# Documentation

Mathlib.Data.Nat.EvenOddRec

# A recursion principle based on even and odd numbers. #

def Nat.evenOddRec {P : Sort u_1} (h0 : P 0) (h_even : (n : ) → P nP (2 * n)) (h_odd : (n : ) → P nP (2 * n + 1)) (n : ) :
P n

Recursion principle on even and odd numbers: if we have P 0, and for all i : ℕ we can 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.

Equations
• One or more equations did not get rendered due to their size.
@[simp]
theorem Nat.evenOddRec_zero (P : Sort u_1) (h0 : P 0) (h_even : (i : ) → P iP (2 * i)) (h_odd : (i : ) → P iP (2 * i + 1)) :
Nat.evenOddRec h0 h_even h_odd 0 = h0
@[simp]
theorem Nat.evenOddRec_even (n : ) (P : Sort u_1) (h0 : P 0) (h_even : (i : ) → P iP (2 * i)) (h_odd : (i : ) → P iP (2 * i + 1)) (H : h_even 0 h0 = h0) :
Nat.evenOddRec h0 h_even h_odd (2 * n) = h_even n (Nat.evenOddRec h0 h_even h_odd n)
@[simp]
theorem Nat.evenOddRec_odd (n : ) (P : Sort u_1) (h0 : P 0) (h_even : (i : ) → P iP (2 * i)) (h_odd : (i : ) → P iP (2 * i + 1)) (H : h_even 0 h0 = h0) :
Nat.evenOddRec h0 h_even h_odd (2 * n + 1) = h_odd n (Nat.evenOddRec h0 h_even h_odd n)