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Proof step 2 from proof modeq-pos-04 (previous step)

0 < j0, 0 < j, j0 ≤ i0 * j, 0 < i0 ⊦ j = (i0 * j - j0) % j + j0 % j, j0 % j = 0

User applied rule insert lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: modeq-pos-03 seq: 0 ≤ i0 * j + j0, 0 < j, j0 < 0, j0 % j ≠ 0 ⊦ (i0 * j + j0) % j = j0 % j + j subst: [i0, j, j0] → [i0, j, ~ j0] flags: current, []

Additional information

Used simplifier rules: mod-neg-01: j ≠ 0 ⊦ ~ i % j = ~(i % j);%mod-neg-01: j ≠ 0 ⊦ ~ i % j = ~(i % j);%neg-15: ⊦ i + ~ j = i - j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%zero-left: ⊦ 0 * i = 0; 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-06: ⊦ 0 - i = ~ i; from specification int-abs%sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%neg-05: ⊦ ~ i < 0 ↔ 0 < i; from specification int-abs%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs%neg-10: ⊦ ~ i = 0 ↔ i = 0; from specification int-abs Used terms: i0, j, ~ j0