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Proof step 81 from proof int2bits-rec-2pot-neg-02 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 82:
sgn(bitsinvert(uint2bits(~(j / 2 ^ i)))) = sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1), i→n(i) + 1 < # uint2bits(~(j % 2 ^ i)), j % 2 ^ i < 0, true ∈ uint2bits(~(j / 2 ^ i)), false ∈ uint2bits(~(j / 2 ^ i)), true ∈ fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest, 0 ≤ i, j < 0, j + 2 ^ i < i, j + 2 ^ i ≤ 0, 0 < i, j / 2 ^ i ≤ 0, j / 2 ^ i < i, i→n(i) < # uint2bits(~(j % 2 ^ i)) ⊦ j % 2 ^ i = 0, bitsinvert(uint2bits(1 - j / 2 ^ i)) +1 = [], fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))) = [], true ' + uint2bits(~(j % 2 ^ i)).rest = fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest, true ∈ uint2bits(~(j % 2 ^ i)).rest

Rule argument

Additional information

Used simplifier rules: uint2bits-empty-01: ⊦ i < 0 → uint2bits(~ i) ≠ [];% ⊦ false ∈ bitsinvert(bits) ↔ true ∈ bits;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + 1 ≤ n ↔ m < n;% ⊦ # bitsinvert(bits) = # bits;% ⊦ m + n = n + m;% ⊦ bitsinvert(bitsinvert(bits)) = bits;%bits ≠ [] ⊦ bitsinvert(bits).rest = bitsinvert(bits.rest);