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Proof step 16 from proof uint2bits-restsgn (previous step)

0 ≤ i ∧ 0 ≤ 2, 2 ^ i < 2 ^ i * 2 ^ 2, j < 2 ^ i * 2 ^ 2, 0 < 2 ^ i, 1 < 2 ^ i, 0 < j, 0 ≤ j, 1 < j, 0 ≤ 2 ^ i, 2 < j, 0 < i, 2 ^ i < j, j < 2 * 2 ^ i, 1 < 2 ^ i * 2 ^ 2, 2 < 2 ^ i * 2 ^ 2, 3 < 2 ^ i * 2 ^ 2, 2 ^ i + 1 < 2 ^ i * 2 ^ 2, 0 < 2 ^ i * 2 ^ 2, 0 ≤ 2 ^ i * 2 ^ 2, ∀ j0, i. i→n(j0) < i→n(j) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ 2 ^ i * 2 ^ 2 = 1 + j, sgn(uint2bits((2 ^ i * 2 ^ 2 - j) / 2).rest + ((2 ^ i * 2 ^ 2 - j) % 2 = 1)') = uint2bits((2 ^ i * 2 ^ 2 - j) / 2).rest + ((2 ^ i * 2 ^ 2 - j) % 2 = 1)'

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 17:
j < 2 * 2 ^ i, 2 ^ i < j, 0 < i, 2 < j, 0 ≤ 2 ^ i, 1 < j, 0 ≤ j, 0 < j, 1 < 2 ^ i, j < 4 * 2 ^ i, 0 ≤ i, 0 < 2 ^ i, ∀ j0, i. i→n(j0) < i→n(j) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ sgn(uint2bits((4 * 2 ^ i - j) / 2).rest + ((4 * 2 ^ i - j) % 2 = 1)') = uint2bits((4 * 2 ^ i - j) / 2).rest + ((4 * 2 ^ i - j) % 2 = 1)', 4 * 2 ^ i = 1 + j

Rule argument

Additional information

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