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open import Relation.Binary.Core
module TreeSort.Impl2.Correctness.Order {A : Set}
(_≤_ : A → A → Set)
(tot≤ : Total _≤_)
(trans≤ : Transitive _≤_) where
open import BBSTree _≤_
open import BBSTree.Properties _≤_ trans≤
open import Data.List
open import Function using (_∘_)
open import List.Sorted _≤_
open import TreeSort.Impl2 _≤_ tot≤
theorem-treeSort-sorted : (xs : List A) → Sorted (flatten (treeSort xs))
theorem-treeSort-sorted = lemma-bbst-sorted ∘ treeSort
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module Issue637 where
data Nat : Set where
zero : Nat
suc : Nat → Nat
infixl 60 _+_
_+_ : Nat → Nat → Nat
zero + n = n
suc m + n = suc (n + m)
data _≡_ {A : Set}(x : A) : A → Set where
refl : x ≡ x
`1 = suc zero
`2 = suc `1
`3 = suc `2
`4 = `2 + `2
`8 = `4 + `4
`16 = `8 + `8
`32 = `16 + `16
`50 = `32 + `16 + `2
`64 = `32 + `32
`100 = `64 + `32 + `4
`200 = `100 + `100
`400 = `200 + `200
`800 = `400 + `400
`1000 = `800 + `200
`2000 = `1000 + `1000
`4000 = `2000 + `2000
prf : `16 ≡ (`4 + (`4 + `8))
prf = refl
infixr 40 _∷_
data Vec : Nat → Set where
[] : Vec zero
_∷_ : ∀ {n} → Nat → Vec n → Vec (suc n)
fromN : ∀ {n} → Nat → Vec n
fromN {zero} _ = []
fromN {suc n} x = x ∷ fromN (suc x)
sum : ∀ {n} → Vec n → Nat
sum [] = zero
sum (n ∷ ns) = n + sum ns
prf₁ : sum (fromN {`100} `1) ≡ (`4000 + `1000 + `50)
prf₁ = refl
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module plfa-exercises.Practice3 where
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; sym; trans; cong)
open Eq.≡-Reasoning using (_≡⟨⟩_; _≡⟨_⟩_; begin_; _∎)
open import Data.Bool using (Bool; true; false; T; _∧_; _∨_; not)
open import Data.Nat using (ℕ; zero; suc; _+_; _*_; _∸_; _≤_; s≤s; z≤n)
open import Data.Nat.Properties using
(+-assoc; +-identityˡ; +-identityʳ; *-assoc; *-identityˡ; *-identityʳ; *-distribʳ-+; +-suc; *-suc; *-comm)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Data.Product using (_×_; ∃; ∃-syntax) renaming (_,_ to ⟨_,_⟩)
open import Function using (_∘_; flip)
open import Level using (Level)
open import plfa.part1.Isomorphism using (_≃_; _⇔_)
--import Data.List.All using (All; []; _∷_)
open import Data.List using (List; []; _∷_; _++_; length; reverse; map; foldr; downFrom; foldl)
open import Axiom.Extensionality.Propositional using (Extensionality)
import Data.Nat.Solver as NatSolver
open import Data.Sum using (_⊎_; inj₁; inj₂)
-- Exercises from plfa Lists
map∘compose₀ : ∀ {A B C : Set} (f : A → B) (g : B → C) (xs : List A) → map (g ∘ f) xs ≡ (map g ∘ map f) xs
map∘compose₀ f g [] = refl
map∘compose₀ f g (x ∷ xs) = cong (g (f x) ∷_) (map∘compose₀ f g xs)
module _ (ext : ∀ {ℓ ℓ′} → Extensionality ℓ ℓ′) where
map∘compose : ∀ {A B C : Set}
(f : A → B) (g : B → C)
→ map (g ∘ f) ≡ (map g ∘ map f)
map∘compose f g = ext (map∘compose₀ f g)
map-++-distribute : ∀ {A B : Set}
(f : A → B) (xs ys : List A)
→ map f (xs ++ ys) ≡ map f xs ++ map f ys
map-++-distribute f [] ys = refl
map-++-distribute f (x ∷ xs) ys = cong (f x ∷_) (map-++-distribute f xs ys)
data Tree (A B : Set) : Set where
leaf : A → Tree A B
node : Tree A B → B → Tree A B → Tree A B
map-Tree : ∀ {A B C D : Set} → (A → C) → (B → D) → Tree A B → Tree C D
map-Tree f g (leaf a) = leaf (f a)
map-Tree f g (node lt b rt) = node (map-Tree f g lt) (g b) (map-Tree f g rt)
--node (leaf 5) (3 ∷ []) (leaf 0)
--map-Tree suc (6 ∷_) (node (leaf 5) (3 ∷ []) (leaf 0))
product : List ℕ → ℕ
product = foldr _*_ 1
_ : product (1 ∷ 2 ∷ 3 ∷ 4 ∷ []) ≡ 24
_ = refl
sum : List ℕ → ℕ
sum = foldr _+_ 0
sum-downFrom : ∀ (n : ℕ) → sum (downFrom n) * 2 ≡ n * (n ∸ 1)
sum-downFrom zero = refl
sum-downFrom 1 = refl
sum-downFrom (suc (suc n))
rewrite
-- sum (downFrom (suc (suc n))) * 2 ≡ suc (suc n) * (suc (suc n) ∸ 1)
-- suc (suc ((n + (n + foldr _+_ 0 (downFrom n))) * 2)) ≡ suc (n + suc (n + n * suc n))
*-distribʳ-+ 2 n (foldr _+_ 0 (downFrom (suc n)))
-- suc (suc (n * 2 + (n + foldr _+_ 0 (downFrom n)) * 2)) ≡ suc (n + suc (n + n * suc n))
| sum-downFrom (suc n)
-- suc (suc (n * 2 + (n + n * n))) ≡ suc (n + suc (n + n * suc n))
| +-suc n (n + n * suc n)
-- suc (suc (n * 2 + (n + n * n))) ≡ suc (suc (n + (n + n * suc n)))
| *-suc n n
-- suc (suc (n * 2 + (n + n * n))) ≡ suc (suc (n + (n + (n + n * n))))
| *-comm n 2
-- suc (suc (n + (n + 0) + (n + n * n))) ≡ suc (suc (n + (n + (n + n * n))))
| +-identityʳ n
-- suc (suc (n + n + (n + n * n))) ≡ suc (suc (n + (n + (n + n * n))))
| +-assoc n n (n + n * n)
-- suc (suc (n + (n + (n + n * n)))) ≡ suc (suc (n + (n + (n + n * n))))
= refl
sum-downFrom` : ∀ (n : ℕ) → sum (downFrom n) * 2 ≡ n * (n ∸ 1)
sum-downFrom` zero = refl
sum-downFrom` 1 = refl
sum-downFrom` (suc (suc n))
rewrite
-- sum (downFrom (suc (suc n))) * 2 ≡ suc (suc n) * (suc (suc n) ∸ 1)
-- suc (suc ((n + (n + foldr _+_ 0 (downFrom n))) * 2)) ≡ suc (n + suc (n + n * suc n))
*-distribʳ-+ 2 n (foldr _+_ 0 (downFrom (suc n)))
-- suc (suc (n * 2 + (n + foldr _+_ 0 (downFrom n)) * 2)) ≡ suc (n + suc (n + n * suc n))
| sum-downFrom` (suc n)
-- suc (suc (n * 2 + (n + n * n))) ≡ suc (n + suc (n + n * suc n))
= simplified n
where
open NatSolver using (module +-*-Solver)
open +-*-Solver using (solve; _:+_; _:*_; _:=_; con)
simplified : ∀ n → 2 + (n * 2 + (n + n * n)) ≡ 1 + (n + (1 + (n + n * (1 + n))))
simplified = solve 1 (λ n → con 2 :+ (n :* con 2 :+ (n :+ n :* n)) := con 1 :+ (n :+ (con 1 :+ (n :+ n :* (con 1 :+ n))))) refl
-- solve 1 (λ n → n :+ con 1 := con 1 :+ n) refl
--open NatSolver using (module +-*-Solver)
--open +-*-Solver using (solve; _:+_; _:*_; _:=_; con; Polynomial)
--
--lemma : ∀ x y → x + y * 1 + 3 ≡ 2 + 1 + y + x
--lemma = solve 2 (λ x y → x :+ y :* con 1 :+ con 3 := con 2 :+ con 1 :+ y :+ x) refl
record IsMonoid {A : Set} (_⊗_ : A → A → A) (e : A) : Set where
field
assoc : ∀ (x y z : A) → (x ⊗ y) ⊗ z ≡ x ⊗ (y ⊗ z)
identityˡ : ∀ (x : A) → e ⊗ x ≡ x
identityʳ : ∀ (x : A) → x ⊗ e ≡ x
open IsMonoid
foldr-monoid : ∀ {A : Set} (_⊗_ : A → A → A) (e : A) → IsMonoid _⊗_ e →
∀ (xs : List A) (y : A) → foldr _⊗_ y xs ≡ foldr _⊗_ e xs ⊗ y
foldr-monoid _⊗_ e ⊗-monoid [] y =
begin
foldr _⊗_ y []
≡⟨⟩
y
≡⟨ sym (identityˡ ⊗-monoid y) ⟩
(e ⊗ y)
≡⟨⟩
foldr _⊗_ e [] ⊗ y
∎
foldr-monoid _⊗_ e ⊗-monoid (x ∷ xs) y =
begin
foldr _⊗_ y (x ∷ xs)
≡⟨⟩
x ⊗ (foldr _⊗_ y xs)
≡⟨ cong (x ⊗_) (foldr-monoid _⊗_ e ⊗-monoid xs y) ⟩
x ⊗ (foldr _⊗_ e xs ⊗ y)
≡⟨ sym (assoc ⊗-monoid x (foldr _⊗_ e xs) y) ⟩
(x ⊗ foldr _⊗_ e xs) ⊗ y
≡⟨⟩
foldr _⊗_ e (x ∷ xs) ⊗ y
∎
--foldl : {A B : Set} (f : B → A → B) → B → List A → B
--foldl f b [] = b
--foldl f b (a ∷ as') = foldl f (f b a) as'
_ : foldl _∸_ 20 (4 ∷ 3 ∷ []) ≡ 13
_ =
begin
foldl _∸_ 20 (4 ∷ 3 ∷ []) ≡⟨⟩
foldl _∸_ (20 ∸ 4) (3 ∷ []) ≡⟨⟩
foldl _∸_ ((20 ∸ 4) ∸ 3) [] ≡⟨⟩
(20 ∸ 4) ∸ 3 ≡⟨⟩
13
∎
_ : foldr _∸_ 20 (4 ∷ 3 ∷ []) ≡ 4
_ =
begin
foldr _∸_ 20 (4 ∷ 3 ∷ []) ≡⟨⟩
4 ∸ foldr _∸_ 20 (3 ∷ []) ≡⟨⟩
4 ∸ (3 ∸ foldr _∸_ 20 []) ≡⟨⟩
4 ∸ (3 ∸ 20) ≡⟨⟩
4 ∸ 0 ≡⟨⟩
4
∎
foldl-monoid : {A : Set}
(_⊗_ : A → A → A) (e : A)
→ IsMonoid _⊗_ e
→ ∀ (y : A) (xs : List A)
→ (y ⊗ foldl _⊗_ e xs) ≡ foldl _⊗_ y xs
foldl-monoid _⊗_ e monoid-⊗ y [] = identityʳ monoid-⊗ y
foldl-monoid _⊗_ e monoid-⊗ y (x ∷ xs) =
begin
y ⊗ foldl _⊗_ e (x ∷ xs) ≡⟨⟩
y ⊗ foldl _⊗_ (e ⊗ x) xs ≡⟨ cong (λ exp → y ⊗ foldl _⊗_ exp xs) (identityˡ monoid-⊗ x) ⟩
y ⊗ foldl _⊗_ x xs ≡⟨ cong (y ⊗_) (sym (foldl-monoid _⊗_ e monoid-⊗ x xs)) ⟩
y ⊗ (x ⊗ foldl _⊗_ e xs) ≡⟨ sym (assoc monoid-⊗ y x (foldl _⊗_ e xs)) ⟩
(y ⊗ x) ⊗ foldl _⊗_ e xs ≡⟨ foldl-monoid _⊗_ e monoid-⊗ (y ⊗ x) xs ⟩
foldl _⊗_ (y ⊗ x) xs ≡⟨⟩
foldl _⊗_ y (x ∷ xs)
∎
foldr-monoid-foldl₀ : ∀ {A : Set} {_⊗_ : A → A → A} {e : A}
→ (xs : List A)
→ IsMonoid _⊗_ e
→ foldr _⊗_ e xs ≡ foldl _⊗_ e xs
foldr-monoid-foldl₀ [] _ = refl
foldr-monoid-foldl₀ {_} {_⊗_} {e} (x ∷ xs) monoid-⊗ =
begin
foldr _⊗_ e (x ∷ xs) ≡⟨⟩
x ⊗ foldr _⊗_ e xs ≡⟨ cong (x ⊗_) (foldr-monoid-foldl₀ xs monoid-⊗) ⟩
x ⊗ foldl _⊗_ e xs ≡⟨ foldl-monoid _⊗_ e monoid-⊗ x xs ⟩
foldl _⊗_ x xs ≡⟨ cong (λ exp → foldl _⊗_ exp xs) (sym (identityˡ monoid-⊗ x)) ⟩
foldl _⊗_ (e ⊗ x) xs ≡⟨⟩
foldl _⊗_ e (x ∷ xs)
∎
module _ (ext : ∀ {ℓ ℓ′} → Extensionality ℓ ℓ′) where
foldr-monoid-foldl : ∀ {A : Set} {_⊗_ : A → A → A} {e : A}
→ IsMonoid _⊗_ e
→ foldr _⊗_ e ≡ foldl _⊗_ e
foldr-monoid-foldl monoid-⊗ = ext (λ xs → foldr-monoid-foldl₀ xs monoid-⊗)
data All {A : Set} (P : A → Set) : List A → Set where
[] : All P []
_∷_ : ∀ {x : A} {xs : List A} → P x → All P xs → All P (x ∷ xs)
pattern [_] z = z ∷ []
pattern [_,_] y z = y ∷ z ∷ []
pattern [_,_,_] x y z = x ∷ y ∷ z ∷ []
pattern [_,_,_,_] w x y z = w ∷ x ∷ y ∷ z ∷ []
pattern [_,_,_,_,_] v w x y z = v ∷ w ∷ x ∷ y ∷ z ∷ []
pattern [_,_,_,_,_,_] u v w x y z = u ∷ v ∷ w ∷ x ∷ y ∷ z ∷ []
_ : All (_≤ 2) [ 0 , 1 , 2 ] -- These brackes desugar to ∷ and [] for List definition
_ = [ z≤n , s≤s z≤n , s≤s (s≤s z≤n) ] -- ∷ and [] from All constructors
++-assoc : ∀ {A : Set} (xs ys zs : List A)
→ (xs ++ ys) ++ zs ≡ xs ++ (ys ++ zs)
++-assoc [] ys zs = refl
++-assoc (x ∷ xs) ys zs =
begin
((x ∷ xs) ++ ys) ++ zs ≡⟨⟩
x ∷ (xs ++ ys) ++ zs ≡⟨ cong (x ∷_) (++-assoc xs ys zs) ⟩
x ∷ xs ++ ys ++ zs ≡⟨⟩
(x ∷ xs) ++ ys ++ zs
∎
length-++ : ∀ {A : Set} (xs ys : List A)
→ length (xs ++ ys) ≡ length xs + length ys
length-++ [] ys = refl
length-++ (x ∷ xs) ys =
begin
length ((x ∷ xs) ++ ys) ≡⟨⟩
suc (length (xs ++ ys)) ≡⟨ cong suc (length-++ xs ys) ⟩
suc (length xs + length ys) ≡⟨⟩
suc (length xs) + length ys ≡⟨⟩
length (x ∷ xs) + length ys
∎
++-[] : ∀ {A : Set} (xs : List A) → xs ++ [] ≡ xs
++-[] [] = refl
++-[] (x ∷ xs) = cong (x ∷_) (++-[] xs)
data Any {A : Set} (P : A → Set) : List A → Set where
here : ∀ {x : A} {xs : List A} → P x → Any P (x ∷ xs)
there : ∀ {x : A} {xs : List A} → Any P xs → Any P (x ∷ xs)
infix 4 _∈_ _∉_
_∈_ : ∀ {A : Set} (x : A) (xs : List A) → Set
x ∈ xs = Any (x ≡_) xs
_∉_ : ∀ {A : Set} (x : A) (xs : List A) → Set
x ∉ xs = ¬ (x ∈ xs)
_ : 0 ∈ [ 0 , 1 , 0 , 2 ]
_ = here refl
--_ = there (there (here refl))
---- Left as exercise for another day
--reverse-inv : ∀ {A : Set} (x : A) (xs : List A)
-- → reverse (x ∷ xs) ≡ reverse xs ++ [ x ]
--reverse-inv _ [] = refl
--reverse-inv y (x ∷ xs) =
-- begin
-- reverse (y ∷ x ∷ xs) ≡⟨⟩
-- foldl _∷`_ [ x , y ] xs ≡⟨ ? ⟩ -- LEFT FOR LATER
-- foldl _∷`_ [ x ] xs ++ [ y ] ≡⟨⟩
-- reverse (x ∷ xs) ++ [ y ]
-- ∎
---- λ y x xs → ≡
---- λ y x xs → foldl (λ y₁ x₁ → x₁ ∷ y₁) [ x , y ] xs ≡ foldl (λ y₁ x₁ → x₁ ∷ y₁) [ x ] xs ++ [ y ]
-- where
-- _∷`_ = flip _∷_
--
--reverse-++-distrib : ∀ {A : Set} (xs ys : List A)
-- → reverse (xs ++ ys) ≡ reverse ys ++ reverse xs
--reverse-++-distrib [] ys = sym (++-[] (reverse ys))
--reverse-++-distrib (x ∷ xs) ys =
-- begin
-- reverse ((x ∷ xs) ++ ys) ≡⟨⟩
-- reverse (x ∷ (xs ++ ys)) ≡⟨ reverse-inv x (xs ++ ys) ⟩
-- reverse (xs ++ ys) ++ [ x ] ≡⟨ cong (_++ [ x ]) (reverse-++-distrib xs ys) ⟩
-- (reverse ys ++ reverse xs) ++ [ x ] ≡⟨ ++-assoc (reverse ys) (reverse xs) [ x ] ⟩
-- reverse ys ++ (reverse xs ++ [ x ]) ≡⟨ sym (cong (reverse ys ++_) (reverse-inv x xs)) ⟩
-- reverse ys ++ reverse (x ∷ xs)
-- ∎
-- --foldl _∷`_ [ x ] (xs ++ ys) ≡⟨ ? ⟩
-- --foldl _∷`_ [] ys ++ foldl _∷`_ [ x ] xs ≡⟨⟩
exercise-823→ : ∀ {A : Set} {z : A} (xs ys : List A)
→ z ∈ (xs ++ ys) → (z ∈ xs) ⊎ (z ∈ ys)
exercise-823→ [] [] ()
exercise-823→ [] _ z∈[]++ys = inj₂ z∈[]++ys
exercise-823→ (_ ∷ _) _ (here refl) = inj₁ (here refl)
exercise-823→ (x ∷ xs) ys (there z∈xs++ys) with exercise-823→ xs ys z∈xs++ys
... | inj₁ z∈xs = inj₁ (there z∈xs)
... | inj₂ z∈ys = inj₂ z∈ys
exercise-823← : ∀ {A : Set} {z : A} (xs ys : List A)
→ (z ∈ xs) ⊎ (z ∈ ys) → z ∈ (xs ++ ys)
exercise-823← [] _ (inj₁ ())
exercise-823← [] _ (inj₂ z∈ys) = z∈ys
exercise-823← (_ ∷ _) [] (inj₂ ())
exercise-823← (_ ∷ _) _ (inj₁ (here z∈xs)) = here z∈xs
exercise-823← (x ∷ xs) ys (inj₁ (there z∈xs)) with exercise-823← xs ys (inj₁ z∈xs)
... | z∈xs++ys = there z∈xs++ys
exercise-823← (_ ∷ xs) ys (inj₂ z∈y∷ys) with exercise-823← xs ys (inj₂ z∈y∷ys)
... | z∈xs++y∷ys = there z∈xs++y∷ys
exercise-823 : ∀ {A : Set} {z : A} (xs ys : List A)
→ (z ∈ xs ++ ys) ⇔ (z ∈ xs ⊎ z ∈ ys)
exercise-823 xs ys = record { to = exercise-823→ xs ys ; from = exercise-823← xs ys }
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{-# OPTIONS --without-K --safe #-}
-- The 'Trivial' instance, with a single arrow between objects
module Categories.Theory.Lawvere.Instance.Triv where
open import Data.Nat using (_*_)
open import Data.Unit.Polymorphic using (⊤; tt)
open import Relation.Binary.PropositionalEquality
using (_≡_; refl; isEquivalence)
open import Categories.Theory.Lawvere using (LawvereTheory)
Triv : ∀ {ℓ} → LawvereTheory ℓ ℓ
Triv = record
{ L = record
{ _⇒_ = λ _ _ → ⊤
; _≈_ = _≡_
; assoc = refl
; sym-assoc = refl
; identityˡ = refl
; identityʳ = refl
; identity² = refl
; equiv = isEquivalence
; ∘-resp-≈ = λ _ _ → refl
}
; T = record
{ terminal = record { ⊤ = 0 ; ⊤-is-terminal = record { ! = tt ; !-unique = λ _ → refl } }
; products = record
{ product = λ {m} {n} → record
{ A×B = m * n
; project₁ = refl
; project₂ = refl
; unique = λ _ _ → refl
}
}
}
; I = record
{ F₁ = λ _ → tt
; identity = refl
; homomorphism = refl
; F-resp-≈ = λ _ → refl
}
; CartF = record
{ F-resp-⊤ = record { ! = tt ; !-unique = λ _ → refl }
; F-resp-× = record { ⟨_,_⟩ = λ _ _ → tt ; project₁ = refl ; project₂ = refl ; unique = λ _ _ → refl }
}
}
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{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9.
Copyright (c) 2021, Oracle and/or its affiliates.
Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl
-}
open import LibraBFT.Prelude
open import LibraBFT.Lemmas
-- TODO-2: The following import should be eliminated and replaced
-- with the necessary module parameters (PK and MetaHonestPK)
open import LibraBFT.Base.PKCS
-- This module brings in the base types used through libra
-- and those necessary for the abstract model.
module LibraBFT.Abstract.Types.EpochConfig
(UID : Set)
(NodeId : Set)
where
open import LibraBFT.Base.Types
ℓ-EC : Level
ℓ-EC = ℓ+1 0ℓ
-- An epoch-configuration carries only simple data about the epoch; the complicated
-- parts will be provided by the System, defined below.
--
-- The reason for the separation is that we should be able to provide
-- an EpochConfig from a single peer state.
record EpochConfig : Set ℓ-EC where
constructor EpochConfig∙new
field
genesisUID : UID
epoch : Epoch
authorsN : ℕ
-- The set of members of this epoch.
Member : Set
Member = Fin authorsN
-- There is a partial isomorphism between NodeIds and the
-- authors participating in this epoch.
field
toNodeId : Member → NodeId
isMember? : NodeId → Maybe Member
nodeid-author-id : ∀{α} → isMember? (toNodeId α) ≡ just α
author-nodeid-id : ∀{nid α} → isMember? nid ≡ just α
→ toNodeId α ≡ nid
getPubKey : Member → PK
PK-inj : ∀ {m1 m2} → getPubKey m1 ≡ getPubKey m2 → m1 ≡ m2
IsQuorum : List Member → Set
bft-assumption : ∀ {xs ys}
→ IsQuorum xs → IsQuorum ys
→ ∃[ α ] (α ∈ xs × α ∈ ys × Meta-Honest-PK (getPubKey α))
open EpochConfig
PK-inj-same-ECs : ∀ {𝓔₁ 𝓔₂ : EpochConfig}{mbr₁ mbr₂}
→ 𝓔₁ ≡ 𝓔₂
→ getPubKey 𝓔₁ mbr₁ ≡ getPubKey 𝓔₂ mbr₂
→ toNodeId 𝓔₁ mbr₁ ≡ toNodeId 𝓔₂ mbr₂
PK-inj-same-ECs {𝓔₁} refl pks≡ = cong (toNodeId 𝓔₁) (PK-inj 𝓔₁ pks≡)
module _ (ec : EpochConfig) where
NodeId-PK-OK : PK → NodeId → Set
NodeId-PK-OK pk pid = ∃[ m ] (toNodeId ec m ≡ pid × getPubKey ec m ≡ pk)
NodeId-PK-OK-injective : ∀ {pk pid1 pid2}
→ NodeId-PK-OK pk pid1
→ NodeId-PK-OK pk pid2
→ pid1 ≡ pid2
NodeId-PK-OK-injective (m1 , pid1 , pk1) (m2 , pid2 , pk2)
rewrite PK-inj ec (trans pk1 (sym pk2)) = trans (sym pid1) pid2
module WithAbsVote (𝓔 : EpochConfig) where
-- The abstract model is connected to the implementaton by means of
-- 'VoteEvidence'. The record module will be parameterized by a
-- v of type 'VoteEvidence 𝓔 UID'; this v will provide evidence
-- that a given author voted for a given block (identified by the UID)
-- on the specified round.
--
-- When it comes time to instantiate the v above concretely, it will
-- be something that states that we have a signature from the specified
-- author voting for the specified block.
record AbsVoteData : Set where
constructor AbsVoteData∙new
field
abs-vRound : Round
abs-vMember : EpochConfig.Member 𝓔
abs-vBlockUID : UID
open AbsVoteData public
AbsVoteData-η : ∀ {r1 r2 : Round} {m1 m2 : EpochConfig.Member 𝓔} {b1 b2 : UID}
→ r1 ≡ r2
→ m1 ≡ m2
→ b1 ≡ b2
→ AbsVoteData∙new r1 m1 b1 ≡ AbsVoteData∙new r2 m2 b2
AbsVoteData-η refl refl refl = refl
VoteEvidence : Set₁
VoteEvidence = AbsVoteData → Set
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module Agda.Builtin.Float where
open import Agda.Builtin.Bool
open import Agda.Builtin.Nat
open import Agda.Builtin.Int
open import Agda.Builtin.String
postulate Float : Set
{-# BUILTIN FLOAT Float #-}
primitive
primFloatEquality : Float → Float → Bool
primFloatLess : Float → Float → Bool
primNatToFloat : Nat → Float
primFloatPlus : Float → Float → Float
primFloatMinus : Float → Float → Float
primFloatTimes : Float → 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
primShowFloat : Float → String
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Coinductive "natural" numbers
------------------------------------------------------------------------
{-# OPTIONS --without-K --guardedness --sized-types #-}
module Codata.Musical.Conat where
open import Codata.Musical.Notation
open import Data.Nat.Base using (ℕ; zero; suc)
open import Function
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as P using (_≡_)
------------------------------------------------------------------------
-- The type
data Coℕ : Set where
zero : Coℕ
suc : (n : ∞ Coℕ) → Coℕ
module Coℕ-injective where
suc-injective : ∀ {m n} → (Coℕ ∋ suc m) ≡ suc n → m ≡ n
suc-injective P.refl = P.refl
------------------------------------------------------------------------
-- Some operations
pred : Coℕ → Coℕ
pred zero = zero
pred (suc n) = ♭ n
fromℕ : ℕ → Coℕ
fromℕ zero = zero
fromℕ (suc n) = suc (♯ fromℕ n)
fromℕ-injective : ∀ {m n} → fromℕ m ≡ fromℕ n → m ≡ n
fromℕ-injective {zero} {zero} eq = P.refl
fromℕ-injective {zero} {suc n} ()
fromℕ-injective {suc m} {zero} ()
fromℕ-injective {suc m} {suc n} eq = P.cong suc (fromℕ-injective (P.cong pred eq))
∞ℕ : Coℕ
∞ℕ = suc (♯ ∞ℕ)
infixl 6 _+_
_+_ : Coℕ → Coℕ → Coℕ
zero + n = n
suc m + n = suc (♯ (♭ m + n))
------------------------------------------------------------------------
-- Equality
data _≈_ : Coℕ → Coℕ → Set where
zero : zero ≈ zero
suc : ∀ {m n} (m≈n : ∞ (♭ m ≈ ♭ n)) → suc m ≈ suc n
module ≈-injective where
suc-injective : ∀ {m n p q} → (suc m ≈ suc n ∋ suc p) ≡ suc q → p ≡ q
suc-injective P.refl = P.refl
setoid : Setoid _ _
setoid = record
{ Carrier = Coℕ
; _≈_ = _≈_
; isEquivalence = record
{ refl = refl
; sym = sym
; trans = trans
}
}
where
refl : Reflexive _≈_
refl {zero} = zero
refl {suc n} = suc (♯ refl)
sym : Symmetric _≈_
sym zero = zero
sym (suc m≈n) = suc (♯ sym (♭ m≈n))
trans : Transitive _≈_
trans zero zero = zero
trans (suc m≈n) (suc n≈k) = suc (♯ trans (♭ m≈n) (♭ n≈k))
------------------------------------------------------------------------
-- Legacy
import Codata.Conat as C
open import Codata.Thunk
import Size
fromMusical : ∀ {i} → Coℕ → C.Conat i
fromMusical zero = C.zero
fromMusical (suc n) = C.suc λ where .force → fromMusical (♭ n)
toMusical : C.Conat Size.∞ → Coℕ
toMusical C.zero = zero
toMusical (C.suc n) = suc (♯ toMusical (n .force))
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module index where
-- A formalisation of Haskell B. Curry’s thesis Grundlagen der Kombinatorischen
-- Logik. See <https://www.jstor.org/stable/2370619> (part 1) and
-- <https://www.jstor.org/stable/2370716> (part 2).
import CombinatoryLogic.Equality
import CombinatoryLogic.Forest
import CombinatoryLogic.Semantics
import CombinatoryLogic.Syntax
-- A formalisation of the theory and problems presented in To Mock a Mockingbird
-- by Raymond Smullyan.
import Mockingbird.Forest
import Mockingbird.Forest.Base
import Mockingbird.Forest.Birds
import Mockingbird.Forest.Combination
import Mockingbird.Forest.Combination.Base
import Mockingbird.Forest.Combination.Properties
import Mockingbird.Forest.Combination.Vec
import Mockingbird.Forest.Combination.Vec.Base
import Mockingbird.Forest.Combination.Vec.Properties
import Mockingbird.Forest.Extensionality
import Mockingbird.Problems.Chapter09
import Mockingbird.Problems.Chapter10
import Mockingbird.Problems.Chapter11
import Mockingbird.Problems.Chapter12
import Mockingbird.Problems.Chapter13
import Mockingbird.Problems.Chapter14
import Mockingbird.Problems.Chapter15
import Mockingbird.Problems.Chapter16
import Mockingbird.Problems.Chapter17
import Mockingbird.Problems.Chapter18
import Mockingbird.Problems.Chapter19
import Mockingbird.Problems.Chapter20
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Membership predicate for fresh lists
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Relation.Binary
module Data.List.Fresh.Membership.Setoid {c ℓ} (S : Setoid c ℓ) where
open import Level using (Level; _⊔_)
open import Data.List.Fresh
open import Data.List.Fresh.Relation.Unary.Any as Any using (Any)
open import Relation.Nullary
open Setoid S renaming (Carrier to A)
private
variable
r : Level
_∈_ : {R : Rel A r} → A → List# A R → Set _
x ∈ xs = Any (x ≈_) xs
_∉_ : {R : Rel A r} → A → List# A R → Set _
x ∉ xs = ¬ (x ∈ xs)
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Bijections on finite sets (i.e. permutations).
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Fin.Permutation where
open import Data.Empty using (⊥-elim)
open import Data.Fin
open import Data.Fin.Properties
import Data.Fin.Permutation.Components as PC
open import Data.Nat using (ℕ; suc; zero)
open import Data.Product using (proj₂)
open import Function.Inverse as Inverse using (_↔_; Inverse; _InverseOf_)
open import Function.Equality using (_⟨$⟩_)
open import Function using (_∘_)
open import Relation.Nullary using (¬_; yes; no)
open import Relation.Nullary.Negation using (contradiction)
open import Relation.Binary.PropositionalEquality as P
using (_≡_; _≢_; refl; trans; sym; →-to-⟶; cong; cong₂)
open P.≡-Reasoning
------------------------------------------------------------------------
-- Types
-- A bijection between finite sets of potentially different sizes.
-- There only exist inhabitants of this type if in fact m = n, however
-- it is often easier to prove the existence of a bijection without
-- first proving that the sets have the same size. Indeed such a
-- bijection is a useful way to prove that the sets are in fact the same
-- size. See '↔-≡' below.
Permutation : ℕ → ℕ → Set
Permutation m n = Fin m ↔ Fin n
Permutation′ : ℕ → Set
Permutation′ n = Permutation n n
------------------------------------------------------------------------
-- Helper functions
permutation : ∀ {m n} (f : Fin m → Fin n) (g : Fin n → Fin m) →
(→-to-⟶ g) InverseOf (→-to-⟶ f) → Permutation m n
permutation f g inv = record
{ to = →-to-⟶ f
; from = →-to-⟶ g
; inverse-of = inv
}
_⟨$⟩ʳ_ : ∀ {m n} → Permutation m n → Fin m → Fin n
_⟨$⟩ʳ_ = _⟨$⟩_ ∘ Inverse.to
_⟨$⟩ˡ_ : ∀ {m n} → Permutation m n → Fin n → Fin m
_⟨$⟩ˡ_ = _⟨$⟩_ ∘ Inverse.from
inverseˡ : ∀ {m n} (π : Permutation m n) {i} → π ⟨$⟩ˡ (π ⟨$⟩ʳ i) ≡ i
inverseˡ π = Inverse.left-inverse-of π _
inverseʳ : ∀ {m n} (π : Permutation m n) {i} → π ⟨$⟩ʳ (π ⟨$⟩ˡ i) ≡ i
inverseʳ π = Inverse.right-inverse-of π _
------------------------------------------------------------------------
-- Example permutations
-- Identity
id : ∀ {n} → Permutation′ n
id = Inverse.id
-- Transpose two indices
transpose : ∀ {n} → Fin n → Fin n → Permutation′ n
transpose i j = permutation (PC.transpose i j) (PC.transpose j i)
record
{ left-inverse-of = λ _ → PC.transpose-inverse _ _
; right-inverse-of = λ _ → PC.transpose-inverse _ _
}
-- Reverse the order of indices
reverse : ∀ {n} → Permutation′ n
reverse = permutation PC.reverse PC.reverse
record
{ left-inverse-of = PC.reverse-involutive
; right-inverse-of = PC.reverse-involutive
}
------------------------------------------------------------------------
-- Operations
-- Composition
_∘ₚ_ : ∀ {m n o} → Permutation m n → Permutation n o → Permutation m o
π₁ ∘ₚ π₂ = π₂ Inverse.∘ π₁
-- Flip
flip : ∀ {m n} → Permutation m n → Permutation n m
flip = Inverse.sym
-- Element removal
--
-- `remove k [0 ↦ i₀, …, k ↦ iₖ, …, n ↦ iₙ]` yields
--
-- [0 ↦ i₀, …, k-1 ↦ iₖ₋₁, k ↦ iₖ₊₁, k+1 ↦ iₖ₊₂, …, n-1 ↦ iₙ]
remove : ∀ {m n} → Fin (suc m) →
Permutation (suc m) (suc n) → Permutation m n
remove {m} {n} i π = permutation to from
record
{ left-inverse-of = left-inverse-of
; right-inverse-of = right-inverse-of
}
where
πʳ = π ⟨$⟩ʳ_
πˡ = π ⟨$⟩ˡ_
permute-≢ : ∀ {i j} → i ≢ j → πʳ i ≢ πʳ j
permute-≢ p = p ∘ (Inverse.injective π)
to-punchOut : ∀ {j : Fin m} → πʳ i ≢ πʳ (punchIn i j)
to-punchOut = permute-≢ (punchInᵢ≢i _ _ ∘ sym)
from-punchOut : ∀ {j : Fin n} → i ≢ πˡ (punchIn (πʳ i) j)
from-punchOut {j} p = punchInᵢ≢i (πʳ i) j (sym (begin
πʳ i ≡⟨ cong πʳ p ⟩
πʳ (πˡ (punchIn (πʳ i) j)) ≡⟨ inverseʳ π ⟩
punchIn (πʳ i) j ∎))
to : Fin m → Fin n
to j = punchOut (to-punchOut {j})
from : Fin n → Fin m
from j = punchOut {j = πˡ (punchIn (πʳ i) j)} from-punchOut
left-inverse-of : ∀ j → from (to j) ≡ j
left-inverse-of j = begin
from (to j) ≡⟨⟩
punchOut {i = i} {πˡ (punchIn (πʳ i) (punchOut to-punchOut))} _ ≡⟨ punchOut-cong′ i (cong πˡ (punchIn-punchOut {i = πʳ i} _)) ⟩
punchOut {i = i} {πˡ (πʳ (punchIn i j))} _ ≡⟨ punchOut-cong i (inverseˡ π) ⟩
punchOut {i = i} {punchIn i j} _ ≡⟨ punchOut-punchIn i ⟩
j ∎
right-inverse-of : ∀ j → to (from j) ≡ j
right-inverse-of j = begin
to (from j) ≡⟨⟩
punchOut {i = πʳ i} {πʳ (punchIn i (punchOut from-punchOut))} _ ≡⟨ punchOut-cong′ (πʳ i) (cong πʳ (punchIn-punchOut {i = i} _)) ⟩
punchOut {i = πʳ i} {πʳ (πˡ (punchIn (πʳ i) j))} _ ≡⟨ punchOut-cong (πʳ i) (inverseʳ π) ⟩
punchOut {i = πʳ i} {punchIn (πʳ i) j} _ ≡⟨ punchOut-punchIn (πʳ i) ⟩
j ∎
------------------------------------------------------------------------
-- Other properties
module _ {m n} (π : Permutation (suc m) (suc n)) where
private
πʳ = π ⟨$⟩ʳ_
πˡ = π ⟨$⟩ˡ_
punchIn-permute : ∀ i j → πʳ (punchIn i j) ≡ punchIn (πʳ i) (remove i π ⟨$⟩ʳ j)
punchIn-permute i j = begin
πʳ (punchIn i j) ≡⟨ sym (punchIn-punchOut {i = πʳ i} _) ⟩
punchIn (πʳ i) (punchOut {i = πʳ i} {πʳ (punchIn i j)} _) ≡⟨⟩
punchIn (πʳ i) (remove i π ⟨$⟩ʳ j) ∎
punchIn-permute′ : ∀ i j → πʳ (punchIn (πˡ i) j) ≡ punchIn i (remove (πˡ i) π ⟨$⟩ʳ j)
punchIn-permute′ i j = begin
πʳ (punchIn (πˡ i) j) ≡⟨ punchIn-permute _ _ ⟩
punchIn (πʳ (πˡ i)) (remove (πˡ i) π ⟨$⟩ʳ j) ≡⟨ cong₂ punchIn (inverseʳ π) refl ⟩
punchIn i (remove (πˡ i) π ⟨$⟩ʳ j) ∎
↔⇒≡ : ∀ {m n} → Permutation m n → m ≡ n
↔⇒≡ {zero} {zero} π = refl
↔⇒≡ {zero} {suc n} π = contradiction (π ⟨$⟩ˡ zero) ¬Fin0
↔⇒≡ {suc m} {zero} π = contradiction (π ⟨$⟩ʳ zero) ¬Fin0
↔⇒≡ {suc m} {suc n} π = cong suc (↔⇒≡ (remove zero π))
fromPermutation : ∀ {m n} → Permutation m n → Permutation′ m
fromPermutation π = P.subst (Permutation _) (sym (↔⇒≡ π)) π
refute : ∀ {m n} → m ≢ n → ¬ Permutation m n
refute m≢n π = contradiction (↔⇒≡ π) m≢n
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-- Andreas, 2016-01-27
-- After complaints of Wolfram Kahl and Aaron Stump
-- we decided it is imported to keep the rewrite behavior
-- that does not rewrite in rewrite terms.
open import Common.Equality
test : ∀{A : Set}{a : A}{f : A → A} (p : f a ≡ a) → f (f a) ≡ a
test p rewrite p = p
-- rewrite should not happen in p itself,
-- otherwise we get p : a ≡ a which is useless.
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module Morphism where
open import Data.Nat
open import Data.Unit
open import Data.Product
open import Relation.Binary.PropositionalEquality
open import Prelude
open import Structure
module _ {A B : Set} (σᴬ : Struct A) (σᴮ : Struct B) where
open Struct σᴬ renaming (ℜ to ℜᴬ ; 𝔉 to 𝔉ᴬ)
open Struct σᴮ renaming (ℜ to ℜᴮ ; 𝔉 to 𝔉ᴮ)
record Hom : Set where
field
f : A → B
ℜ-f : {n : ℕ} → (aⁿ : A ^ n) → ℜᴬ aⁿ → ℜᴮ (𝔉ⁿ f n aⁿ)
𝔉-f : {n : ℕ} → (aⁿ : A ^ n) → f (𝔉ᴬ aⁿ) ≡ 𝔉ᴮ (𝔉ⁿ f n aⁿ)
record Embed : Set where
field
h : Hom
open Hom h
field
f-inj : ∀ {x y} → f x ≡ f y → x ≡ y
ℜ-f' : {n : ℕ} → (aⁿ : A ^ n) → ℜᴮ (𝔉ⁿ f n aⁿ) → ℜᴬ aⁿ
record Iso : Set where
field
e : Embed
open Embed e
open Hom h
field
g : B → A
g-inj : ∀ {x y} → g x ≡ g y → x ≡ y
f-g : ∀ {x} → f (g x) ≡ x
g-f : ∀ {x} → g (f x) ≡ x
module _ {A : Set} (σᴬ : Struct A) where
Endo : Set
Endo = Hom σᴬ σᴬ
Auto : Set
Auto = Iso σᴬ σᴬ
module Examples {A B : Set} (σᴬ : Struct A) (σᴮ : Struct B) (h : Hom σᴬ σᴮ) where
open Struct σᴬ renaming (ℜ to ℜᴬ ; 𝔉 to 𝔉ᴬ)
open Struct σᴮ renaming (ℜ to ℜᴮ ; 𝔉 to 𝔉ᴮ)
open Hom h
hrel : A → A → Set
hrel x y = f x ≡ f y
open import Relation.Binary
hrel-equiv : IsEquivalence hrel
hrel-equiv = record { refl = refl ; sym = sym ; trans = trans }
module _ {x y : A} where
hrel-cong : hrel x y → hrel (𝔉ᴬ (x , tt)) (𝔉ᴬ (y , tt))
hrel-cong p = trans (𝔉-f (x , tt)) (sym (trans (𝔉-f (y , tt)) (sym (cong (λ b → 𝔉ᴮ (b , tt)) p))))
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{-# OPTIONS --warning=error --safe --without-K #-}
open import LogicalFormulae
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
open import Numbers.Naturals.Order.Lemmas
open import Numbers.Naturals.Order.WellFounded
open import Numbers.Primes.PrimeNumbers
open import Semirings.Definition
open import Orders.Total.Definition
open import Orders.WellFounded.Induction
open import Numbers.Naturals.EuclideanAlgorithm
module Numbers.Primes.IntegerFactorisation where
open TotalOrder ℕTotalOrder
-- Represent a factorisation into increasing factors
-- Note that 0 cannot be expressed this way.
record factorisationNonunit (minFactor : ℕ) (a : ℕ) : Set where
inductive
field
1<a : 1 <N a
firstFactor : ℕ
firstFactorNontrivial : 1 <N firstFactor
firstFactorBiggerMin : minFactor ≤N firstFactor
firstFactorDivision : divisionAlgResult firstFactor a
firstFactorDivides : divisionAlgResult.rem firstFactorDivision ≡ 0
firstFactorPrime : Prime firstFactor
otherFactorsNumber : ℕ
otherFactors : ((divisionAlgResult.quot firstFactorDivision ≡ 1) && (otherFactorsNumber ≡ 0)) || (((1 <N divisionAlgResult.quot firstFactorDivision) && (factorisationNonunit firstFactor (divisionAlgResult.quot firstFactorDivision))))
private
lemma : (p : ℕ) → p *N 1 +N 0 ≡ p
lemma p rewrite Semiring.sumZeroRight ℕSemiring (p *N 1) | Semiring.productOneRight ℕSemiring p = refl
lemma' : {a b : ℕ} → a *N zero +N 0 ≡ b → b ≡ zero
lemma' {a} {b} pr rewrite Semiring.sumZeroRight ℕSemiring (a *N zero) | Semiring.productZeroRight ℕSemiring a = equalityCommutative pr
primeFactorisation : {p : ℕ} → (pr : Prime p) → factorisationNonunit 1 p
primeFactorisation {p} record { p>1 = p>1 ; pr = pr } = record {1<a = p>1 ; firstFactor = p ; firstFactorNontrivial = p>1 ; firstFactorBiggerMin = inl p>1 ; firstFactorDivision = record { quot = 1 ; rem = 0 ; pr = lemma p ; remIsSmall = zeroIsValidRem p ; quotSmall = inl (TotalOrder.<Transitive ℕTotalOrder (le zero refl) p>1) } ; firstFactorDivides = refl ; firstFactorPrime = record { p>1 = p>1 ; pr = pr} ; otherFactors = inl record { fst = refl ; snd = refl } ; otherFactorsNumber = 0 }
where
proof : (s : ℕ) → s *N 1 +N 0 ≡ s
proof s rewrite Semiring.sumZeroRight ℕSemiring (s *N 1) | multiplicationNIsCommutative s 1 | Semiring.sumZeroRight ℕSemiring s = refl
twoAsFact : factorisationNonunit 2 2
factorisationNonunit.1<a twoAsFact = succPreservesInequality (succIsPositive 0)
factorisationNonunit.firstFactor twoAsFact = 2
factorisationNonunit.firstFactorNontrivial twoAsFact = succPreservesInequality (succIsPositive 0)
factorisationNonunit.firstFactorBiggerMin twoAsFact = inr refl
factorisationNonunit.firstFactorDivision twoAsFact = record { quot = 1 ; rem = 0 ; remIsSmall = zeroIsValidRem 2 ; pr = refl ; quotSmall = inl (le 1 refl) }
factorisationNonunit.firstFactorDivides twoAsFact = refl
factorisationNonunit.firstFactorPrime twoAsFact = twoIsPrime
factorisationNonunit.otherFactorsNumber twoAsFact = 0
factorisationNonunit.otherFactors twoAsFact = inl record { fst = refl ; snd = refl }
fourFact : factorisationNonunit 1 4
factorisationNonunit.1<a fourFact = succPreservesInequality (succIsPositive 2)
factorisationNonunit.firstFactor fourFact = 2
factorisationNonunit.firstFactorNontrivial fourFact = succPreservesInequality (succIsPositive 0)
factorisationNonunit.firstFactorBiggerMin fourFact = inl (succPreservesInequality (succIsPositive 0))
factorisationNonunit.firstFactorDivision fourFact = record { quot = 2 ; rem = 0 ; remIsSmall = zeroIsValidRem 2 ; pr = refl ; quotSmall = inl (le 1 refl) }
factorisationNonunit.firstFactorDivides fourFact = refl
factorisationNonunit.firstFactorPrime fourFact = twoIsPrime
factorisationNonunit.otherFactorsNumber fourFact = 1
factorisationNonunit.otherFactors fourFact = inr record { fst = succPreservesInequality (succIsPositive 0) ; snd = twoAsFact }
lessLemma : {y : ℕ} → (1 <N y) → (zero <N y)
lessLemma {.(succ (x +N 1))} (le x refl) = succIsPositive (x +N 1)
canReduceFactorBound : {a i j : ℕ} → factorisationNonunit i a → j <N i → factorisationNonunit j a
canReduceFactorBound {a} {i} {j} record { 1<a = 1<a ; firstFactor = firstFactor ; firstFactorNontrivial = firstFactorNontrivial ; firstFactorBiggerMin = inl ff<i ; firstFactorDivision = firstFactorDivision ; firstFactorDivides = firstFactorDivides ; firstFactorPrime = firstFactorPrime ; otherFactors = otherFactors } j<i = record { 1<a = 1<a ; firstFactor = firstFactor ; firstFactorNontrivial = firstFactorNontrivial ; firstFactorBiggerMin = inl (lessTransitive j<i ff<i) ; firstFactorDivision = firstFactorDivision ; firstFactorDivides = firstFactorDivides ; firstFactorPrime = firstFactorPrime ; otherFactors = otherFactors }
canReduceFactorBound {a} {i} {j} record { 1<a = 1<a ; firstFactor = firstFactor ; firstFactorNontrivial = firstFactorNontrivial ; firstFactorBiggerMin = inr ff=i ; firstFactorDivision = firstFactorDivision ; firstFactorDivides = firstFactorDivides ; firstFactorPrime = firstFactorPrime ; otherFactors = otherFactors } j<i = record { 1<a = 1<a ; firstFactor = firstFactor ; firstFactorNontrivial = firstFactorNontrivial ; firstFactorBiggerMin = inl (identityOfIndiscernablesRight _<N_ j<i ff=i) ; firstFactorDivision = firstFactorDivision ; firstFactorDivides = firstFactorDivides ; firstFactorPrime = firstFactorPrime ; otherFactors = otherFactors }
canReduceFactorBound' : {a i j : ℕ} → factorisationNonunit i a → j ≤N i → factorisationNonunit j a
canReduceFactorBound' {a} {i} {j} factA (inl x) = canReduceFactorBound factA x
canReduceFactorBound' {a} {i} {.i} factA (inr refl) = factA
canIncreaseFactorBound : {a i : ℕ} → (fact : factorisationNonunit 1 a) → (∀ x → 1 <N x → x <N i → x ∣ a → False) → (i ≤N factorisationNonunit.firstFactor fact) → factorisationNonunit i a
canIncreaseFactorBound {a} {i} record { 1<a = 1<a ; firstFactor = firstFactor ; firstFactorNontrivial = firstFactorNontrivial ; firstFactorBiggerMin = firstFactorBiggerMin ; firstFactorDivision = firstFactorDivision ; firstFactorDivides = firstFactorDivides ; firstFactorPrime = firstFactorPrime ; otherFactors = otherFactors } noSmaller iSmallEnough = record { 1<a = 1<a ; firstFactor = firstFactor ; firstFactorNontrivial = Prime.p>1 firstFactorPrime ; firstFactorBiggerMin = iSmallEnough ; firstFactorDivision = firstFactorDivision ; firstFactorDivides = firstFactorDivides ; firstFactorPrime = firstFactorPrime ; otherFactors = otherFactors }
-- Get the smallest prime factor of the number
everyNumberHasAPrimeFactor : {a : ℕ} → (1 <N a) → Sg ℕ (λ i → ((i ∣ a) && (1 <N i)) && ((Prime i) && (∀ x → x <N i → 1 <N x → x ∣ a → False)))
everyNumberHasAPrimeFactor {a} 1<a with compositeOrPrime a 1<a
everyNumberHasAPrimeFactor {a} 1<a | inl record { n>1 = n>1 ; divisor = divisor ; dividesN = dividesN ; divisorLessN = divisorLessN ; divisorNot1 = divisorNot1 ; divisorPrime = divisorPrime ; noSmallerDivisors = noSmallerDivisors } = ( divisor , record { fst = record { fst = dividesN ; snd = divisorNot1 } ; snd = record { fst = divisorPrime ; snd = noSmallerDivisors } } )
everyNumberHasAPrimeFactor {a} 1<a | inr x = (a , record { fst = record { fst = aDivA a ; snd = 1<a } ; snd = record { fst = x ; snd = λ y y<a 1<y y|a → irreflexive (<WellDefined (equalityCommutative (Prime.pr x y|a y<a (lessLemma 1<y))) refl 1<y) }} )
lemma2 : {a b c : ℕ} → 1 <N a → 0 <N b → a *N b +N 0 ≡ c → b <N c
lemma2 {zero} {b} {c} 1<a _ pr = exFalso (zeroNeverGreater 1<a)
lemma2 {succ zero} {b} {c} 1<a _ pr = exFalso (lessIrreflexive 1<a)
lemma2 {succ (succ a)} {zero} {zero} 1<a t pr = exFalso (lessIrreflexive t)
lemma2 {succ (succ a)} {zero} {succ c} 1<a t pr = succIsPositive c
lemma2 {succ (succ a)} {succ b} {c} 1<a t pr = le (b +N (a *N succ b)) go
where
assocLemm : (a b c : ℕ) → (a +N b) +N c ≡ (a +N c) +N b
assocLemm a b c rewrite equalityCommutative (Semiring.+Associative ℕSemiring a b c) | Semiring.commutative ℕSemiring b c | Semiring.+Associative ℕSemiring a c b = refl
go : succ ((b +N a *N succ b) +N succ b) ≡ c
go rewrite Semiring.sumZeroRight ℕSemiring (succ (b +N succ (b +N a *N succ b))) | equalityCommutative (assocLemm b (succ b) (a *N succ b)) | Semiring.+Associative ℕSemiring b (succ b) (a *N succ b) = pr
factorIntegerLemma : (x : ℕ) (indHyp : (y : ℕ) (y<x : y <N x) → ((y <N 2) || (factorisationNonunit 1 y))) → ((x <N 2) || (factorisationNonunit 1 x))
factorIntegerLemma zero indHyp = inl (succIsPositive 1)
factorIntegerLemma (succ zero) indHyp = inl (succPreservesInequality (succIsPositive 0))
factorIntegerLemma (succ (succ x)) indHyp with everyNumberHasAPrimeFactor {succ (succ x)} (succPreservesInequality (succIsPositive x))
factorIntegerLemma (succ (succ x)) indHyp | a , record { fst = record { fst = divides record {quot = zero ; rem = .0 ; pr = ssxDivA ; remIsSmall = r } refl ; snd = 1<a }; snd = record { fst = primeA ; snd = smallerFactors } } rewrite Semiring.sumZeroRight ℕSemiring (a *N zero) | multiplicationNIsCommutative a 0 = exFalso (naughtE ssxDivA)
factorIntegerLemma (succ (succ x)) indHyp | a , record { fst = record { fst = divides record {quot = succ zero ; rem = .0 ; pr = ssxDivA ; remIsSmall = r } refl ; snd = 1<a }; snd = record { fst = primeA ; snd = smallerFactors } } = inr record { 1<a = succPreservesInequality (succIsPositive x) ; firstFactor = a ; firstFactorNontrivial = Prime.p>1 primeA ; firstFactorBiggerMin = inl (Prime.p>1 primeA) ; firstFactorDivision = record { quot = 1 ; rem = 0 ; pr = ssxDivA ; remIsSmall = r ; quotSmall = inl (TotalOrder.<Transitive ℕTotalOrder (le zero refl) 1<a) } ; firstFactorDivides = refl ; firstFactorPrime = record { p>1 = Prime.p>1 primeA ; pr = Prime.pr primeA } ; otherFactors = inl record { fst = refl ; snd = refl } ; otherFactorsNumber = 0 }
factorIntegerLemma (succ (succ x)) indHyp | a , record { fst = record { fst = divides record {quot = succ (succ qu) ; rem = .0 ; pr = ssxDivA ; remIsSmall = remSmall } refl ; snd = 1<a }; snd = record { fst = primeA ; snd = smallerFactors } } = inr record { 1<a = succPreservesInequality (succIsPositive x) ; firstFactor = a ; firstFactorNontrivial = Prime.p>1 primeA ; firstFactorBiggerMin = inl (Prime.p>1 primeA) ; firstFactorDivision = record { quot = succ (succ qu) ; rem = 0 ; pr = ssxDivA ; remIsSmall = remSmall ; quotSmall = inl (TotalOrder.<Transitive ℕTotalOrder (le zero refl) 1<a) } ; firstFactorDivides = refl ; firstFactorPrime = record { p>1 = Prime.p>1 primeA ; pr = Prime.pr primeA } ; otherFactors = inr record {fst = succPreservesInequality (succIsPositive qu) ; snd = factNonunit} ; otherFactorsNumber = succ (factorisationNonunit.otherFactorsNumber indHypRes') }
where
indHypRes : ((succ (succ qu)) <N 2) || factorisationNonunit 1 (succ (succ qu))
indHypRes = indHyp (succ (succ qu)) (lemma2 {a} {succ (succ qu)} {succ (succ x)} 1<a (succIsPositive (succ qu)) ssxDivA)
indHypRes' : factorisationNonunit 1 (succ (succ qu))
indHypRes' with indHypRes
indHypRes' | inl y = exFalso (zeroNeverGreater (canRemoveSuccFrom<N (canRemoveSuccFrom<N y)))
indHypRes' | inr y = y
z|ssx : (z : ℕ) → z ∣ succ (succ qu) → z ∣ succ (succ x)
z|ssx z z|ssq = (divisibilityTransitive z|ssq (divides (record { quot = a ; rem = 0 ; pr = identityOfIndiscernablesLeft _≡_ ssxDivA (applyEquality (λ t → t +N 0) (multiplicationNIsCommutative a (succ (succ qu)))) ; remIsSmall = zeroIsValidRem (succ (succ qu)) ; quotSmall = inl (succIsPositive _) }) refl))
factNonunit : factorisationNonunit a (succ (succ qu))
factNonunit with totality a (factorisationNonunit.firstFactor indHypRes')
factNonunit | inl (inl a<ff) = canIncreaseFactorBound indHypRes' (λ z 1<z z<a z|ssq → smallerFactors z z<a 1<z (z|ssx z z|ssq)) (inl a<ff)
factNonunit | inl (inr ff<a) = exFalso (smallerFactors (factorisationNonunit.firstFactor indHypRes') ff<a (factorisationNonunit.firstFactorNontrivial indHypRes') (z|ssx (factorisationNonunit.firstFactor indHypRes') (divides (factorisationNonunit.firstFactorDivision indHypRes') (factorisationNonunit.firstFactorDivides indHypRes'))))
factNonunit | inr ff=a = canIncreaseFactorBound indHypRes' (λ z 1<z z<a z|ssq → smallerFactors z z<a 1<z (divisibilityTransitive z|ssq (divides (record { quot = a ; rem = 0 ; pr = identityOfIndiscernablesLeft _≡_ ssxDivA (applyEquality (λ t → t +N 0) (multiplicationNIsCommutative a (succ (succ qu)))) ; remIsSmall = zeroIsValidRem (succ (succ qu)) ; quotSmall = inl (succIsPositive _) }) refl))) (inr ff=a)
factorInteger : (a : ℕ) → (1 <N a) → factorisationNonunit 1 a
factorInteger a 1<a with (rec <NWellfounded (λ n → (n <N 2) || (factorisationNonunit 1 n))) factorIntegerLemma
... | bl with bl a
factorInteger a 1<a | bl | inl x = exFalso (noIntegersBetweenXAndSuccX 1 1<a x)
factorInteger a 1<a | bl | inr x = x
lessTransLemma : {p i j : ℕ} → p <N i → i ≤N j → p <N j
lessTransLemma {p} {i} {j} p<i (inl x) = <Transitive p<i x
lessTransLemma {p} {i} {j} p<i (inr x) rewrite x = p<i
lemma4' : {quot rem b : ℕ} → (quot +N quot) +N rem ≡ succ b → quot <N succ b
lemma4' {zero} {rem} {b} pr = succIsPositive b
lemma4' {succ quot} {rem} {b} pr rewrite equalityCommutative (Semiring.+Associative ℕSemiring quot (succ quot) rem) = succPreservesInequality (le (quot +N rem) (succInjective (transitivity (applyEquality succ (Semiring.commutative ℕSemiring _ quot)) pr)))
lemma4 : {quot a rem b : ℕ} → (a *N quot +N rem ≡ succ b) → (1 <N a) → (quot <N succ b)
lemma4 {quot} {zero} {rem} {b} pr 1<a = exFalso (zeroNeverGreater 1<a)
lemma4 {quot} {succ zero} {rem} {b} pr 1<a = exFalso (lessIrreflexive 1<a)
lemma4 {quot} {succ (succ zero)} {rem} {b} pr 1<a rewrite Semiring.sumZeroRight ℕSemiring quot = lemma4' pr
lemma4 {quot} {succ (succ (succ a))} {rem} {b} pr 1<a = lemma4 {quot} {succ (succ a)} {quot +N rem} {b} p' (succPreservesInequality (succIsPositive a))
where
p' : (quot +N (quot +N a *N quot)) +N (quot +N rem) ≡ succ b
p' rewrite Semiring.commutative ℕSemiring quot (quot +N (quot +N a *N quot)) | Semiring.+Associative ℕSemiring (quot +N (quot +N a *N quot)) quot rem = pr
noSmallerFactors : {a i p : ℕ} → (factorisationNonunit i a) → (Prime p) → (p <N i) → (p ∣ a) → False
noSmallerFactors {a} {i} {p} factA pPrime p<i p|a with rec <NWellfounded (λ b → (factorisationNonunit i b) → p ∣ b → False)
... | indHyp = (indHyp prf) a factA p|a
where
prf : (x : ℕ) (ind : (y : ℕ) (y<x : y <N x) (factY : factorisationNonunit i y) (p|y : p ∣ y) → False) (factX : factorisationNonunit i x) (p|x : p ∣ x) → False
prf x ind record { 1<a = 1<a ; firstFactor = firstFactor ; firstFactorNontrivial = firstFactorNontrivial ; firstFactorBiggerMin = firstFactorBiggerMin ; firstFactorDivision = firstFactorDivision ; firstFactorDivides = firstFactorDivides ; firstFactorPrime = firstFactorPrime ; otherFactors = (inl record { fst = quotFirstfact=1 ; snd = otherFactorsNumber }) } p|x = exFalso bad
where
x=firstFact : firstFactor *N 1 +N 0 ≡ x
x=firstFact rewrite equalityCommutative firstFactorDivides | equalityCommutative quotFirstfact=1 = divisionAlgResult.pr firstFactorDivision
x=firstFact' : firstFactor ≡ x
x=firstFact' = transitivity (equalityCommutative (lemma firstFactor)) x=firstFact
p|firstFact : p ∣ firstFactor
p|firstFact rewrite x=firstFact' = p|x
p=firstFact : p ≡ firstFactor
p=firstFact = primeDivPrimeImpliesEqual pPrime firstFactorPrime p|firstFact
i<=firstFactor : i ≤N p
i<=firstFactor rewrite p=firstFact = firstFactorBiggerMin
bad : False
bad with i<=firstFactor
... | inl t = TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder t p<i)
... | inr eq rewrite eq = TotalOrder.irreflexive ℕTotalOrder p<i
prf zero ind record { 1<a = 1<a ; firstFactor = firstFactor ; firstFactorNontrivial = firstFactorNontrivial ; firstFactorBiggerMin = firstFactorBiggerMin ; firstFactorDivision = firstFactorDivision ; firstFactorDivides = firstFactorDivides ; firstFactorPrime = firstFactorPrime ; otherFactors = (inr otherFact) } p|x = zeroNeverGreater 1<a
prf (succ x) ind record { 1<a = 1<a ; firstFactor = firstFactor ; firstFactorNontrivial = firstFactorNontrivial ; firstFactorBiggerMin = firstFactorBiggerMin ; firstFactorDivision = firstFactorDivision ; firstFactorDivides = firstFactorDivides ; firstFactorPrime = firstFactorPrime ; otherFactors = (inr otherFact) } p|x = ind (divisionAlgResult.quot firstFactorDivision) (lemma4 {divisionAlgResult.quot firstFactorDivision} {firstFactor} {divisionAlgResult.rem firstFactorDivision} {x} (divisionAlgResult.pr (firstFactorDivision)) (primesAreBiggerThanOne firstFactorPrime)) (canReduceFactorBound' (_&&_.snd otherFact) firstFactorBiggerMin) (p|q p|ffOrQ)
where
succXNotSmaller : succ x <N firstFactor → False
succXNotSmaller = divisorIsSmaller {firstFactor} {x} (divides firstFactorDivision firstFactorDivides)
succXNotSmaller' : firstFactor ≤N succ x
succXNotSmaller' = notSmallerMeansGE succXNotSmaller
inter : firstFactor *N (divisionAlgResult.quot firstFactorDivision) +N divisionAlgResult.rem firstFactorDivision ≡ (succ x)
inter = divisionAlgResult.pr firstFactorDivision
inter' : firstFactor *N (divisionAlgResult.quot firstFactorDivision) +N 0 ≡ (succ x)
inter' rewrite equalityCommutative firstFactorDivides = inter
inter'' : firstFactor *N (divisionAlgResult.quot firstFactorDivision) ≡ (succ x)
inter'' rewrite equalityCommutative (Semiring.sumZeroRight ℕSemiring (firstFactor *N (divisionAlgResult.quot firstFactorDivision))) = inter'
p|ff*q : p ∣ firstFactor *N (divisionAlgResult.quot firstFactorDivision)
p|ff*q rewrite inter'' = p|x
p|ffOrQ : (p ∣ firstFactor) || (p ∣ divisionAlgResult.quot firstFactorDivision)
p|ffOrQ = primesArePrime pPrime p|ff*q
p|ffIsFalse : (p ∣ firstFactor) → False
p|ffIsFalse p|ff = lessIrreflexive (lessTransLemma p<i i<=p)
where
p=ff : p ≡ firstFactor
p=ff = primeDivPrimeImpliesEqual pPrime firstFactorPrime p|ff
i<=p : i ≤N p
i<=p rewrite p=ff = firstFactorBiggerMin
p|q : ((p ∣ firstFactor) || (p ∣ divisionAlgResult.quot firstFactorDivision)) → (p ∣ divisionAlgResult.quot firstFactorDivision)
p|q (inl fls) = exFalso (p|ffIsFalse fls)
p|q (inr res) = res
lemma3 : {a : ℕ} → a ≡ 0 → 1 <N a → False
lemma3 {a} a=0 pr rewrite a=0 = zeroNeverGreater pr
firstFactorUniqueLemma : {i : ℕ} → (a : ℕ) → (1 <N a) → (f1 : factorisationNonunit i a) → (f2 : factorisationNonunit i a) → (factorisationNonunit.firstFactor f1 <N factorisationNonunit.firstFactor f2) → False
firstFactorUniqueLemma {i} a 1<a f1 f2 ff1<ff2 = go
where
p1 = factorisationNonunit.firstFactor f1
rem1 = divisionAlgResult.quot (factorisationNonunit.firstFactorDivision f1)
p2 = factorisationNonunit.firstFactor f2
rem2 = divisionAlgResult.quot (factorisationNonunit.firstFactorDivision f2)
p1<p2 : p1 <N p2
p1<p2 = ff1<ff2
a=p2rem2 : a ≡ p2 *N rem2
a=p2rem2 with divisionAlgResult.pr (factorisationNonunit.firstFactorDivision f2)
... | ff rewrite factorisationNonunit.firstFactorDivides f2 | Semiring.sumZeroRight ℕSemiring (factorisationNonunit.firstFactor f2 *N divisionAlgResult.quot (factorisationNonunit.firstFactorDivision f2)) = equalityCommutative ff
p1|second : (p1 ∣ p2) || (p1 ∣ rem2)
p1|second = primesArePrime {p1} {p2} {rem2} (factorisationNonunit.firstFactorPrime f1) lem
where
lem : p1 ∣ (p2 *N rem2)
lem = identityOfIndiscernablesRight _∣_ (divides (factorisationNonunit.firstFactorDivision f1) (factorisationNonunit.firstFactorDivides f1)) a=p2rem2
p1|second' : ((p1 ∣ p2) || (p1 ∣ rem2)) → ((p1 ≡ p2) || (p1 ∣ rem2))
p1|second' (inl x) = inl (primeDivPrimeImpliesEqual (factorisationNonunit.firstFactorPrime f1) (factorisationNonunit.firstFactorPrime f2) x)
p1|second' (inr x) = inr x
p1|second'' : (p1 ≡ p2) || (p1 ∣ rem2)
p1|second'' = p1|second' p1|second
go : False
go with p1|second''
go | inl x = irreflexive (<WellDefined x refl ff1<ff2)
go | inr x with factorisationNonunit.otherFactors f2
go | inr x | inl record { fst = rem2=1 ; snd = _ } rewrite rem2=1 = exFalso (oneIsNotPrime res)
where
1prime' : Prime p1 ≡ Prime 1
1prime' = applyEquality Prime (oneHasNoDivisors x)
res : Prime 1
res rewrite equalityCommutative 1prime' = (factorisationNonunit.firstFactorPrime f1)
go | inr x | inr p1|rem2 with factorisationNonunit.otherFactors f2
go | inr x | inr p1|rem2 | inl record { fst = rem2=1 ; snd = _ } rewrite rem2=1 = exFalso (oneIsNotPrime res)
where
1prime' : Prime p1 ≡ Prime 1
1prime' = applyEquality Prime (oneHasNoDivisors x)
res : Prime 1
res rewrite equalityCommutative 1prime' = (factorisationNonunit.firstFactorPrime f1)
go | inr x | inr p1|rem2 | inr factorRem2 = noSmallerFactors (_&&_.snd factorRem2) (factorisationNonunit.firstFactorPrime f1) p1<p2 x
firstFactorUnique : {i : ℕ} → (a : ℕ) → (1 <N a) → (f1 : factorisationNonunit i a) → (f2 : factorisationNonunit i a) → (factorisationNonunit.firstFactor f1 ≡ factorisationNonunit.firstFactor f2)
firstFactorUnique {i} a 1<a f1 f2 with totality (factorisationNonunit.firstFactor f1) (factorisationNonunit.firstFactor f2)
firstFactorUnique {i} a 1<a f1 f2 | inl (inl f1<f2) = exFalso (firstFactorUniqueLemma a 1<a f1 f2 f1<f2)
firstFactorUnique {i} a 1<a f1 f2 | inl (inr f2<f1) = exFalso (firstFactorUniqueLemma a 1<a f2 f1 f2<f1)
firstFactorUnique {i} a 1<a f1 f2 | inr x = x
factorListLemma : {i : ℕ} → (a : ℕ) → (1 <N a) → (f1 : factorisationNonunit i a) → (f2 : factorisationNonunit i a) → (divisionAlgResult.quot (factorisationNonunit.firstFactorDivision f2)) ≡ (divisionAlgResult.quot (factorisationNonunit.firstFactorDivision f1))
factorListLemma {i} a 1<a f1 f2 with firstFactorUnique {i} a 1<a f1 f2
... | firstFactEqual = res
where
div1 : divisionAlgResult (factorisationNonunit.firstFactor f1) a
div1 = factorisationNonunit.firstFactorDivision f1
rem1=0 : divisionAlgResult.rem div1 ≡ 0
rem1=0 = factorisationNonunit.firstFactorDivides f1
pr1 : (factorisationNonunit.firstFactor f1) *N (divisionAlgResult.quot div1) +N 0 ≡ a
pr1 rewrite equalityCommutative rem1=0 = divisionAlgResult.pr div1
pr1' : (factorisationNonunit.firstFactor f1) *N (divisionAlgResult.quot div1) ≡ a
pr1' rewrite equalityCommutative (Semiring.sumZeroRight ℕSemiring ((factorisationNonunit.firstFactor f1) *N (divisionAlgResult.quot div1))) = pr1
div2 : divisionAlgResult (factorisationNonunit.firstFactor f2) a
div2 = factorisationNonunit.firstFactorDivision f2
rem2=0 : divisionAlgResult.rem div2 ≡ 0
rem2=0 = factorisationNonunit.firstFactorDivides f2
pr2 : (factorisationNonunit.firstFactor f2) *N (divisionAlgResult.quot div2) +N 0 ≡ a
pr2 rewrite equalityCommutative rem2=0 = divisionAlgResult.pr div2
pr2' : (factorisationNonunit.firstFactor f2) *N (divisionAlgResult.quot div2) ≡ a
pr2' rewrite equalityCommutative (Semiring.sumZeroRight ℕSemiring ((factorisationNonunit.firstFactor f2) *N (divisionAlgResult.quot div2))) = pr2
pr : (factorisationNonunit.firstFactor f2) *N (divisionAlgResult.quot div2) ≡ (factorisationNonunit.firstFactor f1) *N (divisionAlgResult.quot div1)
pr = transitivity pr2' (equalityCommutative pr1')
pr' : (factorisationNonunit.firstFactor f1) *N (divisionAlgResult.quot div2) ≡ (factorisationNonunit.firstFactor f1) *N (divisionAlgResult.quot div1)
pr' = identityOfIndiscernablesLeft _≡_ pr (applyEquality (λ t → t *N divisionAlgResult.quot div2) (equalityCommutative firstFactEqual))
positive : zero <N factorisationNonunit.firstFactor f1
positive = lessTransitive (succIsPositive 0) (factorisationNonunit.firstFactorNontrivial f1)
res : divisionAlgResult.quot div2 ≡ divisionAlgResult.quot div1
res = productCancelsLeft (factorisationNonunit.firstFactor f1) (divisionAlgResult.quot div2) (divisionAlgResult.quot div1) positive pr'
factorListSameLength : {i : ℕ} → (a : ℕ) → (1 <N a) → (f1 : factorisationNonunit i a) → (f2 : factorisationNonunit i a) → (divisionAlgResult.quot (factorisationNonunit.firstFactorDivision f1) ≡ 1) → divisionAlgResult.quot (factorisationNonunit.firstFactorDivision f2) ≡ 1
factorListSameLength {i} a 1<a f1 f2 quot=1 with firstFactorUnique {i} a 1<a f1 f2
... | firstFactEqual with factorListLemma {i} a 1<a f1 f2
... | eq = transitivity eq quot=1
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Streams
------------------------------------------------------------------------
module Data.Stream where
open import Coinduction
open import Data.Colist using (Colist; []; _∷_)
open import Data.Vec using (Vec; []; _∷_)
open import Data.Nat using (ℕ; zero; suc)
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as P using (_≡_)
------------------------------------------------------------------------
-- The type
infixr 5 _∷_
data Stream (A : Set) : Set where
_∷_ : (x : A) (xs : ∞ (Stream A)) → Stream A
{-# IMPORT Data.FFI #-}
{-# COMPILED_DATA Stream Data.FFI.AgdaStream Data.FFI.Cons #-}
------------------------------------------------------------------------
-- Some operations
head : ∀ {A} → Stream A → A
head (x ∷ xs) = x
tail : ∀ {A} → Stream A → Stream A
tail (x ∷ xs) = ♭ xs
map : ∀ {A B} → (A → B) → Stream A → Stream B
map f (x ∷ xs) = f x ∷ ♯ map f (♭ xs)
zipWith : ∀ {A B C} →
(A → B → C) → Stream A → Stream B → Stream C
zipWith _∙_ (x ∷ xs) (y ∷ ys) = (x ∙ y) ∷ ♯ zipWith _∙_ (♭ xs) (♭ ys)
take : ∀ {A} n → Stream A → Vec A n
take zero xs = []
take (suc n) (x ∷ xs) = x ∷ take n (♭ xs)
drop : ∀ {A} → ℕ → Stream A → Stream A
drop zero xs = xs
drop (suc n) (x ∷ xs) = drop n (♭ xs)
repeat : ∀ {A} → A → Stream A
repeat x = x ∷ ♯ repeat x
iterate : ∀ {A} → (A → A) → A → Stream A
iterate f x = x ∷ ♯ iterate f (f x)
-- Interleaves the two streams.
infixr 5 _⋎_
_⋎_ : ∀ {A} → Stream A → Stream A → Stream A
(x ∷ xs) ⋎ ys = x ∷ ♯ (ys ⋎ ♭ xs)
mutual
-- Takes every other element from the stream, starting with the
-- first one.
evens : ∀ {A} → Stream A → Stream A
evens (x ∷ xs) = x ∷ ♯ odds (♭ xs)
-- Takes every other element from the stream, starting with the
-- second one.
odds : ∀ {A} → Stream A → Stream A
odds (x ∷ xs) = evens (♭ xs)
toColist : ∀ {A} → Stream A → Colist A
toColist (x ∷ xs) = x ∷ ♯ toColist (♭ xs)
lookup : ∀ {A} → ℕ → Stream A → A
lookup zero (x ∷ xs) = x
lookup (suc n) (x ∷ xs) = lookup n (♭ xs)
infixr 5 _++_
_++_ : ∀ {A} → Colist A → Stream A → Stream A
[] ++ ys = ys
(x ∷ xs) ++ ys = x ∷ ♯ (♭ xs ++ ys)
------------------------------------------------------------------------
-- Equality and other relations
-- xs ≈ ys means that xs and ys are equal.
infix 4 _≈_
data _≈_ {A} : Stream A → Stream A → Set where
_∷_ : ∀ {x y xs ys}
(x≡ : x ≡ y) (xs≈ : ∞ (♭ xs ≈ ♭ ys)) → x ∷ xs ≈ y ∷ ys
-- x ∈ xs means that x is a member of xs.
