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Proof step 15 from proof product-pot-no-dups (previous step)

j ^ n→i(# rmdup(i0 ' + ints1)) < i0 ^ (1 + n→i(#oc(i0, ints1))) * Π(ints1 -l i0), 0 < min(i0 ' + ints1), i0 ^ (1 + n→i(#oc(i0, ints1))) ≤ j, 0 < i0, i ∈ ints1 → 0 < i ∧ i ^ (n→i(#oc(i, i0 ')) + n→i(#oc(i, ints1))) ≤ j, ∀ i. i ∈ ints1 → 0 < i ∧ i ^ (n→i(#oc(i, i0 ')) + n→i(#oc(i, ints1))) ≤ j, ∀ ints0, j. # ints0 < #(i0 + ints1) → j ^ n→i(# rmdup(ints0)) < Π ints0 ∧ (∀ i. i ∈ ints0 → 0 < i ∧ i ^ n→i(#oc(i, ints0)) ≤ j) → false ⊦ j ^ n→i(# rmdup(ints1 -l i0)) < Π(ints1 -l i0) ∧ (i ∈ ints1 -l i0 → 0 < i ∧ i ^ n→i(#oc(i, ints1 -l i0)) ≤ j)

User applied rule case distinction

And obtained the following new goals

Proof step 16:
j ^ n→i(# rmdup(i0 ' + ints1)) < i0 ^ (1 + n→i(#oc(i0, ints1))) * Π(ints1 -l i0), 0 < min(i0 ' + ints1), i0 ^ (1 + n→i(#oc(i0, ints1))) ≤ j, 0 < i0, i ∈ ints1 → 0 < i ∧ i ^ (n→i(#oc(i, i0 ')) + n→i(#oc(i, ints1))) ≤ j, ∀ i. i ∈ ints1 → 0 < i ∧ i ^ (n→i(#oc(i, i0 ')) + n→i(#oc(i, ints1))) ≤ j, ∀ ints0, j. # ints0 < #(i0 + ints1) → j ^ n→i(# rmdup(ints0)) < Π ints0 ∧ (∀ i. i ∈ ints0 → 0 < i ∧ i ^ n→i(#oc(i, ints0)) ≤ j) → false ⊦ j ^ n→i(# rmdup(ints1 -l i0)) < Π(ints1 -l i0)

Rule argument

pos: right 1

Additional information

Used simplifier rules: in-del-new: ⊦ i ∈ ints -l i0 ↔ i ∈ ints ∧ i ≠ i0; from specification list-del[intlista]%one-01: ⊦ i ≠ i0 → #oc(i, i0 ') = 0; from specification list-perm[intlista]%ax-04: ⊦ i + 0 = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%ocs-del-length: ⊦ i ≠ i0 → #oc(i, ints -l i0) = #oc(i, ints); from specification list-perm[intlista]