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Proof step 13 from proof countFun-pot-01 (previous step)

0 < i2, i2 ≤ i1, i2 ≤ i1, 0 < i2, i2 ≤ i1, 0 < i1, ∀ i3, i4, i, i0. i→n(i3 - i4) < i→n(i1 - i2) → 0 < i4 → countFun(i4, i3, λ j0. binom_coef(i3, j0 - 1) * i ^ j0 * i0 ^ (i3 + 1 - j0)) = countFun(i4, i3, λ j. i * binom_coef(i3, j - 1) * i ^ (j - 1) * i0 ^ (i3 + 1 - j)) ⊦ (binom_coef(i1, i2 - 1) * (i ^ i2 -1 * i) * i0 ^ (i1 + 1 - i2))' + countFun(i2 + 1, i1, λ j0. binom_coef(i1, j0 - 1) * i ^ j0 * i0 ^ (i1 + 1 - j0)) = (i0 ^ (i1 + 1 - i2) * i ^ (i2 - 1) * binom_coef(i1, i2 - 1) * i)' + countFun(i2 + 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: a: ⊦ (ints + ints0) + ints1 = ints + ints0 + ints1; from specification list[intlista]%app-01: ⊦ ints + ints0 = ints + ints1 ↔ ints0 = ints1; from specification list[intlista]%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%sub-pred: ⊦ i -1 = i - 1; from specification int-abs