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Proof step 19 from proof sgn-append-03 (previous step)

bits0 = [], bits0 + bits1 = boolvar1 ' + bits ⊦ bits0 + bits1 = [], boolvar0 ↔ boolvar1, sgn(bits0 + bits1) = sgn((¬ boolvar0)' + bits1), boolvar0 ∈ bits0, ¬ boolvar1 ∈ bits0 ∧ ¬ boolvar1 ∈ bits1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
⊦ sgn(boolvar1 ' + bits) = sgn((¬ boolvar0)' + boolvar1 ' + bits), boolvar0 ↔ boolvar1

Rule argument

Additional information

Used simplifier rules: ⊦ boolvar ∈ boolvar0 ' ↔ boolvar ↔ boolvar0;% ⊦ boolvar ∈ bits + bits0 ↔ ¬ ¬(boolvar ∈ bits ∨ boolvar ∈ bits0);% ⊦ boolvar ' + bits ≠ [];% ⊦ boolvar ' + bits = bits0 → boolvar ∈ bits0; (forward)% ⊦ ¬ boolvar ∈ [];% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ [] + bits = bits;