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Proof step 126 from proof uint2bits-add-high (previous step)

j0 = 1, 1 = (i1 * 2 + j0) + j, 2 ^ (i - 1) + j ≤ 2 ^ i, 2 ^ (i - 1) + j ≤ 0, j < j0, 2 ^ (i - 1) + j < j0, 1 < 2 ^ i, j ≤ 1, j0 < 2, 0 ≤ j0, j < 0, j ≤ 2 ^ i, 0 ≤ j + 2 ^ i, 0 ≤ 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, j < i, 2 ^ (i - 1) + j < 0, 0 < i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) + j < i, j + 1 < i, 2 ^ (i - 1) + j < 2 ^ i, 0 < j + 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 0 < 2 ^ i, 2 ^ (i - 1) + j ≤ j0, 2 ^ (i - 1) ≤ 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) ⊦ j = 0, i = 1, uint2bits(2 ^ (i - 1) + (j - 1) / 2) + (j % 2 ≠ 0)' = sgn(int2bits(j).rest), j + 2 ^ i = 1, ¬ 2 ^ (i - 1) < i1 ∧ ¬ i1 ≤ 2 ^ (i - 2)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 127:
0 = i1 * 2 + j, i1 + j < i, j < ~1, (i1 + j) + 1 < i, i1 + 1 < 2 ^ i, (i1 + j) + 1 < 2 ^ i, i1 < 2 ^ i, i1 + j < 2 ^ i, 2 ^ (i - 1) + j < i1, 2 ^ (i - 1) < i1 * 2, i1 * 2 < 2 ^ i, 0 < i1, j ≤ 1, j ≤ 2 ^ i, 0 ≤ j + 2 ^ i, 0 ≤ 2 ^ i, j < 2 ^ i, j < i, 2 ^ (i - 1) + j < 0, 0 < i, (2 ^ (i - 1) + j) + 1 < i, j + 1 < i, 0 < j + 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 2 ^ (i - 1) ≤ 2 ^ i, 2 ^ (i - 1) + j ≤ 2 ^ i, 2 ^ (i - 1) + j ≤ 0, i1 + j < 0, j + 1 < i1, j < i1, (2 ^ (i - 1) + j) + 1 < i1, j < 0, 1 < 2 ^ i, 0 < 2 ^ i, 2 + j < 2 ^ i, 2 + 2 ^ (i - 1) + j < 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) ⊦ j = 0, i = 1, uint2bits(2 ^ (i - 1) + (j - 1) / 2) + (j % 2 ≠ 0)' = sgn(int2bits(j).rest), j + 2 ^ i = 1, ¬ 2 ^ (i - 1) < i1 ∧ ¬ i1 ≤ 2 ^ (i - 2)

Rule argument

Additional information

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