feat: lemmas about List/Array/Vector lexicographic order (#6423)
This PR adds missing lemmas about lexicographic order on List/Array/Vector.
This commit is contained in:
Kim Morrison
GitHub
parent
7b0b1909f1
commit
e06673e200
6 changed files with 233 additions and 28 deletions
|
|
@ -27,7 +27,7 @@ namespace Array
|
|||
|
||||
/-! ### toList -/
|
||||
|
||||
theorem toList_inj {a b : Array α} : a.toList = b.toList ↔ a = b := by
|
||||
@[simp] theorem toList_inj {a b : Array α} : a.toList = b.toList ↔ a = b := by
|
||||
cases a; cases b; simp
|
||||
|
||||
@[simp] theorem toList_eq_nil_iff (l : Array α) : l.toList = [] ↔ l = #[] := by
|
||||
|
|
|
|||
|
|
@ -11,8 +11,11 @@ namespace Array
|
|||
|
||||
/-! ### Lexicographic ordering -/
|
||||
|
||||
@[simp] theorem lt_toArray [LT α] (l₁ l₂ : List α) : l₁.toArray < l₂.toArray ↔ l₁ < l₂ := Iff.rfl
|
||||
@[simp] theorem le_toArray [LT α] (l₁ l₂ : List α) : l₁.toArray ≤ l₂.toArray ↔ l₁ ≤ l₂ := Iff.rfl
|
||||
@[simp] theorem _root_.List.lt_toArray [LT α] (l₁ l₂ : List α) : l₁.toArray < l₂.toArray ↔ l₁ < l₂ := Iff.rfl
|
||||
@[simp] theorem _root_.List.le_toArray [LT α] (l₁ l₂ : List α) : l₁.toArray ≤ l₂.toArray ↔ l₁ ≤ l₂ := Iff.rfl
|
||||
|
||||
@[simp] theorem lt_toList [LT α] (l₁ l₂ : Array α) : l₁.toList < l₂.toList ↔ l₁ < l₂ := Iff.rfl
|
||||
@[simp] theorem le_toList [LT α] (l₁ l₂ : Array α) : l₁.toList ≤ l₂.toList ↔ l₁ ≤ l₂ := Iff.rfl
|
||||
|
||||
theorem not_lt_iff_ge [LT α] (l₁ l₂ : List α) : ¬ l₁ < l₂ ↔ l₂ ≤ l₁ := Iff.rfl
|
||||
theorem not_le_iff_gt [DecidableEq α] [LT α] [DecidableLT α] (l₁ l₂ : List α) :
|
||||
|
|
@ -59,6 +62,7 @@ protected theorem lt_irrefl [LT α] [Std.Irrefl (· < · : α → α → Prop)]
|
|||
instance ltIrrefl [LT α] [Std.Irrefl (· < · : α → α → Prop)] : Std.Irrefl (α := Array α) (· < ·) where
|
||||
irrefl := Array.lt_irrefl
|
||||
|
||||
@[simp] theorem not_lt_empty [LT α] (l : Array α) : ¬ l < #[] := List.not_lt_nil l.toList
|
||||
@[simp] theorem empty_le [LT α] (l : Array α) : #[] ≤ l := List.nil_le l.toList
|
||||
|
||||
@[simp] theorem le_empty [LT α] (l : Array α) : l ≤ #[] ↔ l = #[] := by
|
||||
|
|
@ -74,13 +78,12 @@ protected theorem le_refl [LT α] [i₀ : Std.Irrefl (· < · : α → α → Pr
|
|||
instance [LT α] [Std.Irrefl (· < · : α → α → Prop)] : Std.Refl (· ≤ · : Array α → Array α → Prop) where
|
||||
refl := Array.le_refl
|
||||
|
||||
protected theorem lt_trans [LT α] [DecidableLT α]
|
||||
protected theorem lt_trans [LT α]
|
||||
[i₁ : Trans (· < · : α → α → Prop) (· < ·) (· < ·)]
|
||||
{l₁ l₂ l₃ : Array α} (h₁ : l₁ < l₂) (h₂ : l₂ < l₃) : l₁ < l₃ :=
|
||||
List.lt_trans h₁ h₂
|
||||
|
