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Proof step 38 from proof Binom-theorem-countFun (previous step)

0 ≤ i1, 0 < i1, ∀ j0, i, i0. i→n(j0) < i→n(i1 + 1) → true → 0 ≤ j0 → (i + i0) ^ j0 = Σ countFun(0, j0, λ j1. binom_coef(j0, j1) * i ^ j1 * i0 ^ (j0 - j1)) ⊦ ~1 = i1, i ^ i1 * i + i0 * (i0 ^ i1 + Σ countFun(1, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0))) = i0 ^ (1 + i1) + Σ countFun(1, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0)) + i ^ (i1 + 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 39:
⊦ i * (j + j0) = i * j + i * j0
Proof step 40:
0 ≤ i1, 0 < i1, ∀ j0, i, i0. i→n(j0) < i→n(i1 + 1) → true → 0 ≤ j0 → (i + i0) ^ j0 = Σ countFun(0, j0, λ j1. binom_coef(j0, j1) * i ^ j1 * i0 ^ (j0 - j1)) ⊦ ~1 = i1, i ^ i1 * i + i0 * i0 ^ i1 + i0 * Σ countFun(1, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) = i0 ^ (1 + i1) + Σ countFun(1, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0)) + i ^ (i1 + 1)

Rule argument

spec: int-mult inst: name: dist-rsplit seq: ⊦ i * (j + j0) = i * j + i * j0 subst: [j0, j, i] → [Σ countFun (1, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)), i0 ^ i1, i0] flags: rewrite

Additional information

Used terms: Σ countFun(1, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)), i0 ^ i1, i0