feat: port Mathlib.Data.List.Duplicate (#1465)

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Co-authored-by: Moritz Firsching <firsching@google.com>
Co-authored-by: Johan Commelin <johan@commelin.net>

Author: mo271

Mathlib.lean

@@ -251,6 +251,7 @@ import Mathlib.Data.List.Card
 import Mathlib.Data.List.Chain
 import Mathlib.Data.List.Count
 import Mathlib.Data.List.Defs
+import Mathlib.Data.List.Duplicate
 import Mathlib.Data.List.Forall2
 import Mathlib.Data.List.Func
 import Mathlib.Data.List.Infix

Mathlib/Data/List/Duplicate.lean

@@ -0,0 +1,165 @@
+/-
+Copyright (c) 2021 Yakov Pechersky. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yakov Pechersky, Chris Hughes
+
+! This file was ported from Lean 3 source module data.list.duplicate
+! leanprover-community/mathlib commit 7b78d1776212a91ecc94cf601f83bdcc46b04213
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Data.List.Nodup
+
+/-!
+# List duplicates
+
+## Main definitions
+
+* `List.Duplicate x l : Prop` is an inductive property that holds when `x` is a duplicate in `l`
+
+## Implementation details
+
+In this file, `x ∈+ l` notation is shorthand for `List.Duplicate x l`.
+
+-/
+
+
+variable {α : Type _}
+
+namespace List
+
+/-- Property that an element `x : α` of `l : list α` can be found in the list more than once. -/
+inductive Duplicate (x : α) : List α → Prop
+  | cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
+  | cons_duplicate {y : α} {l : List α} : Duplicate x l → Duplicate x (y :: l)
+#align list.duplicate List.Duplicate
+
+-- mathport name: «expr ∈+ »
+local infixl:50 " ∈+ " => List.Duplicate
+
+variable {l : List α} {x : α}
+
+theorem Mem.duplicate_cons_self (h : x ∈ l) : x ∈+ x :: l :=
+  Duplicate.cons_mem h
+#align list.mem.duplicate_cons_self List.Mem.duplicate_cons_self
+
+theorem Duplicate.duplicate_cons (h : x ∈+ l) (y : α) : x ∈+ y :: l :=
+  Duplicate.cons_duplicate h
+#align list.duplicate.duplicate_cons List.Duplicate.duplicate_cons
+
+theorem Duplicate.mem (h : x ∈+ l) : x ∈ l :=
+  by
+  induction' h with l' _ y l' _ hm
+  · exact mem_cons_self _ _
+  · exact mem_cons_of_mem _ hm
+#align list.duplicate.mem List.Duplicate.mem
+
+theorem Duplicate.mem_cons_self (h : x ∈+ x :: l) : x ∈ l :=
+  by
+  cases' h with _ h _ _ h
+  · exact h
+  · exact h.mem
+#align list.duplicate.mem_cons_self List.Duplicate.mem_cons_self
+
+@[simp]
+theorem duplicate_cons_self_iff : x ∈+ x :: l ↔ x ∈ l :=
+  ⟨Duplicate.mem_cons_self, Mem.duplicate_cons_self⟩
+#align list.duplicate_cons_self_iff List.duplicate_cons_self_iff
+
+theorem Duplicate.ne_nil (h : x ∈+ l) : l ≠ [] := fun H => (mem_nil_iff x).mp (H ▸ h.mem)
+#align list.duplicate.ne_nil List.Duplicate.ne_nil
+
+@[simp]
+theorem not_duplicate_nil (x : α) : ¬x ∈+ [] := fun H => H.ne_nil rfl
+#align list.not_duplicate_nil List.not_duplicate_nil
+
+theorem Duplicate.ne_singleton (h : x ∈+ l) (y : α) : l ≠ [y] :=
+  by
+  induction' h with l' h z l' h _
+  · simp [ne_nil_of_mem h]
+  · simp [ne_nil_of_mem h.mem]
+#align list.duplicate.ne_singleton List.Duplicate.ne_singleton
+
+@[simp]
+theorem not_duplicate_singleton (x y : α) : ¬x ∈+ [y] := fun H => H.ne_singleton _ rfl
+#align list.not_duplicate_singleton List.not_duplicate_singleton
+
+theorem Duplicate.elim_nil (h : x ∈+ []) : False :=
+  not_duplicate_nil x h
+#align list.duplicate.elim_nil List.Duplicate.elim_nil
+
+theorem Duplicate.elim_singleton {y : α} (h : x ∈+ [y]) : False :=
+  not_duplicate_singleton x y h
+#align list.duplicate.elim_singleton List.Duplicate.elim_singleton
+
+theorem duplicate_cons_iff {y : α} : x ∈+ y :: l ↔ y = x ∧ x ∈ l ∨ x ∈+ l :=
+  by
+  refine' ⟨fun h => _, fun h => _⟩
+  · cases' h with _ hm _ _ hm
+    · exact Or.inl ⟨rfl, hm⟩
+    · exact Or.inr hm
+  · rcases h with (⟨rfl | h⟩ | h)
+    · simpa
+    · exact h.cons_duplicate
+#align list.duplicate_cons_iff List.duplicate_cons_iff
+
+theorem Duplicate.of_duplicate_cons {y : α} (h : x ∈+ y :: l) (hx : x ≠ y) : x ∈+ l := by
+  simpa [duplicate_cons_iff, hx.symm] using h
+#align list.duplicate.of_duplicate_cons List.Duplicate.of_duplicate_cons
+
+theorem duplicate_cons_iff_of_ne {y : α} (hne : x ≠ y) : x ∈+ y :: l ↔ x ∈+ l := by
+  simp [duplicate_cons_iff, hne.symm]
+#align list.duplicate_cons_iff_of_ne List.duplicate_cons_iff_of_ne
+
+theorem Duplicate.mono_sublist {l' : List α} (hx : x ∈+ l) (h : l <+ l') : x ∈+ l' :=
+  by
+  induction' h with l₁ l₂ y _ IH l₁ l₂ y h IH
+  · exact hx
+  · exact (IH hx).duplicate_cons _
+  · rw [duplicate_cons_iff] at hx⊢
+    rcases hx with (⟨rfl, hx⟩ | hx)
+    · simp [h.subset hx]
+    · simp [IH hx]
+#align list.duplicate.mono_sublist List.Duplicate.mono_sublist
+
+/-- The contrapositive of `List.nodup_iff_sublist`. -/
+theorem duplicate_iff_sublist : x ∈+ l ↔ [x, x] <+ l :=
+  by
+  induction' l with y l IH
+  · simp
+  · by_cases hx : x = y
+    · simp [hx, cons_sublist_cons_iff, singleton_sublist]
+    · rw [duplicate_cons_iff_of_ne hx, IH]
+      refine' ⟨sublist_cons_of_sublist y, fun h => _⟩
+      cases h
+      · assumption
+      · contradiction
+#align list.duplicate_iff_sublist List.duplicate_iff_sublist
+
+theorem nodup_iff_forall_not_duplicate : Nodup l ↔ ∀ x : α, ¬x ∈+ l := by
+  simp_rw [nodup_iff_sublist, duplicate_iff_sublist]
+#align list.nodup_iff_forall_not_duplicate List.nodup_iff_forall_not_duplicate
+
+theorem exists_duplicate_iff_not_nodup : (∃ x : α, x ∈+ l) ↔ ¬Nodup l := by
+  simp [nodup_iff_forall_not_duplicate]
+#align list.exists_duplicate_iff_not_nodup List.exists_duplicate_iff_not_nodup
+
+theorem Duplicate.not_nodup (h : x ∈+ l) : ¬Nodup l := fun H =>
+  nodup_iff_forall_not_duplicate.mp H _ h
+#align list.duplicate.not_nodup List.Duplicate.not_nodup
+
+theorem duplicate_iff_two_le_count [DecidableEq α] : x ∈+ l ↔ 2 ≤ count x l := by
+  simp [duplicate_iff_sublist, le_count_iff_replicate_sublist]
+#align list.duplicate_iff_two_le_count List.duplicate_iff_two_le_count
+
+instance decidableDuplicate [DecidableEq α] (x : α) : ∀ l : List α, Decidable (x ∈+ l)
+  | [] => isFalse (not_duplicate_nil x)
+  | y :: l =>
+    match decidableDuplicate x l with
+    | isTrue h => isTrue (h.duplicate_cons y)
+    | isFalse h =>
+      if hx : y = x ∧ x ∈ l then isTrue (hx.left.symm ▸ List.Mem.duplicate_cons_self hx.right)
+      else isFalse (by simpa [duplicate_cons_iff, h] using hx)
+#align list.decidable_duplicate List.decidableDuplicate
+
+end List