mathlib3 documentation

order.min_max

max and min #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This file proves basic properties about maxima and minima on a linear_order.

Tags #

min, max

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@[simp]
theorem le_min_iff {α : Type u} [linear_order α] {a b c : α} :
c ≤ linear_order.min a b ↔ c ≤ a ∧ c ≤ b
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@[simp]
theorem le_max_iff {α : Type u} [linear_order α] {a b c : α} :
a ≤ linear_order.max b c ↔ a ≤ b ∨ a ≤ c
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@[simp]
theorem min_le_iff {α : Type u} [linear_order α] {a b c : α} :
linear_order.min a b ≤ c ↔ a ≤ c ∨ b ≤ c
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@[simp]
theorem max_le_iff {α : Type u} [linear_order α] {a b c : α} :
linear_order.max a b ≤ c ↔ a ≤ c ∧ b ≤ c
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@[simp]
theorem lt_min_iff {α : Type u} [linear_order α] {a b c : α} :
a < linear_order.min b c ↔ a < b ∧ a < c
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@[simp]
theorem lt_max_iff {α : Type u} [linear_order α] {a b c : α} :
a < linear_order.max b c ↔ a < b ∨ a < c
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@[simp]
theorem min_lt_iff {α : Type u} [linear_order α] {a b c : α} :
linear_order.min a b < c ↔ a < c ∨ b < c
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@[simp]
theorem max_lt_iff {α : Type u} [linear_order α] {a b c : α} :
linear_order.max a b < c ↔ a < c ∧ b < c
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theorem max_le_max {α : Type u} [linear_order α] {a b c d : α} :
a ≤ c → b ≤ d → linear_order.max a b ≤ linear_order.max c d
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theorem min_le_min {α : Type u} [linear_order α] {a b c d : α} :
a ≤ c → b ≤ d → linear_order.min a b ≤ linear_order.min c d
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theorem le_max_of_le_left {α : Type u} [linear_order α] {a b c : α} :
a ≤ b → a ≤ linear_order.max b c
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theorem le_max_of_le_right {α : Type u} [linear_order α] {a b c : α} :
a ≤ c → a ≤ linear_order.max b c
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theorem lt_max_of_lt_left {α : Type u} [linear_order α] {a b c : α} (h : a < b) :
a < linear_order.max b c
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theorem lt_max_of_lt_right {α : Type u} [linear_order α] {a b c : α} (h : a < c) :
a < linear_order.max b c
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theorem min_le_of_left_le {α : Type u} [linear_order α] {a b c : α} :
a ≤ c → linear_order.min a b ≤ c
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theorem min_le_of_right_le {α : Type u} [linear_order α] {a b c : α} :
b ≤ c → linear_order.min a b ≤ c
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theorem min_lt_of_left_lt {α : Type u} [linear_order α] {a b c : α} (h : a < c) :
linear_order.min a b < c
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theorem min_lt_of_right_lt {α : Type u} [linear_order α] {a b c : α} (h : b < c) :
linear_order.min a b < c
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theorem max_min_distrib_left {α : Type u} [linear_order α] {a b c : α} :
linear_order.max a (linear_order.min b c) = linear_order.min (linear_order.max a b) (linear_order.max a c)
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theorem max_min_distrib_right {α : Type u} [linear_order α] {a b c : α} :
linear_order.max (linear_order.min a b) c = linear_order.min (linear_order.max a c) (linear_order.max b c)
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theorem min_max_distrib_left {α : Type u} [linear_order α] {a b c : α} :
linear_order.min a (linear_order.max b c) = linear_order.max (linear_order.min a b) (linear_order.min a c)
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theorem min_max_distrib_right {α : Type u} [linear_order α] {a b c : α} :
linear_order.min (linear_order.max a b) c = linear_order.max (linear_order.min a c) (linear_order.min b c)
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theorem min_le_max {α : Type u} [linear_order α] {a b : α} :
linear_order.min a b ≤ linear_order.max a b
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@[simp]
theorem min_eq_left_iff {α : Type u} [linear_order α] {a b : α} :
linear_order.min a b = a ↔ a ≤ b
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@[simp]
theorem min_eq_right_iff {α : Type u} [linear_order α] {a b : α} :
linear_order.min a b = b ↔ b ≤ a
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@[simp]
theorem max_eq_left_iff {α : Type u} [linear_order α] {a b : α} :
linear_order.max a b = a ↔ b ≤ a
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@[simp]
theorem max_eq_right_iff {α : Type u} [linear_order α] {a b : α} :
linear_order.max a b = b ↔ a ≤ b
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theorem min_cases {α : Type u} [linear_order α] (a b : α) :
linear_order.min a b = a ∧ a ≤ b ∨ linear_order.min a b = b ∧ b < a

For elements a and b of a linear order, either min a b = a and a ≤ b, or min a b = b and b < a. Use cases on this lemma to automate linarith in inequalities

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theorem max_cases {α : Type u} [linear_order α] (a b : α) :
linear_order.max a b = a ∧ b ≤ a ∨ linear_order.max a b = b ∧ a < b

For elements a and b of a linear order, either max a b = a and b ≤ a, or max a b = b and a < b. Use cases on this lemma to automate linarith in inequalities

