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Proof step 26 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, Π(ints1 -l i0) ≤ j ^ n→i(# rmdup(ints1 -l 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(i0 ' + ints1)) < j * j ^ n→i(# rmdup(ints1 -l i0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 27:
⊦ i < i0 * j ∧ j ≤ j0 ∧ 0 ≤ i0 → i < i0 * j0
Proof step 28:
¬(j ^ n→i(# rmdup(i0 ' + ints1)) < j * Π(ints1 -l i0) ∧ Π(ints1 -l i0) ≤ j ^ n→i(# rmdup(ints1 -l i0)) ∧ 0 ≤ j), 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, Π(ints1 -l i0) ≤ j ^ n→i(# rmdup(ints1 -l 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(i0 ' + ints1)) < j * j ^ n→i(# rmdup(ints1 -l i0))

Rule argument

spec: int-mult inst: name: pls-pos-02 seq: ⊦ i < i0 * j ∧ j ≤ j0 ∧ 0 ≤ i0 → i < i0 * j0 subst: [j, j0, i0, i] → [Π(ints1 -l i0), j ^ n→i(# rmdup(ints1 -l i0)), j, j ^ n→i(# rmdup(i0 ' + ints1))] flags: rewrite

Additional information

Used terms: Π(ints1 -l i0), j ^ n→i(# rmdup(ints1 -l i0)), j, j ^ n→i(# rmdup(i0 ' + ints1))