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Proof step 121 from proof int2bits-rec-2pot-neg-02 (previous step)

0 < 1 - 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 ⊦ bits2int(bitsinvert(uint2bits(~(j / 2 ^ i)))) = bits2int(int2bits(~(1 - j / 2 ^ i))), sgn(bitsinvert(uint2bits(~(j / 2 ^ i)))) + bitsinvert(fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest) +1 = int2bits(~(1 - j / 2 ^ i)) + fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest, j % 2 ^ i = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 122:
⊦ bits ≠ [] → bits2int(bitsinvert(bits)) = ~ bits2int(bits) - 1
Proof step 123:
uint2bits(~(j / 2 ^ i)) ≠ [], 0 < 1 - 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 ⊦ ~ bits2int(uint2bits(~(j / 2 ^ i))) - 1 = bits2int(int2bits(~(1 - j / 2 ^ i))), sgn(bitsinvert(uint2bits(~(j / 2 ^ i)))) + bitsinvert(fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest) +1 = int2bits(~(1 - j / 2 ^ i)) + fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest, j % 2 ^ i = 0

Rule argument

name: bits2int-inv rotate: F subst: [bits] → [uint2bits(~(j / 2 ^ i))]

Additional information

Used simplifier rules: uint2bits-empty: 0 ≤ i ⊦ uint2bits(i) ≠ [];% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j;% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + 0 = i;% ⊦ i ≤ ~ j ↔ i + j ≤ 0; Used terms: uint2bits(~(j / 2 ^ i))