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Proof step 15 from proof modmod-mult-07 (previous step)

i < 0, 0 < j, i0 < 0, i ≤ 0 ⊦ (i % j * i0) % j = (i * i0) % j, j = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 16:
j ≠ 0 ⊦ i < 0 → (* i) % j = ~(~ i % j)
Proof step 17:
i0 < 0, 0 < j, i < 0, i ≤ 0 ⊦ (~(~ i % j) * i0) % j = (i * i0) % j, j = 0

Rule argument

spec: <toplevel> name: modnum-neg seq: j ≠ 0 ⊦ i < 0 → (* i) % j = ~(~ i % j) subst: [j, i] → [j, i] flags: <all paths>

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s) Used terms: j, i