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Proof step 136 from proof uint2bits-add-high (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 137:
i1 ≤ 2 ^ (i - 2), 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, 0 < 2 ^ (i - 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) ⊦ 2 ^ i = 1 + i1 * 2, uint2bits(2 ^ (i - 1) - (i1 * 2 + 1) / 2) + false ' = sgn(int2bits(~(i1 * 2)).rest), i = 1, i1 = 0

Rule argument

Additional information

Used simplifier rules: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i * j = j * i;