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Proof step 62 from proof int2bits-add-low (previous step)

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

User applied rule insert lemma

And obtained the following new goals

Proof step 63:
j < 0, ¬(∃ i. ~(2 ^ i) = j) ⊦ int2bits(j) = int2bits(2 ^ n→i(# int2bits(~ j)) + j).rest
Proof step 64:
0 ≤ 2 ^ n→i(# int2bits(~(j - 1) - 1)) + (j - 1), 0 ≤ j0 + 2 ^ (i - 1), 0 < 2 ^ (i - 1) + j0, j0 < 0, j < 0, 0 < i, j * 2 ^ i + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, 0 < 2 ^ n→i(# int2bits(~(j - 1) - 1)) + j, 0 ≤ 2 ^ i * (2 ^ n→i(# int2bits(~(j - 1) - 1)) + j) + j0 ⊦ j - 1 < 0 ∧ ¬(∃ i. ~(2 ^ i) = j - 1), j * 2 ^ i + j0 = 0, sgn(int2bits(2 ^ n→i(# int2bits(~(j - 1) - 1)) + (j - 1)).rest + fillsgn(i→n(i), int2bits(j0))) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))
Proof step 66:
int2bits(j - 1) = int2bits(2 ^ n→i(# int2bits(~(j - 1))) + (j - 1)).rest, 0 ≤ 2 ^ n→i(# int2bits(~(j - 1) - 1)) + (j - 1), 0 ≤ j0 + 2 ^ (i - 1), 0 < 2 ^ (i - 1) + j0, j0 < 0, j < 0, 0 < i, j * 2 ^ i + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, 0 < 2 ^ n→i(# int2bits(~(j - 1) - 1)) + j, 0 ≤ 2 ^ i * (2 ^ n→i(# int2bits(~(j - 1) - 1)) + j) + j0 ⊦ j * 2 ^ i + j0 = 0, sgn(int2bits(2 ^ n→i(# int2bits(~(j - 1) - 1)) + (j - 1)).rest + fillsgn(i→n(i), int2bits(j0))) = int2bits(j - 1) + fillsgn(i→n(i), int2bits(j0))

Rule argument

name: int2bits-neg-02 subst: [j] → [j - 1]

Additional information

Used terms: j - 1