diff options
Diffstat (limited to 'Bidir.agda')
| -rw-r--r-- | Bidir.agda | 18 |
1 files changed, 10 insertions, 8 deletions
@@ -18,6 +18,9 @@ open import Data.Vec.Equality using () renaming (module Equality to VecEq) open import Data.Vec.Properties using (tabulate-∘ ; lookup∘tabulate ; map-cong ; map-∘) open import Data.Product using (∃ ; _×_ ; _,_ ; proj₁ ; proj₂) open import Function using (id ; _∘_ ; flip) +open import Function.Equality using (_⟶_ ; _⟨$⟩_) +open import Function.Injection using (module Injection) renaming (Injection to _↪_) +open Injection using (to) open import Relation.Binary.Core using (refl ; _≡_) open import Relation.Binary.Indexed using (_at_) renaming (Setoid to ISetoid) open import Relation.Binary.PropositionalEquality using (cong ; sym ; inspect ; [_] ; trans ; cong₂ ; decSetoid ; module ≡-Reasoning) renaming (setoid to EqSetoid) @@ -25,13 +28,13 @@ open import Relation.Binary using (Setoid ; module Setoid ; module DecSetoid) import Relation.Binary.EqReasoning as EqR import GetTypes -open GetTypes.VecVec using (Get ; module Get) +open GetTypes.PartialVecVec using (Get ; module Get) open import Generic using (mapMV ; mapMV-cong ; mapMV-purity ; sequenceV ; sequence-map ; VecISetoid) open import FinMap import CheckInsert open CheckInsert A import BFF -open BFF.VecBFF A using (assoc ; enumerate ; denumerate ; bff) +open BFF.PartialVecBFF A using (assoc ; enumerate ; denumerate ; bff) open Setoid using () renaming (_≈_ to _∋_≈_) open module A = DecSetoid A using (Carrier) renaming (_≟_ to deq) @@ -125,7 +128,7 @@ lemma-map-denumerate-enumerate (a ∷ as) = cong (_∷_ a) (begin as ∎) where open ≡-Reasoning -theorem-1 : (G : Get) → {m : ℕ} → (s : Vec Carrier m) → bff G s (Get.get G s) ≡ just s +theorem-1 : (G : Get) → {i : Setoid.Carrier (Get.I G)} → (s : Vec Carrier (to (Get.gl₁ G) ⟨$⟩ i)) → bff G s (Get.get G s) ≡ just s theorem-1 G s = begin bff G s (get s) ≡⟨ cong (bff G s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩ @@ -189,9 +192,8 @@ lemma-mapM-successful (x ∷ xs) () | just y | nothing | _ lemma-mapM-successful (x ∷ xs) p | just y | just ys | [ p′ ] with lemma-mapM-successful xs p′ lemma-mapM-successful (x ∷ xs) p | just y | just ys | [ p′ ] | w , pw = y ∷ w , cong (_∷_ (just y)) pw - -lemma-get-mapMV : {A B : Set} {f : A → Maybe B} {n : ℕ} {v : Vec A n} {r : Vec B n} → mapMV f v ≡ just r → (get : Get) → Get.get get <$> mapMV f v ≡ mapMV f (Get.get get v) -lemma-get-mapMV {f = f} {v = v} p G = let w , pw = lemma-mapM-successful v p in begin +lemma-get-mapMV : {A B : Set} {f : A → Maybe B} → (G : Get) → {i : Setoid.Carrier (Get.I G)} {v : Vec A (to (Get.gl₁ G) ⟨$⟩ i)} {r : Vec B (to (Get.gl₁ G) ⟨$⟩ i)} → mapMV f v ≡ just r → Get.get G <$> mapMV f v ≡ mapMV f (Get.get G v) +lemma-get-mapMV {f = f} G {v = v} p = let w , pw = lemma-mapM-successful v p in begin get <$> mapMV f v ≡⟨ cong (_<$>_ get) (sym (sequence-map f v)) ⟩ get <$> (sequenceV (map f v)) @@ -219,7 +221,7 @@ sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (just x≈y VecE sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (just x≈y VecEq.∷-cong xs≈ys) | nothing | nothing | nothing = Setoid.refl (MaybeSetoid (VecISetoid S at _)) sequence-cong {S} (nothing VecEq.∷-cong xs≈ys) = Setoid.refl (MaybeSetoid (VecISetoid S at _)) -theorem-2 : (G : Get) → {m : ℕ} → (v : Vec Carrier (Get.getlen G m)) → (s u : Vec Carrier m) → bff G s v ≡ just u → VecISetoid A.setoid at _ ∋ Get.get G u ≈ v +theorem-2 : (G : Get) → {i : Setoid.Carrier (Get.I G)} → (v : Vec Carrier (Get.gl₂ G ⟨$⟩ i)) → (s u : Vec Carrier (to (Get.gl₁ G) ⟨$⟩ i)) → bff G s v ≡ just u → VecISetoid A.setoid at _ ∋ Get.get G u ≈ v theorem-2 G v s u p with (lemma->>=-just ((flip union (delete-many (Get.get G (enumerate s)) (fromFunc (denumerate s)))) <$> (assoc (Get.get G (enumerate s)) v)) p) theorem-2 G v s u p | h′ , ph′ with (lemma-<$>-just (assoc (Get.get G (enumerate s)) v) ph′) theorem-2 G v s u p | h′ , ph′ | h , ph = drop-just (begin⟨ MaybeSetoid (VecISetoid A.setoid at _) ⟩ @@ -228,7 +230,7 @@ theorem-2 G v s u p | h′ , ph′ | h , ph = drop-just (begin⟨ MaybeSetoid (V get <$> (bff G s v) ≡⟨ cong (_<$>_ get ∘ flip _>>=_ h′↦r ∘ _<$>_ h↦h′) ph ⟩ get <$> mapMV (flip lookupM (h↦h′ h)) s′ - ≡⟨ lemma-get-mapMV (trans (cong (flip _>>=_ h′↦r ∘ _<$>_ h↦h′) (sym ph)) p) G ⟩ + ≡⟨ lemma-get-mapMV G (trans (cong (flip _>>=_ h′↦r ∘ _<$>_ h↦h′) (sym ph)) p) ⟩ mapMV (flip lookupM (h↦h′ h)) (get s′) ≡⟨ sym (sequence-map (flip lookupM (h↦h′ h)) (get s′)) ⟩ sequenceV (map (flip lookupM (h↦h′ h)) (get s′)) |