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Proof step 8 from proof countFun-pot (previous step)

i2 ≤ i1, 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)) ⊦ Σ((i0 * binom_coef(i1, i2) * i ^ i2 * i0 ^ (i1 - i2))' + countFun(i2 + 1, i1, λ j0. i0 * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0))) = Σ countFun(i2, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
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)) = Σ countFun(i2, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0))

Rule argument

Additional information

Used simplifier rules: a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%sum-rec: ⊦ Σ(i ' + ints) = i + Σ ints; from specification intlist