blob: f54e015f75f5be1bf2ae6bf98bd7b407372f77b8 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
|
{-# OPTIONS --without-K --safe #-}
open import Function
open import Relation.Binary
import Relation.Binary.PropositionalEquality as ≡
module Cfe.Language.Construct.Single
{a ℓ} (setoid : Setoid a ℓ)
(≈-trans-bijₗ : ∀ {a b c : Setoid.Carrier setoid}
→ {b≈c : Setoid._≈_ setoid b c}
→ Bijective ≡._≡_ ≡._≡_ (flip (Setoid.trans setoid {a}) b≈c))
(≈-trans-reflₗ : ∀ {a b : Setoid.Carrier setoid} {a≈b : Setoid._≈_ setoid a b}
→ Setoid.trans setoid a≈b (Setoid.refl setoid) ≡.≡ a≈b)
(≈-trans-symₗ : ∀ {a b c : Setoid.Carrier setoid}
→ {a≈b : Setoid._≈_ setoid a b}
→ {a≈c : Setoid._≈_ setoid a c}
→ {b≈c : Setoid._≈_ setoid b c}
→ Setoid.trans setoid a≈b b≈c ≡.≡ a≈c
→ Setoid.trans setoid a≈c (Setoid.sym setoid b≈c) ≡.≡ a≈b)
(≈-trans-transₗ : ∀ {a b c d : Setoid.Carrier setoid}
→ {a≈b : Setoid._≈_ setoid a b}
→ {a≈c : Setoid._≈_ setoid a c}
→ {a≈d : Setoid._≈_ setoid a d}
→ {b≈c : Setoid._≈_ setoid b c}
→ {c≈d : Setoid._≈_ setoid c d}
→ Setoid.trans setoid a≈b b≈c ≡.≡ a≈c
→ Setoid.trans setoid a≈c c≈d ≡.≡ a≈d
→ Setoid.trans setoid a≈b (Setoid.trans setoid b≈c c≈d) ≡.≡ a≈d)
where
open Setoid setoid renaming (Carrier to A)
open import Cfe.Language setoid
open import Data.List
open import Data.List.Relation.Binary.Equality.Setoid setoid
open import Data.Product as Product
open import Level
private
∷-inj : {a b : A} {l₁ l₂ : List A} {a≈b a≈b′ : a ≈ b} {l₁≋l₂ l₁≋l₂′ : l₁ ≋ l₂} → ≡._≡_ {A = a ∷ l₁ ≋ b ∷ l₂} (a≈b ∷ l₁≋l₂) (a≈b′ ∷ l₁≋l₂′) → (a≈b ≡.≡ a≈b′) × (l₁≋l₂ ≡.≡ l₁≋l₂′)
∷-inj ≡.refl = ≡.refl , ≡.refl
≋-trans-injₗ : {x l₁ l₂ : List A} → {l₁≋l₂ : l₁ ≋ l₂} → Injective ≡._≡_ ≡._≡_ (flip (≋-trans {x}) l₁≋l₂)
≋-trans-injₗ {_} {_} {_} {_} {[]} {[]} _ = ≡.refl
≋-trans-injₗ {_} {_} {_} {_ ∷ _} {_ ∷ _} {_ ∷ _} = uncurry (≡.cong₂ _∷_)
∘ Product.map (proj₁ ≈-trans-bijₗ) ≋-trans-injₗ
∘ ∷-inj
≋-trans-surₗ : {x l₁ l₂ : List A} → {l₁≋l₂ : l₁ ≋ l₂} → Surjective {A = x ≋ l₁} ≡._≡_ ≡._≡_ (flip (≋-trans {x}) l₁≋l₂)
≋-trans-surₗ {_} {_} {_} {[]} [] = [] , ≡.refl
≋-trans-surₗ {_} {_} {_} {_ ∷ _} (a≈c ∷ x≋l₂) = Product.zip _∷_ (≡.cong₂ _∷_) (proj₂ ≈-trans-bijₗ a≈c) (≋-trans-surₗ x≋l₂)
≋-trans-reflₗ : {l₁ l₂ : List A} {l₁≋l₂ : l₁ ≋ l₂} → ≋-trans l₁≋l₂ ≋-refl ≡.≡ l₁≋l₂
≋-trans-reflₗ {_} {_} {[]} = ≡.refl
≋-trans-reflₗ {_} {_} {a≈b ∷ l₁≋l₂} = ≡.cong₂ _∷_ ≈-trans-reflₗ ≋-trans-reflₗ
≋-trans-symₗ : {l₁ l₂ l₃ : List A} {l₁≋l₂ : l₁ ≋ l₂} {l₁≋l₃ : l₁ ≋ l₃} {l₂≋l₃ : l₂ ≋ l₃}
→ ≋-trans l₁≋l₂ l₂≋l₃ ≡.≡ l₁≋l₃
→ ≋-trans l₁≋l₃ (≋-sym l₂≋l₃) ≡.≡ l₁≋l₂
≋-trans-symₗ {_} {_} {_} {[]} {[]} {[]} _ = ≡.refl
≋-trans-symₗ {_} {_} {_} {_ ∷ _} {_ ∷ _} {_ ∷ _} = uncurry (≡.cong₂ _∷_)
∘ Product.map ≈-trans-symₗ ≋-trans-symₗ
∘ ∷-inj
≋-trans-transₗ : {l₁ l₂ l₃ l₄ : List A}
→ {l₁≋l₂ : l₁ ≋ l₂} {l₁≋l₃ : l₁ ≋ l₃} {l₁≋l₄ : l₁ ≋ l₄} {l₂≋l₃ : l₂ ≋ l₃} {l₃≋l₄ : l₃ ≋ l₄}
→ ≋-trans l₁≋l₂ l₂≋l₃ ≡.≡ l₁≋l₃
→ ≋-trans l₁≋l₃ l₃≋l₄ ≡.≡ l₁≋l₄
→ ≋-trans l₁≋l₂ (≋-trans l₂≋l₃ l₃≋l₄) ≡.≡ l₁≋l₄
≋-trans-transₗ {l₁≋l₂ = []} {[]} {[]} {[]} {[]} _ _ = ≡.refl
≋-trans-transₗ {l₁≋l₂ = _ ∷ _} {_ ∷ _} {_ ∷ _} {_ ∷ _} {_ ∷ _} = uncurry (≡.cong₂ _∷_)
∘₂ uncurry (Product.zip ≈-trans-transₗ ≋-trans-transₗ)
∘₂ curry (Product.map ∷-inj ∷-inj)
｛_｝ : List A → Language (a ⊔ ℓ) (a ⊔ ℓ)
｛ l ｝ = record
{ 𝕃 = l ≋_
; _≈ᴸ_ = ≡._≡_
; ⤖ = flip ≋-trans
; isLanguage = record
{ ≈ᴸ-isEquivalence = ≡.isEquivalence
; ⤖-cong = λ {_} {_} {l₁≋l₂} → ≡.cong (flip ≋-trans l₁≋l₂)
; ⤖-bijective = ≋-trans-injₗ , ≋-trans-surₗ
; ⤖-refl = ≋-trans-reflₗ
; ⤖-sym = ≋-trans-symₗ
; ⤖-trans = ≋-trans-transₗ
}
}
|