infix 4 _∈_
data _∈_ {A} : A → Stream A → Set where
here : ∀ {x xs} → x ∈ x ∷ xs
there : ∀ {x y xs} (x∈xs : x ∈ ♭ xs) → x ∈ y ∷ xs
-- xs ⊑ ys means that xs is a prefix of ys.
infix 4 _⊑_
data _⊑_ {A} : Colist A → Stream A → Set where
[] : ∀ {ys} → [] ⊑ ys
_∷_ : ∀ x {xs ys} (p : ∞ (♭ xs ⊑ ♭ ys)) → x ∷ xs ⊑ x ∷ ys
------------------------------------------------------------------------
-- Some proofs
setoid : Set → Setoid _ _
setoid A = record
{ Carrier = Stream A
; _≈_ = _≈_ {A}
; isEquivalence = record
{ refl = refl
; sym = sym
; trans = trans
}
}
where
refl : Reflexive _≈_
refl {_ ∷ _} = P.refl ∷ ♯ refl
sym : Symmetric _≈_
sym (x≡ ∷ xs≈) = P.sym x≡ ∷ ♯ sym (♭ xs≈)
trans : Transitive _≈_
trans (x≡ ∷ xs≈) (y≡ ∷ ys≈) = P.trans x≡ y≡ ∷ ♯ trans (♭ xs≈) (♭ ys≈)
head-cong : ∀ {A} {xs ys : Stream A} → xs ≈ ys → head xs ≡ head ys
head-cong (x≡ ∷ _) = x≡
tail-cong : ∀ {A} {xs ys : Stream A} → xs ≈ ys → tail xs ≈ tail ys
tail-cong (_ ∷ xs≈) = ♭ xs≈
map-cong : ∀ {A B} (f : A → B) {xs ys} →
xs ≈ ys → map f xs ≈ map f ys
map-cong f (x≡ ∷ xs≈) = P.cong f x≡ ∷ ♯ map-cong f (♭ xs≈)
zipWith-cong : ∀ {A B C} (_∙_ : A → B → C) {xs xs′ ys ys′} →
xs ≈ xs′ → ys ≈ ys′ →
zipWith _∙_ xs ys ≈ zipWith _∙_ xs′ ys′
zipWith-cong _∙_ (x≡ ∷ xs≈) (y≡ ∷ ys≈) =
P.cong₂ _∙_ x≡ y≡ ∷ ♯ zipWith-cong _∙_ (♭ xs≈) (♭ ys≈)
infixr 5 _⋎-cong_
_⋎-cong_ : ∀ {A} {xs xs′ ys ys′ : Stream A} →
xs ≈ xs′ → ys ≈ ys′ → xs ⋎ ys ≈ xs′ ⋎ ys′
(x ∷ xs≈) ⋎-cong ys≈ = x ∷ ♯ (ys≈ ⋎-cong ♭ xs≈)
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{-# OPTIONS --cubical --safe #-}
module Cubical.Data.BinNat where
open import Cubical.Data.BinNat.BinNat public
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{-# OPTIONS --without-K --safe #-}
open import Categories.Category
open import Categories.Functor hiding (id)
-- Limit of a Cone over a Functor F : J → C
module Categories.Diagram.Limit
{o ℓ e} {o′ ℓ′ e′} {C : Category o ℓ e} {J : Category o′ ℓ′ e′} (F : Functor J C) where
private
module C = Category C
module J = Category J
open C
open HomReasoning
open Functor F
open import Level
open import Data.Product using (proj₂)
open import Categories.Category.Construction.Cones F renaming (Cone⇒ to _⇨_)
open import Categories.Object.Terminal as T hiding (up-to-iso; transport-by-iso)
open import Categories.Morphism.Reasoning C
open import Categories.Morphism C
open import Categories.Morphism Cones as MC using () renaming (_≅_ to _⇔_)
private
variable
K K′ : Cone
A B : J.Obj
X Y Z : Obj
q : K ⇨ K′
-- A Limit is a Terminal object in the category of Cones
-- (This could be unpacked...)
record Limit : Set (o ⊔ ℓ ⊔ e ⊔ o′ ⊔ ℓ′ ⊔ e′) where
field
terminal : Terminal Cones
module terminal = Terminal terminal
open terminal using () renaming (⊤ to limit) public
open Cone limit hiding (apex) renaming (N to apex; ψ to proj; commute to limit-commute) public
rep-cone : ∀ K → K ⇨ limit
rep-cone K = terminal.! {K}
rep : ∀ K → Cone.N K ⇒ apex
rep K = arr
where open _⇨_ (rep-cone K)
unrep : X ⇒ apex → Cone
unrep f = record {
apex = record
{ ψ = λ A → proj A ∘ f
; commute = λ g → pullˡ (limit-commute g)
}
}
conify : (f : X ⇒ apex) → unrep f ⇨ limit
conify f = record
{ arr = f
; commute = refl
}
commute : proj A ∘ rep K ≈ Cone.ψ K A
commute {K = K} = _⇨_.commute (rep-cone K)
unrep-cone : (K ⇨ limit) → Cone
unrep-cone f = unrep (_⇨_.arr f)
g-η : ∀ {f : X ⇒ apex} → rep (unrep f) ≈ f
g-η {f = f} = terminal.!-unique (conify f)
η-cone : Cones [ rep-cone limit ≈ Category.id Cones ]
η-cone = terminal.⊤-id (rep-cone limit)
η : rep limit ≈ id
η = η-cone
rep-cone∘ : Cones [ Cones [ rep-cone K ∘ q ] ≈ rep-cone K′ ]
rep-cone∘ {K = K} {q = q} = Equiv.sym (terminal.!-unique (Cones [ rep-cone K ∘ q ]))
rep∘ : ∀ {q : K′ ⇨ K} → rep K ∘ _⇨_.arr q ≈ rep K′
rep∘ {q = q} = rep-cone∘ {q = q}
rep-cone-self-id : Cones [ rep-cone limit ≈ Cones.id ]
rep-cone-self-id = terminal.!-unique Cones.id
rep-self-id : rep limit ≈ id
rep-self-id = rep-cone-self-id
open Limit
up-to-iso-cone : (L₁ L₂ : Limit) → limit L₁ ⇔ limit L₂
up-to-iso-cone L₁ L₂ = T.up-to-iso Cones (terminal L₁) (terminal L₂)
up-to-iso : (L₁ L₂ : Limit) → apex L₁ ≅ apex L₂
up-to-iso L₁ L₂ = iso-cone⇒iso-apex (up-to-iso-cone L₁ L₂)
transport-by-iso-cone : (L : Limit) → limit L ⇔ K → Limit
transport-by-iso-cone L L⇿K = record
{ terminal = T.transport-by-iso Cones (terminal L) L⇿K
}
transport-by-iso : (L : Limit) → apex L ≅ X → Limit
transport-by-iso L L≅X = transport-by-iso-cone L (proj₂ p)
where p = cone-resp-iso (limit L) L≅X
module _ (X : Obj) (apex₁ : Apex X) (apex₂ : Apex X) (L : Limit) where
private
module apex₁ = Apex apex₁
module apex₂ = Apex apex₂
module L = Limit L
K₁ : Cone
K₁ = record { apex = apex₁ }
module K₁ = Cone K₁
K₂ : Cone
K₂ = record { apex = apex₂ }
module K₂ = Cone K₂
ψ-≈⇒rep-≈ : (∀ A → apex₁.ψ A ≈ apex₂.ψ A) → L.rep K₁ ≈ L.rep K₂
ψ-≈⇒rep-≈ ∀eq = trans (L.terminal.!-unique K₁⇒limit) identityʳ
where K₁⇒K₂ : K₁ ⇨ K₂
K₁⇒K₂ = record
{ arr = id
; commute = λ {X} → trans identityʳ (sym (∀eq X))
}
K₁⇒limit : K₁ ⇨ L.limit
K₁⇒limit = Cones [ L.terminal.! ∘ K₁⇒K₂ ]
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open import Data.Product using ( _,_ )
open import Data.Empty using ( ⊥ )
open import FRP.LTL.ISet.Core using ( ISet ; [_] ; _,_ )
module FRP.LTL.ISet.Empty where
F : ISet
F = [ (λ i → ⊥) , (λ i j i~j → λ ()) , (λ i j i⊑j → λ ()) ]
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module ReducingConstructorsInWith where
data ⊤ : Set where
tt : ⊤
module RegExps where
data RegExp : Set where
_│_ : RegExp -> RegExp -> RegExp
open module R = RegExps
bypassable : (re : RegExp) -> ⊤
bypassable (re₁ │ re₂) with bypassable re₁
bypassable (re₁ │ re₂) | m = m
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module MonoidalCat where
open import Library hiding (_×_)
open import Categories
open import Categories.Products
open import Functors
open import Naturals
record Monoidal {l}{m} : Set (lsuc (l ⊔ m)) where
field C : Cat {l}{m}
open Cat C
open Fun
open NatI
field ⊗ : Fun (C × C) C
I : Obj
I⊗- : Fun C C
I⊗- = functor
(\ A -> OMap ⊗ (I , A))
(\ f -> HMap ⊗ (iden , f))
(fid ⊗)
(trans (cong (\f -> HMap ⊗ (f , _)) (sym idl)) (fcomp ⊗))
field λ' : NatI I⊗- (IdF C)
-⊗I : Fun C C
-⊗I = functor
(\ A -> OMap ⊗ (A , I))
(\ f -> HMap ⊗ (f , iden))
(fid ⊗)
(trans (cong (\f -> HMap ⊗ (_ , f)) (sym idl)) (fcomp ⊗))
field ρ : NatI -⊗I (IdF C)
[-⊗-]⊗- : Fun (C × C × C) C
[-⊗-]⊗- = functor
(λ {(X , Y , Z) → OMap ⊗ (OMap ⊗ (X , Y) , Z)})
(λ {(f , g , h) → HMap ⊗ (HMap ⊗ (f , g) , h)})
(trans (cong (\f -> HMap ⊗ (f , _)) (fid ⊗) ) (fid ⊗))
(trans (cong (\f -> HMap ⊗ (f , _)) (fcomp ⊗)) (fcomp ⊗))
-⊗[-⊗-] : Fun (C × C × C) C
-⊗[-⊗-] = functor
(λ {(X , Y , Z) → OMap ⊗ (X , OMap ⊗ (Y , Z))})
(λ {(f , g , h) → HMap ⊗ (f , HMap ⊗ (g , h))})
(trans (cong (\f -> HMap ⊗ (_ , f)) (fid ⊗) ) (fid ⊗))
(trans (cong (\f -> HMap ⊗ (_ , f)) (fcomp ⊗)) (fcomp ⊗))
field α : NatI [-⊗-]⊗- -⊗[-⊗-]
triangle : ∀{A B} ->
comp (HMap ⊗ (iden {A} , (cmp λ' {B}))) (cmp α {A , I , B})
≅ HMap ⊗ (cmp ρ {A} , iden {B})
square : ∀{A B C D} ->
comp (HMap ⊗ (iden {A} , cmp α {B , C , D}))
(comp (cmp α {A , OMap ⊗ (B , C) , D})
(HMap ⊗ (cmp α {A , B , C} , iden {D})))
≅
comp (cmp α {A , B , OMap ⊗ (C , D)})
(cmp α {OMap ⊗ (A , B) , C , D})
record MonoidalFun {a b c d}(M : Monoidal {a}{b})(M' : Monoidal {c}{d})
: Set (a ⊔ b ⊔ c ⊔ d) where
open Monoidal
open Cat
field F : Fun (C M) (C M')
open Fun
field e : Hom (C M') (I M') (OMap F (I M))
F-⊗'F- : Fun (C M × C M) (C M')
F-⊗'F- = functor
(\ {(A , B) -> OMap (⊗ M') (OMap F A , OMap F B)})
(\ {(f , g) -> HMap (⊗ M') (HMap F f , HMap F g)})
(trans (cong₂ (\f g -> HMap (⊗ M') (f , g)) (fid F) (fid F)) (fid (⊗ M')))
(trans (cong₂ (\f g -> HMap (⊗ M') (f , g))
(fcomp F) (fcomp F)) (fcomp (⊗ M')))
F[-⊗-] : Fun (C M × C M) (C M')
F[-⊗-] = functor
(λ { (A , B) → OMap F (OMap (⊗ M) (A , B)) })
((λ { (f , g) → HMap F (HMap (⊗ M) (f , g)) }))
(trans (cong (HMap F) (fid (⊗ M))) (fid F))
(trans (cong (HMap F) (fcomp (⊗ M))) (fcomp F))
field m : NatT F-⊗'F- F[-⊗-]
field square1 : ∀{A} ->
NatI.cmp (ρ M') {OMap F A}
≅
comp (C M') (HMap F (NatI.cmp (ρ M) {A}))
(comp (C M') (NatT.cmp m {A , I M})
(HMap (⊗ M') (iden (C M') {OMap F A} , e)))
square2 : ∀{B} ->
NatI.cmp (λ' M') {OMap F B}
≅
comp (C M') (HMap F (NatI.cmp (λ' M) {B}))
(comp (C M') (NatT.cmp m {I M , B})
(HMap (⊗ M') (e , iden (C M') {OMap F B})))
hexagon : ∀{A B B'} ->
comp (C M') (HMap F (NatI.cmp (α M) {A , B , B'}))
(comp (C M') (NatT.cmp m {OMap (⊗ M) (A , B) , B'})
(HMap (⊗ M') (NatT.cmp m {A , B} ,
iden (C M') {OMap F B'})))
≅
comp (C M') (NatT.cmp m {A , OMap (⊗ M) (B , B')})
(comp (C M') (HMap (⊗ M') (iden (C M') {OMap F A} ,
NatT.cmp m {B , B'}))
(NatI.cmp (α M') {OMap F A ,
OMap F B ,
OMap F B'}))
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module stateDependentObjects where
open import Size renaming (Size to AgdaSize)
open import Agda.Builtin.Equality
open import Data.Nat.Base as N hiding (_⊔_)
open import Data.Product
open import Data.Vec as Vec using (Vec; []; _∷_; head; tail)
open import Function
open import NativeIO
open import Relation.Binary.PropositionalEquality
open import interactiveProgramsAgda
-- open import SizedIO.Base
{- State dependent interfaces, objects -}
record Interfaceˢ : Set₁ where
field Stateˢ : Set
Methodˢ : (s : Stateˢ) → Set
Resultˢ : (s : Stateˢ) → (m : Methodˢ s) → Set
nextˢ : (s : Stateˢ) → (m : Methodˢ s) → (r : Resultˢ s m)
→ Stateˢ
open Interfaceˢ public
{- State-Dependent Objects -}
record Objectˢ (I : Interfaceˢ) (s : Stateˢ I) : Set where
coinductive
field objectMethod : (m : Methodˢ I s) →
Σ[ r ∈ Resultˢ I s m ] Objectˢ I (nextˢ I s m r)
open Objectˢ public
record IOObjectˢ (Iᵢₒ : IOInterface) (I : Interfaceˢ) (s : Stateˢ I) : Set where
coinductive
field method : (m : Methodˢ I s) →
IO Iᵢₒ ∞ (Σ[ r ∈ Resultˢ I s m ] IOObjectˢ Iᵢₒ I (nextˢ I s m r))
{- Example of a Stack -}
StackStateˢ : Set
StackStateˢ = ℕ
data StackMethodˢ (A : Set) : (n : StackStateˢ) → Set where
push : ∀ {n} → A → StackMethodˢ A n
pop : ∀ {n} → StackMethodˢ A (suc n)
StackResultˢ : (A : Set) → (s : StackStateˢ) → StackMethodˢ A s → Set
StackResultˢ A _ (push _) = Unit
StackResultˢ A _ pop = A
stackNextˢ : ∀ A n (m : StackMethodˢ A n) (r : StackResultˢ A n m) → StackStateˢ
stackNextˢ A n (push x) r = suc n
stackNextˢ A (suc n) pop r = n
StackInterfaceˢ : (A : Set) → Interfaceˢ
Stateˢ (StackInterfaceˢ A) = StackStateˢ
Methodˢ (StackInterfaceˢ A) = StackMethodˢ A
Resultˢ (StackInterfaceˢ A) = StackResultˢ A
nextˢ (StackInterfaceˢ A) = stackNextˢ A
stack : ∀{A}{n : ℕ} (as : Vec A n) → Objectˢ (StackInterfaceˢ A) n
objectMethod (stack as) (push a) = _ , stack (a ∷ as)
objectMethod (stack (a ∷ as)) pop = a , stack as
{- Reasoning about Stateful Objects -}
{- Bisimiliarity -}
module Bisim (I : Interfaceˢ)
(let S = Stateˢ I) (let M = Methodˢ I) (let R = Resultˢ I)
(let next = nextˢ I) (let O = Objectˢ I)
where
data ΣR {A : Set} {B : A → Set} (R : ∀{a} (b b' : B a) → Set)
: (p p' : Σ A B) → Set
where
eqΣ : ∀{a}{b b' : B a} → R b b' → ΣR R (a , b) (a , b')
record _≅_ {s : S} (o o' : O s) : Set where
coinductive
field bisimMethod : (m : M s) →
ΣR (_≅_) (objectMethod o m) (objectMethod o' m)
open _≅_ public
refl≅ : ∀{s} (o : O s) → o ≅ o
bisimMethod (refl≅ o) m = let (r , o') = objectMethod o m
in eqΣ (refl≅ o')
module _ {E : Set} where
private
I = StackInterfaceˢ E
S = Stateˢ I
O = Objectˢ I
open Bisim I
{- Verifying Stack laws -}
pop-after-push : ∀{n} {v : Vec E n} {e : E} →
let st = stack v
(_ , st₁) = objectMethod st (push e)
(e₂ , st₂) = objectMethod st₁ pop
in (e ≡ e₂) × (st ≅ st₂)
pop-after-push = refl , refl≅ _
push-after-pop : ∀{n} {v : Vec E n} {e : E} →
let st = stack (e ∷ v)
(e₁ , st₁) = objectMethod st pop
(_ , st₂) = objectMethod st₁ (push e₁)
in st ≅ st₂
push-after-pop = refl≅ _
{- Bisimilarity of different stack implementations -}
stackFState = ℕ → E
pushStackF : E → stackFState → stackFState
pushStackF e f = λ { 0 → e ;
(suc m) → f m}
popStackFe : stackFState → E
popStackFe f = f 0
popStackFf : stackFState → stackFState
popStackFf f = f ∘ suc
tabulate : ∀ (n : ℕ) → stackFState → Vec E n
tabulate 0 f = []
tabulate (suc n) f = popStackFe f ∷ tabulate n (popStackFf f)
stackF : ∀ (n : ℕ) (f : ℕ → E) → Objectˢ (StackInterfaceˢ E) n
objectMethod (stackF n f) (push e) = _ ,
stackF (suc n) (pushStackF e f)
objectMethod (stackF (suc n) f) pop = popStackFe f ,
stackF n (popStackFf f)
impl-bisim : ∀{n f} v (p : tabulate n f ≡ v) → stackF n f ≅ stack v
bisimMethod (impl-bisim v p) (push e) =
eqΣ (impl-bisim (e ∷ v) (cong (_∷_ e) p))
bisimMethod (impl-bisim (e ∷ v) p) pop rewrite cong head p =
eqΣ (impl-bisim v (cong tail p))
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{-# OPTIONS --without-K #-}
open import Data.Nat.Base using (ℕ)
open import Data.Product
open import HoTT
open import Function.Extensionality
open import Algebra.Group
open import Algebra.Group.Homomorphism
open import Algebra.Group.Isomorphism
open import Function using (id; _∘_ ; flip)
open import Relation.Binary.PropositionalEquality.NP
open import Explore.Core
open import Explore.Properties
import Explore.Explorable as E
open Equivalences
module Explore.Group where
module FromGroupHomomorphism
{ℓ}{A B : Set ℓ}(𝔾+ : Group A)(𝔾* : Group B)
(φ : A → B)(φ-hom : GroupHomomorphism 𝔾+ 𝔾* φ)
where
open Additive-Group 𝔾+
open Multiplicative-Group 𝔾*
module LiftGroupHomomorphism
{i}{I : Set i}
{explore : ∀ {ℓ} → Explore ℓ I}
(explore-ind : ∀ {p ℓ} → ExploreInd {ℓ} p explore)
(g : I → A)
where
open GroupHomomorphism φ-hom
Σᴵ = explore 0# _+_
Πᴵ = explore 1# _*_
lift-hom : φ (Σᴵ g) ≡ Πᴵ (φ ∘ g)
lift-hom = E.FromExploreInd.LiftHom.lift-hom explore-ind _≡_ refl trans 0# _+_ 1# _*_ *= φ g 0-hom-1 hom
module FromExplore
{ℓx} {X : Set ℓx}(z : X)(_⊕_ : X → X → X)
(exploreᴬ : Explore ℓx A)
(exploreᴬ= : ExploreExt exploreᴬ)
(let [⊕] = exploreᴬ z _⊕_)
(O : B → X)
(sui : ∀ {k} → [⊕] (O ∘ φ) ≡ [⊕] (O ∘ _*_ k ∘ φ))
where
+-stable : ∀ {k} → [⊕] (O ∘ φ) ≡ [⊕] (O ∘ φ ∘ _+_ k)
+-stable = Algebra.Group.Homomorphism.Stability.+-stable 𝔾+ 𝔾* φ φ-hom
([⊕] ∘ _∘_ O) (exploreᴬ= _ _ ∘ _∘_ (ap O)) sui
module FromGroupIsomorphism
{ℓa}{ℓb}{A : Set ℓa}{B : Set ℓb}(𝔾+ : Group A)(𝔾* : Group B)
(φ : A → B)(φ-iso : GroupIsomorphism 𝔾+ 𝔾* φ) where
open GroupIsomorphism φ-iso
open Additive-Group 𝔾+
open Multiplicative-Group 𝔾*
module FromBigOp
{ℓx}{X : Set ℓx}(F : BigOp X A)
(F= : ∀ {g₀ g₁ : A → X} → g₀ ≗ g₁ → F g₀ ≡ F g₁)
(O : B → X)
(sui : ∀ {k} → F (O ∘ φ) ≡ F (O ∘ φ ∘ _+_ k))
where
_≈_ : (g₀ g₁ : B → B) → Set ℓx
g₀ ≈ g₁ = F (O ∘ g₀ ∘ φ) ≡ F (O ∘ g₁ ∘ φ)
-- The core of the proof is there:
open Algebra.Group.Isomorphism.Stability 𝔾+ 𝔾* φ φ-iso
id≈k* : ∀ {k} → id ≈ _*_ k
id≈k* = *-stable (F ∘ _∘_ O)
(F= ∘ _∘_ (ap O))
sui
k₀*≈k₁* : ∀ {k₀ k₁} → _*_ k₀ ≈ _*_ k₁
k₀*≈k₁* = ! id≈k* ∙ id≈k*
module FromExplore
{ℓx}{X : Set ℓx}(z : X)(_⊕_ : X → X → X)
(exploreᴬ : Explore ℓx A)
(exploreᴬ= : ExploreExt exploreᴬ)
= FromBigOp (exploreᴬ z _⊕_) (exploreᴬ= z _⊕_)
module FromAdequate-sum
{ℓb}{A : Set}{B : Set ℓb}(𝔾+ : Group A)(𝔾* : Group B)
(φ : A → B)(φ-iso : GroupIsomorphism 𝔾+ 𝔾* φ)
{sum : Sum A}
(open Adequacy _≡_)
(sum-adq : Adequate-sum sum)
{{_ : UA}}{{_ : FunExt}}
(open E.FromAdequate-sum sum-adq)
(O : B → ℕ)
(open Additive-Group 𝔾+)
(open FromGroupIsomorphism 𝔾+ 𝔾* φ φ-iso)
= FromBigOp sum sum-ext O (! (sumStableUnder (_ , +-is-equiv) (O ∘ φ)))
-- -}
-- -}
-- -}
-- -}
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------------------------------------------------------------------------
-- The Agda standard library
--
-- The universe polymorphic unit type and the total relation on unit
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Data.Unit.Polymorphic where
------------------------------------------------------------------------
-- Re-export contents of Base module
open import Data.Unit.Polymorphic.Base public
------------------------------------------------------------------------
-- Re-export query operations
open import Data.Unit.Polymorphic.Properties public using (_≟_)
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open import Relation.Binary using (Decidable; DecSetoid)
open import Level
module CP.Session2 {a} (ChanSetoid : DecSetoid zero a) (Type : Set) where
Chan : Set
Chan = DecSetoid.Carrier ChanSetoid
_≟Chan_ = DecSetoid._≟_ ChanSetoid
_≈Chan_ = DecSetoid._≈_ ChanSetoid
_≉Chan_ = DecSetoid._≉_ ChanSetoid
infixl 5 _,_∶_
data Session : Set a where
_,_∶_ : Session → Chan → Type → Session
_++_ : Session → Session → Session
∅ : Session
-- open import Data.Bool hiding (_≟_)
-- open import Data.Empty
-- open import Data.Unit
open import Relation.Nullary
open import Relation.Nullary.Decidable
-- open import Relation.Binary.PropositionalEquality
infix 4 _∋_
data _∋_ : (Γ : Session) → (x : Chan) → Set a where
here : ∀ {Γ x y A} → x ≈Chan y → Γ , y ∶ A ∋ x
there : ∀ {Γ x y A} → Γ ∋ x → Γ , y ∶ A ∋ x
left : ∀ {Γ Δ x} → Γ ∋ x → Γ ++ Δ ∋ x
right : ∀ {Γ Δ x} → Δ ∋ x → Γ ++ Δ ∋ x
infix 4 _∋?_
_∋?_ : (Γ : Session) → (x : Chan) → Dec (Γ ∋ x)
Γ , y ∶ A ∋? x with x ≟Chan y
... | yes x≈y = yes (here x≈y)
... | no ¬x≈y with Γ ∋? x
... | yes Γ∋x = yes (there Γ∋x)
... | no ¬Γ∋x = no (λ where (here x≈y) → ¬x≈y x≈y
(there Γ∋x) → ¬Γ∋x Γ∋x)
(Γ ++ Δ) ∋? x with Γ ∋? x
... | yes Γ∋x = yes (left Γ∋x)
... | no ¬Γ∋x with Δ ∋? x
... | yes Δ∋x = yes (right Δ∋x)
... | no ¬Δ∋x = no (λ where (left Γ∋x) → ¬Γ∋x Γ∋x
(right Δ∋x) → ¬Δ∋x Δ∋x)
∅ ∋? x = no (λ ())
open import Data.Product
open import Data.Empty using (⊥-elim)
infix 4 _≈_
_≈_ : Session → Session → Set a
Γ ≈ Δ = (∀ x → Γ ∋ x → Δ ∋ x) × (∀ x → Δ ∋ x → Γ ∋ x)
∅∌x : ∀ {x} → ¬ ∅ ∋ x
∅∌x ()
open DecSetoid ChanSetoid hiding (_≈_)
∅-empty : ∀ {Δ x A} → ¬ ∅ ≈ Δ , x ∶ A
∅-empty {Δ} {x} (P , Q) = ∅∌x (Q x (here refl))
<<<<<<< HEAD
swap : ∀ {Γ x y A B} → Γ , x ∶ A , y ∶ B ≈ Γ , y ∶ B , x ∶ A
swap {Γ} {x} {y} {A} {B} = to , from
where
to : ∀ v → Γ , x ∶ A , y ∶ B ∋ v → Γ , y ∶ B , x ∶ A ∋ v
to v (here P) = there (here P)
to v (there (here P)) = here P
to v (there (there P)) = there (there P)
from : ∀ v → Γ , y ∶ B , x ∶ A ∋ v → Γ , x ∶ A , y ∶ B ∋ v
from v (here P) = there (here P)
from v (there (here P)) = here P
from v (there (there P)) = there (there P)
-- contract : ∀ {Γ x y A B} → → Γ , x ∶ A ≈ Γ
-- contract
-- strengthen : ∀ {Γ x A v} → Γ , x ∶ A ∋ v → x ≉ v → Γ ∋ v
-- strengthen (here x≈v) x≉v = ⊥-elim (x≉v (sym x≈v))
-- strengthen (there P) x≉v = P
=======
>>>>>>> 40f82fd098a0c30c87ea0056baa48152aefae1f6
lookup : (Γ : Session) (x : Chan) → Dec (∃[ Δ ] ∃[ A ] (Γ ≈ (Δ , x ∶ A)))
lookup (Γ , y ∶ A) x with x ≟Chan y
... | yes x≈y = yes (Γ , A , (λ where v (here v≈y) → here (trans v≈y (sym x≈y))
v (there Γ∷y∋v) → there Γ∷y∋v)
, λ where v (here v≈x) → here (trans v≈x x≈y)
v (there Γ∷x∋v) → there Γ∷x∋v)
... | no ¬x≈y with lookup Γ x
... | yes (Δ , B , Γ≈Δ,x∶B) = yes (Δ , y ∶ A , B , (λ where v (here v≈y) → there (here v≈y)
<<<<<<< HEAD
v (there Γ∋v) → proj₁ swap v (there (proj₁ Γ≈Δ,x∶B v Γ∋v)))
, λ where v (here v≈x) → there (proj₂ Γ≈Δ,x∶B v (here v≈x))
v (there (here v≈y)) → here v≈y
v (there (there Δ∋v)) → there (proj₂ Γ≈Δ,x∶B v (there Δ∋v)))
... | no ¬P = no (λ (Δ , B , Q) → ¬P (Δ , B , {! !}))
-- (λ where v P → proj₁ Q v (there P))
-- , (λ where v P → {! proj₂ Q v P !})))
=======
v (there Γ∋v) → {! !}) , {! !})
... | no P = {! !}
>>>>>>> 40f82fd098a0c30c87ea0056baa48152aefae1f6
lookup (Γ ++ Δ) x = {! !}
lookup ∅ x = no (λ where (Δ , A , P) → ∅-empty P)
-- ... | yes x≈y = yes (Γ , A , λ v → (λ where (here v≈y) → here (trans v≈y (sym x≈y))
-- (there Γ∋v) → there Γ∋v) -- (λ v≈x → v≉y (trans v≈x x≈y))
-- , λ where (here v≈x) → here (trans v≈x x≈y)
-- (there Γ∋v) → there Γ∋v) -- (λ v≈y → v≉x (trans v≈y (sym x≈y)))
-- ... | no ¬x≈y with lookup Γ x
-- ... | yes (Δ , B , Γ≈Δ,x∶B) = yes (Δ , y ∶ A , B , λ v → (λ where (here v≈y) → there (here v≈y) -- (λ v≈x → ¬x≈y (trans (sym v≈x) v≈y))
-- (there Γ∋v) → lemma-1 (proj₁ (Γ≈Δ,x∶B v) Γ∋v))
-- , λ where (here v≈x) → there (proj₂ (Γ≈Δ,x∶B v) (here v≈x)) -- (λ v≈y → ¬x≈y (trans (sym v≈x) v≈y))
-- (there Δ,y∶A∋y) → ,-weakening y A (proj₂ (Γ≈Δ,x∶B v)) (lemma-1 Δ,y∶A∋y))
-- where
-- lemma-1 : ∀ {Γ x y z A B} → Γ , x ∶ A ∋ z → Γ , y ∶ B , x ∶ A ∋ z
-- lemma-1 (here z≈x) = here z≈x
-- lemma-1 (there Γ∋z) = there (there Γ∋z)
-- ,-weakening : ∀ {Γ Δ v} x A → (Γ ∋ v → Δ ∋ v) → (Γ , x ∶ A ∋ v → Δ , x ∶ A ∋ v)
-- ,-weakening x A f (here v≈x) = here v≈x
-- ,-weakening x A f (there Γ∋v) = there (f Γ∋v)
-- ... | no ¬P = no λ Q → {! !}
-- -- GOAL : Γ , y ∶ A ≉ CTX , x ∶ TYPE
-- -- ¬P Γ ≉ CTX , x : TYPE
-- -- no (λ (Δ , B , Q) → ¬P (Δ , B , {! Q !}))
-- -- λ v → (λ Γ∋v → proj₁ (Q v) (there Γ∋v))
-- -- , (λ Δ,x∶B∋v → {! proj₂ (Q v) Δ,x∶B∋v !})))
-- -- → (λ Γ∋v → proj₁ (Q v) (there Γ∋v))
-- -- , λ Δ,x∶B∋v → {! (proj₂ (Q v) Δ,x∶B∋v) !}))
-- -- where
-- -- (here P) → strengthen (proj₂ (Q v) (here P)) (λ y≈v → ¬x≈y (sym (trans y≈v P)))
-- -- (there P) → {! !}))
-- where
-- -- GOAL : Δ , x ∶ B → Γ
-- -- Q v : Δ , x ∶ B <≈> Γ , y ∶ A
-- -- -- ¬P Δ A = Γ <=> Δ , x ∶ A
-- -- (x₁ : Carrier) →
-- -- Σ (Γ ∋ x₁ → Δ₁ , x ∶ A₁ ∋ x₁) (λ x₂ → Δ₁ , x ∶ A₁ ∋ x₁ → Γ ∋ x₁)))
-- strengthen : ∀ {Γ x A v} → Γ , x ∶ A ∋ v → x ≉ v → Γ ∋ v
-- strengthen (here x≈v) x≉v = ⊥-elim (x≉v (sym x≈v))
-- strengthen (there P) x≉v = P
-- temp : ∀ Δ B v → (Δ ∋ v) → (f : Δ , x ∶ B ∋ v → Γ , y ∶ A ∋ v) → Γ ∋ v
-- temp Δ B v Δ∋v f with v ≟Chan x
-- ... | yes v≈x = strengthen (f (here v≈x)) (λ y≈v → ¬x≈y (sym (trans y≈v v≈x)))
-- ... | no ¬v≈x with y ≟Chan v
-- ... | yes y≈v = {! !}
-- ... | no ¬y≈v = strengthen (f (there Δ∋v)) ¬y≈v
-- -- strengthen (f (there Δ∈v)) λ y≈v → {! !}
-- -- lemma : G
-- lookup (Γ ++ Δ) x = {! !}
-- lookup ∅ x = no (λ where (Δ , A , P) → ∅-empty P)
-- -- _≈?_ : (Γ Δ : Session) → Dec (Γ ≈ Δ)
-- -- Γ ≈? Δ = {! !}
-- -- empty : ∀ {Γ x A} → ¬ (∅ ≈ (Γ , x ∶ A))
-- -- empty {Γ} {x} {A} P with x ≟ x
-- -- ... | no ¬p = {! !}
-- -- ... | yes p = {! !}
-- -- lookup : (Γ : Session) (x : Chan) → Dec (∃[ Δ ] ∃[ A ] (Γ ≈ (Δ , x ∶ A)))
-- -- lookup (Γ , y ∶ A) x = {! !}
-- -- lookup (Γ ++ Δ) x with lookup Γ x
-- -- ... | yes p = {! !}
-- -- ... | no ¬p = {! !}
-- -- lookup ∅ x = no (λ where (Γ , A , P) → {! P x !})
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{-# OPTIONS --safe --without-K #-}
open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; sym)
open import Relation.Nullary using (Dec; yes; no)
open import Relation.Nullary.Decidable using (fromWitness; toWitness)
open import Function using (_∘_)
open import Data.Empty using (⊥-elim)
import Data.Fin as Fin
import Data.Unit as Unit
import Data.Product as Product
import Data.Vec as Vec
import Data.Vec.Relation.Unary.All as All
import Data.Nat as ℕ
open Fin using (Fin; zero; suc)
open Unit using (⊤; tt)
open ℕ using (ℕ)
open Product using (Σ-syntax; ∃-syntax; _,_; _×_; proj₁; proj₂)
open Vec using (Vec; []; _∷_)
open All using (All; []; _∷_)
module PiCalculus.LinearTypeSystem.Algebras where
private
variable
n : ℕ
infixr 100 _²
_² : ∀ {a} → Set a → Set a
A ² = A × A
record Algebra (Q : Set) : Set₁ where
field
0∙ 1∙ : Q
_≔_∙_ : Q → Q → Q → Set
-- Given two operands, we can decide whether a third one exists
∙-computeʳ : ∀ x y → Dec (∃[ z ] (x ≔ y ∙ z))
-- If a third operand exists, it must be unique
∙-unique : ∀ {x x' y z} → x' ≔ y ∙ z → x ≔ y ∙ z → x' ≡ x
∙-uniqueˡ : ∀ {x y y' z} → x ≔ y' ∙ z → x ≔ y ∙ z → y' ≡ y
-- 0 is the minimum
0∙-minˡ : ∀ {y z} → 0∙ ≔ y ∙ z → y ≡ 0∙
-- Neutral element, commutativity, associativity
∙-idˡ : ∀ {x} → x ≔ 0∙ ∙ x
∙-comm : ∀ {x y z} → x ≔ y ∙ z → x ≔ z ∙ y -- no need for right rules
∙-assoc : ∀ {x y z u v} → x ≔ y ∙ z → y ≔ u ∙ v → ∃[ w ] (x ≔ u ∙ w × w ≔ v ∙ z)
ℓ∅ : Q ²
ℓ∅ = 0∙ , 0∙
ℓᵢ : Q ²
ℓᵢ = 1∙ , 0∙
ℓₒ : Q ²
ℓₒ = 0∙ , 1∙
ℓ# : Q ²
ℓ# = 1∙ , 1∙
_≔_∙²_ : Q ² → Q ² → Q ² → Set
(lx , rx) ≔ (ly , ry) ∙² (lz , rz) = (lx ≔ ly ∙ lz) × (rx ≔ ry ∙ rz)
∙-idʳ : ∀ {x} → x ≔ x ∙ 0∙
∙-idʳ = ∙-comm ∙-idˡ
∙-assoc⁻¹ : ∀ {x y z u v} → x ≔ y ∙ z → z ≔ u ∙ v → ∃[ ； ] (x ≔ ； ∙ v × ； ≔ y ∙ u)
∙-assoc⁻¹ a b = let _ , a' , b' = ∙-assoc (∙-comm a) (∙-comm b) in _ , ∙-comm a' , ∙-comm b'
∙-mut-cancel : ∀ {x y y' z} → x ≔ y ∙ z → z ≔ y' ∙ x → x ≡ z
∙-mut-cancel x≔y∙z z≔y'∙x with ∙-assoc⁻¹ x≔y∙z z≔y'∙x
∙-mut-cancel x≔y∙z z≔y'∙x | e , x≔e∙x , e≔y∙y' rewrite ∙-uniqueˡ x≔e∙x ∙-idˡ | 0∙-minˡ e≔y∙y' = ∙-unique x≔y∙z ∙-idˡ
∙²-computeʳ : ∀ x y → Dec (∃[ z ] (x ≔ y ∙² z))
∙²-computeʳ (lx , rx) (ly , ry) with ∙-computeʳ lx ly | ∙-computeʳ rx ry
∙²-computeʳ (lx , rx) (ly , ry) | yes (_ , p) | yes (_ , q) = yes (_ , p , q)
∙²-computeʳ (lx , rx) (ly , ry) | yes p | no ¬q = no λ {(_ , _ , r) → ¬q (_ , r)}
∙²-computeʳ (lx , rx) (ly , ry) | no ¬p | _ = no λ {(_ , l , _) → ¬p (_ , l)}
∙²-unique : ∀ {x x' y z} → x' ≔ y ∙² z → x ≔ y ∙² z → x' ≡ x
∙²-unique {x = _ , _} {x' = _ , _} (ll , rl) (lr , rr)
rewrite ∙-unique ll lr | ∙-unique rl rr = refl
∙²-uniqueˡ : ∀ {x y y' z} → x ≔ y' ∙² z → x ≔ y ∙² z → y' ≡ y
∙²-uniqueˡ {y = _ , _} {y' = _ , _} (ll , lr) (rl , rr)
rewrite ∙-uniqueˡ ll rl | ∙-uniqueˡ lr rr = refl
∙²-idˡ : ∀ {x} → x ≔ (0∙ , 0∙) ∙² x
∙²-idˡ = ∙-idˡ , ∙-idˡ
∙²-comm : ∀ {x y z} → x ≔ y ∙² z → x ≔ z ∙² y
∙²-comm (lx , rx) = ∙-comm lx , ∙-comm rx
∙²-idʳ : ∀ {x} → x ≔ x ∙² (0∙ , 0∙)
∙²-idʳ = ∙²-comm ∙²-idˡ
∙²-assoc : ∀ {x y z u v} → x ≔ y ∙² z → y ≔ u ∙² v → ∃[ w ] (x ≔ u ∙² w × w ≔ v ∙² z)
∙²-assoc (lx , rx) (ly , ry) with ∙-assoc lx ly | ∙-assoc rx ry
∙²-assoc (lx , rx) (ly , ry) | _ , ll , rl | _ , lr , rr = _ , ((ll , lr) , (rl , rr))
∙²-assoc⁻¹ : ∀ {x y z u v} → x ≔ y ∙² z → z ≔ u ∙² v → ∃[ ； ] (x ≔ ； ∙² v × ； ≔ y ∙² u)
∙²-assoc⁻¹ a b = let _ , a' , b' = ∙²-assoc (∙²-comm a) (∙²-comm b) in _ , ∙²-comm a' , ∙²-comm b'
∙²-mut-cancel : ∀ {x y y' z} → x ≔ y ∙² z → z ≔ y' ∙² x → x ≡ z
∙²-mut-cancel {_ , _} (lx , rx) (ly , ry) rewrite ∙-mut-cancel lx ly | ∙-mut-cancel rx ry = refl
record Algebras : Set₁ where
field
Idx : Set
∃Idx : Idx
Usage : Idx → Set
UsageAlgebra : ∀ idx → Algebra (Usage idx)
infixl 40 _-,_
pattern _-,_ xs x = _∷_ x xs
module _ {idx : Idx} where
open Algebra (UsageAlgebra idx) public
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Conversion of _≤_ to _<_
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Relation.Binary
module Relation.Binary.Construct.NonStrictToStrict
{a ℓ₁ ℓ₂} {A : Set a} (_≈_ : Rel A ℓ₁) (_≤_ : Rel A ℓ₂) where
open import Data.Product using (_×_; _,_; proj₁; proj₂)
open import Data.Sum using (inj₁; inj₂)
open import Function using (_∘_; flip)
open import Relation.Nullary using (¬_; yes; no)
------------------------------------------------------------------------
-- _≤_ can be turned into _<_ as follows:
_<_ : Rel A _
x < y = (x ≤ y) × ¬ (x ≈ y)
------------------------------------------------------------------------
-- The converted relations have certain properties
-- (if the original relations have certain other properties)
<⇒≤ : _<_ ⇒ _≤_
<⇒≤ = proj₁
<-irrefl : Irreflexive _≈_ _<_
<-irrefl x≈y (_ , x≉y) = x≉y x≈y
<-trans : IsPartialOrder _≈_ _≤_ → Transitive _<_
<-trans po (x≤y , x≉y) (y≤z , y≉z) =
(trans x≤y y≤z , x≉y ∘ antisym x≤y ∘ trans y≤z ∘ reflexive ∘ Eq.sym)
where open IsPartialOrder po
<-≤-trans : Symmetric _≈_ → Transitive _≤_ → Antisymmetric _≈_ _≤_ →
_≤_ Respectsʳ _≈_ → Trans _<_ _≤_ _<_
<-≤-trans sym trans antisym respʳ (x≤y , x≉y) y≤z =
trans x≤y y≤z , (λ x≈z → x≉y (antisym x≤y (respʳ (sym x≈z) y≤z)))
≤-<-trans : Transitive _≤_ → Antisymmetric _≈_ _≤_ →
_≤_ Respectsˡ _≈_ → Trans _≤_ _<_ _<_
≤-<-trans trans antisym respʳ x≤y (y≤z , y≉z) =
trans x≤y y≤z , (λ x≈z → y≉z (antisym y≤z (respʳ x≈z x≤y)))
<-asym : Antisymmetric _≈_ _≤_ → Asymmetric _<_
<-asym antisym (x≤y , x≉y) (y≤x , _) = x≉y (antisym x≤y y≤x)
<-respˡ-≈ : Transitive _≈_ → _≤_ Respectsˡ _≈_ → _<_ Respectsˡ _≈_
<-respˡ-≈ trans respˡ y≈z (y≤x , y≉x) =
(respˡ y≈z y≤x) , (λ z≈x → y≉x (trans y≈z z≈x))
<-respʳ-≈ : Symmetric _≈_ → Transitive _≈_ →
_≤_ Respectsʳ _≈_ → _<_ Respectsʳ _≈_
<-respʳ-≈ sym trans respʳ {x} {y} {z} y≈z (x≤y , x≉y) =
(respʳ y≈z x≤y) , λ x≈z → x≉y (trans x≈z (sym y≈z))
<-resp-≈ : IsEquivalence _≈_ → _≤_ Respects₂ _≈_ → _<_ Respects₂ _≈_
<-resp-≈ eq (respʳ , respˡ) =
<-respʳ-≈ sym trans respʳ , <-respˡ-≈ trans respˡ
where open IsEquivalence eq
<-trichotomous : Symmetric _≈_ → Decidable _≈_ →
Antisymmetric _≈_ _≤_ → Total _≤_ →
Trichotomous _≈_ _<_
<-trichotomous ≈-sym _≟_ antisym total x y with x ≟ y
... | yes x≈y = tri≈ (<-irrefl x≈y) x≈y (<-irrefl (≈-sym x≈y))
... | no x≉y with total x y
... | inj₁ x≤y = tri< (x≤y , x≉y) x≉y
(x≉y ∘ antisym x≤y ∘ proj₁)
... | inj₂ x≥y = tri> (x≉y ∘ flip antisym x≥y ∘ proj₁) x≉y
(x≥y , x≉y ∘ ≈-sym)
<-decidable : Decidable _≈_ → Decidable _≤_ → Decidable _<_
<-decidable _≟_ _≤?_ x y with x ≟ y | x ≤? y
... | yes x≈y | _ = no (flip proj₂ x≈y)
... | no x≉y | yes x≤y = yes (x≤y , x≉y)
... | no x≉y | no x≰y = no (x≰y ∘ proj₁)
<-isStrictPartialOrder : IsPartialOrder _≈_ _≤_ →
IsStrictPartialOrder _≈_ _<_
<-isStrictPartialOrder po = record
{ isEquivalence = isEquivalence
; irrefl = <-irrefl
; trans = <-trans po
; <-resp-≈ = <-resp-≈ isEquivalence ≤-resp-≈
} where open IsPartialOrder po
<-isStrictTotalOrder₁ : Decidable _≈_ → IsTotalOrder _≈_ _≤_ →
IsStrictTotalOrder _≈_ _<_
<-isStrictTotalOrder₁ ≟ tot = record
{ isEquivalence = isEquivalence
; trans = <-trans isPartialOrder
; compare = <-trichotomous Eq.sym ≟ antisym total
} where open IsTotalOrder tot
<-isStrictTotalOrder₂ : IsDecTotalOrder _≈_ _≤_ →
IsStrictTotalOrder _≈_ _<_
<-isStrictTotalOrder₂ dtot = <-isStrictTotalOrder₁ _≟_ isTotalOrder
where open IsDecTotalOrder dtot
------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.
-- Version 0.16
irrefl = <-irrefl
{-# WARNING_ON_USAGE irrefl
"Warning: irrefl was deprecated in v0.16.
Please use <-irrefl instead."
#-}
trans = <-trans
{-# WARNING_ON_USAGE trans
"Warning: trans was deprecated in v0.16.
Please use <-trans instead."
#-}
antisym⟶asym = <-asym
{-# WARNING_ON_USAGE antisym⟶asym
"Warning: antisym⟶asym was deprecated in v0.16.
Please use <-asym instead."
#-}
decidable = <-decidable
{-# WARNING_ON_USAGE decidable
"Warning: decidable was deprecated in v0.16.
Please use <-decidable instead."
#-}
trichotomous = <-trichotomous
{-# WARNING_ON_USAGE trichotomous
"Warning: trichotomous was deprecated in v0.16.
Please use <-trichotomous instead."
#-}
isPartialOrder⟶isStrictPartialOrder = <-isStrictPartialOrder
{-# WARNING_ON_USAGE isPartialOrder⟶isStrictPartialOrder
"Warning: isPartialOrder⟶isStrictPartialOrder was deprecated in v0.16.
Please use <-isStrictPartialOrder instead."
#-}
isTotalOrder⟶isStrictTotalOrder = <-isStrictTotalOrder₁
{-# WARNING_ON_USAGE isTotalOrder⟶isStrictTotalOrder
"Warning: isTotalOrder⟶isStrictTotalOrder was deprecated in v0.16.
Please use <-isStrictTotalOrder₁ instead."
#-}
isDecTotalOrder⟶isStrictTotalOrder = <-isStrictTotalOrder₂
{-# WARNING_ON_USAGE isDecTotalOrder⟶isStrictTotalOrder
"Warning: isDecTotalOrder⟶isStrictTotalOrder was deprecated in v0.16.
Please use <-isStrictTotalOrder₂ instead."
#-}
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-- Andreas 2012-09-27, reported by Fredrik Forsberg
{-# OPTIONS --sized-types #-}
module Issue701-c where
open import Common.Size
{- If I understand correctly, unconstrained sizes should be resolved to \infty.
If I define -}
data U : {i : Size} -> Set where
c : {i : Size} -> U {↑ i}
data V : {i : Size} -> Set where
d : {i : Size} -> U {∞} -> V {↑ i}
works-with-explicit-infty : {i : Size} -> V {i} -> V {↑ i}
works-with-explicit-infty x = x
-- everything is fine. However, if I leave out {\infty}:
data V' : {i : Size} -> Set where
d : {i : Size} -> U -> V' {↑ i}
fails-if-no-infty : {i : Size} -> V' {i} -> V' {↑ i}
fails-if-no-infty x = x
--.i != ↑ .i of type Size
--when checking that the expression x has type V'
{- V' is not detected as a sized type anymore which seems to break the
promise about unconstrained sizes. Since U is just a non-inductive
argument to d, I wouldn't expect it to influence whether V is a sized
type or not? -}
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open import Relation.Binary.Core using (Decidable ; _≡_)
module Precond (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where
open import Data.Nat using (ℕ)
open import Data.Fin using (Fin ; zero ; suc)
open import Data.Fin.Properties using (_≟_)
open import Data.List using (List ; [] ; _∷_)
open import Level using () renaming (zero to ℓ₀)
import Category.Monad
import Category.Functor
open import Data.Maybe using (Maybe ; nothing ; just ; maybe′)
open Category.Monad.RawMonad {Level.zero} Data.Maybe.monad using (_>>=_)
open Category.Functor.RawFunctor {Level.zero} Data.Maybe.functor using (_<$>_)
open import Data.Vec using (Vec ; [] ; _∷_ ; map ; lookup ; toList)
open import Data.Vec.Properties using (map-cong ; map-∘ ; tabulate-∘)
import Data.List.All
open import Data.List.Any using (here ; there)
open Data.List.Any.Membership-≡ using (_∉_)
open import Data.Maybe using (just)
open import Data.Product using (∃ ; _,_ ; proj₁ ; proj₂)
open import Function using (flip ; _∘_ ; id)
open import Relation.Binary using (Setoid)
open import Relation.Binary.PropositionalEquality using (refl ; cong ; inspect ; [_] ; sym ; decSetoid)
open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎)
open import Relation.Nullary using (yes ; no)
open import Structures using (IsFunctor ; module Shaped ; Shaped)
open import FinMap using (FinMapMaybe ; lookupM ; union ; fromFunc ; empty ; insert ; lemma-lookupM-empty ; delete-many ; lemma-tabulate-∘ ; delete ; lemma-lookupM-delete ; lemma-lookupM-fromFunc ; reshape ; lemma-reshape-id)
import CheckInsert
open CheckInsert (decSetoid deq) using (checkInsert ; lemma-checkInsert-new ; lemma-lookupM-checkInsert-other)
import BFF
import Bidir
open Bidir (decSetoid deq) using (_in-domain-of_ ; lemma-assoc-domain)
import GetTypes
open GetTypes.PartialShapeShape using (Get ; module Get)
open BFF.PartialShapeBFF (decSetoid deq) using (assoc ; enumerate ; denumerate ; bff)
lemma-maybe-just : {A : Set} → (a : A) → (ma : Maybe A) → maybe′ Maybe.just (just a) ma ≡ Maybe.just (maybe′ id a ma)
lemma-maybe-just a (just x) = refl
lemma-maybe-just a nothing = refl
lemma-union-delete-fromFunc : {m n : ℕ} {A : Set} {is : Vec (Fin n) m} {h : FinMapMaybe n A} {g : Fin n → A} → is in-domain-of h → ∃ λ v → union h (delete-many is (fromFunc g)) ≡ fromFunc v
lemma-union-delete-fromFunc {is = []} {h = h} {g = g} p = _ , (lemma-tabulate-∘ (λ f → begin
maybe′ just (lookupM f (fromFunc g)) (lookupM f h)
≡⟨ cong (flip (maybe′ just) (lookupM f h)) (lemma-lookupM-fromFunc g f) ⟩
maybe′ just (just (g f)) (lookupM f h)
≡⟨ lemma-maybe-just (g f) (lookupM f h) ⟩
just (maybe′ id (g f) (lookupM f h)) ∎))
lemma-union-delete-fromFunc {n = n} {is = i ∷ is} {h = h} {g = g} (Data.List.All._∷_ (x , px) ps) = _ , (begin
union h (delete i (delete-many is (fromFunc g)))
≡⟨ lemma-tabulate-∘ inner ⟩
union h (delete-many is (fromFunc g))
≡⟨ proj₂ (lemma-union-delete-fromFunc ps) ⟩
_ ∎)
where inner : (f : Fin n) → maybe′ just (lookupM f (delete i (delete-many is (fromFunc g)))) (lookup f h) ≡ maybe′ just (lookupM f (delete-many is (fromFunc g))) (lookup f h)
inner f with f ≟ i
inner .i | yes refl = begin
maybe′ just (lookupM i (delete i (delete-many is (fromFunc g)))) (lookup i h)
≡⟨ cong (maybe′ just _) px ⟩
just x
≡⟨ cong (maybe′ just _) (sym px) ⟩
maybe′ just (lookupM i (delete-many is (fromFunc g))) (lookup i h) ∎
inner f | no f≢i = cong (flip (maybe′ just) (lookup f h)) (lemma-lookupM-delete (delete-many is (fromFunc g)) f≢i)
module _ (G : Get) where
open Get G
open Shaped ViewShapeT using () renaming (content to contentV)
assoc-enough : {i : I} → (j : I) → (s : SourceContainer Carrier (gl₁ i)) → (v : ViewContainer Carrier (gl₂ j)) → ∃ (λ h → assoc (contentV (get (enumerate SourceShapeT (gl₁ j)))) (contentV v) ≡ just h) → ∃ λ u → bff G j s v ≡ just u
assoc-enough {i} j s v (h , p) = _ , cong (_<$>_ ((λ f → fmapS f t) ∘ flip lookupM) ∘ _<$>_ (flip union (reshape g′ (Shaped.arity SourceShapeT (gl₁ j))))) p
where g′ = delete-many (contentV (get (enumerate SourceShapeT (gl₁ i)))) (fromFunc (denumerate SourceShapeT s))
t = enumerate SourceShapeT (gl₁ j)
module _ (G : Get) where
open Get G
open Shaped ViewShapeT using () renaming (content to contentV)
assoc-enough′ : {i : I} → (s : SourceContainer Carrier (gl₁ i)) → (v : ViewContainer Carrier (gl₂ i)) → ∃ (λ h → assoc (contentV (get (enumerate SourceShapeT (gl₁ i)))) (contentV v) ≡ just h) → ∃ λ u → bff G i s v ≡ just (fmapS just u)
assoc-enough′ {i} s v (h , p) = _ , (begin
bff G i s v
≡⟨ proj₂ (assoc-enough G i s v (h , p)) ⟩
just (fmapS (flip lookupM (union h (reshape g′ (Shaped.arity SourceShapeT (gl₁ i))))) t)
≡⟨ cong just (begin _
≡⟨ cong ((λ f → fmapS f t) ∘ flip lookupM ∘ union h) (lemma-reshape-id g′) ⟩
fmapS (flip lookupM (union h g′)) t
≡⟨ cong ((λ f → fmapS f t) ∘ flip lookupM) (proj₂ wp) ⟩
fmapS (flip lookupM (fromFunc (proj₁ wp))) t
≡⟨ IsFunctor.cong (Shaped.isFunctor SourceShapeT (gl₁ i)) (lemma-lookupM-fromFunc (proj₁ wp)) t ⟩
fmapS (Maybe.just ∘ proj₁ wp) t
≡⟨ IsFunctor.composition (Shaped.isFunctor SourceShapeT (gl₁ i)) just (proj₁ wp) t ⟩
fmapS Maybe.just (fmapS (proj₁ wp) t) ∎) ⟩ _ ∎)
where s′ = enumerate SourceShapeT (gl₁ i)
g = fromFunc (denumerate SourceShapeT s)
g′ = delete-many (contentV (get s′)) g
t = enumerate SourceShapeT (gl₁ i)
wp = lemma-union-delete-fromFunc (lemma-assoc-domain (contentV (get t)) (contentV v) p)
data All-different {A : Set} : List A → Set where
different-[] : All-different []
different-∷ : {x : A} {xs : List A} → x ∉ xs → All-different xs → All-different (x ∷ xs)
lemma-∉-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → {h : FinMapMaybe n Carrier} → assoc is xs ≡ just h → (i ∉ toList is) → lookupM i h ≡ nothing
lemma-∉-lookupM-assoc i [] [] refl i∉is = lemma-lookupM-empty i
lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') ph i∉is with assoc is' xs' | inspect (assoc is') xs'
lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') () i∉is | nothing | [ ph' ]
lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') {h} ph i∉is | just h' | [ ph' ] = begin
lookupM i h
≡⟨ lemma-lookupM-checkInsert-other i i' (i∉is ∘ here) x' h' ph ⟩
lookupM i h'
≡⟨ lemma-∉-lookupM-assoc i is' xs' ph' (i∉is ∘ there) ⟩
nothing ∎
different-assoc : {m n : ℕ} → (u : Vec (Fin n) m) → (v : Vec Carrier m) → All-different (toList u) → ∃ λ h → assoc u v ≡ just h
different-assoc [] [] p = empty , refl
different-assoc (u ∷ us) (v ∷ vs) (different-∷ u∉us diff-us) with different-assoc us vs diff-us
different-assoc (u ∷ us) (v ∷ vs) (different-∷ u∉us diff-us) | h , p' = insert u v h , (begin
assoc (u ∷ us) (v ∷ vs)
≡⟨ refl ⟩
(assoc us vs >>= checkInsert u v)
≡⟨ cong (flip _>>=_ (checkInsert u v)) p' ⟩
checkInsert u v h
≡⟨ lemma-checkInsert-new u v h (lemma-∉-lookupM-assoc u us vs p' u∉us) ⟩
just (insert u v h) ∎)
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module Data.Real.Properties where
open import Data.Real.Base as ℝ
open import Data.Real.Order
open import Data.Real.Abstract.Structures
using (IsOrderedHeytingField; IsArchimedanHeytingField)
open import Assume
open import Algebra using (IsCommutativeRing; CommutativeRing)
open import Algebra.Module using (Module)
open import Algebra.Module.Construct.TensorUnit using (⟨module⟩)
open import Algebra.Module.Normed using (IsNormedModule; NormedModule)
open import Algebra.Apartness using (IsHeytingField; HeytingField)
open import Level using (0ℓ)
open import Data.Product using (_,_)
+-*-isCommutativeRing : IsCommutativeRing _≈_ _+_ _*_ -_ 0.0 1.0
+-*-isCommutativeRing = assume
+-*-commutativeRing : CommutativeRing 0ℓ 0ℓ
+-*-commutativeRing = record { isCommutativeRing = +-*-isCommutativeRing }
+-*-module : Module +-*-commutativeRing 0ℓ 0ℓ
+-*-module = ⟨module⟩
+-*-isHeytingField : IsHeytingField _≈_ _≉_ _+_ _*_ -_ 0.0 1.0
+-*-isHeytingField = assume
+-*-heytingField : HeytingField 0ℓ 0ℓ 0ℓ
+-*-heytingField = record { isHeytingField = +-*-isHeytingField }
+-*-isOrderedHeytingField : IsOrderedHeytingField +-*-heytingField _<_
+-*-isOrderedHeytingField =
record
{ isStrictTotalOrder = <-isStrictTotalOrder
; ordered = assume
}
+-*-isArchimedanHeytingField : IsArchimedanHeytingField +-*-heytingField _<_
+-*-isArchimedanHeytingField =
record
{ dense = λ a b a<b → ((a + b) ÷ 2.0) , assume
; archimedan = λ a b a<0 b<0 → ∣ unsafeM ⌈ b ÷ a ⌉ ∣ , assume
}
where
open import Data.Integer using (∣_∣)
open import Data.Maybe
unsafeM : ∀ {a} {A : Set a} → Maybe A → A
unsafeM (just x) = x
unsafeM nothing = assume
abs-isNormedModule : IsNormedModule +-*-module _≤_ abs
abs-isNormedModule = assume
abs-normedModule : NormedModule +-*-commutativeRing _≤_ 0ℓ 0ℓ
abs-normedModule = record { isNormedModule = abs-isNormedModule }
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module MalformedModuleNameInIMPORT where
{-# IMPORT m #-}
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module Numeral.Natural.TotalOper where
import Lvl
open import Logic.Propositional
open import Logic.Predicate
open import Numeral.Natural hiding (𝐏)
open import Numeral.Natural.Relation.Divisibility
open import Numeral.Natural.Relation.Divisibility.Proofs
open import Numeral.Natural.Relation
open import Numeral.Natural.Relation.Order
open import Numeral.Natural.Relation.Order.Proofs
open import Relator.Equals
open import Relator.Equals.Proofs
-- Total predecessor function (Truncated predecessor)
𝐏 : (n : ℕ) → ⦃ _ : Positive(n) ⦄ → ℕ
𝐏(𝐒(n)) = n
-- Total subtraction (Truncated subtraction)
_−_ : (a : ℕ) → (b : ℕ) → ⦃ _ : a ≥ b ⦄ → ℕ
_−_ a 𝟎 = a
_−_ 𝟎 (𝐒(b)) ⦃ 0≥𝐒b ⦄ with ([<]-to-[≱] ([<]-minimum{b})) (0≥𝐒b)
... | ()
_−_ (𝐒(a)) (𝐒(b)) ⦃ 𝐒b≤𝐒a ⦄ = _−_ a b ⦃ [≤]-without-[𝐒] {b} (𝐒b≤𝐒a) ⦄
-- Total division (Whole number division)
_/₀_ : (a : ℕ) → (b : ℕ) → ⦃ _ : (b ∣ a) ⦄ → ℕ
_/₀_ _ _ ⦃ div ⦄ = divides-quotient div
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{-# OPTIONS --without-K --safe #-}
open import Categories.Category.Groupoid
module Categories.Category.Groupoid.Properties {o ℓ e} (G : Groupoid o ℓ e) where
import Categories.Morphism as Morphism
import Categories.Morphism.Properties as MorphismProps
import Categories.Morphism.Reasoning as MR
open Groupoid G
open Morphism category
open MorphismProps category
open HomReasoning
open MR category
private
variable
A B C : Obj
mono : {f : A ⇒ B} → Mono f
mono = Iso⇒Mono iso
epi : {f : A ⇒ B} → Epi f
epi = Iso⇒Epi iso
id-inverse : id {A = A} ⁻¹ ≈ id
id-inverse = ⟺ identityˡ ○ iso.isoʳ
⁻¹-involutive : {f : A ⇒ B} → f ⁻¹ ⁻¹ ≈ f
⁻¹-involutive {f = f} = begin
f ⁻¹ ⁻¹ ≈⟨ introʳ iso.isoˡ ⟩
f ⁻¹ ⁻¹ ∘ f ⁻¹ ∘ f ≈⟨ sym-assoc ○ elimˡ iso.isoˡ ⟩
f ∎
⁻¹-commute : {f : A ⇒ B} {g : C ⇒ A} → (f ∘ g) ⁻¹ ≈ g ⁻¹ ∘ f ⁻¹
⁻¹-commute {f = f} {g} = epi _ _ ( begin
(f ∘ g) ⁻¹ ∘ f ∘ g ≈⟨ iso.isoˡ ⟩
id ≈˘⟨ iso.isoˡ ⟩
g ⁻¹ ∘ g ≈˘⟨ cancelInner iso.isoˡ ⟩
(g ⁻¹ ∘ f ⁻¹) ∘ f ∘ g ∎ )
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module examplesPaperJFP.Coalgebra where
open import Size
F : Set → Set
mapF : ∀{A B} (f : A → B) → (F A → F B)
--- Dummy implementation to satisfy Agda's positivity checker.
F X = X
mapF f x = f x
S : Set
t : S → F S
data S′ : Set where
S = S′
t x = x
record νF : Set where
coinductive
field force : F νF
open νF using (force)
{-# TERMINATING #-}
unfoldF : ∀{S} (t : S → F S) → (S → νF)
force (unfoldF t s) = mapF (unfoldF t) (t s)
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------------------------------------------------------------------------
-- Semantics
------------------------------------------------------------------------
module RecursiveTypes.Semantics where
open import Codata.Musical.Notation
open import RecursiveTypes.Syntax
open import RecursiveTypes.Substitution
-- The semantics of a recursive type, i.e. its possibly infinite
-- unfolding.
⟦_⟧ : ∀ {n} → Ty n → Tree n
⟦ ⊥ ⟧ = ⊥
⟦ ⊤ ⟧ = ⊤
⟦ var x ⟧ = var x
⟦ σ ⟶ τ ⟧ = ♯ ⟦ σ ⟧ ⟶ ♯ ⟦ τ ⟧
⟦ μ σ ⟶ τ ⟧ = ♯ ⟦ σ [0≔ χ ] ⟧ ⟶ ♯ ⟦ τ [0≔ χ ] ⟧
where χ = μ σ ⟶ τ
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module iff-erasure where
open import Relation.Binary.PropositionalEquality using (_≡_; refl)
open import Data.Empty using (⊥-elim)
-- open import plfa.part1.Isomorphism using (_⇔_)
open import decidable using (Dec; yes; no; Bool; true; false; ⌊_⌋)
-- 同値 (equivalence)
record _⇔_ (A B : Set) : Set where
field
to : A → B
from : B → A
open _⇔_
-- 必要十分条件 (if and only if)
_iff_ : Bool → Bool → Bool
true iff true = true
true iff false = false
false iff true = false
false iff false = true
-- 同値のdecidable
_⇔-dec_ : ∀ {A B : Set} → Dec A → Dec B → Dec (A ⇔ B)
yes a ⇔-dec yes b = yes record { to = λ _ → b ; from = λ _ → a }
yes a ⇔-dec no ¬b = no λ a⇔b → ¬b (to a⇔b a)
no ¬a ⇔-dec yes b = no λ a⇔b → ¬a (from a⇔b b)
no ¬a ⇔-dec no ¬b = yes record { to = λ a → ⊥-elim (¬a a) ; from = λ b → ⊥-elim (¬b b) }
-- erasureを使うことで真偽値の必要十分条件とdecidableの同値が等しくなることの証明
iff-⇔ : ∀ {A B : Set} (x : Dec A) (y : Dec B) → ⌊ x ⌋ iff ⌊ y ⌋ ≡ ⌊ x ⇔-dec y ⌋
iff-⇔ (yes a) (yes b) = refl
iff-⇔ (yes a) (no b) = refl
iff-⇔ (no a) (yes b) = refl
iff-⇔ (no a) (no b) = refl
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{-
The Inductive Version of James Construction
This file contains:
- An inductive family 𝕁, and its direct colimit is equivalence to James;
(KANG Rongji, Feb. 2022)
- The family 𝕁 can be iteratively constructed as pushouts;
- Special cases of 𝕁 n for n = 0, 1 and 2;
- Connectivity of inclusion maps.
This file is the summary of the main results.
The proof is divided into parts and put inside the fold Cubical.HITs.James.Inductive
-}
{-# OPTIONS --safe #-}
module Cubical.HITs.James.Inductive where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Pointed
open import Cubical.Data.Nat
open import Cubical.Data.Unit
open import Cubical.Data.Sigma
open import Cubical.HITs.Wedge
open import Cubical.HITs.Pushout
open import Cubical.HITs.Pushout.PushoutProduct
open import Cubical.HITs.SequentialColimit
open import Cubical.HITs.James.Base
open import Cubical.HITs.James.Inductive.Base
open import Cubical.HITs.James.Inductive.PushoutFormula
renaming (isConnectedIncl to connIncl ; isConnectedInl to connInl)
open import Cubical.HITs.James.Inductive.Reduced
open import Cubical.HITs.James.Inductive.ColimitEquivalence
open import Cubical.Homotopy.Connected
private
variable
ℓ : Level
module JamesInd
(X∙@(X , x₀) : Pointed ℓ) where
-- The family 𝕁 n is equivalence to Brunerie's J n, as will be shown latter.
-- Instead of his inductive procedure, 𝕁 is defined directly as an indexed HIT.
𝕁 : ℕ → Type ℓ
𝕁 = 𝕁ames (X , x₀)
-- This family forms a direct system.
𝕁Seq : Sequence ℓ
𝕁Seq = 𝕁amesSeq (X , x₀)
-- The inductive construction of James is called 𝕁∞.
-- It is the direct colimit of 𝕁 n.
𝕁∞ : Type ℓ
𝕁∞ = Lim→ 𝕁Seq
-- And of course it is equivalent to James.
J≃𝕁∞ : James (X , x₀) ≃ 𝕁∞
J≃𝕁∞ = compEquiv (James≃𝕁Red∞ _) (invEquiv (𝕁ames∞≃𝕁Red∞ _))
-- Special cases of 𝕁 n for n = 0, 1 and 2:
𝕁₀≃Unit : 𝕁 0 ≃ Unit
𝕁₀≃Unit = 𝕁ames0≃ _
𝕁₁≃X : 𝕁 1 ≃ X
𝕁₁≃X = 𝕁ames1≃ _
𝕁₂≃P[X×X←X⋁X→X] : 𝕁 2 ≃ Pushout ⋁↪ fold⋁
𝕁₂≃P[X×X←X⋁X→X] = 𝕁ames2≃ _
-- The following is defined as pushouts of 𝕁 n.
𝕁Push : ℕ → Type ℓ
𝕁Push = 𝕁amesPush (X , x₀)
-- Brunerie uses f and g to denote the following maps, so do I.
module _
{n : ℕ} where
f : 𝕁Push n → X × 𝕁 (1 + n)
f = leftMap _
g : 𝕁Push n → 𝕁 (1 + n)
g = rightMap _
-- Here we show that 𝕁 (n+2) can be made as double pushouts invoving only X, 𝕁 n and 𝕁 (n+1).
-- In particular, our 𝕁 is exactly what Brunerie had defined.
𝕁ₙ₊₂≃Pushout : (n : ℕ) → 𝕁 (2 + n) ≃ Pushout f g
𝕁ₙ₊₂≃Pushout = 𝕁ames2+n≃ _
-- Connectivity of inclusion maps:
module _
(d : ℕ)(conn : isConnected (1 + d) X) where
-- Warning:
-- The connectivity is shifted by 2 from the convention of usual homotopy theory.
-- If X is (d+1)-connected, the transition incl : 𝕁 n → 𝕁 (n+1) will be (n+1)d-connected.
isConnectedIncl : (n : ℕ) → isConnectedFun ((1 + n) · d) (incl {n = n})
isConnectedIncl = connIncl X∙ d conn
-- If X is (d+1)-connected, the inclusion inl : 𝕁 n → 𝕁∞ will be (n+1)d-connected.
inl∞ : (n : ℕ) → 𝕁 n → 𝕁∞
inl∞ _ = inl
isConnectedInl : (n : ℕ) → isConnectedFun ((1 + n) · d) (inl∞ n)
isConnectedInl = connInl X∙ d conn
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{-# OPTIONS --cubical #-}
open import Agda.Builtin.Cubical.Path
open import Agda.Primitive
open import Agda.Primitive.Cubical
variable
a p : Level
A : Set a
P : A → Set p
eq₁ u v x y : A
refl : x ≡ x
refl {x = x} = λ _ → x
subst : (P : A → Set p) → x ≡ y → P x → P y
subst P x≡y p = primTransp (λ i → P (x≡y i)) i0 p
hcong :
(f : (x : A) → P x) (x≡y : x ≡ y) →
PathP (λ i → P (x≡y i)) (f x) (f y)
hcong f x≡y = λ i → f (x≡y i)
dcong :
(f : (x : A) → P x) (x≡y : x ≡ y) →
subst P x≡y (f x) ≡ f y
dcong {P = P} f x≡y = λ i →
primTransp (λ j → P (x≡y (primIMax i j))) i (f (x≡y i))
-- This lemma is due to Anders Mörtberg.
lemma₁ :
{P : I → Set p} {p : P i0} {q : P i1} →
PathP P p q ≡ (primTransp P i0 p ≡ q)
lemma₁ {P = P} {p = p} {q = q} = λ i →
PathP
(λ j → P (primIMax i j))
(primTransp (λ j → P (primIMin i j)) (primINeg i) p)
q
data 𝕊¹ : Set where
base : 𝕊¹
loop : base ≡ base
postulate
Q : 𝕊¹ → Set
b : Q base
ℓ : subst Q loop b ≡ b
lemma₂ :
{eq : x ≡ y} →
subst Q eq u ≡ v →
PathP (λ i → Q (eq i)) u v
lemma₂ = subst (λ A → A → PathP _ _ _) lemma₁ (λ x → x)
lemma₃ : (x : 𝕊¹) → Q x
lemma₃ base = b
lemma₃ (loop i) = lemma₂ ℓ i
postulate
lemma₄ :
(eq₂ : subst Q eq₁ (lemma₃ x) ≡ lemma₃ y) →
hcong lemma₃ eq₁ ≡ lemma₂ eq₂ →
dcong lemma₃ eq₁ ≡ eq₂
works : dcong lemma₃ loop ≡ ℓ
works = lemma₄ ℓ refl
-- "injectivity" of lemma₃ was the problem
should-work-too : dcong lemma₃ loop ≡ ℓ
should-work-too = lemma₄ _ refl
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module _ where
postulate
C : Set → Set
A : Set
i : C A
foo : {X : Set} {{_ : C X}} → X
bar : A
bar = let instance z = i in foo
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-- By default both sized types and constructor-based guardedness are
-- available.
open import Agda.Builtin.Size
record Stream (A : Set) : Set where
coinductive
field
head : A
tail : Stream A
open Stream
repeat : ∀ {A} → A → Stream A
repeat x .head = x
repeat x .tail = repeat x
record Sized-stream (A : Set) (i : Size) : Set where
coinductive
field
head : A
tail : {j : Size< i} → Sized-stream A j
open Sized-stream
postulate
destroy-guardedness : {A : Set} → A → A
repeat-sized : ∀ {A i} → A → Sized-stream A i
repeat-sized x .head = x
repeat-sized x .tail = destroy-guardedness (repeat-sized x)
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{-# OPTIONS --safe --experimental-lossy-unification #-}
module Cubical.Algebra.Polynomials.Multivariate.Equiv.Poly0-A where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Isomorphism
open import Cubical.Data.Nat renaming (_+_ to _+n_; _·_ to _·n_)
open import Cubical.Data.Vec
open import Cubical.Algebra.Ring
open import Cubical.Algebra.CommRing
open import Cubical.Algebra.Polynomials.Univariate.Base
open import Cubical.Algebra.Polynomials.Multivariate.Base
open import Cubical.Algebra.Polynomials.Multivariate.Properties
open import Cubical.Algebra.CommRing.Instances.MultivariatePoly
private variable
ℓ : Level
module Equiv-Poly0-A (A' : CommRing ℓ) where
A = fst A'
cra = snd A'
open CommRingStr cra renaming (is-set to isSetA)
open Nth-Poly-structure A' 0
-----------------------------------------------------------------------------
-- Equivalence
Poly0→A : Poly A' 0 → A
Poly0→A = Poly-Rec-Set.f A' 0 A isSetA
0r
(λ v a → a)
_+_
+Assoc
+Rid
+Comm
(λ _ → refl)
λ _ a b → refl
A→Poly0 : A → Poly A' 0
A→Poly0 a = base [] a
e-sect : (a : A) → Poly0→A (A→Poly0 a) ≡ a
e-sect a = refl
e-retr : (P : Poly A' 0) → A→Poly0 (Poly0→A P) ≡ P
e-retr = Poly-Ind-Prop.f A' 0
(λ P → A→Poly0 (Poly0→A P) ≡ P)
(λ _ → trunc _ _)
(base-0P [])
(λ { [] a → refl })
λ {U V} ind-U ind-V → (sym (base-Poly+ [] (Poly0→A U) (Poly0→A V))) ∙ (cong₂ _Poly+_ ind-U ind-V)
-----------------------------------------------------------------------------
-- Ring homomorphism
map-0P : Poly0→A 0P ≡ 0r
map-0P = refl
Poly0→A-gmorph : (P Q : Poly A' 0) → Poly0→A ( P Poly+ Q) ≡ Poly0→A P + Poly0→A Q
Poly0→A-gmorph P Q = refl
map-1P : Poly0→A 1P ≡ 1r
map-1P = refl
Poly0→A-rmorph : (P Q : Poly A' 0) → Poly0→A ( P Poly* Q) ≡ Poly0→A P · Poly0→A Q
Poly0→A-rmorph = Poly-Ind-Prop.f A' 0
(λ P → (Q : Poly A' 0) → Poly0→A (P Poly* Q) ≡ Poly0→A P · Poly0→A Q)
(λ P p q i Q j → isSetA (Poly0→A (P Poly* Q)) (Poly0→A P · Poly0→A Q) (p Q) (q Q) i j)
(λ Q → sym (RingTheory.0LeftAnnihilates (CommRing→Ring A') (Poly0→A Q)))
(λ v a → Poly-Ind-Prop.f A' 0
(λ P → Poly0→A (base v a Poly* P) ≡ Poly0→A (base v a) · Poly0→A P)
(λ _ → isSetA _ _)
(sym (RingTheory.0RightAnnihilates (CommRing→Ring A') (Poly0→A (base v a))))
(λ v' a' → refl)
λ {U V} ind-U ind-V → (cong₂ _+_ ind-U ind-V) ∙ (sym (·Rdist+ _ _ _)))
λ {U V} ind-U ind-V Q → (cong₂ _+_ (ind-U Q) (ind-V Q)) ∙ (sym (·Ldist+ _ _ _))
-----------------------------------------------------------------------------
-- Ring Equivalence
module _ (A' : CommRing ℓ) where
open Equiv-Poly0-A A'
CRE-Poly0-A : CommRingEquiv (PolyCommRing A' 0) A'
fst CRE-Poly0-A = isoToEquiv is
where
is : Iso (Poly A' 0) (A' .fst)
Iso.fun is = Poly0→A
Iso.inv is = A→Poly0
Iso.rightInv is = e-sect
Iso.leftInv is = e-retr
snd CRE-Poly0-A = makeIsRingHom map-1P Poly0→A-gmorph Poly0→A-rmorph
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{-# OPTIONS --cubical --safe #-}
module Cubical.Relation.Everything where
open import Cubical.Relation.Nullary public
open import Cubical.Relation.Nullary.DecidableEq public
open import Cubical.Relation.Binary public
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open import Issue2229Base public
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-- Andreas, 2019-08-08, issue #3962 reported (+ test case) by guillaumebrunerie
-- Don't lex "{{" as instance braces if followed by "-", as this will confuse Emacs.
