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Proof step 79 from proof int2bits-rec-2pot-neg-02 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ n ≤ # bits → fillsgn(n, bits) = bits subst: [bits, n] → [bitsinvert(uint2bits(~(j % 2 ^ i))), i→n(i) + 1] rotate: F spec: bitopsaux inst: name: fillsgn-longer

Additional information

Used simplifier rules: ⊦ # bitsinvert(bits) = # bits;% ⊦ m + 1 ≤ n ↔ m < n;% ⊦ ¬ m < n ↔ n ≤ m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m < n ∧ n ≤ n0 → m < n0; (forward)% ⊦ ¬ m + n < m;% ⊦ m + n = n + m; Used terms: bitsinvert(uint2bits(~(j % 2 ^ i))), i→n(i) + 1