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standardise use of Properties modules (#2283)
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+292
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src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda

+11-11
Original file line numberDiff line numberDiff line change
@@ -17,8 +17,8 @@ open import Relation.Binary.Bundles using (Setoid)
1717
open import Relation.Binary.Definitions
1818
using (Reflexive; Sym; Trans; Antisym; Symmetric; Transitive)
1919
open import Relation.Binary.Structures using (IsEquivalence)
20-
open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
21-
import Relation.Binary.PropositionalEquality.Properties as P
20+
open import Relation.Binary.PropositionalEquality.Core as using (_≡_)
21+
import Relation.Binary.PropositionalEquality.Properties as
2222

2323
private
2424
variable
@@ -114,16 +114,16 @@ module _ {A : Set a} where
114114
_≈_ = Pointwise _≡_
115115

116116
refl : Reflexive _≈_
117-
refl = reflexive P.refl
117+
refl = reflexive .refl
118118

119119
sym : Symmetric _≈_
120-
sym = symmetric P.sym
120+
sym = symmetric .sym
121121

122122
trans : Transitive _≈_
123-
trans = transitive P.trans
123+
trans = transitive .trans
124124

125125
≈-setoid : Setoid _ _
126-
≈-setoid = setoid (P.setoid A)
126+
≈-setoid = setoid (.setoid A)
127127

128128
------------------------------------------------------------------------
129129
-- Pointwise DSL
@@ -161,15 +161,15 @@ module pw-Reasoning (S : Setoid a ℓ) where
161161

162162
`head : {as bs} `Pointwise as bs as .head ∼ bs .head
163163
`head (`lift rs) = rs .head
164-
`head (`refl eq) = S.reflexive (P.cong head eq)
164+
`head (`refl eq) = S.reflexive (.cong head eq)
165165
`head (`bisim rs) = S.reflexive (rs .head)
166166
`head (`step `rs) = `rs .head
167167
`head (`sym `rs) = S.sym (`head `rs)
168168
`head (`trans `rs₁ `rs₂) = S.trans (`head `rs₁) (`head `rs₂)
169169

170170
`tail : {as bs} `Pointwise as bs `Pointwise (as .tail) (bs .tail)
171171
`tail (`lift rs) = `lift (rs .tail)
172-
`tail (`refl eq) = `refl (P.cong tail eq)
172+
`tail (`refl eq) = `refl (.cong tail eq)
173173
`tail (`bisim rs) = `bisim (rs .tail)
174174
`tail (`step `rs) = `rs .tail
175175
`tail (`sym `rs) = `sym (`tail `rs)
@@ -196,8 +196,8 @@ module pw-Reasoning (S : Setoid a ℓ) where
196196
pattern _≈⟨_⟨_ as bs∼as bs∼cs = `trans {as = as} (`sym (`bisim bs∼as)) bs∼cs
197197
pattern _≡⟨_⟩_ as as∼bs bs∼cs = `trans {as = as} (`refl as∼bs) bs∼cs
198198
pattern _≡⟨_⟨_ as bs∼as bs∼cs = `trans {as = as} (`sym (`refl bs∼as)) bs∼cs
199-
pattern _≡⟨⟩_ as as∼bs = `trans {as = as} (`refl P.refl) as∼bs
200-
pattern _∎ as = `refl {as = as} P.refl
199+
pattern _≡⟨⟩_ as as∼bs = `trans {as = as} (`refl .refl) as∼bs
200+
pattern _∎ as = `refl {as = as} .refl
201201

202202

203203
-- Deprecated v2.0
@@ -226,7 +226,7 @@ module pw-Reasoning (S : Setoid a ℓ) where
226226

227227
module ≈-Reasoning {a} {A : Set a} where
228228

229-
open pw-Reasoning (P.setoid A) public
229+
open pw-Reasoning (.setoid A) public
230230

