249 lines
11 KiB
Text
249 lines
11 KiB
Text
/-
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Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Kim Morrison
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-/
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prelude
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import Init.Data.Vector.Basic
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import Init.Data.Vector.Lemmas
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import Init.Data.Array.Lex.Lemmas
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namespace Vector
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/-! ### Lexicographic ordering -/
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@[simp] theorem lt_toArray [LT α] (l₁ l₂ : Vector α n) : l₁.toArray < l₂.toArray ↔ l₁ < l₂ := Iff.rfl
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@[simp] theorem le_toArray [LT α] (l₁ l₂ : Vector α n) : l₁.toArray ≤ l₂.toArray ↔ l₁ ≤ l₂ := Iff.rfl
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@[simp] theorem lt_toList [LT α] (l₁ l₂ : Vector α n) : l₁.toList < l₂.toList ↔ l₁ < l₂ := Iff.rfl
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@[simp] theorem le_toList [LT α] (l₁ l₂ : Vector α n) : l₁.toList ≤ l₂.toList ↔ l₁ ≤ l₂ := Iff.rfl
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@[simp] theorem mk_lt_mk [LT α] :
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Vector.mk (α := α) (n := n) data₁ size₁ < Vector.mk data₂ size₂ ↔ data₁ < data₂ := Iff.rfl
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@[simp] theorem mk_le_mk [LT α] :
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Vector.mk (α := α) (n := n) data₁ size₁ ≤ Vector.mk data₂ size₂ ↔ data₁ ≤ data₂ := Iff.rfl
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@[simp] theorem mk_lex_mk [BEq α] (lt : α → α → Bool) {l₁ l₂ : Array α} {n₁ : l₁.size = n} {n₂ : l₂.size = n} :
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(Vector.mk l₁ n₁).lex (Vector.mk l₂ n₂) lt = l₁.lex l₂ lt := by
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simp [Vector.lex, Array.lex, n₁, n₂]
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rfl
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@[simp] theorem lex_toArray [BEq α] (lt : α → α → Bool) (l₁ l₂ : Vector α n) :
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l₁.toArray.lex l₂.toArray lt = l₁.lex l₂ lt := by
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cases l₁
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cases l₂
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simp
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@[simp] theorem lex_toList [BEq α] (lt : α → α → Bool) (l₁ l₂ : Vector α n) :
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l₁.toList.lex l₂.toList lt = l₁.lex l₂ lt := by
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rcases l₁ with ⟨⟨l₁⟩, n₁⟩
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rcases l₂ with ⟨⟨l₂⟩, n₂⟩
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simp
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@[simp] theorem lex_empty
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[BEq α] {lt : α → α → Bool} (l₁ : Vector α 0) : l₁.lex #v[] lt = false := by
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cases l₁
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simp_all
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@[simp] theorem singleton_lex_singleton [BEq α] {lt : α → α → Bool} : #v[a].lex #v[b] lt = lt a b := by
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simp only [lex, getElem_mk, List.getElem_toArray, List.getElem_singleton]
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cases lt a b <;> cases a != b <;> simp [Id.run]
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protected theorem lt_irrefl [LT α] [Std.Irrefl (· < · : α → α → Prop)] (l : Vector α n) : ¬ l < l :=
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Array.lt_irrefl l.toArray
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instance ltIrrefl [LT α] [Std.Irrefl (· < · : α → α → Prop)] : Std.Irrefl (α := Vector α n) (· < ·) where
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irrefl := Vector.lt_irrefl
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@[simp] theorem not_lt_empty [LT α] (l : Vector α 0) : ¬ l < #v[] := Array.not_lt_empty l.toArray
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@[simp] theorem empty_le [LT α] (l : Vector α 0) : #v[] ≤ l := Array.empty_le l.toArray
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@[simp] theorem le_empty [LT α] (l : Vector α 0) : l ≤ #v[] ↔ l = #v[] := by
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cases l
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simp
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protected theorem le_refl [LT α] [i₀ : Std.Irrefl (· < · : α → α → Prop)] (l : Vector α n) : l ≤ l :=
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Array.le_refl l.toArray