-- Rather lex "{{-" as "{" "{-".
postulate
A : Set
f : {{_ : A}} → Set
-x : A
B : Set
B = f {{-x}}
C : Set
C = ?
-- WAS: passes parser but confuses emacs, which treats everything following {- as comment.
-- Expected error: Unterminated '{-'
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module Numeral.Finite.Oper.Comparisons where
import Lvl
open import Data.Boolean
import Data.Boolean.Operators
open Data.Boolean.Operators.Programming
open import Functional
open import Numeral.Finite
open import Numeral.Sign
-- Compare
_⋚?_ : ∀{a b} → 𝕟(a) → 𝕟(b) → (−|0|+)
𝟎 ⋚? 𝟎 = 𝟎
𝟎 ⋚? 𝐒(b) = ➖
𝐒(a) ⋚? 𝟎 = ➕
𝐒(a) ⋚? 𝐒(b) = a ⋚? b
-- Equality check
_≡?_ : ∀{a b} → 𝕟(a) → 𝕟(b) → Bool
_≡?_ = elim₃ 𝐹 𝑇 𝐹 ∘₂ (_⋚?_)
-- Non-equality check
_≢?_ : ∀{a b} → 𝕟(a) → 𝕟(b) → Bool
_≢?_ = elim₃ 𝑇 𝐹 𝑇 ∘₂ (_⋚?_)
-- Lesser-than check
_<?_ : ∀{a b} → 𝕟(a) → 𝕟(b) → Bool
_<?_ = elim₃ 𝑇 𝐹 𝐹 ∘₂ (_⋚?_)
-- Lesser-than or equals check
_≤?_ : ∀{a b} → 𝕟(a) → 𝕟(b) → Bool
_≤?_ = elim₃ 𝑇 𝑇 𝐹 ∘₂ (_⋚?_)
-- Greater-than check
_>?_ : ∀{a b} → 𝕟(a) → 𝕟(b) → Bool
_>?_ = elim₃ 𝐹 𝐹 𝑇 ∘₂ (_⋚?_)
-- Greater-than or equals check
_≥?_ : ∀{a b} → 𝕟(a) → 𝕟(b) → Bool
_≥?_ = elim₃ 𝐹 𝑇 𝑇 ∘₂ (_⋚?_)
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{-# OPTIONS --without-K #-}
module Universe.Utility.General where
open import lib.Basics
open import lib.NType2
open import lib.types.Nat hiding (_+_)
open import lib.types.Pi
open import lib.types.Sigma
open import lib.types.Unit
-- A readable notation for the join of universe levels.
infixr 8 _⊔_
_⊔_ : ULevel → ULevel → ULevel
_⊔_ = lmax
-- Important detail: addition operator adjustment.
_+_ : ℕ → ℕ → ℕ
a + b = b lib.types.Nat.+ a -- n + 1 should mean S n (author preference)!
{- Since natural numbers and universe levels use
different types, we require a translation operation. -}
｢_｣ : ℕ → ULevel
｢ 0 ｣ = lzero
｢ S n ｣ = lsucc ｢ n ｣
{- The (function) exponentiation operator. For convenience, we
choose to define it via postcomposition in the recursion step. -}
infix 10 _^_
_^_ : ∀ {i} {A : Set i} → (A → A) → ℕ → A → A
f ^ 0 = idf _
f ^ S n = f ^ n ∘ f
{- The rest of this file contains the remainder of the library type material
that could not yet be merged into the community's HoTT library due to
organizational issues. -}
module _ {i j} {A : Type i} {B : A → Type j} where
module _ (h : is-contr A) where
Π₁-contr : Π A B ≃ B (fst h)
Π₁-contr = Π₁-Unit ∘e equiv-Π-l _ (snd (contr-equiv-Unit h ⁻¹)) ⁻¹
Σ₁-contr : Σ A B ≃ B (fst h)
Σ₁-contr = Σ₁-Unit ∘e equiv-Σ-fst _ (snd (contr-equiv-Unit h ⁻¹)) ⁻¹
module _ (h : (a : A) → is-contr (B a)) where
Σ₂-contr : Σ A B ≃ A
Σ₂-contr = Σ₂-Unit ∘e equiv-Σ-snd (λ _ → contr-equiv-Unit (h _))
Π₂-contr : Π A B ≃ ⊤
Π₂-contr = Π₂-Unit ∘e equiv-Π-r (λ _ → contr-equiv-Unit (h _))
×-comm : ∀ {i j} {A : Type i} {B : Type j} → (A × B) ≃ (B × A)
×-comm = equiv (λ {(a , b) → (b , a)})
(λ {(b , a) → (a , b)})
(λ _ → idp) (λ _ → idp)
Σ-comm-snd : ∀ {i j k} {A : Type i} {B : A → Type j} {C : A → Type k}
→ Σ (Σ A B) (λ ab → C (fst ab)) ≃ Σ (Σ A C) (λ ac → B (fst ac))
Σ-comm-snd {A = A} {B} {C} =
Σ (Σ A B) (λ ab → C (fst ab)) ≃⟨ Σ-assoc ⟩
Σ A (λ a → B a × C a) ≃⟨ equiv-Σ-snd (λ _ → ×-comm) ⟩
Σ A (λ a → C a × B a) ≃⟨ Σ-assoc ⁻¹ ⟩
Σ (Σ A C) (λ ac → B (fst ac)) ≃∎
flip : ∀ {i j k} {A : Type i} {B : Type j} {C : A → B → Type k}
→ ((a : A) (b : B) → C a b) → ((b : B) (a : A) → C a b)
flip = –> (curry-equiv ∘e equiv-Π-l _ (snd ×-comm) ∘e curry-equiv ⁻¹)
module _ {i j k} {A : Type i} {B : Type j} {C : A → B → Type k} where
↓-cst→app-equiv : {x x' : A} {p : x == x'} {u : (b : B) → C x b} {u' : (b : B) → C x' b}
→ ((b : B) → u b == u' b [ (λ x → C x b) ↓ p ]) ≃ (u == u' [ (λ x → (b : B) → C x b) ↓ p ])
↓-cst→app-equiv {p = idp} = equiv λ= app= (! ∘ λ=-η) (λ= ∘ app=-β)
module _ {i j k} {A : Type i} {B : A → Type j} {C : A → Type k} where
↓-cst2-equiv : {x y : A} (p : x == y) {b : C x} {c : C y}
(q : b == c [ C ↓ p ]) {u : B x} {v : B y}
→ (u == v [ B ↓ p ]) ≃ (u == v [ (λ xy → B (fst xy)) ↓ (pair= p q) ])
↓-cst2-equiv idp idp = ide _
module _ {i} {A B : Type i} (e : A ≃ B) {u : A} {v : B} where
↓-idf-ua-equiv : (–> e u == v) ≃ (u == v [ (λ x → x) ↓ (ua e) ])
↓-idf-ua-equiv = to-transp-equiv _ _ ⁻¹ ∘e (_ , pre∙-is-equiv (ap (λ z → z u) (ap coe (ap-idf (ua e)) ∙ ap –> (coe-equiv-β e))))
-- On propositions, equivalence coincides with logical equivalence.
module _ {i j} {A : Type i} {B : Type j} where
prop-equiv' : (B → is-prop A) → (A → is-prop B) → (A → B) → (B → A) → A ≃ B
prop-equiv' h k f g = equiv f g (λ b → prop-has-all-paths (k (g b)) _ _)
(λ a → prop-has-all-paths (h (f a)) _ _)
prop-equiv : is-prop A → is-prop B → (A → B) → (B → A) → A ≃ B
prop-equiv h k f g = prop-equiv' (cst h) (cst k) f g
-- Equivalent types have equivalent truncatedness propositions.
equiv-level : ∀ {i j} {A : Type i} {B : Type j} {n : ℕ₋₂}
→ A ≃ B → has-level n A ≃ has-level n B
equiv-level u = prop-equiv has-level-is-prop
has-level-is-prop
(equiv-preserves-level u)
(equiv-preserves-level (u ⁻¹))
prop-equiv-inhab-to-contr : ∀ {i} {A : Type i} → is-prop A ≃ (A → is-contr A)
prop-equiv-inhab-to-contr = prop-equiv is-prop-is-prop
(Π-level (λ _ → is-contr-is-prop))
(flip inhab-prop-is-contr)
inhab-to-contr-is-prop
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module Data.ByteString.Utf8 where
open import Data.ByteString.Primitive
open import Data.String using (String)
{-# FOREIGN GHC import qualified Data.ByteString #-}
{-# FOREIGN GHC import qualified Data.Text.Encoding #-}
postulate
packStrict : String → ByteStringStrict
unpackStrict : ByteStringStrict → String
{-# COMPILE GHC packStrict = (Data.Text.Encoding.encodeUtf8) #-}
{-# COMPILE GHC unpackStrict = (Data.Text.Encoding.decodeUtf8) #-}
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--------------------------------------------------------------------------------
-- This is part of Agda Inference Systems
{-# OPTIONS --sized-types --guardedness #-}
open import Agda.Builtin.Equality
open import Data.Product
open import Data.Sum
open import Data.Empty
open import Data.Unit
open import Level
open import Size
open import Codata.Thunk
open import Relation.Unary using (_⊆_)
open import Data.Container.Indexed
module is-lib.InfSys.Container {𝓁}(U : Set 𝓁) where
module ISCont {𝓁p} where
ISCont : {𝓁c : Level} → Set _
ISCont {𝓁c} = Container U U 𝓁c 𝓁p
ISContClosed : ∀{𝓁c 𝓁'}(is : ISCont {𝓁c}) → (U → Set 𝓁') → Set _
ISContClosed is P = (⟦ is ⟧ P ⊆ P)
_↾_ : ∀{𝓁c 𝓁'} → ISCont {𝓁c} → (U → Set 𝓁') → ISCont {𝓁c ⊔ 𝓁'}
(is ↾ P) .Command u = is .Command u × P u
(is ↾ P) .Response (c , p) = is .Response c
(is ↾ P) .next (c , p) r = is .next c r
_∪_ : ∀{𝓁c 𝓁c'} → ISCont {𝓁c} → ISCont {𝓁c'} → ISCont {𝓁c ⊔ 𝓁c'}
(is ∪ is') .Command u = is .Command u ⊎ is' .Command u
(is ∪ is') .Response = [ is .Response , is' .Response ]
(is ∪ is') .next = [ is .next , is' .next ]
Ind⟦_⟧ = μ
-- Coinductive interpretation
record CoInd⟦_⟧ {𝓁c : Level}(is : ISCont {𝓁c}) (u : U) : Set (𝓁c ⊔ 𝓁p) where
coinductive
constructor cofold_
field
unfold : ⟦ is ⟧ CoInd⟦ is ⟧ u
-- Sized coinductive interpretation
record CoInd⟦_⟧^ {𝓁c : Level}(is : ISCont {𝓁c}) (i : Size) (u : U) : Set (𝓁c ⊔ 𝓁p) where
coinductive
constructor cofold_
field
unfold : {j : Size< i} → ⟦ is ⟧ (CoInd⟦ is ⟧^ j) u
-- Sized coinductive interpretation (using Thunk)
data SCoInd⟦_⟧ {𝓁c : Level}(is : ISCont {𝓁c}) (u : U) (i : Size) : Set (𝓁c ⊔ 𝓁p) where
sfold : ⟦ is ⟧ (λ u → Thunk (SCoInd⟦ is ⟧ u) i) u → SCoInd⟦ is ⟧ u i
FCoInd⟦_,_⟧ : ∀{𝓁c 𝓁c'} → (I : ISCont {𝓁c}) (C : ISCont {𝓁c'}) → U → Set _
FCoInd⟦ I , C ⟧ = CoInd⟦ I ↾ Ind⟦ I ∪ C ⟧ ⟧
module _ {𝓁c 𝓁'} (is : ISCont {𝓁c}) (P : U → Set 𝓁') (closed : ISContClosed is P) where
open import Data.W.Indexed using (iter)
ind[_] : Ind⟦ is ⟧ ⊆ P
ind[_] = iter is closed
module _ {𝓁c 𝓁'} (is : ISCont {𝓁c}) (P : U → Set 𝓁') (consistent : P ⊆ ⟦ is ⟧ P) where
open CoInd⟦_⟧
coind[] : P ⊆ CoInd⟦ is ⟧
coind[] p .unfold .proj₁ = consistent p .proj₁
coind[] p .unfold .proj₂ r = coind[] (consistent p .proj₂ r)
module _ {𝓁c 𝓁c' 𝓁'}(I : ISCont {𝓁c}) (C : ISCont {𝓁c'}) (P : U → Set 𝓁') (bounded : P ⊆ Ind⟦ I ∪ C ⟧) (consistent : P ⊆ ⟦ I ⟧ P) where
open CoInd⟦_⟧
bounded-coind[] : P ⊆ FCoInd⟦ I , C ⟧
bounded-coind[] p .unfold .proj₁ .proj₁ = consistent p .proj₁
bounded-coind[] p .unfold .proj₁ .proj₂ = bounded p
bounded-coind[] p .unfold .proj₂ r = bounded-coind[] (consistent p .proj₂ r)
module Equivalence {𝓁p 𝓁P 𝓁P'} where
open ISCont {𝓁p}
open import is-lib.InfSys.Base {𝓁} as IS
open IS.MetaRule
open IS.IS
{- Every IS is an EndoContainer -}
C[_] : ∀{𝓁c 𝓁n} → IS {𝓁c} {𝓁p} {𝓁n} U → ISCont {𝓁 ⊔ 𝓁c ⊔ 𝓁n}
C[ is ] .Command u = Σ[ rn ∈ is .Names ] Σ[ c ∈ is .rules rn .Ctx ] u ≡ is .rules rn .conclu c
C[ is ] .Response (rn , c , refl) = is .rules rn .Pos c
C[ is ] .next (rn , c , refl) r = is .rules rn .prems c r
{- Every EndoContainer is an IS -}
IS[_] : ∀{𝓁'} → ISCont {𝓁'} → IS {zero} {𝓁p} {𝓁 ⊔ 𝓁'} U
IS[_] C .Names = Σ[ u ∈ U ] C .Command u
IS[ C ] .rules (u , c) =
record {
Ctx = ⊤ ;
Pos = λ _ → C .Response c ;
prems = λ _ r → C .next c r ;
conclu = λ _ → u }
{- Equivalence -}
isf-to-c : ∀{𝓁c 𝓁n} → {is : IS {𝓁c} {𝓁p} {𝓁n} U}{P : U → Set 𝓁P}
→ ISF[ is ] P ⊆ ⟦ C[ is ] ⟧ P
isf-to-c (rn , c , refl , pr) = (rn , c , refl) , pr
c-to-isf : ∀{𝓁'}{C : ISCont {𝓁'}}{P : U → Set 𝓁P}
→ ⟦ C ⟧ P ⊆ ISF[ IS[ C ] ] P
c-to-isf (c , pr) = (_ , c) , tt , refl , pr
∪-IS-eq : ∀{𝓁c 𝓁n 𝓁n'}{is : IS {𝓁c} {𝓁p} {𝓁n} U}{is' : IS {𝓁c} {𝓁p} {𝓁n'} U}{P : U → Set 𝓁P}
→ ISF[ is IS.∪ is' ] P ⊆ ⟦ C[ is ] ISCont.∪ C[ is' ] ⟧ P
∪-IS-eq (inj₁ rn , c , refl , pr) = inj₁ (rn , c , refl) , pr
∪-IS-eq (inj₂ rn , c , refl , pr) = inj₂ (rn , c , refl) , pr
∪-C-eq : ∀{𝓁1 𝓁2}{c : ISCont {𝓁1}}{c' : ISCont {𝓁2}}{P : U → Set 𝓁P}
→ ⟦ c ISCont.∪ c' ⟧ P ⊆ ISF[ IS[ c ] IS.∪ IS[ c' ] ] P
∪-C-eq (inj₁ c , r) = inj₁ (_ , c) , tt , refl , r
∪-C-eq (inj₂ c , r) = inj₂ (_ , c) , tt , refl , r
⊓-IS-eq : ∀{𝓁c 𝓁n}{is : IS {𝓁c} {𝓁p} {𝓁n} U}{P : U → Set 𝓁P}{P' : U → Set 𝓁P'}
→ ISF[ is ⊓ P ] P' ⊆ ⟦ C[ is ] ↾ P ⟧ P'
⊓-IS-eq (rn , (c , Pu) , refl , pr) = ((rn , c , refl) , Pu) , pr
↾-C-eq : ∀{𝓁c}{c : ISCont {𝓁c}}{P : U → Set 𝓁P}{P' : U → Set 𝓁P'}
→ ⟦ c ↾ P ⟧ P' ⊆ ISF[ IS[ c ] ⊓ P ] P'
↾-C-eq ((c , Pu) , r) = (_ , c) , (tt , Pu) , refl , r | {
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?bug : Set
?bug = ?
-- Andreas, 2014-04-16
-- Issue 1104 reported by Fabien Renaud.
-- Emacs mode got confused by identifiers containing --
-- Problem: {!!} is not turned into hole
bug-- : Set
bug-- = ?
another : Set
another = (-- Senfgurke ?
{!!})-- Noch eine Senfgurke ?
_:--_ : Set → Set → Set
_:--_ = {!!}
bla = let x = Set ;-- bla ?
y = Set in Set
bug? : Set
bug? = {!!}
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{-# OPTIONS --without-K #-}
module hott.topology.loopspace where
open import hott.core
open import hott.types
open import hott.functions
-- The pointed loop space
Ω∙ : ∀{ℓ} → Type● ℓ → Type● ℓ
Ω∙ (A , a) = (a ≡ a , refl)
-- The loops space. It is obtained by suppressing the base point of
-- the corresponding pointed loop space.
Ω : ∀{ℓ} → Type● ℓ → Type ℓ
Ω = space ∘ Ω∙
--
-- The iterated pointed loop space
Ω̂∙ : ∀{ℓ} → ℕ → Type● ℓ → Type● ℓ
Ω̂∙ = iterate Ω∙
-- The iterated loop space. It is obtained by suppressing the base
-- point of the iterated pointed loop space.
Ω̂ : ∀{ℓ} → ℕ → Type● ℓ → Type ℓ
Ω̂ n = space ∘ Ω̂∙ n
-- Some short hands
Ω²∙ : ∀{ℓ} → Type● ℓ → Type● ℓ; Ω²∙ = Ω̂∙ 2
Ω³∙ : ∀{ℓ} → Type● ℓ → Type● ℓ; Ω³∙ = Ω̂∙ 3
Ω⁴∙ : ∀{ℓ} → Type● ℓ → Type● ℓ; Ω⁴∙ = Ω̂∙ 4
Ω⁵∙ : ∀{ℓ} → Type● ℓ → Type● ℓ; Ω⁵∙ = Ω̂∙ 5
Ω⁶∙ : ∀{ℓ} → Type● ℓ → Type● ℓ; Ω⁶∙ = Ω̂∙ 6
Ω⁷∙ : ∀{ℓ} → Type● ℓ → Type● ℓ; Ω⁷∙ = Ω̂∙ 7
Ω⁸∙ : ∀{ℓ} → Type● ℓ → Type● ℓ; Ω⁸∙ = Ω̂∙ 8
Ω⁹∙ : ∀{ℓ} → Type● ℓ → Type● ℓ; Ω⁹∙ = Ω̂∙ 9
Ω² : ∀{ℓ} → Type● ℓ → Type ℓ; Ω² = Ω̂ 2
Ω³ : ∀{ℓ} → Type● ℓ → Type ℓ; Ω³ = Ω̂ 3
Ω⁴ : ∀{ℓ} → Type● ℓ → Type ℓ; Ω⁴ = Ω̂ 4
Ω⁵ : ∀{ℓ} → Type● ℓ → Type ℓ; Ω⁵ = Ω̂ 5
Ω⁶ : ∀{ℓ} → Type● ℓ → Type ℓ; Ω⁶ = Ω̂ 6
Ω⁷ : ∀{ℓ} → Type● ℓ → Type ℓ; Ω⁷ = Ω̂ 7
Ω⁸ : ∀{ℓ} → Type● ℓ → Type ℓ; Ω⁸ = Ω̂ 8
Ω⁹ : ∀{ℓ} → Type● ℓ → Type ℓ; Ω⁹ = Ω̂ 9
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
open import Cubical.Core.Everything
open import Cubical.Foundations.HLevels
module Cubical.Algebra.Group.Construct.Free {ℓ} (Aˢ : hSet ℓ) where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Univalence
open import Cubical.Foundations.Path
open import Cubical.Algebra.Group
open import Cubical.Data.Empty.Polymorphic
open import Cubical.Data.Prod
open import Cubical.Relation.Binary.Reasoning.Equality
A = ⟨ Aˢ ⟩
isSetA = Aˢ .snd
data FreeG : Type ℓ where
inj : A → FreeG
_•_ : Op₂ FreeG
ε : FreeG
_⁻¹ : Op₁ FreeG
•-assoc : Associative _•_
•-identityˡ : LeftIdentity ε _•_
•-identityʳ : RightIdentity ε _•_
•-inverseˡ : LeftInverse ε _⁻¹ _•_
•-inverseʳ : RightInverse ε _⁻¹ _•_
squash : isSet FreeG
•-identity : Identity ε _•_
•-identity = •-identityˡ , •-identityʳ
•-inverse : Inverse ε _⁻¹ _•_
•-inverse = •-inverseˡ , •-inverseʳ
elim : ∀ {ℓ′} {B : FreeG → Type ℓ′} →
(∀ x → isSet (B x)) →
(∀ x → B (inj x)) →
(op : ∀ {x y} → B x → B y → B (x • y)) →
(e : B ε) →
(inv : ∀ {x} → B x → B (x ⁻¹)) →
(∀ {x y z} (a : B x) (b : B y) (c : B z) → PathP (λ i → B (•-assoc x y z i)) (op (op a b) c) (op a (op b c))) →
(∀ {x} (a : B x) → PathP (λ i → B (•-identityˡ x i)) (op e a) a) →
(∀ {x} (a : B x) → PathP (λ i → B (•-identityʳ x i)) (op a e) a) →
(∀ {x} (a : B x) → PathP (λ i → B (•-inverseˡ x i)) (op (inv a) a) e) →
(∀ {x} (a : B x) → PathP (λ i → B (•-inverseʳ x i)) (op a (inv a)) e) →
(x : FreeG) → B x
elim isSetB f op e inv assc idˡ idʳ invˡ invʳ (inj x) = f x
elim isSetB f op e inv assc idˡ idʳ invˡ invʳ (x • y) = op (elim isSetB f op e inv assc idˡ idʳ invˡ invʳ x)
(elim isSetB f op e inv assc idˡ idʳ invˡ invʳ y)
elim isSetB f op e inv assc idˡ idʳ invˡ invʳ ε = e
elim isSetB f op e inv assc idˡ idʳ invˡ invʳ (x ⁻¹) = inv (elim isSetB f op e inv assc idˡ idʳ invˡ invʳ x)
elim isSetB f op e inv assc idˡ idʳ invˡ invʳ (•-assoc x y z i) = assc (elim isSetB f op e inv assc idˡ idʳ invˡ invʳ x)
(elim isSetB f op e inv assc idˡ idʳ invˡ invʳ y)
(elim isSetB f op e inv assc idˡ idʳ invˡ invʳ z) i
elim isSetB f op e inv assc idˡ idʳ invˡ invʳ (•-identityˡ x i) = idˡ (elim isSetB f op e inv assc idˡ idʳ invˡ invʳ x) i
elim isSetB f op e inv assc idˡ idʳ invˡ invʳ (•-identityʳ x i) = idʳ (elim isSetB f op e inv assc idˡ idʳ invˡ invʳ x) i
elim isSetB f op e inv assc idˡ idʳ invˡ invʳ (•-inverseˡ x i) = invˡ (elim isSetB f op e inv assc idˡ idʳ invˡ invʳ x) i
elim isSetB f op e inv assc idˡ idʳ invˡ invʳ (•-inverseʳ x i) = invʳ (elim isSetB f op e inv assc idˡ idʳ invˡ invʳ x) i
elim isSetB f op e inv assc idˡ idʳ invˡ invʳ (squash x y p q i j) =
isOfHLevel→isOfHLevelDep 2 isSetB
(elim isSetB f op e inv assc idˡ idʳ invˡ invʳ x) (elim isSetB f op e inv assc idˡ idʳ invˡ invʳ y)
(cong (elim isSetB f op e inv assc idˡ idʳ invˡ invʳ) p) (cong (elim isSetB f op e inv assc idˡ idʳ invˡ invʳ) q) (squash x y p q) i j
FreeG-isGroup : IsGroup FreeG _•_ ε _⁻¹
FreeG-isGroup = record
{ isMonoid = record
{ isSemigroup = record
{ isMagma = record { is-set = squash }
; assoc = •-assoc
}
; identity = •-identity
}
; inverse = •-inverse
}
FreeGroup : Group ℓ
FreeGroup = record { isGroup = FreeG-isGroup }
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{-# OPTIONS --cubical #-}
module n2o.Network.Http where
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{-
This second-order term syntax was created from the following second-order syntax description:
syntax PDiff | PD
type
* : 0-ary
term
zero : * | 𝟘
add : * * -> * | _⊕_ l20
one : * | 𝟙
mult : * * -> * | _⊗_ l20
neg : * -> * | ⊖_ r50
pd : *.* * -> * | ∂_∣_
theory
(𝟘U⊕ᴸ) a |> add (zero, a) = a
(𝟘U⊕ᴿ) a |> add (a, zero) = a
(⊕A) a b c |> add (add(a, b), c) = add (a, add(b, c))
(⊕C) a b |> add(a, b) = add(b, a)
(𝟙U⊗ᴸ) a |> mult (one, a) = a
(𝟙U⊗ᴿ) a |> mult (a, one) = a
(⊗A) a b c |> mult (mult(a, b), c) = mult (a, mult(b, c))
(⊗D⊕ᴸ) a b c |> mult (a, add (b, c)) = add (mult(a, b), mult(a, c))
(⊗D⊕ᴿ) a b c |> mult (add (a, b), c) = add (mult(a, c), mult(b, c))
(𝟘X⊗ᴸ) a |> mult (zero, a) = zero
(𝟘X⊗ᴿ) a |> mult (a, zero) = zero
(⊖N⊕ᴸ) a |> add (neg (a), a) = zero
(⊖N⊕ᴿ) a |> add (a, neg (a)) = zero
(⊗C) a b |> mult(a, b) = mult(b, a)
(∂⊕) a : * |> x : * |- d0 (add (x, a)) = one
(∂⊗) a : * |> x : * |- d0 (mult(a, x)) = a
(∂C) f : (*,*).* |> x : * y : * |- d1 (d0 (f[x,y])) = d0 (d1 (f[x,y]))
(∂Ch₂) f : (*,*).* g h : *.* |> x : * |- d0 (f[g[x], h[x]]) = add (mult(pd(z. f[z, h[x]], g[x]), d0(g[x])), mult(pd(z. f[g[x], z], h[x]), d0(h[x])))
(∂Ch₁) f g : *.* |> x : * |- d0 (f[g[x]]) = mult (pd (z. f[z], g[x]), d0(g[x]))
-}
module PDiff.Syntax where
open import SOAS.Common
open import SOAS.Context
open import SOAS.Variable
open import SOAS.Families.Core
open import SOAS.Construction.Structure
open import SOAS.ContextMaps.Inductive
open import SOAS.Metatheory.Syntax
open import PDiff.Signature
private
variable
Γ Δ Π : Ctx
α : *T
𝔛 : Familyₛ
-- Inductive term declaration
module PD:Terms (𝔛 : Familyₛ) where
data PD : Familyₛ where
var : ℐ ⇾̣ PD
mvar : 𝔛 α Π → Sub PD Π Γ → PD α Γ
𝟘 : PD * Γ
_⊕_ : PD * Γ → PD * Γ → PD * Γ
𝟙 : PD * Γ
_⊗_ : PD * Γ → PD * Γ → PD * Γ
⊖_ : PD * Γ → PD * Γ
∂_∣_ : PD * (* ∙ Γ) → PD * Γ → PD * Γ
infixl 20 _⊕_
infixl 30 _⊗_
infixr 50 ⊖_
open import SOAS.Metatheory.MetaAlgebra ⅀F 𝔛
PDᵃ : MetaAlg PD
PDᵃ = record
{ 𝑎𝑙𝑔 = λ where
(zeroₒ ⋮ _) → 𝟘
(addₒ ⋮ a , b) → _⊕_ a b
(oneₒ ⋮ _) → 𝟙
(multₒ ⋮ a , b) → _⊗_ a b
(negₒ ⋮ a) → ⊖_ a
(pdₒ ⋮ a , b) → ∂_∣_ a b
; 𝑣𝑎𝑟 = var ; 𝑚𝑣𝑎𝑟 = λ 𝔪 mε → mvar 𝔪 (tabulate mε) }
module PDᵃ = MetaAlg PDᵃ
module _ {𝒜 : Familyₛ}(𝒜ᵃ : MetaAlg 𝒜) where
open MetaAlg 𝒜ᵃ
𝕤𝕖𝕞 : PD ⇾̣ 𝒜
𝕊 : Sub PD Π Γ → Π ~[ 𝒜 ]↝ Γ
𝕊 (t ◂ σ) new = 𝕤𝕖𝕞 t
𝕊 (t ◂ σ) (old v) = 𝕊 σ v
𝕤𝕖𝕞 (mvar 𝔪 mε) = 𝑚𝑣𝑎𝑟 𝔪 (𝕊 mε)
𝕤𝕖𝕞 (var v) = 𝑣𝑎𝑟 v
𝕤𝕖𝕞 𝟘 = 𝑎𝑙𝑔 (zeroₒ ⋮ tt)
𝕤𝕖𝕞 (_⊕_ a b) = 𝑎𝑙𝑔 (addₒ ⋮ 𝕤𝕖𝕞 a , 𝕤𝕖𝕞 b)
𝕤𝕖𝕞 𝟙 = 𝑎𝑙𝑔 (oneₒ ⋮ tt)
𝕤𝕖𝕞 (_⊗_ a b) = 𝑎𝑙𝑔 (multₒ ⋮ 𝕤𝕖𝕞 a , 𝕤𝕖𝕞 b)
𝕤𝕖𝕞 (⊖_ a) = 𝑎𝑙𝑔 (negₒ ⋮ 𝕤𝕖𝕞 a)
𝕤𝕖𝕞 (∂_∣_ a b) = 𝑎𝑙𝑔 (pdₒ ⋮ 𝕤𝕖𝕞 a , 𝕤𝕖𝕞 b)
𝕤𝕖𝕞ᵃ⇒ : MetaAlg⇒ PDᵃ 𝒜ᵃ 𝕤𝕖𝕞
𝕤𝕖𝕞ᵃ⇒ = record
{ ⟨𝑎𝑙𝑔⟩ = λ{ {t = t} → ⟨𝑎𝑙𝑔⟩ t }
; ⟨𝑣𝑎𝑟⟩ = refl
; ⟨𝑚𝑣𝑎𝑟⟩ = λ{ {𝔪 = 𝔪}{mε} → cong (𝑚𝑣𝑎𝑟 𝔪) (dext (𝕊-tab mε)) } }
where
open ≡-Reasoning
⟨𝑎𝑙𝑔⟩ : (t : ⅀ PD α Γ) → 𝕤𝕖𝕞 (PDᵃ.𝑎𝑙𝑔 t) ≡ 𝑎𝑙𝑔 (⅀₁ 𝕤𝕖𝕞 t)
⟨𝑎𝑙𝑔⟩ (zeroₒ ⋮ _) = refl
⟨𝑎𝑙𝑔⟩ (addₒ ⋮ _) = refl
⟨𝑎𝑙𝑔⟩ (oneₒ ⋮ _) = refl
⟨𝑎𝑙𝑔⟩ (multₒ ⋮ _) = refl
⟨𝑎𝑙𝑔⟩ (negₒ ⋮ _) = refl
⟨𝑎𝑙𝑔⟩ (pdₒ ⋮ _) = refl
𝕊-tab : (mε : Π ~[ PD ]↝ Γ)(v : ℐ α Π) → 𝕊 (tabulate mε) v ≡ 𝕤𝕖𝕞 (mε v)
𝕊-tab mε new = refl
𝕊-tab mε (old v) = 𝕊-tab (mε ∘ old) v
module _ (g : PD ⇾̣ 𝒜)(gᵃ⇒ : MetaAlg⇒ PDᵃ 𝒜ᵃ g) where
open MetaAlg⇒ gᵃ⇒
𝕤𝕖𝕞! : (t : PD α Γ) → 𝕤𝕖𝕞 t ≡ g t
𝕊-ix : (mε : Sub PD Π Γ)(v : ℐ α Π) → 𝕊 mε v ≡ g (index mε v)
𝕊-ix (x ◂ mε) new = 𝕤𝕖𝕞! x
𝕊-ix (x ◂ mε) (old v) = 𝕊-ix mε v
𝕤𝕖𝕞! (mvar 𝔪 mε) rewrite cong (𝑚𝑣𝑎𝑟 𝔪) (dext (𝕊-ix mε))
= trans (sym ⟨𝑚𝑣𝑎𝑟⟩) (cong (g ∘ mvar 𝔪) (tab∘ix≈id mε))
𝕤𝕖𝕞! (var v) = sym ⟨𝑣𝑎𝑟⟩
𝕤𝕖𝕞! 𝟘 = sym ⟨𝑎𝑙𝑔⟩
𝕤𝕖𝕞! (_⊕_ a b) rewrite 𝕤𝕖𝕞! a | 𝕤𝕖𝕞! b = sym ⟨𝑎𝑙𝑔⟩
𝕤𝕖𝕞! 𝟙 = sym ⟨𝑎𝑙𝑔⟩
𝕤𝕖𝕞! (_⊗_ a b) rewrite 𝕤𝕖𝕞! a | 𝕤𝕖𝕞! b = sym ⟨𝑎𝑙𝑔⟩
𝕤𝕖𝕞! (⊖_ a) rewrite 𝕤𝕖𝕞! a = sym ⟨𝑎𝑙𝑔⟩
𝕤𝕖𝕞! (∂_∣_ a b) rewrite 𝕤𝕖𝕞! a | 𝕤𝕖𝕞! b = sym ⟨𝑎𝑙𝑔⟩
-- Syntax instance for the signature
PD:Syn : Syntax
PD:Syn = record
{ ⅀F = ⅀F
; ⅀:CS = ⅀:CompatStr
; mvarᵢ = PD:Terms.mvar
; 𝕋:Init = λ 𝔛 → let open PD:Terms 𝔛 in record
{ ⊥ = PD ⋉ PDᵃ
; ⊥-is-initial = record { ! = λ{ {𝒜 ⋉ 𝒜ᵃ} → 𝕤𝕖𝕞 𝒜ᵃ ⋉ 𝕤𝕖𝕞ᵃ⇒ 𝒜ᵃ }
; !-unique = λ{ {𝒜 ⋉ 𝒜ᵃ} (f ⋉ fᵃ⇒) {x = t} → 𝕤𝕖𝕞! 𝒜ᵃ f fᵃ⇒ t } } } }
-- Instantiation of the syntax and metatheory
open Syntax PD:Syn public
open PD:Terms public
open import SOAS.Families.Build public
open import SOAS.Syntax.Shorthands PDᵃ public
open import SOAS.Metatheory PD:Syn public
-- Derived operations
∂₀_ : {𝔛 : Familyₛ} → PD 𝔛 * (* ∙ Γ) → PD 𝔛 * (* ∙ Γ)
∂₀_ {𝔛 = 𝔛} e = ∂ Theory.𝕨𝕜 𝔛 e ∣ x₀
∂₁_ : {𝔛 : Familyₛ} → PD 𝔛 * (* ∙ * ∙ Γ) → PD 𝔛 * (* ∙ * ∙ Γ)
∂₁_ {𝔛 = 𝔛} e = ∂ Theory.𝕨𝕜 𝔛 e ∣ x₁
infix 10 ∂₀_ ∂₁_
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module TruthValue where
open import OscarPrelude
record TruthValue : Set
where
constructor ⟨_⟩
field
truthValue : Bool
open TruthValue public
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module Base.Free.Instance.Identity.Properties where
open import Relation.Binary.PropositionalEquality using (refl; cong)
open import Base.Free using (Free; pure; impure)
open import Base.Free.Instance.Identity renaming (Identity to IdentityF)
open import Base.Isomorphism using (_≃_)
open _≃_
-- The usual `Identity` monad representation an the free version are isomorphic.
data Identity (A : Set) : Set where
Ident : A → Identity A
Identity≃IdentityF : ∀ {A} → Identity A ≃ IdentityF A
to Identity≃IdentityF (Ident x) = pure x
from Identity≃IdentityF (pure x) = Ident x
from∘to Identity≃IdentityF (Ident x) = refl
to∘from Identity≃IdentityF (pure x) = refl
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module Nats.Multiply.Comm where
open import Nats
open import Equality
open import Function
open import Nats.Add.Comm
open import Nats.Add.Assoc
------------------------------------------------------------------------
-- internal stuffs
private
a*0=0*a : ∀ a → a * 0 ≡ 0
a*0=0*a zero = refl
a*0=0*a (suc a) = a*0=0*a a
a+a*b=a*++b : ∀ a b → a + a * b ≡ a * suc b
a+a*b=a*++b zero _ = refl
a+a*b=a*++b (suc a) b
rewrite nat-add-comm b $ a * suc b
| nat-add-comm b $ a * b
| sym $ nat-add-assoc a (a * b) b
| a+a*b=a*++b a b
= refl
a*b=b*a : ∀ a b → a * b ≡ b * a
a*b=b*a zero b
rewrite a*0=0*a b = refl
a*b=b*a (suc a) b
rewrite a*b=b*a a b
| a+a*b=a*++b b a
= refl
------------------------------------------------------------------------
-- public aliases
nat-multiply-comm : ∀ a b → a * b ≡ b * a
nat-multiply-comm = a*b=b*a
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postulate
A : →.B
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{-# OPTIONS --without-K --exact-split #-}
module 13-propositional-truncation where
import 12-function-extensionality
open 12-function-extensionality public
-- Section 13 Propositional truncations, the image of a map, and the replacement axiom
--------------------------------------------------------------------------------
-- Section 13.1 Propositional truncations
-- Definition 13.1.1
type-hom-Prop :
{ l1 l2 : Level} (P : UU-Prop l1) (Q : UU-Prop l2) → UU (l1 ⊔ l2)
type-hom-Prop P Q = type-Prop P → type-Prop Q
hom-Prop :
{ l1 l2 : Level} → UU-Prop l1 → UU-Prop l2 → UU-Prop (l1 ⊔ l2)
hom-Prop P Q =
pair
( type-hom-Prop P Q)
( is-prop-function-type (is-prop-type-Prop Q))
is-prop-type-hom-Prop :
{ l1 l2 : Level} (P : UU-Prop l1) (Q : UU-Prop l2) → is-prop (type-hom-Prop P Q)
is-prop-type-hom-Prop P Q =
is-prop-function-type (is-prop-type-Prop Q)
equiv-Prop :
{ l1 l2 : Level} (P : UU-Prop l1) (Q : UU-Prop l2) → UU (l1 ⊔ l2)
equiv-Prop P Q = (type-Prop P) ≃ (type-Prop Q)
precomp-Prop :
{ l1 l2 l3 : Level} {A : UU l1} (P : UU-Prop l2) →
(A → type-Prop P) → (Q : UU-Prop l3) →
(type-hom-Prop P Q) → (A → type-Prop Q)
precomp-Prop P f Q g = g ∘ f
is-propositional-truncation :
( l : Level) {l1 l2 : Level} {A : UU l1} (P : UU-Prop l2) →
( A → type-Prop P) → UU (lsuc l ⊔ l1 ⊔ l2)
is-propositional-truncation l P f =
(Q : UU-Prop l) → is-equiv (precomp-Prop P f Q)
universal-property-propositional-truncation :
( l : Level) {l1 l2 : Level} {A : UU l1}
(P : UU-Prop l2) (f : A → type-Prop P) → UU (lsuc l ⊔ l1 ⊔ l2)
universal-property-propositional-truncation l {A = A} P f =
(Q : UU-Prop l) (g : A → type-Prop Q) →
is-contr (Σ (type-hom-Prop P Q) (λ h → (h ∘ f) ~ g))
-- Some unnumbered remarks after Definition 13.1.3
universal-property-is-propositional-truncation :
(l : Level) {l1 l2 : Level} {A : UU l1}
(P : UU-Prop l2) (f : A → type-Prop P) →
is-propositional-truncation l P f →
universal-property-propositional-truncation l P f
universal-property-is-propositional-truncation l P f is-ptr-f Q g =
is-contr-equiv'
( Σ (type-hom-Prop P Q) (λ h → Id (h ∘ f) g))
( equiv-tot (λ h → equiv-funext))
( is-contr-map-is-equiv (is-ptr-f Q) g)
map-is-propositional-truncation :
{l1 l2 l3 : Level} {A : UU l1} (P : UU-Prop l2) (f : A → type-Prop P) →
({l : Level} → is-propositional-truncation l P f) →
(Q : UU-Prop l3) (g : A → type-Prop Q) → type-hom-Prop P Q
map-is-propositional-truncation P f is-ptr-f Q g =
pr1
( center
( universal-property-is-propositional-truncation _ P f is-ptr-f Q g))
htpy-is-propositional-truncation :
{l1 l2 l3 : Level} {A : UU l1} (P : UU-Prop l2) (f : A → type-Prop P) →
(is-ptr-f : {l : Level} → is-propositional-truncation l P f) →
(Q : UU-Prop l3) (g : A → type-Prop Q) →
((map-is-propositional-truncation P f is-ptr-f Q g) ∘ f) ~ g
htpy-is-propositional-truncation P f is-ptr-f Q g =
pr2
( center
( universal-property-is-propositional-truncation _ P f is-ptr-f Q g))
is-propositional-truncation-universal-property :
(l : Level) {l1 l2 : Level} {A : UU l1}
(P : UU-Prop l2) (f : A → type-Prop P) →
universal-property-propositional-truncation l P f →
is-propositional-truncation l P f
is-propositional-truncation-universal-property l P f up-f Q =
is-equiv-is-contr-map
( λ g → is-contr-equiv
( Σ (type-hom-Prop P Q) (λ h → (h ∘ f) ~ g))
( equiv-tot (λ h → equiv-funext))
( up-f Q g))
-- Remark 13.1.2
is-propositional-truncation' :
( l : Level) {l1 l2 : Level} {A : UU l1} (P : UU-Prop l2) →
( A → type-Prop P) → UU (lsuc l ⊔ l1 ⊔ l2)
is-propositional-truncation' l {A = A} P f =
(Q : UU-Prop l) → (A → type-Prop Q) → (type-hom-Prop P Q)
is-propositional-truncation-simpl :
{ l1 l2 : Level} {A : UU l1} (P : UU-Prop l2)
( f : A → type-Prop P) →
( (l : Level) → is-propositional-truncation' l P f) →
( (l : Level) → is-propositional-truncation l P f)
is-propositional-truncation-simpl P f up-P l Q =
is-equiv-is-prop
( is-prop-Π (λ x → is-prop-type-Prop Q))
( is-prop-Π (λ x → is-prop-type-Prop Q))
( up-P l Q)
-- Example 13.1.3
--------------------------------------------------------------------------------
-- Section 6.3 Pointed types
-- Definition 6.3.1
UU-pt : (i : Level) → UU (lsuc i)
UU-pt i = Σ (UU i) (λ X → X)
type-UU-pt : {i : Level} → UU-pt i → UU i
type-UU-pt = pr1
pt-UU-pt : {i : Level} (A : UU-pt i) → type-UU-pt A
pt-UU-pt = pr2
-- Definition 6.3.2
_→*_ : {i j : Level} → UU-pt i → UU-pt j → UU-pt (i ⊔ j)
A →* B =
pair
( Σ (type-UU-pt A → type-UU-pt B) (λ f → Id (f (pt-UU-pt A)) (pt-UU-pt B)))
( pair
( const (type-UU-pt A) (type-UU-pt B) (pt-UU-pt B))
( refl))
-- Definition 6.3.3
Ω : {i : Level} → UU-pt i → UU-pt i
Ω A = pair (Id (pt-UU-pt A) (pt-UU-pt A)) refl
-- Definition 6.3.4
iterated-loop-space : {i : Level} → ℕ → UU-pt i → UU-pt i
iterated-loop-space zero-ℕ A = A
iterated-loop-space (succ-ℕ n) A = Ω (iterated-loop-space n A)
--------------------------------------------------------------------------------
is-propositional-truncation-const-star :
{ l1 : Level} (A : UU-pt l1)
( l : Level) → is-propositional-truncation l unit-Prop (const (type-UU-pt A) unit star)
is-propositional-truncation-const-star A =
is-propositional-truncation-simpl
( unit-Prop)
( const (type-UU-pt A) unit star)
( λ l P f → const unit (type-Prop P) (f (pt-UU-pt A)))
-- Example 13.1.4
is-propositional-truncation-id :
{ l1 : Level} (P : UU-Prop l1) →
( l : Level) → is-propositional-truncation l P id
is-propositional-truncation-id P l Q =
is-equiv-id (type-hom-Prop P Q)
-- Proposition 13.1.5
abstract
is-equiv-is-equiv-precomp-Prop :
{l1 l2 : Level} (P : UU-Prop l1) (Q : UU-Prop l2) (f : type-hom-Prop P Q) →
((l : Level) (R : UU-Prop l) →
is-equiv (precomp-Prop Q f R)) → is-equiv f
is-equiv-is-equiv-precomp-Prop P Q f is-equiv-precomp-f =
is-equiv-is-equiv-precomp-subuniverse id (λ l → is-prop) P Q f
is-equiv-precomp-f
triangle-3-for-2-is-ptruncation :
{l1 l2 l3 : Level} {A : UU l1} (P : UU-Prop l2) (P' : UU-Prop l3)
(f : A → type-Prop P) (f' : A → type-Prop P')
(h : type-hom-Prop P P') (H : (h ∘ f) ~ f') →
{l : Level} (Q : UU-Prop l) →
( precomp-Prop P' f' Q) ~
( (precomp-Prop P f Q) ∘ (precomp h (type-Prop Q)))
triangle-3-for-2-is-ptruncation P P' f f' h H Q g =
eq-htpy (λ p → inv (ap g (H p)))
is-equiv-is-ptruncation-is-ptruncation :
{l1 l2 l3 : Level} {A : UU l1} (P : UU-Prop l2) (P' : UU-Prop l3)
(f : A → type-Prop P) (f' : A → type-Prop P')
(h : type-hom-Prop P P') (H : (h ∘ f) ~ f') →
((l : Level) → is-propositional-truncation l P f) →
((l : Level) → is-propositional-truncation l P' f') →
is-equiv h
is-equiv-is-ptruncation-is-ptruncation P P' f f' h H is-ptr-P is-ptr-P' =
is-equiv-is-equiv-precomp-Prop P P' h
( λ l Q →
is-equiv-right-factor
( precomp-Prop P' f' Q)
( precomp-Prop P f Q)
( precomp h (type-Prop Q))
( triangle-3-for-2-is-ptruncation P P' f f' h H Q)
( is-ptr-P l Q)
( is-ptr-P' l Q))
is-ptruncation-is-ptruncation-is-equiv :
{l1 l2 l3 : Level} {A : UU l1} (P : UU-Prop l2) (P' : UU-Prop l3)
(f : A → type-Prop P) (f' : A → type-Prop P')
(h : type-hom-Prop P P') (H : (h ∘ f) ~ f') →
is-equiv h →
((l : Level) → is-propositional-truncation l P f) →
((l : Level) → is-propositional-truncation l P' f')
is-ptruncation-is-ptruncation-is-equiv P P' f f' h H is-equiv-h is-ptr-f l Q =
is-equiv-comp
( precomp-Prop P' f' Q)
( precomp-Prop P f Q)
( precomp h (type-Prop Q))
( triangle-3-for-2-is-ptruncation P P' f f' h H Q)
( is-equiv-precomp-is-equiv h is-equiv-h (type-Prop Q))
( is-ptr-f l Q)
is-ptruncation-is-equiv-is-ptruncation :
{l1 l2 l3 : Level} {A : UU l1} (P : UU-Prop l2) (P' : UU-Prop l3)
(f : A → type-Prop P) (f' : A → type-Prop P')
(h : type-hom-Prop P P') (H : (h ∘ f) ~ f') →
((l : Level) → is-propositional-truncation l P' f') →
is-equiv h →
((l : Level) → is-propositional-truncation l P f)
is-ptruncation-is-equiv-is-ptruncation P P' f f' h H is-ptr-f' is-equiv-h l Q =
is-equiv-left-factor
( precomp-Prop P' f' Q)
( precomp-Prop P f Q)
( precomp h (type-Prop Q))
( triangle-3-for-2-is-ptruncation P P' f f' h H Q)
( is-ptr-f' l Q)
( is-equiv-precomp-is-equiv h is-equiv-h (type-Prop Q))
-- Corollary 13.1.6
is-uniquely-unique-propositional-truncation :
{l1 l2 l3 : Level} {A : UU l1} (P : UU-Prop l2) (P' : UU-Prop l3)
(f : A → type-Prop P) (f' : A → type-Prop P') →
({l : Level} → is-propositional-truncation l P f) →
({l : Level} → is-propositional-truncation l P' f') →
is-contr (Σ (equiv-Prop P P') (λ e → (map-equiv e ∘ f) ~ f'))
is-uniquely-unique-propositional-truncation P P' f f' is-ptr-f is-ptr-f' =
is-contr-total-Eq-substructure
( universal-property-is-propositional-truncation _ P f is-ptr-f P' f')
( is-subtype-is-equiv)
( map-is-propositional-truncation P f is-ptr-f P' f')
( htpy-is-propositional-truncation P f is-ptr-f P' f')
( is-equiv-is-ptruncation-is-ptruncation P P' f f'
( map-is-propositional-truncation P f is-ptr-f P' f')
( htpy-is-propositional-truncation P f is-ptr-f P' f')
( λ l → is-ptr-f)
( λ l → is-ptr-f'))
-- Axiom 13.1.8
postulate type-trunc-Prop : {l : Level} → UU l → UU l
postulate is-prop-type-trunc-Prop : {l : Level} (A : UU l) → is-prop (type-trunc-Prop A)
trunc-Prop : {l : Level} → UU l → UU-Prop l
trunc-Prop A = pair (type-trunc-Prop A) (is-prop-type-trunc-Prop A)
postulate unit-trunc-Prop : {l : Level} (A : UU l) → A → type-Prop (trunc-Prop A)
postulate is-propositional-truncation-trunc-Prop : {l1 l2 : Level} (A : UU l1) → is-propositional-truncation l2 (trunc-Prop A) (unit-trunc-Prop A)
universal-property-trunc-Prop : {l1 l2 : Level} (A : UU l1) →
universal-property-propositional-truncation l2
( trunc-Prop A)
( unit-trunc-Prop A)
universal-property-trunc-Prop A =
universal-property-is-propositional-truncation _
( trunc-Prop A)
( unit-trunc-Prop A)
( is-propositional-truncation-trunc-Prop A)
map-universal-property-trunc-Prop :
{l1 l2 : Level} {A : UU l1} (P : UU-Prop l2) →
(A → type-Prop P) → type-hom-Prop (trunc-Prop A) P
map-universal-property-trunc-Prop {A = A} P f =
map-is-propositional-truncation
( trunc-Prop A)
( unit-trunc-Prop A)
( is-propositional-truncation-trunc-Prop A)
( P)
( f)
-- Proposition 13.1.9
unique-functor-trunc-Prop :
{l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) →
is-contr
( Σ ( type-hom-Prop (trunc-Prop A) (trunc-Prop B))
( λ h → (h ∘ (unit-trunc-Prop A)) ~ ((unit-trunc-Prop B) ∘ f)))
unique-functor-trunc-Prop {l1} {l2} {A} {B} f =
universal-property-trunc-Prop A (trunc-Prop B) ((unit-trunc-Prop B) ∘ f)
functor-trunc-Prop :
{l1 l2 : Level} {A : UU l1} {B : UU l2} →
(A → B) → type-hom-Prop (trunc-Prop A) (trunc-Prop B)
functor-trunc-Prop f =
pr1 (center (unique-functor-trunc-Prop f))
htpy-functor-trunc-Prop :
{ l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) →
( (functor-trunc-Prop f) ∘ (unit-trunc-Prop A)) ~ ((unit-trunc-Prop B) ∘ f)
htpy-functor-trunc-Prop f =
pr2 (center (unique-functor-trunc-Prop f))
htpy-uniqueness-functor-trunc-Prop :
{ l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) →
( h : type-hom-Prop (trunc-Prop A) (trunc-Prop B)) →
( ( h ∘ (unit-trunc-Prop A)) ~ ((unit-trunc-Prop B) ∘ f)) →
(functor-trunc-Prop f) ~ h
htpy-uniqueness-functor-trunc-Prop f h H =
htpy-eq (ap pr1 (contraction (unique-functor-trunc-Prop f) (pair h H)))
id-functor-trunc-Prop :
{ l1 : Level} {A : UU l1} → functor-trunc-Prop (id {A = A}) ~ id
id-functor-trunc-Prop {l1} {A} =
htpy-uniqueness-functor-trunc-Prop id id refl-htpy
comp-functor-trunc-Prop :
{ l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3}
( g : B → C) (f : A → B) →
( functor-trunc-Prop (g ∘ f)) ~
( (functor-trunc-Prop g) ∘ (functor-trunc-Prop f))
comp-functor-trunc-Prop g f =
htpy-uniqueness-functor-trunc-Prop
( g ∘ f)
( (functor-trunc-Prop g) ∘ (functor-trunc-Prop f))
( ( (functor-trunc-Prop g) ·l (htpy-functor-trunc-Prop f)) ∙h
( ( htpy-functor-trunc-Prop g) ·r f))
--------------------------------------------------------------------------------
-- Section 13.2 Propositional truncations as higher inductive types
-- Definition 13.2.1
case-paths-induction-principle-propositional-truncation :
{ l : Level} {l1 l2 : Level} {A : UU l1}
( P : UU-Prop l2) (α : (p q : type-Prop P) → Id p q) (f : A → type-Prop P) →
( B : type-Prop P → UU l) → UU (l ⊔ l2)
case-paths-induction-principle-propositional-truncation P α f B =
(p q : type-Prop P) (x : B p) (y : B q) → Id (tr B (α p q) x) y
induction-principle-propositional-truncation :
(l : Level) {l1 l2 : Level} {A : UU l1}
(P : UU-Prop l2) (α : (p q : type-Prop P) → Id p q) (f : A → type-Prop P) →
UU (lsuc l ⊔ l1 ⊔ l2)
induction-principle-propositional-truncation l {l1} {l2} {A} P α f =
( B : type-Prop P → UU l) →
( g : (x : A) → (B (f x))) →
( β : case-paths-induction-principle-propositional-truncation P α f B) →
Σ ((p : type-Prop P) → B p) (λ h → (x : A) → Id (h (f x)) (g x))
-- Lemma 13.2.2
is-prop-case-paths-induction-principle-propositional-truncation :
{ l : Level} {l1 l2 : Level} {A : UU l1}
( P : UU-Prop l2) (α : (p q : type-Prop P) → Id p q) (f : A → type-Prop P) →
( B : type-Prop P → UU l) →
case-paths-induction-principle-propositional-truncation P α f B →
( p : type-Prop P) → is-prop (B p)
is-prop-case-paths-induction-principle-propositional-truncation P α f B β p =
is-prop-is-contr-if-inh (λ x → pair (tr B (α p p) x) (β p p x))
case-paths-induction-principle-propositional-truncation-is-prop :
{ l : Level} {l1 l2 : Level} {A : UU l1}
( P : UU-Prop l2) (α : (p q : type-Prop P) → Id p q) (f : A → type-Prop P) →
( B : type-Prop P → UU l) →
( (p : type-Prop P) → is-prop (B p)) →
case-paths-induction-principle-propositional-truncation P α f B
case-paths-induction-principle-propositional-truncation-is-prop
P α f B is-prop-B p q x y =
is-prop'-is-prop (is-prop-B q) (tr B (α p q) x) y
-- Definition 13.2.3
dependent-universal-property-propositional-truncation :
( l : Level) {l1 l2 : Level} {A : UU l1}
( P : UU-Prop l2) (f : A → type-Prop P) → UU (lsuc l ⊔ l1 ⊔ l2)
dependent-universal-property-propositional-truncation l {l1} {l2} {A} P f =
( Q : type-Prop P → UU-Prop l) → is-equiv (precomp-Π f (type-Prop ∘ Q))
-- Theorem 13.2.4
abstract
dependent-universal-property-is-propositional-truncation :
{ l1 l2 : Level} {A : UU l1} (P : UU-Prop l2) (f : A → type-Prop P) →
( {l : Level} → is-propositional-truncation l P f) →
( {l : Level} → dependent-universal-property-propositional-truncation l P f)
dependent-universal-property-is-propositional-truncation
{l1} {l2} {A} P f is-ptr-f Q =
is-fiberwise-equiv-is-equiv-toto-is-equiv-base-map
( λ (g : A → type-Prop P) → (x : A) → type-Prop (Q (g x)))
( precomp f (type-Prop P))
( λ h → precomp-Π f (λ p → type-Prop (Q (h p))))
( is-ptr-f P)
( is-equiv-top-is-equiv-bottom-square
( inv-choice-∞
{ C = λ (x : type-Prop P) (p : type-Prop P) → type-Prop (Q p)})
( inv-choice-∞
{ C = λ (x : A) (p : type-Prop P) → type-Prop (Q p)})
( toto
( λ (g : A → type-Prop P) → (x : A) → type-Prop (Q (g x)))
( precomp f (type-Prop P))
( λ h → precomp-Π f (λ p → type-Prop (Q (h p)))))
( precomp f (Σ (type-Prop P) (λ p → type-Prop (Q p))))
( ind-Σ (λ h h' → refl))
( is-equiv-inv-choice-∞)
( is-equiv-inv-choice-∞)
( is-ptr-f (Σ-Prop P Q)))
( id {A = type-Prop P})
dependent-universal-property-trunc-Prop :
{l l1 : Level} (A : UU l1) →
dependent-universal-property-propositional-truncation l
( trunc-Prop A)
( unit-trunc-Prop A)
dependent-universal-property-trunc-Prop A =
dependent-universal-property-is-propositional-truncation
( trunc-Prop A)
( unit-trunc-Prop A)
( is-propositional-truncation-trunc-Prop A)
abstract
is-propositional-truncation-dependent-universal-property :
{ l1 l2 : Level} {A : UU l1} (P : UU-Prop l2) (f : A → type-Prop P) →
( {l : Level} →
dependent-universal-property-propositional-truncation l P f) →
( {l : Level} → is-propositional-truncation l P f)
is-propositional-truncation-dependent-universal-property P f dup-f Q =
dup-f (λ p → Q)
abstract
induction-principle-dependent-universal-property-propositional-truncation :
{ l1 l2 : Level} {A : UU l1} (P : UU-Prop l2) (f : A → type-Prop P) →
( {l : Level} →
dependent-universal-property-propositional-truncation l P f) →
( {l : Level} → induction-principle-propositional-truncation l P
( is-prop'-is-prop (is-prop-type-Prop P)) f)
induction-principle-dependent-universal-property-propositional-truncation
P f dup-f B g α =
tot
( λ h → htpy-eq)
( center
( is-contr-map-is-equiv
( dup-f
( λ p →
pair
( B p)
( is-prop-case-paths-induction-principle-propositional-truncation
( P)
( is-prop'-is-prop (is-prop-type-Prop P))
f B α p)))
( g)))
abstract
dependent-universal-property-induction-principle-propositional-truncation :
{ l1 l2 : Level} {A : UU l1} (P : UU-Prop l2) (f : A → type-Prop P) →
( {l : Level} → induction-principle-propositional-truncation l P
( is-prop'-is-prop (is-prop-type-Prop P)) f) →
( {l : Level} → dependent-universal-property-propositional-truncation l P f)
dependent-universal-property-induction-principle-propositional-truncation
P f ind-f Q =
is-equiv-is-prop
( is-prop-Π (λ p → is-prop-type-Prop (Q p)))
( is-prop-Π (λ a → is-prop-type-Prop (Q (f a))))
( λ g →
pr1
( ind-f
( λ p → type-Prop (Q p))
( g)
( case-paths-induction-principle-propositional-truncation-is-prop
( P)
( is-prop'-is-prop (is-prop-type-Prop P))
( f)
( λ p → type-Prop (Q p))
( λ p → is-prop-type-Prop (Q p)))))
--------------------------------------------------------------------------------
{- We introduce the image inclusion of a map. -}
comp-hom-slice :
{l1 l2 l3 l4 : Level} {X : UU l1} {A : UU l2} {B : UU l3} {C : UU l4}
(f : A → X) (g : B → X) (h : C → X) →
hom-slice g h → hom-slice f g → hom-slice f h
comp-hom-slice f g h j i =
pair ( ( map-hom-slice g h j) ∘
( map-hom-slice f g i))
( ( triangle-hom-slice f g i) ∙h
( (triangle-hom-slice g h j) ·r (map-hom-slice f g i)))
id-hom-slice :
{l1 l2 : Level} {X : UU l1} {A : UU l2} (f : A → X) → hom-slice f f
id-hom-slice f = pair id refl-htpy
is-equiv-hom-slice :
{l1 l2 l3 : Level} {X : UU l1} {A : UU l2} {B : UU l3}
(f : A → X) (g : B → X) → hom-slice f g → UU (l2 ⊔ l3)
is-equiv-hom-slice f g h = is-equiv (map-hom-slice f g h)
precomp-emb :
{ l1 l2 l3 l4 : Level} {X : UU l1} {A : UU l2} (f : A → X)
{B : UU l3} ( i : B ↪ X) (q : hom-slice f (map-emb i)) →
{C : UU l4} ( j : C ↪ X) (r : hom-slice (map-emb i) (map-emb j)) →
hom-slice f (map-emb j)
precomp-emb f i q j r =
pair
( ( map-hom-slice (map-emb i) (map-emb j) r) ∘
( map-hom-slice f (map-emb i) q))
( ( triangle-hom-slice f (map-emb i) q) ∙h
( ( triangle-hom-slice (map-emb i) (map-emb j) r) ·r
( map-hom-slice f (map-emb i) q)))
is-prop-hom-slice :
{ l1 l2 l3 : Level} {X : UU l1} {A : UU l2} (f : A → X) →
{ B : UU l3} (i : B ↪ X) → is-prop (hom-slice f (map-emb i))
is-prop-hom-slice {X = X} f i =
is-prop-is-equiv
( (x : X) → fib f x → fib (map-emb i) x)
( fiberwise-hom-hom-slice f (map-emb i))
( is-equiv-fiberwise-hom-hom-slice f (map-emb i))
( is-prop-Π
( λ x → is-prop-Π
( λ p → is-prop-map-is-emb (map-emb i) (is-emb-map-emb i) x)))
universal-property-image :
( l : Level) {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} (f : A → X) →
{ B : UU l3} (i : B ↪ X) (q : hom-slice f (map-emb i)) →
UU (lsuc l ⊔ l1 ⊔ l2 ⊔ l3)
universal-property-image l {X = X} f i q =
( C : UU l) (j : C ↪ X) → is-equiv (precomp-emb f i q j)
universal-property-image' :
( l : Level) {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} (f : A → X) →
{ B : UU l3} (i : B ↪ X) (q : hom-slice f (map-emb i)) →
UU (lsuc l ⊔ l1 ⊔ l2 ⊔ l3)
universal-property-image' l {X = X} f i q =
( C : UU l) (j : C ↪ X) →
hom-slice f (map-emb j) → hom-slice (map-emb i) (map-emb j)
universal-property-image-universal-property-image' :
( l : Level) {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} (f : A → X) →
{ B : UU l3} (i : B ↪ X) (q : hom-slice f (map-emb i)) →
universal-property-image' l f i q → universal-property-image l f i q
universal-property-image-universal-property-image' l f i q up' C j =
is-equiv-is-prop
( is-prop-hom-slice (map-emb i) j)
( is-prop-hom-slice f j)
( up' C j)
{- Remark 14.4.4 -}
universal-property-image-has-section :
(l : Level) {l1 l2 : Level} {X : UU l1} {A : UU l2} (f : A → X) →
sec f → universal-property-image l f emb-id (pair f refl-htpy)
universal-property-image-has-section l f (pair g H) =
universal-property-image-universal-property-image'
l f emb-id (pair f refl-htpy)
( λ B m h → pair ((pr1 h) ∘ g) ( λ x → (inv (H x)) ∙ (pr2 h (g x))))
universal-property-image-is-emb :
(l : Level) {l1 l2 : Level} {X : UU l1} {A : UU l2} (f : A → X) →
(H : is-emb f) → universal-property-image l f (pair f H) (pair id refl-htpy)
universal-property-image-is-emb l f H =
universal-property-image-universal-property-image'
l f (pair f H) (pair id refl-htpy)
( λ B m h → h)
{- The existence of the image -}
im :
{l1 l2 : Level} {X : UU l1} {A : UU l2} (f : A → X) → UU (l1 ⊔ l2)
im {X = X} {A} f = Σ X (λ x → type-trunc-Prop (fib f x))
inclusion-im :
{l1 l2 : Level} {X : UU l1} {A : UU l2} (f : A → X) → im f → X
inclusion-im f = pr1
is-emb-inclusion-im :
{l1 l2 : Level} {X : UU l1} {A : UU l2} (f : A → X) →
is-emb (inclusion-im f)
is-emb-inclusion-im f =
is-emb-pr1-is-subtype (λ x → is-prop-type-trunc-Prop (fib f x))
emb-im :
{l1 l2 : Level} {X : UU l1} {A : UU l2} (f : A → X) → im f ↪ X
emb-im f = pair (inclusion-im f) (is-emb-inclusion-im f)
map-im :
{l1 l2 : Level} {X : UU l1} {A : UU l2} (f : A → X) → A → im f
map-im f a = pair (f a) (unit-trunc-Prop (fib f (f a)) (pair a refl))
triangle-im :
{l1 l2 : Level} {X : UU l1} {A : UU l2} (f : A → X) →
f ~ (inclusion-im f ∘ map-im f)
triangle-im f a = refl
hom-slice-im :
{l1 l2 : Level} {X : UU l1} {A : UU l2} (f : A → X) →
hom-slice f (inclusion-im f)
hom-slice-im f = pair (map-im f) (triangle-im f)
fiberwise-map-universal-property-im :
{l1 l2 l3 : Level} {X : UU l1} {A : UU l2} {B : UU l3} (f : A → X) →
(m : B ↪ X) (h : hom-slice f (map-emb m)) →
(x : X) → type-trunc-Prop (fib f x) → fib (map-emb m) x
fiberwise-map-universal-property-im f m h x =
map-universal-property-trunc-Prop
{ A = (fib f x)}
( fib-prop-emb m x)
( λ t →
pair ( map-hom-slice f (map-emb m) h (pr1 t))
( ( inv (triangle-hom-slice f (map-emb m) h (pr1 t))) ∙
( pr2 t)))
map-universal-property-im :
{l1 l2 l3 : Level} {X : UU l1} {A : UU l2} {B : UU l3} (f : A → X) →
(m : B ↪ X) (h : hom-slice f (map-emb m)) → im f → B
map-universal-property-im f m h (pair x t) =
pr1 (fiberwise-map-universal-property-im f m h x t)
triangle-universal-property-im :
{l1 l2 l3 : Level} {X : UU l1} {A : UU l2} {B : UU l3} (f : A → X) →
(m : B ↪ X) (h : hom-slice f (map-emb m)) →
inclusion-im f ~ ((map-emb m) ∘ (map-universal-property-im f m h))
triangle-universal-property-im f m h (pair x t) =
inv (pr2 (fiberwise-map-universal-property-im f m h x t))
universal-property-im :
(l : Level) {l1 l2 : Level} {X : UU l1} {A : UU l2} (f : A → X) →
universal-property-image l f (emb-im f) (hom-slice-im f)
universal-property-im l f =
universal-property-image-universal-property-image'
l f (emb-im f) (hom-slice-im f)
( λ B m h →
pair ( map-universal-property-im f m h)
( triangle-universal-property-im f m h))
{- The uniqueness of the image -}
is-equiv-hom-slice-emb :
{l1 l2 l3 : Level} {X : UU l1} {A : UU l2} {B : UU l3}
(f : A ↪ X) (g : B ↪ X) (h : hom-slice (map-emb f) (map-emb g)) →
hom-slice (map-emb g) (map-emb f) →
is-equiv-hom-slice (map-emb f) (map-emb g) h
is-equiv-hom-slice-emb f g h i =
is-equiv-has-inverse
( map-hom-slice (map-emb g) (map-emb f) i)
( λ y →
eq-emb g
( inv
( ( triangle-hom-slice
( map-emb g)
( map-emb f)
( i)
( y)) ∙
( triangle-hom-slice
( map-emb f)
( map-emb g)
( h)
( map-hom-slice (map-emb g) (map-emb f) i y)))))
( λ x →
eq-emb f
( inv
( ( triangle-hom-slice (map-emb f) (map-emb g) h x) ∙
( triangle-hom-slice (map-emb g) (map-emb f) i
( map-hom-slice
( map-emb f)
( map-emb g)
( h)
( x))))))
is-equiv-up-image-up-image :
{l1 l2 l3 l4 : Level} {X : UU l1} {A : UU l2} (f : A → X)
{B : UU l3} (i : B ↪ X) (q : hom-slice f (map-emb i))
{B' : UU l4} (i' : B' ↪ X) (q' : hom-slice f (map-emb i'))
(h : hom-slice (map-emb i) (map-emb i'))
(p : Id (comp-hom-slice f (map-emb i) (map-emb i') h q) q') →
({l : Level} → universal-property-image l f i q) →
({l : Level} → universal-property-image l f i' q') →
is-equiv (map-hom-slice (map-emb i) (map-emb i') h)
is-equiv-up-image-up-image f i q i' q' h p up-i up-i' =
is-equiv-hom-slice-emb i i' h (inv-is-equiv (up-i' _ i) q)
up-image-up-image-is-equiv :
{l1 l2 l3 l4 : Level} {X : UU l1} {A : UU l2} (f : A → X)
{B : UU l3} (i : B ↪ X) (q : hom-slice f (map-emb i))
{B' : UU l4} (i' : B' ↪ X) (q' : hom-slice f (map-emb i'))
(h : hom-slice (map-emb i) (map-emb i'))
(p : Id (comp-hom-slice f (map-emb i) (map-emb i') h q) q') →
is-equiv (map-hom-slice (map-emb i) (map-emb i') h) →
({l : Level} → universal-property-image l f i q) →
({l : Level} → universal-property-image l f i' q')
up-image-up-image-is-equiv f i q i' q' h p is-equiv-h up-i {l} =
universal-property-image-universal-property-image' l f i' q'
( λ C j r →
comp-hom-slice
( map-emb i')
( map-emb i)
( map-emb j)
( inv-is-equiv (up-i C j) r)
( pair
( inv-is-equiv is-equiv-h)
( triangle-section
( map-emb i)
( map-emb i')
( map-hom-slice (map-emb i) (map-emb i') h)
( triangle-hom-slice (map-emb i) (map-emb i') h)
( pair ( inv-is-equiv is-equiv-h)
( issec-inv-is-equiv is-equiv-h)))))
up-image-is-equiv-up-image :
{l1 l2 l3 l4 : Level} {X : UU l1} {A : UU l2} (f : A → X)
{B : UU l3} (i : B ↪ X) (q : hom-slice f (map-emb i))
{B' : UU l4} (i' : B' ↪ X) (q' : hom-slice f (map-emb i'))
(h : hom-slice (map-emb i) (map-emb i'))
(p : Id (comp-hom-slice f (map-emb i) (map-emb i') h q) q') →
({l : Level} → universal-property-image l f i' q') →
is-equiv (map-hom-slice (map-emb i) (map-emb i') h) →
({l : Level} → universal-property-image l f i q)
up-image-is-equiv-up-image f i q i' q' h p up-i' is-equiv-h {l} =
universal-property-image-universal-property-image' l f i q
( λ C j r →
comp-hom-slice
( map-emb i)
( map-emb i')
( map-emb j)
( inv-is-equiv (up-i' C j) r)
( h))
--------------------------------------------------------------------------------
{- Existential quantification -}
exists-Prop :
{l1 l2 : Level} {A : UU l1} (P : A → UU-Prop l2) → UU-Prop (l1 ⊔ l2)
exists-Prop {l1} {l2} {A} P = trunc-Prop (Σ A (λ x → type-Prop (P x)))
exists :
{l1 l2 : Level} {A : UU l1} (P : A → UU-Prop l2) → UU (l1 ⊔ l2)
exists P = type-Prop (exists-Prop P)
is-prop-exists :
{l1 l2 : Level} {A : UU l1} (P : A → UU-Prop l2) → is-prop (exists P)
is-prop-exists P = is-prop-type-Prop (exists-Prop P)
--------------------------------------------------------------------------------
{- Surjective maps -}
-- Definition 13.5.1
is-surjective :
{l1 l2 : Level} {A : UU l1} {B : UU l2} → (A → B) → UU (l1 ⊔ l2)
is-surjective {B = B} f = (y : B) → type-trunc-Prop (fib f y)
-- Proposition 13.5.3
dependent-universal-property-surj :
(l : Level) {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) →
UU ((lsuc l) ⊔ l1 ⊔ l2)
dependent-universal-property-surj l {B = B} f =
(P : B → UU-Prop l) →
is-equiv (λ (h : (b : B) → type-Prop (P b)) x → h (f x))
is-surjective-dependent-universal-property-surj :
{l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) →
({l : Level} → dependent-universal-property-surj l f) →
is-surjective f
is-surjective-dependent-universal-property-surj f dup-surj-f =
inv-is-equiv
( dup-surj-f (λ b → trunc-Prop (fib f b)))
( λ x → unit-trunc-Prop (fib f (f x)) (pair x refl))
square-dependent-universal-property-surj :
{l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (f : A → B) →
(P : B → UU-Prop l3) →
( λ (h : (y : B) → type-Prop (P y)) x → h (f x)) ~
( ( λ h x → h (f x) (pair x refl)) ∘
( ( λ h y → (h y) ∘ (unit-trunc-Prop (fib f y))) ∘
( λ h y → const (type-trunc-Prop (fib f y)) (type-Prop (P y)) (h y))))
square-dependent-universal-property-surj f P = refl-htpy
dependent-universal-property-surj-is-surjective :
{l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) →
is-surjective f →
({l : Level} → dependent-universal-property-surj l f)
dependent-universal-property-surj-is-surjective f is-surj-f P =
is-equiv-comp'
( λ h x → h (f x) (pair x refl))
( ( λ h y → (h y) ∘ (unit-trunc-Prop (fib f y))) ∘
( λ h y → const (type-trunc-Prop (fib f y)) (type-Prop (P y)) (h y)))
( is-equiv-comp'
( λ h y → (h y) ∘ (unit-trunc-Prop (fib f y)))
( λ h y → const (type-trunc-Prop (fib f y)) (type-Prop (P y)) (h y))
( is-equiv-postcomp-Π
( λ y p z → p)
( λ y →
is-equiv-diagonal-is-contr
( is-surj-f y)
( is-contr-is-prop-inh
( is-prop-type-trunc-Prop (fib f y))
( is-surj-f y))
( type-Prop (P y))))
( is-equiv-postcomp-Π
( λ b g → g ∘ (unit-trunc-Prop (fib f b)))
( λ b → is-propositional-truncation-trunc-Prop (fib f b) (P b))))
( is-equiv-map-reduce-Π-fib f ( λ y z → type-Prop (P y)))
-- Theorem 13.5.5
--------------------------------------------------------------------------------
{- Cantor's diagonal argument -}
map-cantor :
{l1 l2 : Level} (X : UU l1) (f : X → (X → UU-Prop l2)) → (X → UU-Prop l2)
map-cantor X f x = neg-Prop (f x x)
iff-eq :
{l1 : Level} {P Q : UU-Prop l1} → Id P Q → P ↔ Q
iff-eq refl = pair id id
no-fixed-points-neg-Prop :
{l1 : Level} (P : UU-Prop l1) → ¬ (P ↔ neg-Prop P)
no-fixed-points-neg-Prop P = no-fixed-points-neg (type-Prop P)
not-in-image-map-cantor :
{l1 l2 : Level} (X : UU l1) (f : X → (X → UU-Prop l2)) →
( t : fib f (map-cantor X f)) → empty
not-in-image-map-cantor X f (pair x α) =
no-fixed-points-neg-Prop (f x x) (iff-eq (htpy-eq α x))
cantor : {l1 l2 : Level} (X : UU l1) (f : X → (X → UU-Prop l2)) →
¬ (is-surjective f)
cantor X f H =
( map-universal-property-trunc-Prop empty-Prop
( not-in-image-map-cantor X f)
( H (map-cantor X f)))
