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feat: Nat.lt_pow_self (#6200)

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This PR upstreams `Nat.lt_pow_self` and `Nat.lt_two_pow` from Mathlib
and uses them to prove the simp theorem `Nat.mod_two_pow`.

This simplifies expressions like `System.Platform.numBits % 2 ^
System.Platform.numBits = System.Platform.numBits`, which is needed for
#6188.
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Mac Malone 2024-11-26 18:42:23 -05:00 • committed by GitHub
parent 3d511a582a
commit 23bec25fce
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src/Init/Data/Nat/Lemmas.lean
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@ -846,6 +846,18 @@ protected theorem pow_lt_pow_iff_pow_mul_le_pow {a n m : Nat} (h : 1 < a) :
rw [←Nat.pow_add_one, Nat.pow_le_pow_iff_right (by omega), Nat.pow_lt_pow_iff_right (by omega)]
omega
protected theorem lt_pow_self {n a : Nat} (h : 1 < a) : n < a ^ n := by
induction n with
| zero => exact Nat.zero_lt_one
| succ _ ih => exact Nat.lt_of_lt_of_le (Nat.add_lt_add_right ih 1) (Nat.pow_lt_pow_succ h)
protected theorem lt_two_pow_self : n < 2 ^ n :=
Nat.lt_pow_self Nat.one_lt_two
@[simp]
protected theorem mod_two_pow_self : n % 2 ^ n = n :=
Nat.mod_eq_of_lt Nat.lt_two_pow_self
@[simp]
theorem two_pow_pred_mul_two (h : 0 < w) :
2 ^ (w - 1) * 2 = 2 ^ w := by