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feat: add Int.emod_sub_emod and Int.sub_emod_emod (#6507)

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This PR adds the subtraction equivalents for `Int.emod_add_emod` (`(a %
n + b) % n = (a + b) % n`) and `Int.add_emod_emod` (`(a + b % n) % n =
(a + b) % n`). These are marked @[simp] like their addition equivalents.

Discussed on Zulip in

https://leanprover.zulipchat.com/#narrow/channel/270676-lean4/topic/Adding.20some.20sub_emod.20lemmas.20to.20DivModLemmas
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Vlad Tsyrklevich 2025-01-08 03:20:43 +01:00 • committed by GitHub
parent 22a799524f
commit 78ed072ab0
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src/Init/Data/Int/DivModLemmas.lean
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@ -534,6 +534,13 @@ theorem mul_emod (a b n : Int) : (a * b) % n = (a % n) * (b % n) % n := by
@[simp] theorem emod_emod (a b : Int) : (a % b) % b = a % b := by
conv => rhs; rw [← emod_add_ediv a b, add_mul_emod_self_left]
@[simp] theorem emod_sub_emod (m n k : Int) : (m % n - k) % n = (m - k) % n :=
Int.emod_add_emod m n (-k)
@[simp] theorem sub_emod_emod (m n k : Int) : (m - n % k) % k = (m - n) % k := by
apply (emod_add_cancel_right (n % k)).mp
rw [Int.sub_add_cancel, Int.add_emod_emod, Int.sub_add_cancel]
theorem sub_emod (a b n : Int) : (a - b) % n = (a % n - b % n) % n := by
apply (emod_add_cancel_right b).mp
rw [Int.sub_add_cancel, ← Int.add_emod_emod, Int.sub_add_cancel, emod_emod]