c |in-ctx (ap i₄∙)))
-- ap from (ap i₄∙ (ap left (H₃₁ c) ∙ glue (f₃₂ c) ∙ ap right (H₃₃ c)))
∙ (∘-ap from i₄∙ (ap left (H₃₁ c) ∙ glue (f₃₂ c) ∙ ap right (H₃₃ c))
∙ ap-∙∙`∘`∘ (from ∘ i₄∙) left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)
∙ (ap-∘ from i₄∙ (glue (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ (ap (ap from) (I₄∙.glue-β (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))))
-- ap (right ∘ left) (H₃₁ c) ∙ ap from (glue (right (f₃₂ c))) ∙ ap (right ∘ right) (H₃₃ c)
∙ (((From.glue-β (right (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ (! (ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)))
∙ ! (ap (ap right) (F₃∙.glue-β c))
∙ ∘-ap right f₃∙ (glue c))
=⟨ coh3 (ap-∘ from (i₄∙ ∘ f₃∙) (glue c))
(ap (ap from) (ap-∘ i₄∙ f₃∙ (glue c)))
(ap (ap from) (F₃∙.glue-β c |in-ctx (ap i₄∙)))
(∘-ap from i₄∙ (ap left (H₃₁ c) ∙ glue (f₃₂ c) ∙ ap right (H₃₃ c)))
_
_ ⟩
(ap-∘ from (i₄∙ ∘ f₃∙) (glue c)
∙ ap (ap from) (ap-∘ i₄∙ f₃∙ (glue c))
∙ ap (ap from) (F₃∙.glue-β c |in-ctx (ap i₄∙))
∙ ∘-ap from i₄∙ (ap left (H₃₁ c) ∙ glue (f₃₂ c) ∙ ap right (H₃₃ c)))
∙ (ap-∙∙`∘`∘ (from ∘ i₄∙) left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)
∙ (ap-∘ from i₄∙ (glue (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ (ap (ap from) (I₄∙.glue-β (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))))
∙ ((From.glue-β (right (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ ! (ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c))
∙ ! (ap (ap right) (F₃∙.glue-β c))
∙ ∘-ap right f₃∙ (glue c))
=⟨ ap-∘-coh from i₄∙ f₃∙ (glue c) (F₃∙.glue-β c)
|in-ctx (λ u → u ∙ (ap-∙∙`∘`∘ (from ∘ i₄∙) left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)
∙ (ap-∘ from i₄∙ (glue (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ (ap (ap from) (I₄∙.glue-β (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))))
∙ ((From.glue-β (right (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ ! (ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c))
∙ ! (ap (ap right) (F₃∙.glue-β c))
∙ ∘-ap right f₃∙ (glue c))) ⟩
(ap-∘ (from ∘ i₄∙) f₃∙ (glue c)
∙ (ap (ap (from ∘ i₄∙)) (F₃∙.glue-β c)))
∙ (ap-∙∙`∘`∘ (from ∘ i₄∙) left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)
∙ (ap-∘ from i₄∙ (glue (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ (ap (ap from) (I₄∙.glue-β (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))))
∙ ((From.glue-β (right (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ ! (ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c))
∙ ! (ap (ap right) (F₃∙.glue-β c))
∙ ∘-ap right f₃∙ (glue c))
=⟨ !-ap-∘-inv right f₃∙ (glue c)
|in-ctx (λ u → (ap-∘ (from ∘ i₄∙) f₃∙ (glue c)
∙ (ap (ap (from ∘ i₄∙)) (F₃∙.glue-β c)))
∙ (ap-∙∙`∘`∘ (from ∘ i₄∙) left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)
∙ (ap-∘ from i₄∙ (glue (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ (ap (ap from) (I₄∙.glue-β (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))))
∙ ((From.glue-β (right (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ ! (ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c))
∙ ! (ap (ap right) (F₃∙.glue-β c))
∙ u)) ⟩
(ap-∘ (from ∘ i₄∙) f₃∙ (glue c)
∙ (ap (ap (from ∘ i₄∙)) (F₃∙.glue-β c)))
∙ (ap-∙∙`∘`∘ (from ∘ i₄∙) left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)
∙ (ap-∘ from i₄∙ (glue (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ (ap (ap from) (I₄∙.glue-β (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))))
∙ ((From.glue-β (right (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ ! (ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c))
∙ ! (ap (ap right) (F₃∙.glue-β c))
∙ ! (ap-∘ right f₃∙ (glue c)))
=⟨ !-∘-ap-inv (from ∘ i₄∙) f₃∙ (glue c)
|in-ctx (λ u → (u
∙ (ap (ap (from ∘ i₄∙)) (F₃∙.glue-β c)))
∙ (ap-∙∙`∘`∘ (from ∘ i₄∙) left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)
∙ (ap-∘ from i₄∙ (glue (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ (ap (ap from) (I₄∙.glue-β (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))))
∙ ((From.glue-β (right (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ ! (ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c))
∙ ! (ap (ap right) (F₃∙.glue-β c))
∙ ! (ap-∘ right f₃∙ (glue c)))) ⟩
(! (∘-ap (from ∘ i₄∙) f₃∙ (glue c))
∙ (ap (ap (from ∘ i₄∙)) (F₃∙.glue-β c)))
∙ (ap-∙∙`∘`∘ (from ∘ i₄∙) left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c)
∙ (ap-∘ from i₄∙ (glue (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ (ap (ap from) (I₄∙.glue-β (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))))
∙ ((From.glue-β (right (f₃₂ c)) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ ! (ap-∙∙`∘`∘ right left right (H₃₁ c) (glue (f₃₂ c)) (H₃₃ c))
∙ ! (ap (ap right) (F₃∙.glue-β c))
∙ ! (ap-∘ right f₃∙ (glue c)))
=⟨ coh4 (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))
(∘-ap (from ∘ i₄∙) f₃∙ (glue c)) _ _ _ _ _ _ _ _ ⟩
! end-lemma1 ∎ where
coh2 : ∀ {i j} {A : Type i} {B : Type j} (f : A → B) {a b c d e : A} {g h k l m n : B}
(p : g == f a) (q : a == b) (r : b == c) (s : c == d) (t : d == e) (u : f e == h) (v : h == k)
(w : k == l) (x : l == m) (y : m == n)
→ p ∙ ap f (_ =⟨ q ⟩ _ =⟨ r ⟩ _ =⟨ s ⟩ _ =⟨ t ⟩ _ ∎) ∙ u ∙ v ∙ (_ =⟨ w ⟩ _ =⟨ x ⟩ _ =⟨ y ⟩ _ ∎)
== (p ∙ ap f q ∙ ap f r) ∙ (ap f s ∙ ap f t ∙ u) ∙ (v ∙ w ∙ x ∙ y)
coh2 f p idp idp idp idp idp idp idp idp idp = ! (∙-unit-r (p ∙ idp))
coh3 : ∀ {i} {A : Type i} {a b c d e f g : A} (p : a == b) (q : b == c) (r : c == d) (s : d == e)
(t : e == f) (u : f == g)
→ (p ∙ q ∙ r) ∙ (s ∙ t) ∙ u == (p ∙ q ∙ r ∙ s) ∙ t ∙ u
coh3 idp idp idp idp idp idp = idp
coh4 : ∀ {i j} {A : Type i} {B : Type j} (h : A → B)
{a b c d : A} {e f g k l m : B} (p : f == e) (q : f == g) (r : g == h a) (s : a == b) (t : b == c)
(u : c == d) (v : k == h d) (w : l == k) (x : m == l)
→ (! p ∙ q) ∙ (r ∙ (s |in-ctx h) ∙ (t |in-ctx h)) ∙ ((u |in-ctx h) ∙ (! v) ∙ (! w) ∙ ! x)
== ! (_ =⟨ x ⟩ _ =⟨ w ⟩ _ =⟨ v ⟩ _ =⟨ ! (_ =⟨ s ⟩ _ =⟨ t ⟩ _ =⟨ u ⟩ _ ∎) |in-ctx h ⟩ _ =⟨ ! r ⟩ _ =⟨ ! q ⟩ _ =⟨ p ⟩ _ ∎)
coh4 h idp idp r idp idp idp v idp idp = ch r v where
ch : ∀ {i} {B : Type i} {a b c : B} (r : a == b) (v : c == b)
→ (r ∙ idp) ∙ ! v ∙ idp == ! (_ =⟨ idp ⟩ _ =⟨ idp ⟩ _ =⟨ v ⟩ _ =⟨ idp ⟩ _ =⟨ ! r ⟩ _ ∎)
ch idp idp = idp
lemma2'-4 =
E∙₂Red.rhs-i c
∙ ((! (From.glue-β (left (f₁₂ c)))) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)))
∙ E₂∙Red.ap-ap-coh-rhs-o c from
∙ ap (ap from) (E₂∙Red.rhs-o c)
∙ ∘-ap from (i₀∙ ∘ f₁∙) (glue c)
=⟨ coh2 (ap from)
(ap-∘ left f₁∙ (glue c))
(F₁∙.glue-β c |in-ctx (ap left))
(ap-∙∙`∘`∘ left left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c))
(! (From.glue-β (left (f₁₂ c))) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)))
(ap-∘ from (i₀∙ ∘ f₁∙) (glue c))
(ap-∘ i₀∙ f₁∙ (glue c))
((F₁∙.glue-β c) |in-ctx (λ u → ap i₀∙ u))
(ap-∙∙`∘`∘ i₀∙ left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c))
((I₀∙.glue-β (f₁₂ c)) |in-ctx (λ u → (ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (right ∘ left) (H₁₃ c))))
(ap-∙∙`∘`∘ from (left ∘ left) (right ∘ left) (H₁₁ c) (glue (left (f₁₂ c))) (H₁₃ c))
(!-ap-∘ i₀∙ f₁∙ (glue c))
(!-∘-ap from (i₀∙ ∘ f₁∙) (glue c)) ⟩
ap-∘ left f₁∙ (glue c)
∙ (F₁∙.glue-β c |in-ctx (ap left))
∙ (ap-∙∙`∘`∘ left left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c))
∙ (! (From.glue-β (left (f₁₂ c))) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)))
∙ ! ((ap-∘ from (i₀∙ ∘ f₁∙) (glue c)
∙ ap (ap from) (ap-∘ i₀∙ f₁∙ (glue c))
∙ ap (ap from) ((F₁∙.glue-β c) |in-ctx (λ u → ap i₀∙ u)))
∙ (ap (ap from) (ap-∙∙`∘`∘ i₀∙ left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c))
∙ ap (ap from) ((I₀∙.glue-β (f₁₂ c)) |in-ctx (λ u → (ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (right ∘ left) (H₁₃ c))))
∙ ap-∙∙`∘`∘ from (left ∘ left) (right ∘ left) (H₁₁ c) (glue (left (f₁₂ c))) (H₁₃ c)))
=⟨ lm |in-ctx (λ u →
ap-∘ left f₁∙ (glue c)
∙ ap (ap left) (F₁∙.glue-β c)
∙ ap-∙∙`∘`∘ left left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c)
∙ (! (From.glue-β (left (f₁₂ c))) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)))
∙ ! u) ⟩
ap-∘ left f₁∙ (glue c)
∙ ap (ap left) (F₁∙.glue-β c)
∙ ap-∙∙`∘`∘ left left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c)
∙ (! (From.glue-β (left (f₁₂ c))) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)))
∙ ! ((ap-∘ (from ∘ i₀∙) f₁∙ (glue c)
∙ ap (ap (from ∘ i₀∙)) (F₁∙.glue-β c))
∙ (ap-∙∙`∘`∘ (from ∘ i₀∙) left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c)
∙ (ap-∘ from i₀∙ (glue (f₁₂ c)) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)))
∙ ((I₀∙.glue-β (f₁₂ c) |in-ctx ap from) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)))))
=⟨ coh3 {f = λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)}
{p = ap-∘ left f₁∙ (glue c)}
{ap (ap left) (F₁∙.glue-β c)}
{ap-∙∙`∘`∘ left left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c)}
{ap-∘ from i₀∙ (glue (f₁₂ c))}
{I₀∙.glue-β (f₁₂ c) |in-ctx ap from}
{From.glue-β (left (f₁₂ c))}
{ap-∙∙`∘`∘ (from ∘ i₀∙) left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c)}
{ap (ap (from ∘ i₀∙)) (F₁∙.glue-β c)}
{ap-∘ (from ∘ i₀∙) f₁∙ (glue c)}
(!-ap-∘ (from ∘ i₀∙) f₁∙ (glue c)) ⟩
end-lemma3 ∎ where
coh2 : ∀ {i j} {A : Type i} {B : Type j} (f : A → B) {a b c d e o : B} (p : a == b) (q : b == c) (r : c == d) (s : d == e)
{g h l m n : A} (y : o == f g) (x : g == h) (w : h == l) (v : l == m) (u : m == n) (t : f n == e)
{x' : h == g} (α : ! x == x') {y' : f g == o} (β : ! y' == y)
→
(_ =⟨ p ⟩ _ =⟨ q ⟩ _ =⟨ r ⟩ _ ∎) ∙ s ∙ ! t ∙ ap f (_ =⟨ ! u ⟩ _ =⟨ ! v ⟩ _ =⟨ ! w ⟩ _ =⟨ x' ⟩ _ ∎) ∙ y'
== p ∙ q ∙ r ∙ s ∙ ! ((y ∙ ap f x ∙ ap f w) ∙ (ap f v ∙ ap f u ∙ t))
coh2 f idp idp idp idp .idp idp idp idp idp idp idp {y' = idp} idp = idp
coh3 : ∀ {i j} {A : Type i} {B : Type j} {f : A → B} {a b c d e g : B} {h k l m : A}
{p : a == b} {q : b == c} {r : c == f m} {v : h == k} {w : k == l} {x : l == m}
{s : d == f h} {t : e == d} {u : g == e} {u' : e == g} (α : ! u == u')
→
p ∙ q ∙ r ∙ (! x |in-ctx f) ∙ ! ((u ∙ t) ∙ (s ∙ (v |in-ctx f) ∙ (w |in-ctx f)))
==
(_ =⟨ p ⟩ _ =⟨ q ⟩ _ =⟨ r ⟩ _ =⟨ ! (_ =⟨ v ⟩ _ =⟨ w ⟩ _ =⟨ x ⟩ _ ∎) |in-ctx f ⟩
_ =⟨ ! s ⟩ _ =⟨ ! t ⟩ _ =⟨ u' ⟩ _ ∎)
coh3 {p = idp} {idp} {r} {idp} {idp} {idp} {s} {idp} {idp} idp = coh' r s where
coh' : ∀ {i} {B : Type i} {a b c : B} (r : a == b) (s : c == b)
→ r ∙ ! (s ∙ idp) == (_ =⟨ idp ⟩ _ =⟨ idp ⟩ _ =⟨ r ⟩ _ =⟨ idp ⟩ _ =⟨ ! s ⟩ _ ∎)
coh' idp idp = idp
lm =
(ap-∘ from (i₀∙ ∘ f₁∙) (glue c)
∙ ap (ap from) (ap-∘ i₀∙ f₁∙ (glue c))
∙ ap (ap from) ((F₁∙.glue-β c) |in-ctx (λ u → ap i₀∙ u)))
∙ (ap (ap from) (ap-∙∙`∘`∘ i₀∙ left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c))
∙ ap (ap from) ((I₀∙.glue-β (f₁₂ c)) |in-ctx (λ u → (ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (right ∘ left) (H₁₃ c))))
∙ ap-∙∙`∘`∘ from (left ∘ left) (right ∘ left) (H₁₁ c) (glue (left (f₁₂ c))) (H₁₃ c))
=⟨ ap-∘-coh2 from i₀∙ f₁∙ (glue c) (F₁∙.glue-β c) |in-ctx (λ u → u ∙ (ap (ap from) (ap-∙∙`∘`∘ i₀∙ left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c))
∙ ap (ap from) ((I₀∙.glue-β (f₁₂ c)) |in-ctx (λ u → (ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (right ∘ left) (H₁₃ c))))
∙ ap-∙∙`∘`∘ from (left ∘ left) (right ∘ left) (H₁₁ c) (glue (left (f₁₂ c))) (H₁₃ c))) ⟩
(ap-∘ (from ∘ i₀∙) f₁∙ (glue c)
∙ ap (ap (from ∘ i₀∙)) (F₁∙.glue-β c)
∙ ap-∘ from i₀∙ (ap left (H₁₁ c) ∙ glue (f₁₂ c) ∙ ap right (H₁₃ c)))
∙ (ap (ap from) (ap-∙∙`∘`∘ i₀∙ left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c))
∙ ap (ap from) ((I₀∙.glue-β (f₁₂ c)) |in-ctx (λ u → (ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (right ∘ left) (H₁₃ c))))
∙ ap-∙∙`∘`∘ from (left ∘ left) (right ∘ left) (H₁₁ c) (glue (left (f₁₂ c))) (H₁₃ c))
=⟨ assoc (ap-∘ (from ∘ i₀∙) f₁∙ (glue c))
(ap (ap (from ∘ i₀∙)) (F₁∙.glue-β c))
(ap-∘ from i₀∙ (ap left (H₁₁ c) ∙ glue (f₁₂ c) ∙ ap right (H₁₃ c)))
(ap (ap from) (ap-∙∙`∘`∘ i₀∙ left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c)))
(ap (ap from) ((I₀∙.glue-β (f₁₂ c)) |in-ctx (λ u → (ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (right ∘ left) (H₁₃ c)))))
(ap-∙∙`∘`∘ from (left ∘ left) (right ∘ left) (H₁₁ c) (glue (left (f₁₂ c))) (H₁₃ c)) ⟩
(ap-∘ (from ∘ i₀∙) f₁∙ (glue c)
∙ ap (ap (from ∘ i₀∙)) (F₁∙.glue-β c))
∙ (ap-∘ from i₀∙ (ap left (H₁₁ c) ∙ glue (f₁₂ c) ∙ ap right (H₁₃ c))
∙ ap (ap from) (ap-∙∙`∘`∘ i₀∙ left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c))
∙ ap (ap from) ((I₀∙.glue-β (f₁₂ c)) |in-ctx (λ u → (ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (right ∘ left) (H₁₃ c))))
∙ ap-∙∙`∘`∘ from (left ∘ left) (right ∘ left) (H₁₁ c) (glue (left (f₁₂ c))) (H₁₃ c))
=⟨ ap-∘-ap-∙∙4`∘`∘-coh from i₀∙ left right (H₁₁ c) (I₀∙.glue-β (f₁₂ c)) (H₁₃ c) |in-ctx (λ u → (ap-∘ (from ∘ i₀∙) f₁∙ (glue c) ∙ ap (ap (from ∘ i₀∙)) (F₁∙.glue-β c)) ∙ u) ⟩
(ap-∘ (from ∘ i₀∙) f₁∙ (glue c)
∙ ap (ap (from ∘ i₀∙)) (F₁∙.glue-β c))
∙ (ap-∙∙`∘`∘ (from ∘ i₀∙) left right (H₁₁ c) (glue (f₁₂ c)) (H₁₃ c)
∙ (ap-∘ from i₀∙ (glue (f₁₂ c)) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)))
∙ ((I₀∙.glue-β (f₁₂ c) |in-ctx ap from) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)))) ∎ where
assoc : ∀ {i} {A : Type i} {a b c d e f g : A} (p : a == b) (q : b == c) (r : c == d) (s : d == e) (t : e == f) (u : f == g)
→ (p ∙ q ∙ r) ∙ (s ∙ t ∙ u) == (p ∙ q) ∙ (r ∙ s ∙ t ∙ u)
assoc idp idp idp idp idp idp = idp
lemma2-2' =
ap□ from (E₂∙Red.coh c (↓-='-out (apd (glue {d = h-v-span}) (glue c))
∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /)
∙□-i/ E₂∙Red.lhs-o c / E₂∙Red.rhs-o c /)
∙□-i/ ap-∘ from (i₄∙ ∘ f₃∙) (glue c) / ∘-ap from (i₀∙ ∘ f₁∙) (glue c) /
=⟨ lemma2-3 |in-ctx (λ u → u ∙□-i/ ap-∘ from (i₄∙ ∘ f₃∙) (glue c) / ∘-ap from (i₀∙ ∘ f₁∙) (glue c) /) ⟩
↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ E∙₂Red.lhs-i c / E∙₂Red.rhs-i c /
∙□-o/ ∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c) / ! (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) /
∙□-i/ (From.glue-β (right (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c))
/ (! (From.glue-β (left (f₁₂ c)))) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)) /
∙□-i/ E₂∙Red.ap-ap-coh-lhs-o c from / E₂∙Red.ap-ap-coh-rhs-o c from /
∙□-i/ ap (ap from) (E₂∙Red.lhs-o c) / ap (ap from) (E₂∙Red.rhs-o c) /
∙□-i/ ap-∘ from (i₄∙ ∘ f₃∙) (glue c) / ∘-ap from (i₀∙ ∘ f₁∙) (glue c) /
=⟨ assoc (↓-='-out (apd (glue {d = v-h-span}) (glue c))) (E∙₂Red.lhs-i c)
((From.glue-β (right (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
(E₂∙Red.ap-ap-coh-lhs-o c from) (ap (ap from) (E₂∙Red.lhs-o c)) (ap-∘ from (i₄∙ ∘ f₃∙) (glue c)) _ _ _ _ _
(∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c)) _ ⟩
↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-o/ ∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c) / ! (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) /
∙□-i/ ap-∘ from (i₄∙ ∘ f₃∙) (glue c)
∙ ap (ap from) (E₂∙Red.lhs-o c)
∙ E₂∙Red.ap-ap-coh-lhs-o c from
∙ ((From.glue-β (right (f₃₂ c))) |in-ctx (λ u → ap (right ∘ left) (H₃₁ c) ∙ u ∙ ap (right ∘ right) (H₃₃ c)))
∙ E∙₂Red.lhs-i c
/ E∙₂Red.rhs-i c
∙ ((! (From.glue-β (left (f₁₂ c)))) |in-ctx (λ u → ap (left ∘ left) (H₁₁ c) ∙ u ∙ ap (left ∘ right) (H₁₃ c)))
∙ E₂∙Red.ap-ap-coh-rhs-o c from
∙ ap (ap from) (E₂∙Red.rhs-o c)
∙ ∘-ap from (i₀∙ ∘ f₁∙) (glue c) /
=⟨ ∙□-i/-rewrite (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-o/ ∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c) / ! (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) /) lemma2'-3 lemma2'-4 ⟩
↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-o/ ∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c) / ! (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) /
∙□-i/ ! end-lemma1 / end-lemma3 / ∎ where
assoc : ∀ {i} {A : Type i} {a b b' c : A} {u u' : a == b} {v v1 v2 v3 v4 v5 : b == c}
{w w1 w2 w3 w4 w5 : a == b'} {x x' : b' == c} (α : (u , v =□ w , x))
(p1 : v1 == v) (p2 : v2 == v1) (p3 : v3 == v2) (p4 : v4 == v3) (p5 : v5 == v4)
(q1 : w == w1) (q2 : w1 == w2) (q3 : w2 == w3) (q4 : w3 == w4) (q5 : w4 == w5)
(r : u' == u) (s : x == x')
→ α ∙□-i/ p1 / q1 / ∙□-o/ r / s / ∙□-i/ p2 / q2 / ∙□-i/ p3 / q3 / ∙□-i/ p4 / q4 / ∙□-i/ p5 / q5 /
== α ∙□-o/ r / s / ∙□-i/ p5 ∙ p4 ∙ p3 ∙ p2 ∙ p1 / q1 ∙ q2 ∙ q3 ∙ q4 ∙ q5 /
assoc α idp idp idp idp idp idp idp idp idp idp idp idp = idp
lemma2-2 =
ap↓ (ap from) (↓-='-in (E₂∙Red.coh c (↓-='-out (apd (glue {d = h-v-span}) (glue c))
∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /)
∙□-i/ E₂∙Red.lhs-o c / E₂∙Red.rhs-o c /))
=⟨ ap↓-↓-='-in-β _ _ from (E₂∙Red.coh c (↓-='-out (apd (glue {d = h-v-span}) (glue c))
∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /)
∙□-i/ E₂∙Red.lhs-o c / E₂∙Red.rhs-o c /) ⟩
↓-='-in ((ap□ from (E₂∙Red.coh c (↓-='-out (apd (glue {d = h-v-span}) (glue c))
∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /)
∙□-i/ E₂∙Red.lhs-o c / E₂∙Red.rhs-o c /))
∙□-i/ ap-∘ from (i₄∙ ∘ f₃∙) (glue c) / ∘-ap from (i₀∙ ∘ f₁∙) (glue c) /)
=⟨ lemma2-2' |in-ctx ↓-='-in ⟩
↓-='-in (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-o/ ∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c) / ! (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) /
∙□-i/ ! end-lemma1 / end-lemma3 /) ∎
lemma2-1 =
apd (ap from ∘ ap to ∘ glue) (glue c)
=⟨ apd-∘' (ap from ∘ ap to) glue (glue c) ⟩
ap↓ (ap from ∘ ap to) (apd (glue {d = v-h-span}) (glue c))
=⟨ ap↓-∘ (ap from) (ap to) (apd glue (glue c)) ⟩
ap↓ (ap from) (ap↓ (ap to) (apd glue (glue c)))
=⟨ to-glue-glue-β c |in-ctx (ap↓ (ap from)) ⟩
ap↓ (ap from) ((↓-='-in (E₂∙Red.coh c (↓-='-out (apd (glue {d = h-v-span}) (glue c))
∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /)
∙□-i/ E₂∙Red.lhs-o c / E₂∙Red.rhs-o c /))
◃/ To.glue-β (left (f₂₁ c)) / ! (To.glue-β (right (f₂₃ c))) /)
=⟨ ap↓-◃/ (ap from) _ (To.glue-β (left (f₂₁ c))) _ ⟩
ap↓ (ap from) (↓-='-in (E₂∙Red.coh c (↓-='-out (apd (glue {d = h-v-span}) (glue c))
∙□-i/ E₂∙Red.lhs-i c / E₂∙Red.rhs-i c /)
∙□-i/ E₂∙Red.lhs-o c / E₂∙Red.rhs-o c /))
◃/ ap (ap from) (To.glue-β (left (f₂₁ c))) / ap (ap from) (! (To.glue-β (right (f₂₃ c)))) /
=⟨ lemma2-2 |in-ctx (λ u → u ◃/ ap (ap from) (To.glue-β (left (f₂₁ c))) / ap (ap from) (! (To.glue-β (right (f₂₃ c)))) /) ⟩
↓-='-in (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-o/ ∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c) / ! (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) /
∙□-i/ ! end-lemma1 / end-lemma3 /)
◃/ ap (ap from) (To.glue-β (left (f₂₁ c))) / ap (ap from) (! (To.glue-β (right (f₂₃ c)))) / ∎
lemma2 =
↓-='-out (apd (ap from ∘ ap to ∘ glue) (glue c))
=⟨ lemma2-1 |in-ctx ↓-='-out ⟩
↓-='-out (↓-='-in (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-o/ ∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c)
/ ! (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) /
∙□-i/ ! end-lemma1
/ end-lemma3 /)
◃/ ap (ap from) (To.glue-β (left (f₂₁ c))) / ap (ap from) (! (To.glue-β (right (f₂₃ c)))) /)
=⟨ thing (↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-o/ ∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c)
/ ! (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) /
∙□-i/ ! end-lemma1
/ end-lemma3 /) (ap (ap from) (To.glue-β (left (f₂₁ c)))) (ap (ap from) (! (To.glue-β (right (f₂₃ c))))) ⟩
↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-o/ ∘-ap from left (glue (f₂₁ c)) ∙ I∙₀.glue-β (f₂₁ c)
/ ! (∘-ap from right (glue (f₂₃ c)) ∙ I∙₄.glue-β (f₂₃ c)) /
∙□-i/ ! end-lemma1
/ end-lemma3 /
∙□-o/ ap (ap from) (To.glue-β (left (f₂₁ c))) / ap (ap from) (! (To.glue-β (right (f₂₃ c)))) /
=⟨ ch (↓-='-out (apd (glue {d = v-h-span}) (glue c))) (∘-ap from left (glue (f₂₁ c)))
(ap-∘ from left (glue (f₂₁ c))) (!-∘-ap from left (glue (f₂₁ c))) (I∙₀.glue-β (f₂₁ c)) (∘-ap from right (glue (f₂₃ c)))
(ap-∘ from right (glue (f₂₃ c))) (!-∘-ap from right (glue (f₂₃ c))) (I∙₄.glue-β (f₂₃ c)) (! end-lemma1) end-lemma3
(ap (ap from) (To.glue-β (left (f₂₁ c)))) (! (To.glue-β (left (f₂₁ c))) |in-ctx ap from) (!-ap (ap from) (To.glue-β (left (f₂₁ c)))) (ap (ap from) (! (To.glue-β (right (f₂₃ c))))) ⟩
↓-='-out (apd (glue {d = v-h-span}) (glue c))
∙□-i/ ! end-lemma1
/ end-lemma3 /
∙□-o/ ! (from-to-g-l (f₂₁ c))
/ from-to-g-r (f₂₃ c) / ∎ where
ch : ∀ {i} {A : Type i} {a b b' c : A} {p₁ p₂ p₃ p₄ : a == b} {q₁ q₂ : b == c} {r₁ r₂ : a == b'}
{s₁ s₂ s₃ s₄ : b' == c} (α : (p₁ , q₁ =□ r₁ , s₁)) (β₁ : p₃ == p₂) (β₁-inv : p₂ == p₃)
(eq : ! β₁ == β₁-inv) (β₂ : p₂ == p₁) (β'₁ : s₃ == s₂)
(β'₁-inv : s₂ == s₃) (eq' : ! β'₁ == β'₁-inv) (β'₂ : s₂ == s₁)
(γ : q₂ == q₁) (γ' : r₁ == r₂) (β₃ : p₄ == p₃) (β₃-inv : p₃ == p₄) (eq₃ : ! β₃ == β₃-inv) (β'₃ : s₃ == s₄)
→
α ∙□-o/ β₁ ∙ β₂ / ! (β'₁ ∙ β'₂) /
∙□-i/ γ / γ' /
∙□-o/ β₃ / β'₃ /
==
α ∙□-i/ γ / γ' /
∙□-o/ ! (_ =⟨ ! β₂ ⟩ _ =⟨ β₁-inv ⟩ _ =⟨ β₃-inv ⟩ _ ∎) / (_ =⟨ ! β'₂ ⟩ _ =⟨ β'₁-inv ⟩ _ =⟨ β'₃ ⟩ _ ∎) /
ch α idp .idp idp idp idp .idp idp idp idp idp idp .idp idp idp = idp
| 56.05315
| 196
| 0.417313
|
cbe41c3079c861fae9c1b24233f3916940c34ee6
| 6,394
|
agda
|
Agda
|
Agda/Gradual Security/dynamic.agda
|
kellino/TypeSystems
|
acf5a153e14a7bdc0c9332fa602fa369fe7add46
|
[
"MIT"
] | 2
|
2016-10-27T08:05:40.000Z
|
2017-05-26T23:06:17.000Z
|
Agda/Gradual Security/dynamic.agda
|
kellino/TypeSystems
|
acf5a153e14a7bdc0c9332fa602fa369fe7add46
|
[
"MIT"
] | null | null | null |
Agda/Gradual Security/dynamic.agda
|
kellino/TypeSystems
|
acf5a153e14a7bdc0c9332fa602fa369fe7add46
|
[
"MIT"
] | null | null | null |
module dynamic where
open import LSsyntax
open import static
open import Relation.Nullary
open import Data.Nat using (ℕ ; _+_)
open import Data.Fin using (Fin; toℕ)
open import Data.Vec using (Vec ; lookup; _∷_; [])
open import Data.Product
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary
open import Data.Bool using (Bool; true ; false)
data _⊓ˡ_ : (ℓ₁ ℓ₂ : Label) → Set where
ℓ⊓✭ : ∀ ℓ → ℓ ⊓ˡ ✭
✭⊓ℓ : ∀ ℓ → ✭ ⊓ˡ ℓ
idℓ : ∀ ℓ → ℓ ⊓ˡ ℓ
_⊓ᵣ_ : ∀ (ℓ₁ ℓ₂ : Label) → Dec (ℓ₁ ⊓ˡ ℓ₂)
⊤ ⊓ᵣ ⊤ = yes (idℓ ⊤)
⊤ ⊓ᵣ ⊥ = no (λ ())
⊤ ⊓ᵣ ✭ = yes (ℓ⊓✭ ⊤)
⊥ ⊓ᵣ ⊤ = no (λ ())
⊥ ⊓ᵣ ⊥ = yes (idℓ ⊥)
⊥ ⊓ᵣ ✭ = yes (ℓ⊓✭ ⊥)
✭ ⊓ᵣ ⊤ = yes (✭⊓ℓ ⊤)
✭ ⊓ᵣ ⊥ = yes (✭⊓ℓ ⊥)
✭ ⊓ᵣ ✭ = yes (ℓ⊓✭ ✭)
-- note that this is not actually correct. The meet of ⊤ and bottom for example should be undefined,
-- but that isn't easy to model directly in Agda. However, if we only call this after induction over
-- the proof of _⊓ᵣ_ it (should) be impossible to call one of the incorrect cases.
recMeet : ∀ (ℓ₁ ℓ₂ : Label) → Label
recMeet ⊤ ⊤ = ⊤
recMeet ⊤ ✭ = ⊤
recMeet ⊥ ⊥ = ⊥
recMeet ⊥ ✭ = ⊥
recMeet ✭ ⊤ = ⊤
recMeet ✭ ⊥ = ⊥
recMeet ✭ ✭ = ✭
recMeet _ _ = ✭ -- fake case
_⊓_ : ∀ (t₁ t₂ : GType) → GType
bool x ⊓ bool x₁ with x ⊓ᵣ x₁
bool x ⊓ bool x₁ | yes p = bool (recMeet x x₁)
bool x ⊓ bool x₁ | no ¬p = err
bool x ⊓ _ = err
(t₁ ⇒ x) t₂ ⊓ (t₃ ⇒ x₁) t₄ with x ⊓ᵣ x₁
(t₁ ⇒ x) t₂ ⊓ (t₃ ⇒ x₁) t₄ | yes p with (t₁ ⊓ t₃) | (t₂ ⊓ t₄)
(t₁ ⇒ x₂) t₂ ⊓ (t₃ ⇒ x₃) t₄ | yes p | (bool x) | (bool x₁) = ((bool x) ⇒ (recMeet x₂ x₃)) (bool x)
-- this is probably not correct, but it keeps things simple.
(t₁ ⇒ x₂) t₂ ⊓ (t₃ ⇒ x₃) t₄ | yes p | (bool x) | ((d ⇒ x₁) d₁) = err
(t₁ ⇒ x₂) t₂ ⊓ (t₃ ⇒ x₃) t₄ | yes p | ((c ⇒ x) c₁) | (bool x₁) = err
(t₁ ⇒ x₂) t₂ ⊓ (t₃ ⇒ x₃) t₄ | yes p | ((c ⇒ x) c₁) | ((d ⇒ x₁) d₁) = err
(t₁ ⇒ x₁) t₂ ⊓ (t₃ ⇒ x₂) t₄ | yes p | (bool x) | err = err
(t₁ ⇒ x₁) t₂ ⊓ (t₃ ⇒ x₂) t₄ | yes p | ((c ⇒ x) c₁) | err = err
(t₁ ⇒ x₁) t₂ ⊓ (t₃ ⇒ x₂) t₄ | yes p | err | (bool x) = err
(t₁ ⇒ x₁) t₂ ⊓ (t₃ ⇒ x₂) t₄ | yes p | err | ((d ⇒ x) d₁) = err
(t₁ ⇒ x) t₂ ⊓ (t₃ ⇒ x₁) t₄ | yes p | err | err = err
(t₁ ⇒ x) t₂ ⊓ (t₃ ⇒ x₁) t₄ | no ¬p = err
_ ⊓ _ = err
I≼ : ∀ (ℓ₁ ℓ₂ : Label) → (Label × Label)
I≼ ⊤ ✭ = ⊤ , ⊤
I≼ ✭ ⊤ = ✭ , ⊤
I≼ ✭ ⊥ = ⊥ , ⊥
I≼ ⊥ ✭ = ⊥ , ✭
I≼ ℓ₁ ℓ₂ = ℓ₁ , ℓ₂
Δ≼ : ∀ (triple : Label × Label × Label) → (Label × Label)
Δ≼ (ℓ₁ , ⊤ , ⊤) = ℓ₁ , ⊤
Δ≼ (ℓ₁ , ⊤ , ✭) = ℓ₁ , ⊤
Δ≼ (⊥ , ⊥ , ℓ₃) = ⊥ , ℓ₃
Δ≼ (⊥ , ✭ , ℓ₃) = ⊥ , ℓ₃
Δ≼ (ℓ₁ , ✭ , ℓ₃) = ℓ₁ , ℓ₃
Δ≼ (ℓ₁ , ℓ₂ , ℓ₃) = ℓ₁ , ℓ₃
Δ< : ∀ (triple : GType × GType × GType) → (GType × GType)
Δ< (bool ℓ₁ , bool ℓ₂ , bool ℓ₃) =
let new = Δ≼ (ℓ₁ , ℓ₂ , ℓ₃) in
bool (proj₁ new) , bool (proj₂ new)
Δ< ((t₁ ⇒ ℓ₁) t′₁ , (t₂ ⇒ ℓ₂) t′₂ , (t₃ ⇒ ℓ₃) t′₃) =
let new = Δ≼ (ℓ₁ , ℓ₂ , ℓ₃) in
((t₁ ⇒ (proj₁ new)) t′₁) , ((t₃ ⇒ (proj₂ new)) t′₃)
Δ< (_ , _ , _) = err , err
interior : ∀ (t : GType) → (GType × GType)
interior (bool ℓ) = (bool ℓ) , (bool ℓ)
interior ((t ⇒ ℓ) t₁) =
let (ℓ₁ , ℓ₂) = I≼ (getLabel t) ℓ in
((setLabel t ℓ₁) ⇒ ℓ₂) t₁ , ((setLabel t ℓ₁) ⇒ ℓ₂) t₁
interior err = err , err
_∘<_ : ∀ (t₁ t₂ : (GType × GType)) → (GType × GType)
(s₁ , s₂₁) ∘< (s₂₂ , s₃) = Δ< (s₁ , (s₂₁ ⊓ s₂₂) , s₃)
dynamicCheck : ∀ {n} (Γ : Ctx n) (t : Term) → Check Γ t
-- variables
dynamicCheck {n} Γ (var v) with fromℕ n v
dynamicCheck {n} Γ (var .(toℕ m)) | yes m = yes (lookup m Γ) (Sx m refl)
dynamicCheck {n} Γ (var .(n + m)) | no m = no
-- literals
dynamicCheck Γ (litBool x ℓ) = yes (bool ℓ) (Sb x ℓ)
-- lambda abstraction
dynamicCheck Γ (lam x t x₁) with dynamicCheck (x ∷ Γ) t
dynamicCheck Γ (lam x .(erase t) ℓ) | yes τ t = yes ((x ⇒ ℓ) τ) (Sλ x ℓ t)
dynamicCheck Γ (lam x t x₁) | no = no
-- logical and
dynamicCheck Γ (t ∧ t₁) with dynamicCheck Γ t | dynamicCheck Γ t₁
dynamicCheck Γ (.(erase t) ∧ .(erase t₁)) | yes τ t | (yes τ₁ t₁) with (interior τ) ∘< (interior τ₁)
dynamicCheck Γ (.(erase t) ∧ .(erase t₁)) | yes τ t | (yes τ₁ t₁) | (bool x , bool x₁) = yes (bool (getLabel τ ~⋎~ getLabel τ₁)) (t S∧ t₁)
dynamicCheck Γ (.(erase t) ∧ .(erase t₁)) | yes τ t | (yes τ₁ t₁) | (_ , _) = no
dynamicCheck Γ (t ∧ t₁) | _ | _ = no
-- logical or
dynamicCheck Γ (t ∨ t₁) with dynamicCheck Γ t | dynamicCheck Γ t₁
dynamicCheck Γ (.(erase t) ∨ .(erase t₁)) | yes τ t | (yes τ₁ t₁) with (interior τ) ∘< (interior τ₁)
dynamicCheck Γ (.(erase t) ∨ .(erase t₁)) | yes τ t | (yes τ₁ t₁) | (bool x , bool x₁) = yes (bool (getLabel τ ~⋎~ getLabel τ₁)) (t S∨ t₁)
dynamicCheck Γ (.(erase t) ∨ .(erase t₁)) | yes τ t | (yes τ₁ t₁) | (_ , _) = no
dynamicCheck Γ (t ∨ t₁) | _ | _ = no
-- application
-- this needs to be doublechecked!
dynamicCheck Γ (t ∙ t₁) with dynamicCheck Γ t | dynamicCheck Γ t₁
dynamicCheck Γ (.(erase t) ∙ .(erase t₁)) | yes ((τ ⇒ ℓ₁) τ₁) t | yes τ₂ t₁ with (interior ((τ ⇒ ℓ₁) τ₁)) ∘< (interior τ₂)
dynamicCheck Γ (.(erase t) ∙ .(erase t₁)) | yes ((τ ⇒ ℓ₁) τ₁) t | (yes τ₂ t₁) | (bool x , bool x₁) = yes (bool (getLabel τ₁ ~⋎~ ℓ₁)) (S∙ t t₁ (yes τ₂ τ))
dynamicCheck Γ (.(erase t) ∙ .(erase t₁)) | yes ((τ ⇒ ℓ₁) τ₁) t | (yes τ₂ t₁) | (bool x , (proj₄ ⇒ x₁) proj₅) = no
dynamicCheck Γ (.(erase t) ∙ .(erase t₁)) | yes ((τ ⇒ ℓ₁) τ₁) t | (yes τ₂ t₁) | ((proj₃ ⇒ x) proj₄ , bool x₁) = no
dynamicCheck Γ (.(erase t) ∙ .(erase t₁)) | yes ((τ ⇒ ℓ₁) τ₁) t | (yes τ₂ t₁) | ((proj₃ ⇒ x) proj₄ , (proj₅ ⇒ x₁) proj₆) = no
dynamicCheck Γ (.(erase t) ∙ .(erase t₁)) | yes ((τ ⇒ ℓ₁) τ₁) t | (yes τ₂ t₁) | (_ , _) = no
dynamicCheck Γ (.(erase t) ∙ .(erase t₁)) | yes _ t | (yes _ t₁) = no
dynamicCheck Γ (.(erase t) ∙ t₁) | yes τ t | no = no
dynamicCheck Γ (t₁ ∙ .(erase t)) | no | yes τ t = no
dynamicCheck Γ (t ∙ t₁) | no | no = no
-- if then else
dynamicCheck Γ (if b then t₁ else t₂) with dynamicCheck Γ b
dynamicCheck Γ (if .(erase b) then t₁ else t₂) | yes τ b with dynamicCheck Γ t₁ | dynamicCheck Γ t₂
dynamicCheck Γ (if .(erase b) then .(erase t₁) else .(erase t₂)) | yes τ b | (yes τ₁ t₁) | (yes τ₂ t₂) with (interior τ₁) ∘< (interior τ₂)
dynamicCheck Γ (if .(erase b) then .(erase t₁) else .(erase t₂)) | yes τ b | (yes τ₁ t₁) | (yes τ₂ t₂) | (bool ℓ , bool ℓ₁) = yes (bool (getLabel (τ₁ :∨: τ₂) ~⋎~ getLabel τ)) (Sif b t₁ t₂)
dynamicCheck Γ (if .(erase b) then .(erase t₁) else .(erase t₂)) | yes τ b | (yes τ₁ t₁) | (yes τ₂ t₂) | (_ , _) = no
dynamicCheck Γ (if .(erase b) then t₁ else t₂) | yes τ b | _ | _ = no
dynamicCheck Γ (if b then t₁ else t₂) | no = no
-- error
dynamicCheck Γ error = no
| 42.626667
| 188
| 0.547545
|
4d14793e7e9fedc62dac9057b602a488b8cdf0ae
| 3,258
|
agda
|
Agda
|
src/Semantics/Iemhoff.agda
|
mietek/imla2017
|
accc6c57390c435728d568ae590a02b2776b8891
|
[
"X11"
] | 17
|
2017-02-27T05:04:55.000Z
|
2021-01-17T13:02:58.000Z
|
src/Semantics/Iemhoff.agda
|
mietek/imla2017
|
accc6c57390c435728d568ae590a02b2776b8891
|
[
"X11"
] | null | null | null |
src/Semantics/Iemhoff.agda
|
mietek/imla2017
|
accc6c57390c435728d568ae590a02b2776b8891
|
[
"X11"
] | null | null | null |
module Semantics.Iemhoff where
open import Syntax public
-- Brilliant Kripke models.
record Model : Set₁ where
infix 3 _⊩ᵅ_
field
World : Set
_≤_ : World → World → Set
refl≤ : ∀ {w} → w ≤ w
trans≤ : ∀ {w w′ w″} → w ≤ w′ → w′ ≤ w″ → w ≤ w″
_R_ : World → World → Set
reflR : ∀ {w} → w R w
transR : ∀ {w w′ w″} → w R w′ → w′ R w″ → w R w″
_⊩ᵅ_ : World → Atom → Set
mono⊩ᵅ : ∀ {w w′ P} → w ≤ w′ → w ⊩ᵅ P → w′ ⊩ᵅ P
_R⨾≤_ : World → World → Set
_R⨾≤_ = _R_ ⨾ _≤_
-- Brilliance.
field
R⨾≤→R : ∀ {w v′} → w R⨾≤ v′ → w R v′
-- Vindication, as a consequence of brilliance.
≤→R : ∀ {w v′} → w ≤ v′ → w R v′
≤→R {w} ψ = R⨾≤→R (w , (reflR , ψ))
open Model {{…}} public
-- Forcing in a particular world of a particular model.
module _ {{_ : Model}} where
infix 3 _⊩_
_⊩_ : World → Type → Set
w ⊩ α P = w ⊩ᵅ P
w ⊩ A ⇒ B = ∀ {w′} → w ≤ w′ → w′ ⊩ A → w′ ⊩ B
w ⊩ □ A = ∀ {v′} → w R v′ → v′ ⊩ A
w ⊩ A ⩕ B = w ⊩ A ∧ w ⊩ B
w ⊩ ⫪ = ⊤
w ⊩ ⫫ = ⊥
w ⊩ A ⩖ B = w ⊩ A ∨ w ⊩ B
infix 3 _⊩⋆_
_⊩⋆_ : World → Stack Type → Set
w ⊩⋆ ∅ = ⊤
w ⊩⋆ Ξ , A = w ⊩⋆ Ξ ∧ w ⊩ A
-- Monotonicity of forcing with respect to constructive accessibility.
module _ {{_ : Model}} where
mono⊩ : ∀ {A w w′} → w ≤ w′ → w ⊩ A → w′ ⊩ A
mono⊩ {α P} ψ s = mono⊩ᵅ ψ s
mono⊩ {A ⇒ B} ψ f = λ ψ′ a → f (trans≤ ψ ψ′) a
mono⊩ {□ A} ψ f = λ ρ → f (transR (≤→R ψ) ρ)
mono⊩ {A ⩕ B} ψ (a , b) = mono⊩ {A} ψ a , mono⊩ {B} ψ b
mono⊩ {⫪} ψ ∙ = ∙
mono⊩ {⫫} ψ ()
mono⊩ {A ⩖ B} ψ (ι₁ a) = ι₁ (mono⊩ {A} ψ a)
mono⊩ {A ⩖ B} ψ (ι₂ b) = ι₂ (mono⊩ {B} ψ b)
mono⊩⋆ : ∀ {Ξ w w′} → w ≤ w′ → w ⊩⋆ Ξ → w′ ⊩⋆ Ξ
mono⊩⋆ {∅} ψ ∙ = ∙
mono⊩⋆ {Ξ , A} ψ (ξ , s) = mono⊩⋆ {Ξ} ψ ξ , mono⊩ {A} ψ s
-- Additional equipment.
module _ {{_ : Model}} where
lookup : ∀ {Ξ A w} → A ∈ Ξ → w ⊩⋆ Ξ → w ⊩ A
lookup top (ξ , s) = s
lookup (pop i) (ξ , s) = lookup i ξ
-- Forcing in all worlds of all models, or semantic entailment.
infix 3 _⊨_
_⊨_ : Context → Type → Set₁
Γ ⁏ Δ ⊨ A = ∀ {{_ : Model}} {w} →
w ⊩⋆ Γ →
(∀ {v′} → w R v′ → v′ ⊩⋆ Δ) →
w ⊩ A
-- Soundness of the semantics with respect to the syntax.
reflect : ∀ {Γ Δ A} → Γ ⁏ Δ ⊢ A → Γ ⁏ Δ ⊨ A
reflect (var i) γ δ = lookup i γ
reflect (mvar i) γ δ = lookup i (δ reflR)
reflect (lam d) γ δ = λ ψ a → reflect d (mono⊩⋆ ψ γ , a)
(λ ρ → δ (transR (≤→R ψ) ρ))
reflect (app d e) γ δ = (reflect d γ δ) refl≤ (reflect e γ δ)
reflect (box d) γ δ = λ ρ → reflect d ∙
(λ ρ′ → δ (transR ρ ρ′))
reflect (unbox d e) γ δ = reflect e γ (λ ρ → δ ρ , (reflect d γ δ) ρ)
reflect (pair d e) γ δ = reflect d γ δ , reflect e γ δ
reflect (fst d) γ δ = π₁ (reflect d γ δ)
reflect (snd d) γ δ = π₂ (reflect d γ δ)
reflect unit γ δ = ∙
reflect (boom d) γ δ = elim⊥ (reflect d γ δ)
reflect (left d) γ δ = ι₁ (reflect d γ δ)
reflect (right d) γ δ = ι₂ (reflect d γ δ)
reflect (case d e f) γ δ = elim∨ (reflect d γ δ) (λ a → reflect e (γ , a) δ)
(λ b → reflect f (γ , b) δ)
| 29.618182
| 76
| 0.440147
|
fbc81c356e88036e1274ad221306cc9c94780a6e
| 8,721
|
agda
|
Agda
|
src/Categories/Bicategory/Bigroupoid.agda
|
turion/agda-categories
|
ad0f94b6cf18d8a448b844b021aeda58e833d152
|
[
"MIT"
] | 5
|
2020-10-07T12:07:53.000Z
|
2020-10-10T21:41:32.000Z
|
src/Categories/Bicategory/Bigroupoid.agda
|
turion/agda-categories
|
ad0f94b6cf18d8a448b844b021aeda58e833d152
|
[
"MIT"
] | null | null | null |
src/Categories/Bicategory/Bigroupoid.agda
|
turion/agda-categories
|
ad0f94b6cf18d8a448b844b021aeda58e833d152
|
[
"MIT"
] | 1
|
2021-11-04T06:54:45.000Z
|
2021-11-04T06:54:45.000Z
|
{-# OPTIONS --without-K --safe #-}
module Categories.Bicategory.Bigroupoid where
open import Level
open import Function using (_$_)
open import Data.Product using (Σ; _,_)
open import Categories.Adjoint.TwoSided using (_⊣⊢_)
open import Categories.Category
open import Categories.Category.Equivalence using (WeakInverse)
import Categories.Category.Equivalence.Properties as EP
open import Categories.Category.Product
open import Categories.Category.Groupoid using (IsGroupoid)
open import Categories.Bicategory
open import Categories.Bicategory.Extras
open import Categories.Functor renaming (id to idF)
open import Categories.Functor.Properties
open import Categories.Functor.Bifunctor.Properties
open import Categories.Functor.Construction.Constant
open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
open import Categories.NaturalTransformation.NaturalIsomorphism using (_≃_; NaturalIsomorphism)
import Categories.Morphism as Mor
import Categories.Morphism.Properties as MP
import Categories.Morphism.Reasoning as MR
-- https://link.springer.com/article/10.1023/A:1011270417127
record IsBigroupoid {o ℓ e t} (C : Bicategory o ℓ e t) : Set (o ⊔ ℓ ⊔ e ⊔ t) where
open Bicategory C public
open Extras C
field
hom-isGroupoid : ∀ A B → IsGroupoid (hom A B)
hom[_,_]⁻¹ : ∀ A B → Functor (hom A B) (hom B A)
cancel : ∀ A B → ⊚ ∘F (hom[ A , B ]⁻¹ ※ idF) ≃ const id₁
cancel′ : ∀ A B → ⊚ ∘F (idF ※ hom[ A , B ]⁻¹) ≃ const id₁
module hom⁻¹ {A B} = Functor (hom[ A , B ]⁻¹)
module cancel {A B} = NaturalIsomorphism (cancel A B)
module cancel′ {A B} = NaturalIsomorphism (cancel′ A B)
infix 13 _⁻¹ _⁻¹′
_⁻¹ : ∀ {A B} → A ⇒₁ B → B ⇒₁ A
_⁻¹ = hom⁻¹.F₀
_⁻¹′ : ∀ {A B} {f g : A ⇒₁ B} → f ⇒₂ g → f ⁻¹ ⇒₂ g ⁻¹
_⁻¹′ = hom⁻¹.F₁
field
pentagon₁ : ∀ {A B} {f : A ⇒₁ B} →
let open Commutation (hom A B) in
[ (f ∘ₕ f ⁻¹) ∘ₕ f ⇒ f ]⟨
associator.from ⇒⟨ f ∘ₕ f ⁻¹ ∘ₕ f ⟩
f ▷ cancel.⇒.η f ⇒⟨ f ∘ₕ id₁ ⟩
unitorʳ.from
≈ cancel′.⇒.η f ◁ f ⇒⟨ id₁ ∘ₕ f ⟩
unitorˡ.from
⟩
pentagon₂ : ∀ {A B} {f : A ⇒₁ B} →
let open Commutation (hom B A) in
[ (f ⁻¹ ∘ₕ f) ∘ₕ f ⁻¹ ⇒ f ⁻¹ ]⟨
associator.from ⇒⟨ f ⁻¹ ∘ₕ f ∘ₕ f ⁻¹ ⟩
f ⁻¹ ▷ cancel′.⇒.η f ⇒⟨ f ⁻¹ ∘ₕ id₁ ⟩
unitorʳ.from
≈ cancel.⇒.η f ◁ f ⁻¹ ⇒⟨ id₁ ∘ₕ f ⁻¹ ⟩
unitorˡ.from
⟩
private
variable
A B : Obj
f g : A ⇒₁ B
α β : f ⇒₂ g
open hom.HomReasoning
open hom.Equiv
module MR′ {A B} where
open MR (hom A B) public
open Mor (hom A B) public
open MP (hom A B) public
open MR′
module ℱ = Functor
cancel-comm : ∀ {α : f ⇒₂ g} → cancel.⇒.η g ∘ᵥ (α ⁻¹′ ⊚₁ α) ≈ cancel.⇒.η f
cancel-comm {α = α} = cancel.⇒.commute α ○ identity₂ˡ
cancel⁻¹-comm : ∀ {α : f ⇒₂ g} → (α ⁻¹′ ⊚₁ α) ∘ᵥ cancel.⇐.η f ≈ cancel.⇐.η g
cancel⁻¹-comm {α = α} = ⟺ (cancel.⇐.commute α) ○ identity₂ʳ
cancel′-comm : ∀ {α : f ⇒₂ g} → cancel′.⇒.η g ∘ᵥ (α ⊚₁ α ⁻¹′) ≈ cancel′.⇒.η f
cancel′-comm {α = α} = cancel′.⇒.commute α ○ identity₂ˡ
cancel′⁻¹-comm : ∀ {α : f ⇒₂ g} → (α ⊚₁ α ⁻¹′) ∘ᵥ cancel′.⇐.η f ≈ cancel′.⇐.η g
cancel′⁻¹-comm {α = α} = ⟺ (cancel′.⇐.commute α) ○ identity₂ʳ
hom⁻¹⁻¹≃id : ∀ {A B} → hom[ B , A ]⁻¹ ∘F hom[ A , B ]⁻¹ ≃ idF
hom⁻¹⁻¹≃id {A} {B} = record
{ F⇒G = ntHelper record
{ η = λ f → (((unitorˡ.from ∘ᵥ cancel.⇒.η (f ⁻¹) ◁ f) ∘ᵥ associator.to) ∘ᵥ f ⁻¹ ⁻¹ ▷ cancel.⇐.η f) ∘ᵥ unitorʳ.to
; commute = λ {f g} α → begin
((((unitorˡ.from ∘ᵥ cancel.⇒.η (g ⁻¹) ◁ g) ∘ᵥ associator.to) ∘ᵥ g ⁻¹ ⁻¹ ▷ cancel.⇐.η g) ∘ᵥ unitorʳ.to) ∘ᵥ α ⁻¹′ ⁻¹′
≈˘⟨ pushʳ ◁-∘ᵥ-λ⁻¹ ⟩
(((unitorˡ.from ∘ᵥ cancel.⇒.η (g ⁻¹) ◁ g) ∘ᵥ associator.to) ∘ᵥ g ⁻¹ ⁻¹ ▷ cancel.⇐.η g) ∘ᵥ ((α ⁻¹′ ⁻¹′ ◁ id₁) ∘ᵥ unitorʳ.to)
≈⟨ center ◁-▷-exchg ⟩
((unitorˡ.from ∘ᵥ cancel.⇒.η (g ⁻¹) ◁ g) ∘ᵥ associator.to) ∘ᵥ (α ⁻¹′ ⁻¹′ ◁ (g ⁻¹ ∘ₕ g) ∘ᵥ f ⁻¹ ⁻¹ ▷ cancel.⇐.η g) ∘ᵥ unitorʳ.to
≈⟨ center (⟺ assoc₂ ○ hom.∘-resp-≈ assoc⁻¹-◁-∘ₕ (ℱ.F-resp-≈ ((f ⁻¹ ⁻¹) ⊚-) (⟺ cancel⁻¹-comm))) ⟩
(unitorˡ.from ∘ᵥ cancel.⇒.η (g ⁻¹) ◁ g) ∘ᵥ ((α ⁻¹′ ⁻¹′ ◁ g ⁻¹ ◁ g ∘ᵥ associator.to) ∘ᵥ f ⁻¹ ⁻¹ ▷ ((α ⁻¹′ ⊚₁ α) ∘ᵥ cancel.⇐.η f)) ∘ᵥ unitorʳ.to
≈⟨ refl⟩∘⟨ (hom.∘-resp-≈ʳ (ℱ.homomorphism ((f ⁻¹ ⁻¹) ⊚-)) ○ center (⊚-assoc.⇐.commute _) ○ center⁻¹ ([ ⊚ ]-merge (⟺ [ ⊚ ]-decompose₁) identity₂ˡ) refl) ⟩∘⟨refl ⟩
(unitorˡ.from ∘ᵥ cancel.⇒.η (g ⁻¹) ◁ g) ∘ᵥ (((α ⁻¹′ ⁻¹′ ⊚₁ α ⁻¹′) ⊚₁ α) ∘ᵥ associator.to ∘ᵥ f ⁻¹ ⁻¹ ▷ cancel.⇐.η f) ∘ᵥ unitorʳ.to
≈˘⟨ assoc₂ ⟩
((unitorˡ.from ∘ᵥ cancel.⇒.η (g ⁻¹) ◁ g) ∘ᵥ (((α ⁻¹′ ⁻¹′ ⊚₁ α ⁻¹′) ⊚₁ α) ∘ᵥ associator.to ∘ᵥ f ⁻¹ ⁻¹ ▷ cancel.⇐.η f)) ∘ᵥ unitorʳ.to
≈⟨ center ([ ⊚ ]-merge cancel-comm identity₂ˡ) ⟩∘⟨refl ⟩
(unitorˡ.from ∘ᵥ cancel.⇒.η (f ⁻¹) ⊚₁ α ∘ᵥ associator.to ∘ᵥ f ⁻¹ ⁻¹ ▷ cancel.⇐.η f) ∘ᵥ unitorʳ.to
≈˘⟨ (assoc₂ ○ assoc₂) ⟩∘⟨refl ⟩
(((unitorˡ.from ∘ᵥ cancel.⇒.η (f ⁻¹) ⊚₁ α) ∘ᵥ associator.to) ∘ᵥ f ⁻¹ ⁻¹ ▷ cancel.⇐.η f) ∘ᵥ unitorʳ.to
≈⟨ (hom.∘-resp-≈ʳ [ ⊚ ]-decompose₂) ⟩∘⟨refl ⟩∘⟨refl ⟩∘⟨refl ⟩
(((unitorˡ.from ∘ᵥ id₁ ▷ α ∘ᵥ cancel.⇒.η (f ⁻¹) ◁ f) ∘ᵥ associator.to) ∘ᵥ f ⁻¹ ⁻¹ ▷ cancel.⇐.η f) ∘ᵥ unitorʳ.to
≈⟨ pullˡ ρ-∘ᵥ-▷ ⟩∘⟨refl ⟩∘⟨refl ⟩∘⟨refl ⟩
((((α ∘ᵥ unitorˡ.from) ∘ᵥ cancel.⇒.η (f ⁻¹) ◁ f) ∘ᵥ associator.to) ∘ᵥ f ⁻¹ ⁻¹ ▷ cancel.⇐.η f) ∘ᵥ unitorʳ.to
≈⟨ (assoc₂ ○ assoc₂ ○ assoc₂ ○ assoc₂) ⟩
α ∘ᵥ unitorˡ.from ∘ᵥ cancel.⇒.η (f ⁻¹) ◁ f ∘ᵥ associator.to ∘ᵥ f ⁻¹ ⁻¹ ▷ cancel.⇐.η f ∘ᵥ unitorʳ.to
≈˘⟨ refl⟩∘⟨ (assoc₂ ○ assoc₂ ○ assoc₂) ⟩
α ∘ᵥ (((unitorˡ.from ∘ᵥ cancel.⇒.η (f ⁻¹) ◁ f) ∘ᵥ associator.to) ∘ᵥ f ⁻¹ ⁻¹ ▷ cancel.⇐.η f) ∘ᵥ unitorʳ.to
∎
}
; F⇐G = ntHelper record
{ η = λ f → unitorʳ.from ∘ᵥ f ⁻¹ ⁻¹ ▷ cancel.⇒.η f ∘ᵥ associator.from ∘ᵥ cancel.⇐.η (f ⁻¹) ◁ f ∘ᵥ unitorˡ.to
; commute = λ {f g} α → begin
(unitorʳ.from ∘ᵥ g ⁻¹ ⁻¹ ▷ cancel.⇒.η g ∘ᵥ associator.from ∘ᵥ cancel.⇐.η (g ⁻¹) ◁ g ∘ᵥ unitorˡ.to) ∘ᵥ α
≈⟨ assoc₂ ○ hom.∘-resp-≈ʳ (assoc₂ ○ hom.∘-resp-≈ʳ (assoc₂ ○ hom.∘-resp-≈ʳ assoc₂)) ⟩
unitorʳ.from ∘ᵥ g ⁻¹ ⁻¹ ▷ cancel.⇒.η g ∘ᵥ associator.from ∘ᵥ cancel.⇐.η (g ⁻¹) ◁ g ∘ᵥ unitorˡ.to ∘ᵥ α
≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ ⟺ ▷-∘ᵥ-ρ⁻¹ ⟩
unitorʳ.from ∘ᵥ g ⁻¹ ⁻¹ ▷ cancel.⇒.η g ∘ᵥ associator.from ∘ᵥ cancel.⇐.η (g ⁻¹) ◁ g ∘ᵥ id₁ ▷ α ∘ᵥ unitorˡ.to
≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ (⟺ [ ⊚ ]-decompose₁ ○ ⊚.F-resp-≈ (⟺ cancel⁻¹-comm , refl)) ⟩
unitorʳ.from ∘ᵥ g ⁻¹ ⁻¹ ▷ cancel.⇒.η g ∘ᵥ associator.from ∘ᵥ (α ⁻¹′ ⁻¹′ ⊚₁ α ⁻¹′ ∘ᵥ cancel.⇐.η (f ⁻¹)) ⊚₁ α ∘ᵥ unitorˡ.to
≈˘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ hom.∘-resp-≈ˡ ([ ⊚ ]-merge refl identity₂ʳ) ⟩
unitorʳ.from ∘ᵥ g ⁻¹ ⁻¹ ▷ cancel.⇒.η g ∘ᵥ associator.from ∘ᵥ ((α ⁻¹′ ⁻¹′ ⊚₁ α ⁻¹′) ⊚₁ α ∘ᵥ cancel.⇐.η (f ⁻¹) ◁ f) ∘ᵥ unitorˡ.to
≈⟨ refl⟩∘⟨ refl⟩∘⟨ center⁻¹ (⊚-assoc.⇒.commute _) refl ⟩
unitorʳ.from ∘ᵥ g ⁻¹ ⁻¹ ▷ cancel.⇒.η g ∘ᵥ (α ⁻¹′ ⁻¹′ ⊚₁ α ⁻¹′ ⊚₁ α ∘ᵥ associator.from) ∘ᵥ cancel.⇐.η (f ⁻¹) ◁ f ∘ᵥ unitorˡ.to
≈⟨ refl⟩∘⟨ (hom.∘-resp-≈ʳ assoc₂ ○ pullˡ ([ ⊚ ]-merge identity₂ˡ cancel-comm)) ⟩
unitorʳ.from ∘ᵥ (α ⁻¹′ ⁻¹′) ⊚₁ (cancel.⇒.η f) ∘ᵥ associator.from ∘ᵥ cancel.⇐.η (f ⁻¹) ◁ f ∘ᵥ unitorˡ.to
≈⟨ refl⟩∘⟨ (hom.∘-resp-≈ˡ [ ⊚ ]-decompose₁ ○ assoc₂) ⟩
unitorʳ.from ∘ᵥ (α ⁻¹′ ⁻¹′) ◁ id₁ ∘ᵥ f ⁻¹ ⁻¹ ▷ cancel.⇒.η f ∘ᵥ associator.from ∘ᵥ cancel.⇐.η (f ⁻¹) ◁ f ∘ᵥ unitorˡ.to
≈⟨ (pullˡ λ-∘ᵥ-◁) ○ assoc₂ ⟩
α ⁻¹′ ⁻¹′ ∘ᵥ unitorʳ.from ∘ᵥ f ⁻¹ ⁻¹ ▷ cancel.⇒.η f ∘ᵥ associator.from ∘ᵥ cancel.⇐.η (f ⁻¹) ◁ f ∘ᵥ unitorˡ.to
∎
}
; iso = λ f → Iso-∘ (Iso-swap (unitʳ.iso _)) $
Iso-∘ ([ (f ⁻¹ ⁻¹) ⊚- ]-resp-Iso (Iso-swap (cancel.iso f))) $
Iso-∘ (Iso-swap associator.iso) $
Iso-∘ ([ -⊚ f ]-resp-Iso (cancel.iso _))
(unitˡ.iso _)
}
hom⁻¹-weakInverse : ∀ {A B} → WeakInverse hom[ A , B ]⁻¹ hom[ B , A ]⁻¹
hom⁻¹-weakInverse = record { F∘G≈id = hom⁻¹⁻¹≃id ; G∘F≈id = hom⁻¹⁻¹≃id }
hom⁻¹-⊣Equivalence : ∀ {A} {B} → hom[ A , B ]⁻¹ ⊣⊢ hom[ B , A ]⁻¹
hom⁻¹-⊣Equivalence {A} {B} = EP.F⊣⊢G (hom⁻¹-weakInverse {A} {B})
-- A bigroupoid is a bicategory that has a bigroupoid structure
record Bigroupoid (o ℓ e t : Level) : Set (suc (o ⊔ ℓ ⊔ e ⊔ t)) where
field
bicategory : Bicategory o ℓ e t
isBigroupoid : IsBigroupoid bicategory
open IsBigroupoid isBigroupoid public
| 51
| 171
| 0.521041
|
cbd1a50e554eb67b80cd328aaf3c8a506cdaad33
| 1,105
|
agda
|
Agda
|
agda/Algebra/Construct/Free/Semilattice/Relation/Unary/All/Dec.agda
|
oisdk/combinatorics-paper
|
3c176d4690566d81611080e9378f5a178b39b851
|
[
"MIT"
] | 6
|
2020-09-11T17:45:41.000Z
|
2021-11-16T08:11:34.000Z
|
agda/Algebra/Construct/Free/Semilattice/Relation/Unary/All/Dec.agda
|
oisdk/combinatorics-paper
|
3c176d4690566d81611080e9378f5a178b39b851
|
[
"MIT"
] | null | null | null |
agda/Algebra/Construct/Free/Semilattice/Relation/Unary/All/Dec.agda
|
oisdk/combinatorics-paper
|
3c176d4690566d81611080e9378f5a178b39b851
|
[
"MIT"
] | 1
|
2021-11-11T12:30:21.000Z
|
2021-11-11T12:30:21.000Z
|
{-# OPTIONS --cubical --safe #-}
module Algebra.Construct.Free.Semilattice.Relation.Unary.All.Dec where
open import Prelude hiding (⊥; ⊤)
open import Algebra.Construct.Free.Semilattice.Eliminators
open import Algebra.Construct.Free.Semilattice.Definition
open import Cubical.Foundations.HLevels
open import Data.Empty.UniversePolymorphic
open import HITs.PropositionalTruncation.Sugar
open import HITs.PropositionalTruncation.Properties
open import HITs.PropositionalTruncation
open import Data.Unit.UniversePolymorphic
open import Algebra.Construct.Free.Semilattice.Relation.Unary.All.Def
open import Relation.Nullary
open import Relation.Nullary.Decidable
open import Relation.Nullary.Decidable.Properties
open import Relation.Nullary.Decidable.Logic
private
variable p : Level
◻′? : ∀ {P : A → Type p} → (∀ x → Dec (P x)) → xs ∈𝒦 A ⇒∥ Dec (◻ P xs) ∥
∥ ◻′? {P = P} P? ∥-prop {xs} = isPropDec (isProp-◻ {P = P} {xs = xs})
∥ ◻′? P? ∥[] = yes tt
∥ ◻′? P? ∥ x ∷ xs ⟨ Pxs ⟩ = map-dec ∣_∣ refute-trunc (P? x) && Pxs
◻? : ∀ {P : A → Type p} → (∀ x → Dec (P x)) → ∀ xs → Dec (◻ P xs)
◻? P? = ∥ ◻′? P? ∥⇓
| 36.833333
| 72
| 0.702262
|
dfb2c053ff8c4285190cf79355c719dc83a1adef
| 1,169
|
agda
|
Agda
|
prototyping/PrettyPrinter.agda
|
FreakingBarbarians/luau
|
5187e64f88953f34785ffe58acd0610ee5041f5f
|
[
"MIT"
] | 1
|
2022-02-11T21:30:17.000Z
|
2022-02-11T21:30:17.000Z
|
prototyping/PrettyPrinter.agda
|
FreakingBarbarians/luau
|
5187e64f88953f34785ffe58acd0610ee5041f5f
|
[
"MIT"
] | null | null | null |
prototyping/PrettyPrinter.agda
|
FreakingBarbarians/luau
|
5187e64f88953f34785ffe58acd0610ee5041f5f
|
[
"MIT"
] | null | null | null |
module PrettyPrinter where
open import Agda.Builtin.IO using (IO)
open import Agda.Builtin.Int using (pos)
open import Agda.Builtin.Unit using (⊤)
open import FFI.IO using (getContents; putStrLn; _>>=_; _>>_)
open import FFI.Data.Aeson using (Value; eitherDecode)
open import FFI.Data.Either using (Left; Right)
open import FFI.Data.String using (String; _++_)
open import FFI.Data.Text.Encoding using (encodeUtf8)
open import FFI.System.Exit using (exitWith; ExitFailure)
open import Luau.Syntax using (Block)
open import Luau.Syntax.FromJSON using (blockFromJSON)
open import Luau.Syntax.ToString using (blockToString)
runBlock : Block → IO ⊤
runBlock block = putStrLn (blockToString block)
runJSON : Value → IO ⊤
runJSON value with blockFromJSON(value)
runJSON value | (Left err) = putStrLn ("Luau error: " ++ err) >> exitWith (ExitFailure (pos 1))
runJSON value | (Right block) = runBlock block
runString : String → IO ⊤
runString txt with eitherDecode (encodeUtf8 txt)
runString txt | (Left err) = putStrLn ("JSON error: " ++ err) >> exitWith (ExitFailure (pos 1))
runString txt | (Right value) = runJSON value
main : IO ⊤
main = getContents >>= runString
| 34.382353
| 95
| 0.745937
|
23424fe57bb0766a8c3107fe5e7e3f6c8d33eda8
| 22,010
|
agda
|
Agda
|
Cubical/HITs/Sn/Properties.agda
|
FernandoLarrain/cubical
|
9acdecfa6437ec455568be4e5ff04849cc2bc13b
|
[
"MIT"
] | 1
|
2022-02-05T01:25:02.000Z
|
2022-02-05T01:25:02.000Z
|
Cubical/HITs/Sn/Properties.agda
|
FernandoLarrain/cubical
|
9acdecfa6437ec455568be4e5ff04849cc2bc13b
|
[
"MIT"
] | null | null | null |
Cubical/HITs/Sn/Properties.agda
|
FernandoLarrain/cubical
|
9acdecfa6437ec455568be4e5ff04849cc2bc13b
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --safe #-}
module Cubical.HITs.Sn.Properties where
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Path
open import Cubical.Foundations.Function
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Transport
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.Univalence
open import Cubical.HITs.S1 renaming (_·_ to _*_)
open import Cubical.Data.Nat hiding (elim)
open import Cubical.Data.Sigma
open import Cubical.HITs.Sn.Base
open import Cubical.HITs.Susp
open import Cubical.HITs.Truncation
-- open import Cubical.Homotopy.Loopspace
open import Cubical.Homotopy.Connected
open import Cubical.HITs.Join
open import Cubical.Data.Bool
private
variable
ℓ : Level
IsoSucSphereSusp : (n : ℕ) → Iso (S₊ (suc n)) (Susp (S₊ n))
IsoSucSphereSusp zero = S¹IsoSuspBool
IsoSucSphereSusp (suc n) = idIso
IsoSucSphereSusp∙ : (n : ℕ)
→ Iso.inv (IsoSucSphereSusp n) north ≡ ptSn (suc n)
IsoSucSphereSusp∙ zero = refl
IsoSucSphereSusp∙ (suc n) = refl
-- Elimination principles for spheres
sphereElim : (n : ℕ) {A : (S₊ (suc n)) → Type ℓ} → ((x : S₊ (suc n)) → isOfHLevel (suc n) (A x))
→ A (ptSn (suc n))
→ (x : S₊ (suc n)) → A x
sphereElim zero hlev pt = toPropElim hlev pt
sphereElim (suc n) hlev pt north = pt
sphereElim (suc n) {A = A} hlev pt south = subst A (merid (ptSn (suc n))) pt
sphereElim (suc n) {A = A} hlev pt (merid a i) =
sphereElim n {A = λ a → PathP (λ i → A (merid a i)) pt (subst A (merid (ptSn (suc n))) pt)}
(λ a → isOfHLevelPathP' (suc n) (hlev south) _ _)
(λ i → transp (λ j → A (merid (ptSn (suc n)) (i ∧ j))) (~ i) pt)
a i
sphereElim2 : ∀ {ℓ} (n : ℕ) {A : (S₊ (suc n)) → (S₊ (suc n)) → Type ℓ}
→ ((x y : S₊ (suc n)) → isOfHLevel (suc n) (A x y))
→ A (ptSn (suc n)) (ptSn (suc n))
→ (x y : S₊ (suc n)) → A x y
sphereElim2 n hlev pt = sphereElim n (λ _ → isOfHLevelΠ (suc n) λ _ → hlev _ _)
(sphereElim n (hlev _ ) pt)
private
compPath-lem : ∀ {ℓ} {A : Type ℓ} {x y z : A} (p : x ≡ y) (q : z ≡ y)
→ PathP (λ i → (p ∙ sym q) i ≡ y) p q
compPath-lem {y = y} p q i j =
hcomp (λ k → λ { (i = i0) → p j
; (i = i1) → q (~ k ∨ j)
; (j = i1) → y })
(p (j ∨ i))
sphereToPropElim : (n : ℕ) {A : (S₊ (suc n)) → Type ℓ} → ((x : S₊ (suc n)) → isProp (A x))
→ A (ptSn (suc n))
→ (x : S₊ (suc n)) → A x
sphereToPropElim zero = toPropElim
sphereToPropElim (suc n) hlev pt north = pt
sphereToPropElim (suc n) {A = A} hlev pt south = subst A (merid (ptSn (suc n))) pt
sphereToPropElim (suc n) {A = A} hlev pt (merid a i) =
isProp→PathP {B = λ i → A (merid a i)} (λ _ → hlev _) pt (subst A (merid (ptSn (suc n))) pt) i
-- Elimination rule for fibrations (x : Sⁿ) → (y : Sᵐ) → A x y of h-Level (n + m).
-- The following principle is just the special case of the "Wedge Connectivity Lemma"
-- for spheres (See Cubical.Homotopy.WedgeConnectivity or chapter 8.6 in the HoTT book).
-- We prove it directly here for three reasons:
-- (i) it should perform better
-- (ii) we get a slightly stronger statement for spheres: one of the homotopies will, by design, be refl
-- (iii) the fact that the two homotopies only differ by (composition with) the homotopy leftFunction(base) ≡ rightFunction(base)
-- is close to trivial
wedgeconFun : (n m : ℕ) {A : (S₊ (suc n)) → (S₊ (suc m)) → Type ℓ}
→ ((x : S₊ (suc n)) (y : S₊ (suc m)) → isOfHLevel ((suc n) + (suc m)) (A x y))
→ (f : (x : _) → A (ptSn (suc n)) x)
→ (g : (x : _) → A x (ptSn (suc m)))
→ (g (ptSn (suc n)) ≡ f (ptSn (suc m)))
→ (x : S₊ (suc n)) (y : S₊ (suc m)) → A x y
wedgeconLeft : (n m : ℕ) {A : (S₊ (suc n)) → (S₊ (suc m)) → Type ℓ}
→ (hLev : ((x : S₊ (suc n)) (y : S₊ (suc m)) → isOfHLevel ((suc n) + (suc m)) (A x y)))
→ (f : (x : _) → A (ptSn (suc n)) x)
→ (g : (x : _) → A x (ptSn (suc m)))
→ (hom : g (ptSn (suc n)) ≡ f (ptSn (suc m)))
→ (x : _) → wedgeconFun n m hLev f g hom (ptSn (suc n)) x ≡ f x
wedgeconRight : (n m : ℕ) {A : (S₊ (suc n)) → (S₊ (suc m)) → Type ℓ}
→ (hLev : ((x : S₊ (suc n)) (y : S₊ (suc m)) → isOfHLevel ((suc n) + (suc m)) (A x y)))
→ (f : (x : _) → A (ptSn (suc n)) x)
→ (g : (x : _) → A x (ptSn (suc m)))
→ (hom : g (ptSn (suc n)) ≡ f (ptSn (suc m)))
→ (x : _) → wedgeconFun n m hLev f g hom x (ptSn (suc m)) ≡ g x
wedgeconFun zero zero {A = A} hlev f g hom = F
where
helper : SquareP (λ i j → A (loop i) (loop j)) (cong f loop) (cong f loop)
(λ i → hcomp (λ k → λ { (i = i0) → hom k
; (i = i1) → hom k })
(g (loop i)))
λ i → hcomp (λ k → λ { (i = i0) → hom k
; (i = i1) → hom k })
(g (loop i))
helper = toPathP (isOfHLevelPathP' 1 (hlev _ _) _ _ _ _)
F : (x y : S¹) → A x y
F base y = f y
F (loop i) base = hcomp (λ k → λ { (i = i0) → hom k
; (i = i1) → hom k })
(g (loop i))
F (loop i) (loop j) = helper i j
wedgeconFun zero (suc m) {A = A} hlev f g hom = F₀
module _ where
transpLemma₀ : (x : S₊ (suc m)) → transport (λ i₁ → A base (merid x i₁)) (g base) ≡ f south
transpLemma₀ x = cong (transport (λ i₁ → A base (merid x i₁)))
hom
∙ (λ i → transp (λ j → A base (merid x (i ∨ j))) i
(f (merid x i)))
pathOverMerid₀ : (x : S₊ (suc m)) → PathP (λ i₁ → A base (merid x i₁))
(g base)
(transport (λ i₁ → A base (merid (ptSn (suc m)) i₁))
(g base))
pathOverMerid₀ x i = hcomp (λ k → λ { (i = i0) → g base
; (i = i1) → (transpLemma₀ x ∙ sym (transpLemma₀ (ptSn (suc m)))) k})
(transp (λ i₁ → A base (merid x (i₁ ∧ i))) (~ i)
(g base))
pathOverMeridId₀ : pathOverMerid₀ (ptSn (suc m)) ≡ λ i → transp (λ i₁ → A base (merid (ptSn (suc m)) (i₁ ∧ i))) (~ i)
(g base)
pathOverMeridId₀ =
(λ j i → hcomp (λ k → λ {(i = i0) → g base
; (i = i1) → rCancel (transpLemma₀ (ptSn (suc m))) j k})
(transp (λ i₁ → A base (merid (ptSn (suc m)) (i₁ ∧ i))) (~ i)
(g base)))
∙ λ j i → hfill (λ k → λ { (i = i0) → g base
; (i = i1) → transport (λ i₁ → A base (merid (ptSn (suc m)) i₁))
(g base)})
(inS (transp (λ i₁ → A base (merid (ptSn (suc m)) (i₁ ∧ i))) (~ i)
(g base))) (~ j)
indStep₀ : (x : _) (a : _) → PathP (λ i → A x (merid a i))
(g x)
(subst (λ y → A x y) (merid (ptSn (suc m)))
(g x))
indStep₀ = wedgeconFun zero m (λ _ _ → isOfHLevelPathP' (2 + m) (hlev _ _) _ _)
pathOverMerid₀
(λ a i → transp (λ i₁ → A a (merid (ptSn (suc m)) (i₁ ∧ i))) (~ i)
(g a))
(sym pathOverMeridId₀)
F₀ : (x : S¹) (y : Susp (S₊ (suc m))) → A x y
F₀ x north = g x
F₀ x south = subst (λ y → A x y) (merid (ptSn (suc m))) (g x)
F₀ x (merid a i) = indStep₀ x a i
wedgeconFun (suc n) m {A = A} hlev f g hom = F₁
module _ where
transpLemma₁ : (x : S₊ (suc n)) → transport (λ i₁ → A (merid x i₁) (ptSn (suc m))) (f (ptSn (suc m))) ≡ g south
transpLemma₁ x = cong (transport (λ i₁ → A (merid x i₁) (ptSn (suc m))))
(sym hom)
∙ (λ i → transp (λ j → A (merid x (i ∨ j)) (ptSn (suc m))) i
(g (merid x i)))
pathOverMerid₁ : (x : S₊ (suc n)) → PathP (λ i₁ → A (merid x i₁) (ptSn (suc m)))
(f (ptSn (suc m)))
(transport (λ i₁ → A (merid (ptSn (suc n)) i₁) (ptSn (suc m)))
(f (ptSn (suc m))))
pathOverMerid₁ x i = hcomp (λ k → λ { (i = i0) → f (ptSn (suc m))
; (i = i1) → (transpLemma₁ x ∙ sym (transpLemma₁ (ptSn (suc n)))) k })
(transp (λ i₁ → A (merid x (i₁ ∧ i)) (ptSn (suc m))) (~ i)
(f (ptSn (suc m))))
pathOverMeridId₁ : pathOverMerid₁ (ptSn (suc n)) ≡ λ i → transp (λ i₁ → A (merid (ptSn (suc n)) (i₁ ∧ i)) (ptSn (suc m))) (~ i)
(f (ptSn (suc m)))
pathOverMeridId₁ =
(λ j i → hcomp (λ k → λ { (i = i0) → f (ptSn (suc m))
; (i = i1) → rCancel (transpLemma₁ (ptSn (suc n))) j k })
(transp (λ i₁ → A (merid (ptSn (suc n)) (i₁ ∧ i)) (ptSn (suc m))) (~ i)
(f (ptSn (suc m)))))
∙ λ j i → hfill (λ k → λ { (i = i0) → f (ptSn (suc m))
; (i = i1) → transport (λ i₁ → A (merid (ptSn (suc n)) i₁) (ptSn (suc m)))
(f (ptSn (suc m))) })
(inS (transp (λ i₁ → A (merid (ptSn (suc n)) (i₁ ∧ i)) (ptSn (suc m))) (~ i)
(f (ptSn (suc m))))) (~ j)
indStep₁ : (a : _) (y : _) → PathP (λ i → A (merid a i) y)
(f y)
(subst (λ x → A x y) (merid (ptSn (suc n)))
(f y))
indStep₁ = wedgeconFun n m (λ _ _ → isOfHLevelPathP' (suc (n + suc m)) (hlev _ _) _ _)
(λ a i → transp (λ i₁ → A (merid (ptSn (suc n)) (i₁ ∧ i)) a) (~ i)
(f a))
pathOverMerid₁
pathOverMeridId₁
F₁ : (x : Susp (S₊ (suc n))) (y : S₊ (suc m)) → A x y
F₁ north y = f y
F₁ south y = subst (λ x → A x y) (merid (ptSn (suc n))) (f y)
F₁ (merid a i) y = indStep₁ a y i
wedgeconRight zero zero {A = A} hlev f g hom = right
where
right : (x : S¹) → _
right base = sym hom
right (loop i) j = hcomp (λ k → λ { (i = i0) → hom (~ j ∧ k)
; (i = i1) → hom (~ j ∧ k)
; (j = i1) → g (loop i) })
(g (loop i))
wedgeconRight zero (suc m) {A = A} hlev f g hom x = refl
wedgeconRight (suc n) m {A = A} hlev f g hom = right
where
lem : (x : _) → indStep₁ n m hlev f g hom x (ptSn (suc m)) ≡ _
lem = wedgeconRight n m (λ _ _ → isOfHLevelPathP' (suc (n + suc m)) (hlev _ _) _ _)
(λ a i → transp (λ i₁ → A (merid (ptSn (suc n)) (i₁ ∧ i)) a) (~ i)
(f a))
(pathOverMerid₁ n m hlev f g hom)
(pathOverMeridId₁ n m hlev f g hom)
right : (x : Susp (S₊ (suc n))) → _ ≡ g x
right north = sym hom
right south = cong (subst (λ x → A x (ptSn (suc m)))
(merid (ptSn (suc n))))
(sym hom)
∙ λ i → transp (λ j → A (merid (ptSn (suc n)) (i ∨ j)) (ptSn (suc m))) i
(g (merid (ptSn (suc n)) i))
right (merid a i) j =
hcomp (λ k → λ { (i = i0) → hom (~ j)
; (i = i1) → transpLemma₁ n m hlev f g hom (ptSn (suc n)) j
; (j = i0) → lem a (~ k) i
; (j = i1) → g (merid a i)})
(hcomp (λ k → λ { (i = i0) → hom (~ j)
; (i = i1) → compPath-lem (transpLemma₁ n m hlev f g hom a) (transpLemma₁ n m hlev f g hom (ptSn (suc n))) k j
; (j = i1) → g (merid a i)})
(hcomp (λ k → λ { (i = i0) → hom (~ j)
; (j = i0) → transp (λ i₂ → A (merid a (i₂ ∧ i)) (ptSn (suc m))) (~ i)
(f (ptSn (suc m)))
; (j = i1) → transp (λ j → A (merid a (i ∧ (j ∨ k))) (ptSn (suc m))) (k ∨ ~ i)
(g (merid a (i ∧ k))) })
(transp (λ i₂ → A (merid a (i₂ ∧ i)) (ptSn (suc m))) (~ i)
(hom (~ j)))))
wedgeconLeft zero zero {A = A} hlev f g hom x = refl
wedgeconLeft zero (suc m) {A = A} hlev f g hom = help
where
left₁ : (x : _) → indStep₀ m hlev f g hom base x ≡ _
left₁ = wedgeconLeft zero m (λ _ _ → isOfHLevelPathP' (2 + m) (hlev _ _) _ _)
(pathOverMerid₀ m hlev f g hom)
(λ a i → transp (λ i₁ → A a (merid (ptSn (suc m)) (i₁ ∧ i))) (~ i)
(g a))
(sym (pathOverMeridId₀ m hlev f g hom))
help : (x : S₊ (suc (suc m))) → _
help north = hom
help south = cong (subst (A base) (merid (ptSn (suc m)))) hom
∙ λ i → transp (λ j → A base (merid (ptSn (suc m)) (i ∨ j))) i
(f (merid (ptSn (suc m)) i))
help (merid a i) j =
hcomp (λ k → λ { (i = i0) → hom j
; (i = i1) → transpLemma₀ m hlev f g hom (ptSn (suc m)) j
; (j = i0) → left₁ a (~ k) i
; (j = i1) → f (merid a i)})
(hcomp (λ k → λ { (i = i0) → hom j
; (i = i1) → compPath-lem (transpLemma₀ m hlev f g hom a)
(transpLemma₀ m hlev f g hom (ptSn (suc m))) k j
; (j = i1) → f (merid a i)})
(hcomp (λ k → λ { (i = i0) → hom j
; (j = i0) → transp (λ i₂ → A base (merid a (i₂ ∧ i))) (~ i)
(g base)
; (j = i1) → transp (λ j → A base (merid a (i ∧ (j ∨ k)))) (k ∨ ~ i)
(f (merid a (i ∧ k)))})
(transp (λ i₂ → A base (merid a (i₂ ∧ i))) (~ i)
(hom j))))
wedgeconLeft (suc n) m {A = A} hlev f g hom _ = refl
---------- Connectedness -----------
sphereConnected : (n : HLevel) → isConnected (suc n) (S₊ n)
sphereConnected n = ∣ ptSn n ∣ , elim (λ _ → isOfHLevelPath (suc n) (isOfHLevelTrunc (suc n)) _ _)
(λ a → sym (spoke ∣_∣ (ptSn n)) ∙ spoke ∣_∣ a)
-- The fact that path spaces of Sn are connected can be proved directly for Sⁿ.
-- (Unfortunately, this does not work for higher paths)
pathIdTruncSⁿ : (n : ℕ) (x y : S₊ (suc n))
→ Path (hLevelTrunc (2 + n) (S₊ (suc n))) ∣ x ∣ ∣ y ∣
→ hLevelTrunc (suc n) (x ≡ y)
pathIdTruncSⁿ n = sphereElim n (λ _ → isOfHLevelΠ (suc n) λ _ → isOfHLevelΠ (suc n) λ _ → isOfHLevelTrunc (suc n))
(sphereElim n (λ _ → isOfHLevelΠ (suc n) λ _ → isOfHLevelTrunc (suc n))
λ _ → ∣ refl ∣)
pathIdTruncSⁿ⁻ : (n : ℕ) (x y : S₊ (suc n))
→ hLevelTrunc (suc n) (x ≡ y)
→ Path (hLevelTrunc (2 + n) (S₊ (suc n))) ∣ x ∣ ∣ y ∣
pathIdTruncSⁿ⁻ n x y = rec (isOfHLevelTrunc (2 + n) _ _)
(J (λ y _ → Path (hLevelTrunc (2 + n) (S₊ (suc n))) ∣ x ∣ ∣ y ∣) refl)
pathIdTruncSⁿretract : (n : ℕ) (x y : S₊ (suc n)) → (p : hLevelTrunc (suc n) (x ≡ y)) → pathIdTruncSⁿ n x y (pathIdTruncSⁿ⁻ n x y p) ≡ p
pathIdTruncSⁿretract n =
sphereElim n (λ _ → isOfHLevelΠ (suc n) λ _ → isOfHLevelΠ (suc n) λ _ → isOfHLevelPath (suc n) (isOfHLevelTrunc (suc n)) _ _)
λ y → elim (λ _ → isOfHLevelPath (suc n) (isOfHLevelTrunc (suc n)) _ _)
(J (λ y p → pathIdTruncSⁿ n (ptSn (suc n)) y (pathIdTruncSⁿ⁻ n (ptSn (suc n)) y ∣ p ∣) ≡ ∣ p ∣)
(cong (pathIdTruncSⁿ n (ptSn (suc n)) (ptSn (suc n))) (transportRefl refl) ∙ pm-help n))
where
pm-help : (n : ℕ) → pathIdTruncSⁿ n (ptSn (suc n)) (ptSn (suc n)) refl ≡ ∣ refl ∣
pm-help zero = refl
pm-help (suc n) = refl
isConnectedPathSⁿ : (n : ℕ) (x y : S₊ (suc n)) → isConnected (suc n) (x ≡ y)
isConnectedPathSⁿ n x y =
isContrRetract
(pathIdTruncSⁿ⁻ n x y)
(pathIdTruncSⁿ n x y)
(pathIdTruncSⁿretract n x y)
((isContr→isProp (sphereConnected (suc n)) ∣ x ∣ ∣ y ∣)
, isProp→isSet (isContr→isProp (sphereConnected (suc n))) _ _ _)
-- Equivalence Sⁿ*Sᵐ≃Sⁿ⁺ᵐ⁺¹
IsoSphereJoin : (n m : ℕ)
→ Iso (join (S₊ n) (S₊ m)) (S₊ (suc (n + m)))
IsoSphereJoin zero m =
compIso join-comm
(compIso (invIso Susp-iso-joinBool)
(invIso (IsoSucSphereSusp m)))
IsoSphereJoin (suc n) m =
compIso (Iso→joinIso
(compIso (pathToIso (cong S₊ (cong suc (+-comm zero n))))
(invIso (IsoSphereJoin n 0)))
idIso)
(compIso (equivToIso joinAssocDirect)
(compIso (Iso→joinIso idIso
(compIso join-comm
(compIso (invIso Susp-iso-joinBool)
(invIso (IsoSucSphereSusp m)))))
(compIso
(IsoSphereJoin n (suc m))
(pathToIso λ i → S₊ (suc (+-suc n m i))))))
IsoSphereJoinPres∙ : (n m : ℕ)
→ Iso.fun (IsoSphereJoin n m) (inl (ptSn n)) ≡ ptSn (suc (n + m))
IsoSphereJoinPres∙ zero zero = refl
IsoSphereJoinPres∙ zero (suc m) = refl
IsoSphereJoinPres∙ (suc n) m =
cong (transport (λ i → S₊ (suc (+-suc n m i))))
(cong (Iso.fun (IsoSphereJoin n (suc m)))
(cong (join→ (idfun (S₊ n))
(λ x →
Iso.inv (IsoSucSphereSusp m)
(Iso.inv Susp-iso-joinBool (join-commFun x))))
(cong (joinAssocDirect {C = S₊ m} .fst)
(cong (inl ∘ (Iso.inv (IsoSphereJoin n 0)))
(transportS∙ (suc n) _ (cong suc (+-comm 0 n)))
∙ cong inl (sym (cong (Iso.inv (IsoSphereJoin n 0))
(IsoSphereJoinPres∙ n 0))
∙ Iso.leftInv (IsoSphereJoin n 0) (inl (ptSn _))))))
∙ IsoSphereJoinPres∙ n (suc m))
∙ transportS∙ _ _ (cong suc (+-suc n m))
where
transportS∙ : (n m : ℕ) (p : n ≡ m) → transport (λ i → S₊ (p i)) (ptSn n)
≡ ptSn _
transportS∙ zero m =
J (λ m p → transport (λ i → S₊ (p i)) true ≡ ptSn m) refl
transportS∙ (suc zero) m =
J (λ m p → transport (λ i → S₊ (p i)) base ≡ ptSn m) refl
transportS∙ (suc (suc n)) m =
J (λ m p → transport (λ i → S₊ (p i)) north ≡ ptSn m) refl
IsoSphereJoin⁻Pres∙ : (n m : ℕ)
→ Iso.inv (IsoSphereJoin n m) (ptSn (suc (n + m))) ≡ inl (ptSn n)
IsoSphereJoin⁻Pres∙ n m =
cong (Iso.inv (IsoSphereJoin n m)) (sym (IsoSphereJoinPres∙ n m))
∙ Iso.leftInv (IsoSphereJoin n m) (inl (ptSn n))
-- Some lemmas on the H
rUnitS¹ : (x : S¹) → x * base ≡ x
rUnitS¹ base = refl
rUnitS¹ (loop i₁) = refl
commS¹ : (a x : S¹) → a * x ≡ x * a
commS¹ = wedgeconFun _ _ (λ _ _ → isGroupoidS¹ _ _)
(sym ∘ rUnitS¹)
rUnitS¹
refl
SuspS¹-hom : (a x : S¹)
→ Path (Path (hLevelTrunc 4 (S₊ 2)) _ _)
(cong ∣_∣ₕ (merid (a * x) ∙ sym (merid base)))
(cong ∣_∣ₕ (merid a ∙ sym (merid base))
∙ (cong ∣_∣ₕ (merid x ∙ sym (merid base))))
SuspS¹-hom = wedgeconFun _ _ (λ _ _ → isOfHLevelTrunc 4 _ _ _ _)
(λ x → lUnit _
∙ cong (_∙ cong ∣_∣ₕ (merid x ∙ sym (merid base)))
(cong (cong ∣_∣ₕ) (sym (rCancel (merid base)))))
(λ x → (λ i → cong ∣_∣ₕ (merid (rUnitS¹ x i) ∙ sym (merid base)))
∙∙ rUnit _
∙∙ cong (cong ∣_∣ₕ (merid x ∙ sym (merid base)) ∙_)
(cong (cong ∣_∣ₕ) (sym (rCancel (merid base)))))
(sym (l (cong ∣_∣ₕ (merid base ∙ sym (merid base)))
(cong (cong ∣_∣ₕ) (sym (rCancel (merid base))))))
where
l : ∀ {ℓ} {A : Type ℓ} {x : A} (p : x ≡ x) (P : refl ≡ p)
→ lUnit p ∙ cong (_∙ p) P ≡ rUnit p ∙ cong (p ∙_) P
l p = J (λ p P → lUnit p ∙ cong (_∙ p) P ≡ rUnit p ∙ cong (p ∙_) P) refl
rCancelS¹ : (x : S¹) → ptSn 1 ≡ x * (invLooper x)
rCancelS¹ base = refl
rCancelS¹ (loop i) j =
hcomp (λ r → λ {(i = i0) → base ; (i = i1) → base ; (j = i0) → base})
base
SuspS¹-inv : (x : S¹) → Path (Path (hLevelTrunc 4 (S₊ 2)) _ _)
(cong ∣_∣ₕ (merid (invLooper x) ∙ sym (merid base)))
(cong ∣_∣ₕ (sym (merid x ∙ sym (merid base))))
SuspS¹-inv x = (lUnit _
∙∙ cong (_∙ cong ∣_∣ₕ (merid (invLooper x) ∙ sym (merid base)))
(sym (lCancel (cong ∣_∣ₕ (merid x ∙ sym (merid base)))))
∙∙ sym (assoc _ _ _))
∙∙ cong (sym (cong ∣_∣ₕ (merid x ∙ sym (merid base))) ∙_) lem
∙∙ (assoc _ _ _
∙∙ cong (_∙ (cong ∣_∣ₕ (sym (merid x ∙ sym (merid base)))))
(lCancel (cong ∣_∣ₕ (merid x ∙ sym (merid base))))
∙∙ sym (lUnit _))
where
lem : cong ∣_∣ₕ (merid x ∙ sym (merid base))
∙ cong ∣_∣ₕ (merid (invLooper x) ∙ sym (merid base))
≡ cong ∣_∣ₕ (merid x ∙ sym (merid base))
∙ cong ∣_∣ₕ (sym (merid x ∙ sym (merid base)))
lem = sym (SuspS¹-hom x (invLooper x))
∙ ((λ i → cong ∣_∣ₕ (merid (rCancelS¹ x (~ i)) ∙ sym (merid base)))
∙ cong (cong ∣_∣ₕ) (rCancel (merid base))) ∙ sym (rCancel _)
| 50.366133
| 138
| 0.449659
|
cb9073a270920e25a56454967eee5d54039f6f54
| 244
|
agda
|
Agda
|
Cubical/Relation/Binary.agda
|
Schippmunk/cubical
|
c345dc0c49d3950dc57f53ca5f7099bb53a4dc3a
|
[
"MIT"
] | null | null | null |
Cubical/Relation/Binary.agda
|
Schippmunk/cubical
|
c345dc0c49d3950dc57f53ca5f7099bb53a4dc3a
|
[
"MIT"
] | null | null | null |
Cubical/Relation/Binary.agda
|
Schippmunk/cubical
|
c345dc0c49d3950dc57f53ca5f7099bb53a4dc3a
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Relation.Binary where
open import Cubical.Relation.Binary.Base public
open import Cubical.Relation.Binary.Properties public
open import Cubical.Relation.Binary.Fiberwise public
| 34.857143
| 53
| 0.807377
|
1814a23ef9e572671b271cf5e5422776a178be23
| 4,092
|
agda
|
Agda
|
src/Categories/Adjoint/RAPL.agda
|
jaykru/agda-categories
|
a4053cf700bcefdf73b857c3352f1eae29382a60
|
[
"MIT"
] | 279
|
2019-06-01T14:36:40.000Z
|
2022-03-22T00:40:14.000Z
|
src/Categories/Adjoint/RAPL.agda
|
seanpm2001/agda-categories
|
d9e4f578b126313058d105c61707d8c8ae987fa8
|
[
"MIT"
] | 236
|
2019-06-01T14:53:54.000Z
|
2022-03-28T14:31:43.000Z
|
src/Categories/Adjoint/RAPL.agda
|
seanpm2001/agda-categories
|
d9e4f578b126313058d105c61707d8c8ae987fa8
|
[
"MIT"
] | 64
|
2019-06-02T16:58:15.000Z
|
2022-03-14T02:00:59.000Z
|
{-# OPTIONS --without-K --safe #-}
open import Categories.Category
open import Categories.Functor
open import Categories.Adjoint
-- Right Adjoint Preserves Limits.
module Categories.Adjoint.RAPL {o o′ ℓ ℓ′ e e′} {C : Category o ℓ e} {D : Category o′ ℓ′ e′}
{L : Functor C D} {R : Functor D C} (L⊣R : L ⊣ R) where
open import Categories.Functor.Properties
import Categories.Morphism.Reasoning as MR
import Categories.Diagram.Limit as Lim
import Categories.Category.Construction.Cones as Con
private
module C = Category C
module D = Category D
module L = Functor L
module R = Functor R
open Adjoint L⊣R
module _ {o″ ℓ″ e″} {J : Category o″ ℓ″ e″} (F : Functor J D) where
private
module F = Functor F
module LF = Lim F
module CF = Con F
RF = R ∘F F
module LRF = Lim RF
module CRF = Con RF
rapl : LF.Limit → LRF.Limit
rapl lim = record
{ terminal = record
{ ⊤ = ⊤
; ⊤-is-terminal = record
{ ! = !
; !-unique = !-unique
}
}
}
where module lim = LF.Limit lim
open lim
⊤ : CRF.Cone
⊤ = record
{ N = R.F₀ apex
; apex = record
{ ψ = λ X → R.F₁ (proj X)
; commute = λ f → [ R ]-resp-∘ (limit-commute f)
}
}
K′ : CRF.Cone → CF.Cone
K′ K = record
{ N = L.F₀ K.N
; apex = record
{ ψ = λ X → counit.η (F.F₀ X) D.∘ L.F₁ (K.ψ X)
; commute = λ {X Y} f → begin
F.F₁ f D.∘ counit.η (F.F₀ X) D.∘ L.F₁ (K.ψ X)
≈˘⟨ pushˡ (counit.commute (F.F₁ f)) ⟩
(counit.η (F.F₀ Y) D.∘ L.F₁ (R.F₁ (F.F₁ f))) D.∘ L.F₁ (K.ψ X)
≈⟨ pullʳ ([ L ]-resp-∘ (K.commute f)) ⟩
counit.η (F.F₀ Y) D.∘ L.F₁ (K.ψ Y)
∎
}
}
where module K = CRF.Cone K
open D.HomReasoning
open MR D
module K′ K = CF.Cone (K′ K)
! : ∀ {K : CRF.Cone} → CRF.Cones [ K , ⊤ ]
! {K} = record
{ arr = R.F₁ (rep (K′ K)) C.∘ unit.η K.N
; commute = commute′
}
where module K = CRF.Cone K
commute′ : ∀ {X} → R.F₁ (proj X) C.∘ R.F₁ (rep (K′ K)) C.∘ unit.η K.N C.≈ K.ψ X
commute′ {X} = begin
R.F₁ (proj X) C.∘ R.F₁ (rep (K′ K)) C.∘ unit.η K.N
≈⟨ pullˡ ([ R ]-resp-∘ commute) ⟩
R.F₁ (K′.ψ K X) C.∘ unit.η K.N
≈⟨ LRadjunct≈id ⟩
K.ψ X
∎
where open C.HomReasoning
open MR C
module ! {K} = CRF.Cone⇒ (! {K})
!-unique : ∀ {K : CRF.Cone} (f : CRF.Cones [ K , ⊤ ]) → CRF.Cones [ ! ≈ f ]
!-unique {K} f =
let open C.HomReasoning
open MR C
in begin
R.F₁ (rep (K′ K)) C.∘ unit.η K.N ≈⟨ R.F-resp-≈ (terminal.!-unique f′) ⟩∘⟨refl ⟩
Ladjunct (Radjunct f.arr) ≈⟨ LRadjunct≈id ⟩
f.arr ∎
where module K = CRF.Cone K
module f = CRF.Cone⇒ f
f′ : CF.Cones [ K′ K , limit ]
f′ = record
{ arr = Radjunct f.arr
; commute = λ {X} → begin
proj X D.∘ Radjunct f.arr ≈˘⟨ pushˡ (counit.commute (proj X)) ⟩
(counit.η (F.F₀ X) D.∘ L.F₁ (R.F₁ (proj X))) D.∘ L.F₁ f.arr ≈˘⟨ pushʳ L.homomorphism ⟩
Radjunct (R.F₁ (proj X) C.∘ f.arr) ≈⟨ Radjunct-resp-≈ f.commute ⟩
Radjunct (K.ψ X) ∎
}
where open D.HomReasoning
open MR D
| 34.386555
| 119
| 0.39956
|
c501c0900987bfa92108700dcf553ea4d5c3af7f
| 227
|
agda
|
Agda
|
test/succeed/Issue561.agda
|
dagit/agda
|
4383a3d20328a6c43689161496cee8eb479aca08
|
[
"MIT"
] | 1
|
2019-11-27T07:26:06.000Z
|
2019-11-27T07:26:06.000Z
|
test/succeed/Issue561.agda
|
dagit/agda
|
4383a3d20328a6c43689161496cee8eb479aca08
|
[
"MIT"
] | null | null | null |
test/succeed/Issue561.agda
|
dagit/agda
|
4383a3d20328a6c43689161496cee8eb479aca08
|
[
"MIT"
] | null | null | null |
module Issue561 where
open import Common.Char
open import Common.Prelude
primitive
primIsDigit : Char → Bool
postulate
IO : Set → Set
return : ∀ {A} → A → IO A
{-# BUILTIN IO IO #-}
main : IO Bool
main = return true
| 13.352941
| 27
| 0.674009
|
dc7766198371bc669abd46421c4534dd983e1836
| 2,469
|
agda
|
Agda
|
notes/FOT/GroupTheory/FormalisationsSL.agda
|
asr/fotc
|
2fc9f2b81052a2e0822669f02036c5750371b72d
|
[
"MIT"
] | 11
|
2015-09-03T20:53:42.000Z
|
2021-09-12T16:09:54.000Z
|
notes/FOT/GroupTheory/FormalisationsSL.agda
|
asr/fotc
|
2fc9f2b81052a2e0822669f02036c5750371b72d
|
[
"MIT"
] | 2
|
2016-10-12T17:28:16.000Z
|
2017-01-01T14:34:26.000Z
|
notes/FOT/GroupTheory/FormalisationsSL.agda
|
asr/fotc
|
2fc9f2b81052a2e0822669f02036c5750371b72d
|
[
"MIT"
] | 3
|
2016-09-19T14:18:30.000Z
|
2018-03-14T08:50:00.000Z
|
------------------------------------------------------------------------------
-- Proving that two group theory formalisations are equivalents
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
-- We prove that group theory axioms based on the signature (G, ·, ε,)
-- (see for example [p. 39, 1]), i.e.
-- ∀ a b c. abc = a(bc)
-- ∀ a. εa = aε = a
-- ∀ a. ∃ a'. a'a = aa' = ε
-- are equivalents to the axioms based on the signature (G, ·, _⁻¹, ε,)
-- (see for example [2,3]), i.e.
-- ∀ a b c. abc = a(bc)
-- ∀ a. εa = aε = a
-- ∀ a. a⁻¹a = aa⁻¹ = ε
-- [1] C. C. Chang and H. J. Keisler. Model Theory, volume 73 of Studies
-- in Logic and the Foundations of Mathematics. North-Holland, 3rd
-- edition, 3rd impression 1992.
-- [2] Agda standard library_0.8.1 (see Algebra/Structures.agda)
-- [3] Coq implementation
-- (http://coq.inria.fr/pylons/contribs/files/GroupTheory/v8.3/GroupTheory.g1.html)
module FOT.GroupTheory.FormalisationsSL where
open import Data.Product
open import Relation.Binary.PropositionalEquality
------------------------------------------------------------------------------
-- NB. We only write the proof for the left-inverse property.
infixl 10 _·_ -- The symbol is '\cdot'.
postulate
G : Set -- The universe
ε : G -- The identity element.
_·_ : G → G → G -- The binary operation.
-- Left-inverse property based on the signature (G, ·, ε,).
leftInverse₁ : Set
leftInverse₁ = ∀ a → Σ G (λ a' → a' · a ≡ ε)
-- Left-inverse property based on the signature (G, ·, _⁻¹, ε,).
infix 11 _⁻¹
postulate _⁻¹ : G → G -- The inverse function.
leftInverse₂ : Set
leftInverse₂ = ∀ a → a ⁻¹ · a ≡ ε
-- From the left-inverse property based on the signature (G, ·, _⁻¹, ε,)
-- to the one based on the signature (G, ·, ε,).
leftInverse₂₋₁ : leftInverse₂ → leftInverse₁
leftInverse₂₋₁ h a = (a ⁻¹) , (h a)
-- From the left-inverse property based on the signature (G, ·, ε,) to
-- the one based on the signature (G, ·, _⁻¹, ε,).
