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Proof step 17 from proof divides-prime (previous step)

0 < i0 * i, i0 < i0 * i, 0 ≤ i0 * i, 2 ≤ i0 * i, i0 ≤ 1, ∀ i1. 1 < i1 ∧ i1 < i0 * i → ¬ i1 | i0 * i ⊦ abs(i) = i0 * i, abs(i) = 1, 0 < i0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 18:
i < 0, i0 < 0, 2 ≤ i0 * i, ∀ i1. 1 < i1 ∧ i1 < i0 * i → ¬ i1 | i0 * i ⊦ ~1 = i0, ~1 = i, i = 0, i0 = 0

Rule argument

Additional information

Used simplifier rules: forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs% ⊦ i < 1 ↔ i ≤ 0; (s)% ⊦ i < 0 → (i < i * j ↔ j < 1); (s)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i < 0 → (0 ≤ i * j ↔ j ≤ 0); (s)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)% ⊦ i * j = 0 ↔ i = 0 ∨ j = 0; (ws)% ⊦ i * j + i = i * (j + 1); (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)%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 ∧ i ≠ 0); (s)%j0 ≠ 0, j ≤ j0 ⊦ i ≤ (* j) → i ≤ (* j0); (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)