[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 27 from proof countFun-shift (previous step)

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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 28:
∀ i3, i4, i, i0. i→n(i3 - 1 - i4) < i→n(i2 + 1 - 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(2 + i2, i2 + 1, λ j. i0 ^ (2 + i2 - j) * i ^ (j - 1) * binom_coef(i2 + 1, j - 1) * i)

Rule argument

Additional information

Used simplifier rules: ler: ⊦ i ≤ i; from specification int-abs%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%sum-empty: ⊦ Σ [] = 0; from specification intlist%countFun-base-01: ⊦ countFun(i + 1, i, intfun) = []; from specification intlist%ax-04: ⊦ i + 0 = i; from specification int-basic2%numsub: ⊦ (* i) + i0 - (* j) = (i - j) + i0; from specification int-abs%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%les: ⊦ i < j → (j ≤ i + 1 ↔ j = i + 1); from specification int-abs