mathlib documentation

order.monotone

Monotonicity #

This file defines (strictly) monotone/antitone functions. Contrary to standard mathematical usage, "monotone"/"mono" here means "increasing", not "increasing or decreasing". We use "antitone"/"anti" to mean "decreasing".

Definitions #

Main theorems #

Implementation notes #

Some of these definitions used to only require has_le α or has_lt α. The advantage of this is unclear and it led to slight elaboration issues. Now, everything requires preorder α and seems to work fine. Related Zulip discussion: https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Order.20diamond/near/254353352.

TODO #

The above theorems are also true in ℕ+, fin n... To make that work, we need succ_order α and succ_archimedean α.

Tags #

monotone, strictly monotone, antitone, strictly antitone, increasing, strictly increasing, decreasing, strictly decreasing

def monotone {α : Type u} {β : Type v} [preorder α] [preorder β] (f : α → β) :
Prop

A function f is monotone if a ≤ b implies f a ≤ f b.

Equations
Instances for monotone
def antitone {α : Type u} {β : Type v} [preorder α] [preorder β] (f : α → β) :
Prop

A function f is antitone if a ≤ b implies f b ≤ f a.

Equations
Instances for antitone
def monotone_on {α : Type u} {β : Type v} [preorder α] [preorder β] (f : α → β) (s : set α) :
Prop

A function f is monotone on s if, for all a, b ∈ s, a ≤ b implies f a ≤ f b.

Equations
def antitone_on {α : Type u} {β : Type v} [preorder α] [preorder β] (f : α → β) (s : set α) :
Prop

A function f is antitone on s if, for all a, b ∈ s, a ≤ b implies f b ≤ f a.

Equations
def strict_mono {α : Type u} {β : Type v} [preorder α] [preorder β] (f : α → β) :
Prop

A function f is strictly monotone if a < b implies f a < f b.

Equations
def strict_anti {α : Type u} {β : Type v} [preorder α] [preorder β] (f : α → β) :
Prop

A function f is strictly antitone if a < b implies f b < f a.

Equations
def strict_mono_on {α : Type u} {β : Type v} [preorder α] [preorder β] (f : α → β) (s : set α) :
Prop

A function f is strictly monotone on s if, for all a, b ∈ s, a < b implies f a < f b.

Equations
def strict_anti_on {α : Type u} {β : Type v} [preorder α] [preorder β] (f : α → β) (s : set α) :
Prop

A function f is strictly antitone on s if, for all a, b ∈ s, a < b implies f b < f a.

Equations

Monotonicity on the dual order #

Strictly, many of the *_on.dual lemmas in this section should use of_dual ⁻¹' s instead of s, but right now this is not possible as set.preimage is not defined yet, and importing it creates an import cycle.

Often, you should not need the rewriting lemmas. Instead, you probably want to add .dual, .dual_left or .dual_right to your monotone/antitone hypothesis.

@[simp]
theorem monotone_comp_of_dual_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :
@[simp]
theorem antitone_comp_of_dual_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :
@[simp]
theorem monotone_to_dual_comp_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :
@[simp]
theorem antitone_to_dual_comp_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :
@[simp]
theorem monotone_on_comp_of_dual_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :
@[simp]
theorem antitone_on_comp_of_dual_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :
@[simp]
theorem monotone_on_to_dual_comp_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :
@[simp]
theorem antitone_on_to_dual_comp_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :
@[simp]
theorem strict_mono_comp_of_dual_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :
@[simp]
theorem strict_anti_comp_of_dual_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :
@[simp]
theorem strict_mono_to_dual_comp_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :
@[simp]
theorem strict_anti_to_dual_comp_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :
@[simp]
theorem strict_mono_on_comp_of_dual_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :
@[simp]
theorem strict_anti_on_comp_of_dual_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :
@[simp]
theorem strict_mono_on_to_dual_comp_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :
@[simp]
theorem strict_anti_on_to_dual_comp_iff {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :
@[protected]
theorem monotone.dual {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (hf : monotone f) :
@[protected]
theorem antitone.dual {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (hf : antitone f) :
@[protected]
theorem monotone_on.dual {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} (hf : monotone_on f s) :
@[protected]
theorem antitone_on.dual {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} (hf : antitone_on f s) :
@[protected]
theorem strict_mono.dual {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (hf : strict_mono f) :
@[protected]
theorem strict_anti.dual {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (hf : strict_anti f) :
@[protected]
theorem strict_mono_on.dual {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} (hf : strict_mono_on f s) :
@[protected]
theorem strict_anti_on.dual {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} (hf : strict_anti_on f s) :
theorem monotone.dual_left {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :

