module Bidir where open import Data.Nat using (ℕ) open import Data.Fin using (Fin) open import Data.Fin.Props using (_≟_) open import Data.Maybe using (Maybe ; nothing ; just ; maybe′) open import Data.List using (List) open import Data.List.Any using (Any ; any ; here ; there) open import Data.List.All using (All) open Data.List.Any.Membership-≡ using (_∈_ ; _∉_) open import Data.Vec using (Vec ; [] ; _∷_ ; toList ; fromList ; map ; tabulate) renaming (lookup to lookupVec) open import Data.Vec.Properties using (tabulate-∘ ; lookup∘tabulate ; map-cong ; map-∘) open import Data.Product using (∃ ; _×_ ; _,_ ; proj₁ ; proj₂) open import Data.Empty using (⊥-elim) open import Function using (id ; _∘_ ; flip) open import Relation.Nullary using (yes ; no ; ¬_) open import Relation.Nullary.Negation using (contradiction) open import Relation.Binary.Core using (_≡_ ; refl) open import Relation.Binary.PropositionalEquality using (cong ; sym ; inspect ; Reveal_is_ ; _≗_ ; trans) open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎) open import FinMap open import CheckInsert open import BFF using (_>>=_ ; fmap) open BFF.VecBFF using (assoc ; enumerate ; denumerate ; bff) lemma-1 : {τ : Set} {m n : ℕ} → (eq : EqInst τ) → (f : Fin n → τ) → (is : Vec (Fin n) m) → assoc eq is (map f is) ≡ just (restrict f (toList is)) lemma-1 eq f [] = refl lemma-1 eq f (i ∷ is′) = begin assoc eq (i ∷ is′) (map f (i ∷ is′)) ≡⟨ refl ⟩ assoc eq is′ (map f is′) >>= checkInsert eq i (f i) ≡⟨ cong (λ m → m >>= checkInsert eq i (f i)) (lemma-1 eq f is′) ⟩ just (restrict f (toList is′)) >>= (checkInsert eq i (f i)) ≡⟨ refl ⟩ checkInsert eq i (f i) (restrict f (toList is′)) ≡⟨ lemma-checkInsert-restrict eq f i (toList is′) ⟩ just (restrict f (toList (i ∷ is′))) ∎ lemma-lookupM-assoc : {A : Set} {m n : ℕ} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (x : A) → (xs : Vec A m) → (h : FinMapMaybe n A) → assoc eq (i ∷ is) (x ∷ xs) ≡ just h → lookupM i h ≡ just x lemma-lookupM-assoc eq i is x xs h p with assoc eq is xs lemma-lookupM-assoc eq i is x xs h () | nothing lemma-lookupM-assoc eq i is x xs h p | just h' = apply-checkInsertProof eq i x h' record { same = λ lookupM≡justx → begin lookupM i h ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym p) (lemma-checkInsert-same eq i x h' lookupM≡justx))) ⟩ lookupM i h' ≡⟨ lookupM≡justx ⟩ just x ∎ ; new = λ lookupM≡nothing → begin lookupM i h ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym p) (lemma-checkInsert-new eq i x h' lookupM≡nothing))) ⟩ lookupM i (insert i x h') ≡⟨ lemma-lookupM-insert i x h' ⟩ just x ∎ ; wrong = λ x' x≢x' lookupM≡justx' → lemma-just≢nothing (trans (sym p) (lemma-checkInsert-wrong eq i x h' x' x≢x' lookupM≡justx')) } lemma-∉-lookupM-assoc : {A : Set} {m n : ℕ} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (xs : Vec A m) → (h : FinMapMaybe n A) → assoc eq is xs ≡ just h → (i ∉ toList is) → lookupM i h ≡ nothing lemma-∉-lookupM-assoc eq 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 eq i (i' ∷ is') (x' ∷ xs') h ph i∉is with i ≟ i' lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h ph i∉is | yes p = contradiction (here p) i∉is lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h ph i∉is | no ¬p with assoc eq is' xs' | inspect (assoc eq is') xs' lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h () i∉is | no ¬p | nothing | Reveal_is_.