231231
infix 4 _≈∞_
232232
_≈∞_ = `Pointwise∞

src/Codata/Sized/Colist/Properties.agda

+54-54
Large diffs are not rendered by default.

src/Data/Char/Properties.agda

+13-13
Original file line numberDiff line numberDiff line change
@@ -27,10 +27,10 @@ open import Relation.Binary.Definitions
2727
import Relation.Binary.Construct.On as On
2828
import Relation.Binary.Construct.Subst.Equality as Subst
2929
import Relation.Binary.Construct.Closure.Reflexive as Refl
30-
import Relation.Binary.Construct.Closure.Reflexive.Properties as Reflₚ
31-
open import Relation.Binary.PropositionalEquality.Core as PropEq
30+
import Relation.Binary.Construct.Closure.Reflexive.Properties as Refl
31+
open import Relation.Binary.PropositionalEquality.Core as
3232
using (_≡_; _≢_; refl; cong; sym; trans; subst)
33-
import Relation.Binary.PropositionalEquality.Properties as PropEq
33+
import Relation.Binary.PropositionalEquality.Properties as
3434

3535
------------------------------------------------------------------------
3636
-- Primitive properties
@@ -59,13 +59,13 @@ _≟_ : Decidable {A = Char} _≡_
5959
x ≟ y = map′ ≈⇒≡ ≈-reflexive (toℕ x ℕ.≟ toℕ y)
6060

6161
setoid : Setoid _ _
62-
setoid = PropEq.setoid Char
62+
setoid = .setoid Char
6363

6464
decSetoid : DecSetoid _ _
65-
decSetoid = PropEq.decSetoid _≟_
65+
decSetoid = .decSetoid _≟_
6666

6767
isDecEquivalence : IsDecEquivalence _≡_
68-
isDecEquivalence = PropEq.isDecEquivalence _≟_
68+
isDecEquivalence = .isDecEquivalence _≟_
6969

7070
------------------------------------------------------------------------
7171
-- Boolean equality test.
@@ -114,11 +114,11 @@ _<?_ = On.decidable toℕ ℕ._<_ ℕ._<?_
114114

115115
<-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<_
116116
<-isStrictPartialOrder = record
117-
{ isEquivalence = PropEq.isEquivalence
117+
{ isEquivalence = .isEquivalence
118118
; irrefl = <-irrefl
119119
; trans = λ {a} {b} {c} <-trans {a} {b} {c}
120-
; <-resp-≈ = (λ {c} PropEq.subst (c <_))
121-
, (λ {c} PropEq.subst (_< c))
120+
; <-resp-≈ = (λ {c} .subst (c <_))
121+
, (λ {c} .subst (_< c))
122122
}
123123

124124
<-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
@@ -142,20 +142,20 @@ _<?_ = On.decidable toℕ ℕ._<_ ℕ._<?_
142142

143143
infix 4 _≤?_
144144
_≤?_ : Decidable _≤_
145-
_≤?_ = Reflₚ.decidable <-cmp
145+
_≤?_ = Refl.decidable <-cmp
146146

147147
≤-reflexive : _≡_ ⇒ _≤_
148148
≤-reflexive = Refl.reflexive
149149

150150
≤-trans : Transitive _≤_
151-
≤-trans = Reflₚ.trans (λ {a} {b} {c} <-trans {a} {b} {c})
151+
≤-trans = Refl.trans (λ {a} {b} {c} <-trans {a} {b} {c})
152152

153153
≤-antisym : Antisymmetric _≡_ _≤_
154-
≤-antisym = Reflₚ.antisym _≡_ refl ℕ.<-asym
154+
≤-antisym = Refl.antisym _≡_ refl ℕ.<-asym
155155

156156
≤-isPreorder : IsPreorder _≡_ _≤_
157157
≤-isPreorder = record
158-
{ isEquivalence = PropEq.isEquivalence
158+
{ isEquivalence = .isEquivalence
159159
; reflexive = ≤-reflexive
160160
; trans = ≤-trans
161161
}

src/Data/Digit/Properties.agda

+4-4
Original file line numberDiff line numberDiff line change
@@ -7,13 +7,13 @@
77
{-# OPTIONS --cubical-compatible --safe #-}
88

