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[Merged by Bors] - feat(Data/Nat/EvenOddRec): add strong recursion principle on even and odd numbers #5932

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[Merged by Bors] - feat(Data/Nat/EvenOddRec): add strong recursion principle on even and odd numbers #5932

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Add Nat.evenOddStrongRec
Multramate Jul 15, 2023
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Multramate Jul 21, 2023
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Merge branch 'master' into Nat.evenOddStrongRec
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18 changes: 14 additions & 4 deletions Mathlib/Data/Nat/EvenOddRec.lean
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Expand Up @@ -3,7 +3,7 @@ Copyright (c) 2022 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Init.Data.Nat.Bitwise
import Mathlib.Data.Nat.Parity
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#align_import data.nat.even_odd_rec from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
/-! # A recursion principle based on even and odd numbers. -/
Expand All @@ -18,7 +18,6 @@ namespace Nat
extend from `P i` to both `P (2 * i)` and `P (2 * i + 1)`, then we have `P n` for all `n : ℕ`.
This is nothing more than a wrapper around `Nat.binaryRec`, to avoid having to switch to
dealing with `bit0` and `bit1`. -/

@[elab_as_elim]
def evenOddRec {P : ℕ → Sort*} (h0 : P 0) (h_even : ∀ n, P n → P (2 * n))
(h_odd : ∀ n, P n → P (2 * n + 1)) (n : ℕ) : P n :=
Expand All @@ -36,7 +35,7 @@ theorem evenOddRec_zero (P : ℕ → Sort*) (h0 : P 0) (h_even : ∀ i, P i →
@[simp]
theorem evenOddRec_even (n : ℕ) (P : ℕ → Sort*) (h0 : P 0) (h_even : ∀ i, P i → P (2 * i))
(h_odd : ∀ i, P i → P (2 * i + 1)) (H : h_even 0 h0 = h0) :
@evenOddRec _ h0 h_even h_odd (2 * n) = h_even n (evenOddRec h0 h_even h_odd n) :=
(2 * n).evenOddRec h0 h_even h_odd = h_even n (evenOddRec h0 h_even h_odd n) :=
have : ∀ a, bit false n = a →
HEq (@evenOddRec _ h0 h_even h_odd a) (h_even n (evenOddRec h0 h_even h_odd n))
| _, rfl => by rw [evenOddRec, binaryRec_eq]; apply eq_rec_heq; exact H
Expand All @@ -46,11 +45,22 @@ theorem evenOddRec_even (n : ℕ) (P : ℕ → Sort*) (h0 : P 0) (h_even : ∀ i
@[simp]
theorem evenOddRec_odd (n : ℕ) (P : ℕ → Sort*) (h0 : P 0) (h_even : ∀ i, P i → P (2 * i))
(h_odd : ∀ i, P i → P (2 * i + 1)) (H : h_even 0 h0 = h0) :
@evenOddRec _ h0 h_even h_odd (2 * n + 1) = h_odd n (evenOddRec h0 h_even h_odd n) :=
(2 * n + 1).evenOddRec h0 h_even h_odd = h_odd n (evenOddRec h0 h_even h_odd n) :=
have : ∀ a, bit true n = a →
HEq (@evenOddRec _ h0 h_even h_odd a) (h_odd n (evenOddRec h0 h_even h_odd n))
| _, rfl => by rw [evenOddRec, binaryRec_eq]; apply eq_rec_heq; exact H
eq_of_heq (this _ (bit1_val _))
#align nat.even_odd_rec_odd Nat.evenOddRec_odd

/-- Strong recursion principle on even and odd numbers: if for all `i : ℕ` we can prove `P (2 * i)`
from `P j` for all `j < 2 * i` and we can prove `P (2 * i + 1)` from `P j` for all `j < 2 * i + 1`,
then we have `P n` for all `n : ℕ`. -/
@[elab_as_elim]
noncomputable def evenOddStrongRec {P : ℕ → Sort*} (n : ℕ)
(h_even : ∀ n : ℕ, (∀ k : ℕ, k < 2 * n → P k) → P (2 * n))
(h_odd : ∀ n : ℕ, (∀ k : ℕ, k < 2 * n + 1 → P k) → P (2 * n + 1)) : P n :=
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n.strongRecOn fun m ih => m.even_or_odd'.choose_spec.by_cases
(fun h => h.symm ▸ h_even m.even_or_odd'.choose <| h ▸ ih)
(fun h => h.symm ▸ h_odd m.even_or_odd'.choose <| h ▸ ih)

end Nat