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Proof step 106 from proof int2bits-rec-2pot-neg-02 (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 107:
sgn(bitsinvert(uint2bits(~(j / 2 ^ i)))) = sgn(bitsinvert(uint2bits(1 - j / 2 ^ i)) +1), 2 ^ i + j % 2 ^ i < i, 2 ^ i + j % 2 ^ i ≤ 0, true ∈ uint2bits(~(j % 2 ^ i)).rest, false ∈ fillsgn(i→n(i) + 1, bitsinvert(uint2bits(~(j % 2 ^ i)))).rest, 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, i→n(i) + 1 < # uint2bits(~(j % 2 ^ i)) ⊦ bitsinvert(uint2bits(1 - j / 2 ^ i)) +1 = [], j % 2 ^ i = 0, fillsgn(i→n(i) + 1, bitsinvert(uint2bits(~(j % 2 ^ i))) +1).rest = fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest, fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))) = [], fillsgn(i→n(i) + 1, bitsinvert(uint2bits(~(j % 2 ^ i)))) = [], bitsinvert(uint2bits(~(j % 2 ^ i))) +1 = []

Rule argument

Additional information

Used simplifier rules: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j;% ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward)% ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ ~ i < j ↔ 0 < j + i;% ⊦ i + 0 = i;% ⊦ i ≤ ~ j ↔ i + j ≤ 0;% ⊦ i < j → ¬ j < i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ ¬ m ≤ n ↔ n < m;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ 0 < i → i→n(i + 1) = i→n(i) + 1;% ⊦ m + n = n + m;