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Proof step 116 from proof uint2bits-add-high (previous step)

j = i1 * 2 + j0 ∧ j0 ≤ 0 ∧ abs(j0) < 2 ∧ 0 < 2, ~1 = j0, uint2bits((j - 1) / 2 + 2 ^ (i - 1)) = sgn(int2bits((j - 1) / 2).rest), 2 ^ (i - 2) < 2 ^ i, 0 < 2 ^ i, 1 < 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 0 < j + 2 ^ i, 2 ^ (i - 1) + j < 2 ^ i, j + 1 < i, 2 ^ (i - 1) + j < i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) < 2 ^ i, 0 < i, 2 ^ (i - 1) + j < 0, j < i, j + 1 < 2 ^ i, j < 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, 0 ≤ 2 ^ i, 0 ≤ j + 2 ^ i, j ≤ 2 ^ i, j < 0, abs(j - 1) / 2 ≤ 2 ^ (i - 1), 2 ^ (i - 2) < abs(j - 1) / 2, abs(j - 1) / 2 < 2 ^ i, 2 ^ (i - 2) < 2 ^ (i - 1), ∀ 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) ⊦ i = 1, j = 0, j = ~1, j + 2 ^ i = 1, uint2bits((j - 1) / 2 + 2 ^ (i - 1)) + true ' = sgn(int2bits((j - 1) / 2).rest + true '), false ∈ int2bits((j - 1) / 2).rest

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 117:
uint2bits(2 ^ (i - 1) + i1 - 1) = sgn(int2bits(i1 - 1).rest), 2 ^ (i - 1) + i1 * 2 < i, abs(i1 - 1) ≤ 2 ^ (i - 1), 2 ^ (i - 2) < abs(i1 - 1), abs(i1 - 1) < 2 ^ i, 0 ≤ i, 1 < i, 0 < i, 1 < 2 ^ i + i1 * 2, 0 < 2 ^ i + i1 * 2, i1 * 2 + 2 ^ (i - 1) ≤ 0, 0 ≤ 2 ^ i, i1 < 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) ⊦ i = 1, i1 = 0, 2 ^ i + i1 * 2 = 2, sgn(int2bits(i1 - 1).rest) + true ' = sgn(int2bits(i1 - 1).rest + true '), false ∈ int2bits(i1 - 1).rest

Rule argument

Additional information

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