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Proof step 124 from proof uint2bits-add-high (previous step)

1 - j = i1 * 2 + j0 ∧ 0 ≤ j0 ∧ j0 < 2 ∧ 0 < 2 ∧ 0 ≤ 1 - j, 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) < i1 ∧ ¬ i1 ≤ 2 ^ (i - 2)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 125:
1 = (i1 * 2 + j0) + j, 2 ^ (i - 1) + j ≤ 2 ^ i, 2 ^ (i - 1) + j ≤ 0, j < j0, 2 ^ (i - 1) + j < j0, 1 < 2 ^ i, j ≤ 1, j0 < 2, 0 ≤ j0, j < 0, j ≤ 2 ^ i, 0 ≤ j + 2 ^ i, 0 ≤ 2 ^ i, (2 ^ (i - 1) + j) + 1 < 2 ^ i, j < 2 ^ i, j + 1 < 2 ^ i, j < i, 2 ^ (i - 1) + j < 0, 0 < i, (2 ^ (i - 1) + j) + 1 < i, 2 ^ (i - 1) + j < i, j + 1 < i, 2 ^ (i - 1) + j < 2 ^ i, 0 < j + 2 ^ i, 2 ^ (i - 1) + 1 < 2 ^ i, 0 < 2 ^ i, 2 ^ (i - 1) + j ≤ j0, 2 ^ (i - 1) ≤ 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 = 0, i = 1, uint2bits(2 ^ (i - 1) + (j - 1) / 2) + (j % 2 ≠ 0)' = sgn(int2bits(j).rest), j + 2 ^ i = 1, ¬ 2 ^ (i - 1) < i1 ∧ ¬ i1 ≤ 2 ^ (i - 2)

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i + j ≤ i + j0 ↔ j ≤ j0;% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)%i0 ≠ 0, i0 ≤ i ⊦ (* i) < j → (* i0) < j;% ⊦ i ≠ j → (i ≤ j ↔ i < j);%j0 ≠ 0, j < j0 ⊦ i < (* j) → i < (* j0);% ⊦ i < 1 ↔ i ≤ 0;% ⊦ i + (* j) < (* j0) ↔ i < j0 - j;% ⊦ i +1 = i + 1;% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i < j ∧ i = i0 → i0 < j; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + 0 = i;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;% ⊦ i - j = i0 ↔ i = i0 + j;