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Proof step 82 from proof int2bits-add-low (previous step)

0 < ~(j * 2 ^ i + j0), 0 ≤ i, 0 < i, j < 0, j0 < 0, 0 < 2 ^ (i - 1) + j0, ~(j * 2 ^ i + j0) < 2 ^ (n→i(i→n(i)) - 1), 0 ≤ 2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0, j * 2 ^ i + j0 < 0 ⊦ sgn(uint2bits(2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 83:
0 ≤ 2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0, j * 2 ^ i + j0 < 0, 0 < i, j < 0, j0 < 0, 0 < 2 ^ (i - 1) + j0, 0 ≤ i, 0 < 2 ^ (i - 1) + j * 2 ^ i + j0 ⊦ sgn(uint2bits(2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

Rule argument

Additional information

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