parameterize Bidir via Carrier and deq

This avoids passing around the decidable equality explicitly.

1 files changed, 81 insertions, 79 deletions

diff --git a/Bidir.agda b/Bidir.agda
index 2567d8f..c3e3273 100644
--- a/Bidir.agda
+++ b/Bidir.agda
@@ -1,4 +1,6 @@
-module Bidir where
+open import Relation.Binary.Core using (Decidable ; _≡_)
+
+module Bidir (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where
 
 open import Data.Nat using (ℕ)
 open import Data.Fin using (Fin)
@@ -15,7 +17,7 @@ open import Data.Empty using (⊥-elim)
 open import Function using (id ; _∘_ ; flip)
 open import Relation.Nullary using (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_ ; _≡⟨_⟩_ ; _∎)
 
@@ -26,126 +28,126 @@ open import CheckInsert
 open import BFF using (_>>=_ ; fmap)
 open BFF.VecBFF using (assoc ; enumerate ; denumerate ; bff)
 
-lemma-1 : {τ : Set} {m n : ℕ} → (eq : EqInst τ) → (f : Fin n → τ) → (is : Vec (Fin n) m) → assoc eq is (map f is) ≡ just (restrict f (toList is))
-lemma-1 eq f []        = refl
-lemma-1 eq f (i ∷ is′) = begin
-  assoc eq (i ∷ is′) (map f (i ∷ is′))
+lemma-1 : {m n : ℕ} → (f : Fin n → Carrier) → (is : Vec (Fin n) m) → assoc deq is (map f is) ≡ just (restrict f (toList is))
+lemma-1 f []        = refl
+lemma-1 f (i ∷ is′) = begin
+  assoc deq (i ∷ is′) (map f (i ∷ is′))
     ≡⟨ refl ⟩
-  assoc eq is′ (map f is′) >>= checkInsert eq i (f i)
-    ≡⟨ cong (λ m → m >>= checkInsert eq i (f i)) (lemma-1 eq f is′) ⟩
-  just (restrict f (toList is′)) >>= (checkInsert eq i (f i))
+  assoc deq 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))
     ≡⟨ refl ⟩
-  checkInsert eq i (f i) (restrict f (toList is′))
-    ≡⟨ lemma-checkInsert-restrict eq f i (toList is′) ⟩
+  checkInsert deq i (f i) (restrict f (toList is′))
+    ≡⟨ lemma-checkInsert-restrict deq f i (toList is′) ⟩
   just (restrict f (toList (i ∷ is′))) ∎
 
-lemma-lookupM-assoc : {A : Set} {m n : ℕ} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (x : A) → (xs : Vec A m) → (h : FinMapMaybe n A) → assoc eq (i ∷ is) (x ∷ xs) ≡ just h → lookupM i h ≡ just x
-lemma-lookupM-assoc eq i is x xs h    p with assoc eq is xs
-lemma-lookupM-assoc eq i is x xs h    () | nothing
-lemma-lookupM-assoc eq i is x xs h    p | just h' = apply-checkInsertProof eq i x h' record
+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 deq (i ∷ is) (x ∷ xs) ≡ just h → lookupM i h ≡ just x
+lemma-lookupM-assoc i is x xs h    p with assoc deq 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
   { same  = λ lookupM≡justx → begin
       lookupM i h
-        ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym p) (lemma-checkInsert-same eq i x h' lookupM≡justx))) ⟩
+        ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym p) (lemma-checkInsert-same deq 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 eq i x h' lookupM≡nothing))) ⟩
+        ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym p) (lemma-checkInsert-new deq 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 eq i x h' x' x≢x' lookupM≡justx'))
+  ; 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'))
   }
 
-lemma-∉-lookupM-assoc : {A : Set} {m n : ℕ} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (xs : Vec A m) → (h : FinMapMaybe n A) → assoc eq is xs ≡ just h → (i ∉ toList is) → lookupM i h ≡ nothing
-lemma-∉-lookupM-assoc eq i []         []         h ph i∉is = begin
+lemma-∉-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc deq is xs ≡ just h → (i ∉ toList is) → lookupM i h ≡ nothing
+lemma-∉-lookupM-assoc i []         []         h ph i∉is = begin
   lookupM i h
     ≡⟨ cong (lookupM i) (sym (lemma-from-just ph)) ⟩
   lookupM i empty
     ≡⟨ lemma-lookupM-empty i ⟩
   nothing ∎
-lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h ph i∉is with assoc eq is' xs' | inspect (assoc eq is') xs'
-lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h () i∉is | nothing | [ ph' ]
-lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h ph i∉is | just h' | [ ph' ] = apply-checkInsertProof eq i' x' h' record {
+lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is with assoc deq is' xs' | inspect (assoc deq 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 {
     same = λ lookupM-i'-h'≡just-x' → begin
       lookupM i h
-        ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym ph) (lemma-checkInsert-same eq i' x' h' lookupM-i'-h'≡just-x'))) ⟩
+        ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym ph) (lemma-checkInsert-same deq i' x' h' lookupM-i'-h'≡just-x'))) ⟩
       lookupM i h'
-        ≡⟨ lemma-∉-lookupM-assoc eq i is' xs' h' ph' (i∉is ∘ there) ⟩
+        ≡⟨ 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 eq i' x' h' lookupM-i'-h'≡nothing))) ⟩
+        ≡⟨ cong (lookupM i)  (lemma-from-just (trans (sym ph) (lemma-checkInsert-new deq 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 eq i is' xs' h' ph' (i∉is ∘ there) ⟩
+        ≡⟨ 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 eq 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 deq 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
 _in-domain-of_ is h = All (λ i → ∃ λ x → lookupM i h ≡ just x) is
 
