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Proof step 8 from proof modmod-add (previous step)

j0 + i0 = i2 * j + j1, 0 ≤ j1, j1 < j, 0 ≤ j0 + i0, 0 ≤ i0, 0 ≤ j0, j0 < j, 0 ≤ i1, 0 < j ⊦ j1 = ((i1 * j + j0) + i0) % j

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: modeq-pos-02 seq: 0 ≤ i0 * j + j0, 0 < j, 0 ≤ j0 ⊦ (i0 * j + j0) % j = j0 % j subst: [j0, j, i0] → [i0 + j0, j, i1] flags: rewrite, current

Additional information

Used simplifier rules: 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%ls-01: ⊦ i + j < i + i0 ↔ j < i0; 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%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 Used terms: i0 + j0, j, i1