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Proof step 29 from proof Binom-theorem-countFun (previous step)

0 ≤ 1 ∧ 1 ≤ i1, 0 < 1 ∧ 1 ≤ 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, 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, j0 - 1) * i ^ j0 * i0 ^ (i1 + 1 - j0)) + Σ countFun(1, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0))) + i ^ (i1 + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 30:
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, 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, j0 - 1) * i ^ j0 * i0 ^ (i1 + 1 - j0)) + Σ countFun(1, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0))) + i ^ (i1 + 1)

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs