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Proof step 45 from proof Binom-theorem-countFun (previous step)

0 ≤ i1, 0 < i1, ∀ j0, i, i0. i→n(j0) < i→n(i1 + 1) → true → 0 ≤ j0 → (i + i0) ^ j0 = Σ countFun(0, j0, λ j1. binom_coef(j0, j1) * i ^ j1 * i0 ^ (j0 - j1)) ⊦ ~1 = i1, i ^ i1 * i + i0 * i0 ^ i1 = i0 ^ (1 + i1) + i ^ (i1 + 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 46:
⊦ 0 < i → j ^ i = j ^ i -1 * j
Proof step 47:
0 < 1 + i1, 0 ≤ i1, 0 < i1, ∀ j0, i, i0. i→n(j0) < i→n(i1 + 1) → true → 0 ≤ j0 → (i + i0) ^ j0 = Σ countFun(0, j0, λ j1. binom_coef(j0, j1) * i ^ j1 * i0 ^ (j0 - j1)) ⊦ ~1 = i1, i ^ i1 * i + i0 * i0 ^ i1 = i0 ^ (1 + i1)-1 * i0 + i ^ (i1 + 1)

Rule argument

spec: int-pot inst: name: Pot-pos seq: ⊦ 0 < i → j ^ i = j ^ i -1 * j subst: [i, j] → [1 + i1, i0] flags: rewrite

Additional information

Used simplifier rules: zero-02: ⊦ 0 < i ∧ 0 < j → 0 < i + j; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: 1 + i1, i0