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Proof step 18 from proof exin2pot (previous step)

j1 < 2, i0 < 2 ^ (i + 1), 0 < i0 * 2 + j1, 2 ^ i < 2 ^ (i + 1), 0 < 2 ^ (i + 1), 0 ≤ j1, 0 ≤ i, 2 ^ i ≤ i0, 0 ≤ i0, ∀ j0. i→n(j0) < i→n(i0 * 2 + j1) → 0 < j0 → (∃ i. 0 ≤ i ∧ 2 ^ i ≤ j0 ∧ j0 < 2 ^ (i + 1)) ⊦ i0 * 2 + j1 = 0, i0 * 2 + j1 = 1, 0 ≤ i + 1 ∧ 2 ^ (i + 1) ≤ i0 * 2 + j1 ∧ i0 * 2 + j1 < 2 ^ (2 + i)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 19:
⊦ 0 < i → j ^ i = j ^ i -1 * j
Proof step 20:
0 < 2 ^ (i + 1)-1 * 2, 2 ^ i < 2 ^ (i + 1)-1 * 2, 0 < i0 * 2 + j1, i0 < 2 ^ (i + 1)-1 * 2, j1 < 2, 0 < i + 1, 0 ≤ i0, 2 ^ i ≤ i0, 0 ≤ i, 0 ≤ j1, ∀ j0. i→n(j0) < i→n(i0 * 2 + j1) → 0 < j0 → (∃ i. 0 ≤ i ∧ 2 ^ i ≤ j0 ∧ j0 < 2 ^ (i + 1)) ⊦ i0 * 2 + j1 = 1, i0 * 2 + j1 = 0, 0 ≤ i + 1 ∧ 2 ^ (i + 1)-1 * 2 ≤ i0 * 2 + j1 ∧ i0 * 2 + j1 < 2 ^ (2 + i)

Rule argument

spec: <toplevel> name: Pot-pos seq: ⊦ 0 < i → j ^ i = j ^ i -1 * j subst: [i, j] → [i + 1, 2] flags: <all paths>

Additional information

Used simplifier rules: ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; (s)% ⊦ ~1 < i ↔ 0 ≤ i; (s)% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i ≤ j → (j < i ↔ false); (as)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: i + 1, 2