99
open import Data.Digit
10-
import Data.Char.Properties as Charₚ
10+
import Data.Char.Properties as Char
1111
open import Data.Nat.Base using (ℕ)
1212
open import Data.Nat.Properties using (_≤?_)
1313
open import Data.Fin.Properties using (inject≤-injective)
1414
open import Data.Product.Base using (_,_; proj₁)
1515
open import Data.Vec.Relation.Unary.Unique.Propositional using (Unique)
16-
import Data.Vec.Relation.Unary.Unique.Propositional.Properties as Uniqueₚ
16+
import Data.Vec.Relation.Unary.Unique.Propositional.Properties as Unique
1717
open import Data.Vec.Relation.Unary.AllPairs using (allPairs?)
1818
open import Relation.Nullary.Decidable.Core using (True; from-yes; ¬?)
1919
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
@@ -22,7 +22,7 @@ open import Function.Base using (_∘_)
2222
module Data.Digit.Properties where
2323

2424
digitCharsUnique : Unique digitChars
25-
digitCharsUnique = from-yes (allPairs? (λ x y ¬? (x Charₚ.≟ y)) digitChars)
25+
digitCharsUnique = from-yes (allPairs? (λ x y ¬? (x Char.≟ y)) digitChars)
2626

2727
module _ (base : ℕ) where
2828
module _ {base≥2 base≥2′ : True (2 ≤? base)} where
@@ -32,4 +32,4 @@ module _ (base : ℕ) where
3232

3333
module _ {base≤16 base≤16′ : True (base ≤? 16)} where
3434
showDigit-injective : (n m : Digit base) showDigit {base} {base≤16} n ≡ showDigit {base} {base≤16′} m n ≡ m
35-
showDigit-injective n m = inject≤-injective _ _ n m ∘ Uniqueₚ.lookup-injective digitCharsUnique _ _
35+
showDigit-injective n m = inject≤-injective _ _ n m ∘ Unique.lookup-injective digitCharsUnique _ _

src/Data/Fin/Induction.agda

+2-2
Original file line numberDiff line numberDiff line change
@@ -16,7 +16,7 @@ import Data.Nat.Properties as ℕ
1616
open import Data.Product.Base using (_,_)
1717
open import Data.Vec.Base as Vec using (Vec; []; _∷_)
1818
open import Data.Vec.Relation.Unary.Linked as Linked using (Linked; [-]; _∷_)
19-
import Data.Vec.Relation.Unary.Linked.Properties as Linkedₚ
19+
import Data.Vec.Relation.Unary.Linked.Properties as Linked
2020
open import Function.Base using (flip; _$_)
2121
open import Induction
2222
open import Induction.WellFounded as WF
@@ -124,7 +124,7 @@ module _ {_≈_ : Rel (Fin n) ℓ} where
124124
pigeon : {xs : Vec (Fin n) n} Linked (flip _⊏_) (i ∷ xs) WellFounded _⊏_
125125
pigeon {xs} i∷xs↑ =
126126
let (i₁ , i₂ , i₁<i₂ , xs[i₁]≡xs[i₂]) = pigeonhole (n<1+n n) (Vec.lookup (i ∷ xs)) in
127-
let xs[i₁]⊏xs[i₂] = Linkedₚ.lookup⁺ (Ord.transitive _⊏_ ⊏.trans) i∷xs↑ i₁<i₂ in
127+
let xs[i₁]⊏xs[i₂] = Linked.lookup⁺ (Ord.transitive _⊏_ ⊏.trans) i∷xs↑ i₁<i₂ in
128128
let xs[i₁]⊏xs[i₁] = ⊏.<-respʳ-≈ (⊏.Eq.reflexive xs[i₁]≡xs[i₂]) xs[i₁]⊏xs[i₂] in
129129
contradiction xs[i₁]⊏xs[i₁] (⊏.irrefl ⊏.Eq.refl)
130130