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theorem min_eq_iff {α : Type u} [linear_order α] {a b c : α} :
linear_order.min a b = c ↔ a = c ∧ a ≤ b ∨ b = c ∧ b ≤ a
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theorem max_eq_iff {α : Type u} [linear_order α] {a b c : α} :
linear_order.max a b = c ↔ a = c ∧ b ≤ a ∨ b = c ∧ a ≤ b
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theorem min_lt_min_left_iff {α : Type u} [linear_order α] {a b c : α} :
linear_order.min a c < linear_order.min b c ↔ a < b ∧ a < c
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theorem min_lt_min_right_iff {α : Type u} [linear_order α] {a b c : α} :
linear_order.min a b < linear_order.min a c ↔ b < c ∧ b < a
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theorem max_lt_max_left_iff {α : Type u} [linear_order α] {a b c : α} :
linear_order.max a c < linear_order.max b c ↔ a < b ∧ c < b
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theorem max_lt_max_right_iff {α : Type u} [linear_order α] {a b c : α} :
linear_order.max a b < linear_order.max a c ↔ b < c ∧ a < c
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@[protected, instance]
def max_idem {α : Type u} [linear_order α] :
is_idempotent α linear_order.max

An instance asserting that max a a = a

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@[protected, instance]
def min_idem {α : Type u} [linear_order α] :
is_idempotent α linear_order.min

An instance asserting that min a a = a

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theorem min_lt_max {α : Type u} [linear_order α] {a b : α} :
linear_order.min a b < linear_order.max a b ↔ a ≠ b
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theorem max_lt_max {α : Type u} [linear_order α] {a b c d : α} (h₁ : a < c) (h₂ : b < d) :
linear_order.max a b < linear_order.max c d
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theorem min_lt_min {α : Type u} [linear_order α] {a b c d : α} (h₁ : a < c) (h₂ : b < d) :
linear_order.min a b < linear_order.min c d
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theorem min_right_comm {α : Type u} [linear_order α] (a b c : α) :
linear_order.min (linear_order.min a b) c = linear_order.min (linear_order.min a c) b
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theorem max.left_comm {α : Type u} [linear_order α] (a b c : α) :
linear_order.max a (linear_order.max b c) = linear_order.max b (linear_order.max a c)
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theorem max.right_comm {α : Type u} [linear_order α] (a b c : α) :
linear_order.max (linear_order.max a b) c = linear_order.max (linear_order.max a c) b
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theorem monotone_on.map_max {α : Type u} {β : Type v} [linear_order α] [linear_order β] {f : α → β} {s : set α} {a b : α} (hf : monotone_on f s) (ha : a ∈ s) (hb : b ∈ s) :
f (linear_order.max a b) = linear_order.max (f a) (f b)
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theorem monotone_on.map_min {α : Type u} {β : Type v} [linear_order α] [linear_order β] {f : α → β} {s : set α} {a b : α} (hf : monotone_on f s) (ha : a ∈ s) (hb : b ∈ s) :
f (linear_order.min a b) = linear_order.min (f a) (f b)
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theorem antitone_on.map_max {α : Type u} {β : Type v} [linear_order α] [linear_order β] {f : α → β} {s : set α} {a b : α} (hf : antitone_on f s) (ha : a ∈ s) (hb : b ∈ s) :
f (linear_order.max a b) = linear_order.min (f a) (f b)
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theorem antitone_on.map_min {α : Type u} {β : Type v} [linear_order α] [linear_order β] {f : α → β} {s : set α} {a b : α} (hf : antitone_on f s) (ha : a ∈ s) (hb : b ∈ s) :
f (linear_order.min a b) = linear_order.max (f a) (f b)
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theorem monotone.map_max {α : Type u} {β : Type v} [linear_order α] [linear_order β] {f : α → β} {a b : α} (hf : monotone f) :
f (linear_order.max a b) = linear_order.max (f a) (f b)
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theorem monotone.map_min {α : Type u} {β : Type v} [linear_order α] [linear_order β] {f : α → β} {a b : α} (hf : monotone f) :
f (linear_order.min a b) = linear_order.min (f a) (f b)
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theorem antitone.map_max {α : Type u} {β : Type v} [linear_order α] [linear_order β] {f : α → β} {a b : α} (hf : antitone f) :
f (linear_order.max a b) = linear_order.min (f a) (f b)
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theorem antitone.map_min {α : Type u} {β : Type v} [linear_order α] [linear_order β] {f : α → β} {a b : α} (hf : antitone f) :
f (linear_order.min a b) = linear_order.max (f a) (f b)
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theorem min_choice {α : Type u} [linear_order α] (a b : α) :
linear_order.min a b = a ∨ linear_order.min a b = b
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theorem max_choice {α : Type u} [linear_order α] (a b : α) :
linear_order.max a b = a ∨ linear_order.max a b = b
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theorem le_of_max_le_left {α : Type u} [linear_order α] {a b c : α} (h : linear_order.max a b ≤ c) :
a ≤ c
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theorem le_of_max_le_right {α : Type u} [linear_order α] {a b c : α} (h : linear_order.max a b ≤ c) :
b ≤ c
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theorem max_commutative {α : Type u} [linear_order α] :
commutative linear_order.max
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theorem max_associative {α : Type u} [linear_order α] :
associative linear_order.max
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theorem max_left_commutative {α : Type u} [linear_order α] :
left_commutative linear_order.max
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theorem min_commutative {α : Type u} [linear_order α] :
commutative linear_order.min
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theorem min_associative {α : Type u} [linear_order α] :
associative linear_order.min
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theorem min_left_commutative {α : Type u} [linear_order α] :
left_commutative linear_order.min