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Proof step 9 from proof le-abs-numerator (previous step)

i < 0, 0 ≤ i ∧ 0 < j → i / j * j ≤ i ⊦ j = 0, 0 ≤ abs(i) + i / j ∧ i / j ≤ abs(i), ¬ i < 0 ∧ ¬ j < 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
0 < i / j + i, i < 0, 0 ≤ i ∧ 0 < j → i / j * j ≤ i ⊦ j = 0

Rule argument

Additional information

Used simplifier rules: zero-05: j ≠ 0 ⊦ i < 0 → i ≤ i / j;%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i ≤ ~ j ↔ i + j ≤ 0; (s)% ⊦ i + 0 = i; (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)% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)