--------------------------------------------------------------------------------
-- Exercises
-- Exercise 13.1
is-propositional-truncation-retract :
{l l1 l2 : Level} {A : UU l1} (P : UU-Prop l2) →
(R : (type-Prop P) retract-of A) →
is-propositional-truncation l P (retraction-retract-of R)
is-propositional-truncation-retract {A = A} P R Q =
is-equiv-is-prop
( is-prop-function-type (is-prop-type-Prop Q))
( is-prop-function-type (is-prop-type-Prop Q))
( λ g → g ∘ (section-retract-of R))
-- Exercise 13.2
is-propositional-truncation-prod :
{l1 l2 l3 l4 : Level}
{A : UU l1} (P : UU-Prop l2) (f : A → type-Prop P)
{A' : UU l3} (P' : UU-Prop l4) (f' : A' → type-Prop P') →
({l : Level} → is-propositional-truncation l P f) →
({l : Level} → is-propositional-truncation l P' f') →
{l : Level} → is-propositional-truncation l (prod-Prop P P') (functor-prod f f')
is-propositional-truncation-prod P f P' f' is-ptr-f is-ptr-f' Q =
is-equiv-top-is-equiv-bottom-square
( ev-pair)
( ev-pair)
( precomp (functor-prod f f') (type-Prop Q))
( λ h a a' → h (f a) (f' a'))
( refl-htpy)
( is-equiv-ev-pair)
( is-equiv-ev-pair)
( is-equiv-comp'
( λ h a a' → h a (f' a'))
( λ h a p' → h (f a) p')
( is-ptr-f (pair (type-hom-Prop P' Q) (is-prop-type-hom-Prop P' Q)))
( is-equiv-postcomp-Π
( λ a g a' → g (f' a'))
( λ a → is-ptr-f' Q)))
equiv-prod-trunc-Prop :
{l1 l2 : Level} (A : UU l1) (A' : UU l2) →
equiv-Prop (trunc-Prop (A × A')) (prod-Prop (trunc-Prop A) (trunc-Prop A'))
equiv-prod-trunc-Prop A A' =
pr1
( center
( is-uniquely-unique-propositional-truncation
( trunc-Prop (A × A'))
( prod-Prop (trunc-Prop A) (trunc-Prop A'))
( unit-trunc-Prop (A × A'))
( functor-prod (unit-trunc-Prop A) (unit-trunc-Prop A'))
( is-propositional-truncation-trunc-Prop (A × A'))
( is-propositional-truncation-prod
( trunc-Prop A)
( unit-trunc-Prop A)
( trunc-Prop A')
( unit-trunc-Prop A')
( is-propositional-truncation-trunc-Prop A)
( is-propositional-truncation-trunc-Prop A'))))
-- Exercise 13.3
-- Exercise 13.3(a)
conj-Prop = prod-Prop
type-conj-Prop :
{l1 l2 : Level} → UU-Prop l1 → UU-Prop l2 → UU (l1 ⊔ l2)
type-conj-Prop P Q = type-Prop (conj-Prop P Q)
disj-Prop :
{l1 l2 : Level} → UU-Prop l1 → UU-Prop l2 → UU-Prop (l1 ⊔ l2)
disj-Prop P Q = trunc-Prop (coprod (type-Prop P) (type-Prop Q))
type-disj-Prop :
{l1 l2 : Level} → UU-Prop l1 → UU-Prop l2 → UU (l1 ⊔ l2)
type-disj-Prop P Q = type-Prop (disj-Prop P Q)
inl-disj-Prop :
{l1 l2 : Level} (P : UU-Prop l1) (Q : UU-Prop l2) →
type-hom-Prop P (disj-Prop P Q)
inl-disj-Prop P Q =
(unit-trunc-Prop (coprod (type-Prop P) (type-Prop Q))) ∘ inl
inr-disj-Prop :
{l1 l2 : Level} (P : UU-Prop l1) (Q : UU-Prop l2) →
type-hom-Prop Q (disj-Prop P Q)
inr-disj-Prop P Q =
(unit-trunc-Prop (coprod (type-Prop P) (type-Prop Q))) ∘ inr
-- Exercise 13.3(b)
ev-disj-Prop :
{l1 l2 l3 : Level} (P : UU-Prop l1) (Q : UU-Prop l2) (R : UU-Prop l3) →
type-hom-Prop
( hom-Prop (disj-Prop P Q) R)
( conj-Prop (hom-Prop P R) (hom-Prop Q R))
ev-disj-Prop P Q R h =
pair (h ∘ (inl-disj-Prop P Q)) (h ∘ (inr-disj-Prop P Q))
inv-ev-disj-Prop :
{l1 l2 l3 : Level} (P : UU-Prop l1) (Q : UU-Prop l2) (R : UU-Prop l3) →
type-hom-Prop
( conj-Prop (hom-Prop P R) (hom-Prop Q R))
( hom-Prop (disj-Prop P Q) R)
inv-ev-disj-Prop P Q R (pair f g) =
map-universal-property-trunc-Prop R (ind-coprod (λ t → type-Prop R) f g)
is-equiv-ev-disj-Prop :
{l1 l2 l3 : Level} (P : UU-Prop l1) (Q : UU-Prop l2) (R : UU-Prop l3) →
is-equiv (ev-disj-Prop P Q R)
is-equiv-ev-disj-Prop P Q R =
is-equiv-is-prop
( is-prop-type-Prop (hom-Prop (disj-Prop P Q) R))
( is-prop-type-Prop (conj-Prop (hom-Prop P R) (hom-Prop Q R)))
( inv-ev-disj-Prop P Q R)
-- Exercise 13.9
map-dn-trunc-Prop :
{l : Level} (A : UU l) → ¬¬ (type-trunc-Prop A) → ¬¬ A
map-dn-trunc-Prop A =
dn-extend (map-universal-property-trunc-Prop (dn-Prop A) intro-dn)
inv-map-dn-trunc-Prop :
{l : Level} (A : UU l) → ¬¬ A → ¬¬ (type-trunc-Prop A)
inv-map-dn-trunc-Prop A =
dn-extend (λ a → intro-dn (unit-trunc-Prop A a))
equiv-dn-trunc-Prop :
{l : Level} (A : UU l) → ¬¬ (type-trunc-Prop A) ≃ ¬¬ A
equiv-dn-trunc-Prop A =
equiv-iff
( dn-Prop (type-trunc-Prop A))
( dn-Prop A)
( pair
( map-dn-trunc-Prop A)
( inv-map-dn-trunc-Prop A))
-- Exercise 13.10
-- The impredicative encoding of the propositional truncation --
impredicative-trunc-Prop :
{l : Level} → UU l → UU-Prop (lsuc l)
impredicative-trunc-Prop {l} A =
Π-Prop
( UU-Prop l)
( λ Q → function-Prop (A → type-Prop Q) Q)
type-impredicative-trunc-Prop :
{l : Level} → UU l → UU (lsuc l)
type-impredicative-trunc-Prop {l} A =
type-Prop (impredicative-trunc-Prop A)
map-impredicative-trunc-Prop :
{l : Level} (A : UU l) →
type-trunc-Prop A → type-impredicative-trunc-Prop A
map-impredicative-trunc-Prop {l} A =
map-universal-property-trunc-Prop
( impredicative-trunc-Prop A)
( λ x Q f → f x)
inv-map-impredicative-trunc-Prop :
{l : Level} (A : UU l) →
type-impredicative-trunc-Prop A → type-trunc-Prop A
inv-map-impredicative-trunc-Prop A H =
H (trunc-Prop A) (unit-trunc-Prop A)
equiv-impredicative-trunc-Prop :
{l : Level} (A : UU l) →
type-trunc-Prop A ≃ type-impredicative-trunc-Prop A
equiv-impredicative-trunc-Prop A =
equiv-iff
( trunc-Prop A)
( impredicative-trunc-Prop A)
( pair
( map-impredicative-trunc-Prop A)
( inv-map-impredicative-trunc-Prop A))
-- The impredicative encoding of conjunction --
impredicative-conj-Prop :
{l1 l2 : Level} → UU-Prop l1 → UU-Prop l2 → UU-Prop (lsuc (l1 ⊔ l2))
impredicative-conj-Prop {l1} {l2} P1 P2 =
Π-Prop
( UU-Prop (l1 ⊔ l2))
( λ Q → function-Prop (type-Prop P1 → (type-Prop P2 → type-Prop Q)) Q)
type-impredicative-conj-Prop :
{l1 l2 : Level} → UU-Prop l1 → UU-Prop l2 → UU (lsuc (l1 ⊔ l2))
type-impredicative-conj-Prop P1 P2 =
type-Prop (impredicative-conj-Prop P1 P2)
map-impredicative-conj-Prop :
{l1 l2 : Level} (P1 : UU-Prop l1) (P2 : UU-Prop l2) →
type-conj-Prop P1 P2 → type-impredicative-conj-Prop P1 P2
map-impredicative-conj-Prop {l1} {l2} P1 P2 (pair p1 p2) Q f =
f p1 p2
inv-map-impredicative-conj-Prop :
{l1 l2 : Level} (P1 : UU-Prop l1) (P2 : UU-Prop l2) →
type-impredicative-conj-Prop P1 P2 → type-conj-Prop P1 P2
inv-map-impredicative-conj-Prop P1 P2 H =
H (conj-Prop P1 P2) (λ p1 p2 → pair p1 p2)
equiv-impredicative-conj-Prop :
{l1 l2 : Level} (P1 : UU-Prop l1) (P2 : UU-Prop l2) →
type-conj-Prop P1 P2 ≃ type-impredicative-conj-Prop P1 P2
equiv-impredicative-conj-Prop P1 P2 =
equiv-iff
( conj-Prop P1 P2)
( impredicative-conj-Prop P1 P2)
( pair
( map-impredicative-conj-Prop P1 P2)
( inv-map-impredicative-conj-Prop P1 P2))
-- The impredicative encoding of disjunction --
impredicative-disj-Prop :
{l1 l2 : Level} → UU-Prop l1 → UU-Prop l2 → UU-Prop (lsuc (l1 ⊔ l2))
impredicative-disj-Prop {l1} {l2} P1 P2 =
Π-Prop
( UU-Prop (l1 ⊔ l2))
( λ Q →
function-Prop
( type-implication-Prop P1 Q)
( function-Prop (type-implication-Prop P2 Q) Q))
type-impredicative-disj-Prop :
{l1 l2 : Level} → UU-Prop l1 → UU-Prop l2 → UU (lsuc (l1 ⊔ l2))
type-impredicative-disj-Prop P1 P2 =
type-Prop (impredicative-disj-Prop P1 P2)
map-impredicative-disj-Prop :
{l1 l2 : Level} (P1 : UU-Prop l1) (P2 : UU-Prop l2) →
type-disj-Prop P1 P2 → type-impredicative-disj-Prop P1 P2
map-impredicative-disj-Prop {l1} {l2} P1 P2 =
map-universal-property-trunc-Prop
( impredicative-disj-Prop P1 P2)
( ind-coprod
( λ x → type-impredicative-disj-Prop P1 P2)
( λ x Q f1 f2 → f1 x)
( λ y Q f1 f2 → f2 y))
inv-map-impredicative-disj-Prop :
{l1 l2 : Level} (P1 : UU-Prop l1) (P2 : UU-Prop l2) →
type-impredicative-disj-Prop P1 P2 → type-disj-Prop P1 P2
inv-map-impredicative-disj-Prop P1 P2 H =
H (disj-Prop P1 P2) (inl-disj-Prop P1 P2) (inr-disj-Prop P1 P2)
equiv-impredicative-disj-Prop :
{l1 l2 : Level} (P1 : UU-Prop l1) (P2 : UU-Prop l2) →
type-disj-Prop P1 P2 ≃ type-impredicative-disj-Prop P1 P2
equiv-impredicative-disj-Prop P1 P2 =
equiv-iff
( disj-Prop P1 P2)
( impredicative-disj-Prop P1 P2)
( pair
( map-impredicative-disj-Prop P1 P2)
( inv-map-impredicative-disj-Prop P1 P2))
-- The impredicative encoding of negation --
impredicative-neg-Prop :
{l : Level} → UU l → UU-Prop (lsuc l)
impredicative-neg-Prop {l} A =
Π-Prop (UU-Prop l) (λ Q → function-Prop A Q)
type-impredicative-neg-Prop :
{l : Level} → UU l → UU (lsuc l)
type-impredicative-neg-Prop A =
type-Prop (impredicative-neg-Prop A)
map-impredicative-neg-Prop :
{l : Level} (A : UU l) →
¬ A → type-impredicative-neg-Prop A
map-impredicative-neg-Prop A f Q a = ex-falso (f a)
inv-map-impredicative-neg-Prop :
{l : Level} (A : UU l) →
type-impredicative-neg-Prop A → ¬ A
inv-map-impredicative-neg-Prop A H a = H (neg-Prop' A) a a
equiv-impredicative-neg-Prop :
{l : Level} (A : UU l) →
¬ A ≃ type-impredicative-neg-Prop A
equiv-impredicative-neg-Prop A =
equiv-iff
( neg-Prop' A)
( impredicative-neg-Prop A)
( pair
( map-impredicative-neg-Prop A)
( inv-map-impredicative-neg-Prop A))
-- The impredicative encoding of existential quantification --
impredicative-exists-Prop :
{l1 l2 : Level} {A : UU l1} (P : A → UU-Prop l2) → UU-Prop (lsuc (l1 ⊔ l2))
impredicative-exists-Prop {l1} {l2} {A} P =
Π-Prop
( UU-Prop (l1 ⊔ l2))
( λ Q → function-Prop ((x : A) → type-Prop (P x) → type-Prop Q) Q)
type-impredicative-exists-Prop :
{l1 l2 : Level} {A : UU l1} (P : A → UU-Prop l2) → UU (lsuc (l1 ⊔ l2))
type-impredicative-exists-Prop P =
type-Prop (impredicative-exists-Prop P)
map-impredicative-exists-Prop :
{l1 l2 : Level} {A : UU l1} (P : A → UU-Prop l2) →
exists P → type-impredicative-exists-Prop P
map-impredicative-exists-Prop {l1} {l2} {A} P =
map-universal-property-trunc-Prop
( impredicative-exists-Prop P)
( ind-Σ (λ x y Q h → h x y))
inv-map-impredicative-exists-Prop :
{l1 l2 : Level} {A : UU l1} (P : A → UU-Prop l2) →
type-impredicative-exists-Prop P → exists P
inv-map-impredicative-exists-Prop {A = A} P H =
H ( exists-Prop P)
( λ x y → unit-trunc-Prop (Σ A (λ x → type-Prop (P x))) (pair x y))
equiv-impredicative-exists-Prop :
{l1 l2 : Level} {A : UU l1} (P : A → UU-Prop l2) →
exists P ≃ type-impredicative-exists-Prop P
equiv-impredicative-exists-Prop P =
equiv-iff
( exists-Prop P)
( impredicative-exists-Prop P)
( pair
( map-impredicative-exists-Prop P)
( inv-map-impredicative-exists-Prop P))
-- The impredicative encoding of the based identity type of a set --
impredicative-based-id-Prop :
{l : Level} (A : UU-Set l) (a x : type-Set A) → UU-Prop (lsuc l)
impredicative-based-id-Prop {l} A a x =
Π-Prop (type-Set A → UU-Prop l) (λ Q → hom-Prop (Q a) (Q x))
type-impredicative-based-id-Prop :
{l : Level} (A : UU-Set l) (a x : type-Set A) → UU (lsuc l)
type-impredicative-based-id-Prop A a x =
type-Prop (impredicative-based-id-Prop A a x)
map-impredicative-based-id-Prop :
{l : Level} (A : UU-Set l) (a x : type-Set A) →
Id a x → type-impredicative-based-id-Prop A a x
map-impredicative-based-id-Prop A a .a refl Q p = p
inv-map-impredicative-based-id-Prop :
{l : Level} (A : UU-Set l) (a x : type-Set A) →
type-impredicative-based-id-Prop A a x → Id a x
inv-map-impredicative-based-id-Prop A a x H =
H (λ x → pair (Id a x) (is-set-type-Set A a x)) refl
equiv-impredicative-based-id-Prop :
{l : Level} (A : UU-Set l) (a x : type-Set A) →
Id a x ≃ type-impredicative-based-id-Prop A a x
equiv-impredicative-based-id-Prop A a x =
equiv-iff
( pair (Id a x) (is-set-type-Set A a x))
( impredicative-based-id-Prop A a x)
( pair
( map-impredicative-based-id-Prop A a x)
( inv-map-impredicative-based-id-Prop A a x))
-- The impredicative encoding of Martin-Löf's identity type of a set --
impredicative-id-Prop :
{l : Level} (A : UU-Set l) (x y : type-Set A) → UU-Prop (lsuc l)
impredicative-id-Prop {l} A x y =
Π-Prop (type-Set A → type-Set A → UU-Prop l)
(λ Q → function-Prop ((a : type-Set A) → type-Prop (Q a a)) (Q x y))
type-impredicative-id-Prop :
{l : Level} (A : UU-Set l) (x y : type-Set A) → UU (lsuc l)
type-impredicative-id-Prop A x y =
type-Prop (impredicative-id-Prop A x y)
map-impredicative-id-Prop :
{l : Level} (A : UU-Set l) (x y : type-Set A) →
Id x y → type-impredicative-id-Prop A x y
map-impredicative-id-Prop A x .x refl Q r = r x
inv-map-impredicative-id-Prop :
{l : Level} (A : UU-Set l ) (x y : type-Set A) →
type-impredicative-id-Prop A x y → Id x y
inv-map-impredicative-id-Prop A x y H =
H (λ a b → pair (Id a b) (is-set-type-Set A a b)) (λ a → refl)
equiv-impredicative-id-Prop :
{l : Level} (A : UU-Set l) (x y : type-Set A) →
Id x y ≃ type-impredicative-id-Prop A x y
equiv-impredicative-id-Prop A x y =
equiv-iff
( pair (Id x y) (is-set-type-Set A x y))
( impredicative-id-Prop A x y)
( pair
( map-impredicative-id-Prop A x y)
( inv-map-impredicative-id-Prop A x y))
-- The impredicative encoding of the based identity type of a 1-type --
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Categories.Structures where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.HLevels using (isSetΣ)
open import Cubical.Data.Sigma
open import Cubical.Categories.Category
_∋_ : ∀ {ℓ} → (A : Type ℓ) → A → A
_∋_ A x = x
private
variable
ℓ ℓ' ℓ'' ℓ''' : Level
𝒞 : Precategory ℓ ℓ'
record StructureOver (𝒞 : Precategory ℓ ℓ') ℓ'' ℓ''' : Type (ℓ-suc (ℓ-max (ℓ-max ℓ ℓ') (ℓ-max ℓ'' ℓ'''))) where
field
Struct : 𝒞 .ob → Type ℓ''
IsHomo : ∀ {x y} → Struct x → Struct y → 𝒞 .hom x y → Type ℓ'''
isPropIsHomo : ∀ {x y α β f} → isProp (IsHomo {x} {y} α β f)
idnIsHomo : ∀ {x} (α : Struct x) → IsHomo α α (𝒞 .idn x)
homoSeq : ∀ {x y z α β γ} {f : 𝒞 .hom x y} {g : 𝒞 .hom y z} →
IsHomo α β f → IsHomo β γ g → IsHomo α γ (𝒞 .seq f g)
_⊆_ : ∀ {x} → Struct x → Struct x → Type ℓ'''
_⊆_ {x} α β = IsHomo α β (𝒞 .idn x)
open StructureOver public renaming (_⊆_ to _[_⊆_])
record StrIsUnivalent {𝒞 : Precategory ℓ ℓ'} (S : StructureOver 𝒞 ℓ'' ℓ''') : Type (ℓ-suc (ℓ-max (ℓ-max ℓ ℓ') (ℓ-max ℓ'' ℓ'''))) where
field
subAntisym : ∀ {x} {α β : S .Struct x} → S [ α ⊆ β ] → S [ β ⊆ α ] → α ≡ β
open StrIsUnivalent public
isPropP : {A : I → Type ℓ} {x : A i0} {y : A i1} → isProp (A i1) → PathP A x y
isPropP Aprop = toPathP (Aprop _ _)
STRUCTURES : (𝒞 : Precategory ℓ ℓ') → StructureOver 𝒞 ℓ'' ℓ''' → Precategory (ℓ-max ℓ ℓ'') (ℓ-max ℓ' ℓ''')
STRUCTURES 𝒞 S .ob = Σ (𝒞 .ob) (S .Struct)
STRUCTURES 𝒞 S .hom (x , α) (y , β) = Σ (𝒞 .hom x y) (S .IsHomo α β)
STRUCTURES 𝒞 S .idn (x , α) = 𝒞 .idn x , S .idnIsHomo α
STRUCTURES 𝒞 S .seq (f , hᶠ) (g , hᵍ) = 𝒞 .seq f g , S .homoSeq hᶠ hᵍ
STRUCTURES 𝒞 S .seq-λ (f , hᶠ) = cong₂ _,_ (𝒞 .seq-λ f) (toPathP (S .isPropIsHomo _ _))
STRUCTURES 𝒞 S .seq-ρ (f , hᶠ) = cong₂ _,_ (𝒞 .seq-ρ f) (toPathP (S .isPropIsHomo _ _))
STRUCTURES 𝒞 S .seq-α (f , hᶠ) (g , hᵍ) (h , hʰ) = cong₂ _,_ (𝒞 .seq-α f g h) (toPathP (S .isPropIsHomo _ _))
STRUCTURESisCat : ⦃ 𝒞-cat : isCategory 𝒞 ⦄ {S : StructureOver 𝒞 ℓ'' ℓ'''} → isCategory (STRUCTURES 𝒞 S)
STRUCTURESisCat {{𝒞-cat}} {S = S} .homIsSet {x = x , α} {y = y , β} = isSetΣ (𝒞-cat .homIsSet) (λ _ → isProp→isSet (S .isPropIsHomo))
SIP : ∀ ⦃ 𝒞-cat : isCategory 𝒞 ⦄ ⦃ 𝒞-univ : isUnivalent 𝒞 ⦄ → (S : StructureOver 𝒞 ℓ'' ℓ''') → ⦃ S-univ : StrIsUnivalent S ⦄ → isUnivalent (STRUCTURES 𝒞 S)
SIP {𝒞 = 𝒞} {{𝒞-cat}} {{𝒞-univ}} S {{S-univ}} .univ (x , α) (y , β) = isoToIsEquiv (iso (pathToIso (x , α) (y , β)) catisoToPath sect ret)
where
catisoToPath : ∀ {x y : 𝒞 .ob} {α : S .Struct x} {β : S .Struct y} → CatIso (STRUCTURES 𝒞 S) (x , α) (y , β) → (x , α) ≡ (y , β)
catisoToPath {x = x} {y} {α} {β} (catiso (f , hᶠ) (g , hᵍ) sec ret) = ΣPathTransport→PathΣ _ _ (x≡y , α≡β x≡y α β (subst (S .IsHomo α β) f≡h hᶠ) (subst (S .IsHomo β α) g≡h⁻¹ hᵍ))
where
x≅y : CatIso 𝒞 x y
x≅y = catiso f g (cong fst sec) (cong fst ret)
x≡y : x ≡ y
x≡y = uva 𝒞-univ x≅y
f≡h : f ≡ pathMor 𝒞 x≡y
f≡h = sym (uvaPathMor 𝒞-univ x≅y)
g≡h⁻¹ : g ≡ pathMor⁻ 𝒞 x≡y
g≡h⁻¹ = sym (uvaPathMor⁻ 𝒞-univ x≅y)
α≡β : ∀ {x y : 𝒞 .ob} (p : x ≡ y)
(α : S .Struct x) (β : S .Struct y)
(h⇒ : S .IsHomo α β (pathMor 𝒞 p)) (h⇐ : S .IsHomo β α (pathMor⁻ 𝒞 p))
→ subst (S .Struct) p α ≡ β
α≡β {x} {y} =
J (λ z q → ∀ (α : S .Struct x) (β : S .Struct z) (h⇒ : S .IsHomo α β (pathMor 𝒞 q)) (h⇐ : S .IsHomo β α (pathMor⁻ 𝒞 q)) → subst (S .Struct) q α ≡ β)
(λ α β h⇒ h⇐ → transportRefl α ∙ subAntisym S-univ (subst (S .IsHomo α β) (pathMorRefl {𝒞 = 𝒞}) h⇒) (subst (S .IsHomo β α) (pathMor⁻Refl {𝒞 = 𝒞}) h⇐))
sect : section (pathToIso (x , α) (y , β)) catisoToPath
sect (catiso h h⁻¹ sec ret) = {! !}
ret : retract (pathToIso (x , α) (y , β)) catisoToPath
ret = J (λ z q → catisoToPath (pathToIso (x , α) z q) ≡ q)
(subst {y = (pathToIso (x , α) (x , α) refl)} (λ x → catisoToPath x ≡ refl) (sym (JRefl (λ z _ → (x , α) ≅ z) ?)))
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------------------------------------------------------------------------
-- The Agda standard library
--
-- The reader monad
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Level
module Category.Monad.Reader {r} (R : Set r) (a : Level) where
open import Function
open import Function.Identity.Categorical as Id using (Identity)
open import Category.Applicative.Indexed
open import Category.Monad.Indexed
open import Category.Monad
open import Data.Unit
private
variable
ℓ : Level
A B I : Set ℓ
------------------------------------------------------------------------
-- Indexed reader
IReaderT : IFun I (r ⊔ a) → IFun I (r ⊔ a)
IReaderT M i j A = R → M i j A
module _ {M : IFun I (r ⊔ a)} where
------------------------------------------------------------------------
-- Indexed reader applicative
ReaderTIApplicative : RawIApplicative M → RawIApplicative (IReaderT M)
ReaderTIApplicative App = record
{ pure = λ x r → pure x
; _⊛_ = λ m n r → m r ⊛ n r
} where open RawIApplicative App
ReaderTIApplicativeZero : RawIApplicativeZero M →
RawIApplicativeZero (IReaderT M)
ReaderTIApplicativeZero App = record
{ applicative = ReaderTIApplicative applicative
; ∅ = const ∅
} where open RawIApplicativeZero App
ReaderTIAlternative : RawIAlternative M → RawIAlternative (IReaderT M)
ReaderTIAlternative Alt = record
{ applicativeZero = ReaderTIApplicativeZero applicativeZero
; _∣_ = λ m n r → m r ∣ n r
} where open RawIAlternative Alt
------------------------------------------------------------------------
-- Indexed reader monad
ReaderTIMonad : RawIMonad M → RawIMonad (IReaderT M)
ReaderTIMonad Mon = record
{ return = λ x r → return x
; _>>=_ = λ m f r → m r >>= flip f r
} where open RawIMonad Mon
ReaderTIMonadZero : RawIMonadZero M → RawIMonadZero (IReaderT M)
ReaderTIMonadZero Mon = record
{ monad = ReaderTIMonad monad
; applicativeZero = ReaderTIApplicativeZero applicativeZero
} where open RawIMonadZero Mon
ReaderTIMonadPlus : RawIMonadPlus M → RawIMonadPlus (IReaderT M)
ReaderTIMonadPlus Mon = record
{ monad = ReaderTIMonad monad
; alternative = ReaderTIAlternative alternative
} where open RawIMonadPlus Mon
------------------------------------------------------------------------
-- Reader monad operations
record RawIMonadReader {I : Set ℓ} (M : IFun I (r ⊔ a))
: Set (ℓ ⊔ suc (r ⊔ a)) where
field
monad : RawIMonad M
reader : ∀ {i} → (R → A) → M i i A
local : ∀ {i j} → (R → R) → M i j A → M i j A
open RawIMonad monad public
ask : ∀ {i} → M i i (Lift (r ⊔ a) R)
ask = reader lift
asks : ∀ {i} → (R → A) → M i i A
asks = reader
ReaderTIMonadReader : {I : Set ℓ} {M : IFun I (r ⊔ a)} →
RawIMonad M → RawIMonadReader (IReaderT M)
ReaderTIMonadReader Mon = record
{ monad = ReaderTIMonad Mon
; reader = λ f r → return (f r)
; local = λ f m → m ∘ f
} where open RawIMonad Mon
------------------------------------------------------------------------
-- Ordinary reader monads
RawMonadReader : (M : Set (r ⊔ a) → Set (r ⊔ a)) → Set _
RawMonadReader M = RawIMonadReader {I = ⊤} (λ _ _ → M)
module RawMonadReader {M} (Mon : RawMonadReader M) where
open RawIMonadReader Mon public
ReaderT : (M : Set (r ⊔ a) → Set (r ⊔ a)) → Set _ → Set _
ReaderT M = IReaderT {I = ⊤} (λ _ _ → M) _ _
ReaderTMonad : ∀ {M} → RawMonad M → RawMonad (ReaderT M)
ReaderTMonad = ReaderTIMonad
ReaderTMonadReader : ∀ {M} → RawMonad M → RawMonadReader (ReaderT M)
ReaderTMonadReader = ReaderTIMonadReader
ReaderTMonadZero : ∀ {M} → RawMonadZero M → RawMonadZero (ReaderT M)
ReaderTMonadZero = ReaderTIMonadZero
ReaderTMonadPlus : ∀ {M} → RawMonadPlus M → RawMonadPlus (ReaderT M)
ReaderTMonadPlus = ReaderTIMonadPlus
Reader : Set (r ⊔ a) → Set (r ⊔ a)
Reader = ReaderT Identity
ReaderMonad : RawMonad Reader
ReaderMonad = ReaderTIMonad Id.monad
ReaderMonadReader : RawMonadReader Reader
ReaderMonadReader = ReaderTIMonadReader Id.monad
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module Data.Real.Diff where
open import Level using (0ℓ)
import Data.Real as ℝ
open ℝ using (ℝ)
open import Data.Real.Properties
import Data.Nat as ℕ
open ℕ using (ℕ; suc; zero; _⊓_; _⊔_)
open import Data.Unit.Polymorphic using (tt; ⊤)
open import Data.Product using (_×_; _,_; proj₁; proj₂)
open import Function using (_∘_; id)
-- TODO should move to _≈_ at some point
open import Relation.Binary.PropositionalEquality hiding ([_])
open ≡-Reasoning
import Data.Vec as V
open V using (Vec; []; _∷_)
import Data.Vec.Recursive as VR
open VR using (_^_; 2+_)
open import Data.Fin using (Fin; zero; suc)
Tower : ℕ → Set
Tower = Vec ℝ
Diff : Set
Diff = ∀ {d n} → Tower d ^ n → Tower d ^ n
Diff2 : Set
Diff2 = ∀ {d n} → Tower d ^ n → Tower d ^ n → Tower d ^ n
-- utility function
infix 2 _!_
_!_ : ∀ {n} → (∀ {d'} → Tower d' ^ n) → ∀ d → Tower d ^ n
x ! d = x {d}
lift : ∀ {n a} {A : Set a} → A → A ^ n
lift {n} x = VR.replicate n x
const' : ∀ {d} → ℝ → Tower d
const' {zero} x = []
const' {suc d} x = x ∷ (const' 0.0)
const : ∀ {n d} → ℝ ^ n → Tower d ^ n
const {n} = VR.map const' n
return' : ∀ {d} → ℝ → Tower (suc d)
return' x = x ∷ const' 1.0
return : ∀ {n d} → ℝ ^ n → Tower (suc d) ^ n
return {n} = VR.map return' n
extract : ∀ {d n} → Tower (suc d) ^ n → ℝ ^ n
extract {n = n} = VR.map V.head n
lop : ∀ {d} → Tower (suc d) → Tower d
lop {zero} _ = []
lop {suc d} (x ∷ xs) = x ∷ lop xs
run : ∀ {m n} (f : Tower 1 ^ m → Tower 1 ^ n) (x : ℝ ^ m) → ℝ ^ n
run f = extract ∘ f ∘ const
-_ : Diff
-_ {n = n} = VR.map (V.map λ x → ℝ.- x) n
infixl 6 _+_ _-_
infixr 9 -_
_+_ _-_ : Diff2
_+_ {n = n} = VR.zipWith (V.zipWith ℝ._+_) n
x - y = x + (- y)
*T : ∀ {d} (x y : Tower d) → Tower d
*T [] _ = []
*T xx@(x ∷ xs) yy@(y ∷ ys) = x ℝ.* y ∷ *T (lop xx) ys + *T (lop yy) xs
infixl 7 _*_
_*_ : Diff2
_*_ {zero} {n} x _ = x
_*_ {suc d} {n} = VR.zipWith *T n
-- directional derivative
du
: ∀ {m n} (f : Tower 2 ^ m → Tower 2 ^ n)
→ Tower 2 ^ m → Tower 1 ^ n
du {n = n} f xs = VR.map V.tail n (f xs)
fins : ∀ n → Fin n ^ n
fins n = VR.tabulate n id
directions : ∀ {d n} → ℝ ^ n → (Tower (suc d) ^ n) ^ n
directions {d} {n} x = VR.map (λ i → go i x) n (fins n)
where
go : ∀ {m} → Fin m → ℝ ^ m → Tower (suc d) ^ m
go {1} zero y = return y
go {2+ m} zero (y , ys) = return y , const ys
go {2+ m} (suc i) (y , ys) = const y , go i ys
outerWith
: ∀ {a b c} {A : Set a} {B : Set b} {C : Set c}
→ (f : A → B → C) → ∀ m n → A ^ m → B ^ n → (C ^ n) ^ m
outerWith f m n rm rn = VR.map (λ x → VR.map (f x) n rn) m rm
-- this is all very likely very slow
-- it's running `du` `m` times...
jacobian grad : ∀ {m n} (f : Tower 2 ^ m → Tower 2 ^ n) → ℝ ^ m → (ℝ ^ n) ^ m
jacobian {m = m} f x = VR.map (extract ∘ du f) m (directions x)
grad = jacobian
-- TODO
-- The second derivative cross terms are not currently clear to me.
-- directions2d : ∀ {d n} → ℝ ^ n → ((Tower (suc d) ^ n) ^ n) ^ n
-- directions2d {d} {n} x = rotations n dirs
-- -- VR.zipWith (λ y ys → {! !}) n (dirs) (rotations dirs)
-- where
-- dirs : (Tower (suc d) ^ n) ^ n
-- dirs = directions x
-- rotations : ∀ m → (Tower (suc d) ^ m) ^ m → ((Tower (suc d) ^ m) ^ m) ^ m
-- rotations = {! !}
-- d2u
-- : ∀ {m n} (f : Tower 3 ^ m → Tower 3 ^ n)
-- → Tower 3 ^ m → Tower 1 ^ n
-- d2u {n = n} f xs = VR.map (V.tail ∘ V.tail) n (f xs)
-- hessian
-- : ∀ {m n} (f : Tower 3 ^ m → Tower 3 ^ n)
-- → ℝ ^ m → ((ℝ ^ n) ^ m) ^ m
-- hessian {m} {n} f x =
-- VR.map (λ dir → VR.map (extract ∘ d2u f) m dir) m (directions2d x)
-- outerWith (λ y z → extract (du (du f) y)) m m (directions x) (directions x)
_>-<_ : (ℝ → ℝ) → (∀ {d'} → Tower d' → Tower d') → ∀ {d} → Tower d → Tower d
(f >-< g) {zero} [] = []
(f >-< g) {suc d} xx@(x ∷ xs) = f x ∷ xs * g (lop xx)
liftF : ∀ {d} (f : Tower d → Tower d) → ∀ {n} → Tower d ^ n → Tower d ^ n
liftF f {n} = VR.map f n
infixl 8 _^^_
_^^_ : ∀ {d n} → Tower d ^ n → (m : ℕ) → Tower d ^ n
x ^^ zero = lift (const 1.0)
x ^^ (suc d) = x * x ^^ d
module Single where
infixr 9 e^_
e^_ log recip sin cos sinh cosh abs sgn : ∀ {d} → Tower d → Tower d
e^ [] = []
e^_ {suc d} xx@(x ∷ xs) = ℝ.e^ x ∷ xs * (e^ lop xx)
log = ℝ.log >-< recip
recip [] = []
recip xx@(x ∷ xs) = 1/x ∷ 1/xx
where
1/x = 1.0 ℝ.÷ x
1/xx = recip (lop xx)
abs = ℝ.abs >-< sgn
sgn [] = []
sgn (x ∷ xs) = if does (0.0 ≤? x) then const 1.0 else const (ℝ.- 1.0)
where
open import Data.Bool using (if_then_else_)
open import Data.Real.Order
open import Relation.Nullary
-- I'm not sure why I have to write these by hand.
sin [] = []
sin xx@(x ∷ xs) = ℝ.sin x ∷ xs * cos (lop xx)
cos [] = []
cos xx@(x ∷ xs) = ℝ.cos x ∷ - xs * sin (lop xx)
sinh [] = []
sinh xx@(x ∷ xs) = ℝ.sinh x ∷ xs * cosh (lop xx)
cosh [] = []
cosh xx@(x ∷ xs) = ℝ.cosh x ∷ xs * sinh (lop xx)
infixr 9 e^_
e^_ log recip sin cos sinh cosh abs sgn : Diff
e^_ = liftF Single.e^_
log = liftF Single.log
recip = liftF Single.recip
abs = liftF Single.abs
sgn = liftF Single.sgn
sin = liftF Single.sin
cos = liftF Single.cos
sinh = liftF Single.sinh
cosh = liftF Single.cosh
infix 8 _**_
_**_ : Diff2
x ** y = e^ (y * log x)
ascend descend
: ∀ {n} (f : Tower 2 ^ n → Tower 2) (δ : ℝ ^ n) (m : ℕ) (x : ℝ ^ n) → ℝ ^ n
ascend f δ zero x = x
ascend {n = n} f δ (suc m) x = ascend f δ m (add x (mul δ (grad f x)))
where
add mul : (x y : ℝ ^ n) → ℝ ^ n
add = VR.zipWith ℝ._+_ n
mul = VR.zipWith ℝ._*_ n
descend f = ascend λ x → - f x
ascend_f=_δ=_steps=_start=_
: ∀ {n} (_ : ⊤ {0ℓ}) (f : Tower 2 ^ n → Tower 2) (δ : ℝ ^ n) (m : ℕ) (x : ℝ ^ n) → ℝ ^ n
ascend_f=_δ=_steps=_start=_ _ = ascend
∶ : ⊤ {0ℓ}
∶ = _
sterling : ℕ → ℝ
sterling n = n' ℝ.* ℝ.log n' ℝ.- n'
where
n' = ℝ.fromℕ n
logPoisson' logPoisson : ∀ {n} → ℕ ^ n → ∀ {d} → Tower d ^ n → Tower d ^ n
-- neglecting the normalization term
logPoisson' {n} k α = const k' * log α - α
where
k' = VR.map ℝ.fromℕ n k
logPoisson {n} k α = logPoisson' k α - const (VR.map sterling n k)
sum : ∀ {d n} → Tower d ^ n → Tower d
sum {d} {n} = VR.foldl (λ _ → Tower d) (const 0.0) id (λ _ x y → x + y) n
where open import Function using (id)
binned : ∀ {d n} → ℕ ^ n → Tower d ^ n → Tower d
binned n = sum ∘ logPoisson' n
test : ∀ {n} → ℝ ^ n → ℝ ^ n
test x =
ascend ∶
f= binned (lift 10)
δ= lift 1.0
steps= 1000
start= x
testgrad : ∀ {n} → ℝ ^ n → ℝ ^ n
testgrad = grad (binned (lift 10))
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------------------------------------------------------------------------
-- INCREMENTAL λ-CALCULUS
--
-- Sets of variables
------------------------------------------------------------------------
module Base.Syntax.Vars
(Type : Set)
where
-- The notion of sets of variables
--
-- This module is calculus-independent.
open import Base.Syntax.Context Type
open import Relation.Binary.PropositionalEquality
open import Data.Unit
open import Data.Sum
open import Data.Bool
-- Sets of variables
open import Base.Data.DependentList
Free : Type → Set
Free _ = Bool
Vars : Context → Set
Vars = DependentList Free
none : {Γ : Context} → Vars Γ
none = tabulate (λ _ → false)
singleton : ∀ {τ Γ} → Var Γ τ → Vars Γ
singleton {Γ = τ • Γ₀} this = true • none
singleton (that x) = false • singleton x
-- Union of variable sets
infixl 6 _∪_ -- just like _+_
_∪_ : ∀ {Γ} → Vars Γ → Vars Γ → Vars Γ
_∪_ = zipWith _∨_
-- Test if a set of variables is empty
empty? : ∀ {Γ} → (vs : Vars Γ) → (vs ≡ none) ⊎ ⊤
empty? ∅ = inj₁ refl
empty? (true • vs) = inj₂ tt
empty? (false • vs) with empty? vs
... | inj₁ vs=∅ = inj₁ (cong₂ _•_ refl vs=∅)
... | inj₂ _ = inj₂ tt
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{-# OPTIONS --cubical --safe #-}
module Inspect where
open import Level
open import Path
record Reveal_·_is_ {A : Type a} {B : A → Type b} (f : (x : A) → B x) (x : A) (y : B x) : Type b where
constructor 〖_〗
field eq : f x ≡ y
inspect : {A : Type a} {B : A → Type b} (f : (x : A) → B x) (x : A) → Reveal f · x is f x
inspect f x = 〖 refl 〗
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open import Prelude
module Implicits.Resolution.Deterministic.Expressiveness where
open import Data.Fin.Substitution
open import Implicits.Syntax
open import Implicits.Syntax.Type.Unification
open import Implicits.Resolution.Ambiguous.Resolution as A
open import Implicits.Resolution.Deterministic.Resolution as D
open import Extensions.ListFirst
module Deterministic⊆Ambiguous where
open FirstLemmas
open import Relation.Unary
soundness : ∀ {ν} {Δ : ICtx ν} {r} → Δ D.⊢ᵣ r → Δ A.⊢ᵣ r
soundness (r-simp r x) = lem x (r-ivar (proj₁ $ first⟶∈ r))
where
lem : ∀ {ν} {a τ} {Δ : ICtx ν} → Δ ⊢ a ↓ τ → Δ A.⊢ᵣ a → Δ A.⊢ᵣ simpl τ
lem (i-simp τ) hyp = hyp
lem (i-iabs ih₁ ih₂) hyp = lem ih₂ (r-iapp hyp (soundness ih₁))
lem (i-tabs b ih) hyp = lem ih (r-tapp b hyp)
soundness (r-iabs _ ih) = r-iabs (soundness ih)
soundness (r-tabs ih) = r-tabs (soundness ih)
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{-# OPTIONS --universe-polymorphism #-}
open import Common.Prelude
renaming (Nat to ℕ; module Nat to ℕ)
using (zero; suc; _+_; _∸_; List; []; _∷_; Bool; true; false)
open import Common.Level
open import Common.Reflect
module TermSplicing where
module Library where
data Box {a} (A : Set a) : Set a where
box : A → Box A
record ⊤ : Set where
constructor tt
infixr 5 _×_
record _×_ (A B : Set) : Set where
constructor _,_
field
proj₁ : A
proj₂ : B
[_] : ∀ {A : Set} → A → List A
[ x ] = x ∷ []
replicate : ∀ {A : Set} → ℕ → A → List A
replicate zero x = []
replicate (suc n) x = x ∷ replicate n x
foldr : ∀ {A B : Set} → (A → B → B) → B → List A → B
foldr c n [] = n
foldr c n (x ∷ xs) = c x (foldr c n xs)
foldl : ∀ {A B : Set} → (A → B → A) → A → List B → A
foldl c n [] = n
foldl c n (x ∷ xs) = foldl c (c n x) xs
reverse : ∀ {A : Set} → List A → List A
reverse = foldl (λ rev x → x ∷ rev) []
length : ∀ {A : Set} → List A → ℕ
length = foldr (λ _ → suc) 0
data Maybe (A : Set) : Set where
nothing : Maybe A
just : A → Maybe A
mapMaybe : ∀ {A B : Set} → (A → B) → Maybe A → Maybe B
mapMaybe f (just x) = just (f x)
mapMaybe f nothing = nothing
when : ∀ {A} → Bool → Maybe A → Maybe A
when true x = x
when false _ = nothing
data _≡_ {a} {A : Set a} (x : A) : A -> Set where
refl : x ≡ x
_→⟨_⟩_ : ∀ (A : Set) (n : ℕ) (B : Set) → Set
A →⟨ zero ⟩ B = B
A →⟨ suc n ⟩ B = A → A →⟨ n ⟩ B
open Library
module ReflectLibrary where
lamᵛ : Term → Term
lamᵛ = lam visible
lamʰ : Term → Term
lamʰ = lam hidden
argᵛʳ : ∀{A} → A → Arg A
argᵛʳ = arg (arginfo visible relevant)
argʰʳ : ∀{A} → A → Arg A
argʰʳ = arg (arginfo hidden relevant)
app` : (Args → Term) → (hrs : List ArgInfo) → Term →⟨ length hrs ⟩ Term
app` f = go [] where
go : List (Arg Term) → (hrs : List ArgInfo) → Term →⟨ length hrs ⟩ Term
go args [] = f (reverse args)
go args (i ∷ hs) = λ t → go (arg i t ∷ args) hs
con` : QName → (hrs : List ArgInfo) → Term →⟨ length hrs ⟩ Term
con` x = app` (con x)
def` : QName → (hrs : List ArgInfo) → Term →⟨ length hrs ⟩ Term
def` x = app` (def x)
var` : ℕ → (hrs : List ArgInfo) → Term →⟨ length hrs ⟩ Term
var` x = app` (var x)
coe : ∀ {A : Set} {z : A} n → (Term →⟨ length (replicate n z) ⟩ Term) → Term →⟨ n ⟩ Term
coe zero t = t
coe (suc n) f = λ t → coe n (f t)
con`ⁿʳ : QName → (n : ℕ) → Term →⟨ n ⟩ Term
con`ⁿʳ x n = coe n (app` (con x) (replicate n (arginfo visible relevant)))
def`ⁿʳ : QName → (n : ℕ) → Term →⟨ n ⟩ Term
def`ⁿʳ x n = coe n (app` (def x) (replicate n (arginfo visible relevant)))
var`ⁿʳ : ℕ → (n : ℕ) → Term →⟨ n ⟩ Term
var`ⁿʳ x n = coe n (app` (var x) (replicate n (arginfo visible relevant)))
sort₀ : Sort
sort₀ = lit 0
sort₁ : Sort
sort₁ = lit 1
`Set₀ : Term
`Set₀ = sort sort₀
el₀ : Term → Type
el₀ = el sort₀
-- Builds a type variable (of type Set₀)
``var₀ : ℕ → Args → Type
``var₀ n args = el₀ (var n args)
``Set₀ : Type
``Set₀ = el sort₁ `Set₀
unEl : Type → Term
unEl (el _ tm) = tm
getSort : Type → Sort
getSort (el s _) = s
unArg : ∀ {A} → Arg A → A
unArg (arg _ a) = a
`Level : Term
`Level = def (quote Level) []
``Level : Type
``Level = el₀ `Level
`sucLevel : Term → Term
`sucLevel = def`ⁿʳ (quote lsuc) 1
sucSort : Sort → Sort
sucSort s = set (`sucLevel (sort s))
ℕ→Level : ℕ → Level
ℕ→Level zero = lzero
ℕ→Level (suc n) = lsuc (ℕ→Level n)
-- Can't match on Levels anymore
-- Level→ℕ : Level → ℕ
-- Level→ℕ zero = zero
-- Level→ℕ (suc n) = suc (Level→ℕ n)
setLevel : Level → Sort
setLevel ℓ = lit 0 -- (Level→ℕ ℓ)
_==_ : QName → QName → Bool
_==_ = primQNameEquality
decodeSort : Sort → Maybe Level
decodeSort (set (con c [])) = when (quote lzero == c) (just lzero)
decodeSort (set (con c (arg (arginfo visible relevant) s ∷ [])))
= when (quote lsuc == c) (mapMaybe lsuc (decodeSort (set s)))
decodeSort (set (sort s)) = decodeSort s
decodeSort (set _) = nothing
decodeSort (lit n) = just (ℕ→Level n)
decodeSort unknown = nothing
_`⊔`_ : Sort → Sort → Sort
s₁ `⊔` s₂ with decodeSort s₁ | decodeSort s₂
... | just n₁ | just n₂ = setLevel (n₁ ⊔ n₂)
... | _ | _ = set (def (quote _⊔_) (argᵛʳ (sort s₁) ∷ argᵛʳ (sort s₂) ∷ []))
Π : Arg Type → Type → Type
Π t u = el (getSort (unArg t) `⊔` getSort u) (pi t u)
Πᵛʳ : Type → Type → Type
Πᵛʳ t u = el (getSort t `⊔` getSort u) (pi (arg (arginfo visible relevant) t) u)
Πʰʳ : Type → Type → Type
Πʰʳ t u = el (getSort t `⊔` getSort u) (pi (arg (arginfo hidden relevant) t) u)
open ReflectLibrary
`ℕ : Term
`ℕ = def (quote ℕ) []
`ℕOk : (unquote `ℕ) ≡ ℕ
`ℕOk = refl
``ℕ : Type
``ℕ = el₀ `ℕ
idℕ : ℕ → ℕ
idℕ = unquote (lamᵛ (var 0 []))
id : (A : Set) → A → A
id = unquote (lamᵛ (lamᵛ (var 0 [])))
idBox : Box ({A : Set} → A → A)
idBox = box (unquote (lamᵛ (var 0 [])))
-- builds a pair
_`,_ : Term → Term → Term
_`,_ = con`ⁿʳ (quote _,_) 2
`tt : Term
`tt = con (quote tt) []
tuple : List Term → Term
tuple = foldr _`,_ `tt
`refl : Term
`refl = con (quote refl) []
`zero : Term
`zero = con (quote ℕ.zero) []
`[] : Term
`[] = con (quote []) []
_`∷_ : (`x `xs : Term) → Term
_`∷_ = con`ⁿʳ (quote _∷_) 2
`var : (`n `args : Term) → Term
`var = con`ⁿʳ (quote var) 2
`lam : (`hiding `args : Term) → Term
`lam = con`ⁿʳ (quote lam) 2
`visible : Term
`visible = con (quote visible) []
`hidden : Term
`hidden = con (quote hidden) []
`[_`] : Term → Term
`[ x `] = x `∷ `[]
quotedTwice : Term
quotedTwice = `lam `visible (`var `zero `[])
unquoteTwice₂ : ℕ → ℕ
unquoteTwice₂ = unquote (unquote quotedTwice)
unquoteTwice : ℕ → ℕ
unquoteTwice x = unquote (unquote (`var `zero `[]))
id₂ : {A : Set} → A → A
id₂ = unquote (lamᵛ (var 0 []))
id₃ : {A : Set} → A → A
id₃ x = unquote (var 0 [])
module Id {A : Set} (x : A) where
x′ : A
x′ = unquote (var 0 [])
k`ℕ : ℕ → Term
k`ℕ zero = `ℕ
k`ℕ (suc n) = unquote (def (quote k`ℕ) [ argᵛʳ (var 0 []) ]) -- k`ℕ n
test : id ≡ (λ A (x : A) → x)
× unquote `Set₀ ≡ Set
× unquote `ℕ ≡ ℕ
× unquote (lamᵛ (var 0 [])) ≡ (λ (x : Set) → x)
× id ≡ (λ A (x : A) → x)
× unquote `tt ≡ tt
× (λ {A} → Id.x′ {A}) ≡ (λ {A : Set} (x : A) → x)
× unquote (pi (argᵛʳ ``Set₀) ``Set₀) ≡ (Set → Set)
× unquoteTwice ≡ (λ (x : ℕ) → x)
× unquote (k`ℕ 42) ≡ ℕ
× ⊤
test = unquote (tuple (replicate n `refl)) where n = 10
Πⁿ : ℕ → Type → Type
Πⁿ zero t = t
Πⁿ (suc n) t = Π (argʰʳ ``Set₀) (Πⁿ n t)
ƛⁿ : Hiding → ℕ → Term → Term
ƛⁿ h zero t = t
ƛⁿ h (suc n) t = lam h (ƛⁿ h n t)
-- projᵢ : Proj i n
-- projᵢ = proj i n
-- Projᵢ = {A₁ ... Ai ... An : Set} → A₁ → ... → Aᵢ → ... → An → Aᵢ
-- projᵢ = λ {A₁ ... Ai ... An} x₁ ... xᵢ ... xn → xᵢ
Proj : (i n : ℕ) → Term
Proj i n = unEl (Πⁿ n (go n)) where
n∸1 = n ∸ 1
go : ℕ → Type
go zero = ``var₀ ((n + n) ∸ i) []
go (suc m) = Π (argᵛʳ (``var₀ n∸1 [])) (go m)
proj : (i n : ℕ) → Term
proj i n = ƛⁿ visible n (var (n ∸ i) [])
projFull : (i n : ℕ) → Term
projFull i n = ƛⁿ hidden n (proj i n)
ℕ→ℕ : Set
ℕ→ℕ = unquote (unEl (Π (argᵛʳ ``ℕ) ``ℕ))
ℕ→ℕOk : ℕ→ℕ ≡ (ℕ → ℕ)
ℕ→ℕOk = refl
``∀A→A : Type
``∀A→A = Π (argᵛʳ ``Set₀) (``var₀ 0 [])
∀A→A : Set₁
∀A→A = unquote (unEl ``∀A→A)
Proj₁¹ : Set₁
Proj₁¹ = unquote (Proj 1 1)
Proj₁² : Set₁
Proj₁² = unquote (Proj 1 2)
Proj₂² : Set₁
Proj₂² = unquote (Proj 2 2)
proj₃⁵ : unquote (Proj 3 5)
proj₃⁵ _ _ x _ _ = x
proj₃⁵′ : Box (unquote (Proj 3 5))
proj₃⁵′ = box (unquote (proj 3 5))
proj₂⁷ : unquote (Proj 2 7)
proj₂⁷ = unquote (proj 2 7)
test-proj : proj₃⁵′ ≡ box (λ _ _ x _ _ → x)
× Proj₁¹ ≡ ({A : Set} → A → A)
× Proj₁² ≡ ({A₁ A₂ : Set} → A₁ → A₂ → A₁)
× Proj₂² ≡ ({A₁ A₂ : Set} → A₁ → A₂ → A₂)
× unquote (Proj 3 5) ≡ ({A₁ A₂ A₃ A₄ A₅ : Set} → A₁ → A₂ → A₃ → A₄ → A₅ → A₃)
× unquote (projFull 1 1) ≡ (λ {A : Set} (x : A) → x)
× unquote (projFull 1 2) ≡ (λ {A₁ A₂ : Set} (x₁ : A₁) (x₂ : A₂) → x₁)
× unquote (projFull 2 2) ≡ (λ {A₁ A₂ : Set} (x₁ : A₁) (x₂ : A₂) → x₂)
× ∀A→A ≡ (∀ (A : Set) → A)
× ⊤
test-proj = unquote (tuple (replicate n `refl)) where n = 9
module Test where
data Squash (A : Set) : Set where
squash : unquote (unEl (Π (arg (arginfo visible irrelevant) (``var₀ 0 [])) (el₀ (def (quote Squash) (argᵛʳ (var 1 []) ∷ [])))))
data Squash (A : Set) : Set where
squash : .A → Squash A
`Squash : Term → Term
`Squash = def`ⁿʳ (quote Squash) 1
squash-type : Type
squash-type = Π (arg (arginfo visible irrelevant) (``var₀ 0 [])) (el₀ (`Squash (var 1 [])))
test-squash : ∀ {A} → (.A → Squash A) ≡ unquote (unEl squash-type)
test-squash = refl
`∀ℓ→Setℓ : Type
`∀ℓ→Setℓ = Πᵛʳ ``Level (el₀ (sort (set (var 0 []))))
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module BHeap {A : Set}(_≤_ : A → A → Set) where
open import Bound.Lower A
open import Bound.Lower.Order _≤_
open import BTree {A} hiding (flatten)
open import Data.Nat hiding (_≤_)
open import Data.List
open import Data.Sum renaming (_⊎_ to _∨_)
open import Relation.Binary
open import Relation.Binary.Core
open import Relation.Binary.PropositionalEquality hiding (trans)
open DecTotalOrder decTotalOrder hiding (refl ; _≤_)
data BHeap : Bound → Set where
lf : {b : Bound}
→ BHeap b
nd : {b : Bound}{x : A}
→ LeB b (val x)
→ (l r : BHeap (val x))
→ BHeap b
forget : {b : Bound} → BHeap b → BTree
forget lf = leaf
forget (nd {x = x} _ l r) = node x (forget l) (forget r)
# : {b : Bound} → BHeap b → ℕ
# lf = zero
# (nd _ l r) = suc (# l + # r)
height : {b : Bound} → BHeap b → ℕ
height lf = zero
height (nd _ l r)
with total (height l) (height r)
... | inj₁ hl≤hr = suc (height r)
... | inj₂ hr≤hl = suc (height l)
merge : {b : Bound} → Total _≤_ → (l r : BHeap b) → BHeap b
merge _ lf r = r
merge _ l lf = l
merge tot≤ (nd {x = x} b≤x l r) (nd {x = x'} b≤x' l' r')
with tot≤ x x'
... | inj₁ x≤x' = nd b≤x (merge tot≤ l r) (nd (lexy x≤x') l' r')
... | inj₂ x'≤x = nd b≤x' (nd (lexy x'≤x) l r) (merge tot≤ l' r')
flatten : {b : Bound}(h : BHeap b) → List A
flatten lf = []
flatten (nd {x = x} b≤x l r) = x ∷ (flatten l ++ flatten r)
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open import SingleSorted.AlgebraicTheory
import SingleSorted.Interpretation as Interpretation
import SingleSorted.SyntacticCategory as SyntacticCategory
import SingleSorted.Substitution as Substitution
module SingleSorted.UniversalInterpretation
{ℓt}
{Σ : Signature}
(T : Theory ℓt Σ) where
open Theory T
open Substitution T
open SyntacticCategory T
-- The universal interpretation in the syntactic category
ℐ : Interpretation.Interpretation Σ cartesian-𝒮
ℐ =
record
{ interp-carrier = ctx-slot
; interp-pow = power-𝒮
; interp-oper = λ f var-var → tm-oper f (λ i → tm-var i)
}
open Interpretation.Interpretation ℐ
-- A term is essentially interpreted by itself
interp-term-self : ∀ {Γ} (t : Term Γ) y → Γ ⊢ interp-term t y ≈ t
interp-term-self (tm-var x) _ = eq-refl
interp-term-self (tm-oper f xs) y = eq-congr (λ i → interp-term-self (xs i) var-var)
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open import Agda.Builtin.Unit
open import Agda.Builtin.Nat
open import Agda.Builtin.List
open import Agda.Builtin.Reflection renaming (bindTC to _>>=_)
open import Agda.Builtin.Equality
macro
macro₁ : Term -> TC ⊤
macro₁ goal = do
u ← quoteTC ((1 + 2) - 3)
u' ← onlyReduceDefs (quote _+_ ∷ []) (normalise u)
qu' ← quoteTC u'
unify qu' goal
test₁ : macro₁ ≡ def (quote _-_)
(arg (arg-info visible relevant) (lit (nat 3)) ∷
arg (arg-info visible relevant) (lit (nat 3)) ∷ [])
test₁ = refl
macro
macro₂ : Term -> TC ⊤
macro₂ goal = do
u ← quoteTC ((1 - 2) + 3)
u' ← dontReduceDefs (quote _+_ ∷ []) (normalise u)
qu' ← quoteTC u'
unify qu' goal
test₂ : macro₂ ≡ def (quote _+_)
(arg (arg-info visible relevant) (lit (nat 0)) ∷
arg (arg-info visible relevant) (lit (nat 3)) ∷ [])
test₂ = refl
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Homomorphism proofs for variables and constants over polynomials
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Tactic.RingSolver.Core.Polynomial.Parameters
module Tactic.RingSolver.Core.Polynomial.Homomorphism.Variables
{r₁ r₂ r₃ r₄}
(homo : Homomorphism r₁ r₂ r₃ r₄)
where
open import Data.Product using (_,_)
open import Data.Vec.Base as Vec using (Vec)
open import Data.Fin using (Fin)
open import Data.List.Kleene
open Homomorphism homo
open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Lemmas homo
open import Tactic.RingSolver.Core.Polynomial.Base (Homomorphism.from homo)
open import Tactic.RingSolver.Core.Polynomial.Reasoning (Homomorphism.to homo)
open import Tactic.RingSolver.Core.Polynomial.Semantics homo
open import Algebra.Operations.Ring rawRing
ι-hom : ∀ {n} (i : Fin n) (Ρ : Vec Carrier n) → ⟦ ι i ⟧ Ρ ≈ Vec.lookup Ρ i
ι-hom i Ρ′ = let (ρ , Ρ) = drop-1 (space≤′n i) Ρ′ in begin
⟦ (κ Raw.1# Δ 1 ∷↓ []) ⊐↓ space≤′n i ⟧ Ρ′ ≈⟨ ⊐↓-hom (κ Raw.1# Δ 1 ∷↓ []) (space≤′n i) Ρ′ ⟩
⅀?⟦ κ Raw.1# Δ 1 ∷↓ [] ⟧ (ρ , Ρ) ≈⟨ ∷↓-hom-s (κ Raw.1#) 0 [] ρ Ρ ⟩
ρ * ⟦ κ Raw.1# ⟧ Ρ ≈⟨ *≫ 1-homo ⟩
ρ * 1# ≈⟨ *-identityʳ ρ ⟩
ρ ≡⟨ drop-1⇒lookup i Ρ′ ⟩
Vec.lookup Ρ′ i ∎
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{-
Holey congruence for propositional equality.
-}
module Holes.Cong.Propositional where
open import Holes.Prelude
open PropEq using (_≡_; refl; cong; sym; trans)
import Holes.Cong.General as Cong
open Cong (quote cong) (quote sym) public
using (cong!)
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data N : Set where
suc : N → N
data Val : N → Set where
valSuc : ∀ n → Val (suc n)
record R : Set where
constructor wrap
field unwrap : N
data W (ft : R) : Set where
immed : (v : Val (R.unwrap ft)) → W ft
test : (fa : R) → W fa → R
test fa (immed (valSuc a)) = fa
postulate
Evaluate : ∀ (ft : R) (P : (w : W ft) → Set) → Set
test₂ : ∀ (fa : R) → Set
test₂ fa = Evaluate fa testw
where
testw : W fa → Set
testw (immed (valSuc a)) = W fa
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module Operations.Combinatorial where
open import Data.Bool using (true; false)
import Data.Bool as B
import Data.Fin as F
open import Data.Nat using (ℕ; suc)
open import Data.Vec using ([]; _∷_)
import Data.Vec as V
open import Relation.Binary.PropositionalEquality
open import Types
open import Eval
not : Closed (𝔹 ⇒ 𝔹)
not = lam (var F.zero refl nand var F.zero refl)
not-prf : ∀ {n} {ctx : Ctx n} {Γ : Env ctx} b → (Γ ⟦ not ⟧) b ≡ B.not b
not-prf true = refl
not-prf false = refl
not⁺ : ∀ n → Closed (𝔹⁺ n ⇒ 𝔹⁺ n)
not⁺ 0 = lam []
not⁺ (suc n) = lam (not ∙ head (var F.zero refl) ∷ not⁺ n ∙ tail (var F.zero refl))
not⁺-prf : ∀ {n m} {ctx : Ctx m} {Γ : Env ctx} bs → (Γ ⟦ not⁺ n ⟧) bs ≡ V.map B.not bs
not⁺-prf [] = refl
not⁺-prf (true ∷ bs) = cong₂ _∷_ refl (not⁺-prf bs)
not⁺-prf (false ∷ bs) = cong₂ _∷_ refl (not⁺-prf bs)
and : Closed (𝔹 ⇒ (𝔹 ⇒ 𝔹))
and = lam (lam (not ∙ ((var (F.suc F.zero) refl) nand (var F.zero refl))))
and-prf : ∀ a b → ([] ⟦ and ⟧) a b ≡ B._∧_ a b
and-prf true true = refl
and-prf true false = refl
and-prf false true = refl
and-prf false false = refl
and⁺ : ∀ n → Closed (𝔹⁺ n ⇒ (𝔹⁺ n ⇒ 𝔹⁺ n))
and⁺ 0 = lam (lam [])
and⁺ (suc n) = lam (lam (x ∷ xs))
where
x = and ∙ head (var (F.suc F.zero) refl) ∙ head (var F.zero refl)
xs = and⁺ n ∙ tail (var (F.suc F.zero) refl) ∙ tail (var F.zero refl)
and⁺-prf : ∀ {n m} {ctx : Ctx m} {Γ : Env ctx} as bs → (Γ ⟦ and⁺ n ⟧) as bs ≡ V.zipWith B._∧_ as bs
and⁺-prf [] [] = refl
and⁺-prf ( true ∷ as) ( true ∷ bs) = cong₂ _∷_ refl (and⁺-prf as bs)
and⁺-prf ( true ∷ as) (false ∷ bs) = cong₂ _∷_ refl (and⁺-prf as bs)
and⁺-prf (false ∷ as) ( true ∷ bs) = cong₂ _∷_ refl (and⁺-prf as bs)
and⁺-prf (false ∷ as) (false ∷ bs) = cong₂ _∷_ refl (and⁺-prf as bs)
or : Closed (𝔹 ⇒ (𝔹 ⇒ 𝔹))
or = lam (lam (l nand r))
where
l = ((var (F.suc F.zero) refl) nand (var (F.suc F.zero) refl))
r = ((var F.zero refl) nand (var F.zero refl))
or-prf : ∀ a b → ([] ⟦ or ⟧) a b ≡ B._∨_ a b
or-prf true true = refl
or-prf true false = refl
or-prf false true = refl
or-prf false false = refl
or⁺ : ∀ n → Closed (𝔹⁺ n ⇒ (𝔹⁺ n ⇒ 𝔹⁺ n))
or⁺ 0 = lam (lam [])
or⁺ (suc n) = lam (lam (x ∷ xs))
where
x = or ∙ head (var (F.suc F.zero) refl) ∙ head (var F.zero refl)
xs = or⁺ n ∙ tail (var (F.suc F.zero) refl) ∙ tail (var F.zero refl)
or⁺-prf : ∀ {n m} {ctx : Ctx m} {Γ : Env ctx} as bs → (Γ ⟦ or⁺ n ⟧) as bs ≡ V.zipWith B._∨_ as bs
or⁺-prf [] [] = refl
or⁺-prf ( true ∷ as) ( true ∷ bs) = cong₂ _∷_ refl (or⁺-prf as bs)
or⁺-prf ( true ∷ as) (false ∷ bs) = cong₂ _∷_ refl (or⁺-prf as bs)
or⁺-prf (false ∷ as) ( true ∷ bs) = cong₂ _∷_ refl (or⁺-prf as bs)
or⁺-prf (false ∷ as) (false ∷ bs) = cong₂ _∷_ refl (or⁺-prf as bs)
xor : Closed (𝔹 ⇒ (𝔹 ⇒ 𝔹))
xor = lam (lam (lam (l nand r) ∙ x))
where
x = (var (F.suc F.zero) refl) nand (var F.zero refl)
l = ((var (F.suc (F.suc F.zero)) refl) nand (var F.zero refl))
r = ((var F.zero refl) nand (var (F.suc F.zero) refl))
xor-prf : ∀ a b → ([] ⟦ xor ⟧) a b ≡ B._xor_ a b
xor-prf true true = refl
xor-prf true false = refl
xor-prf false true = refl
xor-prf false false = refl
xor⁺ : ∀ n → Closed (𝔹⁺ n ⇒ (𝔹⁺ n ⇒ 𝔹⁺ n))
xor⁺ 0 = lam (lam [])
xor⁺ (suc n) = lam (lam (x ∷ xs))
where
x = xor ∙ head (var (F.suc F.zero) refl) ∙ head (var F.zero refl)
xs = xor⁺ n ∙ tail (var (F.suc F.zero) refl) ∙ tail (var F.zero refl)
xor⁺-prf : ∀ {n m} {ctx : Ctx m} {Γ : Env ctx} as bs → (Γ ⟦ xor⁺ n ⟧) as bs ≡ V.zipWith B._xor_ as bs
xor⁺-prf [] [] = refl
xor⁺-prf ( true ∷ as) ( true ∷ bs) = cong₂ _∷_ refl (xor⁺-prf as bs)
xor⁺-prf ( true ∷ as) (false ∷ bs) = cong₂ _∷_ refl (xor⁺-prf as bs)
xor⁺-prf (false ∷ as) ( true ∷ bs) = cong₂ _∷_ refl (xor⁺-prf as bs)
xor⁺-prf (false ∷ as) (false ∷ bs) = cong₂ _∷_ refl (xor⁺-prf as bs)
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{-# OPTIONS --safe --experimental-lossy-unification #-}
module Cubical.Homotopy.Group.Base where
open import Cubical.Homotopy.Loopspace
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Pointed.Homogeneous
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.GroupoidLaws renaming (assoc to ∙assoc)
open import Cubical.Foundations.Path
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Univalence
open import Cubical.Foundations.Function
open import Cubical.Foundations.Transport
open import Cubical.Functions.Morphism
open import Cubical.HITs.SetTruncation
renaming (rec to sRec ; rec2 to sRec2
; elim to sElim ; elim2 to sElim2 ; elim3 to sElim3
; map to sMap)
open import Cubical.HITs.Truncation
renaming (rec to trRec ; elim to trElim ; elim2 to trElim2)
open import Cubical.HITs.Sn
open import Cubical.HITs.Susp renaming (toSusp to σ)
open import Cubical.HITs.S1
open import Cubical.Data.Sigma
open import Cubical.Data.Nat
open import Cubical.Data.Bool
open import Cubical.Data.Unit
open import Cubical.Algebra.Group
open import Cubical.Algebra.Semigroup
open import Cubical.Algebra.Monoid
open Iso
open IsGroup
open IsSemigroup
open IsMonoid
open GroupStr
{- Homotopy group -}
π : ∀ {ℓ} (n : ℕ) (A : Pointed ℓ) → Type ℓ
π n A = ∥ typ ((Ω^ n) A) ∥₂
{- Alternative formulation. This will be given a group structure in
the Properties file -}
π' : ∀ {ℓ} (n : ℕ) (A : Pointed ℓ) → Type ℓ
π' n A = ∥ S₊∙ n →∙ A ∥₂
{- π as a group -}
1π : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} → π n A
1π zero {A = A} = ∣ pt A ∣₂
1π (suc n) = ∣ refl ∣₂
·π : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} → π (suc n) A → π (suc n) A → π (suc n) A
·π n = sRec2 squash₂ λ p q → ∣ p ∙ q ∣₂
-π : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} → π (suc n) A → π (suc n) A
-π n = sMap sym
π-rUnit : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (x : π (suc n) A)
→ (·π n x (1π (suc n))) ≡ x
π-rUnit n = sElim (λ _ → isSetPathImplicit) λ p i → ∣ rUnit p (~ i) ∣₂
π-lUnit : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (x : π (suc n) A)
→ (·π n (1π (suc n)) x) ≡ x
π-lUnit n = sElim (λ _ → isSetPathImplicit) λ p i → ∣ lUnit p (~ i) ∣₂
π-rCancel : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (x : π (suc n) A)
→ (·π n x (-π n x)) ≡ 1π (suc n)
π-rCancel n = sElim (λ _ → isSetPathImplicit) λ p i → ∣ rCancel p i ∣₂
π-lCancel : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (x : π (suc n) A)
→ (·π n (-π n x) x) ≡ 1π (suc n)
π-lCancel n = sElim (λ _ → isSetPathImplicit) λ p i → ∣ lCancel p i ∣₂
π-assoc : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (x y z : π (suc n) A)
→ ·π n x (·π n y z) ≡ ·π n (·π n x y) z
π-assoc n = sElim3 (λ _ _ _ → isSetPathImplicit) λ p q r i → ∣ ∙assoc p q r i ∣₂
π-comm : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (x y : π (suc (suc n)) A)
→ ·π (suc n) x y ≡ ·π (suc n) y x
π-comm n = sElim2 (λ _ _ → isSetPathImplicit) λ p q i → ∣ EH n p q i ∣₂
-- πₙ₊₁
πGr : ∀ {ℓ} (n : ℕ) (A : Pointed ℓ) → Group ℓ
fst (πGr n A) = π (suc n) A
1g (snd (πGr n A)) = 1π (suc n)
GroupStr._·_ (snd (πGr n A)) = ·π n
inv (snd (πGr n A)) = -π n
is-set (isSemigroup (isMonoid (isGroup (snd (πGr n A))))) = squash₂
assoc (isSemigroup (isMonoid (isGroup (snd (πGr n A))))) = π-assoc n
identity (isMonoid (isGroup (snd (πGr n A)))) x = (π-rUnit n x) , (π-lUnit n x)
inverse (isGroup (snd (πGr n A))) x = (π-rCancel n x) , (π-lCancel n x)
-- Group operations on π'.
-- We define the corresponding structure on the untruncated
-- (S₊∙ n →∙ A).
∙Π : ∀ {ℓ} {A : Pointed ℓ} {n : ℕ}
→ (S₊∙ n →∙ A)
→ (S₊∙ n →∙ A)
→ (S₊∙ n →∙ A)
∙Π {A = A} {n = zero} p q = (λ _ → pt A) , refl
fst (∙Π {A = A} {n = suc zero} (f , p) (g , q)) base = pt A
fst (∙Π {A = A} {n = suc zero} (f , p) (g , q)) (loop j) =
((sym p ∙∙ cong f loop ∙∙ p) ∙ (sym q ∙∙ cong g loop ∙∙ q)) j
snd (∙Π {A = A} {n = suc zero} (f , p) (g , q)) = refl
fst (∙Π {A = A} {n = suc (suc n)} (f , p) (g , q)) north = pt A
fst (∙Π {A = A} {n = suc (suc n)} (f , p) (g , q)) south = pt A
fst (∙Π {A = A} {n = suc (suc n)} (f , p) (g , q)) (merid a j) =
((sym p ∙∙ cong f (merid a ∙ sym (merid (ptSn (suc n)))) ∙∙ p)
∙ (sym q ∙∙ cong g (merid a ∙ sym (merid (ptSn (suc n)))) ∙∙ q)) j
snd (∙Π {A = A} {n = suc (suc n)} (f , p) (g , q)) = refl
-Π : ∀ {ℓ} {A : Pointed ℓ} {n : ℕ}
→ (S₊∙ n →∙ A)
→ (S₊∙ n →∙ A)
-Π {n = zero} f = f
fst (-Π {A = A} {n = suc zero} f) base = fst f base
fst (-Π {A = A} {n = suc zero} f) (loop j) = fst f (loop (~ j))
snd (-Π {A = A} {n = suc zero} f) = snd f
fst (-Π {A = A} {n = suc (suc n)} f) north = fst f north
fst (-Π {A = A} {n = suc (suc n)} f) south = fst f north
fst (-Π {A = A} {n = suc (suc n)} f) (merid a j) =
fst f ((merid a ∙ sym (merid (ptSn _))) (~ j))
snd (-Π {A = A} {n = suc (suc n)} f) = snd f
-- to prove that this gives a group structure on π', we first
-- prove that Ωⁿ A ≃ (Sⁿ →∙ A).
