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Proof step 11 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 countFun(0, j, λ j0. binom_coef(j, j0) * i1 ^ j0 * 1 ^ (j - j0)), i1 + 1, j)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
prime(j), ≡mod(i1 ^ j, i1, j), 0 ≤ i1, 0 ≤ j, ∀ 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), i1 + 1, j)

Rule argument

Additional information

Used simplifier rules: forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%one: ⊦ 1 * i = i; from specification int-mult%one-01: ⊦ 1 ^ j = 1; from specification int-pot