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feat: add lemmas about lists #993

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feat: add lemmas about lists #993

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e5dd0d8
feat: add lemmas about lists
chabulhwi Mar 18, 2024
06dd197
feat: add more lemmas about lists
chabulhwi Oct 14, 2024
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18 changes: 18 additions & 0 deletions Batteries/Data/List/Lemmas.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -19,6 +19,15 @@ namespace List

@[deprecated (since := "2024-08-15")] alias isEmpty_iff_eq_nil := isEmpty_iff

/-! ### head and tail -/

theorem head_cons_tail : ∀ (l : List α) (hne : l ≠ []), l.head hne :: l.tail = l
| _::_, _ => rfl

theorem singleton_head_eq_self (l : List α) (hne : l ≠ []) (htl : l.tail = []) :
[l.head hne] = l := by
conv => rhs; rw [← head_cons_tail l hne, htl]

/-! ### next? -/

@[simp] theorem next?_nil : @next? α [] = none := rfl
Expand Down Expand Up @@ -482,6 +491,15 @@ theorem Sublist.erase_diff_erase_sublist {a : α} :

end Diff

/-! ### prefix, suffix, infix -/

theorem singleton_prefix_cons (a) : [a] <+: a :: l :=
(prefix_cons_inj a).mpr nil_prefix

theorem ne_nil_of_not_prefix (h : ¬l₁ <+: l₂) : l₁ ≠ [] := by
intro heq
simp [heq, nil_prefix] at h

/-! ### drop -/

theorem disjoint_take_drop : ∀ {l : List α}, l.Nodup → m ≤ n → Disjoint (l.take m) (l.drop n)
Expand Down