finally parameterize CheckInsert

Also adapt depending modules. Long lines generally become shorter. The
misleading name "EqInst" (hiding the decidability) got discarded.

1 files changed, 48 insertions, 49 deletions

diff --git a/CheckInsert.agda b/CheckInsert.agda
index 40a57d6..01f1302 100644
--- a/CheckInsert.agda
+++ b/CheckInsert.agda
@@ -1,4 +1,6 @@
-module CheckInsert where
+open import Relation.Binary.Core using (Decidable ; _≡_)
+
+module CheckInsert (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where
 
 open import Data.Nat using (ℕ)
 open import Data.Fin using (Fin)
@@ -7,86 +9,83 @@ open import Data.Maybe using (Maybe ; nothing ; just)
 open import Data.List using (List ; [] ; _∷_)
 open import Relation.Nullary using (Dec ; yes ; no)
 open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Binary.Core using (_≡_ ; refl ; _≢_)
+open import Relation.Binary.Core using (refl ; _≢_)
 open import Relation.Binary.PropositionalEquality using (cong ; sym ; inspect ; [_] ; trans)
 open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎)
 
 open import FinMap
 
-EqInst : Set → Set
-EqInst A = (x y : A) → Dec (x ≡ y)
-
-checkInsert : {A : Set} {n : ℕ} → EqInst A → Fin n → A → FinMapMaybe n A → Maybe (FinMapMaybe n A)
-checkInsert eq i b m with lookupM i m
-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)
+checkInsert : {n : ℕ} → Fin n → Carrier → FinMapMaybe n Carrier → Maybe (FinMapMaybe n Carrier)
+checkInsert i b m with lookupM i m
+checkInsert i b m | just c with deq b c
+checkInsert i b m | just .b | yes refl = just m
+checkInsert i b m | just c  | no ¬p    = nothing
+checkInsert i b m | nothing = just (insert i b m)
 
-record checkInsertProof {A : Set} {n : ℕ} (eq : EqInst A) (i : Fin n) (x : A) (m : FinMapMaybe n A) (P : Set) : Set where
+record checkInsertProof {n : ℕ} (i : Fin n) (x : Carrier) (m : FinMapMaybe n Carrier) (P : Set) : Set where
   field
      same : lookupM i m ≡ just x → P
      new : lookupM i m ≡ nothing → P
-     wrong : (x' : A) → x ≢ x' → lookupM i m ≡ just x'  → P
+     wrong : (x' : Carrier) → x ≢ x' → lookupM i m ≡ just x'  → P
 
-apply-checkInsertProof : {A P : Set} {n : ℕ} → (eq : EqInst A) → (i : Fin n) → (x : A) → (m : FinMapMaybe n A) → checkInsertProof eq i x m P → P
-apply-checkInsertProof eq i x m rp with lookupM i m | inspect (lookupM i) m
-apply-checkInsertProof eq i x m rp | just x' | il with eq x x'
-apply-checkInsertProof eq i x m rp | just .x | [ il ] | yes refl = checkInsertProof.same rp il
-apply-checkInsertProof eq i x m rp | just x' | [ il ] | no x≢x' = checkInsertProof.wrong rp x' x≢x' il
-apply-checkInsertProof eq i x m rp | nothing | [ il ] = checkInsertProof.new rp il
+apply-checkInsertProof : {P : Set} {n : ℕ} → (i : Fin n) → (x : Carrier) → (m : FinMapMaybe n Carrier) → checkInsertProof i x m P → P
+apply-checkInsertProof i x m rp with lookupM i m | inspect (lookupM i) m
+apply-checkInsertProof i x m rp | just x' | il with deq x x'
+apply-checkInsertProof i x m rp | just .x | [ il ] | yes refl = checkInsertProof.same rp il
+apply-checkInsertProof i x m rp | just x' | [ il ] | no x≢x' = checkInsertProof.wrong rp x' x≢x' il
+apply-checkInsertProof i x m rp | nothing | [ il ] = checkInsertProof.new rp il
 
-lemma-checkInsert-same : {A : Set} {n : ℕ} → (eq : EqInst A) → (i : Fin n) → (x : A) → (m : FinMapMaybe n A) → lookupM i m ≡ just x → checkInsert eq i x m ≡ just m
-lemma-checkInsert-same eq i x m p with lookupM i m
-lemma-checkInsert-same eq i x m refl | .(just x) with eq x x
-lemma-checkInsert-same eq i x m refl | .(just x) | yes refl = refl
-lemma-checkInsert-same eq i x m refl | .(just x) | no x≢x = contradiction refl x≢x
+lemma-checkInsert-same : {n : ℕ} → (i : Fin n) → (x : Carrier) → (m : FinMapMaybe n Carrier) → lookupM i m ≡ just x → checkInsert i x m ≡ just m
+lemma-checkInsert-same i x m p with lookupM i m
+lemma-checkInsert-same i x m refl | .(just x) with deq x x
+lemma-checkInsert-same i x m refl | .(just x) | yes refl = refl
+lemma-checkInsert-same i x m refl | .(just x) | no x≢x = contradiction refl x≢x
 
-lemma-checkInsert-new : {A : Set} {n : ℕ} → (eq : EqInst A) → (i : Fin n) → (x : A) → (m : FinMapMaybe n A) → lookupM i m ≡ nothing → checkInsert eq i x m ≡ just (insert i x m)
-lemma-checkInsert-new eq i x m p with lookupM i m
-lemma-checkInsert-new eq i x m refl | .nothing = refl
+lemma-checkInsert-new : {n : ℕ} → (i : Fin n) → (x : Carrier) → (m : FinMapMaybe n Carrier) → lookupM i m ≡ nothing → checkInsert i x m ≡ just (insert i x m)
+lemma-checkInsert-new i x m p with lookupM i m
+lemma-checkInsert-new i x m refl | .nothing = refl
 
-lemma-checkInsert-wrong : {A : Set} {n : ℕ} → (eq : EqInst A) → (i : Fin n) → (x : A) → (m : FinMapMaybe n A) → (x' : A) → x ≢ x' → lookupM i m ≡ just x' → checkInsert eq i x m ≡ nothing
-lemma-checkInsert-wrong eq i x m x' d p with lookupM i m
-lemma-checkInsert-wrong eq i x m x' d refl | .(just x') with eq x x'
-lemma-checkInsert-wrong eq i x m x' d refl | .(just x') | yes q = contradiction q d
-lemma-checkInsert-wrong eq i x m x' d refl | .(just x') | no ¬q = refl
+lemma-checkInsert-wrong : {n : ℕ} → (i : Fin n) → (x : Carrier) → (m : FinMapMaybe n Carrier) → (x' : Carrier) → x ≢ x' → lookupM i m ≡ just x' → checkInsert i x m ≡ nothing
+lemma-checkInsert-wrong i x m x' d p with lookupM i m
+lemma-checkInsert-wrong i x m x' d refl | .(just x') with deq x x'
+lemma-checkInsert-wrong i x m x' d refl | .(just x') | yes q = contradiction q d
+lemma-checkInsert-wrong i x m x' d refl | .(just x') | no ¬q = refl
 
-record checkInsertEqualProof {A : Set} {n : ℕ} (eq : EqInst A) (i : Fin n) (x : A) (m : FinMapMaybe n A) (e : Maybe (FinMapMaybe n A)) : Set where
+record checkInsertEqualProof {n : ℕ} (i : Fin n) (x : Carrier) (m : FinMapMaybe n Carrier) (e : Maybe (FinMapMaybe n Carrier)) : Set where
   field
      same : lookupM i m ≡ just x → just m ≡ e
      new : lookupM i m ≡ nothing → just (insert i x m) ≡ e
-     wrong : (x' : A) → x ≢ x' → lookupM i m ≡ just x' → nothing ≡ e
+     wrong : (x' : Carrier) → x ≢ x' → lookupM i m ≡ just x' → nothing ≡ e
 
-lift-checkInsertProof : {A : Set} {n : ℕ} {eq : EqInst A} {i : Fin n} {x : A} {m : FinMapMaybe n A} {e : Maybe (FinMapMaybe n A)} → checkInsertEqualProof eq i x m e → checkInsertProof eq i x m (checkInsert eq i x m ≡ e)
-lift-checkInsertProof {_} {_} {eq} {i} {x} {m} o = record
-  { same  = λ p → trans (lemma-checkInsert-same eq i x m p) (checkInsertEqualProof.same o p)
-  ; new   = λ p → trans (lemma-checkInsert-new eq i x m p) (checkInsertEqualProof.new o p)
-  ; wrong = λ x' q p → trans (lemma-checkInsert-wrong eq i x m x' q p) (checkInsertEqualProof.wrong o x' q p)
+lift-checkInsertProof : {n : ℕ} {i : Fin n} {x : Carrier} {m : FinMapMaybe n Carrier} {e : Maybe (FinMapMaybe n Carrier)} → checkInsertEqualProof i x m e → checkInsertProof i x m (checkInsert i x m ≡ e)
+lift-checkInsertProof {_} {i} {x} {m} o = record
+  { same  = λ p → trans (lemma-checkInsert-same i x m p) (checkInsertEqualProof.same o p)
+  ; new   = λ p → trans (lemma-checkInsert-new i x m p) (checkInsertEqualProof.new o p)
+  ; wrong = λ x' q p → trans (lemma-checkInsert-wrong i x m x' q p) (checkInsertEqualProof.wrong o x' q p)
   }
 
