diff options
Diffstat (limited to 'Precond.agda')
| -rw-r--r-- | Precond.agda | 81 |
1 files changed, 41 insertions, 40 deletions
diff --git a/Precond.agda b/Precond.agda index ebb5412..bc619dc 100644 --- a/Precond.agda +++ b/Precond.agda @@ -18,7 +18,7 @@ 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 (∃ ; _,_ ; proj₂) +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) @@ -26,70 +26,71 @@ open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨ open import Relation.Nullary using (yes ; no) open import Generic using (mapMV ; sequenceV ; sequence-map) -open import FinMap using (FinMapMaybe ; lookupM ; union ; fromFunc ; empty ; insert ; lemma-lookupM-empty ; delete-many ; lemma-tabulate-∘ ; delete ; lemma-lookupM-delete) +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 -open import Bidir (decSetoid deq) using (_in-domain-of_ ; lemma-assoc-domain ; lemma-just-sequence) +import Bidir +open Bidir (decSetoid deq) using (_in-domain-of_ ; lemma-assoc-domain ; lemma-just-sequence) import GetTypes open GetTypes.PartialVecVec using (Get ; module Get) -open BFF.PartialVecBFF (decSetoid deq) using (assoc ; enumerate ; denumerate ; bff) - -lemma-lookup-map-just : {n : ℕ} (f : Fin n) {A : Set} (v : Vec A n) → lookup f (map Maybe.just v) ≡ Maybe.just (lookup f v) -lemma-lookup-map-just zero (x ∷ xs) = refl -lemma-lookup-map-just (suc f) (x ∷ xs) = lemma-lookup-map-just f xs +open BFF.PartialVecBFF (decSetoid deq) using (assoc ; enumerate ; denumerate ; bff ; enumeratel) 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 : Vec A n} → (toList is) in-domain-of h → ∃ λ v → union h (delete-many is (map just g)) ≡ map just v +lemma-union-delete-fromFunc : {m n : ℕ} {A : Set} {is : Vec (Fin n) m} {h : FinMapMaybe n A} {g : Fin n → A} → (toList 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 = _ , (begin - union h (map just g) + union h (fromFunc g) ≡⟨ lemma-tabulate-∘ (λ f → begin - maybe′ just (lookup f (map just g)) (lookup f h) - ≡⟨ cong (flip (maybe′ just) (lookup f h)) (lemma-lookup-map-just f g) ⟩ - maybe′ just (just (lookup f g)) (lookup f h) - ≡⟨ lemma-maybe-just (lookup f g) (lookup f h) ⟩ - just (maybe′ id (lookup f g) (lookup f h)) ∎) ⟩ - tabulate (λ f → just (maybe′ id (lookup f g) (lookup f h))) - ≡⟨ tabulate-∘ just (λ f → maybe′ id (lookup f g) (lookup f h)) ⟩ - map just (tabulate (λ f → maybe′ id (lookup f g) (lookup f h))) ∎) -lemma-union-delete-fromFunc {n = n} {is = i ∷ is} {h = h} {g = g} ((x , px) Data.List.All.∷ ps) = _ , (begin - union h (delete i (delete-many is (map just g))) + 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)) ∎) ⟩ + tabulate (λ f → just (maybe′ id (g f) (lookup f h))) + ≡⟨ tabulate-∘ just (λ f → maybe′ id (g f) (lookup f h)) ⟩ + map just (tabulate (λ f → maybe′ id (g f) (lookup 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 (map just g)) + union h (delete-many is (fromFunc g)) ≡⟨ proj₂ (lemma-union-delete-fromFunc