--
-- In this case we prove the existence of the inverse function.
leftInverse₁₋₂ : leftInverse₁ → Σ (G → G) (λ f → ∀ a → f a · a ≡ ε)
leftInverse₁₋₂ h = f , prf
where
f : G → G -- The inverse function.
f a = proj₁ (h a)
prf : ∀ a → f a · a ≡ ε
prf a = proj₂ (h a)
| 30.109756
| 87
| 0.553665
|
18794b9f90c2bd57cb3b6f18b380480a878f66bb
| 310,255
|
agda
|
Agda
|
agda/Text/Greek/SBLGNT/1Cor.agda
|
scott-fleischman/GreekGrammar
|
915c46c27c7f8aad5907474d8484f2685a4cd6a7
|
[
"MIT"
] | 44
|
2015-05-29T14:48:51.000Z
|
2022-03-06T15:41:57.000Z
|
agda/Text/Greek/SBLGNT/1Cor.agda
|
scott-fleischman/GreekGrammar
|
915c46c27c7f8aad5907474d8484f2685a4cd6a7
|
[
"MIT"
] | 13
|
2015-05-28T20:04:08.000Z
|
2020-09-07T11:58:38.000Z
|
agda/Text/Greek/SBLGNT/1Cor.agda
|
scott-fleischman/GreekGrammar
|
915c46c27c7f8aad5907474d8484f2685a4cd6a7
|
[
"MIT"
] | 5
|
2015-02-27T22:34:13.000Z
|
2017-06-11T11:25:09.000Z
|
module Text.Greek.SBLGNT.1Cor where
open import Data.List
open import Text.Greek.Bible
open import Text.Greek.Script
open import Text.Greek.Script.Unicode
ΠΡΟΣ-ΚΟΡΙΝΘΙΟΥΣ-Α : List (Word)
ΠΡΟΣ-ΚΟΡΙΝΘΙΟΥΣ-Α =
word (Π ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.1.1"
∷ word (κ ∷ ∙λ ∷ η ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.1.1"
∷ word (ἀ ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.1.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.1.1"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.1.1"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.1.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.1.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.1"
∷ word (Σ ∷ ω ∷ σ ∷ θ ∷ έ ∷ ν ∷ η ∷ ς ∷ []) "1Cor.1.1"
∷ word (ὁ ∷ []) "1Cor.1.1"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ς ∷ []) "1Cor.1.1"
∷ word (τ ∷ ῇ ∷ []) "1Cor.1.2"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.1.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.1.2"
∷ word (ἡ ∷ γ ∷ ι ∷ α ∷ σ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.1.2"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1Cor.1.2"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.1.2"
∷ word (τ ∷ ῇ ∷ []) "1Cor.1.2"
∷ word (ο ∷ ὔ ∷ σ ∷ ῃ ∷ []) "1Cor.1.2"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.2"
∷ word (Κ ∷ ο ∷ ρ ∷ ί ∷ ν ∷ θ ∷ ῳ ∷ []) "1Cor.1.2"
∷ word (κ ∷ ∙λ ∷ η ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.1.2"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.1.2"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "1Cor.1.2"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.1.2"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.1.2"
∷ word (ἐ ∷ π ∷ ι ∷ κ ∷ α ∷ ∙λ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.1.2"
∷ word (τ ∷ ὸ ∷ []) "1Cor.1.2"
∷ word (ὄ ∷ ν ∷ ο ∷ μ ∷ α ∷ []) "1Cor.1.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.2"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.1.2"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.1.2"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.1.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.2"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.2"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "1Cor.1.2"
∷ word (τ ∷ ό ∷ π ∷ ῳ ∷ []) "1Cor.1.2"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.1.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.2"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.1.2"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "1Cor.1.3"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.1.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.3"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "1Cor.1.3"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "1Cor.1.3"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.1.3"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.1.3"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.1.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.3"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.1.3"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.1.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.3"
∷ word (Ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῶ ∷ []) "1Cor.1.4"
∷ word (τ ∷ ῷ ∷ []) "1Cor.1.4"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1Cor.1.4"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.1.4"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ο ∷ τ ∷ ε ∷ []) "1Cor.1.4"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1Cor.1.4"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.1.4"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.1.4"
∷ word (τ ∷ ῇ ∷ []) "1Cor.1.4"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ι ∷ []) "1Cor.1.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.4"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.1.4"
∷ word (τ ∷ ῇ ∷ []) "1Cor.1.4"
∷ word (δ ∷ ο ∷ θ ∷ ε ∷ ί ∷ σ ∷ ῃ ∷ []) "1Cor.1.4"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.1.4"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.4"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1Cor.1.4"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.1.4"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.1.5"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.5"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "1Cor.1.5"
∷ word (ἐ ∷ π ∷ ∙λ ∷ ο ∷ υ ∷ τ ∷ ί ∷ σ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.1.5"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.5"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1Cor.1.5"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.5"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "1Cor.1.5"
∷ word (∙λ ∷ ό ∷ γ ∷ ῳ ∷ []) "1Cor.1.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.5"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "1Cor.1.5"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.1.5"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.1.6"
∷ word (τ ∷ ὸ ∷ []) "1Cor.1.6"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.1.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.6"
∷ word (ἐ ∷ β ∷ ε ∷ β ∷ α ∷ ι ∷ ώ ∷ θ ∷ η ∷ []) "1Cor.1.6"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.6"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.1.6"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.1.7"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.1.7"
∷ word (μ ∷ ὴ ∷ []) "1Cor.1.7"
∷ word (ὑ ∷ σ ∷ τ ∷ ε ∷ ρ ∷ ε ∷ ῖ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.1.7"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.7"
∷ word (μ ∷ η ∷ δ ∷ ε ∷ ν ∷ ὶ ∷ []) "1Cor.1.7"
∷ word (χ ∷ α ∷ ρ ∷ ί ∷ σ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.1.7"
∷ word (ἀ ∷ π ∷ ε ∷ κ ∷ δ ∷ ε ∷ χ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.1.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.1.7"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ ά ∷ ∙λ ∷ υ ∷ ψ ∷ ι ∷ ν ∷ []) "1Cor.1.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.7"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.1.7"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.1.7"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.1.7"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.7"
∷ word (ὃ ∷ ς ∷ []) "1Cor.1.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.8"
∷ word (β ∷ ε ∷ β ∷ α ∷ ι ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.1.8"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.1.8"
∷ word (ἕ ∷ ω ∷ ς ∷ []) "1Cor.1.8"
∷ word (τ ∷ έ ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.1.8"
∷ word (ἀ ∷ ν ∷ ε ∷ γ ∷ κ ∷ ∙λ ∷ ή ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.1.8"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.8"
∷ word (τ ∷ ῇ ∷ []) "1Cor.1.8"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "1Cor.1.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.8"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.1.8"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.1.8"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.1.8"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.8"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.1.9"
∷ word (ὁ ∷ []) "1Cor.1.9"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.1.9"
∷ word (δ ∷ ι ∷ []) "1Cor.1.9"
∷ word (ο ∷ ὗ ∷ []) "1Cor.1.9"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ή ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.1.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.1.9"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "1Cor.1.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.9"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "1Cor.1.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.9"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.1.9"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.9"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.1.9"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.1.9"
∷ word (Π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "1Cor.1.10"
∷ word (δ ∷ ὲ ∷ []) "1Cor.1.10"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.1.10"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.1.10"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.1.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.10"
∷ word (ὀ ∷ ν ∷ ό ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.1.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.10"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.1.10"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.1.10"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.1.10"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.10"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.1.10"
∷ word (τ ∷ ὸ ∷ []) "1Cor.1.10"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "1Cor.1.10"
∷ word (∙λ ∷ έ ∷ γ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.1.10"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.1.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.10"
∷ word (μ ∷ ὴ ∷ []) "1Cor.1.10"
∷ word (ᾖ ∷ []) "1Cor.1.10"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.10"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.1.10"
∷ word (σ ∷ χ ∷ ί ∷ σ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.1.10"
∷ word (ἦ ∷ τ ∷ ε ∷ []) "1Cor.1.10"
∷ word (δ ∷ ὲ ∷ []) "1Cor.1.10"
∷ word (κ ∷ α ∷ τ ∷ η ∷ ρ ∷ τ ∷ ι ∷ σ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.1.10"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.10"
∷ word (τ ∷ ῷ ∷ []) "1Cor.1.10"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1Cor.1.10"
∷ word (ν ∷ ο ∷ ῒ ∷ []) "1Cor.1.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.10"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.10"
∷ word (τ ∷ ῇ ∷ []) "1Cor.1.10"
∷ word (α ∷ ὐ ∷ τ ∷ ῇ ∷ []) "1Cor.1.10"
∷ word (γ ∷ ν ∷ ώ ∷ μ ∷ ῃ ∷ []) "1Cor.1.10"
∷ word (ἐ ∷ δ ∷ η ∷ ∙λ ∷ ώ ∷ θ ∷ η ∷ []) "1Cor.1.11"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.1.11"
∷ word (μ ∷ ο ∷ ι ∷ []) "1Cor.1.11"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1Cor.1.11"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.1.11"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.1.11"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.1.11"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.1.11"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.1.11"
∷ word (Χ ∷ ∙λ ∷ ό ∷ η ∷ ς ∷ []) "1Cor.1.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.1.11"
∷ word (ἔ ∷ ρ ∷ ι ∷ δ ∷ ε ∷ ς ∷ []) "1Cor.1.11"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.11"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.1.11"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.1.11"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "1Cor.1.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.1.12"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.1.12"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.1.12"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.1.12"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.1.12"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "1Cor.1.12"
∷ word (Ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.1.12"
∷ word (μ ∷ έ ∷ ν ∷ []) "1Cor.1.12"
∷ word (ε ∷ ἰ ∷ μ ∷ ι ∷ []) "1Cor.1.12"
∷ word (Π ∷ α ∷ ύ ∷ ∙λ ∷ ο ∷ υ ∷ []) "1Cor.1.12"
∷ word (Ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.1.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.1.12"
∷ word (Ἀ ∷ π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ []) "1Cor.1.12"
∷ word (Ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.1.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.1.12"
∷ word (Κ ∷ η ∷ φ ∷ ᾶ ∷ []) "1Cor.1.12"
∷ word (Ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.1.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.1.12"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.12"
∷ word (μ ∷ ε ∷ μ ∷ έ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.1.13"
∷ word (ὁ ∷ []) "1Cor.1.13"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "1Cor.1.13"
∷ word (μ ∷ ὴ ∷ []) "1Cor.1.13"
∷ word (Π ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.1.13"
∷ word (ἐ ∷ σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ώ ∷ θ ∷ η ∷ []) "1Cor.1.13"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "1Cor.1.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.1.13"
∷ word (ἢ ∷ []) "1Cor.1.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.1.13"
∷ word (τ ∷ ὸ ∷ []) "1Cor.1.13"
∷ word (ὄ ∷ ν ∷ ο ∷ μ ∷ α ∷ []) "1Cor.1.13"
∷ word (Π ∷ α ∷ ύ ∷ ∙λ ∷ ο ∷ υ ∷ []) "1Cor.1.13"
∷ word (ἐ ∷ β ∷ α ∷ π ∷ τ ∷ ί ∷ σ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.1.13"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῶ ∷ []) "1Cor.1.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.1.14"
∷ word (ο ∷ ὐ ∷ δ ∷ έ ∷ ν ∷ α ∷ []) "1Cor.1.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.1.14"
∷ word (ἐ ∷ β ∷ ά ∷ π ∷ τ ∷ ι ∷ σ ∷ α ∷ []) "1Cor.1.14"
∷ word (ε ∷ ἰ ∷ []) "1Cor.1.14"
∷ word (μ ∷ ὴ ∷ []) "1Cor.1.14"
∷ word (Κ ∷ ρ ∷ ί ∷ σ ∷ π ∷ ο ∷ ν ∷ []) "1Cor.1.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.14"
∷ word (Γ ∷ ά ∷ ϊ ∷ ο ∷ ν ∷ []) "1Cor.1.14"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.1.15"
∷ word (μ ∷ ή ∷ []) "1Cor.1.15"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.1.15"
∷ word (ε ∷ ἴ ∷ π ∷ ῃ ∷ []) "1Cor.1.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.1.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.1.15"
∷ word (τ ∷ ὸ ∷ []) "1Cor.1.15"
∷ word (ἐ ∷ μ ∷ ὸ ∷ ν ∷ []) "1Cor.1.15"
∷ word (ὄ ∷ ν ∷ ο ∷ μ ∷ α ∷ []) "1Cor.1.15"
∷ word (ἐ ∷ β ∷ α ∷ π ∷ τ ∷ ί ∷ σ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.1.15"
∷ word (ἐ ∷ β ∷ ά ∷ π ∷ τ ∷ ι ∷ σ ∷ α ∷ []) "1Cor.1.16"
∷ word (δ ∷ ὲ ∷ []) "1Cor.1.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.16"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.1.16"
∷ word (Σ ∷ τ ∷ ε ∷ φ ∷ α ∷ ν ∷ ᾶ ∷ []) "1Cor.1.16"
∷ word (ο ∷ ἶ ∷ κ ∷ ο ∷ ν ∷ []) "1Cor.1.16"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ὸ ∷ ν ∷ []) "1Cor.1.16"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.1.16"
∷ word (ο ∷ ἶ ∷ δ ∷ α ∷ []) "1Cor.1.16"
∷ word (ε ∷ ἴ ∷ []) "1Cor.1.16"
∷ word (τ ∷ ι ∷ ν ∷ α ∷ []) "1Cor.1.16"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.1.16"
∷ word (ἐ ∷ β ∷ ά ∷ π ∷ τ ∷ ι ∷ σ ∷ α ∷ []) "1Cor.1.16"
∷ word (ο ∷ ὐ ∷ []) "1Cor.1.17"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.1.17"
∷ word (ἀ ∷ π ∷ έ ∷ σ ∷ τ ∷ ε ∷ ι ∷ ∙λ ∷ έ ∷ ν ∷ []) "1Cor.1.17"
∷ word (μ ∷ ε ∷ []) "1Cor.1.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.1.17"
∷ word (β ∷ α ∷ π ∷ τ ∷ ί ∷ ζ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.1.17"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.1.17"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ζ ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.1.17"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.1.17"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.17"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ ᾳ ∷ []) "1Cor.1.17"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ υ ∷ []) "1Cor.1.17"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.1.17"
∷ word (μ ∷ ὴ ∷ []) "1Cor.1.17"
∷ word (κ ∷ ε ∷ ν ∷ ω ∷ θ ∷ ῇ ∷ []) "1Cor.1.17"
∷ word (ὁ ∷ []) "1Cor.1.17"
∷ word (σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.1.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.17"
∷ word (Ὁ ∷ []) "1Cor.1.18"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "1Cor.1.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.1.18"
∷ word (ὁ ∷ []) "1Cor.1.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.18"
∷ word (σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ο ∷ ῦ ∷ []) "1Cor.1.18"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.1.18"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.1.18"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ∙λ ∷ υ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.1.18"
∷ word (μ ∷ ω ∷ ρ ∷ ί ∷ α ∷ []) "1Cor.1.18"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1Cor.1.18"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.1.18"
∷ word (δ ∷ ὲ ∷ []) "1Cor.1.18"
∷ word (σ ∷ ῳ ∷ ζ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.1.18"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.1.18"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ς ∷ []) "1Cor.1.18"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.1.18"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.1.18"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "1Cor.1.19"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.1.19"
∷ word (Ἀ ∷ π ∷ ο ∷ ∙λ ∷ ῶ ∷ []) "1Cor.1.19"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.1.19"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.1.19"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.1.19"
∷ word (σ ∷ ο ∷ φ ∷ ῶ ∷ ν ∷ []) "1Cor.1.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.19"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.1.19"
∷ word (σ ∷ ύ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.1.19"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.1.19"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.1.19"
∷ word (ἀ ∷ θ ∷ ε ∷ τ ∷ ή ∷ σ ∷ ω ∷ []) "1Cor.1.19"
∷ word (π ∷ ο ∷ ῦ ∷ []) "1Cor.1.20"
∷ word (σ ∷ ο ∷ φ ∷ ό ∷ ς ∷ []) "1Cor.1.20"
∷ word (π ∷ ο ∷ ῦ ∷ []) "1Cor.1.20"
∷ word (γ ∷ ρ ∷ α ∷ μ ∷ μ ∷ α ∷ τ ∷ ε ∷ ύ ∷ ς ∷ []) "1Cor.1.20"
∷ word (π ∷ ο ∷ ῦ ∷ []) "1Cor.1.20"
∷ word (σ ∷ υ ∷ ζ ∷ η ∷ τ ∷ η ∷ τ ∷ ὴ ∷ ς ∷ []) "1Cor.1.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.20"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.1.20"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.1.20"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "1Cor.1.20"
∷ word (ἐ ∷ μ ∷ ώ ∷ ρ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "1Cor.1.20"
∷ word (ὁ ∷ []) "1Cor.1.20"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.1.20"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.1.20"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.1.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.20"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1Cor.1.20"
∷ word (ἐ ∷ π ∷ ε ∷ ι ∷ δ ∷ ὴ ∷ []) "1Cor.1.21"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.1.21"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.21"
∷ word (τ ∷ ῇ ∷ []) "1Cor.1.21"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ ᾳ ∷ []) "1Cor.1.21"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.21"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.1.21"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.1.21"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ []) "1Cor.1.21"
∷ word (ὁ ∷ []) "1Cor.1.21"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.1.21"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.1.21"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.1.21"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.1.21"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.1.21"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "1Cor.1.21"
∷ word (ε ∷ ὐ ∷ δ ∷ ό ∷ κ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.1.21"
∷ word (ὁ ∷ []) "1Cor.1.21"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.1.21"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.1.21"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.1.21"
∷ word (μ ∷ ω ∷ ρ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.1.21"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.21"
∷ word (κ ∷ η ∷ ρ ∷ ύ ∷ γ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.1.21"
∷ word (σ ∷ ῶ ∷ σ ∷ α ∷ ι ∷ []) "1Cor.1.21"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.1.21"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "1Cor.1.21"
∷ word (ἐ ∷ π ∷ ε ∷ ι ∷ δ ∷ ὴ ∷ []) "1Cor.1.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.22"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ῖ ∷ ο ∷ ι ∷ []) "1Cor.1.22"
∷ word (σ ∷ η ∷ μ ∷ ε ∷ ῖ ∷ α ∷ []) "1Cor.1.22"
∷ word (α ∷ ἰ ∷ τ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.1.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.22"
∷ word (Ἕ ∷ ∙λ ∷ ∙λ ∷ η ∷ ν ∷ ε ∷ ς ∷ []) "1Cor.1.22"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.1.22"
∷ word (ζ ∷ η ∷ τ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.1.22"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.1.23"
∷ word (δ ∷ ὲ ∷ []) "1Cor.1.23"
∷ word (κ ∷ η ∷ ρ ∷ ύ ∷ σ ∷ σ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.1.23"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.1.23"
∷ word (ἐ ∷ σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.1.23"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.1.23"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.1.23"
∷ word (σ ∷ κ ∷ ά ∷ ν ∷ δ ∷ α ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.1.23"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.1.23"
∷ word (δ ∷ ὲ ∷ []) "1Cor.1.23"
∷ word (μ ∷ ω ∷ ρ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.1.23"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.1.24"
∷ word (δ ∷ ὲ ∷ []) "1Cor.1.24"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.1.24"
∷ word (κ ∷ ∙λ ∷ η ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.1.24"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.1.24"
∷ word (τ ∷ ε ∷ []) "1Cor.1.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.24"
∷ word (Ἕ ∷ ∙λ ∷ ∙λ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.1.24"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.1.24"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.1.24"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ν ∷ []) "1Cor.1.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.24"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.1.24"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.1.24"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.1.25"
∷ word (τ ∷ ὸ ∷ []) "1Cor.1.25"
∷ word (μ ∷ ω ∷ ρ ∷ ὸ ∷ ν ∷ []) "1Cor.1.25"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.25"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.1.25"
∷ word (σ ∷ ο ∷ φ ∷ ώ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.1.25"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.1.25"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "1Cor.1.25"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1Cor.1.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.25"
∷ word (τ ∷ ὸ ∷ []) "1Cor.1.25"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ὲ ∷ ς ∷ []) "1Cor.1.25"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.25"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.1.25"
∷ word (ἰ ∷ σ ∷ χ ∷ υ ∷ ρ ∷ ό ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.1.25"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.1.25"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "1Cor.1.25"
∷ word (Β ∷ ∙λ ∷ έ ∷ π ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.1.26"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.1.26"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.1.26"
∷ word (κ ∷ ∙λ ∷ ῆ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.1.26"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.1.26"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.1.26"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.1.26"
∷ word (ο ∷ ὐ ∷ []) "1Cor.1.26"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὶ ∷ []) "1Cor.1.26"
∷ word (σ ∷ ο ∷ φ ∷ ο ∷ ὶ ∷ []) "1Cor.1.26"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.1.26"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "1Cor.1.26"
∷ word (ο ∷ ὐ ∷ []) "1Cor.1.26"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὶ ∷ []) "1Cor.1.26"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ τ ∷ ο ∷ ί ∷ []) "1Cor.1.26"
∷ word (ο ∷ ὐ ∷ []) "1Cor.1.26"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὶ ∷ []) "1Cor.1.26"
∷ word (ε ∷ ὐ ∷ γ ∷ ε ∷ ν ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.1.26"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.1.27"
∷ word (τ ∷ ὰ ∷ []) "1Cor.1.27"
∷ word (μ ∷ ω ∷ ρ ∷ ὰ ∷ []) "1Cor.1.27"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.27"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1Cor.1.27"
∷ word (ἐ ∷ ξ ∷ ε ∷ ∙λ ∷ έ ∷ ξ ∷ α ∷ τ ∷ ο ∷ []) "1Cor.1.27"
∷ word (ὁ ∷ []) "1Cor.1.27"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "1Cor.1.27"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.1.27"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ι ∷ σ ∷ χ ∷ ύ ∷ ν ∷ ῃ ∷ []) "1Cor.1.27"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.1.27"
∷ word (σ ∷ ο ∷ φ ∷ ο ∷ ύ ∷ ς ∷ []) "1Cor.1.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.27"
∷ word (τ ∷ ὰ ∷ []) "1Cor.1.27"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ῆ ∷ []) "1Cor.1.27"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.27"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1Cor.1.27"
∷ word (ἐ ∷ ξ ∷ ε ∷ ∙λ ∷ έ ∷ ξ ∷ α ∷ τ ∷ ο ∷ []) "1Cor.1.27"
∷ word (ὁ ∷ []) "1Cor.1.27"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "1Cor.1.27"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.1.27"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ι ∷ σ ∷ χ ∷ ύ ∷ ν ∷ ῃ ∷ []) "1Cor.1.27"
∷ word (τ ∷ ὰ ∷ []) "1Cor.1.27"
∷ word (ἰ ∷ σ ∷ χ ∷ υ ∷ ρ ∷ ά ∷ []) "1Cor.1.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.28"
∷ word (τ ∷ ὰ ∷ []) "1Cor.1.28"
∷ word (ἀ ∷ γ ∷ ε ∷ ν ∷ ῆ ∷ []) "1Cor.1.28"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.28"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1Cor.1.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.28"
∷ word (τ ∷ ὰ ∷ []) "1Cor.1.28"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ θ ∷ ε ∷ ν ∷ η ∷ μ ∷ έ ∷ ν ∷ α ∷ []) "1Cor.1.28"
∷ word (ἐ ∷ ξ ∷ ε ∷ ∙λ ∷ έ ∷ ξ ∷ α ∷ τ ∷ ο ∷ []) "1Cor.1.28"
∷ word (ὁ ∷ []) "1Cor.1.28"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "1Cor.1.28"
∷ word (τ ∷ ὰ ∷ []) "1Cor.1.28"
∷ word (μ ∷ ὴ ∷ []) "1Cor.1.28"
∷ word (ὄ ∷ ν ∷ τ ∷ α ∷ []) "1Cor.1.28"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.1.28"
∷ word (τ ∷ ὰ ∷ []) "1Cor.1.28"
∷ word (ὄ ∷ ν ∷ τ ∷ α ∷ []) "1Cor.1.28"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ ή ∷ σ ∷ ῃ ∷ []) "1Cor.1.28"
∷ word (ὅ ∷ π ∷ ω ∷ ς ∷ []) "1Cor.1.29"
∷ word (μ ∷ ὴ ∷ []) "1Cor.1.29"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "1Cor.1.29"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ []) "1Cor.1.29"
∷ word (σ ∷ ὰ ∷ ρ ∷ ξ ∷ []) "1Cor.1.29"
∷ word (ἐ ∷ ν ∷ ώ ∷ π ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.1.29"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.29"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.1.29"
∷ word (ἐ ∷ ξ ∷ []) "1Cor.1.30"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.1.30"
∷ word (δ ∷ ὲ ∷ []) "1Cor.1.30"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.1.30"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.1.30"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.30"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1Cor.1.30"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.1.30"
∷ word (ὃ ∷ ς ∷ []) "1Cor.1.30"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ή ∷ θ ∷ η ∷ []) "1Cor.1.30"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ []) "1Cor.1.30"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.1.30"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "1Cor.1.30"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.1.30"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ []) "1Cor.1.30"
∷ word (τ ∷ ε ∷ []) "1Cor.1.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.30"
∷ word (ἁ ∷ γ ∷ ι ∷ α ∷ σ ∷ μ ∷ ὸ ∷ ς ∷ []) "1Cor.1.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.1.30"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ύ ∷ τ ∷ ρ ∷ ω ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.1.30"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.1.31"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.1.31"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "1Cor.1.31"
∷ word (Ὁ ∷ []) "1Cor.1.31"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ώ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.1.31"
∷ word (ἐ ∷ ν ∷ []) "1Cor.1.31"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.1.31"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ά ∷ σ ∷ θ ∷ ω ∷ []) "1Cor.1.31"
∷ word (Κ ∷ ἀ ∷ γ ∷ ὼ ∷ []) "1Cor.2.1"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ὼ ∷ ν ∷ []) "1Cor.2.1"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.2.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.2.1"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.2.1"
∷ word (ἦ ∷ ∙λ ∷ θ ∷ ο ∷ ν ∷ []) "1Cor.2.1"
∷ word (ο ∷ ὐ ∷ []) "1Cor.2.1"
∷ word (κ ∷ α ∷ θ ∷ []) "1Cor.2.1"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ ο ∷ χ ∷ ὴ ∷ ν ∷ []) "1Cor.2.1"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ υ ∷ []) "1Cor.2.1"
∷ word (ἢ ∷ []) "1Cor.2.1"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.2.1"
∷ word (κ ∷ α ∷ τ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ω ∷ ν ∷ []) "1Cor.2.1"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.2.1"
∷ word (τ ∷ ὸ ∷ []) "1Cor.2.1"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.2.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.2.1"
∷ word (ο ∷ ὐ ∷ []) "1Cor.2.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.2.2"
∷ word (ἔ ∷ κ ∷ ρ ∷ ι ∷ ν ∷ ά ∷ []) "1Cor.2.2"
∷ word (τ ∷ ι ∷ []) "1Cor.2.2"
∷ word (ε ∷ ἰ ∷ δ ∷ έ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.2.2"
∷ word (ἐ ∷ ν ∷ []) "1Cor.2.2"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.2.2"
∷ word (ε ∷ ἰ ∷ []) "1Cor.2.2"
∷ word (μ ∷ ὴ ∷ []) "1Cor.2.2"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "1Cor.2.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.2.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.2.2"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.2.2"
∷ word (ἐ ∷ σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.2.2"
∷ word (κ ∷ ἀ ∷ γ ∷ ὼ ∷ []) "1Cor.2.3"
∷ word (ἐ ∷ ν ∷ []) "1Cor.2.3"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ί ∷ ᾳ ∷ []) "1Cor.2.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.2.3"
∷ word (ἐ ∷ ν ∷ []) "1Cor.2.3"
∷ word (φ ∷ ό ∷ β ∷ ῳ ∷ []) "1Cor.2.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.2.3"
∷ word (ἐ ∷ ν ∷ []) "1Cor.2.3"
∷ word (τ ∷ ρ ∷ ό ∷ μ ∷ ῳ ∷ []) "1Cor.2.3"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῷ ∷ []) "1Cor.2.3"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ό ∷ μ ∷ η ∷ ν ∷ []) "1Cor.2.3"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.2.3"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.2.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.2.4"
∷ word (ὁ ∷ []) "1Cor.2.4"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "1Cor.2.4"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.2.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.2.4"
∷ word (τ ∷ ὸ ∷ []) "1Cor.2.4"
∷ word (κ ∷ ή ∷ ρ ∷ υ ∷ γ ∷ μ ∷ ά ∷ []) "1Cor.2.4"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.2.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.2.4"
∷ word (ἐ ∷ ν ∷ []) "1Cor.2.4"
∷ word (π ∷ ε ∷ ι ∷ θ ∷ ο ∷ ῖ ∷ []) "1Cor.2.4"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.2.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.2.4"
∷ word (ἐ ∷ ν ∷ []) "1Cor.2.4"
∷ word (ἀ ∷ π ∷ ο ∷ δ ∷ ε ∷ ί ∷ ξ ∷ ε ∷ ι ∷ []) "1Cor.2.4"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.2.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.2.4"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ω ∷ ς ∷ []) "1Cor.2.4"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.2.5"
∷ word (ἡ ∷ []) "1Cor.2.5"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "1Cor.2.5"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.2.5"
∷ word (μ ∷ ὴ ∷ []) "1Cor.2.5"
∷ word (ᾖ ∷ []) "1Cor.2.5"
∷ word (ἐ ∷ ν ∷ []) "1Cor.2.5"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ ᾳ ∷ []) "1Cor.2.5"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "1Cor.2.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.2.5"
∷ word (ἐ ∷ ν ∷ []) "1Cor.2.5"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ []) "1Cor.2.5"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.2.5"
∷ word (Σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.2.6"
∷ word (δ ∷ ὲ ∷ []) "1Cor.2.6"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.2.6"
∷ word (ἐ ∷ ν ∷ []) "1Cor.2.6"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.2.6"
∷ word (τ ∷ ε ∷ ∙λ ∷ ε ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.2.6"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.2.6"
∷ word (δ ∷ ὲ ∷ []) "1Cor.2.6"
∷ word (ο ∷ ὐ ∷ []) "1Cor.2.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.6"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.2.6"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.2.6"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "1Cor.2.6"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.2.6"
∷ word (ἀ ∷ ρ ∷ χ ∷ ό ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.2.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.6"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.2.6"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.2.6"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.2.6"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ ω ∷ ν ∷ []) "1Cor.2.6"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.2.7"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.2.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.2.7"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.2.7"
∷ word (ἐ ∷ ν ∷ []) "1Cor.2.7"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ η ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.2.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.2.7"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ ε ∷ κ ∷ ρ ∷ υ ∷ μ ∷ μ ∷ έ ∷ ν ∷ η ∷ ν ∷ []) "1Cor.2.7"
∷ word (ἣ ∷ ν ∷ []) "1Cor.2.7"
∷ word (π ∷ ρ ∷ ο ∷ ώ ∷ ρ ∷ ι ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.2.7"
∷ word (ὁ ∷ []) "1Cor.2.7"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.2.7"
∷ word (π ∷ ρ ∷ ὸ ∷ []) "1Cor.2.7"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.2.7"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ω ∷ ν ∷ []) "1Cor.2.7"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.2.7"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "1Cor.2.7"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.2.7"
∷ word (ἣ ∷ ν ∷ []) "1Cor.2.8"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.2.8"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.2.8"
∷ word (ἀ ∷ ρ ∷ χ ∷ ό ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.2.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.8"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.2.8"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.2.8"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "1Cor.2.8"
∷ word (ε ∷ ἰ ∷ []) "1Cor.2.8"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.2.8"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "1Cor.2.8"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.2.8"
∷ word (ἂ ∷ ν ∷ []) "1Cor.2.8"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.2.8"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.2.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.2.8"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "1Cor.2.8"
∷ word (ἐ ∷ σ ∷ τ ∷ α ∷ ύ ∷ ρ ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "1Cor.2.8"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.2.9"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.2.9"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "1Cor.2.9"
∷ word (Ἃ ∷ []) "1Cor.2.9"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ὸ ∷ ς ∷ []) "1Cor.2.9"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.2.9"
∷ word (ε ∷ ἶ ∷ δ ∷ ε ∷ ν ∷ []) "1Cor.2.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.2.9"
∷ word (ο ∷ ὖ ∷ ς ∷ []) "1Cor.2.9"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.2.9"
∷ word (ἤ ∷ κ ∷ ο ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.2.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.2.9"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.2.9"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.2.9"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "1Cor.2.9"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.2.9"
∷ word (ἀ ∷ ν ∷ έ ∷ β ∷ η ∷ []) "1Cor.2.9"
∷ word (ὅ ∷ σ ∷ α ∷ []) "1Cor.2.9"
∷ word (ἡ ∷ τ ∷ ο ∷ ί ∷ μ ∷ α ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.2.9"
∷ word (ὁ ∷ []) "1Cor.2.9"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.2.9"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.2.9"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.2.9"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "1Cor.2.9"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.2.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.2.10"
∷ word (ἀ ∷ π ∷ ε ∷ κ ∷ ά ∷ ∙λ ∷ υ ∷ ψ ∷ ε ∷ ν ∷ []) "1Cor.2.10"
∷ word (ὁ ∷ []) "1Cor.2.10"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.2.10"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.2.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.10"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.2.10"
∷ word (τ ∷ ὸ ∷ []) "1Cor.2.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.2.10"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.2.10"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.2.10"
∷ word (ἐ ∷ ρ ∷ α ∷ υ ∷ ν ∷ ᾷ ∷ []) "1Cor.2.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.2.10"
∷ word (τ ∷ ὰ ∷ []) "1Cor.2.10"
∷ word (β ∷ ά ∷ θ ∷ η ∷ []) "1Cor.2.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.10"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.2.10"
∷ word (τ ∷ ί ∷ ς ∷ []) "1Cor.2.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.2.11"
∷ word (ο ∷ ἶ ∷ δ ∷ ε ∷ ν ∷ []) "1Cor.2.11"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "1Cor.2.11"
∷ word (τ ∷ ὰ ∷ []) "1Cor.2.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.11"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "1Cor.2.11"
∷ word (ε ∷ ἰ ∷ []) "1Cor.2.11"
∷ word (μ ∷ ὴ ∷ []) "1Cor.2.11"
∷ word (τ ∷ ὸ ∷ []) "1Cor.2.11"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.2.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.11"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "1Cor.2.11"
∷ word (τ ∷ ὸ ∷ []) "1Cor.2.11"
∷ word (ἐ ∷ ν ∷ []) "1Cor.2.11"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1Cor.2.11"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.2.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.2.11"
∷ word (τ ∷ ὰ ∷ []) "1Cor.2.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.11"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.2.11"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.2.11"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "1Cor.2.11"
∷ word (ε ∷ ἰ ∷ []) "1Cor.2.11"
∷ word (μ ∷ ὴ ∷ []) "1Cor.2.11"
∷ word (τ ∷ ὸ ∷ []) "1Cor.2.11"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.2.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.11"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.2.11"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.2.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.2.12"
∷ word (ο ∷ ὐ ∷ []) "1Cor.2.12"
∷ word (τ ∷ ὸ ∷ []) "1Cor.2.12"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.2.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.12"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1Cor.2.12"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ β ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.2.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.2.12"
∷ word (τ ∷ ὸ ∷ []) "1Cor.2.12"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.2.12"
∷ word (τ ∷ ὸ ∷ []) "1Cor.2.12"
∷ word (ἐ ∷ κ ∷ []) "1Cor.2.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.12"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.2.12"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.2.12"
∷ word (ε ∷ ἰ ∷ δ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.2.12"
∷ word (τ ∷ ὰ ∷ []) "1Cor.2.12"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.2.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.12"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.2.12"
∷ word (χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ θ ∷ έ ∷ ν ∷ τ ∷ α ∷ []) "1Cor.2.12"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.2.12"
∷ word (ἃ ∷ []) "1Cor.2.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.2.13"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.2.13"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.2.13"
∷ word (ἐ ∷ ν ∷ []) "1Cor.2.13"
∷ word (δ ∷ ι ∷ δ ∷ α ∷ κ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.2.13"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ί ∷ ν ∷ η ∷ ς ∷ []) "1Cor.2.13"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.2.13"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.2.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.2.13"
∷ word (ἐ ∷ ν ∷ []) "1Cor.2.13"
∷ word (δ ∷ ι ∷ δ ∷ α ∷ κ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.2.13"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.2.13"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.2.13"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ὰ ∷ []) "1Cor.2.13"
∷ word (σ ∷ υ ∷ γ ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.2.13"
∷ word (Ψ ∷ υ ∷ χ ∷ ι ∷ κ ∷ ὸ ∷ ς ∷ []) "1Cor.2.14"
∷ word (δ ∷ ὲ ∷ []) "1Cor.2.14"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "1Cor.2.14"
∷ word (ο ∷ ὐ ∷ []) "1Cor.2.14"
∷ word (δ ∷ έ ∷ χ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.2.14"
∷ word (τ ∷ ὰ ∷ []) "1Cor.2.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.14"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.2.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.14"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.2.14"
∷ word (μ ∷ ω ∷ ρ ∷ ί ∷ α ∷ []) "1Cor.2.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.2.14"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1Cor.2.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1Cor.2.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.2.14"
∷ word (ο ∷ ὐ ∷ []) "1Cor.2.14"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ τ ∷ α ∷ ι ∷ []) "1Cor.2.14"
∷ word (γ ∷ ν ∷ ῶ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.2.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.2.14"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ῶ ∷ ς ∷ []) "1Cor.2.14"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.2.14"
∷ word (ὁ ∷ []) "1Cor.2.15"
∷ word (δ ∷ ὲ ∷ []) "1Cor.2.15"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ὸ ∷ ς ∷ []) "1Cor.2.15"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ ι ∷ []) "1Cor.2.15"
∷ word (τ ∷ ὰ ∷ []) "1Cor.2.15"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.2.15"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.2.15"
∷ word (δ ∷ ὲ ∷ []) "1Cor.2.15"
∷ word (ὑ ∷ π ∷ []) "1Cor.2.15"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ν ∷ ὸ ∷ ς ∷ []) "1Cor.2.15"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.2.15"
∷ word (τ ∷ ί ∷ ς ∷ []) "1Cor.2.16"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.2.16"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ []) "1Cor.2.16"
∷ word (ν ∷ ο ∷ ῦ ∷ ν ∷ []) "1Cor.2.16"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.2.16"
∷ word (ὃ ∷ ς ∷ []) "1Cor.2.16"
∷ word (σ ∷ υ ∷ μ ∷ β ∷ ι ∷ β ∷ ά ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.2.16"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "1Cor.2.16"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.2.16"
∷ word (δ ∷ ὲ ∷ []) "1Cor.2.16"
∷ word (ν ∷ ο ∷ ῦ ∷ ν ∷ []) "1Cor.2.16"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.2.16"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.2.16"
∷ word (Κ ∷ ἀ ∷ γ ∷ ώ ∷ []) "1Cor.3.1"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.3.1"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.3.1"
∷ word (ἠ ∷ δ ∷ υ ∷ ν ∷ ή ∷ θ ∷ η ∷ ν ∷ []) "1Cor.3.1"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "1Cor.3.1"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.3.1"
∷ word (ὡ ∷ ς ∷ []) "1Cor.3.1"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.3.1"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.3.1"
∷ word (ὡ ∷ ς ∷ []) "1Cor.3.1"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ί ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.3.1"
∷ word (ὡ ∷ ς ∷ []) "1Cor.3.1"
∷ word (ν ∷ η ∷ π ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.3.1"
∷ word (ἐ ∷ ν ∷ []) "1Cor.3.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1Cor.3.1"
∷ word (γ ∷ ά ∷ ∙λ ∷ α ∷ []) "1Cor.3.2"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.3.2"
∷ word (ἐ ∷ π ∷ ό ∷ τ ∷ ι ∷ σ ∷ α ∷ []) "1Cor.3.2"
∷ word (ο ∷ ὐ ∷ []) "1Cor.3.2"
∷ word (β ∷ ρ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.3.2"
∷ word (ο ∷ ὔ ∷ π ∷ ω ∷ []) "1Cor.3.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.3.2"
∷ word (ἐ ∷ δ ∷ ύ ∷ ν ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.3.2"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.3.2"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "1Cor.3.2"
∷ word (ἔ ∷ τ ∷ ι ∷ []) "1Cor.3.2"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "1Cor.3.2"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.3.2"
∷ word (ἔ ∷ τ ∷ ι ∷ []) "1Cor.3.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.3.3"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ι ∷ κ ∷ ο ∷ ί ∷ []) "1Cor.3.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.3.3"
∷ word (ὅ ∷ π ∷ ο ∷ υ ∷ []) "1Cor.3.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.3.3"
∷ word (ἐ ∷ ν ∷ []) "1Cor.3.3"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.3.3"
∷ word (ζ ∷ ῆ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.3.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.3.3"
∷ word (ἔ ∷ ρ ∷ ι ∷ ς ∷ []) "1Cor.3.3"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "1Cor.3.3"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ι ∷ κ ∷ ο ∷ ί ∷ []) "1Cor.3.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.3.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.3.3"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.3.3"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "1Cor.3.3"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "1Cor.3.3"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "1Cor.3.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.3.4"
∷ word (∙λ ∷ έ ∷ γ ∷ ῃ ∷ []) "1Cor.3.4"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.3.4"
∷ word (Ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.3.4"
∷ word (μ ∷ έ ∷ ν ∷ []) "1Cor.3.4"
∷ word (ε ∷ ἰ ∷ μ ∷ ι ∷ []) "1Cor.3.4"
∷ word (Π ∷ α ∷ ύ ∷ ∙λ ∷ ο ∷ υ ∷ []) "1Cor.3.4"
∷ word (ἕ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.3.4"
∷ word (δ ∷ έ ∷ []) "1Cor.3.4"
∷ word (Ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.3.4"
∷ word (Ἀ ∷ π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ []) "1Cor.3.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.3.4"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ί ∷ []) "1Cor.3.4"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.3.4"
∷ word (Τ ∷ ί ∷ []) "1Cor.3.5"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.3.5"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.3.5"
∷ word (Ἀ ∷ π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ς ∷ []) "1Cor.3.5"
∷ word (τ ∷ ί ∷ []) "1Cor.3.5"
∷ word (δ ∷ έ ∷ []) "1Cor.3.5"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.3.5"
∷ word (Π ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.3.5"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.3.5"
∷ word (δ ∷ ι ∷ []) "1Cor.3.5"
∷ word (ὧ ∷ ν ∷ []) "1Cor.3.5"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.3.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.3.5"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ ῳ ∷ []) "1Cor.3.5"
∷ word (ὡ ∷ ς ∷ []) "1Cor.3.5"
∷ word (ὁ ∷ []) "1Cor.3.5"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.3.5"
∷ word (ἔ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "1Cor.3.5"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.3.6"
∷ word (ἐ ∷ φ ∷ ύ ∷ τ ∷ ε ∷ υ ∷ σ ∷ α ∷ []) "1Cor.3.6"
∷ word (Ἀ ∷ π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ς ∷ []) "1Cor.3.6"
∷ word (ἐ ∷ π ∷ ό ∷ τ ∷ ι ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.3.6"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.3.6"
∷ word (ὁ ∷ []) "1Cor.3.6"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.3.6"
∷ word (η ∷ ὔ ∷ ξ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "1Cor.3.6"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.3.7"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "1Cor.3.7"
∷ word (ὁ ∷ []) "1Cor.3.7"
∷ word (φ ∷ υ ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "1Cor.3.7"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1Cor.3.7"
∷ word (τ ∷ ι ∷ []) "1Cor.3.7"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "1Cor.3.7"
∷ word (ὁ ∷ []) "1Cor.3.7"
∷ word (π ∷ ο ∷ τ ∷ ί ∷ ζ ∷ ω ∷ ν ∷ []) "1Cor.3.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.3.7"
∷ word (ὁ ∷ []) "1Cor.3.7"
∷ word (α ∷ ὐ ∷ ξ ∷ ά ∷ ν ∷ ω ∷ ν ∷ []) "1Cor.3.7"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "1Cor.3.7"
∷ word (ὁ ∷ []) "1Cor.3.8"
∷ word (φ ∷ υ ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "1Cor.3.8"
∷ word (δ ∷ ὲ ∷ []) "1Cor.3.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.3.8"
∷ word (ὁ ∷ []) "1Cor.3.8"
∷ word (π ∷ ο ∷ τ ∷ ί ∷ ζ ∷ ω ∷ ν ∷ []) "1Cor.3.8"
∷ word (ἕ ∷ ν ∷ []) "1Cor.3.8"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.3.8"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.3.8"
∷ word (δ ∷ ὲ ∷ []) "1Cor.3.8"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.3.8"
∷ word (ἴ ∷ δ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.3.8"
∷ word (μ ∷ ι ∷ σ ∷ θ ∷ ὸ ∷ ν ∷ []) "1Cor.3.8"
∷ word (∙λ ∷ ή ∷ μ ∷ ψ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.3.8"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.3.8"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.3.8"
∷ word (ἴ ∷ δ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.3.8"
∷ word (κ ∷ ό ∷ π ∷ ο ∷ ν ∷ []) "1Cor.3.8"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.3.9"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.3.9"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.3.9"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ο ∷ ί ∷ []) "1Cor.3.9"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.3.9"
∷ word (γ ∷ ε ∷ ώ ∷ ρ ∷ γ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.3.9"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.3.9"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ή ∷ []) "1Cor.3.9"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.3.9"
∷ word (Κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.3.10"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.3.10"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "1Cor.3.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.3.10"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.3.10"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.3.10"
∷ word (δ ∷ ο ∷ θ ∷ ε ∷ ῖ ∷ σ ∷ ά ∷ ν ∷ []) "1Cor.3.10"
∷ word (μ ∷ ο ∷ ι ∷ []) "1Cor.3.10"
∷ word (ὡ ∷ ς ∷ []) "1Cor.3.10"
∷ word (σ ∷ ο ∷ φ ∷ ὸ ∷ ς ∷ []) "1Cor.3.10"
∷ word (ἀ ∷ ρ ∷ χ ∷ ι ∷ τ ∷ έ ∷ κ ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.3.10"
∷ word (θ ∷ ε ∷ μ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.3.10"
∷ word (ἔ ∷ θ ∷ η ∷ κ ∷ α ∷ []) "1Cor.3.10"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.3.10"
∷ word (δ ∷ ὲ ∷ []) "1Cor.3.10"
∷ word (ἐ ∷ π ∷ ο ∷ ι ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ε ∷ ῖ ∷ []) "1Cor.3.10"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.3.10"
∷ word (δ ∷ ὲ ∷ []) "1Cor.3.10"
∷ word (β ∷ ∙λ ∷ ε ∷ π ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.3.10"
∷ word (π ∷ ῶ ∷ ς ∷ []) "1Cor.3.10"
∷ word (ἐ ∷ π ∷ ο ∷ ι ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ε ∷ ῖ ∷ []) "1Cor.3.10"
∷ word (θ ∷ ε ∷ μ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.3.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.3.11"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.3.11"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.3.11"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ τ ∷ α ∷ ι ∷ []) "1Cor.3.11"
∷ word (θ ∷ ε ∷ ῖ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.3.11"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "1Cor.3.11"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.3.11"
∷ word (κ ∷ ε ∷ ί ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.3.11"
∷ word (ὅ ∷ ς ∷ []) "1Cor.3.11"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.3.11"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "1Cor.3.11"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "1Cor.3.11"
∷ word (ε ∷ ἰ ∷ []) "1Cor.3.12"
∷ word (δ ∷ έ ∷ []) "1Cor.3.12"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.3.12"
∷ word (ἐ ∷ π ∷ ο ∷ ι ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ε ∷ ῖ ∷ []) "1Cor.3.12"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.3.12"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.3.12"
∷ word (θ ∷ ε ∷ μ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.3.12"
∷ word (χ ∷ ρ ∷ υ ∷ σ ∷ ό ∷ ν ∷ []) "1Cor.3.12"
∷ word (ἄ ∷ ρ ∷ γ ∷ υ ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.3.12"
∷ word (∙λ ∷ ί ∷ θ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.3.12"
∷ word (τ ∷ ι ∷ μ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.3.12"
∷ word (ξ ∷ ύ ∷ ∙λ ∷ α ∷ []) "1Cor.3.12"
∷ word (χ ∷ ό ∷ ρ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.3.12"
∷ word (κ ∷ α ∷ ∙λ ∷ ά ∷ μ ∷ η ∷ ν ∷ []) "1Cor.3.12"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.3.13"
∷ word (τ ∷ ὸ ∷ []) "1Cor.3.13"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ν ∷ []) "1Cor.3.13"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ὸ ∷ ν ∷ []) "1Cor.3.13"
∷ word (γ ∷ ε ∷ ν ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.3.13"
∷ word (ἡ ∷ []) "1Cor.3.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.3.13"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ []) "1Cor.3.13"
∷ word (δ ∷ η ∷ ∙λ ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.3.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.3.13"
∷ word (ἐ ∷ ν ∷ []) "1Cor.3.13"
∷ word (π ∷ υ ∷ ρ ∷ ὶ ∷ []) "1Cor.3.13"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ π ∷ τ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.3.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.3.13"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.3.13"
∷ word (τ ∷ ὸ ∷ []) "1Cor.3.13"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ν ∷ []) "1Cor.3.13"
∷ word (ὁ ∷ π ∷ ο ∷ ῖ ∷ ό ∷ ν ∷ []) "1Cor.3.13"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.3.13"
∷ word (τ ∷ ὸ ∷ []) "1Cor.3.13"
∷ word (π ∷ ῦ ∷ ρ ∷ []) "1Cor.3.13"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "1Cor.3.13"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ά ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.3.13"
∷ word (ε ∷ ἴ ∷ []) "1Cor.3.14"
∷ word (τ ∷ ι ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.3.14"
∷ word (τ ∷ ὸ ∷ []) "1Cor.3.14"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ν ∷ []) "1Cor.3.14"
∷ word (μ ∷ ε ∷ ν ∷ ε ∷ ῖ ∷ []) "1Cor.3.14"
∷ word (ὃ ∷ []) "1Cor.3.14"
∷ word (ἐ ∷ π ∷ ο ∷ ι ∷ κ ∷ ο ∷ δ ∷ ό ∷ μ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.3.14"
∷ word (μ ∷ ι ∷ σ ∷ θ ∷ ὸ ∷ ν ∷ []) "1Cor.3.14"
∷ word (∙λ ∷ ή ∷ μ ∷ ψ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.3.14"
∷ word (ε ∷ ἴ ∷ []) "1Cor.3.15"
∷ word (τ ∷ ι ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.3.15"
∷ word (τ ∷ ὸ ∷ []) "1Cor.3.15"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ν ∷ []) "1Cor.3.15"
∷ word (κ ∷ α ∷ τ ∷ α ∷ κ ∷ α ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.3.15"
∷ word (ζ ∷ η ∷ μ ∷ ι ∷ ω ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.3.15"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.3.15"
∷ word (δ ∷ ὲ ∷ []) "1Cor.3.15"
∷ word (σ ∷ ω ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.3.15"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.3.15"
∷ word (δ ∷ ὲ ∷ []) "1Cor.3.15"
∷ word (ὡ ∷ ς ∷ []) "1Cor.3.15"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.3.15"
∷ word (π ∷ υ ∷ ρ ∷ ό ∷ ς ∷ []) "1Cor.3.15"
∷ word (Ο ∷ ὐ ∷ κ ∷ []) "1Cor.3.16"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.3.16"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.3.16"
∷ word (ν ∷ α ∷ ὸ ∷ ς ∷ []) "1Cor.3.16"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.3.16"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.3.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.3.16"
∷ word (τ ∷ ὸ ∷ []) "1Cor.3.16"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.3.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.3.16"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.3.16"
∷ word (ο ∷ ἰ ∷ κ ∷ ε ∷ ῖ ∷ []) "1Cor.3.16"
∷ word (ἐ ∷ ν ∷ []) "1Cor.3.16"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.3.16"
∷ word (ε ∷ ἴ ∷ []) "1Cor.3.17"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.3.17"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.3.17"
∷ word (ν ∷ α ∷ ὸ ∷ ν ∷ []) "1Cor.3.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.3.17"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.3.17"
∷ word (φ ∷ θ ∷ ε ∷ ί ∷ ρ ∷ ε ∷ ι ∷ []) "1Cor.3.17"
∷ word (φ ∷ θ ∷ ε ∷ ρ ∷ ε ∷ ῖ ∷ []) "1Cor.3.17"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.3.17"
∷ word (ὁ ∷ []) "1Cor.3.17"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "1Cor.3.17"
∷ word (ὁ ∷ []) "1Cor.3.17"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.3.17"
∷ word (ν ∷ α ∷ ὸ ∷ ς ∷ []) "1Cor.3.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.3.17"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.3.17"
∷ word (ἅ ∷ γ ∷ ι ∷ ό ∷ ς ∷ []) "1Cor.3.17"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.3.17"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ έ ∷ ς ∷ []) "1Cor.3.17"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.3.17"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.3.17"
∷ word (Μ ∷ η ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.3.18"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.3.18"
∷ word (ἐ ∷ ξ ∷ α ∷ π ∷ α ∷ τ ∷ ά ∷ τ ∷ ω ∷ []) "1Cor.3.18"
∷ word (ε ∷ ἴ ∷ []) "1Cor.3.18"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.3.18"
∷ word (δ ∷ ο ∷ κ ∷ ε ∷ ῖ ∷ []) "1Cor.3.18"
∷ word (σ ∷ ο ∷ φ ∷ ὸ ∷ ς ∷ []) "1Cor.3.18"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.3.18"
∷ word (ἐ ∷ ν ∷ []) "1Cor.3.18"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.3.18"
∷ word (ἐ ∷ ν ∷ []) "1Cor.3.18"
∷ word (τ ∷ ῷ ∷ []) "1Cor.3.18"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ ι ∷ []) "1Cor.3.18"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1Cor.3.18"
∷ word (μ ∷ ω ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.3.18"
∷ word (γ ∷ ε ∷ ν ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "1Cor.3.18"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.3.18"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "1Cor.3.18"
∷ word (σ ∷ ο ∷ φ ∷ ό ∷ ς ∷ []) "1Cor.3.18"
∷ word (ἡ ∷ []) "1Cor.3.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.3.19"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ []) "1Cor.3.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.3.19"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1Cor.3.19"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.3.19"
∷ word (μ ∷ ω ∷ ρ ∷ ί ∷ α ∷ []) "1Cor.3.19"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "1Cor.3.19"
∷ word (τ ∷ ῷ ∷ []) "1Cor.3.19"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1Cor.3.19"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.3.19"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "1Cor.3.19"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.3.19"
∷ word (Ὁ ∷ []) "1Cor.3.19"
∷ word (δ ∷ ρ ∷ α ∷ σ ∷ σ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.3.19"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.3.19"
∷ word (σ ∷ ο ∷ φ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.3.19"
∷ word (ἐ ∷ ν ∷ []) "1Cor.3.19"
∷ word (τ ∷ ῇ ∷ []) "1Cor.3.19"
∷ word (π ∷ α ∷ ν ∷ ο ∷ υ ∷ ρ ∷ γ ∷ ί ∷ ᾳ ∷ []) "1Cor.3.19"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.3.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.3.20"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "1Cor.3.20"
∷ word (Κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.3.20"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ε ∷ ι ∷ []) "1Cor.3.20"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.3.20"
∷ word (δ ∷ ι ∷ α ∷ ∙λ ∷ ο ∷ γ ∷ ι ∷ σ ∷ μ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.3.20"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.3.20"
∷ word (σ ∷ ο ∷ φ ∷ ῶ ∷ ν ∷ []) "1Cor.3.20"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.3.20"
∷ word (ε ∷ ἰ ∷ σ ∷ ὶ ∷ ν ∷ []) "1Cor.3.20"
∷ word (μ ∷ ά ∷ τ ∷ α ∷ ι ∷ ο ∷ ι ∷ []) "1Cor.3.20"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.3.21"
∷ word (μ ∷ η ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.3.21"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ά ∷ σ ∷ θ ∷ ω ∷ []) "1Cor.3.21"
∷ word (ἐ ∷ ν ∷ []) "1Cor.3.21"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.3.21"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.3.21"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.3.21"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.3.21"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.3.21"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.3.22"
∷ word (Π ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.3.22"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.3.22"
∷ word (Ἀ ∷ π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ς ∷ []) "1Cor.3.22"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.3.22"
∷ word (Κ ∷ η ∷ φ ∷ ᾶ ∷ ς ∷ []) "1Cor.3.22"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.3.22"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.3.22"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.3.22"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "1Cor.3.22"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.3.22"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.3.22"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.3.22"
∷ word (ἐ ∷ ν ∷ ε ∷ σ ∷ τ ∷ ῶ ∷ τ ∷ α ∷ []) "1Cor.3.22"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.3.22"
∷ word (μ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "1Cor.3.22"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.3.22"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.3.22"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.3.23"
∷ word (δ ∷ ὲ ∷ []) "1Cor.3.23"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.3.23"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.3.23"
∷ word (δ ∷ ὲ ∷ []) "1Cor.3.23"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.3.23"
∷ word (Ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.4.1"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.4.1"
∷ word (∙λ ∷ ο ∷ γ ∷ ι ∷ ζ ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "1Cor.4.1"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "1Cor.4.1"
∷ word (ὡ ∷ ς ∷ []) "1Cor.4.1"
∷ word (ὑ ∷ π ∷ η ∷ ρ ∷ έ ∷ τ ∷ α ∷ ς ∷ []) "1Cor.4.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.4.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.1"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.4.1"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ η ∷ ρ ∷ ί ∷ ω ∷ ν ∷ []) "1Cor.4.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.4.1"
∷ word (ὧ ∷ δ ∷ ε ∷ []) "1Cor.4.2"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ὸ ∷ ν ∷ []) "1Cor.4.2"
∷ word (ζ ∷ η ∷ τ ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.4.2"
∷ word (ἐ ∷ ν ∷ []) "1Cor.4.2"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.4.2"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ ν ∷ ό ∷ μ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.4.2"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.4.2"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "1Cor.4.2"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.4.2"
∷ word (ε ∷ ὑ ∷ ρ ∷ ε ∷ θ ∷ ῇ ∷ []) "1Cor.4.2"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "1Cor.4.3"
∷ word (δ ∷ ὲ ∷ []) "1Cor.4.3"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.4.3"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ χ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ν ∷ []) "1Cor.4.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.4.3"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.4.3"
∷ word (ὑ ∷ φ ∷ []) "1Cor.4.3"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.4.3"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ ρ ∷ ι ∷ θ ∷ ῶ ∷ []) "1Cor.4.3"
∷ word (ἢ ∷ []) "1Cor.4.3"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.4.3"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ί ∷ ν ∷ η ∷ ς ∷ []) "1Cor.4.3"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "1Cor.4.3"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.4.3"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "1Cor.4.3"
∷ word (ἐ ∷ μ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.4.3"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ω ∷ []) "1Cor.4.3"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ ν ∷ []) "1Cor.4.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.4.4"
∷ word (ἐ ∷ μ ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "1Cor.4.4"
∷ word (σ ∷ ύ ∷ ν ∷ ο ∷ ι ∷ δ ∷ α ∷ []) "1Cor.4.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.4.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.4.4"
∷ word (ἐ ∷ ν ∷ []) "1Cor.4.4"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1Cor.4.4"
∷ word (δ ∷ ε ∷ δ ∷ ι ∷ κ ∷ α ∷ ί ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "1Cor.4.4"
∷ word (ὁ ∷ []) "1Cor.4.4"
∷ word (δ ∷ ὲ ∷ []) "1Cor.4.4"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ω ∷ ν ∷ []) "1Cor.4.4"
∷ word (μ ∷ ε ∷ []) "1Cor.4.4"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ό ∷ ς ∷ []) "1Cor.4.4"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.4.4"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.4.5"
∷ word (μ ∷ ὴ ∷ []) "1Cor.4.5"
∷ word (π ∷ ρ ∷ ὸ ∷ []) "1Cor.4.5"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ο ∷ ῦ ∷ []) "1Cor.4.5"
∷ word (τ ∷ ι ∷ []) "1Cor.4.5"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.4.5"
∷ word (ἕ ∷ ω ∷ ς ∷ []) "1Cor.4.5"
∷ word (ἂ ∷ ν ∷ []) "1Cor.4.5"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ῃ ∷ []) "1Cor.4.5"
∷ word (ὁ ∷ []) "1Cor.4.5"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.4.5"
∷ word (ὃ ∷ ς ∷ []) "1Cor.4.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.5"
∷ word (φ ∷ ω ∷ τ ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.4.5"
∷ word (τ ∷ ὰ ∷ []) "1Cor.4.5"
∷ word (κ ∷ ρ ∷ υ ∷ π ∷ τ ∷ ὰ ∷ []) "1Cor.4.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.4.5"
∷ word (σ ∷ κ ∷ ό ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.4.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.5"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.4.5"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1Cor.4.5"
∷ word (β ∷ ο ∷ υ ∷ ∙λ ∷ ὰ ∷ ς ∷ []) "1Cor.4.5"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.4.5"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ι ∷ ῶ ∷ ν ∷ []) "1Cor.4.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.5"
∷ word (τ ∷ ό ∷ τ ∷ ε ∷ []) "1Cor.4.5"
∷ word (ὁ ∷ []) "1Cor.4.5"
∷ word (ἔ ∷ π ∷ α ∷ ι ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.4.5"
∷ word (γ ∷ ε ∷ ν ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.4.5"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ ῳ ∷ []) "1Cor.4.5"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "1Cor.4.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.4.5"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.4.5"
∷ word (Τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1Cor.4.6"
∷ word (δ ∷ έ ∷ []) "1Cor.4.6"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.4.6"
∷ word (μ ∷ ε ∷ τ ∷ ε ∷ σ ∷ χ ∷ η ∷ μ ∷ ά ∷ τ ∷ ι ∷ σ ∷ α ∷ []) "1Cor.4.6"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.4.6"
∷ word (ἐ ∷ μ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.4.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.6"
∷ word (Ἀ ∷ π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "1Cor.4.6"
∷ word (δ ∷ ι ∷ []) "1Cor.4.6"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.4.6"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.4.6"
∷ word (ἐ ∷ ν ∷ []) "1Cor.4.6"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.4.6"
∷ word (μ ∷ ά ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.4.6"
∷ word (τ ∷ ό ∷ []) "1Cor.4.6"
∷ word (Μ ∷ ὴ ∷ []) "1Cor.4.6"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "1Cor.4.6"
∷ word (ἃ ∷ []) "1Cor.4.6"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "1Cor.4.6"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.4.6"
∷ word (μ ∷ ὴ ∷ []) "1Cor.4.6"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "1Cor.4.6"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "1Cor.4.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.4.6"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "1Cor.4.6"
∷ word (φ ∷ υ ∷ σ ∷ ι ∷ ο ∷ ῦ ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.4.6"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.4.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.4.6"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ []) "1Cor.4.6"
∷ word (τ ∷ ί ∷ ς ∷ []) "1Cor.4.7"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.4.7"
∷ word (σ ∷ ε ∷ []) "1Cor.4.7"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ ι ∷ []) "1Cor.4.7"
∷ word (τ ∷ ί ∷ []) "1Cor.4.7"
∷ word (δ ∷ ὲ ∷ []) "1Cor.4.7"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.4.7"
∷ word (ὃ ∷ []) "1Cor.4.7"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.4.7"
∷ word (ἔ ∷ ∙λ ∷ α ∷ β ∷ ε ∷ ς ∷ []) "1Cor.4.7"
∷ word (ε ∷ ἰ ∷ []) "1Cor.4.7"
∷ word (δ ∷ ὲ ∷ []) "1Cor.4.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.7"
∷ word (ἔ ∷ ∙λ ∷ α ∷ β ∷ ε ∷ ς ∷ []) "1Cor.4.7"
∷ word (τ ∷ ί ∷ []) "1Cor.4.7"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ᾶ ∷ σ ∷ α ∷ ι ∷ []) "1Cor.4.7"
∷ word (ὡ ∷ ς ∷ []) "1Cor.4.7"
∷ word (μ ∷ ὴ ∷ []) "1Cor.4.7"
∷ word (∙λ ∷ α ∷ β ∷ ώ ∷ ν ∷ []) "1Cor.4.7"
∷ word (Ἤ ∷ δ ∷ η ∷ []) "1Cor.4.8"
∷ word (κ ∷ ε ∷ κ ∷ ο ∷ ρ ∷ ε ∷ σ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.4.8"
∷ word (ἐ ∷ σ ∷ τ ∷ έ ∷ []) "1Cor.4.8"
∷ word (ἤ ∷ δ ∷ η ∷ []) "1Cor.4.8"
∷ word (ἐ ∷ π ∷ ∙λ ∷ ο ∷ υ ∷ τ ∷ ή ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.4.8"
∷ word (χ ∷ ω ∷ ρ ∷ ὶ ∷ ς ∷ []) "1Cor.4.8"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.4.8"
∷ word (ἐ ∷ β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.4.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.8"
∷ word (ὄ ∷ φ ∷ ε ∷ ∙λ ∷ ό ∷ ν ∷ []) "1Cor.4.8"
∷ word (γ ∷ ε ∷ []) "1Cor.4.8"
∷ word (ἐ ∷ β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.4.8"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.4.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.8"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.4.8"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.4.8"
∷ word (σ ∷ υ ∷ μ ∷ β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.4.8"
∷ word (δ ∷ ο ∷ κ ∷ ῶ ∷ []) "1Cor.4.9"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.4.9"
∷ word (ὁ ∷ []) "1Cor.4.9"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.4.9"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.4.9"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.4.9"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.4.9"
∷ word (ἐ ∷ σ ∷ χ ∷ ά ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.4.9"
∷ word (ἀ ∷ π ∷ έ ∷ δ ∷ ε ∷ ι ∷ ξ ∷ ε ∷ ν ∷ []) "1Cor.4.9"
∷ word (ὡ ∷ ς ∷ []) "1Cor.4.9"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ α ∷ ν ∷ α ∷ τ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.4.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.4.9"
∷ word (θ ∷ έ ∷ α ∷ τ ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.4.9"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ή ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.4.9"
∷ word (τ ∷ ῷ ∷ []) "1Cor.4.9"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "1Cor.4.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.9"
∷ word (ἀ ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.4.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.9"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.4.9"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.4.10"
∷ word (μ ∷ ω ∷ ρ ∷ ο ∷ ὶ ∷ []) "1Cor.4.10"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.4.10"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ν ∷ []) "1Cor.4.10"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.4.10"
∷ word (δ ∷ ὲ ∷ []) "1Cor.4.10"
∷ word (φ ∷ ρ ∷ ό ∷ ν ∷ ι ∷ μ ∷ ο ∷ ι ∷ []) "1Cor.4.10"
∷ word (ἐ ∷ ν ∷ []) "1Cor.4.10"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1Cor.4.10"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.4.10"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.4.10"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.4.10"
∷ word (δ ∷ ὲ ∷ []) "1Cor.4.10"
∷ word (ἰ ∷ σ ∷ χ ∷ υ ∷ ρ ∷ ο ∷ ί ∷ []) "1Cor.4.10"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.4.10"
∷ word (ἔ ∷ ν ∷ δ ∷ ο ∷ ξ ∷ ο ∷ ι ∷ []) "1Cor.4.10"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.4.10"
∷ word (δ ∷ ὲ ∷ []) "1Cor.4.10"
∷ word (ἄ ∷ τ ∷ ι ∷ μ ∷ ο ∷ ι ∷ []) "1Cor.4.10"
∷ word (ἄ ∷ χ ∷ ρ ∷ ι ∷ []) "1Cor.4.11"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.4.11"
∷ word (ἄ ∷ ρ ∷ τ ∷ ι ∷ []) "1Cor.4.11"
∷ word (ὥ ∷ ρ ∷ α ∷ ς ∷ []) "1Cor.4.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.11"
∷ word (π ∷ ε ∷ ι ∷ ν ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.4.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.11"
∷ word (δ ∷ ι ∷ ψ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.4.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.11"
∷ word (γ ∷ υ ∷ μ ∷ ν ∷ ι ∷ τ ∷ ε ∷ ύ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.4.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.11"
∷ word (κ ∷ ο ∷ ∙λ ∷ α ∷ φ ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1Cor.4.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.11"
∷ word (ἀ ∷ σ ∷ τ ∷ α ∷ τ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.4.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.12"
∷ word (κ ∷ ο ∷ π ∷ ι ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.4.12"
∷ word (ἐ ∷ ρ ∷ γ ∷ α ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.4.12"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "1Cor.4.12"
∷ word (ἰ ∷ δ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "1Cor.4.12"
∷ word (χ ∷ ε ∷ ρ ∷ σ ∷ ί ∷ ν ∷ []) "1Cor.4.12"
∷ word (∙λ ∷ ο ∷ ι ∷ δ ∷ ο ∷ ρ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.4.12"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.4.12"
∷ word (δ ∷ ι ∷ ω ∷ κ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.4.12"
∷ word (ἀ ∷ ν ∷ ε ∷ χ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1Cor.4.12"
∷ word (δ ∷ υ ∷ σ ∷ φ ∷ η ∷ μ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.4.13"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.4.13"
∷ word (ὡ ∷ ς ∷ []) "1Cor.4.13"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ κ ∷ α ∷ θ ∷ ά ∷ ρ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.4.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.4.13"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1Cor.4.13"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ή ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.4.13"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.4.13"
∷ word (π ∷ ε ∷ ρ ∷ ί ∷ ψ ∷ η ∷ μ ∷ α ∷ []) "1Cor.4.13"
∷ word (ἕ ∷ ω ∷ ς ∷ []) "1Cor.4.13"
∷ word (ἄ ∷ ρ ∷ τ ∷ ι ∷ []) "1Cor.4.13"
∷ word (Ο ∷ ὐ ∷ κ ∷ []) "1Cor.4.14"
∷ word (ἐ ∷ ν ∷ τ ∷ ρ ∷ έ ∷ π ∷ ω ∷ ν ∷ []) "1Cor.4.14"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.4.14"
∷ word (γ ∷ ρ ∷ ά ∷ φ ∷ ω ∷ []) "1Cor.4.14"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1Cor.4.14"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.4.14"
∷ word (ὡ ∷ ς ∷ []) "1Cor.4.14"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "1Cor.4.14"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.4.14"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ὰ ∷ []) "1Cor.4.14"
∷ word (ν ∷ ο ∷ υ ∷ θ ∷ ε ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.4.14"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.4.15"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.4.15"
∷ word (μ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.4.15"
∷ word (π ∷ α ∷ ι ∷ δ ∷ α ∷ γ ∷ ω ∷ γ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.4.15"
∷ word (ἔ ∷ χ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.4.15"
∷ word (ἐ ∷ ν ∷ []) "1Cor.4.15"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1Cor.4.15"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.4.15"
∷ word (ο ∷ ὐ ∷ []) "1Cor.4.15"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.4.15"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "1Cor.4.15"
∷ word (ἐ ∷ ν ∷ []) "1Cor.4.15"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.4.15"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1Cor.4.15"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.4.15"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.4.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.4.15"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.4.15"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.4.15"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.4.15"
∷ word (ἐ ∷ γ ∷ έ ∷ ν ∷ ν ∷ η ∷ σ ∷ α ∷ []) "1Cor.4.15"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "1Cor.4.16"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.4.16"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.4.16"
∷ word (μ ∷ ι ∷ μ ∷ η ∷ τ ∷ α ∷ ί ∷ []) "1Cor.4.16"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.4.16"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.4.16"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.4.17"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.4.17"
∷ word (ἔ ∷ π ∷ ε ∷ μ ∷ ψ ∷ α ∷ []) "1Cor.4.17"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.4.17"
∷ word (Τ ∷ ι ∷ μ ∷ ό ∷ θ ∷ ε ∷ ο ∷ ν ∷ []) "1Cor.4.17"
∷ word (ὅ ∷ ς ∷ []) "1Cor.4.17"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1Cor.4.17"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.4.17"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.4.17"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.4.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.17"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.4.17"
∷ word (ἐ ∷ ν ∷ []) "1Cor.4.17"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.4.17"
∷ word (ὃ ∷ ς ∷ []) "1Cor.4.17"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.4.17"
∷ word (ἀ ∷ ν ∷ α ∷ μ ∷ ν ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.4.17"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1Cor.4.17"
∷ word (ὁ ∷ δ ∷ ο ∷ ύ ∷ ς ∷ []) "1Cor.4.17"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.4.17"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1Cor.4.17"
∷ word (ἐ ∷ ν ∷ []) "1Cor.4.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1Cor.4.17"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.4.17"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.4.17"
∷ word (π ∷ α ∷ ν ∷ τ ∷ α ∷ χ ∷ ο ∷ ῦ ∷ []) "1Cor.4.17"
∷ word (ἐ ∷ ν ∷ []) "1Cor.4.17"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "1Cor.4.17"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.4.17"
∷ word (δ ∷ ι ∷ δ ∷ ά ∷ σ ∷ κ ∷ ω ∷ []) "1Cor.4.17"
∷ word (ὡ ∷ ς ∷ []) "1Cor.4.18"
∷ word (μ ∷ ὴ ∷ []) "1Cor.4.18"
∷ word (ἐ ∷ ρ ∷ χ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ []) "1Cor.4.18"
∷ word (δ ∷ έ ∷ []) "1Cor.4.18"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.4.18"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.4.18"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.4.18"
∷ word (ἐ ∷ φ ∷ υ ∷ σ ∷ ι ∷ ώ ∷ θ ∷ η ∷ σ ∷ ά ∷ ν ∷ []) "1Cor.4.18"
∷ word (τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "1Cor.4.18"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.4.19"
∷ word (δ ∷ ὲ ∷ []) "1Cor.4.19"
∷ word (τ ∷ α ∷ χ ∷ έ ∷ ω ∷ ς ∷ []) "1Cor.4.19"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.4.19"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.4.19"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.4.19"
∷ word (ὁ ∷ []) "1Cor.4.19"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.4.19"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ σ ∷ ῃ ∷ []) "1Cor.4.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.4.19"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.4.19"
∷ word (ο ∷ ὐ ∷ []) "1Cor.4.19"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.4.19"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ν ∷ []) "1Cor.4.19"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.4.19"
∷ word (π ∷ ε ∷ φ ∷ υ ∷ σ ∷ ι ∷ ω ∷ μ ∷ έ ∷ ν ∷ ω ∷ ν ∷ []) "1Cor.4.19"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.4.19"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.4.19"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ν ∷ []) "1Cor.4.19"
∷ word (ο ∷ ὐ ∷ []) "1Cor.4.20"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.4.20"
∷ word (ἐ ∷ ν ∷ []) "1Cor.4.20"
∷ word (∙λ ∷ ό ∷ γ ∷ ῳ ∷ []) "1Cor.4.20"
∷ word (ἡ ∷ []) "1Cor.4.20"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ []) "1Cor.4.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.4.20"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.4.20"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.4.20"
∷ word (ἐ ∷ ν ∷ []) "1Cor.4.20"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ []) "1Cor.4.20"
∷ word (τ ∷ ί ∷ []) "1Cor.4.21"
∷ word (θ ∷ έ ∷ ∙λ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.4.21"
∷ word (ἐ ∷ ν ∷ []) "1Cor.4.21"
∷ word (ῥ ∷ ά ∷ β ∷ δ ∷ ῳ ∷ []) "1Cor.4.21"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ω ∷ []) "1Cor.4.21"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.4.21"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.4.21"
∷ word (ἢ ∷ []) "1Cor.4.21"
∷ word (ἐ ∷ ν ∷ []) "1Cor.4.21"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "1Cor.4.21"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ί ∷ []) "1Cor.4.21"
∷ word (τ ∷ ε ∷ []) "1Cor.4.21"
∷ word (π ∷ ρ ∷ α ∷ ΰ ∷ τ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.4.21"
∷ word (Ὅ ∷ ∙λ ∷ ω ∷ ς ∷ []) "1Cor.5.1"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.5.1"
∷ word (ἐ ∷ ν ∷ []) "1Cor.5.1"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.5.1"
∷ word (π ∷ ο ∷ ρ ∷ ν ∷ ε ∷ ί ∷ α ∷ []) "1Cor.5.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.5.1"
∷ word (τ ∷ ο ∷ ι ∷ α ∷ ύ ∷ τ ∷ η ∷ []) "1Cor.5.1"
∷ word (π ∷ ο ∷ ρ ∷ ν ∷ ε ∷ ί ∷ α ∷ []) "1Cor.5.1"
∷ word (ἥ ∷ τ ∷ ι ∷ ς ∷ []) "1Cor.5.1"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "1Cor.5.1"
∷ word (ἐ ∷ ν ∷ []) "1Cor.5.1"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.5.1"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.5.1"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.5.1"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ ά ∷ []) "1Cor.5.1"
∷ word (τ ∷ ι ∷ ν ∷ α ∷ []) "1Cor.5.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.5.1"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.5.1"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.5.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.5.2"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.5.2"
∷ word (π ∷ ε ∷ φ ∷ υ ∷ σ ∷ ι ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.5.2"
∷ word (ἐ ∷ σ ∷ τ ∷ έ ∷ []) "1Cor.5.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.5.2"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "1Cor.5.2"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.5.2"
∷ word (ἐ ∷ π ∷ ε ∷ ν ∷ θ ∷ ή ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.5.2"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.5.2"
∷ word (ἀ ∷ ρ ∷ θ ∷ ῇ ∷ []) "1Cor.5.2"
∷ word (ἐ ∷ κ ∷ []) "1Cor.5.2"
∷ word (μ ∷ έ ∷ σ ∷ ο ∷ υ ∷ []) "1Cor.5.2"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.5.2"
∷ word (ὁ ∷ []) "1Cor.5.2"
∷ word (τ ∷ ὸ ∷ []) "1Cor.5.2"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ν ∷ []) "1Cor.5.2"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.5.2"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ α ∷ ς ∷ []) "1Cor.5.2"
∷ word (Ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.5.3"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.5.3"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.5.3"
∷ word (ἀ ∷ π ∷ ὼ ∷ ν ∷ []) "1Cor.5.3"
∷ word (τ ∷ ῷ ∷ []) "1Cor.5.3"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.5.3"
∷ word (π ∷ α ∷ ρ ∷ ὼ ∷ ν ∷ []) "1Cor.5.3"
∷ word (δ ∷ ὲ ∷ []) "1Cor.5.3"
∷ word (τ ∷ ῷ ∷ []) "1Cor.5.3"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.5.3"
∷ word (ἤ ∷ δ ∷ η ∷ []) "1Cor.5.3"
∷ word (κ ∷ έ ∷ κ ∷ ρ ∷ ι ∷ κ ∷ α ∷ []) "1Cor.5.3"
∷ word (ὡ ∷ ς ∷ []) "1Cor.5.3"
∷ word (π ∷ α ∷ ρ ∷ ὼ ∷ ν ∷ []) "1Cor.5.3"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.5.3"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.5.3"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.5.3"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ρ ∷ γ ∷ α ∷ σ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.5.3"
∷ word (ἐ ∷ ν ∷ []) "1Cor.5.4"
∷ word (τ ∷ ῷ ∷ []) "1Cor.5.4"
∷ word (ὀ ∷ ν ∷ ό ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.5.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.5.4"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.5.4"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.5.4"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.5.4"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ χ ∷ θ ∷ έ ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.5.4"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.5.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.5.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.5.4"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "1Cor.5.4"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.5.4"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "1Cor.5.4"
∷ word (τ ∷ ῇ ∷ []) "1Cor.5.4"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ []) "1Cor.5.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.5.4"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.5.4"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.5.4"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.5.4"
∷ word (π ∷ α ∷ ρ ∷ α ∷ δ ∷ ο ∷ ῦ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.5.5"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.5.5"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.5.5"
∷ word (τ ∷ ῷ ∷ []) "1Cor.5.5"
∷ word (Σ ∷ α ∷ τ ∷ α ∷ ν ∷ ᾷ ∷ []) "1Cor.5.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.5.5"
∷ word (ὄ ∷ ∙λ ∷ ε ∷ θ ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.5.5"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.5.5"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ό ∷ ς ∷ []) "1Cor.5.5"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.5.5"
∷ word (τ ∷ ὸ ∷ []) "1Cor.5.5"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.5.5"
∷ word (σ ∷ ω ∷ θ ∷ ῇ ∷ []) "1Cor.5.5"
∷ word (ἐ ∷ ν ∷ []) "1Cor.5.5"
∷ word (τ ∷ ῇ ∷ []) "1Cor.5.5"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "1Cor.5.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.5.5"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.5.5"
∷ word (Ο ∷ ὐ ∷ []) "1Cor.5.6"
∷ word (κ ∷ α ∷ ∙λ ∷ ὸ ∷ ν ∷ []) "1Cor.5.6"
∷ word (τ ∷ ὸ ∷ []) "1Cor.5.6"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ μ ∷ α ∷ []) "1Cor.5.6"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.5.6"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.5.6"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.5.6"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.5.6"
∷ word (μ ∷ ι ∷ κ ∷ ρ ∷ ὰ ∷ []) "1Cor.5.6"
∷ word (ζ ∷ ύ ∷ μ ∷ η ∷ []) "1Cor.5.6"
∷ word (ὅ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.5.6"
∷ word (τ ∷ ὸ ∷ []) "1Cor.5.6"
∷ word (φ ∷ ύ ∷ ρ ∷ α ∷ μ ∷ α ∷ []) "1Cor.5.6"
∷ word (ζ ∷ υ ∷ μ ∷ ο ∷ ῖ ∷ []) "1Cor.5.6"
∷ word (ἐ ∷ κ ∷ κ ∷ α ∷ θ ∷ ά ∷ ρ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.5.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.5.7"
∷ word (π ∷ α ∷ ∙λ ∷ α ∷ ι ∷ ὰ ∷ ν ∷ []) "1Cor.5.7"
∷ word (ζ ∷ ύ ∷ μ ∷ η ∷ ν ∷ []) "1Cor.5.7"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.5.7"
∷ word (ἦ ∷ τ ∷ ε ∷ []) "1Cor.5.7"
∷ word (ν ∷ έ ∷ ο ∷ ν ∷ []) "1Cor.5.7"
∷ word (φ ∷ ύ ∷ ρ ∷ α ∷ μ ∷ α ∷ []) "1Cor.5.7"
∷ word (κ ∷ α ∷ θ ∷ ώ ∷ ς ∷ []) "1Cor.5.7"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.5.7"
∷ word (ἄ ∷ ζ ∷ υ ∷ μ ∷ ο ∷ ι ∷ []) "1Cor.5.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.5.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.5.7"
∷ word (τ ∷ ὸ ∷ []) "1Cor.5.7"
∷ word (π ∷ ά ∷ σ ∷ χ ∷ α ∷ []) "1Cor.5.7"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.5.7"
∷ word (ἐ ∷ τ ∷ ύ ∷ θ ∷ η ∷ []) "1Cor.5.7"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "1Cor.5.7"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.5.8"
∷ word (ἑ ∷ ο ∷ ρ ∷ τ ∷ ά ∷ ζ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.5.8"
∷ word (μ ∷ ὴ ∷ []) "1Cor.5.8"
∷ word (ἐ ∷ ν ∷ []) "1Cor.5.8"
∷ word (ζ ∷ ύ ∷ μ ∷ ῃ ∷ []) "1Cor.5.8"
∷ word (π ∷ α ∷ ∙λ ∷ α ∷ ι ∷ ᾷ ∷ []) "1Cor.5.8"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "1Cor.5.8"
∷ word (ἐ ∷ ν ∷ []) "1Cor.5.8"
∷ word (ζ ∷ ύ ∷ μ ∷ ῃ ∷ []) "1Cor.5.8"
∷ word (κ ∷ α ∷ κ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.5.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.5.8"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.5.8"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.5.8"
∷ word (ἐ ∷ ν ∷ []) "1Cor.5.8"
∷ word (ἀ ∷ ζ ∷ ύ ∷ μ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.5.8"
∷ word (ε ∷ ἰ ∷ ∙λ ∷ ι ∷ κ ∷ ρ ∷ ι ∷ ν ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "1Cor.5.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.5.8"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "1Cor.5.8"
∷ word (Ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "1Cor.5.9"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.5.9"
∷ word (ἐ ∷ ν ∷ []) "1Cor.5.9"
∷ word (τ ∷ ῇ ∷ []) "1Cor.5.9"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ῇ ∷ []) "1Cor.5.9"
∷ word (μ ∷ ὴ ∷ []) "1Cor.5.9"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ ν ∷ α ∷ μ ∷ ί ∷ γ ∷ ν ∷ υ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.5.9"
∷ word (π ∷ ό ∷ ρ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.5.9"
∷ word (ο ∷ ὐ ∷ []) "1Cor.5.10"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.5.10"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.5.10"
∷ word (π ∷ ό ∷ ρ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.5.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.5.10"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1Cor.5.10"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.5.10"
∷ word (ἢ ∷ []) "1Cor.5.10"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.5.10"
∷ word (π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ έ ∷ κ ∷ τ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.5.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.5.10"
∷ word (ἅ ∷ ρ ∷ π ∷ α ∷ ξ ∷ ι ∷ ν ∷ []) "1Cor.5.10"
∷ word (ἢ ∷ []) "1Cor.5.10"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ο ∷ ∙λ ∷ ά ∷ τ ∷ ρ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.5.10"
∷ word (ἐ ∷ π ∷ ε ∷ ὶ ∷ []) "1Cor.5.10"
∷ word (ὠ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.5.10"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "1Cor.5.10"
∷ word (ἐ ∷ κ ∷ []) "1Cor.5.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.5.10"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1Cor.5.10"
∷ word (ἐ ∷ ξ ∷ ε ∷ ∙λ ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.5.10"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "1Cor.5.11"
∷ word (δ ∷ ὲ ∷ []) "1Cor.5.11"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "1Cor.5.11"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.5.11"
∷ word (μ ∷ ὴ ∷ []) "1Cor.5.11"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ ν ∷ α ∷ μ ∷ ί ∷ γ ∷ ν ∷ υ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.5.11"
∷ word (ἐ ∷ ά ∷ ν ∷ []) "1Cor.5.11"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.5.11"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ς ∷ []) "1Cor.5.11"
∷ word (ὀ ∷ ν ∷ ο ∷ μ ∷ α ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.5.11"
∷ word (ᾖ ∷ []) "1Cor.5.11"
∷ word (π ∷ ό ∷ ρ ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.5.11"
∷ word (ἢ ∷ []) "1Cor.5.11"
∷ word (π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ έ ∷ κ ∷ τ ∷ η ∷ ς ∷ []) "1Cor.5.11"
∷ word (ἢ ∷ []) "1Cor.5.11"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ο ∷ ∙λ ∷ ά ∷ τ ∷ ρ ∷ η ∷ ς ∷ []) "1Cor.5.11"
∷ word (ἢ ∷ []) "1Cor.5.11"
∷ word (∙λ ∷ ο ∷ ί ∷ δ ∷ ο ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.5.11"
∷ word (ἢ ∷ []) "1Cor.5.11"
∷ word (μ ∷ έ ∷ θ ∷ υ ∷ σ ∷ ο ∷ ς ∷ []) "1Cor.5.11"
∷ word (ἢ ∷ []) "1Cor.5.11"
∷ word (ἅ ∷ ρ ∷ π ∷ α ∷ ξ ∷ []) "1Cor.5.11"
∷ word (τ ∷ ῷ ∷ []) "1Cor.5.11"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1Cor.5.11"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "1Cor.5.11"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ σ ∷ θ ∷ ί ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.5.11"
∷ word (τ ∷ ί ∷ []) "1Cor.5.12"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.5.12"
∷ word (μ ∷ ο ∷ ι ∷ []) "1Cor.5.12"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.5.12"
∷ word (ἔ ∷ ξ ∷ ω ∷ []) "1Cor.5.12"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.5.12"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "1Cor.5.12"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.5.12"
∷ word (ἔ ∷ σ ∷ ω ∷ []) "1Cor.5.12"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.5.12"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.5.12"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.5.13"
∷ word (δ ∷ ὲ ∷ []) "1Cor.5.13"
∷ word (ἔ ∷ ξ ∷ ω ∷ []) "1Cor.5.13"
∷ word (ὁ ∷ []) "1Cor.5.13"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.5.13"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ ι ∷ []) "1Cor.5.13"
∷ word (ἐ ∷ ξ ∷ ά ∷ ρ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.5.13"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.5.13"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ὸ ∷ ν ∷ []) "1Cor.5.13"
∷ word (ἐ ∷ ξ ∷ []) "1Cor.5.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.5.13"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.5.13"
∷ word (Τ ∷ ο ∷ ∙λ ∷ μ ∷ ᾷ ∷ []) "1Cor.6.1"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.6.1"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.6.1"
∷ word (π ∷ ρ ∷ ᾶ ∷ γ ∷ μ ∷ α ∷ []) "1Cor.6.1"
∷ word (ἔ ∷ χ ∷ ω ∷ ν ∷ []) "1Cor.6.1"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.6.1"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.6.1"
∷ word (ἕ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.6.1"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.6.1"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.6.1"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.6.1"
∷ word (ἀ ∷ δ ∷ ί ∷ κ ∷ ω ∷ ν ∷ []) "1Cor.6.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.1"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "1Cor.6.1"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.6.1"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.6.1"
∷ word (ἁ ∷ γ ∷ ί ∷ ω ∷ ν ∷ []) "1Cor.6.1"
∷ word (ἢ ∷ []) "1Cor.6.2"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.6.2"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.6.2"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.6.2"
∷ word (ο ∷ ἱ ∷ []) "1Cor.6.2"
∷ word (ἅ ∷ γ ∷ ι ∷ ο ∷ ι ∷ []) "1Cor.6.2"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.6.2"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ν ∷ []) "1Cor.6.2"
∷ word (κ ∷ ρ ∷ ι ∷ ν ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.6.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.2"
∷ word (ε ∷ ἰ ∷ []) "1Cor.6.2"
∷ word (ἐ ∷ ν ∷ []) "1Cor.6.2"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.6.2"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.6.2"
∷ word (ὁ ∷ []) "1Cor.6.2"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.6.2"
∷ word (ἀ ∷ ν ∷ ά ∷ ξ ∷ ι ∷ ο ∷ ί ∷ []) "1Cor.6.2"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.6.2"
∷ word (κ ∷ ρ ∷ ι ∷ τ ∷ η ∷ ρ ∷ ί ∷ ω ∷ ν ∷ []) "1Cor.6.2"
∷ word (ἐ ∷ ∙λ ∷ α ∷ χ ∷ ί ∷ σ ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.6.2"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.6.3"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.6.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.6.3"
∷ word (ἀ ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.6.3"
∷ word (κ ∷ ρ ∷ ι ∷ ν ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.6.3"
∷ word (μ ∷ ή ∷ τ ∷ ι ∷ γ ∷ ε ∷ []) "1Cor.6.3"
∷ word (β ∷ ι ∷ ω ∷ τ ∷ ι ∷ κ ∷ ά ∷ []) "1Cor.6.3"
∷ word (β ∷ ι ∷ ω ∷ τ ∷ ι ∷ κ ∷ ὰ ∷ []) "1Cor.6.4"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.6.4"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.6.4"
∷ word (κ ∷ ρ ∷ ι ∷ τ ∷ ή ∷ ρ ∷ ι ∷ α ∷ []) "1Cor.6.4"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.6.4"
∷ word (ἔ ∷ χ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.6.4"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.6.4"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ θ ∷ ε ∷ ν ∷ η ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.6.4"
∷ word (ἐ ∷ ν ∷ []) "1Cor.6.4"
∷ word (τ ∷ ῇ ∷ []) "1Cor.6.4"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.6.4"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.6.4"
∷ word (κ ∷ α ∷ θ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.6.4"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.6.5"
∷ word (ἐ ∷ ν ∷ τ ∷ ρ ∷ ο ∷ π ∷ ὴ ∷ ν ∷ []) "1Cor.6.5"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.6.5"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "1Cor.6.5"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.6.5"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.6.5"
∷ word (ἔ ∷ ν ∷ ι ∷ []) "1Cor.6.5"
∷ word (ἐ ∷ ν ∷ []) "1Cor.6.5"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.6.5"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.6.5"
∷ word (σ ∷ ο ∷ φ ∷ ὸ ∷ ς ∷ []) "1Cor.6.5"
∷ word (ὃ ∷ ς ∷ []) "1Cor.6.5"
∷ word (δ ∷ υ ∷ ν ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.6.5"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ρ ∷ ῖ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.6.5"
∷ word (ἀ ∷ ν ∷ ὰ ∷ []) "1Cor.6.5"
∷ word (μ ∷ έ ∷ σ ∷ ο ∷ ν ∷ []) "1Cor.6.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.6.5"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ῦ ∷ []) "1Cor.6.5"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.6.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.6.6"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ς ∷ []) "1Cor.6.6"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "1Cor.6.6"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ῦ ∷ []) "1Cor.6.6"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.6.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.6"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.6.6"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.6.6"
∷ word (ἀ ∷ π ∷ ί ∷ σ ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.6.6"
∷ word (ἤ ∷ δ ∷ η ∷ []) "1Cor.6.7"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.6.7"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.6.7"
∷ word (ὅ ∷ ∙λ ∷ ω ∷ ς ∷ []) "1Cor.6.7"
∷ word (ἥ ∷ τ ∷ τ ∷ η ∷ μ ∷ α ∷ []) "1Cor.6.7"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.6.7"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.6.7"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.6.7"
∷ word (κ ∷ ρ ∷ ί ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.6.7"
∷ word (ἔ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.6.7"
∷ word (μ ∷ ε ∷ θ ∷ []) "1Cor.6.7"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.6.7"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.6.7"
∷ word (τ ∷ ί ∷ []) "1Cor.6.7"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "1Cor.6.7"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.6.7"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ε ∷ ῖ ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.6.7"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.6.7"
∷ word (τ ∷ ί ∷ []) "1Cor.6.7"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "1Cor.6.7"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.6.7"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ε ∷ ρ ∷ ε ∷ ῖ ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.6.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.6.8"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.6.8"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "1Cor.6.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.8"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ε ∷ ρ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "1Cor.6.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.8"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.6.8"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ύ ∷ ς ∷ []) "1Cor.6.8"
∷ word (Ἢ ∷ []) "1Cor.6.9"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.6.9"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.6.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.6.9"
∷ word (ἄ ∷ δ ∷ ι ∷ κ ∷ ο ∷ ι ∷ []) "1Cor.6.9"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.6.9"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "1Cor.6.9"
∷ word (ο ∷ ὐ ∷ []) "1Cor.6.9"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ή ∷ σ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.6.9"
∷ word (μ ∷ ὴ ∷ []) "1Cor.6.9"
∷ word (π ∷ ∙λ ∷ α ∷ ν ∷ ᾶ ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.6.9"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "1Cor.6.9"
∷ word (π ∷ ό ∷ ρ ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.6.9"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "1Cor.6.9"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ο ∷ ∙λ ∷ ά ∷ τ ∷ ρ ∷ α ∷ ι ∷ []) "1Cor.6.9"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "1Cor.6.9"
∷ word (μ ∷ ο ∷ ι ∷ χ ∷ ο ∷ ὶ ∷ []) "1Cor.6.9"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "1Cor.6.9"
∷ word (μ ∷ α ∷ ∙λ ∷ α ∷ κ ∷ ο ∷ ὶ ∷ []) "1Cor.6.9"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "1Cor.6.9"
∷ word (ἀ ∷ ρ ∷ σ ∷ ε ∷ ν ∷ ο ∷ κ ∷ ο ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.6.9"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "1Cor.6.10"
∷ word (κ ∷ ∙λ ∷ έ ∷ π ∷ τ ∷ α ∷ ι ∷ []) "1Cor.6.10"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "1Cor.6.10"
∷ word (π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ έ ∷ κ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.6.10"
∷ word (ο ∷ ὐ ∷ []) "1Cor.6.10"
∷ word (μ ∷ έ ∷ θ ∷ υ ∷ σ ∷ ο ∷ ι ∷ []) "1Cor.6.10"
∷ word (ο ∷ ὐ ∷ []) "1Cor.6.10"
∷ word (∙λ ∷ ο ∷ ί ∷ δ ∷ ο ∷ ρ ∷ ο ∷ ι ∷ []) "1Cor.6.10"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1Cor.6.10"
∷ word (ἅ ∷ ρ ∷ π ∷ α ∷ γ ∷ ε ∷ ς ∷ []) "1Cor.6.10"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "1Cor.6.10"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.6.10"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ή ∷ σ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.6.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.11"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ ά ∷ []) "1Cor.6.11"
∷ word (τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "1Cor.6.11"
∷ word (ἦ ∷ τ ∷ ε ∷ []) "1Cor.6.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.6.11"
∷ word (ἀ ∷ π ∷ ε ∷ ∙λ ∷ ο ∷ ύ ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.6.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.6.11"
∷ word (ἡ ∷ γ ∷ ι ∷ ά ∷ σ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.6.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.6.11"
∷ word (ἐ ∷ δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ώ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.6.11"
∷ word (ἐ ∷ ν ∷ []) "1Cor.6.11"
∷ word (τ ∷ ῷ ∷ []) "1Cor.6.11"
∷ word (ὀ ∷ ν ∷ ό ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.6.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.6.11"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.6.11"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.6.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.11"
∷ word (ἐ ∷ ν ∷ []) "1Cor.6.11"
∷ word (τ ∷ ῷ ∷ []) "1Cor.6.11"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.6.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.6.11"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.6.11"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.6.11"
∷ word (Π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.6.12"
∷ word (μ ∷ ο ∷ ι ∷ []) "1Cor.6.12"
∷ word (ἔ ∷ ξ ∷ ε ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.6.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.6.12"
∷ word (ο ∷ ὐ ∷ []) "1Cor.6.12"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.6.12"
∷ word (σ ∷ υ ∷ μ ∷ φ ∷ έ ∷ ρ ∷ ε ∷ ι ∷ []) "1Cor.6.12"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.6.12"
∷ word (μ ∷ ο ∷ ι ∷ []) "1Cor.6.12"
∷ word (ἔ ∷ ξ ∷ ε ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.6.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.6.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.6.12"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.6.12"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ι ∷ α ∷ σ ∷ θ ∷ ή ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.6.12"
∷ word (ὑ ∷ π ∷ ό ∷ []) "1Cor.6.12"
∷ word (τ ∷ ι ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.6.12"
∷ word (τ ∷ ὰ ∷ []) "1Cor.6.13"
∷ word (β ∷ ρ ∷ ώ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.6.13"
∷ word (τ ∷ ῇ ∷ []) "1Cor.6.13"
∷ word (κ ∷ ο ∷ ι ∷ ∙λ ∷ ί ∷ ᾳ ∷ []) "1Cor.6.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.13"
∷ word (ἡ ∷ []) "1Cor.6.13"
∷ word (κ ∷ ο ∷ ι ∷ ∙λ ∷ ί ∷ α ∷ []) "1Cor.6.13"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.6.13"
∷ word (β ∷ ρ ∷ ώ ∷ μ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.6.13"
∷ word (ὁ ∷ []) "1Cor.6.13"
∷ word (δ ∷ ὲ ∷ []) "1Cor.6.13"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.6.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.13"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ η ∷ ν ∷ []) "1Cor.6.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.13"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1Cor.6.13"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.6.13"
∷ word (τ ∷ ὸ ∷ []) "1Cor.6.13"
∷ word (δ ∷ ὲ ∷ []) "1Cor.6.13"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.6.13"
∷ word (ο ∷ ὐ ∷ []) "1Cor.6.13"
∷ word (τ ∷ ῇ ∷ []) "1Cor.6.13"
∷ word (π ∷ ο ∷ ρ ∷ ν ∷ ε ∷ ί ∷ ᾳ ∷ []) "1Cor.6.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.6.13"
∷ word (τ ∷ ῷ ∷ []) "1Cor.6.13"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.6.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.13"
∷ word (ὁ ∷ []) "1Cor.6.13"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.6.13"
∷ word (τ ∷ ῷ ∷ []) "1Cor.6.13"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.6.13"
∷ word (ὁ ∷ []) "1Cor.6.14"
∷ word (δ ∷ ὲ ∷ []) "1Cor.6.14"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.6.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.14"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.6.14"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.6.14"
∷ word (ἤ ∷ γ ∷ ε ∷ ι ∷ ρ ∷ ε ∷ ν ∷ []) "1Cor.6.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.14"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.6.14"
∷ word (ἐ ∷ ξ ∷ ε ∷ γ ∷ ε ∷ ρ ∷ ε ∷ ῖ ∷ []) "1Cor.6.14"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.6.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.6.14"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ω ∷ ς ∷ []) "1Cor.6.14"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.6.14"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.6.15"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.6.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.6.15"
∷ word (τ ∷ ὰ ∷ []) "1Cor.6.15"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.6.15"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.6.15"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "1Cor.6.15"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.6.15"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.6.15"
∷ word (ἄ ∷ ρ ∷ α ∷ ς ∷ []) "1Cor.6.15"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.6.15"
∷ word (τ ∷ ὰ ∷ []) "1Cor.6.15"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "1Cor.6.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.6.15"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.6.15"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ ω ∷ []) "1Cor.6.15"
∷ word (π ∷ ό ∷ ρ ∷ ν ∷ η ∷ ς ∷ []) "1Cor.6.15"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "1Cor.6.15"
∷ word (μ ∷ ὴ ∷ []) "1Cor.6.15"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "1Cor.6.15"
∷ word (ἢ ∷ []) "1Cor.6.16"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.6.16"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.6.16"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.6.16"
∷ word (ὁ ∷ []) "1Cor.6.16"
∷ word (κ ∷ ο ∷ ∙λ ∷ ∙λ ∷ ώ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.6.16"
∷ word (τ ∷ ῇ ∷ []) "1Cor.6.16"
∷ word (π ∷ ό ∷ ρ ∷ ν ∷ ῃ ∷ []) "1Cor.6.16"
∷ word (ἓ ∷ ν ∷ []) "1Cor.6.16"
∷ word (σ ∷ ῶ ∷ μ ∷ ά ∷ []) "1Cor.6.16"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.6.16"
∷ word (Ἔ ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.6.16"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.6.16"
∷ word (φ ∷ η ∷ σ ∷ ί ∷ ν ∷ []) "1Cor.6.16"
∷ word (ο ∷ ἱ ∷ []) "1Cor.6.16"
∷ word (δ ∷ ύ ∷ ο ∷ []) "1Cor.6.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.6.16"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "1Cor.6.16"
∷ word (μ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.6.16"
∷ word (ὁ ∷ []) "1Cor.6.17"
∷ word (δ ∷ ὲ ∷ []) "1Cor.6.17"
∷ word (κ ∷ ο ∷ ∙λ ∷ ∙λ ∷ ώ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.6.17"
∷ word (τ ∷ ῷ ∷ []) "1Cor.6.17"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.6.17"
∷ word (ἓ ∷ ν ∷ []) "1Cor.6.17"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ ά ∷ []) "1Cor.6.17"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.6.17"
∷ word (φ ∷ ε ∷ ύ ∷ γ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.6.18"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.6.18"
∷ word (π ∷ ο ∷ ρ ∷ ν ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "1Cor.6.18"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "1Cor.6.18"
∷ word (ἁ ∷ μ ∷ ά ∷ ρ ∷ τ ∷ η ∷ μ ∷ α ∷ []) "1Cor.6.18"
∷ word (ὃ ∷ []) "1Cor.6.18"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.6.18"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ ῃ ∷ []) "1Cor.6.18"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "1Cor.6.18"
∷ word (ἐ ∷ κ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.6.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.6.18"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ό ∷ ς ∷ []) "1Cor.6.18"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.6.18"
∷ word (ὁ ∷ []) "1Cor.6.18"
∷ word (δ ∷ ὲ ∷ []) "1Cor.6.18"
∷ word (π ∷ ο ∷ ρ ∷ ν ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "1Cor.6.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.6.18"
∷ word (τ ∷ ὸ ∷ []) "1Cor.6.18"
∷ word (ἴ ∷ δ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.6.18"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.6.18"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ά ∷ ν ∷ ε ∷ ι ∷ []) "1Cor.6.18"
∷ word (ἢ ∷ []) "1Cor.6.19"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.6.19"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.6.19"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.6.19"
∷ word (τ ∷ ὸ ∷ []) "1Cor.6.19"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.6.19"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.6.19"
∷ word (ν ∷ α ∷ ὸ ∷ ς ∷ []) "1Cor.6.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.6.19"
∷ word (ἐ ∷ ν ∷ []) "1Cor.6.19"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.6.19"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.6.19"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ό ∷ ς ∷ []) "1Cor.6.19"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.6.19"
∷ word (ο ∷ ὗ ∷ []) "1Cor.6.19"
∷ word (ἔ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.6.19"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "1Cor.6.19"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.6.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.6.19"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.6.19"
∷ word (ἐ ∷ σ ∷ τ ∷ ὲ ∷ []) "1Cor.6.19"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.6.19"
∷ word (ἠ ∷ γ ∷ ο ∷ ρ ∷ ά ∷ σ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.6.20"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.6.20"
∷ word (τ ∷ ι ∷ μ ∷ ῆ ∷ ς ∷ []) "1Cor.6.20"
∷ word (δ ∷ ο ∷ ξ ∷ ά ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.6.20"
∷ word (δ ∷ ὴ ∷ []) "1Cor.6.20"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.6.20"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "1Cor.6.20"
∷ word (ἐ ∷ ν ∷ []) "1Cor.6.20"
∷ word (τ ∷ ῷ ∷ []) "1Cor.6.20"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.6.20"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.6.20"
∷ word (Π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1Cor.7.1"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.1"
∷ word (ὧ ∷ ν ∷ []) "1Cor.7.1"
∷ word (ἐ ∷ γ ∷ ρ ∷ ά ∷ ψ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.7.1"
∷ word (κ ∷ α ∷ ∙λ ∷ ὸ ∷ ν ∷ []) "1Cor.7.1"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ῳ ∷ []) "1Cor.7.1"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ὸ ∷ ς ∷ []) "1Cor.7.1"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.1"
∷ word (ἅ ∷ π ∷ τ ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.7.1"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.7.2"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.2"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1Cor.7.2"
∷ word (π ∷ ο ∷ ρ ∷ ν ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "1Cor.7.2"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.7.2"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.7.2"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.2"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ []) "1Cor.7.2"
∷ word (ἐ ∷ χ ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.7.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.2"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ η ∷ []) "1Cor.7.2"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.7.2"
∷ word (ἴ ∷ δ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.7.2"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ α ∷ []) "1Cor.7.2"
∷ word (ἐ ∷ χ ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.7.2"
∷ word (τ ∷ ῇ ∷ []) "1Cor.7.3"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ὶ ∷ []) "1Cor.7.3"
∷ word (ὁ ∷ []) "1Cor.7.3"
∷ word (ἀ ∷ ν ∷ ὴ ∷ ρ ∷ []) "1Cor.7.3"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.7.3"
∷ word (ὀ ∷ φ ∷ ε ∷ ι ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "1Cor.7.3"
∷ word (ἀ ∷ π ∷ ο ∷ δ ∷ ι ∷ δ ∷ ό ∷ τ ∷ ω ∷ []) "1Cor.7.3"
∷ word (ὁ ∷ μ ∷ ο ∷ ί ∷ ω ∷ ς ∷ []) "1Cor.7.3"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.3"
∷ word (ἡ ∷ []) "1Cor.7.3"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.7.3"
∷ word (τ ∷ ῷ ∷ []) "1Cor.7.3"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ί ∷ []) "1Cor.7.3"
∷ word (ἡ ∷ []) "1Cor.7.4"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.7.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.4"
∷ word (ἰ ∷ δ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.7.4"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.7.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.7.4"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ά ∷ ζ ∷ ε ∷ ι ∷ []) "1Cor.7.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.7.4"
∷ word (ὁ ∷ []) "1Cor.7.4"
∷ word (ἀ ∷ ν ∷ ή ∷ ρ ∷ []) "1Cor.7.4"
∷ word (ὁ ∷ μ ∷ ο ∷ ί ∷ ω ∷ ς ∷ []) "1Cor.7.4"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.4"
∷ word (ὁ ∷ []) "1Cor.7.4"
∷ word (ἀ ∷ ν ∷ ὴ ∷ ρ ∷ []) "1Cor.7.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.4"
∷ word (ἰ ∷ δ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.7.4"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.7.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.7.4"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ά ∷ ζ ∷ ε ∷ ι ∷ []) "1Cor.7.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.7.4"
∷ word (ἡ ∷ []) "1Cor.7.4"
∷ word (γ ∷ υ ∷ ν ∷ ή ∷ []) "1Cor.7.4"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.5"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ε ∷ ρ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "1Cor.7.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.7.5"
∷ word (ε ∷ ἰ ∷ []) "1Cor.7.5"
∷ word (μ ∷ ή ∷ τ ∷ ι ∷ []) "1Cor.7.5"
∷ word (ἂ ∷ ν ∷ []) "1Cor.7.5"
∷ word (ἐ ∷ κ ∷ []) "1Cor.7.5"
∷ word (σ ∷ υ ∷ μ ∷ φ ∷ ώ ∷ ν ∷ ο ∷ υ ∷ []) "1Cor.7.5"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.7.5"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ὸ ∷ ν ∷ []) "1Cor.7.5"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.7.5"
∷ word (σ ∷ χ ∷ ο ∷ ∙λ ∷ ά ∷ σ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.7.5"
∷ word (τ ∷ ῇ ∷ []) "1Cor.7.5"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ ῇ ∷ []) "1Cor.7.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.5"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "1Cor.7.5"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.7.5"
∷ word (τ ∷ ὸ ∷ []) "1Cor.7.5"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "1Cor.7.5"
∷ word (ἦ ∷ τ ∷ ε ∷ []) "1Cor.7.5"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.7.5"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.5"
∷ word (π ∷ ε ∷ ι ∷ ρ ∷ ά ∷ ζ ∷ ῃ ∷ []) "1Cor.7.5"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.7.5"
∷ word (ὁ ∷ []) "1Cor.7.5"
∷ word (Σ ∷ α ∷ τ ∷ α ∷ ν ∷ ᾶ ∷ ς ∷ []) "1Cor.7.5"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.7.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.7.5"
∷ word (ἀ ∷ κ ∷ ρ ∷ α ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.7.5"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.7.5"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.7.6"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.6"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "1Cor.7.6"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.7.6"
∷ word (σ ∷ υ ∷ γ ∷ γ ∷ ν ∷ ώ ∷ μ ∷ η ∷ ν ∷ []) "1Cor.7.6"
∷ word (ο ∷ ὐ ∷ []) "1Cor.7.6"
∷ word (κ ∷ α ∷ τ ∷ []) "1Cor.7.6"
∷ word (ἐ ∷ π ∷ ι ∷ τ ∷ α ∷ γ ∷ ή ∷ ν ∷ []) "1Cor.7.6"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "1Cor.7.7"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.7"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "1Cor.7.7"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.7.7"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.7.7"
∷ word (ὡ ∷ ς ∷ []) "1Cor.7.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.7"
∷ word (ἐ ∷ μ ∷ α ∷ υ ∷ τ ∷ ό ∷ ν ∷ []) "1Cor.7.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.7.7"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.7.7"
∷ word (ἴ ∷ δ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.7.7"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.7.7"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ σ ∷ μ ∷ α ∷ []) "1Cor.7.7"
∷ word (ἐ ∷ κ ∷ []) "1Cor.7.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.7.7"
∷ word (ὁ ∷ []) "1Cor.7.7"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.7.7"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.7.7"
∷ word (ὁ ∷ []) "1Cor.7.7"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.7"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.7.7"
∷ word (Λ ∷ έ ∷ γ ∷ ω ∷ []) "1Cor.7.8"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.8"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.7.8"
∷ word (ἀ ∷ γ ∷ ά ∷ μ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.7.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.8"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "1Cor.7.8"
∷ word (χ ∷ ή ∷ ρ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.7.8"
∷ word (κ ∷ α ∷ ∙λ ∷ ὸ ∷ ν ∷ []) "1Cor.7.8"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.7.8"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.7.8"
∷ word (μ ∷ ε ∷ ί ∷ ν ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.7.8"
∷ word (ὡ ∷ ς ∷ []) "1Cor.7.8"
∷ word (κ ∷ ἀ ∷ γ ∷ ώ ∷ []) "1Cor.7.8"
∷ word (ε ∷ ἰ ∷ []) "1Cor.7.9"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.9"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.7.9"
∷ word (ἐ ∷ γ ∷ κ ∷ ρ ∷ α ∷ τ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.7.9"
∷ word (γ ∷ α ∷ μ ∷ η ∷ σ ∷ ά ∷ τ ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "1Cor.7.9"
∷ word (κ ∷ ρ ∷ ε ∷ ῖ ∷ τ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.7.9"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.7.9"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.7.9"
∷ word (γ ∷ α ∷ μ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "1Cor.7.9"
∷ word (ἢ ∷ []) "1Cor.7.9"
∷ word (π ∷ υ ∷ ρ ∷ ο ∷ ῦ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.7.9"
∷ word (Τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.7.10"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.10"
∷ word (γ ∷ ε ∷ γ ∷ α ∷ μ ∷ η ∷ κ ∷ ό ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.7.10"
∷ word (π ∷ α ∷ ρ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ω ∷ []) "1Cor.7.10"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.7.10"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.7.10"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.7.10"
∷ word (ὁ ∷ []) "1Cor.7.10"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.7.10"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ []) "1Cor.7.10"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "1Cor.7.10"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.7.10"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.10"
∷ word (χ ∷ ω ∷ ρ ∷ ι ∷ σ ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.7.10"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.7.11"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.11"
∷ word (χ ∷ ω ∷ ρ ∷ ι ∷ σ ∷ θ ∷ ῇ ∷ []) "1Cor.7.11"
∷ word (μ ∷ ε ∷ ν ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.7.11"
∷ word (ἄ ∷ γ ∷ α ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.7.11"
∷ word (ἢ ∷ []) "1Cor.7.11"
∷ word (τ ∷ ῷ ∷ []) "1Cor.7.11"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ὶ ∷ []) "1Cor.7.11"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ∙λ ∷ α ∷ γ ∷ ή ∷ τ ∷ ω ∷ []) "1Cor.7.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.11"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ α ∷ []) "1Cor.7.11"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ []) "1Cor.7.11"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.11"
∷ word (ἀ ∷ φ ∷ ι ∷ έ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.7.11"
∷ word (Τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.7.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.12"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.7.12"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "1Cor.7.12"
∷ word (ἐ ∷ γ ∷ ώ ∷ []) "1Cor.7.12"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1Cor.7.12"
∷ word (ὁ ∷ []) "1Cor.7.12"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.7.12"
∷ word (ε ∷ ἴ ∷ []) "1Cor.7.12"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.7.12"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ς ∷ []) "1Cor.7.12"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ []) "1Cor.7.12"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.7.12"
∷ word (ἄ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.7.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.12"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "1Cor.7.12"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ υ ∷ δ ∷ ο ∷ κ ∷ ε ∷ ῖ ∷ []) "1Cor.7.12"
∷ word (ο ∷ ἰ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.7.12"
∷ word (μ ∷ ε ∷ τ ∷ []) "1Cor.7.12"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.12"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.12"
∷ word (ἀ ∷ φ ∷ ι ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.7.12"
∷ word (α ∷ ὐ ∷ τ ∷ ή ∷ ν ∷ []) "1Cor.7.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.13"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.7.13"
∷ word (ε ∷ ἴ ∷ []) "1Cor.7.13"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.7.13"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.7.13"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ α ∷ []) "1Cor.7.13"
∷ word (ἄ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.7.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.13"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.7.13"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ υ ∷ δ ∷ ο ∷ κ ∷ ε ∷ ῖ ∷ []) "1Cor.7.13"
∷ word (ο ∷ ἰ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.7.13"
∷ word (μ ∷ ε ∷ τ ∷ []) "1Cor.7.13"
∷ word (α ∷ ὐ ∷ τ ∷ ῆ ∷ ς ∷ []) "1Cor.7.13"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.13"
∷ word (ἀ ∷ φ ∷ ι ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.7.13"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.7.13"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ α ∷ []) "1Cor.7.13"
∷ word (ἡ ∷ γ ∷ ί ∷ α ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.7.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.7.14"
∷ word (ὁ ∷ []) "1Cor.7.14"
∷ word (ἀ ∷ ν ∷ ὴ ∷ ρ ∷ []) "1Cor.7.14"
∷ word (ὁ ∷ []) "1Cor.7.14"
∷ word (ἄ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.7.14"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.14"
∷ word (τ ∷ ῇ ∷ []) "1Cor.7.14"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ί ∷ []) "1Cor.7.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.14"
∷ word (ἡ ∷ γ ∷ ί ∷ α ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.7.14"
∷ word (ἡ ∷ []) "1Cor.7.14"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.7.14"
∷ word (ἡ ∷ []) "1Cor.7.14"
∷ word (ἄ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.7.14"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.14"
∷ word (τ ∷ ῷ ∷ []) "1Cor.7.14"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ῷ ∷ []) "1Cor.7.14"
∷ word (ἐ ∷ π ∷ ε ∷ ὶ ∷ []) "1Cor.7.14"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "1Cor.7.14"
∷ word (τ ∷ ὰ ∷ []) "1Cor.7.14"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "1Cor.7.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.7.14"
∷ word (ἀ ∷ κ ∷ ά ∷ θ ∷ α ∷ ρ ∷ τ ∷ ά ∷ []) "1Cor.7.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.7.14"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "1Cor.7.14"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.14"
∷ word (ἅ ∷ γ ∷ ι ∷ ά ∷ []) "1Cor.7.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.7.14"
∷ word (ε ∷ ἰ ∷ []) "1Cor.7.15"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.15"
∷ word (ὁ ∷ []) "1Cor.7.15"
∷ word (ἄ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.7.15"
∷ word (χ ∷ ω ∷ ρ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.7.15"
∷ word (χ ∷ ω ∷ ρ ∷ ι ∷ ζ ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "1Cor.7.15"
∷ word (ο ∷ ὐ ∷ []) "1Cor.7.15"
∷ word (δ ∷ ε ∷ δ ∷ ο ∷ ύ ∷ ∙λ ∷ ω ∷ τ ∷ α ∷ ι ∷ []) "1Cor.7.15"
∷ word (ὁ ∷ []) "1Cor.7.15"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ς ∷ []) "1Cor.7.15"
∷ word (ἢ ∷ []) "1Cor.7.15"
∷ word (ἡ ∷ []) "1Cor.7.15"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὴ ∷ []) "1Cor.7.15"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.15"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.7.15"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ύ ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.7.15"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.15"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.15"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ ῃ ∷ []) "1Cor.7.15"
∷ word (κ ∷ έ ∷ κ ∷ ∙λ ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "1Cor.7.15"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.7.15"
∷ word (ὁ ∷ []) "1Cor.7.15"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "1Cor.7.15"
∷ word (τ ∷ ί ∷ []) "1Cor.7.16"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.7.16"
∷ word (ο ∷ ἶ ∷ δ ∷ α ∷ ς ∷ []) "1Cor.7.16"
∷ word (γ ∷ ύ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.7.16"
∷ word (ε ∷ ἰ ∷ []) "1Cor.7.16"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.7.16"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ α ∷ []) "1Cor.7.16"
∷ word (σ ∷ ώ ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.7.16"
∷ word (ἢ ∷ []) "1Cor.7.16"
∷ word (τ ∷ ί ∷ []) "1Cor.7.16"
∷ word (ο ∷ ἶ ∷ δ ∷ α ∷ ς ∷ []) "1Cor.7.16"
∷ word (ἄ ∷ ν ∷ ε ∷ ρ ∷ []) "1Cor.7.16"
∷ word (ε ∷ ἰ ∷ []) "1Cor.7.16"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.7.16"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ []) "1Cor.7.16"
∷ word (σ ∷ ώ ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.7.16"
∷ word (Ε ∷ ἰ ∷ []) "1Cor.7.17"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.17"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ ῳ ∷ []) "1Cor.7.17"
∷ word (ὡ ∷ ς ∷ []) "1Cor.7.17"
∷ word (ἐ ∷ μ ∷ έ ∷ ρ ∷ ι ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.7.17"
∷ word (ὁ ∷ []) "1Cor.7.17"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.7.17"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.7.17"
∷ word (ὡ ∷ ς ∷ []) "1Cor.7.17"
∷ word (κ ∷ έ ∷ κ ∷ ∙λ ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "1Cor.7.17"
∷ word (ὁ ∷ []) "1Cor.7.17"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "1Cor.7.17"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.7.17"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ε ∷ ί ∷ τ ∷ ω ∷ []) "1Cor.7.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.17"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.7.17"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.17"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "1Cor.7.17"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "1Cor.7.17"
∷ word (π ∷ ά ∷ σ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.7.17"
∷ word (δ ∷ ι ∷ α ∷ τ ∷ ά ∷ σ ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.7.17"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ε ∷ τ ∷ μ ∷ η ∷ μ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.7.18"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.7.18"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ή ∷ θ ∷ η ∷ []) "1Cor.7.18"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.18"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ π ∷ ά ∷ σ ∷ θ ∷ ω ∷ []) "1Cor.7.18"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.18"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ ᾳ ∷ []) "1Cor.7.18"
∷ word (κ ∷ έ ∷ κ ∷ ∙λ ∷ η ∷ τ ∷ α ∷ ί ∷ []) "1Cor.7.18"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.7.18"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.18"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ε ∷ μ ∷ ν ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "1Cor.7.18"
∷ word (ἡ ∷ []) "1Cor.7.19"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ὴ ∷ []) "1Cor.7.19"
∷ word (ο ∷ ὐ ∷ δ ∷ έ ∷ ν ∷ []) "1Cor.7.19"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.7.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.19"
∷ word (ἡ ∷ []) "1Cor.7.19"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ α ∷ []) "1Cor.7.19"
∷ word (ο ∷ ὐ ∷ δ ∷ έ ∷ ν ∷ []) "1Cor.7.19"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.7.19"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.7.19"
