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Proof step 51 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)) ⊦ i * Σ countFun(0, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) = 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, λ j1. binom_coef(i1, j1) * i ^ j1 * i0 ^ (i1 - j1)) + i0 * Σ countFun(0, i1, λ j1. binom_coef(i1, j1) * i ^ j1 * i0 ^ (i1 - j1)) = 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)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 52:
⊦ i ≤ j → countFun(i, j, intfun) = countFun(i, j - 1, intfun) + intfun(j) '
Proof step 53:
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)

Rule argument

spec: intlist inst: name: countFun-last seq: ⊦ i ≤ j → countFun(i, j, intfun) = countFun(i, j - 1, intfun) + intfun(j) ' subst: [intfun, j, i] → [λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0), i1, 0] flags: rewrite

Additional information

Used simplifier rules: associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0), i1, 0