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Proof step 9 from proof div-ls-mone (previous step)

i0 * j + j0 < 0, j0 < j, i0 < j, 0 < j, abs(j0) < j, j0 ≤ 0, i0 ≤ 0 ⊦ i0 < ~1 ↔ j * (2 + i0) + j0 ≤ 0, j0 < 0

Heuristic simplifier applied rule simplifier

And closed the goal!

Rule argument

Additional information

Used simplifier rules: i + 1 = i0 ⊦ j < (* i0) → ¬ (* i) < j; (s)% ⊦ i ≤ j ↔ ¬ j < i; (ss)%j0 ≠ 0, j < j0 ⊦ i ≤ (* j) → i < (* j0); (s)%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ i + (* j) ≤ (* j0) ↔ i ≤ j0 - j; (s)% ⊦ 0 < i → (i * j ≤ 0 ↔ j ≤ 0); (s)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2% ⊦ i + 0 = i; (s)% ⊦ i ≤ j → (j ≤ i ↔ i = j); (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ ¬ j < 0 → abs(j) = j; (s)