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Proof step 16 from proof int2bits-add-low (previous step)

1 + # int2bits(~(j * 2 ^ i + j0)) < # int2bits(~((j * 2 ^ i + j0) + 1)), j * 2 ^ i + j0 < ~1, 0 < i, j < 0, j0 < 0, 0 < 2 ^ (i - 1) + j0, j * 2 ^ i + j0 < 0 ⊦ 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 17:
⊦ m + n = n + m
Proof step 18:
# int2bits(~(j * 2 ^ i + j0)) + 1 < # int2bits(~((j * 2 ^ i + j0) + 1)), j * 2 ^ i + j0 < ~1, 0 < i, j < 0, j0 < 0, 0 < 2 ^ (i - 1) + j0, j * 2 ^ i + j0 < 0 ⊦ 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

seq: ⊦ m + n = n + m subst: [n, m] → [# int2bits(~(j * 2 ^ i + j0)), 1] rotate: F spec: nat-basic2 inst: name: c

Additional information

Used terms: # int2bits(~(j * 2 ^ i + j0)), 1