Equivalences involving ℕ

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

@[simp]

theorem Equiv.boolProdNatEquivNat_symm_apply : ∀ (a : ℕ), ↑Equiv.boolProdNatEquivNat.symm a = Nat.boddDiv2 a

@[simp]

theorem Equiv.boolProdNatEquivNat_apply : ∀ (a : Bool × ℕ), ↑Equiv.boolProdNatEquivNat a = Function.uncurry Nat.bit a

def Equiv.boolProdNatEquivNat : Bool × ℕ ≃ ℕ

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

Equations

• Equiv.boolProdNatEquivNat = { toFun := Function.uncurry Nat.bit, invFun := Nat.boddDiv2, left_inv := Equiv.boolProdNatEquivNat.proof_1, right_inv := Equiv.boolProdNatEquivNat.proof_2 }

@[simp]

theorem Equiv.natSumNatEquivNat_symm_apply : ∀ (a : ℕ), ↑Equiv.natSumNatEquivNat.symm a = Bool.rec (Sum.inl (Nat.div2 a)) (Sum.inr (Nat.div2 a)) (Nat.bodd a)

def Equiv.natSumNatEquivNat : ℕ ⊕ ℕ ≃ ℕ

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

Equations

• Equiv.natSumNatEquivNat = (Equiv.boolProdEquivSum ℕ).symm.trans Equiv.boolProdNatEquivNat

@[simp]

theorem Equiv.natSumNatEquivNat_apply : ↑Equiv.natSumNatEquivNat = Sum.elim bit0 bit1

def Equiv.intEquivNat : ℤ ≃ ℕ

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

Equations

• Equiv.intEquivNat = Equiv.intEquivNatSumNat.trans Equiv.natSumNatEquivNat

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

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

Equations

• Equiv.prodEquivOfEquivNat e = Trans.trans (Trans.trans (Equiv.prodCongr e e) Nat.pairEquiv) e.symm