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Proof step 31 from proof countFun-shift (previous step)

i1 ≤ 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))) ⊦ 0 = 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

Rule argument

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

Additional information

Used simplifier rules: sum-empty: ⊦ Σ [] = 0; from specification intlist%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: λ j. i0 ^ (i1 + 1 - j) * i ^ (j - 1) * binom_coef(i1, j - 1) * i, i1, i2 + 1