||||
instance [LT α] [DecidableLT α]
|
||||
[Trans (· < · : α → α → Prop) (· < ·) (· < ·)] :
|
||||
instance [LT α] [Trans (· < · : α → α → Prop) (· < ·) (· < ·)] :
|
||||
Trans (· < · : Array α → Array α → Prop) (· < ·) (· < ·) where
|
||||
trans h₁ h₂ := Array.lt_trans h₁ h₂
|
||||
|
||||
|
|
@ -108,7 +111,7 @@ instance [DecidableEq α] [LT α] [DecidableLT α]
|
|||
Trans (· ≤ · : Array α → Array α → Prop) (· ≤ ·) (· ≤ ·) where
|
||||
trans h₁ h₂ := Array.le_trans h₁ h₂
|
||||
|
||||
protected theorem lt_asymm [DecidableEq α] [LT α] [DecidableLT α]
|
||||
protected theorem lt_asymm [LT α]
|
||||
[i : Std.Asymm (· < · : α → α → Prop)]
|
||||
{l₁ l₂ : Array α} (h : l₁ < l₂) : ¬ l₂ < l₁ := List.lt_asymm h
|
||||
|
||||
|
|
@ -118,13 +121,31 @@ instance [DecidableEq α] [LT α] [DecidableLT α]
|
|||
asymm _ _ := Array.lt_asymm
|
||||
|
||||
protected theorem le_total [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[i : Std.Total (¬ · < · : α → α → Prop)] {l₁ l₂ : Array α} : l₁ ≤ l₂ ∨ l₂ ≤ l₁ :=
|
||||
List.le_total
|
||||
[i : Std.Total (¬ · < · : α → α → Prop)] (l₁ l₂ : Array α) : l₁ ≤ l₂ ∨ l₂ ≤ l₁ :=
|
||||
List.le_total _ _
|
||||
|
||||
@[simp] protected theorem not_lt [LT α]
|
||||
{l₁ l₂ : Array α} : ¬ l₁ < l₂ ↔ l₂ ≤ l₁ := Iff.rfl
|
||||
|
||||
@[simp] protected theorem not_le [DecidableEq α] [LT α] [DecidableLT α]
|
||||
{l₁ l₂ : Array α} : ¬ l₂ ≤ l₁ ↔ l₁ < l₂ := Decidable.not_not
|
||||
|
||||
protected theorem le_of_lt [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[i : Std.Total (¬ · < · : α → α → Prop)]
|
||||
{l₁ l₂ : Array α} (h : l₁ < l₂) : l₁ ≤ l₂ :=
|
||||
List.le_of_lt h
|
||||
|
||||
theorem le_iff_lt_or_eq [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[Std.Irrefl (· < · : α → α → Prop)]
|
||||
[Std.Antisymm (¬ · < · : α → α → Prop)]
|
||||
[Std.Total (¬ · < · : α → α → Prop)]
|
||||
{l₁ l₂ : Array α} : l₁ ≤ l₂ ↔ l₁ < l₂ ∨ l₁ = l₂ := by
|
||||
simpa using List.le_iff_lt_or_eq (l₁ := l₁.toList) (l₂ := l₂.toList)
|
||||
|
||||
instance [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[Std.Total (¬ · < · : α → α → Prop)] :
|
||||
Std.Total (· ≤ · : Array α → Array α → Prop) where
|
||||
total _ _ := Array.le_total
|
||||
total := Array.le_total
|
||||
|
||||
@[simp] theorem lex_eq_true_iff_lt [DecidableEq α] [LT α] [DecidableLT α]
|
||||
{l₁ l₂ : Array α} : lex l₁ l₂ = true ↔ l₁ < l₂ := by
|
||||
|
|
@ -213,4 +234,48 @@ theorem le_iff_exists [DecidableEq α] [LT α] [DecidableLT α]
|
|||
cases l₂
|
||||
simp [List.le_iff_exists]
|
||||
|
||||
theorem append_left_lt [LT α] {l₁ l₂ l₃ : Array α} (h : l₂ < l₃) :
|
||||
l₁ ++ l₂ < l₁ ++ l₃ := by
