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Proof step 56 from proof int2bits-add-low (previous step)

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

User applied rule cut formula

And obtained the following new goals

Proof step 57:
~ j = ~(j - 1) - 1, 0 ≤ j0 + 2 ^ (i - 1), 0 < 2 ^ (i - 1) + j0, j0 < 0, j < 0, 0 < i, j * 2 ^ i + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, 0 < 2 ^ n→i(# int2bits(~ j)) + j, 0 ≤ 2 ^ i * (2 ^ n→i(# int2bits(~ j)) + j) + j0 ⊦ j * 2 ^ i + j0 = 0, sgn(uint2bits(2 ^ n→i(# int2bits(~ j)) + j - 1).rest + fillsgn(i→n(i), int2bits(j0))) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

Rule argument

fma: ~ j = ~(j - 1) - 1

Additional information

Used simplifier rules: ⊦ ~(i - 1) = ~ i + 1;% ⊦ i + ~ j = i - j;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i - j - i0 = i - j + i0;% ⊦ i - j + i = ~ j;% ⊦ ~ i = ~ j ↔ i = j; Used formulas: ~ j = ~(j - 1) - 1