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Proof step 13 from proof mod-pred-neg (previous step)

i0 * j + j0 = (i2 * j + j1) + 1, i2 * j + j1 < 0, j1 < j, i2 < j, i2 * j + j1 < ~1, abs(j0) < j, 0 < j, i0 < j, j0 < j, abs(j1) < j, j1 ≤ 0, i2 ≤ 0, i0 ≤ 0, j0 ≤ 0 ⊦ 1 = j0 + j, j0 = j1 + 1, i0 * j + ~(j1 + 1) = i2 * j + ~ j0 ∧ 0 ≤ ~(j1 + 1) ∧ ~(j1 + 1) < j ∧ 0 ≤ ~ j0 ∧ ~ j0 < j

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 14:
i0 * j + j0 = (i2 * j + j1) + 1, i0 < j + j0, i2 < j + j0, j1 < j + j0, 0 < j + j0, abs(j1) < j, j0 < j, i0 < j, 0 < j, abs(j0) < j, i2 * j + j1 < ~1, i2 < j, j1 < j, i2 * j + j1 < 0, j0 ≤ 0, i0 ≤ 0, i2 ≤ 0, j1 ≤ 0 ⊦ 1 = j0 + j, j0 = j1 + 1, 0 ≤ j + j1 ∧ j1 < 0

Rule argument

Additional information

Used simplifier rules: ⊦ 0 < i → i1 < i1 + i; (s)%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs% ⊦ i ≤ j → ¬ j < i; (s)% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ i + j < j ↔ i < 0; (s)%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ i + 0 = i; (s)% ⊦ ~1 < i ↔ 0 ≤ i; (s)% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; (s)% ⊦ ~ i < j ↔ 0 < j + i; (s)% ⊦ i ≤ ~1 ↔ i < 0; (s)% ⊦ i + (* j) ≤ (* j0) ↔ i ≤ j0 - j; (s)% ⊦ i ≤ ~ j ↔ i + j ≤ 0; (s)% ⊦ i + (j - i0) = i + j - i0; (s)% ⊦ i - j = i0 ↔ i = i0 + j; (s)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2% ⊦ i + ~ j = i - j; (s)