|
||||
cases l₁
|
||||
cases l₂
|
||||
cases l₃
|
||||
simpa using List.append_left_lt h
|
||||
|
||||
theorem append_left_le [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[Std.Irrefl (· < · : α → α → Prop)]
|
||||
[Std.Asymm (· < · : α → α → Prop)]
|
||||
[Std.Antisymm (¬ · < · : α → α → Prop)]
|
||||
{l₁ l₂ l₃ : Array α} (h : l₂ ≤ l₃) :
|
||||
l₁ ++ l₂ ≤ l₁ ++ l₃ := by
|
||||
cases l₁
|
||||
cases l₂
|
||||
cases l₃
|
||||
simpa using List.append_left_le h
|
||||
|
||||
theorem le_append_left [LT α] [Std.Irrefl (· < · : α → α → Prop)]
|
||||
{l₁ l₂ : Array α} : l₁ ≤ l₁ ++ l₂ := by
|
||||
cases l₁
|
||||
cases l₂
|
||||
simpa using List.le_append_left
|
||||
|
||||
theorem map_lt [LT α] [LT β]
|
||||
{l₁ l₂ : Array α} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ < l₂) :
|
||||
map f l₁ < map f l₂ := by
|
||||
cases l₁
|
||||
cases l₂
|
||||
simpa using List.map_lt w h
|
||||
|
||||
theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β] [DecidableLT β]
|
||||
[Std.Irrefl (· < · : α → α → Prop)]
|
||||
[Std.Asymm (· < · : α → α → Prop)]
|
||||
[Std.Antisymm (¬ · < · : α → α → Prop)]
|
||||
[Std.Irrefl (· < · : β → β → Prop)]
|
||||
[Std.Asymm (· < · : β → β → Prop)]
|
||||
[Std.Antisymm (¬ · < · : β → β → Prop)]
|
||||
{l₁ l₂ : Array α} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ ≤ l₂) :
|
||||
map f l₁ ≤ map f l₂ := by
|
||||
cases l₁
|
||||
cases l₂
|
||||
simpa using List.map_le w h
|
||||
|
||||
end Array
|
||||
|
|
|
|||
|
|
@ -5,6 +5,7 @@ Authors: Kim Morrison
|
|||
-/
|
||||
prelude
|
||||
import Init.Data.List.Lemmas
|
||||
import Init.Data.List.Nat.TakeDrop
|
||||
|
||||
namespace List
|
||||
|
||||
|
|
@ -33,6 +34,7 @@ instance ltIrrefl [LT α] [Std.Irrefl (· < · : α → α → Prop)] : Std.Irre
|
|||
|
||||
@[simp] theorem not_lex_nil : ¬Lex r l [] := fun h => nomatch h
|
||||
|
||||
@[simp] theorem not_lt_nil [LT α] (l : List α) : ¬ l < [] := fun h => nomatch h
|
||||
@[simp] theorem nil_le [LT α] (l : List α) : [] ≤ l := fun h => nomatch h
|
||||
|
||||
@[simp] theorem not_nil_lex_iff : ¬Lex r [] l ↔ l = [] := by
|
||||
|
|
@ -59,6 +61,10 @@ theorem cons_lt_cons_iff [LT α] {a b} {l₁ l₂ : List α} :
|
|||
dsimp only [instLT, List.lt]
|
||||
simp [cons_lex_cons_iff]
|
||||
|
||||
@[simp] theorem cons_lt_cons_self [LT α] [i₀ : Std.Irrefl (· < · : α → α → Prop)] {l₁ l₂ : List α} :
|
||||
(a :: l₁) < (a :: l₂) ↔ l₁ < l₂ := by
|
||||
simp [cons_lt_cons_iff, i₀.irrefl]
|
||||
|
||||
theorem not_cons_lex_cons_iff [DecidableEq α] [DecidableRel r] {a b} {l₁ l₂ : List α} :
|
||||
¬ Lex r (a :: l₁) (b :: l₂) ↔ (¬ r a b ∧ a ≠ b) ∨ (¬ r a b ∧ ¬ Lex r l₁ l₂) := by
