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Proof step 29 from proof product-pot-no-dups (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: ls-pos-01 seq: ⊦ 0 < j → (i * j < i0 * j ↔ i < i0) subst: [i0, j, i] → [i0 ^ (1 + n→i(#oc(i0, ints1))), Π(ints1 -l i0), j] flags: rewrite

Additional information

Used simplifier rules: lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs%lst-01: ⊦ ¬ i < j → (i ≠ j ↔ j < i); from specification int-basic1%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%product-min-nlz: ⊦ 0 < min(i ' + ints) → ¬ Π(ints -l i0) < 0; from specification intlist Used terms: i0 ^ (1 + n→i(#oc(i0, ints1))), Π(ints1 -l i0), j