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Proof step 147 from proof uint2bits-add-high (previous step)

i = i2 + 2, 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 ^ i2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 148:
0 ≤ 2 ^ (i2 + 2), 0 < 2 ^ (i2 + 2), 2 + 2 ^ (1 + i2) < 2 ^ (i2 + 2), i1 * 2 ≤ 2 ^ (i2 + 2) + 1, 2 + 2 ^ (1 + i2) < i1 * 2 + 2 ^ (i2 + 2), 2 < i1 * 2 + 2 ^ (i2 + 2), 2 + 2 ^ (1 + i2) < 2 * (i1 + 1) + i2, 2 < 2 * (i1 + 1) + i2, 2 ^ (1 + i2) < i1 * 2 + 2 ^ (i2 + 2), 2 ^ (1 + i2) < i1 * 2, 1 < 2 ^ (i2 + 2), 0 ≤ i1, ~2 < i2, i1 * 2 < 2 ^ (i2 + 2), 0 < i1, 1 + 2 ^ (1 + i2) < i1 * 2, 0 < i1 * 2 + 2 ^ (i2 + 2), 0 ≤ 2 * (i1 + 1) + i2, 0 < 2 * (i1 + 1) + i2, 2 ^ (1 + i2) ≤ 2 ^ (i2 + 2), ∀ i0, j. i→n(i0) < i→n(i2 + 2) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ ~1 = i2, uint2bits(2 ^ (1 + i2) - i1) + ((1 - i1 * 2) % 2 ≠ 0)' = sgn(int2bits(1 - i1 * 2).rest), 2 ^ (i2 + 2) = i1 * 2, ¬ i1 ≤ 2 ^ i2 ∧ ¬ 2 ^ (1 + i2) < i1

Rule argument

Additional information

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