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instance [LT α] [Std.Irrefl (· < · : α → α → Prop)] : Std.Refl (· ≤ · : Vector α n → Vector α n → Prop) where
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refl := Vector.le_refl
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protected theorem lt_trans [LT α]
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[i₁ : Trans (· < · : α → α → Prop) (· < ·) (· < ·)]
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{l₁ l₂ l₃ : Vector α n} (h₁ : l₁ < l₂) (h₂ : l₂ < l₃) : l₁ < l₃ :=
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Array.lt_trans h₁ h₂
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instance [LT α]
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[Trans (· < · : α → α → Prop) (· < ·) (· < ·)] :
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Trans (· < · : Vector α n → Vector α n → Prop) (· < ·) (· < ·) where
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trans h₁ h₂ := Vector.lt_trans h₁ h₂
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protected theorem lt_of_le_of_lt [DecidableEq α] [LT α] [DecidableLT α]
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[i₀ : Std.Irrefl (· < · : α → α → Prop)]
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[i₁ : Std.Asymm (· < · : α → α → Prop)]
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[i₂ : Std.Antisymm (¬ · < · : α → α → Prop)]
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[i₃ : Trans (¬ · < · : α → α → Prop) (¬ · < ·) (¬ · < ·)]
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{l₁ l₂ l₃ : Vector α n} (h₁ : l₁ ≤ l₂) (h₂ : l₂ < l₃) : l₁ < l₃ :=
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Array.lt_of_le_of_lt h₁ h₂
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protected theorem le_trans [DecidableEq α] [LT α] [DecidableLT α]
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[Std.Irrefl (· < · : α → α → Prop)]
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[Std.Asymm (· < · : α → α → Prop)]
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[Std.Antisymm (¬ · < · : α → α → Prop)]
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[Trans (¬ · < · : α → α → Prop) (¬ · < ·) (¬ · < ·)]
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{l₁ l₂ l₃ : Vector α n} (h₁ : l₁ ≤ l₂) (h₂ : l₂ ≤ l₃) : l₁ ≤ l₃ :=
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fun h₃ => h₁ (Vector.lt_of_le_of_lt h₂ h₃)
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instance [DecidableEq α] [LT α] [DecidableLT α]
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[Std.Irrefl (· < · : α → α → Prop)]
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[Std.Asymm (· < · : α → α → Prop)]
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[Std.Antisymm (¬ · < · : α → α → Prop)]
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[Trans (¬ · < · : α → α → Prop) (¬ · < ·) (¬ · < ·)] :
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Trans (· ≤ · : Vector α n → Vector α n → Prop) (· ≤ ·) (· ≤ ·) where
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trans h₁ h₂ := Vector.le_trans h₁ h₂
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protected theorem lt_asymm [LT α]
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[i : Std.Asymm (· < · : α → α → Prop)]
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{l₁ l₂ : Vector α n} (h : l₁ < l₂) : ¬ l₂ < l₁ := Array.lt_asymm h
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instance [DecidableEq α] [LT α] [DecidableLT α]
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[Std.Asymm (· < · : α → α → Prop)] :
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Std.Asymm (· < · : Vector α n → Vector α n → Prop) where
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asymm _ _ := Vector.lt_asymm
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protected theorem le_total [DecidableEq α] [LT α] [DecidableLT α]
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[i : Std.Total (¬ · < · : α → α → Prop)] (l₁ l₂ : Vector α n) : l₁ ≤ l₂ ∨ l₂ ≤ l₁ :=
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Array.le_total _ _
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instance [DecidableEq α] [LT α] [DecidableLT α]
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[Std.Total (¬ · < · : α → α → Prop)] :
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Std.Total (· ≤ · : Vector α n → Vector α n → Prop) where
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total := Vector.le_total
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@[simp] protected theorem not_lt [LT α]
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{l₁ l₂ : Vector α n} : ¬ l₁ < l₂ ↔ l₂ ≤ l₁ := Iff.rfl
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@[simp] protected theorem not_le [DecidableEq α] [LT α] [DecidableLT α]
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{l₁ l₂ : Vector α n} : ¬ l₂ ≤ l₁ ↔ l₁ < l₂ := Decidable.not_not
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protected theorem le_of_lt [DecidableEq α] [LT α] [DecidableLT α]