-- We use the following map
mutual
Ω→SphereMap : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ}
→ typ ((Ω^ n) A) → (S₊∙ n →∙ A)
fst (Ω→SphereMap zero a) false = a
fst (Ω→SphereMap zero {A = A} a) true = pt A
snd (Ω→SphereMap zero a) = refl
fst (Ω→SphereMap (suc zero) {A = A} p) base = pt A
fst (Ω→SphereMap (suc zero) p) (loop i) = p i
snd (Ω→SphereMap (suc zero) p) = refl
fst (Ω→SphereMap (suc (suc n)) {A = A} p) north = pt A
fst (Ω→SphereMap (suc (suc n)) {A = A} p) south = pt A
fst (Ω→SphereMap (suc (suc n)) p) (merid a i) =
(sym (Ω→SphereMapId (suc n) a)
∙∙ (λ i → Ω→SphereMap (suc n) (p i) .fst a)
∙∙ Ω→SphereMapId (suc n) a) i
snd (Ω→SphereMap (suc (suc n)) p) = refl
Ω→SphereMapId : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (a : _)
→ Ω→SphereMap n {A = A} (pt ((Ω^ n) A)) .fst a ≡ pt A
Ω→SphereMapId zero false = refl
Ω→SphereMapId zero true = refl
Ω→SphereMapId (suc zero) base = refl
Ω→SphereMapId (suc zero) (loop i) = refl
Ω→SphereMapId (suc (suc n)) north = refl
Ω→SphereMapId (suc (suc n)) south = refl
Ω→SphereMapId (suc (suc n)) {A = A} (merid a i) j =
∙∙lCancel (Ω→SphereMapId (suc n) {A = A} a) j i
Ω→SphereMapId2 : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ}
→ Ω→SphereMap n {A = A} (pt ((Ω^ n) A)) ≡ ((λ _ → pt A) , refl)
fst (Ω→SphereMapId2 n {A = A} i) a = funExt (Ω→SphereMapId n {A = A}) i a
snd (Ω→SphereMapId2 zero {A = A} i) = refl
snd (Ω→SphereMapId2 (suc zero) {A = A} i) = refl
snd (Ω→SphereMapId2 (suc (suc n)) {A = A} i) = refl
-- Pointed version
Ω→SphereMap∙ : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ}
→ ((Ω^ n) A) →∙ (S₊∙ n →∙ A ∙)
Ω→SphereMap∙ n .fst = Ω→SphereMap n
Ω→SphereMap∙ n .snd = Ω→SphereMapId2 n
-- We define the following maps which will be used to
-- show that Ω→SphereMap is an equivalence
Ω→SphereMapSplit₁ : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ typ ((Ω^ (suc n)) A)
→ typ (Ω (S₊∙ n →∙ A ∙))
Ω→SphereMapSplit₁ n = Ω→ (Ω→SphereMap∙ n) .fst
ΩSphereMap : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ typ (Ω (S₊∙ n →∙ A ∙))
→ (S₊∙ (suc n) →∙ A)
fst (ΩSphereMap {A = A} zero p) base = p i0 .fst false
fst (ΩSphereMap {A = A} zero p) (loop i) = p i .fst false
snd (ΩSphereMap {A = A} zero p) = refl
ΩSphereMap {A = A} (suc n) = fun IsoΩFunSuspFun
-- Functoriality
-- The homogeneity assumption is not necessary but simplifying
isNaturalΩSphereMap : ∀ {ℓ ℓ'} (A : Pointed ℓ) (B : Pointed ℓ')
(homogB : isHomogeneous B) (f : A →∙ B) (n : ℕ)
→ ∀ g → f ∘∙ ΩSphereMap n g ≡ ΩSphereMap n (Ω→ (post∘∙ (S₊∙ n) f) .fst g)
isNaturalΩSphereMap A B homogB f 0 g =
→∙Homogeneous≡ homogB (funExt lem)
where
lem : ∀ x → f .fst (ΩSphereMap 0 g .fst x)
≡ ΩSphereMap 0 (Ω→ (post∘∙ (S₊∙ 0) f) .fst g) .fst x
lem base = f .snd
lem (loop i) j =
hfill
(λ j → λ
{ (i = i0) → post∘∙ _ f .snd j
; (i = i1) → post∘∙ _ f .snd j
})
(inS (f ∘∙ g i))
j .fst false
isNaturalΩSphereMap A B homogB f (n@(suc _)) g =
→∙Homogeneous≡ homogB (funExt lem)
where
lem : ∀ x → f .fst (ΩSphereMap n g .fst x)
≡ ΩSphereMap n (Ω→ (post∘∙ (S₊∙ n) f) .fst g) .fst x
lem north = f .snd
lem south = f .snd
lem (merid a i) j =
hfill
(λ j → λ
{ (i = i0) → post∘∙ _ f .snd j
; (i = i1) → post∘∙ _ f .snd j
})
(inS (f ∘∙ g i))
j .fst a
SphereMapΩ : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ (S₊∙ (suc n) →∙ A)
→ typ (Ω (S₊∙ n →∙ A ∙))
SphereMapΩ {A = A} zero (f , p) =
ΣPathP ((funExt λ { false → sym p ∙∙ cong f loop ∙∙ p
; true → refl})
, refl)
SphereMapΩ {A = A} (suc n) = inv IsoΩFunSuspFun
SphereMapΩIso : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ Iso (S₊∙ (suc n) →∙ A)
(typ (Ω (S₊∙ n →∙ A ∙)))
fun (SphereMapΩIso n) = SphereMapΩ n
inv (SphereMapΩIso n) = ΩSphereMap n
fst (rightInv (SphereMapΩIso zero) f i j) false = rUnit (λ j → fst (f j) false) (~ i) j
fst (rightInv (SphereMapΩIso {A = A} zero) f i j) true = snd (f j) (~ i)
snd (rightInv (SphereMapΩIso {A = A} zero) f i j) k = snd (f j) (~ i ∨ k)
rightInv (SphereMapΩIso (suc n)) = leftInv IsoΩFunSuspFun
leftInv (SphereMapΩIso zero) f =
ΣPathP ((funExt (λ { base → sym (snd f)
; (loop i) j → doubleCompPath-filler
(sym (snd f))
(cong (fst f) loop)
(snd f) (~ j) i}))
, λ i j → snd f (~ i ∨ j))
leftInv (SphereMapΩIso (suc n)) = rightInv IsoΩFunSuspFun
{-
In order to show that Ω→SphereMap is an equivalence, we show that it factors
Ω→SphereMapSplit₁ ΩSphereMap
Ωⁿ⁺¹(Sⁿ →∙ A) ----------------> Ω (Sⁿ →∙ A) -----------> (Sⁿ⁺¹ →∙ A)
-}
Ω→SphereMap-split : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ) (p : typ ((Ω^ (suc n)) A))
→ Ω→SphereMap (suc n) p ≡ ΩSphereMap n (Ω→SphereMapSplit₁ n p)
Ω→SphereMap-split {A = A} zero p =
ΣPathP ((funExt (λ { base → refl
; (loop i) j → lem (~ j) i}))
, refl)
where
lem : funExt⁻ (cong fst (Ω→SphereMapSplit₁ zero p)) false ≡ p
lem = (λ i → funExt⁻ (cong-∙∙ fst (sym (Ω→SphereMapId2 zero))
(cong (Ω→SphereMap zero) p)
(Ω→SphereMapId2 zero) i) false)
∙ sym (rUnit _)
Ω→SphereMap-split {A = A} (suc n) p =
ΣPathP ((funExt (λ { north → refl
; south → refl
; (merid a i) j → lem₂ a j i}))
, refl)
where
lem : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ) (a : S₊ (suc n))
→ Ω→SphereMapId (suc n) {A = A} a
≡ (λ i → fst (Ω→SphereMapId2 (suc n) {A = A} i) a)
lem zero base = refl
lem zero (loop i) = refl
lem (suc n) north = refl
lem (suc n) south = refl
lem (suc n) (merid a i) = refl
lem₂ : (a : S₊ (suc n))
→ ((λ i₁ → Ω→SphereMapId (suc n) {A = A} a (~ i₁))
∙∙ (λ i₁ → Ω→SphereMap (suc n) (p i₁) .fst a)
∙∙ Ω→SphereMapId (suc n) a)
≡ (λ i → Ω→SphereMapSplit₁ (suc n) p i .fst a)
lem₂ a = cong (λ x → sym x
∙∙ funExt⁻ (cong fst (λ i → Ω→SphereMap (suc n) (p i))) a
∙∙ x)
(lem n a)
∙∙ sym (cong-∙∙ (λ x → x a)
(cong fst (λ i → Ω→SphereMapId2 (suc n) (~ i)))
(cong fst (λ i → Ω→SphereMap (suc n) (p i)))
(cong fst (Ω→SphereMapId2 (suc n))))
∙∙ (λ i → funExt⁻ (cong-∙∙ fst (sym (Ω→SphereMapId2 (suc n)))
(cong (Ω→SphereMap (suc n)) p)
(Ω→SphereMapId2 (suc n)) (~ i)) a)
isEquiv-Ω→SphereMap₀ : ∀ {ℓ} {A : Pointed ℓ}
→ isEquiv (Ω→SphereMap 0 {A = A})
isEquiv-Ω→SphereMap₀ {A = A} =
isoToIsEquiv
(iso _ (λ f → fst f false)
(λ f → ΣPathP ((funExt (λ { false → refl ; true → sym (snd f)}))
, λ i j → snd f (~ i ∨ j)))
λ p → refl)
isEquiv-Ω→SphereMap : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ}
→ isEquiv (Ω→SphereMap n {A = A})
isEquiv-Ω→SphereMap zero {A = A} =
(isoToIsEquiv
(iso _ (λ f → fst f false)
(λ f → ΣPathP ((funExt (λ { false → refl
; true → sym (snd f)}))
, λ i j → snd f (~ i ∨ j)))
λ _ → refl))
isEquiv-Ω→SphereMap (suc zero) {A = A} =
isoToIsEquiv (iso _ invFun sec λ p → sym (rUnit p))
where
invFun : S₊∙ 1 →∙ A → typ (Ω A)
invFun (f , p) = sym p ∙∙ cong f loop ∙∙ p
sec : section (Ω→SphereMap 1) invFun
sec (f , p) =
ΣPathP ((funExt (λ { base → sym p
; (loop i) j → doubleCompPath-filler
(sym p) (cong f loop) p (~ j) i}))
, λ i j → p (~ i ∨ j))
isEquiv-Ω→SphereMap (suc (suc n)) =
subst isEquiv (sym (funExt (Ω→SphereMap-split (suc n))))
(snd (compEquiv
((Ω→SphereMapSplit₁ (suc n)) ,
(isEquivΩ→ (Ω→SphereMap (suc n) , Ω→SphereMapId2 (suc n))
(isEquiv-Ω→SphereMap (suc n))))
(invEquiv (isoToEquiv (SphereMapΩIso (suc n))))))
IsoΩSphereMap : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ Iso (typ ((Ω^ n) A)) (S₊∙ n →∙ A)
IsoΩSphereMap n = equivToIso (_ , isEquiv-Ω→SphereMap n)
IsoSphereMapΩ : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ Iso (S₊∙ n →∙ A) (fst ((Ω^ n) A))
IsoSphereMapΩ {A = A} n =
invIso (IsoΩSphereMap n)
SphereMap→Ω : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ S₊∙ n →∙ A → fst ((Ω^ n) A)
SphereMap→Ω n = fun (IsoSphereMapΩ n)
isHom-Ω→SphereMap : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (p q : _)
→ Ω→SphereMap (suc n) {A = A} (p ∙ q)
≡ ∙Π (Ω→SphereMap (suc n) {A = A} p)
(Ω→SphereMap (suc n) {A = A} q)
isHom-Ω→SphereMap zero {A = A} p q =
ΣPathP ((funExt (λ { base → refl
; (loop i) j → (rUnit p j ∙ rUnit q j) i}))
, refl)
isHom-Ω→SphereMap (suc n) {A = A} p q =
ΣPathP ((funExt (λ { north → refl
; south → refl
; (merid a i) j → main a j i}))
, refl)
where
doubleComp-lem : ∀ {ℓ} {A : Type ℓ} {x y : A} (p : x ≡ y) (q r : y ≡ y)
→ (p ∙∙ q ∙∙ sym p) ∙ (p ∙∙ r ∙∙ sym p)
≡ (p ∙∙ (q ∙ r) ∙∙ sym p)
doubleComp-lem p q r i j =
hcomp (λ k → λ { (i = i0) → (doubleCompPath-filler p q (sym p) k
∙ doubleCompPath-filler p r (sym p) k) j
; (i = i1) → doubleCompPath-filler p (q ∙ r) (sym p) k j
; (j = i0) → p (~ k)
; (j = i1) → p (~ k)})
((q ∙ r) j)
lem : (p : typ ((Ω^ (suc (suc n))) A))
→ cong (fst (Ω→SphereMap (suc (suc n)) p)) (merid (ptSn _)) ≡ refl
lem p =
cong (sym (Ω→SphereMapId (suc n) (ptSn _)) ∙∙_∙∙ Ω→SphereMapId (suc n) (ptSn _))
(rUnit _ ∙ (λ j → (λ i → Ω→SphereMap (suc n) {A = A} refl .snd (i ∧ j))
∙∙ (λ i → Ω→SphereMap (suc n) {A = A} (p i) .snd j)
∙∙ λ i → Ω→SphereMap (suc n) {A = A} refl .snd (~ i ∧ j))
∙ ∙∙lCancel _)
∙ ∙∙lCancel _
main : (a : S₊ (suc n))
→ sym (Ω→SphereMapId (suc n) a)
∙∙ funExt⁻ (cong fst (cong (Ω→SphereMap (suc n)) (p ∙ q))) a
∙∙ Ω→SphereMapId (suc n) a
≡ cong (fst (∙Π (Ω→SphereMap (suc (suc n)) p) (Ω→SphereMap (suc (suc n)) q))) (merid a)
main a = (cong (sym (Ω→SphereMapId (suc n) a) ∙∙_∙∙ (Ω→SphereMapId (suc n) a))
(cong-∙ (λ x → Ω→SphereMap (suc n) x .fst a) p q)
∙ sym (doubleComp-lem (sym (Ω→SphereMapId (suc n) a)) _ _))
∙∙ cong₂ _∙_ (sym (cong (cong (fst (Ω→SphereMap (suc (suc n)) p)) (merid a) ∙_)
(cong sym (lem p)) ∙ sym (rUnit _)))
(sym (cong (cong (fst (Ω→SphereMap (suc (suc n)) q)) (merid a) ∙_)
(cong sym (lem q)) ∙ sym (rUnit _)))
∙∙ λ i → (rUnit (cong-∙ (fst (Ω→SphereMap (suc (suc n)) p))
(merid a) (sym (merid (ptSn _))) (~ i)) i)
∙ (rUnit (cong-∙ (fst (Ω→SphereMap (suc (suc n)) q))
(merid a) (sym (merid (ptSn _)))(~ i)) i)
-- The iso is structure preserving
IsoSphereMapΩ-pres∙Π : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ) (f g : S₊∙ (suc n) →∙ A)
→ SphereMap→Ω (suc n) (∙Π f g)
≡ SphereMap→Ω (suc n) f ∙ SphereMap→Ω (suc n) g
IsoSphereMapΩ-pres∙Π n =
morphLemmas.isMorphInv _∙_ ∙Π (Ω→SphereMap (suc n))
(isHom-Ω→SphereMap n)
(SphereMap→Ω (suc n))
(leftInv (IsoSphereMapΩ (suc n)))
(rightInv (IsoSphereMapΩ (suc n)))
-- It is useful to define the ``Group Structure'' on (S₊∙ n →∙ A)
-- before doing it on π'. These will be the equivalents of the
-- usual groupoid laws on Ω A.
1Π : ∀ {ℓ} {A : Pointed ℓ} {n : ℕ} → (S₊∙ n →∙ A)
fst (1Π {A = A}) _ = pt A
snd (1Π {A = A}) = refl
∙Π-rUnit : ∀ {ℓ} {A : Pointed ℓ} {n : ℕ}
→ (f : S₊∙ (suc n) →∙ A)
→ ∙Π f 1Π ≡ f
fst (∙Π-rUnit {A = A} {n = zero} f i) base = snd f (~ i)
fst (∙Π-rUnit {A = A} {n = zero} f i) (loop j) = help i j
where
help : PathP (λ i → snd f (~ i) ≡ snd f (~ i))
(((sym (snd f)) ∙∙ (cong (fst f) loop) ∙∙ snd f)
∙ (refl ∙ refl))
(cong (fst f) loop)
help = (cong ((sym (snd f) ∙∙ cong (fst f) loop ∙∙ snd f) ∙_)
(sym (rUnit refl)) ∙ sym (rUnit _))
◁ λ i j → doubleCompPath-filler (sym (snd f))
(cong (fst f) loop) (snd f) (~ i) j
snd (∙Π-rUnit {A = A} {n = zero} f i) j = snd f (~ i ∨ j)
fst (∙Π-rUnit {A = A} {n = suc n} f i) north = snd f (~ i)
fst (∙Π-rUnit {A = A} {n = suc n} f i) south =
(sym (snd f) ∙ cong (fst f) (merid (ptSn (suc n)))) i
fst (∙Π-rUnit {A = A} {n = suc n} f i) (merid a j) = help i j
where
help : PathP (λ i → snd f (~ i)
≡ (sym (snd f) ∙ cong (fst f) (merid (ptSn (suc n)))) i)
(((sym (snd f))
∙∙ (cong (fst f) (merid a ∙ sym (merid (ptSn (suc n)))))
∙∙ snd f)
∙ (refl ∙ refl))
(cong (fst f) (merid a))
help = (cong (((sym (snd f))
∙∙ (cong (fst f) (merid a ∙ sym (merid (ptSn (suc n)))))
∙∙ snd f) ∙_)
(sym (rUnit refl))
∙ sym (rUnit _))
◁ λ i j → hcomp (λ k →
λ { (j = i0) → snd f (~ i ∧ k)
; (j = i1) → compPath-filler' (sym (snd f))
(cong (fst f) (merid (ptSn (suc n)))) k i
; (i = i0) → doubleCompPath-filler (sym (snd f))
(cong (fst f)
(merid a ∙ sym (merid (ptSn (suc n)))))
(snd f) k j
; (i = i1) → fst f (merid a j)})
(fst f (compPath-filler (merid a)
(sym (merid (ptSn _))) (~ i) j))
snd (∙Π-rUnit {A = A} {n = suc n} f i) j = snd f (~ i ∨ j)
∙Π-lUnit : ∀ {ℓ} {A : Pointed ℓ} {n : ℕ}
→ (f : S₊∙ (suc n) →∙ A)
→ ∙Π 1Π f ≡ f
fst (∙Π-lUnit {n = zero} f i) base = snd f (~ i)
fst (∙Π-lUnit {n = zero} f i) (loop j) = s i j
where
s : PathP (λ i → snd f (~ i) ≡ snd f (~ i))
((refl ∙ refl) ∙ (sym (snd f) ∙∙ cong (fst f) loop ∙∙ snd f))
(cong (fst f) loop)
s = (cong (_∙ (sym (snd f) ∙∙ cong (fst f) loop ∙∙ snd f))
(sym (rUnit refl)) ∙ sym (lUnit _))
◁ λ i j → doubleCompPath-filler (sym (snd f))
(cong (fst f) loop) (snd f) (~ i) j
snd (∙Π-lUnit {n = zero} f i) j = snd f (~ i ∨ j)
fst (∙Π-lUnit {n = suc n} f i) north = snd f (~ i)
fst (∙Π-lUnit {n = suc n} f i) south =
(sym (snd f) ∙ cong (fst f) (merid (ptSn _))) i
fst (∙Π-lUnit {n = suc n} f i) (merid a j) = help i j
where
help : PathP (λ i → snd f (~ i)
≡ (sym (snd f) ∙ cong (fst f) (merid (ptSn (suc n)))) i)
((refl ∙ refl) ∙ ((sym (snd f))
∙∙ (cong (fst f) (merid a ∙ sym (merid (ptSn (suc n)))))
∙∙ snd f))
(cong (fst f) (merid a))
help =
(cong (_∙ ((sym (snd f))
∙∙ (cong (fst f) (merid a ∙ sym (merid (ptSn (suc n)))))
∙∙ snd f))
(sym (rUnit refl))
∙ sym (lUnit _))
◁ λ i j → hcomp (λ k →
λ { (j = i0) → snd f (~ i ∧ k)
; (j = i1) → compPath-filler' (sym (snd f))
(cong (fst f) (merid (ptSn (suc n)))) k i
; (i = i0) → doubleCompPath-filler (sym (snd f))
(cong (fst f) (merid a ∙ sym (merid (ptSn (suc n)))))
(snd f) k j
; (i = i1) → fst f (merid a j)})
(fst f (compPath-filler (merid a) (sym (merid (ptSn _))) (~ i) j))
snd (∙Π-lUnit {n = suc n} f i) j = snd f (~ i ∨ j)
∙Π-rCancel : ∀ {ℓ} {A : Pointed ℓ} {n : ℕ}
→ (f : S₊∙ (suc n) →∙ A)
→ ∙Π f (-Π f) ≡ 1Π
fst (∙Π-rCancel {A = A} {n = zero} f i) base = pt A
fst (∙Π-rCancel {A = A} {n = zero} f i) (loop j) =
rCancel (sym (snd f) ∙∙ cong (fst f) loop ∙∙ snd f) i j
snd (∙Π-rCancel {A = A} {n = zero} f i) = refl
fst (∙Π-rCancel {A = A} {n = suc n} f i) north = pt A
fst (∙Π-rCancel {A = A} {n = suc n} f i) south = pt A
fst (∙Π-rCancel {A = A} {n = suc n} f i) (merid a i₁) = lem i i₁
where
pl = (sym (snd f)
∙∙ cong (fst f) (merid a ∙ sym (merid (ptSn _)))
∙∙ snd f)
lem : pl
∙ ((sym (snd f)
∙∙ cong (fst (-Π f)) (merid a ∙ sym (merid (ptSn _)))
∙∙ snd f)) ≡ refl
lem = cong (pl ∙_) (cong (sym (snd f) ∙∙_∙∙ (snd f))
(cong-∙ (fst (-Π f)) (merid a) (sym (merid (ptSn _)))
∙∙ cong₂ _∙_ refl
(cong (cong (fst f)) (rCancel (merid (ptSn _))))
∙∙ sym (rUnit _)))
∙ rCancel pl
snd (∙Π-rCancel {A = A} {n = suc n} f i) = refl
∙Π-lCancel : ∀ {ℓ} {A : Pointed ℓ} {n : ℕ}
→ (f : S₊∙ (suc n) →∙ A)
→ ∙Π (-Π f) f ≡ 1Π
fst (∙Π-lCancel {A = A} {n = zero} f i) base = pt A
fst (∙Π-lCancel {A = A} {n = zero} f i) (loop j) =
rCancel (sym (snd f) ∙∙ cong (fst f) (sym loop) ∙∙ snd f) i j
fst (∙Π-lCancel {A = A} {n = suc n} f i) north = pt A
fst (∙Π-lCancel {A = A} {n = suc n} f i) south = pt A
fst (∙Π-lCancel {A = A} {n = suc n} f i) (merid a j) = lem i j
where
pl = (sym (snd f)
∙∙ cong (fst f) (merid a ∙ sym (merid (ptSn _)))
∙∙ snd f)
lem : (sym (snd f)
∙∙ cong (fst (-Π f)) (merid a ∙ sym (merid (ptSn _)))
∙∙ snd f) ∙ pl
≡ refl
lem = cong (_∙ pl) (cong (sym (snd f) ∙∙_∙∙ (snd f))
(cong-∙ (fst (-Π f)) (merid a) (sym (merid (ptSn _)))
∙∙ cong₂ _∙_ refl (cong (cong (fst f)) (rCancel (merid (ptSn _))))
∙∙ sym (rUnit _)))
∙ lCancel pl
snd (∙Π-lCancel {A = A} {n = zero} f i) = refl
snd (∙Π-lCancel {A = A} {n = suc n} f i) = refl
∙Π-assoc : ∀ {ℓ} {A : Pointed ℓ} {n : ℕ}
→ (f g h : S₊∙ (suc n) →∙ A)
→ ∙Π f (∙Π g h) ≡ ∙Π (∙Π f g) h
∙Π-assoc {n = n} f g h =
sym (leftInv (IsoSphereMapΩ (suc n)) (∙Π f (∙Π g h)))
∙∙ cong (Ω→SphereMap (suc n)) (IsoSphereMapΩ-pres∙Π n f (∙Π g h)
∙∙ cong (SphereMap→Ω (suc n) f ∙_) (IsoSphereMapΩ-pres∙Π n g h)
∙∙ ∙assoc (SphereMap→Ω (suc n) f) (SphereMap→Ω (suc n) g) (SphereMap→Ω (suc n) h)
∙∙ cong (_∙ SphereMap→Ω (suc n) h) (sym (IsoSphereMapΩ-pres∙Π n f g))
∙∙ sym (IsoSphereMapΩ-pres∙Π n (∙Π f g) h))
∙∙ leftInv (IsoSphereMapΩ (suc n)) (∙Π (∙Π f g) h)
∙Π-comm : ∀ {ℓ} {A : Pointed ℓ} {n : ℕ}
→ (f g : S₊∙ (suc (suc n)) →∙ A)
→ ∙Π f g ≡ ∙Π g f
∙Π-comm {A = A} {n = n} f g =
sym (leftInv (IsoSphereMapΩ (suc (suc n))) (∙Π f g))
∙∙ cong (Ω→SphereMap (suc (suc n))) (IsoSphereMapΩ-pres∙Π (suc n) f g
∙∙ EH _ _ _
∙∙ sym (IsoSphereMapΩ-pres∙Π (suc n) g f))
∙∙ leftInv (IsoSphereMapΩ (suc (suc n))) (∙Π g f)
{- π'' as a group -}
1π' : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} → π' n A
1π' n {A = A} = ∣ 1Π ∣₂
·π' : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} → π' (suc n) A → π' (suc n) A → π' (suc n) A
·π' n = sRec2 squash₂ λ p q → ∣ ∙Π p q ∣₂
-π' : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} → π' (suc n) A → π' (suc n) A
-π' n = sMap -Π
π'-rUnit : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (x : π' (suc n) A)
→ (·π' n x (1π' (suc n))) ≡ x
π'-rUnit n = sElim (λ _ → isSetPathImplicit) λ p i → ∣ ∙Π-rUnit p i ∣₂
π'-lUnit : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (x : π' (suc n) A)
→ (·π' n (1π' (suc n)) x) ≡ x
π'-lUnit n = sElim (λ _ → isSetPathImplicit) λ p i → ∣ ∙Π-lUnit p i ∣₂
π'-rCancel : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (x : π' (suc n) A)
→ (·π' n x (-π' n x)) ≡ 1π' (suc n)
π'-rCancel n = sElim (λ _ → isSetPathImplicit) λ p i → ∣ ∙Π-rCancel p i ∣₂
π'-lCancel : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (x : π' (suc n) A)
→ (·π' n (-π' n x) x) ≡ 1π' (suc n)
π'-lCancel n = sElim (λ _ → isSetPathImplicit) λ p i → ∣ ∙Π-lCancel p i ∣₂
π'-assoc : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (x y z : π' (suc n) A)
→ ·π' n x (·π' n y z) ≡ ·π' n (·π' n x y) z
π'-assoc n = sElim3 (λ _ _ _ → isSetPathImplicit) λ p q r i → ∣ ∙Π-assoc p q r i ∣₂
π'-comm : ∀ {ℓ} (n : ℕ) {A : Pointed ℓ} (x y : π' (suc (suc n)) A)
→ ·π' (suc n) x y ≡ ·π' (suc n) y x
π'-comm n = sElim2 (λ _ _ → isSetPathImplicit) λ p q i → ∣ ∙Π-comm p q i ∣₂
-- We finally get the group definition
π'Gr : ∀ {ℓ} (n : ℕ) (A : Pointed ℓ) → Group ℓ
fst (π'Gr n A) = π' (suc n) A
1g (snd (π'Gr n A)) = 1π' (suc n)
GroupStr._·_ (snd (π'Gr n A)) = ·π' n
inv (snd (π'Gr n A)) = -π' n
is-set (isSemigroup (isMonoid (isGroup (snd (π'Gr n A))))) = squash₂
assoc (isSemigroup (isMonoid (isGroup (snd (π'Gr n A))))) = π'-assoc n
identity (isMonoid (isGroup (snd (π'Gr n A)))) x = (π'-rUnit n x) , (π'-lUnit n x)
inverse (isGroup (snd (π'Gr n A))) x = (π'-rCancel n x) , (π'-lCancel n x)
-- and finally, the Iso
π'Gr≅πGr : ∀ {ℓ} (n : ℕ) (A : Pointed ℓ) → GroupIso (π'Gr n A) (πGr n A)
fst (π'Gr≅πGr n A) = setTruncIso (IsoSphereMapΩ (suc n))
snd (π'Gr≅πGr n A) =
makeIsGroupHom (sElim2 (λ _ _ → isSetPathImplicit)
λ p q i → ∣ IsoSphereMapΩ-pres∙Π n p q i ∣₂)
{- Proof of πₙ(ΩA) = πₙ₊₁(A) -}
Iso-πΩ-π : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ Iso (π n (Ω A)) (π (suc n) A)
Iso-πΩ-π {A = A} n = setTruncIso (invIso (flipΩIso n))
GrIso-πΩ-π : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ GroupIso (πGr n (Ω A)) (πGr (suc n) A)
fst (GrIso-πΩ-π n) = Iso-πΩ-π _
snd (GrIso-πΩ-π n) =
makeIsGroupHom
(sElim2 (λ _ _ → isSetPathImplicit)
λ p q → cong ∣_∣₂ (flipΩIso⁻pres· n p q))
{- Proof that πₙ(A) ≅ πₙ(∥ A ∥ₙ) -}
isContrΩTrunc : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ isContr (typ ((Ω^ n) (hLevelTrunc∙ n A)))
isContrΩTrunc {A = A} zero = isContrUnit*
isContrΩTrunc {A = A} (suc n) =
subst isContr main (isContrΩTrunc {A = Ω A} n)
where
lem₁ : (n : ℕ) → fun (PathIdTruncIso n) (λ _ → ∣ pt A ∣)
≡ snd (hLevelTrunc∙ n (Ω A))
lem₁ zero = refl
lem₁ (suc n) = transportRefl ∣ refl ∣
lem₂ : hLevelTrunc∙ n (Ω A) ≡ (Ω (hLevelTrunc∙ (suc n) A))
lem₂ = sym (ua∙ (isoToEquiv (PathIdTruncIso n))
(lem₁ n))
main : (typ ((Ω^ n) (hLevelTrunc∙ n (Ω A))))
≡ (typ ((Ω^ suc n) (hLevelTrunc∙ (suc n) A)))
main = (λ i → typ ((Ω^ n) (lem₂ i)))
∙ sym (isoToPath (flipΩIso n))
mutual
ΩTruncSwitchFun : ∀ {ℓ} {A : Pointed ℓ} (n m : ℕ) →
(hLevelTrunc∙ (suc (suc m)) ((Ω^ n) A))
→∙ ((Ω^ n) (hLevelTrunc∙ (suc n + suc m) A))
ΩTruncSwitchFun {A = A} n m =
((λ x → transport
(λ i → fst ((Ω^ n) (hLevelTrunc∙ (+-suc n (suc m) i) A)))
(Iso.fun (ΩTruncSwitch {A = A} n (suc (suc m))) x))
, cong (transport
(λ i → fst ((Ω^ n) (hLevelTrunc∙ (+-suc n (suc m) i) A))))
(ΩTruncSwitch∙ n (suc (suc m)))
∙ λ j → transp
(λ i → fst ((Ω^ n) (hLevelTrunc∙ (+-suc n (suc m) (i ∨ j)) A)))
j (snd ((Ω^ n) (hLevelTrunc∙ (+-suc n (suc m) j) A))))
ΩTruncSwitchLem :
∀ {ℓ} {A : Pointed ℓ} (n m : ℕ)
→ Iso
(typ (Ω (hLevelTrunc∙ (suc (suc m)) ((Ω^ n) A))))
(typ ((Ω^ suc n) (hLevelTrunc∙ (suc n + suc m) A)))
ΩTruncSwitchLem {A = A} n m =
(equivToIso
(Ω→ (ΩTruncSwitchFun n m) .fst
, isEquivΩ→ _ (compEquiv (isoToEquiv (ΩTruncSwitch {A = A} n (suc (suc m))))
(transportEquiv
(λ i → typ ((Ω^ n) (hLevelTrunc∙ (+-suc n (suc m) i) A)))) .snd)))
ΩTruncSwitch : ∀ {ℓ} {A : Pointed ℓ} (n m : ℕ)
→ Iso (hLevelTrunc m (fst ((Ω^ n) A)))
(typ ((Ω^ n) (hLevelTrunc∙ (n + m) A)))
ΩTruncSwitch {A = A} n zero =
equivToIso
(invEquiv
(isContr→≃Unit*
(subst isContr
(λ i → (typ ((Ω^ n) (hLevelTrunc∙ (+-comm zero n i) A))))
(isContrΩTrunc n))))
ΩTruncSwitch {A = A} zero (suc m) = idIso
ΩTruncSwitch {A = A} (suc n) (suc m) =
compIso (invIso (PathIdTruncIso _))
(ΩTruncSwitchLem n m)
ΩTruncSwitch∙ : ∀ {ℓ} {A : Pointed ℓ} (n m : ℕ)
→ Iso.fun (ΩTruncSwitch {A = A} n m) (snd (hLevelTrunc∙ m ((Ω^ n) A)))
≡ pt ((Ω^ n) (hLevelTrunc∙ (n + m) A))
ΩTruncSwitch∙ {A = A} n zero =
isContr→isProp
((subst isContr
(λ i → (typ ((Ω^ n) (hLevelTrunc∙ (+-comm zero n i) A))))
(isContrΩTrunc n))) _ _
ΩTruncSwitch∙ {A = A} zero (suc m) = refl
ΩTruncSwitch∙ {A = A} (suc n) (suc m) = ∙∙lCancel _
ΩTruncSwitch-hom : ∀ {ℓ} {A : Pointed ℓ} (n m : ℕ) (p q : _)
→ Iso.fun (ΩTruncSwitch {A = A} (suc n) (suc m)) ∣ p ∙ q ∣
≡ Iso.fun (ΩTruncSwitch {A = A} (suc n) (suc m)) ∣ p ∣
∙ Iso.fun (ΩTruncSwitch {A = A} (suc n) (suc m)) ∣ q ∣
ΩTruncSwitch-hom {A = A} n m p q =
cong (Iso.fun (ΩTruncSwitchLem {A = A} n m))
(cong-∙ ∣_∣ₕ p q)
∙ Ω→pres∙ (ΩTruncSwitchFun n m) (cong ∣_∣ₕ p) (cong ∣_∣ₕ q)
2TruncΩIso : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ Iso (hLevelTrunc 2 (fst ((Ω^ n) A)))
(typ ((Ω^ n) (hLevelTrunc∙ (2 + n) A)))
2TruncΩIso zero = idIso
2TruncΩIso {A = A} (suc n) =
compIso
(ΩTruncSwitch (suc n) 2)
(pathToIso λ i → typ ((Ω^ suc n) (hLevelTrunc∙ (+-comm (suc n) 2 i) A)))
hLevΩ+ : ∀ {ℓ} {A : Pointed ℓ} (n m : ℕ)
→ isOfHLevel (m + n) (typ A) → isOfHLevel n (typ ((Ω^ m) A))
hLevΩ+ n zero p = p
hLevΩ+ {A = A} zero (suc zero) p = refl , λ _ → isProp→isSet p _ _ _ _
hLevΩ+ {A = A} zero (suc (suc zero)) p =
refl , λ y → isOfHLevelSuc 2 p _ _ _ _ refl y
hLevΩ+ {A = A} zero (suc (suc (suc m))) p =
transport
(λ i → isContr (typ (Ω (ua∙
(isoToEquiv (flipΩIso {A = A} (suc m))) (flipΩrefl m) (~ i)))))
(hLevΩ+ {A = Ω A} zero (suc (suc m))
(subst (λ x → isOfHLevel x (typ (Ω A)))
(+-comm zero (suc (suc m)))
(lem (pt A) (pt A))))
where
lem : isOfHLevel (3 + m) (typ A)
lem = subst (λ x → isOfHLevel x (typ A))
(λ i → suc (+-comm (2 + m) zero i)) p
hLevΩ+ {A = A} (suc n) (suc m) p =
subst (isOfHLevel (suc n))
(sym (ua (isoToEquiv (flipΩIso {A = A} m))))
(hLevΩ+ {A = Ω A} (suc n) m
(isOfHLevelPath' (m + suc n) p _ _))
private
isSetΩ : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ (isSet (typ (Ω ((Ω^ n) (hLevelTrunc∙ (suc (suc (suc n))) A)))))
isSetΩ {A = A} zero = isOfHLevelTrunc 3 _ _
isSetΩ {A = A} (suc n) =
hLevΩ+ 2 (suc (suc n))
(transport
(λ i → isOfHLevel (+-comm 2 (2 + n) i) (hLevelTrunc (4 + n) (typ A)))
(isOfHLevelTrunc (suc (suc (suc (suc n))))))
πTruncIso : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ Iso (π n A) (π n (hLevelTrunc∙ (2 + n) A))
πTruncIso {A = A} zero =
compIso (invIso (setTruncIdempotentIso squash₂))
(setTruncIso setTruncTrunc2Iso)
πTruncIso {A = A} (suc n) =
compIso setTruncTrunc2Iso
(compIso
(2TruncΩIso (suc n))
(invIso (setTruncIdempotentIso (isSetΩ n))))
πTruncGroupIso : ∀ {ℓ} {A : Pointed ℓ} (n : ℕ)
→ GroupIso (πGr n A) (πGr n (hLevelTrunc∙ (3 + n) A))
fst (πTruncGroupIso n) = πTruncIso (suc n)
snd (πTruncGroupIso {A = A} n) =
makeIsGroupHom
(sElim2 (λ _ _ → isSetPathImplicit)
λ a b
→ cong (inv (setTruncIdempotentIso (isSetΩ n)))
(cong
(transport
(λ i → typ ((Ω^ suc n) (hLevelTrunc∙ (+-comm (suc n) 2 i) A))))
(ΩTruncSwitch-hom n 1 a b)
∙ transpΩTruncSwitch
(λ w → ((Ω^ n) (hLevelTrunc∙ w A))) (+-comm (suc n) 2) _ _))
where
transpΩTruncSwitch : ∀ {ℓ} (A : ℕ → Pointed ℓ) {n m : ℕ}
(r : n ≡ m) (p q : typ (Ω (A n)))
→ subst (λ n → typ (Ω (A n))) r (p ∙ q)
≡ subst (λ n → typ (Ω (A n))) r p
∙ subst (λ n → typ (Ω (A n))) r q
transpΩTruncSwitch A {n = n} =
J (λ m r → (p q : typ (Ω (A n)))
→ subst (λ n → typ (Ω (A n))) r (p ∙ q)
≡ subst (λ n → typ (Ω (A n))) r p
∙ subst (λ n → typ (Ω (A n))) r q)
λ p q → transportRefl _ ∙ cong₂ _∙_
(sym (transportRefl p)) (sym (transportRefl q))
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{-# OPTIONS --show-implicit #-}
-- {-# OPTIONS --sized-types #-} -- no longer necessary
-- {-# OPTIONS --termination-depth=2 #-} -- no longer necessary
-- {-# OPTIONS -v term:10 #-}
module SizedBTree where
open import Common.Size
module Old where
data BTree (A : Set) : {size : Size} → Set where
leaf : {i : Size} → A → BTree A {↑ i}
node : {i : Size} → BTree A {i} → BTree A {i} → BTree A {↑ i}
map : ∀ {A B i} → (A → B) → BTree A {i} → BTree B {i}
map f (leaf a) = leaf (f a)
map f (node l r) = node (map f l) (map f r)
-- deep matching
deep : ∀ {i A} → BTree A {i} → A
deep (leaf a) = a
deep (node (leaf _) r) = deep r
deep (node (node l r) _) = deep (node l r)
-- nesting
-- Jesper 2015-12-18: The new unifier requires the size argument of the first
-- deep2 to be given explicitly. This is because otherwise the unifier gets
-- stuck on an equation [↑ i₁ =?= _i_46 {i} {A} l r t]. It's not clear whether
-- it is safe to instantiate the meta to [↑ _] here.
deep2 : ∀ {i A} → BTree A {i} → BTree A {i}
deep2 (leaf a) = leaf a
deep2 (node (leaf _) r) = r
deep2 (node (node {i} l r) t) with deep2 {↑ i} (deep2 (node l r))
... | leaf a = leaf a
... | node l2 r2 = deep2 (node l2 r2)
-- OUTDATED: -- increasing the termination count does the job!
module New where
data BTree (A : Set) {i : Size} : Set where
leaf : A → BTree A
node : {i' : Size< i} → BTree A {i'} → BTree A {i'} → BTree A
map : ∀ {A B i} → (A → B) → BTree A {i} → BTree B {i}
map f (leaf a) = leaf (f a)
map f (node l r) = node (map f l) (map f r)
-- deep matching
deep : ∀ {i A} → BTree A {i} → A
deep (leaf a) = a
deep (node (leaf _) r) = deep r
deep (node (node l r) _) = deep (node l r)
-- nesting
deep2 : ∀ {i A} → BTree A {i} → BTree A {i}
deep2 (leaf a) = leaf a
deep2 (node (leaf _) r) = r
deep2 (node (node l r) _) with deep2 (deep2 (node l r))
... | leaf a = leaf a
... | node l2 r2 = deep2 (node l2 r2)
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{-# OPTIONS --safe #-}
open import Relation.Ternary.Separation
module Relation.Ternary.Separation.Allstar
{i} {I : Set i}
{c} {C : Set c} {{rc : RawSep C}} {u} {{sc : IsUnitalSep rc u}}
where
open import Level
open import Data.Product
open import Data.List hiding (concat)
open import Relation.Unary
{- Inductive separating forall over a list -}
module _ {ℓ} where
data Allstar (P : I → Pred C ℓ) : List I → SPred (ℓ ⊔ c ⊔ i) where
nil : ε[ Allstar P [] ]
cons : ∀ {x xs} → ∀[ P x ✴ Allstar P xs ⇒ Allstar P (x ∷ xs) ]
-- not typed well in non-pattern positions
infixr 5 _:⟨_⟩:_
pattern _:⟨_⟩:_ x p xs = cons (x ×⟨ p ⟩ xs)
singleton : ∀ {P x} → ∀[ P x ⇒ Allstar P [ x ] ]
singleton v = cons (v ×⟨ ⊎-idʳ ⟩ nil)
open import Relation.Ternary.Separation.Construct.List I
open import Data.List.Relation.Ternary.Interleaving.Propositional as I
repartition : ∀ {P} {Σ₁ Σ₂ Σ} →
Σ₁ ⊎ Σ₂ ≣ Σ → ∀[ Allstar P Σ ⇒ Allstar P Σ₁ ✴ Allstar P Σ₂ ]
repartition [] nil = nil ×⟨ ⊎-idˡ ⟩ nil
repartition (consˡ σ) (cons (a ×⟨ σ′ ⟩ qx)) =
let
xs ×⟨ σ′′ ⟩ ys = repartition σ qx
_ , τ₁ , τ₂ = ⊎-unassoc σ′ σ′′
in (cons (a ×⟨ τ₁ ⟩ xs)) ×⟨ τ₂ ⟩ ys
repartition (consʳ σ) (cons (a ×⟨ σ′ ⟩ qx)) =
let
xs ×⟨ σ′′ ⟩ ys = repartition σ qx
_ , τ₁ , τ₂ = ⊎-unassoc σ′ (⊎-comm σ′′)
in xs ×⟨ ⊎-comm τ₂ ⟩ (cons (a ×⟨ τ₁ ⟩ ys))
concat : ∀ {P} {Γ₁ Γ₂} → ∀[ Allstar P Γ₁ ✴ Allstar P Γ₂ ⇒ Allstar P (Γ₁ ++ Γ₂) ]
concat (nil ×⟨ s ⟩ env₂) rewrite ⊎-id⁻ˡ s = env₂
concat (cons (v ×⟨ s ⟩ env₁) ×⟨ s' ⟩ env₂) =
let _ , eq₁ , eq₂ = ⊎-assoc s s' in
cons (v ×⟨ eq₁ ⟩ (concat (env₁ ×⟨ eq₂ ⟩ env₂)))
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------------------------------------------------------------------------------
-- Testing the translation of 11-ary predicates symbols
------------------------------------------------------------------------------
{-# OPTIONS --exact-split #-}
{-# OPTIONS --no-sized-types #-}
{-# OPTIONS --no-universe-polymorphism #-}
{-# OPTIONS --without-K #-}
module P11 where
postulate
D : Set
P : D → D → D → D → D → D → D → D → D → D → D → Set
postulate P-refl : ∀ {x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ x₉ x₁₀ x₁₁} →
P x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ x₉ x₁₀ x₁₁ →
P x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ x₉ x₁₀ x₁₁
{-# ATP prove P-refl #-}
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-- The bug documented below was exposed by the fix to issue 274.
{-# OPTIONS --universe-polymorphism #-}
module Issue274 where
postulate
Level : Set
zero : Level
suc : Level → Level
_⊔_ : Level → Level → Level
{-# BUILTIN LEVEL Level #-}
{-# BUILTIN LEVELZERO zero #-}
{-# BUILTIN LEVELSUC suc #-}
{-# BUILTIN LEVELMAX _⊔_ #-}
record Q a : Set (a ⊔ a) where
record R a : Set a where
field q : Q a
A : Set₁
A = Set
postulate
ℓ : Level
r : R (ℓ ⊔ ℓ)
foo : R ℓ
foo = r
-- Issue274.agda:32,7-8
-- ℓ ⊔ ℓ !=< ℓ of type Level
-- when checking that the expression r has type R ℓ
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{-# OPTIONS --without-K --rewriting #-}
open import HoTT
open import cohomology.ChainComplex
open import cohomology.Theory
open import groups.KernelImage
open import cw.CW
module cw.cohomology.ReconstructedFirstCohomologyGroup {i : ULevel} (OT : OrdinaryTheory i) where
open OrdinaryTheory OT
import cw.cohomology.TipCoboundary OT as TC
import cw.cohomology.HigherCoboundary OT as HC
import cw.cohomology.TipAndAugment OT as TAA
open import cw.cohomology.WedgeOfCells OT
open import cw.cohomology.Descending OT
open import cw.cohomology.ReconstructedCochainComplex OT
import cw.cohomology.FirstCohomologyGroup OT as FCG
import cw.cohomology.FirstCohomologyGroupOnDiag OT as FCGD
import cw.cohomology.CohomologyGroupsTooHigh OT as CGTH
private
≤-dec-has-all-paths : {m n : ℕ} → has-all-paths (Dec (m ≤ n))
≤-dec-has-all-paths = prop-has-all-paths (Dec-level ≤-is-prop)
private
abstract
first-cohomology-group-descend : ∀ {n} (⊙skel : ⊙Skeleton {i} (3 + n))
→ cohomology-group (cochain-complex ⊙skel) 1
== cohomology-group (cochain-complex (⊙cw-init ⊙skel)) 1
first-cohomology-group-descend {n = O} ⊙skel
= ap2 (λ δ₁ δ₂ → Ker/Im δ₂ δ₁ (CXₙ/Xₙ₋₁-is-abelian (⊙cw-take (lteSR lteS) ⊙skel) 1))
(coboundary-first-template-descend-from-far {n = 2} ⊙skel (ltSR ltS) ltS)
(coboundary-higher-template-descend-from-one-above ⊙skel)
first-cohomology-group-descend {n = S n} ⊙skel -- n = S n
= ap2 (λ δ₁ δ₂ → Ker/Im δ₂ δ₁ (CXₙ/Xₙ₋₁-is-abelian (⊙cw-take (≤-+-l 1 (lteSR $ lteSR $ inr (O<S n))) ⊙skel) 1))
(coboundary-first-template-descend-from-far {n = 3 + n} ⊙skel (ltSR (ltSR (O<S n))) (<-+-l 1 (ltSR (O<S n))))
(coboundary-higher-template-descend-from-far {n = 3 + n} ⊙skel (<-+-l 1 (ltSR (O<S n))) (<-+-l 2 (O<S n)))
first-cohomology-group-β : ∀ (⊙skel : ⊙Skeleton {i} 2)
→ cohomology-group (cochain-complex ⊙skel) 1
== Ker/Im
(HC.cw-co∂-last ⊙skel)
(TC.cw-co∂-head (⊙cw-init ⊙skel))
(CXₙ/Xₙ₋₁-is-abelian (⊙cw-init ⊙skel) 1)
first-cohomology-group-β ⊙skel
= ap2 (λ δ₁ δ₂ → Ker/Im δ₂ δ₁ (CXₙ/Xₙ₋₁-is-abelian (⊙cw-init ⊙skel) 1))
( coboundary-first-template-descend-from-two ⊙skel
∙ coboundary-first-template-β (⊙cw-init ⊙skel))
(coboundary-higher-template-β ⊙skel)
first-cohomology-group-β-one-below : ∀ (⊙skel : ⊙Skeleton {i} 1)
→ cohomology-group (cochain-complex ⊙skel) 1
== Ker/Im
(cst-hom {H = Lift-group {j = i} Unit-group})
(TC.cw-co∂-head ⊙skel)
(CXₙ/Xₙ₋₁-is-abelian ⊙skel 1)
first-cohomology-group-β-one-below ⊙skel
= ap
(λ δ₁ → Ker/Im
(cst-hom {H = Lift-group {j = i} Unit-group})
δ₁ (CXₙ/Xₙ₋₁-is-abelian ⊙skel 1))
(coboundary-first-template-β ⊙skel)
abstract
first-cohomology-group : ∀ {n} (⊙skel : ⊙Skeleton {i} n)
→ ⊙has-cells-with-choice 0 ⊙skel i
→ C 1 ⊙⟦ ⊙skel ⟧ ≃ᴳ cohomology-group (cochain-complex ⊙skel) 1
first-cohomology-group {n = 0} ⊙skel ac =
CGTH.C-cw-iso-ker/im 1 ltS (TAA.C2×CX₀ ⊙skel 0) ⊙skel ac
first-cohomology-group {n = 1} ⊙skel ac =
coe!ᴳ-iso (first-cohomology-group-β-one-below ⊙skel)
∘eᴳ FCGD.C-cw-iso-ker/im ⊙skel ac
first-cohomology-group {n = 2} ⊙skel ac =
coe!ᴳ-iso (first-cohomology-group-β ⊙skel)
∘eᴳ FCG.C-cw-iso-ker/im ⊙skel ac
first-cohomology-group {n = S (S (S n))} ⊙skel ac =
coe!ᴳ-iso (first-cohomology-group-descend ⊙skel)
∘eᴳ first-cohomology-group (⊙cw-init ⊙skel) (⊙init-has-cells-with-choice ⊙skel ac)
∘eᴳ C-cw-descend-at-lower ⊙skel (<-+-l 1 (O<S n)) ac
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{-# OPTIONS --without-K #-}
open import Base
open import Homotopy.Pointed
open import Homotopy.Connected
module Homotopy.Cover.HomotopyGroupSetIsomorphism {i}
(A⋆ : pType i) (A⋆-is-conn : is-connected⋆ ⟨0⟩ A⋆) where
open pType A⋆ renaming (∣_∣ to A ; ⋆ to a)
open import Algebra.Groups
open import Homotopy.Truncation
open import Homotopy.HomotopyGroups {i}
open import Homotopy.PathTruncation
open import Homotopy.Cover.Def A
open import Homotopy.Cover.Ribbon A⋆
private
module G = group (fundamental-group A⋆)
open import Algebra.GroupSets (fundamental-group A⋆)
{-
Isomorphism between pi1(A)-sets and coverings.
-}
gset⇒covering : gset → covering
gset⇒covering gset[ _ , act , _ ] = cov[ ribbon act , ribbon-is-set ]
covering⇒action : ∀ cov → action (covering.fiber cov a)
covering⇒action cov = act[ tracing cov , (λ _ → refl) , compose-tracing cov ]
covering⇒gset : covering → gset
covering⇒gset cov = let open covering cov in
gset[ fiber a , covering⇒action cov , fiber-is-set a ]
-- The first direction: covering -> gset -> covering
-- From 0-connectedness we can get a (-1)-truncated base path.
-- The challenge is to get this path.
abstract
[base-path] : ∀ a₂ → [ a ≡ a₂ ]
[base-path] = connected-has-all-τ-paths A⋆-is-conn a
-- Part 1: Show that the generated cover (ribbon) is fiberwisely
-- equivalent to the original fiber.
private
module _ (cov : covering) where
-- Suppose that we get the path, we can compute the ribbon easily.
fiber+path⇒ribbon : ∀ a₂ (y : covering.fiber cov a₂) (p : a ≡ a₂)
→ ribbon (covering⇒action cov) a₂
fiber+path⇒ribbon a₂ y p = trace (tracing cov y (proj $ ! p)) (proj p)
abstract
-- Our construction is "constant" with respect to paths.
fiber+path⇒ribbon-is-path-irrelevant : ∀ a₂
(y : covering.fiber cov a₂) (p₁ p₂ : a ≡ a₂)
→ fiber+path⇒ribbon a₂ y p₁ ≡ fiber+path⇒ribbon a₂ y p₂
fiber+path⇒ribbon-is-path-irrelevant .a y p refl =
trace (tracing cov y (proj $ ! p)) (proj p)
≡⟨ paste y (proj $ ! p) (proj p) ⟩
trace y (proj $ (! p ∘ p))
≡⟨ ap (λ x → trace y (proj x)) $ opposite-left-inverse p ⟩∎
trace y refl₀
∎
-- Call the magical factorization library.
open import Homotopy.Extensions.ToPropToConstSet
-- Now we can read the (-1)-truncated path.
fiber+path₋₁⇒ribbon : ∀ a₂ (y : covering.fiber cov a₂)
→ [ a ≡ a₂ ] → ribbon (covering⇒action cov) a₂
fiber+path₋₁⇒ribbon a₂ y = cst-extend
⦃ ribbon-is-set a₂ ⦄
(fiber+path⇒ribbon a₂ y)
(fiber+path⇒ribbon-is-path-irrelevant a₂ y)
-- So the conversion from fiber to ribbon is done.
fiber⇒ribbon : ∀ cov a₂ → covering.fiber cov a₂ → ribbon (covering⇒action cov) a₂
fiber⇒ribbon cov a₂ y = fiber+path₋₁⇒ribbon cov a₂ y $ [base-path] a₂
-- The other direction is easy.
ribbon⇒fiber : ∀ cov a₂ → ribbon (covering⇒action cov) a₂ → covering.fiber cov a₂
ribbon⇒fiber cov a₂ = let open covering cov in
ribbon-rec-nondep a₂ (fiber a₂) ⦃ fiber-is-set a₂ ⦄ (tracing cov) (compose-tracing cov)
private
-- Some routine computations.
abstract
ribbon⇒fiber⇒ribbon : ∀ cov a₂ r → fiber⇒ribbon cov a₂ (ribbon⇒fiber cov a₂ r) ≡ r
ribbon⇒fiber⇒ribbon cov a₂ = ribbon-rec a₂
(λ r → fiber⇒ribbon cov a₂ (ribbon⇒fiber cov a₂ r) ≡ r)
⦃ λ _ → ≡-is-set $ ribbon-is-set a₂ ⦄
(λ y p → []-extend
-- All ugly things will go away when bp = proj bp′
⦃ λ bp → ribbon-is-set a₂
(fiber+path₋₁⇒ribbon cov a₂ (tracing cov y p) bp)
(trace y p) ⦄
(lemma a₂ y p)
([base-path] a₂))
(λ _ _ _ → prop-has-all-paths (ribbon-is-set a₂ _ _) _ _)
where
abstract
lemma : ∀ a₂ (y : covering.fiber cov a) (p : a ≡₀ a₂) (bp : a ≡ a₂)
→ trace {act = covering⇒action cov}
(tracing cov (tracing cov y p) (proj $ ! bp)) (proj bp)
≡ trace y p
lemma .a y p refl =
trace (tracing cov y p) refl₀
≡⟨ paste y p refl₀ ⟩
trace y (p ∘₀ refl₀)
≡⟨ ap (trace y) $ refl₀-right-unit p ⟩∎
trace y p
∎
fiber⇒ribbon⇒fiber : ∀ cov a₂ y → ribbon⇒fiber cov a₂ (fiber⇒ribbon cov a₂ y) ≡ y
fiber⇒ribbon⇒fiber cov a₂ y = let open covering cov in []-extend
-- All ugly things will go away when bp = proj bp′
⦃ λ bp → fiber-is-set a₂
(ribbon⇒fiber cov a₂
(fiber+path₋₁⇒ribbon cov a₂ y bp))
y ⦄
(lemma a₂ y)
([base-path] a₂)
where
abstract
lemma : ∀ a₂ (y : covering.fiber cov a₂) (bp : a ≡ a₂)
→ tracing cov (tracing cov y (proj $ ! bp)) (proj bp)
≡ y
lemma .a y refl = refl
covering⇒gset⇒covering : ∀ cov → gset⇒covering (covering⇒gset cov) ≡ cov
covering⇒gset⇒covering cov = covering-eq $ funext λ a₂
→ eq-to-path $ ribbon⇒fiber cov a₂ , iso-is-eq
(ribbon⇒fiber cov a₂)
(fiber⇒ribbon cov a₂)
(fiber⇒ribbon⇒fiber cov a₂)
(ribbon⇒fiber⇒ribbon cov a₂)
-- The second direction : gset -> covering -> gset
-- Part 2.1: The fiber over the point a is the carrier.
ribbon-a⇒Y : ∀ {Y} {act : action Y} ⦃ _ : is-set Y ⦄ → ribbon act a → Y
ribbon-a⇒Y {Y} {act} ⦃ Y-is-set ⦄ = let open action act in
ribbon-rec-nondep a Y ⦃ Y-is-set ⦄ _∙_ assoc
ribbon-a≃Y : ∀ {Y} {act : action Y} ⦃ _ : is-set Y ⦄ → ribbon act a ≃ Y
ribbon-a≃Y {Y} {act} ⦃ Y-is-set ⦄ = let open action act in
ribbon-a⇒Y ⦃ Y-is-set ⦄ , iso-is-eq _
(λ y → trace y refl₀)
(λ y → right-unit y)
(ribbon-rec a
(λ r → trace (ribbon-a⇒Y ⦃ Y-is-set ⦄ r) refl₀ ≡ r)
⦃ λ _ → ≡-is-set $ ribbon-is-set a ⦄
(λ y p →
trace (y ∙ p) refl₀
≡⟨ paste y p refl₀ ⟩
trace y (p G.∙ refl₀)
≡⟨ ap (trace y) $ G.right-unit p ⟩∎
trace y p
∎)
(λ _ _ _ → prop-has-all-paths (ribbon-is-set a _ _) _ _))
private
-- Some lemmas to simplify the proofs.
trans-eq-∙ : ∀ {Y₁ Y₂ : Set i} (Y≃ : Y₁ ≃ Y₂) (_∙_ : Y₁ → G.carrier → Y₁) (y₂ : Y₂) (g : G.carrier)
→ transport (λ Y → Y → G.carrier → Y) (eq-to-path Y≃) _∙_ y₂ g ≡ (Y≃ ☆ (inverse Y≃ y₂ ∙ g))
trans-eq-∙ = equiv-induction
(λ {Y₁ Y₂ : Set i} (Y≃ : Y₁ ≃ Y₂)
→ ∀ (_∙_ : Y₁ → G.carrier → Y₁) (y₂ : Y₂) (g : G.carrier)
→ transport (λ Y → Y → G.carrier → Y) (eq-to-path Y≃) _∙_ y₂ g ≡ (Y≃ ☆ (inverse Y≃ y₂ ∙ g)))
(λ Y _∙_ y₂ g → ap (λ x → transport (λ Y → Y → G.carrier → Y) x _∙_ y₂ g)
$ path-to-eq-right-inverse refl)
gset⇒covering⇒gset : ∀ gs → covering⇒gset (gset⇒covering gs) ≡ gs
gset⇒covering⇒gset gset[ Y , act , Y-is-set ] =
let
open action act
_⊙_ = tracing cov[ ribbon act , ribbon-is-set {Y} {act} ]
≃Y = ribbon-a≃Y ⦃ Y-is-set ⦄
⇒Y = ribbon-a⇒Y ⦃ Y-is-set ⦄
in gset-eq
(eq-to-path ≃Y)
(funext λ y → funext $ π₀-extend ⦃ λ _ → ≡-is-set Y-is-set ⦄ λ p →
transport (λ Y → Y → G.carrier → Y) (eq-to-path ≃Y) _⊙_ y (proj p)
≡⟨ trans-eq-∙ ≃Y _⊙_ y (proj p) ⟩
⇒Y (transport (ribbon act) p (trace y refl₀))
≡⟨ ap ⇒Y $ trans-trace act p y refl₀ ⟩∎
y ∙ proj p
∎)
-- Finally...
gset-covering-eq : gset ≃ covering
gset-covering-eq = gset⇒covering , iso-is-eq _ covering⇒gset
covering⇒gset⇒covering
gset⇒covering⇒gset
{-
Universality of the covering generated by the fundamental group itself.
-}
-- FIXME What's the established terminology for this?
canonical-gset : gset
canonical-gset = record
{ carrier = G.carrier
; act = record
{ _∙_ = _∘₀_
; right-unit = refl₀-right-unit
; assoc = concat₀-assoc
}
; set = π₀-is-set (a ≡ a)
}
-- FIXME What's the established terminology for this?
canonical-covering : covering
canonical-covering = gset⇒covering canonical-gset
private
module Universality where
open covering canonical-covering
open gset canonical-gset
center′ : Σ A fiber
center′ = (a , trace {act = act} refl₀ refl₀)
center : τ ⟨1⟩ (Σ A fiber)
center = proj center′
private
-- An ugly lemma for this development only
trans-fiber≡cst-proj-Σ-eq : ∀ {i} (P : Set i) (Q : P → Set i)
(a : P) (c : Σ P Q) {b₁ b₂} (p : b₁ ≡ b₂) (q : a ≡ π₁ c)
(r : transport Q q b₁ ≡ π₂ c)
→ transport (λ r → (a , r) ≡₀ c) p (proj $ Σ-eq q r)
≡ proj (Σ-eq q (ap (transport Q q) (! p) ∘ r))
trans-fiber≡cst-proj-Σ-eq P Q a c refl q r = refl
abstract
path-trace-fiber : ∀ {a₂} y (p : a ≡ a₂)
→ transport fiber (! p ∘ ! y) (trace (proj y) (proj p))
≡ trace refl₀ refl₀
path-trace-fiber y refl =
transport fiber (! y) (trace (proj y) refl₀)
≡⟨ trans-trace act (! y) (proj y) refl₀ ⟩
trace (proj y) (proj $ ! y)
≡⟨ paste refl₀ (proj y) (proj $ ! y) ⟩
trace refl₀ (proj $ y ∘ ! y)
≡⟨ ap (trace refl₀ ◯ proj) $ opposite-right-inverse y ⟩∎
trace refl₀ refl₀
∎
path-trace : ∀ {a₂} y p → (a₂ , trace {act = act} y p) ≡₀ center′
path-trace {a₂} =
π₀-extend ⦃ λ y → Π-is-set λ p → π₀-is-set ((a₂ , trace y p) ≡ center′) ⦄
(λ y → π₀-extend ⦃ λ p → π₀-is-set ((a₂ , trace (proj y) p) ≡ center′) ⦄
(λ p → proj $ Σ-eq (! p ∘ ! y) (path-trace-fiber y p)))
abstract
path-paste′ : ∀ {a₂} y loop p
→ transport (λ r → (a₂ , r) ≡₀ center′) (paste (proj y) (proj loop) (proj p))
(path-trace (proj $ y ∘ loop) (proj p))
≡ path-trace (proj y) (proj $ loop ∘ p)
path-paste′ y loop refl =
transport (λ r → (a , r) ≡₀ center′) (paste (proj y) (proj loop) refl₀)
(proj $ Σ-eq (! (y ∘ loop)) (path-trace-fiber (y ∘ loop) refl))
≡⟨ trans-fiber≡cst-proj-Σ-eq A fiber a center′
(paste (proj y) (proj loop) refl₀)
(! (y ∘ loop)) (path-trace-fiber (y ∘ loop) refl) ⟩
proj (Σ-eq (! (y ∘ loop)) _)
≡⟨ ap proj $
ap2 (λ p q → Σ-eq p q)
(! (y ∘ loop)
≡⟨ opposite-concat y loop ⟩
! loop ∘ ! y
≡⟨ ap (λ x → ! x ∘ ! y) $ ! $ refl-right-unit loop ⟩∎
! (loop ∘ refl) ∘ ! y
∎)
(prop-has-all-paths (ribbon-is-set a _ _) _ _) ⟩∎
proj (Σ-eq (! (loop ∘ refl) ∘ ! y) (path-trace-fiber y (loop ∘ refl)))
∎
abstract
path-paste : ∀ {a₂} y loop p
→ transport (λ r → (a₂ , r) ≡₀ center′) (paste y loop p)
(path-trace (y ∘₀ loop) p)
≡ path-trace y (loop ∘₀ p)
path-paste {a₂} =
π₀-extend ⦃ λ y → Π-is-set λ loop → Π-is-set λ p → ≡-is-set $ π₀-is-set _ ⦄
(λ y → π₀-extend ⦃ λ loop → Π-is-set λ p → ≡-is-set $ π₀-is-set _ ⦄
(λ loop → π₀-extend ⦃ λ p → ≡-is-set $ π₀-is-set _ ⦄
(λ p → path-paste′ y loop p)))
path′ : (y : Σ A fiber) → proj {n = ⟨1⟩} y ≡ center
path′ y = τ-path-equiv-path-τ-S {n = ⟨0⟩} ☆
ribbon-rec {act = act} (π₁ y)
(λ r → (π₁ y , r) ≡₀ center′)
⦃ λ r → π₀-is-set ((π₁ y , r) ≡ center′) ⦄
path-trace
path-paste
(π₂ y)
path : (y : τ ⟨1⟩ (Σ A fiber)) → y ≡ center
path = τ-extend {n = ⟨1⟩} ⦃ λ _ → ≡-is-truncated ⟨1⟩ $ τ-is-truncated ⟨1⟩ _ ⦄ path′
canonical-covering-is-universal : is-universal canonical-covering
canonical-covering-is-universal = Universality.center , Universality.path
-- The other direction: If a covering is universal, then the fiber
-- is equivalent to the fundamental group.
module _ (cov : covering) (cov-is-universal : is-universal cov) where
open covering cov
open action (covering⇒action cov)
-- We need a point!
module GiveMeAPoint (center : fiber a) where
-- Goal: fiber a <-> fundamental group
fiber-a⇒fg : fiber a → a ≡₀ a
fiber-a⇒fg y = ap₀ π₁ $ connected-has-all-τ-paths
cov-is-universal (a , center) (a , y)
fg⇒fiber-a : a ≡₀ a → fiber a
fg⇒fiber-a = tracing cov center
fg⇒fiber-a⇒fg : ∀ p → fiber-a⇒fg (fg⇒fiber-a p) ≡ p
fg⇒fiber-a⇒fg = π₀-extend ⦃ λ _ → ≡-is-set $ π₀-is-set _ ⦄ λ p →
ap₀ π₁ (connected-has-all-τ-paths
cov-is-universal (a , center) (a , transport fiber p center))
≡⟨ ap (ap₀ π₁)
$ ! $ π₂ (connected-has-connected-paths cov-is-universal _ _)
(proj $ Σ-eq p refl) ⟩
ap₀ π₁ (proj $ Σ-eq p refl)
≡⟨ ap proj $ base-path-Σ-eq p refl ⟩∎
proj p
∎
fiber-a⇒fg⇒fiber-a : ∀ y → fg⇒fiber-a (fiber-a⇒fg y) ≡ y
fiber-a⇒fg⇒fiber-a y = π₀-extend
⦃ λ p → ≡-is-set {x = tracing cov center (ap₀ π₁ p)} {y = y}
$ fiber-is-set a ⦄
(λ p →
transport fiber (base-path p) center
≡⟨ trans-base-path p ⟩∎
y
∎)
(connected-has-all-τ-paths cov-is-universal (a , center) (a , y))
fiber-a≃fg : fiber a ≃ (a ≡₀ a)
fiber-a≃fg = fiber-a⇒fg , iso-is-eq _ fg⇒fiber-a
fg⇒fiber-a⇒fg fiber-a⇒fg⇒fiber-a
-- This is the best we can obtain, because there is no continuous
-- choice of the center.
[center] : [ fiber a ]
[center] = τ-extend-nondep
⦃ prop-is-gpd []-is-prop ⦄
(λ y → []-extend-nondep
⦃ []-is-prop ⦄
(proj ◯ λ p → transport fiber p (π₂ y))
(connected-has-all-τ-paths A⋆-is-conn (π₁ y) a))
(π₁ cov-is-universal)
-- [ isomorphism between the fiber and the fundamental group ]
-- This is the best we can obtain, because there is no continuous
-- choice of the center.
