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Proof step 51 from proof int2bits-rec-2pot-neg-02 (previous step)

0 < i, j + 2 ^ i ≤ 0, j + 2 ^ i < i, j < 0, 0 ≤ i, true ∈ fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest, j / 2 ^ i < 0 ⊦ uint2bits(~(j / 2 ^ i)).first, sgn(bitsinvert(uint2bits(~(j / 2 ^ i))) + bitsinvert(fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest) +1) = sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, j % 2 ^ i = 0, false ∈ uint2bits(~(j / 2 ^ i))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 52:
⊦ boolvar ∈ bits ↔ (∃ bits0, bits1. bits = bits0 + (boolvar + bits1))
Proof step 53:
0 < i, j + 2 ^ i ≤ 0, j + 2 ^ i < i, j < 0, 0 ≤ i, true ∈ fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest, j / 2 ^ i < 0 ⊦ uint2bits(~(j / 2 ^ i)).first, sgn(bitsinvert(uint2bits(~(j / 2 ^ i))) + bitsinvert(fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest) +1) = sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1) + fillsgn(i→n(i) + 1, int2bits(j % 2 ^ i)).rest, j % 2 ^ i = 0, ∃ bits0, bits1. uint2bits(~(j / 2 ^ i)) = bits0 + (false + bits1)

Rule argument

seq: ⊦ boolvar ∈ bits ↔ (∃ bits0, bits1. bits = bits0 + (boolvar + bits1)) subst: [bits, boolvar] → [uint2bits(~(j / 2 ^ i)), false] rotate: F spec: list inst: bitlista name: In

Additional information

Used terms: uint2bits(~(j / 2 ^ i)), false