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Proof step 68 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 ⊦ bitsinvert(uint2bits(1 - j / 2 ^ i)) +1 = [], j % 2 ^ i = 0, bitsinvert(fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))).rest) +1 = fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 69:
⊦ bits ≠ [] → bitsinvert(bits.rest) = bitsinvert(bits).rest
Proof step 70:
fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i))) ≠ [], 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 ⊦ bitsinvert(uint2bits(1 - j / 2 ^ i)) +1 = [], j % 2 ^ i = 0, bitsinvert(fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i)))).rest +1 = fillsgn(i→n(i) + 1, sgn(bitsinvert(uint2bits(~(j % 2 ^ i))) +1)).rest

Rule argument

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

Additional information

Used simplifier rules: uint2bits-empty-01: ⊦ i < 0 → uint2bits(~ i) ≠ [];% ⊦ fillsgn(n, bits) = [] ↔ bits = [] ∧ n = 0; Used terms: fillsgn(i→n(i) + 1, uint2bits(~(j % 2 ^ i)))