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Proof step 60 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)) ⊦ Σ countFun(0, i1 - 1, λ j0. i * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) = Σ countFun(1, i1, λ j0. i * binom_coef(i1, j0 - 1) * i ^ (j0 - 1) * i0 ^ (i1 - (j0 - 1))), ~1 = i1, i * (Σ countFun(0, i1 - 1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) + i ^ i1) + i0 * (Σ countFun(0, i1 - 1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) + i ^ i1) = i0 ^ (1 + i1) + ( Σ countFun(1, i1, λ j0. i * binom_coef(i1, j0 - 1) * i ^ (j0 - 1) * i0 ^ (i1 - (j0 - 1))) + Σ countFun(1, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0))) + i ^ (i1 + 1)

User applied rule insert lemma

And obtained the following new goals

Rule argument

spec: int-fac inst: name: countFun-shift seq: ⊦ Σ countFun (i2, i1 - 1, λ j0. i * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) = Σ countFun (i2 + 1, i1, λ j0. i * binom_coef(i1, j0 - 1) * i ^ (j0 - 1) * i0 ^ (i1 - (j0 - 1))) subst: [i2, i1, i, i0] → [0, i1, i, i0] flags: current, []

Additional information

Used simplifier rules: add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%sub-sub-04: ⊦ i - (j - j0) = i + j0 - j; from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: 0, i1, i, i0