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Proof step 2 from proof product-positive (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
i * Π ints0 ≤ 0, (i0 ∈ ints0 → 0 < i0) → ¬ Π ints0 ≤ 0, ∀ i1. i1 = i ∨ i1 ∈ ints0 → 0 < i1 ⊦

Rule argument

Additional information

Used simplifier rules: product-app: ⊦ Π(ints + ints0) = Π ints * Π ints0;%product-one: ⊦ Π i ' = i;%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%in-17: ⊦ i ∈ i0 ' ↔ i = i0; from specification list[intlista]%in-append: ⊦ i ∈ ints + ints0 ↔ ¬ ¬(i ∈ ints ∨ i ∈ ints0); from specification list[intlista]%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%c: ⊦ i + ints = i ' + ints; from specification list[intlista]