|
||||
rw [cons_lex_cons_iff, not_or, Decidable.not_and_iff_or_not, and_or_left]
|
||||
|
|
@ -120,7 +126,7 @@ protected theorem le_refl [LT α] [i₀ : Std.Irrefl (· < · : α → α → Pr
|
|||
instance [LT α] [Std.Irrefl (· < · : α → α → Prop)] : Std.Refl (· ≤ · : List α → List α → Prop) where
|
||||
refl := List.le_refl
|
||||
|
||||
theorem lex_trans {r : α → α → Prop} [DecidableRel r]
|
||||
theorem lex_trans {r : α → α → Prop}
|
||||
(lt_trans : ∀ {x y z : α}, r x y → r y z → r x z)
|
||||
(h₁ : Lex r l₁ l₂) (h₂ : Lex r l₂ l₃) : Lex r l₁ l₃ := by
|
||||
induction h₁ generalizing l₃ with
|
||||
|
|
@ -137,14 +143,13 @@ theorem lex_trans {r : α → α → Prop} [DecidableRel r]
|
|||
| .cons ih =>
|
||||
exact List.Lex.cons (ih2 ih)
|
||||
|
||||
protected theorem lt_trans [LT α] [DecidableLT α]
|
||||
protected theorem lt_trans [LT α]
|
||||
[i₁ : Trans (· < · : α → α → Prop) (· < ·) (· < ·)]
|
||||
{l₁ l₂ l₃ : List α} (h₁ : l₁ < l₂) (h₂ : l₂ < l₃) : l₁ < l₃ := by
|
||||
simp only [instLT, List.lt] at h₁ h₂ ⊢
|
||||
exact lex_trans (fun h₁ h₂ => i₁.trans h₁ h₂) h₁ h₂
|
||||
|
||||
instance [LT α] [DecidableLT α]
|
||||
[Trans (· < · : α → α → Prop) (· < ·) (· < ·)] :
|
||||
instance [LT α] [Trans (· < · : α → α → Prop) (· < ·) (· < ·)] :
|
||||
Trans (· < · : List α → List α → Prop) (· < ·) (· < ·) where
|
||||
trans h₁ h₂ := List.lt_trans h₁ h₂
|
||||
|
||||
|
|
@ -197,7 +202,7 @@ instance [DecidableEq α] [LT α] [DecidableLT α]
|
|||
Trans (· ≤ · : List α → List α → Prop) (· ≤ ·) (· ≤ ·) where
|
||||
trans h₁ h₂ := List.le_trans h₁ h₂
|
||||
|
||||
theorem lex_asymm {r : α → α → Prop} [DecidableRel r]
|
||||
theorem lex_asymm {r : α → α → Prop}
|
||||
(h : ∀ {x y : α}, r x y → ¬ r y x) : ∀ {l₁ l₂ : List α}, Lex r l₁ l₂ → ¬ Lex r l₂ l₁
|
||||
| nil, _, .nil => by simp
|
||||
| x :: l₁, y :: l₂, .rel h₁ =>
|
||||
|
|
@ -209,12 +214,11 @@ theorem lex_asymm {r : α → α → Prop} [DecidableRel r]
|
|||
| .rel h₂ => h h₂ h₂
|
||||
| .cons h₂ => lex_asymm h h₁ h₂
|
||||
|
||||
protected theorem lt_asymm [DecidableEq α] [LT α] [DecidableLT α]
|
||||
protected theorem lt_asymm [LT α]
|
||||
[i : Std.Asymm (· < · : α → α → Prop)]
|
||||
{l₁ l₂ : List α} (h : l₁ < l₂) : ¬ l₂ < l₁ := lex_asymm (i.asymm _ _) h
|
||||
|
||||
instance [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[Std.Asymm (· < · : α → α → Prop)] :
|
||||
instance [LT α] [Std.Asymm (· < · : α → α → Prop)] :
|
||||
Std.Asymm (· < · : List α → List α → Prop) where
|
||||
asymm _ _ := List.lt_asymm
|
||||
|
||||
|
|
@ -234,13 +238,43 @@ theorem not_lex_total [DecidableEq α] {r : α → α → Prop} [DecidableRel r]
