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Proof step 12 from proof modmod-mult-07 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
0 < j ⊦ 0 ≤ i ∧ 0 ≤ i0 → (i % j * i0) % j = (i * i0) % j
Proof step 14:
¬(0 < j ∧ 0 ≤ ~ i ∧ 0 ≤ i0), j ≠ 0, j ≠ 0 ∧ i < 0, i < j, i < 0, 0 < j, i + 1 < j ⊦ ~((~ i % j * i0) % j) = (i * i0) % j

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 Used terms: i0, j, ~ i