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Proof step 101 from proof uint2bits-add-high (previous step)

j = i1 * 2 + j0 ∧ j0 ≤ 0 ∧ abs(j0) < 2 ∧ 0 < 2, 2 ^ (i - 1) + (j - 1) / 2 = 0, j0 = 0, abs(j - 1) / 2 = 2 ^ (i - 1), 0 ≤ i, 0 < i, 2 + 2 ^ (i - 1) + j < i, 2 ^ (i - 1) + j < 0, j < 0, j ≤ 2 ^ i, 0 ≤ j + 2 ^ i, 0 ≤ 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, j < i, 0 < j + 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 1 < i, 2 + j < i, (2 ^ (i - 1) + j) + 1 < i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, j < (j - 1) / 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, j + 2 ^ i = 1, j = ~1, j = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 102:
abs(i1 * 2 - 1) / 2 = 2 ^ (i - 1), 2 ^ (i - 1) + i1 = 0, (2 ^ (i - 1) + i1 * 2) + 1 < i, (2 ^ (i - 1) + i1 * 2) + 1 < 2 ^ i, 2 ^ (i - 1) + i1 * 2 < i, 2 ^ (i - 1) + i1 * 2 < 2 ^ i, i1 + 1 < i, i1 < i, i1 < 2 ^ i, 0 ≤ i, 2 * (i1 + 1) + 2 ^ (i - 1) < i, 2 ^ (i - 1) + i1 * 2 < 0, i1 * 2 ≤ 2 ^ i, 0 ≤ i1 * 2 + 2 ^ i, 0 ≤ 2 ^ i, i1 * 2 < 2 ^ i, 1 + i1 * 2 < 2 ^ i, i1 * 2 < i, 0 < i1 * 2 + 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 1 < i, 2 * (i1 + 1) < i, 1 + 2 ^ (i - 1) + i1 * 2 < i, 1 + 2 ^ (i - 1) + i1 * 2 < 2 ^ i, 0 < i, 0 ≤ i1 + 2 ^ i, 0 < 2 ^ (i - 1), 1 < i1 + 2 ^ i, 0 < i1 + 2 ^ i, 2 ^ (i - 1) < i1 + 2 ^ i, i1 + 2 ^ (i - 1) + 1 < 2 ^ i, i1 < 0, i1 + 1 < 2 ^ (i - 1), 1 < 2 ^ i, i1 < 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, i1 * 2 + 2 ^ i = 1, i1 = 0

Rule argument

Additional information

Used simplifier rules: ⊦ 0 ≤ i → (2 < 2 ^ i ↔ 1 < i);% ⊦ i + j < i0 ∧ j0 < j → i + j0 < i0;% ⊦ (* i2) * i + j < i + j0 ↔ (i2 - 1) * i + j < j0;% ⊦ i < 0 ∧ j0 < j ∧ 0 ≤ j → i * j + j0 < 0;% ⊦ i + j < j ↔ i < 0;% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ (* i) < j ∧ 0 < i → 0 ≤ j ∧ 0 < j; (forward)% ⊦ i + j < i0 ∧ ¬ j < j0 → i + j0 < i0;% ⊦ i ≤ i;% ⊦ i < j + 1 ↔ i ≤ j;% ⊦ i - j < i0 ↔ i < j + i0;%0 ≤ j ⊦ 2 ^ j < 2 ^ j0 ↔ j < j0;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ i + j < i + i0 ↔ j < i0;% ⊦ i +1 = i + 1;% ⊦ i < i + j ↔ 0 < j;% ⊦ j < (* i1) * j + i ↔ 0 < (i1 - 1) * j + i;% ⊦ 1 * i = i;% ⊦ i + j < (* i1) * i + j0 ↔ j < (i1 - 1) * i + j0;% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i < j ∧ j = i0 → i < i0; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ i < 0 → (i * j < i ↔ 1 < j);% ⊦ j < 0 → (j * 2 - 1) / 2 = j;% ⊦ 0 < i → (i * j < 0 ↔ j < 0);% ⊦ i * j + i = i * (j + 1);% ⊦ 2 * i = 0 ↔ i = 0;% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 2 * i ≠ ~1;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + 0 = i;