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Proof step 8 from proof mod-zero-06 (previous step)

1 < i / j * j, 0 < i / j * j, i < i / j * j, 0 < i, 0 ≤ i, abs(j) = 1 ∨ abs(j) ≤ i ⊦ j = 0, 0 ≤ i ∧ 0 < j

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
1 < i / j * j, j < 0, 0 < i, i < i / j * j, i / j < 0, 0 ≤ j + i, 0 ≤ i ⊦ ~1 = j

Rule argument

Additional information

Used simplifier rules: div-mone-01: ⊦ i / ~1 = ~ i;%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs% ⊦ ¬ i < i; (s)% ⊦ ~ ~ i = i; (s)% ⊦ i * ~1 = ~ i; (s)% ⊦ ~ i * 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% ⊦ ~ i ≤ j ↔ 0 ≤ 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% ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0; (s)% ⊦ ~ i = j ↔ i + j = 0; (s)% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ i < j → ¬ j < i; (s)%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 < j → (i = j ↔ ¬ j < i); (ss)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)