module Bidir where
-
open import Data.Bool hiding (_≟_)
open import Data.Nat
open import Data.Fin
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)
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
--- /dev/null
+module FinMap where
+
+open import Data.Nat using (â„•)
+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 ∎)
projects / ~helmut / bidiragda.git / commitdiff