|
|||
obtain (_ | _) := not_lex_total h l₁ l₂ <;> contradiction
|
||||
|
||||
protected theorem le_total [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[i : Std.Total (¬ · < · : α → α → Prop)] {l₁ l₂ : List α} : l₁ ≤ l₂ ∨ l₂ ≤ l₁ :=
|
||||
[i : Std.Total (¬ · < · : α → α → Prop)] (l₁ l₂ : List α) : l₁ ≤ l₂ ∨ l₂ ≤ l₁ :=
|
||||
not_lex_total i.total l₂ l₁
|
||||
|
||||
instance [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[Std.Total (¬ · < · : α → α → Prop)] :
|
||||
Std.Total (· ≤ · : List α → List α → Prop) where
|
||||
total _ _ := List.le_total
|
||||
total := List.le_total
|
||||
|
||||
@[simp] protected theorem not_lt [LT α]
|
||||
{l₁ l₂ : List α} : ¬ l₁ < l₂ ↔ l₂ ≤ l₁ := Iff.rfl
|
||||
|
||||
@[simp] protected theorem not_le [DecidableEq α] [LT α] [DecidableLT α]
|
||||
{l₁ l₂ : List α} : ¬ l₂ ≤ l₁ ↔ l₁ < l₂ := Decidable.not_not
|
||||
|
||||
protected theorem le_of_lt [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[i : Std.Total (¬ · < · : α → α → Prop)]
|
||||
{l₁ l₂ : List α} (h : l₁ < l₂) : l₁ ≤ l₂ := by
|
||||
obtain (h' | h') := List.le_total l₁ l₂
|
||||
· exact h'
|
||||
· exfalso
|
||||
exact h' h
|
||||
|
||||
theorem le_iff_lt_or_eq [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[Std.Irrefl (· < · : α → α → Prop)]
|
||||
[Std.Antisymm (¬ · < · : α → α → Prop)]
|
||||
[Std.Total (¬ · < · : α → α → Prop)]
|
||||
{l₁ l₂ : List α} : l₁ ≤ l₂ ↔ l₁ < l₂ ∨ l₁ = l₂ := by
|
||||
constructor
|
||||
· intro h
|
||||
by_cases h' : l₂ ≤ l₁
|
||||
· right
|
||||
apply List.le_antisymm h h'
|
||||
· left
|
||||
exact Decidable.not_not.mp h'
|
||||
· rintro (h | rfl)
|
||||
· exact List.le_of_lt h
|
||||
· exact List.le_refl l₁
|
||||
|
||||
theorem lex_eq_decide_lex [DecidableEq α] (lt : α → α → Bool) :
|
||||
lex l₁ l₂ lt = decide (Lex (fun x y => lt x y) l₁ l₂) := by
|
||||
|
|
@ -427,4 +461,63 @@ theorem le_iff_exists [DecidableEq α] [LT α] [DecidableLT α]
|
|||
· simpa using Std.Asymm.asymm
|
||||
· simpa using Std.Antisymm.antisymm
|
||||
|
||||
theorem append_left_lt [LT α] {l₁ l₂ l₃ : List α} (h : l₂ < l₃) :
|
||||
l₁ ++ l₂ < l₁ ++ l₃ := by
|
||||
induction l₁ with
|
||||
| nil => simp [h]
|
||||
| cons a l₁ ih => simp [cons_lt_cons_iff, ih]
|
||||
|
||||
theorem append_left_le [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[Std.Irrefl (· < · : α → α → Prop)]
|
||||
[Std.Asymm (· < · : α → α → Prop)]
|
||||
[Std.Antisymm (¬ · < · : α → α → Prop)]
|
||||
{l₁ l₂ l₃ : List α} (h : l₂ ≤ l₃) :
|
||||
l₁ ++ l₂ ≤ l₁ ++ l₃ := by
|
||||
induction l₁ with
|
||||
| nil => simp [h]
|
||||
| cons a l₁ ih => simp [cons_le_cons_iff, ih]
|
||||
|
||||
theorem le_append_left [LT α] [Std.Irrefl (· < · : α → α → Prop)]
|
||||
{l₁ l₂ : List α} : l₁ ≤ l₁ ++ l₂ := by
|
||||
intro h
|
||||
match l₁, h with
|
||||
| nil, h => simp at h
|
||||
| cons a l₁, h => exact le_append_left (by simpa using h)
|
||||
|
||||
theorem IsPrefix.le [LT α] [Std.Irrefl (· < · : α → α → Prop)]
|
||||
{l₁ l₂ : List α} (h : l₁ <+: l₂) : l₁ ≤ l₂ := by
|
||||
rcases h with ⟨_, rfl⟩
|
||||
apply le_append_left
|
||||
|
||||
theorem map_lt [LT α] [LT β]
|
||||
{l₁ l₂ : List α} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ < l₂) :
|
||||
map f l₁ < map f l₂ := by
|
||||
match l₁, l₂, h with
|
||||
| nil, nil, h => simp at h
|
||||
| nil, cons b l₂, h => simp
|
||||
| cons a l₁, nil, h => simp at h
|
||||
| cons a l₁, cons _ l₂, .cons h =>
|
||||
simp [cons_lt_cons_iff, map_lt w (by simpa using h)]
|
||||
| cons a l₁, cons b l₂, .rel h =>
|
||||
simp [cons_lt_cons_iff, w, h]
|
||||
|
||||
theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β] [DecidableLT β]
|
||||
[Std.Irrefl (· < · : α → α → Prop)]
|
||||
[Std.Asymm (· < · : α → α → Prop)]
|
||||
[Std.Antisymm (¬ · < · : α → α → Prop)]
|
||||
[Std.Irrefl (· < · : β → β → Prop)]
|
||||
[Std.Asymm (· < · : β → β → Prop)]
|
||||
[Std.Antisymm (¬ · < · : β → β → Prop)]
|
||||
{l₁ l₂ : List α} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ ≤ l₂) :
|
||||
map f l₁ ≤ map f l₂ := by
|
||||
rw [le_iff_exists] at h ⊢
|
||||
obtain (h | ⟨i, h₁, h₂, w₁, w₂⟩) := h
|
||||
· left
|
||||
rw [h]
|
||||
simp
|
||||
· right
|
||||
refine ⟨i, by simpa using h₁, by simpa using h₂, ?_, ?_⟩
|
||||
· simp +contextual [w₁]
|
||||
· simpa using w _ _ w₂
|
||||
|
||||
end List
|
||||
|
|
|
|||
|
|
@ -23,7 +23,7 @@ attribute [local instance] Char.notLTTrans Char.notLTAntisymm Char.notLTTotal
|
|||
|
||||
protected theorem le_trans {a b c : String} : a ≤ b → b ≤ c → a ≤ c := List.le_trans
|
||||
protected theorem lt_trans {a b c : String} : a < b → b < c → a < c := List.lt_trans
|
||||
protected theorem le_total (a b : String) : a ≤ b ∨ b ≤ a := List.le_total
|
||||
protected theorem le_total (a b : String) : a ≤ b ∨ b ≤ a := List.le_total _ _
|
||||
protected theorem le_antisymm {a b : String} : a ≤ b → b ≤ a → a = b := fun h₁ h₂ => String.ext (List.le_antisymm (as := a.data) (bs := b.data) h₁ h₂)
|
||||
protected theorem lt_asymm {a b : String} (h : a < b) : ¬ b < a := List.lt_asymm h
|
||||
protected theorem ne_of_lt {a b : String} (h : a < b) : a ≠ b := by
|
||||
|
|
|
|||
|
|
@ -247,7 +247,7 @@ theorem toArray_mk (a : Array α) (h : a.size = n) : (Vector.mk a h).toArray = a
|
|||
|
