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Proof step 9 from proof product-all-elems-le (previous step)

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)) ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ i * j = j * i
Proof step 11:
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: commutativity seq: ⊦ i * j = j * i subst: [j, i] → [i0, i0 ^ n→i(# ints0)] flags: rewrite

Additional information

Used terms: i0, i0 ^ n→i(# ints0)