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Proof step 9 from proof modmod-mult-04 (previous step)

0 < j0, 0 ≤ j2, j2 < j, 0 ≤ i0, i1 * j + j2 < j * j0, 0 ≤ i1, 0 < j ⊦ j2 = ((i1 + i0 * j0) * j + j2) % j

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: modeq-pos seq: 0 ≤ i0 * j + j0, 0 ≤ j0, j0 < j ⊦ (i0 * j + j0) % j = j0 subst: [j0, j, i0] → [j2, j, i1 + i0 * j0] flags: rewrite, current

Additional information

Used simplifier rules: zero-ge-add: ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 0); from specification int-mult%lels-04: ⊦ i ≤ j → (¬ ¬ j < i ↔ false); from specification int-abs Used terms: j2, j, i1 + i0 * j0