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Proof step 6 from proof product-pot-rec (previous step)

∀ i. i ∈ ints0 → Π ints0 = i ^ n→i(#oc(i, ints0)) * Π(ints0 -l i) ⊦ i0 * Π ints0 = i0 ^ (1 + n→i(#oc(i0, ints0))) * Π(ints0 -l i0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ 0 < i → j ^ i = j ^ i -1 * j
Proof step 8:
0 < n→i(#oc(i0, ints0)) + 1, ∀ i. i ∈ ints0 → Π ints0 = i ^ n→i(#oc(i, ints0)) * Π(ints0 -l i) ⊦ i0 * Π ints0 = (i0 ^ (n→i(#oc(i0, ints0)) + 1)-1 * i0) * Π(ints0 -l i0)

Rule argument

spec: int-pot inst: name: Pot-pos seq: ⊦ 0 < i → j ^ i = j ^ i -1 * j subst: [i, j] → [n→i(#oc(i0, ints0)) + 1, i0] flags: rewrite

Additional information

Used simplifier rules: numls-02: ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; from specification int-abs%mone-06: ⊦ ~1 < i ↔ 0 ≤ i; from specification int-abs%nat2int: ⊦ 0 ≤ n→i(n); 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 Used terms: n→i(#oc(i0, ints0)) + 1, i0