Equivalences involving ℕ

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

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

@[simp]

theorem Equiv.boolProdNatEquivNat_apply : ∀ (a : Bool × ℕ), Equiv.boolProdNatEquivNat a = Function.uncurry Nat.bit 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_symm_apply : ∀ (a : ℕ), Equiv.natSumNatEquivNat.symm a = Bool.rec (Sum.inl a.div2) (Sum.inr a.div2) a.bodd

@[simp]

theorem Equiv.natSumNatEquivNat_apply : ⇑Equiv.natSumNatEquivNat = Sum.elim (fun (x : ℕ) => 2 * x) fun (x : ℕ) => 2 * x + 1

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

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