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Proof step 41 from proof uint2bits-restsgn (previous step)

0 ≤ n→i(# bits0), i0 + 2 ^ n→i(# bits0) * 2 = 2 * 2 ^ i + 1, i0 ≤ 2 * 2 ^ i, i0 * 2 < 4 * 2 ^ i, 2 ^ i < i0 * 2, i0 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ i0, 0 < i0, 1 < i, 1 < i0, 0 < i, 0 ≤ i, 0 ≤ 2 ^ i, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 0) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ sgn(true ' + ((4 * 2 ^ i - i0 * 2) % 2 = 1)') = (true ' + bits0) + ((4 * 2 ^ i - i0 * 2) % 2 = 1)', 4 * 2 ^ i = 1 + i0 * 2, false ∈ bits0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 42:
i0 + 2 ^ n→i(# bits0) * 2 = 2 * 2 ^ i + 1, i0 ≤ 2 * 2 ^ i, i0 * 2 < 4 * 2 ^ i, 2 ^ i < i0 * 2, i0 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ i0, 0 < i0, 1 < i, 1 < i0, 0 < i, 0 ≤ i, 0 ≤ 2 ^ i, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 0) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ sgn(true ' + ((4 * 2 ^ i - i0 * 2) % 2 = 1)') = (true ' + bits0) + ((4 * 2 ^ i - i0 * 2) % 2 = 1)', 4 * 2 ^ i = 1 + i0 * 2, false ∈ bits0

Rule argument

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ 0 ≤ n→i(n);