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Proof step 2 from proof primes-divide (previous step)

∀ i. ¬ dups(ints0) ∧ (∀ j. j ∈ ints0 → prime(j) ∧ j | i) ∧ 0 < i → Π ints0 ≤ i, ¬ dups(i0 + ints0), ∀ j. j ∈ i0 + ints0 → prime(j) ∧ j | i, 0 < i ⊦ Π(i0 + ints0) ≤ i

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
i < i0 * Π ints0, 0 < i, 0 < i0 * Π ints0, 1 < i0 * Π ints0, ∀ i1. (∀ j. j ∈ ints0 → prime(j) ∧ j | i1) ∧ 0 < i1 → Π ints0 ≤ i1, ∀ j. j = i0 ∨ j ∈ ints0 → prime(j) ∧ j | i ⊦ dups(ints0), i0 ∈ ints0

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%transitivity: ⊦ i < i0 ∧ i0 < i1 → i < i1; (forward) from specification oelem[intlista]%forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1%in-17: ⊦ i ∈ i0 ' ↔ i = i0; from specification list[intlista]%in-append: ⊦ i ∈ ints + ints0 ↔ ¬ ¬(i ∈ ints ∨ i ∈ ints0); from specification list[intlista]%dups-cons: ⊦ dups(i ' + ints) ↔ dups(ints) ∨ i ∈ ints; from specification list-dup[intlista]%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%prod-rec: ⊦ Π(i ' + ints) = i * Π ints; from specification intlist%c: ⊦ i + ints = i ' + ints; from specification list[intlista]