||||
@[simp] theorem toArray_mkVector : (mkVector n a).toArray = mkArray n a := rfl
|
||||
|
||||
theorem toArray_inj {v w : Vector α n} : v.toArray = w.toArray ↔ v = w := by
|
||||
@[simp] theorem toArray_inj {v w : Vector α n} : v.toArray = w.toArray ↔ v = w := by
|
||||
cases v
|
||||
cases w
|
||||
simp
|
||||
|
|
|
|||
|
|
@ -57,6 +57,7 @@ protected theorem lt_irrefl [LT α] [Std.Irrefl (· < · : α → α → Prop)]
|
|||
instance ltIrrefl [LT α] [Std.Irrefl (· < · : α → α → Prop)] : Std.Irrefl (α := Vector α n) (· < ·) where
|
||||
irrefl := Vector.lt_irrefl
|
||||
|
||||
@[simp] theorem not_lt_empty [LT α] (l : Vector α 0) : ¬ l < #v[] := Array.not_lt_empty l.toArray
|
||||
@[simp] theorem empty_le [LT α] (l : Vector α 0) : #v[] ≤ l := Array.empty_le l.toArray
|
||||
|
||||
@[simp] theorem le_empty [LT α] (l : Vector α 0) : l ≤ #v[] ↔ l = #v[] := by
|
||||
|
|
@ -69,12 +70,12 @@ protected theorem le_refl [LT α] [i₀ : Std.Irrefl (· < · : α → α → Pr
|
|||
instance [LT α] [Std.Irrefl (· < · : α → α → Prop)] : Std.Refl (· ≤ · : Vector α n → Vector α n → Prop) where
|
||||
refl := Vector.le_refl
|
||||
|
||||
protected theorem lt_trans [LT α] [DecidableLT α]
|
||||
protected theorem lt_trans [LT α]
|
||||
[i₁ : Trans (· < · : α → α → Prop) (· < ·) (· < ·)]
|
||||
{l₁ l₂ l₃ : Vector α n} (h₁ : l₁ < l₂) (h₂ : l₂ < l₃) : l₁ < l₃ :=
|
||||
Array.lt_trans h₁ h₂
|
||||
|
||||
instance [LT α] [DecidableLT α]
|
||||
instance [LT α]
|
||||
[Trans (· < · : α → α → Prop) (· < ·) (· < ·)] :
|
||||
Trans (· < · : Vector α n → Vector α n → Prop) (· < ·) (· < ·) where
|
||||
trans h₁ h₂ := Vector.lt_trans h₁ h₂
|
||||
|
|
@ -103,7 +104,7 @@ instance [DecidableEq α] [LT α] [DecidableLT α]
|
|||
Trans (· ≤ · : Vector α n → Vector α n → Prop) (· ≤ ·) (· ≤ ·) where
|
||||
trans h₁ h₂ := Vector.le_trans h₁ h₂
|
||||
|
||||
protected theorem lt_asymm [DecidableEq α] [LT α] [DecidableLT α]
|
||||
protected theorem lt_asymm [LT α]
|
||||
[i : Std.Asymm (· < · : α → α → Prop)]
|
||||
{l₁ l₂ : Vector α n} (h : l₁ < l₂) : ¬ l₂ < l₁ := Array.lt_asymm h
|
||||
|
||||
|
|
@ -113,13 +114,31 @@ instance [DecidableEq α] [LT α] [DecidableLT α]
|
|||
asymm _ _ := Vector.lt_asymm
|
||||
|
||||
protected theorem le_total [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[i : Std.Total (¬ · < · : α → α → Prop)] {l₁ l₂ : Vector α n} : l₁ ≤ l₂ ∨ l₂ ≤ l₁ :=
|
||||
Array.le_total
|
||||
[i : Std.Total (¬ · < · : α → α → Prop)] (l₁ l₂ : Vector α n) : l₁ ≤ l₂ ∨ l₂ ≤ l₁ :=
|
||||
Array.le_total _ _
|
||||
|
||||
instance [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[Std.Total (¬ · < · : α → α → Prop)] :
|
