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Proof step 151 from proof uint2bits-add-high (previous step)

0 ≤ i2 ∧ 0 ≤ 2, 1 = i + i1 * 2, 0 ≤ 2 ^ i2 * 2 ^ 2, 0 < 2 ^ i2 * 2 ^ 2, 2 + 2 ^ (1 + i2) < 2 ^ i2 * 2 ^ 2, i1 * 2 ≤ 2 ^ i2 * 2 ^ 2 + 1, 2 + 2 ^ (1 + i2) < i1 * 2 + 2 ^ i2 * 2 ^ 2, 2 < i1 * 2 + 2 ^ i2 * 2 ^ 2, 2 + 2 ^ (1 + i2) < 2 * (i1 + 1) + i2, 2 < 2 * (i1 + 1) + i2, 2 ^ (1 + i2) < i1 * 2 + 2 ^ i2 * 2 ^ 2, 2 ^ (1 + i2) < i1 * 2, 1 < 2 ^ i2 * 2 ^ 2, 0 ≤ i1, ~2 < i2, i1 * 2 < 2 ^ i2 * 2 ^ 2, 0 < i1, 1 + 2 ^ (1 + i2) < i1 * 2, 0 < i1 * 2 + 2 ^ i2 * 2 ^ 2, 0 ≤ 2 * (i1 + 1) + i2, 0 < 2 * (i1 + 1) + i2, 2 ^ (1 + i2) ≤ 2 ^ i2 * 2 ^ 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) + (i % 2 ≠ 0)' = sgn(int2bits(i).rest), 2 ^ i2 * 2 ^ 2 = i1 * 2, ¬ i1 ≤ 2 ^ i2 ∧ ¬ 2 ^ (1 + i2) < i1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 152:
1 = i + i1 * 2, i + 1 < i1, i + 1 < 4 * 2 ^ i2 + i1 * 2, i < i1, i < 4 * 2 ^ i2 + i1 * 2, i < 2 ^ i2, i < i2, i < 2 * (i1 + 1) + i2, i < 0, i < ~2, i1 + i < 0, 0 < i1, 0 ≤ i1, 0 ≤ 4 * 2 ^ i2 + i1 * 2, 0 < 4 * 2 ^ i2 + i1 * 2, 2 ^ i2 * 2 < i1, i1 * 2 < 4 * 2 ^ i2, 2 ^ i2 < i1, ~2 * 2 ^ i2 < i1 * 2, 2 < 2 * (i1 + 1) + i2, 2 * (2 ^ i2 + 1) < 2 * (i1 + 1) + i2, 2 < 4 * 2 ^ i2 + i1 * 2, 2 * (2 ^ i2 + 1) < 4 * 2 ^ i2 + i1 * 2, i1 * 2 ≤ 1 + 4 * 2 ^ i2, 2 * (2 ^ i2 + 1) < 4 * 2 ^ i2, 0 ≤ 2 ^ i2, 0 ≤ i2, 0 < 2 * (i1 + 1) + i2, 0 ≤ 2 * (i1 + 1) + i2, 1 < i1, 0 < 2 ^ i2, i < i1 + 4 * 2 ^ i2, i1 + i < 2 ^ i2, i1 + i < i2, i1 + i < 2 * (i1 + 1) + i2, i + 1 < i1 + 4 * 2 ^ i2, (i1 + i) + 1 < 2 ^ i2, i ≤ i1 + i2, ∀ 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) ⊦ 4 * 2 ^ i2 = i1 * 2, uint2bits(2 ^ i2 * 2 - i1) + (i % 2 ≠ 0)' = sgn(int2bits(i).rest), ~1 = i2

Rule argument

Additional information

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