Alias of the reverse direction of antitone_comp_of_dual_iff`.

theorem antitone.dual_left {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :

Alias of the reverse direction of monotone_comp_of_dual_iff`.

theorem monotone.dual_right {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :

Alias of the reverse direction of antitone_to_dual_comp_iff`.

theorem antitone.dual_right {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :

Alias of the reverse direction of monotone_to_dual_comp_iff`.

theorem monotone_on.dual_left {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :

Alias of the reverse direction of antitone_on_comp_of_dual_iff`.

theorem antitone_on.dual_left {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :

Alias of the reverse direction of monotone_on_comp_of_dual_iff`.

theorem monotone_on.dual_right {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :

Alias of the reverse direction of antitone_on_to_dual_comp_iff`.

theorem antitone_on.dual_right {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :

Alias of the reverse direction of monotone_on_to_dual_comp_iff`.

theorem strict_mono.dual_left {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :

Alias of the reverse direction of strict_anti_comp_of_dual_iff`.

theorem strict_anti.dual_left {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :

Alias of the reverse direction of strict_mono_comp_of_dual_iff`.

theorem strict_mono.dual_right {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :

Alias of the reverse direction of strict_anti_to_dual_comp_iff`.

theorem strict_anti.dual_right {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :

Alias of the reverse direction of strict_mono_to_dual_comp_iff`.

theorem strict_mono_on.dual_left {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :

Alias of the reverse direction of strict_anti_on_comp_of_dual_iff`.

theorem strict_anti_on.dual_left {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :

Alias of the reverse direction of strict_mono_on_comp_of_dual_iff`.

theorem strict_mono_on.dual_right {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :

Alias of the reverse direction of strict_anti_on_to_dual_comp_iff`.

theorem strict_anti_on.dual_right {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :

Alias of the reverse direction of strict_mono_on_to_dual_comp_iff`.

Monotonicity in function spaces #

theorem monotone.comp_le_comp_left {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] {f : β → α} {g h : γ → β} (hf : monotone f) (le_gh : g h) :
f g f h
theorem monotone_lam {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder γ] {f : α → β → γ} (hf : ∀ (b : β), monotone (λ (a : α), f a b)) :
theorem monotone_app {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder γ] (f : β → α → γ) (b : β) (hf : monotone (λ (a : α) (b : β), f b a)) :
monotone (f b)
theorem antitone_lam {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder γ] {f : α → β → γ} (hf : ∀ (b : β), antitone (λ (a : α), f a b)) :
theorem antitone_app {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder γ] (f : β → α → γ) (b : β) (hf : antitone (λ (a : α) (b : β), f b a)) :
antitone (f b)
theorem function.monotone_eval {ι : Type u} {α : ι → Type v} [Π (i : ι), preorder (α i)] (i : ι) :

