:: FOMODEL4 semantic presentation begin definitionlet "S" be ($#l1_fomodel1 :::"Language":::); func "S" :::"-sequents"::: -> ($#m1_hidden :::"set"::: ) equals :: FOMODEL4:def 1 "{" (Set ($#k1_domain_1 :::"["::: ) (Set (Var "premises")) "," (Set (Var "conclusion")) ($#k1_domain_1 :::"]"::: ) ) where premises "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S" ")" ), conclusion "is" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S" : (Bool (Set (Var "premises")) "is" ($#v1_finset_1 :::"finite"::: ) ) "}" ; end; :: deftheorem defines :::"-sequents"::: FOMODEL4:def 1 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set ($#k1_domain_1 :::"["::: ) (Set (Var "premises")) "," (Set (Var "conclusion")) ($#k1_domain_1 :::"]"::: ) ) where premises "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Var "S")) ")" ), conclusion "is" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) : (Bool (Set (Var "premises")) "is" ($#v1_finset_1 :::"finite"::: ) ) "}" )); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) -> ($#v1_relat_1 :::"Relation-like"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "x" be ($#m1_hidden :::"set"::: ) ; attr "x" is "S" :::"-sequent-like"::: means :: FOMODEL4:def 2 (Bool "x" ($#r2_hidden :::"in"::: ) (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) )); end; :: deftheorem defines :::"-sequent-like"::: FOMODEL4:def 2 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" (Set (Var "S")) ($#v1_fomodel4 :::"-sequent-like"::: ) ) "iff" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) )) ")" ))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "X" be ($#m1_hidden :::"set"::: ) ; attr "X" is "S" :::"-sequents-like"::: means :: FOMODEL4:def 3 (Bool "X" ($#r1_tarski :::"c="::: ) (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) )); end; :: deftheorem defines :::"-sequents-like"::: FOMODEL4:def 3 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" (Set (Var "S")) ($#v2_fomodel4 :::"-sequents-like"::: ) ) "iff" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) )) ")" ))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster -> "S" ($#v2_fomodel4 :::"-sequents-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )); cluster -> "S" ($#v1_fomodel4 :::"-sequent-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster "S" ($#v1_fomodel4 :::"-sequent-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ); cluster ($#v1_relat_1 :::"Relation-like"::: ) "S" ($#v2_fomodel4 :::"-sequents-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster "S" ($#v1_fomodel4 :::"-sequent-like"::: ) for ($#m1_hidden :::"set"::: ) ; cluster "S" ($#v2_fomodel4 :::"-sequents-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); mode Rule of "S" is ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); mode RuleSet of "S" is ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ")" ); end; registrationlet "A", "B" be ($#m1_hidden :::"set"::: ) ; let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Const "A")) "," (Set (Const "B")) ")" ")" ); cluster (Set ($#k3_tarski :::"union"::: ) "X") -> ($#v1_relat_1 :::"Relation-like"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); cluster (Set ($#k3_tarski :::"union"::: ) "D") -> ($#v1_relat_1 :::"Relation-like"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); func :::"OneStep"::: "D" -> ($#m2_funct_2 :::"Rule":::) "of" "S" means :: FOMODEL4:def 4 (Bool "for" (Set (Var "Seqs")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "Seqs"))) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) (Set "(" (Set "(" ($#k3_tarski :::"union"::: ) "D" ")" ) ($#k7_relat_1 :::".:"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Seqs")) ($#k6_domain_1 :::"}"::: ) ) ")" )))); end; :: deftheorem defines :::"OneStep"::: FOMODEL4:def 4 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "b3")) "being" ($#m2_funct_2 :::"Rule":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_fomodel4 :::"OneStep"::: ) (Set (Var "D")))) "iff" (Bool "for" (Set (Var "Seqs")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) "holds" (Bool (Set (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set (Var "Seqs"))) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) (Set "(" (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "D")) ")" ) ($#k7_relat_1 :::".:"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "Seqs")) ($#k6_domain_1 :::"}"::: ) ) ")" )))) ")" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "m" be ($#m1_hidden :::"Nat":::); func "(" "m" "," "D" ")" :::"-derivables"::: -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 5 (Set ($#k9_funct_7 :::"iter"::: ) "(" (Set "(" ($#k2_fomodel4 :::"OneStep"::: ) "D" ")" ) "," "m" ")" ); end; :: deftheorem defines :::"-derivables"::: FOMODEL4:def 5 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set "(" (Set (Var "m")) "," (Set (Var "D")) ")" ($#k3_fomodel4 :::"-derivables"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_funct_7 :::"iter"::: ) "(" (Set "(" ($#k2_fomodel4 :::"OneStep"::: ) (Set (Var "D")) ")" ) "," (Set (Var "m")) ")" ))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "Seqs1", "Seqs2" be ($#m1_hidden :::"set"::: ) ; attr "Seqs2" is "Seqs1" "," "D" :::"-derivable"::: means :: FOMODEL4:def 6 (Bool "Seqs2" ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" (Set "(" (Set "(" ($#k2_fomodel4 :::"OneStep"::: ) "D" ")" ) ($#k13_lang1 :::"[*]"::: ) ")" ) ($#k7_relset_1 :::".:"::: ) (Set ($#k1_tarski :::"{"::: ) "Seqs1" ($#k1_tarski :::"}"::: ) ) ")" ))); end; :: deftheorem defines :::"-derivable"::: FOMODEL4:def 6 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "Seqs1")) "," (Set (Var "Seqs2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "Seqs2")) "is" (Set (Var "Seqs1")) "," (Set (Var "D")) ($#v3_fomodel4 :::"-derivable"::: ) ) "iff" (Bool (Set (Var "Seqs2")) ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" (Set "(" (Set "(" ($#k2_fomodel4 :::"OneStep"::: ) (Set (Var "D")) ")" ) ($#k13_lang1 :::"[*]"::: ) ")" ) ($#k7_relset_1 :::".:"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "Seqs1")) ($#k1_tarski :::"}"::: ) ) ")" ))) ")" )))); definitionlet "m" be ($#m1_hidden :::"Nat":::); let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "Seqts", "seqt" be ($#m1_hidden :::"set"::: ) ; attr "seqt" is "m" "," "Seqts" "," "D" :::"-derivable"::: means :: FOMODEL4:def 7 (Bool "seqt" ($#r2_hidden :::"in"::: ) (Set (Set "(" "(" "m" "," "D" ")" ($#k3_fomodel4 :::"-derivables"::: ) ")" ) ($#k1_funct_1 :::"."::: ) "Seqts")); end; :: deftheorem defines :::"-derivable"::: FOMODEL4:def 7 : (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "Seqts")) "," (Set (Var "seqt")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) "is" (Set (Var "m")) "," (Set (Var "Seqts")) "," (Set (Var "D")) ($#v4_fomodel4 :::"-derivable"::: ) ) "iff" (Bool (Set (Var "seqt")) ($#r2_hidden :::"in"::: ) (Set (Set "(" "(" (Set (Var "m")) "," (Set (Var "D")) ")" ($#k3_fomodel4 :::"-derivables"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "Seqts")))) ")" ))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); func "D" :::"-iterators"::: -> ($#m1_subset_1 :::"Subset-Family":::) "of" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) equals :: FOMODEL4:def 8 "{" (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" (Set "(" ($#k2_fomodel4 :::"OneStep"::: ) "D" ")" ) "," (Set (Var "mm")) ")" ")" ) where mm "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ; end; :: deftheorem defines :::"-iterators"::: FOMODEL4:def 8 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "D")) ($#k4_fomodel4 :::"-iterators"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k9_funct_7 :::"iter"::: ) "(" (Set "(" ($#k2_fomodel4 :::"OneStep"::: ) (Set (Var "D")) ")" ) "," (Set (Var "mm")) ")" ")" ) where mm "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "R" be ($#m2_funct_2 :::"Rule":::) "of" (Set (Const "S")); attr "R" is :::"isotone"::: means :: FOMODEL4:def 9 (Bool "for" (Set (Var "Seqts1")) "," (Set (Var "Seqts2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "st" (Bool (Bool (Set (Var "Seqts1")) ($#r1_tarski :::"c="::: ) (Set (Var "Seqts2")))) "holds" (Bool (Set "R" ($#k3_funct_2 :::"."::: ) (Set (Var "Seqts1"))) ($#r1_tarski :::"c="::: ) (Set "R" ($#k3_funct_2 :::"."::: ) (Set (Var "Seqts2"))))); end; :: deftheorem defines :::"isotone"::: FOMODEL4:def 9 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "R")) "being" ($#m2_funct_2 :::"Rule":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v5_fomodel4 :::"isotone"::: ) ) "iff" (Bool "for" (Set (Var "Seqts1")) "," (Set (Var "Seqts2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "st" (Bool (Bool (Set (Var "Seqts1")) ($#r1_tarski :::"c="::: ) (Set (Var "Seqts2")))) "holds" (Bool (Set (Set (Var "R")) ($#k3_funct_2 :::"."::: ) (Set (Var "Seqts1"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "R")) ($#k3_funct_2 :::"."::: ) (Set (Var "Seqts2"))))) ")" ))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v5_fomodel4 :::"isotone"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); attr "D" is :::"isotone"::: means :: FOMODEL4:def 10 (Bool "for" (Set (Var "Seqts1")) "," (Set (Var "Seqts2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "Seqts1")) ($#r1_tarski :::"c="::: ) (Set (Var "Seqts2"))) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) "D") & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "Seqts1"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "Seqts2")))) ")" )))); end; :: deftheorem defines :::"isotone"::: FOMODEL4:def 10 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "D")) "is" ($#v6_fomodel4 :::"isotone"::: ) ) "iff" (Bool "for" (Set (Var "Seqts1")) "," (Set (Var "Seqts2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "Seqts1")) ($#r1_tarski :::"c="::: ) (Set (Var "Seqts2"))) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "Seqts1"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "Seqts2")))) ")" )))) ")" ))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "M" be ($#v5_fomodel4 :::"isotone"::: ) ($#m2_funct_2 :::"Rule":::) "of" (Set (Const "S")); cluster (Set ($#k1_tarski :::"{"::: ) "M" ($#k1_tarski :::"}"::: ) ) -> ($#v6_fomodel4 :::"isotone"::: ) for ($#m1_subset_1 :::"RuleSet":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#v4_funct_1 :::"functional"::: ) ($#v6_fomodel4 :::"isotone"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ")" )); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "Seqts" be ($#m1_hidden :::"set"::: ) ; attr "Seqts" is "D" :::"-derivable"::: means :: FOMODEL4:def 11 (Bool "Seqts" "is" (Set ($#k1_xboole_0 :::"{}"::: ) ) "," "D" ($#v3_fomodel4 :::"-derivable"::: ) ); end; :: deftheorem defines :::"-derivable"::: FOMODEL4:def 11 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "Seqts")) "is" (Set (Var "D")) ($#v7_fomodel4 :::"-derivable"::: ) ) "iff" (Bool (Set (Var "Seqts")) "is" (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set (Var "D")) ($#v3_fomodel4 :::"-derivable"::: ) ) ")" )))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); cluster "D" ($#v7_fomodel4 :::"-derivable"::: ) -> (Set ($#k1_xboole_0 :::"{}"::: ) ) "," "D" ($#v3_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k1_xboole_0 :::"{}"::: ) ) "," "D" ($#v3_fomodel4 :::"-derivable"::: ) -> "D" ($#v7_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "Seqts" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_hidden :::"set"::: ) ; cluster "Seqts" "," "D" ($#v3_fomodel4 :::"-derivable"::: ) -> "D" ($#v7_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X", "phi" be ($#m1_hidden :::"set"::: ) ; attr "phi" is "X" "," "D" :::"-provable"::: means :: FOMODEL4:def 12 (Bool "(" (Bool (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) "X" "," "phi" ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) "is" "D" ($#v7_fomodel4 :::"-derivable"::: ) ) "or" (Bool "ex" (Set (Var "seqt")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_tarski :::"c="::: ) "X") & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) "phi") & (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "seqt")) ($#k1_tarski :::"}"::: ) ) "is" "D" ($#v7_fomodel4 :::"-derivable"::: ) ) ")" )) ")" ); end; :: deftheorem defines :::"-provable"::: FOMODEL4:def 12 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "," (Set (Var "phi")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "phi")) "is" (Set (Var "X")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) ) "iff" (Bool "(" (Bool (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "X")) "," (Set (Var "phi")) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) "is" (Set (Var "D")) ($#v7_fomodel4 :::"-derivable"::: ) ) "or" (Bool "ex" (Set (Var "seqt")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "phi"))) & (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "seqt")) ($#k1_tarski :::"}"::: ) ) "is" (Set (Var "D")) ($#v7_fomodel4 :::"-derivable"::: ) ) ")" )) ")" ) ")" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X", "x" be ($#m1_hidden :::"set"::: ) ; redefine attr "x" is "X" "," "D" :::"-provable"::: means :: FOMODEL4:def 13 (Bool "ex" (Set (Var "seqt")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_tarski :::"c="::: ) "X") & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) "x") & (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "seqt")) ($#k1_tarski :::"}"::: ) ) "is" "D" ($#v7_fomodel4 :::"-derivable"::: ) ) ")" )); end; :: deftheorem defines :::"-provable"::: FOMODEL4:def 13 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" (Set (Var "X")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) ) "iff" (Bool "ex" (Set (Var "seqt")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "seqt")) ($#k1_tarski :::"}"::: ) ) "is" (Set (Var "D")) ($#v7_fomodel4 :::"-derivable"::: ) ) ")" )) ")" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "R" be ($#m2_funct_2 :::"Rule":::) "of" (Set (Const "S")); attr "R" is "D" :::"-macro"::: means :: FOMODEL4:def 14 (Bool "for" (Set (Var "Seqs")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool (Set "R" ($#k3_funct_2 :::"."::: ) (Set (Var "Seqs"))) "is" (Set (Var "Seqs")) "," "D" ($#v3_fomodel4 :::"-derivable"::: ) )); end; :: deftheorem defines :::"-macro"::: FOMODEL4:def 14 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "R")) "being" ($#m2_funct_2 :::"Rule":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "R")) "is" (Set (Var "D")) ($#v9_fomodel4 :::"-macro"::: ) ) "iff" (Bool "for" (Set (Var "Seqs")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool (Set (Set (Var "R")) ($#k3_funct_2 :::"."::: ) (Set (Var "Seqs"))) "is" (Set (Var "Seqs")) "," (Set (Var "D")) ($#v3_fomodel4 :::"-derivable"::: ) )) ")" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "Phi" be ($#m1_hidden :::"set"::: ) ; func "(" "Phi" "," "D" ")" :::"-termEq"::: -> ($#m1_hidden :::"set"::: ) equals :: FOMODEL4:def 15 "{" (Set ($#k1_domain_1 :::"["::: ) (Set (Var "t1")) "," (Set (Var "t2")) ($#k1_domain_1 :::"]"::: ) ) where t1, t2 "is" ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" "S" : (Bool (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t2"))) "is" "Phi" "," "D" ($#v8_fomodel4 :::"-provable"::: ) ) "}" ; end; :: deftheorem defines :::"-termEq"::: FOMODEL4:def 15 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "Phi")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set "(" (Set (Var "Phi")) "," (Set (Var "D")) ")" ($#k5_fomodel4 :::"-termEq"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set ($#k1_domain_1 :::"["::: ) (Set (Var "t1")) "," (Set (Var "t2")) ($#k1_domain_1 :::"]"::: ) ) where t1, t2 "is" ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) : (Bool (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t2"))) "is" (Set (Var "Phi")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) ) "}" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "Phi" be ($#m1_hidden :::"set"::: ) ; attr "Phi" is "D" :::"-expanded"::: means :: FOMODEL4:def 16 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) "is" "Phi" "," "D" ($#v8_fomodel4 :::"-provable"::: ) )) "holds" (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ($#r1_tarski :::"c="::: ) "Phi")); end; :: deftheorem defines :::"-expanded"::: FOMODEL4:def 16 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "Phi")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "Phi")) "is" (Set (Var "D")) ($#v10_fomodel4 :::"-expanded"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) "is" (Set (Var "Phi")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) )) "holds" (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "Phi")))) ")" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "x" be ($#m1_hidden :::"set"::: ) ; attr "x" is "S" :::"-null"::: means :: FOMODEL4:def 17 (Bool verum); end; :: deftheorem defines :::"-null"::: FOMODEL4:def 17 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" (Set (Var "S")) ($#v11_fomodel4 :::"-null"::: ) ) "iff" (Bool verum) ")" ))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "Phi" be ($#m1_hidden :::"set"::: ) ; :: original: :::"-termEq"::: redefine func "(" "Phi" "," "D" ")" :::"-termEq"::: -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) "S" ")" ); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "Phi1", "Phi2" be ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Const "S")) ")" ); cluster (Set ($#k4_tarski :::"["::: ) (Set "(" "Phi1" ($#k4_subset_1 :::"\/"::: ) "Phi2" ")" ) "," "phi" ($#k4_tarski :::"]"::: ) ) -> "S" ($#v1_fomodel4 :::"-sequent-like"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "x" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_hidden :::"set"::: ) ; let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); :: original: :::"["::: redefine func :::"[":::"x" "," "phi":::"]"::: -> ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k3_xboole_0 :::"/\"::: ) "S") -> "S" ($#v2_fomodel4 :::"-sequents-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster "S" ($#v11_fomodel4 :::"-null"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster "S" ($#v1_fomodel4 :::"-sequent-like"::: ) -> "S" ($#v11_fomodel4 :::"-null"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster -> "S" ($#v11_fomodel4 :::"-null"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ); end; registrationlet "m" be ($#m1_hidden :::"Nat":::); let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set (Set "(" "(" "m" "," "D" ")" ($#k3_fomodel4 :::"-derivables"::: ) ")" ) ($#k1_funct_1 :::"."::: ) "X") -> "S" ($#v2_fomodel4 :::"-sequents-like"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "Y" be ($#m1_hidden :::"set"::: ) ; let "X" be (Set (Const "S")) ($#v2_fomodel4 :::"-sequents-like"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "X" ($#k3_xboole_0 :::"/\"::: ) "Y") -> "S" ($#v2_fomodel4 :::"-sequents-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "m" be ($#m1_hidden :::"Nat":::); let "X" be ($#m1_hidden :::"set"::: ) ; cluster "m" "," "X" "," "D" ($#v4_fomodel4 :::"-derivable"::: ) -> "S" ($#v1_fomodel4 :::"-sequent-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "Phi1", "Phi2" be ($#m1_hidden :::"set"::: ) ; cluster (Set "Phi1" ($#k6_subset_1 :::"\"::: ) "Phi2") "," "D" ($#v8_fomodel4 :::"-provable"::: ) -> "Phi1" "," "D" ($#v8_fomodel4 :::"-provable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "Phi1", "Phi2" be ($#m1_hidden :::"set"::: ) ; cluster (Set "Phi1" ($#k6_subset_1 :::"\"::: ) "Phi2") "," "D" ($#v8_fomodel4 :::"-provable"::: ) -> (Set "Phi1" ($#k2_xboole_0 :::"\/"::: ) "Phi2") "," "D" ($#v8_fomodel4 :::"-provable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "Phi1", "Phi2" be ($#m1_hidden :::"set"::: ) ; cluster (Set "Phi1" ($#k3_xboole_0 :::"/\"::: ) "Phi2") "," "D" ($#v8_fomodel4 :::"-provable"::: ) -> "Phi1" "," "D" ($#v8_fomodel4 :::"-provable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be ($#m1_hidden :::"set"::: ) ; let "x" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "X")); cluster "x" "," "D" ($#v8_fomodel4 :::"-provable"::: ) -> "X" "," "D" ($#v8_fomodel4 :::"-provable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "premises" be ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Const "S")) ")" ); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) "premises" "," "phi" ($#k4_tarski :::"]"::: ) ) -> "S" ($#v1_fomodel4 :::"-sequent-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi1", "phi2" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set ($#k6_domain_1 :::"{"::: ) "phi1" ($#k6_domain_1 :::"}"::: ) ) "," "phi2" ($#k4_tarski :::"]"::: ) ) -> "S" ($#v1_fomodel4 :::"-sequent-like"::: ) for ($#m1_hidden :::"set"::: ) ; let "phi3" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set ($#k7_domain_1 :::"{"::: ) "phi1" "," "phi2" ($#k7_domain_1 :::"}"::: ) ) "," "phi3" ($#k4_tarski :::"]"::: ) ) -> "S" ($#v1_fomodel4 :::"-sequent-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi1", "phi2" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "Phi" be ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Const "S")) ")" ); cluster (Set ($#k4_tarski :::"["::: ) (Set "(" "Phi" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) "phi1" ($#k6_domain_1 :::"}"::: ) ) ")" ) "," "phi2" ($#k4_tarski :::"]"::: ) ) -> "S" ($#v1_fomodel4 :::"-sequent-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); cluster ($#v4_funct_1 :::"functional"::: ) bbbadV5_FINSET_1() ($#v3_finseq_1 :::"FinSequence-membered"::: ) ($#v1_setfam_1 :::"with_non-empty_elements"::: ) "D" ($#v10_fomodel4 :::"-expanded"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S" ")" )); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); cluster "D" ($#v10_fomodel4 :::"-expanded"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "Seqts" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "seqt" be (Set (Const "S")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) ; pred "seqt" :::"Rule0"::: "Seqts" means :: FOMODEL4:def 18 (Bool (Set "seqt" ($#k2_xtuple_0 :::"`2"::: ) ) ($#r2_hidden :::"in"::: ) (Set "seqt" ($#k1_xtuple_0 :::"`1"::: ) )); pred "seqt" :::"Rule1"::: "Seqts" means :: FOMODEL4:def 19 (Bool "ex" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) "Seqts") & (Bool (Set (Set (Var "y")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_tarski :::"c="::: ) (Set "seqt" ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set "seqt" ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k2_xtuple_0 :::"`2"::: ) )) ")" )); pred "seqt" :::"Rule2"::: "Seqts" means :: FOMODEL4:def 20 (Bool "(" (Bool (Set "seqt" ($#k1_xtuple_0 :::"`1"::: ) ) "is" ($#v1_xboole_0 :::"empty"::: ) ) & (Bool "ex" (Set (Var "t")) "being" ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" "S" "st" (Bool (Set "seqt" ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t"))))) ")" ); pred "seqt" :::"Rule3a"::: "Seqts" means :: FOMODEL4:def 21 (Bool "ex" (Set (Var "t")) "," (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" "S"(Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "Seqts") & (Bool (Set "seqt" ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k1_xtuple_0 :::"`1"::: ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t2")) ")" ) ($#k6_domain_1 :::"}"::: ) ))) & (Bool (Set (Set (Var "x")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t1")))) & (Bool (Set "seqt" ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t2")))) ")" ))); pred "seqt" :::"Rule3b"::: "Seqts" means :: FOMODEL4:def 22 (Bool "ex" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" "S" "st" (Bool "(" (Bool (Set "seqt" ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t2")) ")" ) ($#k6_domain_1 :::"}"::: ) )) & (Bool (Set "seqt" ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t2")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t1")))) ")" )); pred "seqt" :::"Rule3d"::: "Seqts" means :: FOMODEL4:def 23 (Bool "ex" (Set (Var "s")) "being" ($#v5_fomodel1 :::"low-compounding"::: ) ($#m1_subset_1 :::"Element":::) "of" "S"(Bool "ex" (Set (Var "T")) "," (Set (Var "U")) "being" (Set ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "b1")) ")" )) ($#v3_card_1 :::"-element"::: ) ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) "S" ")" ) ($#k2_fomodel0 :::"*"::: ) ) "st" (Bool "(" (Bool (Set (Var "s")) "is" ($#v6_fomodel1 :::"operational"::: ) ) & (Bool (Set "seqt" ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "TT")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" ) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "UU")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" ) ")" ) where j "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "s")) ")" ) ")" )), TT, UU "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "s")) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) : (Bool "(" (Bool (Set (Var "TT")) ($#r1_hidden :::"="::: ) (Set (Var "T"))) & (Bool (Set (Var "UU")) ($#r1_hidden :::"="::: ) (Set (Var "U"))) ")" ) "}" ) & (Bool (Set "seqt" ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "s")) ($#k1_fomodel3 :::"-compound"::: ) (Set (Var "T")) ")" ) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "s")) ($#k1_fomodel3 :::"-compound"::: ) (Set (Var "U")) ")" ))) ")" ))); pred "seqt" :::"Rule3e"::: "Seqts" means :: FOMODEL4:def 24 (Bool "ex" (Set (Var "s")) "being" ($#v7_fomodel1 :::"relational"::: ) ($#m1_subset_1 :::"Element":::) "of" "S"(Bool "ex" (Set (Var "T")) "," (Set (Var "U")) "being" (Set ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "b1")) ")" )) ($#v3_card_1 :::"-element"::: ) ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) "S" ")" ) ($#k2_fomodel0 :::"*"::: ) ) "st" (Bool "(" (Bool (Set "seqt" ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set (Var "s")) ($#k1_fomodel3 :::"-compound"::: ) (Set (Var "T")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) "{" (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "TT")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" ) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "UU")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" ) ")" ) where j "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "s")) ")" ) ")" )), TT, UU "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "s")) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) : (Bool "(" (Bool (Set (Var "TT")) ($#r1_hidden :::"="::: ) (Set (Var "T"))) & (Bool (Set (Var "UU")) ($#r1_hidden :::"="::: ) (Set (Var "U"))) ")" ) "}" )) & (Bool (Set "seqt" ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_fomodel3 :::"-compound"::: ) (Set (Var "U")))) ")" ))); pred "seqt" :::"Rule4"::: "Seqts" means :: FOMODEL4:def 25 (Bool "ex" (Set (Var "l")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" "S"(Bool "ex" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S"(Bool "ex" (Set (Var "t")) "being" ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" "S" "st" (Bool "(" (Bool (Set "seqt" ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" "(" (Set (Var "l")) "," (Set (Var "t")) ")" ($#k37_fomodel3 :::"SubstIn"::: ) (Set (Var "phi")) ")" ) ($#k6_domain_1 :::"}"::: ) )) & (Bool (Set "seqt" ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k31_fomodel2 :::"<*"::: ) (Set (Var "l")) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi")))) ")" )))); pred "seqt" :::"Rule5"::: "Seqts" means :: FOMODEL4:def 26 (Bool "ex" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" "S"(Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set "seqt" ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set (Var "v1")) ($#k31_fomodel2 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "p")) ")" ) ($#k1_tarski :::"}"::: ) ))) & (Bool (Set (Var "v2")) "is" (Set (Set "(" (Set (Var "x")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" "seqt" ($#k2_xtuple_0 :::"`2"::: ) ")" ) ($#k1_tarski :::"}"::: ) )) ($#v7_fomodel2 :::"-absent"::: ) ) & (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "x")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" (Set (Var "v1")) ($#k20_fomodel0 :::"SubstWith"::: ) (Set (Var "v2")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p")) ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) "," (Set "(" "seqt" ($#k2_xtuple_0 :::"`2"::: ) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "Seqts") ")" )))); pred "seqt" :::"Rule6"::: "Seqts" means :: FOMODEL4:def 27 (Bool "ex" (Set (Var "y1")) "," (Set (Var "y2")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "phi1")) "," (Set (Var "phi2")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S" "st" (Bool "(" (Bool (Set (Var "y1")) ($#r2_hidden :::"in"::: ) "Seqts") & (Bool (Set (Var "y2")) ($#r2_hidden :::"in"::: ) "Seqts") & (Bool (Set (Set (Var "y1")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "y2")) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set (Var "y2")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set "seqt" ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set (Var "y1")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")))) & (Bool (Set (Set (Var "y2")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")))) & (Bool (Set "seqt" ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")))) ")" ))); pred "seqt" :::"Rule7"::: "Seqts" means :: FOMODEL4:def 28 (Bool "ex" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "phi1")) "," (Set (Var "phi2")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S" "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) "Seqts") & (Bool (Set (Set (Var "y")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set "seqt" ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set (Var "y")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")))) & (Bool (Set "seqt" ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")))) ")" ))); pred "seqt" :::"Rule8"::: "Seqts" means :: FOMODEL4:def 29 (Bool "ex" (Set (Var "y1")) "," (Set (Var "y2")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "phi")) "," (Set (Var "phi1")) "," (Set (Var "phi2")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S" "st" (Bool "(" (Bool (Set (Var "y1")) ($#r2_hidden :::"in"::: ) "Seqts") & (Bool (Set (Var "y2")) ($#r2_hidden :::"in"::: ) "Seqts") & (Bool (Set (Set (Var "y1")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "y2")) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set (Var "y1")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "phi1"))) & (Bool (Set (Set (Var "y2")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")))) & (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "phi")) ($#k6_domain_1 :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" "seqt" ($#k1_xtuple_0 :::"`1"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y1")) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set "seqt" ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi")))) ")" ))); pred "seqt" :::"Rule9"::: "Seqts" means :: FOMODEL4:def 30 (Bool "ex" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S" "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) "Seqts") & (Bool (Set "seqt" ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "phi"))) & (Bool (Set (Set (Var "y")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set "seqt" ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set (Var "y")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k38_fomodel2 :::"xnot"::: ) (Set "(" ($#k38_fomodel2 :::"xnot"::: ) (Set (Var "phi")) ")" ))) ")" ))); end; :: deftheorem defines :::"Rule0"::: FOMODEL4:def 18 : (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "seqt")) "being" (Set (Var "b2")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) ($#r1_fomodel4 :::"Rule0"::: ) (Set (Var "Seqts"))) "iff" (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) )) ")" )))); :: deftheorem defines :::"Rule1"::: FOMODEL4:def 19 : (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "seqt")) "being" (Set (Var "b2")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) ($#r2_fomodel4 :::"Rule1"::: ) (Set (Var "Seqts"))) "iff" (Bool "ex" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Seqts"))) & (Bool (Set (Set (Var "y")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k2_xtuple_0 :::"`2"::: ) )) ")" )) ")" )))); :: deftheorem defines :::"Rule2"::: FOMODEL4:def 20 : (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "seqt")) "being" (Set (Var "b2")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) ($#r3_fomodel4 :::"Rule2"::: ) (Set (Var "Seqts"))) "iff" (Bool "(" (Bool (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) ) "is" ($#v1_xboole_0 :::"empty"::: ) ) & (Bool "ex" (Set (Var "t")) "being" ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t"))))) ")" ) ")" )))); :: deftheorem defines :::"Rule3a"::: FOMODEL4:def 21 : (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "seqt")) "being" (Set (Var "b2")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) ($#r4_fomodel4 :::"Rule3a"::: ) (Set (Var "Seqts"))) "iff" (Bool "ex" (Set (Var "t")) "," (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S"))(Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Seqts"))) & (Bool (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k1_xtuple_0 :::"`1"::: ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t2")) ")" ) ($#k6_domain_1 :::"}"::: ) ))) & (Bool (Set (Set (Var "x")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t1")))) & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t2")))) ")" ))) ")" )))); :: deftheorem defines :::"Rule3b"::: FOMODEL4:def 22 : (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "seqt")) "being" (Set (Var "b2")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) ($#r5_fomodel4 :::"Rule3b"::: ) (Set (Var "Seqts"))) "iff" (Bool "ex" (Set (Var "t1")) "," (Set (Var "t2")) "being" ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t2")) ")" ) ($#k6_domain_1 :::"}"::: ) )) & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t2")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "t1")))) ")" )) ")" )))); :: deftheorem defines :::"Rule3d"::: FOMODEL4:def 23 : (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "seqt")) "being" (Set (Var "b2")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) ($#r6_fomodel4 :::"Rule3d"::: ) (Set (Var "Seqts"))) "iff" (Bool "ex" (Set (Var "s")) "being" ($#v5_fomodel1 :::"low-compounding"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S"))(Bool "ex" (Set (Var "T")) "," (Set (Var "U")) "being" (Set ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "b4")) ")" )) ($#v3_card_1 :::"-element"::: ) ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) (Set (Var "S")) ")" ) ($#k2_fomodel0 :::"*"::: ) ) "st" (Bool "(" (Bool (Set (Var "s")) "is" ($#v6_fomodel1 :::"operational"::: ) ) & (Bool (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "TT")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" ) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "UU")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" ) ")" ) where j "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "s")) ")" ) ")" )), TT, UU "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "s")) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) : (Bool "(" (Bool (Set (Var "TT")) ($#r1_hidden :::"="::: ) (Set (Var "T"))) & (Bool (Set (Var "UU")) ($#r1_hidden :::"="::: ) (Set (Var "U"))) ")" ) "}" ) & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "s")) ($#k1_fomodel3 :::"-compound"::: ) (Set (Var "T")) ")" ) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "s")) ($#k1_fomodel3 :::"-compound"::: ) (Set (Var "U")) ")" ))) ")" ))) ")" )))); :: deftheorem defines :::"Rule3e"::: FOMODEL4:def 24 : (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "seqt")) "being" (Set (Var "b2")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) ($#r7_fomodel4 :::"Rule3e"::: ) (Set (Var "Seqts"))) "iff" (Bool "ex" (Set (Var "s")) "being" ($#v7_fomodel1 :::"relational"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S"))(Bool "ex" (Set (Var "T")) "," (Set (Var "U")) "being" (Set ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "b4")) ")" )) ($#v3_card_1 :::"-element"::: ) ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) (Set (Var "S")) ")" ) ($#k2_fomodel0 :::"*"::: ) ) "st" (Bool "(" (Bool (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set (Var "s")) ($#k1_fomodel3 :::"-compound"::: ) (Set (Var "T")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) "{" (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "TT")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" ) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "UU")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" ) ")" ) where j "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "s")) ")" ) ")" )), TT, UU "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "s")) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) : (Bool "(" (Bool (Set (Var "TT")) ($#r1_hidden :::"="::: ) (Set (Var "T"))) & (Bool (Set (Var "UU")) ($#r1_hidden :::"="::: ) (Set (Var "U"))) ")" ) "}" )) & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k1_fomodel3 :::"-compound"::: ) (Set (Var "U")))) ")" ))) ")" )))); :: deftheorem defines :::"Rule4"::: FOMODEL4:def 25 : (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "seqt")) "being" (Set (Var "b2")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) ($#r8_fomodel4 :::"Rule4"::: ) (Set (Var "Seqts"))) "iff" (Bool "ex" (Set (Var "l")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S"))(Bool "ex" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S"))(Bool "ex" (Set (Var "t")) "being" ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" "(" (Set (Var "l")) "," (Set (Var "t")) ")" ($#k37_fomodel3 :::"SubstIn"::: ) (Set (Var "phi")) ")" ) ($#k6_domain_1 :::"}"::: ) )) & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k31_fomodel2 :::"<*"::: ) (Set (Var "l")) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi")))) ")" )))) ")" )))); :: deftheorem defines :::"Rule5"::: FOMODEL4:def 26 : (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "seqt")) "being" (Set (Var "b2")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) ($#r9_fomodel4 :::"Rule5"::: ) (Set (Var "Seqts"))) "iff" (Bool "ex" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S"))(Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set (Var "v1")) ($#k31_fomodel2 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "p")) ")" ) ($#k1_tarski :::"}"::: ) ))) & (Bool (Set (Var "v2")) "is" (Set (Set "(" (Set (Var "x")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ")" ) ($#k1_tarski :::"}"::: ) )) ($#v7_fomodel2 :::"-absent"::: ) ) & (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "x")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" (Set (Var "v1")) ($#k20_fomodel0 :::"SubstWith"::: ) (Set (Var "v2")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p")) ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) "," (Set "(" (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Seqts"))) ")" )))) ")" )))); :: deftheorem defines :::"Rule6"::: FOMODEL4:def 27 : (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "seqt")) "being" (Set (Var "b2")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) ($#r10_fomodel4 :::"Rule6"::: ) (Set (Var "Seqts"))) "iff" (Bool "ex" (Set (Var "y1")) "," (Set (Var "y2")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "phi1")) "," (Set (Var "phi2")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "y1")) ($#r2_hidden :::"in"::: ) (Set (Var "Seqts"))) & (Bool (Set (Var "y2")) ($#r2_hidden :::"in"::: ) (Set (Var "Seqts"))) & (Bool (Set (Set (Var "y1")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "y2")) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set (Var "y2")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set (Var "y1")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")))) & (Bool (Set (Set (Var "y2")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")))) & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")))) ")" ))) ")" )))); :: deftheorem defines :::"Rule7"::: FOMODEL4:def 28 : (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "seqt")) "being" (Set (Var "b2")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) ($#r11_fomodel4 :::"Rule7"::: ) (Set (Var "Seqts"))) "iff" (Bool "ex" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "phi1")) "," (Set (Var "phi2")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Seqts"))) & (Bool (Set (Set (Var "y")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set (Var "y")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")))) & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")))) ")" ))) ")" )))); :: deftheorem defines :::"Rule8"::: FOMODEL4:def 29 : (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "seqt")) "being" (Set (Var "b2")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) ($#r12_fomodel4 :::"Rule8"::: ) (Set (Var "Seqts"))) "iff" (Bool "ex" (Set (Var "y1")) "," (Set (Var "y2")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "phi")) "," (Set (Var "phi1")) "," (Set (Var "phi2")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "y1")) ($#r2_hidden :::"in"::: ) (Set (Var "Seqts"))) & (Bool (Set (Var "y2")) ($#r2_hidden :::"in"::: ) (Set (Var "Seqts"))) & (Bool (Set (Set (Var "y1")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "y2")) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set (Var "y1")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "phi1"))) & (Bool (Set (Set (Var "y2")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")))) & (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "phi")) ($#k6_domain_1 :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "y1")) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi")))) ")" ))) ")" )))); :: deftheorem defines :::"Rule9"::: FOMODEL4:def 30 : (Bool "for" (Set (Var "Seqts")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "seqt")) "being" (Set (Var "b2")) ($#v11_fomodel4 :::"-null"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "seqt")) ($#r13_fomodel4 :::"Rule9"::: ) (Set (Var "Seqts"))) "iff" (Bool "ex" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Seqts"))) & (Bool (Set (Set (Var "seqt")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "phi"))) & (Bool (Set (Set (Var "y")) ($#k1_xtuple_0 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "seqt")) ($#k1_xtuple_0 :::"`1"::: ) )) & (Bool (Set (Set (Var "y")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k38_fomodel2 :::"xnot"::: ) (Set "(" ($#k38_fomodel2 :::"xnot"::: ) (Set (Var "phi")) ")" ))) ")" ))) ")" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); func :::"P#0"::: "S" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) means :: FOMODEL4:def 31 (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "seqt")) ($#r1_fomodel4 :::"Rule0"::: ) (Set (Var "Seqts"))) ")" ))); func :::"P#1"::: "S" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) means :: FOMODEL4:def 32 (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "seqt")) ($#r2_fomodel4 :::"Rule1"::: ) (Set (Var "Seqts"))) ")" ))); func :::"P#2"::: "S" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) means :: FOMODEL4:def 33 (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "seqt")) ($#r3_fomodel4 :::"Rule2"::: ) (Set (Var "Seqts"))) ")" ))); func :::"P#3a"::: "S" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) means :: FOMODEL4:def 34 (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "seqt")) ($#r4_fomodel4 :::"Rule3a"::: ) (Set (Var "Seqts"))) ")" ))); func :::"P#3b"::: "S" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) means :: FOMODEL4:def 35 (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "seqt")) ($#r5_fomodel4 :::"Rule3b"::: ) (Set (Var "Seqts"))) ")" ))); func :::"P#3d"::: "S" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) means :: FOMODEL4:def 36 (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "seqt")) ($#r6_fomodel4 :::"Rule3d"::: ) (Set (Var "Seqts"))) ")" ))); func :::"P#3e"::: "S" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) means :: FOMODEL4:def 37 (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "seqt")) ($#r7_fomodel4 :::"Rule3e"::: ) (Set (Var "Seqts"))) ")" ))); func :::"P#4"::: "S" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) means :: FOMODEL4:def 38 (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "seqt")) ($#r8_fomodel4 :::"Rule4"::: ) (Set (Var "Seqts"))) ")" ))); func :::"P#5"::: "S" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) means :: FOMODEL4:def 39 (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "seqt")) ($#r9_fomodel4 :::"Rule5"::: ) (Set (Var "Seqts"))) ")" ))); func :::"P#6"::: "S" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) means :: FOMODEL4:def 40 (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "seqt")) ($#r10_fomodel4 :::"Rule6"::: ) (Set (Var "Seqts"))) ")" ))); func :::"P#7"::: "S" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) means :: FOMODEL4:def 41 (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "seqt")) ($#r11_fomodel4 :::"Rule7"::: ) (Set (Var "Seqts"))) ")" ))); func :::"P#8"::: "S" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) means :: FOMODEL4:def 42 (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "seqt")) ($#r12_fomodel4 :::"Rule8"::: ) (Set (Var "Seqts"))) ")" ))); func :::"P#9"::: "S" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) means :: FOMODEL4:def 43 (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "seqt")) ($#r13_fomodel4 :::"Rule9"::: ) (Set (Var "Seqts"))) ")" ))); end; :: deftheorem defines :::"P#0"::: FOMODEL4:def 31 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k8_fomodel4 :::"P#0"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "seqt")) ($#r1_fomodel4 :::"Rule0"::: ) (Set (Var "Seqts"))) ")" ))) ")" ))); :: deftheorem defines :::"P#1"::: FOMODEL4:def 32 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k9_fomodel4 :::"P#1"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "seqt")) ($#r2_fomodel4 :::"Rule1"::: ) (Set (Var "Seqts"))) ")" ))) ")" ))); :: deftheorem defines :::"P#2"::: FOMODEL4:def 33 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k10_fomodel4 :::"P#2"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "seqt")) ($#r3_fomodel4 :::"Rule2"::: ) (Set (Var "Seqts"))) ")" ))) ")" ))); :: deftheorem defines :::"P#3a"::: FOMODEL4:def 34 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k11_fomodel4 :::"P#3a"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "seqt")) ($#r4_fomodel4 :::"Rule3a"::: ) (Set (Var "Seqts"))) ")" ))) ")" ))); :: deftheorem defines :::"P#3b"::: FOMODEL4:def 35 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k12_fomodel4 :::"P#3b"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "seqt")) ($#r5_fomodel4 :::"Rule3b"::: ) (Set (Var "Seqts"))) ")" ))) ")" ))); :: deftheorem defines :::"P#3d"::: FOMODEL4:def 36 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k13_fomodel4 :::"P#3d"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "seqt")) ($#r6_fomodel4 :::"Rule3d"::: ) (Set (Var "Seqts"))) ")" ))) ")" ))); :: deftheorem defines :::"P#3e"::: FOMODEL4:def 37 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k14_fomodel4 :::"P#3e"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "seqt")) ($#r7_fomodel4 :::"Rule3e"::: ) (Set (Var "Seqts"))) ")" ))) ")" ))); :: deftheorem defines :::"P#4"::: FOMODEL4:def 38 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k15_fomodel4 :::"P#4"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "seqt")) ($#r8_fomodel4 :::"Rule4"::: ) (Set (Var "Seqts"))) ")" ))) ")" ))); :: deftheorem defines :::"P#5"::: FOMODEL4:def 39 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k16_fomodel4 :::"P#5"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "seqt")) ($#r9_fomodel4 :::"Rule5"::: ) (Set (Var "Seqts"))) ")" ))) ")" ))); :: deftheorem defines :::"P#6"::: FOMODEL4:def 40 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k17_fomodel4 :::"P#6"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "seqt")) ($#r10_fomodel4 :::"Rule6"::: ) (Set (Var "Seqts"))) ")" ))) ")" ))); :: deftheorem defines :::"P#7"::: FOMODEL4:def 41 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k18_fomodel4 :::"P#7"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "seqt")) ($#r11_fomodel4 :::"Rule7"::: ) (Set (Var "Seqts"))) ")" ))) ")" ))); :: deftheorem defines :::"P#8"::: FOMODEL4:def 42 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k19_fomodel4 :::"P#8"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "seqt")) ($#r12_fomodel4 :::"Rule8"::: ) (Set (Var "Seqts"))) ")" ))) ")" ))); :: deftheorem defines :::"P#9"::: FOMODEL4:def 43 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k20_fomodel4 :::"P#9"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "Seqts")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )) (Bool "for" (Set (Var "seqt")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "Seqts")) "," (Set (Var "seqt")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "seqt")) ($#r13_fomodel4 :::"Rule9"::: ) (Set (Var "Seqts"))) ")" ))) ")" ))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Const "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Const "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ); func :::"FuncRule"::: "R" -> ($#m2_funct_2 :::"Rule":::) "of" "S" means :: FOMODEL4:def 44 (Bool "for" (Set (Var "inseqs")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "inseqs")) ($#r2_hidden :::"in"::: ) (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "inseqs"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set "S" ($#k1_fomodel4 :::"-sequents"::: ) ) : (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "inseqs")) "," (Set (Var "x")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "R") "}" )); end; :: deftheorem defines :::"FuncRule"::: FOMODEL4:def 44 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) (Bool "for" (Set (Var "b3")) "being" ($#m2_funct_2 :::"Rule":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set (Var "R")))) "iff" (Bool "for" (Set (Var "inseqs")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "inseqs")) ($#r2_hidden :::"in"::: ) (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" )))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "inseqs"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ) : (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "inseqs")) "," (Set (Var "x")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) "}" )) ")" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); func :::"R#0"::: "S" -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 45 (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k8_fomodel4 :::"P#0"::: ) "S" ")" )); func :::"R#1"::: "S" -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 46 (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k9_fomodel4 :::"P#1"::: ) "S" ")" )); func :::"R#2"::: "S" -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 47 (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k10_fomodel4 :::"P#2"::: ) "S" ")" )); func :::"R#3a"::: "S" -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 48 (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k11_fomodel4 :::"P#3a"::: ) "S" ")" )); func :::"R#3b"::: "S" -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 49 (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k12_fomodel4 :::"P#3b"::: ) "S" ")" )); func :::"R#3d"::: "S" -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 50 (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k13_fomodel4 :::"P#3d"::: ) "S" ")" )); func :::"R#3e"::: "S" -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 51 (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k14_fomodel4 :::"P#3e"::: ) "S" ")" )); func :::"R#4"::: "S" -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 52 (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k15_fomodel4 :::"P#4"::: ) "S" ")" )); func :::"R#5"::: "S" -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 53 (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k16_fomodel4 :::"P#5"::: ) "S" ")" )); func :::"R#6"::: "S" -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 54 (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k17_fomodel4 :::"P#6"::: ) "S" ")" )); func :::"R#7"::: "S" -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 55 (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k18_fomodel4 :::"P#7"::: ) "S" ")" )); func :::"R#8"::: "S" -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 56 (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k19_fomodel4 :::"P#8"::: ) "S" ")" )); func :::"R#9"::: "S" -> ($#m2_funct_2 :::"Rule":::) "of" "S" equals :: FOMODEL4:def 57 (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k20_fomodel4 :::"P#9"::: ) "S" ")" )); end; :: deftheorem defines :::"R#0"::: FOMODEL4:def 45 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k22_fomodel4 :::"R#0"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k8_fomodel4 :::"P#0"::: ) (Set (Var "S")) ")" )))); :: deftheorem defines :::"R#1"::: FOMODEL4:def 46 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k23_fomodel4 :::"R#1"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k9_fomodel4 :::"P#1"::: ) (Set (Var "S")) ")" )))); :: deftheorem defines :::"R#2"::: FOMODEL4:def 47 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k24_fomodel4 :::"R#2"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k10_fomodel4 :::"P#2"::: ) (Set (Var "S")) ")" )))); :: deftheorem defines :::"R#3a"::: FOMODEL4:def 48 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k25_fomodel4 :::"R#3a"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k11_fomodel4 :::"P#3a"::: ) (Set (Var "S")) ")" )))); :: deftheorem defines :::"R#3b"::: FOMODEL4:def 49 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k26_fomodel4 :::"R#3b"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k12_fomodel4 :::"P#3b"::: ) (Set (Var "S")) ")" )))); :: deftheorem defines :::"R#3d"::: FOMODEL4:def 50 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k27_fomodel4 :::"R#3d"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k13_fomodel4 :::"P#3d"::: ) (Set (Var "S")) ")" )))); :: deftheorem defines :::"R#3e"::: FOMODEL4:def 51 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k28_fomodel4 :::"R#3e"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k14_fomodel4 :::"P#3e"::: ) (Set (Var "S")) ")" )))); :: deftheorem defines :::"R#4"::: FOMODEL4:def 52 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k29_fomodel4 :::"R#4"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k15_fomodel4 :::"P#4"::: ) (Set (Var "S")) ")" )))); :: deftheorem defines :::"R#5"::: FOMODEL4:def 53 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k30_fomodel4 :::"R#5"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k16_fomodel4 :::"P#5"::: ) (Set (Var "S")) ")" )))); :: deftheorem defines :::"R#6"::: FOMODEL4:def 54 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k31_fomodel4 :::"R#6"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k17_fomodel4 :::"P#6"::: ) (Set (Var "S")) ")" )))); :: deftheorem defines :::"R#7"::: FOMODEL4:def 55 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k32_fomodel4 :::"R#7"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k18_fomodel4 :::"P#7"::: ) (Set (Var "S")) ")" )))); :: deftheorem defines :::"R#8"::: FOMODEL4:def 56 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k33_fomodel4 :::"R#8"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k19_fomodel4 :::"P#8"::: ) (Set (Var "S")) ")" )))); :: deftheorem defines :::"R#9"::: FOMODEL4:def 57 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k34_fomodel4 :::"R#9"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel4 :::"FuncRule"::: ) (Set "(" ($#k20_fomodel4 :::"P#9"::: ) (Set (Var "S")) ")" )))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "t" be ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "t" ")" ) ($#k30_fomodel2 :::"^"::: ) "t" ")" ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) -> (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k24_fomodel4 :::"R#2"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v7_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k24_fomodel4 :::"R#2"::: ) "S") -> ($#v5_fomodel4 :::"isotone"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k26_fomodel4 :::"R#3b"::: ) "S") -> ($#v5_fomodel4 :::"isotone"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "t", "t1", "t2" be ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "premises" be ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Const "S")) ")" ); cluster (Set ($#k4_tarski :::"["::: ) (Set "(" "premises" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "t1" ")" ) ($#k30_fomodel2 :::"^"::: ) "t2" ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) "," (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "t" ")" ) ($#k30_fomodel2 :::"^"::: ) "t2" ")" ) ($#k4_tarski :::"]"::: ) ) -> (Num 1) "," (Set ($#k4_lang1 :::"{"::: ) (Set ($#k1_domain_1 :::"["::: ) "premises" "," (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "t" ")" ) ($#k30_fomodel2 :::"^"::: ) "t1" ")" ) ($#k1_domain_1 :::"]"::: ) ) ($#k4_lang1 :::"}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k25_fomodel4 :::"R#3a"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "t", "t1", "t2" be ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set ($#k7_domain_1 :::"{"::: ) "phi" "," (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "t1" ")" ) ($#k30_fomodel2 :::"^"::: ) "t2" ")" ) ($#k7_domain_1 :::"}"::: ) ) "," (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "t" ")" ) ($#k30_fomodel2 :::"^"::: ) "t2" ")" ) ($#k4_tarski :::"]"::: ) ) -> (Num 1) "," (Set ($#k4_lang1 :::"{"::: ) (Set ($#k1_domain_1 :::"["::: ) (Set ($#k6_domain_1 :::"{"::: ) "phi" ($#k6_domain_1 :::"}"::: ) ) "," (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "t" ")" ) ($#k30_fomodel2 :::"^"::: ) "t1" ")" ) ($#k1_domain_1 :::"]"::: ) ) ($#k4_lang1 :::"}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k25_fomodel4 :::"R#3a"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "Phi" be ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Const "S")) ")" ); cluster (Set ($#k4_tarski :::"["::: ) (Set "(" "Phi" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) "phi" ($#k6_domain_1 :::"}"::: ) ) ")" ) "," "phi" ($#k4_tarski :::"]"::: ) ) -> (Num 1) "," (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k22_fomodel4 :::"R#0"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi1", "phi2" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set ($#k7_domain_1 :::"{"::: ) "phi1" "," "phi2" ($#k7_domain_1 :::"}"::: ) ) "," "phi1" ($#k4_tarski :::"]"::: ) ) -> (Num 1) "," (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k22_fomodel4 :::"R#0"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set ($#k6_domain_1 :::"{"::: ) "phi" ($#k6_domain_1 :::"}"::: ) ) "," "phi" ($#k4_tarski :::"]"::: ) ) -> (Num 1) "," (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k22_fomodel4 :::"R#0"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_domain_1 :::"["::: ) (Set ($#k6_domain_1 :::"{"::: ) "phi" ($#k6_domain_1 :::"}"::: ) ) "," "phi" ($#k1_domain_1 :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) -> (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k22_fomodel4 :::"R#0"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v3_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k22_fomodel4 :::"R#0"::: ) "S") -> ($#v5_fomodel4 :::"isotone"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; cluster (Set ($#k25_fomodel4 :::"R#3a"::: ) "S") -> ($#v5_fomodel4 :::"isotone"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; cluster (Set ($#k27_fomodel4 :::"R#3d"::: ) "S") -> ($#v5_fomodel4 :::"isotone"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; cluster (Set ($#k28_fomodel4 :::"R#3e"::: ) "S") -> ($#v5_fomodel4 :::"isotone"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; let "K1", "K2" be ($#v6_fomodel4 :::"isotone"::: ) ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); cluster (Set "K1" ($#k2_xboole_0 :::"\/"::: ) "K2") -> ($#v6_fomodel4 :::"isotone"::: ) for ($#m1_subset_1 :::"RuleSet":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "t1", "t2" be ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "t1" ")" ) ($#k7_finseq_1 :::"^"::: ) "t2") -> (Set ($#k6_numbers :::"0"::: ) ) ($#v3_fomodel2 :::"-wff"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); let "T", "U" be (Set (Const "m")) ($#v3_card_1 :::"-element"::: ) ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) (Set (Const "S")) ")" ) ($#k2_fomodel0 :::"*"::: ) ); func :::"PairWiseEq"::: "(" "T" "," "U" ")" -> ($#m1_hidden :::"set"::: ) equals :: FOMODEL4:def 58 "{" (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "TT")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" ) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "UU")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" ) ")" ) where j "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k2_finseq_1 :::"Seg"::: ) "m"), TT, UU "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) "m" ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) : (Bool "(" (Bool (Set (Var "TT")) ($#r1_hidden :::"="::: ) "T") & (Bool (Set (Var "UU")) ($#r1_hidden :::"="::: ) "U") ")" ) "}" ; end; :: deftheorem defines :::"PairWiseEq"::: FOMODEL4:def 58 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "T")) "," (Set (Var "U")) "being" (Set (Var "b2")) ($#v3_card_1 :::"-element"::: ) ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) (Set (Var "S")) ")" ) ($#k2_fomodel0 :::"*"::: ) ) "holds" (Bool (Set ($#k35_fomodel4 :::"PairWiseEq"::: ) "(" (Set (Var "T")) "," (Set (Var "U")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "TT")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" ) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set "(" (Set (Var "UU")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" ) ")" ) where j "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m"))), TT, UU "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) : (Bool "(" (Bool (Set (Var "TT")) ($#r1_hidden :::"="::: ) (Set (Var "T"))) & (Bool (Set (Var "UU")) ($#r1_hidden :::"="::: ) (Set (Var "U"))) ")" ) "}" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); let "T1", "T2" be (Set (Const "m")) ($#v3_card_1 :::"-element"::: ) ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) (Set (Const "S")) ")" ) ($#k2_fomodel0 :::"*"::: ) ); :: original: :::"PairWiseEq"::: redefine func :::"PairWiseEq"::: "(" "T1" "," "T2" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S" ")" ); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); let "T", "U" be (Set (Const "m")) ($#v3_card_1 :::"-element"::: ) ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) (Set (Const "S")) ")" ) ($#k2_fomodel0 :::"*"::: ) ); cluster (Set ($#k35_fomodel4 :::"PairWiseEq"::: ) "(" "T" "," "U" ")" ) -> ($#v1_finset_1 :::"finite"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "s" be ($#v7_fomodel1 :::"relational"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); let "T1", "T2" be (Set ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Const "s")) ")" )) ($#v3_card_1 :::"-element"::: ) ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) (Set (Const "S")) ")" ) ($#k2_fomodel0 :::"*"::: ) ); cluster (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set "(" ($#k36_fomodel4 :::"PairWiseEq"::: ) "(" "T1" "," "T2" ")" ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" "s" ($#k1_fomodel3 :::"-compound"::: ) "T1" ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) "," (Set "(" "s" ($#k1_fomodel3 :::"-compound"::: ) "T2" ")" ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) -> (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k28_fomodel4 :::"R#3e"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v3_fomodel4 :::"-derivable"::: ) ; end; definitionlet "m" be ($#m1_hidden :::"Nat":::); let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); attr "D" is "m" :::"-ranked"::: means :: FOMODEL4:def 59 (Bool "(" (Bool (Set ($#k22_fomodel4 :::"R#0"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k24_fomodel4 :::"R#2"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k25_fomodel4 :::"R#3a"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k26_fomodel4 :::"R#3b"::: ) "S") ($#r2_hidden :::"in"::: ) "D") ")" ) if (Bool "m" ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) (Bool "(" (Bool (Set ($#k22_fomodel4 :::"R#0"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k24_fomodel4 :::"R#2"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k25_fomodel4 :::"R#3a"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k26_fomodel4 :::"R#3b"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k27_fomodel4 :::"R#3d"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k28_fomodel4 :::"R#3e"::: ) "S") ($#r2_hidden :::"in"::: ) "D") ")" ) if (Bool "m" ($#r1_hidden :::"="::: ) (Num 1)) (Bool "(" (Bool (Set ($#k22_fomodel4 :::"R#0"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k23_fomodel4 :::"R#1"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k24_fomodel4 :::"R#2"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k25_fomodel4 :::"R#3a"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k26_fomodel4 :::"R#3b"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k27_fomodel4 :::"R#3d"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k28_fomodel4 :::"R#3e"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k29_fomodel4 :::"R#4"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k30_fomodel4 :::"R#5"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k31_fomodel4 :::"R#6"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k32_fomodel4 :::"R#7"::: ) "S") ($#r2_hidden :::"in"::: ) "D") & (Bool (Set ($#k33_fomodel4 :::"R#8"::: ) "S") ($#r2_hidden :::"in"::: ) "D") ")" ) if (Bool "m" ($#r1_hidden :::"="::: ) (Num 2)) otherwise (Bool "D" ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )); end; :: deftheorem defines :::"-ranked"::: FOMODEL4:def 59 : (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "D")) "is" (Set (Var "m")) ($#v12_fomodel4 :::"-ranked"::: ) ) "iff" (Bool "(" (Bool (Set ($#k22_fomodel4 :::"R#0"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k24_fomodel4 :::"R#2"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k25_fomodel4 :::"R#3a"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k26_fomodel4 :::"R#3b"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" ) ")" ) ")" & "(" (Bool (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 1))) "implies" (Bool "(" (Bool (Set (Var "D")) "is" (Set (Var "m")) ($#v12_fomodel4 :::"-ranked"::: ) ) "iff" (Bool "(" (Bool (Set ($#k22_fomodel4 :::"R#0"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k24_fomodel4 :::"R#2"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k25_fomodel4 :::"R#3a"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k26_fomodel4 :::"R#3b"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k27_fomodel4 :::"R#3d"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k28_fomodel4 :::"R#3e"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" ) ")" ) ")" & "(" (Bool (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 2))) "implies" (Bool "(" (Bool (Set (Var "D")) "is" (Set (Var "m")) ($#v12_fomodel4 :::"-ranked"::: ) ) "iff" (Bool "(" (Bool (Set ($#k22_fomodel4 :::"R#0"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k23_fomodel4 :::"R#1"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k24_fomodel4 :::"R#2"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k25_fomodel4 :::"R#3a"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k26_fomodel4 :::"R#3b"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k27_fomodel4 :::"R#3d"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k28_fomodel4 :::"R#3e"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k29_fomodel4 :::"R#4"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k30_fomodel4 :::"R#5"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k31_fomodel4 :::"R#6"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k32_fomodel4 :::"R#7"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k33_fomodel4 :::"R#8"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" ) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) & (Bool (Bool "not" (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 1))) & (Bool (Bool "not" (Set (Var "m")) ($#r1_hidden :::"="::: ) (Num 2)))) "implies" (Bool "(" (Bool (Set (Var "D")) "is" (Set (Var "m")) ($#v12_fomodel4 :::"-ranked"::: ) ) "iff" (Bool (Set (Var "D")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ) ")" ")" )))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Num 1) ($#v12_fomodel4 :::"-ranked"::: ) -> (Set ($#k6_numbers :::"0"::: ) ) ($#v12_fomodel4 :::"-ranked"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ")" )); cluster (Num 2) ($#v12_fomodel4 :::"-ranked"::: ) -> (Num 1) ($#v12_fomodel4 :::"-ranked"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ")" )); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); func "S" :::"-rules"::: -> ($#m1_subset_1 :::"RuleSet":::) "of" "S" equals :: FOMODEL4:def 60 (Set (Set ($#k13_domain_1 :::"{"::: ) (Set "(" ($#k22_fomodel4 :::"R#0"::: ) "S" ")" ) "," (Set "(" ($#k23_fomodel4 :::"R#1"::: ) "S" ")" ) "," (Set "(" ($#k24_fomodel4 :::"R#2"::: ) "S" ")" ) "," (Set "(" ($#k25_fomodel4 :::"R#3a"::: ) "S" ")" ) "," (Set "(" ($#k26_fomodel4 :::"R#3b"::: ) "S" ")" ) "," (Set "(" ($#k27_fomodel4 :::"R#3d"::: ) "S" ")" ) "," (Set "(" ($#k28_fomodel4 :::"R#3e"::: ) "S" ")" ) "," (Set "(" ($#k29_fomodel4 :::"R#4"::: ) "S" ")" ) ($#k13_domain_1 :::"}"::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k9_domain_1 :::"{"::: ) (Set "(" ($#k30_fomodel4 :::"R#5"::: ) "S" ")" ) "," (Set "(" ($#k31_fomodel4 :::"R#6"::: ) "S" ")" ) "," (Set "(" ($#k32_fomodel4 :::"R#7"::: ) "S" ")" ) "," (Set "(" ($#k33_fomodel4 :::"R#8"::: ) "S" ")" ) ($#k9_domain_1 :::"}"::: ) )); end; :: deftheorem defines :::"-rules"::: FOMODEL4:def 60 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set (Set (Var "S")) ($#k37_fomodel4 :::"-rules"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k13_domain_1 :::"{"::: ) (Set "(" ($#k22_fomodel4 :::"R#0"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k23_fomodel4 :::"R#1"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k24_fomodel4 :::"R#2"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k25_fomodel4 :::"R#3a"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k26_fomodel4 :::"R#3b"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k27_fomodel4 :::"R#3d"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k28_fomodel4 :::"R#3e"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k29_fomodel4 :::"R#4"::: ) (Set (Var "S")) ")" ) ($#k13_domain_1 :::"}"::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k9_domain_1 :::"{"::: ) (Set "(" ($#k30_fomodel4 :::"R#5"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k31_fomodel4 :::"R#6"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k32_fomodel4 :::"R#7"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k33_fomodel4 :::"R#8"::: ) (Set (Var "S")) ")" ) ($#k9_domain_1 :::"}"::: ) )))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set "S" ($#k37_fomodel4 :::"-rules"::: ) ) -> (Num 2) ($#v12_fomodel4 :::"-ranked"::: ) for ($#m1_subset_1 :::"RuleSet":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#v4_funct_1 :::"functional"::: ) (Num 2) ($#v12_fomodel4 :::"-ranked"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ")" )); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#v4_funct_1 :::"functional"::: ) (Num 1) ($#v12_fomodel4 :::"-ranked"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ")" )); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#v4_funct_1 :::"functional"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v12_fomodel4 :::"-ranked"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ")" )); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be (Num 1) ($#v12_fomodel4 :::"-ranked"::: ) ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be (Set (Const "D")) ($#v10_fomodel4 :::"-expanded"::: ) ($#m1_hidden :::"set"::: ) ; let "a" be ($#v10_fomodel1 :::"ofAtomicFormula"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); cluster (Set "X" ($#k3_fomodel3 :::"-freeInterpreter"::: ) "a") -> (Set "(" "X" "," "D" ")" ($#k6_fomodel4 :::"-termEq"::: ) ) ($#v2_fomodel3 :::"-respecting"::: ) for ($#m2_fomodel2 :::"Interpreter"::: ) "of" "a" "," (Set ($#k35_fomodel1 :::"AllTermsOf"::: ) "S"); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be (Set ($#k6_numbers :::"0"::: ) ) ($#v12_fomodel4 :::"-ranked"::: ) ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be (Set (Const "D")) ($#v10_fomodel4 :::"-expanded"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "(" "X" "," "D" ")" ($#k5_fomodel4 :::"-termEq"::: ) ) -> ($#v1_partfun1 :::"total"::: ) ($#v3_relat_2 :::"symmetric"::: ) ($#v8_relat_2 :::"transitive"::: ) for ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) "S" ")" ); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#v4_funct_1 :::"functional"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v12_fomodel4 :::"-ranked"::: ) (Num 1) ($#v12_fomodel4 :::"-ranked"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ")" )); end; theorem :: FOMODEL4:1 (Bool "for" (Set (Var "Y")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "D1")) ($#r1_tarski :::"c="::: ) (Set (Var "D2"))) & (Bool "(" (Bool (Set (Var "D2")) "is" ($#v6_fomodel4 :::"isotone"::: ) ) "or" (Bool (Set (Var "D1")) "is" ($#v6_fomodel4 :::"isotone"::: ) ) ")" ) & (Bool (Set (Var "Y")) "is" (Set (Var "X")) "," (Set (Var "D1")) ($#v3_fomodel4 :::"-derivable"::: ) )) "holds" (Bool (Set (Var "Y")) "is" (Set (Var "X")) "," (Set (Var "D2")) ($#v3_fomodel4 :::"-derivable"::: ) )))) ; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "Sq" be (Set (Const "S")) ($#v1_fomodel4 :::"-sequent-like"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k1_tarski :::"{"::: ) "Sq" ($#k1_tarski :::"}"::: ) ) -> "S" ($#v2_fomodel4 :::"-sequents-like"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "SQ1", "SQ2" be (Set (Const "S")) ($#v2_fomodel4 :::"-sequents-like"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "SQ1" ($#k2_xboole_0 :::"\/"::: ) "SQ2") -> "S" ($#v2_fomodel4 :::"-sequents-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "x", "y" be (Set (Const "S")) ($#v1_fomodel4 :::"-sequent-like"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k2_tarski :::"{"::: ) "x" "," "y" ($#k2_tarski :::"}"::: ) ) -> "S" ($#v2_fomodel4 :::"-sequents-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi1", "phi2" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi1" ")" ) "," (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi2" ")" ) ($#k7_domain_1 :::"}"::: ) ) "," (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "phi1" ")" ) ($#k30_fomodel2 :::"^"::: ) "phi2" ")" ) ($#k4_tarski :::"]"::: ) ) -> (Num 1) "," (Set ($#k5_lang1 :::"{"::: ) (Set ($#k1_domain_1 :::"["::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi1" ")" ) "," (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi2" ")" ) ($#k7_domain_1 :::"}"::: ) ) "," (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi1" ")" ) ($#k1_domain_1 :::"]"::: ) ) "," (Set ($#k1_domain_1 :::"["::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi1" ")" ) "," (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi2" ")" ) ($#k7_domain_1 :::"}"::: ) ) "," (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi2" ")" ) ($#k1_domain_1 :::"]"::: ) ) ($#k5_lang1 :::"}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k31_fomodel4 :::"R#6"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi1", "phi2" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set ($#k7_domain_1 :::"{"::: ) "phi1" "," "phi2" ($#k7_domain_1 :::"}"::: ) ) "," "phi2" ($#k4_tarski :::"]"::: ) ) -> (Num 1) "," (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k22_fomodel4 :::"R#0"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FOMODEL4:2 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "SQ")) "being" (Set (Var "b1")) ($#v2_fomodel4 :::"-sequents-like"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Sq")) "being" (Set (Var "b1")) ($#v1_fomodel4 :::"-sequent-like"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "SQ")) "," (Set (Var "Sq")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R")))) "holds" (Bool (Set (Var "Sq")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k21_fomodel4 :::"FuncRule"::: ) (Set (Var "R")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "SQ")))))))) ; theorem :: FOMODEL4:3 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "R")) "being" ($#m2_funct_2 :::"Rule":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "R")) ($#k1_funct_1 :::"."::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "x")) "is" (Num 1) "," (Set (Var "X")) "," (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "R")) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) )))) ; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be ($#m1_hidden :::"set"::: ) ; redefine attr "X" is "D" :::"-expanded"::: means :: FOMODEL4:def 61 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) "is" "X" "," "D" ($#v8_fomodel4 :::"-provable"::: ) )) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X")); end; :: deftheorem defines :::"-expanded"::: FOMODEL4:def 61 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" (Set (Var "D")) ($#v10_fomodel4 :::"-expanded"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) "is" (Set (Var "X")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) )) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) ")" )))); theorem :: FOMODEL4:4 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "phi")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "phi")) "is" (Set (Var "X")) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k22_fomodel4 :::"R#0"::: ) (Set (Var "S")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v8_fomodel4 :::"-provable"::: ) )))) ; theorem :: FOMODEL4:5 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "," (Set (Var "D3")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "SQ1")) "," (Set (Var "SQ2")) "being" (Set (Var "b6")) ($#v2_fomodel4 :::"-sequents-like"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set (Var "D1")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "D2"))) "is" ($#v6_fomodel4 :::"isotone"::: ) ) & (Bool (Set (Set "(" (Set (Var "D1")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "D2")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set (Var "D3"))) "is" ($#v6_fomodel4 :::"isotone"::: ) ) & (Bool (Set (Var "x")) "is" (Set (Var "m")) "," (Set (Var "SQ1")) "," (Set (Var "D1")) ($#v4_fomodel4 :::"-derivable"::: ) ) & (Bool (Set (Var "y")) "is" (Set (Var "m")) "," (Set (Var "SQ2")) "," (Set (Var "D2")) ($#v4_fomodel4 :::"-derivable"::: ) ) & (Bool (Set (Var "z")) "is" (Set (Var "n")) "," (Set ($#k2_tarski :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k2_tarski :::"}"::: ) ) "," (Set (Var "D3")) ($#v4_fomodel4 :::"-derivable"::: ) )) "holds" (Bool (Set (Var "z")) "is" (Set (Set (Var "m")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "n"))) "," (Set (Set (Var "SQ1")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "SQ2"))) "," (Set (Set "(" (Set (Var "D1")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "D2")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set (Var "D3"))) ($#v4_fomodel4 :::"-derivable"::: ) )))))) ; theorem :: FOMODEL4:6 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "y")) "," (Set (Var "X")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "D1")) "is" ($#v6_fomodel4 :::"isotone"::: ) ) & (Bool (Set (Set (Var "D1")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "D2"))) "is" ($#v6_fomodel4 :::"isotone"::: ) ) & (Bool (Set (Var "y")) "is" (Set (Var "m")) "," (Set (Var "X")) "," (Set (Var "D1")) ($#v4_fomodel4 :::"-derivable"::: ) ) & (Bool (Set (Var "z")) "is" (Set (Var "n")) "," (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ) "," (Set (Var "D2")) ($#v4_fomodel4 :::"-derivable"::: ) )) "holds" (Bool (Set (Var "z")) "is" (Set (Set (Var "m")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "n"))) "," (Set (Var "X")) "," (Set (Set (Var "D1")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "D2"))) ($#v4_fomodel4 :::"-derivable"::: ) ))))) ; theorem :: FOMODEL4:7 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x")) "is" (Set (Var "m")) "," (Set (Var "X")) "," (Set (Var "D")) ($#v4_fomodel4 :::"-derivable"::: ) )) "holds" (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) "is" (Set (Var "X")) "," (Set (Var "D")) ($#v3_fomodel4 :::"-derivable"::: ) ))))) ; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k31_fomodel4 :::"R#6"::: ) "S") -> ($#v5_fomodel4 :::"isotone"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; theorem :: FOMODEL4:8 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "D1")) ($#r1_tarski :::"c="::: ) (Set (Var "D2"))) & (Bool "(" (Bool (Set (Var "D1")) "is" ($#v6_fomodel4 :::"isotone"::: ) ) "or" (Bool (Set (Var "D2")) "is" ($#v6_fomodel4 :::"isotone"::: ) ) ")" ) & (Bool (Set (Var "x")) "is" (Set (Var "X")) "," (Set (Var "D1")) ($#v8_fomodel4 :::"-provable"::: ) )) "holds" (Bool (Set (Var "x")) "is" (Set (Var "X")) "," (Set (Var "D2")) ($#v8_fomodel4 :::"-provable"::: ) )))) ; theorem :: FOMODEL4:9 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y"))) & (Bool (Set (Var "x")) "is" (Set (Var "X")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) )) "holds" (Bool (Set (Var "x")) "is" (Set (Var "Y")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) )))) ; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k33_fomodel4 :::"R#8"::: ) "S") -> ($#v5_fomodel4 :::"isotone"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k23_fomodel4 :::"R#1"::: ) "S") -> ($#v5_fomodel4 :::"isotone"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; theorem :: FOMODEL4:10 (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "SQ")) "being" (Set (Var "b2")) ($#v2_fomodel4 :::"-sequents-like"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ) "is" (Set (Var "SQ")) "," (Set (Var "D")) ($#v3_fomodel4 :::"-derivable"::: ) )) "holds" (Bool "ex" (Set (Var "mm")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Var "y")) "is" (Set (Var "mm")) "," (Set (Var "SQ")) "," (Set (Var "D")) ($#v4_fomodel4 :::"-derivable"::: ) )))))) ; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be ($#m1_hidden :::"set"::: ) ; cluster "X" "," "D" ($#v3_fomodel4 :::"-derivable"::: ) -> "S" ($#v2_fomodel4 :::"-sequents-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X", "x" be ($#m1_hidden :::"set"::: ) ; redefine attr "x" is "X" "," "D" :::"-provable"::: means :: FOMODEL4:def 62 (Bool "ex" (Set (Var "H")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "H")) ($#r1_tarski :::"c="::: ) "X") & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "H")) "," "x" ($#k4_tarski :::"]"::: ) ) "is" (Set (Var "m")) "," (Set ($#k1_xboole_0 :::"{}"::: ) ) "," "D" ($#v4_fomodel4 :::"-derivable"::: ) ) ")" ))); end; :: deftheorem defines :::"-provable"::: FOMODEL4:def 62 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" (Set (Var "X")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) ) "iff" (Bool "ex" (Set (Var "H")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "H")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "H")) "," (Set (Var "x")) ($#k4_tarski :::"]"::: ) ) "is" (Set (Var "m")) "," (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set (Var "D")) ($#v4_fomodel4 :::"-derivable"::: ) ) ")" ))) ")" )))); theorem :: FOMODEL4:11 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "D1")) ($#r1_tarski :::"c="::: ) (Set (Var "D2"))) & (Bool "(" (Bool (Set (Var "D2")) "is" ($#v6_fomodel4 :::"isotone"::: ) ) "or" (Bool (Set (Var "D1")) "is" ($#v6_fomodel4 :::"isotone"::: ) ) ")" ) & (Bool (Set (Var "x")) "is" (Set (Var "m")) "," (Set (Var "X")) "," (Set (Var "D1")) ($#v4_fomodel4 :::"-derivable"::: ) )) "holds" (Bool (Set (Var "x")) "is" (Set (Var "m")) "," (Set (Var "X")) "," (Set (Var "D2")) ($#v4_fomodel4 :::"-derivable"::: ) ))))) ; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k32_fomodel4 :::"R#7"::: ) "S") -> ($#v5_fomodel4 :::"isotone"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; theorem :: FOMODEL4:12 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x")) "is" (Set (Var "X")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) )) "holds" (Bool (Set (Var "x")) "is" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")))))) ; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D1" be (Set ($#k6_numbers :::"0"::: ) ) ($#v12_fomodel4 :::"-ranked"::: ) (Num 1) ($#v12_fomodel4 :::"-ranked"::: ) ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be (Set (Const "D1")) ($#v10_fomodel4 :::"-expanded"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "(" "S" "," "X" ")" ($#k5_fomodel3 :::"-freeInterpreter"::: ) ) -> "S" "," (Set ($#k35_fomodel1 :::"AllTermsOf"::: ) "S") ($#v1_fomodel2 :::"-interpreter-like"::: ) (Set "(" "X" "," "D1" ")" ($#k6_fomodel4 :::"-termEq"::: ) ) ($#v3_fomodel3 :::"-respecting"::: ) for"S" "," (Set ($#k35_fomodel1 :::"AllTermsOf"::: ) "S") ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be (Set ($#k6_numbers :::"0"::: ) ) ($#v12_fomodel4 :::"-ranked"::: ) ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be (Set (Const "D")) ($#v10_fomodel4 :::"-expanded"::: ) ($#m1_hidden :::"set"::: ) ; func "D" :::"Henkin"::: "X" -> ($#m1_hidden :::"Function":::) equals :: FOMODEL4:def 63 (Set (Set "(" "(" "S" "," "X" ")" ($#k7_fomodel3 :::"-freeInterpreter"::: ) ")" ) ($#k22_fomodel3 :::"quotient"::: ) (Set "(" "(" "X" "," "D" ")" ($#k6_fomodel4 :::"-termEq"::: ) ")" )); end; :: deftheorem defines :::"Henkin"::: FOMODEL4:def 63 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" (Set ($#k6_numbers :::"0"::: ) ) ($#v12_fomodel4 :::"-ranked"::: ) ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" (Set (Var "b2")) ($#v10_fomodel4 :::"-expanded"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "D")) ($#k38_fomodel4 :::"Henkin"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "(" (Set (Var "S")) "," (Set (Var "X")) ")" ($#k7_fomodel3 :::"-freeInterpreter"::: ) ")" ) ($#k22_fomodel3 :::"quotient"::: ) (Set "(" "(" (Set (Var "X")) "," (Set (Var "D")) ")" ($#k6_fomodel4 :::"-termEq"::: ) ")" )))))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be (Set ($#k6_numbers :::"0"::: ) ) ($#v12_fomodel4 :::"-ranked"::: ) ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be (Set (Const "D")) ($#v10_fomodel4 :::"-expanded"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "D" ($#k38_fomodel4 :::"Henkin"::: ) "X") -> (Set ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S") ($#v4_relat_1 :::"-defined"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D1" be (Set ($#k6_numbers :::"0"::: ) ) ($#v12_fomodel4 :::"-ranked"::: ) (Num 1) ($#v12_fomodel4 :::"-ranked"::: ) ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be (Set (Const "D1")) ($#v10_fomodel4 :::"-expanded"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "D1" ($#k38_fomodel4 :::"Henkin"::: ) "X") -> "S" "," (Set ($#k8_eqrel_1 :::"Class"::: ) (Set "(" "(" "X" "," "D1" ")" ($#k6_fomodel4 :::"-termEq"::: ) ")" )) ($#v1_fomodel2 :::"-interpreter-like"::: ) for ($#m1_hidden :::"Function":::); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D1" be (Set ($#k6_numbers :::"0"::: ) ) ($#v12_fomodel4 :::"-ranked"::: ) (Num 1) ($#v12_fomodel4 :::"-ranked"::: ) ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be (Set (Const "D1")) ($#v10_fomodel4 :::"-expanded"::: ) ($#m1_hidden :::"set"::: ) ; :: original: :::"Henkin"::: redefine func "D1" :::"Henkin"::: "X" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k8_eqrel_1 :::"Class"::: ) (Set "(" "(" "X" "," "D1" ")" ($#k6_fomodel4 :::"-termEq"::: ) ")" ) ")" ) ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S"); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi1", "phi2" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "phi1" ")" ) ($#k7_finseq_1 :::"^"::: ) "phi2") -> (Set ($#k7_domain_1 :::"{"::: ) (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi1" ")" ) "," (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi2" ")" ) ($#k7_domain_1 :::"}"::: ) ) "," (Set (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k22_fomodel4 :::"R#0"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k31_fomodel4 :::"R#6"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) )) ($#v8_fomodel4 :::"-provable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k6_numbers :::"0"::: ) ) ($#v12_fomodel4 :::"-ranked"::: ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) (Set ($#k6_numbers :::"0"::: ) ) ($#v12_fomodel4 :::"-ranked"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ")" )); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "x" be ($#m1_hidden :::"set"::: ) ; attr "x" is "S" :::"-premises-like"::: means :: FOMODEL4:def 64 (Bool "(" (Bool "x" ($#r1_tarski :::"c="::: ) (Set ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S")) & (Bool "x" "is" ($#v1_finset_1 :::"finite"::: ) ) ")" ); end; :: deftheorem defines :::"-premises-like"::: FOMODEL4:def 64 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" (Set (Var "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Var "S")))) & (Bool (Set (Var "x")) "is" ($#v1_finset_1 :::"finite"::: ) ) ")" ) ")" ))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster "S" ($#v13_fomodel4 :::"-premises-like"::: ) -> ($#v1_finset_1 :::"finite"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k1_tarski :::"{"::: ) "phi" ($#k1_tarski :::"}"::: ) ) -> "S" ($#v13_fomodel4 :::"-premises-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "e" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "e" ($#k13_fomodel0 :::"null"::: ) "S") -> "S" ($#v13_fomodel4 :::"-premises-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); cluster "S" ($#v13_fomodel4 :::"-premises-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "X"); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster "S" ($#v13_fomodel4 :::"-premises-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "X" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; cluster -> "S" ($#v13_fomodel4 :::"-premises-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "X"); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "H2", "H1" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; :: original: :::"null"::: redefine func "H1" :::"null"::: "H2" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S" ")" ); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "H" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; let "x" be ($#m1_hidden :::"set"::: ) ; cluster (Set "H" ($#k13_fomodel0 :::"null"::: ) "x") -> "S" ($#v13_fomodel4 :::"-premises-like"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "H1", "H2" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "H1" ($#k2_xboole_0 :::"\/"::: ) "H2") -> "S" ($#v13_fomodel4 :::"-premises-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "H" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) "H" "," "phi" ($#k4_tarski :::"]"::: ) ) -> "S" ($#v1_fomodel4 :::"-sequent-like"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "H1", "H2" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set "(" "H1" ($#k2_xboole_0 :::"\/"::: ) "H2" ")" ) "," "phi" ($#k4_tarski :::"]"::: ) ) -> (Num 1) "," (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) "H1" "," "phi" ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k23_fomodel4 :::"R#1"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "H" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; let "phi", "phi1", "phi2" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set "(" "H" ($#k13_fomodel0 :::"null"::: ) (Set "(" "phi1" ($#k30_fomodel2 :::"^"::: ) "phi2" ")" ) ")" ) "," (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi" ")" ) ($#k4_tarski :::"]"::: ) ) -> (Num 1) "," (Set ($#k2_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" "H" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) "phi" ($#k6_domain_1 :::"}"::: ) ) ")" ) "," "phi1" ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set "(" "H" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) "phi" ($#k6_domain_1 :::"}"::: ) ) ")" ) "," (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "phi1" ")" ) ($#k30_fomodel2 :::"^"::: ) "phi2" ")" ) ($#k4_tarski :::"]"::: ) ) ($#k2_tarski :::"}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k33_fomodel4 :::"R#8"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k13_fomodel0 :::"null"::: ) "S") -> "S" ($#v2_fomodel4 :::"-sequents-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "H" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set "(" "H" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) "phi" ($#k6_domain_1 :::"}"::: ) ) ")" ) "," "phi" ($#k4_tarski :::"]"::: ) ) -> (Num 1) "," (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k22_fomodel4 :::"R#0"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "H" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; let "phi1", "phi2" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set "(" "H" ($#k13_fomodel0 :::"null"::: ) "phi2" ")" ) "," (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi1" ")" ) ($#k4_tarski :::"]"::: ) ) -> (Num 2) "," (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) "H" "," (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "phi1" ")" ) ($#k30_fomodel2 :::"^"::: ) "phi2" ")" ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k22_fomodel4 :::"R#0"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k23_fomodel4 :::"R#1"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k33_fomodel4 :::"R#8"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) )) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "H" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; let "phi1", "phi2" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) "H" "," (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "phi2" ")" ) ($#k30_fomodel2 :::"^"::: ) "phi1" ")" ) ($#k4_tarski :::"]"::: ) ) -> (Num 1) "," (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) "H" "," (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "phi1" ")" ) ($#k30_fomodel2 :::"^"::: ) "phi2" ")" ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k32_fomodel4 :::"R#7"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "H" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; let "phi1", "phi2" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set "(" "H" ($#k13_fomodel0 :::"null"::: ) "phi1" ")" ) "," (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi2" ")" ) ($#k4_tarski :::"]"::: ) ) -> (Num 3) "," (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) "H" "," (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "phi1" ")" ) ($#k30_fomodel2 :::"^"::: ) "phi2" ")" ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set (Set "(" (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k22_fomodel4 :::"R#0"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k23_fomodel4 :::"R#1"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k33_fomodel4 :::"R#8"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k32_fomodel4 :::"R#7"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) )) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "Sq" be (Set (Const "S")) ($#v1_fomodel4 :::"-sequent-like"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "Sq" ($#k1_xtuple_0 :::"`1"::: ) ) -> "S" ($#v13_fomodel4 :::"-premises-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "X" be ($#m1_hidden :::"set"::: ) ; let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); :: original: :::"null"::: redefine func "D" :::"null"::: "X" -> ($#m1_subset_1 :::"RuleSet":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi1", "phi2" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "l1" be ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); let "H" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; let "l2" be ($#v4_fomodel1 :::"literal"::: ) (Set (Set "(" (Set (Const "H")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Const "phi1")) ($#k6_domain_1 :::"}"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Const "phi2")) ($#k6_domain_1 :::"}"::: ) )) ($#v7_fomodel2 :::"-absent"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set "(" (Set "(" "H" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) "l1" ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "phi1" ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ($#k13_fomodel0 :::"null"::: ) "l2" ")" ) "," "phi2" ($#k4_tarski :::"]"::: ) ) -> (Num 1) "," (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" "H" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" "(" "l1" "," "l2" ")" ($#k24_fomodel3 :::"-SymbolSubstIn"::: ) "phi1" ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) "," "phi2" ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k30_fomodel4 :::"R#5"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be ($#m1_hidden :::"set"::: ) ; attr "X" is "D" :::"-inconsistent"::: means :: FOMODEL4:def 65 (Bool "ex" (Set (Var "phi1")) "," (Set (Var "phi2")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S" "st" (Bool "(" (Bool (Set (Var "phi1")) "is" "X" "," "D" ($#v8_fomodel4 :::"-provable"::: ) ) & (Bool (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2"))) "is" "X" "," "D" ($#v8_fomodel4 :::"-provable"::: ) ) ")" )); end; :: deftheorem defines :::"-inconsistent"::: FOMODEL4:def 65 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" (Set (Var "D")) ($#v14_fomodel4 :::"-inconsistent"::: ) ) "iff" (Bool "ex" (Set (Var "phi1")) "," (Set (Var "phi2")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "phi1")) "is" (Set (Var "X")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) ) & (Bool (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2"))) "is" (Set (Var "X")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) ) ")" )) ")" )))); registrationlet "m1" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); let "S" be ($#l1_fomodel1 :::"Language":::); let "H1", "H2" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set "(" (Set "(" "H1" ($#k2_xboole_0 :::"\/"::: ) "H2" ")" ) ($#k13_fomodel0 :::"null"::: ) "m1" ")" ) "," "phi" ($#k4_tarski :::"]"::: ) ) -> "m1" "," (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) "H1" "," "phi" ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k23_fomodel4 :::"R#1"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_funct_1 :::"functional"::: ) ($#v6_fomodel4 :::"isotone"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ")" )); end; theorem :: FOMODEL4:13 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "X")) "is" (Set (Var "D")) ($#v14_fomodel4 :::"-inconsistent"::: ) ) & (Bool (Set (Var "D")) "is" ($#v6_fomodel4 :::"isotone"::: ) ) & (Bool (Set ($#k23_fomodel4 :::"R#1"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k33_fomodel4 :::"R#8"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set ($#k38_fomodel2 :::"xnot"::: ) (Set (Var "phi"))) "is" (Set (Var "X")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) ))))) ; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k30_fomodel4 :::"R#5"::: ) "S") -> ($#v5_fomodel4 :::"isotone"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "l" be ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); let "t" be ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k4_tarski :::"["::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" "(" "l" "," "t" ")" ($#k37_fomodel3 :::"SubstIn"::: ) "phi" ")" ) ($#k6_domain_1 :::"}"::: ) ) "," (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) "l" ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "phi" ")" ) ($#k4_tarski :::"]"::: ) ) -> (Num 1) "," (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k29_fomodel4 :::"R#4"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k29_fomodel4 :::"R#4"::: ) "S") -> ($#v5_fomodel4 :::"isotone"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "X" be ($#m1_hidden :::"set"::: ) ; attr "X" is "S" :::"-witnessed"::: means :: FOMODEL4:def 66 (Bool "for" (Set (Var "l1")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" "S" (Bool "for" (Set (Var "phi1")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S" "st" (Bool (Bool (Set (Set ($#k31_fomodel2 :::"<*"::: ) (Set (Var "l1")) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1"))) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool "ex" (Set (Var "l2")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" "S" "st" (Bool "(" (Bool (Set "(" (Set (Var "l1")) "," (Set (Var "l2")) ")" ($#k24_fomodel3 :::"-SymbolSubstIn"::: ) (Set (Var "phi1"))) ($#r2_hidden :::"in"::: ) "X") & (Bool (Bool "not" (Set (Var "l2")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "phi1"))))) ")" )))); end; :: deftheorem defines :::"-witnessed"::: FOMODEL4:def 66 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" (Set (Var "S")) ($#v15_fomodel4 :::"-witnessed"::: ) ) "iff" (Bool "for" (Set (Var "l1")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "phi1")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Set ($#k31_fomodel2 :::"<*"::: ) (Set (Var "l1")) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1"))) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "ex" (Set (Var "l2")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set "(" (Set (Var "l1")) "," (Set (Var "l2")) ")" ($#k24_fomodel3 :::"-SymbolSubstIn"::: ) (Set (Var "phi1"))) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Bool "not" (Set (Var "l2")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "phi1"))))) ")" )))) ")" ))); theorem :: FOMODEL4:14 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "psi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "D1")) "being" (Set ($#k6_numbers :::"0"::: ) ) ($#v12_fomodel4 :::"-ranked"::: ) (Num 1) ($#v12_fomodel4 :::"-ranked"::: ) ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" (Set (Var "b3")) ($#v10_fomodel4 :::"-expanded"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k23_fomodel4 :::"R#1"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D1"))) & (Bool (Set ($#k29_fomodel4 :::"R#4"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D1"))) & (Bool (Set ($#k31_fomodel4 :::"R#6"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D1"))) & (Bool (Set ($#k32_fomodel4 :::"R#7"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D1"))) & (Bool (Set ($#k33_fomodel4 :::"R#8"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D1"))) & (Bool (Set (Var "X")) "is" (Set (Var "S")) ($#v6_fomodel2 :::"-mincover"::: ) ) & (Bool (Set (Var "X")) "is" (Set (Var "S")) ($#v15_fomodel4 :::"-witnessed"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "D1")) ($#k39_fomodel4 :::"Henkin"::: ) (Set (Var "X")) ")" ) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "psi"))) ($#r1_hidden :::"="::: ) (Num 1)) "iff" (Bool (Set (Var "psi")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" ))))) ; notationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be ($#m1_hidden :::"set"::: ) ; antonym "D" :::"-consistent"::: "X" for "D" :::"-inconsistent"::: ; end; theorem :: FOMODEL4:15 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "X")) "is" (Set (Var "D")) ($#v14_fomodel4 :::"-inconsistent"::: ) )) "holds" (Bool (Set (Var "Y")) "is" (Set (Var "D")) ($#v14_fomodel4 :::"-inconsistent"::: ) ))))) ; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; let "phi" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Const "S"))); func "(" "D" "," "phi" ")" :::"AddAsWitnessTo"::: "X" -> ($#m1_hidden :::"set"::: ) equals :: FOMODEL4:def 67 (Set "X" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" "(" (Set "(" (Set "(" "S" ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) "phi" ")" ) "," "the" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k16_fomodel1 :::"LettersOf"::: ) "S" ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k21_fomodel0 :::"SymbolsOf"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_subset_1 :::"/\"::: ) (Set "(" "X" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k40_fomodel2 :::"head"::: ) "phi" ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) ")" )) ")" ($#k17_fomodel0 :::"-SymbolSubstIn"::: ) (Set "(" ($#k40_fomodel2 :::"head"::: ) "phi" ")" ) ")" ) ($#k1_tarski :::"}"::: ) )) if (Bool "(" (Bool (Set "X" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) "phi" ($#k6_domain_1 :::"}"::: ) )) "is" "D" ($#v14_fomodel4 :::"-consistent"::: ) ) & (Bool (Set (Set "(" ($#k16_fomodel1 :::"LettersOf"::: ) "S" ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k21_fomodel0 :::"SymbolsOf"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_subset_1 :::"/\"::: ) (Set "(" "X" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k40_fomodel2 :::"head"::: ) "phi" ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ) otherwise "X"; end; :: deftheorem defines :::"AddAsWitnessTo"::: FOMODEL4:def 67 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "phi")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Var "S"))) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "phi")) ($#k6_domain_1 :::"}"::: ) )) "is" (Set (Var "D")) ($#v14_fomodel4 :::"-consistent"::: ) ) & (Bool (Set (Set "(" ($#k16_fomodel1 :::"LettersOf"::: ) (Set (Var "S")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k21_fomodel0 :::"SymbolsOf"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_subset_1 :::"/\"::: ) (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k40_fomodel2 :::"head"::: ) (Set (Var "phi")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set "(" (Set (Var "D")) "," (Set (Var "phi")) ")" ($#k42_fomodel4 :::"AddAsWitnessTo"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" "(" (Set "(" (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "phi")) ")" ) "," "the" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" ($#k16_fomodel1 :::"LettersOf"::: ) (Set (Var "S")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k21_fomodel0 :::"SymbolsOf"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_subset_1 :::"/\"::: ) (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k40_fomodel2 :::"head"::: ) (Set (Var "phi")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) ")" )) ")" ($#k17_fomodel0 :::"-SymbolSubstIn"::: ) (Set "(" ($#k40_fomodel2 :::"head"::: ) (Set (Var "phi")) ")" ) ")" ) ($#k1_tarski :::"}"::: ) ))) ")" & "(" (Bool (Bool "(" "not" (Bool (Set (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "phi")) ($#k6_domain_1 :::"}"::: ) )) "is" (Set (Var "D")) ($#v14_fomodel4 :::"-consistent"::: ) ) "or" "not" (Bool (Set (Set "(" ($#k16_fomodel1 :::"LettersOf"::: ) (Set (Var "S")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k21_fomodel0 :::"SymbolsOf"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_subset_1 :::"/\"::: ) (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k40_fomodel2 :::"head"::: ) (Set (Var "phi")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ")" ) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) "implies" (Bool (Set "(" (Set (Var "D")) "," (Set (Var "phi")) ")" ($#k42_fomodel4 :::"AddAsWitnessTo"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) ")" ")" ))))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; let "phi" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Const "S"))); cluster (Set "X" ($#k4_xboole_0 :::"\"::: ) (Set "(" "(" "D" "," "phi" ")" ($#k42_fomodel4 :::"AddAsWitnessTo"::: ) "X" ")" )) -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; let "phi" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Const "S"))); cluster (Set (Set "(" "(" "D" "," "phi" ")" ($#k42_fomodel4 :::"AddAsWitnessTo"::: ) "X" ")" ) ($#k4_xboole_0 :::"\"::: ) "X") -> ($#v1_zfmisc_1 :::"trivial"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "X" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; let "phi" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Const "S"))); :: original: :::"AddAsWitnessTo"::: redefine func "(" "D" "," "phi" ")" :::"AddAsWitnessTo"::: "X" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "X" ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S" ")" ) ")" ); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); attr "D" is :::"Correct"::: means :: FOMODEL4:def 68 (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S" (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "phi")) "is" (Set (Var "X")) "," "D" ($#v8_fomodel4 :::"-provable"::: ) )) "holds" (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S") "st" (Bool (Bool (Set (Var "X")) "is" (Set (Var "I")) ($#v10_fomodel2 :::"-satisfied"::: ) )) "holds" (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Num 1)))))); end; :: deftheorem defines :::"Correct"::: FOMODEL4:def 68 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "D")) "is" ($#v16_fomodel4 :::"Correct"::: ) ) "iff" (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "phi")) "is" (Set (Var "X")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) )) "holds" (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) "st" (Bool (Bool (Set (Var "X")) "is" (Set (Var "I")) ($#v10_fomodel2 :::"-satisfied"::: ) )) "holds" (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Num 1)))))) ")" ))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "t1", "t2" be ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set (Set "(" ($#k34_fomodel1 :::"SubTerms"::: ) (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "t1" ")" ) ($#k30_fomodel2 :::"^"::: ) "t2" ")" ) ")" ) ($#k5_xboole_0 :::"\+\"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) "t1" "," "t2" ($#k10_finseq_1 :::"*>"::: ) )) -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "R" be ($#m2_funct_2 :::"Rule":::) "of" (Set (Const "S")); attr "R" is :::"Correct"::: means :: FOMODEL4:def 69 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" "S" ($#v12_fomodel2 :::"-correct"::: ) )) "holds" (Bool (Set "R" ($#k1_funct_1 :::"."::: ) (Set (Var "X"))) "is" "S" ($#v12_fomodel2 :::"-correct"::: ) )); end; :: deftheorem defines :::"Correct"::: FOMODEL4:def 69 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "R")) "being" ($#m2_funct_2 :::"Rule":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v17_fomodel4 :::"Correct"::: ) ) "iff" (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" (Set (Var "S")) ($#v12_fomodel2 :::"-correct"::: ) )) "holds" (Bool (Set (Set (Var "R")) ($#k1_funct_1 :::"."::: ) (Set (Var "X"))) "is" (Set (Var "S")) ($#v12_fomodel2 :::"-correct"::: ) )) ")" ))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster "S" ($#v1_fomodel4 :::"-sequent-like"::: ) -> "S" ($#v11_fomodel4 :::"-null"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k22_fomodel4 :::"R#0"::: ) "S") -> ($#v17_fomodel4 :::"Correct"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" )) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v17_fomodel4 :::"Correct"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k23_fomodel4 :::"R#1"::: ) "S") -> ($#v17_fomodel4 :::"Correct"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k24_fomodel4 :::"R#2"::: ) "S") -> ($#v17_fomodel4 :::"Correct"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k25_fomodel4 :::"R#3a"::: ) "S") -> ($#v17_fomodel4 :::"Correct"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k26_fomodel4 :::"R#3b"::: ) "S") -> ($#v17_fomodel4 :::"Correct"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k27_fomodel4 :::"R#3d"::: ) "S") -> ($#v17_fomodel4 :::"Correct"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k28_fomodel4 :::"R#3e"::: ) "S") -> ($#v17_fomodel4 :::"Correct"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k29_fomodel4 :::"R#4"::: ) "S") -> ($#v17_fomodel4 :::"Correct"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k30_fomodel4 :::"R#5"::: ) "S") -> ($#v17_fomodel4 :::"Correct"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k31_fomodel4 :::"R#6"::: ) "S") -> ($#v17_fomodel4 :::"Correct"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k32_fomodel4 :::"R#7"::: ) "S") -> ($#v17_fomodel4 :::"Correct"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k33_fomodel4 :::"R#8"::: ) "S") -> ($#v17_fomodel4 :::"Correct"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; theorem :: FOMODEL4:16 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "st" (Bool (Bool "(" "for" (Set (Var "R")) "being" ($#m2_funct_2 :::"Rule":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "R")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Var "R")) "is" ($#v17_fomodel4 :::"Correct"::: ) ) ")" )) "holds" (Bool (Set (Var "D")) "is" ($#v16_fomodel4 :::"Correct"::: ) ))) ; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "R" be ($#v17_fomodel4 :::"Correct"::: ) ($#m2_funct_2 :::"Rule":::) "of" (Set (Const "S")); cluster (Set ($#k1_tarski :::"{"::: ) "R" ($#k1_tarski :::"}"::: ) ) -> ($#v16_fomodel4 :::"Correct"::: ) for ($#m1_subset_1 :::"RuleSet":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set "S" ($#k37_fomodel4 :::"-rules"::: ) ) -> ($#v16_fomodel4 :::"Correct"::: ) for ($#m1_subset_1 :::"RuleSet":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k34_fomodel4 :::"R#9"::: ) "S") -> ($#v5_fomodel4 :::"isotone"::: ) for ($#m2_funct_2 :::"Rule":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "H" be (Set (Const "S")) ($#v13_fomodel4 :::"-premises-like"::: ) ($#m1_hidden :::"set"::: ) ; let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set (Set ($#k4_tarski :::"["::: ) "H" "," "phi" ($#k4_tarski :::"]"::: ) ) ($#k13_fomodel0 :::"null"::: ) (Num 1)) -> (Num 1) "," (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) "H" "," (Set "(" ($#k38_fomodel2 :::"xnot"::: ) (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi" ")" ) ")" ) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k34_fomodel4 :::"R#9"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#v4_fomodel4 :::"-derivable"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k17_fomodel1 :::"AtomicFormulaSymbolsOf"::: ) "S") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV2_PRE_POLY() ($#v15_fomodel1 :::"0wff"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v3_fomodel2 :::"-wff"::: ) ($#v4_fomodel2 :::"wff"::: ) "X" ($#v13_fomodel2 :::"-implied"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k15_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )); end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v4_card_3 :::"countable"::: ) bbbadV2_PRE_POLY() ($#v4_fomodel2 :::"wff"::: ) "X" ($#v13_fomodel2 :::"-implied"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k15_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "X" be ($#m1_hidden :::"set"::: ) ; let "phi" be ($#v4_fomodel2 :::"wff"::: ) (Set (Const "X")) ($#v13_fomodel2 :::"-implied"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k38_fomodel2 :::"xnot"::: ) (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi" ")" )) -> ($#v4_fomodel2 :::"wff"::: ) "X" ($#v13_fomodel2 :::"-implied"::: ) for ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S"; end; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); attr "phi" is "X" :::"-provable"::: means :: FOMODEL4:def 70 (Bool "phi" "is" "X" "," (Set (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k34_fomodel4 :::"R#9"::: ) "S" ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set "(" "S" ($#k37_fomodel4 :::"-rules"::: ) ")" )) ($#v8_fomodel4 :::"-provable"::: ) ); end; :: deftheorem defines :::"-provable"::: FOMODEL4:def 70 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "phi")) "is" (Set (Var "X")) ($#v18_fomodel4 :::"-provable"::: ) ) "iff" (Bool (Set (Var "phi")) "is" (Set (Var "X")) "," (Set (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k34_fomodel4 :::"R#9"::: ) (Set (Var "S")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set (Var "S")) ($#k37_fomodel4 :::"-rules"::: ) ")" )) ($#v8_fomodel4 :::"-provable"::: ) ) ")" )))); begin definitionlet "X" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "num" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Const "S")) ")" ); func "(" "D" "," "num" ")" :::"AddWitnessesTo"::: "X" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "X" ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S" ")" ) ")" ) ")" ) means :: FOMODEL4:def 71 (Bool "(" (Bool (Set it ($#k8_nat_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "X") & (Bool "(" "for" (Set (Var "mm")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k8_nat_1 :::"."::: ) (Set "(" (Set (Var "mm")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set "(" "D" "," (Set "(" "num" ($#k8_nat_1 :::"."::: ) (Set (Var "mm")) ")" ) ")" ($#k43_fomodel4 :::"AddAsWitnessTo"::: ) (Set "(" it ($#k8_nat_1 :::"."::: ) (Set (Var "mm")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"AddWitnessesTo"::: FOMODEL4:def 71 : (Bool "for" (Set (Var "X")) "being" ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "num")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Var "S")) ")" ) ")" ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set "(" (Set (Var "D")) "," (Set (Var "num")) ")" ($#k44_fomodel4 :::"AddWitnessesTo"::: ) (Set (Var "X")))) "iff" (Bool "(" (Bool (Set (Set (Var "b5")) ($#k8_nat_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool "(" "for" (Set (Var "mm")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b5")) ($#k8_nat_1 :::"."::: ) (Set "(" (Set (Var "mm")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set "(" (Set (Var "D")) "," (Set "(" (Set (Var "num")) ($#k8_nat_1 :::"."::: ) (Set (Var "mm")) ")" ) ")" ($#k43_fomodel4 :::"AddAsWitnessTo"::: ) (Set "(" (Set (Var "b5")) ($#k8_nat_1 :::"."::: ) (Set (Var "mm")) ")" ))) ")" ) ")" ) ")" )))))); notationlet "X" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "num" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Const "S")) ")" ); synonym "(" "D" "," "num" ")" :::"addw"::: "X" for "(" "D" "," "num" ")" :::"AddWitnessesTo"::: "X"; end; theorem :: FOMODEL4:17 (Bool "for" (Set (Var "k")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "num")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Var "S")) ")" ) "st" (Bool (Bool (Set (Var "D")) "is" ($#v6_fomodel4 :::"isotone"::: ) ) & (Bool (Set ($#k23_fomodel4 :::"R#1"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k33_fomodel4 :::"R#8"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k24_fomodel4 :::"R#2"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k30_fomodel4 :::"R#5"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set (Set "(" ($#k16_fomodel1 :::"LettersOf"::: ) (Set (Var "S")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k21_fomodel0 :::"SymbolsOf"::: ) (Set "(" (Set (Var "X")) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ) ")" )) "is" ($#v1_finset_1 :::"infinite"::: ) ) & (Bool (Set (Var "X")) "is" (Set (Var "D")) ($#v14_fomodel4 :::"-consistent"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" "(" (Set (Var "D")) "," (Set (Var "num")) ")" ($#k44_fomodel4 :::"addw"::: ) (Set (Var "X")) ")" ) ($#k8_nat_1 :::"."::: ) (Set (Var "k"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" "(" (Set (Var "D")) "," (Set (Var "num")) ")" ($#k44_fomodel4 :::"addw"::: ) (Set (Var "X")) ")" ) ($#k8_nat_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "m")) ")" ))) & (Bool (Set (Set "(" ($#k16_fomodel1 :::"LettersOf"::: ) (Set (Var "S")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k21_fomodel0 :::"SymbolsOf"::: ) (Set "(" (Set "(" (Set "(" "(" (Set (Var "D")) "," (Set (Var "num")) ")" ($#k44_fomodel4 :::"addw"::: ) (Set (Var "X")) ")" ) ($#k8_nat_1 :::"."::: ) (Set (Var "m")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ) ")" )) "is" ($#v1_finset_1 :::"infinite"::: ) ) & (Bool (Set (Set "(" "(" (Set (Var "D")) "," (Set (Var "num")) ")" ($#k44_fomodel4 :::"addw"::: ) (Set (Var "X")) ")" ) ($#k8_nat_1 :::"."::: ) (Set (Var "m"))) "is" (Set (Var "D")) ($#v14_fomodel4 :::"-consistent"::: ) ) ")" )))))) ; definitionlet "X" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "num" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Const "S")) ")" ); func "X" :::"WithWitnessesFrom"::: "(" "D" "," "num" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "X" ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S" ")" ) ")" ) equals :: FOMODEL4:def 72 (Set ($#k3_tarski :::"union"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" "(" "D" "," "num" ")" ($#k44_fomodel4 :::"AddWitnessesTo"::: ) "X" ")" ) ")" )); end; :: deftheorem defines :::"WithWitnessesFrom"::: FOMODEL4:def 72 : (Bool "for" (Set (Var "X")) "being" ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "num")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Var "S")) ")" ) "holds" (Bool (Set (Set (Var "X")) ($#k45_fomodel4 :::"WithWitnessesFrom"::: ) "(" (Set (Var "D")) "," (Set (Var "num")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" "(" (Set (Var "D")) "," (Set (Var "num")) ")" ($#k44_fomodel4 :::"AddWitnessesTo"::: ) (Set (Var "X")) ")" ) ")" ))))))); notationlet "X" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "num" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Const "S")) ")" ); synonym "X" :::"addw"::: "(" "D" "," "num" ")" for "X" :::"WithWitnessesFrom"::: "(" "D" "," "num" ")" ; end; registrationlet "X" be ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "num" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Const "S")) ")" ); cluster (Set "X" ($#k4_xboole_0 :::"\"::: ) (Set "(" "X" ($#k45_fomodel4 :::"addw"::: ) "(" "D" "," "num" ")" ")" )) -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FOMODEL4:18 (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#v4_funct_1 :::"functional"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "num")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Var "S")) ")" ) "st" (Bool (Bool (Set (Var "D")) "is" ($#v6_fomodel4 :::"isotone"::: ) ) & (Bool (Set ($#k23_fomodel4 :::"R#1"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k33_fomodel4 :::"R#8"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k24_fomodel4 :::"R#2"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set ($#k30_fomodel4 :::"R#5"::: ) (Set (Var "S"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set (Set "(" ($#k16_fomodel1 :::"LettersOf"::: ) (Set (Var "S")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k21_fomodel0 :::"SymbolsOf"::: ) (Set "(" (Set (Var "X")) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ) ")" )) "is" ($#v1_finset_1 :::"infinite"::: ) ) & (Bool (Set (Set (Var "X")) ($#k45_fomodel4 :::"addw"::: ) "(" (Set (Var "D")) "," (Set (Var "num")) ")" ) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) "is" (Set (Var "D")) ($#v14_fomodel4 :::"-consistent"::: ) ) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "num"))) ($#r1_hidden :::"="::: ) (Set ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Var "S"))))) "holds" (Bool (Set (Var "Z")) "is" (Set (Var "S")) ($#v15_fomodel4 :::"-witnessed"::: ) )))))) ; begin definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "phi" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Const "S"))); func "(" "D" "," "phi" ")" :::"AddFormulaTo"::: "X" -> ($#m1_hidden :::"set"::: ) equals :: FOMODEL4:def 73 (Set "X" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) "phi" ($#k6_domain_1 :::"}"::: ) )) if (Bool (Bool "not" (Set ($#k38_fomodel2 :::"xnot"::: ) "phi") "is" "X" "," "D" ($#v8_fomodel4 :::"-provable"::: ) )) otherwise (Set "X" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi" ")" ) ($#k6_domain_1 :::"}"::: ) )); end; :: deftheorem defines :::"AddFormulaTo"::: FOMODEL4:def 73 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "phi")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Var "S"))) "holds" (Bool "(" "(" (Bool (Bool (Bool "not" (Set ($#k38_fomodel2 :::"xnot"::: ) (Set (Var "phi"))) "is" (Set (Var "X")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) ))) "implies" (Bool (Set "(" (Set (Var "D")) "," (Set (Var "phi")) ")" ($#k46_fomodel4 :::"AddFormulaTo"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "phi")) ($#k6_domain_1 :::"}"::: ) ))) ")" & "(" (Bool (Bool (Set ($#k38_fomodel2 :::"xnot"::: ) (Set (Var "phi"))) "is" (Set (Var "X")) "," (Set (Var "D")) ($#v8_fomodel4 :::"-provable"::: ) )) "implies" (Bool (Set "(" (Set (Var "D")) "," (Set (Var "phi")) ")" ($#k46_fomodel4 :::"AddFormulaTo"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k38_fomodel2 :::"xnot"::: ) (Set (Var "phi")) ")" ) ($#k6_domain_1 :::"}"::: ) ))) ")" ")" ))))); definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "phi" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Const "S"))); :: original: :::"AddFormulaTo"::: redefine func "(" "D" "," "phi" ")" :::"AddFormulaTo"::: "X" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "X" ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S" ")" ) ")" ); end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "phi" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Const "S"))); cluster (Set "X" ($#k4_xboole_0 :::"\"::: ) (Set "(" "(" "D" "," "phi" ")" ($#k47_fomodel4 :::"AddFormulaTo"::: ) "X" ")" )) -> ($#v1_xboole_0 :::"empty"::: ) ; end; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "num" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Const "S")) ")" ); func "(" "D" "," "num" ")" :::"AddFormulasTo"::: "X" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "X" ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S" ")" ) ")" ) ")" ) means :: FOMODEL4:def 74 (Bool "(" (Bool (Set it ($#k8_nat_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "X") & (Bool "(" "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set it ($#k8_nat_1 :::"."::: ) (Set "(" (Set (Var "m")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set "(" "D" "," (Set "(" "num" ($#k8_nat_1 :::"."::: ) (Set (Var "m")) ")" ) ")" ($#k47_fomodel4 :::"AddFormulaTo"::: ) (Set "(" it ($#k8_nat_1 :::"."::: ) (Set (Var "m")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"AddFormulasTo"::: FOMODEL4:def 74 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "num")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Var "S")) ")" ) ")" ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set "(" (Set (Var "D")) "," (Set (Var "num")) ")" ($#k48_fomodel4 :::"AddFormulasTo"::: ) (Set (Var "X")))) "iff" (Bool "(" (Bool (Set (Set (Var "b5")) ($#k8_nat_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool "(" "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "b5")) ($#k8_nat_1 :::"."::: ) (Set "(" (Set (Var "m")) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set "(" (Set (Var "D")) "," (Set "(" (Set (Var "num")) ($#k8_nat_1 :::"."::: ) (Set (Var "m")) ")" ) ")" ($#k47_fomodel4 :::"AddFormulaTo"::: ) (Set "(" (Set (Var "b5")) ($#k8_nat_1 :::"."::: ) (Set (Var "m")) ")" ))) ")" ) ")" ) ")" )))))); definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "num" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Const "S")) ")" ); func "(" "D" "," "num" ")" :::"CompletionOf"::: "X" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "X" ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S" ")" ) ")" ) equals :: FOMODEL4:def 75 (Set ($#k3_tarski :::"union"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" "(" "D" "," "num" ")" ($#k48_fomodel4 :::"AddFormulasTo"::: ) "X" ")" ) ")" )); end; :: deftheorem defines :::"CompletionOf"::: FOMODEL4:def 75 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "num")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Var "S")) ")" ) "holds" (Bool (Set "(" (Set (Var "D")) "," (Set (Var "num")) ")" ($#k49_fomodel4 :::"CompletionOf"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set "(" "(" (Set (Var "D")) "," (Set (Var "num")) ")" ($#k48_fomodel4 :::"AddFormulasTo"::: ) (Set (Var "X")) ")" ) ")" ))))))); registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "D" be ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Const "S")); let "num" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Const "S")) ")" ); cluster (Set "X" ($#k4_xboole_0 :::"\"::: ) (Set "(" "(" "D" "," "num" ")" ($#k49_fomodel4 :::"CompletionOf"::: ) "X" ")" )) -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FOMODEL4:19 (Bool "for" (Set (Var "y")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k21_fomodel4 :::"FuncRule"::: ) (Set (Var "R")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "X")))) "iff" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "S")) ($#k1_fomodel4 :::"-sequents"::: ) )) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "X")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) ")" ) ")" )))) ; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "r1", "r2" be ($#v5_fomodel4 :::"isotone"::: ) ($#m2_funct_2 :::"Rule":::) "of" (Set (Const "S")); cluster (Set ($#k2_tarski :::"{"::: ) "r1" "," "r2" ($#k2_tarski :::"}"::: ) ) -> ($#v6_fomodel4 :::"isotone"::: ) for ($#m1_subset_1 :::"RuleSet":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "r1", "r2", "r3", "r4" be ($#v5_fomodel4 :::"isotone"::: ) ($#m2_funct_2 :::"Rule":::) "of" (Set (Const "S")); cluster (Set bbbadK2_ENUMSET1("r1" "," "r2" "," "r3" "," "r4")) -> ($#v6_fomodel4 :::"isotone"::: ) for ($#m1_subset_1 :::"RuleSet":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set "S" ($#k37_fomodel4 :::"-rules"::: ) ) -> ($#v6_fomodel4 :::"isotone"::: ) for ($#m1_subset_1 :::"RuleSet":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#v4_funct_1 :::"functional"::: ) ($#v6_fomodel4 :::"isotone"::: ) ($#v16_fomodel4 :::"Correct"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ")" )); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#v4_funct_1 :::"functional"::: ) ($#v6_fomodel4 :::"isotone"::: ) (Num 2) ($#v12_fomodel4 :::"-ranked"::: ) ($#v16_fomodel4 :::"Correct"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "(" "S" ($#k1_fomodel4 :::"-sequents"::: ) ")" ) ")" ) ")" ")" )); end; registrationlet "S" be ($#v1_orders_4 :::"countable"::: ) ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k27_fomodel2 :::"AllFormulasOf"::: ) "S") -> ($#v4_card_3 :::"countable"::: ) ; end; theorem :: FOMODEL4:20 (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#v1_orders_4 :::"countable"::: ) ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"RuleSet":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "D")) "is" (Num 2) ($#v12_fomodel4 :::"-ranked"::: ) ) & (Bool (Set (Var "D")) "is" ($#v6_fomodel4 :::"isotone"::: ) ) & (Bool (Set (Var "D")) "is" ($#v16_fomodel4 :::"Correct"::: ) ) & (Bool (Set (Var "Z")) "is" (Set (Var "D")) ($#v14_fomodel4 :::"-consistent"::: ) ) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Var "S"))))) "holds" (Bool "ex" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) "st" (Bool (Set (Var "Z")) "is" (Set (Var "I")) ($#v10_fomodel2 :::"-satisfied"::: ) )))))) ; theorem :: FOMODEL4:21 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#v1_orders_4 :::"countable"::: ) ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Var "C")))) & (Bool (Set (Var "phi")) "is" (Set (Var "X")) ($#v13_fomodel2 :::"-implied"::: ) )) "holds" (Bool (Set (Var "phi")) "is" (Set (Var "X")) ($#v18_fomodel4 :::"-provable"::: ) )))) ;