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Proof step 17 from proof mod-ls-same-02 (previous step)

i = i0 * j + j0, i < 0, j < j0, j < 0, j + i ≤ 0, j0 ≤ 0, 0 ≤ i0 ⊦ i < j0 ↔ abs(j) = 1 ∨ ¬ 0 < abs(j) + i, j = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 18:
i0 * j + j0 < 0, j < j0, j < 0, j * (i0 + 1) + j0 ≤ 0, j0 ≤ 0, 0 ≤ i0 ⊦ 0 < i0 ↔ ~1 = j ∨ ¬ 0 < j * (i0 - 1) + j0, j = 0

Rule argument

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ 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%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs% ⊦ i * j + i = i * (j + 1); (s)% ⊦ i * j - i = i * (j - 1); (s)% ⊦ i + ~ j = i - j; (s)% ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0; (s)% ⊦ ~ i = j ↔ i + j = 0; (s)% ⊦ i < 0 → abs(i) = ~ i; (s)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ i < 0 → (i * j < 0 ↔ 0 < j); (s)% ⊦ i + j < j ↔ i < 0; (s)