-lemma-assoc-domain : {m n : ℕ} {A : Set} → (eq : EqInst A) → (is : Vec (Fin n) m) → (xs : Vec A m) → (h : FinMapMaybe n A) → assoc eq is xs ≡ just h → (toList is) in-domain-of h
-lemma-assoc-domain eq []         []         h ph = Data.List.All.[]
-lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h ph with assoc eq is' xs' | inspect (assoc eq is') xs'
-lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h () | nothing | [ ph' ]
-lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h ph | just h' | [ ph' ] = apply-checkInsertProof eq i' x' h' record {
+lemma-assoc-domain : {m n : ℕ} → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc deq is xs ≡ just h → (toList is) in-domain-of h
+lemma-assoc-domain []         []         h ph = Data.List.All.[]
+lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h ph with assoc deq is' xs' | inspect (assoc deq 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 {
     same = λ lookupM-i'-h'≡just-x' → Data.List.All._∷_
-      (x' , (trans (cong (lookupM i') (lemma-from-just (trans (sym ph) (lemma-checkInsert-same eq i' x' h' lookupM-i'-h'≡just-x')))) lookupM-i'-h'≡just-x'))
-      (lemma-assoc-domain eq is' xs' h (trans ph' (trans (sym (lemma-checkInsert-same eq i' x' h' lookupM-i'-h'≡just-x')) ph)))
+      (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)))
   ; new  = λ lookupM-i'-h'≡nothing → Data.List.All._∷_
-      (x' , (trans (cong (lookupM i') (lemma-from-just (trans (sym ph) (lemma-checkInsert-new eq 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 deq 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 eq i i' (proj₁ p) x' h' h (proj₂ p) ph)
-        (lemma-assoc-domain eq is' xs' h' ph'))
-  ; wrong = λ x'' x'≢x'' lookupM-i'-h'≡just-x'' → lemma-just≢nothing (trans (sym ph) (lemma-checkInsert-wrong eq i' x' h' x'' x'≢x'' lookupM-i'-h'≡just-x''))
+        (λ {i} p → proj₁ p , lemma-lookupM-checkInsert deq 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''))
   }
 
-lemma-map-lookupM-insert : {m n : ℕ} {A : Set} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (x : A) → (h : FinMapMaybe n A) → i ∉ (toList is) → (toList is) in-domain-of h → map (flip lookupM (insert i x h)) is ≡ map (flip lookupM h) is
-lemma-map-lookupM-insert eq i []         x h i∉is ph = refl
-lemma-map-lookupM-insert eq i (i' ∷ is') x h i∉is ph = begin
+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
+lemma-map-lookupM-insert i []         x h i∉is ph = refl
+lemma-map-lookupM-insert i (i' ∷ is') x h i∉is ph = begin
   lookupM i' (insert i x h) ∷ map (flip lookupM (insert i x h)) is'
     ≡⟨ cong (flip _∷_ (map (flip lookupM (insert i x h)) is')) (sym (lemma-lookupM-insert-other i' i x h (i∉is ∘ here ∘ sym))) ⟩
   lookupM i' h ∷ map (flip lookupM (insert i x h)) is'
-    ≡⟨ cong (_∷_ (lookupM i' h)) (lemma-map-lookupM-insert eq i is' x h (i∉is ∘ there) (Data.List.All.tail ph)) ⟩
+    ≡⟨ 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} {A : Set} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (x : A) → (xs : Vec A m) → (h : FinMapMaybe n A) → (h' : FinMapMaybe n A) → assoc eq is xs ≡ just h' → checkInsert eq i x h' ≡ just h → map (flip lookupM h) is ≡ map (flip lookupM h') is
-lemma-map-lookupM-assoc eq i []         x []         h h' ph' ph = refl
-lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph with any (_≟_ i) (toList (i' ∷ is'))
-lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p with Data.List.All.lookup (lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h' ph') p
-lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p | (x'' , p') with lookupM i h' 
-lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p | (x'' , refl) | .(just x'') with eq x x''
-lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h .h ph' refl | yes p | (.x , refl) | .(just x)  | yes refl = refl
-lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' () | yes p | (x'' , refl) | .(just x'') | no ¬p
-lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | no ¬p rewrite lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h' ph' ¬p = begin
+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 deq 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 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
+lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p | (x'' , p') with lookupM i h' 
+lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p | (x'' , refl) | .(just x'') with deq x x''
+lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h .h ph' refl | yes p | (.x , refl) | .(just x)  | yes refl = refl
+lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' () | yes p | (x'' , refl) | .(just x'') | no ¬p
+lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | no ¬p rewrite lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h' ph' ¬p = begin
   map (flip lookupM h) (i' ∷ is')
     ≡⟨ map-cong (λ i'' → cong (lookupM i'') (lemma-from-just (sym ph))) (i' ∷ is') ⟩
   map (flip lookupM (insert i x h')) (i' ∷ is')
-    ≡⟨ lemma-map-lookupM-insert eq i (i' ∷ is') x h' ¬p (lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h' ph') ⟩
+    ≡⟨ lemma-map-lookupM-insert i (i' ∷ is') x h' ¬p (lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h' ph') ⟩
   map (flip lookupM h') (i' ∷ is') ∎
 