[fiber-a≃fg] : [ fiber a ≃ (a ≡₀ a) ]
[fiber-a≃fg] = []-extend-nondep ⦃ []-is-prop ⦄
(proj ◯ GiveMeAPoint.fiber-a≃fg) [center]
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{-# OPTIONS --safe #-}
module Cubical.HITs.Susp.Properties where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Path
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Pointed.Homogeneous
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Data.Bool
open import Cubical.Data.Sigma
open import Cubical.HITs.Join
open import Cubical.HITs.Susp.Base
open import Cubical.Homotopy.Loopspace
private
variable
ℓ : Level
open Iso
Susp-iso-joinBool : ∀ {ℓ} {A : Type ℓ} → Iso (Susp A) (join A Bool)
fun Susp-iso-joinBool north = inr true
fun Susp-iso-joinBool south = inr false
fun Susp-iso-joinBool (merid a i) = (sym (push a true) ∙ push a false) i
inv Susp-iso-joinBool (inr true ) = north
inv Susp-iso-joinBool (inr false) = south
inv Susp-iso-joinBool (inl _) = north
inv Susp-iso-joinBool (push a true i) = north
inv Susp-iso-joinBool (push a false i) = merid a i
rightInv Susp-iso-joinBool (inr true ) = refl
rightInv Susp-iso-joinBool (inr false) = refl
rightInv Susp-iso-joinBool (inl a) = sym (push a true)
rightInv Susp-iso-joinBool (push a true i) j = push a true (i ∨ ~ j)
rightInv Susp-iso-joinBool (push a false i) j
= hcomp (λ k → λ { (i = i0) → push a true (~ j)
; (i = i1) → push a false k
; (j = i1) → push a false (i ∧ k) })
(push a true (~ i ∧ ~ j))
leftInv Susp-iso-joinBool north = refl
leftInv Susp-iso-joinBool south = refl
leftInv (Susp-iso-joinBool {A = A}) (merid a i) j
= hcomp (λ k → λ { (i = i0) → transp (λ _ → Susp A) (k ∨ j) north
; (i = i1) → transp (λ _ → Susp A) (k ∨ j) (merid a k)
; (j = i1) → merid a (i ∧ k) })
(transp (λ _ → Susp A) j north)
Susp≃joinBool : ∀ {ℓ} {A : Type ℓ} → Susp A ≃ join A Bool
Susp≃joinBool = isoToEquiv Susp-iso-joinBool
Susp≡joinBool : ∀ {ℓ} {A : Type ℓ} → Susp A ≡ join A Bool
Susp≡joinBool = isoToPath Susp-iso-joinBool
congSuspIso : ∀ {ℓ} {A B : Type ℓ} → Iso A B → Iso (Susp A) (Susp B)
fun (congSuspIso is) = suspFun (fun is)
inv (congSuspIso is) = suspFun (inv is)
rightInv (congSuspIso is) north = refl
rightInv (congSuspIso is) south = refl
rightInv (congSuspIso is) (merid a i) j = merid (rightInv is a j) i
leftInv (congSuspIso is) north = refl
leftInv (congSuspIso is) south = refl
leftInv (congSuspIso is) (merid a i) j = merid (leftInv is a j) i
congSuspEquiv : ∀ {ℓ} {A B : Type ℓ} → A ≃ B → Susp A ≃ Susp B
congSuspEquiv {ℓ} {A} {B} h = isoToEquiv (congSuspIso (equivToIso h))
suspToPropElim : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Susp A → Type ℓ'} (a : A)
→ ((x : Susp A) → isProp (B x))
→ B north
→ (x : Susp A) → B x
suspToPropElim a isProp Bnorth north = Bnorth
suspToPropElim {B = B} a isProp Bnorth south = subst B (merid a) Bnorth
suspToPropElim {B = B} a isProp Bnorth (merid a₁ i) =
isOfHLevel→isOfHLevelDep 1 isProp Bnorth (subst B (merid a) Bnorth) (merid a₁) i
suspToPropElim2 : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Susp A → Susp A → Type ℓ'} (a : A)
→ ((x y : Susp A) → isProp (B x y))
→ B north north
→ (x y : Susp A) → B x y
suspToPropElim2 _ _ Bnorth north north = Bnorth
suspToPropElim2 {B = B} a _ Bnorth north south = subst (B north) (merid a) Bnorth
suspToPropElim2 {B = B} a isprop Bnorth north (merid x i) =
isProp→PathP (λ i → isprop north (merid x i))
Bnorth (subst (B north) (merid a) Bnorth) i
suspToPropElim2 {B = B} a _ Bnorth south north = subst (λ x → B x north) (merid a) Bnorth
suspToPropElim2 {B = B} a _ Bnorth south south = subst (λ x → B x x) (merid a) Bnorth
suspToPropElim2 {B = B} a isprop Bnorth south (merid x i) =
isProp→PathP (λ i → isprop south (merid x i))
(subst (λ x → B x north) (merid a) Bnorth)
(subst (λ x → B x x) (merid a) Bnorth) i
suspToPropElim2 {B = B} a isprop Bnorth (merid x i) north =
isProp→PathP (λ i → isprop (merid x i) north)
Bnorth (subst (λ x → B x north) (merid a) Bnorth) i
suspToPropElim2 {B = B} a isprop Bnorth (merid x i) south =
isProp→PathP (λ i → isprop (merid x i) south)
(subst (B north) (merid a) Bnorth)
(subst (λ x → B x x) (merid a) Bnorth) i
suspToPropElim2 {B = B} a isprop Bnorth (merid x i) (merid y j) =
isSet→SquareP (λ i j → isOfHLevelSuc 1 (isprop _ _))
(isProp→PathP (λ i₁ → isprop north (merid y i₁)) Bnorth
(subst (B north) (merid a) Bnorth))
(isProp→PathP (λ i₁ → isprop south (merid y i₁))
(subst (λ x₁ → B x₁ north) (merid a) Bnorth)
(subst (λ x₁ → B x₁ x₁) (merid a) Bnorth))
(isProp→PathP (λ i₁ → isprop (merid x i₁) north) Bnorth
(subst (λ x₁ → B x₁ north) (merid a) Bnorth))
(isProp→PathP (λ i₁ → isprop (merid x i₁) south)
(subst (B north) (merid a) Bnorth)
(subst (λ x₁ → B x₁ x₁) (merid a) Bnorth)) i j
{- Clever proof:
suspToPropElim2 a isProp Bnorth =
suspToPropElim a (λ x → isOfHLevelΠ 1 λ y → isProp x y)
(suspToPropElim a (λ x → isProp north x) Bnorth)
-}
funSpaceSuspIso : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'}
→ Iso (Σ[ x ∈ B ] Σ[ y ∈ B ] (A → x ≡ y)) (Susp A → B)
Iso.fun funSpaceSuspIso (x , y , f) north = x
Iso.fun funSpaceSuspIso (x , y , f) south = y
Iso.fun funSpaceSuspIso (x , y , f) (merid a i) = f a i
Iso.inv funSpaceSuspIso f = (f north) , (f south , (λ x → cong f (merid x)))
Iso.rightInv funSpaceSuspIso f = funExt λ {north → refl
; south → refl
; (merid a i) → refl}
Iso.leftInv funSpaceSuspIso _ = refl
toSusp : (A : Pointed ℓ) → typ A → typ (Ω (Susp∙ (typ A)))
toSusp A x = merid x ∙ merid (pt A) ⁻¹
toSuspPointed : (A : Pointed ℓ) → A →∙ Ω (Susp∙ (typ A))
fst (toSuspPointed A) = toSusp A
snd (toSuspPointed A) = rCancel (merid (pt A))
module _ {ℓ ℓ' : Level} {A : Pointed ℓ} {B : Pointed ℓ'} where
fromSusp→toΩ : Susp∙ (typ A) →∙ B → (A →∙ Ω B)
fst (fromSusp→toΩ f) x = sym (snd f) ∙∙ cong (fst f) (toSusp A x) ∙∙ snd f
snd (fromSusp→toΩ f) =
cong (sym (snd f) ∙∙_∙∙ (snd f))
(cong (cong (fst f))
(rCancel (merid (pt A))))
∙ ∙∙lCancel (snd f)
toΩ→fromSusp : A →∙ Ω B → Susp∙ (typ A) →∙ B
fst (toΩ→fromSusp f) north = pt B
fst (toΩ→fromSusp f) south = pt B
fst (toΩ→fromSusp f) (merid a i) = fst f a i
snd (toΩ→fromSusp f) = refl
ΩSuspAdjointIso : Iso (A →∙ Ω B) (Susp∙ (typ A) →∙ B)
fun ΩSuspAdjointIso = toΩ→fromSusp
inv ΩSuspAdjointIso = fromSusp→toΩ
rightInv ΩSuspAdjointIso f =
ΣPathP (funExt
(λ { north → sym (snd f)
; south → sym (snd f) ∙ cong (fst f) (merid (pt A))
; (merid a i) j →
hcomp (λ k → λ { (i = i0) → snd f (~ j ∧ k)
; (i = i1) → compPath-filler'
(sym (snd f))
(cong (fst f) (merid (pt A))) k j
; (j = i1) → fst f (merid a i)})
(fst f (compPath-filler (merid a) (sym (merid (pt A))) (~ j) i))})
, λ i j → snd f (~ i ∨ j))
leftInv ΩSuspAdjointIso f =
→∙Homogeneous≡ (isHomogeneousPath _ _)
(funExt λ x → sym (rUnit _)
∙ cong-∙ (fst (toΩ→fromSusp f)) (merid x) (sym (merid (pt A)))
∙ cong (fst f x ∙_) (cong sym (snd f))
∙ sym (rUnit _))
IsoΩFunSuspFun : Iso (typ (Ω (A →∙ B ∙))) (Susp∙ (typ A) →∙ B)
IsoΩFunSuspFun = compIso (ΩfunExtIso A B) ΩSuspAdjointIso
-- inversion
invSusp : ∀ {ℓ} {A : Type ℓ} → Susp A → Susp A
invSusp north = south
invSusp south = north
invSusp (merid a i) = merid a (~ i)
invSusp² : ∀ {ℓ} {A : Type ℓ} (x : Susp A) → invSusp (invSusp x) ≡ x
invSusp² north = refl
invSusp² south = refl
invSusp² (merid a i) = refl
invSuspIso : ∀ {ℓ} {A : Type ℓ} → Iso (Susp A) (Susp A)
fun invSuspIso = invSusp
inv invSuspIso = invSusp
rightInv invSuspIso = invSusp²
leftInv invSuspIso = invSusp²
-- Explicit definition of the iso
-- join (Susp A) B ≃ Susp (join A B)
-- for pointed types A and B. This is useful for obtaining a ``nice'' iso
-- join Sⁿ Sᵐ ≃ Sⁿ⁺ᵐ⁺¹
module _ {A B : Pointed ℓ} where
private -- some useful fillers
rinv-filler : (b : typ B) → I → I → I → join (Susp (typ A)) (typ B)
rinv-filler b i j k =
hfill (λ k → λ {(i = i0) → push south b (~ k)
; (i = i1) → push north b (~ k ∨ j)
; (j = i0) → push (merid (pt A) (~ i)) b (~ k)
; (j = i1) → push south b (~ k ∨ i)})
(inS (inr b))
k
suspJoin→joinSuspFiller :
I → I → I → (a : typ A) (b : typ B) → join (Susp (typ A)) (typ B)
suspJoin→joinSuspFiller i j k a b =
hfill (λ k → λ {(i = i0) → push north b (~ k)
; (i = i1) → push south b (~ k)
; (j = i0) → push (merid a i) b (~ k)
; (j = i1) → push (merid (pt A) i) b (~ k)})
(inS (inr b))
k
joinSuspFiller :
I → I → I → (a : typ A) (b : typ B) → Susp (join (typ A) (typ B))
joinSuspFiller i j k a b =
hfill (λ k → λ {(i = i0) → merid (push a b (~ k)) j
; (i = i1) → north
; (j = i0) → north
; (j = i1) → merid (push (pt A) b (~ k)) (~ i)})
(inS (merid (inr b) (~ i ∧ j)))
k
suspJoin→joinSusp : Susp (join (typ A) (typ B)) → join (Susp (typ A)) (typ B)
suspJoin→joinSusp north = inl north
suspJoin→joinSusp south = inl south
suspJoin→joinSusp (merid (inl x) i) = inl ((merid x) i)
suspJoin→joinSusp (merid (inr x) i) = inl (merid (pt A) i)
suspJoin→joinSusp (merid (push a b j) i) = suspJoin→joinSuspFiller i j i1 a b
joinSusp→suspJoin : join (Susp (typ A)) (typ B) → Susp (join (typ A) (typ B))
joinSusp→suspJoin (inl north) = north
joinSusp→suspJoin (inl south) = south
joinSusp→suspJoin (inl (merid a i)) = merid (inl a) i
joinSusp→suspJoin (inr x) = north
joinSusp→suspJoin (push north b i) = north
joinSusp→suspJoin (push south b i) = merid (inl (pt A)) (~ i)
joinSusp→suspJoin (push (merid a j) b i) = joinSuspFiller i j i1 a b
suspJoin→joinSusp→suspJoin : (x : Susp (join (typ A) (typ B)))
→ joinSusp→suspJoin (suspJoin→joinSusp x) ≡ x
suspJoin→joinSusp→suspJoin north = refl
suspJoin→joinSusp→suspJoin south = refl
suspJoin→joinSusp→suspJoin (merid (inl x) i) = refl
suspJoin→joinSusp→suspJoin (merid (inr x) i) j = merid (push (pt A) x j) i
suspJoin→joinSusp→suspJoin (merid (push a b j) i) k =
hcomp (λ r → λ {(i = i0) → north
; (i = i1) → merid (push (snd A) b (k ∧ (~ r ∨ j))) r
; (j = i0) → joinSuspFiller (~ r) i (~ k ∨ r) a b
; (j = i1) → joinSuspFiller (~ r) i (~ k) (pt A) b
; (k = i0) → joinSusp→suspJoin (suspJoin→joinSuspFiller i j r a b)
; (k = i1) → k=i1 i j r})
north
where
k=i1 :
Cube
(λ j r → north)
(λ j r → merid (push (snd A) b (~ r ∨ j)) r)
(λ i r → joinSuspFiller (~ r) i r a b)
(λ i r → merid (inr b) (r ∧ i))
refl
λ i j → merid (push a b j) i
k=i1 i j r =
hcomp (λ k → λ {(i = i0) → north
; (i = i1) → merid (push (snd A) b (~ r ∨ ~ k ∨ j)) r
; (j = i0) → joinSuspFiller (~ r) i (r ∧ k) a b
; (j = i1) → merid (inr b) (r ∧ i)
; (r = i0) → north
; (r = i1) → merid (push a b (~ k ∨ j)) i})
(merid (inr b) (i ∧ r))
joinSusp→suspJoin→joinSusp : (x : join (Susp (typ A)) (typ B))
→ suspJoin→joinSusp (joinSusp→suspJoin x) ≡ x
joinSusp→suspJoin→joinSusp (inl north) = refl
joinSusp→suspJoin→joinSusp (inl south) = refl
joinSusp→suspJoin→joinSusp (inl (merid a i)) = refl
joinSusp→suspJoin→joinSusp (inr x) = push north x
joinSusp→suspJoin→joinSusp (push north b i) j = push north b (j ∧ i)
joinSusp→suspJoin→joinSusp (push south b i) j = rinv-filler b i j i1
joinSusp→suspJoin→joinSusp (push (merid a j) b i) k =
hcomp (λ r → λ { (j = i0) → push north b (k ∧ i)
; (j = i1) → lem i k r
; (i = i0) → suspJoin→joinSuspFiller j (~ r ∧ ~ k) i1 a b
; (i = i1) → push north b k
; (k = i0) → suspJoin→joinSusp (joinSuspFiller i j r a b)
; (k = i1) → push (merid a j) b i})
(hcomp (λ r → λ { (j = i0) → push north b (~ r ∨ (k ∧ i))
; (j = i1) → rinv-filler b i k r
; (i = i0) → suspJoin→joinSuspFiller j (~ k) r a b
; (i = i1) → push north b (~ r ∨ k)
; (k = i0) → push (merid (snd A) (~ i ∧ j)) b (~ r)
; (k = i1) → push (merid a j) b (~ r ∨ i)})
(inr b))
where
lem : Cube (λ k r → inl south)
(λ k r → push north b k)
(λ i r → suspJoin→joinSuspFiller (~ i) r i1 (pt A) b)
(λ i r → push south b i)
(λ i k → rinv-filler b i k i1)
λ i k → rinv-filler b i k i1
lem i k r =
hcomp (λ j → λ { (r = i0) → rinv-filler b i k j
; (r = i1) → rinv-filler b i k j
; (i = i0) → push south b (~ j)
; (i = i1) → push north b (k ∨ ~ j)
; (k = i0) → suspJoin→joinSuspFiller (~ i) r j (pt A) b
; (k = i1) → push south b (i ∨ ~ j)})
(inr b)
Iso-joinSusp-suspJoin :
Iso (join (Susp (typ A)) (typ B)) (Susp (join (typ A) (typ B)))
Iso.fun Iso-joinSusp-suspJoin = joinSusp→suspJoin
Iso.inv Iso-joinSusp-suspJoin = suspJoin→joinSusp
Iso.rightInv Iso-joinSusp-suspJoin = suspJoin→joinSusp→suspJoin
Iso.leftInv Iso-joinSusp-suspJoin = joinSusp→suspJoin→joinSusp
-- interaction between invSusp and toSusp
toSusp-invSusp : (A : Pointed ℓ) (x : Susp (typ A))
→ toSusp (Susp∙ (typ A)) (invSusp x) ≡ sym (toSusp (Susp∙ (typ A)) x)
toSusp-invSusp A north =
cong (toSusp (Susp∙ (typ A))) (sym (merid (snd A)))
∙∙ rCancel (merid north)
∙∙ cong sym (sym (rCancel (merid north)))
toSusp-invSusp A south =
rCancel (merid north)
∙∙ cong sym (sym (rCancel (merid north)))
∙∙ cong sym (cong (toSusp (Susp∙ (typ A))) (merid (pt A)))
toSusp-invSusp A (merid a i) j =
lem (toSusp (Susp∙ (typ A)) north) (toSusp (Susp∙ (typ A)) south)
(sym (rCancel (merid north)))
(cong (toSusp (Susp∙ (typ A))) ((merid (pt A))))
(cong (toSusp (Susp∙ (typ A))) (merid a)) (~ j) i
where
lem : {A : Type ℓ} {x : A} (p q : x ≡ x) (l : refl ≡ p)
(coh r : p ≡ q)
→ PathP (λ i → (cong sym (sym l) ∙∙ l ∙∙ coh) i
≡ (cong sym (sym coh) ∙∙ cong sym (sym l) ∙∙ l) i)
(cong sym r)
(sym r)
lem p q =
J (λ p l → (coh r : p ≡ q)
→ PathP (λ i → (cong sym (sym l) ∙∙ l ∙∙ coh) i
≡ (cong sym (sym coh) ∙∙ cong sym (sym l) ∙∙ l) i)
(cong sym r)
(sym r))
(J (λ q coh → (r : refl ≡ q)
→ PathP (λ i → (refl ∙ coh) i
≡ (cong sym (sym coh) ∙∙ refl ∙∙ refl) i)
(cong sym r)
(sym r))
λ r → flipSquare (sym (rUnit refl)
◁ (flipSquare (sym (sym≡cong-sym r))
▷ rUnit refl)))
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-- With functions are now inlined before termination checking.
module Issue59 where
open import Common.Prelude
open import Common.Equality
module Order (A : Set) (_≤_ : A → A → Bool) where
-- This now termination checks.
merge : List A → List A → List A
merge [] ys = ys
merge xs [] = xs
merge (x ∷ xs) (y ∷ ys) with x ≤ y
merge (x ∷ xs) (y ∷ ys) | false = y ∷ merge (x ∷ xs) ys
merge (x ∷ []) (y ∷ ys) | true = x ∷ y ∷ ys
merge (x ∷ x₁ ∷ xs) (y ∷ ys) | true = x ∷ merge (x₁ ∷ xs) (y ∷ ys)
data Ordering : Nat → Nat → Set where
eq : ∀ n → Ordering n n
lt : ∀ n d → Ordering n (n + suc d)
gt : ∀ n d → Ordering (n + suc d) n
-- Just make sure we didn't mess anything up when there are dot patterns.
-- Andreas, 2013-11-11: But there are no recursive calls in the clauses
-- with dot patterns!
compare : ∀ n m → Ordering n m
compare zero zero = eq zero
compare zero (suc m) = lt zero m
compare (suc n) zero = gt zero n
compare (suc n) (suc m) with compare n m
compare (suc n) (suc .(n + suc d)) | lt .n d = lt (suc n) d
compare (suc .(m + suc d)) (suc m) | gt .m d = gt (suc m) d
compare (suc n) (suc .n) | eq .n = eq (suc n)
-- Rewrite
plus-zero : ∀ n → n + 0 ≡ n
plus-zero zero = refl
plus-zero (suc n) rewrite plus-zero n = refl
plus-suc : ∀ n m → n + suc m ≡ suc n + m
plus-suc zero m = refl
plus-suc (suc n) m rewrite plus-suc n m = refl
commute : ∀ m n → m + n ≡ n + m
commute m zero = plus-zero m
commute m (suc n) rewrite plus-suc m n | commute m n = refl
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{-# OPTIONS --without-K #-}
{-
Favonia: I was trying to generalize OneSkeleton but failed
to achieve what I wanted. Nicolai then told me this HIT
which is suitable for the constancy lemma I was looking for.
This construction should be attributed to Paolo Capriotti
and Nicolai Kraus. [1]
[1] Eliminating Higher Truncations via Constancy
by Paolo Capriotti and Nicolai Kraus
(in preparation/work in progress)
-}
open import HoTT
module experimental.TwoConstancyHIT where
module _ where
private
data #TwoConstancy-aux {i} (A : Type i) : Type i where
#point : A → #TwoConstancy-aux A
data #TwoConstancy {i} (A : Type i) : Type i where
#two-constancy : #TwoConstancy-aux A → (Unit → Unit) → #TwoConstancy A
TwoConstancy : ∀ {i} → Type i → Type i
TwoConstancy = #TwoConstancy
module _ {i} {A : Type i} where
point : A → TwoConstancy A
point a = #two-constancy (#point a) _
postulate -- HIT
link₀ : ∀ a₁ a₂ → point a₁ == point a₂
link₁ : ∀ a₁ a₂ a₃ → link₀ a₁ a₂ ∙' link₀ a₂ a₃ == link₀ a₁ a₃
TwoConstancy-level : is-gpd (TwoConstancy A)
module TwoConstancyElim
{l} {P : TwoConstancy A → Type l}
(p : ∀ x → is-gpd (P x))
(point* : ∀ a → P (point a))
(link₀* : ∀ a₁ a₂ → point* a₁ == point* a₂ [ P ↓ link₀ a₁ a₂ ])
(link₁* : ∀ a₁ a₂ a₃
→ link₀* a₁ a₂ ∙'ᵈ link₀* a₂ a₃
== link₀* a₁ a₃
[ (λ p → point* a₁ == point* a₃ [ P ↓ p ]) ↓ link₁ a₁ a₂ a₃ ]) where
f : Π (TwoConstancy A) P
f = f-aux phantom phantom phantom where
f-aux : Phantom p → Phantom link₀* → Phantom link₁* → Π (TwoConstancy A) P
f-aux phantom phantom phantom (#two-constancy (#point a) _) = point* a
postulate -- HIT
link₀-β : ∀ a₁ a₂ → apd f (link₀ a₁ a₂) == link₀* a₁ a₂
private
lemma : ∀ a₁ a₂ a₃
→ apd f (link₀ a₁ a₂ ∙' link₀ a₂ a₃)
== link₀* a₁ a₂ ∙'ᵈ link₀* a₂ a₃
lemma a₁ a₂ a₃ =
apd f (link₀ a₁ a₂ ∙' link₀ a₂ a₃)
=⟨ apd-∙' f (link₀ a₁ a₂) (link₀ a₂ a₃) ⟩
apd f (link₀ a₁ a₂) ∙'ᵈ apd f (link₀ a₂ a₃)
=⟨ link₀-β a₁ a₂ |in-ctx (λ u → u ∙'ᵈ apd f (link₀ a₂ a₃)) ⟩
link₀* a₁ a₂ ∙'ᵈ apd f (link₀ a₂ a₃)
=⟨ link₀-β a₂ a₃ |in-ctx (λ u → link₀* a₁ a₂ ∙'ᵈ u) ⟩
link₀* a₁ a₂ ∙'ᵈ link₀* a₂ a₃
∎
postulate -- HIT
link₁-β : ∀ a₁ a₂ a₃
→ apd (apd f) (link₁ a₁ a₂ a₃)
== lemma a₁ a₂ a₃ ◃ (link₁* a₁ a₂ a₃ ▹! link₀-β a₁ a₃)
open TwoConstancyElim public using () renaming (f to TwoConstancy-elim)
module TwoConstancyRec
{i} {A : Type i}
{l} {P : Type l}
(p : is-gpd P)
(point* : ∀ a → P)
(link₀* : ∀ a₁ a₂ → point* a₁ == point* a₂)
(link₁* : ∀ a₁ a₂ a₃
→ link₀* a₁ a₂ ∙' link₀* a₂ a₃
== link₀* a₁ a₃) where
private
module M = TwoConstancyElim {A = A}
{l = l} {P = λ _ → P}
(λ _ → p)
point*
(λ a₁ a₂ → ↓-cst-in (link₀* a₁ a₂))
(λ a₁ a₂ a₃
→ ↓-cst-in-∙' (link₀ a₁ a₂) (link₀ a₂ a₃) (link₀* a₁ a₂) (link₀* a₂ a₃)
!◃ ↓-cst-in2 (link₁* a₁ a₂ a₃))
f : TwoConstancy A → P
f = M.f
link₀-β : ∀ a₁ a₂ → ap f (link₀ a₁ a₂) == link₀* a₁ a₂
link₀-β a₁ a₂ = apd=cst-in {f = f} $ M.link₀-β a₁ a₂
private
lemma : ∀ a₁ a₂ a₃
→ ap f (link₀ a₁ a₂ ∙' link₀ a₂ a₃)
== link₀* a₁ a₂ ∙' link₀* a₂ a₃
lemma a₁ a₂ a₃ =
ap f (link₀ a₁ a₂ ∙' link₀ a₂ a₃)
=⟨ ap-∙' f (link₀ a₁ a₂) (link₀ a₂ a₃) ⟩
ap f (link₀ a₁ a₂) ∙' ap f (link₀ a₂ a₃)
=⟨ link₀-β a₁ a₂ |in-ctx (λ u → u ∙' ap f (link₀ a₂ a₃)) ⟩
link₀* a₁ a₂ ∙' ap f (link₀ a₂ a₃)
=⟨ link₀-β a₂ a₃ |in-ctx (λ u → link₀* a₁ a₂ ∙' u) ⟩
link₀* a₁ a₂ ∙' link₀* a₂ a₃
∎
-- I am a lazy person.
postulate -- HIT
link₁-β : ∀ a₁ a₂ a₃
→ ap (ap f) (link₁ a₁ a₂ a₃)
== (lemma a₁ a₂ a₃ ∙ link₁* a₁ a₂ a₃) ∙ (! $ link₀-β a₁ a₃)
open TwoConstancyRec public using () renaming (f to TwoConstancy-rec)
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{-# OPTIONS --cubical --safe #-}
module Cubical.Data.Strict2Group.Explicit.Interface where
open import Cubical.Foundations.Prelude
open import Cubical.Data.Group.Base
open import Cubical.Data.Sigma
open import Cubical.Data.Strict2Group.Explicit.Base
open import Cubical.Data.Strict2Group.Explicit.Notation
module S2GInterface {ℓ : Level} ((strict2groupexp C₀ C₁ s t i ∘ si ti s∘ t∘ isMorph∘ assoc∘ lUnit∘ rUnit∘) : Strict2GroupExp ℓ) where
open S2GBaseNotation C₀ C₁ s t i ∘ public
module Identities1 where
-- to be consistent with the other notation
tarId = ti
srcId = si
-- identity is preserved by id
id1₀≡1₁ : id 1₀ ≡ 1₁
id1₀≡1₁ = morphId {G = C₀} {H = C₁} i
open Identities1 public
module C₁×₀C₁ where
-- the composable morphisms as record type
record Co : Type ℓ where
constructor co
field
g f : TC₁
coh : CohCond g f
-- syntax
𝓁 𝓇 : Co → TC₁
𝓁 = Co.g
𝓇 = Co.f
𝒸 : (gfc : Co) → CohCond (𝓁 gfc) (𝓇 gfc)
𝒸 = Co.coh
-- compose a co object using ∘
⊙ : Co → TC₁
⊙ gfc = ∘ (𝓁 gfc) (𝓇 gfc) (𝒸 gfc)
-- basically ∘, but for the sake of interfacing, we don't want to use ∘
⊙' : (g f : TC₁) → CohCond g f → TC₁
⊙' g f c = ⊙ (co g f c)
-- interface for those names aswell
src⊙ : (gfc : Co) → src (⊙ gfc) ≡ src (𝓇 gfc)
src⊙ (co g f c) = s∘ g f c
tar⊙ : (gfc : Co) → tar (⊙ gfc) ≡ tar (𝓁 gfc)
tar⊙ (co g f c) = t∘ g f c
src⊙' : (g f : TC₁) → (c : CohCond g f) → src (⊙' g f c) ≡ src f
src⊙' g f c = src⊙ (co g f c)
tar⊙' : (g f : TC₁) → (c : CohCond g f) → tar (⊙' g f c) ≡ tar g
tar⊙' g f c = tar⊙ (co g f c)
-- multiplication in C₁×₀C₁
_∙Co_ : (gfc gfc' : Co) → Co
(co g f coh) ∙Co (co g' f' coh') =
co (g ∙₁ g') (f ∙₁ f')
(src∙₁ g g' ∙ cong (_∙₀ src g') coh ∙ cong (tar f ∙₀_) coh' ∙ sym (tar∙₁ f f'))
-- unit element w.r.t. ∙c. Too bad there is no \_c
1c : Co
1c = co 1₁ 1₁ ((cong src (sym id1₀≡1₁)) ∙∙ si 1₀ ∙∙ sym (ti 1₀) ∙ cong tar id1₀≡1₁)
-- the interchange law reformulated using ⊙
isMorph⊙ : (gfc gfc' : Co) → ⊙ (gfc ∙Co gfc') ≡ ⊙ gfc ∙₁ ⊙ gfc'
isMorph⊙ (co _ _ c) (co _ _ c') = isMorph∘ c c'
-- associator notation
assoc⊙' : (h g f : TC₁) → (c : CohCond g f) → (c' : CohCond h g) → ⊙' (⊙' h g c') f ((src⊙' h g c') ∙ c) ≡ ⊙' h (⊙' g f c) (c' ∙ (sym (tar⊙' g f c)))
assoc⊙' h g f c c' = assoc∘ c c'
-- the left and right unit laws reformulated using ⊙
lUnit⊙ : (f : TC₁) → ⊙ (co (id (tar f)) f (srcId (tar f))) ≡ f
lUnit⊙ = lUnit∘
rUnit⊙ : (f : TC₁) → ⊙ (co f (id (src f)) (sym (tarId (src f)))) ≡ f
rUnit⊙ = rUnit∘
-- the path component of f in C₁
ΣC₁p : (f : TC₁) → Type ℓ
ΣC₁p f = Σ[ f' ∈ TC₁ ] (f ≡ f')
private
-- for given g, the type of f that g can be precomposed with
_∘* : TC₁ → Type ℓ
g ∘* = Σ[ f ∈ TC₁ ] (CohCond g f)
-- for given f, the type of g that f can be postcomposed with
*∘_ : TC₁ → Type ℓ
*∘ f = Σ[ g ∈ TC₁ ] (CohCond g f)
-- alternate notation for ∘
-- this is used in ∘*≡ to λ-abstract in cong
_∘*_ : (g : TC₁) (fc : g ∘*) → TC₁
_∘*_ g (f , c) = ∘ g f c
_*∘_ : (f : TC₁) (gc : *∘ f) → TC₁
_*∘_ f (g , c) = ∘ g f c
-- since we have proof irrelevance in C₀ we can show that f ≡ f' → g∘f ≡ g∘f'
∘*≡ : (g : TC₁) → (fc : g ∘*) → (f'p : ΣC₁p (fst fc)) → g ∘* fc ≡ g ∘* ((fst f'p) , snd fc ∙ cong tar (snd f'p))
∘*≡ g fc f'p = cong (g ∘*_) (ΣPathP (snd f'p , isProp→PathP (λ j → Group.setStruc C₀ (src g) (tar (snd f'p j))) (snd fc) (snd fc ∙ cong tar (snd f'p))))
*∘≡ : (f : TC₁) → (gc : *∘ f) → (g'p : ΣC₁p (fst gc)) → f *∘ gc ≡ f *∘ (fst g'p , ((cong src (sym (snd g'p))) ∙ snd gc))
*∘≡ f gc g'p = cong (_*∘_ f) (ΣPathP ((snd g'p) , (isProp→PathP (λ j → Group.setStruc C₀ (src (snd g'p j)) (tar f)) (snd gc) (cong src (sym (snd g'p)) ∙ snd gc))))
-- ⊙ respecs paths on the right
⊙≡ : ((co g f c) : Co) → (f'p : ΣC₁p f) → ⊙ (co g f c) ≡ ⊙ (co g (fst f'p) (c ∙ (cong tar (snd f'p))))
⊙≡ (co g f c) (f' , f≡f') = ∘*≡ g (f , c) (f' , f≡f')
-- ⊙ respects paths on the left
≡⊙ : ((co g f c) : Co) → ((g' , g≡g') : ΣC₁p g) → ⊙ (co g f c) ≡ ⊙ (co g' f (cong src (sym g≡g') ∙ c))
≡⊙ (co g f c) (g' , g≡g') = *∘≡ f (g , c) (g' , g≡g')
-- ⊙ resepcts paths on the coherence condition
⊙≡c : ((co g f c) : Co) → (c' : CohCond g f) → ⊙ (co g f c) ≡ ⊙ (co g f c')
⊙≡c (co g f c) c' = cong (λ z → ⊙ (co g f z)) (Group.setStruc C₀ (src g) (tar f) c c')
-- implicit version of ⊙≡c
⊙≡c~ : {g f : TC₁} (c c' : CohCond g f) → ⊙ (co g f c) ≡ ⊙ (co g f c')
⊙≡c~ {g} {f} c c' = cong (λ z → ⊙ (co g f z)) (Group.setStruc C₀ (src g) (tar f) c c')
-- ⊙ respecting paths on the left also changes the coherence condition so this should be used instead
≡⊙c* : {g g' f : TC₁} (c : CohCond g f) (g≡g' : g ≡ g') (c' : CohCond g' f) → ⊙' g f c ≡ ⊙' g' f c'
≡⊙c* {g} {g'} {f} c g≡g' c' = (≡⊙ (co g f c) (g' , g≡g')) ∙ ⊙≡c~ ((cong src (sym g≡g')) ∙ c) c'
-- ⊙ respecting paths on the right also changes the coherence condition so this should be used instead
⊙≡c* : {g f f' : TC₁} (c : CohCond g f) (f≡f' : f ≡ f') (c' : CohCond g f') → ⊙' g f c ≡ ⊙' g f' c'
⊙≡c* {g} {f} {f'} c f≡f' c' = (⊙≡ (co g f c) (f' , f≡f')) ∙ ⊙≡c~ (c ∙ cong tar f≡f') c'
-- use the left and right unit law with an arbitrary coherence proof c
lUnit⊙c : (f : TC₁) → (c : CohCond (id (tar f)) f) → ⊙ (co (id (tar f)) f c) ≡ f
lUnit⊙c f c = (⊙≡c (co (id (tar f)) f c) (srcId (tar f))) ∙ (lUnit⊙ f)
rUnit⊙c : (f : TC₁) → (c : CohCond f (id (src f))) → ⊙ (co f (id (src f)) c) ≡ f
rUnit⊙c f c = (⊙≡c (co f (id (src f)) c) (sym (tarId (src f)))) ∙ (rUnit⊙ f)
open C₁×₀C₁ public
module Identities2 where
-- source and target of unit element
tar1₁≡1₀ : tar 1₁ ≡ 1₀
tar1₁≡1₀ = morphId {G = C₁} {H = C₀} t
src1₁≡1₀ = morphId {G = C₁} {H = C₀} s
-- taking the source is the same as the target of the identity of the source
src≡tarIdSrc : (f : TC₁) → CohCond f (id (src f))
src≡tarIdSrc f = sym (ti (src f))
open Identities2 public
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open import FRP.LTL.ISet.Core using ( ISet ; ⌈_⌉ ; ⌊_⌋ )
open import FRP.LTL.RSet using ( RSet )
open import FRP.LTL.Time using ( Time ; _≤_ )
module FRP.LTL.ISet.Future where
data Future (A : RSet) (t : Time) : Set where
_,_ : ∀ {u} .(t≤u : t ≤ u) → A u → Future A t
◇ : ISet → ISet
◇ A = ⌈ Future ⌊ A ⌋ ⌉ | {
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module WarningOnUsage2 where
open import WarningOnUsage
λx→x₂ = λx→x
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{-# OPTIONS --without-K --rewriting #-}
open import lib.Basics
open import lib.NConnected
open import lib.NType2
open import lib.types.FunctionSeq
open import lib.types.Span
open import lib.types.Pointed
open import lib.types.Pushout
open import lib.types.PushoutFlip
open import lib.types.PushoutFmap
open import lib.types.PushoutFlattening
open import lib.types.Unit
open import lib.types.Paths
open import lib.types.Pi
open import lib.types.Truncation
open import lib.types.Lift
open import lib.cubical.Cube
open import lib.cubical.Square
-- Suspension is defined as a particular case of pushout
module lib.types.Suspension.Core where
module _ {i} (A : Type i) where
susp-span : Span
susp-span = span Unit Unit A (λ _ → tt) (λ _ → tt)
Susp : Type i
Susp = Pushout susp-span
-- [north'] and [south'] explictly ask for [A]
north' : Susp
north' = left tt
south' : Susp
south' = right tt
module _ {i} {A : Type i} where
north : Susp A
north = north' A
south : Susp A
south = south' A
merid : A → north == south
merid x = glue x
module SuspElim {j} {P : Susp A → Type j} (n : P north)
(s : P south) (p : (x : A) → n == s [ P ↓ merid x ]) where
private
module P = PushoutElim (λ _ → n) (λ _ → s) p
abstract
f : Π (Susp A) P
f = P.f
north-β : f north ↦ n
north-β = P.left-β unit
{-# REWRITE north-β #-}
south-β : f south ↦ s
south-β = P.right-β unit
{-# REWRITE south-β #-}
merid-β : ∀ a → apd f (merid a) == p a
merid-β = P.glue-β
open SuspElim public using () renaming (f to Susp-elim)
module SuspRec {j} {C : Type j} (n s : C) (p : A → n == s) where
private
module P = PushoutRec {d = susp-span A} (λ _ → n) (λ _ → s) p
abstract
f : Susp A → C
f = P.f
north-β : f north ↦ n
north-β = P.left-β unit
{-# REWRITE north-β #-}
south-β : f south ↦ s
south-β = P.right-β unit
{-# REWRITE south-β #-}
merid-β : ∀ a → ap f (merid a) == p a
merid-β = P.glue-β
open SuspRec public using () renaming (f to Susp-rec)
module SuspRecType {j} (n s : Type j) (p : A → n ≃ s)
= PushoutRecType {d = susp-span A} (λ _ → n) (λ _ → s) p
module SuspPathElim
{i} {j} {A : Type i} {B : Type j}
(f g : Susp A → B)
(n : f north == g north)
(s : f south == g south)
(m : ∀ a → Square n (ap f (merid a)) (ap g (merid a)) s)
where
private
module M =
SuspElim
{P = λ sa → f sa == g sa}
n
s
(λ a → ↓-='-from-square (m a))
open M public
merid-square-β : ∀ a → natural-square M.f (merid a) == m a
merid-square-β a = natural-square-β M.f (merid a) (M.merid-β a)
module SuspDoublePathElim
{i} {j} {k} {A : Type i} {B : Type j} {C : Type k}
(f g : Susp A → Susp B → C)
(n-n : f north north == g north north)
(n-s : f north south == g north south)
(s-n : f south north == g south north)
(s-s : f south south == g south south)
(n-m : ∀ b → Square n-n (ap (f north) (merid b)) (ap (g north) (merid b)) n-s)
(s-m : ∀ b → Square s-n (ap (f south) (merid b)) (ap (g south) (merid b)) s-s)
(m-n : ∀ a → Square n-n (ap (λ sa → f sa north) (merid a)) (ap (λ sa → g sa north) (merid a)) s-n)
(m-s : ∀ a → Square n-s (ap (λ sa → f sa south) (merid a)) (ap (λ sa → g sa south) (merid a)) s-s)
(m-m : ∀ a b →
Cube (m-n a)
(m-s a)
(n-m b)
(natural-square (λ sb → ap (λ sa → f sa sb) (merid a)) (merid b))
(natural-square (λ sb → ap (λ sa → g sa sb) (merid a)) (merid b))
(s-m b))
where
private
module N =
SuspElim
{P = λ sb → f north sb == g north sb}
n-n
n-s
(λ b → ↓-='-from-square (n-m b))
n-natural-square : ∀ (b : B) →
natural-square N.f (merid b) == n-m b
n-natural-square b = natural-square-β N.f (merid b) (N.merid-β b)
module S =
SuspElim
{P = λ sb → f south sb == g south sb}
s-n
s-s
(λ b → ↓-='-from-square (s-m b))
s-natural-square : ∀ (b : B) →
natural-square S.f (merid b) == s-m b
s-natural-square b = natural-square-β S.f (merid b) (S.merid-β b)
module M (sb : Susp B) =
SuspElim {A = A}
{P = λ sa → f sa sb == g sa sb}
(N.f sb)
(S.f sb)
(λ a → Susp-elim
{P = λ sb → N.f sb == S.f sb [ (λ sa → f sa sb == g sa sb) ↓ merid a ]}
(↓-='-from-square (m-n a))
(↓-='-from-square (m-s a))
(λ b → ap↓ ↓-='-from-square $
cube-to-↓-square $
cube-shift-back (! (n-natural-square b)) $
cube-shift-front (! (s-natural-square b)) $
m-m a b)
sb)
abstract
p : ∀ sa sb → f sa sb == g sa sb
p sa sb = M.f sb sa
north-north-β : p north north ↦ n-n
north-north-β = N.north-β
{-# REWRITE north-north-β #-}
north-south-β : p north south ↦ n-s
north-south-β = N.south-β
{-# REWRITE north-south-β #-}
south-north-β : p south north ↦ s-n
south-north-β = S.north-β
{-# REWRITE south-north-β #-}
south-south-β : p south south ↦ s-s
south-south-β = S.south-β
{-# REWRITE south-south-β #-}
north-merid-β : ∀ b → apd (p north) (merid b) == ↓-='-from-square (n-m b)
north-merid-β = N.merid-β
north-merid-square-β : ∀ b → natural-square (p north) (merid b) == n-m b
north-merid-square-β b = natural-square-β (p north) (merid b) (north-merid-β b)
south-merid-β : ∀ b → apd (p south) (merid b) == ↓-='-from-square (s-m b)
south-merid-β = S.merid-β
south-merid-square-β : ∀ b → natural-square (p south) (merid b) == s-m b
south-merid-square-β b = natural-square-β (p south) (merid b) (south-merid-β b)
merid-north-β : ∀ a → apd (λ sa → p sa north) (merid a) == ↓-='-from-square (m-n a)
merid-north-β = M.merid-β north
merid-north-square-β : ∀ a → natural-square (λ sa → p sa north) (merid a) == m-n a
merid-north-square-β a =
natural-square-β (λ sa → p sa north) (merid a) (merid-north-β a)
merid-south-β : ∀ a → apd (λ sa → p sa south) (merid a) == ↓-='-from-square (m-s a)
merid-south-β = M.merid-β south
merid-south-square-β : ∀ a → natural-square (λ sa → p sa south) (merid a) == m-s a
merid-south-square-β a =
natural-square-β (λ sa → p sa south) (merid a) (merid-south-β a)
susp-⊙span : ∀ {i} → Ptd i → ⊙Span
susp-⊙span X =
⊙span ⊙Unit ⊙Unit X (⊙cst {X = X}) (⊙cst {X = X})
⊙Susp : ∀ {i} → Type i → Ptd i
⊙Susp A = ⊙[ Susp A , north ]
σloop : ∀ {i} (X : Ptd i) → de⊙ X → north' (de⊙ X) == north' (de⊙ X)
σloop ⊙[ _ , x₀ ] x = merid x ∙ ! (merid x₀)
σloop-pt : ∀ {i} {X : Ptd i} → σloop X (pt X) == idp
σloop-pt {X = ⊙[ _ , x₀ ]} = !-inv-r (merid x₀)
⊙σloop : ∀ {i} (X : Ptd i) → X ⊙→ ⊙[ north' (de⊙ X) == north' (de⊙ X) , idp ]
⊙σloop X = σloop X , σloop-pt
module _ {i j} where
module SuspFmap {A : Type i} {B : Type j} (f : A → B) =
SuspRec north south (merid ∘ f)
Susp-fmap : {A : Type i} {B : Type j} (f : A → B)
→ (Susp A → Susp B)
Susp-fmap = SuspFmap.f
⊙Susp-fmap : {A : Type i} {B : Type j} (f : A → B)
→ ⊙Susp A ⊙→ ⊙Susp B
⊙Susp-fmap f = (Susp-fmap f , idp)
module _ {i} (A : Type i) where
Susp-fmap-idf : ∀ a → Susp-fmap (idf A) a == a
Susp-fmap-idf = Susp-elim idp idp $ λ a →
↓-='-in' (ap-idf (merid a) ∙ ! (SuspFmap.merid-β (idf A) a))
⊙Susp-fmap-idf : ⊙Susp-fmap (idf A) ◃⊙idf =⊙∘ ⊙idf-seq
⊙Susp-fmap-idf = =⊙∘-in (⊙λ=' Susp-fmap-idf idp)
transport-Susp : ∀ {i} {A B : Type i} (p : A == B)
→ transport Susp p ∼ Susp-fmap (coe p)
transport-Susp {i} {A} {.A} p@idp sa = ! (Susp-fmap-idf A sa)
⊙transport-⊙Susp : ∀ {i} {A B : Type i} (p : A == B)
→ ⊙transport ⊙Susp p == ⊙Susp-fmap (coe p)
⊙transport-⊙Susp {i} {A} {.A} p@idp = ! (=⊙∘-out (⊙Susp-fmap-idf A))
module _ {i} {A : Type i} where
Susp-fmap-cst : ∀ {j} {B : Type j} (b : B)
(a : Susp A) → Susp-fmap (cst b) a == north
Susp-fmap-cst b = Susp-elim idp (! (merid b)) $ (λ a →
↓-app=cst-from-square $ SuspFmap.merid-β (cst b) a ∙v⊡ tr-square _)
⊙Susp-fmap-cst : ∀ {j} {Y : Ptd j}
→ ⊙Susp-fmap {A = A} (λ _ → pt Y) == ⊙cst
⊙Susp-fmap-cst = ⊙λ=' (Susp-fmap-cst _) idp
Susp-fmap-∘ : ∀ {i j k} {A : Type i} {B : Type j} {C : Type k} (g : B → C) (f : A → B)
(σ : Susp A) → Susp-fmap (g ∘ f) σ == Susp-fmap g (Susp-fmap f σ)
Susp-fmap-∘ g f = Susp-elim
idp
idp
(λ a → ↓-='-in' $
ap-∘ (Susp-fmap g) (Susp-fmap f) (merid a)
∙ ap (ap (Susp-fmap g)) (SuspFmap.merid-β f a)
∙ SuspFmap.merid-β g (f a)
∙ ! (SuspFmap.merid-β (g ∘ f) a))
⊙Susp-fmap-∘ : ∀ {i j k} {A : Type i} {B : Type j} {C : Type k}
(g : B → C) (f : A → B)
→ ⊙Susp-fmap (g ∘ f) == ⊙Susp-fmap g ⊙∘ ⊙Susp-fmap f
⊙Susp-fmap-∘ g f = ⊙λ=' (Susp-fmap-∘ g f) idp
⊙Susp-fmap-seq : ∀ {i} {A B : Type i} → (A –→ B) → ⊙Susp A ⊙–→ ⊙Susp B
⊙Susp-fmap-seq idf-seq = ⊙idf-seq
⊙Susp-fmap-seq (f ◃∘ fs) = ⊙Susp-fmap f ◃⊙∘ ⊙Susp-fmap-seq fs
abstract
⊙Susp-fmap-seq-∘ : ∀ {i} {A B : Type i} (fs : A –→ B)
→ ⊙Susp-fmap (compose fs) ◃⊙idf =⊙∘ ⊙Susp-fmap-seq fs
⊙Susp-fmap-seq-∘ idf-seq = ⊙Susp-fmap-idf _
⊙Susp-fmap-seq-∘ (f ◃∘ fs) = =⊙∘-in $
⊙Susp-fmap (f ∘ compose fs)
=⟨ ⊙Susp-fmap-∘ f (compose fs) ⟩
⊙Susp-fmap f ⊙∘ ⊙Susp-fmap (compose fs)
=⟨ ap (⊙Susp-fmap f ⊙∘_) (=⊙∘-out (⊙Susp-fmap-seq-∘ fs)) ⟩
⊙Susp-fmap f ⊙∘ ⊙compose (⊙Susp-fmap-seq fs) =∎
⊙Susp-fmap-seq-=⊙∘ : ∀ {i} {A B : Type i} {fs gs : A –→ B}
→ fs =∘ gs
→ ⊙Susp-fmap-seq fs =⊙∘ ⊙Susp-fmap-seq gs
⊙Susp-fmap-seq-=⊙∘ {fs = fs} {gs = gs} p =
⊙Susp-fmap-seq fs
=⊙∘⟨ !⊙∘ (⊙Susp-fmap-seq-∘ fs) ⟩
⊙Susp-fmap (compose fs) ◃⊙idf
=⊙∘₁⟨ ap ⊙Susp-fmap (=∘-out p) ⟩
⊙Susp-fmap (compose gs) ◃⊙idf
=⊙∘⟨ ⊙Susp-fmap-seq-∘ gs ⟩
⊙Susp-fmap-seq gs ∎⊙∘
{- Extract the 'glue component' of a pushout -}
module _ {i j k} {s : Span {i} {j} {k}} where
module ExtractGlue = PushoutRec {d = s} {D = Susp (Span.C s)}
(λ _ → north) (λ _ → south) merid
extract-glue = ExtractGlue.f
module _ {x₀ : Span.A s} where
⊙extract-glue : ⊙[ Pushout s , left x₀ ] ⊙→ ⊙[ Susp (Span.C s) , north ]
⊙extract-glue = extract-glue , idp
module _ {i j} {A : Type i} {B : Type j} (eq : A ≃ B) where
Susp-emap : Susp A ≃ Susp B
Susp-emap =
equiv
(Susp-fmap (–> eq))
(Susp-fmap (<– eq))
(λ sb →
Susp-fmap (–> eq) (Susp-fmap (<– eq) sb)
=⟨ ! (Susp-fmap-∘ (–> eq) (<– eq) sb) ⟩
Susp-fmap ((–> eq) ∘ (<– eq)) sb
=⟨ ap (λ f → Susp-fmap f sb) (λ= (<–-inv-r eq)) ⟩
Susp-fmap (idf B) sb
=⟨ Susp-fmap-idf _ sb ⟩
sb =∎)
(λ sa →
Susp-fmap (<– eq) (Susp-fmap (–> eq) sa)
=⟨ ! (Susp-fmap-∘ (<– eq) (–> eq) sa) ⟩
Susp-fmap ((<– eq) ∘ (–> eq)) sa
=⟨ ap (λ f → Susp-fmap f sa) (λ= (<–-inv-l eq)) ⟩
Susp-fmap (idf A) sa
=⟨ Susp-fmap-idf _ sa ⟩
sa =∎)
private
-- This is to make sure that we did not screw up [Susp-emap].
test₀ : fst Susp-emap == Susp-fmap (fst eq)
test₀ = idp
module _ {i j} {A : Type i} {B : Type j} where
⊙Susp-emap : A ≃ B → ⊙Susp A ⊙≃ ⊙Susp B
⊙Susp-emap eq = ≃-to-⊙≃ (Susp-emap eq) idp
{- Interaction with [Lift] -}
module _ {i j} (A : Type i) where
Susp-Lift-econv : Susp (Lift {j = j} A) ≃ Lift {j = j} (Susp A)
Susp-Lift-econv = lift-equiv ∘e Susp-emap lower-equiv
Susp-Lift-conv : Susp (Lift {j = j} A) == Lift {j = j} (Susp A)
Susp-Lift-conv = ua Susp-Lift-econv
module _ {i j} (A : Type i) where
⊙Susp-Lift-econv : ⊙Susp (Lift {j = j} A) ⊙≃ ⊙Lift {j = j} (⊙Susp A)
⊙Susp-Lift-econv = ⊙lift-equiv {j = j} ⊙∘e ⊙Susp-emap {A = Lift {j = j} A} {B = A} lower-equiv
⊙Susp-Lift-conv : ⊙Susp (Lift {j = j} A) == ⊙Lift {j = j} (⊙Susp A)
⊙Susp-Lift-conv = ⊙ua ⊙Susp-Lift-econv
{- Suspension of an n-connected space is n+1-connected -}
abstract
Susp-conn : ∀ {i} {A : Type i} {n : ℕ₋₂}
→ is-connected n A → is-connected (S n) (Susp A)
Susp-conn {A = A} {n = n} cA = has-level-in
([ north ] ,
Trunc-elim
(Susp-elim
idp
(Trunc-rec (λ a → ap [_] (merid a))
(contr-center cA))
(λ x → Trunc-elim
{P = λ y → idp ==
Trunc-rec (λ a → ap [_] (merid a)) y
[ (λ z → [ north ] == [ z ]) ↓ (merid x) ]}
{{λ _ → ↓-preserves-level ⟨⟩}}
(λ x' → ↓-cst=app-in (∙'-unit-l _ ∙ mers-eq n x x'))
(contr-center cA))))
where
instance _ = cA
mers-eq : ∀ {i} {A : Type i} (n : ℕ₋₂)
{{_ : is-connected n A}} → (x x' : A)
→ ap ([_] {n = S n}) (merid x)
== Trunc-rec {{has-level-apply (Trunc-level {n = S n}) _ _}} (λ a → ap [_] (merid a)) [ x' ]
mers-eq ⟨-2⟩ x x' = contr-has-all-paths _ _
mers-eq {A = A} (S n) x x' =
conn-extend (pointed-conn-out A x)
(λ y → ((ap [_] (merid x) == ap [_] (merid y)) ,
has-level-apply (has-level-apply (Trunc-level {n = S (S n)}) _ _) _ _))
(λ _ → idp) x'
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-- Type interpretation and soundness of typing.
-- Proof of strong normalization for well-typed terms.
module Soundness where
open import Library
open import Terms
open import Substitution
open import SN
open import SN.AntiRename
open import DeclSN using (sn; fromSN)
open import SAT3
-- Type interpretation
⟦_⟧ : (a : Ty) → SAT a
⟦ base ⟧ = ⟦⊥⟧
⟦ a →̂ b ⟧ = ⟦ a ⟧ ⟦→⟧ ⟦ b ⟧
-- Context interpretation (semantic substitutions)
⟦_⟧C : ∀ Γ → ∀ {Δ} (σ : Subst Γ Δ) → Set
⟦ Γ ⟧C σ = ∀ {a} (x : Var Γ a) → σ x ∈ ⟦ a ⟧
Ext : ∀ {a Δ Γ} {t : Tm Δ a} → (𝒕 : t ∈ (⟦ a ⟧)) →
∀ {σ : Subst Γ Δ} (θ : ⟦ Γ ⟧C σ) → ⟦ a ∷ Γ ⟧C (t ∷s σ)
Ext {a} 𝒕 θ (zero) = 𝒕
Ext {a} 𝒕 θ (suc x) = θ x
Rename : ∀ {Δ Δ'} → (ρ : Ren Δ Δ') →
∀ {Γ}{σ : Subst Γ Δ} (θ : ⟦ Γ ⟧C σ) →
⟦ Γ ⟧C (ρ •s σ)
Rename ρ θ {a} x = ↿ SAT.satRename ⟦ a ⟧ ρ (⇃ θ x)
sound : ∀ {a Γ} (t : Tm Γ a) {Δ} {σ : Subst Γ Δ} → (θ : ⟦ Γ ⟧C σ) → subst σ t ∈ (⟦ a ⟧)
sound (var x) θ = θ x
sound (abs t) {σ = σ} θ = ⟦abs⟧ {𝓐 = ⟦ _ ⟧} {𝓑 = ⟦ _ ⟧} (λ ρ {u} 𝑢 →
let open ≡-Reasoning
eq : subst (u ∷s (ρ •s σ)) t ≡ subst0 u (subst (lifts ρ) (subst (lifts σ) t))
eq = begin
subst (u ∷s (ρ •s σ)) t
≡⟨ subst-ext (cons-to-sgs u _) t ⟩
subst (sgs u •s lifts (ρ •s σ)) t
≡⟨ subst-∙ _ _ t ⟩
subst0 u (subst (lifts (ρ •s σ)) t)
≡⟨ ≡.cong (subst0 u) (subst-ext (lifts-∙ ρ σ) t) ⟩
subst0 u (subst (lifts ρ •s lifts σ) t)
≡⟨ ≡.cong (subst0 u) (subst-∙ (lifts ρ) (lifts σ) t) ⟩
subst0 u (subst (lifts ρ) (subst (lifts σ) t))
∎
in (≡.subst (_∈ ⟦ _ ⟧) eq (↿ (⇃ sound t (Ext (↿ (⇃ 𝑢)) ((Rename ρ θ)))))))
sound (app t u) θ = ↿ (⇃ ⟦app⟧ {𝓐 = ⟦ _ ⟧} {𝓑 = ⟦ _ ⟧} (sound t θ) (↿ (⇃ sound u θ)))
-- Identity environment.
id-θ : ∀{Γ} → ⟦ Γ ⟧C ids
id-θ {Γ} {a} x = ⟦var⟧ ⟦ a ⟧ x
-- Any well-typed term inhabits its semantic type.
sound' : ∀ {a Γ} (t : Tm Γ a) → t ∈ ⟦ a ⟧
sound' t rewrite ≡.sym (subst-id {vt = `Tm} t) = sound t id-θ
-- Any well-typed term is strongly normalizing.
strong-normalization : ∀ a {Γ} (t : Tm Γ a) → sn t
strong-normalization a t = fromSN (satSN ⟦ a ⟧ (⇃ sound' t))
-- Q.E.D.
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{-# OPTIONS --safe #-}
module Cubical.Homotopy.Hopf where
open import Cubical.Homotopy.HSpace
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Transport
open import Cubical.Foundations.Function
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Univalence
open import Cubical.Data.Sigma
open import Cubical.Data.Unit
open import Cubical.Data.Int hiding (_·_)
open import Cubical.HITs.Pushout.Flattening
open import Cubical.HITs.Pushout
open import Cubical.HITs.Sn hiding (joinS¹S¹→S³)
open import Cubical.HITs.Susp
open import Cubical.HITs.S1
open import Cubical.HITs.S2
open import Cubical.HITs.S3
open import Cubical.HITs.PropositionalTruncation
renaming (rec to pRec ; elim to pElim)
open import Cubical.HITs.Join
open import Cubical.HITs.Interval
renaming ( zero to I0 ; one to I1 )
open Iso
open HSpace
open AssocHSpace
private
retEq≡secEq : ∀ {ℓ} {A B : Type ℓ} (e : A ≃ B)
→ (x : _) → secEq e (e .fst x) ≡ cong (e .fst) (retEq e x)
retEq≡secEq {A = A} =
EquivJ (λ B e → (x : _) → secEq e (e .fst x) ≡ cong (e .fst) (retEq e x))
λ _ → refl
module Hopf {ℓ : Level} {A : Pointed ℓ} {e : HSpace A}
(e-ass : AssocHSpace e) (conA : ((x y : typ A) → ∥ x ≡ y ∥)) where
isEquiv-μ : (x : typ A) → isEquiv (λ z → (μ e z x))
isEquiv-μ x = pRec (isPropIsEquiv _)
(J (λ x _ → isEquiv (λ z → μ e z x))
(subst isEquiv (funExt (λ z → sym (μᵣ e z)))
(idIsEquiv (typ A))))
(conA (pt A) x)
isEquiv-μ' : (x : typ A) → isEquiv (μ e x)
isEquiv-μ' x =
pRec (isPropIsEquiv _)
(J (λ x _ → isEquiv (μ e x))
(subst isEquiv (funExt (λ x → sym (μₗ e x))) (idIsEquiv (typ A))))
(conA (pt A) x)
μ-eq : (x : typ A) → typ A ≃ typ A
μ-eq x = (λ z → μ e z x) , (isEquiv-μ x)
μ-eq' : (x : typ A) → typ A ≃ typ A
μ-eq' x = μ e x , isEquiv-μ' x
Hopf : Susp (typ A) → Type ℓ
Hopf north = typ A
Hopf south = typ A
Hopf (merid a i₁) = ua (μ-eq a) i₁
TotalSpaceHopfPush : Type _
TotalSpaceHopfPush =
Pushout {A = typ A × typ A} fst λ x → μ e (fst x) (snd x)
TotalSpaceHopfPush→TotalSpace :
TotalSpaceHopfPush → Σ[ x ∈ Susp (typ A) ] Hopf x
TotalSpaceHopfPush→TotalSpace (inl x) = north , x
TotalSpaceHopfPush→TotalSpace (inr x) = south , x
TotalSpaceHopfPush→TotalSpace (push (x , y) i₁) =
merid y i₁ , ua-gluePt (μ-eq y) i₁ x
joinIso₁ : Iso (TotalSpaceHopfPush) (join (typ A) (typ A))
joinIso₁ = theIso
where
F : TotalSpaceHopfPush → join (typ A) (typ A)
F (inl x) = inl x
F (inr x) = inr x
F (push (a , x) i) = push a (μ e a x) i
G : join (typ A) (typ A) → TotalSpaceHopfPush
G (inl x) = inl x
G (inr x) = inr x
G (push a b i) =
(push (a , invEq (μ-eq' a) b) ∙ cong inr (secEq (μ-eq' a) b)) i
s : section F G
s (inl x) = refl
s (inr x) = refl
s (push a b i) j =
hcomp (λ k → λ { (i = i0) → inl a
; (i = i1) → inr (secEq (μ-eq' a) b (j ∨ k))
; (j = i0) → F (compPath-filler
(push (a , invEq (μ-eq' a) b))
(cong inr (secEq (μ-eq' a) b)) k i)
; (j = i1) → push a b i})
(hcomp (λ k → λ { (i = i0) → inl a
; (i = i1) → inr (secEq (μ-eq' a) b (~ k ∨ j))
; (j = i0) → push a (secEq (μ-eq' a) b (~ k)) i
; (j = i1) → push a b i})
(push a b i))
r : retract F G
r (inl x) = refl
r (inr x) = refl
r (push (x , y) i) j =
hcomp (λ k → λ { (i = i0) → inl x
; (i = i1) → inr (μ e x y)
; (j = i0) → (push (x , invEq (μ-eq' x) (μ e x y))
∙ (λ i₁ → inr (retEq≡secEq (μ-eq' x) y (~ k) i₁))) i
; (j = i1) → push (x , y) i})
(hcomp (λ k → λ { (i = i0) → inl x
; (i = i1) → inr (μ e x (retEq (μ-eq' x) y k))
; (j = i1) → push (x , retEq (μ-eq' x) y k) i})
((push (x , invEq (μ-eq' x) (μ e x y))) i))
theIso : Iso TotalSpaceHopfPush (join (typ A) (typ A))
fun theIso = F
inv theIso = G
rightInv theIso = s
leftInv theIso = r
isEquivTotalSpaceHopfPush→TotalSpace :
isEquiv TotalSpaceHopfPush→TotalSpace
isEquivTotalSpaceHopfPush→TotalSpace =
isoToIsEquiv theIso
where
inv' : _ → _
inv' (north , y) = inl y
inv' (south , y) = inr y
inv' (merid a i , y) =
hcomp (λ k → λ { (i = i0) → push (y , a) (~ k)
; (i = i1) → inr y})
(inr (ua-unglue (μ-eq a) i y))
where
pp : PathP (λ i → ua (μ-eq a) i → TotalSpaceHopfPush)
inl inr
pp = ua→ {e = μ-eq a} {B = λ _ → TotalSpaceHopfPush} λ b → push (b , a)
sect : (x : _) → TotalSpaceHopfPush→TotalSpace (inv' x) ≡ x
sect (north , x) = refl
sect (south , x) = refl
sect (merid a i , y) j =
hcomp (λ k → λ { (i = i0) → merid a (~ k ∧ ~ j)
, ua-gluePt (μ-eq a) (~ k ∧ ~ j) y
; (i = i1) → south , y
; (j = i0) →
TotalSpaceHopfPush→TotalSpace
(hfill (λ k → λ { (i = i0) → push (y , a) (~ k)
; (i = i1) → inr y})
(inS (inr (ua-unglue (μ-eq a) i y)))
k)
; (j = i1) → merid a i , y})
((merid a (i ∨ ~ j)) , lem (μ-eq a) i j y)
where
lem : ∀ {ℓ} {A B : Type ℓ} (e : A ≃ B) →
PathP (λ i → PathP (λ j → (y : ua e i) → ua e (i ∨ ~ j))
(λ y → ua-unglue e i y)
λ y → y)
(λ j y → ua-gluePt e (~ j) y)
refl
lem {A = A} {B = B} =
EquivJ (λ B e → PathP (λ i → PathP (λ j → (y : ua e i) → ua e (i ∨ ~ j))
(λ y → ua-unglue e i y)
λ y → y)
(λ j y → ua-gluePt e (~ j) y)
refl)
λ i j a → ua-gluePt (idEquiv B) (i ∨ ~ j) (ua-unglue (idEquiv B) i a)
retr : retract TotalSpaceHopfPush→TotalSpace inv'
retr (inl x) = refl
retr (inr x) = refl
retr (push (x , y) i) j =
hcomp (λ k → λ { (i = i0) → push (x , y) (~ k)
; (i = i1) → inr (μ e x y)
; (j = i1) → push (x , y) (i ∨ ~ k)})
(inr (μ e x y))
theIso : Iso TotalSpaceHopfPush (Σ (Susp (typ A)) Hopf)
fun theIso = TotalSpaceHopfPush→TotalSpace
inv theIso = inv'
rightInv theIso = sect
leftInv theIso = retr
IsoTotalSpaceJoin : Iso (Σ[ x ∈ Susp (typ A) ] Hopf x) (join (typ A) (typ A))
IsoTotalSpaceJoin =
compIso (equivToIso (invEquiv (_ , isEquivTotalSpaceHopfPush→TotalSpace)))
joinIso₁
induced : TotalSpaceHopfPush → Susp (typ A)
induced = fst ∘ TotalSpaceHopfPush→TotalSpace
ua-lem : (x y z : typ A) → (i j : I) → ua (μ-eq y) i
ua-lem x y z i j =
fill (λ k → ua (μ-eq y) i)
(λ j → λ { (i = i0) → μ e z x
; (i = i1) → μ-assoc e-ass z x y j})
(inS (ua-gluePt (μ-eq y) i (μ e z x)))
j
TotalSpaceHopfPush→≃Hopf : (x : TotalSpaceHopfPush) → typ A ≃ Hopf (induced x)
TotalSpaceHopfPush→≃Hopf (inl x) = μ-eq x
TotalSpaceHopfPush→≃Hopf (inr x) = μ-eq x
TotalSpaceHopfPush→≃Hopf (push (x , y) i₁) = pp x y i₁
where
pp : (x y : _) → PathP (λ i → typ A ≃ ua (μ-eq y) i) (μ-eq x) (μ-eq (μ e x y))
pp x y = ΣPathP (P , help)
where
P : PathP (λ z → typ A → ua (μ-eq y) z) (fst (μ-eq x))
(fst (μ-eq (μ e x y)))
P i z = ua-lem x y z i i1
abstract
help : PathP (λ i₂ → isEquiv (P i₂)) (snd (μ-eq x))
(snd (μ-eq (μ e x y)))
help = toPathP (isPropIsEquiv _ _ _)
Push→TotalSpaceHopf : (a : typ A) (x : TotalSpaceHopfPush)
→ Σ[ x ∈ Susp (typ A) ] Hopf x
Push→TotalSpaceHopf a x = (induced x) , fst (TotalSpaceHopfPush→≃Hopf x) a
Push→TotalSpaceHopf-equiv : (a : typ A) → isEquiv (Push→TotalSpaceHopf a)
Push→TotalSpaceHopf-equiv a = pRec (isPropIsEquiv _)
(J (λ a _ → isEquiv (Push→TotalSpaceHopf a))
(subst isEquiv (sym main)
isEquivTotalSpaceHopfPush→TotalSpace))
(conA (pt A) a)
where
lem₁ : (x : _) → fst ((Push→TotalSpaceHopf (pt A)) x)
≡ fst (TotalSpaceHopfPush→TotalSpace x)
lem₁ (inl x) = refl
lem₁ (inr x) = refl
lem₁ (push a i) = refl
lem₂ : (x : _)
→ PathP (λ i → Hopf (lem₁ x i))
(snd ((Push→TotalSpaceHopf (pt A)) x))
(snd (TotalSpaceHopfPush→TotalSpace x))
lem₂ (inl x) = μₗ e x
lem₂ (inr x) = μₗ e x
lem₂ (push (x , y) i) j =
hcomp (λ k → λ {(i = i0) → μₗ e x j
; (i = i1) → μ-assoc-filler e-ass x y j k
; (j = i0) → ua-lem x y (pt A) i k
; (j = i1) → ua-gluePt (μ-eq y) i x})
(ua-gluePt (μ-eq y) i (μₗ e x j))
main : Push→TotalSpaceHopf (pt A) ≡ TotalSpaceHopfPush→TotalSpace
main i x = (lem₁ x i) , (lem₂ x i)
TotalSpaceHopfPush² : Type _
TotalSpaceHopfPush² = Pushout {A = TotalSpaceHopfPush} (λ _ → tt) induced
P : TotalSpaceHopfPush² → Type _
P (inl x) = typ A
P (inr x) = Hopf x
P (push a i) = ua (TotalSpaceHopfPush→≃Hopf a) i
TotalSpacePush² : Type _
TotalSpacePush² = Σ[ x ∈ TotalSpaceHopfPush² ] P x
TotalSpacePush²' : Type _
TotalSpacePush²' =
Pushout {A = typ A × TotalSpaceHopfPush}
{C = Σ[ x ∈ Susp (typ A) ] Hopf x}
fst
λ x → Push→TotalSpaceHopf (fst x) (snd x)
IsoTotalSpacePush²TotalSpacePush²' : Iso TotalSpacePush² TotalSpacePush²'
IsoTotalSpacePush²TotalSpacePush²' =
compIso iso₂ (compIso (equivToIso fl.flatten) iso₁)
where
module fl =
FlatteningLemma (λ _ → tt) induced (λ x → typ A)
Hopf TotalSpaceHopfPush→≃Hopf
iso₁ : Iso (Pushout fl.Σf fl.Σg) TotalSpacePush²'
fun iso₁ (inl x) = inl (snd x)
fun iso₁ (inr x) = inr x
fun iso₁ (push a i) = push ((snd a) , (fst a)) i
inv iso₁ (inl x) = inl (tt , x)
inv iso₁ (inr x) = inr x
inv iso₁ (push a i) = push (snd a , fst a) i
rightInv iso₁ (inl x) = refl
rightInv iso₁ (inr x) = refl
rightInv iso₁ (push a i) = refl
leftInv iso₁ (inl x) = refl
leftInv iso₁ (inr x) = refl
leftInv iso₁ (push a i) = refl
iso₂ : Iso TotalSpacePush² (Σ (Pushout (λ _ → tt) induced) fl.E)
fun iso₂ (inl x , y) = inl x , y
fun iso₂ (inr x , y) = inr x , y
fun iso₂ (push a i , y) = push a i , y
inv iso₂ (inl x , y) = inl x , y
inv iso₂ (inr x , y) = inr x , y
inv iso₂ (push a i , y) = push a i , y
rightInv iso₂ (inl x , snd₁) = refl
rightInv iso₂ (inr x , snd₁) = refl
rightInv iso₂ (push a i , snd₁) = refl
leftInv iso₂ (inl x , snd₁) = refl
leftInv iso₂ (inr x , snd₁) = refl
leftInv iso₂ (push a i , snd₁) = refl
F : TotalSpacePush²'
→ (Pushout {A = typ A × Σ (Susp (typ A)) Hopf} fst snd)
F (inl x) = inl x
F (inr x) = inr x
F (push (x , y) i) = push (x , Push→TotalSpaceHopf x y) i
G : (Pushout {A = typ A × Σ (Susp (typ A)) Hopf} fst snd)
→ TotalSpacePush²'
G (inl x) = inl x
G (inr x) = inr x
G (push (x , y) i) =
hcomp (λ k → λ { (i = i0) → inl x
; (i = i1)
→ inr (secEq (_ , Push→TotalSpaceHopf-equiv x) y k)})
(push (x , invEq (_ , Push→TotalSpaceHopf-equiv x) y) i)
IsoTotalSpacePush²'ΣPush : Iso TotalSpacePush²'
(Pushout {A = typ A × Σ (Susp (typ A)) Hopf} fst snd)
fun IsoTotalSpacePush²'ΣPush = F
inv IsoTotalSpacePush²'ΣPush = G
rightInv IsoTotalSpacePush²'ΣPush (inl x) = refl
rightInv IsoTotalSpacePush²'ΣPush (inr x) = refl
rightInv IsoTotalSpacePush²'ΣPush (push (x , y) i) j =
hcomp (λ k → λ { (i = i0) → inl x
; (i = i1)
→ inr (secEq (_ , Push→TotalSpaceHopf-equiv x) y k)
; (j = i0) → F (
hfill (λ k →
λ { (i = i0) → inl x
; (i = i1)
→ inr (secEq (_
, Push→TotalSpaceHopf-equiv x) y k)})
(inS (push (x
, invEq (_
, Push→TotalSpaceHopf-equiv x) y) i)) k)
; (j = i1)
→ push (x
, (secEq (_
, Push→TotalSpaceHopf-equiv x) y k)) i})
(push (x , (secEq (_ , Push→TotalSpaceHopf-equiv x) y i0)) i)
leftInv IsoTotalSpacePush²'ΣPush (inl x) = refl
leftInv IsoTotalSpacePush²'ΣPush (inr x) = refl
leftInv IsoTotalSpacePush²'ΣPush (push (x , y) i) j =
hcomp (λ k → λ { (i = i0) → inl x
; (i = i1) → inr (secEq (Push→TotalSpaceHopf x
, Push→TotalSpaceHopf-equiv x)
(Push→TotalSpaceHopf x y) (j ∨ k))
; (j = i1) → push (x , y) i})
(hcomp (λ k → λ { (i = i0) → inl x
; (i = i1) → inr (retEq≡secEq
(Push→TotalSpaceHopf x
, Push→TotalSpaceHopf-equiv x)
y (~ k) j)
; (j = i1) → push (x , y) i
; (j = i0) → push (x , invEq
(Push→TotalSpaceHopf x
, Push→TotalSpaceHopf-equiv x)
(Push→TotalSpaceHopf x y)) i})
(push (x , retEq (Push→TotalSpaceHopf x
, Push→TotalSpaceHopf-equiv x) y j) i))
joinIso₂ : Iso TotalSpacePush² (join (typ A) (join (typ A) (typ A)))
joinIso₂ =
compIso IsoTotalSpacePush²TotalSpacePush²'
(compIso IsoTotalSpacePush²'ΣPush
(compIso (equivToIso (joinPushout≃join _ _))
(pathToIso (cong (join (typ A))
(isoToPath IsoTotalSpaceJoin)))))
-- Direct construction of Hopf fibration for S¹
module S¹Hopf where
Border : (x : S¹) → (j : I) → Partial (j ∨ ~ j) (Σ Type₀ (λ T → T ≃ S¹))
Border x j (j = i0) = S¹ , (x ·_) , rotIsEquiv x
Border x j (j = i1) = S¹ , idEquiv S¹
-- Hopf fibration using SuspS¹
HopfSuspS¹ : SuspS¹ → Type₀
HopfSuspS¹ north = S¹
HopfSuspS¹ south = S¹
HopfSuspS¹ (merid x j) = Glue S¹ (Border x j)
-- Hopf fibration using S²
-- TODO : prove that it is equivalent to HopfSuspS¹
HopfS² : S² → Type₀
HopfS² base = S¹
HopfS² (surf i j) = Glue S¹ (λ { (i = i0) → _ , idEquiv S¹
; (i = i1) → _ , idEquiv S¹
; (j = i0) → _ , idEquiv S¹
; (j = i1) → _ , _ , rotIsEquiv (loop i) } )
-- Hopf fibration using more direct definition of the rot equivalence
-- TODO : prove that it is equivalent to HopfSuspS¹
HopfS²' : S² → Type₀
HopfS²' base = S¹
HopfS²' (surf i j) = Glue S¹ (λ { (i = i0) → _ , rotLoopEquiv i0
; (i = i1) → _ , rotLoopEquiv i0
; (j = i0) → _ , rotLoopEquiv i0
; (j = i1) → _ , rotLoopEquiv i } )
-- Total space of the fibration
TotalHopf : Type₀
TotalHopf = Σ SuspS¹ HopfSuspS¹
-- Forward direction
filler-1 : I → (j : I) → (y : S¹) → Glue S¹ (Border y j) → join S¹ S¹
filler-1 i j y x = hfill (λ t → λ { (j = i0) → inl (rotInv-1 x y t)
; (j = i1) → inr x })
(inS (push ((unglue (j ∨ ~ j) x) · invLooper y) (unglue (j ∨ ~ j) x) j)) i
TotalHopf→JoinS¹S¹ : TotalHopf → join S¹ S¹
TotalHopf→JoinS¹S¹ (north , x) = inl x
TotalHopf→JoinS¹S¹ (south , x) = inr x
TotalHopf→JoinS¹S¹ (merid y j , x) = filler-1 i1 j y x
-- Backward direction
JoinS¹S¹→TotalHopf : join S¹ S¹ → TotalHopf
JoinS¹S¹→TotalHopf (inl x) = (north , x)
JoinS¹S¹→TotalHopf (inr x) = (south , x)
JoinS¹S¹→TotalHopf (push y x j) =
(merid (invLooper y · x) j
, glue (λ { (j = i0) → y ; (j = i1) → x }) (rotInv-2 x y j))
-- Now for the homotopies, we will need to fill squares indexed by x y : S¹ with value in S¹
-- Some will be extremeley tough, but happen to be easy when x = y = base
-- therefore, we fill them for x = y = base and then use the connectedness of S¹ × S¹ and
-- the discreteness of ΩS¹ to get general fillers.
-- To proceed with that strategy, we first need a lemma :
-- the sections of the trivial fibration λ (_ : S¹) (_ : S¹) → Int are constant
-- this should be generalized to a constant fibration over a connected space with
-- discrete fiber
fibℤ : S¹ → S¹ → Type₀
fibℤ _ _ = ℤ
S¹→HSet : (A : Type₀) (p : isSet A) (F : S¹ → A) (x : S¹) → F base ≡ F x
S¹→HSet A p F base = refl {x = F base}
S¹→HSet A p F (loop i) = f' i
where
f : PathP (λ i → F base ≡ F (loop i)) refl (cong F loop)
f i = λ j → F (loop (i ∧ j))
L : cong F loop ≡ refl
L = p (F base) (F base) (f i1) refl
f' : PathP (λ i → F base ≡ F (loop i)) (refl {x = F base}) (refl {x = F base})
f' = transport (λ i → PathP (λ j → F base ≡ F (loop j)) refl (L i)) f
constant-loop : (F : S¹ → S¹ → ℤ) → (x y : S¹) → F base base ≡ F x y
constant-loop F x y = L0 ∙ L1
where
p : isSet (S¹ → ℤ)
p = isSetΠ (λ _ → isSetℤ)
L : F base ≡ F x
L = S¹→HSet (S¹ → ℤ) p F x
L0 : F base base ≡ F x base
L0 i = L i base
L1 : F x base ≡ F x y
L1 = S¹→HSet ℤ isSetℤ (F x) y
discretefib : (F : S¹ → S¹ → Type₀) → Type₀
discretefib F = (a : (x y : S¹) → F x y) →
(b : (x y : S¹) → F x y) →
(a base base ≡ b base base) →
(x y : S¹) → a x y ≡ b x y
discretefib-fibℤ : discretefib fibℤ
discretefib-fibℤ a b h x y i =
hcomp (λ t → λ { (i = i0) → constant-loop a x y t
; (i = i1) → constant-loop b x y t })
(h i)
-- first homotopy
assocFiller-3-aux : I → I → I → I → S¹
assocFiller-3-aux x y j i =
hfill (λ t → λ { (i = i0) → rotInv-1 (loop y) (loop (~ y) · loop x) t
; (i = i1) → rotInv-3 (loop y) (loop x) t
; (x = i0) (y = i0) → base
; (x = i0) (y = i1) → base
; (x = i1) (y = i0) → base
; (x = i1) (y = i1) → base })
(inS ((rotInv-2 (loop x) (loop y) i) · (invLooper (loop (~ y) · loop x)))) j
-- assocFiller-3-endpoint is used only in the type of the next function, to specify the
-- second endpoint.
-- However, I only need the first endpoint, but I cannot specify only one of them as is.
-- TODO : use cubical extension types when available to remove assocFiller-3-endpoint
assocFiller-3-endpoint : (x : S¹) → (y : S¹) → y ≡ y
assocFiller-3-endpoint base base i = base
assocFiller-3-endpoint (loop x) base i = assocFiller-3-aux x i0 i1 i
assocFiller-3-endpoint base (loop y) i = assocFiller-3-aux i0 y i1 i
assocFiller-3-endpoint (loop x) (loop y) i = assocFiller-3-aux x y i1 i
assocFiller-3 : (x : S¹) → (y : S¹) →
PathP (λ j → rotInv-1 y (invLooper y · x) j ≡ rotInv-3 y x j)
(λ i → ((rotInv-2 x y i) · (invLooper (invLooper y · x))))
(assocFiller-3-endpoint x y)
assocFiller-3 base base j i = base
assocFiller-3 (loop x) base j i = assocFiller-3-aux x i0 j i
assocFiller-3 base (loop y) j i = assocFiller-3-aux i0 y j i
assocFiller-3 (loop x) (loop y) j i = assocFiller-3-aux x y j i
assoc-3 : (_ y : S¹) → basedΩS¹ y
assoc-3 x y i = assocFiller-3 x y i1 i
fibℤ≡fibAssoc-3 : fibℤ ≡ (λ _ y → basedΩS¹ y)
fibℤ≡fibAssoc-3 i = λ x y → basedΩS¹≡ℤ y (~ i)
discretefib-fibAssoc-3 : discretefib (λ _ y → basedΩS¹ y)
discretefib-fibAssoc-3 =
transp (λ i → discretefib (fibℤ≡fibAssoc-3 i)) i0 discretefib-fibℤ
assocConst-3 : (x y : S¹) → assoc-3 x y ≡ refl
assocConst-3 x y = discretefib-fibAssoc-3 assoc-3 (λ _ _ → refl) refl x y
assocSquare-3 : I → I → S¹ → S¹ → S¹
assocSquare-3 i j x y = hcomp (λ t → λ { (i = i0) → assocFiller-3 x y j i0
; (i = i1) → assocFiller-3 x y j i1
; (j = i0) → assocFiller-3 x y i0 i
; (j = i1) → assocConst-3 x y t i })
(assocFiller-3 x y j i)
filler-3 : I → I → S¹ → S¹ → join S¹ S¹
filler-3 i j y x =
hcomp (λ t → λ { (i = i0) → filler-1 t j (invLooper y · x)
(glue (λ { (j = i0) → y ; (j = i1) → x })
(rotInv-2 x y j))
; (i = i1) → push (rotInv-3 y x t) x j
; (j = i0) → inl (assocSquare-3 i t x y)
; (j = i1) → inr x })
(push ((rotInv-2 x y (i ∨ j)) · (invLooper (invLooper y · x))) (rotInv-2 x y (i ∨ j)) j)
JoinS¹S¹→TotalHopf→JoinS¹S¹ : ∀ x → TotalHopf→JoinS¹S¹ (JoinS¹S¹→TotalHopf x) ≡ x
JoinS¹S¹→TotalHopf→JoinS¹S¹ (inl x) i = inl x
JoinS¹S¹→TotalHopf→JoinS¹S¹ (inr x) i = inr x
JoinS¹S¹→TotalHopf→JoinS¹S¹ (push y x j) i = filler-3 i j y x
-- Second homotopy
-- This HIT is the total space of the Hopf fibration but the ends of SuspS¹ have not been
-- glued together yet — which makes it into a cylinder.
