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Proof step 129 from proof uint2bits-add-high (previous step)

0 = i1 * 2 + ~(i1 * 2), i1 + ~(i1 * 2) < i, ~(i1 * 2) < ~1, (i1 + ~(i1 * 2)) + 1 < i, i1 + 1 < 2 ^ i, (i1 + ~(i1 * 2)) + 1 < 2 ^ i, i1 < 2 ^ i, i1 + ~(i1 * 2) < 2 ^ i, 2 ^ (i - 1) + ~(i1 * 2) < i1, 2 ^ (i - 1) < i1 * 2, i1 * 2 < 2 ^ i, 0 < i1, ~(i1 * 2) ≤ 1, ~(i1 * 2) ≤ 2 ^ i, 0 ≤ ~(i1 * 2) + 2 ^ i, 0 ≤ 2 ^ i, ~(i1 * 2) < 2 ^ i, ~(i1 * 2) < i, 2 ^ (i - 1) + ~(i1 * 2) < 0, 0 < i, (2 ^ (i - 1) + ~(i1 * 2)) + 1 < i, ~(i1 * 2) + 1 < i, 0 < ~(i1 * 2) + 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 2 ^ (i - 1) ≤ 2 ^ i, 2 ^ (i - 1) + ~(i1 * 2) ≤ 2 ^ i, 2 ^ (i - 1) + ~(i1 * 2) ≤ 0, i1 + ~(i1 * 2) < 0, ~(i1 * 2) + 1 < i1, ~(i1 * 2) < i1, (2 ^ (i - 1) + ~(i1 * 2)) + 1 < i1, ~(i1 * 2) < 0, 1 < 2 ^ i, 0 < 2 ^ i, 2 + ~(i1 * 2) < 2 ^ i, 2 + 2 ^ (i - 1) + ~(i1 * 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) ⊦ ~(i1 * 2) = 0, i = 1, uint2bits(2 ^ (i - 1) + (~(i1 * 2) - 1) / 2) + (~(i1 * 2) % 2 ≠ 0)' = sgn(int2bits(~(i1 * 2)).rest), ~(i1 * 2) + 2 ^ i = 1, ¬ 2 ^ (i - 1) < i1 ∧ ¬ i1 ≤ 2 ^ (i - 2)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 130:
0 ≤ i1 * 2 + 2 ^ i, 0 < i1 * 2 + 2 ^ i, 2 ^ (i - 1) ≤ 2 ^ i, 0 ≤ i1 * 2 + i, 0 < i1 * 2 + i, 0 ≤ i1 + 2 ^ i, 0 ≤ i1 + i, 2 ^ (i - 1) + 2 < i1 * 2 + 2 ^ i, 2 < i1 * 2 + 2 ^ i, 0 < 2 ^ i, 1 < 2 ^ i, 2 ^ (i - 1) + 1 < 3 * i1, 2 ^ (i - 1) ≤ i1 * 2, 2 ^ (i - 1) ≤ i1 * 2 + 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 1 < i1 * 2 + i, 2 ^ (i - 1) + 1 < i1 * 2 + i, 0 < i, 0 < i + i1 * 2, 0 < 2 ^ i + i1 * 2, 0 ≤ 2 ^ i, i1 * 2 ≤ 2 ^ i, i1 * 2 < 2 ^ i, 2 ^ (i - 1) < i1 * 2, 2 ^ (i - 1) < 3 * i1, i1 < 2 ^ i, 1 < i1 + 2 ^ i, i1 + 1 < 2 ^ i, 1 < i1 + i, 0 < i1, ∀ 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 + i1 * 2, uint2bits(2 ^ (i - 1) - (i1 * 2 + 1) / 2) + false ' = sgn(int2bits(~(i1 * 2)).rest), i = 1, i1 = 0, ¬ 2 ^ (i - 1) < i1 ∧ ¬ i1 ≤ 2 ^ (i - 2)

Rule argument

Additional information

Used simplifier rules: ⊦ i + j < i0 ∧ j0 < j → i + j0 < i0;% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)% ⊦ ~ i < 0 ↔ 0 < i;% ⊦ 0 < i → (0 < i * j ↔ 0 < j);% ⊦ 0 < i → (i < i * j ↔ 1 < j);% ⊦ i - j ≤ i0 ↔ i ≤ j + i0;% ⊦ i + 0 = i;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ 0 ≤ j1 ∧ j1 < j → (0 ≤ i0 * j + j1 ↔ 0 ≤ i0);% ⊦ ~ i ≤ j ↔ 0 ≤ i + j;% ⊦ (* i0) * i + i = (i0 + 1) * i;% ⊦ i + j < (* i1) * i + j0 ↔ j < (i1 - 1) * i + j0;% ⊦ i + (j - i0) = i + j - i0;% ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ i < j - i0 ↔ i + i0 < j;% ⊦ i + ~1 = i - 1;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ 0 < i ∧ 0 < j → 0 < i + j;% ⊦ 1 * i = i;% ⊦ j < (* i1) * j + i ↔ 0 < (i1 - 1) * j + i;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ (i * 2) % 2 = 0;%j ≠ 0 ⊦ ~ i % j = ~(i % j);% ⊦ i + ~ j = i - j;%j ≠ 0 ⊦ ~ i / j = ~(i / j);% ⊦ ~ i - j = ~(i + j);% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 2 * i = 0 ↔ i = 0;% ⊦ ~ i = 0 ↔ i = 0;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ ~ i + i = 0;% ⊦ i * j = j * i;