Monotonicity hierarchy #

@[protected]
theorem monotone.monotone_on {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (hf : monotone f) (s : set α) :
@[protected]
theorem antitone.antitone_on {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (hf : antitone f) (s : set α) :
theorem monotone_on_univ {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :
theorem antitone_on_univ {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :
@[protected]
theorem strict_mono.strict_mono_on {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (hf : strict_mono f) (s : set α) :
@[protected]
theorem strict_anti.strict_anti_on {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (hf : strict_anti f) (s : set α) :
theorem strict_mono_on_univ {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :
theorem strict_anti_on_univ {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} :
theorem monotone.strict_mono_of_injective {α : Type u} {β : Type v} [preorder α] [partial_order β] {f : α → β} (h₁ : monotone f) (h₂ : function.injective f) :
theorem antitone.strict_anti_of_injective {α : Type u} {β : Type v} [preorder α] [partial_order β] {f : α → β} (h₁ : antitone f) (h₂ : function.injective f) :
theorem monotone_iff_forall_lt {α : Type u} {β : Type v} [partial_order α] [preorder β] {f : α → β} :
monotone f ∀ ⦃a b : α⦄, a < bf a f b
theorem antitone_iff_forall_lt {α : Type u} {β : Type v} [partial_order α] [preorder β] {f : α → β} :
antitone f ∀ ⦃a b : α⦄, a < bf b f a
theorem monotone_on_iff_forall_lt {α : Type u} {β : Type v} [partial_order α] [preorder β] {f : α → β} {s : set α} :
monotone_on f s ∀ ⦃a : α⦄, a s∀ ⦃b : α⦄, b sa < bf a f b
theorem antitone_on_iff_forall_lt {α : Type u} {β : Type v} [partial_order α] [preorder β] {f : α → β} {s : set α} :
antitone_on f s ∀ ⦃a : α⦄, a s∀ ⦃b : α⦄, b sa < bf b f a
@[protected]
theorem strict_mono_on.monotone_on {α : Type u} {β : Type v} [partial_order α] [preorder β] {f : α → β} {s : set α} (hf : strict_mono_on f s) :
@[protected]
theorem strict_anti_on.antitone_on {α : Type u} {β : Type v} [partial_order α] [preorder β] {f : α → β} {s : set α} (hf : strict_anti_on f s) :
@[protected]
theorem strict_mono.monotone {α : Type u} {β : Type v} [partial_order α] [preorder β] {f : α → β} (hf : strict_mono f) :
@[protected]
theorem strict_anti.antitone {α : Type u} {β : Type v} [partial_order α] [preorder β] {f : α → β} (hf : strict_anti f) :

Monotonicity from and to subsingletons #

@[protected]
theorem subsingleton.monotone {α : Type u} {β : Type v} [preorder α] [preorder β] [subsingleton α] (f : α → β) :
@[protected]
theorem subsingleton.antitone {α : Type u} {β : Type v} [preorder α] [preorder β] [subsingleton α] (f : α → β) :
theorem subsingleton.monotone' {α : Type u} {β : Type v} [preorder α] [preorder β] [subsingleton β] (f : α → β) :
theorem subsingleton.antitone' {α : Type u} {β : Type v} [preorder α] [preorder β] [subsingleton β] (f : α → β) :
@[protected]
theorem subsingleton.strict_mono {α : Type u} {β : Type v} [preorder α] [preorder β] [subsingleton α] (f : α → β) :
@[protected]
theorem subsingleton.strict_anti {α : Type u} {β : Type v} [preorder α] [preorder β] [subsingleton α] (f : α → β) :

Miscellaneous monotonicity results #

theorem monotone_id {α : Type u} [preorder α] :
theorem strict_mono_id {α : Type u} [preorder α] :
theorem monotone_const {α : Type u} {β : Type v} [preorder α] [preorder β] {c : β} :
monotone (λ (a : α), c)
theorem antitone_const {α : Type u} {β : Type v} [preorder α] [preorder β] {c : β} :
antitone (λ (a : α), c)
theorem strict_mono_of_le_iff_le {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (h : ∀ (x y : α), x y f x f y) :
theorem injective_of_lt_imp_ne {α : Type u} {β : Type v} [linear_order α] {f : α → β} (h : ∀ (x y : α), x < yf x f y) :
theorem injective_of_le_imp_le {α : Type u} {β : Type v} [partial_order α] [preorder β] (f : α → β) (h : ∀ {x y : α}, f x f yx y) :
theorem strict_mono.is_max_of_apply {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {a : α} (hf : strict_mono f) (ha : is_max (f a)) :
theorem strict_mono.is_min_of_apply {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {a : α} (hf : strict_mono f) (ha : is_min (f a)) :
theorem strict_anti.is_max_of_apply {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {a : α} (hf : strict_anti f) (ha : is_min (f a)) :
theorem strict_anti.is_min_of_apply {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {a : α} (hf : strict_anti f) (ha : is_max (f a)) :
@[protected]
theorem strict_mono.ite' {α : Type u} {β : Type v} [preorder α] [preorder β] {f g : α → β} (hf : strict_mono f) (hg : strict_mono g) {p : α → Prop} [decidable_pred p] (hp : ∀ ⦃x y : α⦄, x < yp yp x) (hfg : ∀ ⦃x y : α⦄, p x¬p yx < yf x < g y) :
strict_mono (λ (x : α), ite (p x) (f x) (g x))
@[protected]
theorem strict_mono.ite {α : Type u} {β : Type v} [preorder α] [preorder β] {f g : α → β} (hf : strict_mono f) (hg : strict_mono g) {p : α → Prop} [decidable_pred p] (hp : ∀ ⦃x y : α⦄, x < yp yp x) (hfg : ∀ (x : α), f x g x) :
strict_mono (λ (x : α), ite (p x) (f x) (g x))
@[protected]
theorem strict_anti.ite' {α : Type u} {β : Type v} [preorder α] [preorder β] {f g : α → β} (hf : strict_anti f) (hg : strict_anti g) {p : α → Prop} [decidable_pred p] (hp : ∀ ⦃x y : α⦄, x < yp yp x) (hfg : ∀ ⦃x y : α⦄, p x¬p yx < yg y < f x) :
strict_anti (λ (x : α), ite (p x) (f x) (g x))
@[protected]
theorem strict_anti.ite {α : Type u} {β : Type v} [preorder α] [preorder β] {f g : α → β} (hf : strict_anti f) (hg : strict_anti g) {p : α → Prop} [decidable_pred p] (hp : ∀ ⦃x y : α⦄, x < yp yp x) (hfg : ∀ (x : α), g x f x) :
strict_anti (λ (x : α), ite (p x) (f x) (g x))