ps) ⟩ map just _ ∎) - where inner : (f : Fin n) → maybe′ just (lookupM f (delete i (delete-many is (map just g)))) (lookup f h) ≡ maybe′ just (lookupM f (delete-many is (map just g))) (lookup f h) + 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 (map just g)))) (lookup i h) + 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 (map just g))) (lookup i h) ∎ - inner f | no f≢i = cong (flip (maybe′ just) (lookup f h)) (lemma-lookupM-delete (delete-many is (map just g)) f≢i) + 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) -assoc-enough : (G : Get) → {i : Get.|I| G} → (s : Vec Carrier (Get.|gl₁| G i)) → (v : Vec Carrier (Get.|gl₂| G i)) → ∃ (λ h → assoc (Get.get G (enumerate s)) v ≡ just h) → ∃ λ u → bff G s v ≡ just u -assoc-enough G s v (h , p) = let w , pw = lemma-union-delete-fromFunc (lemma-assoc-domain (get s′) v h p) in _ , (begin - bff G s v - ≡⟨ cong (flip _>>=_ (flip mapMV s′ ∘ flip lookupM) ∘ _<$>_ (flip union g′)) p ⟩ - mapMV (flip lookupM (union h g′)) s′ - ≡⟨ sym (sequence-map (flip lookupM (union h g′)) s′) ⟩ - sequenceV (map (flip lookupM (union h g′)) s′) - ≡⟨ cong (sequenceV ∘ flip map s′ ∘ flip lookupM) pw ⟩ - sequenceV (map (flip lookupM (map just w)) s′) - ≡⟨ cong sequenceV (map-cong (λ f → lemma-lookup-map-just f w) s′) ⟩ - sequenceV (map (Maybe.just ∘ flip lookup w) s′) - ≡⟨ cong sequenceV (map-∘ just (flip lookup w) s′) ⟩ - sequenceV (map Maybe.just (map (flip lookup w) s′)) - ≡⟨ lemma-just-sequence (map (flip lookup w) s′) ⟩ - just (map (flip lookup w) s′) ∎) +assoc-enough : (G : Get) → {i : Get.|I| G} → (s : Vec Carrier (Get.|gl₁| G i)) → (v : Vec Carrier (Get.|gl₂| G i)) → ∃ (λ h → assoc (Get.get G (enumerate s)) v ≡ just h) → ∃ λ u → bff G i s v ≡ just u +assoc-enough G {i} s v (h , p) = _ , (begin + bff G i s v + ≡⟨ cong (flip _>>=_ (flip mapMV t ∘ flip lookupM) ∘ _<$>_ (flip union (reshape g′ (Get.|gl₁| G i)))) p ⟩ + mapMV (flip lookupM (union h (reshape g′ (Get.|gl₁| G i)))) t + ≡⟨ sym (sequence-map (flip lookupM (union h (reshape g′ (Get.|gl₁| G i)))) t) ⟩ + sequenceV (map (flip lookupM (union h (reshape g′ (Get.|gl₁| G i)))) t) + ≡⟨ cong (sequenceV ∘ flip map t ∘ flip lookupM ∘ union h) (lemma-reshape-id g′) ⟩ + sequenceV (map (flip lookupM (union h g′)) t) + ≡⟨ cong (sequenceV ∘ flip map t ∘ flip lookupM) (proj₂ wp) ⟩ + sequenceV (map (flip lookupM (fromFunc (proj₁ wp))) t) + ≡⟨ cong sequenceV (map-cong (lemma-lookupM-fromFunc (proj₁ wp)) t) ⟩ + sequenceV (map (Maybe.just ∘ proj₁ wp) t) + ≡⟨ cong sequenceV (map-∘ just (proj₁ wp) t) ⟩ + sequenceV (map Maybe.just (map (proj₁ wp) t)) + ≡⟨ lemma-just-sequence (map (proj₁ wp) t) ⟩ + just (map (proj₁ wp) t) ∎) where open Get G s′ = enumerate s g = fromFunc (denumerate s) g′ = delete-many (get s′) g + t = enumeratel (Get.|gl₁| G i) + wp = lemma-union-delete-fromFunc (lemma-assoc-domain (get t) v h p) data All-different {A : Set} : List A → Set where different-[] : All-different [] |