||||
Std.Total (· ≤ · : Vector α n → Vector α n → Prop) where
|
||||
total _ _ := Vector.le_total
|
||||
total := Vector.le_total
|
||||
|
||||
@[simp] protected theorem not_lt [LT α]
|
||||
{l₁ l₂ : Vector α n} : ¬ l₁ < l₂ ↔ l₂ ≤ l₁ := Iff.rfl
|
||||
|
||||
@[simp] protected theorem not_le [DecidableEq α] [LT α] [DecidableLT α]
|
||||
{l₁ l₂ : Vector α n} : ¬ l₂ ≤ l₁ ↔ l₁ < l₂ := Decidable.not_not
|
||||
|
||||
protected theorem le_of_lt [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[i : Std.Total (¬ · < · : α → α → Prop)]
|
||||
{l₁ l₂ : Vector α n} (h : l₁ < l₂) : l₁ ≤ l₂ :=
|
||||
Array.le_of_lt h
|
||||
|
||||
theorem le_iff_lt_or_eq [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[Std.Irrefl (· < · : α → α → Prop)]
|
||||
[Std.Antisymm (¬ · < · : α → α → Prop)]
|
||||
[Std.Total (¬ · < · : α → α → Prop)]
|
||||
{l₁ l₂ : Vector α n} : l₁ ≤ l₂ ↔ l₁ < l₂ ∨ l₁ = l₂ := by
|
||||
simpa using Array.le_iff_lt_or_eq (l₁ := l₁.toArray) (l₂ := l₂.toArray)
|
||||
|
||||
@[simp] theorem lex_eq_true_iff_lt [DecidableEq α] [LT α] [DecidableLT α]
|
||||
{l₁ l₂ : Vector α n} : lex l₁ l₂ = true ↔ l₁ < l₂ := by
|
||||
|
|
@ -199,4 +218,32 @@ theorem le_iff_exists [DecidableEq α] [LT α] [DecidableLT α]
|
|||
rcases l₂ with ⟨l₂, n₂⟩
|
||||
simp [Array.le_iff_exists, ← n₂]
|
||||
|
||||
theorem append_left_lt [LT α] {l₁ : Vector α n} {l₂ l₃ : Vector α m} (h : l₂ < l₃) :
|
||||
l₁ ++ l₂ < l₁ ++ l₃ := by
|
||||
simpa using Array.append_left_lt h
|
||||
|
||||
theorem append_left_le [DecidableEq α] [LT α] [DecidableLT α]
|
||||
[Std.Irrefl (· < · : α → α → Prop)]
|
||||
[Std.Asymm (· < · : α → α → Prop)]
|
||||
[Std.Antisymm (¬ · < · : α → α → Prop)]
|
||||
{l₁ : Vector α n} {l₂ l₃ : Vector α m} (h : l₂ ≤ l₃) :
|
||||
l₁ ++ l₂ ≤ l₁ ++ l₃ := by
|
||||
simpa using Array.append_left_le h
|
||||
|
||||
theorem map_lt [LT α] [LT β]
|
||||
{l₁ l₂ : Vector α n} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ < l₂) :
|
||||
map f l₁ < map f l₂ := by
|
||||
simpa using Array.map_lt w h
|
||||
|
||||
theorem map_le [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β] [DecidableLT β]
|
||||
[Std.Irrefl (· < · : α → α → Prop)]
|
||||
[Std.Asymm (· < · : α → α → Prop)]
|
||||
[Std.Antisymm (¬ · < · : α → α → Prop)]
|
||||
[Std.Irrefl (· < · : β → β → Prop)]
|
||||
[Std.Asymm (· < · : β → β → Prop)]
|
||||
[Std.Antisymm (¬ · < · : β → β → Prop)]
|
||||
{l₁ l₂ : Vector α n} {f : α → β} (w : ∀ x y, x < y → f x < f y) (h : l₁ ≤ l₂) :
|
||||
map f l₁ ≤ map f l₂ := by
|
||||
simpa using Array.map_le w h
|
||||
|
||||
end Vector
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue