diff options
Diffstat (limited to 'Bidir.agda')
| -rw-r--r-- | Bidir.agda | 50 |
1 files changed, 25 insertions, 25 deletions
@@ -26,14 +26,14 @@ open FreeTheorems.VecVec using (get-type ; free-theorem) open import FinMap open import CheckInsert open import BFF using (_>>=_ ; fmap) -open BFF.VecBFF using (assoc ; enumerate ; denumerate ; bff) +open BFF.VecBFF Carrier deq using (assoc ; enumerate ; denumerate ; bff) -lemma-1 : {m n : ℕ} → (f : Fin n → Carrier) → (is : Vec (Fin n) m) → assoc deq is (map f is) ≡ just (restrict f (toList is)) +lemma-1 : {m n : ℕ} → (f : Fin n → Carrier) → (is : Vec (Fin n) m) → assoc is (map f is) ≡ just (restrict f (toList is)) lemma-1 f [] = refl lemma-1 f (i ∷ is′) = begin - assoc deq (i ∷ is′) (map f (i ∷ is′)) + assoc (i ∷ is′) (map f (i ∷ is′)) ≡⟨ refl ⟩ - assoc deq is′ (map f is′) >>= checkInsert deq i (f i) + assoc is′ (map f is′) >>= checkInsert deq i (f i) ≡⟨ cong (λ m → m >>= checkInsert deq i (f i)) (lemma-1 f is′) ⟩ just (restrict f (toList is′)) >>= (checkInsert deq i (f i)) ≡⟨ refl ⟩ @@ -41,8 +41,8 @@ lemma-1 f (i ∷ is′) = begin ≡⟨ lemma-checkInsert-restrict deq f i (toList is′) ⟩ just (restrict f (toList (i ∷ is′))) ∎ -lemma-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (x : Carrier) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc deq (i ∷ is) (x ∷ xs) ≡ just h → lookupM i h ≡ just x -lemma-lookupM-assoc i is x xs h p with assoc deq is xs +lemma-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (x : Carrier) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc (i ∷ is) (x ∷ xs) ≡ just h → lookupM i h ≡ just x +lemma-lookupM-assoc i is x xs h p with assoc is xs lemma-lookupM-assoc i is x xs h () | nothing lemma-lookupM-assoc i is x xs h p | just h' = apply-checkInsertProof deq i x h' record { same = λ lookupM≡justx → begin @@ -60,14 +60,14 @@ lemma-lookupM-assoc i is x xs h p | just h' = apply-checkInsertProof deq i x ; wrong = λ x' x≢x' lookupM≡justx' → lemma-just≢nothing (trans (sym p) (lemma-checkInsert-wrong deq i x h' x' x≢x' lookupM≡justx')) } -lemma-∉-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc deq is xs ≡ just h → (i ∉ toList is) → lookupM i h ≡ nothing +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 [] [] h ph i∉is = begin lookupM i h ≡⟨ cong (lookupM i) (sym (lemma-from-just ph)) ⟩ lookupM i empty ≡⟨ lemma-lookupM-empty i ⟩ nothing ∎ -lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is with assoc deq is' xs' | inspect (assoc deq is') xs' +lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is with assoc is' xs' | inspect (assoc is') xs' lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h () i∉is | nothing | [ ph' ] lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is | just h' | [ ph' ] = apply-checkInsertProof deq i' x' h' record { same = λ lookupM-i'-h'≡just-x' → begin @@ -90,9 +90,9 @@ lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is | just h' | [ ph _in-domain-of_ : {n : ℕ} {A : Set} → (is : List (Fin n)) → (FinMapMaybe n A) → Set _in-domain-of_ is h = All (λ i → ∃ λ x → lookupM i h ≡ just x) is -lemma-assoc-domain : {m n : ℕ} → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc deq is xs ≡ just h → (toList is) in-domain-of h +lemma-assoc-domain : {m n : ℕ} → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc is xs ≡ just h → (toList is) in-domain-of h lemma-assoc-domain [] [] h ph = Data.List.All.[] -lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h ph with assoc deq is' xs' | inspect (assoc deq is') xs' +lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h ph with assoc is' xs' | inspect (assoc is') xs' lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h () | nothing | [ ph' ] lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h ph | just h' | [ ph' ] = apply-checkInsertProof deq i' x' h' record { same = λ lookupM-i'-h'≡just-x' → Data.List.All._∷_ @@ -115,7 +115,7 @@ lemma-map-lookupM-insert i (i' ∷ is') x h i∉is ph = begin ≡⟨ cong (_∷_ (lookupM i' h)) (lemma-map-lookupM-insert i is' x h (i∉is ∘ there) (Data.List.All.tail ph)) ⟩ lookupM i' h ∷ map (flip lookupM h) is' ∎ -lemma-map-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (x : Carrier) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → (h' : FinMapMaybe n Carrier) → assoc deq is xs ≡ just h' → checkInsert deq i x h' ≡ just h → map (flip lookupM h) is ≡ map (flip lookupM h') is +lemma-map-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (x : Carrier) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → (h' : FinMapMaybe n Carrier) → assoc