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Proof step 25 from proof countFun-shift (previous step)

i1 < 2 + i2, i2 < i1, 2 * i1 ≤ 2 * (i2 + 1) + 1, ∀ 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 + 1, i1 - 1, λ j0. i * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) = Σ countFun(2 + i2, i1, λ j. i0 ^ (i1 + 1 - j) * i ^ (j - 1) * binom_coef(i1, j - 1) * i)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 26:
⊦ 2 * i ≤ 2 * j + 1 ↔ i ≤ j
Proof step 27:
i1 < 2 + i2, i2 < i1, i1 ≤ i2 + 1, ∀ 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 + 1, i1 - 1, λ j0. i * binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 - j0)) = Σ countFun(2 + i2, i1, λ j. i0 ^ (i1 + 1 - j) * i ^ (j - 1) * binom_coef(i1, j - 1) * i)

Rule argument

spec: int-mult inst: name: le-two seq: ⊦ 2 * i ≤ 2 * j + 1 ↔ i ≤ j subst: [j, i] → [i2 + 1, i1] flags: rewrite

Additional information

Used terms: i2 + 1, i1