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[Merged by Bors] - feat: port Logic.Equiv.Nat #1227

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[Merged by Bors] - feat: port Logic.Equiv.Nat #1227

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500bbd5
Initial commit
ADedecker Dec 26, 2022
fa20385
Done except deprecated
ADedecker Dec 26, 2022
796b4ed
Use `bit` instead of `bit0` and `bit1`
ADedecker Dec 28, 2022
ef8b93e
Back to `bit0` and `bit1`; better docs
ADedecker Dec 28, 2022
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1 change: 1 addition & 0 deletions Mathlib.lean
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Expand Up @@ -366,6 +366,7 @@ import Mathlib.Logic.Equiv.Defs
import Mathlib.Logic.Equiv.Embedding
import Mathlib.Logic.Equiv.LocalEquiv
import Mathlib.Logic.Equiv.MfldSimpsAttr
import Mathlib.Logic.Equiv.Nat
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Set
import Mathlib.Logic.Function.Basic
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21 changes: 10 additions & 11 deletions Mathlib/Logic/Equiv/Nat.lean
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Expand Up @@ -25,30 +25,30 @@ namespace Equiv

variable {α : Type _}

/-- An equivalence between `bool × ℕ` and `ℕ`, by mapping `(tt, x)` to `2 * x + 1` and `(ff, x)` to
`2 * x`.
-/
/-- An equivalence between `Bool × ℕ` and `ℕ`, by mapping `(true, x)` to `2 * x + 1` and
`(false, x)` to `2 * x`. -/
@[simps]
def boolProdNatEquivNat : Bool × ℕ ≃
ℕ where
toFun := uncurry bit
invFun := boddDiv2
left_inv := fun ⟨b, n⟩ => by simp only [bodd_bit, div2_bit, uncurry_apply_pair, bodd_div2_eq]
right_inv n := by simp only [bit_decomp, bodd_div2_eq, uncurry_apply_pair]
left_inv := fun ⟨b, n⟩ => by simp only [bodd_bit, div2_bit, uncurry_apply_pair, boddDiv2_eq]
right_inv n := by simp only [bit_decomp, boddDiv2_eq, uncurry_apply_pair]
#align equiv.bool_prod_nat_equiv_nat Equiv.boolProdNatEquivNat

/-- An equivalence between `ℕ ⊕ ℕ` and `ℕ`, by mapping `(sum.inl x)` to `2 * x` and `(sum.inr x)` to
/-- An equivalence between `ℕ ⊕ ℕ` and `ℕ`, by mapping `(Sum.inl x)` to `2 * x` and `(Sum.inr x)` to
`2 * x + 1`.
-/
@[simps symmApply]
def natSumNatEquivNat : Sum ℕ ℕ ≃ ℕ :=
@[simps symm_apply]
def natSumNatEquivNat : ℕ ⊕ ℕ ≃ ℕ :=
(boolProdEquivSum ℕ).symm.trans boolProdNatEquivNat
#align equiv.nat_sum_nat_equiv_nat Equiv.natSumNatEquivNat

set_option linter.deprecated false in
@[simp]
theorem nat_sum_nat_equiv_nat_apply : ⇑nat_sum_nat_equiv_nat = Sum.elim bit0 bit1 := by
theorem natSumNatEquivNat_apply : ⇑natSumNatEquivNat = Sum.elim bit0 bit1 := by
ext (x | x) <;> rfl
#align equiv.nat_sum_nat_equiv_nat_apply Equiv.nat_sum_nat_equiv_nat_apply
#align equiv.nat_sum_nat_equiv_nat_apply Equiv.natSumNatEquivNat_apply

/-- An equivalence between `ℤ` and `ℕ`, through `ℤ ≃ ℕ ⊕ ℕ` and `ℕ ⊕ ℕ ≃ ℕ`.
-/
Expand All @@ -63,7 +63,6 @@ def prodEquivOfEquivNat (e : α ≃ ℕ) : α × α ≃ α :=
α × α ≃ ℕ × ℕ := prodCongr e e
_ ≃ ℕ := mkpairEquiv
_ ≃ α := e.symm

#align equiv.prod_equiv_of_equiv_nat Equiv.prodEquivOfEquivNat

end Equiv