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Proof step 19 from proof modmod-mult (previous step)

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, ¬ (i1 * (j1 + i2 * j) + j0 * i2) * j < 0 ∧ ¬ j0 * j1 < 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs% ⊦ i ≤ j → ¬ j < i; (s)% ⊦ 0 ≤ i → (i * j < 0 ↔ j < 0 ∧ i ≠ 0); (s)%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ 0 < i → (i * j < 0 ↔ j < 0); (s)