give a sufficient precondition for theorem-2

If get on Fin results in Vecs whose elements are unique, then theorem-2
can be applied.

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+module Precond where
+
+open import Data.Nat using (ℕ) renaming (zero to nzero ; suc to nsuc)
+open import Data.Fin using (Fin ; zero ; suc)
+open import Data.Vec using (Vec ; [] ; _∷_ ; map ; lookup ; toList)
+open import Data.List.Any using (here ; there)
+open Data.List.Any.Membership-≡ using (_∉_)
+open import Data.Maybe using (just)
+open import Data.Product using (∃ ; _,_)
+open import Function using (flip ; _∘_)
+open import Relation.Binary.Core using (_≡_ ; _≢_)
+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)
+open import BFF using (fmap ; _>>=_)
+open import Bidir using (lemma-∉-lookupM-assoc)
+
+open BFF.VecBFF using (assoc ; enumerate ; denumerate ; bff)
+
+assoc-enough : {getlen : ℕ → ℕ} (get : {A : Set} {n : ℕ} → Vec A n → Vec A (getlen n)) → {B : Set} {m : ℕ} → (eq : EqInst B) → (s : Vec B m) → (v : Vec B (getlen m)) → (h : FinMapMaybe m B) → assoc eq (get (enumerate s)) v ≡ just h → ∃ λ u → bff get eq s v ≡ just u
+assoc-enough get {B} {m} eq s v h p = map (flip lookup (union h g)) s′ , (begin
+  bff get eq s v
+    ≡⟨ refl ⟩
+  fmap (flip map s′ ∘ flip lookup) (fmap (flip union g) (assoc eq (get s′) v))
+    ≡⟨ cong (fmap (flip map s′ ∘ flip lookup)) (cong (fmap (flip union g)) p) ⟩
+  fmap (flip map s′ ∘ flip lookup) (fmap (flip union g) (just h))
+    ≡⟨ refl ⟩
+  just (map (flip lookup (union h g)) s′) ∎)
+    where s′ : Vec (Fin m) m
+          s′ = enumerate s
+          g : FinMap m B
+          g = fromFunc (denumerate s)
+
+all-different : {A : Set} {n : ℕ} → Vec A n → Set
+all-different {_} {n} v = (i : Fin n) → (j : Fin n) → i ≢ j → lookup i v ≢ lookup j v
+
+suc-≡ : {n : ℕ} {i j : Fin n} → (suc i ≡ suc j) → i ≡ j
+suc-≡ refl = refl
+
+different-swap : {A : Set} {n : ℕ} → (a b : A) → (v : Vec A n) → all-different (a ∷ b ∷ v) → all-different (b ∷ a ∷ v)
+different-swap a b v p zero          zero          i≢j li≡lj = i≢j refl
+different-swap a b v p zero          (suc zero)    i≢j li≡lj = p (suc zero) zero (λ ()) li≡lj
+different-swap a b v p zero          (suc (suc j)) i≢j li≡lj = p (suc zero) (suc (suc j)) (λ ()) li≡lj
+different-swap a b v p (suc zero)    zero          i≢j li≡lj = p zero (suc zero) (λ ()) li≡lj
+different-swap a b v p (suc zero)    (suc zero)    i≢j li≡lj = i≢j refl
+different-swap a b v p (suc zero)    (suc (suc j)) i≢j li≡lj = p zero (suc (suc j)) (λ ()) li≡lj
+different-swap a b v p (suc (suc i)) zero          i≢j li≡lj = p (suc (suc i)) (suc zero) (λ ()) li≡lj
+different-swap a b v p (suc (suc i)) (suc zero)    i≢j li≡lj = p (suc (suc i)) zero (λ ()) li≡lj
+different-swap a b v p (suc (suc i)) (suc (suc j)) i≢j li≡lj = p (suc (suc i)) (suc (suc j)) i≢j li≡lj
+
+different-drop : {A : Set} {n : ℕ} → (a : A) → (v : Vec A n) → all-different (a ∷ v) → all-different v
+different-drop a v p i j i≢j = p (suc i) (suc j) (i≢j ∘ suc-≡)
+
+different-∉ : {A : Set} {n : ℕ} → (x : A) (xs : Vec A n) → all-different (x ∷ xs) → x ∉ (toList xs)
+different-∉ x [] p ()
+different-∉ x (y ∷ ys) p (here px) = p zero (suc zero) (λ ()) px
+different-∉ x (y ∷ ys) p (there pxs) = different-∉ x ys (different-drop y (x ∷ ys) (different-swap x y ys p)) pxs
+
+different-assoc : {B : Set} {m n : ℕ} → (eq : EqInst B) → (u : Vec (Fin n) m) → (v : Vec B m) → all-different u → ∃ λ h → assoc eq u v ≡ just h
+different-assoc eq []       []       p = empty , refl
+different-assoc eq (u ∷ us) (v ∷ vs) p with different-assoc eq us vs (λ i j i≢j → p (suc i) (suc j) (i≢j ∘ suc-≡))
+different-assoc eq (u ∷ us) (v ∷ vs) p | h , p' = insert u v h , (begin
+  assoc eq (u ∷ us) (v ∷ vs)
+    ≡⟨ refl ⟩
+  assoc eq us vs >>= checkInsert eq u v
+    ≡⟨ cong (flip _>>=_ (checkInsert eq u v)) p' ⟩
+  checkInsert eq u v h
+    ≡⟨ lemma-checkInsert-new eq u v h (lemma-∉-lookupM-assoc eq u us vs h p' (different-∉ u us p)) ⟩
+  just (insert u v h) ∎)
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