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Proof step 18 from proof intshiftrsign-abs-le (previous step)

i + 1 < 2 ^ band(j, 31), band(j, 31) < 32, i < 0, 0 < band(j, 31), 0 < (2 ^ band(j, 31) - i + 1) / 2 ^ band(j, 31) + i ⊦ band(j, 31) = 0

User applied rule cut formula

And obtained the following new goals

Proof step 19:
2 ^ band(j, 31) - i + 1 = 2 ^ band(j, 31) + ~(i + 1), i + 1 < 2 ^ band(j, 31), band(j, 31) < 32, i < 0, 0 < band(j, 31), 0 < (2 ^ band(j, 31) - i + 1) / 2 ^ band(j, 31) + i ⊦ band(j, 31) = 0

Rule argument

fma: 2 ^ band(j, 31) - i + 1 = 2 ^ band(j, 31) + ~(i + 1)

Additional information

Used simplifier rules: band-commutativity: ⊦ band(i, j) = band(j, i);% ⊦ i + ~ j = i - j;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i - j = i0 ↔ i = i0 + j;% ⊦ i + (j - i) = j; Used formulas: 2 ^ band(j, 31) - i + 1 = 2 ^ band(j, 31) + ~(i + 1)