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Proof step 12 from proof countFun-pot (previous step)

i2 ≤ i1, ∀ i3, i4, i0, i. i→n(i3 - i4) < i→n(i1 - i2) → true → Σ countFun(i4, i3, λ j0. i0 * binom_coef(i3, j0) * i ^ j0 * i0 ^ (i3 - j0)) = Σ countFun(i4, i3, λ j0. binom_coef(i3, j0) * i ^ j0 * i0 ^ (i3 + 1 - j0)) ⊦ i0 * binom_coef(i1, i2) * i ^ i2 * i0 ^ (i1 - i2) + Σ countFun(i2 + 1, i1, λ j0. i0 * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) = binom_coef(i1, i2) * i ^ i2 * i0 ^ (i1 + 1 - i2) + Σ countFun(i2 + 1, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-pot inst: name: Pot-pos seq: ⊦ 0 < i → j ^ i = j ^ i -1 * j subst: [i, j] → [i1 + 1 - i2, i0] flags: rewrite

Additional information

Used simplifier rules: sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; from specification int-abs%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%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs Used terms: i1 + 1 - i2, i0