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Proof step 28 from proof fermat-pos (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%prime-divides-pot-01: ⊦ prime(i) ∧ 0 < j → (i | i0 ^ j ↔ i | i0); from specification int-fac%prime-divides: ⊦ prime(i) → (i | j * j0 ↔ i | j ∨ i | j0); from specification int-gcd%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%succ-01: ⊦ i + 1 ≤ j ↔ i < j; from specification int-abs%sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%leone: ⊦ 1 ≤ i ↔ 0 < i; from specification int-abs