mathlib documentation

core / init.algebra.functions

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theorem min_def {α : Type u} [linear_order α] (a b : α) :
min a b = ite (a ≤ b) a b
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theorem max_def {α : Type u} [linear_order α] (a b : α) :
max a b = ite (b ≤ a) a b
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meta def tactic.interactive.min_tac (a b : interactive.parse (lean.parser.pexpr std.prec.max)) :
tactic unit
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theorem min_le_left {α : Type u} [linear_order α] (a b : α) :
min a b ≤ a
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theorem min_le_right {α : Type u} [linear_order α] (a b : α) :
min a b ≤ b
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theorem le_min {α : Type u} [linear_order α] {a b c : α} (h₁ : c ≤ a) (h₂ : c ≤ b) :
c ≤ min a b
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theorem le_max_left {α : Type u} [linear_order α] (a b : α) :
a ≤ max a b
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theorem le_max_right {α : Type u} [linear_order α] (a b : α) :
b ≤ max a b
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theorem max_le {α : Type u} [linear_order α] {a b c : α} (h₁ : a ≤ c) (h₂ : b ≤ c) :
max a b ≤ c
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theorem eq_min {α : Type u} [linear_order α] {a b c : α} (h₁ : c ≤ a) (h₂ : c ≤ b) (h₃ : ∀ {d : α}, d ≤ a → d ≤ b → d ≤ c) :
c = min a b
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theorem min_comm {α : Type u} [linear_order α] (a b : α) :
min a b = min b a
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theorem min_assoc {α : Type u} [linear_order α] (a b c : α) :
min (min a b) c = min a (min b c)
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theorem min_left_comm {α : Type u} [linear_order α] (a b c : α) :
min a (min b c) = min b (min a c)
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@[simp]
theorem min_self {α : Type u} [linear_order α] (a : α) :
min a a = a
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@[simp]
theorem min_eq_left {α : Type u} [linear_order α] {a b : α} (h : a ≤ b) :
min a b = a
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@[simp]
theorem min_eq_right {α : Type u} [linear_order α] {a b : α} (h : b ≤ a) :
min a b = b
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theorem eq_max {α : Type u} [linear_order α] {a b c : α} (h₁ : a ≤ c) (h₂ : b ≤ c) (h₃ : ∀ {d : α}, a ≤ d → b ≤ d → c ≤ d) :
c = max a b
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theorem max_comm {α : Type u} [linear_order α] (a b : α) :
max a b = max b a
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theorem max_assoc {α : Type u} [linear_order α] (a b c : α) :
max (max a b) c = max a (max b c)
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theorem max_left_comm {α : Type u} [linear_order α] (a b c : α) :
max a (max b c) = max b (max a c)
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@[simp]
theorem max_self {α : Type u} [linear_order α] (a : α) :
max a a = a
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@[simp]
theorem max_eq_left {α : Type u} [linear_order α] {a b : α} (h : b ≤ a) :
max a b = a
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@[simp]
theorem max_eq_right {α : Type u} [linear_order α] {a b : α} (h : a ≤ b) :
max a b = b
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theorem min_eq_left_of_lt {α : Type u} [linear_order α] {a b : α} (h : a < b) :
min a b = a
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theorem min_eq_right_of_lt {α : Type u} [linear_order α] {a b : α} (h : b < a) :
min a b = b
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theorem max_eq_left_of_lt {α : Type u} [linear_order α] {a b : α} (h : b < a) :
max a b = a
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theorem max_eq_right_of_lt {α : Type u} [linear_order α] {a b : α} (h : a < b) :
max a b = b
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theorem lt_min {α : Type u} [linear_order α] {a b c : α} (h₁ : a < b) (h₂ : a < c) :
a < min b c
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theorem max_lt {α : Type u} [linear_order α] {a b c : α} (h₁ : a < c) (h₂ : b < c) :
max a b < c