Monotonicity under composition #

@[protected]
theorem monotone.comp {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} (hg : monotone g) (hf : monotone f) :
theorem monotone.comp_antitone {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} (hg : monotone g) (hf : antitone f) :
@[protected]
theorem antitone.comp {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} (hg : antitone g) (hf : antitone f) :
theorem antitone.comp_monotone {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} (hg : antitone g) (hf : monotone f) :
@[protected]
theorem monotone.iterate {α : Type u} [preorder α] {f : α → α} (hf : monotone f) (n : ) :
@[protected]
theorem monotone.comp_monotone_on {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} {s : set α} (hg : monotone g) (hf : monotone_on f s) :
theorem monotone.comp_antitone_on {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} {s : set α} (hg : monotone g) (hf : antitone_on f s) :
@[protected]
theorem antitone.comp_antitone_on {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} {s : set α} (hg : antitone g) (hf : antitone_on f s) :
theorem antitone.comp_monotone_on {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} {s : set α} (hg : antitone g) (hf : monotone_on f s) :
@[protected]
theorem strict_mono.comp {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} (hg : strict_mono g) (hf : strict_mono f) :
theorem strict_mono.comp_strict_anti {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} (hg : strict_mono g) (hf : strict_anti f) :
@[protected]
theorem strict_anti.comp {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} (hg : strict_anti g) (hf : strict_anti f) :
theorem strict_anti.comp_strict_mono {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} (hg : strict_anti g) (hf : strict_mono f) :
@[protected]
theorem strict_mono.iterate {α : Type u} [preorder α] {f : α → α} (hf : strict_mono f) (n : ) :
@[protected]
theorem strict_mono.comp_strict_mono_on {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} {s : set α} (hg : strict_mono g) (hf : strict_mono_on f s) :
theorem strict_mono.comp_strict_anti_on {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} {s : set α} (hg : strict_mono g) (hf : strict_anti_on f s) :
@[protected]
theorem strict_anti.comp_strict_anti_on {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} {s : set α} (hg : strict_anti g) (hf : strict_anti_on f s) :
theorem strict_anti.comp_strict_mono_on {α : Type u} {β : Type v} {γ : Type w} [preorder α] [preorder β] [preorder γ] {g : β → γ} {f : α → β} {s : set α} (hg : strict_anti g) (hf : strict_mono_on f s) :
theorem list.foldl_monotone {α : Type u} {β : Type v} [preorder α] {f : α → β → α} (H : ∀ (b : β), monotone (λ (a : α), f a b)) (l : list β) :
monotone (λ (a : α), list.foldl f a l)
theorem list.foldr_monotone {α : Type u} {β : Type v} [preorder β] {f : α → β → β} (H : ∀ (a : α), monotone (f a)) (l : list α) :
monotone (λ (b : β), list.foldr f b l)
theorem list.foldl_strict_mono {α : Type u} {β : Type v} [preorder α] {f : α → β → α} (H : ∀ (b : β), strict_mono (λ (a : α), f a b)) (l : list β) :
strict_mono (λ (a : α), list.foldl f a l)
theorem list.foldr_strict_mono {α : Type u} {β : Type v} [preorder β] {f : α → β → β} (H : ∀ (a : α), strict_mono (f a)) (l : list α) :
strict_mono (λ (b : β), list.foldr f b l)