src/Data/Fin/Substitution/Lemmas.agda

+17-17
Original file line numberDiff line numberDiff line change
@@ -12,9 +12,9 @@ open import Data.Fin.Substitution
1212
open import Data.Nat hiding (_⊔_; _/_)
1313
open import Data.Fin.Base using (Fin; zero; suc; lift)
1414
open import Data.Vec.Base
15-
import Data.Vec.Properties as VecProp
15+
import Data.Vec.Properties as Vec
1616
open import Function.Base as Fun using (_∘_; _$_; flip)
17-
open import Relation.Binary.PropositionalEquality.Core as PropEq
17+
open import Relation.Binary.PropositionalEquality.Core as
1818
using (_≡_; refl; sym; cong; cong₂)
1919
open import Relation.Binary.PropositionalEquality.Properties
2020
using (module ≡-Reasoning)
@@ -68,9 +68,9 @@ record Lemmas₀ (T : Pred ℕ ℓ) : Set ℓ where
6868
lookup-map-weaken-↑⋆ zero x = refl
6969
lookup-map-weaken-↑⋆ (suc k) zero = refl
7070
lookup-map-weaken-↑⋆ (suc k) (suc x) {ρ} = begin
71-
lookup (map weaken (map weaken ρ ↑⋆ k)) x ≡⟨ VecProp.lookup-map x weaken (map weaken ρ ↑⋆ k) ⟩
71+
lookup (map weaken (map weaken ρ ↑⋆ k)) x ≡⟨ Vec.lookup-map x weaken (map weaken ρ ↑⋆ k) ⟩
7272
weaken (lookup (map weaken ρ ↑⋆ k) x) ≡⟨ cong weaken (lookup-map-weaken-↑⋆ k x) ⟩
73-
weaken (lookup ((ρ ↑) ↑⋆ k) (lift k suc x)) ≡⟨ sym $ VecProp.lookup-map (lift k suc x) weaken ((ρ ↑) ↑⋆ k) ⟩
73+
weaken (lookup ((ρ ↑) ↑⋆ k) (lift k suc x)) ≡⟨ sym $ Vec.lookup-map (lift k suc x) weaken ((ρ ↑) ↑⋆ k) ⟩
7474
lookup (map weaken ((ρ ↑) ↑⋆ k)) (lift k suc x) ∎
7575

7676
record Lemmas₁ (T : Pred ℕ ℓ) : Setwhere
@@ -85,7 +85,7 @@ record Lemmas₁ (T : Pred ℕ ℓ) : Set ℓ where
8585
lookup ρ x ≡ var y
8686
lookup (map weaken ρ) x ≡ var (suc y)
8787
lookup-map-weaken x {y} {ρ} hyp = begin
88-
lookup (map weaken ρ) x ≡⟨ VecProp.lookup-map x weaken ρ ⟩
88+
lookup (map weaken ρ) x ≡⟨ Vec.lookup-map x weaken ρ ⟩
8989
weaken (lookup ρ x) ≡⟨ cong weaken hyp ⟩
9090
weaken (var y) ≡⟨ weaken-var ⟩
9191
var (suc y) ∎
@@ -153,7 +153,7 @@ record Lemmas₂ (T : Pred ℕ ℓ) : Set ℓ where
153153

154154
lookup-⊙ : x {ρ₁ : Sub T m n} {ρ₂ : Sub T n o}
155155
lookup (ρ₁ ⊙ ρ₂) x ≡ lookup ρ₁ x / ρ₂
156-
lookup-⊙ x {ρ₁} {ρ₂} = VecProp.lookup-map x (λ t t / ρ₂) ρ₁
156+
lookup-⊙ x {ρ₁} {ρ₂} = Vec.lookup-map x (λ t t / ρ₂) ρ₁
157157