-lemma-2 : {τ : Set} {m n : ℕ} → (eq : EqInst τ) → (is : Vec (Fin n) m) → (v : Vec τ m) → (h : FinMapMaybe n τ) → assoc eq is v ≡ just h → map (flip lookupM h) is ≡ map just v
-lemma-2 eq []       []       h p = refl
-lemma-2 eq (i ∷ is) (x ∷ xs) h p with assoc eq is xs | inspect (assoc eq is) xs
-lemma-2 eq (i ∷ is) (x ∷ xs) h () | nothing | _
-lemma-2 eq (i ∷ is) (x ∷ xs) h p | just h' | [ ir ] = begin
+lemma-2 : {m n : ℕ} → (is : Vec (Fin n) m) → (v : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc deq is v ≡ just h → map (flip lookupM h) is ≡ map just v
+lemma-2 []       []       h p = refl
+lemma-2 (i ∷ is) (x ∷ xs) h p with assoc deq is xs | inspect (assoc deq is) xs
+lemma-2 (i ∷ is) (x ∷ xs) h () | nothing | _
+lemma-2 (i ∷ is) (x ∷ xs) h p | just h' | [ ir ] = begin
   map (flip lookupM h) (i ∷ is)
     ≡⟨ refl ⟩
   lookupM i h ∷ map (flip lookupM h) is
-    ≡⟨ cong (flip _∷_ (map (flip lookupM h) is)) (lemma-lookupM-assoc eq i is x xs h (begin
-      assoc eq (i ∷ is) (x ∷ xs)
-        ≡⟨ cong (flip _>>=_ (checkInsert eq i x)) ir ⟩
-      checkInsert eq i x h'
+    ≡⟨ cong (flip _∷_ (map (flip lookupM h) is)) (lemma-lookupM-assoc i is x xs h (begin
+      assoc deq (i ∷ is) (x ∷ xs)
+        ≡⟨ cong (flip _>>=_ (checkInsert deq i x)) ir ⟩
+      checkInsert deq i x h'
         ≡⟨ p ⟩
       just h ∎) ) ⟩
   just x ∷ map (flip lookupM h) is
-    ≡⟨  cong (_∷_ (just x)) (lemma-map-lookupM-assoc eq i is x xs h h' ir p) ⟩
+    ≡⟨  cong (_∷_ (just x)) (lemma-map-lookupM-assoc i is x xs h h' ir p) ⟩
   just x ∷ map (flip lookupM h') is
-    ≡⟨ cong (_∷_ (just x)) (lemma-2 eq is xs h' ir) ⟩
+    ≡⟨ cong (_∷_ (just x)) (lemma-2 is xs h' ir) ⟩
   just x ∷ map just xs
     ≡⟨ refl ⟩
   map just (x ∷ xs) ∎
@@ -165,16 +167,16 @@ lemma-map-denumerate-enumerate (a ∷ as) = cong (_∷_ a) (begin
     ≡⟨ lemma-map-denumerate-enumerate as ⟩
   as ∎)
 
