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Proof step 6 from proof countFun-pot (previous step)

i2 ≤ i1, ∀ i3, i4, i0, i. i→n(i3 - i4) < i→n(i1 - i2) → true → Σ countFun(i4, i3, λ j0. i0 * binom_coef(i3, j0) * i ^ j0 * i0 ^ (i3 - j0)) = Σ countFun(i4, i3, λ j0. binom_coef(i3, j0) * i ^ j0 * i0 ^ (i3 + 1 - j0)) ⊦ Σ countFun(i2, i1, λ j0. i0 * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) = Σ countFun(i2, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ i ≤ j → countFun(i, j, intfun) = intfun(i) ' + countFun(i + 1, j, intfun)
Proof step 8:
i2 ≤ i1, i2 ≤ i1, ∀ i3, i4, i0, i. i→n(i3 - i4) < i→n(i1 - i2) → true → Σ countFun(i4, i3, λ j0. i0 * binom_coef(i3, j0) * i ^ j0 * i0 ^ (i3 - j0)) = Σ countFun(i4, i3, λ j0. binom_coef(i3, j0) * i ^ j0 * i0 ^ (i3 + 1 - j0)) ⊦ Σ((i0 * binom_coef(i1, i2) * i ^ i2 * i0 ^ (i1 - i2))' + countFun(i2 + 1, i1, λ j0. i0 * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0))) = Σ countFun(i2, i1, λ j0. binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0))

Rule argument

spec: intlist inst: name: countFun-rec seq: ⊦ i ≤ j → countFun(i, j, intfun) = intfun(i) ' + countFun(i + 1, j, intfun) subst: [intfun, j, i] → [λ j0. i0 * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0), i1, i2] flags: rewrite

Additional information

Used terms: λ j0. i0 * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0), i1, i2