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.

addition

multiplication

theorem Nat.eq_zero_of_mul_eq_zero {n : ℕ} {m : ℕ} : n * m = 0 → n = 0 ∨ m = 0

properties of inequality

def Nat.ltGeByCases {a : ℕ} {b : ℕ} {C : Sort u} (h₁ : a < b → C) (h₂ : b ≤ a → C) : C

Equations

• Nat.ltGeByCases h₁ h₂ = Decidable.byCases h₁ fun (h : ¬a < b) => h₂ ⋯

def Nat.ltByCases {a : ℕ} {b : ℕ} {C : Sort u} (h₁ : a < b → C) (h₂ : a = b → C) (h₃ : b < a → C) : C

Equations

• Nat.ltByCases h₁ h₂ h₃ = Nat.ltGeByCases h₁ fun (h₁ : b ≤ a) => Nat.ltGeByCases h₃ fun (h : a ≤ b) => h₂ ⋯

bit0/bit1 properties

theorem Nat.bit1_eq_succ_bit0 (n : ℕ) : bit1 n = (bit0 n).succ

theorem Nat.bit1_succ_eq (n : ℕ) : bit1 n.succ = (bit1 n).succ.succ

theorem Nat.bit1_ne_one {n : ℕ} : n ≠ 0 → bit1 n ≠ 1

theorem Nat.bit0_ne_one (n : ℕ) : bit0 n ≠ 1

theorem Nat.bit1_ne_bit0 (n : ℕ) (m : ℕ) : bit1 n ≠ bit0 m

theorem Nat.bit0_ne_bit1 (n : ℕ) (m : ℕ) : bit0 n ≠ bit1 m

theorem Nat.bit0_inj {n : ℕ} {m : ℕ} : bit0 n = bit0 m → n = m

theorem Nat.bit1_inj {n : ℕ} {m : ℕ} : bit1 n = bit1 m → n = m

theorem Nat.bit0_ne {n : ℕ} {m : ℕ} : n ≠ m → bit0 n ≠ bit0 m

theorem Nat.bit1_ne {n : ℕ} {m : ℕ} : n ≠ m → bit1 n ≠ bit1 m

theorem Nat.zero_ne_bit0 {n : ℕ} : n ≠ 0 → 0 ≠ bit0 n

theorem Nat.zero_ne_bit1 (n : ℕ) : 0 ≠ bit1 n

theorem Nat.one_ne_bit0 (n : ℕ) : 1 ≠ bit0 n

theorem Nat.one_ne_bit1 {n : ℕ} : n ≠ 0 → 1 ≠ bit1 n

theorem Nat.one_lt_bit1 {n : ℕ} : n ≠ 0 → 1 < bit1 n

theorem Nat.one_lt_bit0 {n : ℕ} : n ≠ 0 → 1 < bit0 n

theorem Nat.bit0_lt {n : ℕ} {m : ℕ} (h : n < m) : bit0 n < bit0 m

theorem Nat.bit1_lt {n : ℕ} {m : ℕ} (h : n < m) : bit1 n < bit1 m

theorem Nat.bit0_lt_bit1 {n : ℕ} {m : ℕ} (h : n ≤ m) : bit0 n < bit1 m

theorem Nat.bit1_lt_bit0 {n : ℕ} {m : ℕ} : n < m → bit1 n < bit0 m

theorem Nat.one_le_bit1 (n : ℕ) : 1 ≤ bit1 n

theorem Nat.one_le_bit0 (n : ℕ) : n ≠ 0 → 1 ≤ bit0 n

successor and predecessor

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.

theorem Nat.one_eq_succ_zero : 1 = Nat.succ 0

subtraction

Many lemmas are proven more generally in mathlib algebra/order/sub

min

induction principles

def Nat.twoStepInduction {P : ℕ → Sort u} (H1 : P 0) (H2 : P 1) (H3 : (n : ℕ) → P n → P n.succ → P n.succ.succ) (a : ℕ) : P a

Equations

• Nat.twoStepInduction H1 H2 H3 0 = H1
• Nat.twoStepInduction H1 H2 H3 1 = H2
• Nat.twoStepInduction H1 H2 H3 _n.succ.succ = H3 _n (Nat.twoStepInduction H1 H2 H3 _n) (Nat.twoStepInduction H1 H2 H3 _n.succ)

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

• Nat.subInduction H1 H2 H3 0 x = H1 x
• Nat.subInduction H1 H2 H3 _n.succ 0 = H2 _n
• Nat.subInduction H1 H2 H3 n.succ m.succ = H3 n m (Nat.subInduction H1 H2 H3 n m)

theorem Nat.strong_induction_on {p : ℕ → Prop} (n : ℕ) (h : ∀ (n : ℕ), (∀ (m : ℕ), m < n → p m) → p n) : p n

theorem Nat.case_strong_induction_on {p : ℕ → Prop} (a : ℕ) (hz : p 0) (hi : ∀ (n : ℕ), (∀ (m : ℕ), m ≤ n → p m) → p n.succ) : p a

mod

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

div

dvd

find

def Nat.findX {p : ℕ → Prop} [DecidablePred p] (H : ∃ (n : ℕ), p n) : { n : ℕ // p n ∧ ∀ (m : ℕ), m < n → ¬p m }

Equations

• One or more equations did not get rendered due to their size.

def Nat.find {p : ℕ → Prop} [DecidablePred p] (H : ∃ (n : ℕ), p n) : ℕ

If p is a (decidable) predicate on ℕ and hp : ∃ (n : ℕ), p n is a proof that there exists some natural number satisfying p, then Nat.find hp is the smallest natural number satisfying p. Note that Nat.find is protected, meaning that you can't just write find, even if the Nat namespace is open.

The API for Nat.find is:

• Nat.find_spec is the proof that Nat.find hp satisfies p.
• Nat.find_min is the proof that if m < Nat.find hp then m does not satisfy p.
• Nat.find_min' is the proof that if m does satisfy p then Nat.find hp ≤ m.

Equations

• Nat.find H = (Nat.findX H).val

theorem Nat.find_spec {p : ℕ → Prop} [DecidablePred p] (H : ∃ (n : ℕ), p n) : p (Nat.find H)

theorem Nat.find_min {p : ℕ → Prop} [DecidablePred p] (H : ∃ (n : ℕ), p n) {m : ℕ} : m < Nat.find H → ¬p m

theorem Nat.find_min' {p : ℕ → Prop} [DecidablePred p] (H : ∃ (n : ℕ), p n) {m : ℕ} (h : p m) : Nat.find H ≤ m