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Proof step 85 from proof int2bits-rec-2pot-neg-02 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ m ≠ 0 → (mklist(boolvar, m) + bits).rest = mklist(boolvar, m -1) + bits subst: [bits, m, boolvar] → [bitsinvert(uint2bits(~(j % 2 ^ i))), i→n(i) + 1 - # uint2bits(~(j % 2 ^ i)), bitsinvert(uint2bits(~(j % 2 ^ i))).first] rotate: F spec: list-last inst: bitlista name: mk-rest-01

Additional information

Used simplifier rules: uint2bits-onein-01: ⊦ j < 0 → true ∈ uint2bits(~ j);% ⊦ n -1 = n - 1;% ⊦ bitsinvert(bits + bits0) = bitsinvert(bits) + bitsinvert(bits0);% ⊦ bitsinvert(mklist(boolvar, n)) = mklist(¬ boolvar, n);% ⊦ bitsinvert(bitsinvert(bits)) = bits;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ m ≤ n → (n - m = 0 ↔ n = m);% ⊦ m ≠ n → (m ≤ n ↔ m < n);% ⊦ m < n + 1 ↔ ¬ n < m;% ⊦ ¬ m < n ↔ n ≤ m;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ n ≤ m → m + n0 - n = (m - n) + n0;% ⊦ 0 + m = m;% ⊦ (m + 1)-1 = m;% ⊦ boolvar ∈ bits + bits0 ↔ ¬ ¬(boolvar ∈ bits ∨ boolvar ∈ bits0);% ⊦ false ∈ bitsinvert(bits) ↔ true ∈ bits; Used terms: bitsinvert(uint2bits(~(j % 2 ^ i))), i→n(i) + 1 - # uint2bits(~(j % 2 ^ i)), bitsinvert(uint2bits(~(j % 2 ^ i))).first