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Diffstat (limited to 'Precond.agda')
| -rw-r--r-- | Precond.agda | 71 |
1 files changed, 71 insertions, 0 deletions
diff --git a/Precond.agda b/Precond.agda new file mode 100644 index 0000000..f856c57 --- /dev/null +++ b/Precond.agda @@ -0,0 +1,71 @@ +module Precond where + +open import Data.Nat using (ℕ) renaming (zero to nzero ; suc to nsuc) +open import Data.Fin using (Fin ; zero ; suc) +open import Data.Vec using (Vec ; [] ; _∷_ ; map ; lookup ; toList) +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 (∃ ; _,_) +open import Function using (flip ; _∘_) +open import Relation.Binary.Core using (_≡_ ; _≢_) +open import Relation.Binary.PropositionalEquality using (refl ; cong) +open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎) + +open import FinMap using (FinMap ; FinMapMaybe ; union ; fromFunc ; empty ; insert) +open import CheckInsert using (EqInst ; checkInsert ; lemma-checkInsert-new) +open import BFF using (fmap ; _>>=_) +open import Bidir using (lemma-∉-lookupM-assoc) + +open BFF.VecBFF using (assoc ; enumerate ; denumerate ; bff) + +assoc-enough : {getlen : ℕ → ℕ} (get : {A : Set} {n : ℕ} → Vec A n → Vec A (getlen n)) → {B : Set} {m : ℕ} → (eq : EqInst B) → (s : Vec B m) → (v : Vec B (getlen m)) → (h : FinMapMaybe m B) → assoc eq (get (enumerate s)) v ≡ just h → ∃ λ u → bff get eq s v ≡ just u +assoc-enough get {B} {m} eq s v h p = map (flip lookup (union h g)) s′ , (begin + bff get eq s v + ≡⟨ refl ⟩ + fmap (flip map s′ ∘ flip lookup) (fmap (flip union g) (assoc eq (get s′) v)) + ≡⟨ cong (fmap (flip map s′ ∘ flip lookup)) (cong (fmap (flip union g)) p) ⟩ + fmap (flip map s′ ∘ flip lookup) (fmap (flip union g) (just h)) + ≡⟨ refl ⟩ + just (map (flip lookup (union h g)) s′) ∎) + where s′ : Vec (Fin m) m + s′ = enumerate s + g : FinMap m B + g = fromFunc (denumerate s) + +all-different : {A : Set} {n : ℕ} → Vec A n → Set +all-different {_} {n} v = (i : Fin n) → (j : Fin n) → i ≢ j → lookup i v ≢ lookup j v + +suc-≡ : {n : ℕ} {i j : Fin n} → (suc i ≡ suc j) → i ≡ j +suc-≡ refl = refl + +different-swap : {A : Set} {n : ℕ} → (a b : A) → (v : Vec A n) → all-different (a ∷ b ∷ v) → all-different (b ∷ a ∷ v) +different-swap a b v p zero zero i≢j li≡lj = i≢j refl +different-swap a b v p zero (suc zero) i≢j li≡lj = p (suc zero) zero (λ ()) li≡lj +different-swap a b v p zero (suc (suc j)) i≢j li≡lj = p (suc zero) (suc (suc j)) (λ ()) li≡lj +different-swap a b v p (suc zero) zero i≢j li≡lj = p zero (suc zero) (λ ()) li≡lj +different-swap a b v p (suc zero) (suc zero) i≢j li≡lj = i≢j refl +different-swap a b v p (suc zero) (suc (suc j)) i≢j li≡lj = p zero (suc (suc j)) (λ ()) li≡lj +different-swap a b v p (suc (suc i)) zero i≢j li≡lj = p (suc (suc i)) (suc zero) (λ ()) li≡lj +different-swap a b v p (suc (suc i)) (suc zero) i≢j li≡lj = p (suc (suc i)) zero (λ ()) li≡lj +different-swap a b v p (suc (suc i)) (suc (suc j)) i≢j li≡lj = p (suc (suc i)) (suc (suc j)) i≢j li≡lj + +different-drop : {A : Set} {n : ℕ} → (a : A) → (v : Vec A n) → all-different (a ∷ v) → all-different v +different-drop a v p i j i≢j = p (suc i) (suc j) (i≢j ∘ suc-≡) + +different-∉ : {A : Set} {n : ℕ} → (x : A) (xs : Vec A n) → all-different (x ∷ xs) → x ∉ (toList xs) +different-∉ x [] p () +different-∉ x (y ∷ ys) p (here px) = p zero (suc zero) (λ ()) px +different-∉ x (y ∷ ys) p (there pxs) = different-∉ x ys (different-drop y (x ∷ ys) (different-swap x y ys p)) pxs + +different-assoc : {B : Set} {m n : ℕ} → (eq : EqInst B) → (u : Vec (Fin n) m) → (v : Vec B m) → all-different u → ∃ λ h → assoc eq u v ≡ just h +different-assoc eq [] [] p = empty , refl +different-assoc eq (u ∷ us) (v ∷ vs) p with different-assoc eq us vs (λ i j i≢j → p (suc i) (suc j) (i≢j ∘ suc-≡)) +different-assoc eq (u ∷ us) (v ∷ vs) p | h , p' = insert u v h , (begin + assoc eq (u ∷ us) (v ∷ vs) + ≡⟨ refl ⟩ + assoc eq us vs >>= checkInsert eq u v + ≡⟨ cong (flip _>>=_ (checkInsert eq u v)) p' ⟩ + checkInsert eq u v h + ≡⟨ lemma-checkInsert-new eq u v h (lemma-∉-lookupM-assoc eq u us vs h p' (different-∉ u us p)) ⟩ + just (insert u v h) ∎)
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