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Proof step 8 from proof noeth-zero-ls (previous step)

i * j < j, 0 < j, i * j ≤ 0, i ≤ i1, ∀ j0, i0. 0 < j0 ∧ i0 ≠ 0 ∧ abs(i0) < i1 → 0 < abs(i0) * j0 ⊦ i = 0, i < 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
0 < i1, i * j < i, i * j < j, 0 < i, 0 < j, i * j < i1, i * j ≤ 0, i ≤ i1, ∀ j0, i0. 0 < j0 ∧ i0 ≠ 0 ∧ abs(i0) < i1 → 0 < abs(i0) * j0 ⊦

Rule argument

Additional information

Used simplifier rules: 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-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs% ⊦ i < j → ¬ j < i; (s)% ⊦ ¬ i < j → (i = j ↔ ¬ j < i); (ss)