[Merged by Bors] - feat: a + a = 0 ↔ a = 0 in ZMod n for n odd#6086
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[Merged by Bors] - feat: a + a = 0 ↔ a = 0 in ZMod n for n odd#6086
a + a = 0 ↔ a = 0 in ZMod n for n odd#6086Conversation
eric-wieser
reviewed
Jul 24, 2023
eric-wieser
reviewed
Jul 24, 2023
| rw [←mul_two, ←@Nat.cast_two (ZMod n), | ||
| ←ZMod.coe_unitOfCoprime 2 (Nat.prime_two.coprime_iff_not_dvd.mpr hn), Units.mul_left_eq_zero] | ||
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| theorem ne_neg_self (n : ℕ) [hn : Fact ((n : ℕ) % 2 = 1)] {a : ZMod n} (ha : a ≠ 0) : a ≠ -a := by |
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This one shouldn't be using Fact (and probably should also be using Odd), but I guess that's out of scope.
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I agree with Eric that it would be nice to excise this Fact in a future PR.
ocfnash
approved these changes
Jul 31, 2023
| rw [←mul_two, ←@Nat.cast_two (ZMod n), | ||
| ←ZMod.coe_unitOfCoprime 2 (Nat.prime_two.coprime_iff_not_dvd.mpr hn), Units.mul_left_eq_zero] | ||
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| theorem ne_neg_self (n : ℕ) [hn : Fact ((n : ℕ) % 2 = 1)] {a : ZMod n} (ha : a ≠ 0) : a ≠ -a := by |
Contributor
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I agree with Eric that it would be nice to excise this Fact in a future PR.
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a + a = 0 ↔ a = 0 in ZMod n for n odda + a = 0 ↔ a = 0 in ZMod n for n odd
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This PR proves that
a + a = 0 ↔ a = 0, and uses it to golf the proof ofne_neg_self.I removed
le_div_two_iff_lt_negsince it seems to just be a technical auxiliary lemma for the current proof ofne_neg_self, but I can add it back if people think it should be kept around. If we do keep it around, I wonder if it would be better to state it as2 * x.val < n ↔ n < 2 * (-x).val(or something like that) to avoid natural number division.