∷ word (τ ∷ ή ∷ ρ ∷ η ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.7.19"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "1Cor.7.19"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.7.19"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.7.20"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.20"
∷ word (τ ∷ ῇ ∷ []) "1Cor.7.20"
∷ word (κ ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.7.20"
∷ word (ᾗ ∷ []) "1Cor.7.20"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ή ∷ θ ∷ η ∷ []) "1Cor.7.20"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.20"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ ῃ ∷ []) "1Cor.7.20"
∷ word (μ ∷ ε ∷ ν ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.7.20"
∷ word (Δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.7.21"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ή ∷ θ ∷ η ∷ ς ∷ []) "1Cor.7.21"
∷ word (μ ∷ ή ∷ []) "1Cor.7.21"
∷ word (σ ∷ ο ∷ ι ∷ []) "1Cor.7.21"
∷ word (μ ∷ ε ∷ ∙λ ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.7.21"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.7.21"
∷ word (ε ∷ ἰ ∷ []) "1Cor.7.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.21"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ σ ∷ α ∷ ι ∷ []) "1Cor.7.21"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ θ ∷ ε ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.7.21"
∷ word (γ ∷ ε ∷ ν ∷ έ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.7.21"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.7.21"
∷ word (χ ∷ ρ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "1Cor.7.21"
∷ word (ὁ ∷ []) "1Cor.7.22"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.7.22"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.22"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.7.22"
∷ word (κ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.7.22"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.7.22"
∷ word (ἀ ∷ π ∷ ε ∷ ∙λ ∷ ε ∷ ύ ∷ θ ∷ ε ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.7.22"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.7.22"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1Cor.7.22"
∷ word (ὁ ∷ μ ∷ ο ∷ ί ∷ ω ∷ ς ∷ []) "1Cor.7.22"
∷ word (ὁ ∷ []) "1Cor.7.22"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ θ ∷ ε ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.7.22"
∷ word (κ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.7.22"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ό ∷ ς ∷ []) "1Cor.7.22"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.7.22"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.22"
∷ word (τ ∷ ι ∷ μ ∷ ῆ ∷ ς ∷ []) "1Cor.7.23"
∷ word (ἠ ∷ γ ∷ ο ∷ ρ ∷ ά ∷ σ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.7.23"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.23"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.7.23"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ι ∷ []) "1Cor.7.23"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "1Cor.7.23"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.7.24"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.24"
∷ word (ᾧ ∷ []) "1Cor.7.24"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ή ∷ θ ∷ η ∷ []) "1Cor.7.24"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.7.24"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.24"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1Cor.7.24"
∷ word (μ ∷ ε ∷ ν ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.7.24"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "1Cor.7.24"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1Cor.7.24"
∷ word (Π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1Cor.7.25"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.25"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.7.25"
∷ word (π ∷ α ∷ ρ ∷ θ ∷ έ ∷ ν ∷ ω ∷ ν ∷ []) "1Cor.7.25"
∷ word (ἐ ∷ π ∷ ι ∷ τ ∷ α ∷ γ ∷ ὴ ∷ ν ∷ []) "1Cor.7.25"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.7.25"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.7.25"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "1Cor.7.25"
∷ word (γ ∷ ν ∷ ώ ∷ μ ∷ η ∷ ν ∷ []) "1Cor.7.25"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.25"
∷ word (δ ∷ ί ∷ δ ∷ ω ∷ μ ∷ ι ∷ []) "1Cor.7.25"
∷ word (ὡ ∷ ς ∷ []) "1Cor.7.25"
∷ word (ἠ ∷ ∙λ ∷ ε ∷ η ∷ μ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.7.25"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.7.25"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.7.25"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.7.25"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.7.25"
∷ word (ν ∷ ο ∷ μ ∷ ί ∷ ζ ∷ ω ∷ []) "1Cor.7.26"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.7.26"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.7.26"
∷ word (κ ∷ α ∷ ∙λ ∷ ὸ ∷ ν ∷ []) "1Cor.7.26"
∷ word (ὑ ∷ π ∷ ά ∷ ρ ∷ χ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.7.26"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.7.26"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.7.26"
∷ word (ἐ ∷ ν ∷ ε ∷ σ ∷ τ ∷ ῶ ∷ σ ∷ α ∷ ν ∷ []) "1Cor.7.26"
∷ word (ἀ ∷ ν ∷ ά ∷ γ ∷ κ ∷ η ∷ ν ∷ []) "1Cor.7.26"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.7.26"
∷ word (κ ∷ α ∷ ∙λ ∷ ὸ ∷ ν ∷ []) "1Cor.7.26"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ῳ ∷ []) "1Cor.7.26"
∷ word (τ ∷ ὸ ∷ []) "1Cor.7.26"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.7.26"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.7.26"
∷ word (δ ∷ έ ∷ δ ∷ ε ∷ σ ∷ α ∷ ι ∷ []) "1Cor.7.27"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ί ∷ []) "1Cor.7.27"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.27"
∷ word (ζ ∷ ή ∷ τ ∷ ε ∷ ι ∷ []) "1Cor.7.27"
∷ word (∙λ ∷ ύ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.7.27"
∷ word (∙λ ∷ έ ∷ ∙λ ∷ υ ∷ σ ∷ α ∷ ι ∷ []) "1Cor.7.27"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "1Cor.7.27"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ό ∷ ς ∷ []) "1Cor.7.27"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.27"
∷ word (ζ ∷ ή ∷ τ ∷ ε ∷ ι ∷ []) "1Cor.7.27"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ []) "1Cor.7.27"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.7.28"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.28"
∷ word (γ ∷ α ∷ μ ∷ ή ∷ σ ∷ ῃ ∷ ς ∷ []) "1Cor.7.28"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1Cor.7.28"
∷ word (ἥ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.7.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.28"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.7.28"
∷ word (γ ∷ ή ∷ μ ∷ ῃ ∷ []) "1Cor.7.28"
∷ word (ἡ ∷ []) "1Cor.7.28"
∷ word (π ∷ α ∷ ρ ∷ θ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.7.28"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1Cor.7.28"
∷ word (ἥ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ε ∷ ν ∷ []) "1Cor.7.28"
∷ word (θ ∷ ∙λ ∷ ῖ ∷ ψ ∷ ι ∷ ν ∷ []) "1Cor.7.28"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.28"
∷ word (τ ∷ ῇ ∷ []) "1Cor.7.28"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "1Cor.7.28"
∷ word (ἕ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.7.28"
∷ word (ο ∷ ἱ ∷ []) "1Cor.7.28"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ι ∷ []) "1Cor.7.28"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.7.28"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.28"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.7.28"
∷ word (φ ∷ ε ∷ ί ∷ δ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.7.28"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.7.29"
∷ word (δ ∷ έ ∷ []) "1Cor.7.29"
∷ word (φ ∷ η ∷ μ ∷ ι ∷ []) "1Cor.7.29"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.7.29"
∷ word (ὁ ∷ []) "1Cor.7.29"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.7.29"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ σ ∷ τ ∷ α ∷ ∙λ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.7.29"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1Cor.7.29"
∷ word (τ ∷ ὸ ∷ []) "1Cor.7.29"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ὸ ∷ ν ∷ []) "1Cor.7.29"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.7.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.29"
∷ word (ο ∷ ἱ ∷ []) "1Cor.7.29"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.7.29"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ ς ∷ []) "1Cor.7.29"
∷ word (ὡ ∷ ς ∷ []) "1Cor.7.29"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.29"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.7.29"
∷ word (ὦ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.7.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.30"
∷ word (ο ∷ ἱ ∷ []) "1Cor.7.30"
∷ word (κ ∷ ∙λ ∷ α ∷ ί ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.7.30"
∷ word (ὡ ∷ ς ∷ []) "1Cor.7.30"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.30"
∷ word (κ ∷ ∙λ ∷ α ∷ ί ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.7.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.30"
∷ word (ο ∷ ἱ ∷ []) "1Cor.7.30"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.7.30"
∷ word (ὡ ∷ ς ∷ []) "1Cor.7.30"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.30"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.7.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.30"
∷ word (ο ∷ ἱ ∷ []) "1Cor.7.30"
∷ word (ἀ ∷ γ ∷ ο ∷ ρ ∷ ά ∷ ζ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.7.30"
∷ word (ὡ ∷ ς ∷ []) "1Cor.7.30"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.30"
∷ word (κ ∷ α ∷ τ ∷ έ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.7.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.31"
∷ word (ο ∷ ἱ ∷ []) "1Cor.7.31"
∷ word (χ ∷ ρ ∷ ώ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.7.31"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.7.31"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ν ∷ []) "1Cor.7.31"
∷ word (ὡ ∷ ς ∷ []) "1Cor.7.31"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.31"
∷ word (κ ∷ α ∷ τ ∷ α ∷ χ ∷ ρ ∷ ώ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.7.31"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ γ ∷ ε ∷ ι ∷ []) "1Cor.7.31"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.7.31"
∷ word (τ ∷ ὸ ∷ []) "1Cor.7.31"
∷ word (σ ∷ χ ∷ ῆ ∷ μ ∷ α ∷ []) "1Cor.7.31"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.31"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1Cor.7.31"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.7.31"
∷ word (Θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "1Cor.7.32"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.32"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.7.32"
∷ word (ἀ ∷ μ ∷ ε ∷ ρ ∷ ί ∷ μ ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.7.32"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.7.32"
∷ word (ὁ ∷ []) "1Cor.7.32"
∷ word (ἄ ∷ γ ∷ α ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.7.32"
∷ word (μ ∷ ε ∷ ρ ∷ ι ∷ μ ∷ ν ∷ ᾷ ∷ []) "1Cor.7.32"
∷ word (τ ∷ ὰ ∷ []) "1Cor.7.32"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.32"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.7.32"
∷ word (π ∷ ῶ ∷ ς ∷ []) "1Cor.7.32"
∷ word (ἀ ∷ ρ ∷ έ ∷ σ ∷ ῃ ∷ []) "1Cor.7.32"
∷ word (τ ∷ ῷ ∷ []) "1Cor.7.32"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.7.32"
∷ word (ὁ ∷ []) "1Cor.7.33"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.33"
∷ word (γ ∷ α ∷ μ ∷ ή ∷ σ ∷ α ∷ ς ∷ []) "1Cor.7.33"
∷ word (μ ∷ ε ∷ ρ ∷ ι ∷ μ ∷ ν ∷ ᾷ ∷ []) "1Cor.7.33"
∷ word (τ ∷ ὰ ∷ []) "1Cor.7.33"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.33"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1Cor.7.33"
∷ word (π ∷ ῶ ∷ ς ∷ []) "1Cor.7.33"
∷ word (ἀ ∷ ρ ∷ έ ∷ σ ∷ ῃ ∷ []) "1Cor.7.33"
∷ word (τ ∷ ῇ ∷ []) "1Cor.7.33"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ί ∷ []) "1Cor.7.33"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.34"
∷ word (μ ∷ ε ∷ μ ∷ έ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.7.34"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.34"
∷ word (ἡ ∷ []) "1Cor.7.34"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.7.34"
∷ word (ἡ ∷ []) "1Cor.7.34"
∷ word (ἄ ∷ γ ∷ α ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.7.34"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.34"
∷ word (ἡ ∷ []) "1Cor.7.34"
∷ word (π ∷ α ∷ ρ ∷ θ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.7.34"
∷ word (μ ∷ ε ∷ ρ ∷ ι ∷ μ ∷ ν ∷ ᾷ ∷ []) "1Cor.7.34"
∷ word (τ ∷ ὰ ∷ []) "1Cor.7.34"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.34"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.7.34"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.7.34"
∷ word (ᾖ ∷ []) "1Cor.7.34"
∷ word (ἁ ∷ γ ∷ ί ∷ α ∷ []) "1Cor.7.34"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.34"
∷ word (τ ∷ ῷ ∷ []) "1Cor.7.34"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.7.34"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.34"
∷ word (τ ∷ ῷ ∷ []) "1Cor.7.34"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.7.34"
∷ word (ἡ ∷ []) "1Cor.7.34"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.34"
∷ word (γ ∷ α ∷ μ ∷ ή ∷ σ ∷ α ∷ σ ∷ α ∷ []) "1Cor.7.34"
∷ word (μ ∷ ε ∷ ρ ∷ ι ∷ μ ∷ ν ∷ ᾷ ∷ []) "1Cor.7.34"
∷ word (τ ∷ ὰ ∷ []) "1Cor.7.34"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.34"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1Cor.7.34"
∷ word (π ∷ ῶ ∷ ς ∷ []) "1Cor.7.34"
∷ word (ἀ ∷ ρ ∷ έ ∷ σ ∷ ῃ ∷ []) "1Cor.7.34"
∷ word (τ ∷ ῷ ∷ []) "1Cor.7.34"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ί ∷ []) "1Cor.7.34"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.7.35"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.35"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.7.35"
∷ word (τ ∷ ὸ ∷ []) "1Cor.7.35"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.7.35"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.7.35"
∷ word (σ ∷ ύ ∷ μ ∷ φ ∷ ο ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.7.35"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "1Cor.7.35"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1Cor.7.35"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.7.35"
∷ word (β ∷ ρ ∷ ό ∷ χ ∷ ο ∷ ν ∷ []) "1Cor.7.35"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.7.35"
∷ word (ἐ ∷ π ∷ ι ∷ β ∷ ά ∷ ∙λ ∷ ω ∷ []) "1Cor.7.35"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.7.35"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.7.35"
∷ word (τ ∷ ὸ ∷ []) "1Cor.7.35"
∷ word (ε ∷ ὔ ∷ σ ∷ χ ∷ η ∷ μ ∷ ο ∷ ν ∷ []) "1Cor.7.35"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.35"
∷ word (ε ∷ ὐ ∷ π ∷ ά ∷ ρ ∷ ε ∷ δ ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.7.35"
∷ word (τ ∷ ῷ ∷ []) "1Cor.7.35"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.7.35"
∷ word (ἀ ∷ π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ π ∷ ά ∷ σ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.7.35"
∷ word (Ε ∷ ἰ ∷ []) "1Cor.7.36"
∷ word (δ ∷ έ ∷ []) "1Cor.7.36"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.7.36"
∷ word (ἀ ∷ σ ∷ χ ∷ η ∷ μ ∷ ο ∷ ν ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.7.36"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.7.36"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.7.36"
∷ word (π ∷ α ∷ ρ ∷ θ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.7.36"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.36"
∷ word (ν ∷ ο ∷ μ ∷ ί ∷ ζ ∷ ε ∷ ι ∷ []) "1Cor.7.36"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.7.36"
∷ word (ᾖ ∷ []) "1Cor.7.36"
∷ word (ὑ ∷ π ∷ έ ∷ ρ ∷ α ∷ κ ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.7.36"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.36"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.7.36"
∷ word (ὀ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ ε ∷ ι ∷ []) "1Cor.7.36"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.7.36"
∷ word (ὃ ∷ []) "1Cor.7.36"
∷ word (θ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ []) "1Cor.7.36"
∷ word (π ∷ ο ∷ ι ∷ ε ∷ ί ∷ τ ∷ ω ∷ []) "1Cor.7.36"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1Cor.7.36"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ά ∷ ν ∷ ε ∷ ι ∷ []) "1Cor.7.36"
∷ word (γ ∷ α ∷ μ ∷ ε ∷ ί ∷ τ ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "1Cor.7.36"
∷ word (ὃ ∷ ς ∷ []) "1Cor.7.37"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.37"
∷ word (ἕ ∷ σ ∷ τ ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "1Cor.7.37"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.37"
∷ word (τ ∷ ῇ ∷ []) "1Cor.7.37"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ ᾳ ∷ []) "1Cor.7.37"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.37"
∷ word (ἑ ∷ δ ∷ ρ ∷ α ∷ ῖ ∷ ο ∷ ς ∷ []) "1Cor.7.37"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.37"
∷ word (ἔ ∷ χ ∷ ω ∷ ν ∷ []) "1Cor.7.37"
∷ word (ἀ ∷ ν ∷ ά ∷ γ ∷ κ ∷ η ∷ ν ∷ []) "1Cor.7.37"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.7.37"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.37"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.7.37"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1Cor.7.37"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.37"
∷ word (ἰ ∷ δ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.7.37"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.7.37"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.37"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.7.37"
∷ word (κ ∷ έ ∷ κ ∷ ρ ∷ ι ∷ κ ∷ ε ∷ ν ∷ []) "1Cor.7.37"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.37"
∷ word (τ ∷ ῇ ∷ []) "1Cor.7.37"
∷ word (ἰ ∷ δ ∷ ί ∷ ᾳ ∷ []) "1Cor.7.37"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ ᾳ ∷ []) "1Cor.7.37"
∷ word (τ ∷ η ∷ ρ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.7.37"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.7.37"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.37"
∷ word (π ∷ α ∷ ρ ∷ θ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.7.37"
∷ word (κ ∷ α ∷ ∙λ ∷ ῶ ∷ ς ∷ []) "1Cor.7.37"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.7.37"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.7.38"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.38"
∷ word (ὁ ∷ []) "1Cor.7.38"
∷ word (γ ∷ α ∷ μ ∷ ί ∷ ζ ∷ ω ∷ ν ∷ []) "1Cor.7.38"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.7.38"
∷ word (π ∷ α ∷ ρ ∷ θ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.7.38"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.7.38"
∷ word (κ ∷ α ∷ ∙λ ∷ ῶ ∷ ς ∷ []) "1Cor.7.38"
∷ word (π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ []) "1Cor.7.38"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.7.38"
∷ word (ὁ ∷ []) "1Cor.7.38"
∷ word (μ ∷ ὴ ∷ []) "1Cor.7.38"
∷ word (γ ∷ α ∷ μ ∷ ί ∷ ζ ∷ ω ∷ ν ∷ []) "1Cor.7.38"
∷ word (κ ∷ ρ ∷ ε ∷ ῖ ∷ σ ∷ σ ∷ ο ∷ ν ∷ []) "1Cor.7.38"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.7.38"
∷ word (Γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.7.39"
∷ word (δ ∷ έ ∷ δ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.7.39"
∷ word (ἐ ∷ φ ∷ []) "1Cor.7.39"
∷ word (ὅ ∷ σ ∷ ο ∷ ν ∷ []) "1Cor.7.39"
∷ word (χ ∷ ρ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.7.39"
∷ word (ζ ∷ ῇ ∷ []) "1Cor.7.39"
∷ word (ὁ ∷ []) "1Cor.7.39"
∷ word (ἀ ∷ ν ∷ ὴ ∷ ρ ∷ []) "1Cor.7.39"
∷ word (α ∷ ὐ ∷ τ ∷ ῆ ∷ ς ∷ []) "1Cor.7.39"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.7.39"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.39"
∷ word (κ ∷ ο ∷ ι ∷ μ ∷ η ∷ θ ∷ ῇ ∷ []) "1Cor.7.39"
∷ word (ὁ ∷ []) "1Cor.7.39"
∷ word (ἀ ∷ ν ∷ ή ∷ ρ ∷ []) "1Cor.7.39"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ έ ∷ ρ ∷ α ∷ []) "1Cor.7.39"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1Cor.7.39"
∷ word (ᾧ ∷ []) "1Cor.7.39"
∷ word (θ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ []) "1Cor.7.39"
∷ word (γ ∷ α ∷ μ ∷ η ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.7.39"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.7.39"
∷ word (ἐ ∷ ν ∷ []) "1Cor.7.39"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.7.39"
∷ word (μ ∷ α ∷ κ ∷ α ∷ ρ ∷ ι ∷ ω ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "1Cor.7.40"
∷ word (δ ∷ έ ∷ []) "1Cor.7.40"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.7.40"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.7.40"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.7.40"
∷ word (μ ∷ ε ∷ ί ∷ ν ∷ ῃ ∷ []) "1Cor.7.40"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.7.40"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.7.40"
∷ word (ἐ ∷ μ ∷ ὴ ∷ ν ∷ []) "1Cor.7.40"
∷ word (γ ∷ ν ∷ ώ ∷ μ ∷ η ∷ ν ∷ []) "1Cor.7.40"
∷ word (δ ∷ ο ∷ κ ∷ ῶ ∷ []) "1Cor.7.40"
∷ word (δ ∷ ὲ ∷ []) "1Cor.7.40"
∷ word (κ ∷ ἀ ∷ γ ∷ ὼ ∷ []) "1Cor.7.40"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.7.40"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.7.40"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.7.40"
∷ word (Π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1Cor.8.1"
∷ word (δ ∷ ὲ ∷ []) "1Cor.8.1"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.8.1"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ο ∷ θ ∷ ύ ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.8.1"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.8.1"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.8.1"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.8.1"
∷ word (γ ∷ ν ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.8.1"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.8.1"
∷ word (ἡ ∷ []) "1Cor.8.1"
∷ word (γ ∷ ν ∷ ῶ ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.8.1"
∷ word (φ ∷ υ ∷ σ ∷ ι ∷ ο ∷ ῖ ∷ []) "1Cor.8.1"
∷ word (ἡ ∷ []) "1Cor.8.1"
∷ word (δ ∷ ὲ ∷ []) "1Cor.8.1"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1Cor.8.1"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ε ∷ ῖ ∷ []) "1Cor.8.1"
∷ word (ε ∷ ἴ ∷ []) "1Cor.8.2"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.8.2"
∷ word (δ ∷ ο ∷ κ ∷ ε ∷ ῖ ∷ []) "1Cor.8.2"
∷ word (ἐ ∷ γ ∷ ν ∷ ω ∷ κ ∷ έ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.8.2"
∷ word (τ ∷ ι ∷ []) "1Cor.8.2"
∷ word (ο ∷ ὔ ∷ π ∷ ω ∷ []) "1Cor.8.2"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ []) "1Cor.8.2"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.8.2"
∷ word (δ ∷ ε ∷ ῖ ∷ []) "1Cor.8.2"
∷ word (γ ∷ ν ∷ ῶ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.8.2"
∷ word (ε ∷ ἰ ∷ []) "1Cor.8.3"
∷ word (δ ∷ έ ∷ []) "1Cor.8.3"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.8.3"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ᾷ ∷ []) "1Cor.8.3"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.8.3"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "1Cor.8.3"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.8.3"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.8.3"
∷ word (ὑ ∷ π ∷ []) "1Cor.8.3"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.8.3"
∷ word (Π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1Cor.8.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.8.4"
∷ word (β ∷ ρ ∷ ώ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "1Cor.8.4"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.8.4"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.8.4"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ο ∷ θ ∷ ύ ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.8.4"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.8.4"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.8.4"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ ν ∷ []) "1Cor.8.4"
∷ word (ε ∷ ἴ ∷ δ ∷ ω ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.8.4"
∷ word (ἐ ∷ ν ∷ []) "1Cor.8.4"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "1Cor.8.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.8.4"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.8.4"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.8.4"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.8.4"
∷ word (ε ∷ ἰ ∷ []) "1Cor.8.4"
∷ word (μ ∷ ὴ ∷ []) "1Cor.8.4"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "1Cor.8.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.8.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.8.5"
∷ word (ε ∷ ἴ ∷ π ∷ ε ∷ ρ ∷ []) "1Cor.8.5"
∷ word (ε ∷ ἰ ∷ σ ∷ ὶ ∷ ν ∷ []) "1Cor.8.5"
∷ word (∙λ ∷ ε ∷ γ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.8.5"
∷ word (θ ∷ ε ∷ ο ∷ ὶ ∷ []) "1Cor.8.5"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.8.5"
∷ word (ἐ ∷ ν ∷ []) "1Cor.8.5"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ῷ ∷ []) "1Cor.8.5"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.8.5"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.8.5"
∷ word (γ ∷ ῆ ∷ ς ∷ []) "1Cor.8.5"
∷ word (ὥ ∷ σ ∷ π ∷ ε ∷ ρ ∷ []) "1Cor.8.5"
∷ word (ε ∷ ἰ ∷ σ ∷ ὶ ∷ ν ∷ []) "1Cor.8.5"
∷ word (θ ∷ ε ∷ ο ∷ ὶ ∷ []) "1Cor.8.5"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὶ ∷ []) "1Cor.8.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.8.5"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ι ∷ []) "1Cor.8.5"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ί ∷ []) "1Cor.8.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.8.6"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.8.6"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "1Cor.8.6"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.8.6"
∷ word (ὁ ∷ []) "1Cor.8.6"
∷ word (π ∷ α ∷ τ ∷ ή ∷ ρ ∷ []) "1Cor.8.6"
∷ word (ἐ ∷ ξ ∷ []) "1Cor.8.6"
∷ word (ο ∷ ὗ ∷ []) "1Cor.8.6"
∷ word (τ ∷ ὰ ∷ []) "1Cor.8.6"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.8.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.8.6"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.8.6"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.8.6"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "1Cor.8.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.8.6"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "1Cor.8.6"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.8.6"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "1Cor.8.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "1Cor.8.6"
∷ word (δ ∷ ι ∷ []) "1Cor.8.6"
∷ word (ο ∷ ὗ ∷ []) "1Cor.8.6"
∷ word (τ ∷ ὰ ∷ []) "1Cor.8.6"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.8.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.8.6"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.8.6"
∷ word (δ ∷ ι ∷ []) "1Cor.8.6"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.8.6"
∷ word (Ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.8.7"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.8.7"
∷ word (ἐ ∷ ν ∷ []) "1Cor.8.7"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.8.7"
∷ word (ἡ ∷ []) "1Cor.8.7"
∷ word (γ ∷ ν ∷ ῶ ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.8.7"
∷ word (τ ∷ ι ∷ ν ∷ ὲ ∷ ς ∷ []) "1Cor.8.7"
∷ word (δ ∷ ὲ ∷ []) "1Cor.8.7"
∷ word (τ ∷ ῇ ∷ []) "1Cor.8.7"
∷ word (σ ∷ υ ∷ ν ∷ η ∷ θ ∷ ε ∷ ί ∷ ᾳ ∷ []) "1Cor.8.7"
∷ word (ἕ ∷ ω ∷ ς ∷ []) "1Cor.8.7"
∷ word (ἄ ∷ ρ ∷ τ ∷ ι ∷ []) "1Cor.8.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.8.7"
∷ word (ε ∷ ἰ ∷ δ ∷ ώ ∷ ∙λ ∷ ο ∷ υ ∷ []) "1Cor.8.7"
∷ word (ὡ ∷ ς ∷ []) "1Cor.8.7"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ό ∷ θ ∷ υ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.8.7"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.8.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.8.7"
∷ word (ἡ ∷ []) "1Cor.8.7"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ί ∷ δ ∷ η ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.8.7"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.8.7"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ὴ ∷ ς ∷ []) "1Cor.8.7"
∷ word (ο ∷ ὖ ∷ σ ∷ α ∷ []) "1Cor.8.7"
∷ word (μ ∷ ο ∷ ∙λ ∷ ύ ∷ ν ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.8.7"
∷ word (β ∷ ρ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.8.8"
∷ word (δ ∷ ὲ ∷ []) "1Cor.8.8"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.8.8"
∷ word (ο ∷ ὐ ∷ []) "1Cor.8.8"
∷ word (π ∷ α ∷ ρ ∷ α ∷ σ ∷ τ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.8.8"
∷ word (τ ∷ ῷ ∷ []) "1Cor.8.8"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1Cor.8.8"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "1Cor.8.8"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.8.8"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.8.8"
∷ word (φ ∷ ά ∷ γ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.8.8"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.8.8"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "1Cor.8.8"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.8.8"
∷ word (μ ∷ ὴ ∷ []) "1Cor.8.8"
∷ word (φ ∷ ά ∷ γ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.8.8"
∷ word (ὑ ∷ σ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1Cor.8.8"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.8.9"
∷ word (δ ∷ ὲ ∷ []) "1Cor.8.9"
∷ word (μ ∷ ή ∷ []) "1Cor.8.9"
∷ word (π ∷ ω ∷ ς ∷ []) "1Cor.8.9"
∷ word (ἡ ∷ []) "1Cor.8.9"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ []) "1Cor.8.9"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.8.9"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "1Cor.8.9"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ κ ∷ ο ∷ μ ∷ μ ∷ α ∷ []) "1Cor.8.9"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "1Cor.8.9"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.8.9"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ έ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.8.9"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.8.10"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.8.10"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.8.10"
∷ word (ἴ ∷ δ ∷ ῃ ∷ []) "1Cor.8.10"
∷ word (σ ∷ ὲ ∷ []) "1Cor.8.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.8.10"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "1Cor.8.10"
∷ word (γ ∷ ν ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.8.10"
∷ word (ἐ ∷ ν ∷ []) "1Cor.8.10"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ε ∷ ί ∷ ῳ ∷ []) "1Cor.8.10"
∷ word (κ ∷ α ∷ τ ∷ α ∷ κ ∷ ε ∷ ί ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.8.10"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "1Cor.8.10"
∷ word (ἡ ∷ []) "1Cor.8.10"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ί ∷ δ ∷ η ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.8.10"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.8.10"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ο ∷ ῦ ∷ ς ∷ []) "1Cor.8.10"
∷ word (ὄ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.8.10"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ η ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.8.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.8.10"
∷ word (τ ∷ ὸ ∷ []) "1Cor.8.10"
∷ word (τ ∷ ὰ ∷ []) "1Cor.8.10"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ό ∷ θ ∷ υ ∷ τ ∷ α ∷ []) "1Cor.8.10"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.8.10"
∷ word (ἀ ∷ π ∷ ό ∷ ∙λ ∷ ∙λ ∷ υ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.8.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.8.11"
∷ word (ὁ ∷ []) "1Cor.8.11"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ῶ ∷ ν ∷ []) "1Cor.8.11"
∷ word (ἐ ∷ ν ∷ []) "1Cor.8.11"
∷ word (τ ∷ ῇ ∷ []) "1Cor.8.11"
∷ word (σ ∷ ῇ ∷ []) "1Cor.8.11"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.8.11"
∷ word (ὁ ∷ []) "1Cor.8.11"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ς ∷ []) "1Cor.8.11"
∷ word (δ ∷ ι ∷ []) "1Cor.8.11"
∷ word (ὃ ∷ ν ∷ []) "1Cor.8.11"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.8.11"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "1Cor.8.11"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.8.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.8.12"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ά ∷ ν ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.8.12"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.8.12"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.8.12"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.8.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.8.12"
∷ word (τ ∷ ύ ∷ π ∷ τ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.8.12"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.8.12"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.8.12"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ί ∷ δ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.8.12"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ο ∷ ῦ ∷ σ ∷ α ∷ ν ∷ []) "1Cor.8.12"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.8.12"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.8.12"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ά ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.8.12"
∷ word (δ ∷ ι ∷ ό ∷ π ∷ ε ∷ ρ ∷ []) "1Cor.8.13"
∷ word (ε ∷ ἰ ∷ []) "1Cor.8.13"
∷ word (β ∷ ρ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.8.13"
∷ word (σ ∷ κ ∷ α ∷ ν ∷ δ ∷ α ∷ ∙λ ∷ ί ∷ ζ ∷ ε ∷ ι ∷ []) "1Cor.8.13"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.8.13"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ό ∷ ν ∷ []) "1Cor.8.13"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.8.13"
∷ word (ο ∷ ὐ ∷ []) "1Cor.8.13"
∷ word (μ ∷ ὴ ∷ []) "1Cor.8.13"
∷ word (φ ∷ ά ∷ γ ∷ ω ∷ []) "1Cor.8.13"
∷ word (κ ∷ ρ ∷ έ ∷ α ∷ []) "1Cor.8.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.8.13"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.8.13"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ α ∷ []) "1Cor.8.13"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.8.13"
∷ word (μ ∷ ὴ ∷ []) "1Cor.8.13"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.8.13"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ό ∷ ν ∷ []) "1Cor.8.13"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.8.13"
∷ word (σ ∷ κ ∷ α ∷ ν ∷ δ ∷ α ∷ ∙λ ∷ ί ∷ σ ∷ ω ∷ []) "1Cor.8.13"
∷ word (Ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.1"
∷ word (ε ∷ ἰ ∷ μ ∷ ὶ ∷ []) "1Cor.9.1"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ θ ∷ ε ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.9.1"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.1"
∷ word (ε ∷ ἰ ∷ μ ∷ ὶ ∷ []) "1Cor.9.1"
∷ word (ἀ ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.9.1"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "1Cor.9.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "1Cor.9.1"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.9.1"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.9.1"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.9.1"
∷ word (ἑ ∷ ό ∷ ρ ∷ α ∷ κ ∷ α ∷ []) "1Cor.9.1"
∷ word (ο ∷ ὐ ∷ []) "1Cor.9.1"
∷ word (τ ∷ ὸ ∷ []) "1Cor.9.1"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ν ∷ []) "1Cor.9.1"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.9.1"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.9.1"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.9.1"
∷ word (ἐ ∷ ν ∷ []) "1Cor.9.1"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.9.1"
∷ word (ε ∷ ἰ ∷ []) "1Cor.9.2"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.9.2"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.2"
∷ word (ε ∷ ἰ ∷ μ ∷ ὶ ∷ []) "1Cor.9.2"
∷ word (ἀ ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.9.2"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ά ∷ []) "1Cor.9.2"
∷ word (γ ∷ ε ∷ []) "1Cor.9.2"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.9.2"
∷ word (ε ∷ ἰ ∷ μ ∷ ι ∷ []) "1Cor.9.2"
∷ word (ἡ ∷ []) "1Cor.9.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.9.2"
∷ word (σ ∷ φ ∷ ρ ∷ α ∷ γ ∷ ί ∷ ς ∷ []) "1Cor.9.2"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.9.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.9.2"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ῆ ∷ ς ∷ []) "1Cor.9.2"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.9.2"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.9.2"
∷ word (ἐ ∷ ν ∷ []) "1Cor.9.2"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.9.2"
∷ word (Ἡ ∷ []) "1Cor.9.3"
∷ word (ἐ ∷ μ ∷ ὴ ∷ []) "1Cor.9.3"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ []) "1Cor.9.3"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.9.3"
∷ word (ἐ ∷ μ ∷ ὲ ∷ []) "1Cor.9.3"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ο ∷ υ ∷ σ ∷ ί ∷ ν ∷ []) "1Cor.9.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.9.3"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "1Cor.9.3"
∷ word (μ ∷ ὴ ∷ []) "1Cor.9.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.4"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.9.4"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.9.4"
∷ word (φ ∷ α ∷ γ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.9.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.9.4"
∷ word (π ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.9.4"
∷ word (μ ∷ ὴ ∷ []) "1Cor.9.5"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.5"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.9.5"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.9.5"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὴ ∷ ν ∷ []) "1Cor.9.5"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ []) "1Cor.9.5"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ ά ∷ γ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.9.5"
∷ word (ὡ ∷ ς ∷ []) "1Cor.9.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.9.5"
∷ word (ο ∷ ἱ ∷ []) "1Cor.9.5"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ο ∷ ὶ ∷ []) "1Cor.9.5"
∷ word (ἀ ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ι ∷ []) "1Cor.9.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.9.5"
∷ word (ο ∷ ἱ ∷ []) "1Cor.9.5"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ὶ ∷ []) "1Cor.9.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.9.5"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.9.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.9.5"
∷ word (Κ ∷ η ∷ φ ∷ ᾶ ∷ ς ∷ []) "1Cor.9.5"
∷ word (ἢ ∷ []) "1Cor.9.6"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.9.6"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.9.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.9.6"
∷ word (Β ∷ α ∷ ρ ∷ ν ∷ α ∷ β ∷ ᾶ ∷ ς ∷ []) "1Cor.9.6"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.6"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.9.6"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.9.6"
∷ word (μ ∷ ὴ ∷ []) "1Cor.9.6"
∷ word (ἐ ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.9.6"
∷ word (τ ∷ ί ∷ ς ∷ []) "1Cor.9.7"
∷ word (σ ∷ τ ∷ ρ ∷ α ∷ τ ∷ ε ∷ ύ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.9.7"
∷ word (ἰ ∷ δ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.9.7"
∷ word (ὀ ∷ ψ ∷ ω ∷ ν ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.9.7"
∷ word (π ∷ ο ∷ τ ∷ έ ∷ []) "1Cor.9.7"
∷ word (τ ∷ ί ∷ ς ∷ []) "1Cor.9.7"
∷ word (φ ∷ υ ∷ τ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ []) "1Cor.9.7"
∷ word (ἀ ∷ μ ∷ π ∷ ε ∷ ∙λ ∷ ῶ ∷ ν ∷ α ∷ []) "1Cor.9.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.9.7"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.9.7"
∷ word (κ ∷ α ∷ ρ ∷ π ∷ ὸ ∷ ν ∷ []) "1Cor.9.7"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.9.7"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.7"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ε ∷ ι ∷ []) "1Cor.9.7"
∷ word (τ ∷ ί ∷ ς ∷ []) "1Cor.9.7"
∷ word (π ∷ ο ∷ ι ∷ μ ∷ α ∷ ί ∷ ν ∷ ε ∷ ι ∷ []) "1Cor.9.7"
∷ word (π ∷ ο ∷ ί ∷ μ ∷ ν ∷ η ∷ ν ∷ []) "1Cor.9.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.9.7"
∷ word (ἐ ∷ κ ∷ []) "1Cor.9.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.9.7"
∷ word (γ ∷ ά ∷ ∙λ ∷ α ∷ κ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.9.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.9.7"
∷ word (π ∷ ο ∷ ί ∷ μ ∷ ν ∷ η ∷ ς ∷ []) "1Cor.9.7"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.7"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ε ∷ ι ∷ []) "1Cor.9.7"
∷ word (Μ ∷ ὴ ∷ []) "1Cor.9.8"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.9.8"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "1Cor.9.8"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1Cor.9.8"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "1Cor.9.8"
∷ word (ἢ ∷ []) "1Cor.9.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.9.8"
∷ word (ὁ ∷ []) "1Cor.9.8"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.9.8"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1Cor.9.8"
∷ word (ο ∷ ὐ ∷ []) "1Cor.9.8"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "1Cor.9.8"
∷ word (ἐ ∷ ν ∷ []) "1Cor.9.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.9.9"
∷ word (τ ∷ ῷ ∷ []) "1Cor.9.9"
∷ word (Μ ∷ ω ∷ ϋ ∷ σ ∷ έ ∷ ω ∷ ς ∷ []) "1Cor.9.9"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "1Cor.9.9"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "1Cor.9.9"
∷ word (Ο ∷ ὐ ∷ []) "1Cor.9.9"
∷ word (κ ∷ η ∷ μ ∷ ώ ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.9.9"
∷ word (β ∷ ο ∷ ῦ ∷ ν ∷ []) "1Cor.9.9"
∷ word (ἀ ∷ ∙λ ∷ ο ∷ ῶ ∷ ν ∷ τ ∷ α ∷ []) "1Cor.9.9"
∷ word (μ ∷ ὴ ∷ []) "1Cor.9.9"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.9.9"
∷ word (β ∷ ο ∷ ῶ ∷ ν ∷ []) "1Cor.9.9"
∷ word (μ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ []) "1Cor.9.9"
∷ word (τ ∷ ῷ ∷ []) "1Cor.9.9"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1Cor.9.9"
∷ word (ἢ ∷ []) "1Cor.9.10"
∷ word (δ ∷ ι ∷ []) "1Cor.9.10"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.9.10"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.9.10"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "1Cor.9.10"
∷ word (δ ∷ ι ∷ []) "1Cor.9.10"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.9.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.9.10"
∷ word (ἐ ∷ γ ∷ ρ ∷ ά ∷ φ ∷ η ∷ []) "1Cor.9.10"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.9.10"
∷ word (ὀ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ ε ∷ ι ∷ []) "1Cor.9.10"
∷ word (ἐ ∷ π ∷ []) "1Cor.9.10"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ ι ∷ []) "1Cor.9.10"
∷ word (ὁ ∷ []) "1Cor.9.10"
∷ word (ἀ ∷ ρ ∷ ο ∷ τ ∷ ρ ∷ ι ∷ ῶ ∷ ν ∷ []) "1Cor.9.10"
∷ word (ἀ ∷ ρ ∷ ο ∷ τ ∷ ρ ∷ ι ∷ ᾶ ∷ ν ∷ []) "1Cor.9.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.9.10"
∷ word (ὁ ∷ []) "1Cor.9.10"
∷ word (ἀ ∷ ∙λ ∷ ο ∷ ῶ ∷ ν ∷ []) "1Cor.9.10"
∷ word (ἐ ∷ π ∷ []) "1Cor.9.10"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ ι ∷ []) "1Cor.9.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.9.10"
∷ word (μ ∷ ε ∷ τ ∷ έ ∷ χ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.9.10"
∷ word (ε ∷ ἰ ∷ []) "1Cor.9.11"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.9.11"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.9.11"
∷ word (τ ∷ ὰ ∷ []) "1Cor.9.11"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ὰ ∷ []) "1Cor.9.11"
∷ word (ἐ ∷ σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.9.11"
∷ word (μ ∷ έ ∷ γ ∷ α ∷ []) "1Cor.9.11"
∷ word (ε ∷ ἰ ∷ []) "1Cor.9.11"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.9.11"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.9.11"
∷ word (τ ∷ ὰ ∷ []) "1Cor.9.11"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ι ∷ κ ∷ ὰ ∷ []) "1Cor.9.11"
∷ word (θ ∷ ε ∷ ρ ∷ ί ∷ σ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.9.11"
∷ word (ε ∷ ἰ ∷ []) "1Cor.9.12"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ι ∷ []) "1Cor.9.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.9.12"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.9.12"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.9.12"
∷ word (μ ∷ ε ∷ τ ∷ έ ∷ χ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.9.12"
∷ word (ο ∷ ὐ ∷ []) "1Cor.9.12"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.9.12"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.9.12"
∷ word (Ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.9.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.12"
∷ word (ἐ ∷ χ ∷ ρ ∷ η ∷ σ ∷ ά ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1Cor.9.12"
∷ word (τ ∷ ῇ ∷ []) "1Cor.9.12"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.9.12"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ ῃ ∷ []) "1Cor.9.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.9.12"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.9.12"
∷ word (σ ∷ τ ∷ έ ∷ γ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.9.12"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.9.12"
∷ word (μ ∷ ή ∷ []) "1Cor.9.12"
∷ word (τ ∷ ι ∷ ν ∷ α ∷ []) "1Cor.9.12"
∷ word (ἐ ∷ γ ∷ κ ∷ ο ∷ π ∷ ὴ ∷ ν ∷ []) "1Cor.9.12"
∷ word (δ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.9.12"
∷ word (τ ∷ ῷ ∷ []) "1Cor.9.12"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ῳ ∷ []) "1Cor.9.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.9.12"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.9.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.13"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.9.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.9.13"
∷ word (ο ∷ ἱ ∷ []) "1Cor.9.13"
∷ word (τ ∷ ὰ ∷ []) "1Cor.9.13"
∷ word (ἱ ∷ ε ∷ ρ ∷ ὰ ∷ []) "1Cor.9.13"
∷ word (ἐ ∷ ρ ∷ γ ∷ α ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.9.13"
∷ word (τ ∷ ὰ ∷ []) "1Cor.9.13"
∷ word (ἐ ∷ κ ∷ []) "1Cor.9.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.9.13"
∷ word (ἱ ∷ ε ∷ ρ ∷ ο ∷ ῦ ∷ []) "1Cor.9.13"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.9.13"
∷ word (ο ∷ ἱ ∷ []) "1Cor.9.13"
∷ word (τ ∷ ῷ ∷ []) "1Cor.9.13"
∷ word (θ ∷ υ ∷ σ ∷ ι ∷ α ∷ σ ∷ τ ∷ η ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.9.13"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ δ ∷ ρ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.9.13"
∷ word (τ ∷ ῷ ∷ []) "1Cor.9.13"
∷ word (θ ∷ υ ∷ σ ∷ ι ∷ α ∷ σ ∷ τ ∷ η ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.9.13"
∷ word (σ ∷ υ ∷ μ ∷ μ ∷ ε ∷ ρ ∷ ί ∷ ζ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.9.13"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.9.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.9.14"
∷ word (ὁ ∷ []) "1Cor.9.14"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.9.14"
∷ word (δ ∷ ι ∷ έ ∷ τ ∷ α ∷ ξ ∷ ε ∷ ν ∷ []) "1Cor.9.14"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.9.14"
∷ word (τ ∷ ὸ ∷ []) "1Cor.9.14"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.9.14"
∷ word (κ ∷ α ∷ τ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.9.14"
∷ word (ἐ ∷ κ ∷ []) "1Cor.9.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.9.14"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.9.14"
∷ word (ζ ∷ ῆ ∷ ν ∷ []) "1Cor.9.14"
∷ word (Ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.9.15"
∷ word (δ ∷ ὲ ∷ []) "1Cor.9.15"
∷ word (ο ∷ ὐ ∷ []) "1Cor.9.15"
∷ word (κ ∷ έ ∷ χ ∷ ρ ∷ η ∷ μ ∷ α ∷ ι ∷ []) "1Cor.9.15"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ν ∷ ὶ ∷ []) "1Cor.9.15"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.9.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.15"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "1Cor.9.15"
∷ word (δ ∷ ὲ ∷ []) "1Cor.9.15"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1Cor.9.15"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.9.15"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.9.15"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "1Cor.9.15"
∷ word (ἐ ∷ ν ∷ []) "1Cor.9.15"
∷ word (ἐ ∷ μ ∷ ο ∷ ί ∷ []) "1Cor.9.15"
∷ word (κ ∷ α ∷ ∙λ ∷ ὸ ∷ ν ∷ []) "1Cor.9.15"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.9.15"
∷ word (μ ∷ ο ∷ ι ∷ []) "1Cor.9.15"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.9.15"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ α ∷ ν ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.9.15"
∷ word (ἤ ∷ []) "1Cor.9.15"
∷ word (τ ∷ ὸ ∷ []) "1Cor.9.15"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ μ ∷ ά ∷ []) "1Cor.9.15"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.9.15"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.9.15"
∷ word (κ ∷ ε ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.9.15"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.9.16"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.9.16"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ζ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "1Cor.9.16"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.16"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.9.16"
∷ word (μ ∷ ο ∷ ι ∷ []) "1Cor.9.16"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ μ ∷ α ∷ []) "1Cor.9.16"
∷ word (ἀ ∷ ν ∷ ά ∷ γ ∷ κ ∷ η ∷ []) "1Cor.9.16"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.9.16"
∷ word (μ ∷ ο ∷ ι ∷ []) "1Cor.9.16"
∷ word (ἐ ∷ π ∷ ί ∷ κ ∷ ε ∷ ι ∷ τ ∷ α ∷ ι ∷ []) "1Cor.9.16"
∷ word (ο ∷ ὐ ∷ α ∷ ὶ ∷ []) "1Cor.9.16"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.9.16"
∷ word (μ ∷ ο ∷ ί ∷ []) "1Cor.9.16"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.9.16"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.9.16"
∷ word (μ ∷ ὴ ∷ []) "1Cor.9.16"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ σ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "1Cor.9.16"
∷ word (ε ∷ ἰ ∷ []) "1Cor.9.17"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.9.17"
∷ word (ἑ ∷ κ ∷ ὼ ∷ ν ∷ []) "1Cor.9.17"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.9.17"
∷ word (π ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ω ∷ []) "1Cor.9.17"
∷ word (μ ∷ ι ∷ σ ∷ θ ∷ ὸ ∷ ν ∷ []) "1Cor.9.17"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "1Cor.9.17"
∷ word (ε ∷ ἰ ∷ []) "1Cor.9.17"
∷ word (δ ∷ ὲ ∷ []) "1Cor.9.17"
∷ word (ἄ ∷ κ ∷ ω ∷ ν ∷ []) "1Cor.9.17"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.9.17"
∷ word (π ∷ ε ∷ π ∷ ί ∷ σ ∷ τ ∷ ε ∷ υ ∷ μ ∷ α ∷ ι ∷ []) "1Cor.9.17"
∷ word (τ ∷ ί ∷ ς ∷ []) "1Cor.9.18"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.9.18"
∷ word (μ ∷ ο ∷ ύ ∷ []) "1Cor.9.18"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.9.18"
∷ word (ὁ ∷ []) "1Cor.9.18"
∷ word (μ ∷ ι ∷ σ ∷ θ ∷ ό ∷ ς ∷ []) "1Cor.9.18"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.9.18"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.9.18"
∷ word (ἀ ∷ δ ∷ ά ∷ π ∷ α ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.9.18"
∷ word (θ ∷ ή ∷ σ ∷ ω ∷ []) "1Cor.9.18"
∷ word (τ ∷ ὸ ∷ []) "1Cor.9.18"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.9.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.9.18"
∷ word (τ ∷ ὸ ∷ []) "1Cor.9.18"
∷ word (μ ∷ ὴ ∷ []) "1Cor.9.18"
∷ word (κ ∷ α ∷ τ ∷ α ∷ χ ∷ ρ ∷ ή ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.9.18"
∷ word (τ ∷ ῇ ∷ []) "1Cor.9.18"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.9.18"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.9.18"
∷ word (ἐ ∷ ν ∷ []) "1Cor.9.18"
∷ word (τ ∷ ῷ ∷ []) "1Cor.9.18"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ῳ ∷ []) "1Cor.9.18"
∷ word (Ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ θ ∷ ε ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.9.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.9.19"
∷ word (ὢ ∷ ν ∷ []) "1Cor.9.19"
∷ word (ἐ ∷ κ ∷ []) "1Cor.9.19"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.9.19"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.9.19"
∷ word (ἐ ∷ μ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.9.19"
∷ word (ἐ ∷ δ ∷ ο ∷ ύ ∷ ∙λ ∷ ω ∷ σ ∷ α ∷ []) "1Cor.9.19"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.9.19"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.9.19"
∷ word (π ∷ ∙λ ∷ ε ∷ ί ∷ ο ∷ ν ∷ α ∷ ς ∷ []) "1Cor.9.19"
∷ word (κ ∷ ε ∷ ρ ∷ δ ∷ ή ∷ σ ∷ ω ∷ []) "1Cor.9.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.9.20"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ό ∷ μ ∷ η ∷ ν ∷ []) "1Cor.9.20"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.9.20"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.9.20"
∷ word (ὡ ∷ ς ∷ []) "1Cor.9.20"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ῖ ∷ ο ∷ ς ∷ []) "1Cor.9.20"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.9.20"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.9.20"
∷ word (κ ∷ ε ∷ ρ ∷ δ ∷ ή ∷ σ ∷ ω ∷ []) "1Cor.9.20"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.9.20"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.9.20"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "1Cor.9.20"
∷ word (ὡ ∷ ς ∷ []) "1Cor.9.20"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.9.20"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "1Cor.9.20"
∷ word (μ ∷ ὴ ∷ []) "1Cor.9.20"
∷ word (ὢ ∷ ν ∷ []) "1Cor.9.20"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.9.20"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.9.20"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "1Cor.9.20"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.9.20"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.9.20"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.9.20"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "1Cor.9.20"
∷ word (κ ∷ ε ∷ ρ ∷ δ ∷ ή ∷ σ ∷ ω ∷ []) "1Cor.9.20"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.9.21"
∷ word (ἀ ∷ ν ∷ ό ∷ μ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.9.21"
∷ word (ὡ ∷ ς ∷ []) "1Cor.9.21"
∷ word (ἄ ∷ ν ∷ ο ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.9.21"
∷ word (μ ∷ ὴ ∷ []) "1Cor.9.21"
∷ word (ὢ ∷ ν ∷ []) "1Cor.9.21"
∷ word (ἄ ∷ ν ∷ ο ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.9.21"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.9.21"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.9.21"
∷ word (ἔ ∷ ν ∷ ν ∷ ο ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.9.21"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.9.21"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.9.21"
∷ word (κ ∷ ε ∷ ρ ∷ δ ∷ ά ∷ ν ∷ ω ∷ []) "1Cor.9.21"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.9.21"
∷ word (ἀ ∷ ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.9.21"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ό ∷ μ ∷ η ∷ ν ∷ []) "1Cor.9.22"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.9.22"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ έ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.9.22"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ή ∷ ς ∷ []) "1Cor.9.22"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.9.22"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.9.22"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.9.22"
∷ word (κ ∷ ε ∷ ρ ∷ δ ∷ ή ∷ σ ∷ ω ∷ []) "1Cor.9.22"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.9.22"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.9.22"
∷ word (γ ∷ έ ∷ γ ∷ ο ∷ ν ∷ α ∷ []) "1Cor.9.22"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.9.22"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.9.22"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.9.22"
∷ word (τ ∷ ι ∷ ν ∷ ὰ ∷ ς ∷ []) "1Cor.9.22"
∷ word (σ ∷ ώ ∷ σ ∷ ω ∷ []) "1Cor.9.22"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.9.23"
∷ word (δ ∷ ὲ ∷ []) "1Cor.9.23"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ []) "1Cor.9.23"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.9.23"
∷ word (τ ∷ ὸ ∷ []) "1Cor.9.23"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.9.23"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.9.23"
∷ word (σ ∷ υ ∷ γ ∷ κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ὸ ∷ ς ∷ []) "1Cor.9.23"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.9.23"
∷ word (γ ∷ έ ∷ ν ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "1Cor.9.23"
∷ word (Ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.24"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.9.24"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.9.24"
∷ word (ο ∷ ἱ ∷ []) "1Cor.9.24"
∷ word (ἐ ∷ ν ∷ []) "1Cor.9.24"
∷ word (σ ∷ τ ∷ α ∷ δ ∷ ί ∷ ῳ ∷ []) "1Cor.9.24"
∷ word (τ ∷ ρ ∷ έ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.9.24"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.9.24"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.9.24"
∷ word (τ ∷ ρ ∷ έ ∷ χ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.9.24"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "1Cor.9.24"
∷ word (δ ∷ ὲ ∷ []) "1Cor.9.24"
∷ word (∙λ ∷ α ∷ μ ∷ β ∷ ά ∷ ν ∷ ε ∷ ι ∷ []) "1Cor.9.24"
∷ word (τ ∷ ὸ ∷ []) "1Cor.9.24"
∷ word (β ∷ ρ ∷ α ∷ β ∷ ε ∷ ῖ ∷ ο ∷ ν ∷ []) "1Cor.9.24"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.9.24"
∷ word (τ ∷ ρ ∷ έ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.9.24"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.9.24"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ά ∷ β ∷ η ∷ τ ∷ ε ∷ []) "1Cor.9.24"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1Cor.9.25"
∷ word (δ ∷ ὲ ∷ []) "1Cor.9.25"
∷ word (ὁ ∷ []) "1Cor.9.25"
∷ word (ἀ ∷ γ ∷ ω ∷ ν ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.9.25"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.9.25"
∷ word (ἐ ∷ γ ∷ κ ∷ ρ ∷ α ∷ τ ∷ ε ∷ ύ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.9.25"
∷ word (ἐ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.9.25"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.9.25"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.9.25"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.9.25"
∷ word (φ ∷ θ ∷ α ∷ ρ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.9.25"
∷ word (σ ∷ τ ∷ έ ∷ φ ∷ α ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.9.25"
∷ word (∙λ ∷ ά ∷ β ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.9.25"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.9.25"
∷ word (δ ∷ ὲ ∷ []) "1Cor.9.25"
∷ word (ἄ ∷ φ ∷ θ ∷ α ∷ ρ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.9.25"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.9.26"
∷ word (τ ∷ ο ∷ ί ∷ ν ∷ υ ∷ ν ∷ []) "1Cor.9.26"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.9.26"
∷ word (τ ∷ ρ ∷ έ ∷ χ ∷ ω ∷ []) "1Cor.9.26"
∷ word (ὡ ∷ ς ∷ []) "1Cor.9.26"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.26"
∷ word (ἀ ∷ δ ∷ ή ∷ ∙λ ∷ ω ∷ ς ∷ []) "1Cor.9.26"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.9.26"
∷ word (π ∷ υ ∷ κ ∷ τ ∷ ε ∷ ύ ∷ ω ∷ []) "1Cor.9.26"
∷ word (ὡ ∷ ς ∷ []) "1Cor.9.26"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.9.26"
∷ word (ἀ ∷ έ ∷ ρ ∷ α ∷ []) "1Cor.9.26"
∷ word (δ ∷ έ ∷ ρ ∷ ω ∷ ν ∷ []) "1Cor.9.26"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.9.27"
∷ word (ὑ ∷ π ∷ ω ∷ π ∷ ι ∷ ά ∷ ζ ∷ ω ∷ []) "1Cor.9.27"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.9.27"
∷ word (τ ∷ ὸ ∷ []) "1Cor.9.27"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.9.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.9.27"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ α ∷ γ ∷ ω ∷ γ ∷ ῶ ∷ []) "1Cor.9.27"
∷ word (μ ∷ ή ∷ []) "1Cor.9.27"
∷ word (π ∷ ω ∷ ς ∷ []) "1Cor.9.27"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.9.27"
∷ word (κ ∷ η ∷ ρ ∷ ύ ∷ ξ ∷ α ∷ ς ∷ []) "1Cor.9.27"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.9.27"
∷ word (ἀ ∷ δ ∷ ό ∷ κ ∷ ι ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.9.27"
∷ word (γ ∷ έ ∷ ν ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "1Cor.9.27"
∷ word (Ο ∷ ὐ ∷ []) "1Cor.10.1"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "1Cor.10.1"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.10.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.10.1"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.10.1"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.10.1"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.10.1"
∷ word (ο ∷ ἱ ∷ []) "1Cor.10.1"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ ε ∷ ς ∷ []) "1Cor.10.1"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.10.1"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.10.1"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.10.1"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.10.1"
∷ word (ν ∷ ε ∷ φ ∷ έ ∷ ∙λ ∷ η ∷ ν ∷ []) "1Cor.10.1"
∷ word (ἦ ∷ σ ∷ α ∷ ν ∷ []) "1Cor.10.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.1"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.10.1"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.10.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.10.1"
∷ word (θ ∷ α ∷ ∙λ ∷ ά ∷ σ ∷ σ ∷ η ∷ ς ∷ []) "1Cor.10.1"
∷ word (δ ∷ ι ∷ ῆ ∷ ∙λ ∷ θ ∷ ο ∷ ν ∷ []) "1Cor.10.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.2"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.10.2"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.10.2"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.10.2"
∷ word (Μ ∷ ω ∷ ϋ ∷ σ ∷ ῆ ∷ ν ∷ []) "1Cor.10.2"
∷ word (ἐ ∷ β ∷ α ∷ π ∷ τ ∷ ί ∷ σ ∷ α ∷ ν ∷ τ ∷ ο ∷ []) "1Cor.10.2"
∷ word (ἐ ∷ ν ∷ []) "1Cor.10.2"
∷ word (τ ∷ ῇ ∷ []) "1Cor.10.2"
∷ word (ν ∷ ε ∷ φ ∷ έ ∷ ∙λ ∷ ῃ ∷ []) "1Cor.10.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.2"
∷ word (ἐ ∷ ν ∷ []) "1Cor.10.2"
∷ word (τ ∷ ῇ ∷ []) "1Cor.10.2"
∷ word (θ ∷ α ∷ ∙λ ∷ ά ∷ σ ∷ σ ∷ ῃ ∷ []) "1Cor.10.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.3"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.10.3"
∷ word (τ ∷ ὸ ∷ []) "1Cor.10.3"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "1Cor.10.3"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ὸ ∷ ν ∷ []) "1Cor.10.3"
∷ word (β ∷ ρ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.10.3"
∷ word (ἔ ∷ φ ∷ α ∷ γ ∷ ο ∷ ν ∷ []) "1Cor.10.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.4"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.10.4"
∷ word (τ ∷ ὸ ∷ []) "1Cor.10.4"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "1Cor.10.4"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ὸ ∷ ν ∷ []) "1Cor.10.4"
∷ word (ἔ ∷ π ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.10.4"
∷ word (π ∷ ό ∷ μ ∷ α ∷ []) "1Cor.10.4"
∷ word (ἔ ∷ π ∷ ι ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.10.4"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.10.4"
∷ word (ἐ ∷ κ ∷ []) "1Cor.10.4"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ῆ ∷ ς ∷ []) "1Cor.10.4"
∷ word (ἀ ∷ κ ∷ ο ∷ ∙λ ∷ ο ∷ υ ∷ θ ∷ ο ∷ ύ ∷ σ ∷ η ∷ ς ∷ []) "1Cor.10.4"
∷ word (π ∷ έ ∷ τ ∷ ρ ∷ α ∷ ς ∷ []) "1Cor.10.4"
∷ word (ἡ ∷ []) "1Cor.10.4"
∷ word (π ∷ έ ∷ τ ∷ ρ ∷ α ∷ []) "1Cor.10.4"
∷ word (δ ∷ ὲ ∷ []) "1Cor.10.4"
∷ word (ἦ ∷ ν ∷ []) "1Cor.10.4"
∷ word (ὁ ∷ []) "1Cor.10.4"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "1Cor.10.4"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.10.5"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.10.5"
∷ word (ἐ ∷ ν ∷ []) "1Cor.10.5"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.10.5"
∷ word (π ∷ ∙λ ∷ ε ∷ ί ∷ ο ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.10.5"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.10.5"
∷ word (η ∷ ὐ ∷ δ ∷ ό ∷ κ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.10.5"
∷ word (ὁ ∷ []) "1Cor.10.5"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "1Cor.10.5"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ σ ∷ τ ∷ ρ ∷ ώ ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "1Cor.10.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.10.5"
∷ word (ἐ ∷ ν ∷ []) "1Cor.10.5"
∷ word (τ ∷ ῇ ∷ []) "1Cor.10.5"
∷ word (ἐ ∷ ρ ∷ ή ∷ μ ∷ ῳ ∷ []) "1Cor.10.5"
∷ word (Τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1Cor.10.6"
∷ word (δ ∷ ὲ ∷ []) "1Cor.10.6"
∷ word (τ ∷ ύ ∷ π ∷ ο ∷ ι ∷ []) "1Cor.10.6"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.10.6"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ή ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "1Cor.10.6"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.10.6"
∷ word (τ ∷ ὸ ∷ []) "1Cor.10.6"
∷ word (μ ∷ ὴ ∷ []) "1Cor.10.6"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.10.6"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.10.6"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ η ∷ τ ∷ ὰ ∷ ς ∷ []) "1Cor.10.6"
∷ word (κ ∷ α ∷ κ ∷ ῶ ∷ ν ∷ []) "1Cor.10.6"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.10.6"
∷ word (κ ∷ ἀ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.10.6"
∷ word (ἐ ∷ π ∷ ε ∷ θ ∷ ύ ∷ μ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "1Cor.10.6"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "1Cor.10.7"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ο ∷ ∙λ ∷ ά ∷ τ ∷ ρ ∷ α ∷ ι ∷ []) "1Cor.10.7"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.10.7"
∷ word (κ ∷ α ∷ θ ∷ ώ ∷ ς ∷ []) "1Cor.10.7"
∷ word (τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "1Cor.10.7"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.10.7"
∷ word (ὥ ∷ σ ∷ π ∷ ε ∷ ρ ∷ []) "1Cor.10.7"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "1Cor.10.7"
∷ word (Ἐ ∷ κ ∷ ά ∷ θ ∷ ι ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.10.7"
∷ word (ὁ ∷ []) "1Cor.10.7"
∷ word (∙λ ∷ α ∷ ὸ ∷ ς ∷ []) "1Cor.10.7"
∷ word (φ ∷ α ∷ γ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.10.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.7"
∷ word (π ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.10.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.7"
∷ word (ἀ ∷ ν ∷ έ ∷ σ ∷ τ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "1Cor.10.7"
∷ word (π ∷ α ∷ ί ∷ ζ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.10.7"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "1Cor.10.8"
∷ word (π ∷ ο ∷ ρ ∷ ν ∷ ε ∷ ύ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.10.8"
∷ word (κ ∷ α ∷ θ ∷ ώ ∷ ς ∷ []) "1Cor.10.8"
∷ word (τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "1Cor.10.8"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.10.8"
∷ word (ἐ ∷ π ∷ ό ∷ ρ ∷ ν ∷ ε ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "1Cor.10.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.8"
∷ word (ἔ ∷ π ∷ ε ∷ σ ∷ α ∷ ν ∷ []) "1Cor.10.8"
∷ word (μ ∷ ι ∷ ᾷ ∷ []) "1Cor.10.8"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "1Cor.10.8"
∷ word (ε ∷ ἴ ∷ κ ∷ ο ∷ σ ∷ ι ∷ []) "1Cor.10.8"
∷ word (τ ∷ ρ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.10.8"
∷ word (χ ∷ ι ∷ ∙λ ∷ ι ∷ ά ∷ δ ∷ ε ∷ ς ∷ []) "1Cor.10.8"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "1Cor.10.9"
∷ word (ἐ ∷ κ ∷ π ∷ ε ∷ ι ∷ ρ ∷ ά ∷ ζ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.10.9"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.10.9"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ν ∷ []) "1Cor.10.9"
∷ word (κ ∷ α ∷ θ ∷ ώ ∷ ς ∷ []) "1Cor.10.9"
∷ word (τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "1Cor.10.9"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.10.9"
∷ word (ἐ ∷ π ∷ ε ∷ ί ∷ ρ ∷ α ∷ σ ∷ α ∷ ν ∷ []) "1Cor.10.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.9"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.10.9"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.10.9"
∷ word (ὄ ∷ φ ∷ ε ∷ ω ∷ ν ∷ []) "1Cor.10.9"
∷ word (ἀ ∷ π ∷ ώ ∷ ∙λ ∷ ∙λ ∷ υ ∷ ν ∷ τ ∷ ο ∷ []) "1Cor.10.9"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "1Cor.10.10"
∷ word (γ ∷ ο ∷ γ ∷ γ ∷ ύ ∷ ζ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.10.10"
∷ word (κ ∷ α ∷ θ ∷ ά ∷ π ∷ ε ∷ ρ ∷ []) "1Cor.10.10"
∷ word (τ ∷ ι ∷ ν ∷ ὲ ∷ ς ∷ []) "1Cor.10.10"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.10.10"
∷ word (ἐ ∷ γ ∷ ό ∷ γ ∷ γ ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "1Cor.10.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.10"
∷ word (ἀ ∷ π ∷ ώ ∷ ∙λ ∷ ο ∷ ν ∷ τ ∷ ο ∷ []) "1Cor.10.10"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.10.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.10"
∷ word (ὀ ∷ ∙λ ∷ ο ∷ θ ∷ ρ ∷ ε ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.10"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1Cor.10.11"
∷ word (δ ∷ ὲ ∷ []) "1Cor.10.11"
∷ word (τ ∷ υ ∷ π ∷ ι ∷ κ ∷ ῶ ∷ ς ∷ []) "1Cor.10.11"
∷ word (σ ∷ υ ∷ ν ∷ έ ∷ β ∷ α ∷ ι ∷ ν ∷ ε ∷ ν ∷ []) "1Cor.10.11"
∷ word (ἐ ∷ κ ∷ ε ∷ ί ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.10.11"
∷ word (ἐ ∷ γ ∷ ρ ∷ ά ∷ φ ∷ η ∷ []) "1Cor.10.11"
∷ word (δ ∷ ὲ ∷ []) "1Cor.10.11"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.10.11"
∷ word (ν ∷ ο ∷ υ ∷ θ ∷ ε ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.10.11"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.10.11"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.10.11"
∷ word (ο ∷ ὓ ∷ ς ∷ []) "1Cor.10.11"
∷ word (τ ∷ ὰ ∷ []) "1Cor.10.11"
∷ word (τ ∷ έ ∷ ∙λ ∷ η ∷ []) "1Cor.10.11"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.10.11"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ω ∷ ν ∷ []) "1Cor.10.11"
∷ word (κ ∷ α ∷ τ ∷ ή ∷ ν ∷ τ ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "1Cor.10.11"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.10.12"
∷ word (ὁ ∷ []) "1Cor.10.12"
∷ word (δ ∷ ο ∷ κ ∷ ῶ ∷ ν ∷ []) "1Cor.10.12"
∷ word (ἑ ∷ σ ∷ τ ∷ ά ∷ ν ∷ α ∷ ι ∷ []) "1Cor.10.12"
∷ word (β ∷ ∙λ ∷ ε ∷ π ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.10.12"
∷ word (μ ∷ ὴ ∷ []) "1Cor.10.12"
∷ word (π ∷ έ ∷ σ ∷ ῃ ∷ []) "1Cor.10.12"
∷ word (π ∷ ε ∷ ι ∷ ρ ∷ α ∷ σ ∷ μ ∷ ὸ ∷ ς ∷ []) "1Cor.10.13"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.10.13"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.10.13"
∷ word (ε ∷ ἴ ∷ ∙λ ∷ η ∷ φ ∷ ε ∷ ν ∷ []) "1Cor.10.13"
∷ word (ε ∷ ἰ ∷ []) "1Cor.10.13"
∷ word (μ ∷ ὴ ∷ []) "1Cor.10.13"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ι ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.10.13"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.10.13"
∷ word (δ ∷ ὲ ∷ []) "1Cor.10.13"
∷ word (ὁ ∷ []) "1Cor.10.13"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "1Cor.10.13"
∷ word (ὃ ∷ ς ∷ []) "1Cor.10.13"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.10.13"
∷ word (ἐ ∷ ά ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.10.13"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.10.13"
∷ word (π ∷ ε ∷ ι ∷ ρ ∷ α ∷ σ ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.10.13"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "1Cor.10.13"
∷ word (ὃ ∷ []) "1Cor.10.13"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.10.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.10.13"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.10.13"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "1Cor.10.13"
∷ word (τ ∷ ῷ ∷ []) "1Cor.10.13"
∷ word (π ∷ ε ∷ ι ∷ ρ ∷ α ∷ σ ∷ μ ∷ ῷ ∷ []) "1Cor.10.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.13"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.10.13"
∷ word (ἔ ∷ κ ∷ β ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.10.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.13"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.10.13"
∷ word (ὑ ∷ π ∷ ε ∷ ν ∷ ε ∷ γ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.10.13"
∷ word (Δ ∷ ι ∷ ό ∷ π ∷ ε ∷ ρ ∷ []) "1Cor.10.14"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ί ∷ []) "1Cor.10.14"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.10.14"
∷ word (φ ∷ ε ∷ ύ ∷ γ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.10.14"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "1Cor.10.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.10.14"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ο ∷ ∙λ ∷ α ∷ τ ∷ ρ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.10.14"
∷ word (ὡ ∷ ς ∷ []) "1Cor.10.15"
∷ word (φ ∷ ρ ∷ ο ∷ ν ∷ ί ∷ μ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.10.15"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "1Cor.10.15"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ α ∷ τ ∷ ε ∷ []) "1Cor.10.15"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.10.15"
∷ word (ὅ ∷ []) "1Cor.10.15"
∷ word (φ ∷ η ∷ μ ∷ ι ∷ []) "1Cor.10.15"
∷ word (τ ∷ ὸ ∷ []) "1Cor.10.16"
∷ word (π ∷ ο ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.10.16"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.10.16"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.10.16"
∷ word (ὃ ∷ []) "1Cor.10.16"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.10.16"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "1Cor.10.16"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ί ∷ α ∷ []) "1Cor.10.16"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1Cor.10.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.16"
∷ word (α ∷ ἵ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.10.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.16"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.16"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.10.16"
∷ word (ἄ ∷ ρ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.10.16"
∷ word (ὃ ∷ ν ∷ []) "1Cor.10.16"
∷ word (κ ∷ ∙λ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.10.16"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "1Cor.10.16"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ί ∷ α ∷ []) "1Cor.10.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.16"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.10.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.16"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.16"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.10.16"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.10.17"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "1Cor.10.17"
∷ word (ἄ ∷ ρ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.10.17"
∷ word (ἓ ∷ ν ∷ []) "1Cor.10.17"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.10.17"
∷ word (ο ∷ ἱ ∷ []) "1Cor.10.17"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ί ∷ []) "1Cor.10.17"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.10.17"
∷ word (ο ∷ ἱ ∷ []) "1Cor.10.17"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.10.17"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.10.17"
∷ word (ἐ ∷ κ ∷ []) "1Cor.10.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.17"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "1Cor.10.17"
∷ word (ἄ ∷ ρ ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.10.17"
∷ word (μ ∷ ε ∷ τ ∷ έ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.10.17"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.10.18"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.10.18"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ὴ ∷ ∙λ ∷ []) "1Cor.10.18"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.10.18"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "1Cor.10.18"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1Cor.10.18"
∷ word (ο ∷ ἱ ∷ []) "1Cor.10.18"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.10.18"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1Cor.10.18"
∷ word (θ ∷ υ ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.10.18"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ο ∷ ὶ ∷ []) "1Cor.10.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.18"
∷ word (θ ∷ υ ∷ σ ∷ ι ∷ α ∷ σ ∷ τ ∷ η ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.10.18"
∷ word (ε ∷ ἰ ∷ σ ∷ ί ∷ ν ∷ []) "1Cor.10.18"
∷ word (τ ∷ ί ∷ []) "1Cor.10.19"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.10.19"
∷ word (φ ∷ η ∷ μ ∷ ι ∷ []) "1Cor.10.19"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.10.19"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ό ∷ θ ∷ υ ∷ τ ∷ ό ∷ ν ∷ []) "1Cor.10.19"
∷ word (τ ∷ ί ∷ []) "1Cor.10.19"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.10.19"
∷ word (ἢ ∷ []) "1Cor.10.19"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.10.19"
∷ word (ε ∷ ἴ ∷ δ ∷ ω ∷ ∙λ ∷ ό ∷ ν ∷ []) "1Cor.10.19"
∷ word (τ ∷ ί ∷ []) "1Cor.10.19"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.10.19"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.10.20"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.10.20"
∷ word (ἃ ∷ []) "1Cor.10.20"
∷ word (θ ∷ ύ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.10.20"
∷ word (δ ∷ α ∷ ι ∷ μ ∷ ο ∷ ν ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.10.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.20"
∷ word (ο ∷ ὐ ∷ []) "1Cor.10.20"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1Cor.10.20"
∷ word (θ ∷ ύ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.10.20"
∷ word (ο ∷ ὐ ∷ []) "1Cor.10.20"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "1Cor.10.20"
∷ word (δ ∷ ὲ ∷ []) "1Cor.10.20"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.10.20"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.10.20"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.10.20"
∷ word (δ ∷ α ∷ ι ∷ μ ∷ ο ∷ ν ∷ ί ∷ ω ∷ ν ∷ []) "1Cor.10.20"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.10.20"
∷ word (ο ∷ ὐ ∷ []) "1Cor.10.21"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.10.21"
∷ word (π ∷ ο ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.10.21"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.10.21"
∷ word (π ∷ ί ∷ ν ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.10.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.21"
∷ word (π ∷ ο ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.10.21"
∷ word (δ ∷ α ∷ ι ∷ μ ∷ ο ∷ ν ∷ ί ∷ ω ∷ ν ∷ []) "1Cor.10.21"
∷ word (ο ∷ ὐ ∷ []) "1Cor.10.21"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.10.21"
∷ word (τ ∷ ρ ∷ α ∷ π ∷ έ ∷ ζ ∷ η ∷ ς ∷ []) "1Cor.10.21"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.10.21"
∷ word (μ ∷ ε ∷ τ ∷ έ ∷ χ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.10.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.21"
∷ word (τ ∷ ρ ∷ α ∷ π ∷ έ ∷ ζ ∷ η ∷ ς ∷ []) "1Cor.10.21"
∷ word (δ ∷ α ∷ ι ∷ μ ∷ ο ∷ ν ∷ ί ∷ ω ∷ ν ∷ []) "1Cor.10.21"
∷ word (ἢ ∷ []) "1Cor.10.22"
∷ word (π ∷ α ∷ ρ ∷ α ∷ ζ ∷ η ∷ ∙λ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.10.22"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.10.22"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.10.22"
∷ word (μ ∷ ὴ ∷ []) "1Cor.10.22"
∷ word (ἰ ∷ σ ∷ χ ∷ υ ∷ ρ ∷ ό ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ι ∷ []) "1Cor.10.22"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.22"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.10.22"
∷ word (Π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.10.23"
∷ word (ἔ ∷ ξ ∷ ε ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.10.23"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.10.23"
∷ word (ο ∷ ὐ ∷ []) "1Cor.10.23"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.10.23"
∷ word (σ ∷ υ ∷ μ ∷ φ ∷ έ ∷ ρ ∷ ε ∷ ι ∷ []) "1Cor.10.23"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.10.23"
∷ word (ἔ ∷ ξ ∷ ε ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.10.23"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.10.23"
∷ word (ο ∷ ὐ ∷ []) "1Cor.10.23"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.10.23"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ε ∷ ῖ ∷ []) "1Cor.10.23"
∷ word (μ ∷ η ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.10.24"
∷ word (τ ∷ ὸ ∷ []) "1Cor.10.24"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.24"
∷ word (ζ ∷ η ∷ τ ∷ ε ∷ ί ∷ τ ∷ ω ∷ []) "1Cor.10.24"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.10.24"
∷ word (τ ∷ ὸ ∷ []) "1Cor.10.24"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.24"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ []) "1Cor.10.24"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "1Cor.10.25"
∷ word (τ ∷ ὸ ∷ []) "1Cor.10.25"
∷ word (ἐ ∷ ν ∷ []) "1Cor.10.25"
∷ word (μ ∷ α ∷ κ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ῳ ∷ []) "1Cor.10.25"
∷ word (π ∷ ω ∷ ∙λ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.10.25"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.10.25"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ ν ∷ []) "1Cor.10.25"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.10.25"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.10.25"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.10.25"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ί ∷ δ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.10.25"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.26"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.10.26"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.10.26"
∷ word (ἡ ∷ []) "1Cor.10.26"
∷ word (γ ∷ ῆ ∷ []) "1Cor.10.26"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.26"
∷ word (τ ∷ ὸ ∷ []) "1Cor.10.26"
∷ word (π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ μ ∷ α ∷ []) "1Cor.10.26"
∷ word (α ∷ ὐ ∷ τ ∷ ῆ ∷ ς ∷ []) "1Cor.10.26"
∷ word (ε ∷ ἴ ∷ []) "1Cor.10.27"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.10.27"
∷ word (κ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ []) "1Cor.10.27"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.10.27"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.10.27"
∷ word (ἀ ∷ π ∷ ί ∷ σ ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.10.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.27"
∷ word (θ ∷ έ ∷ ∙λ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.10.27"
∷ word (π ∷ ο ∷ ρ ∷ ε ∷ ύ ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.10.27"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "1Cor.10.27"
∷ word (τ ∷ ὸ ∷ []) "1Cor.10.27"
∷ word (π ∷ α ∷ ρ ∷ α ∷ τ ∷ ι ∷ θ ∷ έ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.10.27"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.10.27"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.10.27"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ ν ∷ []) "1Cor.10.27"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.10.27"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.10.27"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.10.27"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ί ∷ δ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.10.27"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.10.28"
∷ word (δ ∷ έ ∷ []) "1Cor.10.28"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.10.28"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.10.28"
∷ word (ε ∷ ἴ ∷ π ∷ ῃ ∷ []) "1Cor.10.28"
∷ word (Τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.10.28"
∷ word (ἱ ∷ ε ∷ ρ ∷ ό ∷ θ ∷ υ ∷ τ ∷ ό ∷ ν ∷ []) "1Cor.10.28"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.10.28"
∷ word (μ ∷ ὴ ∷ []) "1Cor.10.28"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.10.28"
∷ word (δ ∷ ι ∷ []) "1Cor.10.28"
∷ word (ἐ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.10.28"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.10.28"
∷ word (μ ∷ η ∷ ν ∷ ύ ∷ σ ∷ α ∷ ν ∷ τ ∷ α ∷ []) "1Cor.10.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.28"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.10.28"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ί ∷ δ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.10.28"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ί ∷ δ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.10.29"
∷ word (δ ∷ ὲ ∷ []) "1Cor.10.29"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "1Cor.10.29"
∷ word (ο ∷ ὐ ∷ χ ∷ ὶ ∷ []) "1Cor.10.29"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.10.29"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.29"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.10.29"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.10.29"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.29"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ []) "1Cor.10.29"
∷ word (ἱ ∷ ν ∷ α ∷ τ ∷ ί ∷ []) "1Cor.10.29"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.10.29"
∷ word (ἡ ∷ []) "1Cor.10.29"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ ε ∷ ρ ∷ ί ∷ α ∷ []) "1Cor.10.29"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.10.29"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.10.29"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.10.29"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ η ∷ ς ∷ []) "1Cor.10.29"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ι ∷ δ ∷ ή ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "1Cor.10.29"
∷ word (ε ∷ ἰ ∷ []) "1Cor.10.30"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.10.30"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ι ∷ []) "1Cor.10.30"
∷ word (μ ∷ ε ∷ τ ∷ έ ∷ χ ∷ ω ∷ []) "1Cor.10.30"
∷ word (τ ∷ ί ∷ []) "1Cor.10.30"
∷ word (β ∷ ∙λ ∷ α ∷ σ ∷ φ ∷ η ∷ μ ∷ ο ∷ ῦ ∷ μ ∷ α ∷ ι ∷ []) "1Cor.10.30"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "1Cor.10.30"
∷ word (ο ∷ ὗ ∷ []) "1Cor.10.30"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.10.30"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῶ ∷ []) "1Cor.10.30"
∷ word (Ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.10.31"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.10.31"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.10.31"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.10.31"
∷ word (π ∷ ί ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.10.31"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.10.31"
∷ word (τ ∷ ι ∷ []) "1Cor.10.31"
∷ word (π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "1Cor.10.31"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.10.31"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.10.31"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ ν ∷ []) "1Cor.10.31"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.10.31"
∷ word (π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "1Cor.10.31"
∷ word (ἀ ∷ π ∷ ρ ∷ ό ∷ σ ∷ κ ∷ ο ∷ π ∷ ο ∷ ι ∷ []) "1Cor.10.32"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.32"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.10.32"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.10.32"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.32"
∷ word (Ἕ ∷ ∙λ ∷ ∙λ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.10.32"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.10.32"
∷ word (τ ∷ ῇ ∷ []) "1Cor.10.32"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.10.32"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.32"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.10.32"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.10.33"
∷ word (κ ∷ ἀ ∷ γ ∷ ὼ ∷ []) "1Cor.10.33"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.10.33"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.10.33"
∷ word (ἀ ∷ ρ ∷ έ ∷ σ ∷ κ ∷ ω ∷ []) "1Cor.10.33"
∷ word (μ ∷ ὴ ∷ []) "1Cor.10.33"
∷ word (ζ ∷ η ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.10.33"
∷ word (τ ∷ ὸ ∷ []) "1Cor.10.33"
∷ word (ἐ ∷ μ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.10.33"
∷ word (σ ∷ ύ ∷ μ ∷ φ ∷ ο ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.10.33"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.10.33"
∷ word (τ ∷ ὸ ∷ []) "1Cor.10.33"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.10.33"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "1Cor.10.33"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.10.33"
∷ word (σ ∷ ω ∷ θ ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.10.33"
∷ word (μ ∷ ι ∷ μ ∷ η ∷ τ ∷ α ∷ ί ∷ []) "1Cor.11.1"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.11.1"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.11.1"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.11.1"
∷ word (κ ∷ ἀ ∷ γ ∷ ὼ ∷ []) "1Cor.11.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.1"
∷ word (Ἐ ∷ π ∷ α ∷ ι ∷ ν ∷ ῶ ∷ []) "1Cor.11.2"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.2"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.11.2"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.11.2"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.11.2"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.11.2"
∷ word (μ ∷ έ ∷ μ ∷ ν ∷ η ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.11.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.2"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.11.2"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ δ ∷ ω ∷ κ ∷ α ∷ []) "1Cor.11.2"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.11.2"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1Cor.11.2"
∷ word (π ∷ α ∷ ρ ∷ α ∷ δ ∷ ό ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.11.2"
∷ word (κ ∷ α ∷ τ ∷ έ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.11.2"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "1Cor.11.3"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.3"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.11.3"
∷ word (ε ∷ ἰ ∷ δ ∷ έ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.11.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.11.3"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.11.3"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.11.3"
∷ word (ἡ ∷ []) "1Cor.11.3"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ὴ ∷ []) "1Cor.11.3"
∷ word (ὁ ∷ []) "1Cor.11.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "1Cor.11.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.11.3"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ὴ ∷ []) "1Cor.11.3"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.3"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ὸ ∷ ς ∷ []) "1Cor.11.3"
∷ word (ὁ ∷ []) "1Cor.11.3"
∷ word (ἀ ∷ ν ∷ ή ∷ ρ ∷ []) "1Cor.11.3"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ὴ ∷ []) "1Cor.11.3"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.3"
∷ word (ὁ ∷ []) "1Cor.11.3"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "1Cor.11.3"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1Cor.11.4"
∷ word (ἀ ∷ ν ∷ ὴ ∷ ρ ∷ []) "1Cor.11.4"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.11.4"
∷ word (ἢ ∷ []) "1Cor.11.4"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "1Cor.11.4"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.11.4"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ῆ ∷ ς ∷ []) "1Cor.11.4"
∷ word (ἔ ∷ χ ∷ ω ∷ ν ∷ []) "1Cor.11.4"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ι ∷ σ ∷ χ ∷ ύ ∷ ν ∷ ε ∷ ι ∷ []) "1Cor.11.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.11.4"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "1Cor.11.4"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.4"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ []) "1Cor.11.5"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.5"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.11.5"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ ο ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "1Cor.11.5"
∷ word (ἢ ∷ []) "1Cor.11.5"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ύ ∷ ο ∷ υ ∷ σ ∷ α ∷ []) "1Cor.11.5"
∷ word (ἀ ∷ κ ∷ α ∷ τ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ π ∷ τ ∷ ῳ ∷ []) "1Cor.11.5"
∷ word (τ ∷ ῇ ∷ []) "1Cor.11.5"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ῇ ∷ []) "1Cor.11.5"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ι ∷ σ ∷ χ ∷ ύ ∷ ν ∷ ε ∷ ι ∷ []) "1Cor.11.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.11.5"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "1Cor.11.5"
∷ word (α ∷ ὐ ∷ τ ∷ ῆ ∷ ς ∷ []) "1Cor.11.5"
∷ word (ἓ ∷ ν ∷ []) "1Cor.11.5"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.11.5"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.11.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.5"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.5"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "1Cor.11.5"
∷ word (τ ∷ ῇ ∷ []) "1Cor.11.5"
∷ word (ἐ ∷ ξ ∷ υ ∷ ρ ∷ η ∷ μ ∷ έ ∷ ν ∷ ῃ ∷ []) "1Cor.11.5"
∷ word (ε ∷ ἰ ∷ []) "1Cor.11.6"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.11.6"
∷ word (ο ∷ ὐ ∷ []) "1Cor.11.6"
∷ word (κ ∷ α ∷ τ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ π ∷ τ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.11.6"
∷ word (γ ∷ υ ∷ ν ∷ ή ∷ []) "1Cor.11.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.6"
∷ word (κ ∷ ε ∷ ι ∷ ρ ∷ ά ∷ σ ∷ θ ∷ ω ∷ []) "1Cor.11.6"
∷ word (ε ∷ ἰ ∷ []) "1Cor.11.6"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.6"
∷ word (α ∷ ἰ ∷ σ ∷ χ ∷ ρ ∷ ὸ ∷ ν ∷ []) "1Cor.11.6"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ὶ ∷ []) "1Cor.11.6"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.6"
∷ word (κ ∷ ε ∷ ί ∷ ρ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.11.6"
∷ word (ἢ ∷ []) "1Cor.11.6"
∷ word (ξ ∷ υ ∷ ρ ∷ ᾶ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.11.6"
∷ word (κ ∷ α ∷ τ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ υ ∷ π ∷ τ ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "1Cor.11.6"
∷ word (ἀ ∷ ν ∷ ὴ ∷ ρ ∷ []) "1Cor.11.7"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.11.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.11.7"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.11.7"
∷ word (ὀ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ ε ∷ ι ∷ []) "1Cor.11.7"
∷ word (κ ∷ α ∷ τ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ π ∷ τ ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.11.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.11.7"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ή ∷ ν ∷ []) "1Cor.11.7"
∷ word (ε ∷ ἰ ∷ κ ∷ ὼ ∷ ν ∷ []) "1Cor.11.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.7"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "1Cor.11.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.11.7"
∷ word (ὑ ∷ π ∷ ά ∷ ρ ∷ χ ∷ ω ∷ ν ∷ []) "1Cor.11.7"
∷ word (ἡ ∷ []) "1Cor.11.7"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.11.7"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.7"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "1Cor.11.7"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ό ∷ ς ∷ []) "1Cor.11.7"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.11.7"
∷ word (ο ∷ ὐ ∷ []) "1Cor.11.8"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.11.8"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.11.8"
∷ word (ἀ ∷ ν ∷ ὴ ∷ ρ ∷ []) "1Cor.11.8"
∷ word (ἐ ∷ κ ∷ []) "1Cor.11.8"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ό ∷ ς ∷ []) "1Cor.11.8"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.11.8"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.11.8"
∷ word (ἐ ∷ ξ ∷ []) "1Cor.11.8"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ό ∷ ς ∷ []) "1Cor.11.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.11.9"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.11.9"
∷ word (ἐ ∷ κ ∷ τ ∷ ί ∷ σ ∷ θ ∷ η ∷ []) "1Cor.11.9"
∷ word (ἀ ∷ ν ∷ ὴ ∷ ρ ∷ []) "1Cor.11.9"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.11.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.11.9"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ []) "1Cor.11.9"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.11.9"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.11.9"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.11.9"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.11.9"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ α ∷ []) "1Cor.11.9"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.11.10"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.11.10"
∷ word (ὀ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ ε ∷ ι ∷ []) "1Cor.11.10"
∷ word (ἡ ∷ []) "1Cor.11.10"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.11.10"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.11.10"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.11.10"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.11.10"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.11.10"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ῆ ∷ ς ∷ []) "1Cor.11.10"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.11.10"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.11.10"
∷ word (ἀ ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.11.10"
∷ word (π ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "1Cor.11.11"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "1Cor.11.11"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.11.11"
∷ word (χ ∷ ω ∷ ρ ∷ ὶ ∷ ς ∷ []) "1Cor.11.11"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.11.11"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "1Cor.11.11"
∷ word (ἀ ∷ ν ∷ ὴ ∷ ρ ∷ []) "1Cor.11.11"
∷ word (χ ∷ ω ∷ ρ ∷ ὶ ∷ ς ∷ []) "1Cor.11.11"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ὸ ∷ ς ∷ []) "1Cor.11.11"
∷ word (ἐ ∷ ν ∷ []) "1Cor.11.11"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.11.11"
∷ word (ὥ ∷ σ ∷ π ∷ ε ∷ ρ ∷ []) "1Cor.11.12"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.11.12"
∷ word (ἡ ∷ []) "1Cor.11.12"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.11.12"
∷ word (ἐ ∷ κ ∷ []) "1Cor.11.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.12"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ό ∷ ς ∷ []) "1Cor.11.12"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.11.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.12"
∷ word (ὁ ∷ []) "1Cor.11.12"
∷ word (ἀ ∷ ν ∷ ὴ ∷ ρ ∷ []) "1Cor.11.12"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.11.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.11.12"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ό ∷ ς ∷ []) "1Cor.11.12"
∷ word (τ ∷ ὰ ∷ []) "1Cor.11.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.12"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.11.12"
∷ word (ἐ ∷ κ ∷ []) "1Cor.11.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.12"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.11.12"
∷ word (ἐ ∷ ν ∷ []) "1Cor.11.13"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.11.13"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.11.13"
∷ word (κ ∷ ρ ∷ ί ∷ ν ∷ α ∷ τ ∷ ε ∷ []) "1Cor.11.13"
∷ word (π ∷ ρ ∷ έ ∷ π ∷ ο ∷ ν ∷ []) "1Cor.11.13"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1Cor.11.13"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ []) "1Cor.11.13"
∷ word (ἀ ∷ κ ∷ α ∷ τ ∷ α ∷ κ ∷ ά ∷ ∙λ ∷ υ ∷ π ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.11.13"
∷ word (τ ∷ ῷ ∷ []) "1Cor.11.13"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1Cor.11.13"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ ύ ∷ χ ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.11.13"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "1Cor.11.14"
∷ word (ἡ ∷ []) "1Cor.11.14"
∷ word (φ ∷ ύ ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.11.14"
∷ word (α ∷ ὐ ∷ τ ∷ ὴ ∷ []) "1Cor.11.14"
∷ word (δ ∷ ι ∷ δ ∷ ά ∷ σ ∷ κ ∷ ε ∷ ι ∷ []) "1Cor.11.14"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.11.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.11.14"
∷ word (ἀ ∷ ν ∷ ὴ ∷ ρ ∷ []) "1Cor.11.14"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.11.14"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.11.14"
∷ word (κ ∷ ο ∷ μ ∷ ᾷ ∷ []) "1Cor.11.14"
∷ word (ἀ ∷ τ ∷ ι ∷ μ ∷ ί ∷ α ∷ []) "1Cor.11.14"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1Cor.11.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.11.14"
∷ word (γ ∷ υ ∷ ν ∷ ὴ ∷ []) "1Cor.11.15"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.15"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.11.15"
∷ word (κ ∷ ο ∷ μ ∷ ᾷ ∷ []) "1Cor.11.15"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "1Cor.11.15"
∷ word (α ∷ ὐ ∷ τ ∷ ῇ ∷ []) "1Cor.11.15"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.11.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.11.15"
∷ word (ἡ ∷ []) "1Cor.11.15"
∷ word (κ ∷ ό ∷ μ ∷ η ∷ []) "1Cor.11.15"
∷ word (ἀ ∷ ν ∷ τ ∷ ὶ ∷ []) "1Cor.11.15"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ β ∷ ο ∷ ∙λ ∷ α ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.11.15"
∷ word (δ ∷ έ ∷ δ ∷ ο ∷ τ ∷ α ∷ ι ∷ []) "1Cor.11.15"
∷ word (ε ∷ ἰ ∷ []) "1Cor.11.16"
∷ word (δ ∷ έ ∷ []) "1Cor.11.16"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.11.16"
∷ word (δ ∷ ο ∷ κ ∷ ε ∷ ῖ ∷ []) "1Cor.11.16"
∷ word (φ ∷ ι ∷ ∙λ ∷ ό ∷ ν ∷ ε ∷ ι ∷ κ ∷ ο ∷ ς ∷ []) "1Cor.11.16"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.11.16"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.11.16"
∷ word (τ ∷ ο ∷ ι ∷ α ∷ ύ ∷ τ ∷ η ∷ ν ∷ []) "1Cor.11.16"
∷ word (σ ∷ υ ∷ ν ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "1Cor.11.16"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.11.16"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.11.16"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "1Cor.11.16"
∷ word (α ∷ ἱ ∷ []) "1Cor.11.16"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ι ∷ []) "1Cor.11.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.16"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.11.16"
∷ word (Τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.11.17"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.17"
∷ word (π ∷ α ∷ ρ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ω ∷ ν ∷ []) "1Cor.11.17"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.11.17"
∷ word (ἐ ∷ π ∷ α ∷ ι ∷ ν ∷ ῶ ∷ []) "1Cor.11.17"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.11.17"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.11.17"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.11.17"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.17"
∷ word (κ ∷ ρ ∷ ε ∷ ῖ ∷ σ ∷ σ ∷ ο ∷ ν ∷ []) "1Cor.11.17"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.11.17"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.11.17"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.17"
∷ word (ἧ ∷ σ ∷ σ ∷ ο ∷ ν ∷ []) "1Cor.11.17"
∷ word (σ ∷ υ ∷ ν ∷ έ ∷ ρ ∷ χ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.11.17"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.11.18"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.11.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.11.18"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ρ ∷ χ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ω ∷ ν ∷ []) "1Cor.11.18"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.11.18"
∷ word (ἐ ∷ ν ∷ []) "1Cor.11.18"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.11.18"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ ω ∷ []) "1Cor.11.18"
∷ word (σ ∷ χ ∷ ί ∷ σ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.11.18"
∷ word (ἐ ∷ ν ∷ []) "1Cor.11.18"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.11.18"
∷ word (ὑ ∷ π ∷ ά ∷ ρ ∷ χ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.11.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.18"
∷ word (μ ∷ έ ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.11.18"
∷ word (τ ∷ ι ∷ []) "1Cor.11.18"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ω ∷ []) "1Cor.11.18"
∷ word (δ ∷ ε ∷ ῖ ∷ []) "1Cor.11.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.11.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.19"
∷ word (α ∷ ἱ ∷ ρ ∷ έ ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.11.19"
∷ word (ἐ ∷ ν ∷ []) "1Cor.11.19"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.11.19"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.11.19"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.11.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.19"
∷ word (ο ∷ ἱ ∷ []) "1Cor.11.19"
∷ word (δ ∷ ό ∷ κ ∷ ι ∷ μ ∷ ο ∷ ι ∷ []) "1Cor.11.19"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ο ∷ ὶ ∷ []) "1Cor.11.19"
∷ word (γ ∷ έ ∷ ν ∷ ω ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.11.19"
∷ word (ἐ ∷ ν ∷ []) "1Cor.11.19"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.11.19"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ρ ∷ χ ∷ ο ∷ μ ∷ έ ∷ ν ∷ ω ∷ ν ∷ []) "1Cor.11.20"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.11.20"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.11.20"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.11.20"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.20"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "1Cor.11.20"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.11.20"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.11.20"
∷ word (κ ∷ υ ∷ ρ ∷ ι ∷ α ∷ κ ∷ ὸ ∷ ν ∷ []) "1Cor.11.20"
∷ word (δ ∷ ε ∷ ῖ ∷ π ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.11.20"
∷ word (φ ∷ α ∷ γ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.11.20"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.11.21"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.11.21"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.21"
∷ word (ἴ ∷ δ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.11.21"
∷ word (δ ∷ ε ∷ ῖ ∷ π ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.11.21"
∷ word (π ∷ ρ ∷ ο ∷ ∙λ ∷ α ∷ μ ∷ β ∷ ά ∷ ν ∷ ε ∷ ι ∷ []) "1Cor.11.21"
∷ word (ἐ ∷ ν ∷ []) "1Cor.11.21"
∷ word (τ ∷ ῷ ∷ []) "1Cor.11.21"
∷ word (φ ∷ α ∷ γ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.11.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.21"
∷ word (ὃ ∷ ς ∷ []) "1Cor.11.21"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.11.21"
∷ word (π ∷ ε ∷ ι ∷ ν ∷ ᾷ ∷ []) "1Cor.11.21"
∷ word (ὃ ∷ ς ∷ []) "1Cor.11.21"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.21"
∷ word (μ ∷ ε ∷ θ ∷ ύ ∷ ε ∷ ι ∷ []) "1Cor.11.21"
∷ word (μ ∷ ὴ ∷ []) "1Cor.11.22"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.11.22"
∷ word (ο ∷ ἰ ∷ κ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.11.22"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.11.22"
∷ word (ἔ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.11.22"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.11.22"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.22"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.11.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.22"
∷ word (π ∷ ί ∷ ν ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.11.22"
∷ word (ἢ ∷ []) "1Cor.11.22"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.11.22"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.11.22"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.22"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.11.22"
∷ word (κ ∷ α ∷ τ ∷ α ∷ φ ∷ ρ ∷ ο ∷ ν ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "1Cor.11.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.22"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ι ∷ σ ∷ χ ∷ ύ ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.11.22"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.11.22"
∷ word (μ ∷ ὴ ∷ []) "1Cor.11.22"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "1Cor.11.22"
∷ word (τ ∷ ί ∷ []) "1Cor.11.22"
∷ word (ε ∷ ἴ ∷ π ∷ ω ∷ []) "1Cor.11.22"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.11.22"
∷ word (ἐ ∷ π ∷ α ∷ ι ∷ ν ∷ έ ∷ σ ∷ ω ∷ []) "1Cor.11.22"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.11.22"
∷ word (ἐ ∷ ν ∷ []) "1Cor.11.22"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1Cor.11.22"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.11.22"
∷ word (ἐ ∷ π ∷ α ∷ ι ∷ ν ∷ ῶ ∷ []) "1Cor.11.22"
∷ word (Ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.11.23"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.11.23"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ ∙λ ∷ α ∷ β ∷ ο ∷ ν ∷ []) "1Cor.11.23"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "1Cor.11.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.23"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.11.23"
∷ word (ὃ ∷ []) "1Cor.11.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.23"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ δ ∷ ω ∷ κ ∷ α ∷ []) "1Cor.11.23"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.11.23"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.11.23"
∷ word (ὁ ∷ []) "1Cor.11.23"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.11.23"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "1Cor.11.23"
∷ word (ἐ ∷ ν ∷ []) "1Cor.11.23"
∷ word (τ ∷ ῇ ∷ []) "1Cor.11.23"
∷ word (ν ∷ υ ∷ κ ∷ τ ∷ ὶ ∷ []) "1Cor.11.23"
∷ word (ᾗ ∷ []) "1Cor.11.23"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ δ ∷ ί ∷ δ ∷ ε ∷ τ ∷ ο ∷ []) "1Cor.11.23"
∷ word (ἔ ∷ ∙λ ∷ α ∷ β ∷ ε ∷ ν ∷ []) "1Cor.11.23"
∷ word (ἄ ∷ ρ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.11.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.24"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ή ∷ σ ∷ α ∷ ς ∷ []) "1Cor.11.24"
∷ word (ἔ ∷ κ ∷ ∙λ ∷ α ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.11.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.24"
∷ word (ε ∷ ἶ ∷ π ∷ ε ∷ ν ∷ []) "1Cor.11.24"
∷ word (Τ ∷ ο ∷ ῦ ∷ τ ∷ ό ∷ []) "1Cor.11.24"
∷ word (μ ∷ ο ∷ ύ ∷ []) "1Cor.11.24"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.11.24"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.24"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.11.24"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.24"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "1Cor.11.24"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.11.24"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.11.24"
∷ word (π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "1Cor.11.24"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.11.24"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.11.24"
∷ word (ἐ ∷ μ ∷ ὴ ∷ ν ∷ []) "1Cor.11.24"
∷ word (ἀ ∷ ν ∷ ά ∷ μ ∷ ν ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.11.24"
∷ word (ὡ ∷ σ ∷ α ∷ ύ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.11.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.25"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.25"
∷ word (π ∷ ο ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.11.25"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "1Cor.11.25"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.25"
∷ word (δ ∷ ε ∷ ι ∷ π ∷ ν ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "1Cor.11.25"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ ν ∷ []) "1Cor.11.25"
∷ word (Τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.11.25"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.25"
∷ word (π ∷ ο ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.11.25"
∷ word (ἡ ∷ []) "1Cor.11.25"
∷ word (κ ∷ α ∷ ι ∷ ν ∷ ὴ ∷ []) "1Cor.11.25"
∷ word (δ ∷ ι ∷ α ∷ θ ∷ ή ∷ κ ∷ η ∷ []) "1Cor.11.25"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1Cor.11.25"
∷ word (ἐ ∷ ν ∷ []) "1Cor.11.25"
∷ word (τ ∷ ῷ ∷ []) "1Cor.11.25"
∷ word (ἐ ∷ μ ∷ ῷ ∷ []) "1Cor.11.25"
∷ word (α ∷ ἵ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.11.25"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.11.25"
∷ word (π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "1Cor.11.25"
∷ word (ὁ ∷ σ ∷ ά ∷ κ ∷ ι ∷ ς ∷ []) "1Cor.11.25"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.11.25"
∷ word (π ∷ ί ∷ ν ∷ η ∷ τ ∷ ε ∷ []) "1Cor.11.25"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.11.25"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.11.25"
∷ word (ἐ ∷ μ ∷ ὴ ∷ ν ∷ []) "1Cor.11.25"
∷ word (ἀ ∷ ν ∷ ά ∷ μ ∷ ν ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.11.25"
∷ word (ὁ ∷ σ ∷ ά ∷ κ ∷ ι ∷ ς ∷ []) "1Cor.11.26"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.11.26"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.11.26"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ η ∷ τ ∷ ε ∷ []) "1Cor.11.26"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.11.26"
∷ word (ἄ ∷ ρ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.11.26"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.11.26"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.26"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.26"
∷ word (π ∷ ο ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.11.26"
∷ word (π ∷ ί ∷ ν ∷ η ∷ τ ∷ ε ∷ []) "1Cor.11.26"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.11.26"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.11.26"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.26"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.11.26"
∷ word (κ ∷ α ∷ τ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.11.26"
∷ word (ἄ ∷ χ ∷ ρ ∷ ι ∷ []) "1Cor.11.26"
∷ word (ο ∷ ὗ ∷ []) "1Cor.11.26"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ῃ ∷ []) "1Cor.11.26"
∷ word (Ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.11.27"
∷ word (ὃ ∷ ς ∷ []) "1Cor.11.27"
∷ word (ἂ ∷ ν ∷ []) "1Cor.11.27"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ῃ ∷ []) "1Cor.11.27"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.11.27"
∷ word (ἄ ∷ ρ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.11.27"
∷ word (ἢ ∷ []) "1Cor.11.27"
∷ word (π ∷ ί ∷ ν ∷ ῃ ∷ []) "1Cor.11.27"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.27"
∷ word (π ∷ ο ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.11.27"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.27"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.11.27"
∷ word (ἀ ∷ ν ∷ α ∷ ξ ∷ ί ∷ ω ∷ ς ∷ []) "1Cor.11.27"
∷ word (ἔ ∷ ν ∷ ο ∷ χ ∷ ο ∷ ς ∷ []) "1Cor.11.27"
∷ word (ἔ ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.11.27"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.27"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.11.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.27"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.27"
∷ word (α ∷ ἵ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.11.27"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.27"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.11.27"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ α ∷ ζ ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.11.28"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.28"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "1Cor.11.28"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ό ∷ ν ∷ []) "1Cor.11.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.28"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.11.28"
∷ word (ἐ ∷ κ ∷ []) "1Cor.11.28"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.28"
∷ word (ἄ ∷ ρ ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.11.28"
∷ word (ἐ ∷ σ ∷ θ ∷ ι ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.11.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.28"
∷ word (ἐ ∷ κ ∷ []) "1Cor.11.28"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.11.28"
∷ word (π ∷ ο ∷ τ ∷ η ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.11.28"
∷ word (π ∷ ι ∷ ν ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.11.28"
∷ word (ὁ ∷ []) "1Cor.11.29"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.11.29"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ω ∷ ν ∷ []) "1Cor.11.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.29"
∷ word (π ∷ ί ∷ ν ∷ ω ∷ ν ∷ []) "1Cor.11.29"
∷ word (κ ∷ ρ ∷ ί ∷ μ ∷ α ∷ []) "1Cor.11.29"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "1Cor.11.29"
∷ word (ἐ ∷ σ ∷ θ ∷ ί ∷ ε ∷ ι ∷ []) "1Cor.11.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.29"
∷ word (π ∷ ί ∷ ν ∷ ε ∷ ι ∷ []) "1Cor.11.29"
∷ word (μ ∷ ὴ ∷ []) "1Cor.11.29"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ω ∷ ν ∷ []) "1Cor.11.29"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.29"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.11.29"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.11.30"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.11.30"
∷ word (ἐ ∷ ν ∷ []) "1Cor.11.30"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.11.30"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὶ ∷ []) "1Cor.11.30"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.11.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.30"
∷ word (ἄ ∷ ρ ∷ ρ ∷ ω ∷ σ ∷ τ ∷ ο ∷ ι ∷ []) "1Cor.11.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.11.30"
∷ word (κ ∷ ο ∷ ι ∷ μ ∷ ῶ ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.11.30"
∷ word (ἱ ∷ κ ∷ α ∷ ν ∷ ο ∷ ί ∷ []) "1Cor.11.30"
∷ word (ε ∷ ἰ ∷ []) "1Cor.11.31"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.31"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.11.31"
∷ word (δ ∷ ι ∷ ε ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.11.31"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.11.31"
∷ word (ἂ ∷ ν ∷ []) "1Cor.11.31"
∷ word (ἐ ∷ κ ∷ ρ ∷ ι ∷ ν ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1Cor.11.31"
∷ word (κ ∷ ρ ∷ ι ∷ ν ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.11.32"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.32"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.11.32"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.11.32"
∷ word (π ∷ α ∷ ι ∷ δ ∷ ε ∷ υ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1Cor.11.32"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.11.32"
∷ word (μ ∷ ὴ ∷ []) "1Cor.11.32"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "1Cor.11.32"
∷ word (τ ∷ ῷ ∷ []) "1Cor.11.32"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "1Cor.11.32"
∷ word (κ ∷ α ∷ τ ∷ α ∷ κ ∷ ρ ∷ ι ∷ θ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.11.32"
∷ word (Ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.11.33"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.11.33"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.11.33"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ρ ∷ χ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.11.33"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.11.33"
∷ word (τ ∷ ὸ ∷ []) "1Cor.11.33"
∷ word (φ ∷ α ∷ γ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.11.33"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.11.33"
∷ word (ἐ ∷ κ ∷ δ ∷ έ ∷ χ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.11.33"
∷ word (ε ∷ ἴ ∷ []) "1Cor.11.34"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.11.34"
∷ word (π ∷ ε ∷ ι ∷ ν ∷ ᾷ ∷ []) "1Cor.11.34"
∷ word (ἐ ∷ ν ∷ []) "1Cor.11.34"
∷ word (ο ∷ ἴ ∷ κ ∷ ῳ ∷ []) "1Cor.11.34"
∷ word (ἐ ∷ σ ∷ θ ∷ ι ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.11.34"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.11.34"
∷ word (μ ∷ ὴ ∷ []) "1Cor.11.34"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.11.34"
∷ word (κ ∷ ρ ∷ ί ∷ μ ∷ α ∷ []) "1Cor.11.34"
∷ word (σ ∷ υ ∷ ν ∷ έ ∷ ρ ∷ χ ∷ η ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.11.34"
∷ word (Τ ∷ ὰ ∷ []) "1Cor.11.34"
∷ word (δ ∷ ὲ ∷ []) "1Cor.11.34"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ὰ ∷ []) "1Cor.11.34"
∷ word (ὡ ∷ ς ∷ []) "1Cor.11.34"
∷ word (ἂ ∷ ν ∷ []) "1Cor.11.34"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ω ∷ []) "1Cor.11.34"
∷ word (δ ∷ ι ∷ α ∷ τ ∷ ά ∷ ξ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.11.34"
∷ word (Π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1Cor.12.1"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.1"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.12.1"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ῶ ∷ ν ∷ []) "1Cor.12.1"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.12.1"
∷ word (ο ∷ ὐ ∷ []) "1Cor.12.1"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "1Cor.12.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.12.1"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.12.1"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.12.2"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.12.2"
∷ word (ὅ ∷ τ ∷ ε ∷ []) "1Cor.12.2"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "1Cor.12.2"
∷ word (ἦ ∷ τ ∷ ε ∷ []) "1Cor.12.2"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.12.2"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.2"
∷ word (ε ∷ ἴ ∷ δ ∷ ω ∷ ∙λ ∷ α ∷ []) "1Cor.12.2"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.2"
∷ word (ἄ ∷ φ ∷ ω ∷ ν ∷ α ∷ []) "1Cor.12.2"
∷ word (ὡ ∷ ς ∷ []) "1Cor.12.2"
∷ word (ἂ ∷ ν ∷ []) "1Cor.12.2"
∷ word (ἤ ∷ γ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.12.2"
∷ word (ἀ ∷ π ∷ α ∷ γ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.12.2"
∷ word (δ ∷ ι ∷ ὸ ∷ []) "1Cor.12.3"
∷ word (γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ ζ ∷ ω ∷ []) "1Cor.12.3"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.12.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.12.3"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.12.3"
∷ word (ἐ ∷ ν ∷ []) "1Cor.12.3"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.12.3"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.12.3"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "1Cor.12.3"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "1Cor.12.3"
∷ word (Ἀ ∷ ν ∷ ά ∷ θ ∷ ε ∷ μ ∷ α ∷ []) "1Cor.12.3"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "1Cor.12.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.3"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.12.3"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ τ ∷ α ∷ ι ∷ []) "1Cor.12.3"
∷ word (ε ∷ ἰ ∷ π ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.12.3"
∷ word (Κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.12.3"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "1Cor.12.3"
∷ word (ε ∷ ἰ ∷ []) "1Cor.12.3"
∷ word (μ ∷ ὴ ∷ []) "1Cor.12.3"
∷ word (ἐ ∷ ν ∷ []) "1Cor.12.3"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.12.3"
∷ word (ἁ ∷ γ ∷ ί ∷ ῳ ∷ []) "1Cor.12.3"
∷ word (Δ ∷ ι ∷ α ∷ ι ∷ ρ ∷ έ ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.12.4"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.4"
∷ word (χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.12.4"
∷ word (ε ∷ ἰ ∷ σ ∷ ί ∷ ν ∷ []) "1Cor.12.4"
∷ word (τ ∷ ὸ ∷ []) "1Cor.12.4"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.4"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "1Cor.12.4"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.12.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.5"
∷ word (δ ∷ ι ∷ α ∷ ι ∷ ρ ∷ έ ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.12.5"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ι ∷ ῶ ∷ ν ∷ []) "1Cor.12.5"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.12.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.5"
∷ word (ὁ ∷ []) "1Cor.12.5"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.12.5"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.12.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.6"
∷ word (δ ∷ ι ∷ α ∷ ι ∷ ρ ∷ έ ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.12.6"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ η ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.12.6"
∷ word (ε ∷ ἰ ∷ σ ∷ ί ∷ ν ∷ []) "1Cor.12.6"
∷ word (ὁ ∷ []) "1Cor.12.6"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.6"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.12.6"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "1Cor.12.6"
∷ word (ὁ ∷ []) "1Cor.12.6"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ῶ ∷ ν ∷ []) "1Cor.12.6"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.6"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.12.6"
∷ word (ἐ ∷ ν ∷ []) "1Cor.12.6"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.12.6"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ ῳ ∷ []) "1Cor.12.7"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.7"
∷ word (δ ∷ ί ∷ δ ∷ ο ∷ τ ∷ α ∷ ι ∷ []) "1Cor.12.7"
∷ word (ἡ ∷ []) "1Cor.12.7"
∷ word (φ ∷ α ∷ ν ∷ έ ∷ ρ ∷ ω ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.12.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.12.7"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.12.7"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.12.7"
∷ word (τ ∷ ὸ ∷ []) "1Cor.12.7"
∷ word (σ ∷ υ ∷ μ ∷ φ ∷ έ ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.12.7"
∷ word (ᾧ ∷ []) "1Cor.12.8"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.12.8"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.12.8"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.12.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.12.8"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.12.8"
∷ word (δ ∷ ί ∷ δ ∷ ο ∷ τ ∷ α ∷ ι ∷ []) "1Cor.12.8"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "1Cor.12.8"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.12.8"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ῳ ∷ []) "1Cor.12.8"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.8"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "1Cor.12.8"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "1Cor.12.8"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.12.8"
∷ word (τ ∷ ὸ ∷ []) "1Cor.12.8"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "1Cor.12.8"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.12.8"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ ῳ ∷ []) "1Cor.12.9"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "1Cor.12.9"
∷ word (ἐ ∷ ν ∷ []) "1Cor.12.9"
∷ word (τ ∷ ῷ ∷ []) "1Cor.12.9"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1Cor.12.9"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.12.9"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ῳ ∷ []) "1Cor.12.9"
∷ word (χ ∷ α ∷ ρ ∷ ί ∷ σ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.12.9"
∷ word (ἰ ∷ α ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.12.9"
∷ word (ἐ ∷ ν ∷ []) "1Cor.12.9"
∷ word (τ ∷ ῷ ∷ []) "1Cor.12.9"
∷ word (ἑ ∷ ν ∷ ὶ ∷ []) "1Cor.12.9"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.12.9"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ῳ ∷ []) "1Cor.12.10"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.12.10"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ω ∷ ν ∷ []) "1Cor.12.10"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ῳ ∷ []) "1Cor.12.10"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ί ∷ α ∷ []) "1Cor.12.10"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ῳ ∷ []) "1Cor.12.10"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ρ ∷ ί ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.12.10"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.12.10"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ ῳ ∷ []) "1Cor.12.10"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ []) "1Cor.12.10"
∷ word (γ ∷ ∙λ ∷ ω ∷ σ ∷ σ ∷ ῶ ∷ ν ∷ []) "1Cor.12.10"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ῳ ∷ []) "1Cor.12.10"
∷ word (ἑ ∷ ρ ∷ μ ∷ η ∷ ν ∷ ε ∷ ί ∷ α ∷ []) "1Cor.12.10"
∷ word (γ ∷ ∙λ ∷ ω ∷ σ ∷ σ ∷ ῶ ∷ ν ∷ []) "1Cor.12.10"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.12.11"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.11"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1Cor.12.11"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ε ∷ ῖ ∷ []) "1Cor.12.11"
∷ word (τ ∷ ὸ ∷ []) "1Cor.12.11"
∷ word (ἓ ∷ ν ∷ []) "1Cor.12.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.11"
∷ word (τ ∷ ὸ ∷ []) "1Cor.12.11"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "1Cor.12.11"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.12.11"
∷ word (δ ∷ ι ∷ α ∷ ι ∷ ρ ∷ ο ∷ ῦ ∷ ν ∷ []) "1Cor.12.11"
∷ word (ἰ ∷ δ ∷ ί ∷ ᾳ ∷ []) "1Cor.12.11"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ ῳ ∷ []) "1Cor.12.11"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.12.11"
∷ word (β ∷ ο ∷ ύ ∷ ∙λ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.12.11"
∷ word (Κ ∷ α ∷ θ ∷ ά ∷ π ∷ ε ∷ ρ ∷ []) "1Cor.12.12"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.12.12"
∷ word (τ ∷ ὸ ∷ []) "1Cor.12.12"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.12.12"
∷ word (ἕ ∷ ν ∷ []) "1Cor.12.12"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.12.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.12"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "1Cor.12.12"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.12.12"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.12.12"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.12.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.12"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.12"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "1Cor.12.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.12.12"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.12.12"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.12.12"
∷ word (ὄ ∷ ν ∷ τ ∷ α ∷ []) "1Cor.12.12"
∷ word (ἕ ∷ ν ∷ []) "1Cor.12.12"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.12.12"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.12.12"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.12.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.12"
∷ word (ὁ ∷ []) "1Cor.12.12"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "1Cor.12.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.12.13"
∷ word (ἐ ∷ ν ∷ []) "1Cor.12.13"
∷ word (ἑ ∷ ν ∷ ὶ ∷ []) "1Cor.12.13"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.12.13"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.12.13"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.12.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.12.13"
∷ word (ἓ ∷ ν ∷ []) "1Cor.12.13"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.12.13"
∷ word (ἐ ∷ β ∷ α ∷ π ∷ τ ∷ ί ∷ σ ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.12.13"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.12.13"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ῖ ∷ ο ∷ ι ∷ []) "1Cor.12.13"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.12.13"
∷ word (Ἕ ∷ ∙λ ∷ ∙λ ∷ η ∷ ν ∷ ε ∷ ς ∷ []) "1Cor.12.13"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.12.13"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ι ∷ []) "1Cor.12.13"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.12.13"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ θ ∷ ε ∷ ρ ∷ ο ∷ ι ∷ []) "1Cor.12.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.13"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.12.13"
∷ word (ἓ ∷ ν ∷ []) "1Cor.12.13"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.12.13"
∷ word (ἐ ∷ π ∷ ο ∷ τ ∷ ί ∷ σ ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.12.13"
∷ word (Κ ∷ α ∷ ὶ ∷ []) "1Cor.12.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.12.14"
∷ word (τ ∷ ὸ ∷ []) "1Cor.12.14"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.12.14"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.12.14"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.12.14"
∷ word (ἓ ∷ ν ∷ []) "1Cor.12.14"
∷ word (μ ∷ έ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.12.14"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.12.14"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ά ∷ []) "1Cor.12.14"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.12.15"
∷ word (ε ∷ ἴ ∷ π ∷ ῃ ∷ []) "1Cor.12.15"
∷ word (ὁ ∷ []) "1Cor.12.15"
∷ word (π ∷ ο ∷ ύ ∷ ς ∷ []) "1Cor.12.15"
∷ word (Ὅ ∷ τ ∷ ι ∷ []) "1Cor.12.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.12.15"
∷ word (ε ∷ ἰ ∷ μ ∷ ὶ ∷ []) "1Cor.12.15"
∷ word (χ ∷ ε ∷ ί ∷ ρ ∷ []) "1Cor.12.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.12.15"
∷ word (ε ∷ ἰ ∷ μ ∷ ὶ ∷ []) "1Cor.12.15"
∷ word (ἐ ∷ κ ∷ []) "1Cor.12.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.12.15"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.12.15"
∷ word (ο ∷ ὐ ∷ []) "1Cor.12.15"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "1Cor.12.15"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.12.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.12.15"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.12.15"
∷ word (ἐ ∷ κ ∷ []) "1Cor.12.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.12.15"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.12.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.16"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.12.16"
∷ word (ε ∷ ἴ ∷ π ∷ ῃ ∷ []) "1Cor.12.16"
∷ word (τ ∷ ὸ ∷ []) "1Cor.12.16"
∷ word (ο ∷ ὖ ∷ ς ∷ []) "1Cor.12.16"
∷ word (Ὅ ∷ τ ∷ ι ∷ []) "1Cor.12.16"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.12.16"
∷ word (ε ∷ ἰ ∷ μ ∷ ὶ ∷ []) "1Cor.12.16"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ό ∷ ς ∷ []) "1Cor.12.16"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.12.16"
∷ word (ε ∷ ἰ ∷ μ ∷ ὶ ∷ []) "1Cor.12.16"
∷ word (ἐ ∷ κ ∷ []) "1Cor.12.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.12.16"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.12.16"
∷ word (ο ∷ ὐ ∷ []) "1Cor.12.16"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "1Cor.12.16"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.12.16"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.12.16"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.12.16"
∷ word (ἐ ∷ κ ∷ []) "1Cor.12.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.12.16"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.12.16"
∷ word (ε ∷ ἰ ∷ []) "1Cor.12.17"
∷ word (ὅ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.12.17"
∷ word (τ ∷ ὸ ∷ []) "1Cor.12.17"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.12.17"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ό ∷ ς ∷ []) "1Cor.12.17"
∷ word (π ∷ ο ∷ ῦ ∷ []) "1Cor.12.17"
∷ word (ἡ ∷ []) "1Cor.12.17"
∷ word (ἀ ∷ κ ∷ ο ∷ ή ∷ []) "1Cor.12.17"
∷ word (ε ∷ ἰ ∷ []) "1Cor.12.17"
∷ word (ὅ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.12.17"
∷ word (ἀ ∷ κ ∷ ο ∷ ή ∷ []) "1Cor.12.17"
∷ word (π ∷ ο ∷ ῦ ∷ []) "1Cor.12.17"
∷ word (ἡ ∷ []) "1Cor.12.17"
∷ word (ὄ ∷ σ ∷ φ ∷ ρ ∷ η ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.12.17"
∷ word (ν ∷ υ ∷ ν ∷ ὶ ∷ []) "1Cor.12.18"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.18"
∷ word (ὁ ∷ []) "1Cor.12.18"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.12.18"
∷ word (ἔ ∷ θ ∷ ε ∷ τ ∷ ο ∷ []) "1Cor.12.18"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.18"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "1Cor.12.18"
∷ word (ἓ ∷ ν ∷ []) "1Cor.12.18"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.12.18"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.12.18"
∷ word (ἐ ∷ ν ∷ []) "1Cor.12.18"
∷ word (τ ∷ ῷ ∷ []) "1Cor.12.18"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.12.18"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.12.18"
∷ word (ἠ ∷ θ ∷ έ ∷ ∙λ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.12.18"
∷ word (ε ∷ ἰ ∷ []) "1Cor.12.19"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.19"
∷ word (ἦ ∷ ν ∷ []) "1Cor.12.19"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.19"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.12.19"
∷ word (ἓ ∷ ν ∷ []) "1Cor.12.19"
∷ word (μ ∷ έ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.12.19"
∷ word (π ∷ ο ∷ ῦ ∷ []) "1Cor.12.19"
∷ word (τ ∷ ὸ ∷ []) "1Cor.12.19"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.12.19"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "1Cor.12.20"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.20"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.12.20"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.12.20"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "1Cor.12.20"
