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Proof step 13 from proof uint2bits-len-01 (previous step)

~1 ≤ i0, 0 < i1 * 2 + j1, 2 * 2 ^ (i0 + 1) ≤ i1 * 2 + j1, i1 * 2 + j1 < 3 * 2 ^ (i0 + 1), 0 < 2 ^ (i0 + 1), 0 ≤ j1, j1 < 2, 0 ≤ i1, ∀ j0, i. i→n(j0) < i→n(i1 * 2 + j1) → 0 ≤ i ∧ j0 < 3 * 2 ^ i ∧ 2 * 2 ^ i ≤ j0 ∧ 0 < 2 ^ i → # uint2bits(j0) = i→n(i + 3) ⊦ # uint2bits(i1) + 1 = i→n(4 + i0), i1 * 2 + j1 = 1, ~1 ≠ i0 ∧ ~1 ≠ i0 ∧ ¬ 3 * 2 ^ i0 ≤ i1 ∧ ¬ i1 < 2 * 2 ^ i0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j
Proof step 15:
¬ 0 ≤ i0, ~1 ≤ i0, 0 < i1 * 2 + j1, 2 * 2 ^ (i0 + 1) ≤ i1 * 2 + j1, i1 * 2 + j1 < 3 * 2 ^ (i0 + 1), 0 < 2 ^ (i0 + 1), 0 ≤ j1, j1 < 2, 0 ≤ i1, ∀ j0, i. i→n(j0) < i→n(i1 * 2 + j1) → 0 ≤ i ∧ j0 < 3 * 2 ^ i ∧ 2 * 2 ^ i ≤ j0 ∧ 0 < 2 ^ i → # uint2bits(j0) = i→n(i + 3) ⊦ # uint2bits(i1) + 1 = i→n(4 + i0), i1 * 2 + j1 = 1, ~1 ≠ i0 ∧ ~1 ≠ i0 ∧ ¬ 3 * 2 ^ i0 ≤ i1 ∧ ¬ i1 < 2 * 2 ^ i0
Proof step 17:
0 ≤ i0, ~1 ≤ i0, 0 < i1 * 2 + j1, 2 * 2 ^ i0 * 2 ≤ i1 * 2 + j1, i1 * 2 + j1 < 3 * 2 ^ i0 * 2, 0 < 2 ^ i0 * 2, 0 ≤ j1, j1 < 2, 0 ≤ i1, ∀ j0, i. i→n(j0) < i→n(i1 * 2 + j1) → 0 ≤ i ∧ j0 < 3 * 2 ^ i ∧ 2 * 2 ^ i ≤ j0 ∧ 0 < 2 ^ i → # uint2bits(j0) = i→n(i + 3) ⊦ # uint2bits(i1) + 1 = i→n(4 + i0), i1 * 2 + j1 = 1, ~1 ≠ i0 ∧ ~1 ≠ i0 ∧ ¬ 3 * 2 ^ i0 ≤ i1 ∧ ¬ i1 < 2 * 2 ^ i0

Rule argument

seq: ⊦ 0 ≤ i → j ^ (i + 1) = j ^ i * j subst: [i, j] → [i0, 2] rotate: F spec: int-pot inst: name: succ

Additional information

Used terms: i0, 2