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Proof step 22 from proof Binom-theorem-countFun (previous step)

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, Σ countFun(0, i1, λ j1. binom_coef(i1, j1) * i ^ j1 * i0 ^ (i1 - j1)) * (i + i0) = i0 ^ (1 + i1) + Σ countFun(1, i1, λ j0. binom_coef(i1 + 1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0)) + i ^ (i1 + 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-fac inst: name: countFun-add seq: ⊦ 0 < j1 ∧ j1 ≤ i1 → countFun (j1, i1, λ j0. binom_coef(i1 + 1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0)) = countFun (j1, i1, λ j0. binom_coef(i1, j0 - 1) * i ^ j0 * i0 ^ (i1 + 1 - j0) + binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0)) subst: [i0, i, i1, j1] → [i0, i, i1, 1] flags: rewrite, current

Additional information

Used terms: i0, i, i1, 1