sym result of lemma-lookupM-{i,checkI}nsert-other

Typically we have the complex term on the LHS and the simplified term on
the RHS. These lemmata did it otherwise and by symming them, we save two
syms.

3 files changed, 6 insertions, 6 deletions

diff --git a/CheckInsert.agda b/CheckInsert.agda
index 9dc8a60..52dffc4 100644
--- a/CheckInsert.agda
+++ b/CheckInsert.agda
@@ -70,7 +70,7 @@ lemma-lookupM-checkInsert i j x y h h' pl ph'  | ._ | new _ with i ≟ j
 lemma-lookupM-checkInsert i .i x y h h' pl ph' | ._ | new pl' | yes refl = contradiction (trans (sym pl) pl') (λ ())
 lemma-lookupM-checkInsert i j x y h .(insert j y h) pl refl | ._ | new _ | no i≢j = begin
   lookupM i (insert j y h)
-    ≡⟨ sym (lemma-lookupM-insert-other i j y h i≢j) ⟩
+    ≡⟨ lemma-lookupM-insert-other i j y h i≢j ⟩
   lookupM i h
     ≡⟨ pl ⟩
   just x ∎
@@ -78,7 +78,7 @@ lemma-lookupM-checkInsert i j x y h .(insert j y h) pl refl | ._ | new _ | no iâ
 
 lemma-lookupM-checkInsert i j x y h h' pl () | ._ | wrong _ _ _
 
-lemma-lookupM-checkInsert-other : {n : ℕ} → (i j : Fin n) → i ≢ j → (x : Carrier) → (h h' : FinMapMaybe n Carrier) → checkInsert j x h ≡ just h' → lookupM i h ≡ lookupM i h'
+lemma-lookupM-checkInsert-other : {n : ℕ} → (i j : Fin n) → i ≢ j → (x : Carrier) → (h h' : FinMapMaybe n Carrier) → checkInsert j x h ≡ just h' → lookupM i h' ≡ lookupM i h
 lemma-lookupM-checkInsert-other i j i≢j x h h' ph' with lookupM j h
 lemma-lookupM-checkInsert-other i j i≢j x h h' ph' | just y with deq x y
 lemma-lookupM-checkInsert-other i j i≢j x h .h refl | just y | yes x≈y = refl
diff --git a/FinMap.agda b/FinMap.agda
index 05b251e..f9572b8 100644
--- a/FinMap.agda
+++ b/FinMap.agda
@@ -70,7 +70,7 @@ lemma-lookupM-insert : {A : Set} {n : ℕ} → (i : Fin n) → (a : A) → (m :
 lemma-lookupM-insert zero    a (x ∷ xs) = refl
 lemma-lookupM-insert (suc i) a (x ∷ 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 : {A : Set} {n : ℕ} → (i j : Fin n) → (a : A) → (m : FinMapMaybe n A) → i ≢ j → lookupM i (insert j a m) ≡ lookupM i 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
@@ -87,7 +87,7 @@ lemma-lookupM-restrict i f (.i ∷ is) a p | yes refl = just-injective (begin
    just a ∎)
 lemma-lookupM-restrict i f (i' ∷ is) a p | no i≢i' = lemma-lookupM-restrict i f is a (begin
   lookupM i (restrict f is)
-    ≡⟨ lemma-lookupM-insert-other i i' (f i') (restrict f is) i≢i' ⟩
+    ≡⟨ sym (lemma-lookupM-insert-other i i' (f i') (restrict f is) i≢i') ⟩
   lookupM i (insert i' (f i') (restrict f is))
     ≡⟨ p ⟩
   just a ∎)
@@ -123,7 +123,7 @@ lemma-disjoint-union {n} {m} f t = lemma-tabulate-∘ (lemma-inner t)
           lemma-inner (.x ∷ ts) x | yes refl = cong (maybe′ just (lookupM x (delete-many (x ∷ ts) (fromFunc f)))) (lemma-lookupM-insert x (f x) (restrict f (toList ts)))
           lemma-inner (t ∷ ts) x | no ¬p = begin
             maybe′ just (lookupM x (delete-many (t ∷ ts) (fromFunc f))) (lookupM x (restrict f (toList (t ∷ ts))))
-              ≡⟨ cong (maybe′ just (lookupM x (delete-many (t ∷ ts) (fromFunc f)))) (sym (lemma-lookupM-insert-other x t (f t) (restrict f (toList ts)) ¬p)) ⟩
+              ≡⟨ cong (maybe′ just (lookupM x (delete-many (t ∷ ts) (fromFunc f)))) (lemma-lookupM-insert-other x t (f t) (restrict f (toList ts)) ¬p) ⟩
             maybe′ just (lookupM x (delete-many (t ∷ ts) (fromFunc f))) (lookupM x (restrict f (toList ts)))
               ≡⟨ cong (flip (maybe′ just) (lookupM x (restrict f (toList ts)))) (lemma-lookupM-delete (delete-many ts (fromFunc f)) ¬p) ⟩
             maybe′ just (lookupM x (delete-many ts (fromFunc f))) (lookupM x (restrict f (toList ts)))
diff --git a/Precond.agda b/Precond.agda
index 85c955c..8d2eab2 100644
--- a/Precond.agda
+++ b/Precond.agda
@@ -100,7 +100,7 @@ lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is with assoc is' x
 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' ] = begin
   lookupM i h
-    ≡⟨ sym (lemma-lookupM-checkInsert-other i i' (i∉is ∘ here) x' h' h ph) ⟩
+    ≡⟨ lemma-lookupM-checkInsert-other i i' (i∉is ∘ here) x' h' h ph ⟩
   lookupM i h'
     ≡⟨ lemma-∉-lookupM-assoc i is' xs' h' ph' (i∉is ∘ there) ⟩
   nothing ∎