[_] ph' lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h ph i∉is | no ¬p | just h' | Reveal_is_.[_] ph' = apply-checkInsertProof eq i' x' h' record { same = λ lookupM-i'-h'≡just-x' → begin lookupM i h ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym ph) (lemma-checkInsert-same eq i' x' h' lookupM-i'-h'≡just-x'))) ⟩ lookupM i h' ≡⟨ lemma-∉-lookupM-assoc eq i is' xs' h' ph' (i∉is ∘ there) ⟩ nothing ∎ ; new = λ lookupM-i'-h'≡nothing → begin lookupM i h ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym ph) (lemma-checkInsert-new eq i' x' h' lookupM-i'-h'≡nothing))) ⟩ lookupM i (insert i' x' h') ≡⟨ sym (lemma-lookupM-insert-other i i' x' h' ¬p) ⟩ lookupM i h' ≡⟨ lemma-∉-lookupM-assoc eq i is' xs' h' ph' (i∉is ∘ there) ⟩ nothing ∎ ; wrong = λ x'' x'≢x'' lookupM-i'-h'≡just-x'' → lemma-just≢nothing (trans (sym ph) (lemma-checkInsert-wrong eq i' x' h' x'' x'≢x'' lookupM-i'-h'≡just-x'')) } _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} {A : Set} → (eq : EqInst A) → (is : Vec (Fin n) m) → (xs : Vec A m) → (h : FinMapMaybe n A) → assoc eq is xs ≡ just h → (toList is) in-domain-of h lemma-assoc-domain eq [] [] h ph = Data.List.All.[] lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h ph with assoc eq is' xs' | inspect (assoc eq is') xs' lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h () | nothing | ph' lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h ph | just h' | Reveal_is_.[_] ph' = apply-checkInsertProof eq i' x' h' record { same = λ lookupM-i'-h'≡just-x' → Data.List.All._∷_ (x' , (trans (cong (lookupM i') (lemma-from-just (trans (sym ph) (lemma-checkInsert-same eq i' x' h' lookupM-i'-h'≡just-x')))) lookupM-i'-h'≡just-x')) (lemma-assoc-domain eq is' xs' h (trans ph' (trans (sym (lemma-checkInsert-same eq i' x' h' lookupM-i'-h'≡just-x')) ph))) ; new = λ lookupM-i'-h'≡nothing → Data.List.All._∷_ (x' , (trans (cong (lookupM i') (lemma-from-just (trans (sym ph) (lemma-checkInsert-new eq i' x' h' lookupM-i'-h'≡nothing)))) (lemma-lookupM-insert i' x' h'))) (Data.List.All.map (λ {i} p → proj₁ p , lemma-lookupM-checkInsert eq i i' (proj₁ p) x' h' h (proj₂ p) ph) (lemma-assoc-domain eq is' xs' h' ph')) ; wrong = λ x'' x'≢x'' lookupM-i'-h'≡just-x'' → lemma-just≢nothing (trans (sym ph) (lemma-checkInsert-wrong eq i' x' h' x'' x'≢x'' lookupM-i'-h'≡just-x'')) } lemma-map-lookupM-insert : {m n : ℕ} {A : Set} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (x : A) → (h : FinMapMaybe n A) → ¬(i ∈ (toList is)) → (toList is) in-domain-of h → map (flip lookupM (insert i x h)) is ≡ map (flip lookupM h) is lemma-map-lookupM-insert eq i [] x h i∉is ph = refl lemma-map-lookupM-insert eq i (i' ∷ is') x h i∉is ph = begin lookupM i' (insert i x h) ∷ map (flip lookupM (insert i x h)) is' ≡⟨ cong (flip _∷_ (map (flip lookupM (insert i x h)) is')) (sym (lemma-lookupM-insert-other i' i x h (i∉is ∘ here ∘ sym))) ⟩ lookupM i' h ∷ map (flip lookupM (insert i x h)) is' ≡⟨ cong (_∷_ (lookupM i' h)) (lemma-map-lookupM-insert eq 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} {A : Set} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (x : A) → (xs : Vec A m) → (h : FinMapMaybe n A) → (h' : FinMapMaybe n A) → assoc eq is xs ≡ just h' → checkInsert eq i x h' ≡ just h → map (flip lookupM h) is ≡ map (flip lookupM h') is lemma-map-lookupM-assoc eq i [] x [] h h' ph' ph = refl lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph with any (_≟_ i) (toList (i' ∷ is')) lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p with Data.List.All.lookup (lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h' ph') p lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p | (x'' , p') with lookupM i h' lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p | (x'' , refl) | .