-theorem-1 : {getlen : ℕ → ℕ} → (get : get-type getlen) → {m : ℕ} {τ : Set} → (eq : EqInst τ) → (s : Vec τ m) → bff get eq s (get s) ≡ just s
-theorem-1 get eq s = begin
-  bff get eq s (get s)
-    ≡⟨ cong (bff get eq s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩
-  bff get eq s (get (map (denumerate s) (enumerate s)))
-    ≡⟨ cong (bff get eq s) (free-theorem get (denumerate s) (enumerate s)) ⟩
-  bff get eq s (map (denumerate s) (get (enumerate s)))
+theorem-1 : {getlen : ℕ → ℕ} → (get : get-type getlen) → {m : ℕ} → (s : Vec Carrier m) → bff get deq s (get s) ≡ just s
+theorem-1 get s = begin
+  bff get deq s (get s)
+    ≡⟨ cong (bff get deq s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩
+  bff get deq s (get (map (denumerate s) (enumerate s)))
+    ≡⟨ cong (bff get deq s) (free-theorem get (denumerate s) (enumerate s)) ⟩
+  bff get deq s (map (denumerate s) (get (enumerate s)))
     ≡⟨ refl ⟩
-  fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (denumerate s))) (assoc eq (get (enumerate s)) (map (denumerate s) (get (enumerate s)))))
-    ≡⟨ cong (fmap (flip map (enumerate s) ∘ flip lookup) ∘ fmap (flip union (fromFunc (denumerate s)))) (lemma-1 eq (denumerate s) (get (enumerate s))) ⟩
+  fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (denumerate s))) (assoc deq (get (enumerate s)) (map (denumerate s) (get (enumerate s)))))
+    ≡⟨ cong (fmap (flip map (enumerate s) ∘ flip lookup) ∘ fmap (flip union (fromFunc (denumerate s)))) (lemma-1 (denumerate s) (get (enumerate s))) ⟩
   fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (flip lookupVec s))) (just (restrict (denumerate s) (toList (get (enumerate s))))))
     ≡⟨ refl ⟩
   just ((flip map (enumerate s) ∘ flip lookup) (union (restrict (denumerate s) (toList (get (enumerate s)))) (fromFunc (denumerate s))))
@@ -218,16 +220,16 @@ lemma-union-not-used h h' (i ∷ is') p | x , lookupM-i-h≡just-x = begin
     ≡⟨ cong (_∷_ (lookupM i h)) (lemma-union-not-used h h' is' (Data.List.All.tail p)) ⟩
   lookupM i h ∷ map (flip lookupM h) is' ∎
 
-theorem-2 : {getlen : ℕ → ℕ} (get : get-type getlen) → {τ : Set} {m : ℕ} → (eq : EqInst τ) → (v : Vec τ (getlen m)) → (s u : Vec τ m) → bff get eq s v ≡ just u → get u ≡ v
-theorem-2 get eq v s u p with lemma-fmap-just (assoc eq (get (enumerate s)) v) (proj₂ (lemma-fmap-just (fmap (flip union (fromFunc (denumerate s))) (assoc eq (get (enumerate s)) v)) p))
-theorem-2 get eq v s u p | h , ph = begin
+theorem-2 : {getlen : ℕ → ℕ} (get : get-type getlen) → {m : ℕ} → (v : Vec Carrier (getlen m)) → (s u : Vec Carrier m) → bff get deq s v ≡ just u → get u ≡ v
+theorem-2 get v s u p with lemma-fmap-just (assoc deq (get (enumerate s)) v) (proj₂ (lemma-fmap-just (fmap (flip union (fromFunc (denumerate s))) (assoc deq (get (enumerate s)) v)) p))
+theorem-2 get v s u p | h , ph = begin
   get u
     ≡⟨ lemma-from-just (begin
       just (get u)
         ≡⟨ refl ⟩
       fmap get (just u)
         ≡⟨ cong (fmap get) (sym p) ⟩
-      fmap get (bff get eq s v)
+      fmap get (bff get deq s v)
         ≡⟨ cong (fmap get ∘ fmap (flip map (enumerate s) ∘ flip lookup) ∘ fmap (flip union (fromFunc (denumerate s)))) ph ⟩
       fmap get (fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (denumerate s))) (just h)))
        ≡⟨ refl ⟩
@@ -238,9 +240,9 @@ theorem-2 get eq v s u p | h , ph = begin
   map (flip lookup (union h (fromFunc (denumerate s)))) (get (enumerate s))
      ≡⟨ lemma-from-map-just (begin
        map just (map (flip lookup (union h (fromFunc (denumerate s)))) (get (enumerate s)))
-         ≡⟨ lemma-union-not-used h (fromFunc (denumerate s)) (get (enumerate s)) (lemma-assoc-domain eq (get (enumerate s)) v h ph) ⟩
+         ≡⟨ lemma-union-not-used h (fromFunc (denumerate s)) (get (enumerate s)) (lemma-assoc-domain (get (enumerate s)) v h ph) ⟩
        map (flip lookupM h) (get (enumerate s))
-         ≡⟨ lemma-2 eq (get (enumerate s)) v h ph ⟩
+         ≡⟨ lemma-2 (get (enumerate s)) v h ph ⟩
        map just v
          ∎) ⟩
   v ∎