∷ word (ἓ ∷ ν ∷ []) "1Cor.12.20"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.20"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.12.20"
∷ word (ο ∷ ὐ ∷ []) "1Cor.12.21"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ τ ∷ α ∷ ι ∷ []) "1Cor.12.21"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.21"
∷ word (ὁ ∷ []) "1Cor.12.21"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ὸ ∷ ς ∷ []) "1Cor.12.21"
∷ word (ε ∷ ἰ ∷ π ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.12.21"
∷ word (τ ∷ ῇ ∷ []) "1Cor.12.21"
∷ word (χ ∷ ε ∷ ι ∷ ρ ∷ ί ∷ []) "1Cor.12.21"
∷ word (Χ ∷ ρ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "1Cor.12.21"
∷ word (σ ∷ ο ∷ υ ∷ []) "1Cor.12.21"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.12.21"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "1Cor.12.21"
∷ word (ἢ ∷ []) "1Cor.12.21"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "1Cor.12.21"
∷ word (ἡ ∷ []) "1Cor.12.21"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ὴ ∷ []) "1Cor.12.21"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.12.21"
∷ word (π ∷ ο ∷ σ ∷ ί ∷ ν ∷ []) "1Cor.12.21"
∷ word (Χ ∷ ρ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "1Cor.12.21"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.12.21"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.12.21"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "1Cor.12.21"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.12.22"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῷ ∷ []) "1Cor.12.22"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.12.22"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.22"
∷ word (δ ∷ ο ∷ κ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ α ∷ []) "1Cor.12.22"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "1Cor.12.22"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.12.22"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.12.22"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ έ ∷ σ ∷ τ ∷ ε ∷ ρ ∷ α ∷ []) "1Cor.12.22"
∷ word (ὑ ∷ π ∷ ά ∷ ρ ∷ χ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.12.22"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ κ ∷ α ∷ ῖ ∷ ά ∷ []) "1Cor.12.22"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.12.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.23"
∷ word (ἃ ∷ []) "1Cor.12.23"
∷ word (δ ∷ ο ∷ κ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.12.23"
∷ word (ἀ ∷ τ ∷ ι ∷ μ ∷ ό ∷ τ ∷ ε ∷ ρ ∷ α ∷ []) "1Cor.12.23"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.12.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.12.23"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.12.23"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.12.23"
∷ word (τ ∷ ι ∷ μ ∷ ὴ ∷ ν ∷ []) "1Cor.12.23"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "1Cor.12.23"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ί ∷ θ ∷ ε ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.12.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.23"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.23"
∷ word (ἀ ∷ σ ∷ χ ∷ ή ∷ μ ∷ ο ∷ ν ∷ α ∷ []) "1Cor.12.23"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.12.23"
∷ word (ε ∷ ὐ ∷ σ ∷ χ ∷ η ∷ μ ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "1Cor.12.23"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "1Cor.12.23"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.12.23"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.24"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.24"
∷ word (ε ∷ ὐ ∷ σ ∷ χ ∷ ή ∷ μ ∷ ο ∷ ν ∷ α ∷ []) "1Cor.12.24"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.12.24"
∷ word (ο ∷ ὐ ∷ []) "1Cor.12.24"
∷ word (χ ∷ ρ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "1Cor.12.24"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.12.24"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.12.24"
∷ word (ὁ ∷ []) "1Cor.12.24"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.12.24"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ κ ∷ έ ∷ ρ ∷ α ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.12.24"
∷ word (τ ∷ ὸ ∷ []) "1Cor.12.24"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.12.24"
∷ word (τ ∷ ῷ ∷ []) "1Cor.12.24"
∷ word (ὑ ∷ σ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ι ∷ []) "1Cor.12.24"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "1Cor.12.24"
∷ word (δ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.12.24"
∷ word (τ ∷ ι ∷ μ ∷ ή ∷ ν ∷ []) "1Cor.12.24"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.12.25"
∷ word (μ ∷ ὴ ∷ []) "1Cor.12.25"
∷ word (ᾖ ∷ []) "1Cor.12.25"
∷ word (σ ∷ χ ∷ ί ∷ σ ∷ μ ∷ α ∷ []) "1Cor.12.25"
∷ word (ἐ ∷ ν ∷ []) "1Cor.12.25"
∷ word (τ ∷ ῷ ∷ []) "1Cor.12.25"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.12.25"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.12.25"
∷ word (τ ∷ ὸ ∷ []) "1Cor.12.25"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "1Cor.12.25"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "1Cor.12.25"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ω ∷ ν ∷ []) "1Cor.12.25"
∷ word (μ ∷ ε ∷ ρ ∷ ι ∷ μ ∷ ν ∷ ῶ ∷ σ ∷ ι ∷ []) "1Cor.12.25"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.25"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "1Cor.12.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.26"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.12.26"
∷ word (π ∷ ά ∷ σ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.12.26"
∷ word (ἓ ∷ ν ∷ []) "1Cor.12.26"
∷ word (μ ∷ έ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.12.26"
∷ word (σ ∷ υ ∷ μ ∷ π ∷ ά ∷ σ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.12.26"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.12.26"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.26"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "1Cor.12.26"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.12.26"
∷ word (δ ∷ ο ∷ ξ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.12.26"
∷ word (μ ∷ έ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.12.26"
∷ word (σ ∷ υ ∷ γ ∷ χ ∷ α ∷ ί ∷ ρ ∷ ε ∷ ι ∷ []) "1Cor.12.26"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.12.26"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.26"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "1Cor.12.26"
∷ word (Ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.12.27"
∷ word (δ ∷ έ ∷ []) "1Cor.12.27"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.12.27"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.12.27"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.12.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.27"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "1Cor.12.27"
∷ word (ἐ ∷ κ ∷ []) "1Cor.12.27"
∷ word (μ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.12.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.28"
∷ word (ο ∷ ὓ ∷ ς ∷ []) "1Cor.12.28"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.12.28"
∷ word (ἔ ∷ θ ∷ ε ∷ τ ∷ ο ∷ []) "1Cor.12.28"
∷ word (ὁ ∷ []) "1Cor.12.28"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.12.28"
∷ word (ἐ ∷ ν ∷ []) "1Cor.12.28"
∷ word (τ ∷ ῇ ∷ []) "1Cor.12.28"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.12.28"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.12.28"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.12.28"
∷ word (δ ∷ ε ∷ ύ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.12.28"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ ή ∷ τ ∷ α ∷ ς ∷ []) "1Cor.12.28"
∷ word (τ ∷ ρ ∷ ί ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.12.28"
∷ word (δ ∷ ι ∷ δ ∷ α ∷ σ ∷ κ ∷ ά ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.12.28"
∷ word (ἔ ∷ π ∷ ε ∷ ι ∷ τ ∷ α ∷ []) "1Cor.12.28"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.12.28"
∷ word (ἔ ∷ π ∷ ε ∷ ι ∷ τ ∷ α ∷ []) "1Cor.12.28"
∷ word (χ ∷ α ∷ ρ ∷ ί ∷ σ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.12.28"
∷ word (ἰ ∷ α ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.12.28"
∷ word (ἀ ∷ ν ∷ τ ∷ ι ∷ ∙λ ∷ ή ∷ μ ∷ ψ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.12.28"
∷ word (κ ∷ υ ∷ β ∷ ε ∷ ρ ∷ ν ∷ ή ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.12.28"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ []) "1Cor.12.28"
∷ word (γ ∷ ∙λ ∷ ω ∷ σ ∷ σ ∷ ῶ ∷ ν ∷ []) "1Cor.12.28"
∷ word (μ ∷ ὴ ∷ []) "1Cor.12.29"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.12.29"
∷ word (ἀ ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ι ∷ []) "1Cor.12.29"
∷ word (μ ∷ ὴ ∷ []) "1Cor.12.29"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.12.29"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ ῆ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.12.29"
∷ word (μ ∷ ὴ ∷ []) "1Cor.12.29"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.12.29"
∷ word (δ ∷ ι ∷ δ ∷ ά ∷ σ ∷ κ ∷ α ∷ ∙λ ∷ ο ∷ ι ∷ []) "1Cor.12.29"
∷ word (μ ∷ ὴ ∷ []) "1Cor.12.29"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.12.29"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.12.29"
∷ word (μ ∷ ὴ ∷ []) "1Cor.12.30"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.12.30"
∷ word (χ ∷ α ∷ ρ ∷ ί ∷ σ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.12.30"
∷ word (ἔ ∷ χ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.12.30"
∷ word (ἰ ∷ α ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.12.30"
∷ word (μ ∷ ὴ ∷ []) "1Cor.12.30"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.12.30"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.12.30"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.12.30"
∷ word (μ ∷ ὴ ∷ []) "1Cor.12.30"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.12.30"
∷ word (δ ∷ ι ∷ ε ∷ ρ ∷ μ ∷ η ∷ ν ∷ ε ∷ ύ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.12.30"
∷ word (ζ ∷ η ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ε ∷ []) "1Cor.12.31"
∷ word (δ ∷ ὲ ∷ []) "1Cor.12.31"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.31"
∷ word (χ ∷ α ∷ ρ ∷ ί ∷ σ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.12.31"
∷ word (τ ∷ ὰ ∷ []) "1Cor.12.31"
∷ word (μ ∷ ε ∷ ί ∷ ζ ∷ ο ∷ ν ∷ α ∷ []) "1Cor.12.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.12.31"
∷ word (ἔ ∷ τ ∷ ι ∷ []) "1Cor.12.31"
∷ word (κ ∷ α ∷ θ ∷ []) "1Cor.12.31"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "1Cor.12.31"
∷ word (ὁ ∷ δ ∷ ὸ ∷ ν ∷ []) "1Cor.12.31"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.12.31"
∷ word (δ ∷ ε ∷ ί ∷ κ ∷ ν ∷ υ ∷ μ ∷ ι ∷ []) "1Cor.12.31"
∷ word (Ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.13.1"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "1Cor.13.1"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.13.1"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.13.1"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "1Cor.13.1"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "1Cor.13.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.13.1"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.13.1"
∷ word (ἀ ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ω ∷ ν ∷ []) "1Cor.13.1"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "1Cor.13.1"
∷ word (δ ∷ ὲ ∷ []) "1Cor.13.1"
∷ word (μ ∷ ὴ ∷ []) "1Cor.13.1"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "1Cor.13.1"
∷ word (γ ∷ έ ∷ γ ∷ ο ∷ ν ∷ α ∷ []) "1Cor.13.1"
∷ word (χ ∷ α ∷ ∙λ ∷ κ ∷ ὸ ∷ ς ∷ []) "1Cor.13.1"
∷ word (ἠ ∷ χ ∷ ῶ ∷ ν ∷ []) "1Cor.13.1"
∷ word (ἢ ∷ []) "1Cor.13.1"
∷ word (κ ∷ ύ ∷ μ ∷ β ∷ α ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.13.1"
∷ word (ἀ ∷ ∙λ ∷ α ∷ ∙λ ∷ ά ∷ ζ ∷ ο ∷ ν ∷ []) "1Cor.13.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.13.2"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.13.2"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "1Cor.13.2"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "1Cor.13.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.13.2"
∷ word (ε ∷ ἰ ∷ δ ∷ ῶ ∷ []) "1Cor.13.2"
∷ word (τ ∷ ὰ ∷ []) "1Cor.13.2"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ ή ∷ ρ ∷ ι ∷ α ∷ []) "1Cor.13.2"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.13.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.13.2"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "1Cor.13.2"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.13.2"
∷ word (γ ∷ ν ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.13.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.13.2"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.13.2"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "1Cor.13.2"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "1Cor.13.2"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.13.2"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.13.2"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.13.2"
∷ word (ὄ ∷ ρ ∷ η ∷ []) "1Cor.13.2"
∷ word (μ ∷ ε ∷ θ ∷ ι ∷ σ ∷ τ ∷ ά ∷ ν ∷ α ∷ ι ∷ []) "1Cor.13.2"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "1Cor.13.2"
∷ word (δ ∷ ὲ ∷ []) "1Cor.13.2"
∷ word (μ ∷ ὴ ∷ []) "1Cor.13.2"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "1Cor.13.2"
∷ word (ο ∷ ὐ ∷ θ ∷ έ ∷ ν ∷ []) "1Cor.13.2"
∷ word (ε ∷ ἰ ∷ μ ∷ ι ∷ []) "1Cor.13.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.13.3"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.13.3"
∷ word (ψ ∷ ω ∷ μ ∷ ί ∷ σ ∷ ω ∷ []) "1Cor.13.3"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.13.3"
∷ word (τ ∷ ὰ ∷ []) "1Cor.13.3"
∷ word (ὑ ∷ π ∷ ά ∷ ρ ∷ χ ∷ ο ∷ ν ∷ τ ∷ ά ∷ []) "1Cor.13.3"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.13.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.13.3"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.13.3"
∷ word (π ∷ α ∷ ρ ∷ α ∷ δ ∷ ῶ ∷ []) "1Cor.13.3"
∷ word (τ ∷ ὸ ∷ []) "1Cor.13.3"
∷ word (σ ∷ ῶ ∷ μ ∷ ά ∷ []) "1Cor.13.3"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.13.3"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.13.3"
∷ word (κ ∷ α ∷ υ ∷ θ ∷ ή ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.13.3"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "1Cor.13.3"
∷ word (δ ∷ ὲ ∷ []) "1Cor.13.3"
∷ word (μ ∷ ὴ ∷ []) "1Cor.13.3"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "1Cor.13.3"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ ν ∷ []) "1Cor.13.3"
∷ word (ὠ ∷ φ ∷ ε ∷ ∙λ ∷ ο ∷ ῦ ∷ μ ∷ α ∷ ι ∷ []) "1Cor.13.3"
∷ word (Ἡ ∷ []) "1Cor.13.4"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1Cor.13.4"
∷ word (μ ∷ α ∷ κ ∷ ρ ∷ ο ∷ θ ∷ υ ∷ μ ∷ ε ∷ ῖ ∷ []) "1Cor.13.4"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.13.4"
∷ word (ἡ ∷ []) "1Cor.13.4"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1Cor.13.4"
∷ word (ο ∷ ὐ ∷ []) "1Cor.13.4"
∷ word (ζ ∷ η ∷ ∙λ ∷ ο ∷ ῖ ∷ []) "1Cor.13.4"
∷ word (ἡ ∷ []) "1Cor.13.4"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1Cor.13.4"
∷ word (ο ∷ ὐ ∷ []) "1Cor.13.4"
∷ word (π ∷ ε ∷ ρ ∷ π ∷ ε ∷ ρ ∷ ε ∷ ύ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.13.4"
∷ word (ο ∷ ὐ ∷ []) "1Cor.13.4"
∷ word (φ ∷ υ ∷ σ ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.13.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.13.5"
∷ word (ἀ ∷ σ ∷ χ ∷ η ∷ μ ∷ ο ∷ ν ∷ ε ∷ ῖ ∷ []) "1Cor.13.5"
∷ word (ο ∷ ὐ ∷ []) "1Cor.13.5"
∷ word (ζ ∷ η ∷ τ ∷ ε ∷ ῖ ∷ []) "1Cor.13.5"
∷ word (τ ∷ ὰ ∷ []) "1Cor.13.5"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῆ ∷ ς ∷ []) "1Cor.13.5"
∷ word (ο ∷ ὐ ∷ []) "1Cor.13.5"
∷ word (π ∷ α ∷ ρ ∷ ο ∷ ξ ∷ ύ ∷ ν ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.13.5"
∷ word (ο ∷ ὐ ∷ []) "1Cor.13.5"
∷ word (∙λ ∷ ο ∷ γ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.13.5"
∷ word (τ ∷ ὸ ∷ []) "1Cor.13.5"
∷ word (κ ∷ α ∷ κ ∷ ό ∷ ν ∷ []) "1Cor.13.5"
∷ word (ο ∷ ὐ ∷ []) "1Cor.13.6"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ε ∷ ι ∷ []) "1Cor.13.6"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.13.6"
∷ word (τ ∷ ῇ ∷ []) "1Cor.13.6"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ί ∷ ᾳ ∷ []) "1Cor.13.6"
∷ word (σ ∷ υ ∷ γ ∷ χ ∷ α ∷ ί ∷ ρ ∷ ε ∷ ι ∷ []) "1Cor.13.6"
∷ word (δ ∷ ὲ ∷ []) "1Cor.13.6"
∷ word (τ ∷ ῇ ∷ []) "1Cor.13.6"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ ᾳ ∷ []) "1Cor.13.6"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.13.7"
∷ word (σ ∷ τ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "1Cor.13.7"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.13.7"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ []) "1Cor.13.7"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.13.7"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ ζ ∷ ε ∷ ι ∷ []) "1Cor.13.7"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.13.7"
∷ word (ὑ ∷ π ∷ ο ∷ μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1Cor.13.7"
∷ word (Ἡ ∷ []) "1Cor.13.8"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1Cor.13.8"
∷ word (ο ∷ ὐ ∷ δ ∷ έ ∷ π ∷ ο ∷ τ ∷ ε ∷ []) "1Cor.13.8"
∷ word (π ∷ ί ∷ π ∷ τ ∷ ε ∷ ι ∷ []) "1Cor.13.8"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.13.8"
∷ word (δ ∷ ὲ ∷ []) "1Cor.13.8"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ῖ ∷ α ∷ ι ∷ []) "1Cor.13.8"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ η ∷ θ ∷ ή ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.13.8"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.13.8"
∷ word (γ ∷ ∙λ ∷ ῶ ∷ σ ∷ σ ∷ α ∷ ι ∷ []) "1Cor.13.8"
∷ word (π ∷ α ∷ ύ ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.13.8"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.13.8"
∷ word (γ ∷ ν ∷ ῶ ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.13.8"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ η ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.13.8"
∷ word (ἐ ∷ κ ∷ []) "1Cor.13.9"
∷ word (μ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.13.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.13.9"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.13.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.13.9"
∷ word (ἐ ∷ κ ∷ []) "1Cor.13.9"
∷ word (μ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.13.9"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ύ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.13.9"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "1Cor.13.10"
∷ word (δ ∷ ὲ ∷ []) "1Cor.13.10"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ῃ ∷ []) "1Cor.13.10"
∷ word (τ ∷ ὸ ∷ []) "1Cor.13.10"
∷ word (τ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.13.10"
∷ word (τ ∷ ὸ ∷ []) "1Cor.13.10"
∷ word (ἐ ∷ κ ∷ []) "1Cor.13.10"
∷ word (μ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.13.10"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ η ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.13.10"
∷ word (ὅ ∷ τ ∷ ε ∷ []) "1Cor.13.11"
∷ word (ἤ ∷ μ ∷ η ∷ ν ∷ []) "1Cor.13.11"
∷ word (ν ∷ ή ∷ π ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.13.11"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ ∙λ ∷ ο ∷ υ ∷ ν ∷ []) "1Cor.13.11"
∷ word (ὡ ∷ ς ∷ []) "1Cor.13.11"
∷ word (ν ∷ ή ∷ π ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.13.11"
∷ word (ἐ ∷ φ ∷ ρ ∷ ό ∷ ν ∷ ο ∷ υ ∷ ν ∷ []) "1Cor.13.11"
∷ word (ὡ ∷ ς ∷ []) "1Cor.13.11"
∷ word (ν ∷ ή ∷ π ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.13.11"
∷ word (ἐ ∷ ∙λ ∷ ο ∷ γ ∷ ι ∷ ζ ∷ ό ∷ μ ∷ η ∷ ν ∷ []) "1Cor.13.11"
∷ word (ὡ ∷ ς ∷ []) "1Cor.13.11"
∷ word (ν ∷ ή ∷ π ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.13.11"
∷ word (ὅ ∷ τ ∷ ε ∷ []) "1Cor.13.11"
∷ word (γ ∷ έ ∷ γ ∷ ο ∷ ν ∷ α ∷ []) "1Cor.13.11"
∷ word (ἀ ∷ ν ∷ ή ∷ ρ ∷ []) "1Cor.13.11"
∷ word (κ ∷ α ∷ τ ∷ ή ∷ ρ ∷ γ ∷ η ∷ κ ∷ α ∷ []) "1Cor.13.11"
∷ word (τ ∷ ὰ ∷ []) "1Cor.13.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.13.11"
∷ word (ν ∷ η ∷ π ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.13.11"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.13.12"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.13.12"
∷ word (ἄ ∷ ρ ∷ τ ∷ ι ∷ []) "1Cor.13.12"
∷ word (δ ∷ ι ∷ []) "1Cor.13.12"
∷ word (ἐ ∷ σ ∷ ό ∷ π ∷ τ ∷ ρ ∷ ο ∷ υ ∷ []) "1Cor.13.12"
∷ word (ἐ ∷ ν ∷ []) "1Cor.13.12"
∷ word (α ∷ ἰ ∷ ν ∷ ί ∷ γ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.13.12"
∷ word (τ ∷ ό ∷ τ ∷ ε ∷ []) "1Cor.13.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.13.12"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "1Cor.13.12"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.13.12"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "1Cor.13.12"
∷ word (ἄ ∷ ρ ∷ τ ∷ ι ∷ []) "1Cor.13.12"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ω ∷ []) "1Cor.13.12"
∷ word (ἐ ∷ κ ∷ []) "1Cor.13.12"
∷ word (μ ∷ έ ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.13.12"
∷ word (τ ∷ ό ∷ τ ∷ ε ∷ []) "1Cor.13.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.13.12"
∷ word (ἐ ∷ π ∷ ι ∷ γ ∷ ν ∷ ώ ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.13.12"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.13.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.13.12"
∷ word (ἐ ∷ π ∷ ε ∷ γ ∷ ν ∷ ώ ∷ σ ∷ θ ∷ η ∷ ν ∷ []) "1Cor.13.12"
∷ word (ν ∷ υ ∷ ν ∷ ὶ ∷ []) "1Cor.13.13"
∷ word (δ ∷ ὲ ∷ []) "1Cor.13.13"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1Cor.13.13"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "1Cor.13.13"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ ς ∷ []) "1Cor.13.13"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1Cor.13.13"
∷ word (τ ∷ ὰ ∷ []) "1Cor.13.13"
∷ word (τ ∷ ρ ∷ ί ∷ α ∷ []) "1Cor.13.13"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1Cor.13.13"
∷ word (μ ∷ ε ∷ ί ∷ ζ ∷ ω ∷ ν ∷ []) "1Cor.13.13"
∷ word (δ ∷ ὲ ∷ []) "1Cor.13.13"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.13.13"
∷ word (ἡ ∷ []) "1Cor.13.13"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1Cor.13.13"
∷ word (Δ ∷ ι ∷ ώ ∷ κ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.14.1"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.14.1"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "1Cor.14.1"
∷ word (ζ ∷ η ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ε ∷ []) "1Cor.14.1"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.1"
∷ word (τ ∷ ὰ ∷ []) "1Cor.14.1"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ά ∷ []) "1Cor.14.1"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.14.1"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.1"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.14.1"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ύ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.14.1"
∷ word (ὁ ∷ []) "1Cor.14.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.14.2"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "1Cor.14.2"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ ῃ ∷ []) "1Cor.14.2"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.14.2"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.14.2"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ []) "1Cor.14.2"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.14.2"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1Cor.14.2"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1Cor.14.2"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.14.2"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ ε ∷ ι ∷ []) "1Cor.14.2"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.14.2"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.2"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ []) "1Cor.14.2"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ ή ∷ ρ ∷ ι ∷ α ∷ []) "1Cor.14.2"
∷ word (ὁ ∷ []) "1Cor.14.3"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.3"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "1Cor.14.3"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.14.3"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ []) "1Cor.14.3"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "1Cor.14.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.3"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.14.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.3"
∷ word (π ∷ α ∷ ρ ∷ α ∷ μ ∷ υ ∷ θ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.14.3"
∷ word (ὁ ∷ []) "1Cor.14.4"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "1Cor.14.4"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ ῃ ∷ []) "1Cor.14.4"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.14.4"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ε ∷ ῖ ∷ []) "1Cor.14.4"
∷ word (ὁ ∷ []) "1Cor.14.4"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.4"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "1Cor.14.4"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.14.4"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ε ∷ ῖ ∷ []) "1Cor.14.4"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "1Cor.14.5"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.5"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "1Cor.14.5"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.14.5"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.14.5"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.14.5"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.14.5"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.5"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.14.5"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ύ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.14.5"
∷ word (μ ∷ ε ∷ ί ∷ ζ ∷ ω ∷ ν ∷ []) "1Cor.14.5"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.5"
∷ word (ὁ ∷ []) "1Cor.14.5"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "1Cor.14.5"
∷ word (ἢ ∷ []) "1Cor.14.5"
∷ word (ὁ ∷ []) "1Cor.14.5"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "1Cor.14.5"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.14.5"
∷ word (ἐ ∷ κ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.14.5"
∷ word (ε ∷ ἰ ∷ []) "1Cor.14.5"
∷ word (μ ∷ ὴ ∷ []) "1Cor.14.5"
∷ word (δ ∷ ι ∷ ε ∷ ρ ∷ μ ∷ η ∷ ν ∷ ε ∷ ύ ∷ ῃ ∷ []) "1Cor.14.5"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.14.5"
∷ word (ἡ ∷ []) "1Cor.14.5"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ []) "1Cor.14.5"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "1Cor.14.5"
∷ word (∙λ ∷ ά ∷ β ∷ ῃ ∷ []) "1Cor.14.5"
∷ word (Ν ∷ ῦ ∷ ν ∷ []) "1Cor.14.6"
∷ word (δ ∷ έ ∷ []) "1Cor.14.6"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.14.6"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.14.6"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ω ∷ []) "1Cor.14.6"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.14.6"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.14.6"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.14.6"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "1Cor.14.6"
∷ word (τ ∷ ί ∷ []) "1Cor.14.6"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.14.6"
∷ word (ὠ ∷ φ ∷ ε ∷ ∙λ ∷ ή ∷ σ ∷ ω ∷ []) "1Cor.14.6"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.14.6"
∷ word (μ ∷ ὴ ∷ []) "1Cor.14.6"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.14.6"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ή ∷ σ ∷ ω ∷ []) "1Cor.14.6"
∷ word (ἢ ∷ []) "1Cor.14.6"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.6"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ ψ ∷ ε ∷ ι ∷ []) "1Cor.14.6"
∷ word (ἢ ∷ []) "1Cor.14.6"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.6"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.14.6"
∷ word (ἢ ∷ []) "1Cor.14.6"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.6"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ί ∷ ᾳ ∷ []) "1Cor.14.6"
∷ word (ἢ ∷ []) "1Cor.14.6"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.6"
∷ word (δ ∷ ι ∷ δ ∷ α ∷ χ ∷ ῇ ∷ []) "1Cor.14.6"
∷ word (ὅ ∷ μ ∷ ω ∷ ς ∷ []) "1Cor.14.7"
∷ word (τ ∷ ὰ ∷ []) "1Cor.14.7"
∷ word (ἄ ∷ ψ ∷ υ ∷ χ ∷ α ∷ []) "1Cor.14.7"
∷ word (φ ∷ ω ∷ ν ∷ ὴ ∷ ν ∷ []) "1Cor.14.7"
∷ word (δ ∷ ι ∷ δ ∷ ό ∷ ν ∷ τ ∷ α ∷ []) "1Cor.14.7"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.14.7"
∷ word (α ∷ ὐ ∷ ∙λ ∷ ὸ ∷ ς ∷ []) "1Cor.14.7"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.14.7"
∷ word (κ ∷ ι ∷ θ ∷ ά ∷ ρ ∷ α ∷ []) "1Cor.14.7"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.14.7"
∷ word (δ ∷ ι ∷ α ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "1Cor.14.7"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.14.7"
∷ word (φ ∷ θ ∷ ό ∷ γ ∷ γ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.14.7"
∷ word (μ ∷ ὴ ∷ []) "1Cor.14.7"
∷ word (δ ∷ ῷ ∷ []) "1Cor.14.7"
∷ word (π ∷ ῶ ∷ ς ∷ []) "1Cor.14.7"
∷ word (γ ∷ ν ∷ ω ∷ σ ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.7"
∷ word (τ ∷ ὸ ∷ []) "1Cor.14.7"
∷ word (α ∷ ὐ ∷ ∙λ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.14.7"
∷ word (ἢ ∷ []) "1Cor.14.7"
∷ word (τ ∷ ὸ ∷ []) "1Cor.14.7"
∷ word (κ ∷ ι ∷ θ ∷ α ∷ ρ ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.14.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.8"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.14.8"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.14.8"
∷ word (ἄ ∷ δ ∷ η ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.14.8"
∷ word (φ ∷ ω ∷ ν ∷ ὴ ∷ ν ∷ []) "1Cor.14.8"
∷ word (σ ∷ ά ∷ ∙λ ∷ π ∷ ι ∷ γ ∷ ξ ∷ []) "1Cor.14.8"
∷ word (δ ∷ ῷ ∷ []) "1Cor.14.8"
∷ word (τ ∷ ί ∷ ς ∷ []) "1Cor.14.8"
∷ word (π ∷ α ∷ ρ ∷ α ∷ σ ∷ κ ∷ ε ∷ υ ∷ ά ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.14.8"
∷ word (π ∷ ό ∷ ∙λ ∷ ε ∷ μ ∷ ο ∷ ν ∷ []) "1Cor.14.8"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.14.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.9"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.14.9"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.14.9"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.14.9"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ η ∷ ς ∷ []) "1Cor.14.9"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.14.9"
∷ word (μ ∷ ὴ ∷ []) "1Cor.14.9"
∷ word (ε ∷ ὔ ∷ σ ∷ η ∷ μ ∷ ο ∷ ν ∷ []) "1Cor.14.9"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ν ∷ []) "1Cor.14.9"
∷ word (δ ∷ ῶ ∷ τ ∷ ε ∷ []) "1Cor.14.9"
∷ word (π ∷ ῶ ∷ ς ∷ []) "1Cor.14.9"
∷ word (γ ∷ ν ∷ ω ∷ σ ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.9"
∷ word (τ ∷ ὸ ∷ []) "1Cor.14.9"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.14.9"
∷ word (ἔ ∷ σ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.14.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.14.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.14.9"
∷ word (ἀ ∷ έ ∷ ρ ∷ α ∷ []) "1Cor.14.9"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.14.9"
∷ word (τ ∷ ο ∷ σ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1Cor.14.10"
∷ word (ε ∷ ἰ ∷ []) "1Cor.14.10"
∷ word (τ ∷ ύ ∷ χ ∷ ο ∷ ι ∷ []) "1Cor.14.10"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ []) "1Cor.14.10"
∷ word (φ ∷ ω ∷ ν ∷ ῶ ∷ ν ∷ []) "1Cor.14.10"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.14.10"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.10"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "1Cor.14.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.10"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ ν ∷ []) "1Cor.14.10"
∷ word (ἄ ∷ φ ∷ ω ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.14.10"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.14.11"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.14.11"
∷ word (μ ∷ ὴ ∷ []) "1Cor.14.11"
∷ word (ε ∷ ἰ ∷ δ ∷ ῶ ∷ []) "1Cor.14.11"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.14.11"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ν ∷ []) "1Cor.14.11"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.14.11"
∷ word (φ ∷ ω ∷ ν ∷ ῆ ∷ ς ∷ []) "1Cor.14.11"
∷ word (ἔ ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.14.11"
∷ word (τ ∷ ῷ ∷ []) "1Cor.14.11"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ι ∷ []) "1Cor.14.11"
∷ word (β ∷ ά ∷ ρ ∷ β ∷ α ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.14.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.11"
∷ word (ὁ ∷ []) "1Cor.14.11"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "1Cor.14.11"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.11"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "1Cor.14.11"
∷ word (β ∷ ά ∷ ρ ∷ β ∷ α ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.14.11"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.14.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.12"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.14.12"
∷ word (ἐ ∷ π ∷ ε ∷ ὶ ∷ []) "1Cor.14.12"
∷ word (ζ ∷ η ∷ ∙λ ∷ ω ∷ τ ∷ α ∷ ί ∷ []) "1Cor.14.12"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.14.12"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.14.12"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.14.12"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.14.12"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "1Cor.14.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.14.12"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.14.12"
∷ word (ζ ∷ η ∷ τ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "1Cor.14.12"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.14.12"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.14.12"
∷ word (Δ ∷ ι ∷ ὸ ∷ []) "1Cor.14.13"
∷ word (ὁ ∷ []) "1Cor.14.13"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "1Cor.14.13"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ ῃ ∷ []) "1Cor.14.13"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "1Cor.14.13"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.14.13"
∷ word (δ ∷ ι ∷ ε ∷ ρ ∷ μ ∷ η ∷ ν ∷ ε ∷ ύ ∷ ῃ ∷ []) "1Cor.14.13"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.14.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.14.14"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ ύ ∷ χ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "1Cor.14.14"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ ῃ ∷ []) "1Cor.14.14"
∷ word (τ ∷ ὸ ∷ []) "1Cor.14.14"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ ά ∷ []) "1Cor.14.14"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.14.14"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ ύ ∷ χ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.14"
∷ word (ὁ ∷ []) "1Cor.14.14"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.14"
∷ word (ν ∷ ο ∷ ῦ ∷ ς ∷ []) "1Cor.14.14"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.14.14"
∷ word (ἄ ∷ κ ∷ α ∷ ρ ∷ π ∷ ό ∷ ς ∷ []) "1Cor.14.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.14.14"
∷ word (τ ∷ ί ∷ []) "1Cor.14.15"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.14.15"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.14.15"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ ύ ∷ ξ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.14.15"
∷ word (τ ∷ ῷ ∷ []) "1Cor.14.15"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.14.15"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ ύ ∷ ξ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.14.15"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.15"
∷ word (τ ∷ ῷ ∷ []) "1Cor.14.15"
∷ word (ν ∷ ο ∷ ΐ ∷ []) "1Cor.14.15"
∷ word (ψ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "1Cor.14.15"
∷ word (τ ∷ ῷ ∷ []) "1Cor.14.15"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.14.15"
∷ word (ψ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "1Cor.14.15"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.15"
∷ word (τ ∷ ῷ ∷ []) "1Cor.14.15"
∷ word (ν ∷ ο ∷ ΐ ∷ []) "1Cor.14.15"
∷ word (ἐ ∷ π ∷ ε ∷ ὶ ∷ []) "1Cor.14.16"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.14.16"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ῇ ∷ ς ∷ []) "1Cor.14.16"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.14.16"
∷ word (ὁ ∷ []) "1Cor.14.16"
∷ word (ἀ ∷ ν ∷ α ∷ π ∷ ∙λ ∷ η ∷ ρ ∷ ῶ ∷ ν ∷ []) "1Cor.14.16"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.14.16"
∷ word (τ ∷ ό ∷ π ∷ ο ∷ ν ∷ []) "1Cor.14.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.14.16"
∷ word (ἰ ∷ δ ∷ ι ∷ ώ ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.14.16"
∷ word (π ∷ ῶ ∷ ς ∷ []) "1Cor.14.16"
∷ word (ἐ ∷ ρ ∷ ε ∷ ῖ ∷ []) "1Cor.14.16"
∷ word (τ ∷ ὸ ∷ []) "1Cor.14.16"
∷ word (Ἀ ∷ μ ∷ ή ∷ ν ∷ []) "1Cor.14.16"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.14.16"
∷ word (τ ∷ ῇ ∷ []) "1Cor.14.16"
∷ word (σ ∷ ῇ ∷ []) "1Cor.14.16"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ί ∷ ᾳ ∷ []) "1Cor.14.16"
∷ word (ἐ ∷ π ∷ ε ∷ ι ∷ δ ∷ ὴ ∷ []) "1Cor.14.16"
∷ word (τ ∷ ί ∷ []) "1Cor.14.16"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.14.16"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.14.16"
∷ word (ο ∷ ἶ ∷ δ ∷ ε ∷ ν ∷ []) "1Cor.14.16"
∷ word (σ ∷ ὺ ∷ []) "1Cor.14.17"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.14.17"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.14.17"
∷ word (κ ∷ α ∷ ∙λ ∷ ῶ ∷ ς ∷ []) "1Cor.14.17"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.14.17"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.14.17"
∷ word (ὁ ∷ []) "1Cor.14.17"
∷ word (ἕ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.14.17"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.14.17"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.17"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῶ ∷ []) "1Cor.14.18"
∷ word (τ ∷ ῷ ∷ []) "1Cor.14.18"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1Cor.14.18"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.14.18"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.14.18"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.14.18"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.14.18"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "1Cor.14.18"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.14.19"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.19"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.14.19"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "1Cor.14.19"
∷ word (π ∷ έ ∷ ν ∷ τ ∷ ε ∷ []) "1Cor.14.19"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.14.19"
∷ word (τ ∷ ῷ ∷ []) "1Cor.14.19"
∷ word (ν ∷ ο ∷ ΐ ∷ []) "1Cor.14.19"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.14.19"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "1Cor.14.19"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.14.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.19"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.14.19"
∷ word (κ ∷ α ∷ τ ∷ η ∷ χ ∷ ή ∷ σ ∷ ω ∷ []) "1Cor.14.19"
∷ word (ἢ ∷ []) "1Cor.14.19"
∷ word (μ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.14.19"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.14.19"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.19"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ ῃ ∷ []) "1Cor.14.19"
∷ word (Ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.14.20"
∷ word (μ ∷ ὴ ∷ []) "1Cor.14.20"
∷ word (π ∷ α ∷ ι ∷ δ ∷ ί ∷ α ∷ []) "1Cor.14.20"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.14.20"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "1Cor.14.20"
∷ word (φ ∷ ρ ∷ ε ∷ σ ∷ ί ∷ ν ∷ []) "1Cor.14.20"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.14.20"
∷ word (τ ∷ ῇ ∷ []) "1Cor.14.20"
∷ word (κ ∷ α ∷ κ ∷ ί ∷ ᾳ ∷ []) "1Cor.14.20"
∷ word (ν ∷ η ∷ π ∷ ι ∷ ά ∷ ζ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.14.20"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "1Cor.14.20"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.20"
∷ word (φ ∷ ρ ∷ ε ∷ σ ∷ ὶ ∷ ν ∷ []) "1Cor.14.20"
∷ word (τ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ ο ∷ ι ∷ []) "1Cor.14.20"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.14.20"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.21"
∷ word (τ ∷ ῷ ∷ []) "1Cor.14.21"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "1Cor.14.21"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.21"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.14.21"
∷ word (Ἐ ∷ ν ∷ []) "1Cor.14.21"
∷ word (ἑ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.14.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.21"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.21"
∷ word (χ ∷ ε ∷ ί ∷ ∙λ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.14.21"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ ω ∷ ν ∷ []) "1Cor.14.21"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ή ∷ σ ∷ ω ∷ []) "1Cor.14.21"
∷ word (τ ∷ ῷ ∷ []) "1Cor.14.21"
∷ word (∙λ ∷ α ∷ ῷ ∷ []) "1Cor.14.21"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1Cor.14.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.21"
∷ word (ο ∷ ὐ ∷ δ ∷ []) "1Cor.14.21"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.14.21"
∷ word (ε ∷ ἰ ∷ σ ∷ α ∷ κ ∷ ο ∷ ύ ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ί ∷ []) "1Cor.14.21"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.14.21"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "1Cor.14.21"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.14.21"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.14.22"
∷ word (α ∷ ἱ ∷ []) "1Cor.14.22"
∷ word (γ ∷ ∙λ ∷ ῶ ∷ σ ∷ σ ∷ α ∷ ι ∷ []) "1Cor.14.22"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.14.22"
∷ word (σ ∷ η ∷ μ ∷ ε ∷ ῖ ∷ ό ∷ ν ∷ []) "1Cor.14.22"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.14.22"
∷ word (ο ∷ ὐ ∷ []) "1Cor.14.22"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.14.22"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.14.22"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.14.22"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.14.22"
∷ word (ἀ ∷ π ∷ ί ∷ σ ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.14.22"
∷ word (ἡ ∷ []) "1Cor.14.22"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.22"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ί ∷ α ∷ []) "1Cor.14.22"
∷ word (ο ∷ ὐ ∷ []) "1Cor.14.22"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.14.22"
∷ word (ἀ ∷ π ∷ ί ∷ σ ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.14.22"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.14.22"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.14.22"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.14.22"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.14.23"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.14.23"
∷ word (σ ∷ υ ∷ ν ∷ έ ∷ ∙λ ∷ θ ∷ ῃ ∷ []) "1Cor.14.23"
∷ word (ἡ ∷ []) "1Cor.14.23"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ []) "1Cor.14.23"
∷ word (ὅ ∷ ∙λ ∷ η ∷ []) "1Cor.14.23"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.14.23"
∷ word (τ ∷ ὸ ∷ []) "1Cor.14.23"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "1Cor.14.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.23"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.14.23"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.14.23"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.14.23"
∷ word (ε ∷ ἰ ∷ σ ∷ έ ∷ ∙λ ∷ θ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.14.23"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.23"
∷ word (ἰ ∷ δ ∷ ι ∷ ῶ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.23"
∷ word (ἢ ∷ []) "1Cor.14.23"
∷ word (ἄ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ι ∷ []) "1Cor.14.23"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.14.23"
∷ word (ἐ ∷ ρ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.14.23"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.14.23"
∷ word (μ ∷ α ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.14.23"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.14.24"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.24"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.14.24"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ύ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.14.24"
∷ word (ε ∷ ἰ ∷ σ ∷ έ ∷ ∙λ ∷ θ ∷ ῃ ∷ []) "1Cor.14.24"
∷ word (δ ∷ έ ∷ []) "1Cor.14.24"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.14.24"
∷ word (ἄ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.14.24"
∷ word (ἢ ∷ []) "1Cor.14.24"
∷ word (ἰ ∷ δ ∷ ι ∷ ώ ∷ τ ∷ η ∷ ς ∷ []) "1Cor.14.24"
∷ word (ἐ ∷ ∙λ ∷ έ ∷ γ ∷ χ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.24"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.14.24"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.14.24"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.24"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.14.24"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.14.24"
∷ word (τ ∷ ὰ ∷ []) "1Cor.14.25"
∷ word (κ ∷ ρ ∷ υ ∷ π ∷ τ ∷ ὰ ∷ []) "1Cor.14.25"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.14.25"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.14.25"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.14.25"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ὰ ∷ []) "1Cor.14.25"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.25"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.14.25"
∷ word (π ∷ ε ∷ σ ∷ ὼ ∷ ν ∷ []) "1Cor.14.25"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.14.25"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "1Cor.14.25"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ κ ∷ υ ∷ ν ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.14.25"
∷ word (τ ∷ ῷ ∷ []) "1Cor.14.25"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1Cor.14.25"
∷ word (ἀ ∷ π ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ω ∷ ν ∷ []) "1Cor.14.25"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.14.25"
∷ word (Ὄ ∷ ν ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.14.25"
∷ word (ὁ ∷ []) "1Cor.14.25"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.14.25"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.25"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.14.25"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.14.25"
∷ word (Τ ∷ ί ∷ []) "1Cor.14.26"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.14.26"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.14.26"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.14.26"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "1Cor.14.26"
∷ word (σ ∷ υ ∷ ν ∷ έ ∷ ρ ∷ χ ∷ η ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.14.26"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.14.26"
∷ word (ψ ∷ α ∷ ∙λ ∷ μ ∷ ὸ ∷ ν ∷ []) "1Cor.14.26"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.14.26"
∷ word (δ ∷ ι ∷ δ ∷ α ∷ χ ∷ ὴ ∷ ν ∷ []) "1Cor.14.26"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.14.26"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ ά ∷ ∙λ ∷ υ ∷ ψ ∷ ι ∷ ν ∷ []) "1Cor.14.26"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.14.26"
∷ word (γ ∷ ∙λ ∷ ῶ ∷ σ ∷ σ ∷ α ∷ ν ∷ []) "1Cor.14.26"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.14.26"
∷ word (ἑ ∷ ρ ∷ μ ∷ η ∷ ν ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "1Cor.14.26"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1Cor.14.26"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.14.26"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.14.26"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "1Cor.14.26"
∷ word (γ ∷ ι ∷ ν ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "1Cor.14.26"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.14.27"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ ῃ ∷ []) "1Cor.14.27"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.14.27"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ []) "1Cor.14.27"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.14.27"
∷ word (δ ∷ ύ ∷ ο ∷ []) "1Cor.14.27"
∷ word (ἢ ∷ []) "1Cor.14.27"
∷ word (τ ∷ ὸ ∷ []) "1Cor.14.27"
∷ word (π ∷ ∙λ ∷ ε ∷ ῖ ∷ σ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.14.27"
∷ word (τ ∷ ρ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.14.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.27"
∷ word (ἀ ∷ ν ∷ ὰ ∷ []) "1Cor.14.27"
∷ word (μ ∷ έ ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.14.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.27"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "1Cor.14.27"
∷ word (δ ∷ ι ∷ ε ∷ ρ ∷ μ ∷ η ∷ ν ∷ ε ∷ υ ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.14.27"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.14.28"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.28"
∷ word (μ ∷ ὴ ∷ []) "1Cor.14.28"
∷ word (ᾖ ∷ []) "1Cor.14.28"
∷ word (δ ∷ ι ∷ ε ∷ ρ ∷ μ ∷ η ∷ ν ∷ ε ∷ υ ∷ τ ∷ ή ∷ ς ∷ []) "1Cor.14.28"
∷ word (σ ∷ ι ∷ γ ∷ ά ∷ τ ∷ ω ∷ []) "1Cor.14.28"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.28"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.14.28"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "1Cor.14.28"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.28"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ε ∷ ί ∷ τ ∷ ω ∷ []) "1Cor.14.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.28"
∷ word (τ ∷ ῷ ∷ []) "1Cor.14.28"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1Cor.14.28"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ ῆ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.29"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.29"
∷ word (δ ∷ ύ ∷ ο ∷ []) "1Cor.14.29"
∷ word (ἢ ∷ []) "1Cor.14.29"
∷ word (τ ∷ ρ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.14.29"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ε ∷ ί ∷ τ ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "1Cor.14.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.29"
∷ word (ο ∷ ἱ ∷ []) "1Cor.14.29"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ι ∷ []) "1Cor.14.29"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ρ ∷ ι ∷ ν ∷ έ ∷ τ ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "1Cor.14.29"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.14.30"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.30"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ῳ ∷ []) "1Cor.14.30"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ υ ∷ φ ∷ θ ∷ ῇ ∷ []) "1Cor.14.30"
∷ word (κ ∷ α ∷ θ ∷ η ∷ μ ∷ έ ∷ ν ∷ ῳ ∷ []) "1Cor.14.30"
∷ word (ὁ ∷ []) "1Cor.14.30"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.14.30"
∷ word (σ ∷ ι ∷ γ ∷ ά ∷ τ ∷ ω ∷ []) "1Cor.14.30"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.14.31"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.14.31"
∷ word (κ ∷ α ∷ θ ∷ []) "1Cor.14.31"
∷ word (ἕ ∷ ν ∷ α ∷ []) "1Cor.14.31"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.14.31"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.14.31"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.14.31"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.14.31"
∷ word (μ ∷ α ∷ ν ∷ θ ∷ ά ∷ ν ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.14.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.31"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.14.31"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ῶ ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.31"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.32"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.14.32"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.14.32"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ ή ∷ τ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.14.32"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ ά ∷ σ ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.32"
∷ word (ο ∷ ὐ ∷ []) "1Cor.14.33"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.14.33"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.14.33"
∷ word (ἀ ∷ κ ∷ α ∷ τ ∷ α ∷ σ ∷ τ ∷ α ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.14.33"
∷ word (ὁ ∷ []) "1Cor.14.33"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.14.33"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.14.33"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ ς ∷ []) "1Cor.14.33"
∷ word (ὡ ∷ ς ∷ []) "1Cor.14.33"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.33"
∷ word (π ∷ ά ∷ σ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.14.33"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "1Cor.14.33"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "1Cor.14.33"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.14.33"
∷ word (ἁ ∷ γ ∷ ί ∷ ω ∷ ν ∷ []) "1Cor.14.33"
∷ word (Α ∷ ἱ ∷ []) "1Cor.14.34"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ ε ∷ ς ∷ []) "1Cor.14.34"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.34"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "1Cor.14.34"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "1Cor.14.34"
∷ word (σ ∷ ι ∷ γ ∷ ά ∷ τ ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "1Cor.14.34"
∷ word (ο ∷ ὐ ∷ []) "1Cor.14.34"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.14.34"
∷ word (ἐ ∷ π ∷ ι ∷ τ ∷ ρ ∷ έ ∷ π ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.34"
∷ word (α ∷ ὐ ∷ τ ∷ α ∷ ῖ ∷ ς ∷ []) "1Cor.14.34"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.14.34"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.14.34"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ α ∷ σ ∷ σ ∷ έ ∷ σ ∷ θ ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "1Cor.14.34"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.14.34"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.34"
∷ word (ὁ ∷ []) "1Cor.14.34"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.14.34"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "1Cor.14.34"
∷ word (ε ∷ ἰ ∷ []) "1Cor.14.35"
∷ word (δ ∷ έ ∷ []) "1Cor.14.35"
∷ word (τ ∷ ι ∷ []) "1Cor.14.35"
∷ word (μ ∷ α ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.14.35"
∷ word (θ ∷ έ ∷ ∙λ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.14.35"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.35"
∷ word (ο ∷ ἴ ∷ κ ∷ ῳ ∷ []) "1Cor.14.35"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.14.35"
∷ word (ἰ ∷ δ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.14.35"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ α ∷ ς ∷ []) "1Cor.14.35"
∷ word (ἐ ∷ π ∷ ε ∷ ρ ∷ ω ∷ τ ∷ ά ∷ τ ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "1Cor.14.35"
∷ word (α ∷ ἰ ∷ σ ∷ χ ∷ ρ ∷ ὸ ∷ ν ∷ []) "1Cor.14.35"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.14.35"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.14.35"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ὶ ∷ []) "1Cor.14.35"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.14.35"
∷ word (ἐ ∷ ν ∷ []) "1Cor.14.35"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.14.35"
∷ word (ἢ ∷ []) "1Cor.14.36"
∷ word (ἀ ∷ φ ∷ []) "1Cor.14.36"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.14.36"
∷ word (ὁ ∷ []) "1Cor.14.36"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "1Cor.14.36"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.14.36"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.14.36"
∷ word (ἐ ∷ ξ ∷ ῆ ∷ ∙λ ∷ θ ∷ ε ∷ ν ∷ []) "1Cor.14.36"
∷ word (ἢ ∷ []) "1Cor.14.36"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.14.36"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.14.36"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.14.36"
∷ word (κ ∷ α ∷ τ ∷ ή ∷ ν ∷ τ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.14.36"
∷ word (Ε ∷ ἴ ∷ []) "1Cor.14.37"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.14.37"
∷ word (δ ∷ ο ∷ κ ∷ ε ∷ ῖ ∷ []) "1Cor.14.37"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ ή ∷ τ ∷ η ∷ ς ∷ []) "1Cor.14.37"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.14.37"
∷ word (ἢ ∷ []) "1Cor.14.37"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ό ∷ ς ∷ []) "1Cor.14.37"
∷ word (ἐ ∷ π ∷ ι ∷ γ ∷ ι ∷ ν ∷ ω ∷ σ ∷ κ ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.14.37"
∷ word (ἃ ∷ []) "1Cor.14.37"
∷ word (γ ∷ ρ ∷ ά ∷ φ ∷ ω ∷ []) "1Cor.14.37"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.14.37"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.14.37"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.14.37"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1Cor.14.37"
∷ word (ε ∷ ἰ ∷ []) "1Cor.14.38"
∷ word (δ ∷ έ ∷ []) "1Cor.14.38"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.14.38"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ε ∷ ῖ ∷ []) "1Cor.14.38"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.14.38"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.14.39"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.14.39"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.14.39"
∷ word (ζ ∷ η ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ε ∷ []) "1Cor.14.39"
∷ word (τ ∷ ὸ ∷ []) "1Cor.14.39"
∷ word (π ∷ ρ ∷ ο ∷ φ ∷ η ∷ τ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.14.39"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.39"
∷ word (τ ∷ ὸ ∷ []) "1Cor.14.39"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.14.39"
∷ word (μ ∷ ὴ ∷ []) "1Cor.14.39"
∷ word (κ ∷ ω ∷ ∙λ ∷ ύ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.14.39"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ α ∷ ι ∷ ς ∷ []) "1Cor.14.39"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.14.40"
∷ word (δ ∷ ὲ ∷ []) "1Cor.14.40"
∷ word (ε ∷ ὐ ∷ σ ∷ χ ∷ η ∷ μ ∷ ό ∷ ν ∷ ω ∷ ς ∷ []) "1Cor.14.40"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.14.40"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.14.40"
∷ word (τ ∷ ά ∷ ξ ∷ ι ∷ ν ∷ []) "1Cor.14.40"
∷ word (γ ∷ ι ∷ ν ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "1Cor.14.40"
∷ word (Γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ ζ ∷ ω ∷ []) "1Cor.15.1"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.1"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.15.1"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.15.1"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.1"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.15.1"
∷ word (ὃ ∷ []) "1Cor.15.1"
∷ word (ε ∷ ὐ ∷ η ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ι ∷ σ ∷ ά ∷ μ ∷ η ∷ ν ∷ []) "1Cor.15.1"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.15.1"
∷ word (ὃ ∷ []) "1Cor.15.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.1"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ ∙λ ∷ ά ∷ β ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.15.1"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.1"
∷ word (ᾧ ∷ []) "1Cor.15.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.1"
∷ word (ἑ ∷ σ ∷ τ ∷ ή ∷ κ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.15.1"
∷ word (δ ∷ ι ∷ []) "1Cor.15.2"
∷ word (ο ∷ ὗ ∷ []) "1Cor.15.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.2"
∷ word (σ ∷ ῴ ∷ ζ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.15.2"
∷ word (τ ∷ ί ∷ ν ∷ ι ∷ []) "1Cor.15.2"
∷ word (∙λ ∷ ό ∷ γ ∷ ῳ ∷ []) "1Cor.15.2"
∷ word (ε ∷ ὐ ∷ η ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ι ∷ σ ∷ ά ∷ μ ∷ η ∷ ν ∷ []) "1Cor.15.2"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.15.2"
∷ word (ε ∷ ἰ ∷ []) "1Cor.15.2"
∷ word (κ ∷ α ∷ τ ∷ έ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.15.2"
∷ word (ἐ ∷ κ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.15.2"
∷ word (ε ∷ ἰ ∷ []) "1Cor.15.2"
∷ word (μ ∷ ὴ ∷ []) "1Cor.15.2"
∷ word (ε ∷ ἰ ∷ κ ∷ ῇ ∷ []) "1Cor.15.2"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.15.2"
∷ word (Π ∷ α ∷ ρ ∷ έ ∷ δ ∷ ω ∷ κ ∷ α ∷ []) "1Cor.15.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.15.3"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.15.3"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.3"
∷ word (π ∷ ρ ∷ ώ ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.15.3"
∷ word (ὃ ∷ []) "1Cor.15.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.3"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ ∙λ ∷ α ∷ β ∷ ο ∷ ν ∷ []) "1Cor.15.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.15.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.15.3"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "1Cor.15.3"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "1Cor.15.3"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.15.3"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ι ∷ ῶ ∷ ν ∷ []) "1Cor.15.3"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.15.3"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.15.3"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1Cor.15.3"
∷ word (γ ∷ ρ ∷ α ∷ φ ∷ ά ∷ ς ∷ []) "1Cor.15.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.4"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.15.4"
∷ word (ἐ ∷ τ ∷ ά ∷ φ ∷ η ∷ []) "1Cor.15.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.4"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.15.4"
∷ word (ἐ ∷ γ ∷ ή ∷ γ ∷ ε ∷ ρ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.4"
∷ word (τ ∷ ῇ ∷ []) "1Cor.15.4"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "1Cor.15.4"
∷ word (τ ∷ ῇ ∷ []) "1Cor.15.4"
∷ word (τ ∷ ρ ∷ ί ∷ τ ∷ ῃ ∷ []) "1Cor.15.4"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.15.4"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1Cor.15.4"
∷ word (γ ∷ ρ ∷ α ∷ φ ∷ ά ∷ ς ∷ []) "1Cor.15.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.5"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.15.5"
∷ word (ὤ ∷ φ ∷ θ ∷ η ∷ []) "1Cor.15.5"
∷ word (Κ ∷ η ∷ φ ∷ ᾷ ∷ []) "1Cor.15.5"
∷ word (ε ∷ ἶ ∷ τ ∷ α ∷ []) "1Cor.15.5"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.15.5"
∷ word (δ ∷ ώ ∷ δ ∷ ε ∷ κ ∷ α ∷ []) "1Cor.15.5"
∷ word (ἔ ∷ π ∷ ε ∷ ι ∷ τ ∷ α ∷ []) "1Cor.15.6"
∷ word (ὤ ∷ φ ∷ θ ∷ η ∷ []) "1Cor.15.6"
∷ word (ἐ ∷ π ∷ ά ∷ ν ∷ ω ∷ []) "1Cor.15.6"
∷ word (π ∷ ε ∷ ν ∷ τ ∷ α ∷ κ ∷ ο ∷ σ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.15.6"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.15.6"
∷ word (ἐ ∷ φ ∷ ά ∷ π ∷ α ∷ ξ ∷ []) "1Cor.15.6"
∷ word (ἐ ∷ ξ ∷ []) "1Cor.15.6"
∷ word (ὧ ∷ ν ∷ []) "1Cor.15.6"
∷ word (ο ∷ ἱ ∷ []) "1Cor.15.6"
∷ word (π ∷ ∙λ ∷ ε ∷ ί ∷ ο ∷ ν ∷ ε ∷ ς ∷ []) "1Cor.15.6"
∷ word (μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.15.6"
∷ word (ἕ ∷ ω ∷ ς ∷ []) "1Cor.15.6"
∷ word (ἄ ∷ ρ ∷ τ ∷ ι ∷ []) "1Cor.15.6"
∷ word (τ ∷ ι ∷ ν ∷ ὲ ∷ ς ∷ []) "1Cor.15.6"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.6"
∷ word (ἐ ∷ κ ∷ ο ∷ ι ∷ μ ∷ ή ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "1Cor.15.6"
∷ word (ἔ ∷ π ∷ ε ∷ ι ∷ τ ∷ α ∷ []) "1Cor.15.7"
∷ word (ὤ ∷ φ ∷ θ ∷ η ∷ []) "1Cor.15.7"
∷ word (Ἰ ∷ α ∷ κ ∷ ώ ∷ β ∷ ῳ ∷ []) "1Cor.15.7"
∷ word (ε ∷ ἶ ∷ τ ∷ α ∷ []) "1Cor.15.7"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.15.7"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.15.7"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.15.7"
∷ word (ἔ ∷ σ ∷ χ ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.15.8"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.8"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.15.8"
∷ word (ὡ ∷ σ ∷ π ∷ ε ∷ ρ ∷ ε ∷ ὶ ∷ []) "1Cor.15.8"
∷ word (τ ∷ ῷ ∷ []) "1Cor.15.8"
∷ word (ἐ ∷ κ ∷ τ ∷ ρ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.15.8"
∷ word (ὤ ∷ φ ∷ θ ∷ η ∷ []) "1Cor.15.8"
∷ word (κ ∷ ἀ ∷ μ ∷ ο ∷ ί ∷ []) "1Cor.15.8"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.15.9"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.15.9"
∷ word (ε ∷ ἰ ∷ μ ∷ ι ∷ []) "1Cor.15.9"
∷ word (ὁ ∷ []) "1Cor.15.9"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ χ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.15.9"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.15.9"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ω ∷ ν ∷ []) "1Cor.15.9"
∷ word (ὃ ∷ ς ∷ []) "1Cor.15.9"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.15.9"
∷ word (ε ∷ ἰ ∷ μ ∷ ὶ ∷ []) "1Cor.15.9"
∷ word (ἱ ∷ κ ∷ α ∷ ν ∷ ὸ ∷ ς ∷ []) "1Cor.15.9"
∷ word (κ ∷ α ∷ ∙λ ∷ ε ∷ ῖ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.15.9"
∷ word (ἀ ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.15.9"
∷ word (δ ∷ ι ∷ ό ∷ τ ∷ ι ∷ []) "1Cor.15.9"
∷ word (ἐ ∷ δ ∷ ί ∷ ω ∷ ξ ∷ α ∷ []) "1Cor.15.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.15.9"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.15.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.9"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.15.9"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ι ∷ []) "1Cor.15.10"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.10"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.15.10"
∷ word (ε ∷ ἰ ∷ μ ∷ ι ∷ []) "1Cor.15.10"
∷ word (ὅ ∷ []) "1Cor.15.10"
∷ word (ε ∷ ἰ ∷ μ ∷ ι ∷ []) "1Cor.15.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.10"
∷ word (ἡ ∷ []) "1Cor.15.10"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "1Cor.15.10"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.10"
∷ word (ἡ ∷ []) "1Cor.15.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.15.10"
∷ word (ἐ ∷ μ ∷ ὲ ∷ []) "1Cor.15.10"
∷ word (ο ∷ ὐ ∷ []) "1Cor.15.10"
∷ word (κ ∷ ε ∷ ν ∷ ὴ ∷ []) "1Cor.15.10"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ή ∷ θ ∷ η ∷ []) "1Cor.15.10"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.15.10"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ό ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.15.10"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.15.10"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.15.10"
∷ word (ἐ ∷ κ ∷ ο ∷ π ∷ ί ∷ α ∷ σ ∷ α ∷ []) "1Cor.15.10"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.15.10"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.15.10"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.10"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.15.10"
∷ word (ἡ ∷ []) "1Cor.15.10"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "1Cor.15.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.10"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.15.10"
∷ word (ἡ ∷ []) "1Cor.15.10"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "1Cor.15.10"
∷ word (ἐ ∷ μ ∷ ο ∷ ί ∷ []) "1Cor.15.10"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.15.11"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.15.11"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "1Cor.15.11"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "1Cor.15.11"
∷ word (ἐ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.15.11"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.15.11"
∷ word (κ ∷ η ∷ ρ ∷ ύ ∷ σ ∷ σ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.15.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.11"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.15.11"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.15.11"
∷ word (Ε ∷ ἰ ∷ []) "1Cor.15.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.12"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.15.12"
∷ word (κ ∷ η ∷ ρ ∷ ύ ∷ σ ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.12"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.15.12"
∷ word (ἐ ∷ κ ∷ []) "1Cor.15.12"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "1Cor.15.12"
∷ word (ἐ ∷ γ ∷ ή ∷ γ ∷ ε ∷ ρ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.12"
∷ word (π ∷ ῶ ∷ ς ∷ []) "1Cor.15.12"
∷ word (∙λ ∷ έ ∷ γ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.15.12"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.12"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.15.12"
∷ word (τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "1Cor.15.12"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.15.12"
∷ word (ἀ ∷ ν ∷ ά ∷ σ ∷ τ ∷ α ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.15.12"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "1Cor.15.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.15.12"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.15.12"
∷ word (ε ∷ ἰ ∷ []) "1Cor.15.13"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.13"
∷ word (ἀ ∷ ν ∷ ά ∷ σ ∷ τ ∷ α ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.15.13"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "1Cor.15.13"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.15.13"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.15.13"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "1Cor.15.13"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.15.13"
∷ word (ἐ ∷ γ ∷ ή ∷ γ ∷ ε ∷ ρ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.13"
∷ word (ε ∷ ἰ ∷ []) "1Cor.15.14"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.14"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.15.14"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.15.14"
∷ word (ἐ ∷ γ ∷ ή ∷ γ ∷ ε ∷ ρ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.14"
∷ word (κ ∷ ε ∷ ν ∷ ὸ ∷ ν ∷ []) "1Cor.15.14"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "1Cor.15.14"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.14"
∷ word (κ ∷ ή ∷ ρ ∷ υ ∷ γ ∷ μ ∷ α ∷ []) "1Cor.15.14"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.15.14"
∷ word (κ ∷ ε ∷ ν ∷ ὴ ∷ []) "1Cor.15.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.14"
∷ word (ἡ ∷ []) "1Cor.15.14"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "1Cor.15.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.15.14"
∷ word (ε ∷ ὑ ∷ ρ ∷ ι ∷ σ ∷ κ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1Cor.15.15"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.15"
∷ word (ψ ∷ ε ∷ υ ∷ δ ∷ ο ∷ μ ∷ ά ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ε ∷ ς ∷ []) "1Cor.15.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.15"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.15.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.15.15"
∷ word (ἐ ∷ μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ή ∷ σ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.15.15"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.15.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.15"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.15.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.15.15"
∷ word (ἤ ∷ γ ∷ ε ∷ ι ∷ ρ ∷ ε ∷ ν ∷ []) "1Cor.15.15"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.15.15"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ν ∷ []) "1Cor.15.15"
∷ word (ὃ ∷ ν ∷ []) "1Cor.15.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.15.15"
∷ word (ἤ ∷ γ ∷ ε ∷ ι ∷ ρ ∷ ε ∷ ν ∷ []) "1Cor.15.15"
∷ word (ε ∷ ἴ ∷ π ∷ ε ∷ ρ ∷ []) "1Cor.15.15"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "1Cor.15.15"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ο ∷ ὶ ∷ []) "1Cor.15.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.15.15"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.15"
∷ word (ε ∷ ἰ ∷ []) "1Cor.15.16"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.15.16"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ο ∷ ὶ ∷ []) "1Cor.15.16"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.15.16"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.16"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "1Cor.15.16"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.15.16"
∷ word (ἐ ∷ γ ∷ ή ∷ γ ∷ ε ∷ ρ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.16"
∷ word (ε ∷ ἰ ∷ []) "1Cor.15.17"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.15.17"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.15.17"
∷ word (ἐ ∷ γ ∷ ή ∷ γ ∷ ε ∷ ρ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.17"
∷ word (μ ∷ α ∷ τ ∷ α ∷ ί ∷ α ∷ []) "1Cor.15.17"
∷ word (ἡ ∷ []) "1Cor.15.17"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "1Cor.15.17"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.15.17"
∷ word (ἔ ∷ τ ∷ ι ∷ []) "1Cor.15.17"
∷ word (ἐ ∷ σ ∷ τ ∷ ὲ ∷ []) "1Cor.15.17"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.17"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "1Cor.15.17"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "1Cor.15.17"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.15.17"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "1Cor.15.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.18"
∷ word (ο ∷ ἱ ∷ []) "1Cor.15.18"
∷ word (κ ∷ ο ∷ ι ∷ μ ∷ η ∷ θ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.15.18"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.18"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1Cor.15.18"
∷ word (ἀ ∷ π ∷ ώ ∷ ∙λ ∷ ο ∷ ν ∷ τ ∷ ο ∷ []) "1Cor.15.18"
∷ word (ε ∷ ἰ ∷ []) "1Cor.15.19"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.19"
∷ word (τ ∷ ῇ ∷ []) "1Cor.15.19"
∷ word (ζ ∷ ω ∷ ῇ ∷ []) "1Cor.15.19"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ ῃ ∷ []) "1Cor.15.19"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.19"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1Cor.15.19"
∷ word (ἠ ∷ ∙λ ∷ π ∷ ι ∷ κ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.15.19"
∷ word (ἐ ∷ σ ∷ μ ∷ ὲ ∷ ν ∷ []) "1Cor.15.19"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.15.19"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ε ∷ ι ∷ ν ∷ ό ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ι ∷ []) "1Cor.15.19"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.15.19"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "1Cor.15.19"
∷ word (ἐ ∷ σ ∷ μ ∷ έ ∷ ν ∷ []) "1Cor.15.19"
∷ word (Ν ∷ υ ∷ ν ∷ ὶ ∷ []) "1Cor.15.20"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.20"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.15.20"
∷ word (ἐ ∷ γ ∷ ή ∷ γ ∷ ε ∷ ρ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.20"
∷ word (ἐ ∷ κ ∷ []) "1Cor.15.20"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "1Cor.15.20"
∷ word (ἀ ∷ π ∷ α ∷ ρ ∷ χ ∷ ὴ ∷ []) "1Cor.15.20"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.15.20"
∷ word (κ ∷ ε ∷ κ ∷ ο ∷ ι ∷ μ ∷ η ∷ μ ∷ έ ∷ ν ∷ ω ∷ ν ∷ []) "1Cor.15.20"
∷ word (ἐ ∷ π ∷ ε ∷ ι ∷ δ ∷ ὴ ∷ []) "1Cor.15.21"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.15.21"
∷ word (δ ∷ ι ∷ []) "1Cor.15.21"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "1Cor.15.21"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.15.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.21"
∷ word (δ ∷ ι ∷ []) "1Cor.15.21"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "1Cor.15.21"
∷ word (ἀ ∷ ν ∷ ά ∷ σ ∷ τ ∷ α ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.15.21"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "1Cor.15.21"
∷ word (ὥ ∷ σ ∷ π ∷ ε ∷ ρ ∷ []) "1Cor.15.22"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.15.22"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.22"
∷ word (τ ∷ ῷ ∷ []) "1Cor.15.22"
∷ word (Ἀ ∷ δ ∷ ὰ ∷ μ ∷ []) "1Cor.15.22"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.15.22"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ ν ∷ ῄ ∷ σ ∷ κ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.15.22"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.15.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.22"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.22"
∷ word (τ ∷ ῷ ∷ []) "1Cor.15.22"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1Cor.15.22"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.15.22"
∷ word (ζ ∷ ῳ ∷ ο ∷ π ∷ ο ∷ ι ∷ η ∷ θ ∷ ή ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.22"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.15.23"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.23"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.23"
∷ word (τ ∷ ῷ ∷ []) "1Cor.15.23"
∷ word (ἰ ∷ δ ∷ ί ∷ ῳ ∷ []) "1Cor.15.23"
∷ word (τ ∷ ά ∷ γ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.15.23"
∷ word (ἀ ∷ π ∷ α ∷ ρ ∷ χ ∷ ὴ ∷ []) "1Cor.15.23"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "1Cor.15.23"
∷ word (ἔ ∷ π ∷ ε ∷ ι ∷ τ ∷ α ∷ []) "1Cor.15.23"
∷ word (ο ∷ ἱ ∷ []) "1Cor.15.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.23"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.23"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.23"
∷ word (τ ∷ ῇ ∷ []) "1Cor.15.23"
∷ word (π ∷ α ∷ ρ ∷ ο ∷ υ ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.15.23"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.23"
∷ word (ε ∷ ἶ ∷ τ ∷ α ∷ []) "1Cor.15.24"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.24"
∷ word (τ ∷ έ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.15.24"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "1Cor.15.24"
∷ word (π ∷ α ∷ ρ ∷ α ∷ δ ∷ ι ∷ δ ∷ ῷ ∷ []) "1Cor.15.24"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.15.24"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "1Cor.15.24"
∷ word (τ ∷ ῷ ∷ []) "1Cor.15.24"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1Cor.15.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.24"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ί ∷ []) "1Cor.15.24"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "1Cor.15.24"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ ή ∷ σ ∷ ῃ ∷ []) "1Cor.15.24"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "1Cor.15.24"
∷ word (ἀ ∷ ρ ∷ χ ∷ ὴ ∷ ν ∷ []) "1Cor.15.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.24"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "1Cor.15.24"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.15.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.24"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ν ∷ []) "1Cor.15.24"
∷ word (δ ∷ ε ∷ ῖ ∷ []) "1Cor.15.25"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.15.25"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.15.25"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ ν ∷ []) "1Cor.15.25"
∷ word (ἄ ∷ χ ∷ ρ ∷ ι ∷ []) "1Cor.15.25"
∷ word (ο ∷ ὗ ∷ []) "1Cor.15.25"
∷ word (θ ∷ ῇ ∷ []) "1Cor.15.25"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "1Cor.15.25"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.15.25"
∷ word (ἐ ∷ χ ∷ θ ∷ ρ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.15.25"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.15.25"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.15.25"
∷ word (π ∷ ό ∷ δ ∷ α ∷ ς ∷ []) "1Cor.15.25"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.25"
∷ word (ἔ ∷ σ ∷ χ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.15.26"
∷ word (ἐ ∷ χ ∷ θ ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.15.26"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.26"
∷ word (ὁ ∷ []) "1Cor.15.26"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.15.26"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.15.27"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.15.27"
∷ word (ὑ ∷ π ∷ έ ∷ τ ∷ α ∷ ξ ∷ ε ∷ ν ∷ []) "1Cor.15.27"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "1Cor.15.27"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.15.27"
∷ word (π ∷ ό ∷ δ ∷ α ∷ ς ∷ []) "1Cor.15.27"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.27"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "1Cor.15.27"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.27"
∷ word (ε ∷ ἴ ∷ π ∷ ῃ ∷ []) "1Cor.15.27"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.15.27"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.15.27"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ έ ∷ τ ∷ α ∷ κ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.27"
∷ word (δ ∷ ῆ ∷ ∙λ ∷ ο ∷ ν ∷ []) "1Cor.15.27"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.15.27"
∷ word (ἐ ∷ κ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.15.27"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.27"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ ά ∷ ξ ∷ α ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.15.27"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1Cor.15.27"
∷ word (τ ∷ ὰ ∷ []) "1Cor.15.27"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.15.27"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "1Cor.15.28"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.28"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ α ∷ γ ∷ ῇ ∷ []) "1Cor.15.28"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1Cor.15.28"
∷ word (τ ∷ ὰ ∷ []) "1Cor.15.28"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.15.28"
∷ word (τ ∷ ό ∷ τ ∷ ε ∷ []) "1Cor.15.28"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1Cor.15.28"
∷ word (ὁ ∷ []) "1Cor.15.28"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ς ∷ []) "1Cor.15.28"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ α ∷ γ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.28"
∷ word (τ ∷ ῷ ∷ []) "1Cor.15.28"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ ά ∷ ξ ∷ α ∷ ν ∷ τ ∷ ι ∷ []) "1Cor.15.28"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1Cor.15.28"
∷ word (τ ∷ ὰ ∷ []) "1Cor.15.28"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.15.28"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.15.28"
∷ word (ᾖ ∷ []) "1Cor.15.28"
∷ word (ὁ ∷ []) "1Cor.15.28"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.15.28"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.15.28"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.28"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.15.28"
∷ word (Ἐ ∷ π ∷ ε ∷ ὶ ∷ []) "1Cor.15.29"
∷ word (τ ∷ ί ∷ []) "1Cor.15.29"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.15.29"
∷ word (ο ∷ ἱ ∷ []) "1Cor.15.29"
∷ word (β ∷ α ∷ π ∷ τ ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.15.29"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "1Cor.15.29"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.15.29"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "1Cor.15.29"
∷ word (ε ∷ ἰ ∷ []) "1Cor.15.29"
∷ word (ὅ ∷ ∙λ ∷ ω ∷ ς ∷ []) "1Cor.15.29"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ο ∷ ὶ ∷ []) "1Cor.15.29"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.15.29"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.29"
∷ word (τ ∷ ί ∷ []) "1Cor.15.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.29"
∷ word (β ∷ α ∷ π ∷ τ ∷ ί ∷ ζ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.29"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "1Cor.15.29"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.15.29"
∷ word (τ ∷ ί ∷ []) "1Cor.15.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.30"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.15.30"
∷ word (κ ∷ ι ∷ ν ∷ δ ∷ υ ∷ ν ∷ ε ∷ ύ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.15.30"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "1Cor.15.30"
∷ word (ὥ ∷ ρ ∷ α ∷ ν ∷ []) "1Cor.15.30"
∷ word (κ ∷ α ∷ θ ∷ []) "1Cor.15.31"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "1Cor.15.31"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ ν ∷ ῄ ∷ σ ∷ κ ∷ ω ∷ []) "1Cor.15.31"
∷ word (ν ∷ ὴ ∷ []) "1Cor.15.31"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.15.31"
∷ word (ὑ ∷ μ ∷ ε ∷ τ ∷ έ ∷ ρ ∷ α ∷ ν ∷ []) "1Cor.15.31"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.15.31"
∷ word (ἣ ∷ ν ∷ []) "1Cor.15.31"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "1Cor.15.31"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.31"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1Cor.15.31"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.15.31"
∷ word (τ ∷ ῷ ∷ []) "1Cor.15.31"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.15.31"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.15.31"
∷ word (ε ∷ ἰ ∷ []) "1Cor.15.32"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.15.32"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "1Cor.15.32"
∷ word (ἐ ∷ θ ∷ η ∷ ρ ∷ ι ∷ ο ∷ μ ∷ ά ∷ χ ∷ η ∷ σ ∷ α ∷ []) "1Cor.15.32"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.32"
∷ word (Ἐ ∷ φ ∷ έ ∷ σ ∷ ῳ ∷ []) "1Cor.15.32"
∷ word (τ ∷ ί ∷ []) "1Cor.15.32"
∷ word (μ ∷ ο ∷ ι ∷ []) "1Cor.15.32"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.32"
∷ word (ὄ ∷ φ ∷ ε ∷ ∙λ ∷ ο ∷ ς ∷ []) "1Cor.15.32"
∷ word (ε ∷ ἰ ∷ []) "1Cor.15.32"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ο ∷ ὶ ∷ []) "1Cor.15.32"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.15.32"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.32"
∷ word (Φ ∷ ά ∷ γ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.15.32"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.32"
∷ word (π ∷ ί ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.15.32"
∷ word (α ∷ ὔ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.15.32"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.15.32"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ ν ∷ ῄ ∷ σ ∷ κ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.15.32"
∷ word (μ ∷ ὴ ∷ []) "1Cor.15.33"
∷ word (π ∷ ∙λ ∷ α ∷ ν ∷ ᾶ ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.15.33"
∷ word (φ ∷ θ ∷ ε ∷ ί ∷ ρ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.15.33"
∷ word (ἤ ∷ θ ∷ η ∷ []) "1Cor.15.33"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ὰ ∷ []) "1Cor.15.33"
∷ word (ὁ ∷ μ ∷ ι ∷ ∙λ ∷ ί ∷ α ∷ ι ∷ []) "1Cor.15.33"
∷ word (κ ∷ α ∷ κ ∷ α ∷ ί ∷ []) "1Cor.15.33"
∷ word (ἐ ∷ κ ∷ ν ∷ ή ∷ ψ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.15.34"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ί ∷ ω ∷ ς ∷ []) "1Cor.15.34"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.34"
∷ word (μ ∷ ὴ ∷ []) "1Cor.15.34"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ά ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.15.34"
∷ word (ἀ ∷ γ ∷ ν ∷ ω ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.15.34"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.15.34"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.15.34"
∷ word (τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "1Cor.15.34"
∷ word (ἔ ∷ χ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.15.34"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.15.34"
∷ word (ἐ ∷ ν ∷ τ ∷ ρ ∷ ο ∷ π ∷ ὴ ∷ ν ∷ []) "1Cor.15.34"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.15.34"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "1Cor.15.34"
∷ word (Ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.15.35"
∷ word (ἐ ∷ ρ ∷ ε ∷ ῖ ∷ []) "1Cor.15.35"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.15.35"
∷ word (Π ∷ ῶ ∷ ς ∷ []) "1Cor.15.35"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.35"
∷ word (ο ∷ ἱ ∷ []) "1Cor.15.35"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ο ∷ ί ∷ []) "1Cor.15.35"
∷ word (π ∷ ο ∷ ί ∷ ῳ ∷ []) "1Cor.15.35"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.35"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.15.35"
∷ word (ἔ ∷ ρ ∷ χ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.35"
∷ word (ἄ ∷ φ ∷ ρ ∷ ω ∷ ν ∷ []) "1Cor.15.36"
∷ word (σ ∷ ὺ ∷ []) "1Cor.15.36"
∷ word (ὃ ∷ []) "1Cor.15.36"
∷ word (σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.15.36"
∷ word (ο ∷ ὐ ∷ []) "1Cor.15.36"
∷ word (ζ ∷ ῳ ∷ ο ∷ π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.36"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.15.36"
∷ word (μ ∷ ὴ ∷ []) "1Cor.15.36"
∷ word (ἀ ∷ π ∷ ο ∷ θ ∷ ά ∷ ν ∷ ῃ ∷ []) "1Cor.15.36"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.37"
∷ word (ὃ ∷ []) "1Cor.15.37"
∷ word (σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.15.37"
∷ word (ο ∷ ὐ ∷ []) "1Cor.15.37"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.37"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.15.37"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.37"
∷ word (γ ∷ ε ∷ ν ∷ η ∷ σ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.15.37"
∷ word (σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ ε ∷ ι ∷ ς ∷ []) "1Cor.15.37"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.15.37"
∷ word (γ ∷ υ ∷ μ ∷ ν ∷ ὸ ∷ ν ∷ []) "1Cor.15.37"
∷ word (κ ∷ ό ∷ κ ∷ κ ∷ ο ∷ ν ∷ []) "1Cor.15.37"
∷ word (ε ∷ ἰ ∷ []) "1Cor.15.37"
∷ word (τ ∷ ύ ∷ χ ∷ ο ∷ ι ∷ []) "1Cor.15.37"
∷ word (σ ∷ ί ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.15.37"
∷ word (ἤ ∷ []) "1Cor.15.37"
∷ word (τ ∷ ι ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.15.37"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.15.37"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ῶ ∷ ν ∷ []) "1Cor.15.37"
∷ word (ὁ ∷ []) "1Cor.15.38"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.38"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1Cor.15.38"
∷ word (δ ∷ ί ∷ δ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "1Cor.15.38"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1Cor.15.38"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.15.38"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.15.38"
∷ word (ἠ ∷ θ ∷ έ ∷ ∙λ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "1Cor.15.38"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.38"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ ῳ ∷ []) "1Cor.15.38"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.15.38"
∷ word (σ ∷ π ∷ ε ∷ ρ ∷ μ ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.15.38"
∷ word (ἴ ∷ δ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.15.38"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.15.38"
∷ word (ο ∷ ὐ ∷ []) "1Cor.15.39"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ []) "1Cor.15.39"
∷ word (σ ∷ ὰ ∷ ρ ∷ ξ ∷ []) "1Cor.15.39"
∷ word (ἡ ∷ []) "1Cor.15.39"
∷ word (α ∷ ὐ ∷ τ ∷ ὴ ∷ []) "1Cor.15.39"
∷ word (σ ∷ ά ∷ ρ ∷ ξ ∷ []) "1Cor.15.39"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.15.39"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ η ∷ []) "1Cor.15.39"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.15.39"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "1Cor.15.39"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ η ∷ []) "1Cor.15.39"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.39"
∷ word (σ ∷ ὰ ∷ ρ ∷ ξ ∷ []) "1Cor.15.39"
∷ word (κ ∷ τ ∷ η ∷ ν ∷ ῶ ∷ ν ∷ []) "1Cor.15.39"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ η ∷ []) "1Cor.15.39"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.39"
∷ word (σ ∷ ὰ ∷ ρ ∷ ξ ∷ []) "1Cor.15.39"
∷ word (π ∷ τ ∷ η ∷ ν ∷ ῶ ∷ ν ∷ []) "1Cor.15.39"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ η ∷ []) "1Cor.15.39"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.39"
∷ word (ἰ ∷ χ ∷ θ ∷ ύ ∷ ω ∷ ν ∷ []) "1Cor.15.39"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.40"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.15.40"
∷ word (ἐ ∷ π ∷ ο ∷ υ ∷ ρ ∷ ά ∷ ν ∷ ι ∷ α ∷ []) "1Cor.15.40"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.40"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1Cor.15.40"
∷ word (ἐ ∷ π ∷ ί ∷ γ ∷ ε ∷ ι ∷ α ∷ []) "1Cor.15.40"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.15.40"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "1Cor.15.40"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "1Cor.15.40"
∷ word (ἡ ∷ []) "1Cor.15.40"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.15.40"
∷ word (ἐ ∷ π ∷ ο ∷ υ ∷ ρ ∷ α ∷ ν ∷ ί ∷ ω ∷ ν ∷ []) "1Cor.15.40"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "1Cor.15.40"
∷ word (ἑ ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "1Cor.15.40"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.40"
∷ word (ἡ ∷ []) "1Cor.15.40"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.15.40"
∷ word (ἐ ∷ π ∷ ι ∷ γ ∷ ε ∷ ί ∷ ω ∷ ν ∷ []) "1Cor.15.40"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ η ∷ []) "1Cor.15.41"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "1Cor.15.41"
∷ word (ἡ ∷ ∙λ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.15.41"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.41"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ η ∷ []) "1Cor.15.41"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "1Cor.15.41"
∷ word (σ ∷ ε ∷ ∙λ ∷ ή ∷ ν ∷ η ∷ ς ∷ []) "1Cor.15.41"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.41"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ η ∷ []) "1Cor.15.41"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "1Cor.15.41"
∷ word (ἀ ∷ σ ∷ τ ∷ έ ∷ ρ ∷ ω ∷ ν ∷ []) "1Cor.15.41"
∷ word (ἀ ∷ σ ∷ τ ∷ ὴ ∷ ρ ∷ []) "1Cor.15.41"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.15.41"
∷ word (ἀ ∷ σ ∷ τ ∷ έ ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.15.41"
∷ word (δ ∷ ι ∷ α ∷ φ ∷ έ ∷ ρ ∷ ε ∷ ι ∷ []) "1Cor.15.41"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.41"
∷ word (δ ∷ ό ∷ ξ ∷ ῃ ∷ []) "1Cor.15.41"
∷ word (Ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.15.42"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.42"
∷ word (ἡ ∷ []) "1Cor.15.42"
∷ word (ἀ ∷ ν ∷ ά ∷ σ ∷ τ ∷ α ∷ σ ∷ ι ∷ ς ∷ []) "1Cor.15.42"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.15.42"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "1Cor.15.42"
∷ word (σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.42"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.42"
∷ word (φ ∷ θ ∷ ο ∷ ρ ∷ ᾷ ∷ []) "1Cor.15.42"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.42"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.42"
∷ word (ἀ ∷ φ ∷ θ ∷ α ∷ ρ ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.15.42"
∷ word (σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.43"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.43"
∷ word (ἀ ∷ τ ∷ ι ∷ μ ∷ ί ∷ ᾳ ∷ []) "1Cor.15.43"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.43"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.43"
∷ word (δ ∷ ό ∷ ξ ∷ ῃ ∷ []) "1Cor.15.43"
∷ word (σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.43"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.43"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ε ∷ ί ∷ ᾳ ∷ []) "1Cor.15.43"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.43"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.43"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ []) "1Cor.15.43"
∷ word (σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.44"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.15.44"
∷ word (ψ ∷ υ ∷ χ ∷ ι ∷ κ ∷ ό ∷ ν ∷ []) "1Cor.15.44"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.44"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.15.44"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ό ∷ ν ∷ []) "1Cor.15.44"
∷ word (Ε ∷ ἰ ∷ []) "1Cor.15.44"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.15.44"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "1Cor.15.44"
∷ word (ψ ∷ υ ∷ χ ∷ ι ∷ κ ∷ ό ∷ ν ∷ []) "1Cor.15.44"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.15.44"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.44"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ό ∷ ν ∷ []) "1Cor.15.44"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.15.45"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.45"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.45"
∷ word (Ἐ ∷ γ ∷ έ ∷ ν ∷ ε ∷ τ ∷ ο ∷ []) "1Cor.15.45"
∷ word (ὁ ∷ []) "1Cor.15.45"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.15.45"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "1Cor.15.45"
∷ word (Ἀ ∷ δ ∷ ὰ ∷ μ ∷ []) "1Cor.15.45"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.15.45"
∷ word (ψ ∷ υ ∷ χ ∷ ὴ ∷ ν ∷ []) "1Cor.15.45"
∷ word (ζ ∷ ῶ ∷ σ ∷ α ∷ ν ∷ []) "1Cor.15.45"
∷ word (ὁ ∷ []) "1Cor.15.45"
∷ word (ἔ ∷ σ ∷ χ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.15.45"
∷ word (Ἀ ∷ δ ∷ ὰ ∷ μ ∷ []) "1Cor.15.45"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.15.45"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.15.45"
∷ word (ζ ∷ ῳ ∷ ο ∷ π ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ ν ∷ []) "1Cor.15.45"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1Cor.15.46"
∷ word (ο ∷ ὐ ∷ []) "1Cor.15.46"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ν ∷ []) "1Cor.15.46"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.46"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ὸ ∷ ν ∷ []) "1Cor.15.46"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.15.46"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.46"
∷ word (ψ ∷ υ ∷ χ ∷ ι ∷ κ ∷ ό ∷ ν ∷ []) "1Cor.15.46"
∷ word (ἔ ∷ π ∷ ε ∷ ι ∷ τ ∷ α ∷ []) "1Cor.15.46"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.46"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ό ∷ ν ∷ []) "1Cor.15.46"
∷ word (ὁ ∷ []) "1Cor.15.47"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.15.47"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "1Cor.15.47"
∷ word (ἐ ∷ κ ∷ []) "1Cor.15.47"
∷ word (γ ∷ ῆ ∷ ς ∷ []) "1Cor.15.47"
∷ word (χ ∷ ο ∷ ϊ ∷ κ ∷ ό ∷ ς ∷ []) "1Cor.15.47"
∷ word (ὁ ∷ []) "1Cor.15.47"
∷ word (δ ∷ ε ∷ ύ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ς ∷ []) "1Cor.15.47"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "1Cor.15.47"
∷ word (ἐ ∷ ξ ∷ []) "1Cor.15.47"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ο ∷ ῦ ∷ []) "1Cor.15.47"
∷ word (ο ∷ ἷ ∷ ο ∷ ς ∷ []) "1Cor.15.48"
∷ word (ὁ ∷ []) "1Cor.15.48"
∷ word (χ ∷ ο ∷ ϊ ∷ κ ∷ ό ∷ ς ∷ []) "1Cor.15.48"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ι ∷ []) "1Cor.15.48"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.48"
∷ word (ο ∷ ἱ ∷ []) "1Cor.15.48"
∷ word (χ ∷ ο ∷ ϊ ∷ κ ∷ ο ∷ ί ∷ []) "1Cor.15.48"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.48"
∷ word (ο ∷ ἷ ∷ ο ∷ ς ∷ []) "1Cor.15.48"
∷ word (ὁ ∷ []) "1Cor.15.48"
∷ word (ἐ ∷ π ∷ ο ∷ υ ∷ ρ ∷ ά ∷ ν ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.15.48"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ι ∷ []) "1Cor.15.48"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.48"
∷ word (ο ∷ ἱ ∷ []) "1Cor.15.48"
∷ word (ἐ ∷ π ∷ ο ∷ υ ∷ ρ ∷ ά ∷ ν ∷ ι ∷ ο ∷ ι ∷ []) "1Cor.15.48"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.49"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1Cor.15.49"
∷ word (ἐ ∷ φ ∷ ο ∷ ρ ∷ έ ∷ σ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.15.49"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.15.49"
∷ word (ε ∷ ἰ ∷ κ ∷ ό ∷ ν ∷ α ∷ []) "1Cor.15.49"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.49"
∷ word (χ ∷ ο ∷ ϊ ∷ κ ∷ ο ∷ ῦ ∷ []) "1Cor.15.49"
∷ word (φ ∷ ο ∷ ρ ∷ έ ∷ σ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1Cor.15.49"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.49"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.15.49"
∷ word (ε ∷ ἰ ∷ κ ∷ ό ∷ ν ∷ α ∷ []) "1Cor.15.49"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.49"
∷ word (ἐ ∷ π ∷ ο ∷ υ ∷ ρ ∷ α ∷ ν ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.15.49"
∷ word (Τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.15.50"
∷ word (δ ∷ έ ∷ []) "1Cor.15.50"
∷ word (φ ∷ η ∷ μ ∷ ι ∷ []) "1Cor.15.50"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.15.50"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.15.50"
∷ word (σ ∷ ὰ ∷ ρ ∷ ξ ∷ []) "1Cor.15.50"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.50"
∷ word (α ∷ ἷ ∷ μ ∷ α ∷ []) "1Cor.15.50"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "1Cor.15.50"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1Cor.15.50"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "1Cor.15.50"
∷ word (ο ∷ ὐ ∷ []) "1Cor.15.50"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.50"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "1Cor.15.50"
∷ word (ἡ ∷ []) "1Cor.15.50"
∷ word (φ ∷ θ ∷ ο ∷ ρ ∷ ὰ ∷ []) "1Cor.15.50"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.15.50"
∷ word (ἀ ∷ φ ∷ θ ∷ α ∷ ρ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.15.50"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ε ∷ ῖ ∷ []) "1Cor.15.50"
∷ word (ἰ ∷ δ ∷ ο ∷ ὺ ∷ []) "1Cor.15.51"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.15.51"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.15.51"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "1Cor.15.51"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.15.51"
∷ word (ο ∷ ὐ ∷ []) "1Cor.15.51"
∷ word (κ ∷ ο ∷ ι ∷ μ ∷ η ∷ θ ∷ η ∷ σ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1Cor.15.51"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.15.51"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.51"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ α ∷ γ ∷ η ∷ σ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1Cor.15.51"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.52"
∷ word (ἀ ∷ τ ∷ ό ∷ μ ∷ ῳ ∷ []) "1Cor.15.52"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.52"
∷ word (ῥ ∷ ι ∷ π ∷ ῇ ∷ []) "1Cor.15.52"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ο ∷ ῦ ∷ []) "1Cor.15.52"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.52"
∷ word (τ ∷ ῇ ∷ []) "1Cor.15.52"
∷ word (ἐ ∷ σ ∷ χ ∷ ά ∷ τ ∷ ῃ ∷ []) "1Cor.15.52"
∷ word (σ ∷ ά ∷ ∙λ ∷ π ∷ ι ∷ γ ∷ γ ∷ ι ∷ []) "1Cor.15.52"
∷ word (σ ∷ α ∷ ∙λ ∷ π ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "1Cor.15.52"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.15.52"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.52"
∷ word (ο ∷ ἱ ∷ []) "1Cor.15.52"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ο ∷ ὶ ∷ []) "1Cor.15.52"
∷ word (ἐ ∷ γ ∷ ε ∷ ρ ∷ θ ∷ ή ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.52"
∷ word (ἄ ∷ φ ∷ θ ∷ α ∷ ρ ∷ τ ∷ ο ∷ ι ∷ []) "1Cor.15.52"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.52"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.15.52"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ α ∷ γ ∷ η ∷ σ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1Cor.15.52"
∷ word (δ ∷ ε ∷ ῖ ∷ []) "1Cor.15.53"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.15.53"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.53"
∷ word (φ ∷ θ ∷ α ∷ ρ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.15.53"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.15.53"
∷ word (ἐ ∷ ν ∷ δ ∷ ύ ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.15.53"
∷ word (ἀ ∷ φ ∷ θ ∷ α ∷ ρ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.15.53"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.53"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.53"
∷ word (θ ∷ ν ∷ η ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.15.53"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.15.53"
∷ word (ἐ ∷ ν ∷ δ ∷ ύ ∷ σ ∷ α ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.15.53"
∷ word (ἀ ∷ θ ∷ α ∷ ν ∷ α ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.15.53"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "1Cor.15.54"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.54"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.54"
∷ word (φ ∷ θ ∷ α ∷ ρ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.15.54"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.15.54"
∷ word (ἐ ∷ ν ∷ δ ∷ ύ ∷ σ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.54"
∷ word (ἀ ∷ φ ∷ θ ∷ α ∷ ρ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.15.54"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.15.54"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.54"
∷ word (θ ∷ ν ∷ η ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.15.54"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1Cor.15.54"
∷ word (ἐ ∷ ν ∷ δ ∷ ύ ∷ σ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.54"
∷ word (ἀ ∷ θ ∷ α ∷ ν ∷ α ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.15.54"
∷ word (τ ∷ ό ∷ τ ∷ ε ∷ []) "1Cor.15.54"
∷ word (γ ∷ ε ∷ ν ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.15.54"
∷ word (ὁ ∷ []) "1Cor.15.54"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "1Cor.15.54"
∷ word (ὁ ∷ []) "1Cor.15.54"
∷ word (γ ∷ ε ∷ γ ∷ ρ ∷ α ∷ μ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "1Cor.15.54"
∷ word (Κ ∷ α ∷ τ ∷ ε ∷ π ∷ ό ∷ θ ∷ η ∷ []) "1Cor.15.54"
∷ word (ὁ ∷ []) "1Cor.15.54"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.15.54"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.15.54"
∷ word (ν ∷ ῖ ∷ κ ∷ ο ∷ ς ∷ []) "1Cor.15.54"
∷ word (π ∷ ο ∷ ῦ ∷ []) "1Cor.15.55"
∷ word (σ ∷ ο ∷ υ ∷ []) "1Cor.15.55"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ε ∷ []) "1Cor.15.55"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.55"
∷ word (ν ∷ ῖ ∷ κ ∷ ο ∷ ς ∷ []) "1Cor.15.55"
∷ word (π ∷ ο ∷ ῦ ∷ []) "1Cor.15.55"
∷ word (σ ∷ ο ∷ υ ∷ []) "1Cor.15.55"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ε ∷ []) "1Cor.15.55"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.55"
∷ word (κ ∷ έ ∷ ν ∷ τ ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.15.55"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.56"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.56"
∷ word (κ ∷ έ ∷ ν ∷ τ ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.15.56"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.56"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.15.56"
∷ word (ἡ ∷ []) "1Cor.15.56"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "1Cor.15.56"
∷ word (ἡ ∷ []) "1Cor.15.56"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.56"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ μ ∷ ι ∷ ς ∷ []) "1Cor.15.56"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.15.56"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.15.56"
∷ word (ὁ ∷ []) "1Cor.15.56"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "1Cor.15.56"
∷ word (τ ∷ ῷ ∷ []) "1Cor.15.57"
∷ word (δ ∷ ὲ ∷ []) "1Cor.15.57"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1Cor.15.57"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "1Cor.15.57"
∷ word (τ ∷ ῷ ∷ []) "1Cor.15.57"
∷ word (δ ∷ ι ∷ δ ∷ ό ∷ ν ∷ τ ∷ ι ∷ []) "1Cor.15.57"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1Cor.15.57"
∷ word (τ ∷ ὸ ∷ []) "1Cor.15.57"
∷ word (ν ∷ ῖ ∷ κ ∷ ο ∷ ς ∷ []) "1Cor.15.57"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1Cor.15.57"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.57"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.15.57"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.15.57"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.15.57"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.57"
∷ word (Ὥ ∷ σ ∷ τ ∷ ε ∷ []) "1Cor.15.58"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.15.58"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.15.58"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ί ∷ []) "1Cor.15.58"
∷ word (ἑ ∷ δ ∷ ρ ∷ α ∷ ῖ ∷ ο ∷ ι ∷ []) "1Cor.15.58"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.15.58"
∷ word (ἀ ∷ μ ∷ ε ∷ τ ∷ α ∷ κ ∷ ί ∷ ν ∷ η ∷ τ ∷ ο ∷ ι ∷ []) "1Cor.15.58"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.15.58"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.58"
∷ word (τ ∷ ῷ ∷ []) "1Cor.15.58"
∷ word (ἔ ∷ ρ ∷ γ ∷ ῳ ∷ []) "1Cor.15.58"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.15.58"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.15.58"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ο ∷ τ ∷ ε ∷ []) "1Cor.15.58"
∷ word (ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.15.58"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.15.58"
∷ word (ὁ ∷ []) "1Cor.15.58"
∷ word (κ ∷ ό ∷ π ∷ ο ∷ ς ∷ []) "1Cor.15.58"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.15.58"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.15.58"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1Cor.15.58"
∷ word (κ ∷ ε ∷ ν ∷ ὸ ∷ ς ∷ []) "1Cor.15.58"
∷ word (ἐ ∷ ν ∷ []) "1Cor.15.58"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.15.58"
∷ word (Π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1Cor.16.1"
∷ word (δ ∷ ὲ ∷ []) "1Cor.16.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.16.1"
∷ word (∙λ ∷ ο ∷ γ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "1Cor.16.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.16.1"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.16.1"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.16.1"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.16.1"
∷ word (ὥ ∷ σ ∷ π ∷ ε ∷ ρ ∷ []) "1Cor.16.1"
∷ word (δ ∷ ι ∷ έ ∷ τ ∷ α ∷ ξ ∷ α ∷ []) "1Cor.16.1"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "1Cor.16.1"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "1Cor.16.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.16.1"
∷ word (Γ ∷ α ∷ ∙λ ∷ α ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.16.1"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.16.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.16.1"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.16.1"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.16.1"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1Cor.16.2"
∷ word (μ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.16.2"
∷ word (σ ∷ α ∷ β ∷ β ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.16.2"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1Cor.16.2"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.16.2"
∷ word (π ∷ α ∷ ρ ∷ []) "1Cor.16.2"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ῷ ∷ []) "1Cor.16.2"
∷ word (τ ∷ ι ∷ θ ∷ έ ∷ τ ∷ ω ∷ []) "1Cor.16.2"
∷ word (θ ∷ η ∷ σ ∷ α ∷ υ ∷ ρ ∷ ί ∷ ζ ∷ ω ∷ ν ∷ []) "1Cor.16.2"
∷ word (ὅ ∷ []) "1Cor.16.2"
∷ word (τ ∷ ι ∷ []) "1Cor.16.2"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.16.2"
∷ word (ε ∷ ὐ ∷ ο ∷ δ ∷ ῶ ∷ τ ∷ α ∷ ι ∷ []) "1Cor.16.2"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.16.2"
∷ word (μ ∷ ὴ ∷ []) "1Cor.16.2"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "1Cor.16.2"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ω ∷ []) "1Cor.16.2"
∷ word (τ ∷ ό ∷ τ ∷ ε ∷ []) "1Cor.16.2"
∷ word (∙λ ∷ ο ∷ γ ∷ ε ∷ ῖ ∷ α ∷ ι ∷ []) "1Cor.16.2"
∷ word (γ ∷ ί ∷ ν ∷ ω ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.16.2"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "1Cor.16.3"
∷ word (δ ∷ ὲ ∷ []) "1Cor.16.3"
∷ word (π ∷ α ∷ ρ ∷ α ∷ γ ∷ έ ∷ ν ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "1Cor.16.3"
∷ word (ο ∷ ὓ ∷ ς ∷ []) "1Cor.16.3"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.16.3"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ά ∷ σ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.16.3"
∷ word (δ ∷ ι ∷ []) "1Cor.16.3"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "1Cor.16.3"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.16.3"
∷ word (π ∷ έ ∷ μ ∷ ψ ∷ ω ∷ []) "1Cor.16.3"
∷ word (ἀ ∷ π ∷ ε ∷ ν ∷ ε ∷ γ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.16.3"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.16.3"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "1Cor.16.3"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.16.3"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.16.3"
∷ word (Ἰ ∷ ε ∷ ρ ∷ ο ∷ υ ∷ σ ∷ α ∷ ∙λ ∷ ή ∷ μ ∷ []) "1Cor.16.3"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.16.4"
∷ word (δ ∷ ὲ ∷ []) "1Cor.16.4"
∷ word (ἄ ∷ ξ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.16.4"
∷ word (ᾖ ∷ []) "1Cor.16.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.16.4"
∷ word (κ ∷ ἀ ∷ μ ∷ ὲ ∷ []) "1Cor.16.4"
∷ word (π ∷ ο ∷ ρ ∷ ε ∷ ύ ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "1Cor.16.4"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "1Cor.16.4"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "1Cor.16.4"
∷ word (π ∷ ο ∷ ρ ∷ ε ∷ ύ ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.16.4"
∷ word (Ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.16.5"
∷ word (δ ∷ ὲ ∷ []) "1Cor.16.5"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.16.5"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.16.5"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "1Cor.16.5"
∷ word (Μ ∷ α ∷ κ ∷ ε ∷ δ ∷ ο ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "1Cor.16.5"
∷ word (δ ∷ ι ∷ έ ∷ ∙λ ∷ θ ∷ ω ∷ []) "1Cor.16.5"
∷ word (Μ ∷ α ∷ κ ∷ ε ∷ δ ∷ ο ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "1Cor.16.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.16.5"
∷ word (δ ∷ ι ∷ έ ∷ ρ ∷ χ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.16.5"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.16.6"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.16.6"
∷ word (δ ∷ ὲ ∷ []) "1Cor.16.6"
∷ word (τ ∷ υ ∷ χ ∷ ὸ ∷ ν ∷ []) "1Cor.16.6"
∷ word (π ∷ α ∷ ρ ∷ α ∷ μ ∷ ε ∷ ν ∷ ῶ ∷ []) "1Cor.16.6"
∷ word (ἢ ∷ []) "1Cor.16.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.16.6"
∷ word (π ∷ α ∷ ρ ∷ α ∷ χ ∷ ε ∷ ι ∷ μ ∷ ά ∷ σ ∷ ω ∷ []) "1Cor.16.6"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.16.6"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.16.6"
∷ word (μ ∷ ε ∷ []) "1Cor.16.6"
∷ word (π ∷ ρ ∷ ο ∷ π ∷ έ ∷ μ ∷ ψ ∷ η ∷ τ ∷ ε ∷ []) "1Cor.16.6"
∷ word (ο ∷ ὗ ∷ []) "1Cor.16.6"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.16.6"
∷ word (π ∷ ο ∷ ρ ∷ ε ∷ ύ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "1Cor.16.6"
∷ word (ο ∷ ὐ ∷ []) "1Cor.16.7"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "1Cor.16.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.16.7"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.16.7"
∷ word (ἄ ∷ ρ ∷ τ ∷ ι ∷ []) "1Cor.16.7"
∷ word (ἐ ∷ ν ∷ []) "1Cor.16.7"
∷ word (π ∷ α ∷ ρ ∷ ό ∷ δ ∷ ῳ ∷ []) "1Cor.16.7"
∷ word (ἰ ∷ δ ∷ ε ∷ ῖ ∷ ν ∷ []) "1Cor.16.7"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ ζ ∷ ω ∷ []) "1Cor.16.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.16.7"
∷ word (χ ∷ ρ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "1Cor.16.7"
∷ word (τ ∷ ι ∷ ν ∷ ὰ ∷ []) "1Cor.16.7"
∷ word (ἐ ∷ π ∷ ι ∷ μ ∷ ε ∷ ῖ ∷ ν ∷ α ∷ ι ∷ []) "1Cor.16.7"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.16.7"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.16.7"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.16.7"
∷ word (ὁ ∷ []) "1Cor.16.7"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "1Cor.16.7"
∷ word (ἐ ∷ π ∷ ι ∷ τ ∷ ρ ∷ έ ∷ ψ ∷ ῃ ∷ []) "1Cor.16.7"
∷ word (ἐ ∷ π ∷ ι ∷ μ ∷ ε ∷ ν ∷ ῶ ∷ []) "1Cor.16.8"
∷ word (δ ∷ ὲ ∷ []) "1Cor.16.8"
∷ word (ἐ ∷ ν ∷ []) "1Cor.16.8"
∷ word (Ἐ ∷ φ ∷ έ ∷ σ ∷ ῳ ∷ []) "1Cor.16.8"
∷ word (ἕ ∷ ω ∷ ς ∷ []) "1Cor.16.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.16.8"
∷ word (π ∷ ε ∷ ν ∷ τ ∷ η ∷ κ ∷ ο ∷ σ ∷ τ ∷ ῆ ∷ ς ∷ []) "1Cor.16.8"
∷ word (θ ∷ ύ ∷ ρ ∷ α ∷ []) "1Cor.16.9"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1Cor.16.9"
∷ word (μ ∷ ο ∷ ι ∷ []) "1Cor.16.9"
∷ word (ἀ ∷ ν ∷ έ ∷ ῳ ∷ γ ∷ ε ∷ ν ∷ []) "1Cor.16.9"
∷ word (μ ∷ ε ∷ γ ∷ ά ∷ ∙λ ∷ η ∷ []) "1Cor.16.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.16.9"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ή ∷ ς ∷ []) "1Cor.16.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.16.9"
∷ word (ἀ ∷ ν ∷ τ ∷ ι ∷ κ ∷ ε ∷ ί ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "1Cor.16.9"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ί ∷ []) "1Cor.16.9"
∷ word (Ἐ ∷ ὰ ∷ ν ∷ []) "1Cor.16.10"
∷ word (δ ∷ ὲ ∷ []) "1Cor.16.10"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ῃ ∷ []) "1Cor.16.10"
∷ word (Τ ∷ ι ∷ μ ∷ ό ∷ θ ∷ ε ∷ ο ∷ ς ∷ []) "1Cor.16.10"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.16.10"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.16.10"
∷ word (ἀ ∷ φ ∷ ό ∷ β ∷ ω ∷ ς ∷ []) "1Cor.16.10"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "1Cor.16.10"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.16.10"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.16.10"
∷ word (τ ∷ ὸ ∷ []) "1Cor.16.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.16.10"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ν ∷ []) "1Cor.16.10"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.16.10"
∷ word (ἐ ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.16.10"
∷ word (ὡ ∷ ς ∷ []) "1Cor.16.10"
∷ word (κ ∷ ἀ ∷ γ ∷ ώ ∷ []) "1Cor.16.10"
∷ word (μ ∷ ή ∷ []) "1Cor.16.11"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.16.11"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.16.11"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.16.11"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ θ ∷ ε ∷ ν ∷ ή ∷ σ ∷ ῃ ∷ []) "1Cor.16.11"
∷ word (π ∷ ρ ∷ ο ∷ π ∷ έ ∷ μ ∷ ψ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.16.11"
∷ word (δ ∷ ὲ ∷ []) "1Cor.16.11"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.16.11"
∷ word (ἐ ∷ ν ∷ []) "1Cor.16.11"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ ῃ ∷ []) "1Cor.16.11"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.16.11"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ῃ ∷ []) "1Cor.16.11"
∷ word (π ∷ ρ ∷ ό ∷ ς ∷ []) "1Cor.16.11"
∷ word (μ ∷ ε ∷ []) "1Cor.16.11"
∷ word (ἐ ∷ κ ∷ δ ∷ έ ∷ χ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "1Cor.16.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.16.11"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.16.11"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "1Cor.16.11"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.16.11"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ῶ ∷ ν ∷ []) "1Cor.16.11"
∷ word (Π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1Cor.16.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.16.12"
∷ word (Ἀ ∷ π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ []) "1Cor.16.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.16.12"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ῦ ∷ []) "1Cor.16.12"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.16.12"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ κ ∷ ά ∷ ∙λ ∷ ε ∷ σ ∷ α ∷ []) "1Cor.16.12"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "1Cor.16.12"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.16.12"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ῃ ∷ []) "1Cor.16.12"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1Cor.16.12"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.16.12"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "1Cor.16.12"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1Cor.16.12"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ῶ ∷ ν ∷ []) "1Cor.16.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.16.12"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ς ∷ []) "1Cor.16.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1Cor.16.12"
∷ word (ἦ ∷ ν ∷ []) "1Cor.16.12"
∷ word (θ ∷ έ ∷ ∙λ ∷ η ∷ μ ∷ α ∷ []) "1Cor.16.12"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.16.12"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "1Cor.16.12"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ῃ ∷ []) "1Cor.16.12"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.16.12"
∷ word (δ ∷ ὲ ∷ []) "1Cor.16.12"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "1Cor.16.12"
∷ word (ε ∷ ὐ ∷ κ ∷ α ∷ ι ∷ ρ ∷ ή ∷ σ ∷ ῃ ∷ []) "1Cor.16.12"
∷ word (Γ ∷ ρ ∷ η ∷ γ ∷ ο ∷ ρ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "1Cor.16.13"
∷ word (σ ∷ τ ∷ ή ∷ κ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.16.13"
∷ word (ἐ ∷ ν ∷ []) "1Cor.16.13"
∷ word (τ ∷ ῇ ∷ []) "1Cor.16.13"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ι ∷ []) "1Cor.16.13"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ί ∷ ζ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.16.13"
∷ word (κ ∷ ρ ∷ α ∷ τ ∷ α ∷ ι ∷ ο ∷ ῦ ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.16.13"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1Cor.16.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.16.14"
∷ word (ἐ ∷ ν ∷ []) "1Cor.16.14"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "1Cor.16.14"
∷ word (γ ∷ ι ∷ ν ∷ έ ∷ σ ∷ θ ∷ ω ∷ []) "1Cor.16.14"
∷ word (Π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ ῶ ∷ []) "1Cor.16.15"
∷ word (δ ∷ ὲ ∷ []) "1Cor.16.15"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.16.15"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1Cor.16.15"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1Cor.16.15"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1Cor.16.15"
∷ word (ο ∷ ἰ ∷ κ ∷ ί ∷ α ∷ ν ∷ []) "1Cor.16.15"
∷ word (Σ ∷ τ ∷ ε ∷ φ ∷ α ∷ ν ∷ ᾶ ∷ []) "1Cor.16.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.16.15"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1Cor.16.15"
∷ word (ἀ ∷ π ∷ α ∷ ρ ∷ χ ∷ ὴ ∷ []) "1Cor.16.15"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.16.15"
∷ word (Ἀ ∷ χ ∷ α ∷ ΐ ∷ α ∷ ς ∷ []) "1Cor.16.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.16.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1Cor.16.15"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "1Cor.16.15"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.16.15"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.16.15"
∷ word (ἔ ∷ τ ∷ α ∷ ξ ∷ α ∷ ν ∷ []) "1Cor.16.15"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ύ ∷ ς ∷ []) "1Cor.16.15"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1Cor.16.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.16.16"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1Cor.16.16"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ ά ∷ σ ∷ σ ∷ η ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.16.16"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1Cor.16.16"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ύ ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "1Cor.16.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.16.16"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "1Cor.16.16"
∷ word (τ ∷ ῷ ∷ []) "1Cor.16.16"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ι ∷ []) "1Cor.16.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.16.16"
∷ word (κ ∷ ο ∷ π ∷ ι ∷ ῶ ∷ ν ∷ τ ∷ ι ∷ []) "1Cor.16.16"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ω ∷ []) "1Cor.16.17"
∷ word (δ ∷ ὲ ∷ []) "1Cor.16.17"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "1Cor.16.17"
∷ word (τ ∷ ῇ ∷ []) "1Cor.16.17"
∷ word (π ∷ α ∷ ρ ∷ ο ∷ υ ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.16.17"
∷ word (Σ ∷ τ ∷ ε ∷ φ ∷ α ∷ ν ∷ ᾶ ∷ []) "1Cor.16.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.16.17"
∷ word (Φ ∷ ο ∷ ρ ∷ τ ∷ ο ∷ υ ∷ ν ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "1Cor.16.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.16.17"
∷ word (Ἀ ∷ χ ∷ α ∷ ϊ ∷ κ ∷ ο ∷ ῦ ∷ []) "1Cor.16.17"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1Cor.16.17"
∷ word (τ ∷ ὸ ∷ []) "1Cor.16.17"
∷ word (ὑ ∷ μ ∷ έ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "1Cor.16.17"
∷ word (ὑ ∷ σ ∷ τ ∷ έ ∷ ρ ∷ η ∷ μ ∷ α ∷ []) "1Cor.16.17"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ι ∷ []) "1Cor.16.17"
∷ word (ἀ ∷ ν ∷ ε ∷ π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "1Cor.16.17"
∷ word (ἀ ∷ ν ∷ έ ∷ π ∷ α ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "1Cor.16.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1Cor.16.18"
∷ word (τ ∷ ὸ ∷ []) "1Cor.16.18"
∷ word (ἐ ∷ μ ∷ ὸ ∷ ν ∷ []) "1Cor.16.18"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1Cor.16.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.16.18"
∷ word (τ ∷ ὸ ∷ []) "1Cor.16.18"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.16.18"
∷ word (ἐ ∷ π ∷ ι ∷ γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ε ∷ τ ∷ ε ∷ []) "1Cor.16.18"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "1Cor.16.18"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1Cor.16.18"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.16.18"
∷ word (Ἀ ∷ σ ∷ π ∷ ά ∷ ζ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.16.19"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.16.19"
∷ word (α ∷ ἱ ∷ []) "1Cor.16.19"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ι ∷ []) "1Cor.16.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1Cor.16.19"
∷ word (Ἀ ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "1Cor.16.19"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1Cor.16.19"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.16.19"
∷ word (ἐ ∷ ν ∷ []) "1Cor.16.19"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "1Cor.16.19"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1Cor.16.19"
∷ word (Ἀ ∷ κ ∷ ύ ∷ ∙λ ∷ α ∷ ς ∷ []) "1Cor.16.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1Cor.16.19"
∷ word (Π ∷ ρ ∷ ί ∷ σ ∷ κ ∷ α ∷ []) "1Cor.16.19"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "1Cor.16.19"
∷ word (τ ∷ ῇ ∷ []) "1Cor.16.19"
∷ word (κ ∷ α ∷ τ ∷ []) "1Cor.16.19"
∷ word (ο ∷ ἶ ∷ κ ∷ ο ∷ ν ∷ []) "1Cor.16.19"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1Cor.16.19"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "1Cor.16.19"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ ζ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1Cor.16.20"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1Cor.16.20"
∷ word (ο ∷ ἱ ∷ []) "1Cor.16.20"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ὶ ∷ []) "1Cor.16.20"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1Cor.16.20"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "1Cor.16.20"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1Cor.16.20"
∷ word (ἐ ∷ ν ∷ []) "1Cor.16.20"
∷ word (φ ∷ ι ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1Cor.16.20"
∷ word (ἁ ∷ γ ∷ ί ∷ ῳ ∷ []) "1Cor.16.20"
∷ word (Ὁ ∷ []) "1Cor.16.21"
∷ word (ἀ ∷ σ ∷ π ∷ α ∷ σ ∷ μ ∷ ὸ ∷ ς ∷ []) "1Cor.16.21"
∷ word (τ ∷ ῇ ∷ []) "1Cor.16.21"
∷ word (ἐ ∷ μ ∷ ῇ ∷ []) "1Cor.16.21"
∷ word (χ ∷ ε ∷ ι ∷ ρ ∷ ὶ ∷ []) "1Cor.16.21"
∷ word (Π ∷ α ∷ ύ ∷ ∙λ ∷ ο ∷ υ ∷ []) "1Cor.16.21"
∷ word (ε ∷ ἴ ∷ []) "1Cor.16.22"
∷ word (τ ∷ ι ∷ ς ∷ []) "1Cor.16.22"
∷ word (ο ∷ ὐ ∷ []) "1Cor.16.22"
∷ word (φ ∷ ι ∷ ∙λ ∷ ε ∷ ῖ ∷ []) "1Cor.16.22"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1Cor.16.22"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "1Cor.16.22"
∷ word (ἤ ∷ τ ∷ ω ∷ []) "1Cor.16.22"
∷ word (ἀ ∷ ν ∷ ά ∷ θ ∷ ε ∷ μ ∷ α ∷ []) "1Cor.16.22"
∷ word (Μ ∷ α ∷ ρ ∷ ά ∷ ν ∷ α ∷ []) "1Cor.16.22"
∷ word (θ ∷ ά ∷ []) "1Cor.16.22"
∷ word (ἡ ∷ []) "1Cor.16.23"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "1Cor.16.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1Cor.16.23"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "1Cor.16.23"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.16.23"
∷ word (μ ∷ ε ∷ θ ∷ []) "1Cor.16.23"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.16.23"
∷ word (ἡ ∷ []) "1Cor.16.24"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1Cor.16.24"
∷ word (μ ∷ ο ∷ υ ∷ []) "1Cor.16.24"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "1Cor.16.24"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1Cor.16.24"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "1Cor.16.24"
∷ word (ἐ ∷ ν ∷ []) "1Cor.16.24"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1Cor.16.24"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1Cor.16.24"
∷ []
| 45.471933
| 86
| 0.351327
|
1c08b2d9fe68d84d5fb0fa1648c8c509ece8a357
| 6,067
|
agda
|
Agda
|
Agda/GTFL.agda
|
kellino/TypeSystems
|
acf5a153e14a7bdc0c9332fa602fa369fe7add46
|
[
"MIT"
] | 2
|
2016-10-27T08:05:40.000Z
|
2017-05-26T23:06:17.000Z
|
Agda/GTFL.agda
|
kellino/TypeSystems
|
acf5a153e14a7bdc0c9332fa602fa369fe7add46
|
[
"MIT"
] | null | null | null |
Agda/GTFL.agda
|
kellino/TypeSystems
|
acf5a153e14a7bdc0c9332fa602fa369fe7add46
|
[
"MIT"
] | null | null | null |
module GTFL where
open import Data.Nat hiding (_⊓_; erase; _≟_; _≤_)
open import Data.Bool hiding (_≟_)
open import Data.Fin using (Fin; zero; suc; toℕ)
open import Data.Vec
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary
open import Data.Empty
open import Function using (_∘_)
-- | Types
infixr 30 _⇒_
data GType : Set where
nat : GType
bool : GType
_⇒_ : GType → GType → GType
✭ : GType
err : GType -- easier to model this as a type in Agda
-- | Untyped Expressions
data Expr : Set where
litNat : ℕ → Expr
litBool : Bool → Expr
dyn : Expr
err : Expr
var : ℕ → Expr
lam : GType → Expr → Expr
_∙_ : Expr → Expr → Expr
_⊕_ : Expr → Expr → Expr
if_thn_els_ : Expr → Expr → Expr → Expr
Ctx : ℕ → Set
Ctx = Vec GType
infixr 10 _~_
data _~_ {A B : Set} (x : A) (y : B) : Set where
cons : x ~ y
~dom : ∀ (t : GType) → GType
~dom (t ⇒ t₁) = t₁
~dom _ = err
~cod : ∀ (t : GType) → GType
~cod (t ⇒ t₁) = t₁
~cod _ = err
_⊓_ : ∀ (t₁ t₂ : GType) → GType
nat ⊓ nat = nat
bool ⊓ bool = bool
t₁ ⊓ ✭ = t₁
✭ ⊓ t₂ = t₂
(t₁ ⇒ t₂) ⊓ (t₃ ⇒ t₄) = (t₁ ⊓ t₃) ⇒ (t₂ ⊓ t₄)
_ ⊓ _ = err
-- | Typed Terms
data Term {n} (Γ : Ctx n) : GType → Set where
Tx : ∀ {t} (v : Fin n) → t ≡ lookup v Γ → Term Γ t
Tn : ℕ → Term Γ nat
Tb : Bool → Term Γ bool
Tdy : Term Γ ✭
_T∙_ : ∀ {t₁ t₂} → Term Γ t₁ → Term Γ t₂ → t₂ ~ (~dom t₁) → Term Γ (~cod t₁)
_T⊕_ : ∀ {t₁ t₂} → Term Γ t₁ → Term Γ t₂ → (t₁ ~ nat) → (t₂ ~ nat) → Term Γ (t₁ ⊓ t₂)
Tif : ∀ {t₁ t₂ t₃} → Term Γ t₁ → Term Γ t₂ → Term Γ t₃ → (t₁ ~ bool) → Term Γ (t₂ ⊓ t₃)
Tlam : ∀ t₁ {t₂} → Term (t₁ ∷ Γ) t₂ → Term Γ (t₁ ⇒ t₂)
erase : ∀ {n} {Γ : Ctx n} {t} → Term Γ t → Expr
erase (Tx v x) = var (toℕ v)
erase (Tn x) = litNat x
erase (Tb x) = litBool x
erase Tdy = dyn
erase ((term T∙ term₁) _) = (erase term) ∙ (erase term₁)
erase ((term T⊕ term₁) _ _) = (erase term) ⊕ (erase term₁)
erase (Tif b tt ff _) = if erase b thn erase tt els erase ff
erase (Tlam t₁ term) = lam t₁ (erase term)
data Fromℕ (n : ℕ) : ℕ → Set where
yes : (m : Fin n) → Fromℕ n (toℕ m)
no : (m : ℕ) → Fromℕ n (n + m)
fromℕ : ∀ n m → Fromℕ n m
fromℕ zero m = no m
fromℕ (suc n) zero = yes zero
fromℕ (suc n) (suc m) with fromℕ n m
fromℕ (suc n) (suc .(toℕ m)) | yes m = yes (suc m)
fromℕ (suc n) (suc .(n + m)) | no m = no m
data Check {n} (Γ : Ctx n) : Expr → Set where
yes : (τ : GType) (t : Term Γ τ) → Check Γ (erase t)
no : {e : Expr} → Check Γ e
staticCheck : ∀ {n} (Γ : Ctx n) (t : Expr) → Check Γ t
-- | primitives
staticCheck Γ (litNat x) = yes nat (Tn x)
staticCheck Γ (litBool x) = yes bool (Tb x)
staticCheck {n} Γ dyn = yes ✭ Tdy
staticCheck Γ err = no
-- | var lookup
staticCheck {n} Γ (var v) with fromℕ n v
staticCheck {n} Γ (var .(toℕ m)) | yes m = yes (lookup m Γ) (Tx m refl)
staticCheck {n} Γ (var .(n + m)) | no m = no
-- | lambda abstraction
staticCheck Γ (lam x t) with staticCheck (x ∷ Γ) t
staticCheck Γ (lam x .(erase t)) | yes τ t = yes (x ⇒ τ) (Tlam x t) -- double check this
staticCheck Γ (lam x t) | no = no
-- | application
staticCheck Γ (t₁ ∙ t₂) with staticCheck Γ t₁ | staticCheck Γ t₂
staticCheck Γ (.(erase t₁) ∙ .(erase t)) | yes (τ₁ ⇒ τ₂) t₁ | (yes τ t) = yes τ₂ ((t₁ T∙ t) cons)
staticCheck Γ (.(erase t₁) ∙ .(erase t)) | yes _ t₁ | (yes τ t) = no -- not sure about this
staticCheck Γ (t₁ ∙ t₂) | _ | _ = no
-- | addition
staticCheck Γ (t₁ ⊕ t₂) with staticCheck Γ t₁ | staticCheck Γ t₂
staticCheck Γ (.(erase t₁) ⊕ .(erase t)) | yes nat t₁ | (yes nat t) = yes nat ((t₁ T⊕ t) cons cons)
staticCheck Γ (.(erase t₁) ⊕ .(erase t)) | yes ✭ t₁ | (yes nat t) = yes (✭ ⊓ nat) ((t₁ T⊕ t) cons cons)
staticCheck Γ (.(erase t₁) ⊕ .(erase t)) | yes nat t₁ | (yes ✭ t) = yes (nat ⊓ ✭) ((t₁ T⊕ t) cons cons)
staticCheck Γ (.(erase t₁) ⊕ .(erase t)) | yes ✭ t₁ | (yes ✭ t) = yes ✭ ((t₁ T⊕ t) cons cons)
staticCheck Γ (t₁ ⊕ t₂) | _ | _ = no
-- | if ... then ... else
staticCheck Γ (if t thn t₁ els t₂) with staticCheck Γ t
staticCheck Γ (if .(erase t) thn t₁ els t₂) | yes bool t with staticCheck Γ t₁ | staticCheck Γ t₂
staticCheck Γ (if .(erase t₂) thn .(erase t₁) els .(erase t)) | yes bool t₂ | (yes τ₁ t₁) | (yes τ₂ t) = yes (τ₁ ⊓ τ₂) (Tif t₂ t₁ t cons)
staticCheck Γ (if .(erase t₁) thn .(erase t) els t₂) | yes bool t₁ | (yes τ t) | no = no
staticCheck Γ (if .(erase t₂) thn t₁ els .(erase t)) | yes bool t₂ | no | (yes τ t) = no
staticCheck Γ (if .(erase t) thn t₁ els t₂) | yes bool t | no | _ = no
staticCheck Γ (if .(erase t) thn t₁ els t₂) | yes ✭ t with staticCheck Γ t₁ | staticCheck Γ t₂
staticCheck Γ (if .(erase t₂) thn .(erase t₁) els .(erase t)) | yes ✭ t₂ | (yes τ₁ t₁) | (yes τ₂ t) = yes (τ₁ ⊓ τ₂) (Tif t₂ t₁ t cons)
staticCheck Γ (if .(erase t₁) thn .(erase t) els t₂) | yes ✭ t₁ | (yes τ t) | no = no
staticCheck Γ (if .(erase t₂) thn t₁ els .(erase t)) | yes ✭ t₂ | no | (yes τ t) = no
staticCheck Γ (if .(erase t) thn t₁ els t₂) | yes ✭ t | no | no = no
staticCheck Γ (if .(erase t) thn t₁ els t₂) | yes _ t = no
staticCheck Γ (if t thn t₁ els t₂) | no = no
extractType : ∀ {n} {Γ : Ctx n} {t : Expr} → Check Γ t → GType
extractType (yes τ t) = τ
extractType no = err
-- Type Precision
data _⊑_ : GType → GType → Set where
n⊑✭ : nat ⊑ ✭
b⊑✭ : bool ⊑ ✭
⇒⊑ : ∀ (t₁ t₂ : GType) → (t₁ ⇒ t₂) ⊑ ✭
n⊑n : nat ⊑ nat
b⊑b : bool ⊑ bool
✭⊑✭ : ✭ ⊑ ✭
app⊑ : ∀ (t₁ t₂ t₃ t₄ : GType) → t₁ ⊑ t₃ → t₂ ⊑ t₄ → (t₁ ⇒ t₃) ⊑ (t₃ ⇒ t₄)
-- Term Precision
data _≤_ : Expr → Expr → Set where
n≤n : ∀ {n} → litNat n ≤ litNat n
b≤b : ∀ {b} → litBool b ≤ litBool b
n≤✭ : ∀ {n} → litNat n ≤ dyn
b≤✭ : ∀ {b} → litBool b ≤ dyn
d≤d : dyn ≤ dyn
ssG : ∀ {n} {Γ : Ctx n} {e₁ e₂ : Expr} → e₁ ≤ e₂ → extractType (staticCheck Γ e₁) ⊑ extractType (staticCheck Γ e₂)
ssG n≤n = n⊑n
ssG b≤b = b⊑b
ssG n≤✭ = n⊑✭
ssG b≤✭ = b⊑✭
ssG d≤d = ✭⊑✭
| 36.113095
| 137
| 0.54508
|
39e37311d24846dd264f6116a3a5e77d5800531f
| 259
|
agda
|
Agda
|
test/Fail/Issue444.agda
|
mdimjasevic/agda
|
8fb548356b275c7a1e79b768b64511ae937c738b
|
[
"BSD-3-Clause"
] | 1,989
|
2015-01-09T23:51:16.000Z
|
2022-03-30T18:20:48.000Z
|
test/Fail/Issue444.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 4,066
|
2015-01-10T11:24:51.000Z
|
2022-03-31T21:14:49.000Z
|
test/Fail/Issue444.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 371
|
2015-01-03T14:04:08.000Z
|
2022-03-30T19:00:30.000Z
|
-- 2011-09-09, submitted by mokus.4...@gmail.com
-- This bug report wins the first price in the false golfing tournament!
-- {-# OPTIONS -v term:20 #-}
module Issue444 where
data ⊥ : Set where
relevant : .⊥ → ⊥
relevant ()
false : ⊥
false = relevant false
| 19.923077
| 72
| 0.671815
|
0b10affb613c36472ba055500b910fd1b473634e
| 1,416
|
agda
|
Agda
|
test/asset/agda-stdlib-1.0/Axiom/Extensionality/Heterogeneous.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | 5
|
2020-10-07T12:07:53.000Z
|
2020-10-10T21:41:32.000Z
|
test/asset/agda-stdlib-1.0/Axiom/Extensionality/Heterogeneous.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | null | null | null |
test/asset/agda-stdlib-1.0/Axiom/Extensionality/Heterogeneous.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | 1
|
2021-11-04T06:54:45.000Z
|
2021-11-04T06:54:45.000Z
|
------------------------------------------------------------------------
-- The Agda standard library
--
-- Results concerning function extensionality for propositional equality
------------------------------------------------------------------------
{-# OPTIONS --with-K --safe #-}
module Axiom.Extensionality.Heterogeneous where
import Axiom.Extensionality.Propositional as P
open import Function
open import Level
open import Relation.Binary.HeterogeneousEquality.Core
open import Relation.Binary.PropositionalEquality.Core
------------------------------------------------------------------------
-- Function extensionality states that if two functions are
-- propositionally equal for every input, then the functions themselves
-- must be propositionally equal.
