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Proof step 3 from proof countFun-shift (previous step)

∀ i3, i4, i, i0. i→n(i3 - 1 - i4) < i→n(i1 - 1 - i2) → true → Σ countFun(i4, i3 - 1, λ j0. i * binom_coef(i3, j0) * i ^ j0 * i0 ^ (i3 - j0)) = Σ countFun(i4 + 1, i3, λ j0. i * binom_coef(i3, j0 - 1) * i ^ (j0 - 1) * i0 ^ (i3 - (j0 - 1))) ⊦ Σ countFun(i2, i1 - 1, λ j0. i * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) = Σ countFun(i2 + 1, 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 4:
⊦ j < i → countFun(i, j, intfun) = []
Proof step 5:
¬ i1 - 1 < i2, ∀ i3, i4, i, i0. i→n(i3 - 1 - i4) < i→n(i1 - 1 - i2) → true → Σ countFun(i4, i3 - 1, λ j0. i * binom_coef(i3, j0) * i ^ j0 * i0 ^ (i3 - j0)) = Σ countFun(i4 + 1, i3, λ j0. i * binom_coef(i3, j0 - 1) * i ^ (j0 - 1) * i0 ^ (i3 - (j0 - 1))) ⊦ Σ countFun(i2, i1 - 1, λ j0. i * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) = Σ countFun(i2 + 1, i1, λ j. i * binom_coef(i1, j - 1) * i ^ (j - 1) * i0 ^ (i1 + 1 - j))
Proof step 30:
i1 - 1 < i2, ∀ i3, i4, i, i0. i→n(i3 - 1 - i4) < i→n(i1 - 1 - i2) → true → Σ countFun(i4, i3 - 1, λ j0. i * binom_coef(i3, j0) * i ^ j0 * i0 ^ (i3 - j0)) = Σ countFun(i4 + 1, i3, λ j0. i * binom_coef(i3, j0 - 1) * i ^ (j0 - 1) * i0 ^ (i3 - (j0 - 1))) ⊦ Σ [] = Σ countFun(i2 + 1, i1, λ j. i * binom_coef(i1, j - 1) * i ^ (j - 1) * i0 ^ (i1 + 1 - j))

Rule argument

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

Additional information

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