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Proof step 1 from proof countFun-shift (conclusion)

⊦ 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)))

User applied rule induction

And obtained the following new goals

Proof step 2:
true, ∀ 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 j0. i * binom_coef(i1, j0 - 1) * i ^ (j0 - 1) * i0 ^ (i1 - (j0 - 1)))

Rule argument

indvar: i→n(i1 - 1 - i2) wfpred: <

Additional information

Used simplifier rules: <; (wellfounded) Used terms: i→n(i1 - 1 - i2)