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Proof step 78 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(fillsgn(i→n(i) + 1, bitsinvert(uint2bits(~(j % 2 ^ i)))).rest) = fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest, false ∈ fillsgn(i→n(i) + 1, bitsinvert(uint2bits(~(j % 2 ^ i)))).rest

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 79:
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(fillsgn(i→n(i) + 1, 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 ∈ fillsgn(i→n(i) + 1, bitsinvert(uint2bits(~(j % 2 ^ i)))).rest

Rule argument

Additional information

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