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Proof step 13 from proof modmod-mult-03 (previous step)

i < j, i < j0, i < 0, 0 < j, 0 < j0, i + 1 < j, i + 1 < j0 ⊦ (i % (j * j0)) % j = i % j

User applied rule insert lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: modmod-mult-04 seq: ⊦ 0 < j ∧ 0 ≤ i ∧ 0 < j0 → (i % (j * j0)) % j = i % j subst: [j, i, j0] → [j, ~ i, j0] flags: current, []

Additional information

Used simplifier rules: mod-neg-01: j ≠ 0 ⊦ ~ i % j = ~(i % j);%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%zero-neq-01: ⊦ 0 < i → (i * j = 0 ↔ j = 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%zero-left: ⊦ 0 * i = 0; from specification int-mult%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%neg-12: ⊦ ~ i = ~ j ↔ i = j; from specification int-abs Used terms: j, ~ i, j0