Extensionality : (a b : Level) → Set _
Extensionality a b =
{A : Set a} {B₁ B₂ : A → Set b}
{f₁ : (x : A) → B₁ x} {f₂ : (x : A) → B₂ x} →
(∀ x → B₁ x ≡ B₂ x) → (∀ x → f₁ x ≅ f₂ x) → f₁ ≅ f₂
------------------------------------------------------------------------
-- Properties
-- This form of extensionality follows from extensionality for _≡_.
≡-ext⇒≅-ext : ∀ {ℓ₁ ℓ₂} →
P.Extensionality ℓ₁ (suc ℓ₂) →
Extensionality ℓ₁ ℓ₂
≡-ext⇒≅-ext {ℓ₁} {ℓ₂} ext B₁≡B₂ f₁≅f₂ with ext B₁≡B₂
... | refl = ≡-to-≅ $ ext′ (≅-to-≡ ∘ f₁≅f₂)
where
ext′ : P.Extensionality ℓ₁ ℓ₂
ext′ = P.lower-extensionality ℓ₁ (suc ℓ₂) ext
| 34.536585
| 72
| 0.535311
|
cbb00df6757be28a5eeaec0136fa219272accdc4
| 2,644
|
agda
|
Agda
|
examples/AIM6/Path/Lambda.agda
|
mdimjasevic/agda
|
8fb548356b275c7a1e79b768b64511ae937c738b
|
[
"BSD-3-Clause"
] | 1,989
|
2015-01-09T23:51:16.000Z
|
2022-03-30T18:20:48.000Z
|
examples/AIM6/Path/Lambda.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 4,066
|
2015-01-10T11:24:51.000Z
|
2022-03-31T21:14:49.000Z
|
examples/AIM6/Path/Lambda.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 371
|
2015-01-03T14:04:08.000Z
|
2022-03-30T19:00:30.000Z
|
module Lambda where
open import Prelude
open import Star
open import Examples
open import Modal
-- Environments
record TyAlg (ty : Set) : Set where
field
nat : ty
_⟶_ : ty -> ty -> ty
data Ty : Set where
<nat> : Ty
_<⟶>_ : Ty -> Ty -> Ty
freeTyAlg : TyAlg Ty
freeTyAlg = record { nat = <nat>; _⟶_ = _<⟶>_ }
termTyAlg : TyAlg True
termTyAlg = record { nat = _; _⟶_ = \_ _ -> _ }
record TyArrow {ty₁ ty₂ : Set}(T₁ : TyAlg ty₁)(T₂ : TyAlg ty₂) : Set where
field
apply : ty₁ -> ty₂
respNat : apply (TyAlg.nat T₁) == TyAlg.nat T₂
resp⟶ : forall {τ₁ τ₂} ->
apply (TyAlg._⟶_ T₁ τ₁ τ₂) == TyAlg._⟶_ T₂ (apply τ₁) (apply τ₂)
_=Ty=>_ : {ty₁ ty₂ : Set}(T₁ : TyAlg ty₁)(T₂ : TyAlg ty₂) -> Set
_=Ty=>_ = TyArrow
!Ty : {ty : Set}{T : TyAlg ty} -> T =Ty=> termTyAlg
!Ty = record { apply = !
; respNat = refl
; resp⟶ = refl
}
Ctx : Set
Ctx = List Ty
Var : {ty : Set} -> List ty -> ty -> Set
Var Γ τ = Any (_==_ τ) Γ
vzero : {τ : Ty} {Γ : Ctx} -> Var (τ • Γ) τ
vzero = done refl • ε
vsuc : {σ τ : Ty} {Γ : Ctx} -> Var Γ τ -> Var (σ • Γ) τ
vsuc v = step • v
module Term {ty : Set}(T : TyAlg ty) where
private open module TT = TyAlg T
data Tm : List ty -> ty -> Set where
var : forall {Γ τ} -> Var Γ τ -> Tm Γ τ
zz : forall {Γ} -> Tm Γ nat
ss : forall {Γ} -> Tm Γ (nat ⟶ nat)
ƛ : forall {Γ σ τ} -> Tm (σ • Γ) τ -> Tm Γ (σ ⟶ τ)
_$_ : forall {Γ σ τ} -> Tm Γ (σ ⟶ τ) -> Tm Γ σ -> Tm Γ τ
module Eval where
private open module TT = Term freeTyAlg
ty⟦_⟧ : Ty -> Set
ty⟦ <nat> ⟧ = Nat
ty⟦ σ <⟶> τ ⟧ = ty⟦ σ ⟧ -> ty⟦ τ ⟧
Env : Ctx -> Set
Env = All ty⟦_⟧
_[_] : forall {Γ τ} -> Env Γ -> Var Γ τ -> ty⟦ τ ⟧
ρ [ x ] with lookup x ρ
... | result _ refl v = v
⟦_⟧_ : forall {Γ τ} -> Tm Γ τ -> Env Γ -> ty⟦ τ ⟧
⟦ var x ⟧ ρ = ρ [ x ]
⟦ zz ⟧ ρ = zero
⟦ ss ⟧ ρ = suc
⟦ ƛ t ⟧ ρ = \x -> ⟦ t ⟧ (check x • ρ)
⟦ s $ t ⟧ ρ = (⟦ s ⟧ ρ) (⟦ t ⟧ ρ)
module MoreExamples where
private open module TT = TyAlg freeTyAlg
private open module Tm = Term freeTyAlg
open Eval
tm-one : Tm ε nat
tm-one = ss $ zz
tm-id : Tm ε (nat ⟶ nat)
tm-id = ƛ (var (done refl • ε))
tm : Tm ε nat
tm = tm-id $ tm-one
tm-twice : Tm ε ((nat ⟶ nat) ⟶ (nat ⟶ nat))
tm-twice = ƛ (ƛ (f $ (f $ x)))
where Γ : Ctx
Γ = nat • (nat ⟶ nat) • ε
f : Tm Γ (nat ⟶ nat)
f = var (vsuc vzero)
x : Tm Γ nat
x = var vzero
sem : {τ : Ty} -> Tm ε τ -> ty⟦ τ ⟧
sem e = ⟦ e ⟧ ε
one : Nat
one = sem tm
twice : (Nat -> Nat) -> (Nat -> Nat)
twice = sem tm-twice
| 22.40678
| 78
| 0.495461
|
0e76dd38ddd72aaf0d0117659386c518084820db
| 113
|
agda
|
Agda
|
Everything.agda
|
fangyi-zhou/mpst-in-agda
|
3d12eed9d340207d242d70f43c6b34e01d3620de
|
[
"MIT"
] | 1
|
2021-08-14T17:36:53.000Z
|
2021-08-14T17:36:53.000Z
|
Everything.agda
|
fangyi-zhou/mpst-in-agda
|
3d12eed9d340207d242d70f43c6b34e01d3620de
|
[
"MIT"
] | 1
|
2021-08-31T10:15:38.000Z
|
2021-11-24T11:30:17.000Z
|
Everything.agda
|
fangyi-zhou/mpst-in-agda
|
3d12eed9d340207d242d70f43c6b34e01d3620de
|
[
"MIT"
] | null | null | null |
import Common
import Global
import Local
import Projection
import Soundness
import Completeness
import Example
| 11.3
| 19
| 0.858407
|
cbed1b585950c0077ef9ba8e39a52dec9741da31
| 9,678
|
agda
|
Agda
|
lib/Equivalences.agda
|
sattlerc/HoTT-Agda
|
c8fb8da3354fc9e0c430ac14160161759b4c5b37
|
[
"MIT"
] | null | null | null |
lib/Equivalences.agda
|
sattlerc/HoTT-Agda
|
c8fb8da3354fc9e0c430ac14160161759b4c5b37
|
[
"MIT"
] | null | null | null |
lib/Equivalences.agda
|
sattlerc/HoTT-Agda
|
c8fb8da3354fc9e0c430ac14160161759b4c5b37
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --without-K #-}
open import lib.Base
open import lib.PathGroupoid
open import lib.PathFunctor
open import lib.NType
module lib.Equivalences where
{-
We use the half-adjoint definition of equivalences (but this fact should be
invisible to the user of the library). The constructor of the type of
equivalences is [equiv], it takes two maps and the two proofs that the composites
are equal: [equiv to from to-from from-to]
The type of equivalences between two types [A] and [B] can be written either
[A ≃ B] or [Equiv A B].
Given an equivalence [e] : [A ≃ B], you can extract the two maps as follows:
[–> e] : [A → B] and [<– e] : [B → A] (the dash is an en dash)
The proofs that the composites are the identities are [<–-inv-l] and [<–-inv-r].
The identity equivalence on [A] is [ide A], the composition of two equivalences
is [_∘e_] (function composition order) and the inverse of an equivalence is [_⁻¹]
-}
module _ {i} {j} {A : Type i} {B : Type j} where
record is-equiv (f : A → B) : Type (lmax i j)
where
field
g : B → A
f-g : (b : B) → f (g b) == b
g-f : (a : A) → g (f a) == a
adj : (a : A) → ap f (g-f a) == f-g (f a)
{-
In order to prove that something is an equivalence, you have to give an inverse
and a proof that it’s an inverse (you don’t need the adj part).
[is-eq] is a very, very bad name.
-}
is-eq : (f : A → B)
(g : B → A) (f-g : (b : B) → f (g b) == b)
(g-f : (a : A) → g (f a) == a) → is-equiv f
is-eq f g f-g g-f =
record {g = g; f-g = f-g'; g-f = g-f; adj = adj} where
f-g' : (b : B) → f (g b) == b
f-g' b = ! (ap (f ∘ g) (f-g b)) ∙ ap f (g-f (g b)) ∙ f-g b
adj : (a : A) → ap f (g-f a) == f-g' (f a)
adj a =
ap f (g-f a)
=⟨ ! (!-inv-l (ap (f ∘ g) (f-g (f a)))) |in-ctx (λ q → q ∙ ap f (g-f a)) ⟩
(! (ap (f ∘ g) (f-g (f a))) ∙ ap (f ∘ g) (f-g (f a))) ∙ ap f (g-f a)
=⟨ ∙-assoc (! (ap (f ∘ g) (f-g (f a)))) (ap (f ∘ g) (f-g (f a))) _ ⟩
! (ap (f ∘ g) (f-g (f a))) ∙ ap (f ∘ g) (f-g (f a)) ∙ ap f (g-f a)
=⟨ lemma |in-ctx (λ q → ! (ap (f ∘ g) (f-g (f a))) ∙ q) ⟩
! (ap (f ∘ g) (f-g (f a))) ∙ ap f (g-f (g (f a))) ∙ f-g (f a) ∎
where
lemma : ap (f ∘ g) (f-g (f a)) ∙ ap f (g-f a)
== ap f (g-f (g (f a))) ∙ f-g (f a)
lemma =
ap (f ∘ g) (f-g (f a)) ∙ ap f (g-f a)
=⟨ htpy-natural-toid f-g (f a) |in-ctx (λ q → q ∙ ap f (g-f a)) ⟩
f-g (f (g (f a))) ∙ ap f (g-f a)
=⟨ ! (ap-idf (ap f (g-f a))) |in-ctx (λ q → f-g (f (g (f a))) ∙ q) ⟩
f-g (f (g (f a))) ∙ ap (idf B) (ap f (g-f a))
=⟨ ! (htpy-natural f-g (ap f (g-f a))) ⟩
ap (f ∘ g) (ap f (g-f a)) ∙ f-g (f a)
=⟨ ap-∘ f g (ap f (g-f a)) |in-ctx (λ q → q ∙ f-g (f a)) ⟩
ap f (ap g (ap f (g-f a))) ∙ f-g (f a)
=⟨ ∘-ap g f (g-f a) ∙ htpy-natural-toid g-f a
|in-ctx (λ q → ap f q ∙ f-g (f a)) ⟩
ap f (g-f (g (f a))) ∙ f-g (f a) ∎
infix 4 _≃_
_≃_ : ∀ {i j} (A : Type i) (B : Type j) → Type (lmax i j)
A ≃ B = Σ (A → B) is-equiv
Equiv = _≃_
module _ {i} {j} {A : Type i} {B : Type j} where
equiv : (f : A → B) (g : B → A) (f-g : (b : B) → f (g b) == b)
(g-f : (a : A) → g (f a) == a) → A ≃ B
equiv f g f-g g-f = (f , is-eq f g f-g g-f)
–> : (e : A ≃ B) → (A → B)
–> e = fst e
<– : (e : A ≃ B) → (B → A)
<– e = is-equiv.g (snd e)
<–-inv-l : (e : A ≃ B) (a : A)
→ (<– e (–> e a) == a)
<–-inv-l e a = is-equiv.g-f (snd e) a
<–-inv-r : (e : A ≃ B) (b : B)
→ (–> e (<– e b) == b)
<–-inv-r e b = is-equiv.f-g (snd e) b
-- Equivalences are "injective"
equiv-inj : (e : A ≃ B) {x y : A}
→ (–> e x == –> e y → x == y)
equiv-inj e {x} {y} p = ! (<–-inv-l e x) ∙ ap (<– e) p ∙ <–-inv-l e y
idf-is-equiv : ∀ {i} (A : Type i) → is-equiv (idf A)
idf-is-equiv A = is-eq _ (idf A) (λ _ → idp) (λ _ → idp)
ide : ∀ {i} (A : Type i) → A ≃ A
ide A = equiv (idf A) (idf A) (λ _ → idp) (λ _ → idp)
infixr 4 _∘e_
infixr 4 _∘ise_
_∘e_ : ∀ {i j k} {A : Type i} {B : Type j} {C : Type k}
→ B ≃ C → A ≃ B → A ≃ C
e1 ∘e e2 = equiv (–> e1 ∘ –> e2) (<– e2 ∘ <– e1)
(λ c → –> e1 (–> e2 (<– e2 (<– e1 c)))
=⟨ <–-inv-r e2 (<– e1 c) |in-ctx (–> e1) ⟩
–> e1 (<– e1 c) =⟨ <–-inv-r e1 c ⟩
c ∎)
(λ a → <– e2 (<– e1 (–> e1 (–> e2 a)))
=⟨ <–-inv-l e1 (–> e2 a) |in-ctx (<– e2) ⟩
<– e2 (–> e2 a) =⟨ <–-inv-l e2 a ⟩
a ∎)
_∘ise_ : ∀ {i j k} {A : Type i} {B : Type j} {C : Type k}
{f : A → B} {g : B → C}
→ is-equiv g → is-equiv f → is-equiv (g ∘ f)
i1 ∘ise i2 = snd ((_ , i1) ∘e (_ , i2))
_⁻¹ : ∀ {i j} {A : Type i} {B : Type j} → (A ≃ B) → (B ≃ A)
e ⁻¹ = equiv (<– e) (–> e) (<–-inv-l e) (<–-inv-r e)
{- Equational reasoning for equivalences -}
infix 2 _≃∎
infixr 2 _≃⟨_⟩_
_≃⟨_⟩_ : ∀ {i j k} (A : Type i) {B : Type j} {C : Type k} → A ≃ B → B ≃ C → A ≃ C
A ≃⟨ u ⟩ v = v ∘e u
_≃∎ : ∀ {i} (A : Type i) → A ≃ A
_≃∎ = ide
{- lifting is an equivalence -}
lift-equiv : ∀ {i j} {A : Type i} → Lift {j = j} A ≃ A
lift-equiv = equiv lower lift (λ _ → idp) (λ _ → idp)
{- Any contractible type is equivalent to (all liftings of) the unit type -}
module _ {i} {A : Type i} (h : is-contr A) where
contr-equiv-Unit : A ≃ Unit
contr-equiv-Unit = equiv (λ _ → unit) (λ _ → fst h) (λ _ → idp) (snd h)
contr-equiv-LiftUnit : ∀ {j} → A ≃ Lift {j = j} Unit
contr-equiv-LiftUnit = lift-equiv ⁻¹ ∘e contr-equiv-Unit
{- An equivalence induces an equivalence on the path spaces -}
module _ {i j} {A : Type i} {B : Type j} (e : A ≃ B) where
private
abstract
left-inverse : {x y : A} (p : x == y) → equiv-inj e (ap (–> e) p) == p
left-inverse idp = !-inv-l (<–-inv-l e _)
right-inverse : {x y : A} (p : –> e x == –> e y)
→ ap (–> e) (equiv-inj e p) == p
right-inverse {x} {y} p =
ap f (! (g-f x) ∙ ap g p ∙ (g-f y))
=⟨ ap-∙ f (! (g-f x)) (ap g p ∙ (g-f y)) ⟩
ap f (! (g-f x)) ∙ ap f (ap g p ∙ (g-f y))
=⟨ ap-∙ f (ap g p) (g-f y) |in-ctx (λ q → ap f (! (g-f x)) ∙ q) ⟩
ap f (! (g-f x)) ∙ ap f (ap g p) ∙ ap f (g-f y)
=⟨ ∘-ap f g p |in-ctx (λ q → ap f (! (g-f x)) ∙ q ∙ ap f (g-f y)) ⟩
ap f (! (g-f x)) ∙ ap (f ∘ g) p ∙ ap f (g-f y)
=⟨ adj y |in-ctx (λ q → ap f (! (g-f x)) ∙ ap (f ∘ g) p ∙ q) ⟩
ap f (! (g-f x)) ∙ ap (f ∘ g) p ∙ (f-g (f y))
=⟨ ap-! f (g-f x) |in-ctx (λ q → q ∙ ap (f ∘ g) p ∙ (f-g (f y))) ⟩
! (ap f (g-f x)) ∙ ap (f ∘ g) p ∙ (f-g (f y))
=⟨ adj x |in-ctx (λ q → ! q ∙ ap (f ∘ g) p ∙ (f-g (f y))) ⟩
! (f-g (f x)) ∙ ap (f ∘ g) p ∙ (f-g (f y))
=⟨ htpy-natural f-g p |in-ctx (λ q → ! (f-g (f x)) ∙ q) ⟩
! (f-g (f x)) ∙ (f-g (f x)) ∙ ap (idf B) p
=⟨ ! (∙-assoc (! (f-g (f x))) (f-g (f x)) (ap (idf B) p))
∙ ap (λ q → q ∙ ap (idf B) p) (!-inv-l (f-g (f x))) ∙ ap-idf p ⟩
p ∎
where f : A → B
f = fst e
open is-equiv (snd e)
equiv-ap : (x y : A) → (x == y) ≃ (–> e x == –> e y)
equiv-ap x y = equiv (ap (–> e)) (equiv-inj e) right-inverse left-inverse
{- Equivalent types have the same truncation level -}
equiv-preserves-level : ∀ {i j} {A : Type i} {B : Type j} {n : ℕ₋₂} (e : A ≃ B)
→ (has-level n A → has-level n B)
equiv-preserves-level {n = ⟨-2⟩} e (x , p) =
(–> e x , (λ y → ap (–> e) (p _) ∙ <–-inv-r e y))
equiv-preserves-level {n = S n} e c = λ x y →
equiv-preserves-level (equiv-ap (e ⁻¹) x y ⁻¹) (c (<– e x) (<– e y))
{- This is a collection of type equivalences involving basic type formers.
We exclude Empty since Π₁-Empty requires λ=.
-}
module _ {j} {B : Unit → Type j} where
Σ₁-Unit : Σ Unit B ≃ B unit
Σ₁-Unit = equiv (λ {(unit , b) → b}) (λ b → (unit , b)) (λ _ → idp) (λ _ → idp)
Π₁-Unit : Π Unit B ≃ B unit
Π₁-Unit = equiv (λ f → f unit) (λ b _ → b) (λ _ → idp) (λ _ → idp)
module _ {i} {A : Type i} where
Σ₂-Unit : Σ A (λ _ → Unit) ≃ A
Σ₂-Unit = equiv fst (λ a → (a , unit)) (λ _ → idp) (λ _ → idp)
Π₂-Unit : Π A (λ _ → Unit) ≃ Unit
Π₂-Unit = equiv (λ _ → unit) (λ _ _ → unit) (λ _ → idp) (λ _ → idp)
module _ {i j k} {A : Type i} {B : A → Type j} {C : (a : A) → B a → Type k} where
Σ-assoc : Σ (Σ A B) (uncurry C) ≃ Σ A (λ a → Σ (B a) (C a))
Σ-assoc = equiv (λ {((a , b) , c) → (a , (b , c))})
(λ {(a , (b , c)) → ((a , b) , c)}) (λ _ → idp) (λ _ → idp)
curry-equiv : Π (Σ A B) (uncurry C) ≃ Π A (λ a → Π (B a) (C a))
curry-equiv = equiv curry uncurry (λ _ → idp) (λ _ → idp)
{- The type-theoretic axiom of choice -}
choice : Π A (λ a → Σ (B a) (λ b → C a b)) ≃ Σ (Π A B) (λ g → Π A (λ a → C a (g a)))
choice = equiv f g (λ _ → idp) (λ _ → idp)
where f = λ c → ((λ a → fst (c a)) , (λ a → snd (c a)))
g = λ d → (λ a → (fst d a , snd d a))
{- Pre- and post- concatenation are equivalences -}
module _ {i} {A : Type i} {x y z : A} where
pre∙-is-equiv : (p : x == y) → is-equiv (λ (q : y == z) → p ∙ q)
pre∙-is-equiv p = is-eq (λ q → p ∙ q) (λ r → ! p ∙ r) f-g g-f
where f-g : ∀ r → p ∙ ! p ∙ r == r
f-g r = ! (∙-assoc p (! p) r) ∙ ap (λ s → s ∙ r) (!-inv-r p)
g-f : ∀ q → ! p ∙ p ∙ q == q
g-f q = ! (∙-assoc (! p) p q) ∙ ap (λ s → s ∙ q) (!-inv-l p)
post∙-is-equiv : (p : y == z) → is-equiv (λ (q : x == y) → q ∙ p)
post∙-is-equiv p = is-eq (λ q → q ∙ p) (λ r → r ∙ ! p) f-g g-f
where f-g : ∀ r → (r ∙ ! p) ∙ p == r
f-g r = ∙-assoc r (! p) p ∙ ap (λ s → r ∙ s) (!-inv-l p) ∙ ∙-unit-r r
g-f : ∀ q → (q ∙ p) ∙ ! p == q
g-f q = ∙-assoc q p (! p) ∙ ap (λ s → q ∙ s) (!-inv-r p) ∙ ∙-unit-r q
| 37.804688
| 86
| 0.435937
|
c510fd50ecd34a8c867377889594335c9ed23f95
| 2,209
|
agda
|
Agda
|
notes/FOT/Common/FOL/Relation/Binary/PropositionalEquality/TypeTheory.agda
|
asr/fotc
|
2fc9f2b81052a2e0822669f02036c5750371b72d
|
[
"MIT"
] | 11
|
2015-09-03T20:53:42.000Z
|
2021-09-12T16:09:54.000Z
|
notes/FOT/Common/FOL/Relation/Binary/PropositionalEquality/TypeTheory.agda
|
asr/fotc
|
2fc9f2b81052a2e0822669f02036c5750371b72d
|
[
"MIT"
] | 2
|
2016-10-12T17:28:16.000Z
|
2017-01-01T14:34:26.000Z
|
notes/FOT/Common/FOL/Relation/Binary/PropositionalEquality/TypeTheory.agda
|
asr/fotc
|
2fc9f2b81052a2e0822669f02036c5750371b72d
|
[
"MIT"
] | 3
|
2016-09-19T14:18:30.000Z
|
2018-03-14T08:50:00.000Z
|
------------------------------------------------------------------------------
-- Type theory: The identity type
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
-- We can prove the properties of equality used in the formalization
-- of FOTC, from refl and J.
module FOT.Common.FOL.Relation.Binary.PropositionalEquality.TypeTheory where
infix 7 _≡_
postulate
D : Set
_≡_ : D → D → Set
refl : ∀ {x} → x ≡ x
module TypeTheory where
-- Using the type-theoretic eliminator for equality.
postulate J : (C : ∀ x y → x ≡ y → Set) →
(∀ x → (C x x refl)) →
∀ x y → (c : x ≡ y) → C x y c
-- From Thorsten's slides: A short history of equality.
sym : ∀ {x y} → x ≡ y → y ≡ x
sym {x} {y} = J (λ x' y' _ → y' ≡ x') (λ x' → refl) x y
-- From Thorsten's slides: A short history of equality.
trans : ∀ {x y z} → x ≡ y → y ≡ z → x ≡ z
trans {x} {y} {z} = J (λ x' y' _ → y' ≡ z → x' ≡ z) (λ x' pr → pr) x y
subst : (A : D → Set) → ∀ {x y} → x ≡ y → A x → A y
subst A {x} {y} x≡y = J (λ x' y' _ → A x' → A y') (λ x' pr → pr) x y x≡y
module FOL where
-- Using the usual elimination schema for predicate logic.
postulate J : (A : D → Set) → ∀ {x y} → x ≡ y → A x → A y
sym : ∀ {x y} → x ≡ y → y ≡ x
sym {x} {y} x≡y = J (λ y' → y' ≡ x) x≡y refl
trans : ∀ {x y z} → x ≡ y → y ≡ z → x ≡ z
trans {x} {y} {z} x≡y = J (λ y' → y' ≡ z → x ≡ z) x≡y (λ pr → pr)
subst : (A : D → Set) → ∀ {x y} → x ≡ y → A x → A y
subst = J
module ML where
-- Using Martin-Löf elimination ("Hauptsatz ...", 1971).
postulate J : (C : D → D → Set) →
(∀ x → (C x x)) →
∀ x y → x ≡ y → C x y
sym : ∀ {x y} → x ≡ y → y ≡ x
sym {x} {y} = J (λ x' y' → y' ≡ x') (λ x' → refl) x y
trans : ∀ {x y z} → x ≡ y → y ≡ z → x ≡ z
trans {x} {y} {z} = J (λ x' y' → y' ≡ z → x' ≡ z) (λ x' pr → pr) x y
subst : (A : D → Set) → ∀ {x y} → x ≡ y → A x → A y
subst A {x} {y} x≡y = J (λ x' y' → A x' → A y') (λ x' pr → pr) x y x≡y
| 30.680556
| 78
| 0.425985
|
0be547e5eecdd352d11eae6f8fddcac42d4fc6ef
| 115,257
|
agda
|
Agda
|
complexity/complexity-final/submit/Interp.agda
|
benhuds/Agda
|
2404a6ef2688f879bda89860bb22f77664ad813e
|
[
"MIT"
] | 2
|
2016-04-26T20:22:22.000Z
|
2019-08-08T12:27:18.000Z
|
complexity/complexity-final/submit/Interp.agda
|
benhuds/Agda
|
2404a6ef2688f879bda89860bb22f77664ad813e
|
[
"MIT"
] | 1
|
2020-03-23T08:39:04.000Z
|
2020-05-12T00:32:45.000Z
|
complexity/complexity-final/submit/Interp.agda
|
benhuds/Agda
|
2404a6ef2688f879bda89860bb22f77664ad813e
|
[
"MIT"
] | null | null | null |
{- NEW INTERP WITHOUT RREC -}
open import Preliminaries
open import Preorder
open import Complexity
module Interp where
-- interpret complexity types as preorders
[_]t : CTp → PREORDER
[ unit ]t = unit-p
[ nat ]t = Nat , ♭nat-p
[ τ ->c τ₁ ]t = [ τ ]t ->p [ τ₁ ]t
[ τ ×c τ₁ ]t = [ τ ]t ×p [ τ₁ ]t
[ list τ ]t = (List (fst [ τ ]t)) , list-p (snd [ τ ]t)
[ bool ]t = Bool , bool-p
[ C ]t = Nat , nat-p
[ rnat ]t = Nat , nat-p
[_]tm : ∀ {A} → CTpM A → Preorder-max-str (snd [ A ]t)
[ runit ]tm = unit-pM
[ rnat-max ]tm = nat-pM
[ e ×cm e₁ ]tm = axb-pM [ e ]tm [ e₁ ]tm
[ _->cm_ {τ1} e ]tm = mono-pM [ e ]tm
-- interpret contexts as preorders
[_]c : Ctx → PREORDER
[ [] ]c = unit-p
[ τ :: Γ ]c = [ Γ ]c ×p [ τ ]t
lookup : ∀{Γ τ} → τ ∈ Γ → el ([ Γ ]c ->p [ τ ]t)
lookup (i0 {Γ} {τ}) = snd' id
lookup (iS {Γ} {τ} {τ1} x) = comp (fst' id) (lookup x)
interpE : ∀{Γ τ} → Γ |- τ → el ([ Γ ]c ->p [ τ ]t)
sound : ∀ {Γ τ} (e e' : Γ |- τ) → e ≤s e' → PREORDER≤ ([ Γ ]c ->p [ τ ]t) (interpE e) (interpE e')
interpE unit = monotone (λ x → <>) (λ x y x₁ → <>)
interpE 0C = monotone (λ x → Z) (λ x y x₁ → <>)
interpE 1C = monotone (λ x → S Z) (λ x y x₁ → <>)
interpE (plusC e e₁) =
monotone (λ x → Monotone.f (interpE e) x + Monotone.f (interpE e₁) x)
(λ x y x₁ → plus-lem (Monotone.f (interpE e) x) (Monotone.f (interpE e₁) x) (Monotone.f (interpE e) y) (Monotone.f (interpE e₁) y)
(Monotone.is-monotone (interpE e) x y x₁) (Monotone.is-monotone (interpE e₁) x y x₁))
interpE (var x) = lookup x
interpE z = monotone (λ x → Z) (λ x y x₁ → <>)
interpE (s e) = monotone (λ x → S (Monotone.f (interpE e) x)) (λ x y x₁ → Monotone.is-monotone (interpE e) x y x₁)
interpE {Γ} {τ} (rec e e₁ e₂) = comp (pair' id (interpE e)) (♭rec' (interpE e₁) (interpE e₂))
interpE (lam e) = lam' (interpE e)
interpE (app e e₁) = app' (interpE e) (interpE e₁)
interpE (prod e e₁) = pair' (interpE e) (interpE e₁)
interpE (l-proj e) = fst' (interpE e)
interpE (r-proj e) = snd' (interpE e)
interpE nil = nil'
interpE (e ::c e₁) = cons' (interpE e) (interpE e₁)
interpE (listrec e e₁ e₂) = comp (pair' id (interpE e)) (lrec' (interpE e₁) (interpE e₂))
interpE true = monotone (λ x → True) (λ x y x₁ → <>)
interpE false = monotone (λ x → False) (λ x y x₁ → <>)
interpE (letc e e') = app' (lam' (interpE e)) (interpE e')
interpE {Γ} {τ'} (max τ e1 e2) =
monotone (λ x → Preorder-max-str.max [ τ ]tm (Monotone.f (interpE e1) x) (Monotone.f (interpE e2) x))
(λ x y x₁ → Preorder-max-str.max-lub [ τ ]tm (Preorder-max-str.max [ τ ]tm (Monotone.f (interpE e1) y) (Monotone.f (interpE e2) y))
(Monotone.f (interpE e1) x) (Monotone.f (interpE e2) x)
(Preorder-str.trans (snd [ τ' ]t) (Monotone.f (interpE e1) x) (Monotone.f (interpE e1) y)
(Preorder-max-str.max [ τ ]tm (Monotone.f (interpE e1) y) (Monotone.f (interpE e2) y))
(Monotone.is-monotone (interpE e1) x y x₁) (Preorder-max-str.max-l [ τ ]tm (Monotone.f (interpE e1) y) (Monotone.f (interpE e2) y)))
(Preorder-str.trans (snd [ τ' ]t) (Monotone.f (interpE e2) x) (Monotone.f (interpE e2) y)
(Preorder-max-str.max [ τ ]tm (Monotone.f (interpE e1) y) (Monotone.f (interpE e2) y))
(Monotone.is-monotone (interpE e2) x y x₁) (Preorder-max-str.max-r [ τ ]tm (Monotone.f (interpE e1) y) (Monotone.f (interpE e2) y))))
throw-r : ∀ {Γ Γ' τ} → rctx Γ (τ :: Γ') → rctx Γ Γ'
throw-r d = λ x → d (iS x)
interpR : ∀ {Γ Γ'} → rctx Γ Γ' → MONOTONE [ Γ ]c [ Γ' ]c
interpR {Γ' = []} ρ = monotone (λ _ → <>) (λ x y x₁ → <>)
interpR {Γ' = τ :: Γ'} ρ = monotone (λ x → (Monotone.f (interpR (throw-r ρ)) x) , (Monotone.f (lookup (ρ i0)) x))
(λ x y x₁ → (Monotone.is-monotone (interpR (throw-r ρ)) x y x₁) , (Monotone.is-monotone (lookup (ρ i0)) x y x₁))
throw-s : ∀ {Γ Γ' τ} → sctx Γ (τ :: Γ') → sctx Γ Γ'
throw-s d = λ x → d (iS x)
interpS : ∀ {Γ Γ'} → sctx Γ Γ' → el ([ Γ ]c ->p [ Γ' ]c)
interpS {Γ' = []} Θ = monotone (λ _ → <>) (λ x y x₁ → <>)
interpS {Γ' = τ :: Γ'} Θ = monotone (λ x → Monotone.f (interpS (throw-s Θ)) x , Monotone.f (interpE (Θ i0)) x)
(λ x y x₁ → Monotone.is-monotone (interpS (throw-s Θ)) x y x₁ , (Monotone.is-monotone (interpE (Θ i0)) x y x₁))
ren-eq-l-lam : ∀ {Γ Γ' τ1} (ρ : rctx Γ Γ') (k : fst [ Γ ]c) (x : fst [ τ1 ]t)
→ Preorder-str.≤ (snd [ Γ' ]c) (Monotone.f (interpR (throw-r (r-extend {_} {_} {τ1} ρ))) (k , x)) (Monotone.f (interpR ρ) k)
ren-eq-l-lam {Γ' = []} ρ k x = <>
ren-eq-l-lam {Γ' = x :: Γ'} ρ k x₁ =
(ren-eq-l-lam (throw-r ρ) k x₁) ,
(Preorder-str.refl (snd [ x ]t) (Monotone.f (lookup (ρ i0)) k))
ren-eq-l : ∀ {Γ Γ' τ} → (ρ : rctx Γ Γ') → (e : Γ' |- τ) → (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ τ ]t) (Monotone.f (interpE (ren e ρ)) k) (Monotone.f (interpE e) (Monotone.f (interpR ρ) k))
ren-eq-l ρ unit k = <>
ren-eq-l ρ 0C k = <>
ren-eq-l ρ 1C k = <>
ren-eq-l ρ (plusC e e₁) k =
plus-lem (Monotone.f (interpE (ren e ρ)) k) (Monotone.f (interpE (ren e₁ ρ)) k)
(Monotone.f (interpE e) (Monotone.f (interpR ρ) k)) (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k))
(ren-eq-l ρ e k) (ren-eq-l ρ e₁ k)
ren-eq-l {τ = τ} ρ (var i0) k = Preorder-str.refl (snd [ τ ]t) (Monotone.f (lookup (ρ i0)) k)
ren-eq-l {τ = τ} ρ (var (iS x)) k = ren-eq-l (throw-r ρ) (var x) k
ren-eq-l ρ z k = <>
ren-eq-l ρ (s e) k = ren-eq-l ρ e k
ren-eq-l {Γ} {Γ'} {τ = τ} ρ (rec e e₁ e₂) k =
Preorder-str.trans (snd [ τ ]t)
(natrec (Monotone.f (interpE (ren e₁ ρ)) k)
(λ n x₂ → Monotone.f (interpE (ren e₂ (r-extend (r-extend ρ)))) ((k , x₂) , n))
(Monotone.f (interpE (ren e ρ)) k))
(natrec (Monotone.f (interpE (ren e₁ ρ)) k)
(λ n x₂ → Monotone.f (interpE (ren e₂ (r-extend (r-extend ρ)))) ((k , x₂) , n))
(Monotone.f (interpE e) (Monotone.f (interpR ρ) k)))
(natrec (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k))
(λ n x₂ → Monotone.f (interpE e₂) ((Monotone.f (interpR ρ) k , x₂) , n))
(Monotone.f (interpE e) (Monotone.f (interpR ρ) k)))
(♭h-fix-args (interpE (ren e₁ ρ)) (interpE (ren e₂ (r-extend (r-extend ρ))))
(k , (Monotone.f (interpE (ren e ρ)) k))
(k , (Monotone.f (interpE e) (Monotone.f (interpR ρ) k)))
(ren-eq-l ρ e k))
(♭h-cong
(interpE (ren e₁ ρ))
(monotone (λ v → Monotone.f (interpE e₁) (Monotone.f (interpR ρ) v))
(λ x y x₁ →
Monotone.is-monotone (interpE e₁)
(Monotone.f (interpR ρ) x)
(Monotone.f (interpR ρ) y)
(Monotone.is-monotone (interpR ρ) x y x₁)))
(interpE (ren e₂ (r-extend (r-extend ρ))))
(monotone (λ v → Monotone.f (interpE e₂) ((Monotone.f (interpR ρ) (fst (fst v)) , snd (fst v)) , snd v))
(λ x y x₁ →
Monotone.is-monotone (interpE e₂)
((Monotone.f (interpR ρ) (fst (fst x)) , snd (fst x)) , snd x)
((Monotone.f (interpR ρ) (fst (fst y)) , snd (fst y)) , snd y)
(((Monotone.is-monotone (interpR ρ) (fst (fst x)) (fst (fst y)) (fst (fst x₁))) ,
(snd (fst x₁))) ,
(snd x₁))))
(k , (Monotone.f (interpE e) (Monotone.f (interpR ρ) k)))
(λ x → ren-eq-l ρ e₁ x)
(λ x →
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren e₂ (r-extend (r-extend ρ)))) x)
(Monotone.f (interpE e₂) (Monotone.f (interpR {nat :: τ :: Γ} {_ :: _ :: Γ'} (r-extend (r-extend ρ))) x))
(Monotone.f (interpE e₂) ((Monotone.f (interpR ρ) (fst (fst x)) , snd (fst x)) , snd x))
(ren-eq-l (r-extend (r-extend ρ)) e₂ x)
(Monotone.is-monotone (interpE e₂)
(Monotone.f (interpR {nat :: τ :: Γ} {_ :: _ :: Γ'} (r-extend (r-extend ρ))) x)
((Monotone.f (interpR ρ) (fst (fst x)) , snd (fst x)) , snd x)
((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpR (λ x₁ → iS (iS (ρ x₁)))) x)
(Monotone.f (interpR {τ :: Γ} {Γ'} (throw-r (r-extend ρ))) (fst (fst x) , snd (fst x)))
(Monotone.f (interpR ρ) (fst (fst x)))
(ren-eq-l-lam {τ :: Γ} {Γ'} (throw-r (r-extend ρ)) (fst x) (snd x))
(ren-eq-l-lam ρ (fst (fst x)) (snd (fst x))) ,
(Preorder-str.refl (snd [ τ ]t) (snd (fst x)))) ,
(♭nat-refl (snd x))))))
ren-eq-l {Γ} {τ = τ1 ->c τ2} ρ (lam e) k x =
Preorder-str.trans (snd [ τ2 ]t)
(Monotone.f (Monotone.f (interpE (ren (lam e) ρ)) k) x)
(Monotone.f (interpE e) (Monotone.f (interpR (r-extend {_} {_} {τ1} ρ)) (k , x)))
(Monotone.f (interpE e) (Monotone.f (interpR ρ) k , x))
(ren-eq-l (r-extend ρ) e (k , x))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpR (r-extend {_} {_} {τ1} ρ)) (k , x))
(Monotone.f (interpR ρ) k , x)
(ren-eq-l-lam ρ k x ,
(Preorder-str.refl (snd [ τ1 ]t) x)))
ren-eq-l {Γ} {τ = τ} ρ (app e e₁) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (Monotone.f (interpE (ren e ρ)) k) (Monotone.f (interpE (ren e₁ ρ)) k))
(Monotone.f (Monotone.f (interpE (ren e ρ)) k) (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k)))
(Monotone.f (Monotone.f (interpE e) (Monotone.f (interpR ρ) k)) (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k)))
(Monotone.is-monotone (Monotone.f (interpE (ren e ρ)) k) (Monotone.f (interpE (ren e₁ ρ)) k) (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k)) (ren-eq-l ρ e₁ k))
(ren-eq-l ρ e k (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k)))
ren-eq-l ρ (prod e e₁) k = (ren-eq-l ρ e k) , (ren-eq-l ρ e₁ k)
ren-eq-l ρ (l-proj e) k = fst (ren-eq-l ρ e k)
ren-eq-l ρ (r-proj e) k = snd (ren-eq-l ρ e k)
ren-eq-l ρ nil k = <>
ren-eq-l ρ (e ::c e₁) k = (ren-eq-l ρ e k) , (ren-eq-l ρ e₁ k)
ren-eq-l {Γ} {Γ'} {τ = τ} ρ (listrec {.Γ'} {τ'} e e₁ e₂) k =
Preorder-str.trans (snd [ τ ]t)
(lrec (Monotone.f (interpE (ren e ρ)) k)
(Monotone.f (interpE (ren e₁ ρ)) k)
(λ x₁ x₂ x₃ → Monotone.f (interpE (ren e₂ (r-extend (r-extend (r-extend ρ))))) (((k , x₃) , x₂) , x₁)))
(lrec (Monotone.f (interpE e) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE (ren e₁ ρ)) k)
(λ x₁ x₂ x₃ → Monotone.f (interpE (ren e₂ (r-extend (r-extend (r-extend ρ))))) (((k , x₃) , x₂) , x₁)))
(lrec (Monotone.f (interpE e) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k))
(λ x₁ x₂ x₃ → Monotone.f (interpE e₂) (((Monotone.f (interpR ρ) k , x₃) , x₂) , x₁)))
(listrec-fix-args (interpE (ren e₁ ρ)) (interpE (ren e₂ (r-extend (r-extend (r-extend ρ)))))
(k , (Monotone.f (interpE (ren e ρ)) k))
(k , (Monotone.f (interpE e) (Monotone.f (interpR ρ) k)))
((Preorder-str.refl (snd [ Γ ]c) k) , (ren-eq-l ρ e k)))
(lrec-cong
(interpE (ren e₁ ρ))
(monotone (λ k₁ → Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k₁))
(λ x y x₁ →
Monotone.is-monotone (interpE e₁)
(Monotone.f (interpR ρ) x) (Monotone.f (interpR ρ) y)
(Monotone.is-monotone (interpR ρ) x y x₁)))
(interpE (ren e₂ (r-extend (r-extend (r-extend ρ)))))
(monotone (λ x → Monotone.f (interpE e₂) ((((Monotone.f (interpR ρ) (fst (fst (fst x)))) , (snd (fst (fst x)))) , (snd (fst x))) , (snd x)))
(λ x y x₁ → Monotone.is-monotone (interpE e₂)
(((Monotone.f (interpR ρ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x)
(((Monotone.f (interpR ρ) (fst (fst (fst y))) , snd (fst (fst y))) , snd (fst y)) , snd y)
((((Monotone.is-monotone (interpR ρ) (fst (fst (fst x))) (fst (fst (fst y))) (fst (fst (fst x₁)))) ,
(snd (fst (fst x₁)))) ,
(snd (fst x₁))) ,
(snd x₁))))
(k , (Monotone.f (interpE e) (Monotone.f (interpR ρ) k)))
(λ x → ren-eq-l ρ e₁ x)
(λ x →
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren e₂ (r-extend (r-extend (r-extend ρ))))) x)
(Monotone.f (interpE e₂) (Monotone.f (interpR {τ' :: list τ' :: τ :: Γ} {_ :: _ :: _ :: Γ'} (r-extend (r-extend (r-extend ρ)))) x))
(Monotone.f (interpE e₂) (((Monotone.f (interpR ρ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x))
(ren-eq-l (r-extend (r-extend (r-extend ρ))) e₂ x)
(Monotone.is-monotone (interpE e₂)
(Monotone.f (interpR {τ' :: list τ' :: τ :: Γ} {_ :: _ :: _ :: Γ'} (r-extend (r-extend (r-extend ρ)))) x)
(((Monotone.f (interpR ρ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x)
(((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpR (λ x₁ → iS (iS (iS (ρ x₁))))) x)
(Monotone.f (interpR {τ :: Γ} {Γ'} (throw-r (r-extend ρ))) (fst (fst (fst x)) , snd (fst (fst x))))
(Monotone.f (interpR ρ) (fst (fst (fst x))))
(Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpR (λ x₁ → iS (iS (iS (ρ x₁))))) x)
(Monotone.f (interpR (λ x₁ → iS (iS (ρ x₁)))) (fst x))
(Monotone.f (interpR {τ :: Γ} {Γ'} (throw-r (r-extend ρ))) (fst (fst (fst x)) , snd (fst (fst x))))
(ren-eq-l-lam (λ x₁ → iS (iS (ρ x₁))) (fst x) (snd x))
(ren-eq-l-lam (λ x₁ → iS (ρ x₁)) (fst (fst x)) (snd (fst x))))
(ren-eq-l-lam ρ (fst (fst (fst x))) (snd (fst (fst x)))) ,
Preorder-str.refl (snd [ τ ]t) (snd (fst (fst x)))) ,
l-refl (snd [ τ' ]t) (snd (fst x))) ,
Preorder-str.refl (snd [ τ' ]t) (snd x)))))
ren-eq-l ρ true k = <>
ren-eq-l ρ false k = <>
ren-eq-l {Γ} {Γ'} {τ = τ} ρ (letc {.Γ'} {ρ'} e e') k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren e (r-extend ρ)))
(k , Monotone.f (interpE (ren e' ρ)) k))
(Monotone.f (interpE e) (Monotone.f (interpR (r-extend {_} {_} {ρ'} ρ)) (k , Monotone.f (interpE (ren e' ρ)) k)))
(Monotone.f (interpE e) (Monotone.f (interpR ρ) k , Monotone.f (interpE e') (Monotone.f (interpR ρ) k)))
(ren-eq-l (r-extend ρ) e (k , Monotone.f (interpE (ren e' ρ)) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpR (r-extend {_} {_} {ρ'} ρ))
(k , Monotone.f (interpE (ren e' ρ)) k))
(Monotone.f (interpR ρ) k ,
Monotone.f (interpE e') (Monotone.f (interpR ρ) k))
(ren-eq-l-lam ρ k (Monotone.f (interpE (ren e' ρ)) k) , ren-eq-l ρ e' k))
ren-eq-l {τ = τ} ρ (max x e e₁) k =
Preorder-max-str.max-lub [ x ]tm
(Monotone.f (interpE (max x e e₁)) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE (ren e ρ)) k)
(Monotone.f (interpE (ren e₁ ρ)) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren e ρ)) k)
(Monotone.f (interpE e) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE (max x e e₁)) (Monotone.f (interpR ρ) k))
(ren-eq-l ρ e k)
(Preorder-max-str.max-l [ x ]tm (Monotone.f (interpE e) (Monotone.f (interpR ρ) k)) (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k))))
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren e₁ ρ)) k)
(Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE (max x e e₁)) (Monotone.f (interpR ρ) k))
(ren-eq-l ρ e₁ k)
(Preorder-max-str.max-r [ x ]tm (Monotone.f (interpE e) (Monotone.f (interpR ρ) k)) (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k))))
ren-eq-r-lam : ∀ {Γ Γ' τ1} (ρ : rctx Γ Γ') (k : fst [ Γ ]c) (x : fst [ τ1 ]t)
→ Preorder-str.≤ (snd [ Γ' ]c) (Monotone.f (interpR ρ) k) (Monotone.f (interpR (throw-r (r-extend {_} {_} {τ1} ρ))) (k , x))
ren-eq-r-lam {Γ' = []} ρ k x = <>
ren-eq-r-lam {Γ' = x :: Γ'} ρ k x₁ =
(ren-eq-r-lam (throw-r ρ) k x₁) ,
(Preorder-str.refl (snd [ x ]t) (Monotone.f (lookup (ρ i0)) k))
ren-eq-r : ∀ {Γ Γ' τ} → (ρ : rctx Γ Γ') → (e : Γ' |- τ) → (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ τ ]t) (Monotone.f (interpE e) (Monotone.f (interpR ρ) k)) (Monotone.f (interpE (ren e ρ)) k)
ren-eq-r ρ unit k = <>
ren-eq-r ρ 0C k = <>
ren-eq-r ρ 1C k = <>
ren-eq-r ρ (plusC e e₁) k =
plus-lem (Monotone.f (interpE e) (Monotone.f (interpR ρ) k)) (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE (ren e ρ)) k) (Monotone.f (interpE (ren e₁ ρ)) k)
(ren-eq-r ρ e k) (ren-eq-r ρ e₁ k)
ren-eq-r {τ = τ} ρ (var i0) k = Preorder-str.refl (snd [ τ ]t) (Monotone.f (lookup (ρ i0)) k)
ren-eq-r {τ = τ} ρ (var (iS x)) k = ren-eq-r (throw-r ρ) (var x) k
ren-eq-r ρ z k = <>
ren-eq-r ρ (s e) k = ren-eq-r ρ e k
ren-eq-r {Γ} {Γ'} {τ} ρ (rec e e₁ e₂) k =
Preorder-str.trans (snd [ τ ]t)
(natrec (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k))
(λ n x₂ → Monotone.f (interpE e₂) ((Monotone.f (interpR ρ) k , x₂) , n))
(Monotone.f (interpE e) (Monotone.f (interpR ρ) k)))
(natrec (Monotone.f (interpE (ren e₁ ρ)) k)
(λ n x₂ → Monotone.f (interpE (ren e₂ (r-extend (r-extend ρ)))) ((k , x₂) , n))
(Monotone.f (interpE e) (Monotone.f (interpR ρ) k)))
(natrec (Monotone.f (interpE (ren e₁ ρ)) k)
(λ n x₂ → Monotone.f (interpE (ren e₂ (r-extend (r-extend ρ)))) ((k , x₂) , n))
(Monotone.f (interpE (ren e ρ)) k))
(♭h-cong
(monotone (λ v → Monotone.f (interpE e₁) (Monotone.f (interpR ρ) v))
(λ x y x₁ →
Monotone.is-monotone (interpE e₁) (Monotone.f (interpR ρ) x)
(Monotone.f (interpR ρ) y)
(Monotone.is-monotone (interpR ρ) x y x₁)))
(interpE (ren e₁ ρ))
(monotone (λ v → Monotone.f (interpE e₂) ((Monotone.f (interpR ρ) (fst (fst v)) , snd (fst v)) , snd v))
(λ x y x₁ →
Monotone.is-monotone (interpE e₂)
((Monotone.f (interpR ρ) (fst (fst x)) , snd (fst x)) , snd x)
((Monotone.f (interpR ρ) (fst (fst y)) , snd (fst y)) , snd y)
((Monotone.is-monotone (interpR ρ) (fst (fst x)) (fst (fst y)) (fst (fst x₁)) ,
snd (fst x₁)) ,
snd x₁)))
(interpE (ren e₂ (r-extend (r-extend ρ))))
(k , Monotone.f (interpE e) (Monotone.f (interpR ρ) k))
(λ x → ren-eq-r ρ e₁ x)
(λ x →
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e₂) ((Monotone.f (interpR ρ) (fst (fst x)) , snd (fst x)) , snd x))
(Monotone.f (interpE e₂) (Monotone.f (interpR {nat :: τ :: Γ} {_ :: _ :: Γ'} (r-extend (r-extend ρ))) x))
(Monotone.f (interpE (ren e₂ (r-extend (r-extend ρ)))) x)
(Monotone.is-monotone (interpE e₂)
((Monotone.f (interpR ρ) (fst (fst x)) , snd (fst x)) , snd x)
(Monotone.f (interpR {nat :: τ :: Γ} {_ :: _ :: Γ'} (r-extend (r-extend ρ))) x)
(((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpR ρ) (fst (fst x)))
(Monotone.f (interpR {τ :: Γ} {Γ'} (throw-r (r-extend ρ))) (fst (fst x) , snd (fst x)))
(Monotone.f (interpR (λ x₁ → iS (iS (ρ x₁)))) x)
(ren-eq-r-lam ρ (fst (fst x)) (snd (fst x)))
(ren-eq-r-lam {τ :: Γ} {Γ'} (throw-r (r-extend ρ)) (fst x) (snd x))) ,
(Preorder-str.refl (snd [ τ ]t) (snd (fst x)))) ,
(♭nat-refl (snd x))))
(ren-eq-r (r-extend (r-extend ρ)) e₂ x)))
(♭h-fix-args (interpE (ren e₁ ρ))
(interpE (ren e₂ (r-extend (r-extend ρ))))
(k , Monotone.f (interpE e) (Monotone.f (interpR ρ) k))
(k , Monotone.f (interpE (ren e ρ)) k) (ren-eq-r ρ e k))
ren-eq-r {Γ} {τ = τ1 ->c τ2} ρ (lam e) k x =
Preorder-str.trans (snd [ τ2 ]t)
(Monotone.f (interpE e) (Monotone.f (interpR ρ) k , x))
(Monotone.f (interpE e) (Monotone.f (interpR (r-extend {_} {_} {τ1} ρ)) (k , x)))
(Monotone.f (Monotone.f (interpE (ren (lam e) ρ)) k) x)
((Monotone.is-monotone (interpE e)
(Monotone.f (interpR ρ) k , x)
(Monotone.f (interpR (r-extend {_} {_} {τ1} ρ)) (k , x))
(ren-eq-r-lam ρ k x ,
(Preorder-str.refl (snd [ τ1 ]t) x))))
(ren-eq-r (r-extend ρ) e (k , x))
ren-eq-r {Γ} {τ = τ} ρ (app e e₁) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (Monotone.f (interpE e) (Monotone.f (interpR ρ) k)) (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k)))
(Monotone.f (Monotone.f (interpE (ren e ρ)) k) (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k)))
(Monotone.f (Monotone.f (interpE (ren e ρ)) k) (Monotone.f (interpE (ren e₁ ρ)) k))
(ren-eq-r ρ e k (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k)))
(Monotone.is-monotone (Monotone.f (interpE (ren e ρ)) k) (Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k)) (Monotone.f (interpE (ren e₁ ρ)) k) (ren-eq-r ρ e₁ k))
ren-eq-r ρ (prod e e₁) k = (ren-eq-r ρ e k) , (ren-eq-r ρ e₁ k)
ren-eq-r ρ (l-proj e) k = fst (ren-eq-r ρ e k)
ren-eq-r ρ (r-proj e) k = snd (ren-eq-r ρ e k)
ren-eq-r ρ nil k = <>
ren-eq-r ρ (e ::c e₁) k = (ren-eq-r ρ e k) , (ren-eq-r ρ e₁ k)
ren-eq-r {Γ} {Γ'} {τ} ρ (listrec {.Γ'} {τ'} e e₁ e₂) k =
Preorder-str.trans (snd [ τ ]t)
(lrec (Monotone.f (interpE e) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k))
(λ x₁ x₂ x₃ → Monotone.f (interpE e₂) (((Monotone.f (interpR ρ) k , x₃) , x₂) , x₁)))
(lrec (Monotone.f (interpE e) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE (ren e₁ ρ)) k)
(λ x₁ x₂ x₃ → Monotone.f (interpE (ren e₂ (r-extend (r-extend (r-extend ρ))))) (((k , x₃) , x₂) , x₁)))
(lrec (Monotone.f (interpE (ren e ρ)) k)
(Monotone.f (interpE (ren e₁ ρ)) k)
(λ x₁ x₂ x₃ → Monotone.f (interpE (ren e₂ (r-extend (r-extend (r-extend ρ))))) (((k , x₃) , x₂) , x₁)))
(lrec-cong
(monotone (λ k₁ → Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k₁))
(λ x y x₁ →
Monotone.is-monotone (interpE e₁)
(Monotone.f (interpR ρ) x)
(Monotone.f (interpR ρ) y)
(Monotone.is-monotone (interpR ρ) x y x₁)))
(interpE (ren e₁ ρ))
(monotone (λ x → Monotone.f (interpE e₂) (((Monotone.f (interpR ρ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x))
(λ x y x₁ →
Monotone.is-monotone (interpE e₂)
(((Monotone.f (interpR ρ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x)
(((Monotone.f (interpR ρ) (fst (fst (fst y))) , snd (fst (fst y))) , snd (fst y)) , snd y)
(((Monotone.is-monotone (interpR ρ)
(fst (fst (fst x))) (fst (fst (fst y))) (fst (fst (fst x₁))) ,
snd (fst (fst x₁))) ,
snd (fst x₁)) ,
snd x₁)))
(interpE (ren e₂ (r-extend (r-extend (r-extend ρ)))))
(k , (Monotone.f (interpE e) (Monotone.f (interpR ρ) k)))
(λ x → ren-eq-r ρ e₁ x)
(λ x →
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e₂) (((Monotone.f (interpR ρ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x))
(Monotone.f (interpE e₂) (Monotone.f (interpR {τ' :: list τ' :: τ :: Γ} {_ :: _ :: _ :: Γ'} (r-extend (r-extend (r-extend ρ)))) x))
(Monotone.f (interpE (ren e₂ (r-extend (r-extend (r-extend ρ))))) x)
(Monotone.is-monotone (interpE e₂)
(((Monotone.f (interpR ρ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x)
(Monotone.f (interpR {τ' :: list τ' :: τ :: Γ} {_ :: _ :: _ :: Γ'} (r-extend (r-extend (r-extend ρ)))) x)
((((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpR ρ) (fst (fst (fst x))))
(Monotone.f (interpR {τ :: Γ} {Γ'} (throw-r (r-extend ρ))) (fst (fst (fst x)) , snd (fst (fst x))))
(Monotone.f (interpR (λ x₁ → iS (iS (iS (ρ x₁))))) x)
(ren-eq-r-lam ρ (fst (fst (fst x))) (snd (fst (fst x))))
(Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpR {τ :: Γ} {Γ'} (throw-r (r-extend ρ))) (fst (fst (fst x)) , snd (fst (fst x))))
(Monotone.f (interpR (λ x₁ → iS (iS (ρ x₁)))) (fst x))
(Monotone.f (interpR (λ x₁ → iS (iS (iS (ρ x₁))))) x)
(ren-eq-r-lam (λ x₁ → iS (ρ x₁)) (fst (fst x)) (snd (fst x)))
(ren-eq-r-lam (λ x₁ → iS (iS (ρ x₁))) (fst x) (snd x)))) ,
(Preorder-str.refl (snd [ τ ]t) (snd (fst (fst x))))) ,
(l-refl (snd [ τ' ]t) (snd (fst x)))) ,
(Preorder-str.refl (snd [ τ' ]t) (snd x))))
(ren-eq-r (r-extend (r-extend (r-extend ρ))) e₂ x)))
(listrec-fix-args (interpE (ren e₁ ρ)) (interpE (ren e₂ (r-extend (r-extend (r-extend ρ)))))
(k , Monotone.f (interpE e) (Monotone.f (interpR ρ) k))
(k , Monotone.f (interpE (ren e ρ)) k)
(Preorder-str.refl (snd [ Γ ]c) k , ren-eq-r ρ e k))
ren-eq-r ρ true k = <>
ren-eq-r ρ false k = <>
ren-eq-r {Γ} {Γ'} {τ = τ} ρ (letc {.Γ'} {ρ'} e e') k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e) (Monotone.f (interpR ρ) k , Monotone.f (interpE e') (Monotone.f (interpR ρ) k)))
(Monotone.f (interpE e) (Monotone.f (interpR (r-extend {_} {_} {ρ'} ρ)) (k , Monotone.f (interpE (ren e' ρ)) k)))
(Monotone.f (interpE (ren e (r-extend ρ)))
(k , Monotone.f (interpE (ren e' ρ)) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpR ρ) k ,
Monotone.f (interpE e') (Monotone.f (interpR ρ) k))
(Monotone.f (interpR (r-extend {_} {_} {ρ'} ρ))
(k , Monotone.f (interpE (ren e' ρ)) k))
(ren-eq-r-lam ρ k (Monotone.f (interpE (ren e' ρ)) k) , ren-eq-r ρ e' k))
(ren-eq-r (r-extend ρ) e (k , Monotone.f (interpE (ren e' ρ)) k))
ren-eq-r {τ = τ} ρ (max x e e₁) k =
Preorder-max-str.max-lub [ x ]tm
(Monotone.f (interpE (ren (max x e e₁) ρ)) k)
(Monotone.f (interpE e) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k))
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE (ren e ρ)) k)
(Monotone.f (interpE (ren (max x e e₁) ρ)) k)
(ren-eq-r ρ e k)
(Preorder-max-str.max-l [ x ]tm (Monotone.f (interpE (ren e ρ)) k) (Monotone.f (interpE (ren e₁ ρ)) k)))
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e₁) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE (ren e₁ ρ)) k)
(Monotone.f (interpE (ren (max x e e₁) ρ)) k)
(ren-eq-r ρ e₁ k)
(Preorder-max-str.max-r [ x ]tm (Monotone.f (interpE (ren e ρ)) k) (Monotone.f (interpE (ren e₁ ρ)) k)))
ids-lem-l : ∀ {Γ} (k : fst [ Γ ]c) → Preorder-str.≤ (snd [ Γ ]c) (Monotone.f (interpR {Γ} {Γ} (λ x₂ → x₂)) k) k
ids-lem-l {[]} k = <>
ids-lem-l {x :: Γ} (k1 , k2) =
(Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpR {x :: Γ} {Γ} (throw-r (λ x₂ → x₂))) (k1 , k2))
(Monotone.f (interpR {Γ} {Γ} (λ x₂ → x₂)) k1)
k1
(ren-eq-l-lam {Γ} {Γ} (λ x₂ → x₂) k1 k2)
(ids-lem-l {Γ} k1)) ,
(Preorder-str.refl (snd [ x ]t) k2)
subst-eq-l-lam : ∀ {Γ Γ' τ1} (Θ : sctx Γ Γ') (k : fst [ Γ ]c) (x : fst [ τ1 ]t)
→ Preorder-str.≤ (snd [ Γ' ]c) (Monotone.f (interpS (throw-s (s-extend {_} {_} {τ1} Θ))) (k , x)) (Monotone.f (interpS Θ) k)
subst-eq-l-lam {Γ' = []} Θ k x = <>
subst-eq-l-lam {Γ} {Γ' = x :: Γ'} {τ1} Θ k x₁ =
(subst-eq-l-lam (throw-s Θ) k x₁) ,
Preorder-str.trans (snd [ x ]t)
(Monotone.f (interpE (ren (Θ i0) iS)) (k , x₁))
(Monotone.f (interpE (Θ i0)) (Monotone.f (interpR {τ1 :: Γ} {Γ} iS) (k , x₁)))
(snd (Monotone.f (interpS Θ) k))
(ren-eq-l iS (Θ i0) (k , x₁))
(Monotone.is-monotone (interpE (Θ i0)) (Monotone.f (interpR {τ1 :: Γ} {Γ} iS) (k , x₁)) k
(Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpR {τ1 :: Γ} {Γ} iS) (k , x₁))
(Monotone.f (interpR {Γ} {Γ} (λ x₂ → x₂)) k)
k
(ren-eq-l-lam {Γ} {Γ} (λ x₂ → x₂) k x₁)
(ids-lem-l {Γ} k)))
subst-eq-l : ∀ {Γ Γ' τ} → (Θ : sctx Γ Γ') → (e : Γ' |- τ) → (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ τ ]t) (Monotone.f (interpE (subst e Θ)) k) (Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
subst-eq-l Θ unit k = <>
subst-eq-l Θ 0C k = <>
subst-eq-l Θ 1C k = <>
subst-eq-l Θ (plusC e e₁) k =
plus-lem (Monotone.f (interpE (subst e Θ)) k) (Monotone.f (interpE (subst e₁ Θ)) k)
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k)) (Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k))
(subst-eq-l Θ e k) (subst-eq-l Θ e₁ k)
subst-eq-l {τ = τ} Θ (var i0) k = Preorder-str.refl (snd [ τ ]t) (Monotone.f (interpE (Θ i0)) k)
subst-eq-l {τ = τ} Θ (var (iS x)) k = subst-eq-l (throw-s Θ) (var x) k
subst-eq-l Θ z k = <>
subst-eq-l Θ (s e) k = subst-eq-l Θ e k
subst-eq-l {Γ} {Γ'} {τ} Θ (rec e e₁ e₂) k =
Preorder-str.trans (snd [ τ ]t)
(natrec (Monotone.f (interpE (subst e₁ Θ)) k)
(λ n x₂ → Monotone.f (interpE (subst e₂ (s-extend (s-extend Θ)))) ((k , x₂) , n))
(Monotone.f (interpE (subst e Θ)) k))
(natrec (Monotone.f (interpE (subst e₁ Θ)) k)
(λ n x₂ → Monotone.f (interpE (subst e₂ (s-extend (s-extend Θ)))) ((k , x₂) , n))
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k)))
(natrec (Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k))
(λ n x₂ → Monotone.f (interpE e₂) ((Monotone.f (interpS Θ) k , x₂) , n))
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k)))
(♭h-fix-args (interpE (subst e₁ Θ)) (interpE (subst e₂ (s-extend (s-extend Θ))))
(k , Monotone.f (interpE (subst e Θ)) k)
(k , Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(subst-eq-l Θ e k))
(♭h-cong
(interpE (subst e₁ Θ))
(monotone (λ v → Monotone.f (interpE e₁) (Monotone.f (interpS Θ) v))
(λ x y x₁ →
Monotone.is-monotone (interpE e₁) (Monotone.f (interpS Θ) x) (Monotone.f (interpS Θ) y) (Monotone.is-monotone (interpS Θ) x y x₁)))
(interpE (subst e₂ (s-extend (s-extend Θ))))
(monotone (λ v → Monotone.f (interpE e₂) ((Monotone.f (interpS Θ) (fst (fst v)) , snd (fst v)) , snd v))
(λ x y x₁ →
Monotone.is-monotone (interpE e₂)
((Monotone.f (interpS Θ) (fst (fst x)) , snd (fst x)) , snd x)
((Monotone.f (interpS Θ) (fst (fst y)) , snd (fst y)) , snd y)
((Monotone.is-monotone (interpS Θ) (fst (fst x)) (fst (fst y)) (fst (fst x₁)) ,
snd (fst x₁)) ,
snd x₁)))
(k , Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(λ x → subst-eq-l Θ e₁ x)
(λ x →
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e₂ (s-extend (s-extend Θ)))) x)
(Monotone.f (interpE e₂) (Monotone.f (interpS {nat :: τ :: Γ} {_ :: _ :: Γ'} (s-extend (s-extend Θ))) x))
(Monotone.f (interpE e₂) ((Monotone.f (interpS Θ) (fst (fst x)) , snd (fst x)) , snd x))
(subst-eq-l (s-extend (s-extend Θ)) e₂ x)
(Monotone.is-monotone (interpE e₂)
(Monotone.f (interpS {nat :: τ :: Γ} {_ :: _ :: Γ'} (s-extend (s-extend Θ))) x)
((Monotone.f (interpS Θ) (fst (fst x)) , snd (fst x)) , snd x)
(((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS (λ x₁ → ren (ren (Θ x₁) iS) iS)) x)
(Monotone.f (interpS (λ x₁ → ren (Θ x₁) iS)) (fst x))
(Monotone.f (interpS Θ) (fst (fst x)))
(subst-eq-l-lam {τ :: Γ} {Γ'} (λ x₁ → ren (Θ x₁) iS) (fst x) (snd x))
(subst-eq-l-lam Θ (fst (fst x)) (snd (fst x)))) ,
(Preorder-str.refl (snd [ τ ]t) (snd (fst x)))) ,
(♭nat-refl (snd x))))))
subst-eq-l {Γ} {τ = τ1 ->c τ2} Θ (lam e) k x =
Preorder-str.trans (snd [ τ2 ]t)
(Monotone.f (Monotone.f (interpE (subst (lam e) Θ)) k) x)
(Monotone.f (interpE e) (Monotone.f (interpS (s-extend {_} {_} {τ1} Θ)) (k , x)))
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k , x))
(subst-eq-l (s-extend Θ) e (k , x))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpS (s-extend {_} {_} {τ1} Θ)) (k , x))
(Monotone.f (interpS Θ) k , x)
(subst-eq-l-lam Θ k x ,
(Preorder-str.refl (snd [ τ1 ]t) x)))
subst-eq-l {Γ} {τ = τ} Θ (app e e₁) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (Monotone.f (interpE (subst e Θ)) k) (Monotone.f (interpE (subst e₁ Θ)) k))
(Monotone.f (Monotone.f (interpE (subst e Θ)) k) (Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k)))
(Monotone.f (Monotone.f (interpE e) (Monotone.f (interpS Θ) k)) (Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k)))
(Monotone.is-monotone (Monotone.f (interpE (subst e Θ)) k)
(Monotone.f (interpE (subst e₁ Θ)) k) (Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k)) (subst-eq-l Θ e₁ k))
(subst-eq-l Θ e k (Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k)))
subst-eq-l Θ (prod e e₁) k = (subst-eq-l Θ e k) , (subst-eq-l Θ e₁ k)
subst-eq-l Θ (l-proj e) k = fst (subst-eq-l Θ e k)
subst-eq-l Θ (r-proj e) k = snd (subst-eq-l Θ e k)
subst-eq-l Θ nil k = <>
subst-eq-l Θ (e ::c e₁) k = (subst-eq-l Θ e k) , (subst-eq-l Θ e₁ k)
subst-eq-l {Γ} {Γ'} {τ} Θ (listrec {.Γ'} {τ'} e e₁ e₂) k =
Preorder-str.trans (snd [ τ ]t)
(lrec (Monotone.f (interpE (subst e Θ)) k)
(Monotone.f (interpE (subst e₁ Θ)) k)
(λ x₁ x₂ x₃ → Monotone.f (interpE (subst e₂ (s-extend (s-extend (s-extend Θ))))) (((k , x₃) , x₂) , x₁)))
(lrec (Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (subst e₁ Θ)) k)
(λ x₁ x₂ x₃ → Monotone.f (interpE (subst e₂ (s-extend (s-extend (s-extend Θ))))) (((k , x₃) , x₂) , x₁)))
(lrec (Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k))
(λ x₁ x₂ x₃ → Monotone.f (interpE e₂) (((Monotone.f (interpS Θ) k , x₃) , x₂) , x₁)))
(listrec-fix-args (interpE (subst e₁ Θ)) (interpE (subst e₂ (s-extend (s-extend (s-extend Θ)))))
(k , Monotone.f (interpE (subst e Θ)) k)
(k , Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(Preorder-str.refl (snd [ Γ ]c) k , subst-eq-l Θ e k))
(lrec-cong
(interpE (subst e₁ Θ))
(monotone (λ k₁ → Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k₁))
(λ x y x₁ → Monotone.is-monotone (interpE e₁) (Monotone.f (interpS Θ) x) (Monotone.f (interpS Θ) y) (Monotone.is-monotone (interpS Θ) x y x₁)))
(interpE (subst e₂ (s-extend (s-extend (s-extend Θ)))))
(monotone (λ x → Monotone.f (interpE e₂) (((Monotone.f (interpS Θ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x))
(λ x y x₁ →
Monotone.is-monotone (interpE e₂)
(((Monotone.f (interpS Θ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x)
(((Monotone.f (interpS Θ) (fst (fst (fst y))) , snd (fst (fst y))) , snd (fst y)) , snd y)
(((Monotone.is-monotone (interpS Θ) (fst (fst (fst x))) (fst (fst (fst y))) (fst (fst (fst x₁))) ,
snd (fst (fst x₁))) ,
snd (fst x₁)) ,
snd x₁)))
(k , Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(λ x → subst-eq-l Θ e₁ x)
(λ x →
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e₂ (s-extend (s-extend (s-extend Θ))))) x)
(Monotone.f (interpE e₂) (Monotone.f (interpS {τ' :: list τ' :: τ :: Γ} {_ :: _ :: _ :: Γ'} (s-extend (s-extend (s-extend Θ)))) x))
(Monotone.f (interpE e₂) (((Monotone.f (interpS Θ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x))
(subst-eq-l (s-extend (s-extend (s-extend Θ))) e₂ x)
(Monotone.is-monotone (interpE e₂)
(Monotone.f (interpS {τ' :: list τ' :: τ :: Γ} {_ :: _ :: _ :: Γ'} (s-extend (s-extend (s-extend Θ)))) x)
(((Monotone.f (interpS Θ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x)
((((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS (λ x₁ → ren (ren (ren (Θ x₁) iS) iS) iS)) x)
(Monotone.f (interpS {τ :: Γ} {Γ'} (throw-s (s-extend Θ))) (fst (fst x)))
(Monotone.f (interpS Θ) (fst (fst (fst x))))
(Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS (λ x₁ → ren (ren (ren (Θ x₁) iS) iS) iS)) x)
(Monotone.f (interpS (λ x₁ → ren (ren (Θ x₁) iS) iS)) (fst x))
(Monotone.f (interpS {τ :: Γ} {Γ'} (throw-s (s-extend Θ))) (fst (fst x)))
(subst-eq-l-lam (λ x₁ → ren (ren (Θ x₁) iS) iS) (fst x) (snd x))
(subst-eq-l-lam (λ x₁ → ren (Θ x₁) iS) (fst (fst x)) (snd (fst x))))
(subst-eq-l-lam {Γ} {Γ'} Θ (fst (fst (fst x))) (snd (fst (fst x))))) ,
(Preorder-str.refl (snd [ τ ]t) (snd (fst (fst x))))) ,
(l-refl (snd [ τ' ]t) (snd (fst x)))) ,
(Preorder-str.refl (snd [ τ' ]t) (snd x))))))
subst-eq-l Θ true k = <>
subst-eq-l Θ false k = <>
subst-eq-l {Γ} {Γ'} {τ = τ} Θ (letc {.Γ'} {ρ'} e e') k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (s-extend Θ))) (k , Monotone.f (interpE (subst e' Θ)) k))
(Monotone.f (interpE e) (Monotone.f (interpS (s-extend {_} {_} {ρ'} Θ)) (k , Monotone.f (interpE (subst e' Θ)) k)))
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k , Monotone.f (interpE e') (Monotone.f (interpS Θ) k)))
(subst-eq-l (s-extend Θ) e (k , Monotone.f (interpE (subst e' Θ)) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpS (s-extend {_} {_} {ρ'} Θ)) (k , Monotone.f (interpE (subst e' Θ)) k))
(Monotone.f (interpS Θ) k , Monotone.f (interpE e') (Monotone.f (interpS Θ) k))
((subst-eq-l-lam Θ k (Monotone.f (interpE (subst e' Θ)) k)) , (subst-eq-l Θ e' k)))
subst-eq-l {τ = τ} Θ (max x e e₁) k =
Preorder-max-str.max-lub [ x ]tm
(Monotone.f (interpE (max x e e₁)) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (subst e Θ)) k)
(Monotone.f (interpE (subst e₁ Θ)) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e Θ)) k)
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (max x e e₁)) (Monotone.f (interpS Θ) k))
(subst-eq-l Θ e k)
(Preorder-max-str.max-l [ x ]tm (Monotone.f (interpE e) (Monotone.f (interpS Θ) k)) (Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k))))
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e₁ Θ)) k)
(Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (max x e e₁)) (Monotone.f (interpS Θ) k))
(subst-eq-l Θ e₁ k)
(Preorder-max-str.max-r [ x ]tm (Monotone.f (interpE e) (Monotone.f (interpS Θ) k)) (Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k))))
ids-lem-r : ∀ {Γ} (k : fst [ Γ ]c) → Preorder-str.≤ (snd [ Γ ]c) k (Monotone.f (interpR {Γ} {Γ} (λ x₂ → x₂)) k)
ids-lem-r {[]} k = <>
ids-lem-r {x :: Γ} (k1 , k2) =
(Preorder-str.trans (snd [ Γ ]c)
k1
(Monotone.f (interpR {Γ} {Γ} (λ x₂ → x₂)) k1)
(Monotone.f (interpR {x :: Γ} {Γ} (throw-r (λ x₂ → x₂))) (k1 , k2))
(ids-lem-r {Γ} k1)
(ren-eq-r-lam {Γ} {Γ} (λ x₂ → x₂) k1 k2)) ,
(Preorder-str.refl (snd [ x ]t) k2)
subst-eq-r-lam : ∀ {Γ Γ' τ1} (Θ : sctx Γ Γ') (k : fst [ Γ ]c) (x : fst [ τ1 ]t)
→ Preorder-str.≤ (snd [ Γ' ]c) (Monotone.f (interpS Θ) k) (Monotone.f (interpS (throw-s (s-extend {_} {_} {τ1} Θ))) (k , x))
subst-eq-r-lam {Γ' = []} Θ k x = <>
subst-eq-r-lam {Γ} {Γ' = x :: Γ'} {τ1} Θ k x₁ =
(subst-eq-r-lam (throw-s Θ) k x₁) ,
Preorder-str.trans (snd [ x ]t)
(snd (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (Θ i0)) (Monotone.f (interpR {τ1 :: Γ} {Γ} iS) (k , x₁)))
(Monotone.f (interpE (ren (Θ i0) iS)) (k , x₁))
(Monotone.is-monotone (interpE (Θ i0))
k
(Monotone.f (interpR {τ1 :: Γ} {Γ} iS) (k , x₁))
(Preorder-str.trans (snd [ Γ ]c)
k
(Monotone.f (interpR {Γ} {Γ} (λ x₂ → x₂)) k)
(Monotone.f (interpR {τ1 :: Γ} {Γ} iS) (k , x₁))
(ids-lem-r {Γ} k)
(ren-eq-r-lam {Γ} {Γ} (λ x₂ → x₂) k x₁)))
(ren-eq-r iS (Θ i0) (k , x₁))
subst-eq-r : ∀ {Γ Γ' τ} → (Θ : sctx Γ Γ') → (e : Γ' |- τ) → (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ τ ]t) (Monotone.f (interpE e) (Monotone.f (interpS Θ) k)) (Monotone.f (interpE (subst e Θ)) k)
subst-eq-r Θ unit k = <>
subst-eq-r Θ 0C k = <>
subst-eq-r Θ 1C k = <>
subst-eq-r Θ (plusC e e₁) k =
plus-lem (Monotone.f (interpE e) (Monotone.f (interpS Θ) k)) (Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (subst e Θ)) k) (Monotone.f (interpE (subst e₁ Θ)) k)
(subst-eq-r Θ e k) (subst-eq-r Θ e₁ k)
subst-eq-r {τ = τ} Θ (var i0) k = Preorder-str.refl (snd [ τ ]t) (Monotone.f (interpE (Θ i0)) k)
subst-eq-r {τ = τ} Θ (var (iS x)) k = subst-eq-r (throw-s Θ) (var x) k
subst-eq-r Θ z k = <>
subst-eq-r Θ (s e) k = subst-eq-r Θ e k
subst-eq-r {Γ} {Γ'} {τ} Θ (rec e e₁ e₂) k =
Preorder-str.trans (snd [ τ ]t)
(natrec (Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k))
(λ n x₂ → Monotone.f (interpE e₂) ((Monotone.f (interpS Θ) k , x₂) , n))
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k)))
(natrec (Monotone.f (interpE (subst e₁ Θ)) k)
(λ n x₂ → Monotone.f (interpE (subst e₂ (s-extend (s-extend Θ)))) ((k , x₂) , n))
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k)))
(natrec (Monotone.f (interpE (subst e₁ Θ)) k)
(λ n x₂ → Monotone.f (interpE (subst e₂ (s-extend (s-extend Θ)))) ((k , x₂) , n))
(Monotone.f (interpE (subst e Θ)) k))
(♭h-cong
(monotone (λ v → Monotone.f (interpE e₁) (Monotone.f (interpS Θ) v))
(λ x y x₁ → Monotone.is-monotone (interpE e₁) (Monotone.f (interpS Θ) x) (Monotone.f (interpS Θ) y) (Monotone.is-monotone (interpS Θ) x y x₁)))
(interpE (subst e₁ Θ))
(monotone (λ v → Monotone.f (interpE e₂) ((Monotone.f (interpS Θ) (fst (fst v)) , snd (fst v)) , snd v))
(λ x y x₁ →
Monotone.is-monotone (interpE e₂)
((Monotone.f (interpS Θ) (fst (fst x)) , snd (fst x)) , snd x)
((Monotone.f (interpS Θ) (fst (fst y)) , snd (fst y)) , snd y)
((Monotone.is-monotone (interpS Θ) (fst (fst x)) (fst (fst y)) (fst (fst x₁)) ,
snd (fst x₁)) ,
snd x₁)))
(interpE (subst e₂ (s-extend (s-extend Θ))))
(k , (Monotone.f (interpE e) (Monotone.f (interpS Θ) k)))
(λ x → subst-eq-r Θ e₁ x)
(λ x →
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e₂) ((Monotone.f (interpS Θ) (fst (fst x)) , snd (fst x)) , snd x))
(Monotone.f (interpE e₂) (Monotone.f (interpS {nat :: τ :: Γ} {_ :: _ :: Γ'} (s-extend (s-extend Θ))) x))
(Monotone.f (interpE (subst e₂ (s-extend (s-extend Θ)))) x)
(Monotone.is-monotone (interpE e₂)
((Monotone.f (interpS Θ) (fst (fst x)) , snd (fst x)) , snd x)
(Monotone.f (interpS {nat :: τ :: Γ} {_ :: _ :: Γ'} (s-extend (s-extend Θ))) x)
(((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS Θ) (fst (fst x)))
(Monotone.f (interpS (λ x₁ → ren (Θ x₁) iS)) (fst x))
(Monotone.f (interpS (λ x₁ → ren (ren (Θ x₁) iS) iS)) x)
(subst-eq-r-lam Θ (fst (fst x)) (snd (fst x)))
(subst-eq-r-lam {τ :: Γ} {Γ'} (λ x₁ → ren (Θ x₁) iS) (fst x) (snd x))) ,
(Preorder-str.refl (snd [ τ ]t) (snd (fst x)))) ,
(♭nat-refl (snd x))))
(subst-eq-r (s-extend (s-extend Θ)) e₂ x)))
(♭h-fix-args (interpE (subst e₁ Θ))
(interpE (subst e₂ (s-extend (s-extend Θ))))
(k , Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(k , Monotone.f (interpE (subst e Θ)) k)
(subst-eq-r Θ e k))
subst-eq-r {Γ} {τ = τ1 ->c τ2} Θ (lam e) k x =
Preorder-str.trans (snd [ τ2 ]t)
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k , x))
(Monotone.f (interpE e) (Monotone.f (interpS (s-extend {_} {_} {τ1} Θ)) (k , x)))
(Monotone.f (Monotone.f (interpE (subst (lam e) Θ)) k) x)
((Monotone.is-monotone (interpE e)
(Monotone.f (interpS Θ) k , x)
(Monotone.f (interpS (s-extend {_} {_} {τ1} Θ)) (k , x))
(subst-eq-r-lam Θ k x ,
(Preorder-str.refl (snd [ τ1 ]t) x))))
(subst-eq-r (s-extend Θ) e (k , x))
subst-eq-r {Γ} {τ = τ} Θ (app e e₁) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (Monotone.f (interpE e) (Monotone.f (interpS Θ) k)) (Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k)))
(Monotone.f (Monotone.f (interpE (subst e Θ)) k) (Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k)))
(Monotone.f (Monotone.f (interpE (subst e Θ)) k) (Monotone.f (interpE (subst e₁ Θ)) k))
(subst-eq-r Θ e k (Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k)))
(Monotone.is-monotone (Monotone.f (interpE (subst e Θ)) k)
(Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k)) (Monotone.f (interpE (subst e₁ Θ)) k) (subst-eq-r Θ e₁ k))
subst-eq-r Θ (prod e e₁) k = (subst-eq-r Θ e k) , (subst-eq-r Θ e₁ k)
subst-eq-r Θ (l-proj e) k = fst (subst-eq-r Θ e k)
subst-eq-r Θ (r-proj e) k = snd (subst-eq-r Θ e k)
subst-eq-r Θ nil k = <>
subst-eq-r Θ (e ::c e₁) k = (subst-eq-r Θ e k) , (subst-eq-r Θ e₁ k)
subst-eq-r {Γ} {Γ'} {τ} Θ (listrec {.Γ'} {τ'} e e₁ e₂) k =
Preorder-str.trans (snd [ τ ]t)
(lrec (Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k))
(λ x₁ x₂ x₃ → Monotone.f (interpE e₂) (((Monotone.f (interpS Θ) k , x₃) , x₂) , x₁)))
(lrec (Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (subst e₁ Θ)) k)
(λ x₁ x₂ x₃ → Monotone.f (interpE (subst e₂ (s-extend (s-extend (s-extend Θ))))) (((k , x₃) , x₂) , x₁)))
(lrec (Monotone.f (interpE (subst e Θ)) k)
(Monotone.f (interpE (subst e₁ Θ)) k)
(λ x₁ x₂ x₃ → Monotone.f (interpE (subst e₂ (s-extend (s-extend (s-extend Θ))))) (((k , x₃) , x₂) , x₁)))
(lrec-cong
(monotone (λ k₁ → Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k₁))
(λ x y x₁ → Monotone.is-monotone (interpE e₁) (Monotone.f (interpS Θ) x) (Monotone.f (interpS Θ) y) (Monotone.is-monotone (interpS Θ) x y x₁)))
(interpE (subst e₁ Θ))
(monotone (λ x → Monotone.f (interpE e₂) (((Monotone.f (interpS Θ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x))
(λ x y x₁ →
Monotone.is-monotone (interpE e₂)
(((Monotone.f (interpS Θ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x)
(((Monotone.f (interpS Θ) (fst (fst (fst y))) , snd (fst (fst y))) , snd (fst y)) , snd y)
(((Monotone.is-monotone (interpS Θ)
(fst (fst (fst x))) (fst (fst (fst y))) (fst (fst (fst x₁))) , snd (fst (fst x₁))) , snd (fst x₁)) , snd x₁)))
(interpE (subst e₂ (s-extend (s-extend (s-extend Θ)))))
(k , (Monotone.f (interpE e) (Monotone.f (interpS Θ) k)))
(λ x → subst-eq-r Θ e₁ x)
(λ x →
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e₂) (((Monotone.f (interpS Θ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x))
(Monotone.f (interpE e₂) (Monotone.f (interpS {τ' :: list τ' :: τ :: Γ} {_ :: _ :: _ :: Γ'} (s-extend (s-extend (s-extend Θ)))) x))
(Monotone.f (interpE (subst e₂ (s-extend (s-extend (s-extend Θ))))) x)
(Monotone.is-monotone (interpE e₂)
(((Monotone.f (interpS Θ) (fst (fst (fst x))) , snd (fst (fst x))) , snd (fst x)) , snd x)
(Monotone.f (interpS {τ' :: list τ' :: τ :: Γ} {_ :: _ :: _ :: Γ'} (s-extend (s-extend (s-extend Θ)))) x)
((((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS Θ) (fst (fst (fst x))))
(Monotone.f (interpS {τ :: Γ} {Γ'} (throw-s (s-extend Θ))) (fst (fst x)))
(Monotone.f (interpS (λ x₁ → ren (ren (ren (Θ x₁) iS) iS) iS)) x)
(subst-eq-r-lam {Γ} {Γ'} Θ (fst (fst (fst x))) (snd (fst (fst x))))
(Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS {τ :: Γ} {Γ'} (throw-s (s-extend Θ))) (fst (fst x)))
(Monotone.f (interpS (λ x₁ → ren (ren (Θ x₁) iS) iS)) (fst x))
(Monotone.f (interpS (λ x₁ → ren (ren (ren (Θ x₁) iS) iS) iS)) x)
(subst-eq-r-lam (λ x₁ → ren (Θ x₁) iS) (fst (fst x)) (snd (fst x)))
(subst-eq-r-lam (λ x₁ → ren (ren (Θ x₁) iS) iS) (fst x) (snd x)))) ,
(Preorder-str.refl (snd [ τ ]t) (snd (fst (fst x))))) ,
(l-refl (snd [ τ' ]t) (snd (fst x)))) ,
(Preorder-str.refl (snd [ τ' ]t) (snd x))))
(subst-eq-r (s-extend (s-extend (s-extend Θ))) e₂ x)))
(listrec-fix-args (interpE (subst e₁ Θ)) (interpE (subst e₂ (s-extend (s-extend (s-extend Θ)))))
(k , Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(k , Monotone.f (interpE (subst e Θ)) k)
(Preorder-str.refl (snd [ Γ ]c) k , subst-eq-r Θ e k))
subst-eq-r Θ true k = <>
subst-eq-r Θ false k = <>
subst-eq-r {Γ} {Γ'} {τ = τ} Θ (letc {.Γ'} {ρ'} e e') k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k , Monotone.f (interpE e') (Monotone.f (interpS Θ) k)))
(Monotone.f (interpE e) (Monotone.f (interpS (s-extend {_} {_} {ρ'} Θ)) (k , Monotone.f (interpE (subst e' Θ)) k)))
(Monotone.f (interpE (subst e (s-extend Θ))) (k , Monotone.f (interpE (subst e' Θ)) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpS Θ) k , Monotone.f (interpE e') (Monotone.f (interpS Θ) k))
(Monotone.f (interpS (s-extend {_} {_} {ρ'} Θ)) (k , Monotone.f (interpE (subst e' Θ)) k))
((subst-eq-r-lam Θ k (Monotone.f (interpE (subst e' Θ)) k)) , (subst-eq-r Θ e' k)))
(subst-eq-r (s-extend Θ) e (k , Monotone.f (interpE (subst e' Θ)) k))
subst-eq-r {τ = τ} Θ (max x e e₁) k =
Preorder-max-str.max-lub [ x ]tm
(Monotone.f (interpE (subst (max x e e₁) Θ)) k)
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k))
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (subst e Θ)) k)
(Monotone.f (interpE (subst (max x e e₁) Θ)) k)
(subst-eq-r Θ e k)
(Preorder-max-str.max-l [ x ]tm (Monotone.f (interpE (subst e Θ)) k) (Monotone.f (interpE (subst e₁ Θ)) k)))
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e₁) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (subst e₁ Θ)) k)
(Monotone.f (interpE (subst (max x e e₁) Θ)) k)
(subst-eq-r Θ e₁ k)
(Preorder-max-str.max-r [ x ]tm (Monotone.f (interpE (subst e Θ)) k) (Monotone.f (interpE (subst e₁ Θ)) k)))
interp-rr-l : ∀ {Γ Γ' Γ''} (ρ1 : rctx Γ Γ') (ρ2 : rctx Γ' Γ'') (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ Γ'' ]c)
(Monotone.f (interpR ρ2) (Monotone.f (interpR ρ1) k))
(Monotone.f (interpR (λ x → ρ1 (ρ2 x))) k)
interp-rr-l {Γ'' = []} ρ1 ρ2 k = <>
interp-rr-l {Γ'' = x :: Γ''} ρ1 ρ2 k = (interp-rr-l ρ1 (throw-r ρ2) k) , (ren-eq-r ρ1 (var (ρ2 i0)) k)
interp-rr-r : ∀ {Γ Γ' Γ''} (ρ1 : rctx Γ Γ') (ρ2 : rctx Γ' Γ'') (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ Γ'' ]c)
(Monotone.f (interpR (λ x → ρ1 (ρ2 x))) k)
(Monotone.f (interpR ρ2) (Monotone.f (interpR ρ1) k))
interp-rr-r {Γ'' = []} ρ1 ρ2 k = <>
interp-rr-r {Γ'' = x :: Γ''} ρ1 ρ2 k = (interp-rr-r ρ1 (throw-r ρ2) k) , (ren-eq-l ρ1 (var (ρ2 i0)) k)
interp-rs-l : ∀ {Γ Γ' Γ''} (ρ : rctx Γ Γ') (Θ : sctx Γ' Γ'') (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ Γ'' ]c)
(Monotone.f (interpS Θ) (Monotone.f (interpR ρ) k))
(Monotone.f (interpS (λ x → ren (Θ x) ρ)) k)
interp-rs-l {Γ'' = []} ρ Θ k = <>
interp-rs-l {Γ'' = x :: Γ''} ρ Θ k = (interp-rs-l ρ (throw-s Θ) k) , (ren-eq-r ρ (Θ i0) k)
interp-rs-r : ∀ {Γ Γ' Γ''} (ρ : rctx Γ Γ') (Θ : sctx Γ' Γ'') (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ Γ'' ]c)
(Monotone.f (interpS (λ x → ren (Θ x) ρ)) k)
(Monotone.f (interpS Θ) (Monotone.f (interpR ρ) k))
interp-rs-r {Γ'' = []} ρ Θ k = <>
interp-rs-r {Γ'' = x :: Γ''} ρ Θ k = (interp-rs-r ρ (throw-s Θ) k) , (ren-eq-l ρ (Θ i0) k)
interp-sr-l : ∀ {Γ Γ' Γ''} (Θ : sctx Γ Γ') (ρ : rctx Γ' Γ'') (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ Γ'' ]c)
(Monotone.f (interpR ρ) (Monotone.f (interpS Θ) k))
(Monotone.f (interpS (λ x → Θ (ρ x))) k)
interp-sr-l {Γ'' = []} Θ ρ k = <>
interp-sr-l {Γ'' = x :: Γ''} Θ ρ k = (interp-sr-l Θ (throw-r ρ) k) , (subst-eq-r Θ (var (ρ i0)) k)
interp-sr-r : ∀ {Γ Γ' Γ''} (Θ : sctx Γ Γ') (ρ : rctx Γ' Γ'') (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ Γ'' ]c)
(Monotone.f (interpS (λ x → Θ (ρ x))) k)
(Monotone.f (interpR ρ) (Monotone.f (interpS Θ) k))
interp-sr-r {Γ'' = []} Θ ρ k = <>
interp-sr-r {Γ'' = x :: Γ''} Θ ρ k = (interp-sr-r Θ (throw-r ρ) k) , (subst-eq-l Θ (var (ρ i0)) k)
interp-ss-l : ∀ {Γ Γ' Γ''} (Θ1 : sctx Γ Γ') (Θ2 : sctx Γ' Γ'') (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ Γ'' ]c)
(Monotone.f (interpS (λ x → subst (Θ2 x) Θ1)) k)
(Monotone.f (interpS Θ2) (Monotone.f (interpS Θ1) k))
interp-ss-l {Γ'' = []} Θ1 Θ2 k = <>
interp-ss-l {Γ'' = x :: Γ''} Θ1 Θ2 k = (interp-ss-l Θ1 (throw-s Θ2) k) , (subst-eq-l Θ1 (Θ2 i0) k)
interp-ss-r : ∀ {Γ Γ' Γ''} (Θ1 : sctx Γ Γ') (Θ2 : sctx Γ' Γ'') (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ Γ'' ]c)
(Monotone.f (interpS Θ2) (Monotone.f (interpS Θ1) k))
(Monotone.f (interpS (λ x → subst (Θ2 x) Θ1)) k)
interp-ss-r {Γ'' = []} Θ1 Θ2 k = <>
interp-ss-r {Γ'' = x :: Γ''} Θ1 Θ2 k = (interp-ss-r Θ1 (throw-s Θ2) k) , (subst-eq-r Θ1 (Θ2 i0) k)
lam-s-lem : ∀ {Γ} (k : fst [ Γ ]c) → Preorder-str.≤ (snd [ Γ ]c) (Monotone.f (interpS {Γ} {Γ} ids) k) k
lam-s-lem {[]} k = <>
lam-s-lem {x :: Γ} (k1 , k2) =
(Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpS {x :: Γ} {Γ} (throw-s ids)) (k1 , k2))
(Monotone.f (interpS {Γ} {Γ} ids) k1)
k1
(subst-eq-l-lam {Γ} {Γ} ids k1 (Monotone.f (interpE {x :: Γ} {x} (ids i0)) (k1 , k2)))
(lam-s-lem {Γ} k1)) ,
(Preorder-str.refl (snd [ x ]t) k2)
lam-s-lem-r : ∀ {Γ} (k : fst [ Γ ]c) → Preorder-str.≤ (snd [ Γ ]c) k (Monotone.f (interpS {Γ} {Γ} ids) k)
lam-s-lem-r {[]} k = <>
lam-s-lem-r {x :: Γ} (k1 , k2) =
(Preorder-str.trans (snd [ Γ ]c)
k1
(Monotone.f (interpS {Γ} {Γ} ids) k1)
(Monotone.f (interpS {x :: Γ} {Γ} (throw-s ids)) (k1 , k2))
(lam-s-lem-r {Γ} k1)
(subst-eq-r-lam {Γ} {Γ} ids k1 (Monotone.f (interpE {x :: Γ} {x} (ids i0)) (k1 , k2)))) ,
(Preorder-str.refl (snd [ x ]t) k2)
interp-subst-comp-l : ∀ {Γ Γ' τ'} (Θ : sctx Γ Γ') (v : Γ |- τ') (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ Γ' ]c)
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v))) k)
(Monotone.f (interpS (λ x → Θ x)) k)
interp-subst-comp-l {Γ' = []} Θ v k = <>
interp-subst-comp-l {Γ} {Γ' = x :: Γ'} {τ'} Θ v k =
(interp-subst-comp-l (throw-s Θ) v k) ,
(Preorder-str.trans (snd [ x ]t)
(Monotone.f (interpE (subst (ren (Θ i0) iS) (lem3' ids v))) k)
(Monotone.f (interpE (ren (Θ i0) iS)) (Monotone.f (interpS (lem3' ids v)) k))
(Monotone.f (interpE (Θ i0)) k)
(subst-eq-l (lem3' ids v) (ren (Θ i0) iS) k)
(Preorder-str.trans (snd [ x ]t)
(Monotone.f (interpE (ren (Θ i0) iS)) (Monotone.f (interpS (lem3' ids v)) k))
(Monotone.f (interpE (Θ i0)) (Monotone.f (interpR {τ' :: Γ} {Γ} iS) (Monotone.f (interpS {Γ} {τ' :: Γ} (lem3' ids v)) k)))
(Monotone.f (interpE (Θ i0)) k)
(ren-eq-l iS (Θ i0) (Monotone.f (interpS (lem3' ids v)) k))
(Monotone.is-monotone (interpE (Θ i0))
(Monotone.f (interpR {τ' :: Γ} {Γ} iS) (Monotone.f (interpS {Γ} {τ' :: Γ} (lem3' ids v)) k))
k
(Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpR {τ' :: Γ} {Γ} iS) (Monotone.f (interpS {Γ} {τ' :: Γ} (lem3' ids v)) k))
(Monotone.f (interpR {Γ} {Γ} (λ x₂ → x₂)) k)
k
(Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpR {τ' :: Γ} {Γ} iS) (Monotone.f (interpS {Γ} {τ' :: Γ} (lem3' ids v)) k))
(Monotone.f (interpR {τ' :: Γ} {Γ} iS) (k , Monotone.f (interpE v) k))
(Monotone.f (interpR {Γ} {Γ} (λ x₂ → x₂)) k)
(Monotone.is-monotone (interpR {τ' :: Γ} {Γ} iS)
(Monotone.f (interpS {Γ} {τ' :: Γ} (lem3' ids v)) k)
(k , Monotone.f (interpE v) k)
(lam-s-lem {Γ} k , (Preorder-str.refl (snd [ τ' ]t) (Monotone.f (interpE v) k))))
(ren-eq-l-lam {Γ} {Γ} {τ'} (λ x₂ → x₂) k (Monotone.f (interpE v) k)))
(ids-lem-l {Γ} k)))))
interp-subst-comp-r : ∀ {Γ Γ' τ'} (Θ : sctx Γ Γ') (v : Γ |- τ') (k : fst [ Γ ]c)
→ Preorder-str.≤ (snd [ Γ' ]c)
(Monotone.f (interpS (λ x → Θ x)) k)
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v))) k)
interp-subst-comp-r {Γ' = []} Θ v k = <>
interp-subst-comp-r {Γ} {Γ' = x :: Γ'} {τ'} Θ v k =
(interp-subst-comp-r (throw-s Θ) v k) ,
Preorder-str.trans (snd [ x ]t)
(Monotone.f (interpE (Θ i0)) k)
(Monotone.f (interpE (ren (Θ i0) iS)) (Monotone.f (interpS (lem3' ids v)) k))
(Monotone.f (interpE (subst (ren (Θ i0) iS) (lem3' ids v))) k)
(Preorder-str.trans (snd [ x ]t)
(Monotone.f (interpE (Θ i0)) k)
(Monotone.f (interpE (Θ i0)) (Monotone.f (interpR {τ' :: Γ} {Γ} iS) (Monotone.f (interpS {Γ} {τ' :: Γ} (lem3' ids v)) k)))
(Monotone.f (interpE (ren (Θ i0) iS)) (Monotone.f (interpS (lem3' ids v)) k))
(Monotone.is-monotone (interpE (Θ i0))
k
(Monotone.f (interpR {τ' :: Γ} {Γ} iS) (Monotone.f (interpS {Γ} {τ' :: Γ} (lem3' ids v)) k))
(Preorder-str.trans (snd [ Γ ]c)
k
(Monotone.f (interpR {Γ} {Γ} (λ x₂ → x₂)) k)
(Monotone.f (interpR {τ' :: Γ} {Γ} iS) (Monotone.f (interpS {Γ} {τ' :: Γ} (lem3' ids v)) k))
(ids-lem-r {Γ} k)
(Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpR {Γ} {Γ} (λ x₂ → x₂)) k)
(Monotone.f (interpR {τ' :: Γ} {Γ} iS) (k , Monotone.f (interpE v) k))
(Monotone.f (interpR {τ' :: Γ} {Γ} iS) (Monotone.f (interpS {Γ} {τ' :: Γ} (lem3' ids v)) k))
(ren-eq-r-lam {Γ} {Γ} {τ'} (λ x₂ → x₂) k (Monotone.f (interpE v) k))
(Monotone.is-monotone (interpR {τ' :: Γ} {Γ} iS)
(k , Monotone.f (interpE v) k)
(Monotone.f (interpS {Γ} {τ' :: Γ} (lem3' ids v)) k)
(lam-s-lem-r {Γ} k , Preorder-str.refl (snd [ τ' ]t) (Monotone.f (interpE v) k))))))
(ren-eq-r iS (Θ i0) (Monotone.f (interpS (lem3' ids v)) k)))
(subst-eq-r (lem3' ids v) (ren (Θ i0) iS) k)
interp-subst-comp2-l : ∀ {Γ Γ' τ' τ''} (Θ : sctx Γ Γ') (k : fst [ Γ ]c) (v1 : Γ |- τ') (v2 : Γ |- τ'')
→ Preorder-str.≤ (snd [ Γ' ]c)
(Monotone.f (interpS {τ' :: τ'' :: Γ} {Γ'} (λ x → ren (ren (Θ x) iS) iS))
((Monotone.f (interpS {Γ} {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v2))) k)
interp-subst-comp2-l {Γ' = []} Θ k v1 v2 = <>
interp-subst-comp2-l {Γ} {Γ' = x :: Γ'} {τ'} {τ''} Θ k v1 v2 =
(interp-subst-comp2-l (throw-s Θ) k v1 v2) ,
Preorder-str.trans (snd [ x ]t)
(Monotone.f (interpE (ren (ren (Θ i0) iS) iS)) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpE (ren (Θ i0) iS))
(Monotone.f (interpR {τ' :: τ'' :: Γ} iS) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k)))
(Monotone.f (interpE (subst (ren (Θ i0) iS) (lem3' ids v2))) k)
(ren-eq-l iS (ren (Θ i0) iS) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Preorder-str.trans (snd [ x ]t)
(Monotone.f (interpE (ren (Θ i0) iS))
(Monotone.f (interpR {τ' :: τ'' :: Γ} iS) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k)))
(Monotone.f (interpE (ren (Θ i0) iS)) (Monotone.f (interpS (lem3' ids v2)) k))
(Monotone.f (interpE (subst (ren (Θ i0) iS) (lem3' ids v2))) k)
(Monotone.is-monotone (interpE (ren (Θ i0) iS))
(Monotone.f (interpR {τ' :: τ'' :: Γ} iS) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpS (lem3' ids v2)) k)
(Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpR {τ' :: τ'' :: Γ} (λ x₁ → iS (iS x₁))) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpR {Γ} {Γ} (λ x₁ → x₁)) (Monotone.f (interpS {Γ} ids) k))
(Monotone.f (interpS {Γ} ids) k)
(Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpR {τ' :: τ'' :: Γ} (λ x₁ → iS (iS x₁))) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpR {τ'' :: Γ} (throw-r (r-extend (λ x₁ → x₁)))) (Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k))
(Monotone.f (interpR {Γ} {Γ} (λ x₁ → x₁)) (Monotone.f (interpS {Γ} ids) k))
(ren-eq-l-lam {τ'' :: Γ} (λ x₁ → iS x₁) (Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) (Monotone.f (interpE v1) k))
(ren-eq-l-lam {Γ} {Γ} {τ''} (λ x₁ → x₁) (Monotone.f (interpS {Γ} ids) k) (Monotone.f (interpE v2) k)))
(interp-sr-l {Γ} ids (λ x₁ → x₁) k) ,
Preorder-str.refl (snd [ τ'' ]t) (Monotone.f (interpE v2) k)))
(subst-eq-r (lem3' ids v2) (ren (Θ i0) iS) k))
interp-subst-comp2-r : ∀ {Γ Γ' τ' τ''} (Θ : sctx Γ Γ') (k : fst [ Γ ]c) (v1 : Γ |- τ') (v2 : Γ |- τ'')
→ Preorder-str.≤ (snd [ Γ' ]c)
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v2))) k)
(Monotone.f (interpS {τ' :: τ'' :: Γ} {Γ'} (λ x → ren (ren (Θ x) iS) iS))
((Monotone.f (interpS {Γ} {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
interp-subst-comp2-r {Γ' = []} Θ k v1 v2 = <>
interp-subst-comp2-r {Γ} {Γ' = x :: Γ'} {τ'} {τ''} Θ k v1 v2 =
(interp-subst-comp2-r (throw-s Θ) k v1 v2) ,
(Preorder-str.trans (snd [ x ]t)
(Monotone.f (interpE (subst (ren (Θ i0) iS) (lem3' ids v2))) k)
(Monotone.f (interpE (ren (Θ i0) iS))
(Monotone.f (interpR {τ' :: τ'' :: Γ} iS) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k)))
(Monotone.f (interpE (ren (ren (Θ i0) iS) iS))
((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Preorder-str.trans (snd [ x ]t)
(Monotone.f (interpE (subst (ren (Θ i0) iS) (lem3' ids v2))) k)
(Monotone.f (interpE (ren (Θ i0) iS)) (Monotone.f (interpS (lem3' ids v2)) k))
(Monotone.f (interpE (ren (Θ i0) iS))
(Monotone.f (interpR {τ' :: τ'' :: Γ} iS) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k)))
(subst-eq-l (lem3' ids v2) (ren (Θ i0) iS) k)
(Monotone.is-monotone (interpE (ren (Θ i0) iS))
(Monotone.f (interpS (lem3' ids v2)) k)
(Monotone.f (interpR {τ' :: τ'' :: Γ} iS) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
((Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpS {Γ} ids) k)
(Monotone.f (interpR {Γ} {Γ} (λ x₁ → x₁)) (Monotone.f (interpS {Γ} ids) k))
(Monotone.f (interpR {τ' :: τ'' :: Γ} (λ x₁ → iS (iS x₁))) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(interp-sr-r {Γ} ids (λ x₁ → x₁) k)
(Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpR {Γ} {Γ} (λ x₁ → x₁)) (Monotone.f (interpS {Γ} ids) k))
(Monotone.f (interpR {τ'' :: Γ} (throw-r (r-extend (λ x₁ → x₁)))) (Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k))
(Monotone.f (interpR {τ' :: τ'' :: Γ} (λ x₁ → iS (iS x₁))) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(ren-eq-r-lam {Γ} {Γ} {τ''} (λ x₁ → x₁) (Monotone.f (interpS {Γ} ids) k) (Monotone.f (interpE v2) k))
(ren-eq-r-lam {τ'' :: Γ} (λ x₁ → iS x₁) (Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) (Monotone.f (interpE v1) k)))) ,
(Preorder-str.refl (snd [ τ'' ]t) (Monotone.f (interpE v2) k)))))
(ren-eq-r iS (ren (Θ i0) iS) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k)))
interp-subst-comp3-l : ∀ {Γ Γ' τ1 τ2 τ3} (Θ : sctx Γ Γ') (k : fst [ Γ ]c) (v3 : Γ |- τ3) (v2 : Γ |- τ2) (v1 : Γ |- τ1)
→ Preorder-str.≤ (snd [ Γ' ]c)
(Monotone.f (interpS {τ1 :: τ2 :: τ3 :: Γ} {Γ'} (λ x → ren (ren (ren (Θ x) iS) iS) iS))
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpS {τ2 :: τ3 :: Γ} {Γ'} (λ x → ren (ren (Θ x) iS) iS))
((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k))
interp-subst-comp3-l {Γ' = []} Θ k v3 v2 v1 = <>
interp-subst-comp3-l {Γ} {Γ' = x :: Γ'} {τ1} {τ2} {τ3} Θ k v3 v2 v1 =
(interp-subst-comp3-l (throw-s Θ) k v3 v2 v1) ,
(Preorder-str.trans (snd [ x ]t)
(Monotone.f (interpE (ren (ren (ren (Θ i0) iS) iS) iS))
(((Monotone.f (interpS {Γ} {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpE (ren (ren (Θ i0) iS) iS))
(Monotone.f (interpR {τ1 :: τ2 :: τ3 :: Γ} {τ2 :: τ3 :: Γ} iS)
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k)))
(Monotone.f (interpE (ren (ren (Θ i0) iS) iS))
((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k))
(ren-eq-l iS (ren (ren (Θ i0) iS) iS) (((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.is-monotone (interpE (ren (ren (Θ i0) iS) iS))
(Monotone.f (interpR {τ1 :: τ2 :: τ3 :: Γ} {τ2 :: τ3 :: Γ} iS)
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k)
((Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpR {τ1 :: τ2 :: τ3 :: Γ} {Γ} (λ x₁ → iS (iS (iS x₁))))
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpR {Γ} (λ x₁ → x₁)) (Monotone.f (interpS {Γ} ids) k))
(Monotone.f (interpS {Γ} ids) k)
(Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpR {τ1 :: τ2 :: τ3 :: Γ} {Γ} (λ x₁ → iS (iS (iS x₁))))
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpR {τ3 :: Γ} {Γ} (throw-r (r-extend (λ x₁ → x₁)))) (Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k))
(Monotone.f (interpR {Γ} (λ x₁ → x₁)) (Monotone.f (interpS {Γ} ids) k))
(Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpR {τ1 :: τ2 :: τ3 :: Γ} {Γ} (λ x₁ → iS (iS (iS x₁))))
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpR {τ2 :: τ3 :: Γ} (throw-r (r-extend iS)))
((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k))
(Monotone.f (interpR {τ3 :: Γ} {Γ} (throw-r (r-extend (λ x₁ → x₁)))) (Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k))
(fst (ren-eq-l-lam {τ2 :: τ3 :: Γ} {τ3 :: Γ} {τ1} (λ x₁ → iS x₁)
((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) (Monotone.f (interpE v1) k)))
(ren-eq-l-lam {τ3 :: Γ} {Γ} {τ2} (λ x₁ → iS x₁) (Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) (Monotone.f (interpE v2) k)))
(ren-eq-l-lam {Γ} {Γ} {τ3} (λ x₁ → x₁) (Monotone.f (interpS {Γ} ids) k) (Monotone.f (interpE v3) k)))
(interp-sr-l {Γ} ids (λ x₁ → x₁) k) ,
Preorder-str.refl (snd [ τ3 ]t) (Monotone.f (interpE v3) k)) ,
Preorder-str.refl (snd [ τ2 ]t) (Monotone.f (interpE v2) k))))
interp-subst-comp3-r : ∀ {Γ Γ' τ1 τ2 τ3} (Θ : sctx Γ Γ') (k : fst [ Γ ]c) (v3 : Γ |- τ3) (v2 : Γ |- τ2) (v1 : Γ |- τ1)
→ Preorder-str.≤ (snd [ Γ' ]c)
(Monotone.f (interpS {τ2 :: τ3 :: Γ} {Γ'} (λ x → ren (ren (Θ x) iS) iS))
((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k))
(Monotone.f (interpS {τ1 :: τ2 :: τ3 :: Γ} {Γ'} (λ x → ren (ren (ren (Θ x) iS) iS) iS))
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
interp-subst-comp3-r {Γ' = []} Θ k v3 v2 v1 = <>
interp-subst-comp3-r {Γ} {Γ' = x :: Γ'} {τ1} {τ2} {τ3} Θ k v3 v2 v1 =
(interp-subst-comp3-r (throw-s Θ) k v3 v2 v1) ,
Preorder-str.trans (snd [ x ]t)
(Monotone.f (interpE (ren (ren (Θ i0) iS) iS))
((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k))
(Monotone.f (interpE (ren (ren (Θ i0) iS) iS))
(Monotone.f (interpR {τ1 :: τ2 :: τ3 :: Γ} {τ2 :: τ3 :: Γ} iS)
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k)))
(Monotone.f (interpE (ren (ren (ren (Θ i0) iS) iS) iS))
(((Monotone.f (interpS {Γ} {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.is-monotone (interpE (ren (ren (Θ i0) iS) iS))
((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k)
(Monotone.f (interpR {τ1 :: τ2 :: τ3 :: Γ} {τ2 :: τ3 :: Γ} iS)
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(((Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpS {Γ} ids) k)
(Monotone.f (interpR {Γ} (λ x₁ → x₁)) (Monotone.f (interpS {Γ} ids) k))
(Monotone.f (interpR {τ1 :: τ2 :: τ3 :: Γ} {Γ} (λ x₁ → iS (iS (iS x₁))))
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(interp-sr-r {Γ} ids (λ x₁ → x₁) k)
(Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpR {Γ} (λ x₁ → x₁)) (Monotone.f (interpS {Γ} ids) k))
(Monotone.f (interpR {τ3 :: Γ} {Γ} (throw-r (r-extend (λ x₁ → x₁)))) (Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k))
(Monotone.f (interpR {τ1 :: τ2 :: τ3 :: Γ} {Γ} (λ x₁ → iS (iS (iS x₁))))
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(ren-eq-r-lam {Γ} {Γ} {τ3} (λ x₁ → x₁) (Monotone.f (interpS {Γ} ids) k) (Monotone.f (interpE v3) k))
(Preorder-str.trans (snd [ Γ ]c)
(Monotone.f (interpR {τ3 :: Γ} {Γ} (throw-r (r-extend (λ x₁ → x₁)))) (Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k))
(Monotone.f (interpR {τ2 :: τ3 :: Γ} (throw-r (r-extend iS)))
((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k))
(Monotone.f (interpR {τ1 :: τ2 :: τ3 :: Γ} {Γ} (λ x₁ → iS (iS (iS x₁))))
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(ren-eq-r-lam {τ3 :: Γ} {Γ} {τ2} (λ x₁ → iS x₁) (Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) (Monotone.f (interpE v2) k))
(fst (ren-eq-r-lam {τ2 :: τ3 :: Γ} {τ3 :: Γ} {τ1} (λ x₁ → iS x₁)
((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) (Monotone.f (interpE v1) k)))))) ,
(Preorder-str.refl (snd [ τ3 ]t) (Monotone.f (interpE v3) k))) ,
(Preorder-str.refl (snd [ τ2 ]t) (Monotone.f (interpE v2) k))))
(ren-eq-r iS (ren (ren (Θ i0) iS) iS) (((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
s-cong2-lem : ∀ {Γ Γ'} (Θ Θ' : sctx Γ Γ') (k : fst [ Γ ]c)
(x : (τ₁ : CTp) (x₁ : τ₁ ∈ Γ') → Preorder-str.≤ (snd [ τ₁ ]t) (Monotone.f (interpE (Θ x₁)) k) (Monotone.f (interpE (Θ' x₁)) k))
→ Preorder-str.≤ (snd [ Γ' ]c) (Monotone.f (interpS Θ) k) (Monotone.f (interpS Θ') k)
s-cong2-lem {Γ' = []} Θ Θ' x k = <>
s-cong2-lem {Γ' = x :: Γ'} Θ Θ' x₁ k = (s-cong2-lem (throw-s Θ) (throw-s Θ') x₁ (λ τ₁ x₂ → k τ₁ (iS x₂))) , k x i0
sound {_} {τ} e .e refl-s k = Preorder-str.refl (snd [ τ ]t) (Monotone.f (interpE e) k)
sound {Γ} {τ} e e' (trans-s {.Γ} {.τ} {.e} {e''} {.e'} d d₁) k =
Preorder-str.trans (snd [ τ ]t) (Monotone.f (interpE e) k) (Monotone.f (interpE e'') k) (Monotone.f (interpE e') k) (sound e e'' d k) (sound e'' e' d₁ k)
sound {_} {τ} e .e (cong-refl Refl) k = Preorder-str.refl (snd [ τ ]t) (Monotone.f (interpE e) k)
sound {_} {._} ._ ._ (lt {._}) k = <>
sound {_} {τ} (letc e e') .(app (lam e) e') letc-app-l k = Preorder-str.refl (snd [ τ ]t) (Monotone.f (interpE e) (k , Monotone.f (interpE e') k))
sound {_} {τ} (app (lam e) e') .(letc e e') letc-app-r k = Preorder-str.refl (snd [ τ ]t) (Monotone.f (interpE e) (k , Monotone.f (interpE e') k))
sound .(plusC 0C e') e' +-unit-l k = Preorder-str.refl (snd [ C ]t) (Monotone.f (interpE e') k)
sound e .(plusC 0C e) +-unit-l' k = Preorder-str.refl (snd [ C ]t) (Monotone.f (interpE e) k)
sound {_} {.C} .(plusC e' 0C) e' +-unit-r k = +-unit (Monotone.f (interpE e') k)
sound e .(plusC e 0C) +-unit-r' k = plus-lem' (Monotone.f (interpE e) k) (Monotone.f (interpE e) k) Z (nat-refl (Monotone.f (interpE e) k))
sound {Γ} {.C} ._ ._ (+-assoc {.Γ} {e1} {e2} {e3}) k = plus-assoc (Monotone.f (interpE e1) k) (Monotone.f (interpE e2) k) (Monotone.f (interpE e3) k)
sound {Γ} {.C} ._ ._ (+-assoc' {.Γ} {e1} {e2} {e3}) k = plus-assoc' (Monotone.f (interpE e1) k) (Monotone.f (interpE e2) k) (Monotone.f (interpE e3) k)
sound {Γ} {.C} ._ ._ (refl-+ {.Γ} {e0} {e1}) k = +-comm (Monotone.f (interpE e0) k) (Monotone.f (interpE e1) k)
sound {Γ} {C} ._ ._ (cong-+ {.Γ} {e0} {e1} {e0'} {e1'} d d₁) k = --also called plus-s. should really delete this rule so we don't have duplicates
plus-lem (Monotone.f (interpE e0) k) (Monotone.f (interpE e1) k) (Monotone.f (interpE e0') k) (Monotone.f (interpE e1') k)
(sound e0 e0' d k) (sound e1 e1' d₁ k)
sound ._ ._ (cong-suc d) k = sound _ _ d k
sound ._ ._ (cong-prod d d₁) k = (sound _ _ d k) , (sound _ _ d₁ k)
sound {Γ} {τ} ._ ._ (cong-lproj {.Γ} {.τ} {_} {e} {e'} d) k = fst (sound e e' d k)
sound {Γ} {τ} ._ ._ (cong-rproj {.Γ} {_} {.τ} {e} {e'} d) k = snd (sound e e' d k)
sound ._ ._ (cong-lam d) k x = sound _ _ d (k , x)
sound {Γ} {τ} ._ ._ (cong-app {.Γ} {τ'} {.τ} {e} {e'} {e1} d) k = sound e e' d k (Monotone.f (interpE e1) k)
