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Proof step 119 from proof uint2bits-add-high (previous step)

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 ≠ 0 ∧ i ≠ 1 ∧ ¬ abs(j - 1) / 2 ≤ 2 ^ (i - 2) ∧ ¬ 2 ^ (i - 1) < abs(j - 1) / 2, 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 120:
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, 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, ∀ 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) + (j % 2 ≠ 0)' = sgn(int2bits(j).rest), i = 1, j = 0, ¬ 2 ^ (i - 1) < abs(j - 1) / 2 ∧ ¬ abs(j - 1) / 2 ≤ 2 ^ (i - 2)

Rule argument

Additional information

Used simplifier rules: int2bits-zero: ⊦ int2bits(0) = false ';% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < 0 → (~2 < i ↔ i = ~1);% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0;% ⊦ i < 1 ↔ i ≤ 0;% ⊦ i + (* j) < (* j0) ↔ i < j0 - j;% ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0;% ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0;% ⊦ ¬ 2 ^ i < 0;% ⊦ i + j < i0 ∧ ¬ j < j0 → i + j0 < i0;% ⊦ 2 ^ i = 1 ↔ i = 0;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ bits + boolvar ' ≠ [];% ⊦ sgn([]) = [];% ⊦ boolvar '.rest = [];% ⊦ i + 0 = i;% ⊦ i ≠ j → (i ≤ j ↔ i < j);% ⊦ i ≤ i;% ⊦ i < j + 1 ↔ i ≤ j;% ⊦ i - j < i0 ↔ i < j + i0;%0 ≤ j ⊦ 2 ^ j < 2 ^ j0 ↔ j < j0;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ 1 ≤ i ↔ 0 < i;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ j < 0 → (j % 2 ≠ ~1 ↔ j % 2 = 0);