feat: Port Data.List.Lemmas (#1351)

Mathlib.lean

@@ -237,6 +237,7 @@ import Mathlib.Data.List.Card
 import Mathlib.Data.List.Chain
 import Mathlib.Data.List.Defs
 import Mathlib.Data.List.Func
+import Mathlib.Data.List.Lemmas
 import Mathlib.Data.List.Lex
 import Mathlib.Data.List.Nodup
 import Mathlib.Data.List.Pairwise

Mathlib/Data/List/Lemmas.lean

@@ -0,0 +1,74 @@
+/-
+Copyright (c) 2021 Yakov Pechersky. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yakov Pechersky, Yury Kudryashov
+
+! This file was ported from Lean 3 source module data.list.lemmas
+! leanprover-community/mathlib commit 6d0adfa76594f304b4650d098273d4366edeb61b
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Data.Set.Function
+import Mathlib.Data.List.Basic
+
+/-! # Some lemmas about lists involving sets
+
+Split out from `Data.List.Basic` to reduce its dependencies.
+-/
+
+
+open List
+
+variable {α β : Type _}
+
+namespace List
+
+theorem range_map (f : α → β) : Set.range (map f) = { l | ∀ x ∈ l, x ∈ Set.range f } :=
+  by
+  refine'
+    Set.Subset.antisymm
+      (Set.range_subset_iff.2 fun l => forall_mem_map_iff.2 fun y _ => Set.mem_range_self _)
+      fun l hl => _
+  induction' l with a l ihl; · exact ⟨[], rfl⟩
+  rcases ihl fun x hx => hl x <| subset_cons _ _ hx with ⟨l, rfl⟩
+  rcases hl a (mem_cons_self _ _) with ⟨a, rfl⟩
+  exact ⟨a :: l, map_cons _ _ _⟩
+#align list.range_map List.range_map
+
+theorem range_map_coe (s : Set α) : Set.range (map ((↑) : s → α)) = { l | ∀ x ∈ l, x ∈ s } := by
+  rw [range_map, Subtype.range_coe]
+#align list.range_map_coe List.range_map_coe
+
+--Porting note: Waiting for `lift` tactic
+-- /-- If each element of a list can be lifted to some type, then the whole list can be
+-- lifted to this type. -/
+-- instance canLift (c) (p) [CanLift α β c p] :
+--     CanLift (List α) (List β) (List.map c) fun l => ∀ x ∈ l, p x
+--     where
+--     prf l H := by
+--       rw [← Set.mem_range, range_map]
+--       exact fun a ha => CanLift.prf a (H a ha)
+-- #align list.can_lift List.canLift
+
+theorem injOn_insertNth_index_of_not_mem (l : List α) (x : α) (hx : x ∉ l) :
+    Set.InjOn (fun k => insertNth k x l) { n | n ≤ l.length } := by
+  induction' l with hd tl IH
+  · intro n hn m hm _
+    simp only [Set.mem_singleton_iff, Set.setOf_eq_eq_singleton,
+      length, nonpos_iff_eq_zero] at hn hm
+    simp [hn, hm]
+  · intro n hn m hm h
+    simp only [length, Set.mem_setOf_eq] at hn hm
+    simp only [mem_cons, not_or] at hx
+    cases n <;> cases m
+    · rfl
+    · simp [hx.left] at h
+    · simp [Ne.symm hx.left] at h
+    · simp only [true_and_iff, eq_self_iff_true, insertNth_succ_cons] at h
+      rw [Nat.succ_inj']
+      refine' IH hx.right _ _ (by injection h)
+      · simpa [Nat.succ_le_succ_iff] using hn
+      · simpa [Nat.succ_le_succ_iff] using hm
+#align list.inj_on_insert_nth_index_of_not_mem List.injOn_insertNth_index_of_not_mem
+
+end List