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Proof step 105 from proof uint2bits-add-high (previous step)

abs(~(2 ^ (i - 1)) * 2 - 1) / 2 = 2 ^ (i - 1), (2 ^ (i - 1) + ~(2 ^ (i - 1)) * 2) + 1 < i, (2 ^ (i - 1) + ~(2 ^ (i - 1)) * 2) + 1 < 2 ^ i, 2 ^ (i - 1) + ~(2 ^ (i - 1)) * 2 < i, 2 ^ (i - 1) + ~(2 ^ (i - 1)) * 2 < 2 ^ i, ~(2 ^ (i - 1)) + 1 < i, ~(2 ^ (i - 1)) < i, ~(2 ^ (i - 1)) < 2 ^ i, 0 ≤ i, 2 * (~(2 ^ (i - 1)) + 1) + 2 ^ (i - 1) < i, 2 ^ (i - 1) + ~(2 ^ (i - 1)) * 2 < 0, ~(2 ^ (i - 1)) * 2 ≤ 2 ^ i, 0 ≤ ~(2 ^ (i - 1)) * 2 + 2 ^ i, 0 ≤ 2 ^ i, ~(2 ^ (i - 1)) * 2 < 2 ^ i, 1 + ~(2 ^ (i - 1)) * 2 < 2 ^ i, ~(2 ^ (i - 1)) * 2 < i, 0 < ~(2 ^ (i - 1)) * 2 + 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 1 < i, 2 * (~(2 ^ (i - 1)) + 1) < i, 1 + 2 ^ (i - 1) + ~(2 ^ (i - 1)) * 2 < i, 1 + 2 ^ (i - 1) + ~(2 ^ (i - 1)) * 2 < 2 ^ i, 0 < i, 0 ≤ ~(2 ^ (i - 1)) + 2 ^ i, 0 < 2 ^ (i - 1), 1 < ~(2 ^ (i - 1)) + 2 ^ i, 0 < ~(2 ^ (i - 1)) + 2 ^ i, 2 ^ (i - 1) < ~(2 ^ (i - 1)) + 2 ^ i, ~(2 ^ (i - 1)) + 2 ^ (i - 1) + 1 < 2 ^ i, ~(2 ^ (i - 1)) < 0, ~(2 ^ (i - 1)) + 1 < 2 ^ (i - 1), 1 < 2 ^ i, ~(2 ^ (i - 1)) < 2 ^ (i - 1), 0 < 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, ~(2 ^ (i - 1)) * 2 + 2 ^ i = 1, ~(2 ^ (i - 1)) = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 106:
(2 ^ (i - 1) * 2 + 1) / 2 = 2 ^ (i - 1), 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ (i - 1) * 2 + 2 ^ i, 0 < 2 ^ (i - 1) * 2 + 2 ^ i, 0 ≤ 2 ^ (i - 1) + i, 1 < 2 ^ (i - 1) + 2 ^ (i - 1), 2 ^ (i - 1) + 2 ^ (i - 1) < 2 ^ i, 0 < 2 ^ (i - 1), 2 ^ (i - 1) ≤ 2 ^ i, 0 < i, 2 * (1 - 2 ^ (i - 1)) < i, 2 ^ (i - 1) + 1 < 2 ^ i, 2 ^ (i - 1) * 2 < 2 ^ i, 0 < i + 2 ^ (i - 1) * 2, 1 < 2 ^ (i - 1) * 2 + 2 ^ i, 0 < 2 ^ i + 2 ^ (i - 1) * 2, 0 ≤ 2 ^ i, 2 ^ (i - 1) * 2 ≤ 2 ^ i, 2 ^ (i - 1) + 2 * (1 - 2 ^ (i - 1)) < i, 0 ≤ i, 1 < 2 ^ (i - 1) + i, 0 ≤ 2 ^ (i - 1) + 2 ^ (i - 1), 1 < 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) ⊦ 2 ^ (i - 1) = 0, 2 ^ i = 1 + 2 ^ (i - 1) * 2, i = 1

Rule argument

Additional information

Used simplifier rules: ⊦ 0 ≤ i → (2 < 2 ^ i ↔ 1 < i);% ⊦ i < j ∧ j = i0 → i < i0; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)% ⊦ ~ i < 0 ↔ 0 < i;% ⊦ ~ i + i = 0;% ⊦ i ≤ i;% ⊦ i < j + 1 ↔ i ≤ j;%0 ≤ j ⊦ 2 ^ j < 2 ^ j0 ↔ j < j0;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ i + j < i0 ∧ ¬ j < j0 → i + j0 < i0;% ⊦ i < j - i0 ↔ i + i0 < j;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ ~ i ≤ j ↔ 0 ≤ i + j;% ⊦ 0 < i → (i < i * j ↔ 1 < j);% ⊦ i + 0 = i;% ⊦ ~ i < j ↔ 0 < j + i;%i0 ≠ 0, i0 ≤ i ⊦ (* i) < j → (* i0) < j;% ⊦ 0 < i ∧ 0 < j → 0 < i + j;% ⊦ j < (* i1) * j + i ↔ 0 < (i1 - 1) * j + i;% ⊦ i < j + i0 ∧ j < j0 → i < j0 + i0;% ⊦ 1 * i = i;% ⊦ i + j < (* i1) * i + j0 ↔ j < (i1 - 1) * i + j0;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i + (j - i0) = i + j - i0;% ⊦ ~ i = 0 ↔ i = 0;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ i + ~ j = i - j;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;%0 ≤ j, 0 ≤ j0 ⊦ 0 < i → abs(i * (* j) + (* j0)) = i * j + j0;% ⊦ abs(~ i) = abs(i);% ⊦ ~ i - j = ~(i + j);% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ ~ i * j = ~(i * j);