Documentation

Mathlib.Logic.Equiv.Nat

Equivalences involving #

This file defines some additional constructive equivalences using Encodable and the pairing function on .

An equivalence between Bool × ℕ and , by mapping (true, x) to 2 * x + 1 and (false, x) to 2 * x.

Equations
@[simp]
theorem Equiv.boolProdNatEquivNat_symm_apply (a✝ : ) :
boolProdNatEquivNat.symm a✝ = a✝.boddDiv2

An equivalence between ℕ ⊕ ℕ and , by mapping (Sum.inl x) to 2 * x and (Sum.inr x) to 2 * x + 1.

Equations
@[simp]
theorem Equiv.natSumNatEquivNat_symm_apply (a✝ : ) :
natSumNatEquivNat.symm a✝ = Bool.rec (Sum.inl a✝.div2) (Sum.inr a✝.div2) a✝.bodd
@[simp]
theorem Equiv.natSumNatEquivNat_apply :
natSumNatEquivNat = Sum.elim (fun (x : ) => 2 * x) fun (x : ) => 2 * x + 1

An equivalence between and , through ℤ ≃ ℕ ⊕ ℕ and ℕ ⊕ ℕ ≃ ℕ.

Equations
def Equiv.prodEquivOfEquivNat {α : Type u_1} (e : α ) :
α × α α

An equivalence between α × α and α, given that there is an equivalence between α and .

Equations