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feat: add lemmas about List.IsPrefix (#5448)

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Add iff version of `List.IsPrefix.getElem`, and `eq_of_length_le`
variants of `List.IsInfix.eq_of_length, List.IsPrefix.eq_of_length,
List.IsSuffix.eq_of_length`
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Daniel Weber 2024-09-26 09:58:40 +03:00 • committed by GitHub
parent 0196bca784
commit 3d1ac7cfa2
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27
src/Init/Data/List/Sublist.lean
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@ -725,12 +725,21 @@ theorem infix_iff_suffix_prefix {l₁ l₂ : List α} : l₁ <:+: l₂ ↔ ∃ t
theorem IsInfix.eq_of_length (h : l₁ <:+: l₂) : l₁.length = l₂.length → l₁ = l₂ :=
h.sublist.eq_of_length
theorem IsInfix.eq_of_length_le (h : l₁ <:+: l₂) : l₂.length ≤ l₁.length → l₁ = l₂ :=
h.sublist.eq_of_length_le
theorem IsPrefix.eq_of_length (h : l₁ <+: l₂) : l₁.length = l₂.length → l₁ = l₂ :=
h.sublist.eq_of_length
theorem IsPrefix.eq_of_length_le (h : l₁ <+: l₂) : l₂.length ≤ l₁.length → l₁ = l₂ :=
h.sublist.eq_of_length_le
theorem IsSuffix.eq_of_length (h : l₁ <:+ l₂) : l₁.length = l₂.length → l₁ = l₂ :=
h.sublist.eq_of_length
theorem IsSuffix.eq_of_length_le (h : l₁ <:+ l₂) : l₂.length ≤ l₁.length → l₁ = l₂ :=
h.sublist.eq_of_length_le
theorem prefix_of_prefix_length_le :
∀ {l₁ l₂ l₃ : List α}, l₁ <+: l₃ → l₂ <+: l₃ → length l₁ ≤ length l₂ → l₁ <+: l₂
| [], l₂, _, _, _, _ => nil_prefix
@ -829,6 +838,24 @@ theorem isPrefix_iff : l₁ <+: l₂ ↔ ∀ i (h : i < l₁.length), l₂[i]? =
rw (config := {occs := .pos [2]}) [← Nat.and_forall_add_one]
simp [Nat.succ_lt_succ_iff, eq_comm]
theorem isPrefix_iff_getElem {l₁ l₂ : List α} :
l₁ <+: l₂ ↔ ∃ (h : l₁.length ≤ l₂.length), ∀ x (hx : x < l₁.length),
l₁[x] = l₂[x]'(Nat.lt_of_lt_of_le hx h) where
mp h := ⟨h.length_le, fun _ _ ↦ h.getElem _⟩
mpr h := by
obtain ⟨hl, h⟩ := h
induction l₂ generalizing l₁ with
| nil =>
simpa using hl
| cons _ _ tail_ih =>
cases l₁ with
| nil =>
exact nil_prefix
| cons _ _ =>
simp only [length_cons, Nat.add_le_add_iff_right, Fin.getElem_fin] at hl h
simp only [cons_prefix_cons]
exact ⟨h 0 (zero_lt_succ _), tail_ih hl fun a ha ↦ h a.succ (succ_lt_succ ha)⟩
-- See `Init.Data.List.Nat.Sublist` for `isSuffix_iff` and `ifInfix_iff`.
theorem isPrefix_filterMap_iff {β} {f : α → Option β} {l₁ : List α} {l₂ : List β} :