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Proof step 20 from proof modmod-mult (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-basic2 name: pls-01 seq: ¬ i < 0, ¬ j < 0 ⊦ ¬ i + j < 0 subst: [j, i] → [j0 * i2, i1 * (j1 + i2 * j)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ 0 ≤ i → (i * j < 0 ↔ j < 0 ∧ i ≠ 0); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i ≤ j → ¬ j < i; (s)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i + j < j ↔ i < 0; (s)% ⊦ 0 < i → (i * j < 0 ↔ j < 0); (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: j0 * i2, i1 * (j1 + i2 * j)