Monotonicity in linear orders #

theorem monotone.reflect_lt {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : monotone f) {a b : α} (h : f a < f b) :
a < b
theorem antitone.reflect_lt {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : antitone f) {a b : α} (h : f a < f b) :
b < a
theorem monotone_on.reflect_lt {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} {s : set α} (hf : monotone_on f s) {a b : α} (ha : a s) (hb : b s) (h : f a < f b) :
a < b
theorem antitone_on.reflect_lt {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} {s : set α} (hf : antitone_on f s) {a b : α} (ha : a s) (hb : b s) (h : f a < f b) :
b < a
theorem strict_mono_on.le_iff_le {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} {s : set α} (hf : strict_mono_on f s) {a b : α} (ha : a s) (hb : b s) :
f a f b a b
theorem strict_anti_on.le_iff_le {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} {s : set α} (hf : strict_anti_on f s) {a b : α} (ha : a s) (hb : b s) :
f a f b b a
theorem strict_mono_on.lt_iff_lt {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} {s : set α} (hf : strict_mono_on f s) {a b : α} (ha : a s) (hb : b s) :
f a < f b a < b
theorem strict_anti_on.lt_iff_lt {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} {s : set α} (hf : strict_anti_on f s) {a b : α} (ha : a s) (hb : b s) :
f a < f b b < a
theorem strict_mono.le_iff_le {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : strict_mono f) {a b : α} :
f a f b a b
theorem strict_anti.le_iff_le {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : strict_anti f) {a b : α} :
f a f b b a
theorem strict_mono.lt_iff_lt {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : strict_mono f) {a b : α} :
f a < f b a < b
theorem strict_anti.lt_iff_lt {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : strict_anti f) {a b : α} :
f a < f b b < a
@[protected]
theorem strict_mono_on.compares {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} {s : set α} (hf : strict_mono_on f s) {a b : α} (ha : a s) (hb : b s) {o : ordering} :
o.compares (f a) (f b) o.compares a b
@[protected]
theorem strict_anti_on.compares {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} {s : set α} (hf : strict_anti_on f s) {a b : α} (ha : a s) (hb : b s) {o : ordering} :
o.compares (f a) (f b) o.compares b a
@[protected]
theorem strict_mono.compares {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : strict_mono f) {a b : α} {o : ordering} :
o.compares (f a) (f b) o.compares a b
@[protected]
theorem strict_anti.compares {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : strict_anti f) {a b : α} {o : ordering} :
o.compares (f a) (f b) o.compares b a
theorem strict_mono.injective {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : strict_mono f) :
theorem strict_anti.injective {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : strict_anti f) :
theorem strict_mono.maximal_of_maximal_image {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : strict_mono f) {a : α} (hmax : ∀ (p : β), p f a) (x : α) :
x a
theorem strict_mono.minimal_of_minimal_image {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : strict_mono f) {a : α} (hmin : ∀ (p : β), f a p) (x : α) :
a x
theorem strict_anti.minimal_of_maximal_image {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : strict_anti f) {a : α} (hmax : ∀ (p : β), p f a) (x : α) :
a x
theorem strict_anti.maximal_of_minimal_image {α : Type u} {β : Type v} [linear_order α] [preorder β] {f : α → β} (hf : strict_anti f) {a : α} (hmin : ∀ (p : β), f a p) (x : α) :
x a
theorem monotone.strict_mono_iff_injective {α : Type u} {β : Type v} [linear_order α] [partial_order β] {f : α → β} (hf : monotone f) :
theorem antitone.strict_anti_iff_injective {α : Type u} {β : Type v} [linear_order α] [partial_order β] {f : α → β} (hf : antitone f) :

