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Proof step 10 from proof countFun-add (previous step)

j1 ≤ i1, j1 ≤ i1, j1 ≤ i1, 0 < j1, 0 < i1, ∀ i2, j, i, i0. i→n(i2 - j) < i→n(i1 - j1) → true → 0 < j ∧ j ≤ i2 → countFun(j, i2, λ j0. binom_coef(i2 + 1, j0) * i ^ j0 * i0 ^ (i2 + 1 - j0)) = countFun(j, i2, λ j0. binom_coef(i2, j0 - 1) * i ^ j0 * i0 ^ (i2 + 1 - j0) + binom_coef(i2, j0) * i ^ j0 * i0 ^ (i2 + 1 - j0)) ⊦ (binom_coef(i1 + 1, j1) * i ^ j1 * i0 ^ (i1 + 1 - j1))' + countFun(j1 + 1, i1, λ j0. binom_coef(i1 + 1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0)) = (binom_coef(i1, j1 - 1) * i ^ j1 * i0 ^ (i1 + 1 - j1) + binom_coef(i1, j1) * i ^ j1 * i0 ^ (i1 + 1 - j1))' + countFun(j1 + 1, i1, λ j0. binom_coef(i1, j0 - 1) * i ^ j0 * i0 ^ (i1 + 1 - j0) + binom_coef(i1, j0) * i ^ j0 * i0 ^ (i1 + 1 - j0))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-fac inst: name: coef-rec-add-02 seq: ⊦ 0 < j ∧ j ≤ i + 1 → binom_coef(i + 1, j) = binom_coef(i, j - 1) + binom_coef(i, j) subst: [j, i] → [j1, i1] flags: rewrite, current

Additional information

Used simplifier rules: le-add-03: ⊦ i ≤ i0 ∧ j0 ≥ 0 → i ≤ i0 + j0; 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 Used terms: j1, i1