split Bidir.agda to FinMap.agda

1 files changed, 2 insertions, 87 deletions

diff --git a/Bidir.agda b/Bidir.agda
index 0bc6e20..9604e00 100644
--- a/Bidir.agda
+++ b/Bidir.agda
@@ -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