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Proof step 2 from proof uint2bits-add-high (previous step)

true, ∀ 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 ∧ 0 < i ∧ 2 ^ (i - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i → uint2bits(j + 2 ^ i) = sgn(int2bits(j).rest)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
2 ^ (i - 1) + j < 2 ^ i, j + 1 < i, 0 < 2 ^ 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, 0 ≤ j + 2 ^ i, j < i, j + 1 < 2 ^ i, j < 2 ^ i, (2 ^ (i - 1) + j) + 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) ⊦ uint2bits(j + 2 ^ i) = sgn(int2bits(j).rest)

Rule argument

Additional information

Used simplifier rules: ⊦ i < i + j ↔ 0 < j;% ⊦ i +1 = i + 1;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j < i + i0 ↔ j < i0;% ⊦ i < j ∧ j < j0 → i < j0; (forward)% ⊦ i < j ∧ j < i0 → i +1 < i0; (forward)% ⊦ i < j ∧ j ≤ j0 → i < j0; (forward)% ⊦ ~ i ≤ j ↔ 0 ≤ i + j;% ⊦ i < ~ j ↔ i + j < 0;% ⊦ i < 0 → abs(i) = ~ i;