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Proof step 15 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)) ⊦ Σ countFun(i2 + 1, i1, λ j0. i0 * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) = Σ countFun(i2 + 1, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0))

User applied rule apply induction

And obtained the following new goals

Proof step 16:
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)) ⊦ i→n(i1 - i2 + 1) < i→n(i1 - i2), Σ countFun(i2 + 1, i1, λ j0. i0 * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) = Σ countFun(i2 + 1, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0))

Rule argument

subst: [i3, i4, i0, i] → [i1, i2 + 1, i0, i]

Additional information

Used simplifier rules: associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: i1, i2 + 1, i0, i