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Proof step 8 from proof product-all-elems-le (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
i0 ^ n→i(# ints0) * i0 < i * Π ints0, i ≤ 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)) ⊦

Rule argument

Additional information

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