finally parameterize CheckInsert

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

4 files changed, 83 insertions, 80 deletions

diff --git a/BFF.agda b/BFF.agda
index b7502ce..2888a3d 100644
--- a/BFF.agda
+++ b/BFF.agda
@@ -9,7 +9,7 @@ open import Function using (id ; _∘_ ; flip)
 open import Relation.Binary.Core using (Decidable ; _≡_)
 
 open import FinMap
-open import CheckInsert
+import CheckInsert
 import FreeTheorems
 
 _>>=_ : {A B : Set} → Maybe A → (A → Maybe B) → Maybe B
@@ -20,10 +20,11 @@ fmap f = maybe′ (λ a → just (f a)) nothing
 
 module ListBFF (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where
   open FreeTheorems.ListList public using (get-type)
+  open CheckInsert Carrier deq
 
   assoc : {n : ℕ} → List (Fin n) → List Carrier → Maybe (FinMapMaybe n Carrier)
   assoc []       []       = just empty
-  assoc (i ∷ is) (b ∷ bs) = (assoc is bs) >>= (checkInsert deq i b)
+  assoc (i ∷ is) (b ∷ bs) = (assoc is bs) >>= (checkInsert i b)
   assoc _        _        = nothing
 
   enumerate : (l : List Carrier) → List (Fin (length l))
@@ -41,10 +42,11 @@ module ListBFF (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where
 
 module VecBFF (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where
   open FreeTheorems.VecVec public using (get-type)
+  open CheckInsert Carrier deq
 
   assoc : {n m : ℕ} → Vec (Fin n) m → Vec Carrier m → Maybe (FinMapMaybe n Carrier)
   assoc []V       []V       = just empty
-  assoc (i ∷V is) (b ∷V bs) = (assoc is bs) >>= (checkInsert deq i b)
+  assoc (i ∷V is) (b ∷V bs) = (assoc is bs) >>= (checkInsert i b)
 
   enumerate : {n : ℕ} → Vec Carrier n → Vec (Fin n) n
   enumerate _ = tabulate id
diff --git a/Bidir.agda b/Bidir.agda
index 1b68e60..437dccf 100644
--- a/Bidir.agda
+++ b/Bidir.agda
@@ -24,7 +24,8 @@ open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨
 import FreeTheorems
 open FreeTheorems.VecVec using (get-type ; free-theorem)
 open import FinMap
-open import CheckInsert
+import CheckInsert
+open CheckInsert Carrier deq
 open import BFF using (_>>=_ ; fmap)
 open BFF.VecBFF Carrier deq using (assoc ; enumerate ; denumerate ; bff)
 
@@ -33,31 +34,31 @@ lemma-1 f []        = refl
 lemma-1 f (i ∷ is′) = begin
   assoc (i ∷ is′) (map f (i ∷ is′))
     ≡⟨ refl ⟩
-  assoc is′ (map f is′) >>= checkInsert deq i (f i)
-    ≡⟨ cong (λ m → m >>= checkInsert deq i (f i)) (lemma-1 f is′) ⟩
-  just (restrict f (toList is′)) >>= (checkInsert deq i (f i))
+  assoc is′ (map f is′) >>= checkInsert i (f i)
+    ≡⟨ cong (λ m → m >>= checkInsert i (f i)) (lemma-1 f is′) ⟩
+  just (restrict f (toList is′)) >>= (checkInsert i (f i))
     ≡⟨ refl ⟩
-  checkInsert deq i (f i) (restrict f (toList is′))
-    ≡⟨ lemma-checkInsert-restrict deq f i (toList is′) ⟩
+  checkInsert i (f i) (restrict f (toList is′))
+    ≡⟨ lemma-checkInsert-restrict f i (toList is′) ⟩
   just (restrict f (toList (i ∷ is′))) ∎
 
 lemma-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (x : Carrier) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc (i ∷ is) (x ∷ xs) ≡ just h → lookupM i h ≡ just x
 lemma-lookupM-assoc i is x xs h    p with assoc is xs
 lemma-lookupM-assoc i is x xs h    () | nothing
-lemma-lookupM-assoc i is x xs h    p | just h' = apply-checkInsertProof deq i x h' record
+lemma-lookupM-assoc i is x xs h    p | just h' = apply-checkInsertProof i x h' record
   { same  = λ lookupM≡justx → begin
       lookupM i h
-        ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym p) (lemma-checkInsert-same deq i x h' lookupM≡justx))) ⟩
+        ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym p) (lemma-checkInsert-same i x h' lookupM≡justx))) ⟩
       lookupM i h'
         ≡⟨ lookupM≡justx ⟩
       just x ∎
   ; new   = λ lookupM≡nothing → begin
       lookupM i h
-        ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym p) (lemma-checkInsert-new deq i x h' lookupM≡nothing))) ⟩
+        ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym p) (lemma-checkInsert-new i x h' lookupM≡nothing))) ⟩
       lookupM i (insert i x h')
         ≡⟨ lemma-lookupM-insert i x h' ⟩
       just x ∎
-  ; wrong = λ x' x≢x' lookupM≡justx' → lemma-just≢nothing (trans (sym p) (lemma-checkInsert-wrong deq i x h' x' x≢x' lookupM≡justx'))
+  ; wrong = λ x' x≢x' lookupM≡justx' → lemma-just≢nothing (trans (sym p) (lemma-checkInsert-wrong i x h' x' x≢x' lookupM≡justx'))
   }
 
