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Proof step 7 from proof product-positive (previous step)

0 < i, Π ints0 ≤ 0, (i0 ∈ ints0 → 0 < i0) → ¬ Π ints0 ≤ 0, ∀ i1. i1 = i ∨ i1 ∈ ints0 → 0 < i1 ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
i0 ∈ ints0, 0 < i, Π ints0 ≤ 0, Π ints0 < i, i0 ≤ 0, i0 < i, ∀ i1. i1 ∈ ints0 → 0 < i1 ⊦

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%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%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs