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Proof step 27 from proof mod-ls-same-01 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 28:
i0 < i0 * j + j0, j0 + j < 0, 0 < i0 * j + j0, j < 0, 0 ≤ j0, i0 ≤ 0 ⊦ i0 * j + j0 = 0, i0 < 0 ↔ ¬ (i0 * j + j0) + j < 0, ~1 = j, j = 1

Rule argument

Additional information

Used simplifier rules: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i + j < j ↔ i < 0; (s)%forward-le-05: ⊦ 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%forward-lels: ⊦ 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-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 ↔ i + j < 0; (s)% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (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 → (0 < i * j ↔ j < 0); (s)% ⊦ i < i + j ↔ 0 < j; (s)