From 07ff0900aae330cbab35f33154691f0f5b1409ac Mon Sep 17 00:00:00 2001
From: Leonardo de Moura <leonardo@microsoft.com>
Date: Wed, 8 Apr 2015 19:02:35 -0700
Subject: [PATCH] feat(library/data/list): add permutation theorems for union
 and insert

---
 library/data/list/basic.lean |  5 ++++
 library/data/list/perm.lean  | 55 ++++++++++++++++++++++++++++++++++++
 2 files changed, 60 insertions(+)

diff --git a/library/data/list/basic.lean b/library/data/list/basic.lean
index 90c79990f..96b37d4c8 100644
--- a/library/data/list/basic.lean
+++ b/library/data/list/basic.lean
@@ -188,6 +188,11 @@ iff.rfl
 theorem mem_or_mem_of_mem_cons {x y : T} {l : list T} : x ∈ y::l → x = y ∨ x ∈ l :=
 assume h, h
 
+theorem mem_of_mem_cons_of_mem {a b : T} {l : list T} : a ∈ b::l → b ∈ l → a ∈ l :=
+assume ainbl binl, or.elim (mem_or_mem_of_mem_cons ainbl)
+  (λ aeqb : a = b, by rewrite [aeqb]; exact binl)
+  (λ ainl : a ∈ l, ainl)
+
 theorem mem_or_mem_of_mem_append {x : T} {s t : list T} : x ∈ s ++ t → x ∈ s ∨ x ∈ t :=
 list.induction_on s or.inr
   (take y s,
diff --git a/library/data/list/perm.lean b/library/data/list/perm.lean
index ff89b9f01..86c64de24 100644
--- a/library/data/list/perm.lean
+++ b/library/data/list/perm.lean
@@ -77,6 +77,9 @@ assume p, perm.induction_on p
       (assume ainl : a ∈ l, or.inr (or.inr ainl))))
   (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ ainl₁, r₂ (r₁ ainl₁))
 
+theorem not_mem_perm {a : A} {l₁ l₂ : list A} : l₁ ~ l₂ → a ∉ l₁ → a ∉ l₂ :=
+assume p nainl₁ ainl₂, absurd (mem_perm (symm p) ainl₂) nainl₁
+
 theorem perm_app_left {l₁ l₂ : list A} (t₁ : list A) : l₁ ~ l₂ → (l₁++t₁) ~ (l₂++t₁) :=
 assume p, perm.induction_on p
   !refl
@@ -565,4 +568,56 @@ assume p, perm_induction_on p
             exact (xswap x y r)
           end)))
   (λ t₁ t₂ t₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
+
+section perm_union
+variable [H : decidable_eq A]
+include H
+
+theorem perm_union_left {l₁ l₂ : list A} (t₁ : list A) : l₁ ~ l₂ → (union l₁ t₁) ~ (union l₂ t₁) :=
+assume p, perm.induction_on p
+  (by rewrite [union_nil]; exact !refl)
+  (λ x l₁ l₂ p₁ r₁, by_cases
+     (λ xint₁  : x ∈ t₁, by rewrite [*union_cons_of_mem _ xint₁]; exact r₁)
+     (λ nxint₁ : x ∉ t₁, by rewrite [*union_cons_of_not_mem _ nxint₁]; exact (skip _ r₁)))
+  (λ x y l, by_cases
+    (λ yint  : y ∈ t₁, by_cases
+      (λ xint  : x ∈ t₁,
+        by rewrite [*union_cons_of_mem _ xint, *union_cons_of_mem _ yint, *union_cons_of_mem _ xint]; exact !refl)
+      (λ nxint : x ∉ t₁,
+        by rewrite [*union_cons_of_mem _ yint, *union_cons_of_not_mem _ nxint, union_cons_of_mem _ yint]; exact !refl))
+    (λ nyint : y ∉ t₁, by_cases
+      (λ xint  : x ∈ t₁,
+        by rewrite [*union_cons_of_mem _ xint, *union_cons_of_not_mem _ nyint, union_cons_of_mem _ xint]; exact !refl)
+      (λ nxint : x ∉ t₁,
+        by rewrite [*union_cons_of_not_mem _ nxint, *union_cons_of_not_mem _ nyint, union_cons_of_not_mem _ nxint]; exact !swap)))
+  (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
+
+theorem perm_union_right (l : list A) {t₁ t₂ : list A} : t₁ ~ t₂ → (union l t₁) ~ (union l t₂) :=
+list.induction_on l
+  (λ p, by rewrite [*union_nil]; exact p)
+  (λ x xs r p, by_cases
+    (λ xint₁  : x ∈ t₁,
+      assert xint₂ : x ∈ t₂, from mem_perm p xint₁,
+      by rewrite [union_cons_of_mem _ xint₁, union_cons_of_mem _ xint₂]; exact (r p))
+    (λ nxint₁ : x ∉ t₁,
+      assert nxint₂ : x ∉ t₂, from not_mem_perm p nxint₁,
+      by rewrite [union_cons_of_not_mem _ nxint₁, union_cons_of_not_mem _ nxint₂]; exact (skip _ (r p))))
+
+theorem perm_union {l₁ l₂ t₁ t₂ : list A} : l₁ ~ l₂ → t₁ ~ t₂ → (union l₁ t₁) ~ (union l₂ t₂) :=
+assume p₁ p₂, trans (perm_union_left t₁ p₁) (perm_union_right l₂ p₂)
+end perm_union
+
+section perm_insert
+variable [H : decidable_eq A]
+include H
+
+theorem perm_insert (a : A) {l₁ l₂ : list A} : l₁ ~ l₂ → (insert a l₁) ~ (insert a l₂) :=
+assume p, by_cases
+ (λ ainl₁  : a ∈ l₁,
+   assert ainl₂ : a ∈ l₂, from mem_perm p ainl₁,
+   by rewrite [insert_eq_of_mem ainl₁, insert_eq_of_mem ainl₂]; exact p)
+ (λ nainl₁ : a ∉ l₁,
+   assert nainl₂ : a ∉ l₂, from not_mem_perm p nainl₁,
+   by rewrite [insert_eq_of_non_mem nainl₁, insert_eq_of_non_mem nainl₂]; exact (skip _ p))
+end perm_insert
 end perm