Note about Mathlib/Init/

The files in Mathlib/Init are leftovers from the port from Mathlib3. (They contain content moved from lean3 itself that Mathlib needed but was not moved to lean4.)

We intend to move all the content of these files out into the main Mathlib directory structure. Contributions assisting with this are appreciated.

#align statements without corresponding declarations (i.e. because the declaration is in Batteries or Lean) can be left here. These will be deleted soon so will not significantly delay deleting otherwise empty Init files.

Basic lemmas about linear orders.

The contents of this file came from init.algebra.functions in Lean 3, and it would be good to find everything a better home.

theorem min_def {α : Type u} [LinearOrder α] (a : α) (b : α) : min a b = if a ≤ b then a else b

theorem max_def {α : Type u} [LinearOrder α] (a : α) (b : α) : max a b = if a ≤ b then b else a

theorem min_le_left {α : Type u} [LinearOrder α] (a : α) (b : α) : min a b ≤ a

theorem min_le_right {α : Type u} [LinearOrder α] (a : α) (b : α) : min a b ≤ b

theorem le_min {α : Type u} [LinearOrder α] {a : α} {b : α} {c : α} (h₁ : c ≤ a) (h₂ : c ≤ b) : c ≤ min a b

theorem le_max_left {α : Type u} [LinearOrder α] (a : α) (b : α) : a ≤ max a b

theorem le_max_right {α : Type u} [LinearOrder α] (a : α) (b : α) : b ≤ max a b

theorem max_le {α : Type u} [LinearOrder α] {a : α} {b : α} {c : α} (h₁ : a ≤ c) (h₂ : b ≤ c) : max a b ≤ c

theorem eq_min {α : Type u} [LinearOrder α] {a : α} {b : α} {c : α} (h₁ : c ≤ a) (h₂ : c ≤ b) (h₃ : ∀ {d : α}, d ≤ a → d ≤ b → d ≤ c) : c = min a b

theorem min_comm {α : Type u} [LinearOrder α] (a : α) (b : α) : min a b = min b a

theorem min_assoc {α : Type u} [LinearOrder α] (a : α) (b : α) (c : α) : min (min a b) c = min a (min b c)

theorem min_left_comm {α : Type u} [LinearOrder α] (a : α) (b : α) (c : α) : min a (min b c) = min b (min a c)

@[simp]

theorem min_self {α : Type u} [LinearOrder α] (a : α) : min a a = a

theorem min_eq_left {α : Type u} [LinearOrder α] {a : α} {b : α} (h : a ≤ b) : min a b = a

theorem min_eq_right {α : Type u} [LinearOrder α] {a : α} {b : α} (h : b ≤ a) : min a b = b

theorem eq_max {α : Type u} [LinearOrder α] {a : α} {b : α} {c : α} (h₁ : a ≤ c) (h₂ : b ≤ c) (h₃ : ∀ {d : α}, a ≤ d → b ≤ d → c ≤ d) : c = max a b

theorem max_comm {α : Type u} [LinearOrder α] (a : α) (b : α) : max a b = max b a

theorem max_assoc {α : Type u} [LinearOrder α] (a : α) (b : α) (c : α) : max (max a b) c = max a (max b c)

theorem max_left_comm {α : Type u} [LinearOrder α] (a : α) (b : α) (c : α) : max a (max b c) = max b (max a c)

@[simp]

theorem max_self {α : Type u} [LinearOrder α] (a : α) : max a a = a

theorem max_eq_left {α : Type u} [LinearOrder α] {a : α} {b : α} (h : b ≤ a) : max a b = a

theorem max_eq_right {α : Type u} [LinearOrder α] {a : α} {b : α} (h : a ≤ b) : max a b = b

theorem min_eq_left_of_lt {α : Type u} [LinearOrder α] {a : α} {b : α} (h : a < b) : min a b = a

theorem min_eq_right_of_lt {α : Type u} [LinearOrder α] {a : α} {b : α} (h : b < a) : min a b = b

theorem max_eq_left_of_lt {α : Type u} [LinearOrder α] {a : α} {b : α} (h : b < a) : max a b = a

theorem max_eq_right_of_lt {α : Type u} [LinearOrder α] {a : α} {b : α} (h : a < b) : max a b = b

theorem lt_min {α : Type u} [LinearOrder α] {a : α} {b : α} {c : α} (h₁ : a < b) (h₂ : a < c) : a < min b c

theorem max_lt {α : Type u} [LinearOrder α] {a : α} {b : α} {c : α} (h₁ : a < c) (h₂ : b < c) : max a b < c