-- This allows to write compositions that do not properly match at the endpoints. However,
-- I suspect it is unnecessary. TODO : do without PseudoHopf
PseudoHopf : Type₀
PseudoHopf = (S¹ × Interval) × S¹
PseudoHopf-π1 : PseudoHopf → S¹
PseudoHopf-π1 ((y , _) , _) = y
PseudoHopf-π2 : PseudoHopf → S¹
PseudoHopf-π2 (_ , x) = x
assocFiller-4-aux : I → I → I → I → S¹
assocFiller-4-aux x y j i =
hfill (λ t → λ { (i = i0) → ((invLooper (loop y · loop x · loop (~ y))) · (loop y · loop x))
· (rotInv-1 (loop x) (loop y) t)
; (i = i1) → (rotInv-4 (loop y) (loop y · loop x) (~ t)) · loop x
; (x = i0) (y = i0) → base
; (x = i0) (y = i1) → base
; (x = i1) (y = i0) → base
; (x = i1) (y = i1) → base })
(inS (rotInv-2 (loop y · loop x) (loop y · loop x · loop (~ y)) i)) j
-- See assocFiller-3-endpoint
-- TODO : use cubical extension types when available to remove assocFiller-4-endpoint
assocFiller-4-endpoint : (x y : S¹) → basedΩS¹ (((invLooper (y · x · invLooper y)) · (y · x)) · x)
assocFiller-4-endpoint base base i = base
assocFiller-4-endpoint (loop x) base i = assocFiller-4-aux x i0 i1 i
assocFiller-4-endpoint base (loop y) i = assocFiller-4-aux i0 y i1 i
assocFiller-4-endpoint (loop x) (loop y) i = assocFiller-4-aux x y i1 i
assocFiller-4 : (x y : S¹) →
PathP (λ j → ((invLooper (y · x · invLooper y)) · (y · x)) · (rotInv-1 x y j) ≡ (rotInv-4 y (y · x) (~ j)) · x)
(λ i → (rotInv-2 (y · x) (y · x · invLooper y) i))
(assocFiller-4-endpoint x y)
assocFiller-4 base base j i = base
assocFiller-4 (loop x) base j i = assocFiller-4-aux x i0 j i
assocFiller-4 base (loop y) j i = assocFiller-4-aux i0 y j i
assocFiller-4 (loop x) (loop y) j i = assocFiller-4-aux x y j i
assoc-4 : (x y : S¹) → basedΩS¹ (((invLooper (y · x · invLooper y)) · (y · x)) · x)
assoc-4 x y i = assocFiller-4 x y i1 i
fibℤ≡fibAssoc-4 : fibℤ ≡ (λ x y → basedΩS¹ (((invLooper (y · x · invLooper y)) · (y · x)) · x))
fibℤ≡fibAssoc-4 i = λ x y → basedΩS¹≡ℤ (((invLooper (y · x · invLooper y)) · (y · x)) · x) (~ i)
discretefib-fibAssoc-4 : discretefib (λ x y → basedΩS¹ (((invLooper (y · x · invLooper y)) · (y · x)) · x))
discretefib-fibAssoc-4 =
transp (λ i → discretefib (fibℤ≡fibAssoc-4 i)) i0 discretefib-fibℤ
assocConst-4 : (x y : S¹) → assoc-4 x y ≡ refl
assocConst-4 x y = discretefib-fibAssoc-4 assoc-4 (λ _ _ → refl) refl x y
assocSquare-4 : I → I → S¹ → S¹ → S¹
assocSquare-4 i j x y =
hcomp (λ t → λ { (i = i0) → assocFiller-4 x y j i0
; (i = i1) → assocFiller-4 x y j i1
; (j = i0) → assocFiller-4 x y i0 i
; (j = i1) → assocConst-4 x y t i })
(assocFiller-4 x y j i)
filler-4-0 : (_ j : I) → (y : S¹) → Glue S¹ (Border y j) → PseudoHopf
filler-4-0 i j y x =
let x' = unglue (j ∨ ~ j) x in
hfill (λ t → λ { (j = i0) → ((invLooper (y · x · invLooper y) · (y · x) , I0)
, invLooper (y · x · invLooper y) · (y · x) · (rotInv-1 x y t))
; (j = i1) → ((invLooper (x · invLooper y) · x , I1) , x) })
(inS ((invLooper (x' · invLooper y) · x' , seg j) , rotInv-2 x' (x' · invLooper y) j)) i
filler-4-1 : (_ j : I) → (y : S¹) → Glue S¹ (Border y j) → PseudoHopf
filler-4-1 i j y x =
let x' = unglue (j ∨ ~ j) x in
hfill (λ t → λ { (j = i0) → ((invLooper (y · x · invLooper y) · (y · x) , I0)
, (rotInv-4 y (y · x) (~ t)) · x)
; (j = i1) → ((invLooper (x · invLooper y) · x , I1) , x) })
(inS ((invLooper (x' · invLooper y) · x' , seg j) , unglue (j ∨ ~ j) x)) i
filler-4-2 : (_ j : I) → (y : S¹) → Glue S¹ (Border y j) → TotalHopf
filler-4-2 i j y x =
let x' = unglue (j ∨ ~ j) x in
hcomp (λ t → λ { (i = i0) → JoinS¹S¹→TotalHopf (filler-1 t j y x)
; (i = i1) → (merid (PseudoHopf-π1 (filler-4-0 t j y x)) j
, glue (λ { (j = i0) → rotInv-1 x y t ; (j = i1) → x })
(PseudoHopf-π2 (filler-4-0 t j y x)))
; (j = i0) → (north , rotInv-1 x y t)
; (j = i1) → (south , x) })
(merid (invLooper (x' · invLooper y) · x') j
, glue (λ { (j = i0) → y · x · invLooper y ; (j = i1) → x }) (rotInv-2 x' (x' · invLooper y) j))
filler-4-3 : (_ j : I) → (y : S¹) → Glue S¹ (Border y j) → PseudoHopf
filler-4-3 i j y x =
let x' = unglue (j ∨ ~ j) x in
hcomp (λ t → λ { (i = i0) → filler-4-0 t j y x
; (i = i1) → filler-4-1 t j y x
; (j = i0) → ((invLooper (y · x · invLooper y) · (y · x) , I0) , assocSquare-4 i t x y)
; (j = i1) → ((invLooper (x · invLooper y) · x , I1) , x) })
((invLooper (x' · invLooper y) · x' , seg j) , rotInv-2 x' (x' · invLooper y) (i ∨ j))
filler-4-4 : (_ j : I) → (y : S¹) → Glue S¹ (Border y j) → PseudoHopf
filler-4-4 i j y x =
let x' = unglue (j ∨ ~ j) x in
hcomp (λ t → λ { (i = i0) → filler-4-1 t j y x
; (i = i1) → ((y , seg j) , unglue (j ∨ ~ j) x)
; (j = i0) → ((rotInv-4 y (y · x) i , I0)
, (rotInv-4 y (y · x) (i ∨ ~ t)) · x)
; (j = i1) → ((rotInv-4 y x i , I1) , x) })
((rotInv-4 y x' i , seg j) , x')
filler-4-5 : (_ j : I) → (y : S¹) → Glue S¹ (Border y j) → TotalHopf
filler-4-5 i j y x =
hcomp (λ t → λ { (i = i0) → filler-4-2 (~ t) j y x
; (i = i1) → (merid (PseudoHopf-π1 (filler-4-4 t j y x)) j
, glue (λ { (j = i0) → x ; (j = i1) → x })
(PseudoHopf-π2 (filler-4-4 t j y x)))
; (j = i0) → (north , x)
; (j = i1) → (south , x) })
(merid (PseudoHopf-π1 (filler-4-3 i j y x)) j
, glue (λ { (j = i0) → x ; (j = i1) → x }) (PseudoHopf-π2 (filler-4-3 i j y x)))
TotalHopf→JoinS¹S¹→TotalHopf : ∀ x → JoinS¹S¹→TotalHopf (TotalHopf→JoinS¹S¹ x) ≡ x
TotalHopf→JoinS¹S¹→TotalHopf (north , x) i = (north , x)
TotalHopf→JoinS¹S¹→TotalHopf (south , x) i = (south , x)
TotalHopf→JoinS¹S¹→TotalHopf (merid y j , x) i = filler-4-5 i j y x
JoinS¹S¹≡TotalHopf : join S¹ S¹ ≡ TotalHopf
JoinS¹S¹≡TotalHopf = isoToPath (iso JoinS¹S¹→TotalHopf
TotalHopf→JoinS¹S¹
TotalHopf→JoinS¹S¹→TotalHopf
JoinS¹S¹→TotalHopf→JoinS¹S¹)
S³≡TotalHopf : S³ ≡ TotalHopf
S³≡TotalHopf = S³≡joinS¹S¹ ∙ JoinS¹S¹≡TotalHopf
open Iso
IsoS³TotalHopf : Iso (S₊ 3) TotalHopf
fun IsoS³TotalHopf x = JoinS¹S¹→TotalHopf (S³→joinS¹S¹ (inv IsoS³S3 x))
inv IsoS³TotalHopf x = fun IsoS³S3 (joinS¹S¹→S³ (TotalHopf→JoinS¹S¹ x))
rightInv IsoS³TotalHopf x =
cong (JoinS¹S¹→TotalHopf ∘ S³→joinS¹S¹)
(leftInv IsoS³S3 (joinS¹S¹→S³ (TotalHopf→JoinS¹S¹ x)))
∙∙ cong JoinS¹S¹→TotalHopf
(joinS¹S¹→S³→joinS¹S¹ (TotalHopf→JoinS¹S¹ x))
∙∙ TotalHopf→JoinS¹S¹→TotalHopf x
leftInv IsoS³TotalHopf x =
cong (fun IsoS³S3 ∘ joinS¹S¹→S³)
(JoinS¹S¹→TotalHopf→JoinS¹S¹ (S³→joinS¹S¹ (inv IsoS³S3 x)))
∙∙ cong (fun IsoS³S3) (S³→joinS¹S¹→S³ (inv IsoS³S3 x))
∙∙ Iso.rightInv IsoS³S3 x
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{- Voevodsky's proof that univalence implies funext -}
{-# OPTIONS --cubical --safe #-}
module Cubical.Experiments.FunExtFromUA where
open import Cubical.Core.Everything
open import Cubical.Foundations.Everything
variable
ℓ ℓ' : Level
_∼_ : {X : Type ℓ} {A : X → Type ℓ'} → (f g : (x : X) → A x) → Type (ℓ-max ℓ ℓ')
f ∼ g = ∀ x → f x ≡ g x
funext : ∀ ℓ ℓ' → Type (ℓ-suc(ℓ-max ℓ ℓ'))
funext ℓ ℓ' = {X : Type ℓ} {Y : Type ℓ'} {f g : X → Y} → f ∼ g → f ≡ g
elimEquivFun' : ∀ {ℓ} (P : (A B : Type ℓ) → (A → B) → Type ℓ)
→ (r : (B : Type ℓ) → P B B (λ x → x))
→ (A B : Type ℓ) → (e : A ≃ B) → P A B (e .fst)
elimEquivFun' P r A B = elimEquivFun B (λ A → P A B) (r B) A
pre-comp-is-equiv : (X Y : Type ℓ) (f : X → Y) (Z : Type ℓ)
→ isEquiv f
→ isEquiv (λ (g : Y → Z) → g ∘ f)
pre-comp-is-equiv {ℓ} X Y f Z e = elimEquivFun' P r X Y (f , e)
where
P : (X Y : Type ℓ) → (X → Y) → Type ℓ
P X Y f = isEquiv (λ (g : Y → Z) → g ∘ f)
r : (B : Type ℓ) → P B B (λ x → x)
r B = idIsEquiv (B → Z)
leftCancellable : {X : Type ℓ} {Y : Type ℓ'} → (X → Y) → Type (ℓ-max ℓ ℓ')
leftCancellable f = ∀ {x x'} → f x ≡ f x' → x ≡ x'
equivLC : {X : Type ℓ} {Y : Type ℓ'} (f : X → Y) → isEquiv f → leftCancellable f
equivLC f e {x} {x'} p i = hcomp (λ j → \ {(i = i0) → secEq (f , e) x j ;
(i = i1) → secEq (f , e) x' j})
(invEq (f , e) (p i))
univalence-gives-funext : funext ℓ' ℓ
univalence-gives-funext {ℓ'} {ℓ} {X} {Y} {f₀} {f₁} = γ
where
Δ = Σ[ y₀ ∈ Y ] Σ[ y₁ ∈ Y ] y₀ ≡ y₁
δ : Y → Δ
δ y = (y , y , refl)
π₀ π₁ : Δ → Y
π₀ (y₀ , y₁ , p) = y₀
π₁ (y₀ , y₁ , p) = y₁
δ-is-equiv : isEquiv δ
δ-is-equiv = isoToIsEquiv (iso δ π₀ ε η)
where
η : (y : Y) → π₀ (δ y) ≡ y
η y = refl
ε : (d : Δ) → δ (π₀ d) ≡ d
ε (y₀ , y₁ , p) i = y₀ , p i , λ j → p (i ∧ j)
φ : (Δ → Y) → (Y → Y)
φ π = π ∘ δ
e : isEquiv φ
e = pre-comp-is-equiv Y Δ δ Y δ-is-equiv
p : φ π₀ ≡ φ π₁
p = refl
q : π₀ ≡ π₁
q = equivLC φ e p
g : (h : f₀ ∼ f₁) (π : Δ → Y) (x : X) → Y
g = λ h π x → π (f₀ x , f₁ x , h x)
γ : f₀ ∼ f₁ → f₀ ≡ f₁
γ h = cong (g h) q
γ' : f₀ ∼ f₁ → f₀ ≡ f₁
γ' h = f₀ ≡⟨ refl ⟩
(λ x → f₀ x) ≡⟨ refl ⟩
(λ x → π₀ (f₀ x , f₁ x , h x)) ≡⟨ cong (g h) q ⟩
(λ x → π₁ (f₀ x , f₁ x , h x)) ≡⟨ refl ⟩
(λ x → f₁ x) ≡⟨ refl ⟩
f₁ ∎
{- Experiment testing univalence via funext -}
private
data ℕ : Type₀ where
zero : ℕ
succ : ℕ → ℕ
f g : ℕ → ℕ
f n = n
g zero = zero
g (succ n) = succ (g n)
h : (n : ℕ) → f n ≡ g n
h zero = refl
h (succ n) = cong succ (h n)
p : f ≡ g
p = univalence-gives-funext h
five : ℕ
five = succ (succ (succ (succ (succ zero))))
a : Σ ℕ (λ n → f n ≡ five)
a = five , refl
b : Σ ℕ (λ n → g n ≡ five)
b = subst (λ - → Σ ℕ (λ n → - n ≡ five)) p a
c : fst b ≡ five
c = refl
{- It works, fast. -}
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{-# OPTIONS --cubical #-}
module _ where
-- Test case by Ulf Norell, 16/09/2020
open import Agda.Primitive.Cubical renaming (primIMin to _∧_)
open import Agda.Builtin.Cubical.Path using (_≡_)
data Nat : Set where
zero : Nat
suc : Nat → Nat
record Pos : Set where
constructor 1+_
field unpos : Nat
open Pos
Pos→Nat : Pos → Nat
Pos→Nat (1+ n) = suc n
variable
A : Set
x : A
refl : x ≡ x
refl {x = x} i = x
id : Pos → Pos
id n = n
-- (i ∧ j) in the system caused a mishandling of de Bruijn indexes.
foo : (n : Pos) (i j : I) → Nat
foo n i j = primHComp
(λ k → primPOr (i ∧ j) (i ∧ j)
(λ _ → suc (n .unpos)) (λ _ → suc (n .unpos)))
(suc (n .unpos))
-- The test triggers normalization to make sure the `primHComp` in
-- `foo` reduces properly.
-- v foil syntactic equality check
test : ∀ n i j → foo n i j ≡ foo (id n) i j
test n i j = refl
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------------------------------------------------------------------------
-- Inductively defined beta-eta-equality
------------------------------------------------------------------------
import Level
open import Data.Universe
module README.DependentlyTyped.Beta-Eta
(Uni₀ : Universe Level.zero Level.zero)
where
open import Data.Product renaming (curry to c)
open import Function hiding (_∋_)
import README.DependentlyTyped.Term as Term; open Term Uni₀
import README.DependentlyTyped.Term.Substitution as S; open S Uni₀
import Relation.Binary.PropositionalEquality as P
open P.≡-Reasoning
-- βη-equality.
infixl 9 _·_
infix 4 _≈_
infixr 2 _≈⟨_⟩_
data _≈_ : ∀ {Γ₁ σ₁} → Γ₁ ⊢ σ₁ → ∀ {Γ₂ σ₂} → Γ₂ ⊢ σ₂ → Set where
-- β and η.
β : ∀ {Γ sp₁ sp₂} {σ : IType Γ (π sp₁ sp₂)}
(t₁ : Γ ▻ fst σ ⊢ snd σ) (t₂ : Γ ⊢ fst σ) →
ƛ t₁ · t₂ ≈ t₁ /⊢ sub t₂
η : ∀ {Γ sp₁ sp₂ σ} (t : Γ ⊢ (π sp₁ sp₂ , σ)) →
ƛ ((t /⊢ wk[ fst σ ]) · var zero) ≈ t
-- The relation is an equivalence (reflexivity is proved below).
sym : ∀ {Γ₁ σ₁} {t₁ : Γ₁ ⊢ σ₁}
{Γ₂ σ₂} {t₂ : Γ₂ ⊢ σ₂}
(t₁≈t₂ : t₁ ≈ t₂) → t₂ ≈ t₁
_≈⟨_⟩_ : ∀ {Γ₁ σ₁} (t₁ : Γ₁ ⊢ σ₁)
{Γ₂ σ₂} {t₂ : Γ₂ ⊢ σ₂}
{Γ₃ σ₃} {t₃ : Γ₃ ⊢ σ₃}
(t₁≈t₂ : t₁ ≈ t₂) (t₂≈t₃ : t₂ ≈ t₃) → t₁ ≈ t₃
-- The relation is a congruence.
var : ∀ {Γ₁ σ₁} {x₁ : Γ₁ ∋ σ₁}
{Γ₂ σ₂} {x₂ : Γ₂ ∋ σ₂}
(x₁≅x₂ : x₁ ≅-∋ x₂) → var x₁ ≈ var x₂
ƛ : ∀ {Γ₁ σ₁ τ₁} {t₁ : Γ₁ ▻ σ₁ ⊢ τ₁}
{Γ₂ σ₂ τ₂} {t₂ : Γ₂ ▻ σ₂ ⊢ τ₂}
(t₁≈t₂ : t₁ ≈ t₂) → ƛ t₁ ≈ ƛ t₂
_·_ : ∀ {Γ₁ sp₁₁ sp₂₁ σ₁}
{t₁₁ : Γ₁ ⊢ π sp₁₁ sp₂₁ , σ₁} {t₂₁ : Γ₁ ⊢ fst σ₁}
{Γ₂ sp₁₂ sp₂₂ σ₂}
{t₁₂ : Γ₂ ⊢ π sp₁₂ sp₂₂ , σ₂} {t₂₂ : Γ₂ ⊢ fst σ₂}
(t₁₁≈t₁₂ : t₁₁ ≈ t₁₂) (t₂₁≈t₂₂ : t₂₁ ≈ t₂₂) →
t₁₁ · t₂₁ ≈ t₁₂ · t₂₂
-- Reflexivity.
infix 3 _□
_□ : ∀ {Γ σ} (t : Γ ⊢ σ) → t ≈ t
var x □ = var P.refl
ƛ t □ = ƛ (t □)
t₁ · t₂ □ = (t₁ □) · (t₂ □)
abstract
-- βη-equal terms have the same semantics.
≈-sound : ∀ {Γ₁ σ₁} {t₁ : Γ₁ ⊢ σ₁} {Γ₂ σ₂} {t₂ : Γ₂ ⊢ σ₂} →
t₁ ≈ t₂ → ⟦ t₁ ⟧ ≅-Value ⟦ t₂ ⟧
≈-sound (β t₁ t₂) = begin
[ c ⟦ t₁ ⟧ ˢ ⟦ t₂ ⟧ ] ≡⟨ corresponds (app (sub t₂)) t₁ ⟩
[ ⟦ t₁ /⊢ sub t₂ ⟧ ] ∎
≈-sound (η t) = begin
[ c (⟦ t /⊢ wk ⟧ ˢ lookup zero) ] ≡⟨ curry-cong $ ˢ-cong (P.sym $ corresponds (app wk) t)
(P.refl {x = [ lookup zero ]}) ⟩
[ c ((⟦ t ⟧ /̂Val ŵk) ˢ lookup zero) ] ≡⟨ P.refl ⟩
[ ⟦ t ⟧ ] ∎
≈-sound (sym t₁≈t₂) = P.sym $ ≈-sound t₁≈t₂
≈-sound (t₁ ≈⟨ t₁≈t₂ ⟩ t₂≈t₃) = P.trans (≈-sound t₁≈t₂) (≈-sound t₂≈t₃)
≈-sound (var P.refl) = P.refl
≈-sound (ƛ t₁≈t₂) = curry-cong (≈-sound t₁≈t₂)
≈-sound (t₁₁≈t₁₂ · t₂₁≈t₂₂) = ˢ-cong (≈-sound t₁₁≈t₁₂) (≈-sound t₂₁≈t₂₂)
-- βη-equal terms have identical contexts.
≈-⇒-≅-Ctxt : ∀ {Γ₁ σ₁} {t₁ : Γ₁ ⊢ σ₁} {Γ₂ σ₂} {t₂ : Γ₂ ⊢ σ₂} →
t₁ ≈ t₂ → Γ₁ ≅-Ctxt Γ₂
≈-⇒-≅-Ctxt t₁≈t₂ = P.cong [Value].Γ $ ≈-sound t₁≈t₂
-- βη-equal terms have identical types.
≈-⇒-≅-Type : ∀ {Γ₁ σ₁} {t₁ : Γ₁ ⊢ σ₁} {Γ₂ σ₂} {t₂ : Γ₂ ⊢ σ₂} →
t₁ ≈ t₂ → σ₁ ≅-Type σ₂
≈-⇒-≅-Type {t₁ = t₁} {t₂ = t₂} t₁≈t₂
with ⟦ t₂ ⟧ | ≈-sound t₁≈t₂
... | .(⟦ t₁ ⟧) | P.refl = P.refl
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-- Andreas, 2012-10-19 issue #719 blame correct module
-- Andreas, 2017-07-28 point to correct binding site ("as A")
module Issue719 where
import Common.Size as A
module M where
private open module A = M
-- WAS:
-- Duplicate definition of module A. Previous definition of module A
-- at .../Common/Size.agda:7,15-19
-- when scope checking the declaration
-- open module A = M
-- EXPECTED:
-- Duplicate definition of module A. Previous definition of module A
-- at .../Issue719.agda:6,25-26
-- when scope checking the declaration
-- open module A = M
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{-# OPTIONS --without-K --safe #-}
open import Categories.Category.Core using (Category)
module Categories.Diagram.Coequalizer {o ℓ e} (𝒞 : Category o ℓ e) where
open Category 𝒞
open HomReasoning
open Equiv
open import Categories.Morphism 𝒞
open import Categories.Morphism.Reasoning 𝒞
open import Level
open import Function using (_$_)
private
variable
A B C : Obj
h i j k : A ⇒ B
record IsCoequalizer {E} (f g : A ⇒ B) (arr : B ⇒ E) : Set (o ⊔ ℓ ⊔ e) where
field
equality : arr ∘ f ≈ arr ∘ g
coequalize : {h : B ⇒ C} → h ∘ f ≈ h ∘ g → E ⇒ C
universal : {h : B ⇒ C} {eq : h ∘ f ≈ h ∘ g} → h ≈ coequalize eq ∘ arr
unique : {h : B ⇒ C} {i : E ⇒ C} {eq : h ∘ f ≈ h ∘ g} → h ≈ i ∘ arr → i ≈ coequalize eq
unique′ : (eq eq′ : h ∘ f ≈ h ∘ g) → coequalize eq ≈ coequalize eq′
unique′ eq eq′ = unique universal
id-coequalize : id ≈ coequalize equality
id-coequalize = unique (⟺ identityˡ)
coequalize-resp-≈ : ∀ {eq : h ∘ f ≈ h ∘ g} {eq′ : i ∘ f ≈ i ∘ g} →
h ≈ i → coequalize eq ≈ coequalize eq′
coequalize-resp-≈ {h = h} {i = i} {eq = eq} {eq′ = eq′} h≈i = unique $ begin
i ≈˘⟨ h≈i ⟩
h ≈⟨ universal ⟩
coequalize eq ∘ arr ∎
coequalize-resp-≈′ : (eq : h ∘ f ≈ h ∘ g) → (eq′ : i ∘ f ≈ i ∘ g) →
h ≈ i → j ≈ coequalize eq → k ≈ coequalize eq′ → j ≈ k
coequalize-resp-≈′ {j = j} {k = k} eq eq′ h≈i eqj eqk = begin
j ≈⟨ eqj ⟩
coequalize eq ≈⟨ coequalize-resp-≈ h≈i ⟩
coequalize eq′ ≈˘⟨ eqk ⟩
k ∎
-- This could be proved via duality, but is easier to just write by hand,
-- as it makes the dependency graph a lot cleaner.
IsCoequalizer⇒Epi : IsCoequalizer h i j → Epi j
IsCoequalizer⇒Epi coeq _ _ eq =
coequalize-resp-≈′ (extendˡ equality) (extendˡ equality) eq (unique refl) (unique refl)
where
open IsCoequalizer coeq
record Coequalizer (f g : A ⇒ B) : Set (o ⊔ ℓ ⊔ e) where
field
{obj} : Obj
arr : B ⇒ obj
isCoequalizer : IsCoequalizer f g arr
open IsCoequalizer isCoequalizer public
Coequalizer⇒Epi : (e : Coequalizer h i) → Epi (Coequalizer.arr e)
Coequalizer⇒Epi coeq = IsCoequalizer⇒Epi isCoequalizer
where
open Coequalizer coeq
-- Proving this via duality arguments is kind of annoying, as ≅ does not behave nicely in
-- concert with op.
up-to-iso : (coe₁ coe₂ : Coequalizer h i) → Coequalizer.obj coe₁ ≅ Coequalizer.obj coe₂
up-to-iso coe₁ coe₂ = record
{ from = repack coe₁ coe₂
; to = repack coe₂ coe₁
; iso = record
{ isoˡ = repack-cancel coe₂ coe₁
; isoʳ = repack-cancel coe₁ coe₂
}
}
where
open Coequalizer
repack : (coe₁ coe₂ : Coequalizer h i) → obj coe₁ ⇒ obj coe₂
repack coe₁ coe₂ = coequalize coe₁ (equality coe₂)
repack∘ : (coe₁ coe₂ coe₃ : Coequalizer h i) → repack coe₂ coe₃ ∘ repack coe₁ coe₂ ≈ repack coe₁ coe₃
repack∘ coe₁ coe₂ coe₃ = unique coe₁ (⟺ (glueTrianglesˡ (⟺ (universal coe₂)) (⟺ (universal coe₁)))) -- unique e₃ (⟺ (glueTrianglesʳ (⟺ (universal e₃)) (⟺ (universal e₂))))
repack-cancel : (e₁ e₂ : Coequalizer h i) → repack e₁ e₂ ∘ repack e₂ e₁ ≈ id
repack-cancel coe₁ coe₂ = repack∘ coe₂ coe₁ coe₂ ○ ⟺ (id-coequalize coe₂)
IsCoequalizer⇒Coequalizer : IsCoequalizer h i k → Coequalizer h i
IsCoequalizer⇒Coequalizer {k = k} is-coe = record
{ arr = k
; isCoequalizer = is-coe
}
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module Numeral.Natural.Relation.Order.Proofs where
import Lvl
open import Data.Tuple as Tuple using (_⨯_ ; _,_)
open import Functional
open import Logic
open import Logic.Propositional
open import Logic.Propositional.Theorems
open import Logic.Predicate
open import Numeral.Natural
open import Numeral.Natural.Oper
open import Numeral.Natural.Oper.Proofs
open import Numeral.Natural.Induction
open import Numeral.Natural.Relation.Order
open import Relator.Equals
open import Relator.Equals.Proofs
open import Structure.Relator
import Structure.Relator.Names as Names
open import Structure.Function
open import Structure.Function.Domain
open import Structure.Operator.Properties
open import Structure.Relator.Ordering
open import Structure.Relator.Properties
open import Structure.Relator.Properties.Proofs
open import Syntax.Transitivity
open import Type.Properties.MereProposition
open import Type
-- TODO: A method for pattern matching: https://stackoverflow.com/questions/20682013/agda-why-am-i-unable-to-pattern-match-on-refl
instance
[≤]-succ-injectivity : ∀{x y} → Injective(succ{x}{y})
Injective.proof [≤]-succ-injectivity [≡]-intro = [≡]-intro
instance
[≤]-mereProposition : ∀{x y} → MereProposition(x ≤ y)
MereProposition.uniqueness [≤]-mereProposition {min} {min} = [≡]-intro
MereProposition.uniqueness [≤]-mereProposition {succ x} {succ y} = congruence₁(succ) (MereProposition.uniqueness [≤]-mereProposition {x}{y})
instance
[≤]-minimum = \{y} → _≤_.min {y}
[≤]-with-[𝐒] = \{x}{y} ⦃ xy ⦄ → _≤_.succ {x}{y} xy
[<]-minimum = \{y} → succ([≤]-minimum {y})
[≡]-to-[≤] : ∀{x y : ℕ} → (x ≡ y) → (x ≤ y)
[≡]-to-[≤] {𝟎} {_} _ = [≤]-minimum
[≡]-to-[≤] {𝐒(x)}{𝐒(y)} [≡]-intro = succ([≡]-to-[≤] {x}{y} [≡]-intro)
[≡]-to-[≥] : ∀{x y : ℕ} → (x ≡ y) → (x ≥ y)
[≡]-to-[≥] = [≡]-to-[≤] ∘ symmetry(_≡_)
[≰]-to-[≢] : ∀{x y : ℕ} → (x ≰ y) → (x ≢ y)
[≰]-to-[≢] = contrapositiveᵣ [≡]-to-[≤]
[≱]-to-[≢] : ∀{x y : ℕ} → (x ≱ y) → (x ≢ y)
[≱]-to-[≢] = contrapositiveᵣ [≡]-to-[≥]
[≤][0]ᵣ : ∀{x : ℕ} → (x ≤ 0) → (x ≡ 0)
[≤][0]ᵣ {𝟎} (_) = [≡]-intro
[≤][0]ᵣ {𝐒(_)} ()
[≤][0]ᵣ-negation : ∀{x : ℕ} → (𝐒(x) ≰ 0)
[≤][0]ᵣ-negation ()
[≤]-successor : ∀{x y : ℕ} → (x ≤ y) → (x ≤ 𝐒(y))
[≤]-successor {𝟎} {_} (_) = [≤]-minimum
[≤]-successor {𝐒(x)}{𝟎} ()
[≤]-successor {𝐒(x)}{𝐒(y)} (succ proof) = succ([≤]-successor {x}{y} (proof))
[≤]-predecessor : ∀{x y : ℕ} → (𝐒(x) ≤ y) → (x ≤ y)
[≤]-predecessor {x} {𝟎} ()
[≤]-predecessor {𝟎} {𝐒(y)} (_) = [≤]-minimum
[≤]-predecessor {𝐒(x)}{𝐒(y)} (succ proof) = succ([≤]-predecessor {x}{y} (proof))
[≤]-without-[𝐒] : ∀{x y : ℕ} → (𝐒(x) ≤ 𝐒(y)) → (x ≤ y)
[≤]-without-[𝐒] (succ proof) = proof
[≤][𝐒]ₗ : ∀{x : ℕ} → ¬(𝐒(x) ≤ x)
[≤][𝐒]ₗ {𝟎} (1≤0) = [≤][0]ᵣ-negation{0}(1≤0)
[≤][𝐒]ₗ {𝐒(n)} (SSn≤Sn) = [≤][𝐒]ₗ {n} ([≤]-without-[𝐒] {𝐒(n)}{n} (SSn≤Sn))
instance
[≤]-reflexivity : Reflexivity (_≤_)
Reflexivity.proof([≤]-reflexivity) = [≡]-to-[≤] [≡]-intro
instance
[≤]-transitivity : Transitivity (_≤_)
Transitivity.proof([≤]-transitivity) = proof where
proof : Names.Transitivity (_≤_)
proof {𝟎} {_} {_} (_)(_) = [≤]-minimum
proof {𝐒(a)}{𝐒(b)}{𝐒(c)} (succ proofₗ) (succ proofᵣ ) =
succ(proof {a}{b}{c} (proofₗ) (proofᵣ))
instance
[≤]-antisymmetry : Antisymmetry (_≤_) (_≡_)
Antisymmetry.proof([≤]-antisymmetry) = proof where
proof : Names.Antisymmetry (_≤_) (_≡_)
proof {𝟎} {𝟎} (_) (_) = [≡]-intro
proof {𝐒(_)} {𝟎} ()
proof {𝟎} {𝐒(_)} (_) ()
proof {𝐒(a)} {𝐒(b)} (succ proofₗ) (succ proofᵣ) =
[≡]-with(𝐒) (proof {a}{b} proofₗ proofᵣ)
instance
[≤]-totality : ConverseTotal(_≤_)
ConverseTotal.proof([≤]-totality) = proof where
proof : Names.ConverseTotal(_≤_)
proof {𝟎} {𝟎} = [∨]-introₗ ([≡]-to-[≤] [≡]-intro)
proof {𝐒(a)}{𝟎} = [∨]-introᵣ ([≤]-minimum)
proof {𝟎} {𝐒(b)} = [∨]-introₗ ([≤]-minimum)
proof {𝐒(a)}{𝐒(b)} = [∨]-elim ([∨]-introₗ ∘ (proof ↦ [≤]-with-[𝐒] {a}{b} ⦃ proof ⦄)) ([∨]-introᵣ ∘ (proof ↦ [≤]-with-[𝐒] {b}{a} ⦃ proof ⦄)) (proof {a}{b})
instance
[≤]-weakPartialOrder : Weak.PartialOrder (_≤_) (_≡_)
[≤]-weakPartialOrder = record{}
instance
[≤]-weakTotalOrder : Weak.TotalOrder (_≤_) (_≡_)
[≤]-weakTotalOrder = record{}
instance
[≥]-reflexivity : Reflexivity (_≥_)
Reflexivity.proof([≥]-reflexivity) = Reflexivity.proof([≤]-reflexivity)
instance
[≥]-transitivity : Transitivity (_≥_)
Transitivity.proof([≥]-transitivity) = swap(Transitivity.proof([≤]-transitivity))
instance
[≥]-antisymmetry : Antisymmetry (_≥_) (_≡_)
Antisymmetry.proof([≥]-antisymmetry) = swap(Antisymmetry.proof([≤]-antisymmetry))
instance
[≥]-totality : ConverseTotal(_≥_)
ConverseTotal.proof([≥]-totality) = ConverseTotal.proof([≤]-totality)
instance
[≥]-weakPartialOrder : Weak.PartialOrder (_≥_) (_≡_)
[≥]-weakPartialOrder = record{}
instance
[≥]-weakTotalOrder : Weak.TotalOrder (_≥_) (_≡_)
[≥]-weakTotalOrder = record{}
[≥]-to-[≮] : ∀{a b : ℕ} → (a ≮ b) ← (a ≥ b)
[≥]-to-[≮] {a}{b} (b≤a) (𝐒a≤b) = [≤][𝐒]ₗ (transitivity(_≤_) {𝐒(a)}{b}{a} (𝐒a≤b) (b≤a))
[≤]-to-[≯] : ∀{a b : ℕ} → (a ≯ b) ← (a ≤ b)
[≤]-to-[≯] {a}{b} (a≤b) (𝐒b≤a) = [≥]-to-[≮] {b}{a} (a≤b) (𝐒b≤a)
[>]-to-[≰] : ∀{a b : ℕ} → (a ≰ b) ← (a > b)
[>]-to-[≰] {a}{b} (𝐒b≤a) (a≤b) = [≤]-to-[≯] (a≤b) (𝐒b≤a)
[<]-to-[≱] : ∀{a b : ℕ} → (a ≱ b) ← (a < b)
[<]-to-[≱] {a}{b} (𝐒a≤b) (b≤a) = [≥]-to-[≮] (b≤a) (𝐒a≤b)
[<]-to-[≢] : ∀{a b : ℕ} → (a < b) → (a ≢ b)
[<]-to-[≢] = [≱]-to-[≢] ∘ [<]-to-[≱]
[>]-to-[≢] : ∀{a b : ℕ} → (a > b) → (a ≢ b)
[>]-to-[≢] = [≰]-to-[≢] ∘ [>]-to-[≰]
[<][0]ᵣ : ∀{x : ℕ} → (x ≮ 0)
[<][0]ᵣ = [≤][0]ᵣ-negation
instance
[<]-irreflexivity : Irreflexivity (_<_)
Irreflexivity.proof([<]-irreflexivity) = [≤][𝐒]ₗ
instance
[<]-transitivity : Transitivity (_<_)
Transitivity.proof([<]-transitivity) {x}{y}{z} (l) (r) = Transitivity.proof([≤]-transitivity) {𝐒(x)} {𝐒(y)} {z} ([≤]-successor (l)) (r)
instance
[<]-asymmetry : Asymmetry (_<_)
Asymmetry.proof([<]-asymmetry) (l) (r) = Irreflexivity.proof([<]-irreflexivity) (Transitivity.proof([<]-transitivity) (l) (r))
instance
[<]-converseTrichotomy : ConverseTrichotomy(_<_)(_≡_)
ConverseTrichotomy.proof [<]-converseTrichotomy = p where
p : Names.ConverseTrichotomy(_<_)(_≡_)
p {𝟎} {𝟎} = [∨]-introₗ ([∨]-introᵣ [≡]-intro)
p {𝟎} {𝐒 y} = [∨]-introₗ ([∨]-introₗ [≤]-with-[𝐒])
p {𝐒 x} {𝟎} = [∨]-introᵣ [≤]-with-[𝐒]
p {𝐒 x} {𝐒 y} with p {x} {y}
... | [∨]-introₗ ([∨]-introₗ (succ xy)) = [∨]-introₗ ([∨]-introₗ (succ (succ xy)))
... | [∨]-introₗ ([∨]-introᵣ [≡]-intro) = [∨]-introₗ ([∨]-introᵣ [≡]-intro)
... | [∨]-introᵣ (succ xy) = [∨]-introᵣ (succ (succ xy))
instance
[<]-strictPartialOrder : Strict.PartialOrder (_<_)
[<]-strictPartialOrder = record{}
instance
[<]-strictTotalOrder : Strict.TotalOrder (_<_)(_≡_)
[<]-strictTotalOrder = record{}
instance
[>]-irreflexivity : Irreflexivity (_>_)
Irreflexivity.proof([>]-irreflexivity) = Irreflexivity.proof([<]-irreflexivity)
instance
[>]-transitivity : Transitivity (_>_)
Transitivity.proof([>]-transitivity) = swap(Transitivity.proof([<]-transitivity))
instance
[>]-asymmetry : Asymmetry (_>_)
Asymmetry.proof([>]-asymmetry) = swap(Asymmetry.proof([<]-asymmetry))
instance
[>]-strictOrder : Strict.PartialOrder (_>_)
[>]-strictOrder = record{}
[<]-of-[𝐒] : ∀{x : ℕ} → (x < 𝐒(x))
[<]-of-[𝐒] = reflexivity(_≤_)
[<]-of-[𝟎][𝐒] : ∀{x : ℕ} → (𝟎 < 𝐒(x))
[<]-of-[𝟎][𝐒] {𝟎} = [<]-of-[𝐒]
[<]-of-[𝟎][𝐒] {𝐒 x} = succ([≤]-minimum)
instance
[≤]-of-[𝐒] : ∀{x : ℕ} → (x ≤ 𝐒(x))
[≤]-of-[𝐒] = [≤]-successor(reflexivity(_≤_))
[<][≢]-equivalence : ∀{x} → (x > 0) ↔ (x ≢ 0)
[<][≢]-equivalence {x} = [↔]-intro (l{x}) (r{x}) where
l : ∀{x} → (x > 0) ← (x ≢ 0)
l{𝟎} (x≢𝟎) = [⊥]-elim((x≢𝟎)([≡]-intro))
l{𝐒(x)} (𝐒x≢𝟎) = succ([≤]-minimum)
r : ∀{x} → (x > 0) → (x ≢ 0)
r{𝟎} ()
r{𝐒(x)} (𝟏≤𝐒x) (𝐒x≡𝟎) with [≡]-substitutionᵣ (𝐒x≡𝟎) {expr ↦ 1 ≤ expr} (𝟏≤𝐒x)
... | ()
[≤]-to-[<][≡] : ∀{a b : ℕ} → (a ≤ b) → (a < b)∨(a ≡ b)
[≤]-to-[<][≡] {𝟎} {𝟎} ([≤]-minimum) = [∨]-introᵣ([≡]-intro)
[≤]-to-[<][≡] {𝟎} {𝐒(b)} ([≤]-minimum) = [∨]-introₗ([<]-minimum)
[≤]-to-[<][≡] {𝐒(a)}{𝐒(b)} (succ(a≤b)) with [≤]-to-[<][≡] {a}{b} (a≤b)
... | [∨]-introₗ(a<b) = [∨]-introₗ(succ(a<b))
... | [∨]-introᵣ(a≡b) = [∨]-introᵣ([≡]-with(𝐒) (a≡b))
[≮][≢]-to-[≰] : ∀{a b : ℕ} → (a ≮ b) → (a ≢ b) → (a ≰ b)
[≮][≢]-to-[≰] (a≮b) (a≢b) (a≤b) with [≤]-to-[<][≡] (a≤b)
... | [∨]-introₗ (a<b) = a≮b a<b
... | [∨]-introᵣ (a≡b) = a≢b a≡b
[<][≡]-to-[≤] : ∀{a b : ℕ} → (a < b)∨(a ≡ b) → (a ≤ b)
[<][≡]-to-[≤] {a} {.a} ([∨]-introᵣ([≡]-intro)) = [≡]-to-[≤] ([≡]-intro)
[<][≡]-to-[≤] {a} {b} ([∨]-introₗ(a<b)) = [≤]-predecessor (a<b)
instance
[<][≤]-sub : (_<_) ⊆₂ (_≤_)
[<][≤]-sub = intro [≤]-predecessor
instance
[>][≥]-sub : (_>_) ⊆₂ (_≥_)
[>][≥]-sub = intro(sub₂(_<_)(_≤_))
[≰]-to-[≮] : ∀{x y : ℕ} → (x ≰ y) → (x ≮ y)
[≰]-to-[≮] = contrapositiveᵣ (sub₂(_<_)(_≤_))
[≥]-to-[>][≡] : ∀{a b : ℕ} → (a ≥ b) → (a > b)∨(a ≡ b)
[≥]-to-[>][≡] {a}{b} (proof) with [≤]-to-[<][≡] {b}{a} (proof)
... | [∨]-introₗ(a<b) = [∨]-introₗ(a<b)
... | [∨]-introᵣ(b≡a) = [∨]-introᵣ(symmetry(_≡_) (b≡a))
[<]-trichotomy : ∀{x y} → (x < y) ∨ (x ≡ y) ∨ (x > y)
[<]-trichotomy {x}{y} with converseTotal(_≤_) ⦃ [≤]-totality ⦄
[<]-trichotomy {x}{y} | [∨]-introₗ x≤y with [≤]-to-[<][≡] {x}{y} x≤y
[<]-trichotomy {x}{y} | [∨]-introₗ x≤y | [∨]-introₗ x<y = [∨]-introₗ ([∨]-introₗ x<y)
[<]-trichotomy {x}{y} | [∨]-introₗ x≤y | [∨]-introᵣ x≡y = [∨]-introₗ ([∨]-introᵣ x≡y)
[<]-trichotomy {x}{y} | [∨]-introᵣ y≤x with [≥]-to-[>][≡] {x}{y} y≤x
[<]-trichotomy {x}{y} | [∨]-introᵣ y≤x | [∨]-introₗ y<x = [∨]-introᵣ y<x
[<]-trichotomy {x}{y} | [∨]-introᵣ y≤x | [∨]-introᵣ y≡x = [∨]-introₗ ([∨]-introᵣ y≡x)
[≤][>]-dichotomy : ∀{x y} → (x ≤ y) ∨ (x > y)
[≤][>]-dichotomy {x}{y} with [<]-trichotomy {x}{y}
[≤][>]-dichotomy {x} {y} | [∨]-introₗ ([∨]-introₗ x<y) = [∨]-introₗ(sub₂(_<_)(_≤_) x<y)
[≤][>]-dichotomy {x} {y} | [∨]-introₗ ([∨]-introᵣ x≡y) = [∨]-introₗ(sub₂(_≡_)(_≤_) x≡y)
[≤][>]-dichotomy {x} {y} | [∨]-introᵣ x>y = [∨]-introᵣ(x>y)
[<][≥]-dichotomy : ∀{x y} → (x < y) ∨ (x ≥ y)
[<][≥]-dichotomy {x}{y} = [∨]-symmetry([≤][>]-dichotomy {y}{x})
[≯][≢]-to-[≱] : ∀{a b : ℕ} → (a ≯ b) → (a ≢ b) → (a ≱ b)
[≯][≢]-to-[≱] (a≯b) (a≢b) (a≥b) with [≥]-to-[>][≡] (a≥b)
... | [∨]-introₗ (a>b) = a≯b a>b
... | [∨]-introᵣ (a≡b) = a≢b a≡b
[>][≡]-to-[≥] : ∀{a b : ℕ} → (a > b)∨(a ≡ b) → (a ≥ b)
[>][≡]-to-[≥] {a}{b} ([∨]-introₗ(a<b)) = [<][≡]-to-[≤] {b}{a} ([∨]-introₗ(a<b))
[>][≡]-to-[≥] {a}{b} ([∨]-introᵣ(b≡a)) = [<][≡]-to-[≤] {b}{a} ([∨]-introᵣ(symmetry(_≡_)(b≡a)))
[>]-to-[≥] : ∀{a b : ℕ} → (a > b) → (a ≥ b)
[>]-to-[≥] {a}{b} (a<b) = [<][≡]-to-[≤] {b}{a} ([∨]-introₗ(a<b))
[≱]-to-[≯] : ∀{x y : ℕ} → (x ≱ y) → (x ≯ y)
[≱]-to-[≯] = contrapositiveᵣ [>]-to-[≥]
[≮][≯]-to-[≡] : ∀{a b : ℕ} → (a ≮ b) → (a ≯ b) → (a ≡ b)
[≮][≯]-to-[≡] {a}{b} (a≮b) (a≯b) with [<]-trichotomy {a}{b}
... | [∨]-introₗ ([∨]-introₗ a<b) = [⊥]-elim(a≮b a<b)
... | [∨]-introₗ ([∨]-introᵣ a≡b) = a≡b
... | [∨]-introᵣ b<a = [⊥]-elim(a≯b b<a)
[≮][≢][≯]-not : ∀{a b : ℕ} → (a ≮ b) → (a ≢ b) → (a ≯ b) → ⊥
[≮][≢][≯]-not {a}{b} (a≮b) (a≢b) (a≯b) with [<]-trichotomy {a}{b}
... | [∨]-introₗ ([∨]-introₗ a<b) = a≮b a<b
... | [∨]-introₗ ([∨]-introᵣ a≡b) = a≢b a≡b
... | [∨]-introᵣ b<a = a≯b b<a
[≰][≯]-not : ∀{a b : ℕ} → (a ≰ b) → (a ≯ b) → ⊥
[≰][≯]-not {a}{b} (a≰b) (a≯b) = [≮][≢][≯]-not ([≰]-to-[≮] a≰b) ([≰]-to-[≢] a≰b) (a≯b)
[≮][≱]-not : ∀{a b : ℕ} → (a ≮ b) → (a ≱ b) → ⊥
[≮][≱]-not {a}{b} (a≮b) (a≱b) = [≮][≢][≯]-not (a≮b) ([≱]-to-[≢] a≱b) ([≱]-to-[≯] a≱b)
[<]-non-zero-existence : ∀{a b : ℕ} → (a < b) → (𝟎 < b)
[<]-non-zero-existence (succ _) = [<]-of-[𝟎][𝐒]
[≢]-to-[<]-of-0ᵣ : ∀{n} → (n ≢ 0) → (0 < n)
[≢]-to-[<]-of-0ᵣ {𝟎} p with () ← p [≡]-intro
[≢]-to-[<]-of-0ᵣ {𝐒 n} p = succ min
[≤][≢]-to-[<] : ∀{a b : ℕ} → (a ≤ b) → (a ≢ b) → (a < b)
[≤][≢]-to-[<] {.𝟎} {b} min ne = [≢]-to-[<]-of-0ᵣ (ne ∘ symmetry(_≡_))
[≤][≢]-to-[<] {.(𝐒 _)} {.(𝐒 _)} (succ lt) ne = succ([≤][≢]-to-[<] lt (ne ∘ congruence₁(𝐒)))
instance
[≤][≡]-subtransitivityₗ : Subtransitivityₗ(_≤_)(_≡_)
[≤][≡]-subtransitivityₗ = subrelation-transitivity-to-subtransitivityₗ
instance
[≤][≡]-subtransitivityᵣ : Subtransitivityᵣ(_≤_)(_≡_)
[≤][≡]-subtransitivityᵣ = subrelation-transitivity-to-subtransitivityᵣ
instance
[≥][≡]-subtransitivityₗ : Subtransitivityₗ(_≥_)(_≡_)
[≥][≡]-subtransitivityₗ = subrelation-transitivity-to-subtransitivityₗ
instance
[≥][≡]-subtransitivityᵣ : Subtransitivityᵣ(_≥_)(_≡_)
[≥][≡]-subtransitivityᵣ = subrelation-transitivity-to-subtransitivityᵣ
instance
[<][≤]-subtransitivityₗ : Subtransitivityₗ(_≤_)(_<_)
[<][≤]-subtransitivityₗ = subrelation-transitivity-to-subtransitivityₗ
instance
[<][≤]-subtransitivityᵣ : Subtransitivityᵣ(_≤_)(_<_)
[<][≤]-subtransitivityᵣ = subrelation-transitivity-to-subtransitivityᵣ
instance
[>][≥]-subtransitivityₗ : Subtransitivityₗ(_≥_)(_>_)
[>][≥]-subtransitivityₗ = subrelation-transitivity-to-subtransitivityₗ
instance
[>][≥]-subtransitivityᵣ : Subtransitivityᵣ(_≥_)(_>_)
[>][≥]-subtransitivityᵣ = subrelation-transitivity-to-subtransitivityᵣ
instance
[>][≡]-subtransitivityₗ : Subtransitivityₗ(_>_)(_≡_)
Subtransitivityₗ.proof [>][≡]-subtransitivityₗ = substitute₂ₗ(_>_) ∘ symmetry(_≡_)
instance
[>][≡]-subtransitivityᵣ : Subtransitivityᵣ(_>_)(_≡_)
Subtransitivityᵣ.proof [>][≡]-subtransitivityᵣ = swap(substitute₂ᵣ(_>_))
instance
[<][≡]-subtransitivityₗ : Subtransitivityₗ(_<_)(_≡_)
Subtransitivityₗ.proof [<][≡]-subtransitivityₗ = substitute₂ₗ(_<_) ∘ symmetry(_≡_)
instance
[<][≡]-subtransitivityᵣ : Subtransitivityᵣ(_<_)(_≡_)
Subtransitivityᵣ.proof [<][≡]-subtransitivityᵣ = swap(substitute₂ᵣ(_<_))
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-- Andreas, 2014-05-17
open import Common.Prelude
open import Common.Equality
postulate
bla : ∀ x → x ≡ zero
P : Nat → Set
p : P zero
f : ∀ x → P x
f x rewrite bla {!!} = {!!}
-- Expected: two interaction points!
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-- Andreas, 2018-11-03, issue #3364
-- Andreas, 2019-02-23, issue #3457
--
-- Better error when trying to import with new qualified module name.
open import Agda.Builtin.Nat as Builtin.Nat
-- WAS: Error:
-- Not in scope:
-- as at ...
-- when scope checking as
-- NOW: Warning
-- `as' must be followed by an identifier; a qualified name is not allowed here
-- when scope checking the declaration
-- open import Agda.Builtin.Nat as Builtin.Nat
import Agda.Builtin.Sigma as .as
-- `as' must be followed by an identifier
-- when scope checking the declaration
-- import Agda.Builtin.Sigma as .as
import Agda.Builtin.String as _
-- `as' must be followed by an identifier; an underscore is not allowed here
-- when scope checking the declaration
-- import Agda.Builtin.String as _
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module _ where
record Semiring (A : Set) : Set where
infixl 6 _+_
field _+_ : A → A → A
open Semiring {{...}} public
infix 4 _≡_
postulate
Nat Bool : Set
_≡_ : Nat → Nat → Set
refl : ∀ {x} → x ≡ x
to : ∀ {x} (y : Nat) → x ≡ y
trans : {x y z : Nat} → x ≡ y → y ≡ z → x ≡ z
instance _ : Semiring Nat
_ : Semiring Bool
bad : (a b c : Nat) → a + b ≡ c + a
bad a b c =
trans (to (a + c)) refl
-- Should complain about a != c when checking refl, not a missing instance Semiring Nat.
