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Proof step 33 from proof fermat-pos (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 34:
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)

Rule argument

Additional information

Used simplifier rules: numleledel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j; from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%le-add-03: ⊦ i ≤ i0 ∧ j0 ≥ 0 → i ≤ i0 + j0; from specification int-abs%zero-02: ⊦ 0 < i ∧ 0 < j → 0 < i + j; from specification int-basic2%numls-02: ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; from specification int-abs%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%numle-02: ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0; from specification int-abs