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Proof step 107 from proof uint2bits-add-high (previous step)

2 ^ (i - 1) = 2 ^ i0, j % 2 = 0, 2 ^ (i - 1) + (j - 1) / 2 = 0, 2 ^ (i - 1) + j < i0, 2 ^ (i - 1) + j < i, 2 ^ (i - 2) < 2 ^ (i - 1), j + 1 < i0, 0 < i0, j < i0, 2 ^ (i - 2) < 2 ^ i, 0 < 2 ^ i, 1 < 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 0 < j + 2 ^ i, j + 1 < i, 0 < i, j < i, j + 1 < 2 ^ i, j < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ j + 2 ^ i, j ≤ 2 ^ i, j < 0, abs(j - 1) / 2 ≤ 2 ^ (i - 1), 2 ^ (i - 2) < abs(j - 1) / 2, abs(j - 1) / 2 < 2 ^ i, 2 ^ (i - 1) + j < 2 ^ i, 2 ^ (i - 1) < 2 ^ i, 2 ^ (i - 1) + j < 0, j < (j - 1) / 2, (2 ^ (i - 1) + j) + 1 < i0, (2 ^ (i - 1) + j) + 1 < 2 ^ i, (2 ^ (i - 1) + j) + 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) ⊦ abs(j - 1) / 2 = 2 ^ (i - 1), i = 1, j = 0, j = ~1, j + 2 ^ i = 1, i0 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 108:
2 ^ i0 + (j - 1) / 2 = 0, j % 2 = 0, 2 ^ (i - 1) = 2 ^ i0, j < 0, j ≤ 2 ^ i, 0 ≤ j + 2 ^ i, 0 ≤ 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, j < i, 0 < i, j + 1 < i, 0 < j + 2 ^ i, 2 ^ i0 + 1 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, j < i0, 0 < i0, j + 1 < i0, 2 ^ i0 + j < i, 2 ^ i0 + j < i0, (2 ^ i0 + j) + 1 < i, (2 ^ i0 + j) + 1 < 2 ^ i, (2 ^ i0 + j) + 1 < i0, j < (j - 1) / 2, 2 ^ i0 + j < 0, 2 ^ i0 + j < 2 ^ i, abs(j - 1) / 2 < 2 ^ i, 2 ^ (i - 2) < abs(j - 1) / 2, abs(j - 1) / 2 < 2 ^ i0, 2 ^ (i - 2) + 1 < 2 ^ i0, ∀ 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) ⊦ i0 = 0, j + 2 ^ i = 1, j = ~1, j = 0, i = 1, abs(j - 1) / 2 = 2 ^ i0

Rule argument

Additional information

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