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Proof step 22 from proof int2bits-add-low (previous step)

0 < 2 ^ (i - 1) + j0, j0 < 0, j < 0, 0 < i, i < n→i(# int2bits(~(j * 2 ^ i + j0))), j * 2 ^ i + j0 < 0, 0 ≤ 2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + 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)), j * 2 ^ i + j0 = 0

User applied rule cut formula

And obtained the following new goals

Proof step 23:
2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0 = (j + 2 ^ (n→i(# int2bits(~(j * 2 ^ i + j0))) - i)) * 2 ^ i + j0, 0 < 2 ^ (i - 1) + j0, j0 < 0, j < 0, 0 < i, i < n→i(# int2bits(~(j * 2 ^ i + j0))), j * 2 ^ i + j0 < 0, 0 ≤ 2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + 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)), j * 2 ^ i + j0 = 0
Proof step 74:
0 < 2 ^ (i - 1) + j0, j0 < 0, j < 0, 0 < i, i < n→i(# int2bits(~(j * 2 ^ i + j0))), j * 2 ^ i + j0 < 0, 0 ≤ 2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0 ⊦ 2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0 = (j + 2 ^ (n→i(# int2bits(~(j * 2 ^ i + j0))) - i)) * 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)), j * 2 ^ i + j0 = 0

Rule argument

fma: 2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0 = (j + 2 ^ (n→i(# int2bits(~(j * 2 ^ i + j0))) - i)) * 2 ^ i + j0

Additional information

Used formulas: 2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0 = (j + 2 ^ (n→i(# int2bits(~(j * 2 ^ i + j0))) - i)) * 2 ^ i + j0