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Proof step 5 from proof product-positive (previous step)

¬ 0 < i, 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 6:
i ≤ 0, (i0 ∈ ints0 → 0 < i0) → ¬ Π ints0 ≤ 0, ∀ i1. i1 = i ∨ i1 ∈ ints0 → 0 < i1 ⊦ i ≠ 0 ∧ ¬ 0 ≤ Π ints0

Rule argument

Additional information

Used simplifier rules: forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%zero-le-03: ⊦ i ≤ 0 → (i * j ≤ 0 ↔ ¬ ¬(i = 0 ∨ 0 ≤ j)); from specification int-mult%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs