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.

Lemmas about booleans

These are the lemmas about booleans which were present in core Lean 3. See also the file Mathlib.Data.Bool.Basic which contains lemmas about booleans from mathlib 3.

theorem Bool.true_eq_false_eq_False : ¬true = false

theorem Bool.false_eq_true_eq_False : ¬false = true

theorem Bool.eq_false_eq_not_eq_true (b : Bool) : (¬b = true) = (b = false)

theorem Bool.eq_true_eq_not_eq_false (b : Bool) : (¬b = false) = (b = true)

theorem Bool.eq_false_of_not_eq_true {b : Bool} : ¬b = true → b = false

theorem Bool.eq_true_of_not_eq_false {b : Bool} : ¬b = false → b = true

theorem Bool.and_eq_true_eq_eq_true_and_eq_true (a : Bool) (b : Bool) : ((a && b) = true) = (a = true ∧ b = true)

theorem Bool.or_eq_true_eq_eq_true_or_eq_true (a : Bool) (b : Bool) : ((a || b) = true) = (a = true ∨ b = true)

theorem Bool.not_eq_true_eq_eq_false (a : Bool) : ((!a) = true) = (a = false)

theorem Bool.and_eq_false_eq_eq_false_or_eq_false (a : Bool) (b : Bool) : ((a && b) = false) = (a = false ∨ b = false)

theorem Bool.or_eq_false_eq_eq_false_and_eq_false (a : Bool) (b : Bool) : ((a || b) = false) = (a = false ∧ b = false)

theorem Bool.not_eq_false_eq_eq_true (a : Bool) : ((!a) = false) = (a = true)

theorem Bool.coe_false : (false = true) = False

theorem Bool.coe_true : (true = true) = True

theorem Bool.coe_sort_false : (false = true) = False

theorem Bool.coe_sort_true : (true = true) = True

theorem Bool.decide_iff (p : Prop) [d : Decidable p] : decide p = true ↔ p

theorem Bool.decide_true {p : Prop} [Decidable p] : p → decide p = true

theorem Bool.of_decide_true {p : Prop} [Decidable p] : decide p = true → p

theorem Bool.bool_iff_false {b : Bool} : ¬b = true ↔ b = false

theorem Bool.bool_eq_false {b : Bool} : ¬b = true → b = false

theorem Bool.decide_false_iff (p : Prop) : ∀ {x : Decidable p}, decide p = false ↔ ¬p

theorem Bool.decide_false {p : Prop} [Decidable p] : ¬p → decide p = false

theorem Bool.of_decide_false {p : Prop} [Decidable p] : decide p = false → ¬p

theorem Bool.decide_congr {p : Prop} {q : Prop} [Decidable p] [Decidable q] (h : p ↔ q) : decide p = decide q

@[deprecated Bool.or_eq_true_iff]

theorem Bool.coe_or_iff (x : Bool) (y : Bool) : (x || y) = true ↔ x = true ∨ y = true

Alias of Bool.or_eq_true_iff.

@[deprecated Bool.and_eq_true_iff]

theorem Bool.coe_and_iff (x : Bool) (y : Bool) : (x && y) = true ↔ x = true ∧ y = true

Alias of Bool.and_eq_true_iff.

theorem Bool.coe_xor_iff (a : Bool) (b : Bool) : xor a b = true ↔ Xor' (a = true) (b = true)