daa4fd9955
This PR reviews the implicitness of arguments across List/Array/Vector, generally trying to make arguments implicit where possible, although sometimes correcting propositional arguments which were incorrectly implicit to explicit.
255 lines
11 KiB
Text
255 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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set_option linter.listVariables true -- Enforce naming conventions for `List`/`Array`/`Vector` variables.
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set_option linter.indexVariables true -- Enforce naming conventions for index variables.
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namespace Vector
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/-! ### Lexicographic ordering -/
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@[simp] theorem lt_toArray [LT α] {xs ys : Vector α n} : xs.toArray < ys.toArray ↔ xs < ys := Iff.rfl
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@[simp] theorem le_toArray [LT α] {xs ys : Vector α n} : xs.toArray ≤ ys.toArray ↔ xs ≤ ys := Iff.rfl
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@[simp] theorem lt_toList [LT α] {xs ys : Vector α n} : xs.toList < ys.toList ↔ xs < ys := Iff.rfl
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@[simp] theorem le_toList [LT α] {xs ys : Vector α n} : xs.toList ≤ ys.toList ↔ xs ≤ ys := Iff.rfl
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protected theorem not_lt_iff_ge [LT α] {xs ys : Vector α n} : ¬ xs < ys ↔ ys ≤ xs := Iff.rfl
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protected theorem not_le_iff_gt [DecidableEq α] [LT α] [DecidableLT α] {xs ys : Vector α n} :
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¬ xs ≤ ys ↔ ys < xs :=
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Decidable.not_not
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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} {xs ys : Array α} {n₁ : xs.size = n} {n₂ : ys.size = n} :
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(Vector.mk xs n₁).lex (Vector.mk ys n₂) lt = xs.lex ys 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} {xs ys : Vector α n} :
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xs.toArray.lex ys.toArray lt = xs.lex ys lt := by
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cases xs
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cases ys
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simp
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@[simp] theorem lex_toList [BEq α] {lt : α → α → Bool} {xs ys : Vector α n} :
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xs.toList.lex ys.toList lt = xs.lex ys lt := by
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rcases xs with ⟨xs, n₁⟩
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rcases ys with ⟨ys, n₂⟩
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simp
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@[simp] theorem lex_empty
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[BEq α] {lt : α → α → Bool} (xs : Vector α 0) : xs.lex #v[] lt = false := by
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cases xs
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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)] (xs : Vector α n) : ¬ xs < xs :=
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Array.lt_irrefl xs.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] (xs : Vector α 0) : ¬ xs < #v[] := Array.not_lt_empty xs.toArray
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@[simp] theorem empty_le [LT α] (xs : Vector α 0) : #v[] ≤ xs := Array.empty_le xs.toArray
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@[simp] theorem le_empty [LT α] {xs : Vector α 0} : xs ≤ #v[] ↔ xs = #v[] := by
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cases xs
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simp
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protected theorem le_refl [LT α] [i₀ : Std.Irrefl (· < · : α → α → Prop)] (xs : Vector α n) : xs ≤ xs :=
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Array.le_refl xs.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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{xs ys zs : Vector α n} (h₁ : xs < ys) (h₂ : ys < zs) : xs < zs :=
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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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{xs ys zs : Vector α n} (h₁ : xs ≤ ys) (h₂ : ys < zs) : xs < zs :=
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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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{xs ys zs : Vector α n} (h₁ : xs ≤ ys) (h₂ : ys ≤ zs) : xs ≤ zs :=
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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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{xs ys : Vector α n} (h : xs < ys) : ¬ ys < xs := 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)] (xs ys : Vector α n) : xs ≤ ys ∨ ys ≤ xs :=
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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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{xs ys : Vector α n} : ¬ xs < ys ↔ ys ≤ xs := Iff.rfl
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@[simp] protected theorem not_le [DecidableEq α] [LT α] [DecidableLT α]
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{xs ys : Vector α n} : ¬ ys ≤ xs ↔ xs < ys := 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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{xs ys : Vector α n} (h : xs < ys) : xs ≤ ys :=
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Array.le_of_lt h
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protected 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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{xs ys : Vector α n} : xs ≤ ys ↔ xs < ys ∨ xs = ys := by
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simpa using Array.le_iff_lt_or_eq (xs := xs.toArray) (ys := ys.toArray)
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@[simp] theorem lex_eq_true_iff_lt [DecidableEq α] [LT α] [DecidableLT α]
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{xs ys : Vector α n} : lex xs ys = true ↔ xs < ys := by
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cases xs
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cases ys
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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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{xs ys : Vector α n} : lex xs ys = false ↔ ys ≤ xs := by
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cases xs
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cases ys
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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 xs ys => decidable_of_iff (lex xs ys = 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 xs ys => decidable_of_iff (lex ys xs = false) lex_eq_false_iff_ge
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/--
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`xs` is lexicographically less than `ys` if
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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) {xs ys : Vector α n} :
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lex xs ys lt = true ↔
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(∃ (i : Nat) (h : i < n), (∀ j, (hj : j < i) → xs[j] == ys[j]) ∧ lt xs[i] ys[i]) := by
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rcases xs with ⟨xs, n₁⟩
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rcases ys with ⟨ys, 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` 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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{xs ys : Vector α n} :
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lex xs ys lt = false ↔
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(ys.isEqv xs (· == ·)) ∨
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(∃ (i : Nat) (h : i < n),(∀ j, (hj : j < i) → xs[j] == ys[j]) ∧ lt ys[i] xs[i]) := by
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rcases xs with ⟨xs, rfl⟩
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rcases ys with ⟨ys, n₂⟩
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simp_all [Array.lex_eq_false_iff_exists, n₂]
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protected theorem lt_iff_exists [DecidableEq α] [LT α] [DecidableLT α] {xs ys : Vector α n} :
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xs < ys ↔
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(∃ (i : Nat) (h : i < n), (∀ j, (hj : j < i) → xs[j] = ys[j]) ∧ xs[i] < ys[i]) := by
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cases xs
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cases ys
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simp_all [Array.lt_iff_exists]
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protected 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)] {xs ys : Vector α n} :
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xs ≤ ys ↔
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(xs = ys) ∨
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(∃ (i : Nat) (h : i < n), (∀ j, (hj : j < i) → xs[j] = ys[j]) ∧ xs[i] < ys[i]) := by
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rcases xs with ⟨xs, rfl⟩
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rcases ys with ⟨ys, n₂⟩
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simp [Array.le_iff_exists, ← n₂]
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theorem append_left_lt [LT α] {xs : Vector α n} {ys ys' : Vector α m} (h : ys < ys') :
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xs ++ ys < xs ++ ys' := 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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{xs : Vector α n} {ys ys' : Vector α m} (h : ys ≤ ys') :
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xs ++ ys ≤ xs ++ ys' := by
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simpa using Array.append_left_le h
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protected theorem map_lt [LT α] [LT β]
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{xs ys : Vector α n} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : xs < ys) :
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map f xs < map f ys := by
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simpa using Array.map_lt w h
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protected 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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{xs ys : Vector α n} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : xs ≤ ys) :
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map f xs ≤ map f ys := by
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simpa using Array.map_le w h
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end Vector
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