Progress towards UFDs (#88)

Lists/Permutations.agda

@@ -0,0 +1,45 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Sets.EquivalenceRelations
+open import Functions
+open import Lists.Definition
+open import Lists.Fold.Fold
+open import Lists.Concat
+open import Lists.Length
+open import Setoids.Setoids
+open import Maybe
+open import Numbers.Naturals.Semiring
+
+module Lists.Permutations {a b : _} {A : Set a} (S : Setoid {a} {b} A) (decidable : (x y : A) → (Setoid._∼_ S x y) || ((Setoid._∼_ S x y) → False)) where
+
+open Setoid S
+open Equivalence eq
+
+indexOf : (a : A) → (l : List A) → Maybe ℕ
+indexOf a [] = no
+indexOf a (x :: l) with decidable a x
+... | inl eq = yes 0
+... | inr noneq = mapMaybe succ (indexOf a l)
+
+eltAt : (n : ℕ) → (l : List A) → Maybe A
+eltAt n [] = no
+eltAt zero (x :: l) = yes x
+eltAt (succ n) (x :: l) = eltAt n l
+
+indexOfIsIndexOf : (a : A) → (l : List A) → {n : ℕ} → indexOf a l ≡ yes n → Sg A (λ b → (eltAt n l ≡ yes b) && (b ∼ a))
+indexOfIsIndexOf a (x :: l) {n} ind with decidable a x
+indexOfIsIndexOf a (x :: l) {.0} refl | inl a=x = x , (refl ,, symmetric a=x)
+indexOfIsIndexOf a (x :: l) {zero} ind | inr a!=x with indexOf a l
+indexOfIsIndexOf a (x :: l) {zero} () | inr a!=x | no
+indexOfIsIndexOf a (x :: l) {zero} () | inr a!=x | yes _
+indexOfIsIndexOf a (x :: l) {succ n} ind | inr a!=x with mapMaybePreservesYes ind
+... | m , (pr ,, z=n) = indexOfIsIndexOf a l (transitivity pr (applyEquality yes (succInjective z=n)))
+
+Permutation : (List A) → (List A) → Set
+Permutation [] [] = True
+Permutation [] (x :: l2) = False
+Permutation (x :: l1) [] = False
+Permutation (a :: as) (b :: bs) with decidable a b
+Permutation (a :: as) (b :: bs) | inl a=b = Permutation as bs
+Permutation (a :: as) (b :: bs) | inr a!=b = {!!}