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Proof step 15 from proof mod-pred-neg (previous step)

i0 * j + j0 = (i2 * j + j1) + 1, j1 = 0, i2 * j + j1 < 0, j1 < j, i2 < j, i2 * j + j1 < ~1, abs(j0) < j, 0 < j, i0 < j, j0 < j, abs(j1) < j, 0 < j + j0, j1 < j + j0, i2 < j + j0, i0 < j + j0, j1 ≤ 0, i2 ≤ 0, i0 ≤ 0, j0 ≤ 0 ⊦ 1 = j0 + j, j0 = j1 + 1, 0 ≤ j + j1 ∧ 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)%numlsf: ⊦ i < (* j) ∧ j < 0 → i < 0; (forward) from specification int-basic1%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs% ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; (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)% ⊦ i + 0 = i; (s)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2