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Proof step 11 from proof product-all-elems-le (previous step)

i0 * i0 ^ n→i(# ints0) < 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)) ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
i ≤ i0, j ≤ j0, 0 ≤ i, 0 ≤ j ⊦ ¬ i0 * j0 < i * j
Proof step 13:
¬(i ≤ i0 ∧ Π ints0 ≤ i0 ^ n→i(# ints0) ∧ 0 ≤ i ∧ 0 ≤ Π ints0), i0 * i0 ^ n→i(# ints0) < 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

spec: int-mult inst: name: pnls-pos seq: i ≤ i0, j ≤ j0, 0 ≤ i, 0 ≤ j ⊦ ¬ i0 * j0 < i * j subst: [j, i, j0, i0] → [Π ints0, i, i0 ^ n→i(# ints0), i0] flags: rewrite

Additional information

Used terms: Π ints0, i, i0 ^ n→i(# ints0), i0