158158
lookup-⨀ : x (ρs : Subs T m n)
159159
lookup (⨀ ρs) x ≡ var x /✶ ρs
@@ -239,8 +239,8 @@ record Lemmas₃ (T : Pred ℕ ℓ) : Set ℓ where
239239

240240
⊙-id :: Sub T m n} ρ ⊙ id ≡ ρ
241241
⊙-id {ρ = ρ} = begin
242-
map (λ t t / id) ρ ≡⟨ VecProp.map-cong id-vanishes ρ ⟩
243-
map Fun.id ρ ≡⟨ VecProp.map-id ρ ⟩
242+
map (λ t t / id) ρ ≡⟨ Vec.map-cong id-vanishes ρ ⟩
243+
map Fun.id ρ ≡⟨ Vec.map-id ρ ⟩
244244
ρ ∎
245245

246246
open Lemmas₂ lemmas₂ public hiding (wk-⊙-sub′)
@@ -264,13 +264,13 @@ record Lemmas₄ (T : Pred ℕ ℓ) : Set ℓ where
264264
ρ₁ ↑ ⊙ ρ₂ ↑ ∎
265265
where
266266
lemma = begin
267-
map weaken (map (λ t t / ρ₂) ρ₁) ≡⟨ sym (VecProp.map-∘ _ _ _) ⟩
268-
map (λ t weaken (t / ρ₂)) ρ₁ ≡⟨ VecProp.map-cong (λ t begin
267+
map weaken (map (λ t t / ρ₂) ρ₁) ≡⟨ sym (Vec.map-∘ _ _ _) ⟩
268+
map (λ t weaken (t / ρ₂)) ρ₁ ≡⟨ Vec.map-cong (λ t begin
269269
weaken (t / ρ₂) ≡⟨ sym /-wk ⟩
270270
t / ρ₂ / wk ≡⟨ hyp t ⟩
271271
t / wk / ρ₂ ↑ ≡⟨ cong₂ _/_ /-wk refl ⟩
272272
weaken t / ρ₂ ↑ ∎) ρ₁ ⟩
273-
map (λ t weaken t / ρ₂ ↑) ρ₁ ≡⟨ VecProp.map-∘ _ _ _ ⟩
273+
map (λ t weaken t / ρ₂ ↑) ρ₁ ≡⟨ Vec.map-∘ _ _ _ ⟩
274274
map (λ t t / ρ₂ ↑) (map weaken ρ₁) ∎
275275

276276
↑⋆-distrib′ : {ρ₁ : Sub T m n} {ρ₂ : Sub T n o}
@@ -284,7 +284,7 @@ record Lemmas₄ (T : Pred ℕ ℓ) : Set ℓ where
284284

285285
map-weaken :: Sub T m n} map weaken ρ ≡ ρ ⊙ wk
286286
map-weaken {ρ = ρ} = begin
287-
map weaken ρ ≡⟨ VecProp.map-cong (λ _ sym /-wk) ρ ⟩
287+
map weaken ρ ≡⟨ Vec.map-cong (λ _ sym /-wk) ρ ⟩
288288
map (λ t t / wk) ρ ≡⟨ refl ⟩
289289
ρ ⊙ wk ∎
290290

@@ -324,8 +324,8 @@ record Lemmas₄ (T : Pred ℕ ℓ) : Set ℓ where
324324
⊙-assoc : {ρ₁ : Sub T m n} {ρ₂ : Sub T n o} {ρ₃ : Sub T o p}
325325
ρ₁ ⊙ (ρ₂ ⊙ ρ₃) ≡ (ρ₁ ⊙ ρ₂) ⊙ ρ₃
326326
⊙-assoc {ρ₁ = ρ₁} {ρ₂} {ρ₃} = begin
327-
map (λ t t / ρ₂ ⊙ ρ₃) ρ₁ ≡⟨ VecProp.map-cong /-⊙ ρ₁ ⟩
328-
map (λ t t / ρ₂ / ρ₃) ρ₁ ≡⟨ VecProp.map-∘ _ _ _ ⟩
327+
map (λ t t / ρ₂ ⊙ ρ₃) ρ₁ ≡⟨ Vec.map-cong /-⊙ ρ₁ ⟩
328+
map (λ t t / ρ₂ / ρ₃) ρ₁ ≡⟨ Vec.map-∘ _ _ _ ⟩
329329
map (λ t t / ρ₃) (map (λ t t / ρ₂) ρ₁) ∎
330330

