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Proof step 24 from proof product-pot-no-dups (previous step)

j ^ n→i(# rmdup(i0 ' + ints1)) < j ^ (n→i(# rmdup(ints1 -l i0)) + 1), 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)), j ≤ 1, i0 ^ (1 + n→i(#oc(i0, ints1))) ≤ 1, 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 ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs inst: name: le-rec seq: ⊦ i ≠ j → (i ≤ j ↔ i ≤ j - 1) subst: [j, i] → [1, j] flags: rewrite

Additional information

Used simplifier rules: forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%pot-lez: ⊦ 0 < i → (i ^ j ≤ 0 ↔ i ^ j = 0); from specification int-pot%pot-zero-01: ⊦ 0 < i → (i ^ j = 0 ↔ j < 0 ∧ 1 < i); from specification int-pot%nat2int-06: ⊦ ¬ n→i(n) + 1 < 0; from specification intnat%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%one-01: ⊦ 1 ^ j = 1; from specification int-pot Used terms: 1, j