diff options
Diffstat (limited to 'Bidir.agda')
| -rw-r--r-- | Bidir.agda | 39 |
1 files changed, 21 insertions, 18 deletions
@@ -25,7 +25,7 @@ open import Relation.Binary using (Setoid ; module Setoid ; module DecSetoid) import Relation.Binary.EqReasoning as EqR import FreeTheorems -open FreeTheorems.VecVec using (get-type ; free-theorem) +open FreeTheorems.VecVec using (Get ; module Get) open import Generic using (mapMV ; mapMV-cong ; mapMV-purity ; sequenceV ; sequence-map ; VecISetoid) open import FinMap import CheckInsert @@ -125,13 +125,13 @@ lemma-map-denumerate-enumerate (a ∷ as) = cong (_∷_ a) (begin as ∎) where open ≡-Reasoning -theorem-1 : {getlen : ℕ → ℕ} → (get : get-type getlen) → {m : ℕ} → (s : Vec Carrier m) → bff get s (get s) ≡ just s -theorem-1 get s = begin - bff get s (get s) - ≡⟨ cong (bff get s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩ - bff get s (get (map (denumerate s) (enumerate s))) - ≡⟨ cong (bff get s) (free-theorem get (denumerate s) (enumerate s)) ⟩ - bff get s (map (denumerate s) (get (enumerate s))) +theorem-1 : (G : Get) → {m : ℕ} → (s : Vec Carrier m) → 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)) ⟩ + bff G s (get (map (denumerate s) (enumerate s))) + ≡⟨ cong (bff G s) (free-theorem (denumerate s) (enumerate s)) ⟩ + bff G s (map (denumerate s) (get (enumerate s))) ≡⟨ refl ⟩ (h′↦r ∘ h↦h′) (assoc (get (enumerate s)) (map (denumerate s) (get (enumerate s)))) ≡⟨ cong (h′↦r ∘ h↦h′) (lemma-1 (denumerate s) (get (enumerate s))) ⟩ @@ -151,6 +151,7 @@ theorem-1 get s = begin ≡⟨ cong just (lemma-map-denumerate-enumerate s) ⟩ just s ∎ where open ≡-Reasoning + open Get G h↦h′ = _<$>_ (flip union (delete-many (get (enumerate s)) (fromFunc (denumerate s)))) h′↦r = flip _>>=_ (flip mapMV (enumerate s) ∘ flip lookupVec) @@ -189,8 +190,8 @@ lemma-mapM-successful (x ∷ xs) p | just y | just ys | [ p′ ] with l lemma-mapM-successful (x ∷ xs) p | just y | just ys | [ p′ ] | w , pw = y ∷ w , cong (_∷_ (just y)) pw -lemma-get-mapMV : {A B : Set} {f : A → Maybe B} {n : ℕ} {v : Vec A n} {r : Vec B n} → mapMV f v ≡ just r → {getlen : ℕ → ℕ} (get : get-type getlen) → get <$> mapMV f v ≡ mapMV f (get v) -lemma-get-mapMV {f = f} {v = v} p get = let w , pw = lemma-mapM-successful v p in begin +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 <$> mapMV f v ≡ mapMV f (Get.get get v) +lemma-get-mapMV {f = f} {v = v} p G = let w , pw = lemma-mapM-successful v p in begin get <$> mapMV f v ≡⟨ cong (_<$>_ get) (sym (sequence-map f v)) ⟩ get <$> (sequenceV (map f v)) @@ -200,15 +201,16 @@ lemma-get-mapMV {f = f} {v = v} p get = let w , pw = lemma-mapM-successful v p i get <$> just w ≡⟨ sym (lemma-just-sequence (get w)) ⟩ sequenceV (map just (get w)) - ≡⟨ cong sequenceV (sym (free-theorem get just w)) ⟩ + ≡⟨ cong sequenceV (sym (free-theorem just w)) ⟩ sequenceV (get (map just w)) ≡⟨ cong (sequenceV ∘ get) (sym pw) ⟩ sequenceV (get (map f v)) - ≡⟨ cong sequenceV (free-theorem get f v) ⟩ + ≡⟨ cong sequenceV (free-theorem f v) ⟩ sequenceV (map f (get v)) ≡⟨ sequence-map f (get v) ⟩ mapMV f (get v) ∎ where open ≡-Reasoning + open Get G sequence-cong : {S : Setoid ℓ₀ ℓ₀} {n : ℕ} {m₁ m₂ : Setoid.Carrier (VecISetoid (MaybeSetoid S) at n)} → VecISetoid (MaybeSetoid S) at _ ∋ m₁ ≈ m₂ → MaybeSetoid (VecISetoid S at n) ∋ sequenceV m₁ ≈ sequenceV m₂ sequence-cong {S} VecEq.[]-cong = Setoid.refl (MaybeSetoid (VecISetoid S at _)) @@ -217,16 +219,16 @@ sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (just x≈y VecE sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (just x≈y VecEq.∷-cong xs≈ys) | nothing | nothing | nothing = Setoid.refl (MaybeSetoid (VecISetoid S at _)) sequence-cong {S} (nothing VecEq.∷-cong xs≈ys) = Setoid.refl (MaybeSetoid (VecISetoid S at _)) -theorem-2 : {getlen : ℕ → ℕ} (get : get-type getlen) → {m : ℕ} → (v : Vec Carrier (getlen m)) → (s u : Vec Carrier m) → bff get s v ≡ just u → VecISetoid A.setoid at _ ∋ get u ≈ v -theorem-2 get v s u p with (lemma->>=-just ((flip union (delete-many (get (enumerate s)) (fromFunc (denumerate s)))) <$> (assoc (get (enumerate s)) v)) p) -theorem-2 get v s u p | h′ , ph′ with (lemma-<$>-just (assoc (get (enumerate s)) v) ph′) -theorem-2 get v s u p | h′ , ph′ | h , ph = drop-just (begin⟨ MaybeSetoid (VecISetoid A.setoid at _) ⟩ +theorem-2 : (G : Get) → {m : ℕ} → (v : Vec Carrier (Get.getlen G m)) → (s u : Vec Carrier m) → bff G s v ≡ just u → VecISetoid A.setoid at _ ∋ Get.get G u ≈ v +theorem-2 G v s u p with (lemma->>=-just ((flip union (delete-many (Get.get G (enumerate s)) (fromFunc (denumerate s)))) <$> (assoc (Get.get G (enumerate s)) v)) p) +theorem-2 G v s u p | h′ , ph′ with (lemma-<$>-just (assoc (Get.get G (enumerate s)) v) ph′) +theorem-2 G v s u p | h′ , ph′ | h , ph = drop-just (begin⟨ MaybeSetoid (VecISetoid A.setoid at _) ⟩ get <$> (just u) ≡⟨ cong (_<$>_ get) (sym p) ⟩ - get <$> (bff get s 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 ∘ _<$>_ h↦h′) (sym ph)) p) get ⟩ + ≡⟨ lemma-get-mapMV (trans (cong (flip _>>=_ h′↦r ∘ _<$>_ h↦h′) (sym ph)) p) G ⟩ 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′)) @@ -237,6 +239,7 @@ theorem-2 get v s u p | h′ , ph′ | h , ph = drop-just (begin⟨ MaybeSetoid ≡⟨ lemma-just-sequence v ⟩ just v ∎) where open SetoidReasoning + open Get G s′ = enumerate s g = fromFunc (denumerate s) g′ = delete-many (get s′) g |