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Proof step 9 from proof fermat-pos (previous step)

0 ≤ j, prime(j), ≡mod(i1 ^ j, i1, j), 0 ≤ i1, ∀ i0, j. i→n(i0) < i→n(i1 + 1) → prime(j) ∧ 0 < i0 → ≡mod(i0 ^ j, i0, j) ⊦ ≡mod(Σ binom_list(i1, 1, j), i1 + 1, j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ binom_list(i, i0, j) = countFun(0, j, λ j0. binom_coef(j, j0) * i ^ j0 * i0 ^ (j - j0))
Proof step 11:
0 ≤ j, prime(j), ≡mod(i1 ^ j, i1, j), 0 ≤ i1, ∀ i0, j. i→n(i0) < i→n(i1 + 1) → prime(j) ∧ 0 < i0 → ≡mod(i0 ^ j, i0, j) ⊦ ≡mod(Σ countFun(0, j, λ j0. binom_coef(j, j0) * i1 ^ j0 * 1 ^ (j - j0)), i1 + 1, j)

Rule argument

spec: int-fac inst: name: binomlist-def2 seq: ⊦ binom_list(i, i0, j) = countFun (0, j, λ j0. binom_coef(j, j0) * i ^ j0 * i0 ^ (j - j0)) subst: [j, i0, i] → [j, 1, i1] flags: rewrite

Additional information

Used terms: j, 1, i1