diff options
Diffstat (limited to 'Precond.agda')
| -rw-r--r-- | Precond.agda | 47 |
1 files changed, 24 insertions, 23 deletions
diff --git a/Precond.agda b/Precond.agda index cc81db4..07775ac 100644 --- a/Precond.agda +++ b/Precond.agda @@ -1,4 +1,5 @@ -open import Relation.Binary.Core using (Decidable ; _≡_) +open import Relation.Binary.Core using (Decidable) +open import Relation.Binary.PropositionalEquality using (_≡_) module Precond (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where @@ -21,29 +22,29 @@ 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.Binary.PropositionalEquality as P using (inspect ; [_]) +open P.≡-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) +open CheckInsert (P.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) +open Bidir (P.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) +open BFF.PartialShapeBFF (P.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-maybe-just a (just x) = P.refl +lemma-maybe-just a nothing = P.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) ⟩ + ≡⟨ P.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)) ∎)) @@ -55,20 +56,20 @@ lemma-union-delete-fromFunc {n = n} {is = i ∷ is} {h = h} {g = g} (Data.List.A _ ∎) 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 + inner .i | yes P.refl = begin maybe′ just (lookupM i (delete i (delete-many is (fromFunc g)))) (lookup i h) - ≡⟨ cong (maybe′ just _) px ⟩ + ≡⟨ P.cong (maybe′ just _) px ⟩ just x - ≡⟨ cong (maybe′ just _) (sym px) ⟩ + ≡⟨ P.cong (maybe′ just _) (P.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) + inner f | no f≢i = P.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 + assoc-enough {i} j s v (h , p) = _ , 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) @@ -81,10 +82,10 @@ module _ (G : Get) where 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′) ⟩ + ≡⟨ P.cong just (begin _ + ≡⟨ P.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) ⟩ + ≡⟨ P.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 @@ -101,9 +102,9 @@ data All-different {A : Set} : List A → Set where 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 [] [] P.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 ⟩ @@ -112,13 +113,13 @@ lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') {h} ph i∉is | just h' | [ 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 [] [] p = empty , P.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 ⟩ + ≡⟨ P.refl ⟩ (assoc us vs >>= checkInsert u v) - ≡⟨ cong (flip _>>=_ (checkInsert u v)) p' ⟩ + ≡⟨ P.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) ∎) |