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Proof step 145 from proof uint2bits-add-high (previous step)

j = 1 - i1 * 2, 1 = i1 * 2 + j, 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) + 1 < i, 2 ^ (i - 1) + j < i, j + 1 < i, 2 ^ (i - 1) + 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, 1 < 2 ^ i, j ≤ 1, 0 < i, 0 < j + 2 ^ i, 0 < 2 ^ i, j < 0, 2 ^ (i - 1) + j < 0, ∀ 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 146:
0 ≤ i1 * 2 + i, 0 < i1 * 2 + i, 0 < i1 * 2 + 2 ^ i, 2 ^ (i - 1) + 1 < i1 * 2, 0 < i1, 0 < 2 ^ i, i1 * 2 < 2 ^ i, 0 < i, 0 ≤ i1, 1 < 2 ^ i, 2 ^ (i - 1) < i1 * 2, 2 ^ (i - 1) < i1 * 2 + 2 ^ i, 2 < i1 * 2 + i, 2 ^ (i - 1) + 2 < i1 * 2 + i, 2 < i1 * 2 + 2 ^ i, 2 ^ (i - 1) + 2 < i1 * 2 + 2 ^ i, 0 ≤ 2 ^ i, i1 * 2 ≤ 2 ^ i + 1, 2 + 2 ^ (i - 1) < 2 ^ i, 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) ⊦ 2 ^ i = i1 * 2, uint2bits(2 ^ (i - 1) - i1) + ((1 - i1 * 2) % 2 ≠ 0)' = sgn(int2bits(1 - i1 * 2).rest), i = 1, ¬ 2 ^ (i - 1) < i1 ∧ ¬ i1 ≤ 2 ^ (i - 2)

Rule argument

Additional information

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