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Proof step 82 from proof uint2bits-add-high (previous step)

uint2bits(2 ^ (i - 1) - 2 ^ i0) = false ', j0 = ~(2 ^ i0), j0 * 2 ≤ 2 ^ i, 0 ≤ j0 * 2 + 2 ^ i, 0 ≤ 2 ^ i, (2 ^ (i - 1) + j0 * 2) + 1 < 2 ^ i, j0 * 2 < 2 ^ i, j0 * 2 + 1 < 2 ^ i, j0 * 2 < i, 2 ^ (i - 1) + j0 * 2 < 0, 0 ≤ i, (2 ^ (i - 1) + j0 * 2) + 1 < i, 2 ^ (i - 1) + j0 * 2 < i, j0 * 2 + 1 < i, 2 ^ (i - 1) + j0 * 2 < 2 ^ i, 0 < j0 * 2 + 2 ^ i, i ≤ i0, 1 < 2 ^ i, 0 < 2 ^ i, 1 < i, 0 < i, 0 < i0, j0 * 2 < i0, 2 ^ (i - 1) + j0 * 2 < i0, j0 * 2 + 1 < i0, (2 ^ (i - 1) + j0 * 2) + 1 < i0, abs(j0 * 2 - 1) / 2 < 2 ^ i, 2 ^ (i - 2) < abs(j0 * 2 - 1) / 2, abs(j0 * 2 - 1) / 2 ≤ 2 ^ (i - 1), j0 < 0, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ i = 1, i0 = 0, j0 * 2 + 2 ^ i = 1, j0 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 83:
uint2bits(2 ^ (i - 1) - 2 ^ i0) = false ', ~(2 ^ i0) * 2 ≤ 2 ^ i, 0 ≤ ~(2 ^ i0) * 2 + 2 ^ i, 0 ≤ 2 ^ i, (2 ^ (i - 1) + ~(2 ^ i0) * 2) + 1 < 2 ^ i, ~(2 ^ i0) * 2 < 2 ^ i, ~(2 ^ i0) * 2 + 1 < 2 ^ i, ~(2 ^ i0) * 2 < i, 2 ^ (i - 1) + ~(2 ^ i0) * 2 < 0, 0 ≤ i, (2 ^ (i - 1) + ~(2 ^ i0) * 2) + 1 < i, 2 ^ (i - 1) + ~(2 ^ i0) * 2 < i, ~(2 ^ i0) * 2 + 1 < i, 2 ^ (i - 1) + ~(2 ^ i0) * 2 < 2 ^ i, 0 < ~(2 ^ i0) * 2 + 2 ^ i, i ≤ i0, 1 < 2 ^ i, 0 < 2 ^ i, 1 < i, 0 < i, 0 < i0, ~(2 ^ i0) * 2 < i0, 2 ^ (i - 1) + ~(2 ^ i0) * 2 < i0, ~(2 ^ i0) * 2 + 1 < i0, (2 ^ (i - 1) + ~(2 ^ i0) * 2) + 1 < i0, abs(~(2 ^ i0) * 2 - 1) / 2 < 2 ^ i, 2 ^ (i - 2) < abs(~(2 ^ i0) * 2 - 1) / 2, abs(~(2 ^ i0) * 2 - 1) / 2 ≤ 2 ^ (i - 1), ~(2 ^ i0) < 0, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ ~(2 ^ i0) = 0, i = 1, i0 = 0, 2 ^ i = 1 + 2 ^ i0 * 2

Rule argument

Additional information

Used simplifier rules: ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + ~ j = i - j;% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ ~ i * j = ~(i * j);