sound {Γ} {τ} ._ ._ (ren-cong {.Γ} {Γ'} {.τ} {e1} {e2} {ρ} d) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren e1 ρ)) k)
(Monotone.f (interpE e1) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE (ren e2 ρ)) k)
(ren-eq-l ρ e1 k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e1) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE e2) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE (ren e2 ρ)) k)
(sound e1 e2 d (Monotone.f (interpR ρ) k))
(ren-eq-r ρ e2 k))
sound {Γ} {τ} ._ ._ (subst-cong {.Γ} {Γ'} {.τ} {e1} {e2} {Θ} d) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 Θ)) k)
(Monotone.f (interpE e1) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (subst e2 Θ)) k)
(subst-eq-l Θ e1 k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e1) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE e2) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (subst e2 Θ)) k)
(sound e1 e2 d (Monotone.f (interpS Θ) k))
(subst-eq-r Θ e2 k))
sound {Γ} {τ} ._ ._ (subst-cong2 {.Γ} {Γ'} {.τ} {Θ} {Θ'} {e} x) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e Θ)) k)
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (subst e Θ')) k) (subst-eq-l Θ e k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE e) (Monotone.f (interpS Θ') k))
(Monotone.f (interpE (subst e Θ')) k)
(Monotone.is-monotone (interpE e) (Monotone.f (interpS Θ) k)
(Monotone.f (interpS Θ') k) (s-cong2-lem Θ Θ' k (λ τ1 x1 → sound _ _ (x _ x1) k)))
(subst-eq-r Θ' e k))
sound {Γ} {τ} ._ ._ (cong-rec {.Γ} {.τ} {e} {e'} {e0} {e1} d) k =
♭h-fix-args (interpE e0) (interpE e1) (k , Monotone.f (interpE e) k) (k , Monotone.f (interpE e') k) (sound e e' d k)
sound {Γ} {τ} ._ ._ (cong-listrec {.Γ} {τ'} {.τ} {e} {e'} {e0} {e1} d) k =
listrec-fix-args (interpE e0) (interpE e1) (k , (Monotone.f (interpE e) k)) (k , Monotone.f (interpE e') k) ((Preorder-str.refl (snd [ Γ ]c) k) , (sound e e' d k))
sound {Γ} {τ} ._ ._ (lam-s {.Γ} {τ'} {.τ} {e} {e2}) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (q e2))) k)
(Monotone.f (interpE e) (Monotone.f (interpS (q e2)) k))
(Monotone.f (interpE e) (k , Monotone.f (interpE e2) k))
(subst-eq-l (q e2) e k)
(Monotone.is-monotone (interpE e)
(Monotone.f (interpS (q e2)) k)
(k , Monotone.f (interpE e2) k)
(lam-s-lem {Γ} k , (Preorder-str.refl (snd [ τ' ]t) (Monotone.f (interpE e2) k))))
sound {Γ} {τ} e ._ (l-proj-s {.Γ}) k = Preorder-str.refl (snd [ τ ]t) (Monotone.f (interpE e) k)
sound {Γ} {τ} e ._ (r-proj-s {.Γ}) k = Preorder-str.refl (snd [ τ ]t) (Monotone.f (interpE e) k)
sound {_} {τ} e ._ rec-steps-z k = Preorder-str.refl (snd [ τ ]t) (Monotone.f (interpE e) k)
sound {Γ} {τ} ._ ._ (rec-steps-s {.Γ} {.τ} {e} {e0} {e1}) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (lem3' (lem3' ids (rec e e0 e1)) e))) k)
(Monotone.f (interpE e1) (Monotone.f (interpS (lem3' (lem3' ids (rec e e0 e1)) e)) k))
(Monotone.f (interpE e1)
((k , natrec (Monotone.f (interpE e0) k) (λ n x₂ → Monotone.f (interpE e1) ((k , x₂) , n)) (Monotone.f (interpE e) k)) , Monotone.f (interpE e) k))
(subst-eq-l (lem3' (lem3' ids (rec e e0 e1)) e) e1 k)
(Monotone.is-monotone (interpE e1)
(Monotone.f (interpS (lem3' (lem3' ids (rec e e0 e1)) e)) k)
((k , natrec (Monotone.f (interpE e0) k)
(λ n x₂ → Monotone.f (interpE e1) ((k , x₂) , n)) (Monotone.f (interpE e) k)) , Monotone.f (interpE e) k)
((lam-s-lem {Γ} k ,
(Preorder-str.refl (snd [ τ ]t) (natrec (Monotone.f (interpE e0) k) (λ n x₂ → Monotone.f (interpE e1) ((k , x₂) , n)) (Monotone.f (interpE e) k)))) ,
(♭nat-refl (Monotone.f (interpE e) k))))
sound {Γ} {τ} e ._ listrec-steps-nil k = Preorder-str.refl (snd [ τ ]t) (Monotone.f (interpE e) k)
sound {Γ} {τ} ._ ._ (listrec-steps-cons {.Γ} {τ'} {.τ} {h} {t} {e0} {e1}) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (lem3' (lem3' (lem3' ids (listrec t e0 e1)) t) h))) k)
(Monotone.f (interpE e1) (Monotone.f (interpS (lem3' (lem3' (lem3' ids (listrec t e0 e1)) t) h)) k))
(Monotone.f (interpE e1)
(((k ,
lrec (Monotone.f (interpE t) k) (Monotone.f (interpE e0) k) (λ x₁ x₂ x₃ → Monotone.f (interpE e1) (((k , x₃) , x₂) , x₁))) ,
Monotone.f (interpE t) k) ,
Monotone.f (interpE h) k))
(subst-eq-l (lem3' (lem3' (lem3' ids (listrec t e0 e1)) t) h) e1 k)
(Monotone.is-monotone (interpE e1)
(Monotone.f (interpS (lem3' (lem3' (lem3' ids (listrec t e0 e1)) t) h)) k)
(((k , lrec (Monotone.f (interpE t) k) (Monotone.f (interpE e0) k) (λ x₁ x₂ x₃ → Monotone.f (interpE e1) (((k , x₃) , x₂) , x₁))) ,
Monotone.f (interpE t) k) , Monotone.f (interpE h) k)
(((lam-s-lem {Γ} k ,
(Preorder-str.refl (snd [ τ ]t) (lrec (Monotone.f (interpE t) k) (Monotone.f (interpE e0) k) (λ x₁ x₂ x₃ → Monotone.f (interpE e1) (((k , x₃) , x₂) , x₁))))) ,
(l-refl (snd [ τ' ]t) (Monotone.f (interpE t) k))) ,
(Preorder-str.refl (snd [ τ' ]t) (Monotone.f (interpE h) k))))
sound {Γ} {τ} .(ren (ren e ρ2) ρ1) ._ (ren-comp-l ρ1 ρ2 e) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren (ren e ρ2) ρ1)) k)
(Monotone.f (interpE (ren e ρ2)) (Monotone.f (interpR ρ1) k))
(Monotone.f (interpE (ren e (ρ1 ∙rr ρ2))) k)
(ren-eq-l ρ1 (ren e ρ2) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren e ρ2)) (Monotone.f (interpR ρ1) k))
(Monotone.f (interpE e) (Monotone.f (interpR (ρ1 ∙rr ρ2)) k))
(Monotone.f (interpE (ren e (ρ1 ∙rr ρ2))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren e ρ2)) (Monotone.f (interpR ρ1) k))
(Monotone.f (interpE e) (Monotone.f (interpR ρ2) (Monotone.f (interpR ρ1) k)))
(Monotone.f (interpE e) (Monotone.f (interpR (ρ1 ∙rr ρ2)) k))
(ren-eq-l ρ2 e (Monotone.f (interpR ρ1) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpR ρ2) (Monotone.f (interpR ρ1) k))
(Monotone.f (interpR (ρ1 ∙rr ρ2)) k)
(interp-rr-l ρ1 ρ2 k)))
(ren-eq-r (ρ1 ∙rr ρ2) e k))
sound {Γ} {τ} ._ .(ren (ren e ρ2) ρ1) (ren-comp-r ρ1 ρ2 e) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren e (ρ1 ∙rr ρ2))) k)
(Monotone.f (interpE (ren e ρ2)) (Monotone.f (interpR ρ1) k))
(Monotone.f (interpE (ren (ren e ρ2) ρ1)) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren e (ρ1 ∙rr ρ2))) k)
(Monotone.f (interpE e) (Monotone.f (interpR (ρ1 ∙rr ρ2)) k))
(Monotone.f (interpE (ren e ρ2)) (Monotone.f (interpR ρ1) k))
(ren-eq-l (ρ1 ∙rr ρ2) e k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e) (Monotone.f (interpR (ρ1 ∙rr ρ2)) k))
(Monotone.f (interpE e) (Monotone.f (interpR ρ2) (Monotone.f (interpR ρ1) k)))
(Monotone.f (interpE (ren e ρ2)) (Monotone.f (interpR ρ1) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpR (ρ1 ∙rr ρ2)) k)
(Monotone.f (interpR ρ2) (Monotone.f (interpR ρ1) k))
(interp-rr-r ρ1 ρ2 k))
(ren-eq-r ρ2 e (Monotone.f (interpR ρ1) k))))
(ren-eq-r ρ1 (ren e ρ2) k)
sound {Γ} {τ} e ._ (subst-id-l .e) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e) k)
(Monotone.f (interpE e) (Monotone.f (interpS {Γ} {Γ} ids) k))
(Monotone.f (interpE (subst e ids)) k)
(Monotone.is-monotone (interpE e) k (Monotone.f (interpS {Γ} {Γ} ids) k) (lam-s-lem-r {Γ} k))
(subst-eq-r ids e k)
sound {Γ} {τ} ._ e' (subst-id-r .e') k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e' ids)) k)
(Monotone.f (interpE e') (Monotone.f (interpS {Γ} {Γ} ids) k))
(Monotone.f (interpE e') k)
(subst-eq-l ids e' k)
(Monotone.is-monotone (interpE e') (Monotone.f (interpS {Γ} {Γ} ids) k) k (lam-s-lem {Γ} k))
sound {Γ} {τ} .(ren (subst e Θ) ρ) ._ (subst-rs-l ρ Θ e) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren (subst e Θ) ρ)) k)
(Monotone.f (interpE (subst e Θ)) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE (subst e (ρ rs Θ))) k)
(ren-eq-l ρ (subst e Θ) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e Θ)) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE e) (Monotone.f (interpS Θ) (Monotone.f (interpR ρ) k)))
(Monotone.f (interpE (subst e (ρ rs Θ))) k)
(subst-eq-l Θ e (Monotone.f (interpR ρ) k))
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e) (Monotone.f (interpS Θ) (Monotone.f (interpR ρ) k)))
(Monotone.f (interpE e) (Monotone.f (interpS (ρ rs Θ)) k))
(Monotone.f (interpE (subst e (ρ rs Θ))) k)
(Monotone.is-monotone (interpE e)
(Monotone.f (interpS Θ) (Monotone.f (interpR ρ) k))
(Monotone.f (interpS (ρ rs Θ)) k)
(interp-rs-l ρ Θ k))
(subst-eq-r (ρ rs Θ) e k)))
sound {Γ} {τ} ._ .(ren (subst e Θ) ρ) (subst-rs-r ρ Θ e) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (ρ rs Θ))) k)
(Monotone.f (interpE (subst e Θ)) (Monotone.f (interpR ρ) k))
(Monotone.f (interpE (ren (subst e Θ) ρ)) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (ρ rs Θ))) k)
(Monotone.f (interpE e) (Monotone.f (interpS Θ) (Monotone.f (interpR ρ) k)))
(Monotone.f (interpE (subst e Θ)) (Monotone.f (interpR ρ) k))
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (ρ rs Θ))) k)
(Monotone.f (interpE e) (Monotone.f (interpS (ρ rs Θ)) k))
(Monotone.f (interpE e) (Monotone.f (interpS Θ) (Monotone.f (interpR ρ) k)))
(subst-eq-l (ρ rs Θ) e k)
(Monotone.is-monotone (interpE e)
(Monotone.f (interpS (ρ rs Θ)) k)
(Monotone.f (interpS Θ) (Monotone.f (interpR ρ) k))
(interp-rs-r ρ Θ k)))
(subst-eq-r Θ e (Monotone.f (interpR ρ) k)))
(ren-eq-r ρ (subst e Θ) k)
sound {Γ} {τ} .(subst (ren e ρ) Θ) ._ (subst-sr-l Θ ρ e) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst (ren e ρ) Θ)) k)
(Monotone.f (interpE (ren e ρ)) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (subst e (Θ sr ρ))) k)
(subst-eq-l Θ (ren e ρ) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren e ρ)) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE e) (Monotone.f (interpS (Θ sr ρ)) k))
(Monotone.f (interpE (subst e (Θ sr ρ))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (ren e ρ)) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE e) (Monotone.f (interpR ρ) (Monotone.f (interpS Θ) k)))
(Monotone.f (interpE e) (Monotone.f (interpS (Θ sr ρ)) k))
(ren-eq-l ρ e (Monotone.f (interpS Θ) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpR ρ) (Monotone.f (interpS Θ) k))
(Monotone.f (interpS (Θ sr ρ)) k)
(interp-sr-l Θ ρ k)))
(subst-eq-r (Θ sr ρ) e k))
sound {Γ} {τ} ._ .(subst (ren e ρ) Θ) (subst-sr-r Θ ρ e) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (Θ sr ρ))) k)
(Monotone.f (interpE (ren e ρ)) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (subst (ren e ρ) Θ)) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (Θ sr ρ))) k)
(Monotone.f (interpE e) (Monotone.f (interpS (Θ sr ρ)) k))
(Monotone.f (interpE (ren e ρ)) (Monotone.f (interpS Θ) k))
(subst-eq-l (Θ sr ρ) e k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e) (Monotone.f (interpS (Θ sr ρ)) k))
(Monotone.f (interpE e) (Monotone.f (interpR ρ) (Monotone.f (interpS Θ) k)))
(Monotone.f (interpE (ren e ρ)) (Monotone.f (interpS Θ) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpS (Θ sr ρ)) k)
(Monotone.f (interpR ρ) (Monotone.f (interpS Θ) k))
(interp-sr-r Θ ρ k))
(ren-eq-r ρ e (Monotone.f (interpS Θ) k))))
(subst-eq-r Θ (ren e ρ) k)
sound {Γ} {τ} ._ .(subst (subst e Θ2) Θ1) (subst-ss-l Θ1 Θ2 e) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (Θ1 ss Θ2))) k)
(Monotone.f (interpE e) (Monotone.f (interpS (Θ1 ss Θ2)) k))
(Monotone.f (interpE (subst (subst e Θ2) Θ1)) k)
(subst-eq-l (Θ1 ss Θ2) e k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e) (Monotone.f (interpS (Θ1 ss Θ2)) k))
(Monotone.f (interpE (subst e Θ2)) (Monotone.f (interpS Θ1) k))
(Monotone.f (interpE (subst (subst e Θ2) Θ1)) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e) (Monotone.f (interpS (Θ1 ss Θ2)) k))
(Monotone.f (interpE e) (Monotone.f (interpS Θ2) (Monotone.f (interpS Θ1) k)))
(Monotone.f (interpE (subst e Θ2)) (Monotone.f (interpS Θ1) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpS (Θ1 ss Θ2)) k)
(Monotone.f (interpS Θ2) (Monotone.f (interpS Θ1) k))
(interp-ss-l Θ1 Θ2 k))
(subst-eq-r Θ2 e (Monotone.f (interpS Θ1) k)))
(subst-eq-r Θ1 (subst e Θ2) k))
sound {Γ} {τ} .(subst (subst e Θ2) Θ1) ._ (subst-ss-r Θ1 Θ2 e) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst (subst e Θ2) Θ1)) k)
(Monotone.f (interpE e) (Monotone.f (interpS (Θ1 ss Θ2)) k))
(Monotone.f (interpE (subst e (Θ1 ss Θ2))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst (subst e Θ2) Θ1)) k)
(Monotone.f (interpE (subst e Θ2)) (Monotone.f (interpS Θ1) k))
(Monotone.f (interpE e) (Monotone.f (interpS (Θ1 ss Θ2)) k))
(subst-eq-l Θ1 (subst e Θ2) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e Θ2)) (Monotone.f (interpS Θ1) k))
(Monotone.f (interpE e) (Monotone.f (interpS Θ2) (Monotone.f (interpS Θ1) k)))
(Monotone.f (interpE e) (Monotone.f (interpS (Θ1 ss Θ2)) k))
(subst-eq-l Θ2 e (Monotone.f (interpS Θ1) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpS Θ2) (Monotone.f (interpS Θ1) k))
(Monotone.f (interpS (Θ1 ss Θ2)) k)
(interp-ss-r Θ1 Θ2 k))))
(subst-eq-r (Θ1 ss Θ2) e k)
sound {Γ} {τ} ._ .(subst e (lem3' Θ v)) (subst-compose-l {.Γ} {Γ'} {τ'} Θ v e) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst (subst e (s-extend Θ)) (q v))) k)
(Monotone.f (interpE (subst e (s-extend Θ))) (Monotone.f (interpS (q v)) k))
(Monotone.f (interpE (subst e (lem3' Θ v))) k)
(subst-eq-l (q v) (subst e (s-extend Θ)) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (s-extend Θ))) (Monotone.f (interpS (q v)) k))
(Monotone.f (interpE e) (Monotone.f (interpS (lem3' Θ v)) k))
(Monotone.f (interpE (subst e (lem3' Θ v))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (s-extend Θ))) (Monotone.f (interpS (q v)) k))
(Monotone.f (interpE e) (Monotone.f (interpS {τ' :: Γ} {τ' :: Γ'} (s-extend {Γ} {Γ'} Θ)) (Monotone.f (interpS {Γ} {τ' :: Γ} (q v)) k)))
(Monotone.f (interpE e) (Monotone.f (interpS (lem3' Θ v)) k))
(subst-eq-l (s-extend Θ) e (Monotone.f (interpS (q v)) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpS {τ' :: Γ} {τ' :: Γ'} (s-extend {Γ} {Γ'} Θ)) (Monotone.f (interpS {Γ} {τ' :: Γ} (q v)) k))
(Monotone.f (interpS (lem3' Θ v)) k)
(Preorder-str.trans (snd [ Γ' ]c)
(fst (Monotone.f (interpS (s-extend Θ)) (Monotone.f (interpS (q v)) k)))
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v))) k)
(Monotone.f (interpS Θ) k)
(fst (interp-ss-r (q v) (s-extend Θ) k))
(interp-subst-comp-l Θ v k) ,
Preorder-str.refl (snd [ τ' ]t) (Monotone.f (interpE v) k))))
(subst-eq-r (lem3' Θ v) e k))
sound {Γ} {τ} .(subst e (lem3' Θ v)) ._ (subst-compose-r {.Γ} {Γ'} {τ'} Θ v e) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (lem3' Θ v))) k)
(Monotone.f (interpE (subst e (s-extend Θ))) (Monotone.f (interpS (q v)) k))
(Monotone.f (interpE (subst (subst e (s-extend Θ)) (q v))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (lem3' Θ v))) k)
(Monotone.f (interpE e) (Monotone.f (interpS (lem3' Θ v)) k))
(Monotone.f (interpE (subst e (s-extend Θ))) (Monotone.f (interpS (q v)) k))
(subst-eq-l (lem3' Θ v) e k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e) (Monotone.f (interpS (lem3' Θ v)) k))
(Monotone.f (interpE e) (Monotone.f (interpS {τ' :: Γ} {τ' :: Γ'} (s-extend {Γ} {Γ'} Θ)) (Monotone.f (interpS {Γ} {τ' :: Γ} (q v)) k)))
(Monotone.f (interpE (subst e (s-extend Θ))) (Monotone.f (interpS (q v)) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpS (lem3' Θ v)) k)
(Monotone.f (interpS {τ' :: Γ} {τ' :: Γ'} (s-extend {Γ} {Γ'} Θ)) (Monotone.f (interpS {Γ} {τ' :: Γ} (q v)) k))
((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS Θ) k)
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v))) k)
(fst (Monotone.f (interpS (s-extend Θ)) (Monotone.f (interpS (q v)) k)))
(interp-subst-comp-r Θ v k)
(fst (interp-ss-l (q v) (s-extend Θ) k))) ,
(Preorder-str.refl (snd [ τ' ]t) (Monotone.f (interpE v) k))))
(subst-eq-r (s-extend Θ) e (Monotone.f (interpS (q v)) k))))
(subst-eq-r (q v) (subst e (s-extend Θ)) k)
sound {Γ} {τ} ._ .(subst e1 (lem3' (lem3' Θ v2) v1)) (subst-compose2-l {.Γ} {Γ'} {.τ} {τ'} {τ''} Θ e1 v1 v2) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst (subst e1 (s-extend (s-extend Θ))) (lem4 v1 v2))) k)
(Monotone.f (interpE (subst e1 (s-extend (s-extend Θ)))) (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.f (interpE (subst e1 (lem4' Θ v1 v2))) k)
(subst-eq-l (lem4 v1 v2) (subst e1 (s-extend (s-extend Θ))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (s-extend (s-extend Θ)))) (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.f (interpE e1) (Monotone.f (interpS (lem4' Θ v1 v2)) k))
(Monotone.f (interpE (subst e1 (lem4' Θ v1 v2))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (s-extend (s-extend Θ)))) (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.f (interpE e1) (Monotone.f (interpS {τ' :: τ'' :: Γ} {τ' :: τ'' :: Γ'} (s-extend (s-extend Θ))) (Monotone.f (interpS (lem4 v1 v2)) k)))
(Monotone.f (interpE e1) (Monotone.f (interpS (lem4' Θ v1 v2)) k))
(subst-eq-l (s-extend (s-extend Θ)) e1 (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.is-monotone (interpE e1)
(Monotone.f (interpS {τ' :: τ'' :: Γ} {τ' :: τ'' :: Γ'} (s-extend (s-extend Θ))) (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.f (interpS (lem4' Θ v1 v2)) k)
((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS (λ x → ren (ren (Θ x) iS) iS)) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v2))) k)
(Monotone.f (interpS Θ) k)
(interp-subst-comp2-l Θ k v1 v2)
(interp-subst-comp-l Θ v2 k) ,
Preorder-str.refl (snd [ τ'' ]t) (Monotone.f (interpE v2) k)) ,
Preorder-str.refl (snd [ τ' ]t) (Monotone.f (interpE v1) k))))
(subst-eq-r (lem4' Θ v1 v2) e1 k))
sound {Γ} {τ} .(subst e1 (lem3' (lem3' Θ v2) v1)) ._ (subst-compose2-r {.Γ} {Γ'} {.τ} {τ'} {τ''} Θ e1 v1 v2) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (lem4' Θ v1 v2))) k)
(Monotone.f (interpE (subst e1 (s-extend (s-extend Θ)))) (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.f (interpE (subst (subst e1 (s-extend (s-extend Θ))) (lem4 v1 v2))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (lem4' Θ v1 v2))) k)
(Monotone.f (interpE e1) (Monotone.f (interpS (lem4' Θ v1 v2)) k))
(Monotone.f (interpE (subst e1 (s-extend (s-extend Θ)))) (Monotone.f (interpS (lem4 v1 v2)) k))
(subst-eq-l (lem4' Θ v1 v2) e1 k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e1) (Monotone.f (interpS (lem4' Θ v1 v2)) k))
(Monotone.f (interpE e1) (Monotone.f (interpS {τ' :: τ'' :: Γ} {τ' :: τ'' :: Γ'} (s-extend (s-extend Θ))) (Monotone.f (interpS (lem4 v1 v2)) k)))
(Monotone.f (interpE (subst e1 (s-extend (s-extend Θ)))) (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.is-monotone (interpE e1)
(Monotone.f (interpS (lem4' Θ v1 v2)) k)
(Monotone.f (interpS {τ' :: τ'' :: Γ} {τ' :: τ'' :: Γ'} (s-extend (s-extend Θ))) (Monotone.f (interpS (lem4 v1 v2)) k))
(((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS Θ) k)
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v2))) k)
(Monotone.f (interpS (λ x → ren (ren (Θ x) iS) iS)) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(interp-subst-comp-r Θ v2 k)
(interp-subst-comp2-r Θ k v1 v2)) ,
(Preorder-str.refl (snd [ τ'' ]t) (Monotone.f (interpE v2) k))) ,
(Preorder-str.refl (snd [ τ' ]t) (Monotone.f (interpE v1) k))))
(subst-eq-r (s-extend (s-extend Θ)) e1 (Monotone.f (interpS (lem4 v1 v2)) k))))
(subst-eq-r (lem4 v1 v2) (subst e1 (s-extend (s-extend Θ))) k)
sound {Γ} {τ} ._ .(subst e1 (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ))) (subst-compose3-l {.Γ} {Γ'} {.τ} {τ'} {τ''} Θ e1 v1 v2) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst (subst e1 (lem3' (lem3' ids v2) v1)) Θ)) k)
(Monotone.f (interpE (subst e1 (lem3' (lem3' ids v2) v1))) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (subst e1 (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ)))) k)
(subst-eq-l Θ (subst e1 (lem3' (lem3' ids v2) v1)) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (lem3' (lem3' ids v2) v1))) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE e1) (Monotone.f (interpS (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ))) k))
(Monotone.f (interpE (subst e1 (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ)))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (lem3' (lem3' ids v2) v1))) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE e1) (Monotone.f (interpS (lem3' (lem3' ids v2) v1)) (Monotone.f (interpS Θ) k)))
(Monotone.f (interpE e1) (Monotone.f (interpS (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ))) k))
(subst-eq-l (lem3' (lem3' ids v2) v1) e1 (Monotone.f (interpS Θ) k))
(Monotone.is-monotone (interpE e1)
(Monotone.f (interpS (lem3' (lem3' ids v2) v1)) (Monotone.f (interpS Θ) k))
(Monotone.f (interpS (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ))) k)
(((interp-ss-r Θ ids k) ,
(subst-eq-r Θ v2 k)) ,
(subst-eq-r Θ v1 k))))
(subst-eq-r (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ)) e1 k))
sound {Γ} {τ} .(subst e1 (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ))) ._ (subst-compose3-r {.Γ} {Γ'} {.τ} {τ'} {τ''} Θ e1 v1 v2) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ)))) k)
(Monotone.f (interpE (subst e1 (lem3' (lem3' ids v2) v1))) (Monotone.f (interpS Θ) k))
(Monotone.f (interpE (subst (subst e1 (lem3' (lem3' ids v2) v1)) Θ)) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ)))) k)
(Monotone.f (interpE e1) (Monotone.f (interpS (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ))) k))
(Monotone.f (interpE (subst e1 (lem3' (lem3' ids v2) v1))) (Monotone.f (interpS Θ) k))
(subst-eq-l (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ)) e1 k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e1) (Monotone.f (interpS (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ))) k))
(Monotone.f (interpE e1) (Monotone.f (interpS (lem3' (lem3' ids v2) v1)) (Monotone.f (interpS Θ) k)))
(Monotone.f (interpE (subst e1 (lem3' (lem3' ids v2) v1))) (Monotone.f (interpS Θ) k))
(Monotone.is-monotone (interpE e1)
(Monotone.f (interpS (lem3' (lem3' Θ (subst v2 Θ)) (subst v1 Θ))) k)
(Monotone.f (interpS (lem3' (lem3' ids v2) v1)) (Monotone.f (interpS Θ) k))
(((interp-ss-l Θ ids k) ,
(subst-eq-l Θ v2 k)) ,
(subst-eq-l Θ v1 k)))
(subst-eq-r (lem3' (lem3' ids v2) v1) e1 (Monotone.f (interpS Θ) k))))
(subst-eq-r Θ (subst e1 (lem3' (lem3' ids v2) v1)) k)
sound {Γ} {τ} ._ .(subst e1 (lem3' (lem3' Θ v2) v1)) (subst-compose4-l {.Γ} {Γ'} Θ v1 v2 e1) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst (subst e1 (s-extend (s-extend Θ))) (lem4 v1 v2))) k)
(Monotone.f (interpE (subst e1 (s-extend (s-extend Θ)))) (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.f (interpE (subst e1 (lem4' Θ v1 v2))) k)
(subst-eq-l (lem4 v1 v2) (subst e1 (s-extend (s-extend Θ))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (s-extend (s-extend Θ)))) (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.f (interpE e1) (Monotone.f (interpS (lem4' Θ v1 v2)) k))
(Monotone.f (interpE (subst e1 (lem4' Θ v1 v2))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (s-extend (s-extend Θ)))) (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.f (interpE e1) (Monotone.f (interpS {nat :: τ :: Γ} {nat :: τ :: Γ'} (s-extend (s-extend Θ))) (Monotone.f (interpS (lem4 v1 v2)) k)))
(Monotone.f (interpE e1) (Monotone.f (interpS (lem4' Θ v1 v2)) k))
(subst-eq-l (s-extend (s-extend Θ)) e1 (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.is-monotone (interpE e1)
(Monotone.f (interpS {nat :: τ :: Γ} {nat :: τ :: Γ'} (s-extend (s-extend Θ))) (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.f (interpS (lem4' Θ v1 v2)) k)
((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS (λ x → ren (ren (Θ x) iS) iS)) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v2))) k)
(Monotone.f (interpS Θ) k)
(interp-subst-comp2-l Θ k v1 v2)
(interp-subst-comp-l Θ v2 k) ,
Preorder-str.refl (snd [ τ ]t) (Monotone.f (interpE v2) k)) ,
Preorder-str.refl (snd [ nat ]t) (Monotone.f (interpE v1) k))))
(subst-eq-r (lem4' Θ v1 v2) e1 k))
sound {Γ} {τ} .(subst e1 (lem3' (lem3' Θ v2) v1)) ._ (subst-compose4-r {.Γ} {Γ'} Θ v1 v2 e1) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (lem4' Θ v1 v2))) k)
(Monotone.f (interpE (subst e1 (s-extend (s-extend Θ)))) (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.f (interpE (subst (subst e1 (s-extend (s-extend Θ))) (lem4 v1 v2))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e1 (lem4' Θ v1 v2))) k)
(Monotone.f (interpE e1) (Monotone.f (interpS (lem4' Θ v1 v2)) k))
(Monotone.f (interpE (subst e1 (s-extend (s-extend Θ)))) (Monotone.f (interpS (lem4 v1 v2)) k))
(subst-eq-l (lem4' Θ v1 v2) e1 k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e1) (Monotone.f (interpS (lem4' Θ v1 v2)) k))
(Monotone.f (interpE e1) (Monotone.f (interpS {nat :: τ :: Γ} {nat :: τ :: Γ'} (s-extend (s-extend Θ))) (Monotone.f (interpS (lem4 v1 v2)) k)))
(Monotone.f (interpE (subst e1 (s-extend (s-extend Θ)))) (Monotone.f (interpS (lem4 v1 v2)) k))
(Monotone.is-monotone (interpE e1)
(Monotone.f (interpS (lem4' Θ v1 v2)) k)
(Monotone.f (interpS {nat :: τ :: Γ} {nat :: τ :: Γ'} (s-extend (s-extend Θ))) (Monotone.f (interpS (lem4 v1 v2)) k))
(((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS Θ) k)
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v2))) k)
(Monotone.f (interpS (λ x → ren (ren (Θ x) iS) iS)) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(interp-subst-comp-r Θ v2 k)
(interp-subst-comp2-r Θ k v1 v2)) ,
(Preorder-str.refl (snd [ τ ]t) (Monotone.f (interpE v2) k))) ,
(Preorder-str.refl (snd [ nat ]t) (Monotone.f (interpE v1) k))))
(subst-eq-r (s-extend (s-extend Θ)) e1 (Monotone.f (interpS (lem4 v1 v2)) k))))
(subst-eq-r (lem4 v1 v2) (subst e1 (s-extend (s-extend Θ))) k)
sound {Γ} {τ} ._ .(subst e (lem3' (lem3' (lem3' Θ v3) v2) v1)) (subst-compose5-l {.Γ} {Γ'} {.τ} {τ1} {τ2} {τ3} Θ e v1 v2 v3) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst (subst e (s-extend (s-extend (s-extend Θ)))) (lem3' (lem3' (lem3' ids v3) v2) v1))) k)
(Monotone.f (interpE (subst e (s-extend (s-extend (s-extend Θ))))) (Monotone.f (interpS (lem3' (lem3' (lem3' ids v3) v2) v1)) k))
(Monotone.f (interpE (subst e (lem3' (lem3' (lem3' Θ v3) v2) v1))) k)
(subst-eq-l (lem3' (lem3' (lem3' ids v3) v2) v1) (subst e (s-extend (s-extend (s-extend Θ)))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (s-extend (s-extend (s-extend Θ))))) (Monotone.f (interpS (lem3' (lem3' (lem3' ids v3) v2) v1)) k))
(Monotone.f (interpE e) (Monotone.f (interpS (lem3' (lem3' (lem3' Θ v3) v2) v1)) k))
(Monotone.f (interpE (subst e (lem3' (lem3' (lem3' Θ v3) v2) v1))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (s-extend (s-extend (s-extend Θ))))) (Monotone.f (interpS (lem3' (lem3' (lem3' ids v3) v2) v1)) k))
(Monotone.f (interpE e) (Monotone.f (interpS {τ1 :: τ2 :: τ3 :: Γ} {τ1 :: τ2 :: τ3 :: Γ'} (s-extend (s-extend (s-extend Θ))))
(Monotone.f (interpS {Γ} {τ1 :: τ2 :: τ3 :: Γ} (lem3' (lem3' (lem3' ids v3) v2) v1)) k)))
(Monotone.f (interpE e) (Monotone.f (interpS (lem3' (lem3' (lem3' Θ v3) v2) v1)) k))
(subst-eq-l (s-extend (s-extend (s-extend Θ))) e (Monotone.f (interpS (lem3' (lem3' (lem3' ids v3) v2) v1)) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpS {τ1 :: τ2 :: τ3 :: Γ} {τ1 :: τ2 :: τ3 :: Γ'} (s-extend (s-extend (s-extend Θ))))
(Monotone.f (interpS {Γ} {τ1 :: τ2 :: τ3 :: Γ} (lem3' (lem3' (lem3' ids v3) v2) v1)) k))
(Monotone.f (interpS (lem3' (lem3' (lem3' Θ v3) v2) v1)) k)
((((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS (λ x → ren (ren (ren (Θ x) iS) iS) iS)) (((Monotone.f (interpS {Γ} ids) k ,
Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v3))) k)
(Monotone.f (interpS Θ) k)
(Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS (λ x → ren (ren (ren (Θ x) iS) iS) iS)) (((Monotone.f (interpS {Γ} ids) k ,
Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(Monotone.f (interpS (λ x → ren (ren (Θ x) iS) iS)) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k))
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v3))) k)
(interp-subst-comp3-l Θ k v3 v2 v1)
(interp-subst-comp2-l Θ k v2 v3))
(interp-subst-comp-l Θ v3 k)) ,
(Preorder-str.refl (snd [ τ3 ]t) (Monotone.f (interpE v3) k))) ,
(Preorder-str.refl (snd [ τ2 ]t) (Monotone.f (interpE v2) k))) ,
(Preorder-str.refl (snd [ τ1 ]t) (Monotone.f (interpE v1) k)))))
(subst-eq-r (lem3' (lem3' (lem3' Θ v3) v2) v1) e k))
sound {Γ} {τ} .(subst e (lem3' (lem3' (lem3' Θ v3) v2) v1)) ._ (subst-compose5-r {.Γ} {Γ'} {.τ} {τ1} {τ2} {τ3} Θ e v1 v2 v3) k =
Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (lem3' (lem3' (lem3' Θ v3) v2) v1))) k)
(Monotone.f (interpE (subst e (s-extend (s-extend (s-extend Θ))))) (Monotone.f (interpS (lem3' (lem3' (lem3' ids v3) v2) v1)) k))
(Monotone.f (interpE (subst (subst e (s-extend (s-extend (s-extend Θ)))) (lem3' (lem3' (lem3' ids v3) v2) v1))) k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE (subst e (lem3' (lem3' (lem3' Θ v3) v2) v1))) k)
(Monotone.f (interpE e) (Monotone.f (interpS (lem3' (lem3' (lem3' Θ v3) v2) v1)) k))
(Monotone.f (interpE (subst e (s-extend (s-extend (s-extend Θ))))) (Monotone.f (interpS (lem3' (lem3' (lem3' ids v3) v2) v1)) k))
(subst-eq-l (lem3' (lem3' (lem3' Θ v3) v2) v1) e k)
(Preorder-str.trans (snd [ τ ]t)
(Monotone.f (interpE e) (Monotone.f (interpS (lem3' (lem3' (lem3' Θ v3) v2) v1)) k))
(Monotone.f (interpE e) (Monotone.f (interpS {τ1 :: τ2 :: τ3 :: Γ} {τ1 :: τ2 :: τ3 :: Γ'} (s-extend (s-extend (s-extend Θ))))
(Monotone.f (interpS {Γ} {τ1 :: τ2 :: τ3 :: Γ} (lem3' (lem3' (lem3' ids v3) v2) v1)) k)))
(Monotone.f (interpE (subst e (s-extend (s-extend (s-extend Θ))))) (Monotone.f (interpS (lem3' (lem3' (lem3' ids v3) v2) v1)) k))
(Monotone.is-monotone (interpE e)
(Monotone.f (interpS (lem3' (lem3' (lem3' Θ v3) v2) v1)) k)
(Monotone.f (interpS {τ1 :: τ2 :: τ3 :: Γ} {τ1 :: τ2 :: τ3 :: Γ'} (s-extend (s-extend (s-extend Θ))))
(Monotone.f (interpS {Γ} {τ1 :: τ2 :: τ3 :: Γ} (lem3' (lem3' (lem3' ids v3) v2) v1)) k))
((((Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS Θ) k)
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v3))) k)
(Monotone.f (interpS (λ x → ren (ren (ren (Θ x) iS) iS) iS))
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(interp-subst-comp-r Θ v3 k)
(Preorder-str.trans (snd [ Γ' ]c)
(Monotone.f (interpS (λ x → subst (ren (Θ x) iS) (lem3' ids v3))) k)
(Monotone.f (interpS (λ x → ren (ren (Θ x) iS) iS)) ((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k))
(Monotone.f (interpS (λ x → ren (ren (ren (Θ x) iS) iS) iS))
(((Monotone.f (interpS {Γ} ids) k , Monotone.f (interpE v3) k) , Monotone.f (interpE v2) k) , Monotone.f (interpE v1) k))
(interp-subst-comp2-r Θ k v2 v3)
(interp-subst-comp3-r Θ k v3 v2 v1))) ,
(Preorder-str.refl (snd [ τ3 ]t) (Monotone.f (interpE v3) k))) ,
(Preorder-str.refl (snd [ τ2 ]t) (Monotone.f (interpE v2) k))) ,
(Preorder-str.refl (snd [ τ1 ]t) (Monotone.f (interpE v1) k))))
(subst-eq-r (s-extend (s-extend (s-extend Θ))) e (Monotone.f (interpS (lem3' (lem3' (lem3' ids v3) v2) v1)) k))))
(subst-eq-r (lem3' (lem3' (lem3' ids v3) v2) v1) (subst e (s-extend (s-extend (s-extend Θ)))) k)
| 66.622543
| 172
| 0.510199
|
0b4d117d685182d62c12f31b02d1603938b14df1
| 1,498
|
agda
|
Agda
|
agda-stdlib/src/Data/Product/Properties/WithK.agda
|
DreamLinuxer/popl21-artifact
|
fb380f2e67dcb4a94f353dbaec91624fcb5b8933
|
[
"MIT"
] | 5
|
2020-10-07T12:07:53.000Z
|
2020-10-10T21:41:32.000Z
|
agda-stdlib/src/Data/Product/Properties/WithK.agda
|
DreamLinuxer/popl21-artifact
|
fb380f2e67dcb4a94f353dbaec91624fcb5b8933
|
[
"MIT"
] | null | null | null |
agda-stdlib/src/Data/Product/Properties/WithK.agda
|
DreamLinuxer/popl21-artifact
|
fb380f2e67dcb4a94f353dbaec91624fcb5b8933
|
[
"MIT"
] | 1
|
2021-11-04T06:54:45.000Z
|
2021-11-04T06:54:45.000Z
|
------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties, related to products, that rely on the K rule
------------------------------------------------------------------------
{-# OPTIONS --with-K --safe #-}
module Data.Product.Properties.WithK where
open import Data.Bool.Base
open import Data.Product
open import Data.Product.Properties using (,-injectiveˡ)
open import Function
open import Relation.Binary using (Decidable)
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary.Reflects
open import Relation.Nullary using (Dec; _because_; yes; no)
open import Relation.Nullary.Decidable using (map′)
------------------------------------------------------------------------
-- Equality
module _ {a b} {A : Set a} {B : Set b} where
,-injective : ∀ {a c : A} {b d : B} → (a , b) ≡ (c , d) → a ≡ c × b ≡ d
,-injective refl = refl , refl
module _ {a b} {A : Set a} {B : A → Set b} where
,-injectiveʳ : ∀ {a} {b c : B a} → (Σ A B ∋ (a , b)) ≡ (a , c) → b ≡ c
,-injectiveʳ refl = refl
-- Note: this is not an instance of `_×-dec_`, because we need `x` and `y`
-- to have the same type before we can test them for equality.
≡-dec : Decidable _≡_ → (∀ {a} → Decidable {A = B a} _≡_) →
Decidable {A = Σ A B} _≡_
≡-dec dec₁ dec₂ (a , x) (b , y) with dec₁ a b
... | false because [a≢b] = no (invert [a≢b] ∘ ,-injectiveˡ)
... | yes refl = map′ (cong (a ,_)) ,-injectiveʳ (dec₂ x y)
| 36.536585
| 76
| 0.537383
|
50f283f9d705302f2154004ab015a36ff247adf3
| 2,617
|
agda
|
Agda
|
theorems/cw/cohomology/reconstructed/HigherCoboundaryGrid.agda
|
AntoineAllioux/HoTT-Agda
|
1037d82edcf29b620677a311dcfd4fc2ade2faa6
|
[
"MIT"
] | 294
|
2015-01-09T16:23:23.000Z
|
2022-03-20T13:54:45.000Z
|
theorems/cw/cohomology/reconstructed/HigherCoboundaryGrid.agda
|
AntoineAllioux/HoTT-Agda
|
1037d82edcf29b620677a311dcfd4fc2ade2faa6
|
[
"MIT"
] | 31
|
2015-03-05T20:09:00.000Z
|
2021-10-03T19:15:25.000Z
|
theorems/cw/cohomology/reconstructed/HigherCoboundaryGrid.agda
|
AntoineAllioux/HoTT-Agda
|
1037d82edcf29b620677a311dcfd4fc2ade2faa6
|
[
"MIT"
] | 50
|
2015-01-10T01:48:08.000Z
|
2022-02-14T03:03:25.000Z
|
{-# OPTIONS --without-K --rewriting #-}
open import HoTT
open import groups.Exactness
open import groups.ExactSequence
open import cw.CW
open import cohomology.Theory
module cw.cohomology.reconstructed.HigherCoboundaryGrid {i} (OT : OrdinaryTheory i)
{n} (⊙skel : ⊙Skeleton {i} (S (S n))) (ac : ⊙has-cells-with-choice 0 ⊙skel i) where
open OrdinaryTheory OT
open import cw.cohomology.WedgeOfCells OT
open import cw.cohomology.grid.PtdMap (⊙cw-incl-last (⊙cw-init ⊙skel)) (⊙cw-incl-last ⊙skel)
open import cw.cohomology.reconstructed.HigherCoboundary OT ⊙skel
import cw.cohomology.grid.LongExactSequence
private
module GLES n = cw.cohomology.grid.LongExactSequence
cohomology-theory n (⊙cw-incl-last (⊙cw-init ⊙skel)) (⊙cw-incl-last ⊙skel)
{-
Xn --> X(n+1) -----> X(n+2)
| | |
v v v
1 -> X(n+1)/n ---> X(n+2)/n
| this |
v one v
1 -----> X(n+2)/(n+1)
-}
private
n≤SSn : n ≤ S (S n)
n≤SSn = inr (ltSR ltS)
private
-- separate lemmas to speed up the type checking
abstract
lemma₀-exact₀ : is-exact
(C-fmap (ℕ-to-ℤ (S n)) Z/X-to-Z/Y)
(C-fmap (ℕ-to-ℤ (S n)) Y/X-to-Z/X)
lemma₀-exact₀ = exact-seq-index 2 $ GLES.C-grid-cofiber-exact-seq (ℕ-to-ℤ n)
lemma₀-exact₁ : is-exact (C-fmap (ℕ-to-ℤ (S n)) Y/X-to-Z/X) cw-co∂-last
lemma₀-exact₁ = exact-seq-index 0 $ GLES.C-grid-cofiber-exact-seq (ℕ-to-ℤ (S n))
lemma₀-trivial : is-trivialᴳ (C (ℕ-to-ℤ (S n)) Z/Y)
lemma₀-trivial = CXₙ/Xₙ₋₁-<-is-trivial ⊙skel ltS ac
Ker-cw-co∂-last : C (ℕ-to-ℤ (S n)) (⊙Cofiber (⊙cw-incl-tail n≤SSn ⊙skel))
≃ᴳ Ker.grp cw-co∂-last
Ker-cw-co∂-last = Exact2.G-trivial-implies-H-iso-ker
lemma₀-exact₀ lemma₀-exact₁ lemma₀-trivial
private
-- separate lemmas to speed up the type checking
abstract
lemma₁-exact₀ : is-exact cw-co∂-last (C-fmap (ℕ-to-ℤ (S (S n))) Z/X-to-Z/Y)
lemma₁-exact₀ = exact-seq-index 1 $ GLES.C-grid-cofiber-exact-seq (ℕ-to-ℤ (S n))
lemma₁-exact₁ : is-exact
(C-fmap (ℕ-to-ℤ (S (S n))) Z/X-to-Z/Y)
(C-fmap (ℕ-to-ℤ (S (S n))) Y/X-to-Z/X)
lemma₁-exact₁ = exact-seq-index 2 $ GLES.C-grid-cofiber-exact-seq (ℕ-to-ℤ (S n))
lemma₁-trivial : is-trivialᴳ (C (ℕ-to-ℤ (S (S n))) Y/X)
lemma₁-trivial = CXₙ/Xₙ₋₁->-is-trivial (⊙cw-init ⊙skel) ltS
(⊙init-has-cells-with-choice ⊙skel ac)
Coker-cw-co∂-last : CokerCo∂ ≃ᴳ C (ℕ-to-ℤ (S (S n))) (⊙Cofiber (⊙cw-incl-tail n≤SSn ⊙skel))
Coker-cw-co∂-last = Exact2.L-trivial-implies-coker-iso-K
lemma₁-exact₀ lemma₁-exact₁ (CXₙ/Xₙ₋₁-is-abelian ⊙skel (ℕ-to-ℤ (S (S n)))) lemma₁-trivial
| 36.347222
| 92
| 0.620558
|
0e33fab0106ddf44389bd36e97afde92a4c40204
| 11,639
|
agda
|
Agda
|
Cubical/Algebra/Ring/QuotientRing.agda
|
guilhermehas/cubical
|
ce3120d3f8d692847b2744162bcd7a01f0b687eb
|
[
"MIT"
] | 1
|
2021-10-31T17:32:49.000Z
|
2021-10-31T17:32:49.000Z
|
Cubical/Algebra/Ring/QuotientRing.agda
|
guilhermehas/cubical
|
ce3120d3f8d692847b2744162bcd7a01f0b687eb
|
[
"MIT"
] | null | null | null |
Cubical/Algebra/Ring/QuotientRing.agda
|
guilhermehas/cubical
|
ce3120d3f8d692847b2744162bcd7a01f0b687eb
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --safe #-}
module Cubical.Algebra.Ring.QuotientRing where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Structure
open import Cubical.Foundations.Powerset using (_∈_; _⊆_; ⊆-extensionality) -- \in, \sub=
open import Cubical.Data.Sigma using (Σ≡Prop)
open import Cubical.Relation.Binary
open import Cubical.HITs.SetQuotients.Base renaming (_/_ to _/ₛ_)
open import Cubical.HITs.SetQuotients.Properties
open import Cubical.Algebra.Ring
open import Cubical.Algebra.Ring.Ideal
open import Cubical.Algebra.Ring.Kernel
open import Cubical.Algebra.CommRingSolver.Reflection
private
variable
ℓ ℓ' : Level
module _ (R' : Ring ℓ) (I : ⟨ R' ⟩ → hProp ℓ) (I-isIdeal : isIdeal R' I) where
open RingStr (snd R')
private R = ⟨ R' ⟩
open isIdeal I-isIdeal
open RingTheory R'
R/I : Type ℓ
R/I = R /ₛ (λ x y → x - y ∈ I)
private
homogeneity : ∀ (x a b : R)
→ (a - b ∈ I)
→ (x + a) - (x + b) ∈ I
homogeneity x a b p = subst (λ u → u ∈ I) (translatedDifference x a b) p
isSetR/I : isSet R/I
isSetR/I = squash/
[_]/I : (a : R) → R/I
[ a ]/I = [ a ]
lemma : (x y a : R)
→ x - y ∈ I
→ [ x + a ]/I ≡ [ y + a ]/I
lemma x y a x-y∈I = eq/ (x + a) (y + a) (subst (λ u → u ∈ I) calculate x-y∈I)
where calculate : x - y ≡ (x + a) - (y + a)
calculate =
x - y ≡⟨ translatedDifference a x y ⟩
((a + x) - (a + y)) ≡⟨ cong (λ u → u - (a + y)) (+Comm _ _) ⟩
((x + a) - (a + y)) ≡⟨ cong (λ u → (x + a) - u) (+Comm _ _) ⟩
((x + a) - (y + a)) ∎
pre-+/I : R → R/I → R/I
pre-+/I x = elim
(λ _ → squash/)
(λ y → [ x + y ])
λ y y' diffrenceInIdeal
→ eq/ (x + y) (x + y') (homogeneity x y y' diffrenceInIdeal)
pre-+/I-DescendsToQuotient : (x y : R) → (x - y ∈ I)
→ pre-+/I x ≡ pre-+/I y
pre-+/I-DescendsToQuotient x y x-y∈I i r = pointwise-equal r i
where
pointwise-equal : ∀ (u : R/I)
→ pre-+/I x u ≡ pre-+/I y u
pointwise-equal = elimProp (λ u → isSetR/I (pre-+/I x u) (pre-+/I y u))
(λ a → lemma x y a x-y∈I)
_+/I_ : R/I → R/I → R/I
x +/I y = (elim R/I→R/I-isSet pre-+/I pre-+/I-DescendsToQuotient x) y
where
R/I→R/I-isSet : R/I → isSet (R/I → R/I)
R/I→R/I-isSet _ = isSetΠ (λ _ → squash/)
-- Note that _+/I_ reduces in this case:
_ : (x y : R) → [ x ] +/I [ y ] ≡ [ x + y ]
_ = λ x y → refl
+/I-comm : (x y : R/I) → x +/I y ≡ y +/I x
+/I-comm = elimProp2 (λ _ _ → squash/ _ _) eq
where eq : (x y : R) → [ x ] +/I [ y ] ≡ [ y ] +/I [ x ]
eq x y i = [ +Comm x y i ]
+/I-assoc : (x y z : R/I) → x +/I (y +/I z) ≡ (x +/I y) +/I z
+/I-assoc = elimProp3 (λ _ _ _ → squash/ _ _) eq
where eq : (x y z : R) → [ x ] +/I ([ y ] +/I [ z ]) ≡ ([ x ] +/I [ y ]) +/I [ z ]
eq x y z i = [ +Assoc x y z i ]
0/I : R/I
0/I = [ 0r ]
1/I : R/I
1/I = [ 1r ]
-/I : R/I → R/I
-/I = elim (λ _ → squash/) (λ x' → [ - x' ]) eq
where
eq : (x y : R) → (x - y ∈ I) → [ - x ] ≡ [ - y ]
eq x y x-y∈I = eq/ (- x) (- y) (subst (λ u → u ∈ I) eq' (isIdeal.-closed I-isIdeal x-y∈I))
where
eq' = - (x + (- y)) ≡⟨ sym (-Dist _ _) ⟩
(- x) - (- y) ∎
+/I-rinv : (x : R/I) → x +/I (-/I x) ≡ 0/I
+/I-rinv = elimProp (λ x → squash/ _ _) eq
where
eq : (x : R) → [ x ] +/I (-/I [ x ]) ≡ 0/I
eq x i = [ +Rinv x i ]
+/I-rid : (x : R/I) → x +/I 0/I ≡ x
+/I-rid = elimProp (λ x → squash/ _ _) eq
where
eq : (x : R) → [ x ] +/I 0/I ≡ [ x ]
eq x i = [ +Rid x i ]
_·/I_ : R/I → R/I → R/I
_·/I_ =
elim (λ _ → isSetΠ (λ _ → squash/))
(λ x → left· x)
eq'
where
eq : (x y y' : R) → (y - y' ∈ I) → [ x · y ] ≡ [ x · y' ]
eq x y y' y-y'∈I = eq/ _ _
(subst (λ u → u ∈ I)
(x · (y - y') ≡⟨ ·Rdist+ _ _ _ ⟩
((x · y) + x · (- y')) ≡⟨ cong (λ u → (x · y) + u)
(-DistR· x y') ⟩
(x · y) - (x · y') ∎)
(isIdeal.·-closedLeft I-isIdeal x y-y'∈I))
left· : (x : R) → R/I → R/I
left· x = elim (λ y → squash/)
(λ y → [ x · y ])
(eq x)
eq' : (x x' : R) → (x - x' ∈ I) → left· x ≡ left· x'
eq' x x' x-x'∈I i y = elimProp (λ y → squash/ (left· x y) (left· x' y))
(λ y → eq′ y)
y i
where
eq′ : (y : R) → left· x [ y ] ≡ left· x' [ y ]
eq′ y = eq/ (x · y) (x' · y)
(subst (λ u → u ∈ I)
((x - x') · y ≡⟨ ·Ldist+ x (- x') y ⟩
x · y + (- x') · y ≡⟨ cong
(λ u → x · y + u)
(-DistL· x' y) ⟩
x · y - x' · y ∎)
(isIdeal.·-closedRight I-isIdeal y x-x'∈I))
-- more or less copy paste from '+/I' - this is preliminary anyway
·/I-assoc : (x y z : R/I) → x ·/I (y ·/I z) ≡ (x ·/I y) ·/I z
·/I-assoc = elimProp3 (λ _ _ _ → squash/ _ _) eq
where eq : (x y z : R) → [ x ] ·/I ([ y ] ·/I [ z ]) ≡ ([ x ] ·/I [ y ]) ·/I [ z ]
eq x y z i = [ ·Assoc x y z i ]
·/I-lid : (x : R/I) → 1/I ·/I x ≡ x
·/I-lid = elimProp (λ x → squash/ _ _) eq
where
eq : (x : R) → 1/I ·/I [ x ] ≡ [ x ]
eq x i = [ ·Lid x i ]
·/I-rid : (x : R/I) → x ·/I 1/I ≡ x
·/I-rid = elimProp (λ x → squash/ _ _) eq
where
eq : (x : R) → [ x ] ·/I 1/I ≡ [ x ]
eq x i = [ ·Rid x i ]
/I-ldist : (x y z : R/I) → (x +/I y) ·/I z ≡ (x ·/I z) +/I (y ·/I z)
/I-ldist = elimProp3 (λ _ _ _ → squash/ _ _) eq
where
eq : (x y z : R) → ([ x ] +/I [ y ]) ·/I [ z ] ≡ ([ x ] ·/I [ z ]) +/I ([ y ] ·/I [ z ])
eq x y z i = [ ·Ldist+ x y z i ]
/I-rdist : (x y z : R/I) → x ·/I (y +/I z) ≡ (x ·/I y) +/I (x ·/I z)
/I-rdist = elimProp3 (λ _ _ _ → squash/ _ _) eq
where
eq : (x y z : R) → [ x ] ·/I ([ y ] +/I [ z ]) ≡ ([ x ] ·/I [ y ]) +/I ([ x ] ·/I [ z ])
eq x y z i = [ ·Rdist+ x y z i ]
asRing : Ring ℓ
asRing = makeRing 0/I 1/I _+/I_ _·/I_ -/I isSetR/I
+/I-assoc +/I-rid +/I-rinv +/I-comm
·/I-assoc ·/I-rid ·/I-lid /I-rdist /I-ldist
_/_ : (R : Ring ℓ) → (I : IdealsIn R) → Ring ℓ
R / (I , IisIdeal) = asRing R I IisIdeal
[_]/I : {R : Ring ℓ} {I : IdealsIn R} → (a : ⟨ R ⟩) → ⟨ R / I ⟩
[ a ]/I = [ a ]
quotientHom : (R : Ring ℓ) → (I : IdealsIn R) → RingHom R (R / I)
fst (quotientHom R I) = [_]
IsRingHom.pres0 (snd (quotientHom R I)) = refl
IsRingHom.pres1 (snd (quotientHom R I)) = refl
IsRingHom.pres+ (snd (quotientHom R I)) _ _ = refl
IsRingHom.pres· (snd (quotientHom R I)) _ _ = refl
IsRingHom.pres- (snd (quotientHom R I)) _ = refl
module UniversalProperty (R : Ring ℓ) (I : IdealsIn R) where
open RingStr ⦃...⦄
open RingTheory ⦃...⦄
Iₛ = fst I
private
instance
_ = R
_ = snd R
module _ {S : Ring ℓ'} (φ : RingHom R S) where
open IsRingHom
open RingHomTheory φ
private
instance
_ = S
_ = snd S
f = fst φ
module φ = IsRingHom (snd φ)
{-
We do not use the kernel ideal, since it is *not* an ideal in R,
if S is from a different universe. Instead, the condition, that
Iₛ is contained in the kernel of φ is rephrased explicitly.
-}
inducedHom : ((x : ⟨ R ⟩) → x ∈ Iₛ → φ $ x ≡ 0r) → RingHom (R / I) S
fst (inducedHom Iₛ⊆kernel) =
elim
(λ _ → isSetRing S)
f
λ r₁ r₂ r₁-r₂∈I → equalByDifference (f r₁) (f r₂)
(f r₁ - f r₂ ≡⟨ cong (λ u → f r₁ + u) (sym (φ.pres- _)) ⟩
f r₁ + f (- r₂) ≡⟨ sym (φ.pres+ _ _) ⟩
f (r₁ - r₂) ≡⟨ Iₛ⊆kernel (r₁ - r₂) r₁-r₂∈I ⟩
0r ∎)
pres0 (snd (inducedHom Iₛ⊆kernel)) = φ.pres0
pres1 (snd (inducedHom Iₛ⊆kernel)) = φ.pres1
pres+ (snd (inducedHom Iₛ⊆kernel)) =
elimProp2 (λ _ _ → isSetRing S _ _) φ.pres+
pres· (snd (inducedHom Iₛ⊆kernel)) =
elimProp2 (λ _ _ → isSetRing S _ _) φ.pres·
pres- (snd (inducedHom Iₛ⊆kernel)) =
elimProp (λ _ → isSetRing S _ _) φ.pres-
solution : (p : ((x : ⟨ R ⟩) → x ∈ Iₛ → φ $ x ≡ 0r))
→ (x : ⟨ R ⟩) → inducedHom p $ [ x ] ≡ φ $ x
solution p x = refl
unique : (p : ((x : ⟨ R ⟩) → x ∈ Iₛ → φ $ x ≡ 0r))
→ (ψ : RingHom (R / I) S) → (ψIsSolution : (x : ⟨ R ⟩) → ψ $ [ x ] ≡ φ $ x)
→ (x : ⟨ R ⟩) → ψ $ [ x ] ≡ inducedHom p $ [ x ]
unique p ψ ψIsSolution x = ψIsSolution x
{-
Show that the kernel of the quotient map
π : R ─→ R/I
is the given ideal I.
-}
module idealIsKernel {R : Ring ℓ} (I : IdealsIn R) where
open RingStr (snd R)
open isIdeal (snd I)
open BinaryRelation.isEquivRel
private
π = quotientHom R I
x-0≡x : (x : ⟨ R ⟩) → x - 0r ≡ x
x-0≡x x =
x - 0r ≡⟨ cong (x +_) (RingTheory.0Selfinverse R) ⟩
x + 0r ≡⟨ +Rid x ⟩
x ∎
I⊆ker : fst I ⊆ kernel π
I⊆ker x x∈I = eq/ _ _ (subst (_∈ fst I) (sym (x-0≡x x)) x∈I)
private
_~_ : Rel ⟨ R ⟩ ⟨ R ⟩ ℓ
x ~ y = x - y ∈ fst I
~IsPropValued : BinaryRelation.isPropValued _~_
~IsPropValued x y = snd (fst I (x - y))
-- _~_ is an equivalence relation.
-- Note: This could be proved in the general setting of a subgroup of a group.
-[x-y]≡y-x : {x y : ⟨ R ⟩} → - (x - y) ≡ y - x
-[x-y]≡y-x {x} {y} =
- (x - y) ≡⟨ sym (-Dist _ _) ⟩
- x + - (- y) ≡⟨ cong (- x +_) (-Idempotent _) ⟩
- x + y ≡⟨ +Comm _ _ ⟩
y - x ∎
where open RingTheory R
x-y+y-z≡x-z : {x y z : ⟨ R ⟩} → (x - y) + (y - z) ≡ x - z
x-y+y-z≡x-z {x} {y} {z} =
(x + - y) + (y + - z) ≡⟨ +Assoc _ _ _ ⟩
((x + - y) + y) + - z ≡⟨ cong (_+ - z) (sym (+Assoc _ _ _)) ⟩
(x + (- y + y)) + - z ≡⟨ cong (λ -y+y → (x + -y+y) + - z) (+Linv _) ⟩
(x + 0r) + - z ≡⟨ cong (_+ - z) (+Rid _) ⟩
x - z ∎
~IsEquivRel : BinaryRelation.isEquivRel _~_
reflexive ~IsEquivRel x = subst (_∈ fst I) (sym (+Rinv x)) 0r-closed
symmetric ~IsEquivRel x y x~y = subst (_∈ fst I) -[x-y]≡y-x (-closed x~y)
transitive ~IsEquivRel x y z x~y y~z = subst (_∈ fst I) x-y+y-z≡x-z (+-closed x~y y~z)
ker⊆I : kernel π ⊆ fst I
ker⊆I x x∈ker = subst (_∈ fst I) (x-0≡x x) x-0∈I
where
x-0∈I : x - 0r ∈ fst I
x-0∈I = effective ~IsPropValued ~IsEquivRel x 0r x∈ker
kernel≡I : {R : Ring ℓ} (I : IdealsIn R)
→ kernelIdeal (quotientHom R I) ≡ I
kernel≡I {R = R} I = Σ≡Prop (isPropIsIdeal R) (⊆-extensionality _ _ (ker⊆I , I⊆ker))
where open idealIsKernel I
| 36.258567
| 98
| 0.414383
|
109002dc1342b8c8da5975110a189ccb2e33089b
| 6,804
|
agda
|
Agda
|
Cubical/Structures/Relational/Function.agda
|
guilhermehas/cubical
|
ce3120d3f8d692847b2744162bcd7a01f0b687eb
|
[
"MIT"
] | 1
|
2021-10-31T17:32:49.000Z
|
2021-10-31T17:32:49.000Z
|
Cubical/Structures/Relational/Function.agda
|
guilhermehas/cubical
|
ce3120d3f8d692847b2744162bcd7a01f0b687eb
|
[
"MIT"
] | null | null | null |
Cubical/Structures/Relational/Function.agda
|
guilhermehas/cubical
|
ce3120d3f8d692847b2744162bcd7a01f0b687eb
|
[
"MIT"
] | null | null | null |
{-
Index a structure T a positive structure S: X ↦ S X → T X
-}
{-# OPTIONS --safe #-}
module Cubical.Structures.Relational.Function where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Function
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Structure
open import Cubical.Foundations.RelationalStructure
open import Cubical.Foundations.Univalence
open import Cubical.Functions.FunExtEquiv
open import Cubical.Data.Sigma
open import Cubical.Relation.Binary.Base
open import Cubical.Relation.ZigZag.Base
open import Cubical.HITs.SetQuotients
open import Cubical.HITs.PropositionalTruncation as Trunc
open import Cubical.Structures.Function
private
variable
ℓ ℓ₁ ℓ₁' ℓ₁'' ℓ₂ ℓ₂' ℓ₂'' : Level
FunctionRelStr : {S : Type ℓ → Type ℓ₁} {T : Type ℓ → Type ℓ₂}
→ StrRel S ℓ₁' → StrRel T ℓ₂' → StrRel (FunctionStructure S T) (ℓ-max ℓ₁ (ℓ-max ℓ₁' ℓ₂'))
FunctionRelStr ρ₁ ρ₂ R f g =
∀ {x y} → ρ₁ R x y → ρ₂ R (f x) (g y)
open BinaryRelation
open isEquivRel
private
composeWith[_] : {A : Type ℓ} (R : EquivPropRel A ℓ)
→ compPropRel (R .fst) (quotientPropRel (R .fst .fst)) .fst ≡ graphRel [_]
composeWith[_] R =
funExt₂ λ a t →
hPropExt squash₁ (squash/ _ _)
(Trunc.rec (squash/ _ _) (λ {(b , r , p) → eq/ a b r ∙ p }))
(λ p → ∣ a , R .snd .reflexive a , p ∣₁)
[_]∙[_]⁻¹ : {A : Type ℓ} (R : EquivPropRel A ℓ)
→ compPropRel (quotientPropRel (R .fst .fst)) (invPropRel (quotientPropRel (R .fst .fst))) .fst
≡ R .fst .fst
[_]∙[_]⁻¹ R =
funExt₂ λ a b →
hPropExt squash₁ (R .fst .snd a b)
(Trunc.rec (R .fst .snd a b)
(λ {(c , p , q) → effective (R .fst .snd) (R .snd) a b (p ∙ sym q)}))
(λ r → ∣ _ , eq/ a b r , refl ∣₁)
functionSuitableRel : {S : Type ℓ → Type ℓ₁} {T : Type ℓ → Type ℓ₂}
{ρ₁ : StrRel S ℓ₁'} {ρ₂ : StrRel T ℓ₂'}
(θ₁ : SuitableStrRel S ρ₁)
→ PositiveStrRel θ₁
→ SuitableStrRel T ρ₂
→ SuitableStrRel (FunctionStructure S T) (FunctionRelStr ρ₁ ρ₂)
functionSuitableRel {S = S} {T = T} {ρ₁ = ρ₁} {ρ₂} θ₁ σ₁ θ₂ .quo (X , f) R h =
final
where
ref : (s : S X) → ρ₁ (R .fst .fst) s s
ref = posRelReflexive σ₁ R
[f] : S X / ρ₁ (R .fst .fst) → T (X / R .fst .fst)
[f] [ s ] = θ₂ .quo (X , f s) R (h (ref s)) .fst .fst
[f] (eq/ s₀ s₁ r i) =
cong fst
(θ₂ .quo (X , f s₀) R (h (ref s₀)) .snd
( [f] [ s₁ ]
, subst (λ R' → ρ₂ R' (f s₀) ([f] [ s₁ ]))
(composeWith[_] R)
(θ₂ .transitive (R .fst) (quotientPropRel (R .fst .fst))
(h r)
(θ₂ .quo (X , f s₁) R (h (ref s₁)) .fst .snd))
))
i
[f] (squash/ _ _ p q j i) =
θ₂ .set squash/ _ _ (cong [f] p) (cong [f] q) j i
relLemma : (s : S X) (t : S X)
→ ρ₁ (graphRel [_]) s (funIsEq (σ₁ .quo R) [ t ])
→ ρ₂ (graphRel [_]) (f s) ([f] [ t ])
relLemma s t r =
subst (λ R' → ρ₂ R' (f s) ([f] [ t ]))
(composeWith[_] R)
(θ₂ .transitive (R .fst) (quotientPropRel (R .fst .fst))
(h r')
(θ₂ .quo (X , f t) R (h (ref t)) .fst .snd))
where
r' : ρ₁ (R .fst .fst) s t
r' =
subst (λ R' → ρ₁ R' s t) ([_]∙[_]⁻¹ R)
(θ₁ .transitive
(quotientPropRel (R .fst .fst))
(invPropRel (quotientPropRel (R .fst .fst)))
r
(θ₁ .symmetric (quotientPropRel (R .fst .fst))
(subst
(λ t' → ρ₁ (graphRel [_]) t' (funIsEq (σ₁ .quo R) [ t ]))
(σ₁ .act .actStrId t)
(σ₁ .act .actRel eq/ t t (ref t)))))
quoRelLemma : (s : S X) (t : S X / ρ₁ (R .fst .fst))
→ ρ₁ (graphRel [_]) s (funIsEq (σ₁ .quo R) t)
→ ρ₂ (graphRel [_]) (f s) ([f] t)
quoRelLemma s =
elimProp (λ _ → isPropΠ λ _ → θ₂ .prop (λ _ _ → squash/ _ _) _ _)
(relLemma s)
final : Σ (Σ _ _) _
final .fst .fst = [f] ∘ invIsEq (σ₁ .quo R)
final .fst .snd {s} {t} r =
quoRelLemma s
(invIsEq (σ₁ .quo R) t)
(subst (ρ₁ (graphRel [_]) s) (sym (secIsEq (σ₁ .quo R) t)) r)
final .snd (f' , c) =
Σ≡Prop
(λ _ → isPropImplicitΠ λ s →
isPropImplicitΠ λ t →
isPropΠ λ _ → θ₂ .prop (λ _ _ → squash/ _ _) _ _)
(funExt λ s → contractorLemma (invIsEq (σ₁ .quo R) s) ∙ cong f' (secIsEq (σ₁ .quo R) s))
where
contractorLemma : (s : S X / ρ₁ (R .fst .fst))
→ [f] s ≡ f' (funIsEq (σ₁ .quo R) s)
contractorLemma =
elimProp
(λ _ → θ₂ .set squash/ _ _)
(λ s →
cong fst
(θ₂ .quo (X , f s) R (h (ref s)) .snd
( f' (funIsEq (σ₁ .quo R) [ s ])
, c
(subst
(λ s' → ρ₁ (graphRel [_]) s' (funIsEq (σ₁ .quo R) [ s ]))
(σ₁ .act .actStrId s)
(σ₁ .act .actRel eq/ s s (ref s)))
)))
functionSuitableRel {ρ₁ = ρ₁} {ρ₂} θ₁ σ θ₂ .symmetric R h r =
θ₂ .symmetric R (h (θ₁ .symmetric (invPropRel R) r))
functionSuitableRel {ρ₁ = ρ₁} {ρ₂} θ₁ σ θ₂ .transitive R R' h h' rr' =
Trunc.rec
(θ₂ .prop (λ _ _ → squash₁) _ _)
(λ {(_ , r , r') → θ₂ .transitive R R' (h r) (h' r')})
(σ .detransitive R R' rr')
functionSuitableRel {ρ₁ = ρ₁} {ρ₂} θ₁ σ θ₂ .set setX =
isSetΠ λ _ → θ₂ .set setX
functionSuitableRel {ρ₁ = ρ₁} {ρ₂} θ₁ σ θ₂ .prop propR f g =
isPropImplicitΠ λ _ →
isPropImplicitΠ λ _ →
isPropΠ λ _ →
θ₂ .prop propR _ _
functionRelMatchesEquiv : {S : Type ℓ → Type ℓ₁} {T : Type ℓ → Type ℓ₂}
(ρ₁ : StrRel S ℓ₁') {ι₁ : StrEquiv S ℓ₁''}
(ρ₂ : StrRel T ℓ₂') {ι₂ : StrEquiv T ℓ₂''}
→ StrRelMatchesEquiv ρ₁ ι₁
→ StrRelMatchesEquiv ρ₂ ι₂
→ StrRelMatchesEquiv (FunctionRelStr ρ₁ ρ₂) (FunctionEquivStr ι₁ ι₂)
functionRelMatchesEquiv ρ₁ ρ₂ μ₁ μ₂ (X , f) (Y , g) e =
equivImplicitΠCod (equivImplicitΠCod (equiv→ (μ₁ _ _ e) (μ₂ _ _ e)))
functionRelMatchesEquiv+ : {S : Type ℓ → Type ℓ₁} {T : Type ℓ → Type ℓ₂}
(ρ₁ : StrRel S ℓ₁') (α₁ : EquivAction S)
(ρ₂ : StrRel T ℓ₂') (ι₂ : StrEquiv T ℓ₂'')
→ StrRelMatchesEquiv ρ₁ (EquivAction→StrEquiv α₁)
→ StrRelMatchesEquiv ρ₂ ι₂
→ StrRelMatchesEquiv (FunctionRelStr ρ₁ ρ₂) (FunctionEquivStr+ α₁ ι₂)
functionRelMatchesEquiv+ ρ₁ α₁ ρ₂ ι₂ μ₁ μ₂ (X , f) (Y , g) e =
compEquiv
(functionRelMatchesEquiv ρ₁ ρ₂ μ₁ μ₂ (X , f) (Y , g) e)
(isoToEquiv isom)
where
open Iso
isom : Iso
(FunctionEquivStr (EquivAction→StrEquiv α₁) ι₂ (X , f) (Y , g) e)
(FunctionEquivStr+ α₁ ι₂ (X , f) (Y , g) e)
isom .fun h s = h refl
isom .inv k {x} = J (λ y _ → ι₂ (X , f x) (Y , g y) e) (k x)
isom .rightInv k i x = JRefl (λ y _ → ι₂ (X , f x) (Y , g y) e) (k x) i
isom .leftInv h =
implicitFunExt λ {x} →
implicitFunExt λ {y} →
funExt λ p →
J (λ y p → isom .inv (isom .fun h) p ≡ h p)
(funExt⁻ (isom .rightInv (isom .fun h)) x)
p
| 35.4375
| 99
| 0.559818
|
0b33568ca3f83047721432ed58c38564fd5172ad
| 2,012
|
agda
|
Agda
|
autotests/input/test.agda
|
danipozo/syntax-highlighting
|
4da852ec232411be5abc065c0f2ee21fdb016008
|
[
"MIT"
] | null | null | null |
autotests/input/test.agda
|
danipozo/syntax-highlighting
|
4da852ec232411be5abc065c0f2ee21fdb016008
|
[
"MIT"
] | null | null | null |
autotests/input/test.agda
|
danipozo/syntax-highlighting
|
4da852ec232411be5abc065c0f2ee21fdb016008
|
[
"MIT"
] | null | null | null |
-- Agda Sample File
-- https://github.com/agda/agda/blob/master/examples/syntax/highlighting/Test.agda
-- This test file currently lacks module-related stuff.
{- Nested
{- comment. -} -}
module Test where
infix 12 _!
infixl 7 _+_ _-_
infixr 2 -_
postulate x : Set
f : (Set -> Set -> Set) -> Set
f _*_ = x * x
data ℕ : Set where
zero : ℕ
suc : ℕ -> ℕ
_+_ : ℕ -> ℕ -> ℕ
zero + n = n
suc m + n = suc (m + n)
postulate _-_ : ℕ -> ℕ -> ℕ
-_ : ℕ -> ℕ
- n = n
_! : ℕ -> ℕ
zero ! = suc zero
suc n ! = n - n !
record Equiv {a : Set} (_≈_ : a -> a -> Set) : Set where
field
refl : forall x -> x ≈ x
sym : {x y : a} -> x ≈ y -> y ≈ x
_`trans`_ : forall {x y z} -> x ≈ y -> y ≈ z -> x ≈ z
data _≡_ {a : Set} (x : a) : a -> Set where
refl : x ≡ x
subst : forall {a x y} ->
(P : a -> Set) -> x ≡ y -> P x -> P y
subst {x = x} .{y = x} _ refl p = p
Equiv-≡ : forall {a} -> Equiv {a} _≡_
Equiv-≡ {a} =
record { refl = \_ -> refl
; sym = sym
; _`trans`_ = _`trans`_
}
where
sym : {x y : a} -> x ≡ y -> y ≡ x
sym refl = refl
_`trans`_ : {x y z : a} -> x ≡ y -> y ≡ z -> x ≡ z
refl `trans` refl = refl
postulate
String : Set
Char : Set
Float : Set
data Int : Set where
pos : ℕ → Int
negsuc : ℕ → Int
{-# BUILTIN STRING String #-}
{-# BUILTIN CHAR Char #-}
{-# BUILTIN FLOAT Float #-}
{-# BUILTIN NATURAL ℕ #-}
{-# BUILTIN INTEGER Int #-}
{-# BUILTIN INTEGERPOS pos #-}
{-# BUILTIN INTEGERNEGSUC negsuc #-}
data [_] (a : Set) : Set where
[] : [ a ]
_∷_ : a -> [ a ] -> [ a ]
{-# BUILTIN LIST [_] #-}
{-# BUILTIN NIL [] #-}
{-# BUILTIN CONS _∷_ #-}
primitive
primStringToList : String -> [ Char ]
string : [ Char ]
string = primStringToList "∃ apa"
char : Char
char = '∀'
anotherString : String
anotherString = "¬ be\
\pa"
nat : ℕ
nat = 45
float : Float
float = 45.0e-37
-- Testing qualified names.
open import Test
Eq = Test.Equiv {Test.ℕ}
| 17.80531
| 82
| 0.503479
|
0bcf99ac81b46e3be72a334dedb0d5c34bf5d081
| 59,178
|
agda
|
Agda
|
src/Equivalence/Erased.agda
|
nad/equality
|
402b20615cfe9ca944662380d7b2d69b0f175200
|
[
"MIT"
] | 3
|
2020-05-21T22:58:50.000Z
|
2021-09-02T17:18:15.000Z
|
src/Equivalence/Erased.agda
|
nad/equality
|
402b20615cfe9ca944662380d7b2d69b0f175200
|
[
"MIT"
] | null | null | null |
src/Equivalence/Erased.agda
|
nad/equality
|
402b20615cfe9ca944662380d7b2d69b0f175200
|
[
"MIT"
] | null | null | null |
------------------------------------------------------------------------
-- Equivalences with erased "proofs"
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Equality
module Equivalence.Erased
{reflexive} (eq : ∀ {a p} → Equality-with-J a p reflexive) where
open Derived-definitions-and-properties eq
open import Logical-equivalence using (_⇔_)
open import Prelude as P hiding (id; [_,_]) renaming (_∘_ to _⊚_)
open import Bijection eq using (_↔_)
open import Equivalence eq as Eq using (_≃_; Is-equivalence)
import Equivalence.Contractible-preimages eq as CP
open import Equivalence.Erased.Contractible-preimages eq as ECP
using (_⁻¹ᴱ_; Contractibleᴱ)
import Equivalence.Half-adjoint eq as HA
open import Erased.Level-1 eq as Erased
hiding (module []-cong; module []-cong₁; module []-cong₂-⊔)
open import Function-universe eq as F
hiding (id; _∘_; inverse; from-isomorphism;
step-↔; _↔⟨⟩_; _□; finally-↔; $⟨_⟩_)
open import H-level eq as H-level
open import H-level.Closure eq
import Nat eq as Nat
open import Preimage eq as Preimage using (_⁻¹_)
open import Surjection eq as Surjection using (_↠_; Split-surjective)
open import Tactic.Sigma-cong eq
open import Univalence-axiom eq
private
variable
a b d ℓ ℓ₁ ℓ₂ q : Level
A B C D : Type a
c k k′ p x y : A
P Q : A → Type p
f g : (x : A) → P x
------------------------------------------------------------------------
-- Some basic stuff
open import Equivalence.Erased.Basics eq public
private
variable
A≃B : A ≃ᴱ B
------------------------------------------------------------------------
-- More conversion lemmas
-- In an erased context Is-equivalence and Is-equivalenceᴱ are
-- pointwise equivalent.
@0 Is-equivalence≃Is-equivalenceᴱ :
{A : Type a} {B : Type b} {f : A → B} →
Is-equivalence f ≃ Is-equivalenceᴱ f
Is-equivalence≃Is-equivalenceᴱ {f = f} =
(∃ λ f⁻¹ → HA.Proofs f f⁻¹) F.↔⟨ (∃-cong λ _ → F.inverse $ erased Erased↔) ⟩□
(∃ λ f⁻¹ → Erased (HA.Proofs f f⁻¹)) □
_ :
_≃_.to Is-equivalence≃Is-equivalenceᴱ p ≡
Is-equivalence→Is-equivalenceᴱ p
_ = refl _
@0 _ :
_≃_.from Is-equivalence≃Is-equivalenceᴱ p ≡
Is-equivalenceᴱ→Is-equivalence p
_ = refl _
-- In an erased context _≃_ and _≃ᴱ_ are pointwise equivalent.
@0 ≃≃≃ᴱ : (A ≃ B) ≃ (A ≃ᴱ B)
≃≃≃ᴱ {A = A} {B = B} =
A ≃ B F.↔⟨ Eq.≃-as-Σ ⟩
(∃ λ f → Is-equivalence f) ↝⟨ (∃-cong λ _ → Is-equivalence≃Is-equivalenceᴱ) ⟩
(∃ λ f → Is-equivalenceᴱ f) F.↔⟨ Eq.inverse ≃ᴱ-as-Σ ⟩□
A ≃ᴱ B □
_ : _≃_.to ≃≃≃ᴱ p ≡ ≃→≃ᴱ p
_ = refl _
@0 _ : _≃_.from ≃≃≃ᴱ p ≡ ≃ᴱ→≃ p
_ = refl _
-- A variant of F.from-isomorphism with erased type arguments.
from-isomorphism :
{@0 A : Type a} {@0 B : Type b} →
A ↔[ k ] B → A ≃ᴱ B
from-isomorphism {k = equivalence} = ≃→≃ᴱ
from-isomorphism {k = bijection} = λ A↔B →
let record
{ surjection = record
{ logical-equivalence = record
{ to = to
; from = from
}
}
} = A↔B
in
↔→≃ᴱ
to
from
(_↔_.right-inverse-of A↔B)
(_↔_.left-inverse-of A↔B)
------------------------------------------------------------------------
-- "Equational" reasoning combinators with erased type arguments
infix -1 finally-≃ᴱ finally-↔
infix -1 _□
infixr -2 step-≃ᴱ step-↔ _↔⟨⟩_
infix -3 $⟨_⟩_
-- For an explanation of why step-≃ᴱ and step-↔ are defined in this
-- way, see Equality.step-≡.
step-≃ᴱ :
(@0 A : Type a) {@0 B : Type b} {@0 C : Type c} →
B ≃ᴱ C → A ≃ᴱ B → A ≃ᴱ C
step-≃ᴱ _ = _∘_
syntax step-≃ᴱ A B≃ᴱC A≃ᴱB = A ≃ᴱ⟨ A≃ᴱB ⟩ B≃ᴱC
step-↔ :
(@0 A : Type a) {@0 B : Type b} {@0 C : Type c} →
B ≃ᴱ C → A ↔[ k ] B → A ≃ᴱ C
step-↔ _ B≃ᴱC A↔B = step-≃ᴱ _ B≃ᴱC (from-isomorphism A↔B)
syntax step-↔ A B≃ᴱC A↔B = A ↔⟨ A↔B ⟩ B≃ᴱC
_↔⟨⟩_ : (@0 A : Type a) {@0 B : Type b} → A ≃ᴱ B → A ≃ᴱ B
_ ↔⟨⟩ A≃ᴱB = A≃ᴱB
_□ : (@0 A : Type a) → A ≃ᴱ A
A □ = id
finally-≃ᴱ : (@0 A : Type a) (@0 B : Type b) → A ≃ᴱ B → A ≃ᴱ B
finally-≃ᴱ _ _ A≃ᴱB = A≃ᴱB
syntax finally-≃ᴱ A B A≃ᴱB = A ≃ᴱ⟨ A≃ᴱB ⟩□ B □
finally-↔ : (@0 A : Type a) (@0 B : Type b) → A ↔[ k ] B → A ≃ᴱ B
finally-↔ _ _ A↔B = from-isomorphism A↔B
syntax finally-↔ A B A↔B = A ↔⟨ A↔B ⟩□ B □
$⟨_⟩_ : {@0 A : Type a} {@0 B : Type b} → A → A ≃ᴱ B → B
$⟨ a ⟩ A≃ᴱB = _≃ᴱ_.to A≃ᴱB a
------------------------------------------------------------------------
-- Is-equivalenceᴱ is sometimes propositional
-- In an erased context Is-equivalenceᴱ f is a proposition (assuming
-- extensionality).
--
-- See also Is-equivalenceᴱ-propositional-for-Erased below.
@0 Is-equivalenceᴱ-propositional :
{A : Type a} {B : Type b} →
Extensionality (a ⊔ b) (a ⊔ b) →
(f : A → B) → Is-proposition (Is-equivalenceᴱ f)
Is-equivalenceᴱ-propositional ext f =
H-level.respects-surjection
(_≃_.surjection $ Is-equivalence≃Is-equivalenceᴱ)
1
(Eq.propositional ext f)
------------------------------------------------------------------------
-- Even more conversion lemmas, and a related result
-- Is-equivalenceᴱ f is logically equivalent to ECP.Is-equivalenceᴱ f.
Is-equivalenceᴱ⇔Is-equivalenceᴱ-CP :
{@0 A : Type a} {@0 B : Type b} {@0 f : A → B} →
Is-equivalenceᴱ f ⇔ ECP.Is-equivalenceᴱ f
Is-equivalenceᴱ⇔Is-equivalenceᴱ-CP {f = f} =
record { to = to; from = from }
where
to : Is-equivalenceᴱ f → ECP.Is-equivalenceᴱ f
to eq y =
(proj₁₀ eq y , [ erased (proj₂ $ proj₁ eq′) ])
, [ erased (proj₂ eq′) ]
where
@0 eq′ : Contractibleᴱ (f ⁻¹ᴱ y)
eq′ =
ECP.Is-equivalence→Is-equivalenceᴱ
(_⇔_.to HA.Is-equivalence⇔Is-equivalence-CP $
Is-equivalenceᴱ→Is-equivalence eq)
y
from : ECP.Is-equivalenceᴱ f → Is-equivalenceᴱ f
from eq =
proj₁₀ ⊚ proj₁₀ ⊚ eq
, [ erased $ proj₂ $
Is-equivalence→Is-equivalenceᴱ $
_⇔_.from HA.Is-equivalence⇔Is-equivalence-CP $
ECP.Is-equivalenceᴱ→Is-equivalence eq
]
-- Is-equivalenceᴱ f is equivalent (with erased proofs) to
-- ECP.Is-equivalenceᴱ f (assuming extensionality).
Is-equivalenceᴱ≃ᴱIs-equivalenceᴱ-CP :
{@0 A : Type a} {@0 B : Type b} {@0 f : A → B} →
@0 Extensionality (a ⊔ b) (a ⊔ b) →
Is-equivalenceᴱ f ≃ᴱ ECP.Is-equivalenceᴱ f
Is-equivalenceᴱ≃ᴱIs-equivalenceᴱ-CP ext =
let record { to = to; from = from } =
Is-equivalenceᴱ⇔Is-equivalenceᴱ-CP
in
⇔→≃ᴱ
(Is-equivalenceᴱ-propositional ext _)
(ECP.Is-equivalenceᴱ-propositional ext _)
to
from
-- When proving that a function is an equivalence (with erased proofs)
-- one can assume that the codomain is inhabited.
[inhabited→Is-equivalenceᴱ]→Is-equivalenceᴱ :
{@0 A : Type a} {@0 B : Type b} {@0 f : A → B} →
(B → Is-equivalenceᴱ f) → Is-equivalenceᴱ f
[inhabited→Is-equivalenceᴱ]→Is-equivalenceᴱ hyp =
let record { to = to; from = from } =
Is-equivalenceᴱ⇔Is-equivalenceᴱ-CP
in
from (λ x → to (hyp x) x)
------------------------------------------------------------------------
-- Some preservation lemmas
-- A variant of _×-cong_ for _≃ᴱ_. Note that all the type arguments
-- are erased.
infixr 2 _×-cong-≃ᴱ_
_×-cong-≃ᴱ_ :
{@0 A : Type a} {@0 B : Type b} {@0 C : Type c} {@0 D : Type d} →
A ≃ᴱ C → B ≃ᴱ D → (A × B) ≃ᴱ (C × D)
A≃ᴱC ×-cong-≃ᴱ B≃ᴱD = ↔→≃ᴱ
(Σ-map (_≃ᴱ_.to A≃ᴱC) (_≃ᴱ_.to B≃ᴱD))
(Σ-map (_≃ᴱ_.from A≃ᴱC) (_≃ᴱ_.from B≃ᴱD))
(λ _ →
cong₂ _,_
(_≃ᴱ_.right-inverse-of A≃ᴱC _)
(_≃ᴱ_.right-inverse-of B≃ᴱD _))
(λ _ →
cong₂ _,_
(_≃ᴱ_.left-inverse-of A≃ᴱC _)
(_≃ᴱ_.left-inverse-of B≃ᴱD _))
-- A variant of ∃-cong for _≃ᴱ_. Note that all the type arguments are
-- erased.
∃-cong-≃ᴱ :
{@0 A : Type a} {@0 P : A → Type p} {@0 Q : A → Type q} →
(∀ x → P x ≃ᴱ Q x) → ∃ P ≃ᴱ ∃ Q
∃-cong-≃ᴱ P≃ᴱQ = ↔→≃ᴱ
(λ (x , y) → x , _≃ᴱ_.to (P≃ᴱQ x) y)
(λ (x , y) → x , _≃ᴱ_.from (P≃ᴱQ x) y)
(λ (x , y) → cong (x ,_) $ _≃ᴱ_.right-inverse-of (P≃ᴱQ x) y)
(λ (x , y) → cong (x ,_) $ _≃ᴱ_.left-inverse-of (P≃ᴱQ x) y)
-- A preservation lemma related to Σ.
--
-- Note that the third argument is not marked as erased. The from
-- argument of [≃]→≃ᴱ (which Agda can infer in this case, at least at
-- the time of writing) is included explicitly to show where the third
-- argument is used in a (potentially) non-erased context.
--
-- See also Σ-cong-≃ᴱ-Erased below.
Σ-cong-≃ᴱ :
{@0 A : Type a} {@0 P : A → Type p}
(f : A → B) (g : B → A) →
(∀ x → f (g x) ≡ x) →
@0 (∀ x → g (f x) ≡ x) →
(∀ x → P x ≃ᴱ Q (f x)) →
Σ A P ≃ᴱ Σ B Q
Σ-cong-≃ᴱ {Q = Q} f g f-g g-f P≃Q =
[≃]→≃ᴱ
{from = λ (x , y) →
g x
, _≃ᴱ_.from (P≃Q (g x)) (subst Q (sym (f-g x)) y)}
([proofs] (Σ-cong (Eq.↔→≃ f g f-g g-f) (≃ᴱ→≃ ⊚ P≃Q)))
-- Another preservation lemma related to Σ.
--
-- See also Σ-cong-contra-≃ᴱ-Erased below.
Σ-cong-contra-≃ᴱ :
{@0 B : Type b} {@0 Q : B → Type q}
(f : B → A) (g : A → B) →
(∀ x → f (g x) ≡ x) →
@0 (∀ x → g (f x) ≡ x) →
(∀ x → P (f x) ≃ᴱ Q x) →
Σ A P ≃ᴱ Σ B Q
Σ-cong-contra-≃ᴱ f g f-g g-f P≃Q =
inverse $ Σ-cong-≃ᴱ f g f-g g-f (inverse ⊚ P≃Q)
-- Yet another preservation lemma related to Σ.
Σ-cong-≃ᴱ′ :
{@0 A : Type a} {@0 B : Type b}
{@0 P : A → Type p} {@0 Q : B → Type q}
(A≃ᴱB : A ≃ᴱ B)
(P→Q : ∀ x → P x → Q (_≃ᴱ_.to A≃ᴱB x))
(Q→P : ∀ x → Q x → P (_≃ᴱ_.from A≃ᴱB x))
(@0 eq : ∀ x → Is-equivalence (P→Q x)) →
@0 (∀ x y →
Q→P x y ≡
_≃_.from Eq.⟨ P→Q (_≃ᴱ_.from A≃ᴱB x) , eq (_≃ᴱ_.from A≃ᴱB x) ⟩
(subst Q (sym (_≃ᴱ_.right-inverse-of A≃ᴱB x)) y)) →
Σ A P ≃ᴱ Σ B Q
Σ-cong-≃ᴱ′ {A = A} {B = B} {P = P} {Q = Q} A≃B P→Q Q→P eq hyp =
[≃]→≃ᴱ ([proofs] ΣAP≃ΣBQ)
where
@0 ΣAP≃ΣBQ : Σ A P ≃ Σ B Q
ΣAP≃ΣBQ =
Eq.with-other-inverse
(Σ-cong (≃ᴱ→≃ A≃B) (λ x → Eq.⟨ P→Q x , eq x ⟩))
(λ (x , y) → _≃ᴱ_.from A≃B x , Q→P x y)
(λ (x , y) → cong (_ ,_) (sym (hyp x y)))
-- Three preservation lemmas related to Π.
--
-- See also Π-cong-≃ᴱ′-≃ᴱ, Π-cong-≃ᴱ′-≃ᴱ′, Π-cong-≃ᴱ-Erased and
-- Π-cong-contra-≃ᴱ-Erased below.
Π-cong-≃ᴱ :
{@0 A : Type a} {B : Type b} {@0 P : A → Type p} {Q : B → Type q} →
@0 Extensionality (a ⊔ b) (p ⊔ q) →
(f : A → B) (g : B → A) →
(∀ x → f (g x) ≡ x) →
@0 (∀ x → g (f x) ≡ x) →
(∀ x → P x ≃ᴱ Q (f x)) →
((x : A) → P x) ≃ᴱ ((x : B) → Q x)
Π-cong-≃ᴱ {Q = Q} ext f g f-g g-f P≃Q =
[≃]→≃ᴱ
{to = λ h x → subst Q (f-g x) (_≃ᴱ_.to (P≃Q (g x)) (h (g x)))}
([proofs] (Π-cong ext {B₂ = Q} (Eq.↔→≃ f g f-g g-f) (≃ᴱ→≃ ⊚ P≃Q)))
Π-cong-contra-≃ᴱ :
{A : Type a} {@0 B : Type b} {P : A → Type p} {@0 Q : B → Type q} →
@0 Extensionality (a ⊔ b) (p ⊔ q) →
(f : B → A) (g : A → B) →
(∀ x → f (g x) ≡ x) →
@0 (∀ x → g (f x) ≡ x) →
(∀ x → P (f x) ≃ᴱ Q x) →
((x : A) → P x) ≃ᴱ ((x : B) → Q x)
Π-cong-contra-≃ᴱ ext f g f-g g-f P≃Q =
inverse $ Π-cong-≃ᴱ ext f g f-g g-f (inverse ⊚ P≃Q)
Π-cong-≃ᴱ′ :
{@0 A : Type a} {@0 B : Type b}
{@0 P : A → Type p} {@0 Q : B → Type q} →
@0 Extensionality (a ⊔ b) (p ⊔ q) →
(A≃ᴱB : A ≃ᴱ B)
(P→Q : ∀ x → P (_≃ᴱ_.from A≃ᴱB x) → Q x)
(Q→P : ∀ x → Q (_≃ᴱ_.to A≃ᴱB x) → P x)
(@0 eq : ∀ x → Is-equivalence (Q→P x)) →
@0 ((f : (x : A) → P x) (y : B) →
let x = _≃ᴱ_.from A≃ᴱB y in
P→Q y (f x) ≡
subst Q (_≃ᴱ_.right-inverse-of A≃ᴱB y)
(_≃_.from Eq.⟨ Q→P x , eq x ⟩ (f x))) →
((x : A) → P x) ≃ᴱ ((x : B) → Q x)
Π-cong-≃ᴱ′ {a = a} {p = p} {A = A} {B = B} {P = P} {Q = Q}
ext A≃B P→Q Q→P eq hyp =
[≃]→≃ᴱ ([proofs] ΠAP≃ΠBQ)
where
@0 ΠAP≃ΠBQ : ((x : A) → P x) ≃ ((x : B) → Q x)
ΠAP≃ΠBQ =
Eq.with-other-function
(Π-cong ext (≃ᴱ→≃ A≃B) (λ x → Eq.inverse Eq.⟨ Q→P x , eq x ⟩))
(λ f x → P→Q x (f (_≃ᴱ_.from A≃B x)))
(λ f → apply-ext (lower-extensionality a p ext) λ x →
sym (hyp f x))
-- A variant of ∀-cong for _≃ᴱ_.
∀-cong-≃ᴱ :
{@0 A : Type a} {@0 P : A → Type p} {@0 Q : A → Type q} →
@0 Extensionality a (p ⊔ q) →
(∀ x → P x ≃ᴱ Q x) →
((x : A) → P x) ≃ᴱ ((x : A) → Q x)
∀-cong-≃ᴱ ext P≃Q = [≃]→≃ᴱ ([proofs] (∀-cong ext (≃ᴱ→≃ ⊚ P≃Q)))
-- Is-equivalenceᴱ f is equivalent (with erased proofs) to
-- Is-equivalenceᴱ g if f and g are pointwise equal (assuming
-- extensionality).
--
-- See also Is-equivalenceᴱ-cong below.
Is-equivalenceᴱ-cong-≃ᴱ :
{@0 A : Type a} {@0 B : Type b} {@0 f g : A → B} →
@0 Extensionality (a ⊔ b) (a ⊔ b) →
@0 (∀ x → f x ≡ g x) →
Is-equivalenceᴱ f ≃ᴱ Is-equivalenceᴱ g
Is-equivalenceᴱ-cong-≃ᴱ ext f≡g =
∃-cong-≃ᴱ λ _ → Erased-cong-≃ᴱ (≃→≃ᴱ $ Proofs-cong ext f≡g)
-- The _≃ᴱ_ operator preserves equivalences with erased proofs
-- (assuming extensionality).
≃ᴱ-cong :
{@0 A : Type a} {@0 B : Type b} {@0 C : Type c} {@0 D : Type d} →
@0 Extensionality (a ⊔ b ⊔ c ⊔ d) (a ⊔ b ⊔ c ⊔ d) →
A ≃ᴱ B → C ≃ᴱ D → (A ≃ᴱ C) ≃ᴱ (B ≃ᴱ D)
≃ᴱ-cong {A = A} {B = B} {C = C} {D = D} ext A≃B C≃D =
[≃]→≃ᴱ ([proofs] lemma)
where
@0 lemma : (A ≃ᴱ C) ≃ (B ≃ᴱ D)
lemma =
A ≃ᴱ C ↝⟨ F.inverse ≃≃≃ᴱ ⟩
A ≃ C ↝⟨ Eq.≃-preserves ext (≃ᴱ→≃ A≃B) (≃ᴱ→≃ C≃D) ⟩
B ≃ D ↝⟨ ≃≃≃ᴱ ⟩□
B ≃ᴱ D □
-- A variant of ↑-cong for _≃ᴱ_.
↑-cong-≃ᴱ :
{@0 B : Type b} {@0 C : Type c} →
B ≃ᴱ C → ↑ a B ≃ᴱ ↑ a C
↑-cong-≃ᴱ B≃ᴱC = ↔→≃ᴱ
(λ (lift x) → lift (_≃ᴱ_.to B≃ᴱC x))
(λ (lift x) → lift (_≃ᴱ_.from B≃ᴱC x))
(λ _ → cong lift (_≃ᴱ_.right-inverse-of B≃ᴱC _))
(λ _ → cong lift (_≃ᴱ_.left-inverse-of B≃ᴱC _))
------------------------------------------------------------------------
-- Variants of some lemmas from Function-universe
-- A variant of drop-⊤-left-Σ.
--
-- See also drop-⊤-left-Σ-≃ᴱ-Erased below.
drop-⊤-left-Σ-≃ᴱ :
{@0 A : Type a} {P : A → Type p}
(A≃⊤ : A ≃ᴱ ⊤) →
(∀ x y → P x ≃ᴱ P y) →
Σ A P ≃ᴱ P (_≃ᴱ_.from A≃⊤ tt)
drop-⊤-left-Σ-≃ᴱ {A = A} {P = P} A≃⊤ P≃P =
Σ A P ≃ᴱ⟨ Σ-cong-≃ᴱ
_
(_≃ᴱ_.from A≃⊤)
refl
(_≃ᴱ_.left-inverse-of A≃⊤)
(λ _ → P≃P _ _) ⟩
Σ ⊤ (λ x → P (_≃ᴱ_.from A≃⊤ x)) ↔⟨ Σ-left-identity ⟩□
P (_≃ᴱ_.from A≃⊤ tt) □
-- A variant of drop-⊤-left-Π.
--
-- See also drop-⊤-left-Π-≃ᴱ-Erased below.
drop-⊤-left-Π-≃ᴱ :
{@0 A : Type a} {P : A → Type p} →
@0 Extensionality a p →
(A≃⊤ : A ≃ᴱ ⊤) →
(∀ x y → P x ≃ᴱ P y) →
((x : A) → P x) ≃ᴱ P (_≃ᴱ_.from A≃⊤ tt)
drop-⊤-left-Π-≃ᴱ {A = A} {P = P} ext A≃⊤ P≃P =
((x : A) → P x) ≃ᴱ⟨ Π-cong-≃ᴱ
ext
_
(_≃ᴱ_.from A≃⊤)
refl
(_≃ᴱ_.left-inverse-of A≃⊤)
(λ _ → P≃P _ _) ⟩
((x : ⊤) → P (_≃ᴱ_.from A≃⊤ x)) ↔⟨ Π-left-identity ⟩□
P (_≃ᴱ_.from A≃⊤ tt) □
------------------------------------------------------------------------
-- Lemmas relating equality between equivalences (with erased proofs)
-- to equality between the forward directions of the equivalences
-- In an erased context two equivalences are equal if the underlying
-- functions are equal (assuming extensionality).
--
-- See also to≡to→≡-Erased below.
@0 to≡to→≡ :
{A : Type a} {B : Type b} {p q : A ≃ᴱ B} →
Extensionality (a ⊔ b) (a ⊔ b) →
_≃ᴱ_.to p ≡ _≃ᴱ_.to q → p ≡ q
to≡to→≡ ext p≡q =
_≃_.to (Eq.≃-≡ (Eq.inverse ≃≃≃ᴱ))
(Eq.lift-equality ext p≡q)
-- A variant of ≃-to-≡↔≡.
@0 to≡to≃≡ :
{A : Type a} {B : Type b} {p q : A ≃ᴱ B} →
Extensionality (a ⊔ b) (a ⊔ b) →
(∀ x → _≃ᴱ_.to p x ≡ _≃ᴱ_.to q x) ≃ (p ≡ q)
to≡to≃≡ {p = p} {q = q} ext =
(∀ x → _≃ᴱ_.to p x ≡ _≃ᴱ_.to q x) F.↔⟨⟩
(∀ x → _≃_.to (_≃_.from ≃≃≃ᴱ p) x ≡ _≃_.to (_≃_.from ≃≃≃ᴱ q) x) F.↔⟨ ≃-to-≡↔≡ ext ⟩
_≃_.from ≃≃≃ᴱ p ≡ _≃_.from ≃≃≃ᴱ q ↝⟨ Eq.≃-≡ (Eq.inverse ≃≃≃ᴱ) ⟩□
p ≡ q □
------------------------------------------------------------------------
-- A variant of _≃ᴱ_
-- Half adjoint equivalences with certain erased proofs.
private
module Dummy where
infix 4 _≃ᴱ′_
record _≃ᴱ′_ (A : Type a) (B : Type b) : Type (a ⊔ b) where
field
to : A → B
from : B → A
@0 to-from : ∀ x → to (from x) ≡ x
from-to : ∀ x → from (to x) ≡ x
@0 to-from-to : ∀ x → cong to (from-to x) ≡ to-from (to x)
open Dummy public using (_≃ᴱ′_) hiding (module _≃ᴱ′_)
-- Note that the type arguments A and B are erased. This is not the
-- case for the record module Dummy._≃ᴱ′_.
module _≃ᴱ′_ {@0 A : Type a} {@0 B : Type b} (A≃B : A ≃ᴱ′ B) where
-- Variants of the projections.
to : A → B
to = let record { to = to } = A≃B in to
from : B → A
from = let record { from = from } = A≃B in from
@0 to-from : ∀ x → to (from x) ≡ x
to-from = Dummy._≃ᴱ′_.to-from A≃B
from-to : ∀ x → from (to x) ≡ x
from-to = let record { from-to = from-to } = A≃B in from-to
@0 to-from-to : ∀ x → cong to (from-to x) ≡ to-from (to x)
to-from-to = Dummy._≃ᴱ′_.to-from-to A≃B
-- Half adjoint equivalences with certain erased proofs are
-- equivalences with erased proofs.
equivalence-with-erased-proofs : A ≃ᴱ B
equivalence-with-erased-proofs =
⟨ to , (from , [ to-from , from-to , to-from-to ]) ⟩₀
-- A coherence property.
@0 from-to-from : ∀ x → cong from (to-from x) ≡ from-to (from x)
from-to-from = _≃ᴱ_.right-left-lemma equivalence-with-erased-proofs
-- Data corresponding to the erased proofs of an equivalence with
-- certain erased proofs.
record Erased-proofs′
{A : Type a} {B : Type b}
(to : A → B) (from : B → A)
(from-to : ∀ x → from (to x) ≡ x) :
Type (a ⊔ b) where
field
to-from : ∀ x → to (from x) ≡ x
to-from-to : ∀ x → cong to (from-to x) ≡ to-from (to x)
-- Extracts "erased proofs" from a regular equivalence.
[proofs′] :
{@0 A : Type a} {@0 B : Type b}
(A≃B : A ≃ B) →
Erased-proofs′ (_≃_.to A≃B) (_≃_.from A≃B) (_≃_.left-inverse-of A≃B)
[proofs′] A≃B .Erased-proofs′.to-from =
let record { is-equivalence = _ , to-from , _ } = A≃B in
to-from
[proofs′] A≃B .Erased-proofs′.to-from-to =
let record { is-equivalence = _ , _ , _ , to-from-to } = A≃B in
to-from-to
-- Converts two functions, one proof and some erased proofs to an
-- equivalence with certain erased proofs.
[≃]→≃ᴱ′ :
{@0 A : Type a} {@0 B : Type b}
{to : A → B} {from : B → A} {from-to : ∀ x → from (to x) ≡ x} →
@0 Erased-proofs′ to from from-to →
A ≃ᴱ′ B
[≃]→≃ᴱ′ {to = to} {from = from} {from-to = from-to} ep = λ where
.Dummy._≃ᴱ′_.to → to
.Dummy._≃ᴱ′_.from → from
.Dummy._≃ᴱ′_.to-from → ep .Erased-proofs′.to-from
.Dummy._≃ᴱ′_.from-to → from-to
.Dummy._≃ᴱ′_.to-from-to → ep .Erased-proofs′.to-from-to
-- A function with a quasi-inverse with one proof and one erased proof
-- can be turned into an equivalence with certain erased proofs.
↔→≃ᴱ′ :
{@0 A : Type a} {@0 B : Type b}
(f : A → B) (g : B → A) →
@0 (∀ x → f (g x) ≡ x) →
(∀ x → g (f x) ≡ x) →
A ≃ᴱ′ B
↔→≃ᴱ′ {A = A} {B = B} to from to-from from-to =
[≃]→≃ᴱ′ ([proofs′] equiv)
where
@0 equiv : A ≃ B
equiv =
Eq.⟨ to
, HA.↔→Is-equivalenceˡ (record
{ surjection = record
{ logical-equivalence = record
{ to = to
; from = from
}
; right-inverse-of = to-from
}
; left-inverse-of = from-to
})
⟩
-- If f is an equivalence with certain erased proofs, then there is an
-- equivalence with certain erased proofs from x ≡ y to f x ≡ f y.
≡≃ᴱ′to≡to :
(A≃ᴱ′B : A ≃ᴱ′ B) →
(x ≡ y) ≃ᴱ′ (_≃ᴱ′_.to A≃ᴱ′B x ≡ _≃ᴱ′_.to A≃ᴱ′B y)
≡≃ᴱ′to≡to {x = x} {y = y} A≃ᴱ′B =
↔→≃ᴱ′
(_↠_.from ≡↠≡)
(_↠_.to ≡↠≡)
(λ eq →
_↠_.from ≡↠≡ (_↠_.to ≡↠≡ eq) ≡⟨⟩
cong to (trans (sym (from-to x)) (trans (cong from eq) (from-to y))) ≡⟨ cong-trans _ _ _ ⟩
trans (cong to (sym (from-to x)))
(cong to (trans (cong from eq) (from-to y))) ≡⟨ cong₂ trans
(cong-sym _ _)
(cong-trans _ _ _) ⟩
trans (sym (cong to (from-to x)))
(trans (cong to (cong from eq)) (cong to (from-to y))) ≡⟨ cong₂ (λ p q → trans (sym p) (trans (cong to (cong from eq)) q))
(to-from-to _)
(to-from-to _) ⟩
trans (sym (to-from (to x)))
(trans (cong to (cong from eq)) (to-from (to y))) ≡⟨⟩
_↠_.to ≡↠≡′ (_↠_.from ≡↠≡′ eq) ≡⟨ _↠_.right-inverse-of ≡↠≡′ eq ⟩∎
eq ∎)
(_↠_.right-inverse-of ≡↠≡)
where
open _≃ᴱ′_ A≃ᴱ′B
≡↠≡ : (to x ≡ to y) ↠ (x ≡ y)
≡↠≡ = Surjection.↠-≡ (record
{ logical-equivalence = record
{ to = from
; from = to
}
; right-inverse-of = from-to
})
@0 ≡↠≡′ : ∀ {x y} → (from x ≡ from y) ↠ (x ≡ y)
≡↠≡′ = Surjection.↠-≡ (record
{ logical-equivalence = record
{ to = to
; from = from
}
; right-inverse-of = to-from
})
-- If f is an equivalence with certain erased proofs, then x ≡ y is
-- equivalent (with erased proofs) to f x ≡ f y.
--
-- See also to≡to≃ᴱ≡-Erased below.
≡≃ᴱto≡to :
(A≃ᴱ′B : A ≃ᴱ′ B) →
(x ≡ y) ≃ᴱ (_≃ᴱ′_.to A≃ᴱ′B x ≡ _≃ᴱ′_.to A≃ᴱ′B y)
≡≃ᴱto≡to = _≃ᴱ′_.equivalence-with-erased-proofs ⊚ ≡≃ᴱ′to≡to
-- Two preservation lemmas related to Π.
Π-cong-≃ᴱ′-≃ᴱ :
{@0 A : Type a} {B : Type b} {@0 P : A → Type p} {Q : B → Type q} →
@0 Extensionality (a ⊔ b) (p ⊔ q) →
(B≃A : B ≃ᴱ′ A) →
(∀ x → P x ≃ᴱ Q (_≃ᴱ′_.from B≃A x)) →
((x : A) → P x) ≃ᴱ ((x : B) → Q x)
Π-cong-≃ᴱ′-≃ᴱ ext B≃A =
Π-cong-≃ᴱ
ext
(_≃ᴱ′_.from B≃A)
(_≃ᴱ′_.to B≃A)
(_≃ᴱ′_.from-to B≃A)
(_≃ᴱ′_.to-from B≃A)
Π-cong-≃ᴱ′-≃ᴱ′ :
{A : Type a} {@0 B : Type b} {P : A → Type p} {@0 Q : B → Type q} →
Extensionality (a ⊔ b) (p ⊔ q) →
(A≃B : A ≃ᴱ′ B) →
(∀ x → P (_≃ᴱ′_.from A≃B x) ≃ᴱ′ Q x) →
((x : A) → P x) ≃ᴱ′ ((x : B) → Q x)
Π-cong-≃ᴱ′-≃ᴱ′
{a = a} {b = b} {p = p} {q = q} {A = A} {B = B} {P = P} {Q = Q}
ext A≃B P≃Q =
↔→≃ᴱ′
(λ f x → _≃ᴱ′_.to (P≃Q x) (f (_≃ᴱ′_.from A≃B x)))
(λ f x →
subst P (_≃ᴱ′_.from-to A≃B x)
(_≃ᴱ′_.from (P≃Q (_≃ᴱ′_.to A≃B x)) (f (_≃ᴱ′_.to A≃B x))))
(λ f → apply-ext (lower-extensionality a p ext) λ x →
_≃ᴱ′_.to (P≃Q x)
(subst P (_≃ᴱ′_.from-to A≃B (_≃ᴱ′_.from A≃B x))
(_≃ᴱ′_.from (P≃Q (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x)))
(f (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x))))) ≡⟨ cong (_≃ᴱ′_.to (P≃Q x) ⊚ flip (subst P) _) $ sym $
_≃ᴱ′_.from-to-from A≃B _ ⟩
_≃ᴱ′_.to (P≃Q x)
(subst P (cong (_≃ᴱ′_.from A≃B) (_≃ᴱ′_.to-from A≃B x))
(_≃ᴱ′_.from (P≃Q (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x)))
(f (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x))))) ≡⟨ elim¹
(λ {y} eq →
_≃ᴱ′_.to (P≃Q y)
(subst P (cong (_≃ᴱ′_.from A≃B) eq)
(_≃ᴱ′_.from (P≃Q (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x)))
(f (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x))))) ≡
f y)
(
_≃ᴱ′_.to (P≃Q (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x)))
(subst P (cong (_≃ᴱ′_.from A≃B) (refl _))
(_≃ᴱ′_.from (P≃Q (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x)))
(f (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x))))) ≡⟨ cong (_≃ᴱ′_.to (P≃Q (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x)))) $
trans (cong (flip (subst P) _) $ cong-refl _) $
subst-refl _ _ ⟩
_≃ᴱ′_.to (P≃Q (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x)))
(_≃ᴱ′_.from (P≃Q (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x)))
(f (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x)))) ≡⟨ _≃ᴱ′_.to-from (P≃Q _) _ ⟩∎
f (_≃ᴱ′_.to A≃B (_≃ᴱ′_.from A≃B x)) ∎)
_ ⟩∎
f x ∎)
(λ f → apply-ext (lower-extensionality b q ext) λ x →
subst P (_≃ᴱ′_.from-to A≃B x)
(_≃ᴱ′_.from (P≃Q (_≃ᴱ′_.to A≃B x))
(_≃ᴱ′_.to (P≃Q (_≃ᴱ′_.to A≃B x))
(f (_≃ᴱ′_.from A≃B (_≃ᴱ′_.to A≃B x))))) ≡⟨ elim¹
(λ {y} eq →
subst P eq
(_≃ᴱ′_.from (P≃Q (_≃ᴱ′_.to A≃B x))
(_≃ᴱ′_.to (P≃Q (_≃ᴱ′_.to A≃B x))
(f (_≃ᴱ′_.from A≃B (_≃ᴱ′_.to A≃B x))))) ≡
f y)
(
subst P (refl _)
(_≃ᴱ′_.from (P≃Q (_≃ᴱ′_.to A≃B x))
(_≃ᴱ′_.to (P≃Q (_≃ᴱ′_.to A≃B x))
(f (_≃ᴱ′_.from A≃B (_≃ᴱ′_.to A≃B x))))) ≡⟨ subst-refl _ _ ⟩
_≃ᴱ′_.from (P≃Q (_≃ᴱ′_.to A≃B x))
(_≃ᴱ′_.to (P≃Q (_≃ᴱ′_.to A≃B x))
(f (_≃ᴱ′_.from A≃B (_≃ᴱ′_.to A≃B x)))) ≡⟨ _≃ᴱ′_.from-to (P≃Q _) _ ⟩∎
f (_≃ᴱ′_.from A≃B (_≃ᴱ′_.to A≃B x)) ∎)
_ ⟩∎
f x ∎)
------------------------------------------------------------------------
-- Some results related to Contractibleᴱ
-- Two types that are contractible (with erased proofs) are equivalent
-- (with erased proofs).
Contractibleᴱ→≃ᴱ :
{@0 A : Type a} {@0 B : Type b} →
Contractibleᴱ A → Contractibleᴱ B → A ≃ᴱ B
Contractibleᴱ→≃ᴱ (a , [ irrA ]) (b , [ irrB ]) = ↔→≃ᴱ
(const b)
(const a)
irrB
irrA
-- There is a logical equivalence between Contractibleᴱ A and A ≃ᴱ ⊤.
Contractibleᴱ⇔≃ᴱ⊤ :
{@0 A : Type a} →
Contractibleᴱ A ⇔ A ≃ᴱ ⊤
Contractibleᴱ⇔≃ᴱ⊤ = record
{ to = flip Contractibleᴱ→≃ᴱ Contractibleᴱ-⊤
; from = λ A≃⊤ →
ECP.Contractibleᴱ-respects-surjection
(_≃ᴱ_.from A≃⊤)
(λ a → tt
, (_≃ᴱ_.from A≃⊤ tt ≡⟨⟩
_≃ᴱ_.from A≃⊤ (_≃ᴱ_.to A≃⊤ a) ≡⟨ _≃ᴱ_.left-inverse-of A≃⊤ _ ⟩∎
a ∎))
Contractibleᴱ-⊤
}
where
Contractibleᴱ-⊤ = ECP.Contractible→Contractibleᴱ ⊤-contractible
-- There is an equivalence with erased proofs between Contractibleᴱ A
-- and A ≃ᴱ ⊤ (assuming extensionality).
Contractibleᴱ≃ᴱ≃ᴱ⊤ :
{@0 A : Type a} →
@0 Extensionality a a →
Contractibleᴱ A ≃ᴱ (A ≃ᴱ ⊤)
Contractibleᴱ≃ᴱ≃ᴱ⊤ ext =
let record { to = to; from = from } = Contractibleᴱ⇔≃ᴱ⊤ in
↔→≃ᴱ
to
from
(λ _ → to≡to→≡ ext (refl _))
(λ _ → ECP.Contractibleᴱ-propositional ext _ _)
-- If an inhabited type comes with an erased proof of
-- propositionality, then it is equivalent (with erased proofs) to the
-- unit type.
inhabited→Is-proposition→≃ᴱ⊤ :
{@0 A : Type a} →
A → @0 Is-proposition A → A ≃ᴱ ⊤
inhabited→Is-proposition→≃ᴱ⊤ x prop =
let record { to = to } = Contractibleᴱ⇔≃ᴱ⊤ in
to (ECP.inhabited→Is-proposition→Contractibleᴱ x prop)
-- Contractibleᴱ commutes with _×_ (up to _≃ᴱ_, assuming
-- extensionality).
Contractibleᴱ-commutes-with-× :
{@0 A : Type a} {@0 B : Type b} →
@0 Extensionality (a ⊔ b) (a ⊔ b) →
Contractibleᴱ (A × B) ≃ᴱ (Contractibleᴱ A × Contractibleᴱ B)
Contractibleᴱ-commutes-with-× {A = A} {B = B} ext =
[≃]→≃ᴱ ([proofs] lemma)
where
@0 lemma : _
lemma =
Contractibleᴱ (A × B) ↝⟨ F.inverse ECP.Contractible≃Contractibleᴱ ⟩
Contractible (A × B) ↝⟨ Contractible-commutes-with-× ext ⟩
(Contractible A × Contractible B) ↝⟨ ECP.Contractible≃Contractibleᴱ ×-cong
ECP.Contractible≃Contractibleᴱ ⟩□
(Contractibleᴱ A × Contractibleᴱ B) □
------------------------------------------------------------------------
-- Groupoid laws and related properties
module Groupoid where
-- In an erased context the groupoid laws hold for id and _∘_.
--
-- TODO: Is it possible to prove the first three results in a
-- non-erased context?
@0 left-identity :
{A : Type a} {B : Type b} →
Extensionality (a ⊔ b) (a ⊔ b) →
(f : A ≃ᴱ B) → id ∘ f ≡ f
left-identity ext _ = to≡to→≡ ext (refl _)
@0 right-identity :
{A : Type a} {B : Type b} →
Extensionality (a ⊔ b) (a ⊔ b) →
(f : A ≃ᴱ B) → f ∘ id ≡ f
right-identity ext _ = to≡to→≡ ext (refl _)
@0 associativity :
{A : Type a} {D : Type d} →
Extensionality (a ⊔ d) (a ⊔ d) →
(f : C ≃ᴱ D) (g : B ≃ᴱ C) (h : A ≃ᴱ B) →
f ∘ (g ∘ h) ≡ (f ∘ g) ∘ h
associativity ext _ _ _ = to≡to→≡ ext (refl _)
@0 left-inverse :
{A : Type a} →
Extensionality a a →
(f : A ≃ᴱ B) → inverse f ∘ f ≡ id
left-inverse ext f =
to≡to→≡ ext $ apply-ext ext $
_≃_.left-inverse-of (≃ᴱ→≃ f)
@0 right-inverse :
{B : Type b} →
Extensionality b b →
(f : A ≃ᴱ B) → f ∘ inverse f ≡ id
right-inverse ext f =
to≡to→≡ ext $ apply-ext ext $
_≃_.right-inverse-of (≃ᴱ→≃ f)
-- Inverse is a logical equivalence.
inverse-logical-equivalence :
{@0 A : Type a} {@0 B : Type b} →
A ≃ᴱ B ⇔ B ≃ᴱ A
inverse-logical-equivalence = record
{ to = inverse
; from = inverse
}
-- Inverse is an equivalence with erased proofs (assuming
-- extensionality).
inverse-equivalence :
{@0 A : Type a} {@0 B : Type b} →
@0 Extensionality (a ⊔ b) (a ⊔ b) →
(A ≃ᴱ B) ≃ᴱ (B ≃ᴱ A)
inverse-equivalence ext = ↔→≃ᴱ
inverse
inverse
(λ _ → to≡to→≡ ext (refl _))
(λ _ → to≡to→≡ ext (refl _))
------------------------------------------------------------------------
-- Some results that depend on univalence
-- A variant of ≃⇒≡.
@0 ≃ᴱ→≡ :
{A B : Type a} →
Univalence a →
A ≃ᴱ B → A ≡ B
≃ᴱ→≡ univ = ≃⇒≡ univ ⊚ ≃ᴱ→≃
-- A variant of ≡≃≃.
@0 ≡≃≃ᴱ :
{A B : Type a} →
Univalence a →
(A ≡ B) ≃ (A ≃ᴱ B)
≡≃≃ᴱ {A = A} {B = B} univ =
Eq.with-other-function
(A ≡ B ↝⟨ ≡≃≃ univ ⟩
A ≃ B ↝⟨ ≃≃≃ᴱ ⟩□
A ≃ᴱ B □)
(≡⇒↝ _)
(elim₁ (λ eq → ≃→≃ᴱ (≡⇒≃ eq) ≡ ≡⇒↝ _ eq)
(≃→≃ᴱ (≡⇒≃ (refl _)) ≡⟨ cong ≃→≃ᴱ ≡⇒≃-refl ⟩
≃→≃ᴱ Eq.id ≡⟨⟩
id ≡⟨ sym ≡⇒↝-refl ⟩∎
≡⇒↝ _ (refl _) ∎))
@0 _ :
{univ : Univalence a} →
_≃_.from (≡≃≃ᴱ {A = A} {B = B} univ) ≡ ≃ᴱ→≡ univ
_ = refl _
-- A variant of ≃⇒≡-id.
@0 ≃ᴱ→≡-id :
{A : Type a} →
Extensionality a a →
(univ : Univalence a) →
≃ᴱ→≡ univ id ≡ refl A
≃ᴱ→≡-id ext univ =
≃⇒≡ univ (≃ᴱ→≃ id) ≡⟨ cong (≃⇒≡ univ) $ Eq.lift-equality ext (refl _) ⟩
≃⇒≡ univ Eq.id ≡⟨ ≃⇒≡-id univ ⟩∎
refl _ ∎
-- A variant of ≃⇒≡-inverse.
@0 ≃ᴱ→≡-inverse :
Extensionality a a →
(univ : Univalence a)
(A≃B : A ≃ᴱ B) →
≃ᴱ→≡ univ (inverse A≃B) ≡ sym (≃ᴱ→≡ univ A≃B)
≃ᴱ→≡-inverse ext univ A≃B =
≃⇒≡ univ (≃ᴱ→≃ (inverse A≃B)) ≡⟨ cong (≃⇒≡ univ) $ Eq.lift-equality ext (refl _) ⟩
≃⇒≡ univ (Eq.inverse (≃ᴱ→≃ A≃B)) ≡⟨ ≃⇒≡-inverse univ ext _ ⟩∎
sym (≃⇒≡ univ (≃ᴱ→≃ A≃B)) ∎
-- A variant of ≃⇒≡-∘.
