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Proof step 29 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, 0 < 2 ^ n→i(# int2bits(~ j)) + j, 0 ≤ (2 ^ n→i(# int2bits(~ j)) + j) * 2 ^ i + j0, 0 ≤ j0 + 2 ^ i ⊦ sgn(sgn(uint2bits(2 ^ n→i(# int2bits(~ j)) + j - 1) + fillsgn(i→n(i), int2bits(j0).rest)).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 30:
⊦ j < 0 → int2bits(j) = sgn(int2bits(2 ^ n→i(# int2bits(~ j)) + j).rest)
Proof step 31:
j - 1 < 0, j < 0, j0 < 0, 0 < i, 2 ^ (i - 1) + j0 < 0, 0 < 2 ^ i + j0, 0 ≤ i, j * 2 ^ i + j0 < 0, 0 < 2 ^ n→i(# int2bits(~ j)) + j, 0 ≤ (2 ^ n→i(# int2bits(~ j)) + j) * 2 ^ i + j0, 0 ≤ j0 + 2 ^ i ⊦ sgn(sgn(uint2bits(2 ^ n→i(# int2bits(~ j)) + j - 1) + fillsgn(i→n(i), int2bits(j0).rest)).rest) = sgn(int2bits(2 ^ n→i(# int2bits(~(j - 1))) + (j - 1)).rest) + fillsgn(i→n(i), int2bits(j0).rest)

Rule argument

name: int2bits-neg-03 rotate: F subst: [j] → [j - 1]

Additional information

Used simplifier rules: ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i < 1 ↔ i ≤ 0;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i; Used terms: j - 1