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Proof step 53 from proof Binom-theorem-countFun (previous step)

0 ≤ i1, 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)) ⊦ i * Σ(countFun(0, i1 - 1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) + (binom_coef(i1, i1) * i ^ i1 * i0 ^ (i1 - i1))') = i ^ i1 * i + Σ countFun(1, i1, λ j0. binom_coef(i1, j0 - 1) * i ^ j0 * i0 ^ (i1 + 1 - j0)), ~1 = i1, i * Σ(countFun(0, i1 - 1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) + (binom_coef(i1, i1) * i ^ i1 * i0 ^ (i1 - i1))') + i0 * Σ(countFun(0, i1 - 1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) + (binom_coef(i1, i1) * i ^ i1 * i0 ^ (i1 - i1))') = 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

Rule argument

Additional information

Used simplifier rules: coef-same: ⊦ 0 ≤ j → binom_coef(j, j) = 1;%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%eq-dist: ⊦ (i + j) * i0 = j0 + j * i0 ↔ i * i0 = j0; from specification int-mult%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%sum-one: ⊦ Σ i ' = i; from specification intlist%sum-app: ⊦ Σ(ints + ints0) = Σ ints + Σ ints0; from specification intlist%one: ⊦ 1 * i = i; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%Pot-zero: ⊦ j ^ 0 = 1; from specification int-pot%sub-sub-03: ⊦ i - i = 0; from specification int-abs