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Proof step 19 from proof uint2bits-restsgn (previous step)

j = i0 * 2 + j1 ∧ 0 ≤ j1 ∧ j1 < 2 ∧ 0 < 2, j1 = 0, 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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
i0 * 2 < 4 * 2 ^ i, 0 ≤ 2 ^ i, 2 ^ i < i0 * 2, i0 < 2 ^ i, 1 < 2 ^ i, 0 < 2 ^ i, 0 ≤ i0, 0 < i0, 1 < i, 1 < i0, 0 < i, 0 ≤ i, ∀ j0, i. i→n(j0) < i→n(i0 * 2 + 0) → 0 < i ∧ 0 ≤ j0 ∧ 2 ^ i < j0 ∧ j0 < 2 * 2 ^ i → sgn(uint2bits(2 ^ (i + 2) - j0).rest) = uint2bits(2 ^ (i + 2) - j0).rest ⊦ 4 * 2 ^ i = 1 + i0 * 2, sgn(uint2bits((4 * 2 ^ i - i0 * 2) / 2).rest + ((4 * 2 ^ i - i0 * 2) % 2 = 1)') = uint2bits((4 * 2 ^ i - i0 * 2) / 2).rest + ((4 * 2 ^ i - i0 * 2) % 2 = 1)'

Rule argument

Additional information

Used simplifier rules: ⊦ 0 < i → (i < i * j ↔ 1 < j);% ⊦ 1 * i = i;% ⊦ i + j < (* i1) * i ↔ j < (i1 - 1) * i;% ⊦ i +1 = i + 1;% ⊦ 0 ≤ i → (2 < 2 ^ i ↔ 1 < i);% ⊦ (* 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)% ⊦ 0 < i → (0 < i * j ↔ 0 < j);% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j);% ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0);% ⊦ i * j = j * i;% ⊦ i * j * j0 = (i * j) * j0;% ⊦ 0 < i → (i * j < i * j0 ↔ j < j0);% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + 0 = i;