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Proof step 19 from proof product-all-elems-le (previous step)

i0 * i0 ^ n→i(# ints0) < i * Π ints0, i ≤ i0, 0 < i, 0 < i0, 0 ≤ i, 0 ≤ i0, ∀ i1. i1 ∈ ints0 → 0 < i1 ∧ i1 ≤ i0, ∀ i1. ¬ i1 ^ n→i(# ints0) < Π ints0 ∨ (∃ i. ¬(i ∈ ints0 → 0 < i ∧ i ≤ i1)) ⊦ 0 ≤ Π ints0, 0 ≤ Π ints0 ∧ Π ints0 ≤ i0 ^ n→i(# ints0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
Π ints0 < 0, 0 ≤ i0, 0 ≤ i, 0 < i0, 0 < i, i ≤ i0, i0 * i0 ^ n→i(# ints0) < i * Π ints0, ∀ i1. ¬ i1 ^ n→i(# ints0) < Π ints0 ∨ (∃ i. ¬(i ∈ ints0 → 0 < i ∧ i ≤ i1)), ∀ i1. i1 ∈ ints0 → 0 < i1 ∧ i1 ≤ i0 ⊦

Rule argument

Additional information

Used simplifier rules: forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%transitivity: ⊦ i < i0 ∧ i0 < i1 → i < i1; (forward) from specification oelem[intlista]%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs