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Proof step 84 from proof int2bits-rec-2pot-neg-02 (previous step)

# bitsinvert(uint2bits(~(j % 2 ^ i))) ≤ i→n(i) + 1 ∧ bitsinvert(uint2bits(~(j % 2 ^ i))) ≠ [], sgn(bitsinvert(uint2bits(~(j / 2 ^ i)))) = sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1), j / 2 ^ i < i, j / 2 ^ i ≤ 0, 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, false ∈ uint2bits(~(j / 2 ^ i)), true ∈ uint2bits(~(j / 2 ^ i)), j % 2 ^ i < 0 ⊦ fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))) = [], bitsinvert(uint2bits(1 - j / 2 ^ i)) +1 = [], j % 2 ^ i = 0, true ' + bitsinvert ((mklist(bitsinvert(uint2bits(~(j % 2 ^ i))).first, i→n(i) + 1 - # bitsinvert(uint2bits(~(j % 2 ^ i)))) + bitsinvert(uint2bits(~(j % 2 ^ i))) ).rest) = fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest, false ∈ (mklist(bitsinvert(uint2bits(~(j % 2 ^ i))).first, i→n(i) + 1 - # bitsinvert(uint2bits(~(j % 2 ^ i)))) + bitsinvert(uint2bits(~(j % 2 ^ i)))).rest

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 85:
sgn(bitsinvert(uint2bits(~(j / 2 ^ i)))) = sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1), j / 2 ^ i < i, j / 2 ^ i ≤ 0, 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, false ∈ uint2bits(~(j / 2 ^ i)), true ∈ uint2bits(~(j / 2 ^ i)), j % 2 ^ i < 0, # uint2bits(~(j % 2 ^ i)) ≤ i→n(i) + 1 ⊦ fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))) = [], bitsinvert(uint2bits(1 - j / 2 ^ i)) +1 = [], j % 2 ^ i = 0, true ' + bitsinvert ((mklist(bitsinvert(uint2bits(~(j % 2 ^ i))).first, i→n(i) + 1 - # uint2bits(~(j % 2 ^ i))) + bitsinvert(uint2bits(~(j % 2 ^ i)))).rest) = fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest, false ∈ (mklist(bitsinvert(uint2bits(~(j % 2 ^ i))).first, i→n(i) + 1 - # uint2bits(~(j % 2 ^ i))) + bitsinvert(uint2bits(~(j % 2 ^ i)))).rest

Rule argument

Additional information

Used simplifier rules: uint2bits-empty-01: ⊦ i < 0 → uint2bits(~ i) ≠ [];% ⊦ bitsinvert(bits) = [] ↔ bits = [];% ⊦ # bitsinvert(bits) = # bits;