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Proof step 51 from proof uint2bits-add-high (previous step)

(j - 1) / 2 < 0 ∧ 0 < i - 1 ∧ 2 ^ (i - 1 - 1) < abs((j - 1) / 2) ∧ abs((j - 1) / 2) ≤ 2 ^ (i - 1) → uint2bits((j - 1) / 2 + 2 ^ (i - 1)) = sgn(int2bits((j - 1) / 2).rest), j ≤ 2 ^ i, j ≤ 0, 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, j < 0, 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, ∀ 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) ⊦ j + 2 ^ i = 1, uint2bits(2 ^ (i - 1) + (j - 1) / 2) + (~1 = j % 2)' = sgn(int2bits(j).rest)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 52:
j ≤ 2 ^ i, j ≤ 0, 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, j < 0, 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 ≠ 0 ∧ i ≠ 1 ∧ ¬ abs(j - 1) / 2 ≤ 2 ^ (i - 2) ∧ ¬ 2 ^ (i - 1) < abs(j - 1) / 2 → uint2bits((j - 1) / 2 + 2 ^ (i - 1)) = sgn(int2bits((j - 1) / 2).rest), ∀ 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) ⊦ j + 2 ^ i = 1, uint2bits(2 ^ (i - 1) + (j - 1) / 2) + (~1 = j % 2)' = sgn(int2bits(j).rest)

Rule argument

Additional information

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