Documentation

Mathlib.Init.Data.Nat.Lemmas

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.

multiplication

source
@[deprecated Nat.mul_eq_zero]
theorem Nat.eq_zero_of_mul_eq_zero {m : ℕ} {n : ℕ} :
n * m = 0 → n = 0 ∨ m = 0

Alias of the forward direction of Nat.mul_eq_zero.

successor and predecessor

source
@[deprecated]
def Nat.discriminate {B : Sort u} {n : ℕ} (H1 : n = 0 → B) (H2 : (m : ℕ) → n = m.succ → B) :
B
Equations
  • One or more equations did not get rendered due to their size.
Instances For
    source
    @[deprecated]
    theorem Nat.one_eq_succ_zero :
    1 = Nat.succ 0

    induction principles

    source
    @[deprecated]
    def Nat.subInduction {P : ℕ → ℕ → Sort u} (H1 : (m : ℕ) → P 0 m) (H2 : (n : ℕ) → P n.succ 0) (H3 : (n m : ℕ) → P n m → P n.succ m.succ) (n : ℕ) (m : ℕ) :
    P n m
    Equations
    Instances For

      mod

      source
      @[deprecated]
      theorem Nat.cond_decide_mod_two (x : ℕ) [d : Decidable (x % 2 = 1)] :
      (bif decide (x % 2 = 1) then 1 else 0) = x % 2