@0 ≃ᴱ→≡-∘ :
Extensionality a a →
(univ : Univalence a)
(A≃B : A ≃ᴱ B) (B≃C : B ≃ᴱ C) →
≃ᴱ→≡ univ (B≃C ∘ A≃B) ≡ trans (≃ᴱ→≡ univ A≃B) (≃ᴱ→≡ univ B≃C)
≃ᴱ→≡-∘ ext univ A≃B B≃C =
≃⇒≡ univ (≃ᴱ→≃ (B≃C ∘ A≃B)) ≡⟨ cong (≃⇒≡ univ) $ Eq.lift-equality ext (refl _) ⟩
≃⇒≡ univ (≃ᴱ→≃ B≃C Eq.∘ ≃ᴱ→≃ A≃B) ≡⟨ ≃⇒≡-∘ univ ext _ _ ⟩
trans (≃⇒≡ univ (≃ᴱ→≃ A≃B)) (≃⇒≡ univ (≃ᴱ→≃ B≃C)) ∎
-- Singletons expressed using equivalences with erased proofs instead
-- of equalities are equivalent (with erased proofs) to the unit type
-- (assuming extensionality and univalence).
singleton-with-≃ᴱ-≃ᴱ-⊤ :
∀ a {B : Type b} →
@0 Extensionality (a ⊔ b) (a ⊔ b) →
@0 Univalence (a ⊔ b) →
(∃ λ (A : Type (a ⊔ b)) → A ≃ᴱ B) ≃ᴱ ⊤
singleton-with-≃ᴱ-≃ᴱ-⊤ {b = b} a {B = B} ext univ =
[≃]→≃ᴱ ([proofs] lemma)
where
@0 lemma : (∃ λ (A : Type (a ⊔ b)) → A ≃ᴱ B) ≃ ⊤
lemma =
(∃ λ (A : Type (a ⊔ b)) → A ≃ᴱ B) ↝⟨ (∃-cong λ _ → F.inverse ≃≃≃ᴱ) ⟩
(∃ λ (A : Type (a ⊔ b)) → A ≃ B) F.↔⟨ singleton-with-≃-↔-⊤ {a = a} ext univ ⟩□
⊤ □
other-singleton-with-≃ᴱ-≃ᴱ-⊤ :
∀ b {A : Type a} →
@0 Extensionality (a ⊔ b) (a ⊔ b) →
@0 Univalence (a ⊔ b) →
(∃ λ (B : Type (a ⊔ b)) → A ≃ᴱ B) ≃ᴱ ⊤
other-singleton-with-≃ᴱ-≃ᴱ-⊤ b {A = A} ext univ =
(∃ λ B → A ≃ᴱ B) ≃ᴱ⟨ (∃-cong λ _ → inverse-equivalence ext) ⟩
(∃ λ B → B ≃ᴱ A) ≃ᴱ⟨ singleton-with-≃ᴱ-≃ᴱ-⊤ b ext univ ⟩□
⊤ □
-- Variants of the two lemmas above.
singleton-with-Π-≃ᴱ-≃ᴱ-⊤ :
{A : Type a} {Q : A → Type q} →
@0 Extensionality a (lsuc q) →
@0 Univalence q →
(∃ λ (P : A → Type q) → ∀ x → P x ≃ᴱ Q x) ≃ᴱ ⊤
singleton-with-Π-≃ᴱ-≃ᴱ-⊤ {a = a} {q = q} {A = A} {Q = Q} ext univ =
[≃]→≃ᴱ ([proofs] lemma)
where
@0 ext′ : Extensionality a q
ext′ = lower-extensionality lzero _ ext
@0 lemma : (∃ λ (P : A → Type q) → ∀ x → P x ≃ᴱ Q x) ≃ ⊤
lemma =
(∃ λ (P : A → Type q) → ∀ x → P x ≃ᴱ Q x) ↝⟨ (∃-cong λ _ → ∀-cong ext′ λ _ → F.inverse ≃≃≃ᴱ) ⟩
(∃ λ (P : A → Type q) → ∀ x → P x ≃ Q x) F.↔⟨ singleton-with-Π-≃-≃-⊤ ext univ ⟩□
⊤ □
other-singleton-with-Π-≃ᴱ-≃ᴱ-⊤ :
{A : Type a} {P : A → Type p} →
@0 Extensionality (a ⊔ p) (lsuc p) →
@0 Univalence p →
(∃ λ (Q : A → Type p) → ∀ x → P x ≃ᴱ Q x) ≃ᴱ ⊤
other-singleton-with-Π-≃ᴱ-≃ᴱ-⊤ {a = a} {p = p} {A = A} {P = P}
ext univ =
(∃ λ (Q : A → Type p) → ∀ x → P x ≃ᴱ Q x) ≃ᴱ⟨ (∃-cong λ _ → ∀-cong-≃ᴱ ext₁ λ _ → inverse-equivalence ext₂) ⟩
(∃ λ (Q : A → Type p) → ∀ x → Q x ≃ᴱ P x) ≃ᴱ⟨ singleton-with-Π-≃ᴱ-≃ᴱ-⊤ ext₃ univ ⟩□
⊤ □
where
@0 ext₁ : Extensionality a p
ext₁ = lower-extensionality p _ ext
@0 ext₂ : Extensionality p p
ext₂ = lower-extensionality a _ ext
@0 ext₃ : Extensionality a (lsuc p)
ext₃ = lower-extensionality p lzero ext
-- ∃ Contractibleᴱ is equivalent (with erased proofs) to the unit type
-- (assuming extensionality and univalence).
∃Contractibleᴱ≃ᴱ⊤ :
@0 Extensionality a a →
@0 Univalence a →
(∃ λ (A : Type a) → Contractibleᴱ A) ≃ᴱ ⊤
∃Contractibleᴱ≃ᴱ⊤ ext univ =
(∃ λ A → Contractibleᴱ A) ≃ᴱ⟨ (∃-cong λ _ → Contractibleᴱ≃ᴱ≃ᴱ⊤ ext) ⟩
(∃ λ A → A ≃ᴱ ⊤) ≃ᴱ⟨ singleton-with-≃ᴱ-≃ᴱ-⊤ _ ext univ ⟩□
⊤ □
------------------------------------------------------------------------
-- Some simplification lemmas
-- Two simplification lemmas for id.
right-inverse-of-id :
_≃ᴱ_.right-inverse-of id x ≡ refl x
right-inverse-of-id = refl _
@0 left-inverse-of-id :
_≃ᴱ_.left-inverse-of id x ≡ refl x
left-inverse-of-id {x = x} =
left-inverse-of x ≡⟨⟩
left-inverse-of (P.id x) ≡⟨ sym $ right-left-lemma x ⟩
cong P.id (right-inverse-of x) ≡⟨ sym $ cong-id _ ⟩
right-inverse-of x ≡⟨ right-inverse-of-id ⟩∎
refl x ∎
where
open _≃ᴱ_ id
-- Two simplification lemmas for inverse.
@0 right-inverse-of∘inverse :
(A≃B : A ≃ᴱ B) →
_≃ᴱ_.right-inverse-of (inverse A≃B) x ≡
_≃ᴱ_.left-inverse-of A≃B x
right-inverse-of∘inverse _ = refl _
@0 left-inverse-of∘inverse :
(A≃B : A ≃ᴱ B) →
_≃ᴱ_.left-inverse-of (inverse A≃B) x ≡
_≃ᴱ_.right-inverse-of A≃B x
left-inverse-of∘inverse {A = A} {B = B} {x = x} A≃B =
subst (λ x → _≃ᴱ_.left-inverse-of (inverse A≃B) x ≡
right-inverse-of x)
(right-inverse-of x)
(_≃ᴱ_.left-inverse-of (inverse A≃B) (to (from x)) ≡⟨ sym $ _≃ᴱ_.right-left-lemma (inverse A≃B) (from x) ⟩
cong to (_≃ᴱ_.right-inverse-of (inverse A≃B) (from x)) ≡⟨ cong (cong to) $ right-inverse-of∘inverse A≃B ⟩
cong to (left-inverse-of (from x)) ≡⟨ left-right-lemma (from x) ⟩∎
right-inverse-of (to (from x)) ∎)
where
open _≃ᴱ_ A≃B
-- Two simplification lemmas for subst.
to-subst :
{eq : x ≡ y} {f : P x ≃ᴱ Q x} →
_≃ᴱ_.to (subst (λ x → P x ≃ᴱ Q x) eq f) ≡
subst (λ x → P x → Q x) eq (_≃ᴱ_.to f)
to-subst {P = P} {Q = Q} {eq = eq} {f = f} = elim¹
(λ eq →
_≃ᴱ_.to (subst (λ x → P x ≃ᴱ Q x) eq f) ≡
subst (λ x → P x → Q x) eq (_≃ᴱ_.to f))
(_≃ᴱ_.to (subst (λ x → P x ≃ᴱ Q x) (refl _) f) ≡⟨ cong _≃ᴱ_.to $ subst-refl _ _ ⟩
_≃ᴱ_.to f ≡⟨ sym $ subst-refl _ _ ⟩∎
subst (λ x → P x → Q x) (refl _) (_≃ᴱ_.to f) ∎)
eq
from-subst :
{eq : x ≡ y} {f : P x ≃ᴱ Q x} →
_≃ᴱ_.from (subst (λ x → P x ≃ᴱ Q x) eq f) ≡
subst (λ x → Q x → P x) eq (_≃ᴱ_.from f)
from-subst {P = P} {Q = Q} {eq = eq} {f = f} = elim¹
(λ eq →
_≃ᴱ_.from (subst (λ x → P x ≃ᴱ Q x) eq f) ≡
subst (λ x → Q x → P x) eq (_≃ᴱ_.from f))
(_≃ᴱ_.from (subst (λ x → P x ≃ᴱ Q x) (refl _) f) ≡⟨ cong _≃ᴱ_.from $ subst-refl _ _ ⟩
_≃ᴱ_.from f ≡⟨ sym $ subst-refl _ _ ⟩∎
subst (λ x → Q x → P x) (refl _) (_≃ᴱ_.from f) ∎)
eq
------------------------------------------------------------------------
-- The two-out-of-three properties
-- If f and g are equivalences with erased proofs, then g ⊚ f is also
-- an equivalence with erased proofs.
12→3 :
{@0 A : Type a} {@0 B : Type b} {@0 C : Type c}
{@0 f : A → B} {@0 g : B → C} →
Is-equivalenceᴱ f → Is-equivalenceᴱ g → Is-equivalenceᴱ (g ⊚ f)
12→3 p q =
proj₁₀ p ⊚ proj₁₀ q
, [ _≃ᴱ_.is-equivalence (⟨ _ , q ⟩₀ ∘ ⟨ _ , p ⟩₀) .proj₂ .erased ]
-- If g and g ⊚ f are equivalences with erased proofs, then f is
-- also an equivalence with erased proofs.
23→1 :
{@0 A : Type a} {@0 B : Type b} {@0 C : Type c}
{@0 f : A → B} {g : B → C} →
@0 Is-equivalenceᴱ g → Is-equivalenceᴱ (g ⊚ f) → Is-equivalenceᴱ f
23→1 {f = f} {g = g} q r =
let record { to = to } =
Is-equivalenceᴱ-cong-⇔ λ x →
_≃ᴱ_.from ⟨ g , q ⟩ (g (f x)) ≡⟨ _≃ᴱ_.left-inverse-of ⟨ g , q ⟩ (f x) ⟩∎
f x ∎
in
to ( proj₁₀ r ⊚ g
, [ _≃ᴱ_.is-equivalence (inverse ⟨ _ , q ⟩₀ ∘ ⟨ _ , r ⟩₀)
.proj₂ .erased
]
)
-- If g ⊚ f and f are equivalences with erased proofs, then g is
-- also an equivalence with erased proofs.
31→2 :
{@0 A : Type a} {@0 B : Type b} {@0 C : Type c}
{f : A → B} {@0 g : B → C} →
Is-equivalenceᴱ (g ⊚ f) → @0 Is-equivalenceᴱ f → Is-equivalenceᴱ g
31→2 {f = f} {g = g} r p =
let record { to = to } =
Is-equivalenceᴱ-cong-⇔ λ x →
g (f (_≃ᴱ_.from ⟨ f , p ⟩ x)) ≡⟨ cong g (_≃ᴱ_.right-inverse-of ⟨ f , p ⟩ x) ⟩∎
g x ∎
in
to ( f ⊚ proj₁₀ r
, [ _≃ᴱ_.is-equivalence (⟨ _ , r ⟩₀ ∘ inverse ⟨ _ , p ⟩₀)
.proj₂ .erased
]
)
-- Some consequences of the two-out-of-three properties.
Is-equivalenceᴱ⇔Is-equivalenceᴱ-∘ˡ :
{@0 A : Type a} {@0 B : Type b} {@0 C : Type c}
{f : B → C} {@0 g : A → B} →
Is-equivalenceᴱ f →
Is-equivalenceᴱ g ⇔ Is-equivalenceᴱ (f ⊚ g)
Is-equivalenceᴱ⇔Is-equivalenceᴱ-∘ˡ f-eq = record
{ to = flip 12→3 f-eq
; from = 23→1 f-eq
}
Is-equivalenceᴱ⇔Is-equivalenceᴱ-∘ʳ :
{@0 A : Type a} {@0 B : Type b} {@0 C : Type c}
{@0 f : B → C} {g : A → B} →
Is-equivalenceᴱ g →
Is-equivalenceᴱ f ⇔ Is-equivalenceᴱ (f ⊚ g)
Is-equivalenceᴱ⇔Is-equivalenceᴱ-∘ʳ g-eq = record
{ to = 12→3 g-eq
; from = λ f∘g-eq → 31→2 f∘g-eq g-eq
}
Is-equivalenceᴱ≃ᴱIs-equivalenceᴱ-∘ˡ :
{@0 A : Type a} {@0 B : Type b} {@0 C : Type c}
{f : B → C} {@0 g : A → B} →
@0 Extensionality (a ⊔ b ⊔ c) (a ⊔ b ⊔ c) →
Is-equivalenceᴱ f →
Is-equivalenceᴱ g ≃ᴱ Is-equivalenceᴱ (f ⊚ g)
Is-equivalenceᴱ≃ᴱIs-equivalenceᴱ-∘ˡ {b = b} {c = c} ext f-eq = ⇔→≃ᴱ
(Is-equivalenceᴱ-propositional (lower-extensionality c c ext) _)
(Is-equivalenceᴱ-propositional (lower-extensionality b b ext) _)
(flip 12→3 f-eq)
(23→1 f-eq)
Is-equivalenceᴱ≃ᴱIs-equivalenceᴱ-∘ʳ :
{@0 A : Type a} {@0 B : Type b} {@0 C : Type c}
{@0 f : B → C} {g : A → B} →
@0 Extensionality (a ⊔ b ⊔ c) (a ⊔ b ⊔ c) →
Is-equivalenceᴱ g →
Is-equivalenceᴱ f ≃ᴱ Is-equivalenceᴱ (f ⊚ g)
Is-equivalenceᴱ≃ᴱIs-equivalenceᴱ-∘ʳ {a = a} {b = b} ext g-eq = ⇔→≃ᴱ
(Is-equivalenceᴱ-propositional (lower-extensionality a a ext) _)
(Is-equivalenceᴱ-propositional (lower-extensionality b b ext) _)
(12→3 g-eq)
(λ f∘g-eq → 31→2 f∘g-eq g-eq)
------------------------------------------------------------------------
-- Results that depend on an axiomatisation of []-cong (for a single
-- universe level)
module []-cong₁ (ax : []-cong-axiomatisation ℓ) where
open Erased.[]-cong₁ ax
----------------------------------------------------------------------
-- More preservation lemmas
-- Equivalences with erased proofs are in some cases preserved by Σ
-- (assuming extensionality). Note the type of Q.
Σ-cong-≃ᴱ-Erased :
{@0 A : Type a} {@0 B : Type ℓ}
{@0 P : A → Type p} {Q : @0 B → Type q}
(A≃B : A ≃ᴱ B) →
(∀ x → P x ≃ᴱ Q (_≃ᴱ_.to A≃B x)) →
Σ A P ≃ᴱ Σ B (λ x → Q x)
Σ-cong-≃ᴱ-Erased {A = A} {B = B} {P = P} {Q = Q} A≃B P≃Q =
[≃]→≃ᴱ ([proofs] ΣAP≃ΣBQ)
where
@0 ΣAP≃ΣBQ : Σ A P ≃ Σ B (λ x → Q x)
ΣAP≃ΣBQ =
Eq.with-other-inverse
(Σ-cong (≃ᴱ→≃ A≃B) (λ x → ≃ᴱ→≃ (P≃Q x)))
(λ (x , y) →
_≃ᴱ_.from A≃B x
, _≃ᴱ_.from (P≃Q (_≃ᴱ_.from A≃B x))
(substᴱ Q (sym (_≃ᴱ_.right-inverse-of A≃B x)) y))
(λ (x , y) →
cong (λ y → _ , _≃ᴱ_.from (P≃Q (_≃ᴱ_.from A≃B x)) y) (
subst (λ x → Q x) (sym (_≃ᴱ_.right-inverse-of A≃B x)) y ≡⟨ sym substᴱ≡subst ⟩∎
substᴱ Q (sym (_≃ᴱ_.right-inverse-of A≃B x)) y ∎))
-- A variant of Σ-cong-≃ᴱ-Erased.
Σ-cong-contra-≃ᴱ-Erased :
{@0 A : Type ℓ} {@0 B : Type b}
{P : @0 A → Type p} {@0 Q : B → Type q}
(B≃A : B ≃ᴱ A) →
(∀ x → P (_≃ᴱ_.to B≃A x) ≃ᴱ Q x) →
Σ A (λ x → P x) ≃ᴱ Σ B Q
Σ-cong-contra-≃ᴱ-Erased {P = P} {Q = Q} B≃A P≃Q = ↔→≃ᴱ
(λ (x , y) →
_≃ᴱ_.from B≃A x
, _≃ᴱ_.to (P≃Q (_≃ᴱ_.from B≃A x))
(substᴱ P (sym (_≃ᴱ_.right-inverse-of B≃A x)) y))
(λ (x , y) → _≃ᴱ_.to B≃A x , _≃ᴱ_.from (P≃Q x) y)
(λ (x , y) → Σ-≡,≡→≡
(_≃ᴱ_.left-inverse-of B≃A x)
(subst Q (_≃ᴱ_.left-inverse-of B≃A x)
(_≃ᴱ_.to (P≃Q _)
(substᴱ P (sym (_≃ᴱ_.right-inverse-of B≃A _))
(_≃ᴱ_.from (P≃Q x) y))) ≡⟨ cong (λ eq → subst Q (_≃ᴱ_.left-inverse-of B≃A x) (_≃ᴱ_.to (P≃Q _) eq))
substᴱ≡subst ⟩
subst Q (_≃ᴱ_.left-inverse-of B≃A x)
(_≃ᴱ_.to (P≃Q _)
(subst (λ x → P x) (sym (_≃ᴱ_.right-inverse-of B≃A _))
(_≃ᴱ_.from (P≃Q x) y))) ≡⟨ cong (λ eq → subst Q (_≃ᴱ_.left-inverse-of B≃A x)
(_≃ᴱ_.to (P≃Q _)
(subst (λ x → P x) (sym eq) _))) $ sym $
_≃ᴱ_.left-right-lemma B≃A _ ⟩
subst Q (_≃ᴱ_.left-inverse-of B≃A x)
(_≃ᴱ_.to (P≃Q (_≃ᴱ_.from B≃A (_≃ᴱ_.to B≃A x)))
(subst (λ x → P x)
(sym (cong (_≃ᴱ_.to B≃A) (_≃ᴱ_.left-inverse-of B≃A _)))
(_≃ᴱ_.from (P≃Q x) y))) ≡⟨ elim₁
(λ eq → subst Q eq
(_≃ᴱ_.to (P≃Q _)
(subst (λ x → P x)
(sym (cong (_≃ᴱ_.to B≃A) eq))
(_≃ᴱ_.from (P≃Q x) y))) ≡
y)
(
subst Q (refl _)
(_≃ᴱ_.to (P≃Q x)
(subst (λ x → P x)
(sym (cong (_≃ᴱ_.to B≃A) (refl _)))
(_≃ᴱ_.from (P≃Q x) y))) ≡⟨ subst-refl _ _ ⟩
_≃ᴱ_.to (P≃Q x)
(subst (λ x → P x)
(sym (cong (_≃ᴱ_.to B≃A) (refl _)))
(_≃ᴱ_.from (P≃Q x) y)) ≡⟨ cong (λ eq → _≃ᴱ_.to (P≃Q _) (subst (λ x → P x) (sym eq) _)) $
cong-refl _ ⟩
_≃ᴱ_.to (P≃Q x)
(subst (λ x → P x)
(sym (refl _)) (_≃ᴱ_.from (P≃Q x) y)) ≡⟨ cong (λ eq → _≃ᴱ_.to (P≃Q _) (subst (λ x → P x) eq _))
sym-refl ⟩
_≃ᴱ_.to (P≃Q x)
(subst (λ x → P x)
(refl _) (_≃ᴱ_.from (P≃Q x) y)) ≡⟨ cong (λ eq → _≃ᴱ_.to (P≃Q _) eq) $
subst-refl _ _ ⟩
_≃ᴱ_.to (P≃Q x) (_≃ᴱ_.from (P≃Q x) y) ≡⟨ _≃ᴱ_.right-inverse-of (P≃Q x) _ ⟩∎
y ∎)
(_≃ᴱ_.left-inverse-of B≃A x) ⟩∎
y ∎))
(λ (x , y) → Σ-≡,≡→≡
(_≃ᴱ_.right-inverse-of B≃A x)
(subst (λ x → P x) (_≃ᴱ_.right-inverse-of B≃A x)
(_≃ᴱ_.from (P≃Q _)
(_≃ᴱ_.to (P≃Q _)
(substᴱ P (sym (_≃ᴱ_.right-inverse-of B≃A _)) y))) ≡⟨ cong (subst (λ x → P x) (_≃ᴱ_.right-inverse-of B≃A x)) $
_≃ᴱ_.left-inverse-of (P≃Q _) _ ⟩
subst (λ x → P x) (_≃ᴱ_.right-inverse-of B≃A x)
(substᴱ P (sym (_≃ᴱ_.right-inverse-of B≃A _)) y) ≡⟨ cong (subst (λ x → P x) (_≃ᴱ_.right-inverse-of B≃A x))
substᴱ≡subst ⟩
subst (λ x → P x) (_≃ᴱ_.right-inverse-of B≃A x)
(subst (λ x → P x) (sym (_≃ᴱ_.right-inverse-of B≃A _)) y) ≡⟨ subst-subst-sym _ _ _ ⟩∎
y ∎))
-- Equivalences with erased proofs are in some cases preserved by Π
-- (assuming extensionality). Note the type of Q.
Π-cong-≃ᴱ-Erased :
{@0 A : Type a} {@0 B : Type ℓ}
{@0 P : A → Type p} {Q : @0 B → Type q} →
@0 Extensionality (a ⊔ ℓ) (p ⊔ q) →
(A≃B : A ≃ᴱ B) →
(∀ x → P x ≃ᴱ Q (_≃ᴱ_.to A≃B x)) →
((x : A) → P x) ≃ᴱ ((x : B) → Q x)
Π-cong-≃ᴱ-Erased
{a = a} {p = p} {A = A} {B = B} {P = P} {Q = Q} ext A≃B P≃Q =
[≃]→≃ᴱ ([proofs] ΠAP≃ΠBQ)
where
@0 ΠAP≃ΠBQ : ((x : A) → P x) ≃ ((x : B) → Q x)
ΠAP≃ΠBQ =
Eq.with-other-function
(Π-cong ext (≃ᴱ→≃ A≃B) (λ x → ≃ᴱ→≃ (P≃Q x)))
(λ f x → substᴱ Q
(_≃ᴱ_.right-inverse-of A≃B x)
(_≃ᴱ_.to (P≃Q (_≃ᴱ_.from A≃B x))
(f (_≃ᴱ_.from A≃B x))))
(λ f → apply-ext (lower-extensionality a p ext) λ x →
subst (λ x → Q x) (_≃ᴱ_.right-inverse-of A≃B x)
(_≃ᴱ_.to (P≃Q (_≃ᴱ_.from A≃B x)) (f (_≃ᴱ_.from A≃B x))) ≡⟨ sym substᴱ≡subst ⟩∎
substᴱ Q
(_≃ᴱ_.right-inverse-of A≃B x)
(_≃ᴱ_.to (P≃Q (_≃ᴱ_.from A≃B x)) (f (_≃ᴱ_.from A≃B x))) ∎)
-- A variant of Π-cong-≃ᴱ-Erased.
Π-cong-contra-≃ᴱ-Erased :
{@0 A : Type ℓ} {@0 B : Type b}
{P : @0 A → Type p} {@0 Q : B → Type q} →
@0 Extensionality (b ⊔ ℓ) (p ⊔ q) →
(B≃A : B ≃ᴱ A) →
(∀ x → P (_≃ᴱ_.to B≃A x) ≃ᴱ Q x) →
((x : A) → P x) ≃ᴱ ((x : B) → Q x)
Π-cong-contra-≃ᴱ-Erased
{b = b} {q = q} {A = A} {B = B} {P = P} {Q = Q} ext B≃A P≃Q =
[≃]→≃ᴱ ([proofs] ΠAP≃ΠBQ)
where
@0 ΠAP≃ΠBQ : ((x : A) → P x) ≃ ((x : B) → Q x)
ΠAP≃ΠBQ =
Eq.with-other-inverse
(Π-cong-contra ext (≃ᴱ→≃ B≃A) (λ x → ≃ᴱ→≃ (P≃Q x)))
(λ f x → substᴱ P
(_≃ᴱ_.right-inverse-of B≃A x)
(_≃ᴱ_.from (P≃Q (_≃ᴱ_.from B≃A x))
(f (_≃ᴱ_.from B≃A x))))
(λ f → apply-ext (lower-extensionality b q ext) λ x →
subst (λ x → P x) (_≃ᴱ_.right-inverse-of B≃A x)
(_≃ᴱ_.from (P≃Q (_≃ᴱ_.from B≃A x)) (f (_≃ᴱ_.from B≃A x))) ≡⟨ sym substᴱ≡subst ⟩∎
substᴱ P
(_≃ᴱ_.right-inverse-of B≃A x)
(_≃ᴱ_.from (P≃Q (_≃ᴱ_.from B≃A x)) (f (_≃ᴱ_.from B≃A x))) ∎)
----------------------------------------------------------------------
-- Variants of some lemmas from Function-universe
-- A variant of drop-⊤-left-Σ.
drop-⊤-left-Σ-≃ᴱ-Erased :
{@0 A : Type ℓ} {P : @0 A → Type p} →
(A≃⊤ : A ≃ᴱ ⊤) → Σ A (λ x → P x) ≃ᴱ P (_≃ᴱ_.from A≃⊤ tt)
drop-⊤-left-Σ-≃ᴱ-Erased {A = A} {P = P} A≃⊤ =
Σ A (λ x → P x) ≃ᴱ⟨ inverse $ Σ-cong-≃ᴱ-Erased (inverse A≃⊤) (λ _ → F.id) ⟩
Σ ⊤ (λ x → P (_≃ᴱ_.from A≃⊤ x)) ↔⟨ Σ-left-identity ⟩□
P (_≃ᴱ_.from A≃⊤ tt) □
-- A variant of drop-⊤-left-Π.
drop-⊤-left-Π-≃ᴱ-Erased :
{@0 A : Type ℓ} {P : @0 A → Type p} →
@0 Extensionality ℓ p →
(A≃⊤ : A ≃ᴱ ⊤) →
((x : A) → P x) ≃ᴱ P (_≃ᴱ_.from A≃⊤ tt)
drop-⊤-left-Π-≃ᴱ-Erased {A = A} {P = P} ext A≃⊤ =
((x : A) → P x) ≃ᴱ⟨ Π-cong-contra-≃ᴱ-Erased ext (inverse A≃⊤) (λ _ → F.id) ⟩
((x : ⊤) → P (_≃ᴱ_.from A≃⊤ x)) ↔⟨ Π-left-identity ⟩□
P (_≃ᴱ_.from A≃⊤ tt) □
----------------------------------------------------------------------
-- A variant of a lemma proved above
-- If f is an equivalence (with erased proofs) from Erased A to B,
-- then x ≡ y is equivalent (with erased proofs) to f x ≡ f y.
to≡to≃ᴱ≡-Erased :
∀ {@0 A : Type ℓ} {x y}
(A≃B : Erased A ≃ᴱ B) →
(_≃ᴱ_.to A≃B x ≡ _≃ᴱ_.to A≃B y) ≃ᴱ (x ≡ y)
to≡to≃ᴱ≡-Erased {B = B} {A = A} {x = x} {y = y} A≃B =
[≃]→≃ᴱ ([proofs] ≡≃≡)
where
@0 ≡≃≡ : (_≃ᴱ_.to A≃B x ≡ _≃ᴱ_.to A≃B y) ≃ (x ≡ y)
≡≃≡ =
Eq.with-other-function
(Eq.≃-≡ (≃ᴱ→≃ A≃B))
(λ eq →
x ≡⟨ sym $ []-cong [ cong erased (_≃ᴱ_.left-inverse-of A≃B x) ] ⟩
_≃ᴱ_.from A≃B (_≃ᴱ_.to A≃B x) ≡⟨ cong (_≃ᴱ_.from A≃B) eq ⟩
_≃ᴱ_.from A≃B (_≃ᴱ_.to A≃B y) ≡⟨ []-cong [ cong erased (_≃ᴱ_.left-inverse-of A≃B y) ] ⟩∎
y ∎)
(λ eq →
let f = _≃ᴱ_.left-inverse-of A≃B in
trans (sym (f x)) (trans (cong (_≃ᴱ_.from A≃B) eq) (f y)) ≡⟨ cong₂ (λ p q → trans (sym p) (trans (cong (_≃ᴱ_.from A≃B) eq) q))
(sym $ _≃_.right-inverse-of ≡≃[]≡[] _)
(sym $ _≃_.right-inverse-of ≡≃[]≡[] _) ⟩∎
trans (sym ([]-cong [ cong erased (f x) ]))
(trans (cong (_≃ᴱ_.from A≃B) eq)
([]-cong [ cong erased (f y) ])) ∎)
------------------------------------------------------------------------
-- Results that follow if the []-cong axioms hold for the maximum of
-- two universe levels (as well as for the two universe levels)
module []-cong₂-⊔
(ax₁ : []-cong-axiomatisation ℓ₁)
(ax₂ : []-cong-axiomatisation ℓ₂)
(ax : []-cong-axiomatisation (ℓ₁ ⊔ ℓ₂))
where
open Erased-cong ax ax
open Erased.[]-cong₁ ax
open Erased.[]-cong₂-⊔ ax₁ ax₂ ax
open []-cong₁ ax
----------------------------------------------------------------------
-- Another preservation lemma
-- Is-equivalenceᴱ f is equivalent to Is-equivalenceᴱ g if f and g
-- are pointwise equal (assuming extensionality).
Is-equivalenceᴱ-cong :
{A : Type ℓ₁} {B : Type ℓ₂} {@0 f g : A → B} →
@0 Extensionality? k (ℓ₁ ⊔ ℓ₂) (ℓ₁ ⊔ ℓ₂) →
@0 (∀ x → f x ≡ g x) →
Is-equivalenceᴱ f ↝[ k ] Is-equivalenceᴱ g
Is-equivalenceᴱ-cong {f = f} {g = g} ext f≡g =
generalise-erased-ext?
(Is-equivalenceᴱ-cong-⇔ f≡g)
(λ ext →
(∃ λ f⁻¹ → Erased (HA.Proofs f f⁻¹)) F.↔⟨ (∃-cong λ _ → Erased-cong-≃ (Proofs-cong ext f≡g)) ⟩□
(∃ λ f⁻¹ → Erased (HA.Proofs g f⁻¹)) □)
ext
----------------------------------------------------------------------
-- More conversion lemmas
-- Some equivalences relating Is-equivalenceᴱ to Is-equivalence.
--
-- See also Is-equivalenceᴱ↔Is-equivalence below.
Erased-Is-equivalenceᴱ≃Erased-Is-equivalence :
{@0 A : Type ℓ₁} {@0 B : Type ℓ₂} {@0 f : A → B} →
Erased (Is-equivalenceᴱ f) ≃ Erased (Is-equivalence f)
Erased-Is-equivalenceᴱ≃Erased-Is-equivalence {f = f} =
Erased (∃ λ f⁻¹ → Erased (HA.Proofs f f⁻¹)) ↝⟨ Erased-cong-≃ (∃-cong λ _ → Eq.↔⇒≃ $ erased Erased↔) ⟩□
Erased (∃ λ f⁻¹ → HA.Proofs f f⁻¹) □
Erased-Is-equivalence≃Is-equivalenceᴱ :
{@0 A : Type ℓ₁} {B : Type ℓ₂} {@0 f : Erased A → B} →
Erased (Is-equivalence f) ≃ Is-equivalenceᴱ f
Erased-Is-equivalence≃Is-equivalenceᴱ {A = A} {B = B} {f = f} =
Erased (Is-equivalence f) F.↔⟨⟩
Erased (∃ λ (f⁻¹ : B → Erased A) → HA.Proofs f f⁻¹) F.↔⟨ Erased-cong-↔ (F.inverse $ Σ-cong-id →≃→Erased) ⟩
Erased (∃ λ (f⁻¹ : B → A) → HA.Proofs f ([_]→ ⊚ f⁻¹)) F.↔⟨ Erased-Σ↔Σ ⟩
(∃ λ (f⁻¹ : Erased (B → A)) →
Erased (HA.Proofs f (λ x → map (_$ x) f⁻¹))) ↝⟨ (F.Σ-cong Erased-Π↔Π λ _ → F.id) ⟩
(∃ λ (f⁻¹ : B → Erased A) → Erased (HA.Proofs f f⁻¹)) F.↔⟨⟩
Is-equivalenceᴱ f F.□
where
@0 →≃→Erased : (B → A) ≃ (B → Erased A)
→≃→Erased = Eq.↔→≃
(λ f x → [ f x ])
(λ f x → erased (f x))
refl
refl
----------------------------------------------------------------------
-- Variants of some lemmas proved above
-- Is-equivalenceᴱ f is a proposition if the domain of f is Erased A
-- (assuming extensionality).
Is-equivalenceᴱ-propositional-for-Erased :
{@0 A : Type ℓ₁} {B : Type ℓ₂} {@0 f : Erased A → B} →
@0 Extensionality (ℓ₁ ⊔ ℓ₂) (ℓ₁ ⊔ ℓ₂) →
Is-proposition (Is-equivalenceᴱ f)
Is-equivalenceᴱ-propositional-for-Erased {f = f} ext =
F.$⟨ H-level-Erased 1 (Eq.propositional ext _) ⟩
Is-proposition (Erased (Is-equivalence f)) ↝⟨ H-level-cong _ 1 Erased-Is-equivalence≃Is-equivalenceᴱ ⦂ (_ → _) ⟩□
Is-proposition (Is-equivalenceᴱ f) □
-- A variant of to≡to→≡ that is not defined in an erased context.
-- Note that one side of the equivalence is Erased A.
to≡to→≡-Erased :
{@0 A : Type ℓ₁} {B : Type ℓ₂} {p q : Erased A ≃ᴱ B} →
@0 Extensionality (ℓ₁ ⊔ ℓ₂) (ℓ₁ ⊔ ℓ₂) →
_≃ᴱ_.to p ≡ _≃ᴱ_.to q → p ≡ q
to≡to→≡-Erased {p = ⟨ f , f-eq ⟩} {q = ⟨ g , g-eq ⟩} ext f≡g =
elim (λ {f g} f≡g → ∀ f-eq g-eq → ⟨ f , f-eq ⟩ ≡ ⟨ g , g-eq ⟩)
(λ f _ _ →
cong ⟨ f ,_⟩
(Is-equivalenceᴱ-propositional-for-Erased ext _ _))
f≡g f-eq g-eq
----------------------------------------------------------------------
-- More lemmas
-- An equivalence relating Is-equivalenceᴱ to Is-equivalence.
Is-equivalenceᴱ↔Is-equivalence :
{@0 A : Type ℓ₁} {@0 B : Type ℓ₂} {@0 f : A → B} →
Is-equivalenceᴱ (map f) ↝[ ℓ₁ ⊔ ℓ₂ ∣ ℓ₁ ⊔ ℓ₂ ]
Is-equivalence (map f)
Is-equivalenceᴱ↔Is-equivalence {f = f} =
generalise-ext?-prop
(Is-equivalenceᴱ (map f) ↝⟨ Is-equivalenceᴱ⇔Is-equivalenceᴱ-CP ⟩
(∀ y → Contractibleᴱ (map f ⁻¹ᴱ y)) F.↔⟨⟩
(∀ y → Contractibleᴱ (∃ λ x → Erased ([ f (erased x) ] ≡ y))) ↝⟨ (∀-cong _ λ _ → ECP.[]-cong₂.Contractibleᴱ-cong ax ax _ (Eq.↔⇒≃ $ F.inverse Erased-Σ↔Σ)) ⟩
(∀ y → Contractibleᴱ (Erased (∃ λ x → [ f x ] ≡ y))) ↝⟨ (∀-cong _ λ _ → ECP.[]-cong₁.Contractibleᴱ-Erased↔Contractible-Erased ax _) ⟩
(∀ y → Contractible (Erased (∃ λ x → [ f x ] ≡ y))) ↝⟨ (∀-cong _ λ _ → H-level-cong _ 0 Erased-Σ↔Σ) ⟩
(∀ y → Contractible (∃ λ x → Erased (map f x ≡ y))) F.↔⟨⟩
(∀ y → Contractible (map f ⁻¹ᴱ y)) ↝⟨ (∀-cong _ λ _ → H-level-cong _ 0 $ ECP.[]-cong₁.⁻¹ᴱ[]↔⁻¹[] ax₂) ⟩
(∀ y → Contractible (map f ⁻¹ y)) ↝⟨ inverse-ext? Is-equivalence≃Is-equivalence-CP _ ⟩□
Is-equivalence (map f) □)
(λ ext → Is-equivalenceᴱ-propositional-for-Erased ext)
(λ ext → Eq.propositional ext _)
-- Erased "commutes" with Is-equivalenceᴱ (assuming extensionality).
Erased-Is-equivalenceᴱ↔Is-equivalenceᴱ :
{@0 A : Type ℓ₁} {@0 B : Type ℓ₂} {@0 f : A → B} →
Erased (Is-equivalenceᴱ f) ↝[ ℓ₁ ⊔ ℓ₂ ∣ ℓ₁ ⊔ ℓ₂ ]ᴱ
Is-equivalenceᴱ (map f)
Erased-Is-equivalenceᴱ↔Is-equivalenceᴱ {f = f} ext =
Erased (Is-equivalenceᴱ f) F.↔⟨ Erased-Is-equivalenceᴱ≃Erased-Is-equivalence ⟩
Erased (Is-equivalence f) F.↔⟨ F.inverse Erased-Erased↔Erased ⟩
Erased (Erased (Is-equivalence f)) ↝⟨ Erased-cong? Erased-Is-equivalence↔Is-equivalence ext ⟩
Erased (Is-equivalence (map f)) F.↔⟨ Erased-Is-equivalence≃Is-equivalenceᴱ ⟩□
Is-equivalenceᴱ (map f) □
------------------------------------------------------------------------
-- Results that depend on an axiomatisation of []-cong (for all
-- universe levels)
module []-cong (ax : ∀ {ℓ} → []-cong-axiomatisation ℓ) where
private
open module BC₁ {ℓ} =
[]-cong₁ (ax {ℓ = ℓ})
public
open module BC₂ {ℓ₁ ℓ₂} =
[]-cong₂-⊔ (ax {ℓ = ℓ₁}) (ax {ℓ = ℓ₂}) (ax {ℓ = ℓ₁ ⊔ ℓ₂})
public
| 36.484587
| 163
| 0.449559
|
c518be7bd996d56750cf8fb1093dea9faa4f41a9
| 4,521
|
agda
|
Agda
|
Cubical/Experiments/Problem.agda
|
maxdore/cubical
|
ef62b84397396d48135d73ba7400b71c721ddc94
|
[
"MIT"
] | null | null | null |
Cubical/Experiments/Problem.agda
|
maxdore/cubical
|
ef62b84397396d48135d73ba7400b71c721ddc94
|
[
"MIT"
] | null | null | null |
Cubical/Experiments/Problem.agda
|
maxdore/cubical
|
ef62b84397396d48135d73ba7400b71c721ddc94
|
[
"MIT"
] | 1
|
2021-03-12T20:08:45.000Z
|
2021-03-12T20:08:45.000Z
|
-- An example of something where normalization is surprisingly slow
{-# OPTIONS --safe #-}
module Cubical.Experiments.Problem where
open import Cubical.Foundations.Prelude
open import Cubical.Data.Int
open import Cubical.HITs.S1
open import Cubical.HITs.S2
open import Cubical.HITs.S3
open import Cubical.HITs.Join
open import Cubical.HITs.Hopf
ptType : Type _
ptType = Σ Type₀ \ A → A
pt : (A : ptType) → A .fst
pt A = A .snd
S¹pt : ptType
S¹pt = (S¹ , base)
S²pt : ptType
S²pt = (S² , base)
S³pt : ptType
S³pt = (S³ , base)
joinpt : ptType
joinpt = (join S¹ S¹ , inl base)
Ω : (A : ptType) → ptType
Ω A = Path _ (pt A) (pt A) , refl
Ω² : (A : ptType) → ptType
Ω² A = Ω (Ω A)
Ω³ : (A : ptType) → ptType
Ω³ A = Ω² (Ω A)
α : join S¹ S¹ → S²
α (inl _) = base
α (inr _) = base
α (push x y i) = (merid y ∙ merid x) i
where
merid : S¹ → Path S² base base
merid base = refl
merid (loop i) = λ j → surf i j
-- The tests
test0To2 : Ω³ S³pt .fst
test0To2 i j k = surf i j k
f3 : Ω³ S³pt .fst → Ω³ joinpt .fst
f3 p i j k = S³→joinS¹S¹ (p i j k)
test0To3 : Ω³ joinpt .fst
test0To3 = f3 test0To2
f4 : Ω³ joinpt .fst → Ω³ S²pt .fst
f4 p i j k = α (p i j k)
test0To4 : Ω³ S²pt .fst
test0To4 = f4 test0To3
innerpath : ∀ i j → HopfS² (test0To4 i j i1)
innerpath i j = transp (λ k → HopfS² (test0To4 i j k)) i0 base
-- C-c C-n problem uses a lot of memory
problem : pos 0 ≡ pos 0
problem i = transp (λ j → helix (innerpath i j)) i0 (pos 0)
-- Lots of tests: (thanks Evan!)
winding2 : Path (Path S² base base) refl refl → Int
winding2 p = winding (λ j → transp (λ i → HopfS² (p i j)) i0 base)
test0 : Int
test0 = winding2 (λ i j → surf i j)
test1 : Int
test1 = winding2 (λ i j → surf j i)
test2 : Int
test2 = winding2 (λ i j → hcomp (λ _ → λ { (i = i0) → base ; (i = i1) → base ; (j = i0) → base ; (j = i1) → base}) (surf i j))
test3 : Int
test3 = winding2 (λ i j → hcomp (λ k → λ { (i = i0) → surf j k ; (i = i1) → base ; (j = i0) → base ; (j = i1) → base}) base)
test4 : Int
test4 = winding2 (λ i j → hcomp (λ k → λ { (i = i0) → surf j k ; (i = i1) → base ; (j = i0) → base ; (j = i1) → base}) base)
test5 : Int
test5 = winding2 (λ i j → hcomp (λ k → λ { (i = i0) → base ; (i = i1) → base ; (j = i0) → surf k i ; (j = i1) → base}) base)
test6 : Int
test6 = winding2 (λ i j → hcomp (λ k → λ { (i = i0) → base ; (i = i1) → base ; (j = i0) → base ; (j = i1) → surf k i}) base)
test7 : Int
test7 = winding2 (λ i j → hcomp (λ k → λ { (i = i0) → base ; (i = i1) → surf j k ; (j = i0) → base ; (j = i1) → base}) (surf i j))
test8 : Int
test8 = winding2 (λ i j → hcomp (λ k → λ { (i = i0) → base ; (i = i1) → base ; (j = i0) → surf k i ; (j = i1) → base}) (surf i j))
test9 : Int
test9 = winding2 (λ i j → hcomp (λ k → λ { (i = i0) → base ; (i = i1) → base ; (j = i0) → base ; (j = i1) → surf k i}) (surf i j))
test10 : Int
test10 = winding2 (λ i j → hcomp (λ k → λ { (i = i0) → surf j k ; (i = i1) → base ; (j = i0) → base ; (j = i1) → base}) (surf i j))
-- Tests using HopfS²'
winding2' : Path (Path S² base base) refl refl → Int
winding2' p = winding (λ j → transp (λ i → HopfS²' (p i j)) i0 base)
test0' : Int
test0' = winding2' (λ i j → surf i j)
test1' : Int
test1' = winding2' (λ i j → surf j i)
test2' : Int
test2' = winding2' (λ i j → hcomp (λ _ → λ { (i = i0) → base ; (i = i1) → base ; (j = i0) → base ; (j = i1) → base}) (surf i j))
test3' : Int
test3' = winding2' (λ i j → hcomp (λ k → λ { (i = i0) → surf j k ; (i = i1) → base ; (j = i0) → base ; (j = i1) → base}) base)
test4' : Int
test4' = winding2' (λ i j → hcomp (λ k → λ { (i = i0) → surf j k ; (i = i1) → base ; (j = i0) → base ; (j = i1) → base}) base)
test5' : Int
test5' = winding2' (λ i j → hcomp (λ k → λ { (i = i0) → base ; (i = i1) → base ; (j = i0) → surf k i ; (j = i1) → base}) base)
test6' : Int
test6' = winding2' (λ i j → hcomp (λ k → λ { (i = i0) → base ; (i = i1) → base ; (j = i0) → base ; (j = i1) → surf k i}) base)
test7' : Int
test7' = winding2' (λ i j → hcomp (λ k → λ { (i = i0) → base ; (i = i1) → surf j k ; (j = i0) → base ; (j = i1) → base}) (surf i j))
test8' : Int
test8' = winding2' (λ i j → hcomp (λ k → λ { (i = i0) → base ; (i = i1) → base ; (j = i0) → surf k i ; (j = i1) → base}) (surf i j))
test9' : Int
test9' = winding2' (λ i j → hcomp (λ k → λ { (i = i0) → base ; (i = i1) → base ; (j = i0) → base ; (j = i1) → surf k i}) (surf i j))
test10' : Int
test10' = winding2' (λ i j → hcomp (λ k → λ { (i = i0) → surf j k ; (i = i1) → base ; (j = i0) → base ; (j = i1) → base}) (surf i j))
| 30.342282
| 133
| 0.54457
|
1896bd25e99bf749dfc3cdc646040e1978b196c4
| 310
|
agda
|
Agda
|
Cubical/Structures/Everything.agda
|
borsiemir/cubical
|
cefeb3669ffdaea7b88ae0e9dd258378418819ca
|
[
"MIT"
] | null | null | null |
Cubical/Structures/Everything.agda
|
borsiemir/cubical
|
cefeb3669ffdaea7b88ae0e9dd258378418819ca
|
[
"MIT"
] | null | null | null |
Cubical/Structures/Everything.agda
|
borsiemir/cubical
|
cefeb3669ffdaea7b88ae0e9dd258378418819ca
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --cubical --safe #-}
module Cubical.Structures.Everything where
open import Cubical.Structures.Pointed public
open import Cubical.Structures.InftyMagma public
open import Cubical.Structures.Monoid public
open import Cubical.Structures.Queue public
open import Cubical.Structures.TypeEqvTo public
| 31
| 48
| 0.832258
|
1cda814c4381efa8dc1fc490bd39b17f4459abf3
| 347
|
agda
|
Agda
|
Cubical/HITs/Sn/Base.agda
|
kiana-S/univalent-foundations
|
8bdb0766260489f9c89a14f4c0f2ad26e5190dec
|
[
"MIT"
] | null | null | null |
Cubical/HITs/Sn/Base.agda
|
kiana-S/univalent-foundations
|
8bdb0766260489f9c89a14f4c0f2ad26e5190dec
|
[
"MIT"
] | null | null | null |
Cubical/HITs/Sn/Base.agda
|
kiana-S/univalent-foundations
|
8bdb0766260489f9c89a14f4c0f2ad26e5190dec
|
[
"MIT"
] | 1
|
2021-11-22T02:02:01.000Z
|
2021-11-22T02:02:01.000Z
|
{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.HITs.Sn.Base where
open import Cubical.HITs.Susp
open import Cubical.Data.Nat
open import Cubical.Data.NatMinusOne
open import Cubical.Data.Empty
open import Cubical.Foundations.Prelude
S : ℕ₋₁ → Type₀
S neg1 = ⊥
S (ℕ→ℕ₋₁ n) = Susp (S (-1+ n))
S₊ : ℕ → Type₀
S₊ n = S (ℕ→ℕ₋₁ n)
| 21.6875
| 50
| 0.691643
|
1cc035de0942e4202307dd0087205c59bc02bba2
| 2,197
|
agda
|
Agda
|
lib/Haskell/Prim/Applicative.agda
|
JonathanBrouwer/agda2hs
|
dcf63cc7ce51a325a97ac58bdd0aeace24c08b15
|
[
"MIT"
] | 55
|
2020-10-20T13:36:25.000Z
|
2022-03-26T21:57:56.000Z
|
lib/Haskell/Prim/Applicative.agda
|
SNU-2D/agda2hs
|
160478a51bc78b0fdab07b968464420439f9fed6
|
[
"MIT"
] | 63
|
2020-10-22T05:19:27.000Z
|
2022-02-25T15:47:30.000Z
|
lib/Haskell/Prim/Applicative.agda
|
SNU-2D/agda2hs
|
160478a51bc78b0fdab07b968464420439f9fed6
|
[
"MIT"
] | 18
|
2020-10-21T22:19:09.000Z
|
2022-03-12T11:42:52.000Z
|
module Haskell.Prim.Applicative where
open import Haskell.Prim
open import Haskell.Prim.Either
open import Haskell.Prim.Foldable
open import Haskell.Prim.Functor
open import Haskell.Prim.List
open import Haskell.Prim.Maybe
open import Haskell.Prim.Monoid
open import Haskell.Prim.Tuple
--------------------------------------------------
-- Applicative
record Applicative (f : Set → Set) : Set₁ where
infixl 4 _<*>_
field
pure : a → f a
_<*>_ : f (a → b) → f a → f b
overlap ⦃ super ⦄ : Functor f
_<*_ : f a → f b → f a
x <* y = const <$> x <*> y
_*>_ : f a → f b → f b
x *> y = const id <$> x <*> y
open Applicative ⦃ ... ⦄ public
{-# COMPILE AGDA2HS Applicative existing-class #-}
instance
iApplicativeList : Applicative List
iApplicativeList .pure x = x ∷ []
iApplicativeList ._<*>_ fs xs = concatMap (λ f → map f xs) fs
iApplicativeMaybe : Applicative Maybe
iApplicativeMaybe .pure = Just
iApplicativeMaybe ._<*>_ (Just f) (Just x) = Just (f x)
iApplicativeMaybe ._<*>_ _ _ = Nothing
iApplicativeEither : Applicative (Either a)
iApplicativeEither .pure = Right
iApplicativeEither ._<*>_ (Right f) (Right x) = Right (f x)
iApplicativeEither ._<*>_ (Left e) _ = Left e
iApplicativeEither ._<*>_ _ (Left e) = Left e
iApplicativeFun : Applicative (λ b → a → b)
iApplicativeFun .pure = const
iApplicativeFun ._<*>_ f g x = f x (g x)
iApplicativeTuple₂ : ⦃ Monoid a ⦄ → Applicative (a ×_)
iApplicativeTuple₂ .pure x = mempty , x
iApplicativeTuple₂ ._<*>_ (a , f) (b , x) = a <> b , f x
iApplicativeTuple₃ : ⦃ Monoid a ⦄ → ⦃ Monoid b ⦄ → Applicative (a × b ×_)
iApplicativeTuple₃ .pure x = mempty , mempty , x
iApplicativeTuple₃ ._<*>_ (a , b , f) (a₁ , b₁ , x) = a <> a₁ , b <> b₁ , f x
iApplicativeTuple₄ : ⦃ Monoid a ⦄ → ⦃ Monoid b ⦄ → ⦃ Monoid c ⦄ →
Applicative (λ d → Tuple (a ∷ b ∷ c ∷ d ∷ []))
iApplicativeTuple₄ .pure x = mempty ∷ mempty ∷ mempty ∷ x ∷ []
iApplicativeTuple₄ ._<*>_ (a ∷ b ∷ c ∷ f ∷ []) (a₁ ∷ b₁ ∷ c₁ ∷ x ∷ []) =
a <> a₁ ∷ b <> b₁ ∷ c <> c₁ ∷ f x ∷ []
| 32.791045
| 89
| 0.571234
|
0b57f294421e7f1ebfdcd826af907270b6ff629d
| 1,827
|
agda
|
Agda
|
Cubical/Categories/Instances/Cospan.agda
|
barrettj12/cubical
|
7b41b9171f90473efc98487cb2ea7a4d02320cb2
|
[
"MIT"
] | null | null | null |
Cubical/Categories/Instances/Cospan.agda
|
barrettj12/cubical
|
7b41b9171f90473efc98487cb2ea7a4d02320cb2
|
[
"MIT"
] | null | null | null |
Cubical/Categories/Instances/Cospan.agda
|
barrettj12/cubical
|
7b41b9171f90473efc98487cb2ea7a4d02320cb2
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --safe #-}
module Cubical.Categories.Instances.Cospan where
open import Cubical.Foundations.Prelude
open import Cubical.Categories.Category
open import Cubical.Data.Unit
open import Cubical.Data.Empty
open Category
data 𝟛 : Type ℓ-zero where
⓪ : 𝟛
① : 𝟛
② : 𝟛
CospanCat : Category ℓ-zero ℓ-zero
CospanCat .ob = 𝟛
CospanCat .Hom[_,_] ⓪ ① = Unit
CospanCat .Hom[_,_] ② ① = Unit
CospanCat .Hom[_,_] ⓪ ⓪ = Unit
CospanCat .Hom[_,_] ① ① = Unit
CospanCat .Hom[_,_] ② ② = Unit
CospanCat .Hom[_,_] _ _ = ⊥
CospanCat ._⋆_ {x = ⓪} {⓪} {⓪} f g = tt
CospanCat ._⋆_ {x = ①} {①} {①} f g = tt
CospanCat ._⋆_ {x = ②} {②} {②} f g = tt
CospanCat ._⋆_ {x = ⓪} {①} {①} f g = tt
CospanCat ._⋆_ {x = ②} {①} {①} f g = tt
CospanCat ._⋆_ {x = ⓪} {⓪} {①} f g = tt
CospanCat ._⋆_ {x = ②} {②} {①} f g = tt
CospanCat .id {⓪} = tt
CospanCat .id {①} = tt
CospanCat .id {②} = tt
CospanCat .⋆IdL {⓪} {①} _ = refl
CospanCat .⋆IdL {②} {①} _ = refl
CospanCat .⋆IdL {⓪} {⓪} _ = refl
CospanCat .⋆IdL {①} {①} _ = refl
CospanCat .⋆IdL {②} {②} _ = refl
CospanCat .⋆IdR {⓪} {①} _ = refl
CospanCat .⋆IdR {②} {①} _ = refl
CospanCat .⋆IdR {⓪} {⓪} _ = refl
CospanCat .⋆IdR {①} {①} _ = refl
CospanCat .⋆IdR {②} {②} _ = refl
CospanCat .⋆Assoc {⓪} {⓪} {⓪} {⓪} _ _ _ = refl
CospanCat .⋆Assoc {⓪} {⓪} {⓪} {①} _ _ _ = refl
CospanCat .⋆Assoc {⓪} {⓪} {①} {①} _ _ _ = refl
CospanCat .⋆Assoc {⓪} {①} {①} {①} _ _ _ = refl
CospanCat .⋆Assoc {①} {①} {①} {①} _ _ _ = refl
CospanCat .⋆Assoc {②} {②} {②} {②} _ _ _ = refl
CospanCat .⋆Assoc {②} {②} {②} {①} _ _ _ = refl
CospanCat .⋆Assoc {②} {②} {①} {①} _ _ _ = refl
CospanCat .⋆Assoc {②} {①} {①} {①} _ _ _ = refl
CospanCat .isSetHom {⓪} {⓪} = isSetUnit
CospanCat .isSetHom {⓪} {①} = isSetUnit
CospanCat .isSetHom {①} {①} = isSetUnit
CospanCat .isSetHom {②} {①} = isSetUnit
CospanCat .isSetHom {②} {②} = isSetUnit
| 28.107692
| 48
| 0.600438
|
4aa23e0fbe94e4d76482277d911dd5de26031376
| 278
|
agda
|
Agda
|
test/Fail/Issue2386-b.agda
|
mdimjasevic/agda
|
8fb548356b275c7a1e79b768b64511ae937c738b
|
[
"BSD-3-Clause"
] | 1,989
|
2015-01-09T23:51:16.000Z
|
2022-03-30T18:20:48.000Z
|
test/Fail/Issue2386-b.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 4,066
|
2015-01-10T11:24:51.000Z
|
2022-03-31T21:14:49.000Z
|
test/Fail/Issue2386-b.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 371
|
2015-01-03T14:04:08.000Z
|
2022-03-30T19:00:30.000Z
|
-- Andreas, 2017-01-12, issue #2386
-- Should be rejected:
data Eq (A : Set) : (x y : A) → Set where
refl : (x y : A) → Eq A x y
{-# BUILTIN EQUALITY Eq #-}
-- Expected error:
-- Wrong type of constructor of BUILTIN EQUALITY
-- when checking the pragma BUILTIN EQUALITY Eq
| 23.166667
| 48
| 0.647482
|
1c08acd0c6c817ac34d1f755abb12b7e783c7b83
| 2,986
|
agda
|
Agda
|
test/asset/agda-stdlib-1.0/Data/Nat/GeneralisedArithmetic.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | null | null | null |
test/asset/agda-stdlib-1.0/Data/Nat/GeneralisedArithmetic.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | null | null | null |
test/asset/agda-stdlib-1.0/Data/Nat/GeneralisedArithmetic.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | null | null | null |
------------------------------------------------------------------------
-- The Agda standard library
--
-- A generalisation of the arithmetic operations
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Nat.GeneralisedArithmetic where
open import Data.Nat
open import Data.Nat.Properties
open import Function using (_∘′_; _∘_; id)
open import Relation.Binary.PropositionalEquality
open ≡-Reasoning
module _ {a} {A : Set a} where
fold : A → (A → A) → ℕ → A
fold z s zero = z
fold z s (suc n) = s (fold z s n)
add : (0# : A) (1+ : A → A) → ℕ → A → A
add 0# 1+ n z = fold z 1+ n
mul : (0# : A) (1+ : A → A) → (+ : A → A → A) → (ℕ → A → A)
mul 0# 1+ _+_ n x = fold 0# (λ s → x + s) n
-- Properties
module _ {a} {A : Set a} where
fold-+ : ∀ {s : A → A} {z : A} →
∀ m {n} → fold z s (m + n) ≡ fold (fold z s n) s m
fold-+ zero = refl
fold-+ {s = s} (suc m) = cong s (fold-+ m)
fold-k : ∀ {s : A → A} {z : A} {k} m →
fold k (s ∘′_) m z ≡ fold (k z) s m
fold-k zero = refl
fold-k {s = s} (suc m) = cong s (fold-k m)
fold-* : ∀ {s : A → A} {z : A} m {n} →
fold z s (m * n) ≡ fold z (fold id (s ∘_) n) m
fold-* zero = refl
fold-* {s = s} {z} (suc m) {n} = let +n = fold id (s ∘′_) n in begin
fold z s (n + m * n) ≡⟨ fold-+ n ⟩
fold (fold z s (m * n)) s n ≡⟨ cong (λ z → fold z s n) (fold-* m) ⟩
fold (fold z +n m) s n ≡⟨ sym (fold-k n) ⟩
fold z +n (suc m) ∎
fold-pull : ∀ {s : A → A} {z : A} (g : A → A → A) (p : A)
(eqz : g z p ≡ p)
(eqs : ∀ l → s (g l p) ≡ g (s l) p) →
∀ m → fold p s m ≡ g (fold z s m) p
fold-pull _ _ eqz _ zero = sym eqz
fold-pull {s = s} {z} g p eqz eqs (suc m) = begin
s (fold p s m) ≡⟨ cong s (fold-pull g p eqz eqs m) ⟩
s (g (fold z s m) p) ≡⟨ eqs (fold z s m) ⟩
g (s (fold z s m)) p ∎
id-is-fold : ∀ m → fold zero suc m ≡ m
id-is-fold zero = refl
id-is-fold (suc m) = cong suc (id-is-fold m)
+-is-fold : ∀ m {n} → fold n suc m ≡ m + n
+-is-fold zero = refl
+-is-fold (suc m) = cong suc (+-is-fold m)
*-is-fold : ∀ m {n} → fold zero (n +_) m ≡ m * n
*-is-fold zero = refl
*-is-fold (suc m) {n} = cong (n +_) (*-is-fold m)
^-is-fold : ∀ {m} n → fold 1 (m *_) n ≡ m ^ n
^-is-fold zero = refl
^-is-fold {m} (suc n) = cong (m *_) (^-is-fold n)
*+-is-fold : ∀ m n {p} → fold p (n +_) m ≡ m * n + p
*+-is-fold m n {p} = begin
fold p (n +_) m ≡⟨ fold-pull _+_ p refl
(λ l → sym (+-assoc n l p)) m ⟩
fold 0 (n +_) m + p ≡⟨ cong (_+ p) (*-is-fold m) ⟩
m * n + p ∎
^*-is-fold : ∀ m n {p} → fold p (m *_) n ≡ m ^ n * p
^*-is-fold m n {p} = begin
fold p (m *_) n ≡⟨ fold-pull _*_ p (*-identityˡ p)
(λ l → sym (*-assoc m l p)) n ⟩
fold 1 (m *_) n * p ≡⟨ cong (_* p) (^-is-fold n) ⟩
m ^ n * p ∎
| 32.813187
| 72
| 0.428667
|
500c87d6e9b55d7b2f874ee8ea3d7760586bf40a
| 2,262
|
agda
|
Agda
|
Experiment/FingerTree/Common.agda
|
rei1024/agda-misc
|
37200ea91d34a6603d395d8ac81294068303f577
|
[
"MIT"
] | 3
|
2020-04-07T17:49:42.000Z
|
2020-04-21T00:03:43.000Z
|
Experiment/FingerTree/Common.agda
|
rei1024/agda-misc
|
37200ea91d34a6603d395d8ac81294068303f577
|
[
"MIT"
] | null | null | null |
Experiment/FingerTree/Common.agda
|
rei1024/agda-misc
|
37200ea91d34a6603d395d8ac81294068303f577
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --without-K --safe #-}
module Experiment.FingerTree.Common where
open import Level renaming (zero to lzero ; suc to lsuc)
open import Algebra
open import Data.Product
open import Function.Core
open import Function.Endomorphism.Propositional
open import Data.Nat hiding (_⊔_)
import Data.Nat.Properties as ℕₚ
foldr-to-foldMap : ∀ {a b e} {F : Set a → Set a} →
(∀ {A : Set a} {B : Set b} → (A → B → B) → B → F A → B) →
∀ {A : Set a} (M : Monoid b e) → (A → Monoid.Carrier M) → F A → Monoid.Carrier M
foldr-to-foldMap foldr M f xs = foldr (λ x m → Monoid._∙_ M (f x) m) (Monoid.ε M) xs
foldMap-to-foldr : ∀ {a b} {F : Set a → Set a} →
(∀ {A : Set a} (M : Monoid b b) → (A → Monoid.Carrier M) → F A → Monoid.Carrier M) →
∀ {A : Set a} {B : Set b} → (A → B → B) → B → F A → B
foldMap-to-foldr foldMap {B = B} f e xs = foldMap (∘-id-monoid B) f xs e
dual : ∀ {c e} → Monoid c e → Monoid c e
dual m = record
{ Carrier = Carrier
; _≈_ = _≈_
; _∙_ = flip _∙_
; ε = ε
; isMonoid = record
{ isSemigroup = record
{ isMagma = record
{ isEquivalence = isEquivalence
; ∙-cong = λ x≈y u≈v → ∙-cong u≈v x≈y
}
; assoc = λ x y z → sym $ assoc z y x
}
; identity = identityʳ , identityˡ
}
}
where open Monoid m
foldMap-to-foldl : ∀ {a b} {F : Set a → Set a} →
(∀ {A : Set a} (M : Monoid b b) → (A → Monoid.Carrier M) → F A → Monoid.Carrier M) →
∀ {A : Set a} {B : Set b} → (B → A → B) → B → F A → B
foldMap-to-foldl foldMap {B = B} f e xs = foldMap (dual (∘-id-monoid B)) (flip f) xs e
record RawFoldable {a} (F : Set a → Set a) : Set (lsuc a) where
field
foldMap : ∀ {A : Set a} (M : Monoid a a) → (A → Monoid.Carrier M) → F A → Monoid.Carrier M
fold : (M : Monoid a a) → F (Monoid.Carrier M) → Monoid.Carrier M
fold M = foldMap M id
foldr : ∀ {A B : Set a} → (A → B → B) → B → F A → B
foldr {A} {B} f e xs = foldMap (∘-id-monoid B) f xs e
foldl : ∀ {A B : Set a} → (B → A → B) → B → F A → B
foldl {A} {B} f e xs = foldMap (dual (∘-id-monoid B)) (flip f) xs e
fromFoldr : ∀ {a} {F : Set a → Set a} →
(∀ {A B : Set a} → (A → B → B) → B → F A → B) → RawFoldable {a} F
fromFoldr foldr = record
{ foldMap = foldr-to-foldMap foldr
}
| 35.34375
| 94
| 0.544651
|
507aafb6a34889b87312d56a6709ff15a460757a
| 8,285
|
agda
|
Agda
|
Cubical/Data/Nat/Order.agda
|
cangiuli/cubical
|
d103ec455d41cccf9b13a4803e7d3cf462e00067
|
[
"MIT"
] | null | null | null |
Cubical/Data/Nat/Order.agda
|
cangiuli/cubical
|
d103ec455d41cccf9b13a4803e7d3cf462e00067
|
[
"MIT"
] | 1
|
2022-01-27T02:07:48.000Z
|
2022-01-27T02:07:48.000Z
|
Cubical/Data/Nat/Order.agda
|
cangiuli/cubical
|
d103ec455d41cccf9b13a4803e7d3cf462e00067
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --cubical --no-import-sorts --no-exact-split --safe #-}
module Cubical.Data.Nat.Order where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function
open import Cubical.Foundations.HLevels
open import Cubical.Data.Empty as ⊥
open import Cubical.Data.Sigma
open import Cubical.Data.Sum as ⊎
open import Cubical.Data.Nat.Base
open import Cubical.Data.Nat.Properties
open import Cubical.Induction.WellFounded
open import Cubical.Relation.Nullary
infix 4 _≤_ _<_
_≤_ : ℕ → ℕ → Type₀
m ≤ n = Σ[ k ∈ ℕ ] k + m ≡ n
_<_ : ℕ → ℕ → Type₀
m < n = suc m ≤ n
data Trichotomy (m n : ℕ) : Type₀ where
lt : m < n → Trichotomy m n
eq : m ≡ n → Trichotomy m n
gt : n < m → Trichotomy m n
private
variable
k l m n : ℕ
private
witness-prop : ∀ j → isProp (j + m ≡ n)
witness-prop {m} {n} j = isSetℕ (j + m) n
m≤n-isProp : isProp (m ≤ n)
m≤n-isProp {m} {n} (k , p) (l , q)
= Σ≡Prop witness-prop lemma
where
lemma : k ≡ l
lemma = inj-+m (p ∙ (sym q))
zero-≤ : 0 ≤ n
zero-≤ {n} = n , +-zero n
suc-≤-suc : m ≤ n → suc m ≤ suc n
suc-≤-suc (k , p) = k , (+-suc k _) ∙ (cong suc p)
≤-+k : m ≤ n → m + k ≤ n + k
≤-+k {m} {k = k} (i , p)
= i , +-assoc i m k ∙ cong (_+ k) p
≤-k+ : m ≤ n → k + m ≤ k + n
≤-k+ {m} {n} {k}
= subst (_≤ k + n) (+-comm m k)
∘ subst (m + k ≤_) (+-comm n k)
∘ ≤-+k
pred-≤-pred : suc m ≤ suc n → m ≤ n
pred-≤-pred (k , p) = k , injSuc ((sym (+-suc k _)) ∙ p)
≤-refl : m ≤ m
≤-refl = 0 , refl
≤-suc : m ≤ n → m ≤ suc n
≤-suc (k , p) = suc k , cong suc p
≤-predℕ : predℕ n ≤ n
≤-predℕ {zero} = ≤-refl
≤-predℕ {suc n} = ≤-suc ≤-refl
≤-trans : k ≤ m → m ≤ n → k ≤ n
≤-trans {k} {m} {n} (i , p) (j , q) = i + j , l2 ∙ (l1 ∙ q)
where
l1 : j + i + k ≡ j + m
l1 = (sym (+-assoc j i k)) ∙ (cong (j +_) p)
l2 : i + j + k ≡ j + i + k
l2 = cong (_+ k) (+-comm i j)
≤-antisym : m ≤ n → n ≤ m → m ≡ n
≤-antisym {m} (i , p) (j , q) = (cong (_+ m) l3) ∙ p
where
l1 : j + i + m ≡ m
l1 = (sym (+-assoc j i m)) ∙ ((cong (j +_) p) ∙ q)
l2 : j + i ≡ 0
l2 = m+n≡n→m≡0 l1
l3 : 0 ≡ i
l3 = sym (snd (m+n≡0→m≡0×n≡0 l2))
≤-k+-cancel : k + m ≤ k + n → m ≤ n
≤-k+-cancel {k} {m} (l , p) = l , inj-m+ (sub k m ∙ p)
where
sub : ∀ k m → k + (l + m) ≡ l + (k + m)
sub k m = +-assoc k l m ∙ cong (_+ m) (+-comm k l) ∙ sym (+-assoc l k m)
≤-+k-cancel : m + k ≤ n + k → m ≤ n
≤-+k-cancel {m} {k} {n} (l , p) = l , cancelled
where
cancelled : l + m ≡ n
cancelled = inj-+m (sym (+-assoc l m k) ∙ p)
≤-·k : m ≤ n → m · k ≤ n · k
≤-·k {m} {n} {k} (d , r) = d · k , reason where
reason : d · k + m · k ≡ n · k
reason = d · k + m · k ≡⟨ ·-distribʳ d m k ⟩
(d + m) · k ≡⟨ cong (_· k) r ⟩
n · k ∎
<-k+-cancel : k + m < k + n → m < n
<-k+-cancel {k} {m} {n} = ≤-k+-cancel ∘ subst (_≤ k + n) (sym (+-suc k m))
¬-<-zero : ¬ m < 0
¬-<-zero (k , p) = snotz ((sym (+-suc k _)) ∙ p)
¬m<m : ¬ m < m
¬m<m {m} = ¬-<-zero ∘ ≤-+k-cancel {k = m}
≤0→≡0 : n ≤ 0 → n ≡ 0
≤0→≡0 {zero} ineq = refl
≤0→≡0 {suc n} ineq = ⊥.rec (¬-<-zero ineq)
predℕ-≤-predℕ : m ≤ n → (predℕ m) ≤ (predℕ n)
predℕ-≤-predℕ {zero} {zero} ineq = ≤-refl
predℕ-≤-predℕ {zero} {suc n} ineq = zero-≤
predℕ-≤-predℕ {suc m} {zero} ineq = ⊥.rec (¬-<-zero ineq)
predℕ-≤-predℕ {suc m} {suc n} ineq = pred-≤-pred ineq
¬m+n<m : ¬ m + n < m
¬m+n<m {m} {n} = ¬-<-zero ∘ <-k+-cancel ∘ subst (m + n <_) (sym (+-zero m))
<-weaken : m < n → m ≤ n
<-weaken (k , p) = suc k , sym (+-suc k _) ∙ p
≤<-trans : l ≤ m → m < n → l < n
≤<-trans p = ≤-trans (suc-≤-suc p)
<≤-trans : l < m → m ≤ n → l < n
<≤-trans = ≤-trans
<-trans : l < m → m < n → l < n
<-trans p = ≤<-trans (<-weaken p)
<-asym : m < n → ¬ n ≤ m
<-asym m<n = ¬m<m ∘ <≤-trans m<n
<-+k : m < n → m + k < n + k
<-+k p = ≤-+k p
<-k+ : m < n → k + m < k + n
<-k+ {m} {n} {k} p = subst (λ km → km ≤ k + n) (+-suc k m) (≤-k+ p)
<-·sk : m < n → m · suc k < n · suc k
<-·sk {m} {n} {k} (d , r) = (d · suc k + k) , reason where
reason : (d · suc k + k) + suc (m · suc k) ≡ n · suc k
reason = (d · suc k + k) + suc (m · suc k) ≡⟨ sym (+-assoc (d · suc k) k _) ⟩
d · suc k + (k + suc (m · suc k)) ≡[ i ]⟨ d · suc k + +-suc k (m · suc k) i ⟩
d · suc k + suc m · suc k ≡⟨ ·-distribʳ d (suc m) (suc k) ⟩
(d + suc m) · suc k ≡⟨ cong (_· suc k) r ⟩
n · suc k ∎
Trichotomy-suc : Trichotomy m n → Trichotomy (suc m) (suc n)
Trichotomy-suc (lt m<n) = lt (suc-≤-suc m<n)
Trichotomy-suc (eq m=n) = eq (cong suc m=n)
Trichotomy-suc (gt n<m) = gt (suc-≤-suc n<m)
_≟_ : ∀ m n → Trichotomy m n
zero ≟ zero = eq refl
zero ≟ suc n = lt (n , +-comm n 1)
suc m ≟ zero = gt (m , +-comm m 1)
suc m ≟ suc n = Trichotomy-suc (m ≟ n)
<-split : m < suc n → (m < n) ⊎ (m ≡ n)
<-split {n = zero} = inr ∘ snd ∘ m+n≡0→m≡0×n≡0 ∘ snd ∘ pred-≤-pred
<-split {zero} {suc n} = λ _ → inl (suc-≤-suc zero-≤)
<-split {suc m} {suc n} = ⊎.map suc-≤-suc (cong suc) ∘ <-split ∘ pred-≤-pred
private
acc-suc : Acc _<_ n → Acc _<_ (suc n)
acc-suc a
= acc λ y y<sn
→ case <-split y<sn of λ
{ (inl y<n) → access a y y<n
; (inr y≡n) → subst _ (sym y≡n) a
}
<-wellfounded : WellFounded _<_
<-wellfounded zero = acc λ _ → ⊥.rec ∘ ¬-<-zero
<-wellfounded (suc n) = acc-suc (<-wellfounded n)
<→≢ : n < m → ¬ n ≡ m
<→≢ {n} {m} p q = ¬m<m (subst (_< m) q p)
module _
(b₀ : ℕ)
(P : ℕ → Type₀)
(base : ∀ n → n < suc b₀ → P n)
(step : ∀ n → P n → P (suc b₀ + n))
where
open WFI (<-wellfounded)
private
dichotomy : ∀ b n → (n < b) ⊎ (Σ[ m ∈ ℕ ] n ≡ b + m)
dichotomy b n
= case n ≟ b return (λ _ → (n < b) ⊎ (Σ[ m ∈ ℕ ] n ≡ b + m)) of λ
{ (lt o) → inl o
; (eq p) → inr (0 , p ∙ sym (+-zero b))
; (gt (m , p)) → inr (suc m , sym p ∙ +-suc m b ∙ +-comm (suc m) b)
}
dichotomy<≡ : ∀ b n → (n<b : n < b) → dichotomy b n ≡ inl n<b
dichotomy<≡ b n n<b
= case dichotomy b n return (λ d → d ≡ inl n<b) of λ
{ (inl x) → cong inl (m≤n-isProp x n<b)
; (inr (m , p)) → ⊥.rec (<-asym n<b (m , sym (p ∙ +-comm b m)))
}
dichotomy+≡ : ∀ b m n → (p : n ≡ b + m) → dichotomy b n ≡ inr (m , p)
dichotomy+≡ b m n p
= case dichotomy b n return (λ d → d ≡ inr (m , p)) of λ
{ (inl n<b) → ⊥.rec (<-asym n<b (m , +-comm m b ∙ sym p))
; (inr (m' , q))
→ cong inr (Σ≡Prop (λ x → isSetℕ n (b + x)) (inj-m+ {m = b} (sym q ∙ p)))
}
b = suc b₀
lemma₁ : ∀{x y z} → x ≡ suc z + y → y < x
lemma₁ {y = y} {z} p = z , +-suc z y ∙ sym p
subStep : (n : ℕ) → (∀ m → m < n → P m) → (n < b) ⊎ (Σ[ m ∈ ℕ ] n ≡ b + m) → P n
subStep n _ (inl l) = base n l
subStep n rec (inr (m , p))
= transport (cong P (sym p)) (step m (rec m (lemma₁ p)))
wfStep : (n : ℕ) → (∀ m → m < n → P m) → P n
wfStep n rec = subStep n rec (dichotomy b n)
wfStepLemma₀ : ∀ n (n<b : n < b) rec → wfStep n rec ≡ base n n<b
wfStepLemma₀ n n<b rec = cong (subStep n rec) (dichotomy<≡ b n n<b)
wfStepLemma₁ : ∀ n rec → wfStep (b + n) rec ≡ step n (rec n (lemma₁ refl))
wfStepLemma₁ n rec
= cong (subStep (b + n) rec) (dichotomy+≡ b n (b + n) refl)
∙ transportRefl _
+induction : ∀ n → P n
+induction = induction wfStep
+inductionBase : ∀ n → (l : n < b) → +induction n ≡ base n l
+inductionBase n l = induction-compute wfStep n ∙ wfStepLemma₀ n l _
+inductionStep : ∀ n → +induction (b + n) ≡ step n (+induction n)
+inductionStep n = induction-compute wfStep (b + n) ∙ wfStepLemma₁ n _
module <-Reasoning where
-- TODO: would it be better to mirror the way it is done in the agda-stdlib?
infixr 2 _<⟨_⟩_ _≤<⟨_⟩_ _≤⟨_⟩_ _<≤⟨_⟩_ _≡<⟨_⟩_ _≡≤⟨_⟩_ _<≡⟨_⟩_ _≤≡⟨_⟩_
_<⟨_⟩_ : ∀ k → k < n → n < m → k < m
_ <⟨ p ⟩ q = <-trans p q
_≤<⟨_⟩_ : ∀ k → k ≤ n → n < m → k < m
_ ≤<⟨ p ⟩ q = ≤<-trans p q
_≤⟨_⟩_ : ∀ k → k ≤ n → n ≤ m → k ≤ m
_ ≤⟨ p ⟩ q = ≤-trans p q
_<≤⟨_⟩_ : ∀ k → k < n → n ≤ m → k < m
_ <≤⟨ p ⟩ q = <≤-trans p q
_≡≤⟨_⟩_ : ∀ k → k ≡ l → l ≤ m → k ≤ m
_ ≡≤⟨ p ⟩ q = subst (λ k → k ≤ _) (sym p) q
_≡<⟨_⟩_ : ∀ k → k ≡ l → l < m → k < m
_ ≡<⟨ p ⟩ q = _ ≡≤⟨ cong suc p ⟩ q
_≤≡⟨_⟩_ : ∀ k → k ≤ l → l ≡ m → k ≤ m
_ ≤≡⟨ p ⟩ q = subst (λ l → _ ≤ l) q p
_<≡⟨_⟩_ : ∀ k → k < l → l ≡ m → k < m
_ <≡⟨ p ⟩ q = _ ≤≡⟨ p ⟩ q
| 28.968531
| 88
| 0.462523
|
18cd5779a5b14b1846a7202d3379de355baacbc1
| 527
|
agda
|
Agda
|
test/interaction/Issue2803.agda
|
asr/eagda
|
7220bebfe9f64297880ecec40314c0090018fdd0
|
[
"BSD-3-Clause"
] | 1
|
2016-03-17T01:45:59.000Z
|
2016-03-17T01:45:59.000Z
|
test/interaction/Issue2803.agda
|
asr/eagda
|
7220bebfe9f64297880ecec40314c0090018fdd0
|
[
"BSD-3-Clause"
] | null | null | null |
test/interaction/Issue2803.agda
|
asr/eagda
|
7220bebfe9f64297880ecec40314c0090018fdd0
|
[
"BSD-3-Clause"
] | 1
|
2019-03-05T20:02:38.000Z
|
2019-03-05T20:02:38.000Z
|
-- Andreas, 2017-11-12, issue #2803
-- Problem: names of hidden variable patterns
-- can get lost during case splitting.
-- They actually get lost already during lhs type checking,
-- but it is noticed only when printed back to the user
-- during case splitting.
-- {-# OPTIONS -v tc.lhs:40 #-}
record HFun (A B : Set) : Set where
field apply : {a : A} → B
postulate A : Set
test : HFun A (A → A)
HFun.apply test {β} = {!!} -- C-c C-c
-- YIELDS:
-- HFun.apply test {a} x = ?
-- EXPECTED:
-- HFun.apply test {β} x = ?
| 21.08
| 59
| 0.633776
|
506bdc7fcd489f39ad12e65c999ea85e55446ae5
| 1,681
|
agda
|
Agda
|
Cats/Util/Reflection.agda
|
alessio-b-zak/cats
|
a3b69911c4c6ec380ddf6a0f4510d3a755734b86
|
[
"MIT"
] | null | null | null |
Cats/Util/Reflection.agda
|
alessio-b-zak/cats
|
a3b69911c4c6ec380ddf6a0f4510d3a755734b86
|
[
"MIT"
] | null | null | null |
Cats/Util/Reflection.agda
|
alessio-b-zak/cats
|
a3b69911c4c6ec380ddf6a0f4510d3a755734b86
|
[
"MIT"
] | null | null | null |
module Cats.Util.Reflection where
open import Reflection public
open import Data.List using ([])
open import Data.Unit using (⊤)
open import Function using (_∘_)
open import Level using (zero ; Lift)
open import Cats.Util.Monad using (RawMonad ; _>>=_ ; _>>_ ; return ; mapM′)
instance
tcMonad : ∀ {l} → RawMonad {l} TC
tcMonad = record
{ return = returnTC
; _>>=_ = bindTC
}
pattern argH x = arg (arg-info hidden relevant) x
pattern argD x = arg (arg-info visible relevant) x
pattern defD x = def x []
fromArg : ∀ {A} → Arg A → A
fromArg (arg _ x) = x
fromAbs : ∀ {A} → Abs A → A
fromAbs (abs _ x) = x
blockOnAnyMeta-clause : Clause → TC (Lift zero ⊤)
-- This may or may not loop if there are metas in the input term that cannot be
-- solved when this tactic is called.
{-# TERMINATING #-}
blockOnAnyMeta : Term → TC (Lift zero ⊤)
blockOnAnyMeta (var x args) = mapM′ (blockOnAnyMeta ∘ fromArg) args
blockOnAnyMeta (con c args) = mapM′ (blockOnAnyMeta ∘ fromArg) args
blockOnAnyMeta (def f args) = mapM′ (blockOnAnyMeta ∘ fromArg) args
blockOnAnyMeta (lam v t) = blockOnAnyMeta (fromAbs t)
blockOnAnyMeta (pat-lam cs args) = do
mapM′ blockOnAnyMeta-clause cs
mapM′ (blockOnAnyMeta ∘ fromArg) args
blockOnAnyMeta (pi a b) = do
blockOnAnyMeta (fromArg a)
blockOnAnyMeta (fromAbs b)
blockOnAnyMeta (sort (set t)) = blockOnAnyMeta t
blockOnAnyMeta (sort (lit n)) = return _
blockOnAnyMeta (sort unknown) = return _
blockOnAnyMeta (lit l) = return _
blockOnAnyMeta (meta x _) = blockOnMeta x
blockOnAnyMeta unknown = return _
blockOnAnyMeta-clause (clause ps t) = blockOnAnyMeta t
blockOnAnyMeta-clause (absurd-clause ps) = return _
| 28.491525
| 79
| 0.702558
|
3914404ba70d090a90d46cc2444d16f2b9167834
| 7,326
|
agda
|
Agda
|
test/Succeed/Issue854.agda
|
pthariensflame/agda
|
222c4c64b2ccf8e0fc2498492731c15e8fef32d4
|
[
"BSD-3-Clause"
] | 3
|
2015-03-28T14:51:03.000Z
|
2015-12-07T20:14:00.000Z
|
test/Succeed/Issue854.agda
|
Blaisorblade/Agda
|
802a28aa8374f15fe9d011ceb80317fdb1ec0949
|
[
"BSD-3-Clause"
] | null | null | null |
test/Succeed/Issue854.agda
|
Blaisorblade/Agda
|
802a28aa8374f15fe9d011ceb80317fdb1ec0949
|
[
"BSD-3-Clause"
] | null | null | null |
-- 2013-06-15 Andreas, issue reported by Stevan Andjelkovic
module Issue854 where
infixr 1 _⊎_
infixr 2 _×_
infixr 4 _,_
infix 4 _≡_
data ⊥ : Set where
⊥-elim : {A : Set} → ⊥ → A
⊥-elim ()
record ⊤ : Set where
constructor tt
data Bool : Set where
true false : Bool
data ℕ : Set where
zero : ℕ
suc : (n : ℕ) → ℕ
data Maybe (A : Set) : Set where
nothing : Maybe A
just : (x : A) → Maybe A
data _≡_ {A : Set} (x : A) : A → Set where
refl : x ≡ x
data _⊎_ (A : Set) (B : Set) : Set where
inj₁ : (x : A) → A ⊎ B
inj₂ : (y : B) → A ⊎ B
[_,_] : ∀ {A : Set} {B : Set} {C : A ⊎ B → Set} →
((x : A) → C (inj₁ x)) → ((x : B) → C (inj₂ x)) →
((x : A ⊎ B) → C x)
[ f , g ] (inj₁ x) = f x
[ f , g ] (inj₂ y) = g y
[_,_]₁ : ∀ {A : Set} {B : Set} {C : A ⊎ B → Set₁} →
((x : A) → C (inj₁ x)) → ((x : B) → C (inj₂ x)) →
((x : A ⊎ B) → C x)
[ f , g ]₁ (inj₁ x) = f x
[ f , g ]₁ (inj₂ y) = g y
record Σ (A : Set) (B : A → Set) : Set where
constructor _,_
field
proj₁ : A
proj₂ : B proj₁
open Σ public
_×_ : Set → Set → Set
A × B = Σ A λ _ → B
uncurry₁ : {A : Set} {B : A → Set} {C : Σ A B → Set₁} →
((x : A) → (y : B x) → C (x , y)) →
((p : Σ A B) → C p)
uncurry₁ f (x , y) = f x y
------------------------------------------------------------------------
infix 5 _◃_
infixr 1 _⊎C_
infixr 2 _×C_
record Container : Set₁ where
constructor _◃_
field
Shape : Set
Position : Shape → Set
open Container public
⟦_⟧ : Container → (Set → Set)
⟦ S ◃ P ⟧ X = Σ S λ s → P s → X
idC : Container
idC = ⊤ ◃ λ _ → ⊤
constC : Set → Container
constC X = X ◃ λ _ → ⊥
𝟘 = constC ⊥
𝟙 = constC ⊤
_⊎C_ : Container → Container → Container
S ◃ P ⊎C S′ ◃ P′ = (S ⊎ S′) ◃ [ P , P′ ]₁
_×C_ : Container → Container → Container
S ◃ P ×C S′ ◃ P′ = (S × S′) ◃ uncurry₁ (λ s s′ → P s ⊎ P′ s′)
data μ (C : Container) : Set where
⟨_⟩ : ⟦ C ⟧ (μ C) → μ C
_⋆C_ : Container → Set → Container
C ⋆C X = constC X ⊎C C
_⋆_ : Container → Set → Set
C ⋆ X = μ (C ⋆C X)
AlgIter : Container → Set → Set
AlgIter C X = ⟦ C ⟧ X → X
iter : ∀ {C X} → AlgIter C X → μ C → X
iter φ ⟨ s , k ⟩ = φ (s , λ p → iter φ (k p))
AlgRec : Container → Set → Set
AlgRec C X = ⟦ C ⟧ (μ C × X) → X
rec : ∀ {C X} → AlgRec C X → μ C → X
rec φ ⟨ s , k ⟩ = φ (s , λ p → (k p , rec φ (k p)))
return : ∀ {C X} → X → C ⋆ X
return x = ⟨ inj₁ x , ⊥-elim ⟩
do : ∀ {C X} → ⟦ C ⟧ (C ⋆ X) → C ⋆ X
do (s , k) = ⟨ inj₂ s , k ⟩
_>>=_ : ∀ {C X Y} → C ⋆ X → (X → C ⋆ Y) → C ⋆ Y
_>>=_ {C}{X}{Y} m k = iter φ m
where
φ : AlgIter (C ⋆C X) (C ⋆ Y)
φ (inj₁ x , _) = k x
φ (inj₂ s , k) = do (s , k)
------------------------------------------------------------------------
_↠_ : Set → Set → Container
I ↠ O = I ◃ λ _ → O
State : Set → Container
State S = (⊤ ↠ S) -- get
⊎C (S ↠ ⊤) -- put
get : ∀ {S} → State S ⋆ S
get = do (inj₁ tt , return)
put : ∀ {S} → S → State S ⋆ ⊤
put s = do (inj₂ s , return)
Homo : Container → Set → Set → Container → Set → Set
Homo Σ X I Σ′ Y = AlgRec (Σ ⋆C X) (I → Σ′ ⋆ Y)
Pseudohomo : Container → Set → Set → Container → Set → Set
Pseudohomo Σ X I Σ′ Y =
⟦ Σ ⋆C X ⟧ (((Σ ⊎C Σ′) ⋆ X) × (I → Σ′ ⋆ Y)) → I → Σ′ ⋆ Y
state : ∀ {Σ S X} → Pseudohomo (State S) X S Σ (X × S)
state (inj₁ x , _) = λ s → return (x , s) -- return
state (inj₂ (inj₁ _) , k) = λ s → proj₂ (k s) s -- get
state (inj₂ (inj₂ s) , k) = λ _ → proj₂ (k tt) s -- put
Abort : Container
Abort = ⊤ ↠ ⊥
aborting : ∀ {X} → Abort ⋆ X
aborting = do (tt , ⊥-elim)
abort : ∀ {Σ X} → Pseudohomo Abort X ⊤ Σ (Maybe X)
abort (inj₁ x , _) _ = return (just x) -- return
abort (inj₂ _ , _) _ = return nothing -- abort
------------------------------------------------------------------------
record _⇒_ (C C′ : Container) : Set where
field
shape : Shape C → Shape C′
position : ∀ {s} → Position C′ (shape s) → Position C s
open _⇒_ public
idMorph : ∀ {C} → C ⇒ C
idMorph = record { shape = λ s → s; position = λ p → p }
inlMorph : ∀ {C C′ : Container} → C ⇒ (C ⊎C C′)
inlMorph = record
{ shape = inj₁
; position = λ p → p
}
swapMorph : ∀ {C C′} → (C ⊎C C′) ⇒ (C′ ⊎C C)
swapMorph {C}{C′}= record
{ shape = sh
; position = λ {s} p → pos {s} p
}
where
sh : Shape C ⊎ Shape C′ → Shape C′ ⊎ Shape C
sh (inj₁ s) = inj₂ s
sh (inj₂ s′) = inj₁ s′
pos : ∀ {s} → Position (C′ ⊎C C) (sh s) → Position (C ⊎C C′) s
pos {inj₁ s} p = p
pos {inj₂ s′} p′ = p′
⟪_⟫ : ∀ {C C′ X} → C ⇒ C′ → ⟦ C ⟧ X → ⟦ C′ ⟧ X
⟪ m ⟫ xs = shape m (proj₁ xs) , λ p′ → proj₂ xs (position m p′)
⟪_⟫Homo : ∀ {C C′ X} → C ⇒ C′ → Homo C X ⊤ C′ X
⟪ m ⟫Homo (inj₁ x , _) _ = return x
⟪ m ⟫Homo (inj₂ s , k) _ = let (s′ , k′) = ⟪ m ⟫ (s , k)
in do (s′ , λ p′ → proj₂ (k′ p′) tt)
natural : ∀ {C C′ X} → C ⇒ C′ → C ⋆ X → C′ ⋆ X
natural f m = rec ⟪ f ⟫Homo m tt
inl : ∀ {C C′ X} → C ⋆ X → (C ⊎C C′) ⋆ X
inl = natural inlMorph
squeeze : ∀ {Σ Σ′ X} → ((Σ ⊎C Σ′) ⊎C Σ′) ⋆ X → (Σ ⊎C Σ′) ⋆ X
squeeze = natural m
where
m = record
{ shape = [ (λ x → x) , inj₂ ]
; position = λ { {inj₁ x} p → p ; {inj₂ x} p → p}
}
lift : ∀ {Σ Σ′ X Y I} → Pseudohomo Σ X I Σ′ Y →
Pseudohomo (Σ ⊎C Σ′) X I Σ′ Y
lift φ (inj₁ x , _) i = φ (inj₁ x , ⊥-elim) i
lift φ (inj₂ (inj₁ s) , k) i = φ (inj₂ s , λ p →
let (w , ih) = k p in squeeze w , ih) i
lift φ (inj₂ (inj₂ s′) , k′) i = do (s′ , λ p′ → proj₂ (k′ p′) i)
weaken : ∀ {Σ Σ′ Σ″ Σ‴ X Y I} → Homo Σ′ X I Σ″ Y →
Σ ⇒ Σ′ → Σ″ ⇒ Σ‴ → Homo Σ X I Σ‴ Y
weaken {Σ}{Σ′}{Σ″}{Σ‴}{X}{Y} φ f g (s , k) i = w‴
where
w : Σ ⋆ X
w = ⟨ s , (λ p → proj₁ (k p)) ⟩
w′ : Σ′ ⋆ X
w′ = natural f w
w″ : Σ″ ⋆ Y
w″ = rec φ w′ i
w‴ : Σ‴ ⋆ Y
w‴ = natural g w″
⌈_⌉Homo : ∀ {Σ Σ′ X Y I} → Pseudohomo Σ X I Σ′ Y → Homo Σ X I Σ′ Y
⌈ φ ⌉Homo (inj₁ x , _) = φ (inj₁ x , ⊥-elim)
⌈ φ ⌉Homo (inj₂ s , k) = φ (inj₂ s , λ p → let (w , ih) = k p
in inl w , ih)
run : ∀ {Σ Σ′ Σ″ Σ‴ X Y I} → Pseudohomo Σ X I Σ′ Y →
Σ″ ⇒ (Σ ⊎C Σ′) → Σ′ ⇒ Σ‴ →
Σ″ ⋆ X → I → Σ‴ ⋆ Y
run φ p q = rec (weaken ⌈ lift φ ⌉Homo p q)
------------------------------------------------------------------------
prog : (State ℕ ⊎C Abort) ⋆ Bool
prog =
⟨ inj₂ (inj₁ (inj₁ tt)) , (λ n → -- get >>= λ n →
⟨ inj₂ (inj₁ (inj₂ (suc n))) , (λ _ → -- put (suc n)
⟨ inj₂ (inj₂ tt) , (λ _ → -- aborting
return true) ⟩) ⟩) ⟩
progA : State ℕ ⋆ Maybe Bool
progA = run abort swapMorph idMorph prog tt
progS : ℕ → Abort ⋆ (Bool × ℕ)
progS = run state idMorph idMorph prog
progAS : ℕ → 𝟘 ⋆ (Maybe Bool × ℕ)
progAS = run state inlMorph idMorph progA
progSA : ℕ → 𝟘 ⋆ Maybe (Bool × ℕ)
progSA n = run abort inlMorph idMorph (progS n) tt
testSA : progSA zero ≡ return nothing
testSA = refl
testAS : progAS zero ≡ return (nothing , suc zero)
testAS = refl
-- The last statement seemed to make the type checker loop.
-- But it just created huge terms during the conversion check
-- and never finished.
-- These terms contained many projection redexes
-- (projection applied to record value).
-- After changing the strategy, such that these redexes are,
-- like beta-redexes, removed immediately in internal syntax,
-- the code checks instantaneously.
| 25.262069
| 76
| 0.464646
|
236350e5e7da8d87a297770766f564b57f3997a9
| 973
|
agda
|
Agda
|
src/Integer/Difference/Properties.agda
|
kcsmnt0/numbers
|
67ea7b96228c592daf79e800ebe4a7c12ed7221e
|
[
"MIT"
] | 9
|
2019-05-20T01:29:41.000Z
|
2020-01-16T07:16:26.000Z
|
src/Integer/Difference/Properties.agda
|
kcsmnt0/numbers
|
67ea7b96228c592daf79e800ebe4a7c12ed7221e
|
[
"MIT"
] | null | null | null |
src/Integer/Difference/Properties.agda
|
kcsmnt0/numbers
|
67ea7b96228c592daf79e800ebe4a7c12ed7221e
|
[
"MIT"
] | null | null | null |
module Integer.Difference.Properties where
open import Data.Product as Σ
open import Data.Product.Relation.Pointwise.NonDependent
open import Data.Unit
open import Equality
open import Integer.Difference
open import Natural as ℕ using (ℕ; zero; suc)
open import Quotient as /
open import Relation.Binary
open import Syntax
open Equality.FunctionProperties
+-comm : Commutative {A = ℤ} _+_
+-comm =
⟦⟧-≗₂ uip _ _ λ where
(a – b) (c – d) →
cong ⟦_⟧ (cong₂ _–_ (⟨ ℕ.+-comm a c ⟩) (⟨ ℕ.+-comm d b ⟩))
+-identityˡ : LeftIdentity {A = ℤ} 0 _+_
+-identityˡ =
⟦⟧-≗ uip _ _ λ where
(a – b) →
cong (λ z → ⟦ a – z ⟧) ⟨ ℕ.+-identityʳ b ⟩
+-identityʳ : RightIdentity {A = ℤ} 0 _+_
+-identityʳ =
⟦⟧-≗ uip _ _ λ where
(a – b) →
cong (λ z → ⟦ z – b ⟧) ⟨ ℕ.+-identityʳ a ⟩
+-assoc : Associative {A = ℤ} _+_
+-assoc =
⟦⟧-≗₃ uip _ _
λ where
(a – b) (c – d) (e – f) →
cong ⟦_⟧ (cong₂ _–_ (sym (ℕ.+-assoc a _ _)) (ℕ.+-assoc f _ _))
| 24.948718
| 70
| 0.589928
|
cbd5d9bfd15a108d592d4f72afbfe97628ae1923
| 1,295
|
agda
|
Agda
|
src/data/lib/prim/Agda/Builtin/Float.agda
|
vlopezj/agda
|
ff4d89e75970cf27599fb9f572bd43c9455cbb56
|
[
"BSD-3-Clause"
] | 2
|
2019-10-29T09:40:30.000Z
|
2020-09-20T00:28:57.000Z
|
src/data/lib/prim/Agda/Builtin/Float.agda
|
vikfret/agda
|
49ad0b3f0d39c01bc35123478b857e702b29fb9d
|
[
"BSD-3-Clause"
] | 3
|
2018-11-14T15:31:44.000Z
|
2019-04-01T19:39:26.000Z
|
src/data/lib/prim/Agda/Builtin/Float.agda
|
vikfret/agda
|
49ad0b3f0d39c01bc35123478b857e702b29fb9d
|
[
"BSD-3-Clause"
] | 1
|
2021-04-01T18:30:09.000Z
|
2021-04-01T18:30:09.000Z
|
{-# OPTIONS --without-K --safe --no-sized-types --no-guardedness #-}
module Agda.Builtin.Float where
open import Agda.Builtin.Bool
open import Agda.Builtin.Nat
open import Agda.Builtin.Int
open import Agda.Builtin.Word
open import Agda.Builtin.String
postulate Float : Set
{-# BUILTIN FLOAT Float #-}
primitive
primFloatToWord64 : Float → Word64
primFloatEquality : Float → Float → Bool
primFloatLess : Float → Float → Bool
primFloatNumericalEquality : Float → Float → Bool
primFloatNumericalLess : Float → Float → Bool
primNatToFloat : Nat → Float
primFloatPlus : Float → Float → Float
primFloatMinus : Float → Float → Float
primFloatTimes : Float → Float → Float
primFloatNegate : Float → Float
primFloatDiv : Float → Float → Float
primFloatSqrt : Float → Float
primRound : Float → Int
primFloor : Float → Int
primCeiling : Float → Int
primExp : Float → Float
primLog : Float → Float
primSin : Float → Float
primCos : Float → Float
primTan : Float → Float
primASin : Float → Float
primACos : Float → Float
primATan : Float → Float
primATan2 : Float → Float → Float
primShowFloat : Float → String
| 32.375
| 68
| 0.636293
|
0b1dfbb1991d9ca6211a1be6f28d42678048a95c
| 658
|
agda
|
Agda
|
Rings/Units/Lemmas.agda
|
Smaug123/agdaproofs
|
0f4230011039092f58f673abcad8fb0652e6b562
|
[
"MIT"
] | 4
|
2019-08-08T12:44:19.000Z
|
2022-01-28T06:04:15.000Z
|
Rings/Units/Lemmas.agda
|
Smaug123/agdaproofs
|
0f4230011039092f58f673abcad8fb0652e6b562
|
[
"MIT"
] | 14
|
2019-01-06T21:11:59.000Z
|
2020-04-11T11:03:39.000Z
|
Rings/Units/Lemmas.agda
|
Smaug123/agdaproofs
|
0f4230011039092f58f673abcad8fb0652e6b562
|
[
"MIT"
] | 1
|
2021-11-29T13:23:07.000Z
|
2021-11-29T13:23:07.000Z
|
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Rings.Definition
module Rings.Units.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
open import Rings.Units.Definition R
open import Rings.Ideals.Definition R
open Ring R
open Setoid S
open Equivalence eq
unitImpliesGeneratedIdealEverything : {x : A} → Unit x → {y : A} → generatedIdealPred x y
unitImpliesGeneratedIdealEverything {x} (a , xa=1) {y} = (a * y) , transitive *Associative (transitive (*WellDefined xa=1 reflexive) identIsIdent)
| 32.9
| 146
| 0.720365
|
d06f465e805d28fe8e84b04cc488a8e8cfb85c6c
| 2,469
|
agda
|
Agda
|
PiFrac/Syntax.agda
|
DreamLinuxer/popl21-artifact
|
fb380f2e67dcb4a94f353dbaec91624fcb5b8933
|
[
"MIT"
] | 5
|
2020-10-07T12:07:53.000Z
|
2020-10-10T21:41:32.000Z
|
PiFrac/Syntax.agda
|
DreamLinuxer/popl21-artifact
|
fb380f2e67dcb4a94f353dbaec91624fcb5b8933
|
[
"MIT"
] | null | null | null |
PiFrac/Syntax.agda
|
DreamLinuxer/popl21-artifact
|
fb380f2e67dcb4a94f353dbaec91624fcb5b8933
|
[
"MIT"
] | 1
|
2021-11-04T06:54:45.000Z
|
2021-11-04T06:54:45.000Z
|
module PiFrac.Syntax where
open import Data.Empty
open import Data.Unit
open import Data.Sum
open import Data.Product
infixr 12 _×ᵤ_
infixr 11 _+ᵤ_
infixr 50 _⨾_
infixr 10 _↔_
infixr 70 _⊕_
infixr 70 _⊗_
infix 99 !_
infix 99 𝟙/_
mutual
-- Types
data 𝕌 : Set where
𝟘 : 𝕌
𝟙 : 𝕌
_+ᵤ_ : 𝕌 → 𝕌 → 𝕌
_×ᵤ_ : 𝕌 → 𝕌 → 𝕌
𝟙/_ : {t : 𝕌} → ⟦ t ⟧ → 𝕌
data ◯ : Set where
↻ : ◯
⟦_⟧ : (A : 𝕌) → Set
⟦ 𝟘 ⟧ = ⊥
⟦ 𝟙 ⟧ = ⊤
⟦ t₁ +ᵤ t₂ ⟧ = ⟦ t₁ ⟧ ⊎ ⟦ t₂ ⟧
⟦ t₁ ×ᵤ t₂ ⟧ = ⟦ t₁ ⟧ × ⟦ t₂ ⟧
⟦ 𝟙/ v ⟧ = ◯
-- Combinators
data _↔_ : 𝕌 → 𝕌 → Set where
unite₊l : {t : 𝕌} → 𝟘 +ᵤ t ↔ t
uniti₊l : {t : 𝕌} → t ↔ 𝟘 +ᵤ t
swap₊ : {t₁ t₂ : 𝕌} → t₁ +ᵤ t₂ ↔ t₂ +ᵤ t₁
assocl₊ : {t₁ t₂ t₃ : 𝕌} → t₁ +ᵤ (t₂ +ᵤ t₃) ↔ (t₁ +ᵤ t₂) +ᵤ t₃
assocr₊ : {t₁ t₂ t₃ : 𝕌} → (t₁ +ᵤ t₂) +ᵤ t₃ ↔ t₁ +ᵤ (t₂ +ᵤ t₃)
unite⋆l : {t : 𝕌} → 𝟙 ×ᵤ t ↔ t
uniti⋆l : {t : 𝕌} → t ↔ 𝟙 ×ᵤ t
swap⋆ : {t₁ t₂ : 𝕌} → t₁ ×ᵤ t₂ ↔ t₂ ×ᵤ t₁
assocl⋆ : {t₁ t₂ t₃ : 𝕌} → t₁ ×ᵤ (t₂ ×ᵤ t₃) ↔ (t₁ ×ᵤ t₂) ×ᵤ t₃
assocr⋆ : {t₁ t₂ t₃ : 𝕌} → (t₁ ×ᵤ t₂) ×ᵤ t₃ ↔ t₁ ×ᵤ (t₂ ×ᵤ t₃)
absorbr : {t : 𝕌} → 𝟘 ×ᵤ t ↔ 𝟘
factorzl : {t : 𝕌} → 𝟘 ↔ 𝟘 ×ᵤ t
dist : {t₁ t₂ t₃ : 𝕌} → (t₁ +ᵤ t₂) ×ᵤ t₃ ↔ (t₁ ×ᵤ t₃) +ᵤ (t₂ ×ᵤ t₃)
factor : {t₁ t₂ t₃ : 𝕌} → (t₁ ×ᵤ t₃) +ᵤ (t₂ ×ᵤ t₃) ↔ (t₁ +ᵤ t₂) ×ᵤ t₃
id↔ : {t : 𝕌} → t ↔ t
_⨾_ : {t₁ t₂ t₃ : 𝕌} → (t₁ ↔ t₂) → (t₂ ↔ t₃) → (t₁ ↔ t₃)
_⊕_ : {t₁ t₂ t₃ t₄ : 𝕌} → (t₁ ↔ t₃) → (t₂ ↔ t₄) → (t₁ +ᵤ t₂ ↔ t₃ +ᵤ t₄)
_⊗_ : {t₁ t₂ t₃ t₄ : 𝕌} → (t₁ ↔ t₃) → (t₂ ↔ t₄) → (t₁ ×ᵤ t₂ ↔ t₃ ×ᵤ t₄)
ηₓ : {t : 𝕌} (v : ⟦ t ⟧) → 𝟙 ↔ t ×ᵤ 𝟙/ v
εₓ : {t : 𝕌} (v : ⟦ t ⟧) → t ×ᵤ 𝟙/ v ↔ 𝟙
-- Some useful combinators
unite⋆r : {t : 𝕌} → t ×ᵤ 𝟙 ↔ t
unite⋆r = swap⋆ ⨾ unite⋆l
uniti⋆r : {t : 𝕌} → t ↔ t ×ᵤ 𝟙
uniti⋆r = uniti⋆l ⨾ swap⋆
distl : {t₁ t₂ t₃ : 𝕌} → t₁ ×ᵤ (t₂ +ᵤ t₃) ↔ (t₁ ×ᵤ t₂) +ᵤ (t₁ ×ᵤ t₃)
distl = swap⋆ ⨾ dist ⨾ (swap⋆ ⊕ swap⋆)
factorl : {t₁ t₂ t₃ : 𝕌 } → (t₁ ×ᵤ t₂) +ᵤ (t₁ ×ᵤ t₃) ↔ t₁ ×ᵤ (t₂ +ᵤ t₃)
factorl = (swap⋆ ⊕ swap⋆) ⨾ factor ⨾ swap⋆
-- Inverses of combinators
!_ : {A B : 𝕌} → A ↔ B → B ↔ A
! unite₊l = uniti₊l
! uniti₊l = unite₊l
! swap₊ = swap₊
! assocl₊ = assocr₊
! assocr₊ = assocl₊
! unite⋆l = uniti⋆l
! uniti⋆l = unite⋆l
! swap⋆ = swap⋆
! assocl⋆ = assocr⋆
! assocr⋆ = assocl⋆
! absorbr = factorzl
! factorzl = absorbr
! dist = factor
! factor = dist
! id↔ = id↔
! (c₁ ⨾ c₂) = ! c₂ ⨾ ! c₁
! (c₁ ⊕ c₂) = (! c₁) ⊕ (! c₂)
! (c₁ ⊗ c₂) = (! c₁) ⊗ (! c₂)
! (ηₓ v) = εₓ v
! (εₓ v) = ηₓ v
| 26.548387
| 78
| 0.447955
|
1804b500385fe2204f3a067f494b348f28225094
| 116
|
agda
|
Agda
|
test/succeed/Issue1110a.agda
|
larrytheliquid/agda
|
477c8c37f948e6038b773409358fd8f38395f827
|
[
"MIT"
] | 1
|
2018-10-10T17:08:44.000Z
|
2018-10-10T17:08:44.000Z
|
test/succeed/Issue1110a.agda
|
masondesu/agda
|
70c8a575c46f6a568c7518150a1a64fcd03aa437
|
[
"MIT"
] | null | null | null |
test/succeed/Issue1110a.agda
|
masondesu/agda
|
70c8a575c46f6a568c7518150a1a64fcd03aa437
|
[
"MIT"
] | 1
|
2022-03-12T11:35:18.000Z
|
2022-03-12T11:35:18.000Z
|
-- Andreas, 2014-05-17
open import Common.Prelude
open import Common.Equality
test : Nat
test rewrite refl = zero
| 14.5
| 27
| 0.758621
|
fb2ee7f35f762a584d9bbceffb67ede02193fe90
| 453
|
agda
|
Agda
|
test/Succeed/Issue4172-without-K.agda
|
mdimjasevic/agda
|
8fb548356b275c7a1e79b768b64511ae937c738b
|
[
"BSD-3-Clause"
] | 1,989
|
2015-01-09T23:51:16.000Z
|
2022-03-30T18:20:48.000Z
|
test/Succeed/Issue4172-without-K.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 4,066
|
2015-01-10T11:24:51.000Z
|
2022-03-31T21:14:49.000Z
|
test/Succeed/Issue4172-without-K.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 371
|
2015-01-03T14:04:08.000Z
|
2022-03-30T19:00:30.000Z
|
{-# OPTIONS --without-K #-}
record Erased (A : Set) : Set where
constructor [_]
field
@0 erased : A
open Erased
data W (A : Set) (B : A → Set) : Set where
sup : (x : A) → (B x → W A B) → W A B
lemma :
{A : Set} {B : A → Set} →
Erased (W A B) → W (Erased A) (λ x → Erased (B (erased x)))
lemma [ sup x f ] = sup [ x ] λ ([ y ]) → lemma [ f y ]
data ⊥ : Set where
data E : Set where
c : E → E
magic : @0 E → ⊥
magic (c e) = magic e
| 18.12
| 61
| 0.501104
|
4a0dd0759916ab5c4b63cbc9edd250183f6fee43
| 334
|
agda
|
Agda
|
src/sets/finite/hlevel.agda
|
pcapriotti/agda-base
|
bbbc3bfb2f80ad08c8e608cccfa14b83ea3d258c
|
[
"BSD-3-Clause"
] | 20
|
2015-06-12T12:20:17.000Z
|
2022-02-01T11:25:54.000Z
|
src/sets/finite/hlevel.agda
|
pcapriotti/agda-base
|
bbbc3bfb2f80ad08c8e608cccfa14b83ea3d258c
|
[
"BSD-3-Clause"
] | 4
|
2015-02-02T14:32:16.000Z
|
2016-10-26T11:57:26.000Z
|
src/sets/finite/hlevel.agda
|
pcapriotti/agda-base
|
bbbc3bfb2f80ad08c8e608cccfa14b83ea3d258c
|
[
"BSD-3-Clause"
] | 4
|
2015-02-02T12:17:00.000Z
|
2019-05-04T19:31:00.000Z
|
{-# OPTIONS --without-K #-}
module sets.finite.level where
open import sum
open import function.isomorphism.core
open import hott.level.core
open import hott.level.closure
open import hott.level.sets
open import sets.finite.core
finite-h2 : ∀ {i}{A : Set i} → IsFinite A → h 2 A
finite-h2 (n , fA) = iso-level (sym≅ fA) (fin-set n)
| 25.692308
| 52
| 0.718563
|
50f93afc872c91cc48a3769e77d0582876753799
| 1,785
|
agda
|
Agda
|
src/Tactic/Nat/Coprime.agda
|
L-TChen/agda-prelude
|
158d299b1b365e186f00d8ef5b8c6844235ee267
|
[
"MIT"
] | 111
|
2015-01-05T11:28:15.000Z
|
2022-02-12T23:29:26.000Z
|
src/Tactic/Nat/Coprime.agda
|
L-TChen/agda-prelude
|
158d299b1b365e186f00d8ef5b8c6844235ee267
|
[
"MIT"
] | 59
|
2016-02-09T05:36:44.000Z
|
2022-01-14T07:32:36.000Z
|
src/Tactic/Nat/Coprime.agda
|
L-TChen/agda-prelude
|
158d299b1b365e186f00d8ef5b8c6844235ee267
|
[
"MIT"
] | 24
|
2015-03-12T18:03:45.000Z
|
2021-04-22T06:10:41.000Z
|
-- Tactic for proving coprimality.
-- Finds Coprime hypotheses in the context.
module Tactic.Nat.Coprime where
import Agda.Builtin.Nat as Builtin
open import Prelude
open import Control.Monad.Zero
open import Control.Monad.State
open import Container.List
open import Container.Traversable
open import Numeric.Nat
open import Tactic.Reflection
open import Tactic.Reflection.Parse
open import Tactic.Reflection.Quote
open import Tactic.Nat.Coprime.Problem
open import Tactic.Nat.Coprime.Decide
open import Tactic.Nat.Coprime.Reflect
private
Proof : Problem → Env → Set
Proof Q ρ with canProve Q
... | true = ⟦ Q ⟧p ρ
... | false = ⊤
erasePrf : ∀ Q {ρ} → ⟦ Q ⟧p ρ → ⟦ Q ⟧p ρ
erasePrf ([] ⊨ a ⋈ b) Ξ = eraseEquality Ξ
erasePrf (ψ ∷ Γ ⊨ φ) Ξ = λ H → erasePrf (Γ ⊨ φ) (Ξ H)
proof : ∀ Q ρ → Proof Q ρ
proof Q ρ with canProve Q | sound Q
... | true | prf = erasePrf Q (prf refl ρ)
... | false | _ = _
-- For the error message
unquoteE : List Term → Exp → Term
unquoteE ρ (atom x) = fromMaybe (lit (nat 0)) (index ρ x)
unquoteE ρ (e ⊗ e₁) = def₂ (quote _*_) (unquoteE ρ e) (unquoteE ρ e₁)
unquoteF : List Term → Formula → Term
unquoteF ρ (a ⋈ b) = def₂ (quote Coprime) (unquoteE ρ a) (unquoteE ρ b)
macro
auto-coprime : Tactic
auto-coprime ?hole = withNormalisation true $ do
goal ← inferType ?hole
ensureNoMetas goal
cxt ← reverse <$> getContext
(_ , Hyps , Q) , ρ ← runParse (parseProblem (map unArg cxt) goal)
unify ?hole (def (quote proof) $ map vArg (` Q ∷ quotedEnv ρ ∷ Hyps))
<|> do
case Q of λ where
(Γ ⊨ φ) → typeErrorFmt "Cannot prove %t from %e"
(unquoteF ρ φ)
(punctuate (strErr "and") (map (termErr ∘ unquoteF ρ) Γ))
| 30.775862
| 90
| 0.633613
|
18058a54590e7181a30a4866610a2f027f835e67
| 2,179
|
agda
|
Agda
|
test/interaction/Highlighting.agda
|
vlopezj/agda
|
ff4d89e75970cf27599fb9f572bd43c9455cbb56
|
[
"BSD-3-Clause"
] | 1
|
2019-09-27T06:54:44.000Z
|
2019-09-27T06:54:44.000Z
|
test/interaction/Highlighting.agda
|
vlopezj/agda
|
ff4d89e75970cf27599fb9f572bd43c9455cbb56
|
[
"BSD-3-Clause"
] | 3
|
2018-11-14T15:31:44.000Z
|
2019-04-01T19:39:26.000Z
|
test/interaction/Highlighting.agda
|
vlopezj/agda
|
ff4d89e75970cf27599fb9f572bd43c9455cbb56
|
[
"BSD-3-Clause"
] | null | null | null |
module Highlighting where
Set-one : Set₂
Set-one = Set₁
record R (A : Set) : Set-one where
constructor con
field X : Set
F : Set → Set → Set
F A B = B
field P : F A X → Set
-- highlighting of non-terminating definition
Q : F A X → Set
Q = Q
postulate P : _
open import Highlighting.M using (ℕ) renaming
( _+_ to infixl 5 _⊕_
; _*_ to infixl 7 _⊗_
)
data D (A : Set) : Set-one where
d : let X = D in X A
postulate _+_ _×_ : Set → Set → Set
infixl 4 _×_ _+_
-- Issue #2140: the operators should be highlighted also in the
-- fixity declaration.
-- Issue #3120, jump-to-definition for record field tags
-- in record expressions and patterns.
anR : ∀ A → R A
anR A = record { X = A ; P = λ _ → A }
idR : ∀ A → R A → R A
idR A r@record { X = X; P = P } = record r { X = X }
record S (A : Set) : Set where
field
X : A
idR' : ∀ A → R A → R A
idR' A r@record { X = X; P = P } = record r { X = X }
open S
bla : ∀{A} → A → S A
bla x .X = x
-- Issue #3825: highlighting of unsolved metas in record{M} expressions
record R₂ (A : Set) : Set where
field
impl : {a : A} → A
module M {A : Set} where
impl : {a : A} → A -- yellow should not be here
impl {a} = a
r₂ : ∀{A} → R₂ A
r₂ = record {M} -- just because there is an unsolved meta here
-- End issue #3825
-- Issue #3855: highlighting of quantity attributes.
-- @0 and @erased should be highlighted as symbols.
idPoly0 : {@0 A : Set} → A → A
idPoly0 x = x
idPolyE : {@erased A : Set} → A → A
idPolyE x = x
-- Issue #3989: Shadowed repeated variables in telescopes should by
-- default /not/ be highlighted.
