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Proof step 18 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, 2 ^ i * (2 ^ n→i(# int2bits(~ j)) + j) + j0 < 0, 0 < 2 ^ n→i(# int2bits(~ j)) + j ⊦ 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 19:
⊦ i * j = j * i
Proof step 20:
j < 0, j0 < 0, 0 < i, 2 ^ (i - 1) + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, j * 2 ^ i + j0 < 0, (2 ^ n→i(# int2bits(~ j)) + j) * 2 ^ i + j0 < 0, 0 < 2 ^ n→i(# int2bits(~ j)) + j ⊦ sgn(int2bits((2 ^ n→i(# int2bits(~ j)) + j) * 2 ^ i + j0).rest) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0).rest)

Rule argument

seq: ⊦ i * j = j * i subst: [j, i] → [2 ^ n→i(# int2bits(~ j)) + j, 2 ^ i] rotate: F spec: int-mult inst: name: commutativity

Additional information

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