Monotonicity in and #

theorem nat.rel_of_forall_rel_succ_of_le_of_lt {β : Type v} (r : β → β → Prop) [is_trans β r] {f : → β} {a : } (h : ∀ (n : ), a nr (f n) (f (n + 1))) ⦃b c : (hab : a b) (hbc : b < c) :
r (f b) (f c)
theorem nat.rel_of_forall_rel_succ_of_le_of_le {β : Type v} (r : β → β → Prop) [is_refl β r] [is_trans β r] {f : → β} {a : } (h : ∀ (n : ), a nr (f n) (f (n + 1))) ⦃b c : (hab : a b) (hbc : b c) :
r (f b) (f c)
theorem nat.rel_of_forall_rel_succ_of_lt {β : Type v} (r : β → β → Prop) [is_trans β r] {f : → β} (h : ∀ (n : ), r (f n) (f (n + 1))) ⦃a b : (hab : a < b) :
r (f a) (f b)
theorem nat.rel_of_forall_rel_succ_of_le {β : Type v} (r : β → β → Prop) [is_refl β r] [is_trans β r] {f : → β} (h : ∀ (n : ), r (f n) (f (n + 1))) ⦃a b : (hab : a b) :
r (f a) (f b)
theorem monotone_nat_of_le_succ {α : Type u} [preorder α] {f : → α} (hf : ∀ (n : ), f n f (n + 1)) :
theorem antitone_nat_of_succ_le {α : Type u} [preorder α] {f : → α} (hf : ∀ (n : ), f (n + 1) f n) :
theorem strict_mono_nat_of_lt_succ {α : Type u} [preorder α] {f : → α} (hf : ∀ (n : ), f n < f (n + 1)) :
theorem strict_anti_nat_of_succ_lt {α : Type u} [preorder α] {f : → α} (hf : ∀ (n : ), f (n + 1) < f n) :
theorem int.rel_of_forall_rel_succ_of_lt {β : Type v} (r : β → β → Prop) [is_trans β r] {f : → β} (h : ∀ (n : ), r (f n) (f (n + 1))) ⦃a b : (hab : a < b) :
r (f a) (f b)
theorem int.rel_of_forall_rel_succ_of_le {β : Type v} (r : β → β → Prop) [is_refl β r] [is_trans β r] {f : → β} (h : ∀ (n : ), r (f n) (f (n + 1))) ⦃a b : (hab : a b) :
r (f a) (f b)
theorem monotone_int_of_le_succ {α : Type u} [preorder α] {f : → α} (hf : ∀ (n : ), f n f (n + 1)) :
theorem antitone_int_of_succ_le {α : Type u} [preorder α] {f : → α} (hf : ∀ (n : ), f (n + 1) f n) :
theorem strict_mono_int_of_lt_succ {α : Type u} [preorder α] {f : → α} (hf : ∀ (n : ), f n < f (n + 1)) :
theorem strict_anti_int_of_succ_lt {α : Type u} [preorder α] {f : → α} (hf : ∀ (n : ), f (n + 1) < f n) :
theorem monotone.ne_of_lt_of_lt_nat {α : Type u} [preorder α] {f : → α} (hf : monotone f) (n : ) {x : α} (h1 : f n < x) (h2 : x < f (n + 1)) (a : ) :
f a x

If f is a monotone function from to a preorder such that x lies between f n and f (n + 1), then x doesn't lie in the range of f.

theorem antitone.ne_of_lt_of_lt_nat {α : Type u} [preorder α] {f : → α} (hf : antitone f) (n : ) {x : α} (h1 : f (n + 1) < x) (h2 : x < f n) (a : ) :
f a x

If f is an antitone function from to a preorder such that x lies between f (n + 1) and f n, then x doesn't lie in the range of f.

theorem monotone.ne_of_lt_of_lt_int {α : Type u} [preorder α] {f : → α} (hf : monotone f) (n : ) {x : α} (h1 : f n < x) (h2 : x < f (n + 1)) (a : ) :
f a x

If f is a monotone function from to a preorder and x lies between f n and f (n + 1), then x doesn't lie in the range of f.

theorem antitone.ne_of_lt_of_lt_int {α : Type u} [preorder α] {f : → α} (hf : antitone f) (n : ) {x : α} (h1 : f (n + 1) < x) (h2 : x < f n) (a : ) :
f a x

If f is an antitone function from to a preorder and x lies between f (n + 1) and f n, then x doesn't lie in the range of f.

theorem strict_mono.id_le {φ : } (h : strict_mono φ) (n : ) :
n φ n
theorem subtype.mono_coe {α : Type u} [preorder α] (t : set α) :
theorem subtype.strict_mono_coe {α : Type u} [preorder α] (t : set α) :
theorem monotone_fst {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] :
theorem monotone_snd {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] :