331331
map-weaken-⊙-sub : : Sub T m n} {t} map weaken ρ ⊙ sub t ≡ ρ
@@ -560,7 +560,7 @@ record TermLemmas (T : ℕ → Set) : Set₁ where
560560
( x lookup ρ₂ x ≡ T.var (f x))
561561
map T.var ρ₁ ≡ ρ₂
562562
map-var≡ {ρ₁ = ρ₁} {ρ₂ = ρ₂} {f = f} hyp₁ hyp₂ = extensionality λ x
563-
lookup (map T.var ρ₁) x ≡⟨ VecProp.lookup-map x _ ρ₁ ⟩
563+
lookup (map T.var ρ₁) x ≡⟨ Vec.lookup-map x _ ρ₁ ⟩
564564
T.var (lookup ρ₁ x) ≡⟨ cong T.var $ hyp₁ x ⟩
565565
T.var (f x) ≡⟨ sym $ hyp₂ x ⟩
566566
lookup ρ₂ x ∎
@@ -577,15 +577,15 @@ record TermLemmas (T : ℕ → Set) : Set₁ where
577577
↑≡↑ :: Sub Fin m n} map T.var (ρ VarSubst.↑) ≡ map T.var ρ T.↑
578578
↑≡↑ {ρ = ρ} = map-var≡
579579
(VarLemmas.lookup-↑⋆ (lookup ρ) (λ _ refl) 1)
580-
(lookup-↑⋆ (lookup ρ) (λ _ VecProp.lookup-map _ _ ρ) 1)
580+
(lookup-↑⋆ (lookup ρ) (λ _ Vec.lookup-map _ _ ρ) 1)
581581

582582
/Var≡/ : : Sub Fin m n} {t} t /Var ρ ≡ t T./ map T.var ρ
583583
/Var≡/ {ρ = ρ} {t = t} =
584584
/✶-↑✶ (ε ▻ ρ) (ε ▻ map T.var ρ)
585585
(λ k x
586586
T.var x /Var ρ VarSubst.↑⋆ k ≡⟨ app-var ⟩
587587
T.var (lookup (ρ VarSubst.↑⋆ k) x) ≡⟨ cong T.var $ VarLemmas.lookup-↑⋆ _ (λ _ refl) k _ ⟩
588-
T.var (lift k (VarSubst._/ ρ) x) ≡⟨ sym $ lookup-↑⋆ _ (λ _ VecProp.lookup-map _ _ ρ) k _ ⟩
588+
T.var (lift k (VarSubst._/ ρ) x) ≡⟨ sym $ lookup-↑⋆ _ (λ _ Vec.lookup-map _ _ ρ) k _ ⟩
589589
lookup (map T.var ρ T.↑⋆ k) x ≡⟨ sym app-var ⟩
590590
T.var x T./ map T.var ρ T.↑⋆ k ∎)
591591
zero t

