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Proof step 7 from proof fermat-pos (previous step)

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((i1 + 1) ^ j, i1 + 1, j)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ 0 ≤ j → (i + i0) ^ j = Σ binom_list(i, i0, j)
Proof step 9:
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)

Rule argument

spec: int-fac inst: name: Binom-theorem seq: ⊦ 0 ≤ j → (i + i0) ^ j = Σ binom_list(i, i0, j) subst: [j, i0, i] → [j, 1, i1] flags: rewrite

Additional information

Used simplifier rules: lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%prime-not-neg: ⊦ j < 0 → ¬ prime(j); from specification int-gcd Used terms: j, 1, i1