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Proof step 16 from proof exin2pot (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 17:
j = i0 * 2 + j1, 2 ^ i < 2 ^ (i + 1), 0 < j, i0 < 2 ^ (i + 1), j1 < 2, 0 < 2, 2 ^ i ≤ i0, 0 ≤ i, 0 ≤ j1, 0 ≤ j, ∀ j0. i→n(j0) < i→n(j) → 0 < j0 → (∃ i. 0 ≤ i ∧ 2 ^ i ≤ j0 ∧ j0 < 2 ^ (i + 1)) ⊦ j = 1, j = 0, 0 ≤ i + 1 ∧ 2 ^ (i + 1) ≤ j ∧ j < 2 ^ (2 + i)

Rule argument

spec: int-div name: div-elim-pos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j) subst: [i, j, i0, j0] → [j, 2, i0, j1]

Additional information

Used simplifier rules: ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i < j → ¬ j < i; (s)%⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j); (elim) Used terms: j, 2, i0, j1