(just x'') with eq x x'' lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h .h ph' refl | yes p | (.x , refl) | .(just x) | yes refl = refl lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' () | yes p | (x'' , refl) | .(just x'') | no ¬p lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | no ¬p with lookupM i h' | lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h' ph' ¬p lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | no ¬p | .nothing | refl = begin map (flip lookupM h) (i' ∷ is') ≡⟨ map-cong (λ i'' → cong (lookupM i'') (lemma-from-just (sym ph))) (i' ∷ is') ⟩ map (flip lookupM (insert i x h')) (i' ∷ is') ≡⟨ lemma-map-lookupM-insert eq i (i' ∷ is') x h' ¬p (lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h' ph') ⟩ map (flip lookupM h') (i' ∷ is') ∎ lemma-2 : {τ : Set} {m n : ℕ} → (eq : EqInst τ) → (is : Vec (Fin n) m) → (v : Vec τ m) → (h : FinMapMaybe n τ) → assoc eq is v ≡ just h → map (flip lookupM h) is ≡ map just v lemma-2 eq [] [] h p = refl lemma-2 eq (i ∷ is) (x ∷ xs) h p with assoc eq is xs | inspect (assoc eq is) xs lemma-2 eq (i ∷ is) (x ∷ xs) h () | nothing | _ lemma-2 eq (i ∷ is) (x ∷ xs) h p | just h' | Reveal_is_.[_] 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 eq i is x xs h (begin assoc eq (i ∷ is) (x ∷ xs) ≡⟨ cong (flip _>>=_ (checkInsert eq i x)) ir ⟩ checkInsert eq i x h' ≡⟨ p ⟩ just h ∎) ) ⟩ just x ∷ map (flip lookupM h) is ≡⟨ cong (_∷_ (just x)) (lemma-map-lookupM-assoc eq i is x xs h h' ir p) ⟩ just x ∷ map (flip lookupM h') is ≡⟨ cong (_∷_ (just x)) (lemma-2 eq is xs h' ir) ⟩ just x ∷ map just xs ≡⟨ refl ⟩ map just (x ∷ xs) ∎ postulate free-theorem-list-list : {β γ : Set} → {getlen : ℕ → ℕ} → (get : {α : Set} {n : ℕ} → Vec α n → Vec α (getlen n)) → (f : β → γ) → {m : ℕ } → (v : Vec β m) → get (map f v) ≡ map f (get v) lemma-map-denumerate-enumerate : {m : ℕ} {A : Set} → (as : Vec A 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) ≡⟨ cong (map (flip lookupVec (a ∷ as))) (tabulate-∘ Fin.suc id) ⟩ map (flip lookupVec (a ∷ as)) (map Fin.suc (tabulate id)) ≡⟨ refl ⟩ map (flip lookupVec (a ∷ as)) (map Fin.suc (enumerate as)) ≡⟨ sym (map-∘ _ _ (enumerate as)) ⟩ map (flip lookupVec (a ∷ as) ∘ Fin.suc) (enumerate as) ≡⟨ refl ⟩ map (denumerate as) (enumerate as) ≡⟨ lemma-map-denumerate-enumerate as ⟩ as ∎) theorem-1 : {getlen : ℕ → ℕ} → (get : {α : Set} {n : ℕ} → Vec α n → Vec α (getlen n)) → {m : ℕ} {τ : Set} → (eq : EqInst τ) → (s : Vec τ m) → bff get eq s (get s) ≡ just s theorem-1 get eq s = begin bff get eq s (get s) ≡⟨ cong (bff get eq s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩ bff get eq s (get (map (denumerate s) (enumerate s))) ≡⟨ cong (bff get eq s) (free-theorem-list-list get (denumerate s) (enumerate s)) ⟩ bff get eq s (map (denumerate s) (get (enumerate s))) ≡⟨ refl ⟩ fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (denumerate s))) (assoc eq (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 eq (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 ⟩ just ((flip map (enumerate s) ∘ flip lookup) (union (restrict (denumerate s) (toList (get (enumerate s)))) (fromFunc (denumerate s)))) ≡⟨ cong just (cong (flip map (enumerate s) ∘ flip lookup) (lemma-union-restrict (denumerate s) (toList (get (enumerate s))))) ⟩ just ((flip map (enumerate s) ∘ flip lookup) (fromFunc (denumerate s))) ≡⟨ refl ⟩ just (map (flip lookup (fromFunc (denumerate s))) (enumerate s)) ≡⟨ cong just (map-cong (lookup∘tabulate (denumerate s)) (enumerate s)) ⟩ just (map (denumerate s) (enumerate s)) ≡⟨ cong just (lemma-map-denumerate-enumerate s) ⟩ just s ∎ lemma-fmap-just : {A B : Set} → {f : A → B} {b : B} → (ma : Maybe A) → fmap f ma ≡ just b → ∃ λ a → ma ≡ just a lemma-fmap-just (just x) fmap-f-ma≡just-b = x , refl lemma-fmap-just nothing () ∷-injective : {A : Set} {n : ℕ} {x y : A} {xs ys : Vec A n} → (x ∷ xs) ≡ (y ∷ ys) → x ≡ y × xs ≡ ys ∷-injective refl = refl , refl lemma-from-map-just : {A : Set} {n : ℕ} → {xs ys : Vec A n} → map Maybe.just xs ≡ map Maybe.just ys → xs ≡ ys lemma-from-map-just {xs = []} {ys = []} p = refl lemma-from-map-just {xs = x ∷ xs'} {ys = y ∷ ys'} p with ∷-injective p lemma-from-map-just {xs = x ∷ xs'} {ys = .x ∷ ys'} p | refl , p' = cong (_∷_ x) (lemma-from-map-just p') lemma-union-not-used : {m n : ℕ} {A : Set} (h : FinMapMaybe n A) → (h' : FinMap n A) → (is : Vec (Fin n) m) → (toList is) in-domain-of h → map just (map (flip lookup (union h h')) is) ≡ map (flip lookupM h) is lemma-union-not-used h h' [] p = refl lemma-union-not-used h h' (i ∷ is') p with Data.List.All.head p lemma-union-not-used h h' (i ∷ is') p | x , lookupM-i-h≡just-x = begin just (lookup i (union h h')) ∷ map just (map (flip lookup (union h h')) is') ≡⟨ cong (flip _∷_ (map just (map (flip lookup (union h h')) is'))) (begin just (lookup i (union h h')) ≡⟨ cong just (lookup∘tabulate (λ j → maybe′ id (lookup j h') (lookupM j h)) i) ⟩ just (maybe′ id (lookup i h') (lookupM i h)) ≡⟨ cong just (cong (maybe′ id (lookup i h')) lookupM-i-h≡just-x) ⟩ just (maybe′ id (lookup i h') (just x)) ≡⟨ refl ⟩ just x ≡⟨ sym lookupM-i-h≡just-x ⟩ lookupM i h ∎) ⟩ lookupM i h ∷ map just (map (flip lookup (union h h')) is') ≡⟨ 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 : {α : Set} {n : ℕ} → Vec α n → Vec α (getlen n)) → {τ : Set} {m : ℕ} → (eq : EqInst τ) → (v : Vec τ (getlen m)) → (s u : Vec τ m) → bff get eq s v ≡ just u → get u ≡ v theorem-2 get eq v s u p with lemma-fmap-just (assoc eq (get (enumerate s)) v) (proj₂ (lemma-fmap-just (fmap (flip union (fromFunc (denumerate s))) (assoc eq (get (enumerate s)) v)) p)) theorem-2 get eq v s u p | h , ph = begin get u ≡⟨ lemma-from-just (begin just (get u) ≡⟨ refl ⟩ fmap get (just u) ≡⟨ cong (fmap get) (sym p) ⟩ fmap get (bff get eq 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 ⟩ just (get (map (flip lookup (union h (fromFunc (denumerate s)))) (enumerate s))) ∎) ⟩ get (map (flip lookup (union h (fromFunc (denumerate s)))) (enumerate s)) ≡⟨ free-theorem-list-list get (flip lookup (union h (fromFunc (denumerate s)))) (enumerate s) ⟩ map (flip lookup (union h (fromFunc (denumerate s)))) (get (enumerate s)) ≡⟨ lemma-from-map-just (begin map just (map (flip lookup (union h (fromFunc (denumerate s)))) (get (enumerate s))) ≡⟨ lemma-union-not-used h (fromFunc (denumerate s)) (get (enumerate s)) (lemma-assoc-domain eq (get (enumerate s)) v h ph) ⟩ map (flip lookupM h) (get (enumerate s)) ≡⟨ lemma-2 eq (get (enumerate s)) v h ph ⟩ map just v ∎) ⟩ v ∎