Definition codeFormula := codeFormula LNN codeLNTFunction codeLNNRelation.Definition codeSysPrf :=
  codeSysPrf LNN codeLNTFunction codeLNNRelation codeArityLNTF codeArityLNNR
    codeArityLNTFIsPR codeArityLNNRIsPR repT v0.Definition codeSysPrfCorrect1 :=
  codeSysPrfCorrect1 LNN codeLNTFunction codeLNNRelation codeArityLNTF
    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNNRIsPR
    codeArityLNNRIsCorrect1 T extendsNN T repT v0 freeVarRepT expressT1.Definition codeSysPrfCorrect2 :=
  codeSysPrfCorrect2 LNN codeLNTFunction codeLNNRelation codeArityLNTF
    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNNRIsPR
    codeArityLNNRIsCorrect1 T extendsNN T repT v0 freeVarRepT expressT2.Definition codeSysPrfCorrect3 :=
  codeSysPrfCorrect3 LNN codeLNTFunction codeLNNRelation codeArityLNTF
    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1
    codeArityLNTFIsCorrect2 codeArityLNNRIsPR codeArityLNNRIsCorrect1
    codeArityLNNRIsCorrect2 codeLNTFunctionInj codeLNNRelationInj T extendsNN.Definition codePrf := codePrf LNN codeLNTFunction codeLNNRelation.Definition codeSysPrfNot :=
  codeSysPrfNot LNN codeLNTFunction codeLNNRelation codeArityLNTF
    codeArityLNNR codeArityLNTFIsPR codeArityLNNRIsPR repT v0.Definition freeVarCodeSysPrfN :=
  freeVarCodeSysPrfN LNN codeLNTFunction codeLNNRelation codeArityLNTF
    codeArityLNNR codeArityLNTFIsPR codeArityLNNRIsPR repT v0 freeVarRepT.Definition codeSysPrfNCorrect1 :=
  codeSysPrfNCorrect1 LNN codeLNTFunction codeLNNRelation codeArityLNTF
    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNNRIsPR
    codeArityLNNRIsCorrect1 T extendsNN T repT v0 freeVarRepT expressT1.Definition codeSysPrfNCorrect2 :=
  codeSysPrfNCorrect2 LNN codeLNTFunction codeLNNRelation codeArityLNTF
    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1 codeArityLNNRIsPR
    codeArityLNNRIsCorrect1 T extendsNN T repT v0 freeVarRepT expressT2.Definition codeSysPrfNCorrect3 :=
  codeSysPrfNCorrect3 LNN codeLNTFunction codeLNNRelation codeArityLNTF
    codeArityLNNR codeArityLNTFIsPR codeArityLNTFIsCorrect1
    codeArityLNTFIsCorrect2 codeArityLNNRIsPR codeArityLNNRIsCorrect1
    codeArityLNNRIsCorrect2 codeLNTFunctionInj codeLNNRelationInj T extendsNN
    repT v0 freeVarRepT.

Lemma searchProof : (forall x : Formula, mem _ T x \/ ~ mem _ T x) -> forall (a b : Formula) (A : Formulas) (p : Prf LNN A a), (exists B : Formulas, (exists q : Prf LNN B b, codePrf _ _ q < S (codePrf _ _ p) /\ (forall x : Formula, In x B -> mem _ T x))) \/ (forall (B : Formulas) (q : Prf LNN B b), codePrf _ _ q < S (codePrf _ _ p) -> exists g : Formula, In g B /\ ~ mem _ T g).