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Proof step 13 from proof int2bits-add-low (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ i < 0 → # int2bits(i) = # int2bits(~ i - 1)
Proof step 15:
j * 2 ^ i + j0 < 0, 1 + # int2bits(~(j * 2 ^ i + j0)) < # int2bits(~(j * 2 ^ i + j0) - 1), j * 2 ^ i + j0 < 0, 0 < 2 ^ (i - 1) + j0, j0 < 0, j < 0, 0 < i, j * 2 ^ i + j0 < ~1 ⊦ sgn(int2bits(2 ^ n→i(# int2bits(~(j * 2 ^ i + j0))) + j * 2 ^ i + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

Rule argument

name: int2bits-len-neg-01 rotate: F subst: [i] → [j * 2 ^ i + j0]

Additional information

Used terms: j * 2 ^ i + j0