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module Text.Greek.SBLGNT.Col where
open import Data.List
open import Text.Greek.Bible
open import Text.Greek.Script
open import Text.Greek.Script.Unicode
ΠΡΟΣ-ΚΟΛΟΣΣΑΕΙΣ : List (Word)
ΠΡΟΣ-ΚΟΛΟΣΣΑΕΙΣ =
word (Π ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Col.1.1"
∷ word (ἀ ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ς ∷ []) "Col.1.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Col.1.1"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Col.1.1"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Col.1.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.1.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.1"
∷ word (Τ ∷ ι ∷ μ ∷ ό ∷ θ ∷ ε ∷ ο ∷ ς ∷ []) "Col.1.1"
∷ word (ὁ ∷ []) "Col.1.1"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ς ∷ []) "Col.1.1"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.2"
∷ word (ἐ ∷ ν ∷ []) "Col.1.2"
∷ word (Κ ∷ ο ∷ ∙λ ∷ ο ∷ σ ∷ σ ∷ α ∷ ῖ ∷ ς ∷ []) "Col.1.2"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Col.1.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.2"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.2"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.2"
∷ word (ἐ ∷ ν ∷ []) "Col.1.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Col.1.2"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Col.1.2"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.1.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.2"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "Col.1.2"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Col.1.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.1.2"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "Col.1.2"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.1.2"
∷ word (Ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "Col.1.3"
∷ word (τ ∷ ῷ ∷ []) "Col.1.3"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Col.1.3"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὶ ∷ []) "Col.1.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.3"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Col.1.3"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.1.3"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Col.1.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.3"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ο ∷ τ ∷ ε ∷ []) "Col.1.3"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "Col.1.3"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.1.3"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.1.3"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.1.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.4"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.1.4"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.1.4"
∷ word (ἐ ∷ ν ∷ []) "Col.1.4"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Col.1.4"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Col.1.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.4"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "Col.1.4"
∷ word (ἣ ∷ ν ∷ []) "Col.1.4"
∷ word (ἔ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "Col.1.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.1.4"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Col.1.4"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Col.1.4"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "Col.1.4"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Col.1.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.5"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ α ∷ []) "Col.1.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.5"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ ε ∷ ι ∷ μ ∷ έ ∷ ν ∷ η ∷ ν ∷ []) "Col.1.5"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.1.5"
∷ word (ἐ ∷ ν ∷ []) "Col.1.5"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.5"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.5"
∷ word (ἣ ∷ ν ∷ []) "Col.1.5"
∷ word (π ∷ ρ ∷ ο ∷ η ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Col.1.5"
∷ word (ἐ ∷ ν ∷ []) "Col.1.5"
∷ word (τ ∷ ῷ ∷ []) "Col.1.5"
∷ word (∙λ ∷ ό ∷ γ ∷ ῳ ∷ []) "Col.1.5"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.1.5"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Col.1.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.5"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ο ∷ υ ∷ []) "Col.1.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.6"
∷ word (π ∷ α ∷ ρ ∷ ό ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Col.1.6"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.1.6"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.1.6"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Col.1.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.6"
∷ word (ἐ ∷ ν ∷ []) "Col.1.6"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "Col.1.6"
∷ word (τ ∷ ῷ ∷ []) "Col.1.6"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "Col.1.6"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "Col.1.6"
∷ word (κ ∷ α ∷ ρ ∷ π ∷ ο ∷ φ ∷ ο ∷ ρ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "Col.1.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.6"
∷ word (α ∷ ὐ ∷ ξ ∷ α ∷ ν ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "Col.1.6"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Col.1.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.6"
∷ word (ἐ ∷ ν ∷ []) "Col.1.6"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.1.6"
∷ word (ἀ ∷ φ ∷ []) "Col.1.6"
∷ word (ἧ ∷ ς ∷ []) "Col.1.6"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "Col.1.6"
∷ word (ἠ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Col.1.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.6"
∷ word (ἐ ∷ π ∷ έ ∷ γ ∷ ν ∷ ω ∷ τ ∷ ε ∷ []) "Col.1.6"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.6"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Col.1.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.6"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.1.6"
∷ word (ἐ ∷ ν ∷ []) "Col.1.6"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ ᾳ ∷ []) "Col.1.6"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Col.1.7"
∷ word (ἐ ∷ μ ∷ ά ∷ θ ∷ ε ∷ τ ∷ ε ∷ []) "Col.1.7"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Col.1.7"
∷ word (Ἐ ∷ π ∷ α ∷ φ ∷ ρ ∷ ᾶ ∷ []) "Col.1.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.7"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.7"
∷ word (σ ∷ υ ∷ ν ∷ δ ∷ ο ∷ ύ ∷ ∙λ ∷ ο ∷ υ ∷ []) "Col.1.7"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.1.7"
∷ word (ὅ ∷ ς ∷ []) "Col.1.7"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.1.7"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Col.1.7"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Col.1.7"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.1.7"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ς ∷ []) "Col.1.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.7"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.7"
∷ word (ὁ ∷ []) "Col.1.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.8"
∷ word (δ ∷ η ∷ ∙λ ∷ ώ ∷ σ ∷ α ∷ ς ∷ []) "Col.1.8"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.1.8"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.8"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.1.8"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "Col.1.8"
∷ word (ἐ ∷ ν ∷ []) "Col.1.8"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Col.1.8"
∷ word (Δ ∷ ι ∷ ὰ ∷ []) "Col.1.9"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Col.1.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.9"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Col.1.9"
∷ word (ἀ ∷ φ ∷ []) "Col.1.9"
∷ word (ἧ ∷ ς ∷ []) "Col.1.9"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "Col.1.9"
∷ word (ἠ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Col.1.9"
∷ word (ο ∷ ὐ ∷ []) "Col.1.9"
∷ word (π ∷ α ∷ υ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Col.1.9"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Col.1.9"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.1.9"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.1.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.9"
∷ word (α ∷ ἰ ∷ τ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.1.9"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Col.1.9"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ω ∷ θ ∷ ῆ ∷ τ ∷ ε ∷ []) "Col.1.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.9"
∷ word (ἐ ∷ π ∷ ί ∷ γ ∷ ν ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Col.1.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.9"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Col.1.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.9"
∷ word (ἐ ∷ ν ∷ []) "Col.1.9"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "Col.1.9"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ ᾳ ∷ []) "Col.1.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.9"
∷ word (σ ∷ υ ∷ ν ∷ έ ∷ σ ∷ ε ∷ ι ∷ []) "Col.1.9"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ῇ ∷ []) "Col.1.9"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Col.1.10"
∷ word (ἀ ∷ ξ ∷ ί ∷ ω ∷ ς ∷ []) "Col.1.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.10"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Col.1.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.1.10"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "Col.1.10"
∷ word (ἀ ∷ ρ ∷ ε ∷ σ ∷ κ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "Col.1.10"
∷ word (ἐ ∷ ν ∷ []) "Col.1.10"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "Col.1.10"
∷ word (ἔ ∷ ρ ∷ γ ∷ ῳ ∷ []) "Col.1.10"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ῷ ∷ []) "Col.1.10"
∷ word (κ ∷ α ∷ ρ ∷ π ∷ ο ∷ φ ∷ ο ∷ ρ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.1.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.10"
∷ word (α ∷ ὐ ∷ ξ ∷ α ∷ ν ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.1.10"
∷ word (τ ∷ ῇ ∷ []) "Col.1.10"
∷ word (ἐ ∷ π ∷ ι ∷ γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "Col.1.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.10"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.1.10"
∷ word (ἐ ∷ ν ∷ []) "Col.1.11"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "Col.1.11"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ []) "Col.1.11"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ μ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.1.11"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Col.1.11"
∷ word (τ ∷ ὸ ∷ []) "Col.1.11"
∷ word (κ ∷ ρ ∷ ά ∷ τ ∷ ο ∷ ς ∷ []) "Col.1.11"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.1.11"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Col.1.11"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.11"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.1.11"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ ν ∷ []) "Col.1.11"
∷ word (ὑ ∷ π ∷ ο ∷ μ ∷ ο ∷ ν ∷ ὴ ∷ ν ∷ []) "Col.1.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.11"
∷ word (μ ∷ α ∷ κ ∷ ρ ∷ ο ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Col.1.11"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Col.1.11"
∷ word (χ ∷ α ∷ ρ ∷ ᾶ ∷ ς ∷ []) "Col.1.11"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.1.12"
∷ word (τ ∷ ῷ ∷ []) "Col.1.12"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὶ ∷ []) "Col.1.12"
∷ word (τ ∷ ῷ ∷ []) "Col.1.12"
∷ word (ἱ ∷ κ ∷ α ∷ ν ∷ ώ ∷ σ ∷ α ∷ ν ∷ τ ∷ ι ∷ []) "Col.1.12"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.1.12"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.1.12"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.12"
∷ word (μ ∷ ε ∷ ρ ∷ ί ∷ δ ∷ α ∷ []) "Col.1.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.12"
∷ word (κ ∷ ∙λ ∷ ή ∷ ρ ∷ ο ∷ υ ∷ []) "Col.1.12"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.1.12"
∷ word (ἁ ∷ γ ∷ ί ∷ ω ∷ ν ∷ []) "Col.1.12"
∷ word (ἐ ∷ ν ∷ []) "Col.1.12"
∷ word (τ ∷ ῷ ∷ []) "Col.1.12"
∷ word (φ ∷ ω ∷ τ ∷ ί ∷ []) "Col.1.12"
∷ word (ὃ ∷ ς ∷ []) "Col.1.13"
∷ word (ἐ ∷ ρ ∷ ρ ∷ ύ ∷ σ ∷ α ∷ τ ∷ ο ∷ []) "Col.1.13"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.1.13"
∷ word (ἐ ∷ κ ∷ []) "Col.1.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.1.13"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Col.1.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.13"
∷ word (σ ∷ κ ∷ ό ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Col.1.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.13"
∷ word (μ ∷ ε ∷ τ ∷ έ ∷ σ ∷ τ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Col.1.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.1.13"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.13"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "Col.1.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.13"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "Col.1.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.1.13"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ς ∷ []) "Col.1.13"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.13"
∷ word (ἐ ∷ ν ∷ []) "Col.1.14"
∷ word (ᾧ ∷ []) "Col.1.14"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Col.1.14"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.14"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ύ ∷ τ ∷ ρ ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Col.1.14"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.14"
∷ word (ἄ ∷ φ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Col.1.14"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.1.14"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ι ∷ ῶ ∷ ν ∷ []) "Col.1.14"
∷ word (ὅ ∷ ς ∷ []) "Col.1.15"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.1.15"
∷ word (ε ∷ ἰ ∷ κ ∷ ὼ ∷ ν ∷ []) "Col.1.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.15"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.1.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.15"
∷ word (ἀ ∷ ο ∷ ρ ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "Col.1.15"
∷ word (π ∷ ρ ∷ ω ∷ τ ∷ ό ∷ τ ∷ ο ∷ κ ∷ ο ∷ ς ∷ []) "Col.1.15"
∷ word (π ∷ ά ∷ σ ∷ η ∷ ς ∷ []) "Col.1.15"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Col.1.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Col.1.16"
∷ word (ἐ ∷ ν ∷ []) "Col.1.16"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Col.1.16"
∷ word (ἐ ∷ κ ∷ τ ∷ ί ∷ σ ∷ θ ∷ η ∷ []) "Col.1.16"
∷ word (τ ∷ ὰ ∷ []) "Col.1.16"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.1.16"
∷ word (ἐ ∷ ν ∷ []) "Col.1.16"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.16"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.16"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Col.1.16"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.1.16"
∷ word (γ ∷ ῆ ∷ ς ∷ []) "Col.1.16"
∷ word (τ ∷ ὰ ∷ []) "Col.1.16"
∷ word (ὁ ∷ ρ ∷ α ∷ τ ∷ ὰ ∷ []) "Col.1.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.16"
∷ word (τ ∷ ὰ ∷ []) "Col.1.16"
∷ word (ἀ ∷ ό ∷ ρ ∷ α ∷ τ ∷ α ∷ []) "Col.1.16"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "Col.1.16"
∷ word (θ ∷ ρ ∷ ό ∷ ν ∷ ο ∷ ι ∷ []) "Col.1.16"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "Col.1.16"
∷ word (κ ∷ υ ∷ ρ ∷ ι ∷ ό ∷ τ ∷ η ∷ τ ∷ ε ∷ ς ∷ []) "Col.1.16"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "Col.1.16"
∷ word (ἀ ∷ ρ ∷ χ ∷ α ∷ ὶ ∷ []) "Col.1.16"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "Col.1.16"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ι ∷ []) "Col.1.16"
∷ word (τ ∷ ὰ ∷ []) "Col.1.16"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.1.16"
∷ word (δ ∷ ι ∷ []) "Col.1.16"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.1.16"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Col.1.16"
∷ word (ἔ ∷ κ ∷ τ ∷ ι ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "Col.1.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.17"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ς ∷ []) "Col.1.17"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.1.17"
∷ word (π ∷ ρ ∷ ὸ ∷ []) "Col.1.17"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Col.1.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.17"
∷ word (τ ∷ ὰ ∷ []) "Col.1.17"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.1.17"
∷ word (ἐ ∷ ν ∷ []) "Col.1.17"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Col.1.17"
∷ word (σ ∷ υ ∷ ν ∷ έ ∷ σ ∷ τ ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "Col.1.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.18"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ς ∷ []) "Col.1.18"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.1.18"
∷ word (ἡ ∷ []) "Col.1.18"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ὴ ∷ []) "Col.1.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.18"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Col.1.18"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.1.18"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Col.1.18"
∷ word (ὅ ∷ ς ∷ []) "Col.1.18"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.1.18"
∷ word (ἀ ∷ ρ ∷ χ ∷ ή ∷ []) "Col.1.18"
∷ word (π ∷ ρ ∷ ω ∷ τ ∷ ό ∷ τ ∷ ο ∷ κ ∷ ο ∷ ς ∷ []) "Col.1.18"
∷ word (ἐ ∷ κ ∷ []) "Col.1.18"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.1.18"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Col.1.18"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Col.1.18"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Col.1.18"
∷ word (ἐ ∷ ν ∷ []) "Col.1.18"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "Col.1.18"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "Col.1.18"
∷ word (π ∷ ρ ∷ ω ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "Col.1.18"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Col.1.19"
∷ word (ἐ ∷ ν ∷ []) "Col.1.19"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Col.1.19"
∷ word (ε ∷ ὐ ∷ δ ∷ ό ∷ κ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Col.1.19"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "Col.1.19"
∷ word (τ ∷ ὸ ∷ []) "Col.1.19"
∷ word (π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ μ ∷ α ∷ []) "Col.1.19"
∷ word (κ ∷ α ∷ τ ∷ ο ∷ ι ∷ κ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Col.1.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.20"
∷ word (δ ∷ ι ∷ []) "Col.1.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.20"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ τ ∷ α ∷ ∙λ ∷ ∙λ ∷ ά ∷ ξ ∷ α ∷ ι ∷ []) "Col.1.20"
∷ word (τ ∷ ὰ ∷ []) "Col.1.20"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.1.20"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.1.20"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "Col.1.20"
∷ word (ε ∷ ἰ ∷ ρ ∷ η ∷ ν ∷ ο ∷ π ∷ ο ∷ ι ∷ ή ∷ σ ∷ α ∷ ς ∷ []) "Col.1.20"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Col.1.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.20"
∷ word (α ∷ ἵ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Col.1.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.20"
∷ word (σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ο ∷ ῦ ∷ []) "Col.1.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.20"
∷ word (δ ∷ ι ∷ []) "Col.1.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.20"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "Col.1.20"
∷ word (τ ∷ ὰ ∷ []) "Col.1.20"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Col.1.20"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.1.20"
∷ word (γ ∷ ῆ ∷ ς ∷ []) "Col.1.20"
∷ word (ε ∷ ἴ ∷ τ ∷ ε ∷ []) "Col.1.20"
∷ word (τ ∷ ὰ ∷ []) "Col.1.20"
∷ word (ἐ ∷ ν ∷ []) "Col.1.20"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.20"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.21"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.1.21"
∷ word (π ∷ ο ∷ τ ∷ ε ∷ []) "Col.1.21"
∷ word (ὄ ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Col.1.21"
∷ word (ἀ ∷ π ∷ η ∷ ∙λ ∷ ∙λ ∷ ο ∷ τ ∷ ρ ∷ ι ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ ς ∷ []) "Col.1.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.21"
∷ word (ἐ ∷ χ ∷ θ ∷ ρ ∷ ο ∷ ὺ ∷ ς ∷ []) "Col.1.21"
∷ word (τ ∷ ῇ ∷ []) "Col.1.21"
∷ word (δ ∷ ι ∷ α ∷ ν ∷ ο ∷ ί ∷ ᾳ ∷ []) "Col.1.21"
∷ word (ἐ ∷ ν ∷ []) "Col.1.21"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.21"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ι ∷ ς ∷ []) "Col.1.21"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.21"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.21"
∷ word (ν ∷ υ ∷ ν ∷ ὶ ∷ []) "Col.1.22"
∷ word (δ ∷ ὲ ∷ []) "Col.1.22"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ τ ∷ η ∷ ∙λ ∷ ∙λ ∷ ά ∷ γ ∷ η ∷ τ ∷ ε ∷ []) "Col.1.22"
∷ word (ἐ ∷ ν ∷ []) "Col.1.22"
∷ word (τ ∷ ῷ ∷ []) "Col.1.22"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Col.1.22"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.1.22"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Col.1.22"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.22"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Col.1.22"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.22"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "Col.1.22"
∷ word (π ∷ α ∷ ρ ∷ α ∷ σ ∷ τ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Col.1.22"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.1.22"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "Col.1.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.22"
∷ word (ἀ ∷ μ ∷ ώ ∷ μ ∷ ο ∷ υ ∷ ς ∷ []) "Col.1.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.22"
∷ word (ἀ ∷ ν ∷ ε ∷ γ ∷ κ ∷ ∙λ ∷ ή ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Col.1.22"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ ν ∷ ώ ∷ π ∷ ι ∷ ο ∷ ν ∷ []) "Col.1.22"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.22"
∷ word (ε ∷ ἴ ∷ []) "Col.1.23"
∷ word (γ ∷ ε ∷ []) "Col.1.23"
∷ word (ἐ ∷ π ∷ ι ∷ μ ∷ έ ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "Col.1.23"
∷ word (τ ∷ ῇ ∷ []) "Col.1.23"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ι ∷ []) "Col.1.23"
∷ word (τ ∷ ε ∷ θ ∷ ε ∷ μ ∷ ε ∷ ∙λ ∷ ι ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Col.1.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.23"
∷ word (ἑ ∷ δ ∷ ρ ∷ α ∷ ῖ ∷ ο ∷ ι ∷ []) "Col.1.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.23"
∷ word (μ ∷ ὴ ∷ []) "Col.1.23"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ κ ∷ ι ∷ ν ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.1.23"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Col.1.23"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.1.23"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ ο ∷ ς ∷ []) "Col.1.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.23"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ο ∷ υ ∷ []) "Col.1.23"
∷ word (ο ∷ ὗ ∷ []) "Col.1.23"
∷ word (ἠ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Col.1.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.23"
∷ word (κ ∷ η ∷ ρ ∷ υ ∷ χ ∷ θ ∷ έ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Col.1.23"
∷ word (ἐ ∷ ν ∷ []) "Col.1.23"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "Col.1.23"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "Col.1.23"
∷ word (τ ∷ ῇ ∷ []) "Col.1.23"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Col.1.23"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Col.1.23"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ό ∷ ν ∷ []) "Col.1.23"
∷ word (ο ∷ ὗ ∷ []) "Col.1.23"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ό ∷ μ ∷ η ∷ ν ∷ []) "Col.1.23"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Col.1.23"
∷ word (Π ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Col.1.23"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ς ∷ []) "Col.1.23"
∷ word (Ν ∷ ῦ ∷ ν ∷ []) "Col.1.24"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ω ∷ []) "Col.1.24"
∷ word (ἐ ∷ ν ∷ []) "Col.1.24"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.24"
∷ word (π ∷ α ∷ θ ∷ ή ∷ μ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "Col.1.24"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Col.1.24"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.1.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.24"
∷ word (ἀ ∷ ν ∷ τ ∷ α ∷ ν ∷ α ∷ π ∷ ∙λ ∷ η ∷ ρ ∷ ῶ ∷ []) "Col.1.24"
∷ word (τ ∷ ὰ ∷ []) "Col.1.24"
∷ word (ὑ ∷ σ ∷ τ ∷ ε ∷ ρ ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Col.1.24"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.1.24"
∷ word (θ ∷ ∙λ ∷ ί ∷ ψ ∷ ε ∷ ω ∷ ν ∷ []) "Col.1.24"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.24"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.24"
∷ word (ἐ ∷ ν ∷ []) "Col.1.24"
∷ word (τ ∷ ῇ ∷ []) "Col.1.24"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ί ∷ []) "Col.1.24"
∷ word (μ ∷ ο ∷ υ ∷ []) "Col.1.24"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Col.1.24"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.24"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Col.1.24"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.24"
∷ word (ὅ ∷ []) "Col.1.24"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.1.24"
∷ word (ἡ ∷ []) "Col.1.24"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ []) "Col.1.24"
∷ word (ἧ ∷ ς ∷ []) "Col.1.25"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ό ∷ μ ∷ η ∷ ν ∷ []) "Col.1.25"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Col.1.25"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ς ∷ []) "Col.1.25"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Col.1.25"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.25"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Col.1.25"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.25"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.1.25"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.25"
∷ word (δ ∷ ο ∷ θ ∷ ε ∷ ῖ ∷ σ ∷ ά ∷ ν ∷ []) "Col.1.25"
∷ word (μ ∷ ο ∷ ι ∷ []) "Col.1.25"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.1.25"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.1.25"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ῶ ∷ σ ∷ α ∷ ι ∷ []) "Col.1.25"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Col.1.25"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ν ∷ []) "Col.1.25"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.25"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.1.25"
∷ word (τ ∷ ὸ ∷ []) "Col.1.26"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Col.1.26"
∷ word (τ ∷ ὸ ∷ []) "Col.1.26"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ ε ∷ κ ∷ ρ ∷ υ ∷ μ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "Col.1.26"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Col.1.26"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.1.26"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ω ∷ ν ∷ []) "Col.1.26"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.26"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Col.1.26"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.1.26"
∷ word (γ ∷ ε ∷ ν ∷ ε ∷ ῶ ∷ ν ∷ []) "Col.1.26"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Col.1.26"
∷ word (δ ∷ ὲ ∷ []) "Col.1.26"
∷ word (ἐ ∷ φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ώ ∷ θ ∷ η ∷ []) "Col.1.26"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.26"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Col.1.26"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.26"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "Col.1.27"
∷ word (ἠ ∷ θ ∷ έ ∷ ∙λ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Col.1.27"
∷ word (ὁ ∷ []) "Col.1.27"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Col.1.27"
∷ word (γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ σ ∷ α ∷ ι ∷ []) "Col.1.27"
∷ word (τ ∷ ί ∷ []) "Col.1.27"
∷ word (τ ∷ ὸ ∷ []) "Col.1.27"
∷ word (π ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ς ∷ []) "Col.1.27"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.1.27"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Col.1.27"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.1.27"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ η ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Col.1.27"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "Col.1.27"
∷ word (ἐ ∷ ν ∷ []) "Col.1.27"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.1.27"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Col.1.27"
∷ word (ὅ ∷ []) "Col.1.27"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.1.27"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Col.1.27"
∷ word (ἐ ∷ ν ∷ []) "Col.1.27"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.1.27"
∷ word (ἡ ∷ []) "Col.1.27"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ὶ ∷ ς ∷ []) "Col.1.27"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.1.27"
∷ word (δ ∷ ό ∷ ξ ∷ η ∷ ς ∷ []) "Col.1.27"
∷ word (ὃ ∷ ν ∷ []) "Col.1.28"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Col.1.28"
∷ word (κ ∷ α ∷ τ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Col.1.28"
∷ word (ν ∷ ο ∷ υ ∷ θ ∷ ε ∷ τ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.1.28"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.1.28"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Col.1.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.28"
∷ word (δ ∷ ι ∷ δ ∷ ά ∷ σ ∷ κ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.1.28"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.1.28"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Col.1.28"
∷ word (ἐ ∷ ν ∷ []) "Col.1.28"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "Col.1.28"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ ᾳ ∷ []) "Col.1.28"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Col.1.28"
∷ word (π ∷ α ∷ ρ ∷ α ∷ σ ∷ τ ∷ ή ∷ σ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Col.1.28"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.1.28"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Col.1.28"
∷ word (τ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ ο ∷ ν ∷ []) "Col.1.28"
∷ word (ἐ ∷ ν ∷ []) "Col.1.28"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Col.1.28"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.1.29"
∷ word (ὃ ∷ []) "Col.1.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.1.29"
∷ word (κ ∷ ο ∷ π ∷ ι ∷ ῶ ∷ []) "Col.1.29"
∷ word (ἀ ∷ γ ∷ ω ∷ ν ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Col.1.29"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Col.1.29"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.29"
∷ word (ἐ ∷ ν ∷ έ ∷ ρ ∷ γ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Col.1.29"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.1.29"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.1.29"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ η ∷ ν ∷ []) "Col.1.29"
∷ word (ἐ ∷ ν ∷ []) "Col.1.29"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Col.1.29"
∷ word (ἐ ∷ ν ∷ []) "Col.1.29"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ []) "Col.1.29"
∷ word (Θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "Col.2.1"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Col.2.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.2.1"
∷ word (ε ∷ ἰ ∷ δ ∷ έ ∷ ν ∷ α ∷ ι ∷ []) "Col.2.1"
∷ word (ἡ ∷ ∙λ ∷ ί ∷ κ ∷ ο ∷ ν ∷ []) "Col.2.1"
∷ word (ἀ ∷ γ ∷ ῶ ∷ ν ∷ α ∷ []) "Col.2.1"
∷ word (ἔ ∷ χ ∷ ω ∷ []) "Col.2.1"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Col.2.1"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.2.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.1"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.2.1"
∷ word (ἐ ∷ ν ∷ []) "Col.2.1"
∷ word (Λ ∷ α ∷ ο ∷ δ ∷ ι ∷ κ ∷ ε ∷ ί ∷ ᾳ ∷ []) "Col.2.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.1"
∷ word (ὅ ∷ σ ∷ ο ∷ ι ∷ []) "Col.2.1"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "Col.2.1"
∷ word (ἑ ∷ ό ∷ ρ ∷ α ∷ κ ∷ α ∷ ν ∷ []) "Col.2.1"
∷ word (τ ∷ ὸ ∷ []) "Col.2.1"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ ω ∷ π ∷ ό ∷ ν ∷ []) "Col.2.1"
∷ word (μ ∷ ο ∷ υ ∷ []) "Col.2.1"
∷ word (ἐ ∷ ν ∷ []) "Col.2.1"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ί ∷ []) "Col.2.1"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Col.2.2"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ ∙λ ∷ η ∷ θ ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "Col.2.2"
∷ word (α ∷ ἱ ∷ []) "Col.2.2"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ι ∷ []) "Col.2.2"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Col.2.2"
∷ word (σ ∷ υ ∷ μ ∷ β ∷ ι ∷ β ∷ α ∷ σ ∷ θ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.2.2"
∷ word (ἐ ∷ ν ∷ []) "Col.2.2"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "Col.2.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.2"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.2.2"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "Col.2.2"
∷ word (π ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ς ∷ []) "Col.2.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.2.2"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ φ ∷ ο ∷ ρ ∷ ί ∷ α ∷ ς ∷ []) "Col.2.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.2.2"
∷ word (σ ∷ υ ∷ ν ∷ έ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Col.2.2"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.2.2"
∷ word (ἐ ∷ π ∷ ί ∷ γ ∷ ν ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Col.2.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.2.2"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ η ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Col.2.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.2.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.2.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.2.2"
∷ word (ἐ ∷ ν ∷ []) "Col.2.3"
∷ word (ᾧ ∷ []) "Col.2.3"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "Col.2.3"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.2.3"
∷ word (ο ∷ ἱ ∷ []) "Col.2.3"
∷ word (θ ∷ η ∷ σ ∷ α ∷ υ ∷ ρ ∷ ο ∷ ὶ ∷ []) "Col.2.3"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.2.3"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ς ∷ []) "Col.2.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.3"
∷ word (γ ∷ ν ∷ ώ ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "Col.2.3"
∷ word (ἀ ∷ π ∷ ό ∷ κ ∷ ρ ∷ υ ∷ φ ∷ ο ∷ ι ∷ []) "Col.2.3"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Col.2.4"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Col.2.4"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Col.2.4"
∷ word (μ ∷ η ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "Col.2.4"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.2.4"
∷ word (π ∷ α ∷ ρ ∷ α ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ ζ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Col.2.4"
∷ word (ἐ ∷ ν ∷ []) "Col.2.4"
∷ word (π ∷ ι ∷ θ ∷ α ∷ ν ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ ᾳ ∷ []) "Col.2.4"
∷ word (ε ∷ ἰ ∷ []) "Col.2.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Col.2.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.5"
∷ word (τ ∷ ῇ ∷ []) "Col.2.5"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "Col.2.5"
∷ word (ἄ ∷ π ∷ ε ∷ ι ∷ μ ∷ ι ∷ []) "Col.2.5"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Col.2.5"
∷ word (τ ∷ ῷ ∷ []) "Col.2.5"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Col.2.5"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Col.2.5"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.2.5"
∷ word (ε ∷ ἰ ∷ μ ∷ ι ∷ []) "Col.2.5"
∷ word (χ ∷ α ∷ ί ∷ ρ ∷ ω ∷ ν ∷ []) "Col.2.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.5"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ω ∷ ν ∷ []) "Col.2.5"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.2.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.2.5"
∷ word (τ ∷ ά ∷ ξ ∷ ι ∷ ν ∷ []) "Col.2.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.5"
∷ word (τ ∷ ὸ ∷ []) "Col.2.5"
∷ word (σ ∷ τ ∷ ε ∷ ρ ∷ έ ∷ ω ∷ μ ∷ α ∷ []) "Col.2.5"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.2.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.2.5"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Col.2.5"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Col.2.5"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.2.5"
∷ word (Ὡ ∷ ς ∷ []) "Col.2.6"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Col.2.6"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ ∙λ ∷ ά ∷ β ∷ ε ∷ τ ∷ ε ∷ []) "Col.2.6"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Col.2.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Col.2.6"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "Col.2.6"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Col.2.6"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Col.2.6"
∷ word (ἐ ∷ ν ∷ []) "Col.2.6"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Col.2.6"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Col.2.6"
∷ word (ἐ ∷ ρ ∷ ρ ∷ ι ∷ ζ ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Col.2.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.7"
∷ word (ἐ ∷ π ∷ ο ∷ ι ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.2.7"
∷ word (ἐ ∷ ν ∷ []) "Col.2.7"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Col.2.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.7"
∷ word (β ∷ ε ∷ β ∷ α ∷ ι ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.2.7"
∷ word (τ ∷ ῇ ∷ []) "Col.2.7"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ι ∷ []) "Col.2.7"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Col.2.7"
∷ word (ἐ ∷ δ ∷ ι ∷ δ ∷ ά ∷ χ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Col.2.7"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ε ∷ ύ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.2.7"
∷ word (ἐ ∷ ν ∷ []) "Col.2.7"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Col.2.7"
∷ word (Β ∷ ∙λ ∷ έ ∷ π ∷ ε ∷ τ ∷ ε ∷ []) "Col.2.8"
∷ word (μ ∷ ή ∷ []) "Col.2.8"
∷ word (τ ∷ ι ∷ ς ∷ []) "Col.2.8"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.2.8"
∷ word (ἔ ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "Col.2.8"
∷ word (ὁ ∷ []) "Col.2.8"
∷ word (σ ∷ υ ∷ ∙λ ∷ α ∷ γ ∷ ω ∷ γ ∷ ῶ ∷ ν ∷ []) "Col.2.8"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Col.2.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.2.8"
∷ word (φ ∷ ι ∷ ∙λ ∷ ο ∷ σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ς ∷ []) "Col.2.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.8"
∷ word (κ ∷ ε ∷ ν ∷ ῆ ∷ ς ∷ []) "Col.2.8"
∷ word (ἀ ∷ π ∷ ά ∷ τ ∷ η ∷ ς ∷ []) "Col.2.8"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Col.2.8"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.2.8"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ δ ∷ ο ∷ σ ∷ ι ∷ ν ∷ []) "Col.2.8"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.2.8"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "Col.2.8"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Col.2.8"
∷ word (τ ∷ ὰ ∷ []) "Col.2.8"
∷ word (σ ∷ τ ∷ ο ∷ ι ∷ χ ∷ ε ∷ ῖ ∷ α ∷ []) "Col.2.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.2.8"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "Col.2.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.8"
∷ word (ο ∷ ὐ ∷ []) "Col.2.8"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Col.2.8"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ν ∷ []) "Col.2.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Col.2.9"
∷ word (ἐ ∷ ν ∷ []) "Col.2.9"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Col.2.9"
∷ word (κ ∷ α ∷ τ ∷ ο ∷ ι ∷ κ ∷ ε ∷ ῖ ∷ []) "Col.2.9"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "Col.2.9"
∷ word (τ ∷ ὸ ∷ []) "Col.2.9"
∷ word (π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ μ ∷ α ∷ []) "Col.2.9"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.2.9"
∷ word (θ ∷ ε ∷ ό ∷ τ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "Col.2.9"
∷ word (σ ∷ ω ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ῶ ∷ ς ∷ []) "Col.2.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.10"
∷ word (ἐ ∷ σ ∷ τ ∷ ὲ ∷ []) "Col.2.10"
∷ word (ἐ ∷ ν ∷ []) "Col.2.10"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Col.2.10"
∷ word (π ∷ ε ∷ π ∷ ∙λ ∷ η ∷ ρ ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Col.2.10"
∷ word (ὅ ∷ ς ∷ []) "Col.2.10"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.2.10"
∷ word (ἡ ∷ []) "Col.2.10"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ὴ ∷ []) "Col.2.10"
∷ word (π ∷ ά ∷ σ ∷ η ∷ ς ∷ []) "Col.2.10"
∷ word (ἀ ∷ ρ ∷ χ ∷ ῆ ∷ ς ∷ []) "Col.2.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.10"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Col.2.10"
∷ word (ἐ ∷ ν ∷ []) "Col.2.11"
∷ word (ᾧ ∷ []) "Col.2.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.11"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ ε ∷ τ ∷ μ ∷ ή ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Col.2.11"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῇ ∷ []) "Col.2.11"
∷ word (ἀ ∷ χ ∷ ε ∷ ι ∷ ρ ∷ ο ∷ π ∷ ο ∷ ι ∷ ή ∷ τ ∷ ῳ ∷ []) "Col.2.11"
∷ word (ἐ ∷ ν ∷ []) "Col.2.11"
∷ word (τ ∷ ῇ ∷ []) "Col.2.11"
∷ word (ἀ ∷ π ∷ ε ∷ κ ∷ δ ∷ ύ ∷ σ ∷ ε ∷ ι ∷ []) "Col.2.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.2.11"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Col.2.11"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.2.11"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ό ∷ ς ∷ []) "Col.2.11"
∷ word (ἐ ∷ ν ∷ []) "Col.2.11"
∷ word (τ ∷ ῇ ∷ []) "Col.2.11"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῇ ∷ []) "Col.2.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.2.11"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.2.11"
∷ word (σ ∷ υ ∷ ν ∷ τ ∷ α ∷ φ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.2.12"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Col.2.12"
∷ word (ἐ ∷ ν ∷ []) "Col.2.12"
∷ word (τ ∷ ῷ ∷ []) "Col.2.12"
∷ word (β ∷ α ∷ π ∷ τ ∷ ι ∷ σ ∷ μ ∷ ῷ ∷ []) "Col.2.12"
∷ word (ἐ ∷ ν ∷ []) "Col.2.12"
∷ word (ᾧ ∷ []) "Col.2.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.12"
∷ word (σ ∷ υ ∷ ν ∷ η ∷ γ ∷ έ ∷ ρ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Col.2.12"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Col.2.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.2.12"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Col.2.12"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.2.12"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Col.2.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.2.12"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.2.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.2.12"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ α ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Col.2.12"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Col.2.12"
∷ word (ἐ ∷ κ ∷ []) "Col.2.12"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Col.2.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.13"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.2.13"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ο ∷ ὺ ∷ ς ∷ []) "Col.2.13"
∷ word (ὄ ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Col.2.13"
∷ word (ἐ ∷ ν ∷ []) "Col.2.13"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.2.13"
∷ word (π ∷ α ∷ ρ ∷ α ∷ π ∷ τ ∷ ώ ∷ μ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "Col.2.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.13"
∷ word (τ ∷ ῇ ∷ []) "Col.2.13"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Col.2.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.2.13"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Col.2.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.2.13"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ζ ∷ ω ∷ ο ∷ π ∷ ο ∷ ί ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Col.2.13"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.2.13"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Col.2.13"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Col.2.13"
∷ word (χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Col.2.13"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.2.13"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.2.13"
∷ word (τ ∷ ὰ ∷ []) "Col.2.13"
∷ word (π ∷ α ∷ ρ ∷ α ∷ π ∷ τ ∷ ώ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Col.2.13"
∷ word (ἐ ∷ ξ ∷ α ∷ ∙λ ∷ ε ∷ ί ∷ ψ ∷ α ∷ ς ∷ []) "Col.2.14"
∷ word (τ ∷ ὸ ∷ []) "Col.2.14"
∷ word (κ ∷ α ∷ θ ∷ []) "Col.2.14"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.2.14"
∷ word (χ ∷ ε ∷ ι ∷ ρ ∷ ό ∷ γ ∷ ρ ∷ α ∷ φ ∷ ο ∷ ν ∷ []) "Col.2.14"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.2.14"
∷ word (δ ∷ ό ∷ γ ∷ μ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "Col.2.14"
∷ word (ὃ ∷ []) "Col.2.14"
∷ word (ἦ ∷ ν ∷ []) "Col.2.14"
∷ word (ὑ ∷ π ∷ ε ∷ ν ∷ α ∷ ν ∷ τ ∷ ί ∷ ο ∷ ν ∷ []) "Col.2.14"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.2.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.14"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Col.2.14"
∷ word (ἦ ∷ ρ ∷ κ ∷ ε ∷ ν ∷ []) "Col.2.14"
∷ word (ἐ ∷ κ ∷ []) "Col.2.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.2.14"
∷ word (μ ∷ έ ∷ σ ∷ ο ∷ υ ∷ []) "Col.2.14"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ η ∷ ∙λ ∷ ώ ∷ σ ∷ α ∷ ς ∷ []) "Col.2.14"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Col.2.14"
∷ word (τ ∷ ῷ ∷ []) "Col.2.14"
∷ word (σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ῷ ∷ []) "Col.2.14"
∷ word (ἀ ∷ π ∷ ε ∷ κ ∷ δ ∷ υ ∷ σ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Col.2.15"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Col.2.15"
∷ word (ἀ ∷ ρ ∷ χ ∷ ὰ ∷ ς ∷ []) "Col.2.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.15"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Col.2.15"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ σ ∷ ί ∷ α ∷ ς ∷ []) "Col.2.15"
∷ word (ἐ ∷ δ ∷ ε ∷ ι ∷ γ ∷ μ ∷ ά ∷ τ ∷ ι ∷ σ ∷ ε ∷ ν ∷ []) "Col.2.15"
∷ word (ἐ ∷ ν ∷ []) "Col.2.15"
∷ word (π ∷ α ∷ ρ ∷ ρ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "Col.2.15"
∷ word (θ ∷ ρ ∷ ι ∷ α ∷ μ ∷ β ∷ ε ∷ ύ ∷ σ ∷ α ∷ ς ∷ []) "Col.2.15"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Col.2.15"
∷ word (ἐ ∷ ν ∷ []) "Col.2.15"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Col.2.15"
∷ word (Μ ∷ ὴ ∷ []) "Col.2.16"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Col.2.16"
∷ word (τ ∷ ι ∷ ς ∷ []) "Col.2.16"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.2.16"
∷ word (κ ∷ ρ ∷ ι ∷ ν ∷ έ ∷ τ ∷ ω ∷ []) "Col.2.16"
∷ word (ἐ ∷ ν ∷ []) "Col.2.16"
∷ word (β ∷ ρ ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "Col.2.16"
∷ word (ἢ ∷ []) "Col.2.16"
∷ word (ἐ ∷ ν ∷ []) "Col.2.16"
∷ word (π ∷ ό ∷ σ ∷ ε ∷ ι ∷ []) "Col.2.16"
∷ word (ἢ ∷ []) "Col.2.16"
∷ word (ἐ ∷ ν ∷ []) "Col.2.16"
∷ word (μ ∷ έ ∷ ρ ∷ ε ∷ ι ∷ []) "Col.2.16"
∷ word (ἑ ∷ ο ∷ ρ ∷ τ ∷ ῆ ∷ ς ∷ []) "Col.2.16"
∷ word (ἢ ∷ []) "Col.2.16"
∷ word (ν ∷ ο ∷ υ ∷ μ ∷ η ∷ ν ∷ ί ∷ α ∷ ς ∷ []) "Col.2.16"
∷ word (ἢ ∷ []) "Col.2.16"
∷ word (σ ∷ α ∷ β ∷ β ∷ ά ∷ τ ∷ ω ∷ ν ∷ []) "Col.2.16"
∷ word (ἅ ∷ []) "Col.2.17"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.2.17"
∷ word (σ ∷ κ ∷ ι ∷ ὰ ∷ []) "Col.2.17"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.2.17"
∷ word (μ ∷ ε ∷ ∙λ ∷ ∙λ ∷ ό ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Col.2.17"
∷ word (τ ∷ ὸ ∷ []) "Col.2.17"
∷ word (δ ∷ ὲ ∷ []) "Col.2.17"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "Col.2.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.2.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.2.17"
∷ word (μ ∷ η ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "Col.2.18"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.2.18"
∷ word (κ ∷ α ∷ τ ∷ α ∷ β ∷ ρ ∷ α ∷ β ∷ ε ∷ υ ∷ έ ∷ τ ∷ ω ∷ []) "Col.2.18"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ ν ∷ []) "Col.2.18"
∷ word (ἐ ∷ ν ∷ []) "Col.2.18"
∷ word (τ ∷ α ∷ π ∷ ε ∷ ι ∷ ν ∷ ο ∷ φ ∷ ρ ∷ ο ∷ σ ∷ ύ ∷ ν ∷ ῃ ∷ []) "Col.2.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.18"
∷ word (θ ∷ ρ ∷ η ∷ σ ∷ κ ∷ ε ∷ ί ∷ ᾳ ∷ []) "Col.2.18"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.2.18"
∷ word (ἀ ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ω ∷ ν ∷ []) "Col.2.18"
∷ word (ἃ ∷ []) "Col.2.18"
∷ word (ἑ ∷ ό ∷ ρ ∷ α ∷ κ ∷ ε ∷ ν ∷ []) "Col.2.18"
∷ word (ἐ ∷ μ ∷ β ∷ α ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "Col.2.18"
∷ word (ε ∷ ἰ ∷ κ ∷ ῇ ∷ []) "Col.2.18"
∷ word (φ ∷ υ ∷ σ ∷ ι ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Col.2.18"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Col.2.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.2.18"
∷ word (ν ∷ ο ∷ ὸ ∷ ς ∷ []) "Col.2.18"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.2.18"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Col.2.18"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.2.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.19"
∷ word (ο ∷ ὐ ∷ []) "Col.2.19"
∷ word (κ ∷ ρ ∷ α ∷ τ ∷ ῶ ∷ ν ∷ []) "Col.2.19"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.2.19"
∷ word (κ ∷ ε ∷ φ ∷ α ∷ ∙λ ∷ ή ∷ ν ∷ []) "Col.2.19"
∷ word (ἐ ∷ ξ ∷ []) "Col.2.19"
∷ word (ο ∷ ὗ ∷ []) "Col.2.19"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "Col.2.19"
∷ word (τ ∷ ὸ ∷ []) "Col.2.19"
∷ word (σ ∷ ῶ ∷ μ ∷ α ∷ []) "Col.2.19"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Col.2.19"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.2.19"
∷ word (ἁ ∷ φ ∷ ῶ ∷ ν ∷ []) "Col.2.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.19"
∷ word (σ ∷ υ ∷ ν ∷ δ ∷ έ ∷ σ ∷ μ ∷ ω ∷ ν ∷ []) "Col.2.19"
∷ word (ἐ ∷ π ∷ ι ∷ χ ∷ ο ∷ ρ ∷ η ∷ γ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "Col.2.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.19"
∷ word (σ ∷ υ ∷ μ ∷ β ∷ ι ∷ β ∷ α ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "Col.2.19"
∷ word (α ∷ ὔ ∷ ξ ∷ ε ∷ ι ∷ []) "Col.2.19"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.2.19"
∷ word (α ∷ ὔ ∷ ξ ∷ η ∷ σ ∷ ι ∷ ν ∷ []) "Col.2.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.2.19"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.2.19"
∷ word (Ε ∷ ἰ ∷ []) "Col.2.20"
∷ word (ἀ ∷ π ∷ ε ∷ θ ∷ ά ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "Col.2.20"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Col.2.20"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Col.2.20"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Col.2.20"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.2.20"
∷ word (σ ∷ τ ∷ ο ∷ ι ∷ χ ∷ ε ∷ ί ∷ ω ∷ ν ∷ []) "Col.2.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.2.20"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "Col.2.20"
∷ word (τ ∷ ί ∷ []) "Col.2.20"
∷ word (ὡ ∷ ς ∷ []) "Col.2.20"
∷ word (ζ ∷ ῶ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.2.20"
∷ word (ἐ ∷ ν ∷ []) "Col.2.20"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "Col.2.20"
∷ word (δ ∷ ο ∷ γ ∷ μ ∷ α ∷ τ ∷ ί ∷ ζ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Col.2.20"
∷ word (Μ ∷ ὴ ∷ []) "Col.2.21"
∷ word (ἅ ∷ ψ ∷ ῃ ∷ []) "Col.2.21"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "Col.2.21"
∷ word (γ ∷ ε ∷ ύ ∷ σ ∷ ῃ ∷ []) "Col.2.21"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "Col.2.21"
∷ word (θ ∷ ί ∷ γ ∷ ῃ ∷ ς ∷ []) "Col.2.21"
∷ word (ἅ ∷ []) "Col.2.22"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.2.22"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.2.22"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.2.22"
∷ word (φ ∷ θ ∷ ο ∷ ρ ∷ ὰ ∷ ν ∷ []) "Col.2.22"
∷ word (τ ∷ ῇ ∷ []) "Col.2.22"
∷ word (ἀ ∷ π ∷ ο ∷ χ ∷ ρ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "Col.2.22"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Col.2.22"
∷ word (τ ∷ ὰ ∷ []) "Col.2.22"
∷ word (ἐ ∷ ν ∷ τ ∷ ά ∷ ∙λ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Col.2.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.22"
∷ word (δ ∷ ι ∷ δ ∷ α ∷ σ ∷ κ ∷ α ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "Col.2.22"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.2.22"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "Col.2.22"
∷ word (ἅ ∷ τ ∷ ι ∷ ν ∷ ά ∷ []) "Col.2.23"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.2.23"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ν ∷ []) "Col.2.23"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Col.2.23"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "Col.2.23"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ α ∷ ς ∷ []) "Col.2.23"
∷ word (ἐ ∷ ν ∷ []) "Col.2.23"
∷ word (ἐ ∷ θ ∷ ε ∷ ∙λ ∷ ο ∷ θ ∷ ρ ∷ η ∷ σ ∷ κ ∷ ί ∷ ᾳ ∷ []) "Col.2.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.23"
∷ word (τ ∷ α ∷ π ∷ ε ∷ ι ∷ ν ∷ ο ∷ φ ∷ ρ ∷ ο ∷ σ ∷ ύ ∷ ν ∷ ῃ ∷ []) "Col.2.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.2.23"
∷ word (ἀ ∷ φ ∷ ε ∷ ι ∷ δ ∷ ί ∷ ᾳ ∷ []) "Col.2.23"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Col.2.23"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Col.2.23"
∷ word (ἐ ∷ ν ∷ []) "Col.2.23"
∷ word (τ ∷ ι ∷ μ ∷ ῇ ∷ []) "Col.2.23"
∷ word (τ ∷ ι ∷ ν ∷ ι ∷ []) "Col.2.23"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Col.2.23"
∷ word (π ∷ ∙λ ∷ η ∷ σ ∷ μ ∷ ο ∷ ν ∷ ὴ ∷ ν ∷ []) "Col.2.23"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.2.23"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ό ∷ ς ∷ []) "Col.2.23"
∷ word (Ε ∷ ἰ ∷ []) "Col.3.1"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Col.3.1"
∷ word (σ ∷ υ ∷ ν ∷ η ∷ γ ∷ έ ∷ ρ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Col.3.1"
∷ word (τ ∷ ῷ ∷ []) "Col.3.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Col.3.1"
∷ word (τ ∷ ὰ ∷ []) "Col.3.1"
∷ word (ἄ ∷ ν ∷ ω ∷ []) "Col.3.1"
∷ word (ζ ∷ η ∷ τ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Col.3.1"
∷ word (ο ∷ ὗ ∷ []) "Col.3.1"
∷ word (ὁ ∷ []) "Col.3.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "Col.3.1"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.3.1"
∷ word (ἐ ∷ ν ∷ []) "Col.3.1"
∷ word (δ ∷ ε ∷ ξ ∷ ι ∷ ᾷ ∷ []) "Col.3.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.3.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.3.1"
∷ word (κ ∷ α ∷ θ ∷ ή ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Col.3.1"
∷ word (τ ∷ ὰ ∷ []) "Col.3.2"
∷ word (ἄ ∷ ν ∷ ω ∷ []) "Col.3.2"
∷ word (φ ∷ ρ ∷ ο ∷ ν ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Col.3.2"
∷ word (μ ∷ ὴ ∷ []) "Col.3.2"
∷ word (τ ∷ ὰ ∷ []) "Col.3.2"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Col.3.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.3.2"
∷ word (γ ∷ ῆ ∷ ς ∷ []) "Col.3.2"
∷ word (ἀ ∷ π ∷ ε ∷ θ ∷ ά ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "Col.3.3"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Col.3.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.3"
∷ word (ἡ ∷ []) "Col.3.3"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "Col.3.3"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.3.3"
∷ word (κ ∷ έ ∷ κ ∷ ρ ∷ υ ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Col.3.3"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Col.3.3"
∷ word (τ ∷ ῷ ∷ []) "Col.3.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Col.3.3"
∷ word (ἐ ∷ ν ∷ []) "Col.3.3"
∷ word (τ ∷ ῷ ∷ []) "Col.3.3"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Col.3.3"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "Col.3.4"
∷ word (ὁ ∷ []) "Col.3.4"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Col.3.4"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ω ∷ θ ∷ ῇ ∷ []) "Col.3.4"
∷ word (ἡ ∷ []) "Col.3.4"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "Col.3.4"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.3.4"
∷ word (τ ∷ ό ∷ τ ∷ ε ∷ []) "Col.3.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.4"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Col.3.4"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Col.3.4"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Col.3.4"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ω ∷ θ ∷ ή ∷ σ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Col.3.4"
∷ word (ἐ ∷ ν ∷ []) "Col.3.4"
∷ word (δ ∷ ό ∷ ξ ∷ ῃ ∷ []) "Col.3.4"
∷ word (Ν ∷ ε ∷ κ ∷ ρ ∷ ώ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Col.3.5"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Col.3.5"
∷ word (τ ∷ ὰ ∷ []) "Col.3.5"
∷ word (μ ∷ έ ∷ ∙λ ∷ η ∷ []) "Col.3.5"
∷ word (τ ∷ ὰ ∷ []) "Col.3.5"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Col.3.5"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.3.5"
∷ word (γ ∷ ῆ ∷ ς ∷ []) "Col.3.5"
∷ word (π ∷ ο ∷ ρ ∷ ν ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "Col.3.5"
∷ word (ἀ ∷ κ ∷ α ∷ θ ∷ α ∷ ρ ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Col.3.5"
∷ word (π ∷ ά ∷ θ ∷ ο ∷ ς ∷ []) "Col.3.5"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Col.3.5"
∷ word (κ ∷ α ∷ κ ∷ ή ∷ ν ∷ []) "Col.3.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.3.5"
∷ word (π ∷ ∙λ ∷ ε ∷ ο ∷ ν ∷ ε ∷ ξ ∷ ί ∷ α ∷ ν ∷ []) "Col.3.5"
∷ word (ἥ ∷ τ ∷ ι ∷ ς ∷ []) "Col.3.5"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "Col.3.5"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ο ∷ ∙λ ∷ α ∷ τ ∷ ρ ∷ ί ∷ α ∷ []) "Col.3.5"
∷ word (δ ∷ ι ∷ []) "Col.3.6"
∷ word (ἃ ∷ []) "Col.3.6"
∷ word (ἔ ∷ ρ ∷ χ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Col.3.6"
∷ word (ἡ ∷ []) "Col.3.6"
∷ word (ὀ ∷ ρ ∷ γ ∷ ὴ ∷ []) "Col.3.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.3.6"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.3.6"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Col.3.6"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Col.3.6"
∷ word (υ ∷ ἱ ∷ ο ∷ ὺ ∷ ς ∷ []) "Col.3.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.3.6"
∷ word (ἀ ∷ π ∷ ε ∷ ι ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Col.3.6"
∷ word (ἐ ∷ ν ∷ []) "Col.3.7"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "Col.3.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.7"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Col.3.7"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ ε ∷ π ∷ α ∷ τ ∷ ή ∷ σ ∷ α ∷ τ ∷ έ ∷ []) "Col.3.7"
∷ word (π ∷ ο ∷ τ ∷ ε ∷ []) "Col.3.7"
∷ word (ὅ ∷ τ ∷ ε ∷ []) "Col.3.7"
∷ word (ἐ ∷ ζ ∷ ῆ ∷ τ ∷ ε ∷ []) "Col.3.7"
∷ word (ἐ ∷ ν ∷ []) "Col.3.7"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "Col.3.7"
∷ word (ν ∷ υ ∷ ν ∷ ὶ ∷ []) "Col.3.8"
∷ word (δ ∷ ὲ ∷ []) "Col.3.8"
∷ word (ἀ ∷ π ∷ ό ∷ θ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Col.3.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.8"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Col.3.8"
∷ word (τ ∷ ὰ ∷ []) "Col.3.8"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.3.8"
∷ word (ὀ ∷ ρ ∷ γ ∷ ή ∷ ν ∷ []) "Col.3.8"
∷ word (θ ∷ υ ∷ μ ∷ ό ∷ ν ∷ []) "Col.3.8"
∷ word (κ ∷ α ∷ κ ∷ ί ∷ α ∷ ν ∷ []) "Col.3.8"
∷ word (β ∷ ∙λ ∷ α ∷ σ ∷ φ ∷ η ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Col.3.8"
∷ word (α ∷ ἰ ∷ σ ∷ χ ∷ ρ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ ν ∷ []) "Col.3.8"
∷ word (ἐ ∷ κ ∷ []) "Col.3.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.3.8"
∷ word (σ ∷ τ ∷ ό ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Col.3.8"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.3.8"
∷ word (μ ∷ ὴ ∷ []) "Col.3.9"
∷ word (ψ ∷ ε ∷ ύ ∷ δ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Col.3.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.3.9"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Col.3.9"
∷ word (ἀ ∷ π ∷ ε ∷ κ ∷ δ ∷ υ ∷ σ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.3.9"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Col.3.9"
∷ word (π ∷ α ∷ ∙λ ∷ α ∷ ι ∷ ὸ ∷ ν ∷ []) "Col.3.9"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Col.3.9"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Col.3.9"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Col.3.9"
∷ word (π ∷ ρ ∷ ά ∷ ξ ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Col.3.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.3.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.10"
∷ word (ἐ ∷ ν ∷ δ ∷ υ ∷ σ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.3.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Col.3.10"
∷ word (ν ∷ έ ∷ ο ∷ ν ∷ []) "Col.3.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Col.3.10"
∷ word (ἀ ∷ ν ∷ α ∷ κ ∷ α ∷ ι ∷ ν ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "Col.3.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.3.10"
∷ word (ἐ ∷ π ∷ ί ∷ γ ∷ ν ∷ ω ∷ σ ∷ ι ∷ ν ∷ []) "Col.3.10"
∷ word (κ ∷ α ∷ τ ∷ []) "Col.3.10"
∷ word (ε ∷ ἰ ∷ κ ∷ ό ∷ ν ∷ α ∷ []) "Col.3.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.3.10"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ α ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Col.3.10"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "Col.3.10"
∷ word (ὅ ∷ π ∷ ο ∷ υ ∷ []) "Col.3.11"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Col.3.11"
∷ word (ἔ ∷ ν ∷ ι ∷ []) "Col.3.11"
∷ word (Ἕ ∷ ∙λ ∷ ∙λ ∷ η ∷ ν ∷ []) "Col.3.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.11"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ῖ ∷ ο ∷ ς ∷ []) "Col.3.11"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ὴ ∷ []) "Col.3.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.11"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ α ∷ []) "Col.3.11"
∷ word (β ∷ ά ∷ ρ ∷ β ∷ α ∷ ρ ∷ ο ∷ ς ∷ []) "Col.3.11"
∷ word (Σ ∷ κ ∷ ύ ∷ θ ∷ η ∷ ς ∷ []) "Col.3.11"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Col.3.11"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ θ ∷ ε ∷ ρ ∷ ο ∷ ς ∷ []) "Col.3.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Col.3.11"
∷ word (τ ∷ ὰ ∷ []) "Col.3.11"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.3.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.11"
∷ word (ἐ ∷ ν ∷ []) "Col.3.11"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "Col.3.11"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "Col.3.11"
∷ word (Ἐ ∷ ν ∷ δ ∷ ύ ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Col.3.12"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Col.3.12"
∷ word (ὡ ∷ ς ∷ []) "Col.3.12"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ε ∷ κ ∷ τ ∷ ο ∷ ὶ ∷ []) "Col.3.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.3.12"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.3.12"
∷ word (ἅ ∷ γ ∷ ι ∷ ο ∷ ι ∷ []) "Col.3.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.12"
∷ word (ἠ ∷ γ ∷ α ∷ π ∷ η ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Col.3.12"
∷ word (σ ∷ π ∷ ∙λ ∷ ά ∷ γ ∷ χ ∷ ν ∷ α ∷ []) "Col.3.12"
∷ word (ο ∷ ἰ ∷ κ ∷ τ ∷ ι ∷ ρ ∷ μ ∷ ο ∷ ῦ ∷ []) "Col.3.12"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ό ∷ τ ∷ η ∷ τ ∷ α ∷ []) "Col.3.12"
∷ word (τ ∷ α ∷ π ∷ ε ∷ ι ∷ ν ∷ ο ∷ φ ∷ ρ ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Col.3.12"
∷ word (π ∷ ρ ∷ α ∷ ΰ ∷ τ ∷ η ∷ τ ∷ α ∷ []) "Col.3.12"
∷ word (μ ∷ α ∷ κ ∷ ρ ∷ ο ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Col.3.12"
∷ word (ἀ ∷ ν ∷ ε ∷ χ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.3.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ω ∷ ν ∷ []) "Col.3.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.13"
∷ word (χ ∷ α ∷ ρ ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.3.13"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.3.13"
∷ word (ἐ ∷ ά ∷ ν ∷ []) "Col.3.13"
∷ word (τ ∷ ι ∷ ς ∷ []) "Col.3.13"
∷ word (π ∷ ρ ∷ ό ∷ ς ∷ []) "Col.3.13"
∷ word (τ ∷ ι ∷ ν ∷ α ∷ []) "Col.3.13"
∷ word (ἔ ∷ χ ∷ ῃ ∷ []) "Col.3.13"
∷ word (μ ∷ ο ∷ μ ∷ φ ∷ ή ∷ ν ∷ []) "Col.3.13"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Col.3.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.13"
∷ word (ὁ ∷ []) "Col.3.13"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "Col.3.13"
∷ word (ἐ ∷ χ ∷ α ∷ ρ ∷ ί ∷ σ ∷ α ∷ τ ∷ ο ∷ []) "Col.3.13"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.3.13"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Col.3.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.13"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Col.3.13"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Col.3.14"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "Col.3.14"
∷ word (δ ∷ ὲ ∷ []) "Col.3.14"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "Col.3.14"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.3.14"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "Col.3.14"
∷ word (ὅ ∷ []) "Col.3.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.3.14"
∷ word (σ ∷ ύ ∷ ν ∷ δ ∷ ε ∷ σ ∷ μ ∷ ο ∷ ς ∷ []) "Col.3.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.3.14"
∷ word (τ ∷ ε ∷ ∙λ ∷ ε ∷ ι ∷ ό ∷ τ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "Col.3.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.15"
∷ word (ἡ ∷ []) "Col.3.15"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "Col.3.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.3.15"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.3.15"
∷ word (β ∷ ρ ∷ α ∷ β ∷ ε ∷ υ ∷ έ ∷ τ ∷ ω ∷ []) "Col.3.15"
∷ word (ἐ ∷ ν ∷ []) "Col.3.15"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Col.3.15"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Col.3.15"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.3.15"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.3.15"
∷ word (ἣ ∷ ν ∷ []) "Col.3.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.15"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ή ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Col.3.15"
∷ word (ἐ ∷ ν ∷ []) "Col.3.15"
∷ word (ἑ ∷ ν ∷ ὶ ∷ []) "Col.3.15"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Col.3.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.15"
∷ word (ε ∷ ὐ ∷ χ ∷ ά ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ι ∷ []) "Col.3.15"
∷ word (γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Col.3.15"
∷ word (ὁ ∷ []) "Col.3.16"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "Col.3.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.3.16"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.3.16"
∷ word (ἐ ∷ ν ∷ ο ∷ ι ∷ κ ∷ ε ∷ ί ∷ τ ∷ ω ∷ []) "Col.3.16"
∷ word (ἐ ∷ ν ∷ []) "Col.3.16"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.3.16"
∷ word (π ∷ ∙λ ∷ ο ∷ υ ∷ σ ∷ ί ∷ ω ∷ ς ∷ []) "Col.3.16"
∷ word (ἐ ∷ ν ∷ []) "Col.3.16"
∷ word (π ∷ ά ∷ σ ∷ ῃ ∷ []) "Col.3.16"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ ᾳ ∷ []) "Col.3.16"
∷ word (δ ∷ ι ∷ δ ∷ ά ∷ σ ∷ κ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.3.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.16"
∷ word (ν ∷ ο ∷ υ ∷ θ ∷ ε ∷ τ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.3.16"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Col.3.16"
∷ word (ψ ∷ α ∷ ∙λ ∷ μ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.3.16"
∷ word (ὕ ∷ μ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "Col.3.16"
∷ word (ᾠ ∷ δ ∷ α ∷ ῖ ∷ ς ∷ []) "Col.3.16"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ α ∷ ῖ ∷ ς ∷ []) "Col.3.16"
∷ word (ἐ ∷ ν ∷ []) "Col.3.16"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ι ∷ []) "Col.3.16"
∷ word (ᾄ ∷ δ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.3.16"
∷ word (ἐ ∷ ν ∷ []) "Col.3.16"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Col.3.16"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Col.3.16"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.3.16"
∷ word (τ ∷ ῷ ∷ []) "Col.3.16"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Col.3.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.17"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "Col.3.17"
∷ word (ὅ ∷ []) "Col.3.17"
∷ word (τ ∷ ι ∷ []) "Col.3.17"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Col.3.17"
∷ word (π ∷ ο ∷ ι ∷ ῆ ∷ τ ∷ ε ∷ []) "Col.3.17"
∷ word (ἐ ∷ ν ∷ []) "Col.3.17"
∷ word (∙λ ∷ ό ∷ γ ∷ ῳ ∷ []) "Col.3.17"
∷ word (ἢ ∷ []) "Col.3.17"
∷ word (ἐ ∷ ν ∷ []) "Col.3.17"
∷ word (ἔ ∷ ρ ∷ γ ∷ ῳ ∷ []) "Col.3.17"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.3.17"
∷ word (ἐ ∷ ν ∷ []) "Col.3.17"
∷ word (ὀ ∷ ν ∷ ό ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Col.3.17"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Col.3.17"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Col.3.17"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.3.17"
∷ word (τ ∷ ῷ ∷ []) "Col.3.17"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Col.3.17"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὶ ∷ []) "Col.3.17"
∷ word (δ ∷ ι ∷ []) "Col.3.17"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.3.17"
∷ word (Α ∷ ἱ ∷ []) "Col.3.18"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ ε ∷ ς ∷ []) "Col.3.18"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ ά ∷ σ ∷ σ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Col.3.18"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.3.18"
∷ word (ἀ ∷ ν ∷ δ ∷ ρ ∷ ά ∷ σ ∷ ι ∷ ν ∷ []) "Col.3.18"
∷ word (ὡ ∷ ς ∷ []) "Col.3.18"
∷ word (ἀ ∷ ν ∷ ῆ ∷ κ ∷ ε ∷ ν ∷ []) "Col.3.18"
∷ word (ἐ ∷ ν ∷ []) "Col.3.18"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Col.3.18"
∷ word (ο ∷ ἱ ∷ []) "Col.3.19"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ ε ∷ ς ∷ []) "Col.3.19"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ᾶ ∷ τ ∷ ε ∷ []) "Col.3.19"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Col.3.19"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ῖ ∷ κ ∷ α ∷ ς ∷ []) "Col.3.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.19"
∷ word (μ ∷ ὴ ∷ []) "Col.3.19"
∷ word (π ∷ ι ∷ κ ∷ ρ ∷ α ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Col.3.19"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Col.3.19"
∷ word (α ∷ ὐ ∷ τ ∷ ά ∷ ς ∷ []) "Col.3.19"
∷ word (Τ ∷ ὰ ∷ []) "Col.3.20"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Col.3.20"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ύ ∷ ε ∷ τ ∷ ε ∷ []) "Col.3.20"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.3.20"
∷ word (γ ∷ ο ∷ ν ∷ ε ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Col.3.20"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Col.3.20"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.3.20"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Col.3.20"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Col.3.20"
∷ word (ε ∷ ὐ ∷ ά ∷ ρ ∷ ε ∷ σ ∷ τ ∷ ό ∷ ν ∷ []) "Col.3.20"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.3.20"
∷ word (ἐ ∷ ν ∷ []) "Col.3.20"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Col.3.20"
∷ word (ο ∷ ἱ ∷ []) "Col.3.21"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ ε ∷ ς ∷ []) "Col.3.21"
∷ word (μ ∷ ὴ ∷ []) "Col.3.21"
∷ word (ἐ ∷ ρ ∷ ε ∷ θ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ ε ∷ []) "Col.3.21"
∷ word (τ ∷ ὰ ∷ []) "Col.3.21"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Col.3.21"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.3.21"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Col.3.21"
∷ word (μ ∷ ὴ ∷ []) "Col.3.21"
∷ word (ἀ ∷ θ ∷ υ ∷ μ ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "Col.3.21"
∷ word (ο ∷ ἱ ∷ []) "Col.3.22"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ι ∷ []) "Col.3.22"
∷ word (ὑ ∷ π ∷ α ∷ κ ∷ ο ∷ ύ ∷ ε ∷ τ ∷ ε ∷ []) "Col.3.22"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Col.3.22"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.3.22"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.3.22"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Col.3.22"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Col.3.22"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ ι ∷ ς ∷ []) "Col.3.22"
∷ word (μ ∷ ὴ ∷ []) "Col.3.22"
∷ word (ἐ ∷ ν ∷ []) "Col.3.22"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ο ∷ δ ∷ ο ∷ υ ∷ ∙λ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Col.3.22"
∷ word (ὡ ∷ ς ∷ []) "Col.3.22"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ά ∷ ρ ∷ ε ∷ σ ∷ κ ∷ ο ∷ ι ∷ []) "Col.3.22"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Col.3.22"
∷ word (ἐ ∷ ν ∷ []) "Col.3.22"
∷ word (ἁ ∷ π ∷ ∙λ ∷ ό ∷ τ ∷ η ∷ τ ∷ ι ∷ []) "Col.3.22"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "Col.3.22"
∷ word (φ ∷ ο ∷ β ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.3.22"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Col.3.22"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Col.3.22"
∷ word (ὃ ∷ []) "Col.3.23"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Col.3.23"
∷ word (π ∷ ο ∷ ι ∷ ῆ ∷ τ ∷ ε ∷ []) "Col.3.23"
∷ word (ἐ ∷ κ ∷ []) "Col.3.23"
∷ word (ψ ∷ υ ∷ χ ∷ ῆ ∷ ς ∷ []) "Col.3.23"
∷ word (ἐ ∷ ρ ∷ γ ∷ ά ∷ ζ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Col.3.23"
∷ word (ὡ ∷ ς ∷ []) "Col.3.23"
∷ word (τ ∷ ῷ ∷ []) "Col.3.23"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Col.3.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.23"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Col.3.23"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "Col.3.23"
∷ word (ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "Col.3.24"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Col.3.24"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Col.3.24"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Col.3.24"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ή ∷ μ ∷ ψ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Col.3.24"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.3.24"
∷ word (ἀ ∷ ν ∷ τ ∷ α ∷ π ∷ ό ∷ δ ∷ ο ∷ σ ∷ ι ∷ ν ∷ []) "Col.3.24"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Col.3.24"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ ς ∷ []) "Col.3.24"
∷ word (τ ∷ ῷ ∷ []) "Col.3.24"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Col.3.24"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Col.3.24"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ ε ∷ τ ∷ ε ∷ []) "Col.3.24"
∷ word (ὁ ∷ []) "Col.3.25"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Col.3.25"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ῶ ∷ ν ∷ []) "Col.3.25"
∷ word (κ ∷ ο ∷ μ ∷ ί ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Col.3.25"
∷ word (ὃ ∷ []) "Col.3.25"
∷ word (ἠ ∷ δ ∷ ί ∷ κ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Col.3.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.3.25"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Col.3.25"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.3.25"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ω ∷ π ∷ ο ∷ ∙λ ∷ η ∷ μ ∷ ψ ∷ ί ∷ α ∷ []) "Col.3.25"
∷ word (ο ∷ ἱ ∷ []) "Col.4.1"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ι ∷ []) "Col.4.1"
∷ word (τ ∷ ὸ ∷ []) "Col.4.1"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ ο ∷ ν ∷ []) "Col.4.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.1"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.4.1"
∷ word (ἰ ∷ σ ∷ ό ∷ τ ∷ η ∷ τ ∷ α ∷ []) "Col.4.1"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.4.1"
∷ word (δ ∷ ο ∷ ύ ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "Col.4.1"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ χ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Col.4.1"
∷ word (ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "Col.4.1"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Col.4.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.1"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Col.4.1"
∷ word (ἔ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "Col.4.1"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Col.4.1"
∷ word (ἐ ∷ ν ∷ []) "Col.4.1"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ῷ ∷ []) "Col.4.1"
∷ word (Τ ∷ ῇ ∷ []) "Col.4.2"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ ῇ ∷ []) "Col.4.2"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ κ ∷ α ∷ ρ ∷ τ ∷ ε ∷ ρ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Col.4.2"
∷ word (γ ∷ ρ ∷ η ∷ γ ∷ ο ∷ ρ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.4.2"
∷ word (ἐ ∷ ν ∷ []) "Col.4.2"
∷ word (α ∷ ὐ ∷ τ ∷ ῇ ∷ []) "Col.4.2"
∷ word (ἐ ∷ ν ∷ []) "Col.4.2"
∷ word (ε ∷ ὐ ∷ χ ∷ α ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ί ∷ ᾳ ∷ []) "Col.4.2"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.4.3"
∷ word (ἅ ∷ μ ∷ α ∷ []) "Col.4.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.3"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "Col.4.3"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.4.3"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Col.4.3"
∷ word (ὁ ∷ []) "Col.4.3"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Col.4.3"
∷ word (ἀ ∷ ν ∷ ο ∷ ί ∷ ξ ∷ ῃ ∷ []) "Col.4.3"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.4.3"
∷ word (θ ∷ ύ ∷ ρ ∷ α ∷ ν ∷ []) "Col.4.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.4.3"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ υ ∷ []) "Col.4.3"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Col.4.3"
∷ word (τ ∷ ὸ ∷ []) "Col.4.3"
∷ word (μ ∷ υ ∷ σ ∷ τ ∷ ή ∷ ρ ∷ ι ∷ ο ∷ ν ∷ []) "Col.4.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.4.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.4.3"
∷ word (δ ∷ ι ∷ []) "Col.4.3"
∷ word (ὃ ∷ []) "Col.4.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.3"
∷ word (δ ∷ έ ∷ δ ∷ ε ∷ μ ∷ α ∷ ι ∷ []) "Col.4.3"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Col.4.4"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ώ ∷ σ ∷ ω ∷ []) "Col.4.4"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Col.4.4"
∷ word (ὡ ∷ ς ∷ []) "Col.4.4"
∷ word (δ ∷ ε ∷ ῖ ∷ []) "Col.4.4"
∷ word (μ ∷ ε ∷ []) "Col.4.4"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Col.4.4"
∷ word (Ἐ ∷ ν ∷ []) "Col.4.5"
∷ word (σ ∷ ο ∷ φ ∷ ί ∷ ᾳ ∷ []) "Col.4.5"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Col.4.5"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Col.4.5"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Col.4.5"
∷ word (ἔ ∷ ξ ∷ ω ∷ []) "Col.4.5"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Col.4.5"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ὸ ∷ ν ∷ []) "Col.4.5"
∷ word (ἐ ∷ ξ ∷ α ∷ γ ∷ ο ∷ ρ ∷ α ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Col.4.5"
∷ word (ὁ ∷ []) "Col.4.6"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "Col.4.6"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.4.6"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ο ∷ τ ∷ ε ∷ []) "Col.4.6"
∷ word (ἐ ∷ ν ∷ []) "Col.4.6"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ι ∷ []) "Col.4.6"
∷ word (ἅ ∷ ∙λ ∷ α ∷ τ ∷ ι ∷ []) "Col.4.6"
∷ word (ἠ ∷ ρ ∷ τ ∷ υ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "Col.4.6"
∷ word (ε ∷ ἰ ∷ δ ∷ έ ∷ ν ∷ α ∷ ι ∷ []) "Col.4.6"
∷ word (π ∷ ῶ ∷ ς ∷ []) "Col.4.6"
∷ word (δ ∷ ε ∷ ῖ ∷ []) "Col.4.6"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.4.6"
∷ word (ἑ ∷ ν ∷ ὶ ∷ []) "Col.4.6"
∷ word (ἑ ∷ κ ∷ ά ∷ σ ∷ τ ∷ ῳ ∷ []) "Col.4.6"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ ρ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Col.4.6"
∷ word (Τ ∷ ὰ ∷ []) "Col.4.7"
∷ word (κ ∷ α ∷ τ ∷ []) "Col.4.7"
∷ word (ἐ ∷ μ ∷ ὲ ∷ []) "Col.4.7"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.4.7"
∷ word (γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "Col.4.7"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.4.7"
∷ word (Τ ∷ υ ∷ χ ∷ ι ∷ κ ∷ ὸ ∷ ς ∷ []) "Col.4.7"
∷ word (ὁ ∷ []) "Col.4.7"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ὸ ∷ ς ∷ []) "Col.4.7"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ς ∷ []) "Col.4.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.7"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Col.4.7"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ς ∷ []) "Col.4.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.7"
∷ word (σ ∷ ύ ∷ ν ∷ δ ∷ ο ∷ υ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Col.4.7"
∷ word (ἐ ∷ ν ∷ []) "Col.4.7"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Col.4.7"
∷ word (ὃ ∷ ν ∷ []) "Col.4.8"
∷ word (ἔ ∷ π ∷ ε ∷ μ ∷ ψ ∷ α ∷ []) "Col.4.8"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Col.4.8"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.4.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.4.8"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Col.4.8"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Col.4.8"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Col.4.8"
∷ word (γ ∷ ν ∷ ῶ ∷ τ ∷ ε ∷ []) "Col.4.8"
∷ word (τ ∷ ὰ ∷ []) "Col.4.8"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "Col.4.8"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.4.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.8"
∷ word (π ∷ α ∷ ρ ∷ α ∷ κ ∷ α ∷ ∙λ ∷ έ ∷ σ ∷ ῃ ∷ []) "Col.4.8"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Col.4.8"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "Col.4.8"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.4.8"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Col.4.9"
∷ word (Ὀ ∷ ν ∷ η ∷ σ ∷ ί ∷ μ ∷ ῳ ∷ []) "Col.4.9"
∷ word (τ ∷ ῷ ∷ []) "Col.4.9"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Col.4.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.9"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ῷ ∷ []) "Col.4.9"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ῷ ∷ []) "Col.4.9"
∷ word (ὅ ∷ ς ∷ []) "Col.4.9"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Col.4.9"
∷ word (ἐ ∷ ξ ∷ []) "Col.4.9"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.4.9"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Col.4.9"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.4.9"
∷ word (γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ σ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Col.4.9"
∷ word (τ ∷ ὰ ∷ []) "Col.4.9"
∷ word (ὧ ∷ δ ∷ ε ∷ []) "Col.4.9"
∷ word (Ἀ ∷ σ ∷ π ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Col.4.10"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.4.10"
∷ word (Ἀ ∷ ρ ∷ ί ∷ σ ∷ τ ∷ α ∷ ρ ∷ χ ∷ ο ∷ ς ∷ []) "Col.4.10"
∷ word (ὁ ∷ []) "Col.4.10"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ ι ∷ χ ∷ μ ∷ ά ∷ ∙λ ∷ ω ∷ τ ∷ ό ∷ ς ∷ []) "Col.4.10"
∷ word (μ ∷ ο ∷ υ ∷ []) "Col.4.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.10"
∷ word (Μ ∷ ᾶ ∷ ρ ∷ κ ∷ ο ∷ ς ∷ []) "Col.4.10"
∷ word (ὁ ∷ []) "Col.4.10"
∷ word (ἀ ∷ ν ∷ ε ∷ ψ ∷ ι ∷ ὸ ∷ ς ∷ []) "Col.4.10"
∷ word (Β ∷ α ∷ ρ ∷ ν ∷ α ∷ β ∷ ᾶ ∷ []) "Col.4.10"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "Col.4.10"
∷ word (ο ∷ ὗ ∷ []) "Col.4.10"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ β ∷ ε ∷ τ ∷ ε ∷ []) "Col.4.10"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ά ∷ ς ∷ []) "Col.4.10"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Col.4.10"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ῃ ∷ []) "Col.4.10"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Col.4.10"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.4.10"
∷ word (δ ∷ έ ∷ ξ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Col.4.10"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "Col.4.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.11"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "Col.4.11"
∷ word (ὁ ∷ []) "Col.4.11"
∷ word (∙λ ∷ ε ∷ γ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Col.4.11"
∷ word (Ἰ ∷ ο ∷ ῦ ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "Col.4.11"
∷ word (ο ∷ ἱ ∷ []) "Col.4.11"
∷ word (ὄ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Col.4.11"
∷ word (ἐ ∷ κ ∷ []) "Col.4.11"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῆ ∷ ς ∷ []) "Col.4.11"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ι ∷ []) "Col.4.11"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ι ∷ []) "Col.4.11"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ο ∷ ὶ ∷ []) "Col.4.11"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Col.4.11"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.4.11"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "Col.4.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.4.11"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.4.11"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "Col.4.11"
∷ word (ἐ ∷ γ ∷ ε ∷ ν ∷ ή ∷ θ ∷ η ∷ σ ∷ ά ∷ ν ∷ []) "Col.4.11"
∷ word (μ ∷ ο ∷ ι ∷ []) "Col.4.11"
∷ word (π ∷ α ∷ ρ ∷ η ∷ γ ∷ ο ∷ ρ ∷ ί ∷ α ∷ []) "Col.4.11"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Col.4.12"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.4.12"
∷ word (Ἐ ∷ π ∷ α ∷ φ ∷ ρ ∷ ᾶ ∷ ς ∷ []) "Col.4.12"
∷ word (ὁ ∷ []) "Col.4.12"
∷ word (ἐ ∷ ξ ∷ []) "Col.4.12"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.4.12"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Col.4.12"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Col.4.12"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ο ∷ τ ∷ ε ∷ []) "Col.4.12"
∷ word (ἀ ∷ γ ∷ ω ∷ ν ∷ ι ∷ ζ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Col.4.12"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Col.4.12"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.4.12"
∷ word (ἐ ∷ ν ∷ []) "Col.4.12"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Col.4.12"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ υ ∷ χ ∷ α ∷ ῖ ∷ ς ∷ []) "Col.4.12"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Col.4.12"
∷ word (σ ∷ τ ∷ α ∷ θ ∷ ῆ ∷ τ ∷ ε ∷ []) "Col.4.12"
∷ word (τ ∷ έ ∷ ∙λ ∷ ε ∷ ι ∷ ο ∷ ι ∷ []) "Col.4.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.12"
∷ word (π ∷ ε ∷ π ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ φ ∷ ο ∷ ρ ∷ η ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Col.4.12"
∷ word (ἐ ∷ ν ∷ []) "Col.4.12"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "Col.4.12"
∷ word (θ ∷ ε ∷ ∙λ ∷ ή ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Col.4.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Col.4.12"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Col.4.12"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ῶ ∷ []) "Col.4.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Col.4.13"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Col.4.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Col.4.13"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "Col.4.13"
∷ word (π ∷ ο ∷ ∙λ ∷ ὺ ∷ ν ∷ []) "Col.4.13"
∷ word (π ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Col.4.13"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Col.4.13"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.4.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.13"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.4.13"
∷ word (ἐ ∷ ν ∷ []) "Col.4.13"
∷ word (Λ ∷ α ∷ ο ∷ δ ∷ ι ∷ κ ∷ ε ∷ ί ∷ ᾳ ∷ []) "Col.4.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.13"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.4.13"
∷ word (ἐ ∷ ν ∷ []) "Col.4.13"
∷ word (Ἱ ∷ ε ∷ ρ ∷ α ∷ π ∷ ό ∷ ∙λ ∷ ε ∷ ι ∷ []) "Col.4.13"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Col.4.14"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.4.14"
∷ word (Λ ∷ ο ∷ υ ∷ κ ∷ ᾶ ∷ ς ∷ []) "Col.4.14"
∷ word (ὁ ∷ []) "Col.4.14"
∷ word (ἰ ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "Col.4.14"
∷ word (ὁ ∷ []) "Col.4.14"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ὸ ∷ ς ∷ []) "Col.4.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.14"
∷ word (Δ ∷ η ∷ μ ∷ ᾶ ∷ ς ∷ []) "Col.4.14"
∷ word (ἀ ∷ σ ∷ π ∷ ά ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Col.4.15"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Col.4.15"
∷ word (ἐ ∷ ν ∷ []) "Col.4.15"
∷ word (Λ ∷ α ∷ ο ∷ δ ∷ ι ∷ κ ∷ ε ∷ ί ∷ ᾳ ∷ []) "Col.4.15"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ὺ ∷ ς ∷ []) "Col.4.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.15"
∷ word (Ν ∷ ύ ∷ μ ∷ φ ∷ α ∷ ν ∷ []) "Col.4.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.15"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.4.15"
∷ word (κ ∷ α ∷ τ ∷ []) "Col.4.15"
∷ word (ο ∷ ἶ ∷ κ ∷ ο ∷ ν ∷ []) "Col.4.15"
∷ word (α ∷ ὐ ∷ τ ∷ ῆ ∷ ς ∷ []) "Col.4.15"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Col.4.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.16"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "Col.4.16"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ ν ∷ ω ∷ σ ∷ θ ∷ ῇ ∷ []) "Col.4.16"
∷ word (π ∷ α ∷ ρ ∷ []) "Col.4.16"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Col.4.16"
∷ word (ἡ ∷ []) "Col.4.16"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ή ∷ []) "Col.4.16"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Col.4.16"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Col.4.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.16"
∷ word (ἐ ∷ ν ∷ []) "Col.4.16"
∷ word (τ ∷ ῇ ∷ []) "Col.4.16"
∷ word (Λ ∷ α ∷ ο ∷ δ ∷ ι ∷ κ ∷ έ ∷ ω ∷ ν ∷ []) "Col.4.16"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ᾳ ∷ []) "Col.4.16"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ ν ∷ ω ∷ σ ∷ θ ∷ ῇ ∷ []) "Col.4.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.16"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.4.16"
∷ word (ἐ ∷ κ ∷ []) "Col.4.16"
∷ word (Λ ∷ α ∷ ο ∷ δ ∷ ι ∷ κ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Col.4.16"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Col.4.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.16"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Col.4.16"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ ν ∷ ῶ ∷ τ ∷ ε ∷ []) "Col.4.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Col.4.17"
∷ word (ε ∷ ἴ ∷ π ∷ α ∷ τ ∷ ε ∷ []) "Col.4.17"
∷ word (Ἀ ∷ ρ ∷ χ ∷ ί ∷ π ∷ π ∷ ῳ ∷ []) "Col.4.17"
∷ word (Β ∷ ∙λ ∷ έ ∷ π ∷ ε ∷ []) "Col.4.17"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Col.4.17"
∷ word (δ ∷ ι ∷ α ∷ κ ∷ ο ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "Col.4.17"
∷ word (ἣ ∷ ν ∷ []) "Col.4.17"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ ∙λ ∷ α ∷ β ∷ ε ∷ ς ∷ []) "Col.4.17"
∷ word (ἐ ∷ ν ∷ []) "Col.4.17"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Col.4.17"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Col.4.17"
∷ word (α ∷ ὐ ∷ τ ∷ ὴ ∷ ν ∷ []) "Col.4.17"
∷ word (π ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ῖ ∷ ς ∷ []) "Col.4.17"
∷ word (Ὁ ∷ []) "Col.4.18"
∷ word (ἀ ∷ σ ∷ π ∷ α ∷ σ ∷ μ ∷ ὸ ∷ ς ∷ []) "Col.4.18"
∷ word (τ ∷ ῇ ∷ []) "Col.4.18"
∷ word (ἐ ∷ μ ∷ ῇ ∷ []) "Col.4.18"
∷ word (χ ∷ ε ∷ ι ∷ ρ ∷ ὶ ∷ []) "Col.4.18"
∷ word (Π ∷ α ∷ ύ ∷ ∙λ ∷ ο ∷ υ ∷ []) "Col.4.18"
∷ word (μ ∷ ν ∷ η ∷ μ ∷ ο ∷ ν ∷ ε ∷ ύ ∷ ε ∷ τ ∷ έ ∷ []) "Col.4.18"
∷ word (μ ∷ ο ∷ υ ∷ []) "Col.4.18"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Col.4.18"
∷ word (δ ∷ ε ∷ σ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.4.18"
∷ word (ἡ ∷ []) "Col.4.18"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Col.4.18"
∷ word (μ ∷ ε ∷ θ ∷ []) "Col.4.18"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Col.4.18"
∷ []
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module ExCoinduction where
open import Coinduction
open import Relation.Binary.PropositionalEquality
open import Data.Stream
open import Data.Nat
open import Data.Bool
data Loc : Set where
A : Loc
DONE : Loc
record Values : Set where
field
x : ℕ
δ : ℕ
k : ℕ
data _Π_ (A B : Set) : Set where
<_,_> : (a : Loc) → (b : Values) → A Π B
-- The computation of x in the state
-- The state machine step function
funₓ : ℕ → ℕ → ℕ → ℕ
funₓ x slope δ = x + δ * slope
step : (Loc Π Values) → (Loc Π Values)
step < A , b > with (X >= 10)
where
_>=_ : ℕ → ℕ → Bool
zero >= zero = true
zero >= suc y = false
suc x >= zero = false
suc x >= suc y = x >= y
_==_ : ℕ → ℕ → Bool
zero == zero = true
zero == suc y = false
suc x == zero = false
suc x == suc y = x == y
X : ℕ
X = funₓ (Values.x b) 1 (Values.δ b)
step < A , b > | false = < A , record { x = X; δ = Values.δ b ; k = Values.k b + 1 } >
where
X : ℕ
X = funₓ (Values.x b) 1 (Values.δ b)
step < A , b > | true = < DONE , record { x = X; δ = Values.δ b ; k = Values.k b + 1 } >
where
X : ℕ
X = funₓ (Values.x b) 1 (Values.δ b)
step < DONE , b > = < DONE , b > -- Just remain in this state forever
-- Make a stream of runFSM
--
f' : ∀ st → Stream (Loc Π Values)
f' st = st ∷ ♯ (f' (step st))
from : Stream (Loc Π Values)
from = f' (< A , (record { x = zero ; δ = 1 ; k = zero }) >)
thm : from ≈ from
thm = refl ∷ ♯ (refl ∷ (♯ (refl ∷ (♯ (refl ∷ (♯ (refl ∷ (♯ (refl ∷ (♯ (refl ∷ (♯ (refl ∷ (♯ (refl ∷
(♯ (refl ∷ ♯ (t'))))))))))))))))))
where
y : Stream (Loc Π Values)
y = f' (< DONE , record { x = 10 ; δ = 1 ; k = 10 } > )
t' : y ≈ y
t' = refl ∷ (♯ t')
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{-# OPTIONS --without-K #-}
open import lib.Basics
open import lib.Relation
module lib.types.SetQuotient where
module _ {i} {A : Type i} {j} where
private
data #SetQuotient-aux (R : Rel A j) : Type i where
#q[_] : A → #SetQuotient-aux R
data #SetQuotient (R : Rel A j) : Type i where
#setquot : #SetQuotient-aux R → (Unit → Unit) → #SetQuotient R
SetQuotient : (R : Rel A j) → Type i
SetQuotient = #SetQuotient
module _ {R : Rel A j} where
q[_] : (a : A) → SetQuotient R
q[ a ] = #setquot #q[ a ] _
postulate -- HIT
quot-rel : {a₁ a₂ : A} → R a₁ a₂ → q[ a₁ ] == q[ a₂ ]
postulate -- HIT
SetQuotient-level : is-set (SetQuotient R)
SetQuotient-is-set = SetQuotient-level
module SetQuotElim {k} {P : SetQuotient R → Type k}
(p : (x : SetQuotient R) → is-set (P x)) (q[_]* : (a : A) → P q[ a ])
(rel* : ∀ {a₁ a₂} (r : R a₁ a₂) → q[ a₁ ]* == q[ a₂ ]* [ P ↓ quot-rel r ]) where
f : Π (SetQuotient R) P
f = f-aux phantom phantom where
f-aux : Phantom p
→ Phantom {A = ∀ {a₁ a₂} (r : R a₁ a₂) → _} rel*
→ Π (SetQuotient R) P
f-aux phantom phantom (#setquot #q[ a ] _) = q[ a ]*
postulate -- HIT
quot-rel-β : ∀ {a₁ a₂} (r : R a₁ a₂) → apd f (quot-rel r) == rel* r
open SetQuotElim public renaming (f to SetQuot-elim)
module SetQuotRec {i} {A : Type i} {j} {R : Rel A j} {k} {B : Type k} (p : is-set B)
(q[_]* : A → B) (rel* : ∀ {a₁ a₂} (r : R a₁ a₂) → q[ a₁ ]* == q[ a₂ ]*) where
private
module M = SetQuotElim (λ x → p) q[_]* (λ {a₁ a₂} r → ↓-cst-in (rel* r))
f : SetQuotient R → B
f = M.f
open SetQuotRec public renaming (f to SetQuot-rec)
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-- Andreas, 2020-02-15
-- Test case by Jesper to prevent regressions when fixing #3541.
-- Jesper, 2019-09-12: The fix of #3541 introduced a regression: the
-- index of the equality type is treated as a positive argument.
postulate X : Set
module EqualityAsPredicate where
data _≡_ (A : Set) : Set → Set where
refl : A ≡ A
data D : Set where
c : D ≡ X → D
data E : Set where
c : X ≡ E → E
module EqualityAsRelation where
data _≡_ : Set → Set → Set where
refl : ∀{A} → A ≡ A
data D : Set where
c : D ≡ X → D
data E : Set where
c : X ≡ E → E
-- Expected:
-- Positivity checker rejects D and E in both variants.
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Categories.NaturalTransformation.Properties where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Univalence
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism renaming (iso to iIso)
open import Cubical.Data.Sigma
open import Cubical.Categories.Category
open import Cubical.Categories.Functor.Base
open import Cubical.Categories.Morphism renaming (isIso to isIsoC)
open import Cubical.Categories.NaturalTransformation.Base
private
variable
ℓC ℓC' ℓD ℓD' : Level
open isIsoC
open NatIso
open NatTrans
open Precategory
open Functor
module _ {C : Precategory ℓC ℓC'} {D : Precategory ℓD ℓD'} where
private
_⋆ᴰ_ : ∀ {x y z} (f : D [ x , y ]) (g : D [ y , z ]) → D [ x , z ]
f ⋆ᴰ g = f ⋆⟨ D ⟩ g
-- natural isomorphism is symmetric
symNatIso : ∀ {F G : Functor C D}
→ F ≅ᶜ G
→ G ≅ᶜ F
symNatIso η .trans .N-ob x = η .nIso x .inv
symNatIso η .trans .N-hom _ = sqLL η
symNatIso η .nIso x .inv = η .trans .N-ob x
symNatIso η .nIso x .sec = η .nIso x .ret
symNatIso η .nIso x .ret = η .nIso x .sec
-- Properties
-- path helpers
module NatTransP where
module _ {F G : Functor C D} where
open Iso
-- same as Sigma version
NatTransΣ = Σ[ ob ∈ ((x : C .ob) → D [(F .F-ob x) , (G .F-ob x)]) ]
({x y : _ } (f : C [ x , y ]) → (F .F-hom f) ⋆ᴰ (ob y) ≡ (ob x) ⋆ᴰ (G .F-hom f))
NatTransIsoΣ : Iso (NatTrans F G) NatTransΣ
NatTransIsoΣ .fun (natTrans N-ob N-hom) = N-ob , N-hom
NatTransIsoΣ .inv (N-ob , N-hom) = (natTrans N-ob N-hom)
NatTransIsoΣ .rightInv _ = refl
NatTransIsoΣ .leftInv _ = refl
NatTrans≡Σ = ua (isoToEquiv NatTransIsoΣ)
-- introducing paths
NatTrans-≡-intro : ∀ {αo βo : N-ob-Type F G}
{αh : N-hom-Type F G αo}
{βh : N-hom-Type F G βo}
→ (p : αo ≡ βo)
→ PathP (λ i → ({x y : C .ob} (f : C [ x , y ]) → (F .F-hom f) ⋆ᴰ (p i y) ≡ (p i x) ⋆ᴰ (G .F-hom f))) αh βh
→ natTrans {F = F} {G} αo αh ≡ natTrans βo βh
NatTrans-≡-intro p q i = natTrans (p i) (q i)
module _ {F G : Functor C D} {α β : NatTrans F G} where
open Iso
private
αOb = α .N-ob
βOb = β .N-ob
αHom = α .N-hom
βHom = β .N-hom
-- path between natural transformations is the same as a pair of paths (between ob and hom)
NTPathIsoPathΣ : Iso (α ≡ β)
(Σ[ p ∈ (αOb ≡ βOb) ]
(PathP (λ i → ({x y : _} (f : _) → F ⟪ f ⟫ ⋆ᴰ (p i y) ≡ (p i x) ⋆ᴰ G ⟪ f ⟫))
αHom
βHom))
NTPathIsoPathΣ .fun p = (λ i → p i .N-ob) , (λ i → p i .N-hom)
NTPathIsoPathΣ .inv (po , ph) i = record { N-ob = po i ; N-hom = ph i }
NTPathIsoPathΣ .rightInv pσ = refl
NTPathIsoPathΣ .leftInv p = refl
NTPath≃PathΣ = isoToEquiv NTPathIsoPathΣ
NTPath≡PathΣ = ua NTPath≃PathΣ
module _ ⦃ isCatD : isCategory D ⦄ where
open NatTransP
-- if the target category has hom Sets, then any natural transformation is a set
isSetNat : ∀ {F G : Functor C D}
→ isSet (NatTrans F G)
isSetNat {F} {G} α β p1 p2 i = comp (λ i → NTPath≡PathΣ {F = F} {G} {α} {β} (~ i))
(λ j → λ {(i = i0) → transport-filler NTPath≡PathΣ p1 (~ j) ;
(i = i1) → transport-filler NTPath≡PathΣ p2 (~ j)})
(p1Σ≡p2Σ i)
where
αOb = α .N-ob
βOb = β .N-ob
αHom = α .N-hom
βHom = β .N-hom
-- convert to sigmas so we can reason about constituent paths separately
p1Σ : Σ[ p ∈ (αOb ≡ βOb) ]
(PathP (λ i → ({x y : _} (f : _) → F ⟪ f ⟫ ⋆ᴰ (p i y) ≡ (p i x) ⋆ᴰ G ⟪ f ⟫))
αHom
βHom)
p1Σ = transport NTPath≡PathΣ p1
p2Σ : Σ[ p ∈ (αOb ≡ βOb) ]
(PathP (λ i → ({x y : _} (f : _) → F ⟪ f ⟫ ⋆ᴰ (p i y) ≡ (p i x) ⋆ᴰ G ⟪ f ⟫))
αHom
βHom)
p2Σ = transport NTPath≡PathΣ p2
-- type aliases
typeN-ob = (x : C .ob) → D [(F .F-ob x) , (G .F-ob x)]
typeN-hom : typeN-ob → Type _
typeN-hom ϕ = {x y : C .ob} (f : C [ x , y ]) → (F .F-hom f) ⋆ᴰ (ϕ y) ≡ (ϕ x) ⋆ᴰ (G .F-hom f)
-- the Ob function is a set
isSetN-ob : isSet ((x : C .ob) → D [(F .F-ob x) , (G .F-ob x)])
isSetN-ob = isOfHLevelΠ 2 λ _ → isCatD .isSetHom
-- the Hom function is a set
isSetN-hom : (ϕ : typeN-ob) → isSet (typeN-hom ϕ)
isSetN-hom γ = isProp→isSet (isPropImplicitΠ λ x → isPropImplicitΠ λ y → isPropΠ λ f → isCatD .isSetHom _ _)
-- in fact it's a dependent Set, which we need because N-hom depends on N-ob
isSetN-homP : isOfHLevelDep 2 (λ γ → {x y : C .ob} (f : C [ x , y ]) → (F .F-hom f) ⋆ᴰ (γ y) ≡ (γ x) ⋆ᴰ (G .F-hom f))
isSetN-homP = isOfHLevel→isOfHLevelDep 2 isSetN-hom
-- components of the equality
p1Ob≡p2Ob : fst p1Σ ≡ fst p2Σ
p1Ob≡p2Ob = isSetN-ob _ _ (fst p1Σ) (fst p2Σ)
p1Hom≡p2Hom : PathP (λ i → PathP (λ j → typeN-hom (p1Ob≡p2Ob i j)) αHom βHom)
(snd p1Σ) (snd p2Σ)
p1Hom≡p2Hom = isSetN-homP _ _ (snd p1Σ) (snd p2Σ) p1Ob≡p2Ob
p1Σ≡p2Σ : p1Σ ≡ p2Σ
p1Σ≡p2Σ = ΣPathP (p1Ob≡p2Ob , p1Hom≡p2Hom)
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