src/Data/Integer/Properties.agda

+3-3
Original file line numberDiff line numberDiff line change
@@ -24,7 +24,7 @@ open import Data.Nat.Solver
2424
open import Data.Product.Base using (proj₁; proj₂; _,_; _×_)
2525
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
2626
open import Data.Sign as Sign using (Sign) renaming (_*_ to _𝕊*_)
27-
import Data.Sign.Properties as 𝕊ₚ
27+
import Data.Sign.Properties as Sign
2828
open import Function.Base using (_∘_; _$_; id)
2929
open import Level using (0ℓ)
3030
open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
@@ -1597,7 +1597,7 @@ abs-* i j = abs-◃ _ _
15971597
*-cancelʳ-≡ : i j k .{{_ : NonZero k}} i * k ≡ j * k i ≡ j
15981598
*-cancelʳ-≡ i j k eq with sign-cong′ eq
15991599
... | inj₁ s[ik]≡s[jk] = ◃-cong
1600-
(𝕊ₚ.*-cancelʳ-≡ (sign k) (sign i) (sign j) s[ik]≡s[jk])
1600+
(Sign.*-cancelʳ-≡ (sign k) (sign i) (sign j) s[ik]≡s[jk])
16011601
(ℕ.*-cancelʳ-≡ ∣ i ∣ ∣ j ∣ _ (abs-cong eq))
16021602
... | inj₂ (∣ik∣≡0 , ∣jk∣≡0) = trans
16031603
(∣i∣≡0⇒i≡0 (ℕ.m*n≡0⇒m≡0 _ _ ∣ik∣≡0))
@@ -1709,7 +1709,7 @@ neg-distribʳ-* i j = begin
17091709
◃-distrib-* s t zero (suc n) = refl
17101710
◃-distrib-* s t (suc m) zero =
17111711
trans
1712-
(cong₂ _◃_ (𝕊ₚ.*-comm s t) (ℕ.*-comm m 0))
1712+
(cong₂ _◃_ (Sign.*-comm s t) (ℕ.*-comm m 0))
17131713
(*-comm (t ◃ zero) (s ◃ suc m))
17141714
◃-distrib-* s t (suc m) (suc n) =
17151715
sym (cong₂ _◃_

src/Data/List/Countdown.agda

+6-6
Original file line numberDiff line numberDiff line change
@@ -29,10 +29,10 @@ open import Data.Sum.Properties
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open import Relation.Nullary.Reflects using (invert)
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open import Relation.Nullary
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open import Relation.Nullary.Decidable using (dec-true; dec-false)
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open import Relation.Binary.PropositionalEquality.Core as PropEq
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open import Relation.Binary.PropositionalEquality.Core as
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using (_≡_; _≢_; refl; cong)
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import Relation.Binary.PropositionalEquality.Properties as PropEq
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open PropEq.≡-Reasoning
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import Relation.Binary.PropositionalEquality.Properties as
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open .≡-Reasoning
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private
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open module D = DecSetoid D
@@ -124,12 +124,12 @@ record _⊕_ (counted : List Elem) (n : ℕ) : Set where
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-- A countdown can be initialised by proving that Elem is finite.
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empty : {n} Injection D.setoid (PropEq.setoid (Fin n)) [] ⊕ n
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empty : {n} Injection D.setoid (.setoid (Fin n)) [] ⊕ n
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empty inj =
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record { kind = inj₂ ∘ to
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; injective = λ {x} {y} {i} eq₁ eq₂ injective (begin
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to x ≡⟨ inj₂-injective eq₁ ⟩
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i ≡⟨ PropEq.sym $ inj₂-injective eq₂ ⟩
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i ≡⟨ .sym $ inj₂-injective eq₂ ⟩
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to y ∎)
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}
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where open Injection inj
@@ -199,7 +199,7 @@ insert {counted} {n} counted⊕1+n x x∉counted =
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inj eq₁ eq₂ | no _ | no _ | inj₂ i | inj₂ _ | inj₂ _ | _ | _ | hlp =
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hlp _ refl refl $
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punchOut-injective {i = i} _ _ $
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(PropEq.trans (inj₂-injective eq₁) (PropEq.sym (inj₂-injective eq₂)))
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(.trans (inj₂-injective eq₁) (.sym (inj₂-injective eq₂)))
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-- Counts an element if it has not already been counted.
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