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Proof step 95 from proof uint2bits-restsgn (previous step)

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

User applied rule cut formula

And obtained the following new goals

Proof step 96:
4 * 2 ^ i - i0 * 2 + 1 = 2 * (2 * 2 ^ i - i0) - 1, 0 < 2 ^ i, 1 < 2 ^ i, i0 < 2 ^ i, 2 ^ i ≤ i0 * 2, 0 < i, 0 ≤ 2 ^ i, i0 * 2 + 1 < 4 * 2 ^ i, 0 ≤ i, 0 ≤ i0, 0 < i0, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 1) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ 4 * 2 ^ i = 2 * (i0 + 1), sgn(uint2bits((4 * 2 ^ i - i0 * 2 + 1) / 2).rest + ((4 * 2 ^ i - i0 * 2 + 1) % 2 = 1)') = uint2bits((4 * 2 ^ i - i0 * 2 + 1) / 2).rest + ((4 * 2 ^ i - i0 * 2 + 1) % 2 = 1)'

Rule argument

fma: 4 * 2 ^ i - i0 * 2 + 1 = 2 * (2 * 2 ^ i - i0) - 1

Additional information

Used simplifier rules: ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ i + (j - i) = j;% ⊦ i * j + i * j0 = i * (j + j0);% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ i * j = j * i;% ⊦ 0 < i → (i * j = i * j0 ↔ j = j0); Used formulas: 4 * 2 ^ i - i0 * 2 + 1 = 2 * (2 * 2 ^ i - i0) - 1