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

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ 0 ≤ i → int2bits(i) = uint2bits(i)
Proof step 15:
¬ 0 ≤ 2 ^ i * (2 ^ n→i(# int2bits(~ j)) + j) + j0, j < 0, j0 < 0, 0 < i, 2 ^ (i - 1) + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, j * 2 ^ i + j0 < 0 ⊦ sgn(int2bits(2 ^ i * (2 ^ n→i(# int2bits(~ j)) + j) + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0).rest)
Proof step 23:
0 ≤ 2 ^ i * (2 ^ n→i(# int2bits(~ j)) + j) + j0, j < 0, j0 < 0, 0 < i, 2 ^ (i - 1) + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, j * 2 ^ i + j0 < 0 ⊦ sgn(uint2bits(2 ^ i * (2 ^ n→i(# int2bits(~ j)) + j) + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0).rest)

Rule argument

name: int2bits-pos rotate: F subst: [i] → [2 ^ i * (2 ^ n→i(# int2bits(~ j)) + j) + j0]

Additional information

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