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Proof step 7 from proof countFun-pot-01 (previous step)

0 < i2, i2 ≤ i1, 0 < i1, ∀ i3, i4, i, i0. i→n(i3 - i4) < i→n(i1 - i2) → 0 < i4 → countFun(i4, i3, λ j0. binom_coef(i3, j0 - 1) * i ^ j0 * i0 ^ (i3 + 1 - j0)) = countFun(i4, i3, λ j. i * binom_coef(i3, j - 1) * i ^ (j - 1) * i0 ^ (i3 + 1 - j)) ⊦ countFun(i2, i1, λ j0. binom_coef(i1, j0 - 1) * i ^ j0 * i0 ^ (i1 + 1 - j0)) = countFun(i2, i1, λ j. i * binom_coef(i1, j - 1) * i ^ (j - 1) * i0 ^ (i1 + 1 - j))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ i ≤ j → countFun(i, j, intfun) = intfun(i) ' + countFun(i + 1, j, intfun)
Proof step 9:
i2 ≤ i1, 0 < i2, i2 ≤ i1, 0 < i1, ∀ i3, i4, i, i0. i→n(i3 - i4) < i→n(i1 - i2) → 0 < i4 → countFun(i4, i3, λ j0. binom_coef(i3, j0 - 1) * i ^ j0 * i0 ^ (i3 + 1 - j0)) = countFun(i4, i3, λ j. i * binom_coef(i3, j - 1) * i ^ (j - 1) * i0 ^ (i3 + 1 - j)) ⊦ (binom_coef(i1, i2 - 1) * i ^ i2 * i0 ^ (i1 + 1 - i2))' + countFun(i2 + 1, i1, λ j0. binom_coef(i1, j0 - 1) * i ^ j0 * i0 ^ (i1 + 1 - j0)) = countFun(i2, i1, λ j. i * binom_coef(i1, j - 1) * i ^ (j - 1) * i0 ^ (i1 + 1 - j))

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. binom_coef(i1, j0 - 1) * i ^ j0 * i0 ^ (i1 + 1 - j0), i1, i2] flags: rewrite

Additional information

Used terms: λ j0. binom_coef(i1, j0 - 1) * i ^ j0 * i0 ^ (i1 + 1 - j0), i1, i2