is xs ≡ just h' → checkInsert deq i x h' ≡ just h → map (flip lookupM h) is ≡ map (flip lookupM h') is lemma-map-lookupM-assoc i [] x [] h h' ph' ph = refl lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' ph with any (_≟_ i) (toList (i' ∷ is')) lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p with Data.List.All.lookup (lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h' ph') p @@ -130,16 +130,16 @@ lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | no ¬p rewri ≡⟨ lemma-map-lookupM-insert i (i' ∷ is') x h' ¬p (lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h' ph') ⟩ map (flip lookupM h') (i' ∷ is') ∎ -lemma-2 : {m n : ℕ} → (is : Vec (Fin n) m) → (v : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc deq is v ≡ just h → map (flip lookupM h) is ≡ map just v +lemma-2 : {m n : ℕ} → (is : Vec (Fin n) m) → (v : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc is v ≡ just h → map (flip lookupM h) is ≡ map just v lemma-2 [] [] h p = refl -lemma-2 (i ∷ is) (x ∷ xs) h p with assoc deq is xs | inspect (assoc deq is) xs +lemma-2 (i ∷ is) (x ∷ xs) h p with assoc is xs | inspect (assoc is) xs lemma-2 (i ∷ is) (x ∷ xs) h () | nothing | _ lemma-2 (i ∷ is) (x ∷ xs) h p | just h' | [ ir ] = begin map (flip lookupM h) (i ∷ is) ≡⟨ refl ⟩ lookupM i h ∷ map (flip lookupM h) is ≡⟨ cong (flip _∷_ (map (flip lookupM h) is)) (lemma-lookupM-assoc i is x xs h (begin - assoc deq (i ∷ is) (x ∷ xs) + assoc (i ∷ is) (x ∷ xs) ≡⟨ cong (flip _>>=_ (checkInsert deq i x)) ir ⟩ checkInsert deq i x h' ≡⟨ p ⟩ @@ -152,7 +152,7 @@ lemma-2 (i ∷ is) (x ∷ xs) h p | just h' | [ ir ] = begin ≡⟨ refl ⟩ map just (x ∷ xs) ∎ -lemma-map-denumerate-enumerate : {m : ℕ} {A : Set} → (as : Vec A m) → map (denumerate as) (enumerate as) ≡ as +lemma-map-denumerate-enumerate : {m : ℕ} → (as : Vec Carrier m) → map (denumerate as) (enumerate as) ≡ as lemma-map-denumerate-enumerate [] = refl lemma-map-denumerate-enumerate (a ∷ as) = cong (_∷_ a) (begin map (flip lookupVec (a ∷ as)) (tabulate Fin.suc) @@ -167,15 +167,15 @@ lemma-map-denumerate-enumerate (a ∷ as) = cong (_∷_ a) (begin ≡⟨ lemma-map-denumerate-enumerate as ⟩ as ∎) -theorem-1 : {getlen : ℕ → ℕ} → (get : get-type getlen) → {m : ℕ} → (s : Vec Carrier m) → bff get deq s (get s) ≡ just s +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 deq s (get s) - ≡⟨ cong (bff get deq s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩ - bff get deq s (get (map (denumerate s) (enumerate s))) - ≡⟨ cong (bff get deq s) (free-theorem get (denumerate s) (enumerate s)) ⟩ - bff get deq s (map (denumerate s) (get (enumerate s))) + 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))) ≡⟨ refl ⟩ - fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (denumerate s))) (assoc deq (get (enumerate s)) (map (denumerate s) (get (enumerate s))))) + fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (denumerate s))) (assoc (get (enumerate s)) (map (denumerate s) (get (enumerate s))))) ≡⟨ cong (fmap (flip map (enumerate s) ∘ flip lookup) ∘ fmap (flip union (fromFunc (denumerate s)))) (lemma-1 (denumerate s) (get (enumerate s))) ⟩ fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (flip lookupVec s))) (just (restrict (denumerate s) (toList (get (enumerate s)))))) ≡⟨ refl ⟩ @@ -220,8 +220,8 @@ lemma-union-not-used h h' (i ∷ is') p | x , lookupM-i-h≡just-x = begin ≡⟨ cong (_∷_ (lookupM i h)) (lemma-union-not-used h h' is' (Data.List.All.tail p)) ⟩ lookupM i h ∷ map (flip lookupM h) is' ∎ -theorem-2 : {getlen : ℕ → ℕ} (get : get-type getlen) → {m : ℕ} → (v : Vec Carrier (getlen m)) → (s u : Vec Carrier m) → bff get deq s v ≡ just u → get u ≡ v -theorem-2 get v s u p with lemma-fmap-just (assoc deq (get (enumerate s)) v) (proj₂ (lemma-fmap-just (fmap (flip union (fromFunc (denumerate s))) (assoc deq (get (enumerate s)) v)) p)) +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 → get u ≡ v +theorem-2 get v s u p with lemma-fmap-just (assoc (get (enumerate s)) v) (proj₂ (lemma-fmap-just (fmap (flip union (fromFunc (denumerate s))) (assoc (get (enumerate s)) v)) p)) theorem-2 get v s u p | h , ph = begin get u ≡⟨ lemma-from-just (begin @@ -229,7 +229,7 @@ theorem-2 get v s u p | h , ph = begin ≡⟨ refl ⟩ fmap get (just u) ≡⟨ cong (fmap get) (sym p) ⟩ - fmap get (bff get deq s v) + fmap get (bff get s v) ≡⟨ cong (fmap get ∘ fmap (flip map (enumerate s) ∘ flip lookup) ∘ fmap (flip union (fromFunc (denumerate s)))) ph ⟩ fmap get (fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (denumerate s))) (just h))) ≡⟨ refl ⟩ |