diff options
| -rw-r--r-- | Bidir.agda | 89 | ||||
| -rw-r--r-- | FinMap.agda | 97 |
2 files changed, 99 insertions, 87 deletions
@@ -1,5 +1,4 @@ module Bidir where - open import Data.Bool hiding (_≟_) open import Data.Nat open import Data.Fin @@ -15,44 +14,14 @@ open import Relation.Binary.Core open import Relation.Binary.PropositionalEquality open Relation.Binary.PropositionalEquality.≡-Reasoning +open import FinMap + _>>=_ : {A B : Set} → Maybe A → (A → Maybe B) → Maybe B _>>=_ = flip (flip maybe′ nothing) fmap : {A B : Set} → (A → B) → Maybe A → Maybe B fmap f = maybe′ (λ a → just (f a)) nothing -module FinMap where - - FinMapMaybe : ℕ → Set → Set - FinMapMaybe n A = Vec (Maybe A) n - - lookupM : {A : Set} {n : ℕ} → Fin n → FinMapMaybe n A → Maybe A - lookupM = lookupVec - - insert : {A : Set} {n : ℕ} → Fin n → A → FinMapMaybe n A → FinMapMaybe n A - insert f a m = m [ f ]≔ (just a) - - empty : {A : Set} {n : ℕ} → FinMapMaybe n A - empty = replicate nothing - - fromAscList : {A : Set} {n : ℕ} → List (Fin n × A) → FinMapMaybe n A - fromAscList [] = empty - fromAscList ((f , a) ∷ xs) = insert f a (fromAscList xs) - - FinMap : ℕ → Set → Set - FinMap n A = Vec A n - - lookup : {A : Set} {n : ℕ} → Fin n → FinMap n A → A - lookup = lookupVec - - fromFunc : {A : Set} {n : ℕ} → (Fin n → A) → FinMap n A - fromFunc = tabulate - - union : {A : Set} {n : ℕ} → FinMapMaybe n A → FinMap n A → FinMap n A - union m1 m2 = tabulate (λ f → maybe′ id (lookup f m2) (lookupM f m1)) - -open FinMap - EqInst : Set → Set EqInst A = (x y : A) → Dec (x ≡ y) @@ -62,65 +31,11 @@ checkInsert eq i b m | just c with eq b c checkInsert eq i b m | just .b | yes refl = just m checkInsert eq i b m | just c | no ¬p = nothing checkInsert eq i b m | nothing = just (insert i b m) - assoc : {A : Set} {n : ℕ} → EqInst A → List (Fin n) → List A → Maybe (FinMapMaybe n A) assoc _ [] [] = just empty assoc eq (i ∷ is) (b ∷ bs) = (assoc eq is bs) >>= (checkInsert eq i b) assoc _ _ _ = nothing -generate : {A : Set} {n : ℕ} → (Fin n → A) → List (Fin n) → FinMapMaybe n A -generate f is = fromAscList (zip is (map f is)) - -lemma-insert-same : {τ : Set} {n : ℕ} → (m : FinMapMaybe n τ) → (f : Fin n) → (a : τ) → lookupM f m ≡ just a → m ≡ insert f a m -lemma-insert-same [] () a p -lemma-insert-same (.(just a) ∷ xs) zero a refl = refl -lemma-insert-same (x ∷ xs) (suc i) a p = cong (_∷_ x) (lemma-insert-same xs i a p) - -lemma-lookupM-empty : {A : Set} {n : ℕ} → (i : Fin n) → lookupM {A} i empty ≡ nothing -lemma-lookupM-empty zero = refl -lemma-lookupM-empty (suc i) = lemma-lookupM-empty i - -lemma-from-just : {A : Set} → {x y : A} → _≡_ {_} {Maybe A} (just x) (just y) → x ≡ y -lemma-from-just refl = refl - -lemma-lookupM-insert : {A : Set} {n : ℕ} → (i : Fin n) → (a : A) → (m : FinMapMaybe n A) → lookupM i (insert i a m) ≡ just a -lemma-lookupM-insert zero _ (_ ∷ _) = refl -lemma-lookupM-insert (suc i) a (_ ∷ xs) = lemma-lookupM-insert i a xs - -lemma-lookupM-insert-other : {A : Set} {n : ℕ} → (i j : Fin n) → (a : A) → (m : FinMapMaybe n A) → ¬(i ≡ j) → lookupM i m ≡ lookupM i (insert j a m) -lemma-lookupM-insert-other zero zero a m p = contradiction refl p -lemma-lookupM-insert-other zero (suc j) a (x ∷ xs) p = refl -lemma-lookupM-insert-other (suc i) zero a (x ∷ xs) p = refl -lemma-lookupM-insert-other (suc i) (suc j) a (x ∷ xs) p = lemma-lookupM-insert-other i j a xs (contraposition (cong suc) p) - -lemma-lookupM-generate : {A : Set} {n : ℕ} → (i : Fin n) → (f : Fin n → A) → (is : List (Fin n)) → (a : A) → lookupM i (generate f is) ≡ just a → f i ≡ a -lemma-lookupM-generate {A} i f [] a p with begin - just a - ≡⟨ sym p ⟩ - lookupM i (generate f []) - ≡⟨ refl ⟩ - lookupM i empty - ≡⟨ lemma-lookupM-empty i ⟩ - nothing ∎ -lemma-lookupM-generate i f [] a p | () -lemma-lookupM-generate i f (i' ∷ is) a p with i ≟F i' -lemma-lookupM-generate i f (.i ∷ is) a p | yes refl = lemma-from-just (begin - just (f i) - ≡⟨ sym (lemma-lookupM-insert i (f i) (generate f is)) ⟩ - lookupM i (insert i (f i) (generate f is)) - ≡⟨ refl ⟩ - lookupM i (generate f (i ∷ is)) - ≡⟨ p ⟩ - just a ∎) -lemma-lookupM-generate i f (i' ∷ is) a p | no ¬p2 = lemma-lookupM-generate i f is a (begin - lookupM i (generate f is) - ≡⟨ lemma-lookupM-insert-other i i' (f i') (generate f is) ¬p2 ⟩ - lookupM i (insert i' (f i') (generate f is)) - ≡⟨ refl ⟩ - lookupM i (generate f (i' ∷ is)) - ≡⟨ p ⟩ - just a ∎) - lemma-checkInsert-generate : {τ : Set} {n : ℕ} → (eq : EqInst τ) → (f : Fin n → τ) → (i : Fin n) → (is : List (Fin n)) → checkInsert eq i (f i) (generate f is) ≡ just (generate f (i ∷ is)) lemma-checkInsert-generate eq f i is with lookupM i (generate f is) | inspect (lookupM i) (generate f is) lemma-checkInsert-generate eq f i is | nothing | _ = refl diff --git a/FinMap.agda b/FinMap.agda new file mode 100644 index 0000000..a099e4f --- /dev/null +++ b/FinMap.agda @@ -0,0 +1,97 @@ +module FinMap where + +open import Data.Nat using (ℕ) +open import Data.Maybe using (Maybe ; just ; nothing ; maybe′) +open import Data.Fin using (Fin ; zero ; suc) +open import Data.Fin.Props using (_≟_) +open import Data.Vec using (Vec ; [] ; _∷_ ; _[_]≔_ ; replicate ; tabulate) renaming (lookup to lookupVec) +open import Data.List using (List ; [] ; _∷_ ; map ; zip) +open import Data.Product using (_×_ ; _,_) +open import Function using (id) +open import Relation.Nullary using (¬_ ; yes ; no) +open import Relation.Nullary.Negation using (contradiction ; contraposition) +open import Relation.Binary.Core using (_≡_ ; refl) +open import Relation.Binary.PropositionalEquality using (cong ; sym) +open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎) + +FinMapMaybe : ℕ → Set → Set +FinMapMaybe n A = Vec (Maybe A) n + +lookupM : {A : Set} {n : ℕ} → Fin n → FinMapMaybe n A → Maybe A +lookupM = lookupVec + +insert : {A : Set} {n : ℕ} → Fin n → A → FinMapMaybe n A → FinMapMaybe n A +insert f a m = m [ f ]≔ (just a) + +empty : {A : Set} {n : ℕ} → FinMapMaybe n A +empty = replicate nothing + +fromAscList : {A : Set} {n : ℕ} → List (Fin n × A) → FinMapMaybe n A +fromAscList [] = empty +fromAscList ((f , a) ∷ xs) = insert f a (fromAscList xs) + +FinMap : ℕ → Set → Set +FinMap n A = Vec A n + +lookup : {A : Set} {n : ℕ} → Fin n → FinMap n A → A +lookup = lookupVec + +fromFunc : {A : Set} {n : ℕ} → (Fin n → A) → FinMap n A +fromFunc = tabulate + +union : {A : Set} {n : ℕ} → FinMapMaybe n A → FinMap n A → FinMap n A +union m1 m2 = tabulate (λ f → maybe′ id (lookup f m2) (lookupM f m1)) + +generate : {A : Set} {n : ℕ} → (Fin n → A) → List (Fin n) → FinMapMaybe n A +generate f is = fromAscList (zip is (map f is)) + + +lemma-insert-same : {τ : Set} {n : ℕ} → (m : FinMapMaybe n τ) → (f : Fin n) → (a : τ) → lookupM f m ≡ just a → m ≡ insert f a m +lemma-insert-same [] () a p +lemma-insert-same (.(just a) ∷ xs) zero a refl = refl +lemma-insert-same (x ∷ xs) (suc i) a p = cong (_∷_ x) (lemma-insert-same xs i a p) + +lemma-lookupM-empty : {A : Set} {n : ℕ} → (i : Fin n) → lookupM {A} i empty ≡ nothing +lemma-lookupM-empty zero = refl +lemma-lookupM-empty (suc i) = lemma-lookupM-empty i + +lemma-lookupM-insert : {A : Set} {n : ℕ} → (i : Fin n) → (a : A) → (m : FinMapMaybe n A) → lookupM i (insert i a m) ≡ just a +lemma-lookupM-insert zero _ (_ ∷ _) = refl +lemma-lookupM-insert (suc i) a (_ ∷ xs) = lemma-lookupM-insert i a xs + +lemma-lookupM-insert-other : {A : Set} {n : ℕ} → (i j : Fin n) → (a : A) → (m : FinMapMaybe n A) → ¬(i ≡ j) → lookupM i m ≡ lookupM i (insert j a m) +lemma-lookupM-insert-other zero zero a m p = contradiction refl p +lemma-lookupM-insert-other zero (suc j) a (x ∷ xs) p = refl +lemma-lookupM-insert-other (suc i) zero a (x ∷ xs) p = refl +lemma-lookupM-insert-other (suc i) (suc j) a (x ∷ xs) p = lemma-lookupM-insert-other i j a xs (contraposition (cong suc) p) + +lemma-from-just : {A : Set} → {x y : A} → _≡_ {_} {Maybe A} (just x) (just y) → x ≡ y +lemma-from-just refl = refl + +lemma-lookupM-generate : {A : Set} {n : ℕ} → (i : Fin n) → (f : Fin n → A) → (is : List (Fin n)) → (a : A) → lookupM i (generate f is) ≡ just a → f i ≡ a +lemma-lookupM-generate {A} i f [] a p with begin + just a + ≡⟨ sym p ⟩ + lookupM i (generate f []) + ≡⟨ refl ⟩ + lookupM i empty + ≡⟨ lemma-lookupM-empty i ⟩ + nothing ∎ +lemma-lookupM-generate i f [] a p | () +lemma-lookupM-generate i f (i' ∷ is) a p with i ≟ i' +lemma-lookupM-generate i f (.i ∷ is) a p | yes refl = lemma-from-just (begin + just (f i) + ≡⟨ sym (lemma-lookupM-insert i (f i) (generate f is)) ⟩ + lookupM i (insert i (f i) (generate f is)) + ≡⟨ refl ⟩ + lookupM i (generate f (i ∷ is)) + ≡⟨ p ⟩ + just a ∎) +lemma-lookupM-generate i f (i' ∷ is) a p | no ¬p2 = lemma-lookupM-generate i f is a (begin + lookupM i (generate f is) + ≡⟨ lemma-lookupM-insert-other i i' (f i') (generate f is) ¬p2 ⟩ + lookupM i (insert i' (f i') (generate f is)) + ≡⟨ refl ⟩ + lookupM i (generate f (i' ∷ is)) + ≡⟨ p ⟩ + just a ∎) |