Issue-3989 : (A A : Set) → Set
Issue-3989 _ A = A
-- Issue #4356.
open import Agda.Builtin.Sigma
Issue-4356₁ : Σ Set (λ _ → Set) → Σ Set (λ _ → Set)
Issue-4356₁ = λ P@(A , B) → P
Issue-4356₂ : Σ Set (λ _ → Set) → Set
Issue-4356₂ = λ (A , B) → A
Issue-4356₃ : Σ Set (λ _ → Set) → Σ Set (λ _ → Set)
Issue-4356₃ P = let Q@(A , B) = P in Q
Issue-4356₄ : Σ Set (λ _ → Set) → Set
Issue-4356₄ P = let (A , B) = P in B
Issue-4356₅ : Σ Set (λ _ → Set) → Σ Set (λ _ → Set)
Issue-4356₅ P@(A , B) = P
Issue-4356₆ : Σ Set (λ _ → Set) → Set
Issue-4356₆ (A , B) = B
| 20.175926
| 71
| 0.590179
|
0b3838e12fd78adcab5f30797e826f529601ec05
| 1,423
|
agda
|
Agda
|
test/asset/agda-stdlib-1.0/Data/Product/Function/Dependent/Propositional/WithK.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | 5
|
2020-10-07T12:07:53.000Z
|
2020-10-10T21:41:32.000Z
|
test/asset/agda-stdlib-1.0/Data/Product/Function/Dependent/Propositional/WithK.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | null | null | null |
test/asset/agda-stdlib-1.0/Data/Product/Function/Dependent/Propositional/WithK.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | 1
|
2021-11-04T06:54:45.000Z
|
2021-11-04T06:54:45.000Z
|
------------------------------------------------------------------------
-- The Agda standard library
--
-- Dependent product combinators for propositional equality
-- preserving functions
------------------------------------------------------------------------
{-# OPTIONS --with-K --safe #-}
module Data.Product.Function.Dependent.Propositional.WithK where
open import Data.Product
open import Data.Product.Function.Dependent.Setoid
open import Data.Product.Relation.Binary.Pointwise.Dependent
open import Data.Product.Relation.Binary.Pointwise.Dependent.WithK
open import Function.Equality using (_⟨$⟩_)
open import Function.Injection as Inj using (_↣_; module Injection)
open import Function.Inverse as Inv using (_↔_; module Inverse)
import Relation.Binary.HeterogeneousEquality as H
------------------------------------------------------------------------
-- Combinator for Injection
module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
{b₁ b₂} {B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂}
where
↣ : ∀ (A₁↣A₂ : A₁ ↣ A₂) →
(∀ {x} → B₁ x ↣ B₂ (Injection.to A₁↣A₂ ⟨$⟩ x)) →
Σ A₁ B₁ ↣ Σ A₂ B₂
↣ A₁↣A₂ B₁↣B₂ =
Inverse.injection Pointwise-≡↔≡ ⟨∘⟩
injection (H.indexedSetoid B₂) A₁↣A₂
(Inverse.injection (H.≡↔≅ B₂) ⟨∘⟩
B₁↣B₂ ⟨∘⟩
Inverse.injection (Inv.sym (H.≡↔≅ B₁))) ⟨∘⟩
Inverse.injection (Inv.sym Pointwise-≡↔≡)
where open Inj using () renaming (_∘_ to _⟨∘⟩_)
| 36.487179
| 72
| 0.567814
|
d02a88f9e15475a57f9b2bee7ef392c2396794b7
| 767
|
agda
|
Agda
|
test/Succeed/ReflectionBlockOnMeta.agda
|
mdimjasevic/agda
|
8fb548356b275c7a1e79b768b64511ae937c738b
|
[
"BSD-3-Clause"
] | 1,989
|
2015-01-09T23:51:16.000Z
|
2022-03-30T18:20:48.000Z
|
test/Succeed/ReflectionBlockOnMeta.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 4,066
|
2015-01-10T11:24:51.000Z
|
2022-03-31T21:14:49.000Z
|
test/Succeed/ReflectionBlockOnMeta.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 371
|
2015-01-03T14:04:08.000Z
|
2022-03-30T19:00:30.000Z
|
module _ where
open import Common.Prelude hiding (_>>=_; _<$>_)
open import Common.Reflection
infixl 8 _<$>_
_<$>_ : ∀ {a b} {A : Set a} {B : Set b} → (A → B) → TC A → TC B
f <$> m = m >>= λ x → returnTC (f x)
macro
default : Tactic
default hole =
inferType hole >>= λ goal →
reduce goal >>= λ
{ (def (quote Nat) []) → unify hole (lit (nat 42))
; (def (quote Bool) []) → unify hole (con (quote false) [])
; (meta x _) → catchTC (blockOnMeta x) (typeError (strErr "impossible" ∷ []))
-- check that the block isn't caught
; _ → typeError (strErr "No default" ∷ [])
}
aNat : Nat
aNat = default
aBool : Bool
aBool = default
alsoNat : Nat
soonNat : _
soonNat = default
alsoNat = soonNat
| 21.914286
| 81
| 0.552803
|
cb61c71be18784b45fd4f5e79fd40a66892b2e9d
| 163
|
agda
|
Agda
|
test/Fail/IrrelevantTelescope.agda
|
mdimjasevic/agda
|
8fb548356b275c7a1e79b768b64511ae937c738b
|
[
"BSD-3-Clause"
] | 1,989
|
2015-01-09T23:51:16.000Z
|
2022-03-30T18:20:48.000Z
|
test/Fail/IrrelevantTelescope.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 4,066
|
2015-01-10T11:24:51.000Z
|
2022-03-31T21:14:49.000Z
|
test/Fail/IrrelevantTelescope.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 371
|
2015-01-03T14:04:08.000Z
|
2022-03-30T19:00:30.000Z
|
-- Andreas, 2011-04-07
module IrrelevantTelescope where
data Wrap .(A : Set) : Set where
wrap : A -> Wrap A
-- cannot use A, because it is declared irrelevant
| 20.375
| 50
| 0.699387
|
0e936c0f5ba702daa17575236c64d15fe80f765f
| 2,992
|
agda
|
Agda
|
src/data/lib/prim/Agda/Builtin/Cubical/HCompU.agda
|
vlopezj/agda
|
ff4d89e75970cf27599fb9f572bd43c9455cbb56
|
[
"BSD-3-Clause"
] | 2
|
2019-10-29T09:40:30.000Z
|
2020-09-20T00:28:57.000Z
|
src/data/lib/prim/Agda/Builtin/Cubical/HCompU.agda
|
vikfret/agda
|
49ad0b3f0d39c01bc35123478b857e702b29fb9d
|
[
"BSD-3-Clause"
] | 3
|
2018-11-14T15:31:44.000Z
|
2019-04-01T19:39:26.000Z
|
src/data/lib/prim/Agda/Builtin/Cubical/HCompU.agda
|
vikfret/agda
|
49ad0b3f0d39c01bc35123478b857e702b29fb9d
|
[
"BSD-3-Clause"
] | 1
|
2021-04-01T18:30:09.000Z
|
2021-04-01T18:30:09.000Z
|
{-# OPTIONS --cubical --safe --no-sized-types --no-guardedness #-}
module Agda.Builtin.Cubical.HCompU where
open import Agda.Primitive
open import Agda.Builtin.Sigma
open import Agda.Primitive.Cubical renaming (primINeg to ~_; primIMax to _∨_; primIMin to _∧_;
primHComp to hcomp; primTransp to transp; primComp to comp;
itIsOne to 1=1)
open import Agda.Builtin.Cubical.Path
open import Agda.Builtin.Cubical.Sub renaming (Sub to _[_↦_]; primSubOut to outS; inc to inS)
module Helpers where
-- Homogeneous filling
hfill : ∀ {ℓ} {A : Set ℓ} {φ : I}
(u : ∀ i → Partial φ A)
(u0 : A [ φ ↦ u i0 ]) (i : I) → A
hfill {φ = φ} u u0 i =
hcomp (λ j → \ { (φ = i1) → u (i ∧ j) 1=1
; (i = i0) → outS u0 })
(outS u0)
-- Heterogeneous filling defined using comp
fill : ∀ {ℓ : I → Level} (A : ∀ i → Set (ℓ i)) {φ : I}
(u : ∀ i → Partial φ (A i))
(u0 : A i0 [ φ ↦ u i0 ]) →
∀ i → A i
fill A {φ = φ} u u0 i =
comp (λ j → A (i ∧ j))
(λ j → \ { (φ = i1) → u (i ∧ j) 1=1
; (i = i0) → outS u0 })
(outS {φ = φ} u0)
module _ {ℓ} {A : Set ℓ} where
refl : {x : A} → x ≡ x
refl {x = x} = λ _ → x
sym : {x y : A} → x ≡ y → y ≡ x
sym p = λ i → p (~ i)
cong : ∀ {ℓ'} {B : A → Set ℓ'} {x y : A}
(f : (a : A) → B a) (p : x ≡ y)
→ PathP (λ i → B (p i)) (f x) (f y)
cong f p = λ i → f (p i)
isContr : ∀ {ℓ} → Set ℓ → Set ℓ
isContr A = Σ A \ x → (∀ y → x ≡ y)
fiber : ∀ {ℓ ℓ'} {A : Set ℓ} {B : Set ℓ'} (f : A → B) (y : B) → Set (ℓ ⊔ ℓ')
fiber {A = A} f y = Σ A \ x → f x ≡ y
open Helpers
primitive
prim^glueU : {la : Level} {φ : I} {T : I → Partial φ (Set la)}
{A : Set la [ φ ↦ T i0 ]} →
PartialP φ (T i1) → outS A → hcomp T (outS A)
prim^unglueU : {la : Level} {φ : I} {T : I → Partial φ (Set la)}
{A : Set la [ φ ↦ T i0 ]} →
hcomp T (outS A) → outS A
transpProof : ∀ {l} → (e : I → Set l) → (φ : I) → (a : Partial φ (e i0)) → (b : e i1 [ φ ↦ (\ o → transp e i0 (a o)) ] ) → fiber (transp e i0) (outS b)
transpProof e φ a b = f , \ j → comp e (\ i → \ { (φ = i1) → transp (\ j → e (j ∧ i)) (~ i) (a 1=1)
; (j = i0) → transp (\ j → e (j ∧ i)) (~ i) f
; (j = i1) → g (~ i) })
f
where
g = fill (\ i → e (~ i)) (\ i → \ { (φ = i1) → transp (\ j → e (j ∧ ~ i)) i (a 1=1); (φ = i0) → transp (\ j → e (~ j ∨ ~ i)) (~ i) (outS b) }) (inS (outS b))
f = comp (\ i → e (~ i)) (\ i → \ { (φ = i1) → transp (\ j → e (j ∧ ~ i)) i (a 1=1); (φ = i0) → transp (\ j → e (~ j ∨ ~ i)) (~ i) (outS b) }) (outS b)
{-# BUILTIN TRANSPPROOF transpProof #-}
| 41.555556
| 163
| 0.397727
|
0b486031f6a63887363c8e9ac7ba4a14dbcf80fb
| 148
|
agda
|
Agda
|
test/Succeed/AnonymousModuleWithParameter.agda
|
mdimjasevic/agda
|
8fb548356b275c7a1e79b768b64511ae937c738b
|
[
"BSD-3-Clause"
] | 1,989
|
2015-01-09T23:51:16.000Z
|
2022-03-30T18:20:48.000Z
|
test/Succeed/AnonymousModuleWithParameter.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 4,066
|
2015-01-10T11:24:51.000Z
|
2022-03-31T21:14:49.000Z
|
test/Succeed/AnonymousModuleWithParameter.agda
|
Seanpm2001-languages/agda
|
9911f73061e21a87fad76c662463257afe02c861
|
[
"BSD-2-Clause"
] | 371
|
2015-01-03T14:04:08.000Z
|
2022-03-30T19:00:30.000Z
|
module _ where
-- open import Common.Prelude
module Id (A : Set) where
id : A → A
id x = x
module _ (A : Set) where
open Id A
id2 = id
| 10.571429
| 29
| 0.594595
|
0e9620ffdc9685b75134717e36cd4ec666cbf8b8
| 3,481
|
agda
|
Agda
|
test/Succeed/ReflectTC.agda
|
AntoineAllioux/agda
|
68ec2312961776e415c99d2839e41a92ffe464db
|
[
"BSD-3-Clause"
] | null | null | null |
test/Succeed/ReflectTC.agda
|
AntoineAllioux/agda
|
68ec2312961776e415c99d2839e41a92ffe464db
|
[
"BSD-3-Clause"
] | null | null | null |
test/Succeed/ReflectTC.agda
|
AntoineAllioux/agda
|
68ec2312961776e415c99d2839e41a92ffe464db
|
[
"BSD-3-Clause"
] | null | null | null |
-- Building some simple tactics using the reflected type checking monad.
module _ where
open import Common.Reflection
open import Common.Prelude hiding (_>>=_)
open import Common.Equality
open import Agda.Builtin.Sigma
-- Some helpers --
quotegoal : (Type → Tactic) → Tactic
quotegoal tac hole =
inferType hole >>= λ goal →
reduce goal >>= λ goal →
tac goal hole
case_of_ : ∀ {a b} {A : Set a} {B : Set b} → A → (A → B) → B
case x of f = f x
replicateTC : {A : Set} → Nat → TC A → TC (List A)
replicateTC zero m = returnTC []
replicateTC (suc n) m = m >>= λ x → replicateTC n m >>= λ xs → returnTC (x ∷ xs)
mapTC! : ∀ {A : Set} → (A → TC ⊤) → List A → TC ⊤
mapTC! f [] = returnTC _
mapTC! f (x ∷ xs) = f x >>= λ _ → mapTC! f xs
mapTC!r : ∀ {A} → (A → TC ⊤) → List A → TC ⊤
mapTC!r f [] = returnTC _
mapTC!r f (x ∷ xs) = mapTC! f xs >>= λ _ → f x
visibleArity : QName → TC Nat
visibleArity q = getType q >>= λ t → returnTC (typeArity t)
where
typeArity : Type → Nat
typeArity (pi (arg (argInfo visible _) _) (abs _ b)) = suc (typeArity b)
typeArity (pi _ (abs _ b)) = typeArity b
typeArity _ = 0
newMeta! : TC Term
newMeta! = newMeta unknown
absurdLam : Term
absurdLam = extLam (absurdClause
(("()" , arg (argInfo visible relevant) unknown) ∷ [])
(arg (argInfo visible relevant) (absurd 0) ∷ [])
∷ []) []
-- Simple assumption tactic --
assumption-tac : Nat → Nat → Tactic
assumption-tac x 0 _ = typeError (strErr "No assumption matched" ∷ [])
assumption-tac x (suc n) hole =
catchTC (unify hole (var x []))
(assumption-tac (suc x) n hole)
macro
assumption : Tactic
assumption hole = getContext >>= λ Γ → assumption-tac 0 (length Γ) hole
test-assumption : ∀ {A B : Set} → A → B → A
test-assumption x y = assumption
test-assumption₂ : ∀ {A B : Set} → A → B → _
test-assumption₂ x y = assumption -- will pick y
-- Solving a goal using only constructors --
tryConstructors : Nat → List QName → Tactic
constructors-tac : Nat → Type → Tactic
constructors-tac zero _ _ = typeError (strErr "Search depth exhausted" ∷ [])
constructors-tac (suc n) (def d vs) hole =
getDefinition d >>= λ def →
case def of λ
{ (dataDef _ cs) → tryConstructors n cs hole
; _ → returnTC _ }
constructors-tac _ (pi a b) hole = give absurdLam hole
constructors-tac _ _ hole = returnTC _
tryConstructors n [] hole = typeError (strErr "No matching constructor term" ∷ [])
tryConstructors n (c ∷ cs) hole =
visibleArity c >>= λ ar →
catchTC (replicateTC ar newMeta! >>= λ vs →
unify hole (con c (map (arg (argInfo visible relevant)) vs)) >>= λ _ →
mapTC!r (quotegoal (constructors-tac n)) vs)
(tryConstructors n cs hole)
macro
constructors : Tactic
constructors = quotegoal (constructors-tac 10)
data Any {A : Set} (P : A → Set) : List A → Set where
zero : ∀ {x xs} → P x → Any P (x ∷ xs)
suc : ∀ {x xs} → Any P xs → Any P (x ∷ xs)
infix 1 _∈_
_∈_ : ∀ {A : Set} → A → List A → Set
x ∈ xs = Any (x ≡_) xs
data Dec (A : Set) : Set where
yes : A → Dec A
no : (A → ⊥) → Dec A
test₁ : 3 ∈ 1 ∷ 2 ∷ 3 ∷ []
test₁ = constructors
test₂ : Dec (2 + 3 ≡ 5)
test₂ = constructors
test₃ : Dec (2 + 2 ≡ 5)
test₃ = constructors
data Singleton (n : Nat) : Set where
it : (m : Nat) → m ≡ n → Singleton n
test₄ : Singleton 5
test₄ = constructors -- this works because we solve goals right to left (picking refl before m)
| 29.252101
| 95
| 0.608733
|
fb4f33e6375f8d4a7ccbcfd1ade82bde97ccc7f7
| 5,217
|
agda
|
Agda
|
combinators.agda
|
heades/AUGL
|
b33c6a59d664aed46cac8ef77d34313e148fecc2
|
[
"MIT"
] | null | null | null |
combinators.agda
|
heades/AUGL
|
b33c6a59d664aed46cac8ef77d34313e148fecc2
|
[
"MIT"
] | null | null | null |
combinators.agda
|
heades/AUGL
|
b33c6a59d664aed46cac8ef77d34313e148fecc2
|
[
"MIT"
] | null | null | null |
module combinators where
open import bool
open import bool-thms2
import closures
open import eq
open import list
open import list-thms
open import nat
open import nat-thms
open import product
open import product-thms
open import sum
open import string
open import termination
data comb : Set where
S : comb
K : comb
app : comb → comb → comb
size : comb → ℕ
size S = 1
size K = 1
size (app a b) = suc (size a + size b)
data _↝_ : comb → comb → Set where
↝K : (a b : comb) → (app (app K a) b) ↝ a
↝S : (a b c : comb) → (app (app (app S a) b) c) ↝ (app (app a c) (app b c))
↝Cong1 : {a a' : comb} (b : comb) → a ↝ a' → (app a b) ↝ (app a' b)
↝Cong2 : (a : comb) {b b' : comb} → b ↝ b' → (app a b) ↝ (app a b')
Sfree : comb → 𝔹
Sfree S = ff
Sfree K = tt
Sfree (app a b) = Sfree a && Sfree b
Sfree-↝-size> : ∀{a b : comb} → Sfree a ≡ tt → a ↝ b → size a > size b ≡ tt
Sfree-↝-size> p (↝K a b) = ≤<-trans {size a} (≤+1 (size a) (size b))
(<+2 {size a + size b} {2})
Sfree-↝-size> () (↝S a b c)
Sfree-↝-size> p (↝Cong1{a}{a'} b u) with &&-elim{Sfree a} p
Sfree-↝-size> p (↝Cong1{a}{a'} b u) | p1 , _ = <+mono2 {size a'} (Sfree-↝-size> p1 u)
Sfree-↝-size> p (↝Cong2 a u) with &&-elim{Sfree a} p
Sfree-↝-size> p (↝Cong2 a u) | _ , p2 = <+mono1{size a} (Sfree-↝-size> p2 u)
↝-preserves-Sfree : ∀{a b : comb} → Sfree a ≡ tt → a ↝ b → Sfree b ≡ tt
↝-preserves-Sfree p (↝K a b) = fst (&&-elim p)
↝-preserves-Sfree () (↝S a b c)
↝-preserves-Sfree p (↝Cong1 b u) with &&-elim p
↝-preserves-Sfree p (↝Cong1 b u) | p1 , p2 = &&-intro (↝-preserves-Sfree p1 u) p2
↝-preserves-Sfree p (↝Cong2 a u) with &&-elim{Sfree a} p
↝-preserves-Sfree p (↝Cong2 a u) | p1 , p2 = &&-intro p1 (↝-preserves-Sfree p2 u)
Sfree-comb : Set
Sfree-comb = Σ comb (λ a → Sfree a ≡ tt)
↝-Sfree-comb : Sfree-comb → Sfree-comb → Set
↝-Sfree-comb (a , _) (b , _) = a ↝ b
size-Sfree-comb : Sfree-comb → ℕ
size-Sfree-comb (a , _) = size a
decrease-size : ∀ {x y : Sfree-comb} → ↝-Sfree-comb x y → size-Sfree-comb x > size-Sfree-comb y ≡ tt
decrease-size{a , u}{b , _} p = Sfree-↝-size> u p
open measure{A = Sfree-comb} ↝-Sfree-comb (λ x y → x > y ≡ tt) size-Sfree-comb decrease-size
measure-decreases : ∀(a : Sfree-comb) → ↓ ↝-Sfree-comb a
measure-decreases a = measure-↓ (↓-> (size-Sfree-comb a))
Sfree-terminatesh : ∀{a : comb}{p : Sfree a ≡ tt} → ↓ ↝-Sfree-comb (a , p) → ↓ _↝_ a
Sfree-terminatesh{a}{p} (pf↓ f) = pf↓ h
where h : {y : comb} → a ↝ y → ↓ _↝_ y
h{y} u = Sfree-terminatesh (f {y , ↝-preserves-Sfree p u} u)
Sfree-terminates : ∀(a : comb) → Sfree a ≡ tt → ↓ _↝_ a
Sfree-terminates a p = Sfree-terminatesh (measure-decreases (a , p))
data varcomb : Set where
S : varcomb
K : varcomb
app : varcomb → varcomb → varcomb
var : (s : string) → varcomb
λ* : (s : string) → varcomb → varcomb
λ* s S = app K S
λ* s K = app K K
λ* s (app c1 c2) = app (app S (λ* s c1)) (λ* s c2)
λ* s (var s') = if (s =string s') then (app (app S K) K) else (app K (var s'))
subst : varcomb → string → varcomb → varcomb
subst c s S = S
subst c s K = K
subst c s (app c1 c2) = app (subst c s c1) (subst c s c2)
subst c s (var s') = if (s =string s') then c else var s'
data _↝vc_ : varcomb → varcomb → Set where
↝K : (a b : varcomb) → (app (app K a) b) ↝vc a
↝S : (a b c : varcomb) → (app (app (app S a) b) c) ↝vc (app (app a c) (app b c))
↝Cong1 : {a a' : varcomb} (b : varcomb) → a ↝vc a' → (app a b) ↝vc (app a' b)
↝Cong2 : (a : varcomb) {b b' : varcomb} → b ↝vc b' → (app a b) ↝vc (app a b')
-- open closures.basics _↝vc_
-- _↝vc+_ : varcomb → varcomb → Set
-- _↝vc+_ = tc
-- id↝ : ∀ (a : varcomb) → app (app (app S K) K) a ↝vc+ a
-- id↝ a = (tc-trans (tc-step (↝S K K a)) (tc-step (↝K a (app K a))))
-- trans-Cong1 : ∀{a a' : varcomb} (b : varcomb) → a ↝vc+ a' → (app a b) ↝vc+ (app a' b)
-- trans-Cong1 b (tc-trans d1 d2) = (tc-trans (trans-Cong1 b d1) (trans-Cong1 b d2))
-- trans-Cong1 b (tc-step d) = tc-step (↝Cong1 b d)
-- trans-Cong2 : ∀(a : varcomb) {b b' : varcomb} → b ↝vc+ b' → (app a b) ↝vc+ (app a b')
-- trans-Cong2 a (tc-trans d1 d2) = (tc-trans (trans-Cong2 a d1) (trans-Cong2 a d2))
-- trans-Cong2 a (tc-step d) = tc-step (↝Cong2 a d)
-- contains-var : string → varcomb → 𝔹
-- contains-var s S = ff
-- contains-var s K = ff
-- contains-var s (app c1 c2) = contains-var s c1 || contains-var s c2
-- contains-var s (var s') = s =string s'
-- λ*-binds : ∀(s : string)(v : varcomb) → contains-var s (λ* s v) ≡ ff
-- λ*-binds s S = refl
-- λ*-binds s K = refl
-- λ*-binds s (app c1 c2) rewrite λ*-binds s c1 | λ*-binds s c2 = refl
-- λ*-binds s (var s') with keep (s =string s')
-- λ*-binds s (var s') | tt , p rewrite p = refl
-- λ*-binds s (var s') | ff , p rewrite p = p
-- λ*-↝ : ∀ (v1 v2 : varcomb)(s : string) → (app (λ* s v1) v2) ↝vc+ (subst v2 s v1)
-- λ*-↝ S v2 s = tc-step (↝K S v2)
-- λ*-↝ K v2 s = tc-step (↝K K v2)
-- λ*-↝ (app c1 c2) v2 s =
-- (tc-trans (tc-step (↝S (λ* s c1) (λ* s c2) v2))
-- (tc-trans (trans-Cong1 (app (λ* s c2) v2) (λ*-↝ c1 v2 s))
-- (trans-Cong2 (subst v2 s c1) (λ*-↝ c2 v2 s))))
-- λ*-↝ (var s') v2 s with s =string s'
-- λ*-↝ (var s') v2 s | tt = id↝ v2
-- λ*-↝ (var s') v2 s | ff = tc-step (↝K (var s') v2)
| 36.229167
| 100
| 0.556642
|
18df341da36d566d78e8c81528bd3e7f95dc6b53
| 13,118
|
agda
|
Agda
|
nicolai/thesis/HHHUU-ComplicatedTypes.agda
|
nicolaikraus/HoTT-Agda
|
939a2d83e090fcc924f69f7dfa5b65b3b79fe633
|
[
"MIT"
] | 1
|
2021-06-30T00:17:55.000Z
|
2021-06-30T00:17:55.000Z
|
nicolai/thesis/HHHUU-ComplicatedTypes.agda
|
nicolaikraus/HoTT-Agda
|
939a2d83e090fcc924f69f7dfa5b65b3b79fe633
|
[
"MIT"
] | null | null | null |
nicolai/thesis/HHHUU-ComplicatedTypes.agda
|
nicolaikraus/HoTT-Agda
|
939a2d83e090fcc924f69f7dfa5b65b3b79fe633
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --without-K #-}
{- Here, we derive our main theorem: there is a type in the n-th universe
that is not an n-type, implying the n-th universe is not n-truncated.
The n-th universe restricted to n-types is hence a 'strict' n-type.
For this, we first derive local-global looping in a modular way.
A technical point worth noting is that Agda does not implement
cumulative universes. Since that means the crucial steps in our
transformations (where we pass between universes uning univalence)
can not be expressed using equality without resorting to explicit lifting,
we decide to explicitely uses equivalences (and pointed equivalences,
respectively) instead where possible.
As a drawback, we have to use lemmata showing preservation of
(pointed) equivalences of various (pointed) type constructions,
a parametricity property derived for free
from univalence-induced equalities. -}
module HHHUU-ComplicatedTypes where
open import lib.Basics hiding (_⊔_)
open import lib.Equivalences2
open import lib.NType2
open import lib.types.Bool
open import lib.types.Nat hiding (_+_)
open import lib.types.Paths
open import lib.types.Sigma
open import lib.types.Pi
open import lib.types.TLevel
open import Preliminaries
open import Pointed
open import UniverseOfNTypes
-- The argument that (Type lzero) is not a set is standard.
-- We omit it and the argument that (Type (S lzero)) is not a 1-Type:
-- they are both special cases of the general theorem we prove.
-- The general theorem is formalised in this file.
--Definition 7.3.1
-- We have fibered notions of the loop space contruction and n-truncatedness.
module _ {i} {X : Type• i} {j} where
{- Note that the definition of the family of path types differs slightly from
that in the thesis, which would correspond to transport P p x == x.
We use dependent paths since this follows the design of the HoTT
community's Agda library. There is no actual difference;
both types are equivalent. -}
Ω̃ : Fam• X j → Fam• (Ω X) j
Ω̃ (P , x) = ((λ p → x == x [ P ↓ p ]) , idp)
fam-has-level : ℕ₋₂ → Fam• X j → Type (i ⊔ j)
fam-has-level n Q = (a : base X) → has-level n (fst Q a)
{- == Pointed dependent sums ==
Pointed families as defined above enable us to introduce a Σ-connective
for pointed types. Because of the abstract nature of some of our lemmata, we
give Σ• in its uncurried form and first define the type of its parameter. -}
Σ•-param : ∀ i j → Type (lsucc (i ⊔ j))
Σ•-param i j = Σ (Type• i) (λ X → Fam• X j)
module _ {i j} where
Ω-Σ•-param : Σ•-param i j → Σ•-param i j
Ω-Σ•-param (X , W) = (Ω X , Ω̃ W)
-- Definition 7.3.2
Σ• : Σ•-param i j → Type• (i ⊔ j)
Σ• (X , Q) = (Σ (base X) (fst Q) , (pt X , snd Q))
-- Lemma 7.3.3
{- Commutativity of pointed dependent sums and the loop space construction
will become an important technical tool, enabling us to work at a more
abstract level later on. -}
Ω-Σ•-comm : (R : Σ•-param i j) → Ω (Σ• R) ≃• Σ• (Ω-Σ•-param R)
Ω-Σ•-comm _ = (=Σ-eqv _ _ , idp) ⁻¹•
{- Pointed dependent products.
This is not quite the equivalent of Π for pointed types: the domain parameter
is just an ordinary non-pointed type. However, to enable our goal of
abstractly working with pointed types, defining this notion is useful. -}
Π•-param : ∀ i j → Type (lsucc (i ⊔ j))
Π•-param i j = Σ (Type i) (λ A → A → Type• j)
module _ {i j} where
Ω-Π•-param : Π•-param i j → Π•-param i j
Ω-Π•-param (A , F) = (A , Ω ∘ F)
-- Definition 7.3.4
Π• : Π•-param i j → Type• (i ⊔ j)
Π• (A , Y) = (Π A (base ∘ Y) , pt ∘ Y)
-- Lemma 7.3.5
{- Pointed dependent products and loop space construction on
its codomain parameter commute as well. -}
Ω-Π•-comm : (R : Π•-param i j) → Ω (Π• R) ≃• Π• (Ω-Π•-param R)
Ω-Π•-comm _ = (app=-equiv , idp)
Ω^-Π•-comm : (C : Type i) (F : C → Type• j) (n : ℕ)
→ (Ω ^ n) (Π• (C , F)) ≃• Π• (C , ((Ω ^ n) ∘ F))
Ω^-Π•-comm C F 0 = ide• _
Ω^-Π•-comm C F (S n) = Ω^-Π•-comm C _ n ∘e• equiv-Ω^ n (Ω-Π•-comm _)
equiv-Π• : ∀ {i₀ i₁ j₀ j₁} {R₀ : Π•-param i₀ j₀} {R₁ : Π•-param i₁ j₁}
→ Σ (fst R₀ ≃ fst R₁) (λ u → ∀ a → snd R₀ (<– u a) ≃• snd R₁ a)
→ Π• R₀ ≃• Π• R₁
equiv-Π• (u , v) = (equiv-Π u (fst ∘ v) , λ= (snd ∘ v))
-- Lemma 7.4.1
-- In an n-th loop space, we can forget components of truncation level n.
forget-Ω^-Σ•₂ : ∀ {i j} {X : Type• i} (Q : Fam• X j) (n : ℕ)
→ fam-has-level (n -2) Q → (Ω ^ n) (Σ• (X , Q)) ≃• (Ω ^ n) X
forget-Ω^-Σ•₂ {X = X} Q O h = (Σ₂-contr h , idp)
forget-Ω^-Σ•₂ {i} {X = X} Q (S n) h =
(Ω ^ (S n)) (Σ• (X , Q)) ≃•⟨ equiv-Ω^ n (Ω-Σ•-comm _) ⟩
(Ω ^ n) (Σ• (Ω X , Ω̃ Q)) ≃•⟨ forget-Ω^-Σ•₂ {i} _ n (λ _ → ↓-level h) ⟩
(Ω ^ (S n)) X ≃•∎
-- Lemma 7.4.2
{- Our Local-global looping principle.
We would like to state this principle in the form of
Ωⁿ⁺¹ (Type i , A) == ∀• a → Ωⁿ (A , a)
for n ≥ 1. Unfortunately, the two sides have different universe
levels since (Type i , A) lives in Type• (suc i) instead of Type• i.
Morally, this is outbalanced by the extra Ω on the left-hand side,
which might help explain on an intuitive level why
the n-th universe ends up being not an n-type.
The reason why the univalence principle (A ≡ B) ≃ (A ≃ B)
cannot be written as (A ≡ B) ≡ (A ≃ B) is precisely the same. -}
module _ {i} {A : Type i} where
-- The degenerate pre-base case carries around a propositional component.
Ω-Type : Ω (Type i , A) ≃• Σ• (Π• (A , λ a → (A , a))
, (is-equiv , idf-is-equiv _))
Ω-Type =
Ω (Type i , A)
≃•⟨ ide• _ ⟩
((A == A) , idp)
≃•⟨ ua-equiv• ⁻¹• ⟩
((A ≃ A) , ide _)
≃•⟨ ide• _ ⟩
((Σ (A → A) is-equiv) , (idf _ , idf-is-equiv _))
≃•⟨ ide• _ ⟩
Σ• (Π• (A , λ a → (A , a)) , (is-equiv , idf-is-equiv _))
≃•∎
-- The base case.
Ω²-Type : (Ω ^ 2) (Type i , A) ≃• Π• (A , λ a → Ω (A , a))
Ω²-Type =
(Ω ^ 2) (Type i , A)
≃•⟨ equiv-Ω Ω-Type ⟩
Ω (Σ• (Π• (A , λ a → (A , a)) , (is-equiv , idf-is-equiv _)))
≃•⟨ forget-Ω^-Σ•₂ {i} _ 1 (is-equiv-is-prop ∘ _) ⟩
Ω (Π• (A , λ a → (A , a)))
≃•⟨ Ω-Π•-comm _ ⟩
Π• (A , λ a → Ω (A , a))
≃•∎
-- The general case follows by permuting Ω and Π• repeatedly.
Ω^-Type : (n : ℕ) → (Ω ^ (n + 2)) (Type i , A)
≃• Π• (A , λ a → (Ω ^ (n + 1)) (A , a))
Ω^-Type n = Ω^-Π•-comm _ _ n ∘e• equiv-Ω^ n Ω²-Type
-- Lemma 7.4.3 is taken from the library
-- lib.NType2._-Type-level_
{- The pointed family P (see thesis).
It takes an n-type A and returns the space of (n+1)-loops
with basepoint A in U_n^n (the n-th universe restricted to n-types).
This crucial restriction to n-types implies it is just a set. -}
module _ (n : ℕ) (A : ⟨ n ⟩ -Type ｢ n ｣) where
-- Definition of P and
-- Corollary 7.4.4
P : ⟨0⟩ -Type• ｢ n + 1 ｣
P = Ω^-≤' (n + 1) q where
q : ⟨ n + 1 ⟩ -Type• ｢ n + 1 ｣
q = –> Σ-comm-snd (((⟨ n ⟩ -Type-≤ ｢ n ｣) , A))
-- Forgetting about the truncation level, we may present P as follows:
Q : Type• ｢ n + 1 ｣
Q = (Ω ^ (n + 1)) (Type ｢ n ｣ , fst A)
P-is-Q : fst P ≃• Q
P-is-Q = equiv-Ω^ n (forget-Ω^-Σ•₂ _ 1 (λ _ → has-level-is-prop))
-- Definition of the type 'Loop' and
-- Lemma 7.4.5
{- The type 'Loop' of (images of) n-loops in U_(n-1)^(n-1) is
just the dependent sum over P except for the special case n ≡ 0,
where we take U_(-1)^(-1) (and hence Loop) to be the booleans.
The boilerplate with raise-≤T is just to verify that Loop n is
n-truncated.
The bulk of the rest of this module consists of showing
the n-th universe is not n-truncated at basepoint Loop n,
i.e. that Q n (Loop n) is not contractible.
Warning: The indexing of Loop starts at -1 in the thesis,
but we use natural numbers here (starting at 0),
thus everything is shifted by one. -}
Loop : (n : ℕ) → ⟨ n ⟩ -Type ｢ n ｣
Loop 0 = (Bool , Bool-is-set)
Loop (S n) = Σ-≤ (⟨ n ⟩ -Type-≤ ｢ n ｣) (λ A →
raise-≤T {n = ⟨ n + 1 ⟩} (≤T-+2+-l ⟨0⟩ (-2≤T _))
(fst (<– Σ-comm-snd (P n A))))
-- Lemma 7.4.6, preparations
-- The base case is given in Section 7.2 or the thesis.
-- It is done as usual (there is a non-trivial automorphism on booleans).
-- Let us go slowly.
module negation where
-- Negation.
~ : Bool → Bool
~ = λ {true → false; false → true}
-- Negation is an equivalence.
e : Bool ≃ Bool
e = equiv ~ ~ inv inv where
inv = λ {true → idp; false → idp}
base-case : ¬ (is-contr• (Q 0 (Loop 0)))
base-case c = Bool-false≠true false-is-true where
-- Negation being equal to the identity yields a contradiction.
false-is-true =
false =⟨ ! (coe-β e true) ⟩
coe (ua e) true =⟨ ap (λ p → coe p true) (! (c (ua e))) ⟩
coe idp true =⟨ idp ⟩
true ∎
-- Let us now turn towards the successor case.
module _ (m : ℕ) where
-- We expand the type we are later going to assume contractible.
Q-L-is-… =
Q (m + 1) (Loop (m + 1))
≃•⟨ ide• _ ⟩
(Ω ^ (m + 2)) (_ , ⟦ Loop (m + 1) ⟧)
≃•⟨ (Ω^-Type m) ⟩
Π• (_ , λ {(A , q) → Ω ^ (m + 1) $ (⟦ Loop (m + 1) ⟧ , (A , q))})
≃•⟨ ide• _ ⟩
Π• (_ , λ {(A , q) → Ω ^ (m + 1) $ Σ• ((_ , A) , (base ∘ fst ∘ P m , q))})
≃•∎
-- What we really want is to arrive at contractibility of E (m ≥ 1)...
E = Π• (⟦ Loop (m + 1) ⟧ , λ {(A , q) → Ω ^ (m + 1) $ ((⟨ m ⟩ -Type ｢ m ｣) , A)})
-- ...or at least show that the following element f of E is trivial (m ≡ 0).
f : base E
f (_ , q) = q
-- We want to use our assumption of contractibility of Q (n + 1) (Loop (n + 1))
-- to show that f is trivial, i.e. constant with value the basepoint.
f-is-trivial : (m : ℕ) → is-contr• (Q (m + 1) (Loop (m + 1))) → f m == pt (E m)
-- m ≡ 0
f-is-trivial 0 c = ap (λ f' → fst ∘ f')
(! (–> (equiv-is-contr• …-is-E') c f')) where
-- This is almost E, except for the additional component
-- specifying that the first component p should commute with q.
E' = Π• (_ , λ {(A , q) → (Σ (A == A) (λ p → q == q [ (λ x → x == x) ↓ p ])
, (idp , idp))})
-- This "almost" E comes from Q 1 (Loop 1), hence can be shown contractible.
…-is-E' : Q 1 (Loop 1) ≃• E'
…-is-E' = equiv-Π• (ide _ , Ω-Σ•-comm ∘ _) ∘e• Q-L-is-… 0
-- Fortunately, f can be 'extended' to an element f' of E',
-- and triviality of f' implies triviality of f.
f' = λ {(_ , q) → (q , ↓-idf=idf-in (∙=∙' q q))}
-- m ≥ 1: We can show Q (k + 2) (Loop (k + 2)) ≃ E (k + 1),
-- thus E is contractible, hence f trivial.
f-is-trivial (S k) c = ! (–> (equiv-is-contr• (…-is-E ∘e• Q-L-is-… (k + 1)))
c
(f (k + 1))) where
…-is-E : _ ≃• E (k + 1)
…-is-E = equiv-Π• (ide _ , equiv-Ω^ k ∘ (λ {(A , q) → forget-Ω^-Σ•₂
{｢ k + 2 ｣} (base ∘ fst ∘ P (k + 1) , q) 2
(snd ∘ P (k + 1))}))
-- Lemma 7.4.6, part 1
-- Our main lemma: like in the thesis, but in negative form.
-- This is sufficient to prove our results, and easier to formalise.
main : (n : ℕ) → ¬ (is-contr• (Q n (Loop n)))
main 0 = negation.base-case
main (S m) c = main m step where
{- We know Q (m + 1) (Loop (m + 1)) is contractible,
use that to show that the above f is trivial,
deduce f (Loop m , p) ≡ p is trivial for all p in P m (Loop m),
which implies P m (Loop m) is contractible.
But this is just another form of Q m (Loop m),
so the conclusion follows by induction hypothesis. -}
step : is-contr• (Q m (Loop m))
step = –> (equiv-is-contr• (P-is-Q m (Loop m)))
(λ q → app= (! (f-is-trivial m c)) (Loop m , q))
-- Lemma 7.4.6, part 2
-- Alternate form of the main lemma
main' : (n : ℕ) → ¬ (is-contr• ((Ω ^ (n + 1)) ((⟨ n ⟩ -Type ｢ n ｣) , Loop n )))
main' n = main n ∘ –> (equiv-is-contr• (P-is-Q n (Loop n)))
-- Small helper thingy
helpy : ∀ {i} {n : ℕ} {X : Type• i}
→ has-level• (n -1) X → is-contr• ((Ω ^ n) X)
helpy {n = n} {X} = <– contr•-equiv-prop
∘ trunc-many n
∘ transport (λ k → has-level• (k -2) X)
(+-comm 1 n)
-- Main theorems now fall out as corollaries.
module _ (n : ℕ) where
{- Recall that L n is n-truncated.
We also know it is not (n-1)-truncated, it is thus a 'strict' n-type. -}
Loop-is-not-trunc : ¬ (has-level (n -1) ⟦ Loop n ⟧)
Loop-is-not-trunc = main n ∘ helpy ∘ (λ t → universe-=-level t t)
-- Theorem 7.4.7
-- The n-th universe is not n-truncated.
Type-is-not-trunc : ¬ (has-level ⟨ n ⟩ (Type ｢ n ｣))
Type-is-not-trunc = main n ∘ helpy
-- Theorem 7.4.8
-- MAIN RESULT:
-- The n-th universe restricted to n-types is a 'strict' (n+1)-type.
-- We do not repeat that it is (n+1)-truncated; this is formalised above
-- (7.4.3). Instead, we only show that it is not an n-type.
Type-≤-is-not-trunc : ¬ (has-level ⟨ n ⟩ (⟨ n ⟩ -Type ｢ n ｣))
Type-≤-is-not-trunc = main' n ∘ helpy
| 38.696165
| 83
| 0.556182
|
fbc6e85e277c088e7ee01846e78967a2bd4dac7b
| 527
|
agda
|
Agda
|
agda-stdlib/src/Relation/Binary/Morphism.agda
|
DreamLinuxer/popl21-artifact
|
fb380f2e67dcb4a94f353dbaec91624fcb5b8933
|
[
"MIT"
] | 5
|
2020-10-07T12:07:53.000Z
|
2020-10-10T21:41:32.000Z
|
agda-stdlib/src/Relation/Binary/Morphism.agda
|
DreamLinuxer/popl21-artifact
|
fb380f2e67dcb4a94f353dbaec91624fcb5b8933
|
[
"MIT"
] | null | null | null |
agda-stdlib/src/Relation/Binary/Morphism.agda
|
DreamLinuxer/popl21-artifact
|
fb380f2e67dcb4a94f353dbaec91624fcb5b8933
|
[
"MIT"
] | 1
|
2021-11-04T06:54:45.000Z
|
2021-11-04T06:54:45.000Z
|
------------------------------------------------------------------------
-- The Agda standard library
--
-- Order morphisms
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Relation.Binary.Core
module Relation.Binary.Morphism where
------------------------------------------------------------------------
-- Re-export contents of morphisms
open import Relation.Binary.Morphism.Definitions public
open import Relation.Binary.Morphism.Structures public
| 29.277778
| 72
| 0.455408
|
18799c47c9e19eda51739af019dd48a28726d10c
| 3,520
|
agda
|
Agda
|
Cubical/Categories/Functor/Properties.agda
|
FernandoLarrain/cubical
|
9acdecfa6437ec455568be4e5ff04849cc2bc13b
|
[
"MIT"
] | 1
|
2022-03-05T00:29:41.000Z
|
2022-03-05T00:29:41.000Z
|
Cubical/Categories/Functor/Properties.agda
|
FernandoLarrain/cubical
|
9acdecfa6437ec455568be4e5ff04849cc2bc13b
|
[
"MIT"
] | null | null | null |
Cubical/Categories/Functor/Properties.agda
|
FernandoLarrain/cubical
|
9acdecfa6437ec455568be4e5ff04849cc2bc13b
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --safe #-}
module Cubical.Categories.Functor.Properties where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function renaming (_∘_ to _◍_)
open import Cubical.Foundations.GroupoidLaws using (lUnit; rUnit; assoc; cong-∙)
open import Cubical.Categories.Category
open import Cubical.Categories.Functor.Base
private
variable
ℓ ℓ' ℓ'' : Level
B C D E : Category ℓ ℓ'
open Category
open Functor
{-
x ---p--- x'
⇓ᵍ
g x' ---q--- y
⇓ʰ
h y ---r--- z
The path from `h (g x) ≡ z` obtained by
1. first applying cong to p and composing with q; then applying cong again and composing with r
2. first applying cong to q and composing with r; then applying a double cong to p and precomposing
are path equal.
-}
congAssoc : ∀ {X : Type ℓ} {Y : Type ℓ'} {Z : Type ℓ''} (g : X → Y) (h : Y → Z) {x x' : X} {y : Y} {z : Z}
→ (p : x ≡ x') (q : g x' ≡ y) (r : h y ≡ z)
→ cong (h ◍ g) p ∙ (cong h q ∙ r) ≡ cong h (cong g p ∙ q) ∙ r
congAssoc g h p q r
= cong (h ◍ g) p ∙ (cong h q ∙ r)
≡⟨ assoc _ _ _ ⟩
((cong (h ◍ g) p) ∙ (cong h q)) ∙ r
≡⟨ refl ⟩
(cong h (cong g p) ∙ (cong h q)) ∙ r
≡⟨ cong (_∙ r) (sym (cong-∙ h _ _)) ⟩
cong h (cong g p ∙ q) ∙ r
∎
-- composition is associative
F-assoc : {F : Functor B C} {G : Functor C D} {H : Functor D E}
→ H ∘F (G ∘F F) ≡ (H ∘F G) ∘F F
F-assoc {F = F} {G} {H} i .F-ob x = H ⟅ G ⟅ F ⟅ x ⟆ ⟆ ⟆
F-assoc {F = F} {G} {H} i .F-hom f = H ⟪ G ⟪ F ⟪ f ⟫ ⟫ ⟫
F-assoc {F = F} {G} {H} i .F-id {x} = congAssoc (G ⟪_⟫) (H ⟪_⟫) (F .F-id {x}) (G .F-id {F ⟅ x ⟆}) (H .F-id) (~ i)
F-assoc {F = F} {G} {H} i .F-seq f g = congAssoc (G ⟪_⟫) (H ⟪_⟫) (F .F-seq f g) (G .F-seq _ _) (H .F-seq _ _) (~ i)
-- Results about functors
module _ {F : Functor C D} where
-- the identity is the identity
F-lUnit : F ∘F 𝟙⟨ C ⟩ ≡ F
F-lUnit i .F-ob x = F ⟅ x ⟆
F-lUnit i .F-hom f = F ⟪ f ⟫
F-lUnit i .F-id {x} = lUnit (F .F-id) (~ i)
F-lUnit i .F-seq f g = lUnit (F .F-seq f g) (~ i)
F-rUnit : 𝟙⟨ D ⟩ ∘F F ≡ F
F-rUnit i .F-ob x = F ⟅ x ⟆
F-rUnit i .F-hom f = F ⟪ f ⟫
F-rUnit i .F-id {x} = rUnit (F .F-id) (~ i)
F-rUnit i .F-seq f g = rUnit (F .F-seq f g) (~ i)
-- functors preserve commutative diagrams (specificallysqures here)
preserveCommF : ∀ {x y z w} {f : C [ x , y ]} {g : C [ y , w ]} {h : C [ x , z ]} {k : C [ z , w ]}
→ f ⋆⟨ C ⟩ g ≡ h ⋆⟨ C ⟩ k
→ (F ⟪ f ⟫) ⋆⟨ D ⟩ (F ⟪ g ⟫) ≡ (F ⟪ h ⟫) ⋆⟨ D ⟩ (F ⟪ k ⟫)
preserveCommF {f = f} {g = g} {h = h} {k = k} eq
= (F ⟪ f ⟫) ⋆⟨ D ⟩ (F ⟪ g ⟫)
≡⟨ sym (F .F-seq _ _) ⟩
F ⟪ f ⋆⟨ C ⟩ g ⟫
≡⟨ cong (F ⟪_⟫) eq ⟩
F ⟪ h ⋆⟨ C ⟩ k ⟫
≡⟨ F .F-seq _ _ ⟩
(F ⟪ h ⟫) ⋆⟨ D ⟩ (F ⟪ k ⟫)
∎
-- functors preserve isomorphisms
preserveIsosF : ∀ {x y} → CatIso C x y → CatIso D (F ⟅ x ⟆) (F ⟅ y ⟆)
preserveIsosF {x} {y} (catiso f f⁻¹ sec' ret') =
catiso
g g⁻¹
-- sec
( (g⁻¹ ⋆⟨ D ⟩ g)
≡⟨ sym (F .F-seq f⁻¹ f) ⟩
F ⟪ f⁻¹ ⋆⟨ C ⟩ f ⟫
≡⟨ cong (F .F-hom) sec' ⟩
F ⟪ C .id ⟫
≡⟨ F .F-id ⟩
D .id
∎ )
-- ret
( (g ⋆⟨ D ⟩ g⁻¹)
≡⟨ sym (F .F-seq f f⁻¹) ⟩
F ⟪ f ⋆⟨ C ⟩ f⁻¹ ⟫
≡⟨ cong (F .F-hom) ret' ⟩
F ⟪ C .id ⟫
≡⟨ F .F-id ⟩
D .id
∎ )
where
x' : D .ob
x' = F ⟅ x ⟆
y' : D .ob
y' = F ⟅ y ⟆
g : D [ x' , y' ]
g = F ⟪ f ⟫
g⁻¹ : D [ y' , x' ]
g⁻¹ = F ⟪ f⁻¹ ⟫
| 30.08547
| 116
| 0.444602
|
0b1644efecf2ce86c8fda916cdf633ebe87d966f
| 1,211
|
agda
|
Agda
|
test/asset/agda-stdlib-1.0/Data/Sum/Properties.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | null | null | null |
test/asset/agda-stdlib-1.0/Data/Sum/Properties.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | null | null | null |
test/asset/agda-stdlib-1.0/Data/Sum/Properties.agda
|
omega12345/agda-mode
|
0debb886eb5dbcd38dbeebd04b34cf9d9c5e0e71
|
[
"MIT"
] | null | null | null |
------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties of sums (disjoint unions)
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Sum.Properties where
open import Data.Sum
open import Function
open import Relation.Binary using (Decidable)
open import Relation.Binary.PropositionalEquality
open import Relation.Nullary using (yes; no)
module _ {a b} {A : Set a} {B : Set b} where
inj₁-injective : ∀ {x y} → (A ⊎ B ∋ inj₁ x) ≡ inj₁ y → x ≡ y
inj₁-injective refl = refl
inj₂-injective : ∀ {x y} → (A ⊎ B ∋ inj₂ x) ≡ inj₂ y → x ≡ y
inj₂-injective refl = refl
≡-dec : Decidable _≡_ → Decidable _≡_ → Decidable {A = A ⊎ B} _≡_
≡-dec dec₁ dec₂ (inj₁ x) (inj₁ y) with dec₁ x y
... | yes refl = yes refl
... | no x≢y = no (x≢y ∘ inj₁-injective)
≡-dec dec₁ dec₂ (inj₁ x) (inj₂ y) = no λ()
≡-dec dec₁ dec₂ (inj₂ x) (inj₁ y) = no λ()
≡-dec dec₁ dec₂ (inj₂ x) (inj₂ y) with dec₂ x y
... | yes refl = yes refl
... | no x≢y = no (x≢y ∘ inj₂-injective)
swap-involutive : swap {A = A} {B} ∘ swap ≗ id
swap-involutive = [ (λ _ → refl) , (λ _ → refl) ]
| 32.72973
| 72
| 0.535095
|
0bb83334bf3f71e22db5c830c21da758dfd5f495
| 702
|
agda
|
Agda
|
test/interaction/ExtendedLambdaCase.agda
|
masondesu/agda
|
70c8a575c46f6a568c7518150a1a64fcd03aa437
|
[
"MIT"
] | 1
|
2018-10-10T17:08:44.000Z
|
2018-10-10T17:08:44.000Z
|
test/interaction/ExtendedLambdaCase.agda
|
np/agda-git-experiment
|
20596e9dd9867166a64470dd24ea68925ff380ce
|
[
"MIT"
] | null | null | null |
test/interaction/ExtendedLambdaCase.agda
|
np/agda-git-experiment
|
20596e9dd9867166a64470dd24ea68925ff380ce
|
[
"MIT"
] | null | null | null |
module ExtendedLambdaCase where
data Bool : Set where
true false : Bool
data Void : Set where
foo : Bool -> Bool -> Bool -> Bool
foo = λ { x → λ { y z → {!!} } }
module parameterised {A : Set}(B : A -> Set) where
data Bar : (Bool -> Bool) -> Set where
baz : (t : Void) -> Bar λ { x → {!!} }
-- with hidden argument
data Bar' : (Bool -> Bool) -> Set where
baz' : {t : Void} -> (t' : Void) -> Bar' λ { x' → {!!} }
baz : Bool -> {w : Bool} -> Bool
baz = λ { z {w} → {!!} }
another-short-name : {A : Set} -> (A -> {x : A} -> A -> A)
another-short-name = {! λ { a {x} b → a } !}
f : Set
f = (y : Bool) -> parameterised.Bar {Bool}(λ _ → Void) (λ { true → true ; false → false })
| 22.645161
| 91
| 0.501425
|
187f3a04e11ea97a1012eb187f89cf12169bfd5f
| 3,348
|
agda
|
Agda
|
src/Categories/Functor/Hom.agda
|
glittershark/agda-categories
|
2128fab9e8d341364cbf784bb17c547bf73891de
|
[
"MIT"
] | null | null | null |
src/Categories/Functor/Hom.agda
|
glittershark/agda-categories
|
2128fab9e8d341364cbf784bb17c547bf73891de
|
[
"MIT"
] | null | null | null |
src/Categories/Functor/Hom.agda
|
glittershark/agda-categories
|
2128fab9e8d341364cbf784bb17c547bf73891de
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --without-K --safe #-}
module Categories.Functor.Hom where
-- The Hom Functor from C.op × C to Setoids,
-- the two 1-argument version fixing one object
-- and some notation for the version where the category must be made explicit
open import Data.Product
open import Function using () renaming (_∘_ to _∙_)
open import Categories.Category
open import Categories.Functor hiding (id)
open import Categories.Functor.Properties
open import Categories.Functor.Bifunctor
open import Categories.Category.Instance.Setoids
import Categories.Morphism.Reasoning as MR
open import Relation.Binary using (Setoid)
module Hom {o ℓ e} (C : Category o ℓ e) where
open Category C
open MR C
Hom[-,-] : Bifunctor (Category.op C) C (Setoids ℓ e)
Hom[-,-] = record
{ F₀ = F₀′
; F₁ = λ where
(f , g) → record
{ _⟨$⟩_ = λ h → g ∘ h ∘ f
; cong = ∘-resp-≈ʳ ∙ ∘-resp-≈ˡ
}
; identity = identity′
; homomorphism = homomorphism′
; F-resp-≈ = F-resp-≈′
}
where F₀′ : Obj × Obj → Setoid ℓ e
F₀′ (A , B) = hom-setoid {A} {B}
open HomReasoning
identity′ : {A : Obj × Obj} {x y : uncurry _⇒_ A} → x ≈ y → id ∘ x ∘ id ≈ y
identity′ {A} {x} {y} x≈y = begin
id ∘ x ∘ id ≈⟨ identityˡ ⟩
x ∘ id ≈⟨ identityʳ ⟩
x ≈⟨ x≈y ⟩
y ∎
homomorphism′ : ∀ {X Y Z : Σ Obj (λ x → Obj)}
{f : proj₁ Y ⇒ proj₁ X × proj₂ X ⇒ proj₂ Y}
{g : proj₁ Z ⇒ proj₁ Y × proj₂ Y ⇒ proj₂ Z}
{x y : proj₁ X ⇒ proj₂ X} →
x ≈ y →
(proj₂ g ∘ proj₂ f) ∘ x ∘ proj₁ f ∘ proj₁ g ≈
proj₂ g ∘ (proj₂ f ∘ y ∘ proj₁ f) ∘ proj₁ g
homomorphism′ {f = f₁ , f₂} {g₁ , g₂} {x} {y} x≈y = begin
(g₂ ∘ f₂) ∘ x ∘ f₁ ∘ g₁ ≈⟨ refl⟩∘⟨ sym-assoc ⟩
(g₂ ∘ f₂) ∘ (x ∘ f₁) ∘ g₁ ≈⟨ pullʳ (pullˡ (∘-resp-≈ʳ (∘-resp-≈ˡ x≈y))) ⟩
g₂ ∘ (f₂ ∘ y ∘ f₁) ∘ g₁ ∎
F-resp-≈′ : ∀ {A B : Σ Obj (λ x → Obj)}
{f g : Σ (proj₁ B ⇒ proj₁ A) (λ x → proj₂ A ⇒ proj₂ B)} →
Σ (proj₁ f ≈ proj₁ g) (λ x → proj₂ f ≈ proj₂ g) →
{x y : proj₁ A ⇒ proj₂ A} →
x ≈ y → proj₂ f ∘ x ∘ proj₁ f ≈ proj₂ g ∘ y ∘ proj₁ g
F-resp-≈′ {f = f₁ , f₂} {g₁ , g₂} (f₁≈g₁ , f₂≈g₂) {x} {y} x≈y = begin
f₂ ∘ x ∘ f₁ ≈⟨ f₂≈g₂ ⟩∘⟨ x≈y ⟩∘⟨ f₁≈g₁ ⟩
g₂ ∘ y ∘ g₁ ∎
open Functor Hom[-,-]
open Equiv
open HomReasoning
Hom[_,-] : Obj → Functor C (Setoids ℓ e)
Hom[_,-] = appˡ Hom[-,-]
Hom[-,_] : Obj → Contravariant C (Setoids ℓ e)
Hom[-,_] = appʳ Hom[-,-]
Hom[_,_] : Obj → Obj → Setoid ℓ e
Hom[ A , B ] = hom-setoid {A} {B}
-- Notation for when the ambient Category must be specified explicitly.
module _ {o ℓ e} (C : Category o ℓ e) where
open Category C
open Hom C
Hom[_][-,-] : Bifunctor (Category.op C) C (Setoids ℓ e)
Hom[_][-,-] = Hom[-,-]
Hom[_][_,-] : Obj → Functor C (Setoids ℓ e)
Hom[_][_,-] B = Hom[ B ,-]
Hom[_][-,_] : Obj → Contravariant C (Setoids ℓ e)
Hom[_][-,_] B = Hom[-, B ]
Hom[_][_,_] : Obj → Obj → Setoid ℓ e
Hom[_][_,_] A B = hom-setoid {A} {B}
| 34.515464
| 85
| 0.48178
|
0b9e1c7cd0cfa68166517c4b05dbad6e50bd2a9f
| 863
|
agda
|
Agda
|
part1/lists/map-compose.agda
|
akiomik/plfa-solutions
|
df7722b88a9b3dfde320a690b78c4c1ef8c7c547
|
[
"Apache-2.0"
] | 1
|
2020-07-07T09:42:22.000Z
|
2020-07-07T09:42:22.000Z
|
part1/lists/map-compose.agda
|
akiomik/plfa-solutions
|
df7722b88a9b3dfde320a690b78c4c1ef8c7c547
|
[
"Apache-2.0"
] | null | null | null |
part1/lists/map-compose.agda
|
akiomik/plfa-solutions
|
df7722b88a9b3dfde320a690b78c4c1ef8c7c547
|
[
"Apache-2.0"
] | null | null | null |
module map-compose where
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; cong)
open Eq.≡-Reasoning
open import Function using (_∘_)
open import lists using (List; []; _∷_; map)
postulate
-- 外延性の公理
extensionality : ∀ {A B : Set} {f g : A → B}
→ (∀ (x : A) → f x ≡ g x)
-----------------------
→ f ≡ g
-- 外延性の公理を用いた証明のための補助定理
lemma : ∀ {A B C : Set} → (f : A → B) → (g : B → C) → (x : List A)
→ map (g ∘ f) x ≡ (map g ∘ map f) x
lemma f g [] = refl
lemma f g (x ∷ xs) =
begin
map (g ∘ f) (x ∷ xs)
≡⟨⟩
(g ∘ f) x ∷ map (g ∘ f) xs
≡⟨ cong ((g ∘ f) x ∷_) (lemma f g xs) ⟩
(g ∘ f) x ∷ (map g ∘ map f) xs
≡⟨⟩
(map g ∘ map f) (x ∷ xs)
∎
-- mapの分配法則の証明
map-compose : ∀ {A B C : Set} → (f : A → B) → (g : B → C)
→ map (g ∘ f) ≡ map g ∘ map f
map-compose f g = extensionality (lemma f g)
| 24.657143
| 66
| 0.491309
|
0bfa2ec4e3695485ad828b1e8527b60af9027404
| 2,138
|
agda
|
Agda
|
ConfluenceParallel.agda
|
iwilare/church-rosser
|
2fa17f7738cc7da967375be928137adc4be38696
|
[
"MIT"
] | 5
|
2020-06-02T07:27:54.000Z
|
2021-11-22T01:43:09.000Z
|
ConfluenceParallel.agda
|
iwilare/church-rosser
|
2fa17f7738cc7da967375be928137adc4be38696
|
[
"MIT"
] | null | null | null |
ConfluenceParallel.agda
|
iwilare/church-rosser
|
2fa17f7738cc7da967375be928137adc4be38696
|
[
"MIT"
] | null | null | null |
open import Data.Product using (∃; ∃-syntax; _×_; _,_)
open import DeBruijn
open import Parallel
open import Beta
par-diamond : ∀ {n} {M N N′ : Term n}
→ M ⇉ N → M ⇉ N′
-----------------------
→ ∃[ L ] (N ⇉ L × N′ ⇉ L)
par-diamond (⇉-c {x = x}) ⇉-c = # x , ⇉-c , ⇉-c
par-diamond (⇉-ƛ p1) (⇉-ƛ p2)
with par-diamond p1 p2
... | L′ , p3 , p4 =
ƛ L′ , ⇉-ƛ p3 , ⇉-ƛ p4
par-diamond (⇉-ξ p1 p3) (⇉-ξ p2 p4)
with par-diamond p1 p2
... | L₃ , p5 , p6
with par-diamond p3 p4
... | M₃ , p7 , p8 =
L₃ · M₃ , ⇉-ξ p5 p7 , ⇉-ξ p6 p8
par-diamond (⇉-ξ (⇉-ƛ p1) p3) (⇉-β p2 p4)
with par-diamond p1 p2
... | N₃ , p5 , p6
with par-diamond p3 p4
... | M₃ , p7 , p8 =
N₃ [ M₃ ] , ⇉-β p5 p7 , sub-par p6 p8
par-diamond (⇉-β p1 p3) (⇉-ξ (⇉-ƛ p2) p4)
with par-diamond p1 p2
... | N₃ , p5 , p6
with par-diamond p3 p4
... | M₃ , p7 , p8 =
N₃ [ M₃ ] , sub-par p5 p7 , ⇉-β p6 p8
par-diamond (⇉-β p1 p3) (⇉-β p2 p4)
with par-diamond p1 p2
... | N₃ , p5 , p6
with par-diamond p3 p4
... | M₃ , p7 , p8 =
N₃ [ M₃ ] , sub-par p5 p7 , sub-par p6 p8
strip : ∀ {n} {M A B : Term n}
→ M ⇉ A → M ⇉* B
------------------------
→ ∃[ N ] (A ⇉* N × B ⇉ N)
strip {A = A} M⇉A (M ∎) = A , (A ∎) , M⇉A
strip {A = A} M⇉A (M ⇉⟨ M⇉M′ ⟩ M′⇉*B)
with par-diamond M⇉A M⇉M′
... | N , A⇉N , M′⇉N
with strip M′⇉N M′⇉*B
... | N′ , N⇉*N′ , B⇉N′ =
N′ , (A ⇉⟨ A⇉N ⟩ N⇉*N′) , B⇉N′
par-confluence : ∀ {n} {M A B : Term n}
→ M ⇉* A → M ⇉* B
------------------------
→ ∃[ N ] (A ⇉* N × B ⇉* N)
par-confluence {B = B} (M ∎) M⇉*B = B , M⇉*B , (B ∎)
par-confluence {B = B} (M ⇉⟨ M⇉A ⟩ A⇉*A′) M⇉*B
with strip M⇉A M⇉*B
... | N , A⇉*N , B⇉N
with par-confluence A⇉*A′ A⇉*N
... | N′ , A′⇉*N′ , N⇉*N′ =
N′ , A′⇉*N′ , (B ⇉⟨ B⇉N ⟩ N⇉*N′)
confluence : ∀ {n} {M A B : Term n}
→ M —↠ A → M —↠ B
------------------------
→ ∃[ N ] (A —↠ N × B —↠ N)
confluence M—↠A M—↠B
with par-confluence (betas-pars M—↠A) (betas-pars M—↠B)
... | N , A⇉*N , B⇉*N =
N , pars-betas A⇉*N , pars-betas B⇉*N
| 27.766234
| 59
| 0.404116
|
fb0f463d8c66e649dd22831688e8dacbad044ff7
| 17,183
|
agda
|
Agda
|
lib/cubical/Square.agda
|
UlrikBuchholtz/HoTT-Agda
|
f8fa68bf753d64d7f45556ca09d0da7976709afa
|
[
"MIT"
] | null | null | null |
lib/cubical/Square.agda
|
UlrikBuchholtz/HoTT-Agda
|
f8fa68bf753d64d7f45556ca09d0da7976709afa
|
[
"MIT"
] | null | null | null |
lib/cubical/Square.agda
|
UlrikBuchholtz/HoTT-Agda
|
f8fa68bf753d64d7f45556ca09d0da7976709afa
|
[
"MIT"
] | null | null | null |
{-# OPTIONS --without-K #-}
open import lib.Base
open import lib.PathGroupoid
open import lib.PathOver
module lib.cubical.Square where
data Square {i} {A : Type i} {a₀₀ : A} : {a₀₁ a₁₀ a₁₁ : A}
→ a₀₀ == a₀₁ → a₀₀ == a₁₀ → a₀₁ == a₁₁ → a₁₀ == a₁₁ → Type i
where
ids : Square idp idp idp idp
hid-square : ∀ {i} {A : Type i} {a₀₀ a₀₁ : A} {p : a₀₀ == a₀₁}
→ Square p idp idp p
hid-square {p = idp} = ids
vid-square : ∀ {i} {A : Type i} {a₀₀ a₁₀ : A} {p : a₀₀ == a₁₀}
→ Square idp p p idp
vid-square {p = idp} = ids
square-to-disc : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
→ Square p₀₋ p₋₀ p₋₁ p₁₋
→ p₀₋ ∙ p₋₁ == p₋₀ ∙ p₁₋
square-to-disc ids = idp
disc-to-square : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
→ p₀₋ ∙ p₋₁ == p₋₀ ∙ p₁₋
→ Square p₀₋ p₋₀ p₋₁ p₁₋
disc-to-square {p₀₋ = idp} {p₋₀ = idp} {p₋₁ = idp} {p₁₋ = .idp} idp = ids
square-to-disc-β : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(α : p₀₋ ∙ p₋₁ == p₋₀ ∙ p₁₋)
→ square-to-disc (disc-to-square {p₀₋ = p₀₋} {p₋₀ = p₋₀} α) == α
square-to-disc-β {p₀₋ = idp} {p₋₀ = idp} {p₋₁ = idp} {p₁₋ = .idp} idp = idp
disc-to-square-β : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ disc-to-square (square-to-disc sq) == sq
disc-to-square-β ids = idp
ap-square : ∀ {i j} {A : Type i} {B : Type j} (f : A → B)
{a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
→ Square p₀₋ p₋₀ p₋₁ p₁₋
→ Square (ap f p₀₋) (ap f p₋₀) (ap f p₋₁) (ap f p₁₋)
ap-square f ids = ids
ap-square-hid : ∀ {i j} {A : Type i} {B : Type j} {f : A → B}
{a₀ a₁ : A} {p : a₀ == a₁}
→ ap-square f (hid-square {p = p}) == hid-square
ap-square-hid {p = idp} = idp
ap-square-vid : ∀ {i j} {A : Type i} {B : Type j} {f : A → B}
{a₀ a₁ : A} {p : a₀ == a₁}
→ ap-square f (vid-square {p = p}) == vid-square
ap-square-vid {p = idp} = idp
module _ {i} {A : Type i} where
horiz-degen-square : {a a' : A} {p q : a == a'}
→ p == q → Square p idp idp q
horiz-degen-square {p = idp} α = disc-to-square α
horiz-degen-path : {a a' : A} {p q : a == a'}
→ Square p idp idp q → p == q
horiz-degen-path {p = idp} sq = square-to-disc sq
horiz-degen-path-β : {a a' : A} {p q : a == a'} (α : p == q)
→ horiz-degen-path (horiz-degen-square α) == α
horiz-degen-path-β {p = idp} α = square-to-disc-β α
horiz-degen-square-β : {a a' : A} {p q : a == a'} (sq : Square p idp idp q)
→ horiz-degen-square (horiz-degen-path sq) == sq
horiz-degen-square-β {p = idp} sq = disc-to-square-β sq
vert-degen-square : {a a' : A} {p q : a == a'}
→ p == q → Square idp p q idp
vert-degen-square {p = idp} α = disc-to-square (! α)
vert-degen-path : {a a' : A} {p q : a == a'}
→ Square idp p q idp → p == q
vert-degen-path {p = idp} sq = ! (square-to-disc sq)
vert-degen-path-β : {a a' : A} {p q : a == a'} (α : p == q)
→ vert-degen-path (vert-degen-square α) == α
vert-degen-path-β {p = idp} α = ap ! (square-to-disc-β (! α)) ∙ !-! α
vert-degen-square-β : {a a' : A} {p q : a == a'} (sq : Square idp p q idp)
→ vert-degen-square (vert-degen-path sq) == sq
vert-degen-square-β {p = idp} sq =
ap disc-to-square (!-! (square-to-disc sq)) ∙ disc-to-square-β sq
horiz-degen-square-idp : {a a' : A} {p : a == a'}
→ horiz-degen-square (idp {a = p}) == hid-square
horiz-degen-square-idp {p = idp} = idp
vert-degen-square-idp : {a a' : A} {p : a == a'}
→ vert-degen-square (idp {a = p}) == vid-square
vert-degen-square-idp {p = idp} = idp
{- Flipping squares -}
module _ {i} {A : Type i} where
square-symmetry : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
→ Square p₀₋ p₋₀ p₋₁ p₁₋ → Square p₋₀ p₀₋ p₁₋ p₋₁
square-symmetry ids = ids
square-sym-inv : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ square-symmetry (square-symmetry sq) == sq
square-sym-inv ids = idp
ap-square-symmetry : ∀ {i j} {A : Type i} {B : Type j} (f : A → B)
{a₀₀ a₀₁ a₁₀ a₁₁ : A} {p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}
{p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ ap-square f (square-symmetry sq) == square-symmetry (ap-square f sq)
ap-square-symmetry f ids = idp
{- Alternate induction principles -}
square-left-J : ∀ {i j} {A : Type i} {a₀₀ a₀₁ : A} {p₀₋ : a₀₀ == a₀₁}
(P : {a₁₀ a₁₁ : A} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ Type j)
(r : P hid-square)
{a₁₀ a₁₁ : A} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ P sq
square-left-J P r ids = r
square-top-J : ∀ {i j} {A : Type i} {a₀₀ a₁₀ : A} {p₋₀ : a₀₀ == a₁₀}
(P : {a₀₁ a₁₁ : A} {p₀₋ : a₀₀ == a₀₁} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ Type j)
(r : P vid-square)
{a₀₁ a₁₁ : A} {p₀₋ : a₀₀ == a₀₁} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ P sq
square-top-J P r ids = r
square-bot-J : ∀ {i j} {A : Type i} {a₀₁ a₁₁ : A} {p₋₁ : a₀₁ == a₁₁}
(P : {a₀₀ a₁₀ : A} {p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ Type j)
(r : P vid-square)
{a₀₀ a₁₀ : A} {p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ P sq
square-bot-J P r ids = r
square-right-J : ∀ {i j} {A : Type i} {a₁₀ a₁₁ : A} {p₁₋ : a₁₀ == a₁₁}
(P : {a₀₀ a₀₁ : A} {p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ Type j)
(r : P hid-square)
{a₀₀ a₀₁ : A} {p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ P sq
square-right-J P r ids = r
module _ where
private
lemma : ∀ {i j} {A : Type i} {a₀ : A}
(P : {a₁ : A} {p q : a₀ == a₁} → p == q → Type j)
(r : P (idp {a = idp}))
{a₁ : A} {p q : a₀ == a₁} (α : p == q)
→ P α
lemma P r {p = idp} idp = r
horiz-degen-J : ∀ {i j} {A : Type i} {a₀ : A}
(P : {a₁ : A} {p q : a₀ == a₁} → Square p idp idp q → Type j)
(r : P ids)
{a₁ : A} {p q : a₀ == a₁} (sq : Square p idp idp q)
→ P sq
horiz-degen-J P r sq = transport P
(horiz-degen-square-β sq)
(lemma (P ∘ horiz-degen-square) r (horiz-degen-path sq))
vert-degen-J : ∀ {i j} {A : Type i} {a₀ : A}
(P : {a₁ : A} {p q : a₀ == a₁} → Square idp p q idp → Type j)
(r : P ids)
{a₁ : A} {p q : a₀ == a₁} (sq : Square idp p q idp)
→ P sq
vert-degen-J P r sq = transport P
(vert-degen-square-β sq)
(lemma (P ∘ vert-degen-square) r (vert-degen-path sq))
{- Square filling -}
module _ {i} {A : Type i} where
fill-square-left : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
(p₋₀ : a₀₀ == a₁₀) (p₋₁ : a₀₁ == a₁₁) (p₁₋ : a₁₀ == a₁₁)
→ Σ (a₀₀ == a₀₁) (λ p₀₋ → Square p₀₋ p₋₀ p₋₁ p₁₋)
fill-square-left idp idp p = (p , hid-square)
fill-square-top : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
(p₀₋ : a₀₀ == a₀₁) (p₋₁ : a₀₁ == a₁₁) (p₁₋ : a₁₀ == a₁₁)
→ Σ (a₀₀ == a₁₀) (λ p₋₀ → Square p₀₋ p₋₀ p₋₁ p₁₋)
fill-square-top idp p idp = (p , vid-square)
fill-square-bot : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
(p₀₋ : a₀₀ == a₀₁) (p₋₀ : a₀₀ == a₁₀) (p₁₋ : a₁₀ == a₁₁)
→ Σ (a₀₁ == a₁₁) (λ p₋₁ → Square p₀₋ p₋₀ p₋₁ p₁₋)
fill-square-bot idp p idp = (p , vid-square)
fill-square-right : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
(p₀₋ : a₀₀ == a₀₁) (p₋₀ : a₀₀ == a₁₀) (p₋₁ : a₀₁ == a₁₁)
→ Σ (a₁₀ == a₁₁) (λ p₁₋ → Square p₀₋ p₋₀ p₋₁ p₁₋)
fill-square-right p idp idp = (p , hid-square)
module _ {i j} {A : Type i} {B : Type j} {f g : A → B} where
↓-='-to-square : {x y : A} {p : x == y} {u : f x == g x} {v : f y == g y}
→ u == v [ (λ z → f z == g z) ↓ p ]
→ Square u (ap f p) (ap g p) v
↓-='-to-square {p = idp} q = horiz-degen-square q
↓-='-from-square : {x y : A} {p : x == y} {u : f x == g x} {v : f y == g y}
→ Square u (ap f p) (ap g p) v
→ u == v [ (λ z → f z == g z) ↓ p ]
↓-='-from-square {p = idp} sq = horiz-degen-path sq
module _ {i j} {A : Type i} {B : Type j} {f : A → B} {b : B} where
↓-cst=app-from-square : {x y : A} {p : x == y}
{u : b == f x} {v : b == f y}
→ Square u idp (ap f p) v
→ u == v [ (λ z → b == f z) ↓ p ]
↓-cst=app-from-square {p = idp} sq = horiz-degen-path sq
↓-cst=app-to-square : {x y : A} {p : x == y}
{u : b == f x} {v : b == f y}
→ u == v [ (λ z → b == f z) ↓ p ]
→ Square u idp (ap f p) v
↓-cst=app-to-square {p = idp} sq = horiz-degen-square sq
↓-app=cst-from-square : {x y : A} {p : x == y}
{u : f x == b} {v : f y == b}
→ Square u (ap f p) idp v
→ u == v [ (λ z → f z == b) ↓ p ]
↓-app=cst-from-square {p = idp} sq = horiz-degen-path sq
↓-app=cst-to-square : {x y : A} {p : x == y}
{u : f x == b} {v : f y == b}
→ u == v [ (λ z → f z == b) ↓ p ]
→ Square u (ap f p) idp v
↓-app=cst-to-square {p = idp} sq = horiz-degen-square sq
module _ {i j} {A : Type i} {B : Type j} (g : B → A) (f : A → B) where
↓-∘=idf-from-square : {x y : A} {p : x == y}
{u : g (f x) == x} {v : g (f y) == y}
→ Square u (ap g (ap f p)) p v
→ (u == v [ (λ z → g (f z) == z) ↓ p ])
↓-∘=idf-from-square {p = idp} sq = horiz-degen-path sq
↓-∘=idf-to-square : {x y : A} {p : x == y}
{u : g (f x) == x} {v : g (f y) == y}
→ (u == v [ (λ z → g (f z) == z) ↓ p ])
→ Square u (ap g (ap f p)) p v
↓-∘=idf-to-square {p = idp} q = horiz-degen-square q
module _ {i j} {A : Type i} {B : Type j} where
natural-square : {f₁ f₂ : A → B} (p : ∀ a → f₁ a == f₂ a)
{a₁ a₂ : A} (q : a₁ == a₂)
→ Square (p a₁) (ap f₁ q) (ap f₂ q) (p a₂)
natural-square p idp = hid-square
natural-square-idp : {f₁ : A → B} {a₁ a₂ : A} (q : a₁ == a₂)
→ natural-square {f₁ = f₁} (λ _ → idp) q == vid-square
natural-square-idp idp = idp
{- Used for getting square equivalents of glue-β terms -}
natural-square-β : {f₁ f₂ : A → B} (p : (a : A) → f₁ a == f₂ a)
{x y : A} (q : x == y)
{sq : Square (p x) (ap f₁ q) (ap f₂ q) (p y)}
→ apd p q == ↓-='-from-square sq
→ natural-square p q == sq
natural-square-β _ idp α =
! horiz-degen-square-idp ∙ ap horiz-degen-square α ∙ horiz-degen-square-β _
_⊡v_ : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ a₀₂ a₁₂ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
{q₀₋ : a₀₁ == a₀₂} {q₋₂ : a₀₂ == a₁₂} {q₁₋ : a₁₁ == a₁₂}
→ Square p₀₋ p₋₀ p₋₁ p₁₋ → Square q₀₋ p₋₁ q₋₂ q₁₋
→ Square (p₀₋ ∙ q₀₋) p₋₀ q₋₂ (p₁₋ ∙ q₁₋)
ids ⊡v sq = sq
_⊡v'_ : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ a₀₂ a₁₂ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
{q₀₋ : a₀₁ == a₀₂} {q₋₂ : a₀₂ == a₁₂} {q₁₋ : a₁₁ == a₁₂}
→ Square p₀₋ p₋₀ p₋₁ p₁₋ → Square q₀₋ p₋₁ q₋₂ q₁₋
→ Square (p₀₋ ∙' q₀₋) p₋₀ q₋₂ (p₁₋ ∙' q₁₋)
sq ⊡v' ids = sq
_∙v⊡_ : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ p₋₀' : a₀₀ == a₁₀}
{p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
→ p₋₀ == p₋₀'
→ Square p₀₋ p₋₀' p₋₁ p₁₋
→ Square p₀₋ p₋₀ p₋₁ p₁₋
idp ∙v⊡ sq = sq
_⊡v∙_ : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₋₀ : a₀₀ == a₁₀} {p₀₋ : a₀₀ == a₀₁}
{p₋₁ p₋₁' : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
→ Square p₀₋ p₋₀ p₋₁ p₁₋
→ p₋₁ == p₋₁'
→ Square p₀₋ p₋₀ p₋₁' p₁₋
sq ⊡v∙ idp = sq
_⊡h_ : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ a₂₀ a₂₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
{q₋₀ : a₁₀ == a₂₀} {q₋₁ : a₁₁ == a₂₁} {q₂₋ : a₂₀ == a₂₁}
→ Square p₀₋ p₋₀ p₋₁ p₁₋
→ Square p₁₋ q₋₀ q₋₁ q₂₋
→ Square p₀₋ (p₋₀ ∙ q₋₀) (p₋₁ ∙ q₋₁) q₂₋
ids ⊡h sq = sq
_⊡h'_ : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ a₂₀ a₂₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
{q₋₀ : a₁₀ == a₂₀} {q₋₁ : a₁₁ == a₂₁} {q₂₋ : a₂₀ == a₂₁}
→ Square p₀₋ p₋₀ p₋₁ p₁₋
→ Square p₁₋ q₋₀ q₋₁ q₂₋
→ Square p₀₋ (p₋₀ ∙' q₋₀) (p₋₁ ∙' q₋₁) q₂₋
sq ⊡h' ids = sq
_∙h⊡_ : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ p₀₋' : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}
{p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
→ p₀₋ == p₀₋'
→ Square p₀₋' p₋₀ p₋₁ p₁₋
→ Square p₀₋ p₋₀ p₋₁ p₁₋
idp ∙h⊡ sq = sq
_⊡h∙_ : ∀ {i} {A : Type i} {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}
{p₋₁ : a₀₁ == a₁₁} {p₁₋ p₁₋' : a₁₀ == a₁₁}
→ Square p₀₋ p₋₀ p₋₁ p₁₋
→ p₁₋ == p₁₋'
→ Square p₀₋ p₋₀ p₋₁ p₁₋'
sq ⊡h∙ idp = sq
infixr 80 _⊡v_ _∙v⊡_
_⊡h_ _∙h⊡_
_⊡h'_
infixr 80 _⊡v∙_ _⊡h∙_
module _ {i} {A : Type i} where
!□h : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}
{p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
→ Square p₀₋ p₋₀ p₋₁ p₁₋
→ Square p₁₋ (! p₋₀) (! p₋₁) p₀₋
!□h ids = ids
!□v : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}
{p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
→ Square p₀₋ p₋₀ p₋₁ p₁₋
→ Square (! p₀₋) p₋₁ p₋₀ (! p₁₋)
!□v ids = ids
module _ {i} {A : Type i} where
{- TODO rest of these -}
⊡h-unit-l : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ hid-square ⊡h sq == sq
⊡h-unit-l ids = idp
⊡h-unit-r : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ sq ⊡h hid-square == ∙-unit-r _ ∙v⊡ sq ⊡v∙ ! (∙-unit-r _)
⊡h-unit-r ids = idp
⊡h'-unit-l : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ hid-square ⊡h' sq == ∙'-unit-l _ ∙v⊡ sq ⊡v∙ ! (∙'-unit-l _)
⊡h'-unit-l ids = idp
⊡h-unit-l-unique : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀} {p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq' : Square p₀₋ idp idp p₀₋) (sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ sq' ⊡h sq == sq
→ sq' == hid-square
⊡h-unit-l-unique sq' ids p = ! (⊡h-unit-r sq') ∙ p
module _ {i} {A : Type i} where
!□h-inv-l : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}
{p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ (!□h sq) ⊡h sq == !-inv-l p₋₀ ∙v⊡ hid-square ⊡v∙ ! (!-inv-l p₋₁)
!□h-inv-l ids = idp
!□h-inv-r : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}
{p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ sq ⊡h (!□h sq) == !-inv-r p₋₀ ∙v⊡ hid-square ⊡v∙ ! (!-inv-r p₋₁)
!□h-inv-r ids = idp
!□v-inv-l : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}
{p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ (!□v sq) ⊡v sq == !-inv-l p₀₋ ∙h⊡ vid-square ⊡h∙ ! (!-inv-l p₁₋)
!□v-inv-l ids = idp
!□v-inv-r : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}
{p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
(sq : Square p₀₋ p₋₀ p₋₁ p₁₋)
→ sq ⊡v (!□v sq) == !-inv-r p₀₋ ∙h⊡ vid-square ⊡h∙ ! (!-inv-r p₁₋)
!□v-inv-r ids = idp
module _ {i} {A : Type i} where
square-left-unique : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ p₀₋' : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}
{p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
→ Square p₀₋ p₋₀ p₋₁ p₁₋ → Square p₀₋' p₋₀ p₋₁ p₁₋
→ p₀₋ == p₀₋'
square-left-unique {p₋₀ = idp} {p₋₁ = idp} sq₁ sq₂ =
horiz-degen-path (sq₁ ⊡h (!□h sq₂))
square-top-unique : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ p₋₀' : a₀₀ == a₁₀}
{p₋₁ : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
→ Square p₀₋ p₋₀ p₋₁ p₁₋ → Square p₀₋ p₋₀' p₋₁ p₁₋
→ p₋₀ == p₋₀'
square-top-unique {p₀₋ = idp} {p₁₋ = idp} sq₁ sq₂ =
vert-degen-path (sq₁ ⊡v (!□v sq₂))
square-bot-unique : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}
{p₋₁ p₋₁' : a₀₁ == a₁₁} {p₁₋ : a₁₀ == a₁₁}
→ Square p₀₋ p₋₀ p₋₁ p₁₋ → Square p₀₋ p₋₀ p₋₁' p₁₋
→ p₋₁ == p₋₁'
square-bot-unique {p₀₋ = idp} {p₁₋ = idp} sq₁ sq₂ =
vert-degen-path ((!□v sq₁) ⊡v sq₂)
square-right-unique : {a₀₀ a₀₁ a₁₀ a₁₁ : A}
{p₀₋ : a₀₀ == a₀₁} {p₋₀ : a₀₀ == a₁₀}
{p₋₁ : a₀₁ == a₁₁} {p₁₋ p₁₋' : a₁₀ == a₁₁}
→ Square p₀₋ p₋₀ p₋₁ p₁₋ → Square p₀₋ p₋₀ p₋₁ p₁₋'
→ p₁₋ == p₁₋'
square-right-unique {p₋₀ = idp} {p₋₁ = idp} sq₁ sq₂ =
horiz-degen-path ((!□h sq₁) ⊡h sq₂)
module _ {i} {A : Type i} where
connection : {a₀ a₁ : A} {q : a₀ == a₁}
→ Square idp idp q q
connection {q = idp} = ids
connection2 : {a₀ a₁ a₂ : A} {p : a₀ == a₁} {q : a₁ == a₂}
→ Square p p q q
connection2 {p = idp} {q = idp} = ids
lb-square : {a₀ a₁ : A} (p : a₀ == a₁)
→ Square p idp (! p) idp
lb-square idp = ids
bl-square : {a₀ a₁ : A} (p : a₀ == a₁)
→ Square (! p) idp p idp
bl-square idp = ids
rt-square : {a₀ a₁ : A} (p : a₀ == a₁)
→ Square idp (! p) idp p
rt-square idp = ids
tr-square : {a₀ a₁ : A} (p : a₀ == a₁)
→ Square idp p idp (! p)
tr-square idp = ids
lt-square : {a₀ a₁ : A} (p : a₀ == a₁)
→ Square p p idp idp
lt-square idp = ids
| 34.504016
| 79
| 0.479311
|
18798a4d0d6b00bfd8a0ed7501fbb54af323fd0d
| 9,650
|
agda
|
Agda
|
src/Categories/Category/Monoidal/Properties.agda
|
yourboynico/agda-categories
|
6a087c592dbe58fc4bd9d02e1be9b94a9e138aca
|
[
"MIT"
] | 279
|
2019-06-01T14:36:40.000Z
|
2022-03-22T00:40:14.000Z
|
src/Categories/Category/Monoidal/Properties.agda
|
seanpm2001/agda-categories
|
d9e4f578b126313058d105c61707d8c8ae987fa8
|
[
"MIT"
] | 236
|
2019-06-01T14:53:54.000Z
|
2022-03-28T14:31:43.000Z
|
src/Categories/Category/Monoidal/Properties.agda
|
seanpm2001/agda-categories
|
d9e4f578b126313058d105c61707d8c8ae987fa8
|
[
"MIT"
] | 64
|
2019-06-02T16:58:15.000Z
|
2022-03-14T02:00:59.000Z
|
{-# OPTIONS --without-K --safe #-}
open import Categories.Category
import Categories.Category.Monoidal as M
-- Properties of Monoidal Categories
module Categories.Category.Monoidal.Properties
{o ℓ e} {C : Category o ℓ e} (MC : M.Monoidal C) where
open import Data.Product using (_,_; Σ; uncurry′)
open Category C
open M.Monoidal MC
open import Categories.Category.Monoidal.Utilities MC
import Categories.Category.Monoidal.Reasoning as MonR
open import Categories.Category.Construction.Core C as Core using (Core)
open import Categories.Category.Product using (Product)
open import Categories.Functor using (Functor)
open import Categories.Functor.Bifunctor
open import Categories.Functor.Properties
open import Categories.Morphism.Isomorphism C
using (elim-triangleˡ′; triangle-prism; cut-squareʳ)
import Categories.Morphism.Reasoning as MR
open import Categories.NaturalTransformation.NaturalIsomorphism.Properties
using (push-eq)
private
module C = Category C
variable
A B : Obj
open Core.Shorthands
⊗-iso : Bifunctor Core Core Core
⊗-iso = record
{ F₀ = uncurry′ _⊗₀_
; F₁ = λ where (f , g) → f ⊗ᵢ g
; identity = refl⊗refl≃refl
; homomorphism = ⌞ homomorphism ⌟
; F-resp-≈ = λ where (⌞ eq₁ ⌟ , ⌞ eq₂ ⌟) → ⌞ F-resp-≈ (eq₁ , eq₂) ⌟
}
where open Functor ⊗
_⊗ᵢ- : Obj → Functor Core Core
X ⊗ᵢ- = appˡ ⊗-iso X
-⊗ᵢ_ : Obj → Functor Core Core
-⊗ᵢ X = appʳ ⊗-iso X
-- Coherence laws due to Mac Lane (1963) that were subsequently proven
-- admissible by Max Kelly (1964). See
-- https://ncatlab.org/nlab/show/monoidal+category#other_coherence_conditions
-- for more details.
module Kelly's where
open Functor
open Shorthands
open Commutation C
open Commutationᵢ
private
variable
f f′ g h h′ i i′ j k : A ≅ B
module _ {X Y : Obj} where
open HomReasoningᵢ
-- TS: following three isos commute
ua : unit ⊗₀ (unit ⊗₀ X) ⊗₀ Y ≅ unit ⊗₀ unit ⊗₀ X ⊗₀ Y
ua = idᵢ ⊗ᵢ associator
u[λY] : unit ⊗₀ (unit ⊗₀ X) ⊗₀ Y ≅ unit ⊗₀ X ⊗₀ Y
u[λY] = idᵢ ⊗ᵢ unitorˡ ⊗ᵢ idᵢ
uλ : unit ⊗₀ unit ⊗₀ X ⊗₀ Y ≅ unit ⊗₀ X ⊗₀ Y
uλ = idᵢ ⊗ᵢ unitorˡ
-- setups
perimeter : [ ((unit ⊗₀ unit) ⊗₀ X) ⊗₀ Y ≅ unit ⊗₀ X ⊗₀ Y ]⟨
(unitorʳ ⊗ᵢ idᵢ) ⊗ᵢ idᵢ ≅⟨ (unit ⊗₀ X) ⊗₀ Y ⟩
associator
≈ associator ≅⟨ (unit ⊗₀ unit) ⊗₀ X ⊗₀ Y ⟩
associator ≅⟨ unit ⊗₀ unit ⊗₀ X ⊗₀ Y ⟩
uλ
⟩
perimeter = ⟺ (glue◃◽′ triangle-iso
(⟺ ⌞ Equiv.trans assoc-commute-from
(∘-resp-≈ˡ (F-resp-≈ ⊗ (Equiv.refl , identity ⊗))) ⌟))
where open MR Core
[uλ]Y : (unit ⊗₀ (unit ⊗₀ X)) ⊗₀ Y ≅ (unit ⊗₀ X) ⊗₀ Y
[uλ]Y = (idᵢ ⊗ᵢ unitorˡ) ⊗ᵢ idᵢ
aY : ((unit ⊗₀ unit) ⊗₀ X) ⊗₀ Y ≅ (unit ⊗₀ unit ⊗₀ X) ⊗₀ Y
aY = associator ⊗ᵢ idᵢ
[ρX]Y : ((unit ⊗₀ unit) ⊗₀ X) ⊗₀ Y ≅ (unit ⊗₀ X) ⊗₀ Y
[ρX]Y = (unitorʳ ⊗ᵢ idᵢ) ⊗ᵢ idᵢ
tri : [uλ]Y ∘ᵢ aY ≈ᵢ [ρX]Y
tri = ⌞ [ appʳ ⊗ Y ]-resp-∘ triangle ⌟
sq : associator ∘ᵢ [uλ]Y ≈ᵢ u[λY] ∘ᵢ associator
sq = ⌞ assoc-commute-from ⌟
-- proofs
perimeter′ : [ ((unit ⊗₀ unit) ⊗₀ X) ⊗₀ Y ≅ unit ⊗₀ X ⊗₀ Y ]⟨
(unitorʳ ⊗ᵢ idᵢ) ⊗ᵢ idᵢ ≅⟨ (unit ⊗₀ X) ⊗₀ Y ⟩
associator
≈ aY ≅⟨ (unit ⊗₀ (unit ⊗₀ X)) ⊗₀ Y ⟩
associator ≅⟨ unit ⊗₀ (unit ⊗₀ X) ⊗₀ Y ⟩
ua ≅⟨ unit ⊗₀ unit ⊗₀ X ⊗₀ Y ⟩
uλ
⟩
perimeter′ = begin
associator ∘ᵢ (unitorʳ ⊗ᵢ idᵢ) ⊗ᵢ idᵢ ≈⟨ perimeter ⟩
uλ ∘ᵢ associator ∘ᵢ associator ≈˘⟨ refl⟩∘⟨ pentagon-iso ⟩
uλ ∘ᵢ ua ∘ᵢ associator ∘ᵢ aY ∎
top-face : uλ ∘ᵢ ua ≈ᵢ u[λY]
top-face = elim-triangleˡ′ (⟺ perimeter′) (glue◽◃ (⟺ sq) tri)
where open MR Core
coherence-iso₁ : [ (unit ⊗₀ X) ⊗₀ Y ≅ X ⊗₀ Y ]⟨
associator ≅⟨ unit ⊗₀ X ⊗₀ Y ⟩
unitorˡ
≈ unitorˡ ⊗ᵢ idᵢ
⟩
coherence-iso₁ = triangle-prism top-face square₁ square₂ square₃
where square₁ : [ unit ⊗₀ X ⊗₀ Y ≅ unit ⊗₀ X ⊗₀ Y ]⟨
unitorˡ ⁻¹ ∘ᵢ unitorˡ
≈ idᵢ ⊗ᵢ unitorˡ ∘ᵢ unitorˡ ⁻¹
⟩
square₁ = ⌞ unitorˡ-commute-to ⌟
square₂ : [ (unit ⊗₀ X) ⊗₀ Y ≅ unit ⊗₀ unit ⊗₀ X ⊗₀ Y ]⟨
unitorˡ ⁻¹ ∘ᵢ associator
≈ idᵢ ⊗ᵢ associator ∘ᵢ unitorˡ ⁻¹
⟩
square₂ = ⌞ unitorˡ-commute-to ⌟
square₃ : [ (unit ⊗₀ X) ⊗₀ Y ≅ unit ⊗₀ X ⊗₀ Y ]⟨
unitorˡ ⁻¹ ∘ᵢ unitorˡ ⊗ᵢ idᵢ
≈ idᵢ ⊗ᵢ unitorˡ ⊗ᵢ idᵢ ∘ᵢ unitorˡ ⁻¹
⟩
square₃ = ⌞ unitorˡ-commute-to ⌟
coherence₁ : [ (unit ⊗₀ X) ⊗₀ Y ⇒ X ⊗₀ Y ]⟨
α⇒ ⇒⟨ unit ⊗₀ X ⊗₀ Y ⟩
λ⇒
≈ λ⇒ ⊗₁ id
⟩
coherence₁ = from-≈ coherence-iso₁
coherence-inv₁ : [ X ⊗₀ Y ⇒ (unit ⊗₀ X) ⊗₀ Y ]⟨
λ⇐ ⇒⟨ unit ⊗₀ X ⊗₀ Y ⟩
α⇐
≈ λ⇐ ⊗₁ id
⟩
coherence-inv₁ = to-≈ coherence-iso₁
-- another coherence property
-- TS : the following three commute
ρu : ((X ⊗₀ Y) ⊗₀ unit) ⊗₀ unit ≅ (X ⊗₀ Y) ⊗₀ unit
ρu = unitorʳ ⊗ᵢ idᵢ
au : ((X ⊗₀ Y) ⊗₀ unit) ⊗₀ unit ≅ (X ⊗₀ Y ⊗₀ unit) ⊗₀ unit
au = associator ⊗ᵢ idᵢ
[Xρ]u : (X ⊗₀ Y ⊗₀ unit) ⊗₀ unit ≅ (X ⊗₀ Y) ⊗₀ unit
[Xρ]u = (idᵢ ⊗ᵢ unitorʳ) ⊗ᵢ idᵢ
perimeter″ : [ ((X ⊗₀ Y) ⊗₀ unit) ⊗₀ unit ≅ X ⊗₀ Y ⊗₀ unit ]⟨
associator ≅⟨ (X ⊗₀ Y) ⊗₀ unit ⊗₀ unit ⟩
associator ≅⟨ X ⊗₀ Y ⊗₀ unit ⊗₀ unit ⟩
idᵢ ⊗ᵢ idᵢ ⊗ᵢ unitorˡ
≈ ρu ≅⟨ (X ⊗₀ Y) ⊗₀ unit ⟩
associator
⟩
perimeter″ = glue▹◽ triangle-iso (⟺ ⌞
Equiv.trans (∘-resp-≈ʳ (F-resp-≈ ⊗ (Equiv.sym (identity ⊗) , Equiv.refl)))
assoc-commute-from ⌟)
where open MR Core
perimeter‴ : [ ((X ⊗₀ Y) ⊗₀ unit) ⊗₀ unit ≅ X ⊗₀ Y ⊗₀ unit ]⟨
associator ⊗ᵢ idᵢ ≅⟨ (X ⊗₀ (Y ⊗₀ unit)) ⊗₀ unit ⟩
(associator ≅⟨ X ⊗₀ (Y ⊗₀ unit) ⊗₀ unit ⟩
idᵢ ⊗ᵢ associator ≅⟨ X ⊗₀ Y ⊗₀ unit ⊗₀ unit ⟩
idᵢ ⊗ᵢ idᵢ ⊗ᵢ unitorˡ)
≈ ρu ≅⟨ (X ⊗₀ Y) ⊗₀ unit ⟩
associator
⟩
perimeter‴ = let α = associator in let λλ = unitorˡ in begin
(idᵢ ⊗ᵢ idᵢ ⊗ᵢ λλ ∘ᵢ idᵢ ⊗ᵢ α ∘ᵢ α) ∘ᵢ α ⊗ᵢ idᵢ ≈⟨ ⌞ assoc ⌟ ⟩
idᵢ ⊗ᵢ idᵢ ⊗ᵢ λλ ∘ᵢ (idᵢ ⊗ᵢ α ∘ᵢ α) ∘ᵢ α ⊗ᵢ idᵢ ≈⟨ refl⟩∘⟨ ⌞ assoc ⌟ ⟩
idᵢ ⊗ᵢ idᵢ ⊗ᵢ λλ ∘ᵢ idᵢ ⊗ᵢ α ∘ᵢ α ∘ᵢ α ⊗ᵢ idᵢ ≈⟨ refl⟩∘⟨ pentagon-iso ⟩
idᵢ ⊗ᵢ idᵢ ⊗ᵢ λλ ∘ᵢ α ∘ᵢ α ≈⟨ perimeter″ ⟩
α ∘ᵢ ρu ∎
top-face′ : [Xρ]u ∘ᵢ au ≈ᵢ ρu
top-face′ = cut-squareʳ perimeter‴ (⟺ (glue◃◽′ tri′ (⟺ ⌞ assoc-commute-from ⌟)))
where open MR Core
tri′ : [ X ⊗₀ (Y ⊗₀ unit) ⊗₀ unit ≅ X ⊗₀ Y ⊗₀ unit ]⟨
(idᵢ ⊗ᵢ idᵢ ⊗ᵢ unitorˡ ∘ᵢ idᵢ ⊗ᵢ associator)
≈ idᵢ ⊗ᵢ unitorʳ ⊗ᵢ idᵢ
⟩
tri′ = ⌞ [ X ⊗- ]-resp-∘ triangle ⌟
coherence-iso₂ : [ (X ⊗₀ Y) ⊗₀ unit ≅ X ⊗₀ Y ]⟨
idᵢ ⊗ᵢ unitorʳ ∘ᵢ associator
≈ unitorʳ
⟩
coherence-iso₂ = triangle-prism top-face′ square₁ square₂ ⌞ unitorʳ-commute-to ⌟
where square₁ : [ X ⊗₀ Y ⊗₀ unit ≅ (X ⊗₀ Y) ⊗₀ unit ]⟨
unitorʳ ⁻¹ ∘ᵢ idᵢ ⊗ᵢ unitorʳ
≈ (idᵢ ⊗ᵢ unitorʳ) ⊗ᵢ idᵢ ∘ᵢ unitorʳ ⁻¹
⟩
square₁ = ⌞ unitorʳ-commute-to ⌟
square₂ : [ (X ⊗₀ Y) ⊗₀ unit ≅ (X ⊗₀ Y ⊗₀ unit) ⊗₀ unit ]⟨
unitorʳ ⁻¹ ∘ᵢ associator
≈ associator ⊗ᵢ idᵢ ∘ᵢ unitorʳ ⁻¹
⟩
square₂ = ⌞ unitorʳ-commute-to ⌟
coherence₂ : [ (X ⊗₀ Y) ⊗₀ unit ⇒ X ⊗₀ Y ]⟨
α⇒ ⇒⟨ X ⊗₀ (Y ⊗₀ unit) ⟩
id ⊗₁ ρ⇒
≈ ρ⇒
⟩
coherence₂ = from-≈ coherence-iso₂
coherence-inv₂ : [ X ⊗₀ Y ⇒ (X ⊗₀ Y) ⊗₀ unit ]⟨
id ⊗₁ ρ⇐ ⇒⟨ X ⊗₀ (Y ⊗₀ unit) ⟩
α⇐
≈ ρ⇐
⟩
coherence-inv₂ = to-≈ coherence-iso₂
-- A third coherence condition (Lemma 2.3)
coherence₃ : [ unit ⊗₀ unit ⇒ unit ]⟨ λ⇒ ≈ ρ⇒ ⟩
coherence₃ = push-eq unitorˡ-naturalIsomorphism (begin
C.id ⊗₁ λ⇒ ≈˘⟨ cancelʳ associator.isoʳ ⟩
(C.id ⊗₁ λ⇒ ∘ α⇒) ∘ α⇐ ≈⟨ triangle ⟩∘⟨refl ⟩
ρ⇒ ⊗₁ C.id ∘ α⇐ ≈⟨ unitor-coherenceʳ ⟩∘⟨refl ⟩
ρ⇒ ∘ α⇐ ≈˘⟨ coherence₂ ⟩∘⟨refl ⟩
(C.id ⊗₁ ρ⇒ ∘ α⇒) ∘ α⇐ ≈⟨ cancelʳ associator.isoʳ ⟩
C.id ⊗₁ ρ⇒ ∎)
where
open MR C hiding (push-eq)
open C.HomReasoning
coherence-iso₃ : [ unit ⊗₀ unit ≅ unit ]⟨ unitorˡ ≈ unitorʳ ⟩
coherence-iso₃ = ⌞ coherence₃ ⌟
coherence-inv₃ : [ unit ⇒ unit ⊗₀ unit ]⟨ λ⇐ ≈ ρ⇐ ⟩
coherence-inv₃ = to-≈ coherence-iso₃
open Kelly's public using
( coherence₁; coherence-iso₁; coherence-inv₁
; coherence₂; coherence-iso₂; coherence-inv₂
; coherence₃; coherence-iso₃; coherence-inv₃
)
| 35.740741
| 102
| 0.466425
|
dff68fef36206686e9a8c4353b825d307c9df640
| 2,282
|
agda
|
Agda
|
core/lib/types/IteratedSuspension.agda
|
mikeshulman/HoTT-Agda
|
e7d663b63d89f380ab772ecb8d51c38c26952dbb
|
[
"MIT"
] | null | null | null |
core/lib/types/IteratedSuspension.agda
|
mikeshulman/HoTT-Agda
|
e7d663b63d89f380ab772ecb8d51c38c26952dbb
|
[
"MIT"
] | null | null | null |
core/lib/types/IteratedSuspension.agda
|
mikeshulman/HoTT-Agda
|
e7d663b63d89f380ab772ecb8d51c38c26952dbb
|
[
"MIT"
] | 1
|
2018-12-26T21:31:57.000Z
|
2018-12-26T21:31:57.000Z
|
{-# OPTIONS --without-K --rewriting #-}
open import lib.Basics
open import lib.NConnected
open import lib.types.Bool
open import lib.types.Nat
open import lib.types.TLevel
open import lib.types.Suspension
module lib.types.IteratedSuspension where
⊙Susp^ : ∀ {i} (n : ℕ) → Ptd i → Ptd i
⊙Susp^ O X = X
⊙Susp^ (S n) X = ⊙Susp (⊙Susp^ n X)
abstract
⊙Susp^-conn : ∀ {i} (n : ℕ) {X : Ptd i} {m : ℕ₋₂}
→ is-connected m (de⊙ X) → is-connected (⟨ n ⟩₋₂ +2+ m) (de⊙ (⊙Susp^ n X))
⊙Susp^-conn O cX = cX
⊙Susp^-conn (S n) cX = Susp-conn (⊙Susp^-conn n cX)
⊙Susp^-+ : ∀ {i} (m n : ℕ) {X : Ptd i}
→ ⊙Susp^ m (⊙Susp^ n X) == ⊙Susp^ (m + n) X
⊙Susp^-+ O n = idp
⊙Susp^-+ (S m) n = ap ⊙Susp (⊙Susp^-+ m n)
⊙Susp^-fmap : ∀ {i j} (n : ℕ) {X : Ptd i} {Y : Ptd j}
→ X ⊙→ Y → ⊙Susp^ n X ⊙→ ⊙Susp^ n Y
⊙Susp^-fmap O f = f
⊙Susp^-fmap (S n) f = ⊙Susp-fmap (⊙Susp^-fmap n f)
⊙Susp^-fmap-idf : ∀ {i} (n : ℕ) (X : Ptd i)
→ ⊙Susp^-fmap n (⊙idf X) == ⊙idf (⊙Susp^ n X)
⊙Susp^-fmap-idf O X = idp
⊙Susp^-fmap-idf (S n) X =
ap ⊙Susp-fmap (⊙Susp^-fmap-idf n X) ∙ ⊙Susp-fmap-idf (⊙Susp^ n X)
⊙Susp^-fmap-cst : ∀ {i j} (n : ℕ) {X : Ptd i} {Y : Ptd j}
→ ⊙Susp^-fmap n (⊙cst {X = X} {Y = Y}) == ⊙cst
⊙Susp^-fmap-cst O = idp
⊙Susp^-fmap-cst (S n) = ap ⊙Susp-fmap (⊙Susp^-fmap-cst n)
∙ (⊙Susp-fmap-cst {X = ⊙Susp^ n _})
⊙Susp^-fmap-∘ : ∀ {i j k} (n : ℕ) {X : Ptd i} {Y : Ptd j} {Z : Ptd k}
(g : Y ⊙→ Z) (f : X ⊙→ Y)
→ ⊙Susp^-fmap n (g ⊙∘ f) == ⊙Susp^-fmap n g ⊙∘ ⊙Susp^-fmap n f
⊙Susp^-fmap-∘ O g f = idp
⊙Susp^-fmap-∘ (S n) g f =
ap ⊙Susp-fmap (⊙Susp^-fmap-∘ n g f)
∙ ⊙Susp-fmap-∘ (⊙Susp^-fmap n g) (⊙Susp^-fmap n f)
⊙Susp^-Susp-split-iso : ∀ {i} (n : ℕ) (X : Ptd i)
→ ⊙Susp^ (S n) X ⊙≃ ⊙Susp^ n (⊙Susp X)
⊙Susp^-Susp-split-iso O X = ⊙ide _
⊙Susp^-Susp-split-iso (S n) X = ⊙Susp-emap (⊙Susp^-Susp-split-iso n X)
⊙Sphere : (n : ℕ) → Ptd₀
⊙Sphere n = ⊙Susp^ n ⊙Bool
Sphere : (n : ℕ) → Type₀
Sphere n = de⊙ (⊙Sphere n)
abstract
Sphere-conn : ∀ (n : ℕ) → is-connected ⟨ n ⟩₋₁ (Sphere n)
Sphere-conn 0 = inhab-conn true
Sphere-conn (S n) = Susp-conn (Sphere-conn n)
-- favonia: [S¹] has its own elim rules in Circle.agda.
⊙S⁰ = ⊙Sphere 0
⊙S¹ = ⊙Sphere 1
⊙S² = ⊙Sphere 2
⊙S³ = ⊙Sphere 3
S⁰ = Sphere 0
S¹ = Sphere 1
S² = Sphere 2
S³ = Sphere 3
| 29.25641
| 78
| 0.532428
|
4d9fcdd4c267536aad667ade0858f53562932f86
| 6,969
|
agda
|
Agda
|
proglangs-learning/Agda/plfa-exercises/Practice2.agda
|
helq/old_code
|
a432faf1b340cb379190a2f2b11b997b02d1cd8d
|
[
"CC0-1.0"
] | null | null | null |
proglangs-learning/Agda/plfa-exercises/Practice2.agda
|
helq/old_code
|
a432faf1b340cb379190a2f2b11b997b02d1cd8d
|
[
"CC0-1.0"
] | 4
|
2020-03-10T19:20:21.000Z
|
2021-06-07T15:39:48.000Z
|
proglangs-learning/Agda/plfa-exercises/Practice2.agda
|
helq/old_code
|
a432faf1b340cb379190a2f2b11b997b02d1cd8d
|
[
"CC0-1.0"
] | null | null | null |
module plfa-exercises.Practice2 where
-- Trying exercises:
-- 5.2 pp 340
-- 5.7 pp 386
-- 6.1 pp 423
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; subst)
open import Data.Nat using (ℕ; zero; suc; _+_; _*_)
open import Relation.Nullary using (¬_)
open import Data.Product using (_×_; proj₁; ∃-syntax) renaming (_,_ to ⟨_,_⟩)
open import Data.Sum using (_⊎_; inj₁; inj₂)
--open import plfa.part1.Isomorphism using (_≃_; extensionality)
∀-elim : ∀ {A : Set} {B : A → Set} → (L : ∀ (x : A) → B x) → (M : A) → B M
∀-elim l m = l m
--totype : ℕ → Set
--totype 0 = ℕ
--totype 1 = 4 ≡ 2 + 2
--totype _ = ℕ
-- λ noidea → ∀-elim {ℕ} {totype} noidea 4
------- Preliminary proofs -------
modus-tollens : ∀ {A B : Set}
→ (A → B)
-------------
→ (¬ B → ¬ A)
modus-tollens a→b = λ{¬b → λ{a → ¬b (a→b a)}}
----------------------------------
postulate
dne : ∀ {A : Set} → ¬ ¬ A → A
--data Σ (A : Set) (B : A → Set) : Set where
-- ⟨_,_⟩ : (x : A) → B x → Σ A B
--
--Σ-syntax = Σ
--infix 2 Σ-syntax
--syntax Σ-syntax A (λ x → B) = Σ[ x ∈ A ] B
--
--∃ : ∀ {A : Set} (B : A → Set) → Set
--∃ {A} B = Σ A B
--
--∃-syntax = ∃
--syntax ∃-syntax (λ x → B) = ∃[ x ] B
--⟨,⟩-syntax : ∀ {A : Set} {B : A → Set} (x : A) → B x → Σ A B
--⟨,⟩-syntax = ⟨_,_⟩
----⟨,⟩-syntax = Σ.⟨_,_⟩
--syntax ⟨,⟩-syntax x p = the-proof-for x is p
--∃-elim : ∀ {A : Set} {B : A → Set} {C : Set}
-- → (∀ x → B x → C)
-- → ∃[ x ] B x
-- ---------------
-- → C
--∃-elim f ⟨ x , y ⟩ = f x y
record _⇔_ (A B : Set) : Set where
field
to : A → B
from : B → A
---------------------- Athena book exercises ----------------------
exercise531 : ∀ {A : Set} {R : A → A → Set}
→ (∀ (x y : A) → R x y)
→ (∀ (x : A) → R x x)
exercise531 R x = R x x
exercise532 : ∀ {A : Set} (x : A) → ∃[ y ] (x ≡ y)
exercise532 e = ⟨ e , refl ⟩
exercise533 : ∀ {A : Set} {P Q S : A → Set}
→ (∀ {x} → (P x ⊎ Q x) → S x)
→ (∃[ y ] (Q y))
-------------------------
→ (∃[ y ] (S y))
exercise533 Px⊎Qx→Sx ⟨ y , qy ⟩ =
let
-- P x ⊎ Q x
py⊎qy = inj₂ qy
in ⟨ y , Px⊎Qx→Sx py⊎qy ⟩
exercise534 : ∀ {A : Set} {P Q S : A → Set}
→ (∃[ y ] (P y × Q y))
→ (∀ {y} → P y → S y)
-------------------------
→ (∃[ y ] (S y × Q y))
exercise534 ⟨ y , ⟨ py , qy ⟩ ⟩ py→sy = ⟨ y , ⟨ py→sy py , qy ⟩ ⟩
exercise535 : ∀ {A : Set} {P Q : A → Set}
→ (¬ ∃[ x ] (Q x))
→ (∀ {x} → P x → Q x)
-------------------------
→ (¬ ∃[ x ] (P x))
exercise535 ¬∃x-qx ∀x→px→qx = λ{ ∃x-px@(⟨ x , px ⟩) → ¬∃x-qx ⟨ x , ∀x→px→qx px ⟩ }
exercise536 : ∀ {A : Set} {P Q S : A → Set}
→ (∀ {y} → P y → Q y)
→ (∃[ y ] (S y × ¬ Q y))
------------------------
→ (∃[ y ] (S y × ¬ P y))
exercise536 ∀y→py→qy ⟨ y , ⟨ sy , ¬qy ⟩ ⟩ = ⟨ y , ⟨ sy , modus-tollens ∀y→py→qy ¬qy ⟩ ⟩
exercise537 : ∀ {A : Set} {P Q : A → Set} {R : A → A → Set}
→ (∀ {x} → R x x → P x)
→ (∃[ x ] (P x) → ¬ ∃[ x ] (Q x))
-------------------------------
→ ((∀ {x} → Q x) → ¬ ∃[ x ] (R x x))
exercise537 ∀x→rxx→px ∃x-px→¬∃x-qx ∀x→qx = λ{∃x-rxx@(⟨ x , rxx ⟩) → ∃x-px→¬∃x-qx ⟨ x , ∀x→rxx→px rxx ⟩ ⟨ x , ∀x→qx {x} ⟩ }
exercise538 : ∀ {A : Set} {P Q : A → Set}
→ (∃[ x ] (P x ⊎ Q x)) ⇔ (∃[ x ] (P x) ⊎ ∃[ x ] (Q x))
exercise538 = record { to = to ; from = from }
where
to : ∀ {A : Set} {P Q : A → Set}
→ ∃[ x ] (P x ⊎ Q x)
-----------------------------
→ ∃[ x ] (P x) ⊎ ∃[ x ] (Q x)
to ⟨ x , (inj₁ px) ⟩ = inj₁ ⟨ x , px ⟩
to ⟨ x , (inj₂ qx) ⟩ = inj₂ ⟨ x , qx ⟩
from : ∀ {A : Set} {P Q : A → Set}
→ ∃[ x ] (P x) ⊎ ∃[ x ] (Q x)
-----------------------------
→ ∃[ x ] (P x ⊎ Q x)
from (inj₁ ⟨ x , px ⟩) = ⟨ x , inj₁ px ⟩
from (inj₂ ⟨ x , qx ⟩) = ⟨ x , inj₂ qx ⟩
------------------
exercise571 : ∀ {A : Set} {P Q : A → Set}
→ (∀ {x} → P x ⇔ Q x)
→ (∀ {x} → P x) ⇔ (∀ {x} → Q x)
exercise571 px⇔qx =
record
{ to = λ{px → (_⇔_.to px⇔qx) px}
; from = λ{qx → (_⇔_.from px⇔qx) qx}
}
--exercise571 ∀x→px⇔qx =
-- record
-- { to = λ{∀x→px → (_⇔_.to (∀x→px⇔qx {x})) (∀x→px {x})}
-- ; from = λ{∀x→qx → (_⇔_.from (∀x→px⇔qx {x})) (∀x→qx {x})}
-- }
exercise572 : ∀ {A : Set} {B : Set} {Q S : B → Set} {R T : B → B → Set}
→ (∃[ y ] (R y y × A))
→ (∃[ y ] (Q y × T y y))
→ (∀ y → A × Q y → ¬ S y)
→ (∃[ y ] (¬ S y × T y y))
exercise572 ⟨ _ , ⟨ _ , a ⟩ ⟩ ⟨ y , ⟨ qy , tyy ⟩ ⟩ ∀y→a×qy→¬sy = ⟨ y , ⟨ ∀y→a×qy→¬sy y ⟨ a , qy ⟩ , tyy ⟩ ⟩
-- This is fucking false!!!
--postulate
-- existence : {A : Set} {P : A → Set}
-- → (∀ x → P x)
-- ----------------
-- → (∃[ x ] (P x))
-- This cannot be proved! Take the empty set as an example. For any function
-- and relation the ∀'s are trivially true, but there is no element that
-- actually fulfills the function or the relation
--exercise574 : ∀ {A : Set} {F : A → A} {R : A → A → Set}
-- → (∀ x → R x x)
-- → (∀ x → F x ≡ F (F x))
-- -----------------------
-- → (∃[ y ] (R y (F y)))
--exercise574 ∀x→rxx ∀x→fx≡ffx = ⟨ ? , ? ⟩
exercise574 : ∀ {F : ℕ → ℕ} {R : ℕ → ℕ → Set}
→ (∀ x → R x x)
→ (∀ x → F x ≡ F (F x))
-----------------------
→ (∃[ y ] (R y (F y)))
exercise574 {f} {r} ∀x→rxx ∀x→fx≡ffx =
let y = f 0
y≡fy = ∀x→fx≡ffx 0 -- F 0 ≡ F (F 0) => y ≡ F y
ryy = ∀x→rxx y -- R (F 0) (F 0) => R y y
ryfy = subst (r y) y≡fy ryy -- R (F 0) (F (F 0)) => R y (F y)
in ⟨ y , ryfy ⟩
------------------
exercise61a : ∀ {A B : Set}
→ (B ⊎ (A → B))
→ A
---------------
→ B
exercise61a (inj₁ b) _ = b
exercise61a (inj₂ a→b) a = a→b a
exercise61b : ∀ {A B C : Set}
→ (¬ B → ¬ C)
→ ((A × B) ⊎ ¬ ¬ C)
-------------------
→ B
exercise61b ¬b→¬c (inj₁ ⟨ a , b ⟩) = b
exercise61b ¬b→¬c (inj₂ ¬¬c) = dne ((modus-tollens ¬b→¬c) ¬¬c)
¬¬ : ∀ {A : Set}
→ A
→ ¬ ¬ A
¬¬ a = λ{¬a → ¬a a}
-- What is the difference between: ∀ x → P x → ∀ y → R y → L and (∀ x → P x) → (∀ y → R y) → L
---- Can the following be proved without Double Negation Elimination (dne)?
--lemma₁ : ∀ {A B : Set}
-- → (¬ B → ¬ A)
-- → A
-- -------------------
-- → B
--lemma₁ = ?
--
--lemma₂ : ∀ {A B : Set}
-- → (¬ B → ¬ A)
-- → ¬ ¬ A
-- -------------------
-- → B
--lemma₂ = ?
--
---- `lemma₁` seems not to imply `dne` but it can't be proven without it.
----
---- The answer is NO (for the second one)! Because from it we can prove `dne`!
---- `dne` cannot be proved in vanila Agda.
--dne₁ : ∀ {A : Set} → ¬ ¬ A → A
--dne₁ ¬¬a = lemma₁ ? ?
--
--dne₂ : ∀ {A : Set} → ¬ ¬ A → A
--dne₂ ¬¬a = lemma₂ (λ{x → x}) ¬¬a
| 29.529661
| 122
| 0.367485
|
dfa9cfd2366dd48ecbe94752c07d238aeae6e8c8
| 1,591
|
agda
|
Agda
|
archive/agda-1/LiteralSequent.agda
|
m0davis/oscar
|
52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb
|
[
"RSA-MD"
] | null | null | null |
archive/agda-1/LiteralSequent.agda
|
m0davis/oscar
|
52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb
|
[
"RSA-MD"
] | 1
|
2019-04-29T00:35:04.000Z
|
2019-05-11T23:33:04.000Z
|
archive/agda-1/LiteralSequent.agda
|
m0davis/oscar
|
52e1cdbdee54d9a8eaee04ee518a0d7f61d25afb
|
[
"RSA-MD"
] | null | null | null |
{-# OPTIONS --allow-unsolved-metas #-}
module LiteralSequent where
open import Sequent
open import IsLiteralSequent
record LiteralSequent : Set
where
constructor ⟨_⟩
field
{sequent} : Sequent
isLiteralSequent : IsLiteralSequent sequent
open LiteralSequent public
open import OscarPrelude
private
module _ where
pattern ⟪_,_⟫ h s = ⟨_⟩ {h} s
pattern ⟪_⟫ h = (⟨_⟩ {h} _)
instance EqLiteralSequent : Eq LiteralSequent
Eq._==_ EqLiteralSequent ⟪ Φ₁ ⟫ ⟪ Φ₂ ⟫ with Φ₁ ≟ Φ₂
Eq._==_ EqLiteralSequent ⟨ !Φ₁ ⟩ ⟨ !Φ₂ ⟩ | yes refl with !Φ₁ ≟ !Φ₂
Eq._==_ EqLiteralSequent _ _ | yes refl | yes refl = yes refl
Eq._==_ EqLiteralSequent ⟨ Φ₁ ⟩ ⟨ Φ₂ ⟩ | yes refl | no !Φ₁≢!Φ₂ = no λ {refl → !Φ₁≢!Φ₂ refl}
Eq._==_ EqLiteralSequent ⟨ Φ₁ ⟩ ⟨ Φ₂ ⟩ | no Φ₁≢Φ₂ = no λ {refl → Φ₁≢Φ₂ refl}
module _ where
open import HasNegation
open import IsLiteralFormula
instance HasNegationLiteralSequent : HasNegation LiteralSequent
HasNegation.~ HasNegationLiteralSequent ⟨ atomic 𝑃 τs ╱ φˢs ⟩ = ⟨ logical 𝑃 τs ╱ φˢs ⟩
HasNegation.~ HasNegationLiteralSequent ⟨ logical 𝑃 τs ╱ φˢs ⟩ = ⟨ atomic 𝑃 τs ╱ φˢs ⟩
open import 𝓐ssertion
instance 𝓐ssertionLiteralSequent : 𝓐ssertion LiteralSequent
𝓐ssertionLiteralSequent = record {}
open import HasSatisfaction
instance HasSatisfactionLiteralSequent : HasSatisfaction LiteralSequent
HasSatisfaction._⊨_ HasSatisfactionLiteralSequent I Φ = I ⊨ sequent Φ
open import HasDecidableValidation
instance HasDecidableValidationLiteralSequent : HasDecidableValidation LiteralSequent
HasDecidableValidationLiteralSequent = {!!}
| 29.462963
| 93
| 0.735387
|
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