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Proof step 89 from proof uint2bits-add-high (previous step)

¬(~(2 ^ i0) * 2 ≤ 0 ∧ ~(2 ^ i0) * 2 ≠ 0), uint2bits(2 ^ (i - 1) - 2 ^ i0) = false ', ~(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, 0 ≤ 2 ^ i0 * 2 + 2 ^ i, 2 ^ i0 * 2 ≤ 2 ^ i, 0 ≤ 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i0 * 2 + 2 ^ i, ∀ 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, 2 ^ i = 1 + 2 ^ i0 * 2, ~(2 ^ i0) * 2 - 1 < 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 90:
uint2bits(2 ^ (i - 1) - 2 ^ i0) = false ', 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, 0 ≤ 2 ^ i0 * 2 + 2 ^ i, 2 ^ i0 * 2 ≤ 2 ^ i, 0 ≤ 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i0 * 2 + 2 ^ i, 0 < 2 ^ i + 2 ^ i0 * 2, 1 < 2 ^ i0 * 2 + 2 ^ i, 0 < i + 2 ^ i0 * 2, ∀ 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, 2 ^ i = 1 + 2 ^ i0 * 2, ~(2 ^ i0) * 2 - 1 < 0, ~(2 ^ i0) * 2 ≤ 0 ∧ ~(2 ^ i0) * 2 ≠ 0

Rule argument

Additional information

Used simplifier rules: ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i + ~ j = i - j;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ ~ i * j = ~(i * j);