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Proof step 34 from proof fermat-pos (previous step)

0 < i2, 0 ≤ i2, 0 < j1, j1 ≤ i2, 0 ≤ i1, ≡mod(i1 ^ (i2 + 1), i1, i2 + 1), prime(i2 + 1), 0 ≤ j1, ∀ i0, j. i→n(i0) < i→n(i1 + 1) → prime(j) ∧ 0 < i0 → ≡mod(i0 ^ j, i0, j) ⊦ gcd(i2 + 1, j1) = 1, i2 + 1 | binom_coef(i2 + 1, j1), i2 + 1 | i1, i2 + 1 | Σ countFun(1, i2, λ j0. i1 ^ j0 * binom_coef(i2 + 1, j0)), ≡mod(Σ countFun(1, i2, λ j0. i1 ^ j0 * binom_coef(i2 + 1, j0)) + i1 ^ (i2 + 1), i1, i2 + 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 35:
⊦ gcd(i, j) = gcd(j, i)
Proof step 36:
0 < i2, 0 ≤ i2, 0 < j1, j1 ≤ i2, 0 ≤ i1, ≡mod(i1 ^ (i2 + 1), i1, i2 + 1), prime(i2 + 1), 0 ≤ j1, ∀ i0, j. i→n(i0) < i→n(i1 + 1) → prime(j) ∧ 0 < i0 → ≡mod(i0 ^ j, i0, j) ⊦ gcd(j1, i2 + 1) = 1, i2 + 1 | binom_coef(i2 + 1, j1), i2 + 1 | i1, i2 + 1 | Σ countFun(1, i2, λ j0. i1 ^ j0 * binom_coef(i2 + 1, j0)), ≡mod(Σ countFun(1, i2, λ j0. i1 ^ j0 * binom_coef(i2 + 1, j0)) + i1 ^ (i2 + 1), i1, i2 + 1)

Rule argument

spec: int-gcd inst: name: gcd-com seq: ⊦ gcd(i, j) = gcd(j, i) subst: [j, i] → [j1, i2 + 1] flags: rewrite

Additional information

Used terms: j1, i2 + 1