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Proof step 19 from proof modmod-mult-07 (previous step)

j ≠ 0 ∧ i < 0, i0 < j, i0 < 0, i + 1 < j, 0 < j, i < 0, i < j, i0 + 1 < j ⊦ (~ i % j * ~ i0) % j = (i * i0) % j, j = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: modmod-mult-08 seq: 0 < j ⊦ 0 ≤ i ∧ 0 ≤ i0 → (i % j * i0) % j = (i * i0) % j subst: [i0, j, i] → [~ i0, j, ~ i] flags: rewrite, current

Additional information

Used simplifier rules: mod-neg-01: j ≠ 0 ⊦ ~ i % j = ~(i % j);%minus-left: ⊦ ~ i * j = ~(i * j); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%neg-02: ⊦ ~ ~ i = i; from specification int-abs%negle-07: ⊦ i ≤ ~ j ↔ i + j ≤ 0; from specification int-abs%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%zero-gels-03: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs Used terms: ~ i0, j, ~ i