-lemma-checkInsert-restrict : {τ : Set} {n : ℕ} → (eq : EqInst τ) → (f : Fin n → τ) → (i : Fin n) → (is : List (Fin n)) → checkInsert eq i (f i) (restrict f is) ≡ just (restrict f (i ∷ is))
-lemma-checkInsert-restrict {τ} eq f i is = apply-checkInsertProof eq i (f i) (restrict f is) (lift-checkInsertProof record
+lemma-checkInsert-restrict : {n : ℕ} → (f : Fin n → Carrier) → (i : Fin n) → (is : List (Fin n)) → checkInsert i (f i) (restrict f is) ≡ just (restrict f (i ∷ is))
+lemma-checkInsert-restrict f i is = apply-checkInsertProof i (f i) (restrict f is) (lift-checkInsertProof record
   { same  = λ lookupM≡justx → cong just (lemma-insert-same (restrict f is) i (f i) lookupM≡justx)
   ; new   = λ lookupM≡nothing → refl
   ; wrong = λ x' x≢x' lookupM≡justx' → contradiction (lemma-lookupM-restrict i f is x' lookupM≡justx') x≢x'
   })
 
-lemma-lookupM-checkInsert : {A : Set} {n : ℕ} → (eq : EqInst A) → (i j : Fin n) → (x y : A) → (h h' : FinMapMaybe n A) → lookupM i h ≡ just x → checkInsert eq j y h ≡ just h' → lookupM i h' ≡ just x
-lemma-lookupM-checkInsert eq i j x y h h' pl ph' with lookupM j h | inspect (lookupM j) h
-lemma-lookupM-checkInsert eq i j x y h .(insert j y h) pl refl | nothing | pl' with i ≟ j
-lemma-lookupM-checkInsert eq i .i x y h .(insert i y h) pl refl | nothing | [ pl' ] | yes refl = lemma-just≢nothing (begin just x ≡⟨ sym pl ⟩ lookupM i h ≡⟨ pl' ⟩ (nothing ∎))
-lemma-lookupM-checkInsert eq i j x y h .(insert j y h) pl refl | nothing | pl' | no ¬p = begin
+lemma-lookupM-checkInsert : {n : ℕ} → (i j : Fin n) → (x y : Carrier) → (h h' : FinMapMaybe n Carrier) → lookupM i h ≡ just x → checkInsert j y h ≡ just h' → lookupM i h' ≡ just x
+lemma-lookupM-checkInsert i j x y h h' pl ph' with lookupM j h | inspect (lookupM j) h
+lemma-lookupM-checkInsert i j x y h .(insert j y h) pl refl | nothing | pl' with i ≟ j
+lemma-lookupM-checkInsert i .i x y h .(insert i y h) pl refl | nothing | [ pl' ] | yes refl = lemma-just≢nothing (begin just x ≡⟨ sym pl ⟩ lookupM i h ≡⟨ pl' ⟩ (nothing ∎))
+lemma-lookupM-checkInsert i j x y h .(insert j y h) pl refl | nothing | pl' | no ¬p = begin
   lookupM i (insert j y h)
     ≡⟨ sym (lemma-lookupM-insert-other i j y h ¬p) ⟩
   lookupM i h
     ≡⟨ pl ⟩
   just x ∎
-lemma-lookupM-checkInsert eq i j x y h h' pl ph' | just z | pl' with eq y z
-lemma-lookupM-checkInsert eq i j x y h h' pl ph' | just .y | pl' | yes refl = begin
+lemma-lookupM-checkInsert i j x y h h' pl ph' | just z | pl' with deq y z
+lemma-lookupM-checkInsert i j x y h h' pl ph' | just .y | pl' | yes refl = begin
   lookupM i h'
     ≡⟨ cong (lookupM i) (lemma-from-just (sym ph')) ⟩
   lookupM i h
     ≡⟨ pl ⟩
   just x ∎
-lemma-lookupM-checkInsert eq i j x y h h' pl () | just z | pl' | no ¬p
+lemma-lookupM-checkInsert i j x y h h' pl () | just z | pl' | no ¬p