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Proof step 5 from proof product-pot-no-dups (previous step)

0 < min(ints), j ^ n→i(# rmdup(ints)) < Π ints, ∀ i. i ∈ ints → 0 < i ∧ i ^ n→i(#oc(i, ints)) ≤ j, ∀ ints0, j. # ints0 < # ints → j ^ n→i(# rmdup(ints0)) < Π ints0 ∧ (∀ i. i ∈ ints0 → 0 < i ∧ i ^ n→i(#oc(i, ints0)) ≤ j) → false ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ i ∈ ints → Π ints = i ^ n→i(#oc(i, ints)) * Π(ints -l i)
Proof step 7:
ints.first ∈ ints, 0 < min(ints), j ^ n→i(# rmdup(ints)) < ints.first ^ n→i(#oc(ints.first, ints)) * Π(ints -l ints.first), ∀ i. i ∈ ints → 0 < i ∧ i ^ n→i(#oc(i, ints)) ≤ j, ∀ ints0, j. # ints0 < # ints → j ^ n→i(# rmdup(ints0)) < Π ints0 ∧ (∀ i. i ∈ ints0 → 0 < i ∧ i ^ n→i(#oc(i, ints0)) ≤ j) → false ⊦

Rule argument

spec: inst: name: product-pot-rec seq: ⊦ i ∈ ints → Π ints = i ^ n→i(#oc(i, ints)) * Π(ints -l i) subst: [i, ints] → [ints.first, ints] flags: rewrite, current

Additional information

Used simplifier rules: in-08: ⊦ ints.first ∈ ints ↔ ints ≠ []; from specification list[intlista]%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%min-empty: ⊦ min([]) = 0; from specification intlist Used terms: ints.first, ints