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[i : Std.Total (¬ · < · : α → α → Prop)]
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{l₁ l₂ : Vector α n} (h : l₁ < l₂) : l₁ ≤ l₂ :=
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Array.le_of_lt h
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theorem le_iff_lt_or_eq [DecidableEq α] [LT α] [DecidableLT α]
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[Std.Irrefl (· < · : α → α → Prop)]
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[Std.Antisymm (¬ · < · : α → α → Prop)]
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[Std.Total (¬ · < · : α → α → Prop)]
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{l₁ l₂ : Vector α n} : l₁ ≤ l₂ ↔ l₁ < l₂ ∨ l₁ = l₂ := by
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simpa using Array.le_iff_lt_or_eq (l₁ := l₁.toArray) (l₂ := l₂.toArray)
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@[simp] theorem lex_eq_true_iff_lt [DecidableEq α] [LT α] [DecidableLT α]
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{l₁ l₂ : Vector α n} : lex l₁ l₂ = true ↔ l₁ < l₂ := by
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cases l₁
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cases l₂
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simp
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@[simp] theorem lex_eq_false_iff_ge [DecidableEq α] [LT α] [DecidableLT α]
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{l₁ l₂ : Vector α n} : lex l₁ l₂ = false ↔ l₂ ≤ l₁ := by
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cases l₁
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cases l₂
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simp [Array.not_lt_iff_ge]
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instance [DecidableEq α] [LT α] [DecidableLT α] : DecidableLT (Vector α n) :=
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fun l₁ l₂ => decidable_of_iff (lex l₁ l₂ = true) lex_eq_true_iff_lt
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instance [DecidableEq α] [LT α] [DecidableLT α] : DecidableLE (Vector α n) :=
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fun l₁ l₂ => decidable_of_iff (lex l₂ l₁ = false) lex_eq_false_iff_ge
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/--
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`l₁` is lexicographically less than `l₂` if either
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- `l₁` is pairwise equivalent under `· == ·` to `l₂.take l₁.size`,
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and `l₁` is shorter than `l₂` or
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- there exists an index `i` such that
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- for all `j < i`, `l₁[j] == l₂[j]` and
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- `l₁[i] < l₂[i]`
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-/
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theorem lex_eq_true_iff_exists [BEq α] (lt : α → α → Bool) {l₁ l₂ : Vector α n} :
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lex l₁ l₂ lt = true ↔
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(∃ (i : Nat) (h : i < n), (∀ j, (hj : j < i) → l₁[j] == l₂[j]) ∧ lt l₁[i] l₂[i]) := by
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rcases l₁ with ⟨l₁, n₁⟩
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rcases l₂ with ⟨l₂, n₂⟩
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simp [Array.lex_eq_true_iff_exists, n₁, n₂]
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/--
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`l₁` is *not* lexicographically less than `l₂`
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(which you might think of as "`l₂` is lexicographically greater than or equal to `l₁`"") if either
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- `l₁` is pairwise equivalent under `· == ·` to `l₂.take l₁.length` or
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- there exists an index `i` such that
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- for all `j < i`, `l₁[j] == l₂[j]` and
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- `l₂[i] < l₁[i]`
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This formulation requires that `==` and `lt` are compatible in the following senses:
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- `==` is symmetric
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(we unnecessarily further assume it is transitive, to make use of the existing typeclasses)
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- `lt` is irreflexive with respect to `==` (i.e. if `x == y` then `lt x y = false`
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- `lt` is asymmmetric (i.e. `lt x y = true → lt y x = false`)
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- `lt` is antisymmetric with respect to `==` (i.e. `lt x y = false → lt y x = false → x == y`)
