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Proof step 12 from proof fermat-pos (previous step)

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)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
⊦ i ≤ j → countFun(i, j, intfun) = intfun(i) ' + countFun(i + 1, j, intfun)
Proof step 14:
0 ≤ j, 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((i1 ^ 0 * binom_coef(j, 0))' + countFun(0 + 1, j, λ j0. i1 ^ j0 * binom_coef(j, j0))), i1 + 1, j)

Rule argument

spec: intlist inst: name: countFun-rec seq: ⊦ i ≤ j → countFun(i, j, intfun) = intfun(i) ' + countFun(i + 1, j, intfun) subst: [intfun, j, i] → [λ j0. i1 ^ j0 * binom_coef(j, j0), j, 0] flags: rewrite

Additional information

Used simplifier rules: commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: λ j0. i1 ^ j0 * binom_coef(j, j0), j, 0