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Proof step 11 from proof countFun-shift (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: succ-01: ⊦ i + 1 ≤ j ↔ i < j; from specification int-abs%add-in: ⊦ i + j = i + i0 ↔ j = i0; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult%a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%sum-rec: ⊦ Σ(i ' + ints) = i + Σ ints; from specification intlist%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%ax-04: ⊦ i + 0 = i; from specification int-basic2%numsub: ⊦ (* i) + i0 - (* j) = (i - j) + i0; from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%sub-add-06: ⊦ i + j - i0 + j = i - i0; from specification int-abs