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Proof step 115 from proof int2bits-rec-2pot-neg-02 (previous step)

2 ^ i ≠ 0, j % 2 ^ i < 0, 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, 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 = []

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

seq: ⊦ 0 < j → ¬ j ≤ i % j subst: [i, j] → [~ j, 2 ^ i] rotate: F spec: int-div inst: name: mod-les

Additional information

Used simplifier rules: j ≠ 0 ⊦ ~ i % j = ~(i % j);% ⊦ 0 ≤ j → 2 ^ j ≠ 0;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + j = j + i;% ⊦ i ≤ ~ j ↔ i + j ≤ 0;% ⊦ 0 < 2 ^ i ↔ 0 ≤ i;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ i < j → ¬ j < i; Used terms: ~ j, 2 ^ i