 lemma-∉-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc is xs ≡ just h → (i ∉ toList is) → lookupM i h ≡ nothing
@@ -69,22 +70,22 @@ lemma-∉-lookupM-assoc i []         []         h ph i∉is = begin
   nothing ∎
 lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is with assoc is' xs' | inspect (assoc is') xs'
 lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h () i∉is | nothing | [ ph' ]
-lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is | just h' | [ ph' ] = apply-checkInsertProof deq i' x' h' record {
+lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is | just h' | [ ph' ] = apply-checkInsertProof i' x' h' record {
     same = λ lookupM-i'-h'≡just-x' → begin
       lookupM i h
-        ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym ph) (lemma-checkInsert-same deq i' x' h' lookupM-i'-h'≡just-x'))) ⟩
+        ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym ph) (lemma-checkInsert-same i' x' h' lookupM-i'-h'≡just-x'))) ⟩
       lookupM i h'
         ≡⟨ lemma-∉-lookupM-assoc i is' xs' h' ph' (i∉is ∘ there) ⟩
       nothing ∎
   ; new = λ lookupM-i'-h'≡nothing → begin
       lookupM i h
-        ≡⟨ cong (lookupM i)  (lemma-from-just (trans (sym ph) (lemma-checkInsert-new deq i' x' h' lookupM-i'-h'≡nothing))) ⟩
+        ≡⟨ cong (lookupM i)  (lemma-from-just (trans (sym ph) (lemma-checkInsert-new i' x' h' lookupM-i'-h'≡nothing))) ⟩
       lookupM i (insert i' x' h')
         ≡⟨ sym (lemma-lookupM-insert-other i i' x' h' (i∉is ∘ here)) ⟩
       lookupM i h'
         ≡⟨ lemma-∉-lookupM-assoc i is' xs' h' ph' (i∉is ∘ there) ⟩
       nothing ∎
-  ; wrong = λ x'' x'≢x'' lookupM-i'-h'≡just-x'' → lemma-just≢nothing (trans (sym ph) (lemma-checkInsert-wrong deq i' x' h' x'' x'≢x'' lookupM-i'-h'≡just-x''))
+  ; wrong = λ x'' x'≢x'' lookupM-i'-h'≡just-x'' → lemma-just≢nothing (trans (sym ph) (lemma-checkInsert-wrong i' x' h' x'' x'≢x'' lookupM-i'-h'≡just-x''))
   }
 
 _in-domain-of_ : {n : ℕ} {A : Set} → (is : List (Fin n)) → (FinMapMaybe n A) → Set
@@ -94,16 +95,16 @@ lemma-assoc-domain : {m n : ℕ} → (is : Vec (Fin n) m) → (xs : Vec Carrier
 lemma-assoc-domain []         []         h ph = Data.List.All.[]
 lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h ph with assoc is' xs' | inspect (assoc is') xs'
 lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h () | nothing | [ ph' ]
-lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h ph | just h' | [ ph' ] = apply-checkInsertProof deq i' x' h' record {
+lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h ph | just h' | [ ph' ] = apply-checkInsertProof i' x' h' record {
     same = λ lookupM-i'-h'≡just-x' → Data.List.All._∷_
-      (x' , (trans (cong (lookupM i') (lemma-from-just (trans (sym ph) (lemma-checkInsert-same deq i' x' h' lookupM-i'-h'≡just-x')))) lookupM-i'-h'≡just-x'))
-      (lemma-assoc-domain is' xs' h (trans ph' (trans (sym (lemma-checkInsert-same deq i' x' h' lookupM-i'-h'≡just-x')) ph)))
+      (x' , (trans (cong (lookupM i') (lemma-from-just (trans (sym ph) (lemma-checkInsert-same i' x' h' lookupM-i'-h'≡just-x')))) lookupM-i'-h'≡just-x'))
+      (lemma-assoc-domain is' xs' h (trans ph' (trans (sym (lemma-checkInsert-same i' x' h' lookupM-i'-h'≡just-x')) ph)))
   ; new  = λ lookupM-i'-h'≡nothing → Data.List.All._∷_
-      (x' , (trans (cong (lookupM i') (lemma-from-just (trans (sym ph) (lemma-checkInsert-new deq i' x' h' lookupM-i'-h'≡nothing)))) (lemma-lookupM-insert i' x' h')))
+      (x' , (trans (cong (lookupM i') (lemma-from-just (trans (sym ph) (lemma-checkInsert-new i' x' h' lookupM-i'-h'≡nothing)))) (lemma-lookupM-insert i' x' h')))
       (Data.List.All.map
-        (λ {i} p → proj₁ p , lemma-lookupM-checkInsert deq i i' (proj₁ p) x' h' h (proj₂ p) ph)
+        (λ {i} p → proj₁ p , lemma-lookupM-checkInsert i i' (proj₁ p) x' h' h (proj₂ p) ph)
         (lemma-assoc-domain is' xs' h' ph'))
-  ; wrong = λ x'' x'≢x'' lookupM-i'-h'≡just-x'' → lemma-just≢nothing (trans (sym ph) (lemma-checkInsert-wrong deq i' x' h' x'' x'≢x'' lookupM-i'-h'≡just-x''))
+  ; wrong = λ x'' x'≢x'' lookupM-i'-h'≡just-x'' → lemma-just≢nothing (trans (sym ph) (lemma-checkInsert-wrong i' x' h' x'' x'≢x'' lookupM-i'-h'≡just-x''))
   }
 