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-/
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theorem lex_eq_false_iff_exists [BEq α] [PartialEquivBEq α] (lt : α → α → Bool)
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(lt_irrefl : ∀ x y, x == y → lt x y = false)
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(lt_asymm : ∀ x y, lt x y = true → lt y x = false)
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(lt_antisymm : ∀ x y, lt x y = false → lt y x = false → x == y)
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{l₁ l₂ : Vector α n} :
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lex l₁ l₂ lt = false ↔
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(l₂.isEqv l₁ (· == ·)) ∨
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(∃ (i : Nat) (h : i < n),(∀ j, (hj : j < i) → l₁[j] == l₂[j]) ∧ lt l₂[i] l₁[i]) := by
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rcases l₁ with ⟨l₁, rfl⟩
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rcases l₂ with ⟨l₂, n₂⟩
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simp_all [Array.lex_eq_false_iff_exists, n₂]
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theorem lt_iff_exists [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : Vector α n} :
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l₁ < l₂ ↔
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(∃ (i : Nat) (h : i < n), (∀ j, (hj : j < i) → l₁[j] = l₂[j]) ∧ l₁[i] < l₂[i]) := by
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cases l₁
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cases l₂
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simp_all [Array.lt_iff_exists]
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theorem le_iff_exists [DecidableEq α] [LT α] [DecidableLT α]
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[Std.Irrefl (· < · : α → α → Prop)]
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[Std.Asymm (· < · : α → α → Prop)]
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[Std.Antisymm (¬ · < · : α → α → Prop)] {l₁ l₂ : Vector α n} :
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l₁ ≤ l₂ ↔
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(l₁ = l₂) ∨
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(∃ (i : Nat) (h : i < n), (∀ j, (hj : j < i) → l₁[j] = l₂[j]) ∧ l₁[i] < l₂[i]) := by
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rcases l₁ with ⟨l₁, rfl⟩
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rcases l₂ with ⟨l₂, n₂⟩
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simp [Array.le_iff_exists, ← n₂]
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theorem append_left_lt [LT α] {l₁ : Vector α n} {l₂ l₃ : Vector α m} (h : l₂ < l₃) :
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l₁ ++ l₂ < l₁ ++ l₃ := by
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simpa using Array.append_left_lt h
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theorem append_left_le [DecidableEq α] [LT α] [DecidableLT α]
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[Std.Irrefl (· < · : α → α → Prop)]
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[Std.Asymm (· < · : α → α → Prop)]
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[Std.Antisymm (¬ · < · : α → α → Prop)]
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{l₁ : Vector α n} {l₂ l₃ : Vector α m} (h : l₂ ≤ l₃) :
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l₁ ++ l₂ ≤ l₁ ++ l₃ := by
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simpa using Array.append_left_le h
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theorem map_lt [LT α] [LT β]
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{l₁ l₂ : Vector α n} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ < l₂) :
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map f l₁ < map f l₂ := by
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simpa using Array.map_lt w h
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theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β] [DecidableLT β]
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[Std.Irrefl (· < · : α → α → Prop)]
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[Std.Asymm (· < · : α → α → Prop)]
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[Std.Antisymm (¬ · < · : α → α → Prop)]
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[Std.Irrefl (· < · : β → β → Prop)]
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[Std.Asymm (· < · : β → β → Prop)]
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[Std.Antisymm (¬ · < · : β → β → Prop)]
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{l₁ l₂ : Vector α n} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ ≤ l₂) :
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map f l₁ ≤ map f l₂ := by
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simpa using Array.map_le w h
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end Vector
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