 lemma-map-lookupM-insert : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (x : Carrier) → (h : FinMapMaybe n Carrier) → i ∉ (toList is) → (toList is) in-domain-of h → map (flip lookupM (insert i x h)) is ≡ map (flip lookupM h) is
@@ -115,7 +116,7 @@ lemma-map-lookupM-insert i (i' ∷ is') x h i∉is ph = begin
     ≡⟨ cong (_∷_ (lookupM i' h)) (lemma-map-lookupM-insert i is' x h (i∉is ∘ there) (Data.List.All.tail ph)) ⟩
   lookupM i' h ∷ map (flip lookupM h) is' ∎
 
-lemma-map-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (x : Carrier) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → (h' : FinMapMaybe n Carrier) → assoc is xs ≡ just h' → checkInsert deq i x h' ≡ just h → map (flip lookupM h) is ≡ map (flip lookupM h') is
+lemma-map-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (x : Carrier) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → (h' : FinMapMaybe n Carrier) → assoc is xs ≡ just h' → checkInsert i x h' ≡ just h → map (flip lookupM h) is ≡ map (flip lookupM h') is
 lemma-map-lookupM-assoc i []         x []         h h' ph' ph = refl
 lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' ph with any (_≟_ i) (toList (i' ∷ is'))
 lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p with Data.List.All.lookup (lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h' ph') p
@@ -140,8 +141,8 @@ lemma-2 (i ∷ is) (x ∷ xs) h p | just h' | [ ir ] = begin
   lookupM i h ∷ map (flip lookupM h) is
     ≡⟨ cong (flip _∷_ (map (flip lookupM h) is)) (lemma-lookupM-assoc i is x xs h (begin
       assoc (i ∷ is) (x ∷ xs)
-        ≡⟨ cong (flip _>>=_ (checkInsert deq i x)) ir ⟩
-      checkInsert deq i x h'
+        ≡⟨ cong (flip _>>=_ (checkInsert i x)) ir ⟩
+      checkInsert i x h'
         ≡⟨ p ⟩
       just h ∎) ) ⟩
   just x ∷ map (flip lookupM h) is
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
diff --git a/Precond.agda b/Precond.agda
index 5011e41..a6d2d37 100644
--- a/Precond.agda
+++ b/Precond.agda
@@ -15,7 +15,8 @@ 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)
+import CheckInsert
+open CheckInsert Carrier deq using (checkInsert ; lemma-checkInsert-new)
 open import BFF using (fmap ; _>>=_)
 import Bidir
 open Bidir Carrier deq using (lemma-∉-lookupM-assoc)
@@ -67,8 +68,8 @@ different-assoc (u ∷ us) (v ∷ vs) p with different-assoc us vs (λ i j i≢j
 different-assoc (u ∷ us) (v ∷ vs) p | h , p' = insert u v h , (begin
   assoc (u ∷ us) (v ∷ vs)
     ≡⟨ refl ⟩
-  assoc us vs >>= checkInsert deq u v
-    ≡⟨ cong (flip _>>=_ (checkInsert deq u v)) p' ⟩
-  checkInsert deq u v h
-    ≡⟨ lemma-checkInsert-new deq u v h (lemma-∉-lookupM-assoc u us vs h p' (different-∉ u us p)) ⟩
+  assoc us vs >>= checkInsert u v
+    ≡⟨ cong (flip _>>=_ (checkInsert u v)) p' ⟩
+  checkInsert u v h
+    ≡⟨ lemma-checkInsert-new u v h (lemma-∉-lookupM-assoc u us vs h p' (different-∉ u us p)) ⟩
   just (insert u v h) ∎)