:: FOMODEL2 semantic presentation begin definitionlet "S" be ($#l1_fomodel1 :::"Language":::); :: original: :::"TheNorSymbOf"::: redefine func :::"TheNorSymbOf"::: "S" -> ($#m1_subset_1 :::"Element":::) "of" "S"; end; definitionlet "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; func "U" :::"-deltaInterpreter"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Num 2) ($#k4_finseq_2 :::"-tuples_on"::: ) "U" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) equals :: FOMODEL2:def 1 (Set ($#k9_fomodel0 :::"chi"::: ) "(" (Set "(" (Set "(" "U" ($#k11_monoid_0 :::"-concatenation"::: ) ")" ) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set "(" (Num 1) ($#k4_finseq_2 :::"-tuples_on"::: ) "U" ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k4_finseq_2 :::"-tuples_on"::: ) "U" ")" ) ")" ); end; :: deftheorem defines :::"-deltaInterpreter"::: FOMODEL2:def 1 : (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "U")) ($#k2_fomodel2 :::"-deltaInterpreter"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_fomodel0 :::"chi"::: ) "(" (Set "(" (Set "(" (Set (Var "U")) ($#k11_monoid_0 :::"-concatenation"::: ) ")" ) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set "(" (Num 1) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Var "U")) ")" ) ")" ) ")" ) "," (Set "(" (Num 2) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Var "U")) ")" ) ")" ))); definitionlet "X" be ($#m1_hidden :::"set"::: ) ; :: original: :::"id"::: redefine func :::"id"::: "X" -> ($#m1_subset_1 :::"Equivalence_Relation":::) "of" "X"; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#v10_fomodel1 :::"ofAtomicFormula"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); mode :::"Interpreter"::: "of" "s" "," "U" -> ($#m1_hidden :::"set"::: ) means :: FOMODEL2:def 2 (Bool it "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) "s" ")" ) ")" ) ($#k4_finseq_2 :::"-tuples_on"::: ) "U" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) )) if (Bool "s" "is" ($#v7_fomodel1 :::"relational"::: ) ) otherwise (Bool it "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) "s" ")" ) ")" ) ($#k4_finseq_2 :::"-tuples_on"::: ) "U" ")" ) "," "U"); end; :: deftheorem defines :::"Interpreter"::: FOMODEL2:def 2 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#v10_fomodel1 :::"ofAtomicFormula"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "b4")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "s")) "is" ($#v7_fomodel1 :::"relational"::: ) )) "implies" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m1_fomodel2 :::"Interpreter"::: ) "of" (Set (Var "s")) "," (Set (Var "U"))) "iff" (Bool (Set (Var "b4")) "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "s")) ")" ) ")" ) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Var "U")) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) )) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "s")) "is" ($#v7_fomodel1 :::"relational"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m1_fomodel2 :::"Interpreter"::: ) "of" (Set (Var "s")) "," (Set (Var "U"))) "iff" (Bool (Set (Var "b4")) "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set (Var "s")) ")" ) ")" ) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Var "U")) ")" ) "," (Set (Var "U"))) ")" ) ")" ")" ))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#v10_fomodel1 :::"ofAtomicFormula"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); :: original: :::"Interpreter"::: redefine mode :::"Interpreter"::: "of" "s" "," "U" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) "s" ")" ) ")" ) ($#k4_finseq_2 :::"-tuples_on"::: ) "U" ")" ) "," (Set "(" "U" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#v8_fomodel1 :::"termal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); cluster -> "U" ($#v5_relat_1 :::"-valued"::: ) for ($#m1_fomodel2 :::"Interpreter"::: ) "of" "s" "," "U"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#v4_fomodel1 :::"literal"::: ) -> ($#v9_fomodel1 :::"own"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S"); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; mode :::"Interpreter"::: "of" "S" "," "U" -> ($#m1_hidden :::"Function":::) means :: FOMODEL2:def 3 (Bool "for" (Set (Var "s")) "being" ($#v9_fomodel1 :::"own"::: ) ($#m1_subset_1 :::"Element":::) "of" "S" "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) "is" ($#m2_fomodel2 :::"Interpreter"::: ) "of" (Set (Var "s")) "," "U")); end; :: deftheorem defines :::"Interpreter"::: FOMODEL2:def 3 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m3_fomodel2 :::"Interpreter"::: ) "of" (Set (Var "S")) "," (Set (Var "U"))) "iff" (Bool "for" (Set (Var "s")) "being" ($#v9_fomodel1 :::"own"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) "is" ($#m2_fomodel2 :::"Interpreter"::: ) "of" (Set (Var "s")) "," (Set (Var "U")))) ")" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_hidden :::"Function":::); attr "f" is "S" "," "U" :::"-interpreter-like"::: means :: FOMODEL2:def 4 (Bool "(" (Bool "f" "is" ($#m3_fomodel2 :::"Interpreter"::: ) "of" "S" "," "U") & (Bool "f" "is" ($#v1_funcop_1 :::"Function-yielding"::: ) ) ")" ); end; :: deftheorem defines :::"-interpreter-like"::: FOMODEL2:def 4 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" (Set (Var "S")) "," (Set (Var "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ) "iff" (Bool "(" (Bool (Set (Var "f")) "is" ($#m3_fomodel2 :::"Interpreter"::: ) "of" (Set (Var "S")) "," (Set (Var "U"))) & (Bool (Set (Var "f")) "is" ($#v1_funcop_1 :::"Function-yielding"::: ) ) ")" ) ")" )))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) "S" "," "U" ($#v1_fomodel2 :::"-interpreter-like"::: ) -> ($#v1_funcop_1 :::"Function-yielding"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#v9_fomodel1 :::"own"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); cluster -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_fomodel2 :::"Interpreter"::: ) "of" "s" "," "U"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) "S" "," "U" ($#v1_fomodel2 :::"-interpreter-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); let "s" be ($#v9_fomodel1 :::"own"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); :: original: :::"."::: redefine func "I" :::"."::: "s" -> ($#m2_fomodel2 :::"Interpreter"::: ) "of" "s" "," "U"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); let "x" be ($#v9_fomodel1 :::"own"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); let "f" be ($#m2_fomodel2 :::"Interpreter"::: ) "of" (Set (Const "x")) "," (Set (Const "U")); cluster (Set "I" ($#k1_funct_4 :::"+*"::: ) (Set "(" "x" ($#k16_funcop_1 :::".-->"::: ) "f" ")" )) -> "S" "," "U" ($#v1_fomodel2 :::"-interpreter-like"::: ) ; end; definitionlet "f" be ($#m1_hidden :::"Function":::); let "x", "y" be ($#m1_hidden :::"set"::: ) ; func "(" "x" "," "y" ")" :::"ReassignIn"::: "f" -> ($#m1_hidden :::"Function":::) equals :: FOMODEL2:def 5 (Set "f" ($#k1_funct_4 :::"+*"::: ) (Set "(" "x" ($#k16_funcop_1 :::".-->"::: ) (Set "(" (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k16_funcop_1 :::".-->"::: ) "y" ")" ) ")" )); end; :: deftheorem defines :::"ReassignIn"::: FOMODEL2:def 5 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set "(" (Set (Var "x")) "," (Set (Var "y")) ")" ($#k5_fomodel2 :::"ReassignIn"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set (Var "x")) ($#k16_funcop_1 :::".-->"::: ) (Set "(" (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k16_funcop_1 :::".-->"::: ) (Set (Var "y")) ")" ) ")" ))))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); let "x" be ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); let "u" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "U")); cluster (Set "(" "x" "," "u" ")" ($#k5_fomodel2 :::"ReassignIn"::: ) "I") -> "S" "," "U" ($#v1_fomodel2 :::"-interpreter-like"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "Y" be ($#m1_hidden :::"set"::: ) ; let "X", "Z" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) -> ($#v1_funcop_1 :::"Function-yielding"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) "X" "," (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "Y" "," "Z" ")" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "X", "Y", "Z" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) "X" ($#v4_relat_1 :::"-defined"::: ) (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "Y" "," "Z" ")" ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v1_funcop_1 :::"Function-yielding"::: ) bbbadV2_FUNCOP_1() for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) "X" "," (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" "Y" "," "Z" ")" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; definitionlet "f" be ($#v1_funcop_1 :::"Function-yielding"::: ) ($#m1_hidden :::"Function":::); let "g" be ($#m1_hidden :::"Function":::); func :::"^^^":::"g" "," "f":::"__"::: -> ($#m1_hidden :::"Function":::) means :: FOMODEL2:def 6 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set "g" ($#k3_relat_1 :::"*"::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"^^^"::: FOMODEL2:def 6 : (Bool "for" (Set (Var "f")) "being" ($#v1_funcop_1 :::"Function-yielding"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "g")) "," (Set (Var "b3")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k6_fomodel2 :::"^^^"::: ) (Set (Var "g")) "," (Set (Var "f")) ($#k6_fomodel2 :::"__"::: ) )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" ) ")" ))); registrationlet "f" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_hidden :::"Function":::); let "g" be ($#m1_hidden :::"Function":::); cluster (Set ($#k6_fomodel2 :::"^^^"::: ) "g" "," "f" ($#k6_fomodel2 :::"__"::: ) ) -> ($#v1_xboole_0 :::"empty"::: ) ; end; definitionlet "f" be ($#v1_funcop_1 :::"Function-yielding"::: ) ($#m1_hidden :::"Function":::); let "g" be ($#m1_hidden :::"Function":::); func :::"^^^":::"f" "," "g":::"__"::: -> ($#m1_hidden :::"Function":::) means :: FOMODEL2:def 7 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "f" ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "g" ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" "g" ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"^^^"::: FOMODEL2:def 7 : (Bool "for" (Set (Var "f")) "being" ($#v1_funcop_1 :::"Function-yielding"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "g")) "," (Set (Var "b3")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k7_fomodel2 :::"^^^"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k7_fomodel2 :::"__"::: ) )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" ) ")" ))); registrationlet "f" be ($#v1_funcop_1 :::"Function-yielding"::: ) ($#m1_hidden :::"Function":::); let "g" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_hidden :::"Function":::); cluster (Set ($#k7_fomodel2 :::"^^^"::: ) "f" "," "g" ($#k7_fomodel2 :::"__"::: ) ) -> ($#v1_xboole_0 :::"empty"::: ) ; end; registrationlet "X" be ($#v3_finseq_1 :::"FinSequence-membered"::: ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) "X" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) -> ($#v1_funcop_1 :::"Function-yielding"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "E", "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "p" be (Set (Const "D")) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"FinSequence":::); let "h" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "D")) "," (Set (Const "E")); cluster (Set "h" ($#k3_relat_1 :::"(*)"::: ) "p") -> (Set ($#k3_finseq_1 :::"len"::: ) "p") ($#v3_card_1 :::"-element"::: ) for ($#m1_hidden :::"FinSequence":::); end; registrationlet "X", "Y" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "Y")); let "p" be (Set (Const "X")) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"FinSequence":::); cluster (Set "f" ($#k3_relat_1 :::"(*)"::: ) "p") -> ($#v1_finseq_1 :::"FinSequence-like"::: ) ; end; registrationlet "E", "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "n" be ($#m1_hidden :::"Nat":::); let "p" be (Set (Const "D")) ($#v5_relat_1 :::"-valued"::: ) (Set (Const "n")) ($#v3_card_1 :::"-element"::: ) ($#m1_hidden :::"FinSequence":::); let "h" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "D")) "," (Set (Const "E")); cluster (Set "h" ($#k3_relat_1 :::"(*)"::: ) "p") -> "n" ($#v3_card_1 :::"-element"::: ) for ($#m2_finseq_1 :::"FinSequence"::: ) "of" "E"; end; theorem :: FOMODEL2:1 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "t0")) "being" (Set ($#k6_numbers :::"0"::: ) ) ($#v14_fomodel1 :::"-termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool (Set (Var "t0")) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "t0")) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))) ; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "u" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "U")); let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); func "(" "I" "," "u" ")" :::"-TermEval"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) "S" ")" ) "," "U" ")" ")" ) means :: FOMODEL2:def 8 (Bool "(" (Bool (Set it ($#k10_funct_5 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) "S" ")" ) ($#k8_funcop_1 :::"-->"::: ) "u")) & (Bool "(" "for" (Set (Var "mm")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k10_funct_5 :::"."::: ) (Set "(" (Set (Var "mm")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_fomodel2 :::"^^^"::: ) (Set "(" "I" ($#k3_relat_1 :::"*"::: ) (Set "(" "S" ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ")" ) "," (Set ($#k6_fomodel2 :::"^^^"::: ) (Set "(" it ($#k10_funct_5 :::"."::: ) (Set (Var "mm")) ")" ) "," (Set "(" "S" ($#k36_fomodel1 :::"-subTerms"::: ) ")" ) ($#k6_fomodel2 :::"__"::: ) ) ($#k7_fomodel2 :::"__"::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"-TermEval"::: FOMODEL2:def 8 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "U")) (Bool "for" (Set (Var "I")) "being" (Set (Var "b1")) "," (Set (Var "b2")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) (Set (Var "S")) ")" ) "," (Set (Var "U")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set "(" (Set (Var "I")) "," (Set (Var "u")) ")" ($#k8_fomodel2 :::"-TermEval"::: ) )) "iff" (Bool "(" (Bool (Set (Set (Var "b5")) ($#k10_funct_5 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) (Set (Var "S")) ")" ) ($#k8_funcop_1 :::"-->"::: ) (Set (Var "u")))) & (Bool "(" "for" (Set (Var "mm")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b5")) ($#k10_funct_5 :::"."::: ) (Set "(" (Set (Var "mm")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_fomodel2 :::"^^^"::: ) (Set "(" (Set (Var "I")) ($#k3_relat_1 :::"*"::: ) (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ")" ) "," (Set ($#k6_fomodel2 :::"^^^"::: ) (Set "(" (Set (Var "b5")) ($#k10_funct_5 :::"."::: ) (Set (Var "mm")) ")" ) "," (Set "(" (Set (Var "S")) ($#k36_fomodel1 :::"-subTerms"::: ) ")" ) ($#k6_fomodel2 :::"__"::: ) ) ($#k7_fomodel2 :::"__"::: ) )) ")" ) ")" ) ")" )))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); let "t" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k35_fomodel1 :::"AllTermsOf"::: ) (Set (Const "S"))); func "I" :::"-TermEval"::: "t" -> ($#m1_subset_1 :::"Element"::: ) "of" "U" means :: FOMODEL2:def 9 (Bool "for" (Set (Var "u1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "U" (Bool "for" (Set (Var "mm")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool "t" ($#r2_hidden :::"in"::: ) (Set (Set "(" "S" ($#k39_fomodel1 :::"-termsOfMaxDepth"::: ) ")" ) ($#k8_nat_1 :::"."::: ) (Set (Var "mm"))))) "holds" (Bool it ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" "(" "I" "," (Set (Var "u1")) ")" ($#k8_fomodel2 :::"-TermEval"::: ) ")" ) ($#k10_funct_5 :::"."::: ) (Set "(" (Set (Var "mm")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k3_funct_2 :::"."::: ) "t")))); end; :: deftheorem defines :::"-TermEval"::: FOMODEL2:def 9 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" (Set (Var "b1")) "," (Set (Var "b2")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "t")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k35_fomodel1 :::"AllTermsOf"::: ) (Set (Var "S"))) (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "U")) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set (Set (Var "I")) ($#k9_fomodel2 :::"-TermEval"::: ) (Set (Var "t")))) "iff" (Bool "for" (Set (Var "u1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "U")) (Bool "for" (Set (Var "mm")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set (Set "(" (Set (Var "S")) ($#k39_fomodel1 :::"-termsOfMaxDepth"::: ) ")" ) ($#k8_nat_1 :::"."::: ) (Set (Var "mm"))))) "holds" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" "(" (Set (Var "I")) "," (Set (Var "u1")) ")" ($#k8_fomodel2 :::"-TermEval"::: ) ")" ) ($#k10_funct_5 :::"."::: ) (Set "(" (Set (Var "mm")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "t")))))) ")" )))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); func "I" :::"-TermEval"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) "S" ")" ) "," "U" means :: FOMODEL2:def 10 (Bool "for" (Set (Var "t")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k35_fomodel1 :::"AllTermsOf"::: ) "S") "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set "I" ($#k9_fomodel2 :::"-TermEval"::: ) (Set (Var "t"))))); end; :: deftheorem defines :::"-TermEval"::: FOMODEL2:def 10 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" (Set (Var "b1")) "," (Set (Var "b2")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k35_fomodel1 :::"AllTermsOf"::: ) (Set (Var "S")) ")" ) "," (Set (Var "U")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "I")) ($#k10_fomodel2 :::"-TermEval"::: ) )) "iff" (Bool "for" (Set (Var "t")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k35_fomodel1 :::"AllTermsOf"::: ) (Set (Var "S"))) "holds" (Bool (Set (Set (Var "b4")) ($#k3_funct_2 :::"."::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "I")) ($#k9_fomodel2 :::"-TermEval"::: ) (Set (Var "t"))))) ")" ))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); func "I" :::"==="::: -> ($#m1_hidden :::"Function":::) equals :: FOMODEL2:def 11 (Set "I" ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set "(" ($#k18_fomodel1 :::"TheEqSymbOf"::: ) "S" ")" ) ($#k16_funcop_1 :::".-->"::: ) (Set "(" "U" ($#k2_fomodel2 :::"-deltaInterpreter"::: ) ")" ) ")" )); end; :: deftheorem defines :::"==="::: FOMODEL2:def 11 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" (Set (Var "b1")) "," (Set (Var "b2")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set (Var "I")) ($#k11_fomodel2 :::"==="::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "I")) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set "(" ($#k18_fomodel1 :::"TheEqSymbOf"::: ) (Set (Var "S")) ")" ) ($#k16_funcop_1 :::".-->"::: ) (Set "(" (Set (Var "U")) ($#k2_fomodel2 :::"-deltaInterpreter"::: ) ")" ) ")" )))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); let "x" be ($#m1_hidden :::"set"::: ) ; attr "x" is "I" :::"-extension"::: means :: FOMODEL2:def 12 (Bool "x" ($#r1_hidden :::"="::: ) (Set "I" ($#k11_fomodel2 :::"==="::: ) )); end; :: deftheorem defines :::"-extension"::: FOMODEL2:def 12 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" (Set (Var "b1")) "," (Set (Var "b2")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" (Set (Var "I")) ($#v2_fomodel2 :::"-extension"::: ) ) "iff" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "I")) ($#k11_fomodel2 :::"==="::: ) )) ")" ))))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); cluster (Set "I" ($#k11_fomodel2 :::"==="::: ) ) -> "I" ($#v2_fomodel2 :::"-extension"::: ) for ($#m1_hidden :::"Function":::); cluster "I" ($#v2_fomodel2 :::"-extension"::: ) -> ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) "I" ($#v2_fomodel2 :::"-extension"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); cluster (Set "I" ($#k11_fomodel2 :::"==="::: ) ) -> "S" "," "U" ($#v1_fomodel2 :::"-interpreter-like"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); let "f" be (Set (Const "I")) ($#v2_fomodel2 :::"-extension"::: ) ($#m1_hidden :::"Function":::); let "s" be ($#v10_fomodel1 :::"ofAtomicFormula"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); :: original: :::"."::: redefine func "f" :::"."::: "s" -> ($#m2_fomodel2 :::"Interpreter"::: ) "of" "s" "," "U"; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); let "phi" be ($#v15_fomodel1 :::"0wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); func "I" :::"-AtomicEval"::: "phi" -> ($#m1_hidden :::"set"::: ) equals :: FOMODEL2:def 13 (Set (Set "(" (Set "(" "I" ($#k11_fomodel2 :::"==="::: ) ")" ) ($#k12_fomodel2 :::"."::: ) (Set "(" (Set "(" "S" ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) "phi" ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" "I" ($#k10_fomodel2 :::"-TermEval"::: ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k34_fomodel1 :::"SubTerms"::: ) "phi" ")" ) ")" )); end; :: deftheorem defines :::"-AtomicEval"::: FOMODEL2:def 13 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" (Set (Var "b1")) "," (Set (Var "b2")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "phi")) "being" ($#v15_fomodel1 :::"0wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "I")) ($#k13_fomodel2 :::"-AtomicEval"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "I")) ($#k11_fomodel2 :::"==="::: ) ")" ) ($#k12_fomodel2 :::"."::: ) (Set "(" (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "phi")) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "I")) ($#k10_fomodel2 :::"-TermEval"::: ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k34_fomodel1 :::"SubTerms"::: ) (Set (Var "phi")) ")" ) ")" ))))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); let "phi" be ($#v15_fomodel1 :::"0wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); :: original: :::"-AtomicEval"::: redefine func "I" :::"-AtomicEval"::: "phi" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); cluster (Set "I" ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S" ")" )) -> (Set ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set "(" "U" ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "(" "U" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ")" ) ($#v5_relat_1 :::"-valued"::: ) for ($#m1_hidden :::"Function":::); cluster (Set "I" ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S" ")" )) -> "S" "," "U" ($#v1_fomodel2 :::"-interpreter-like"::: ) for ($#m1_hidden :::"Function":::); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); cluster (Set "I" ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S" ")" )) -> (Set ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S") ($#v4_relat_1 :::"-defined"::: ) ($#v1_partfun1 :::"total"::: ) for(Set ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S") ($#v4_relat_1 :::"-defined"::: ) ($#m1_hidden :::"Relation":::); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; func "U" :::"-InterpretersOf"::: "S" -> ($#m1_hidden :::"set"::: ) equals :: FOMODEL2:def 14 "{" (Set (Var "f")) where f "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S" ")" ) "," (Set "(" ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set "(" "U" ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "(" "U" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ")" ")" ) ")" ) : (Bool (Set (Var "f")) "is" "S" "," "U" ($#v1_fomodel2 :::"-interpreter-like"::: ) ) "}" ; end; :: deftheorem defines :::"-InterpretersOf"::: FOMODEL2:def 14 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "U")) ($#k15_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "f")) where f "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set "(" (Set (Var "U")) ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "(" (Set (Var "U")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ")" ")" ) ")" ) : (Bool (Set (Var "f")) "is" (Set (Var "S")) "," (Set (Var "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ) "}" ))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; :: original: :::"-InterpretersOf"::: redefine func "U" :::"-InterpretersOf"::: "S" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S" ")" ) "," (Set "(" ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set "(" "U" ($#k3_finseq_2 :::"*"::: ) ")" ) "," (Set "(" "U" ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) ")" ")" ) ")" ")" ); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster (Set "U" ($#k15_fomodel2 :::"-InterpretersOf"::: ) "S") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster -> "S" "," "U" ($#v1_fomodel2 :::"-interpreter-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S"); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; func "S" :::"-TruthEval"::: "U" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S" ")" ) "," (Set "(" ($#k32_fomodel1 :::"AtomicFormulasOf"::: ) "S" ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) means :: FOMODEL2:def 15 (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S") (Bool "for" (Set (Var "phi")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k32_fomodel1 :::"AtomicFormulasOf"::: ) "S") "holds" (Bool (Set it ($#k2_binop_1 :::"."::: ) "(" (Set (Var "I")) "," (Set (Var "phi")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "I")) ($#k14_fomodel2 :::"-AtomicEval"::: ) (Set (Var "phi")))))); end; :: deftheorem defines :::"-TruthEval"::: FOMODEL2:def 15 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k32_fomodel1 :::"AtomicFormulasOf"::: ) (Set (Var "S")) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k17_fomodel2 :::"-TruthEval"::: ) (Set (Var "U")))) "iff" (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) (Bool "for" (Set (Var "phi")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k32_fomodel1 :::"AtomicFormulasOf"::: ) (Set (Var "S"))) "holds" (Bool (Set (Set (Var "b3")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "I")) "," (Set (Var "phi")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "I")) ($#k14_fomodel2 :::"-AtomicEval"::: ) (Set (Var "phi")))))) ")" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S")) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Const "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ); let "phi" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Const "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )); func "f" :::"-ExFunctor"::: "(" "I" "," "phi" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) equals :: FOMODEL2:def 16 (Set ($#k8_margrel1 :::"TRUE"::: ) ) if (Bool "ex" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "U"(Bool "ex" (Set (Var "v")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" "S" "st" (Bool "(" (Bool (Set "phi" ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "v"))) & (Bool (Set "f" ($#k1_binop_1 :::"."::: ) "(" (Set "(" "(" (Set (Var "v")) "," (Set (Var "u")) ")" ($#k5_fomodel2 :::"ReassignIn"::: ) "I" ")" ) "," (Set "(" "phi" ($#k1_rfinseq :::"/^"::: ) (Num 1) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" ))) otherwise (Set ($#k7_margrel1 :::"FALSE"::: ) ); end; :: deftheorem defines :::"-ExFunctor"::: FOMODEL2:def 16 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~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"))) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S")) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) (Bool "for" (Set (Var "phi")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "holds" (Bool "(" "(" (Bool (Bool "ex" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "U"))(Bool "ex" (Set (Var "v")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Set (Var "phi")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "v"))) & (Bool (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set "(" "(" (Set (Var "v")) "," (Set (Var "u")) ")" ($#k5_fomodel2 :::"ReassignIn"::: ) (Set (Var "I")) ")" ) "," (Set "(" (Set (Var "phi")) ($#k1_rfinseq :::"/^"::: ) (Num 1) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )))) "implies" (Bool (Set (Set (Var "f")) ($#k18_fomodel2 :::"-ExFunctor"::: ) "(" (Set (Var "I")) "," (Set (Var "phi")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" & "(" (Bool (Bool "(" "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "U")) (Bool "for" (Set (Var "v")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "holds" (Bool "(" "not" (Bool (Set (Set (Var "phi")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "v"))) "or" "not" (Bool (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set "(" "(" (Set (Var "v")) "," (Set (Var "u")) ")" ($#k5_fomodel2 :::"ReassignIn"::: ) (Set (Var "I")) ")" ) "," (Set "(" (Set (Var "phi")) ($#k1_rfinseq :::"/^"::: ) (Num 1) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )) ")" )) "implies" (Bool (Set (Set (Var "f")) ($#k18_fomodel2 :::"-ExFunctor"::: ) "(" (Set (Var "I")) "," (Set (Var "phi")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) ")" ")" )))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "g" be ($#m2_rfunct_3 :::"Element"::: ) "of" (Set ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S")) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Const "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ); func :::"ExIterator"::: "g" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S" ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) means :: FOMODEL2:def 17 (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S") (Bool "for" (Set (Var "y")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) it)) "iff" (Bool "ex" (Set (Var "v")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" "S"(Bool "ex" (Set (Var "w")) "being" ($#m2_subset_1 :::"string":::) "of" "S" "st" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "w")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "g")) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "v")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "w")))) ")" ))) ")" )) ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S") (Bool "for" (Set (Var "y")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "st" (Bool (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set "g" ($#k18_fomodel2 :::"-ExFunctor"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"ExIterator"::: FOMODEL2:def 17 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "being" ($#m2_rfunct_3 :::"Element"::: ) "of" (Set ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S")) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (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 :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k19_fomodel2 :::"ExIterator"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) (Bool "for" (Set (Var "y")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b4")))) "iff" (Bool "ex" (Set (Var "v")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S"))(Bool "ex" (Set (Var "w")) "being" ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "w")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "v")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "w")))) ")" ))) ")" )) ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) (Bool "for" (Set (Var "y")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "st" (Bool (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b4"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k18_fomodel2 :::"-ExFunctor"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ))) ")" ) ")" ) ")" ))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S")) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Const "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ); let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); let "phi" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Const "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )); func "f" :::"-NorFunctor"::: "(" "I" "," "phi" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) equals :: FOMODEL2:def 18 (Set ($#k8_margrel1 :::"TRUE"::: ) ) if (Bool "ex" (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "st" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) "I" "," (Set (Var "w1")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool (Set "f" ($#k1_binop_1 :::"."::: ) "(" "I" "," (Set (Var "w1")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) & (Bool (Set "f" ($#k1_binop_1 :::"."::: ) "(" "I" "," (Set (Var "w2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) & (Bool "phi" ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "w1")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "w2")))) ")" )) otherwise (Set ($#k7_margrel1 :::"FALSE"::: ) ); end; :: deftheorem defines :::"-NorFunctor"::: FOMODEL2:def 18 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (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 :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) (Bool "for" (Set (Var "phi")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "holds" (Bool "(" "(" (Bool (Bool "ex" (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "st" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "I")) "," (Set (Var "w1")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "I")) "," (Set (Var "w1")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "I")) "," (Set (Var "w2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) & (Bool (Set (Var "phi")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "w1")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "w2")))) ")" ))) "implies" (Bool (Set (Set (Var "f")) ($#k20_fomodel2 :::"-NorFunctor"::: ) "(" (Set (Var "I")) "," (Set (Var "phi")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" & "(" (Bool (Bool "(" "for" (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "holds" (Bool "(" "not" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "I")) "," (Set (Var "w1")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) "or" "not" (Bool (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "I")) "," (Set (Var "w1")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) "or" "not" (Bool (Set (Set (Var "f")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "I")) "," (Set (Var "w2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) "or" "not" (Bool (Set (Var "phi")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "w1")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "w2")))) ")" ) ")" )) "implies" (Bool (Set (Set (Var "f")) ($#k20_fomodel2 :::"-NorFunctor"::: ) "(" (Set (Var "I")) "," (Set (Var "phi")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) ")" ")" )))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "g" be ($#m2_rfunct_3 :::"Element"::: ) "of" (Set ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S")) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Const "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ); func :::"NorIterator"::: "g" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S" ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) means :: FOMODEL2:def 19 (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S") (Bool "for" (Set (Var "y")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) it)) "iff" (Bool "ex" (Set (Var "phi1")) "," (Set (Var "phi2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "st" (Bool "(" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "phi2")))) & (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "phi1")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "g")) & (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "phi2")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "g")) ")" )) ")" )) ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S") (Bool "for" (Set (Var "y")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "st" (Bool (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set "g" ($#k20_fomodel2 :::"-NorFunctor"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"NorIterator"::: FOMODEL2:def 19 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "being" ($#m2_rfunct_3 :::"Element"::: ) "of" (Set ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (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 :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S")) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k21_fomodel2 :::"NorIterator"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) (Bool "for" (Set (Var "y")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b4")))) "iff" (Bool "ex" (Set (Var "phi1")) "," (Set (Var "phi2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "st" (Bool "(" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "phi2")))) & (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "phi1")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "phi2")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) ")" )) ")" )) ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) (Bool "for" (Set (Var "y")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) "st" (Bool (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b4"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k20_fomodel2 :::"-NorFunctor"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ))) ")" ) ")" ) ")" ))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; func "(" "S" "," "U" ")" :::"-TruthEval"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S" ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ")" ) means :: FOMODEL2:def 20 (Bool "(" (Bool (Set it ($#k8_nat_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set "S" ($#k17_fomodel2 :::"-TruthEval"::: ) "U")) & (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_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k19_fomodel2 :::"ExIterator"::: ) (Set "(" it ($#k8_nat_1 :::"."::: ) (Set (Var "mm")) ")" ) ")" ) ($#k7_funct_4 :::"+*"::: ) (Set "(" ($#k21_fomodel2 :::"NorIterator"::: ) (Set "(" it ($#k8_nat_1 :::"."::: ) (Set (Var "mm")) ")" ) ")" ) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" it ($#k8_nat_1 :::"."::: ) (Set (Var "mm")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"-TruthEval"::: FOMODEL2:def 20 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S")) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set "(" (Set (Var "S")) "," (Set (Var "U")) ")" ($#k22_fomodel2 :::"-TruthEval"::: ) )) "iff" (Bool "(" (Bool (Set (Set (Var "b3")) ($#k8_nat_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k17_fomodel2 :::"-TruthEval"::: ) (Set (Var "U")))) & (Bool "(" "for" (Set (Var "mm")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b3")) ($#k8_nat_1 :::"."::: ) (Set "(" (Set (Var "mm")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k19_fomodel2 :::"ExIterator"::: ) (Set "(" (Set (Var "b3")) ($#k8_nat_1 :::"."::: ) (Set (Var "mm")) ")" ) ")" ) ($#k7_funct_4 :::"+*"::: ) (Set "(" ($#k21_fomodel2 :::"NorIterator"::: ) (Set "(" (Set (Var "b3")) ($#k8_nat_1 :::"."::: ) (Set (Var "mm")) ")" ) ")" ) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set (Var "b3")) ($#k8_nat_1 :::"."::: ) (Set (Var "mm")) ")" ))) ")" ) ")" ) ")" )))); theorem :: FOMODEL2:2 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" (Set (Var "b1")) "," (Set (Var "b2")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set (Var "I")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) (Set (Var "S")) ")" )) ($#r2_hidden :::"in"::: ) (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))))))) ; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; func "(" "S" "," "U" ")" :::"-TruthEval"::: "m" -> ($#m2_rfunct_3 :::"Element"::: ) "of" (Set ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S" ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) means :: FOMODEL2:def 21 (Bool "for" (Set (Var "mm")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool "m" ($#r1_hidden :::"="::: ) (Set (Var "mm")))) "holds" (Bool it ($#r1_hidden :::"="::: ) (Set (Set "(" "(" "S" "," "U" ")" ($#k22_fomodel2 :::"-TruthEval"::: ) ")" ) ($#k8_nat_1 :::"."::: ) (Set (Var "mm"))))); end; :: deftheorem defines :::"-TruthEval"::: FOMODEL2:def 21 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b4")) "being" ($#m2_rfunct_3 :::"Element"::: ) "of" (Set ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S")) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set "(" (Set (Var "S")) "," (Set (Var "U")) ")" ($#k23_fomodel2 :::"-TruthEval"::: ) (Set (Var "m")))) "iff" (Bool "for" (Set (Var "mm")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set (Var "mm")))) "holds" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set "(" "(" (Set (Var "S")) "," (Set (Var "U")) ")" ($#k22_fomodel2 :::"-TruthEval"::: ) ")" ) ($#k8_nat_1 :::"."::: ) (Set (Var "mm"))))) ")" ))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "m" be ($#m1_hidden :::"Nat":::); let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); func "(" "I" "," "m" ")" :::"-TruthEval"::: -> ($#m2_rfunct_3 :::"Element"::: ) "of" (Set ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ) equals :: FOMODEL2:def 22 (Set (Set "(" ($#k1_funct_5 :::"curry"::: ) (Set "(" "(" "S" "," "U" ")" ($#k23_fomodel2 :::"-TruthEval"::: ) "m" ")" ) ")" ) ($#k1_funct_1 :::"."::: ) "I"); end; :: deftheorem defines :::"-TruthEval"::: FOMODEL2:def 22 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) "holds" (Bool (Set "(" (Set (Var "I")) "," (Set (Var "m")) ")" ($#k24_fomodel2 :::"-TruthEval"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_funct_5 :::"curry"::: ) (Set "(" "(" (Set (Var "S")) "," (Set (Var "U")) ")" ($#k23_fomodel2 :::"-TruthEval"::: ) (Set (Var "m")) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "I")))))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); func "S" :::"-formulasOfMaxDepth"::: "m" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) means :: FOMODEL2:def 23 (Bool "for" (Set (Var "U")) "being" ($#~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") (Bool "for" (Set (Var "mm")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool "m" ($#r1_hidden :::"="::: ) (Set (Var "mm")))) "holds" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" "(" (Set (Var "I")) "," (Set (Var "mm")) ")" ($#k24_fomodel2 :::"-TruthEval"::: ) ")" )))))); end; :: deftheorem defines :::"-formulasOfMaxDepth"::: FOMODEL2:def 23 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) (Set (Var "m")))) "iff" (Bool "for" (Set (Var "U")) "being" ($#~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"))) (Bool "for" (Set (Var "mm")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set (Var "mm")))) "holds" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" "(" (Set (Var "I")) "," (Set (Var "mm")) ")" ($#k24_fomodel2 :::"-TruthEval"::: ) ")" )))))) ")" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); let "w" be ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); attr "w" is "m" :::"-wff"::: means :: FOMODEL2:def 24 (Bool "w" ($#r2_hidden :::"in"::: ) (Set "S" ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) "m")); end; :: deftheorem defines :::"-wff"::: FOMODEL2:def 24 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "w")) "being" ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "w")) "is" (Set (Var "m")) ($#v3_fomodel2 :::"-wff"::: ) ) "iff" (Bool (Set (Var "w")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "S")) ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) (Set (Var "m")))) ")" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "w" be ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); attr "w" is :::"wff"::: means :: FOMODEL2:def 25 (Bool "ex" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "w" "is" (Set (Var "m")) ($#v3_fomodel2 :::"-wff"::: ) )); end; :: deftheorem defines :::"wff"::: FOMODEL2:def 25 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "w")) "being" ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "w")) "is" ($#v4_fomodel2 :::"wff"::: ) ) "iff" (Bool "ex" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set (Var "w")) "is" (Set (Var "m")) ($#v3_fomodel2 :::"-wff"::: ) )) ")" ))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k6_numbers :::"0"::: ) ) ($#v3_fomodel2 :::"-wff"::: ) -> ($#v15_fomodel1 :::"0wff"::: ) 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 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )); cluster ($#v15_fomodel1 :::"0wff"::: ) -> (Set ($#k6_numbers :::"0"::: ) ) ($#v3_fomodel2 :::"-wff"::: ) 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 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )); let "m" be ($#m1_hidden :::"Nat":::); cluster "m" ($#v3_fomodel2 :::"-wff"::: ) -> ($#v4_fomodel2 :::"wff"::: ) 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 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )); let "n" be ($#m1_hidden :::"Nat":::); cluster (Set "m" ($#k1_nat_1 :::"+"::: ) (Set "(" (Set ($#k6_numbers :::"0"::: ) ) ($#k4_nat_1 :::"*"::: ) "n" ")" )) ($#v3_fomodel2 :::"-wff"::: ) -> (Set "m" ($#k2_xcmplx_0 :::"+"::: ) "n") ($#v3_fomodel2 :::"-wff"::: ) 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 "m" be ($#m1_hidden :::"Nat":::); cluster ($#~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"::: ) bbbadV2_PRE_POLY() "m" ($#v3_fomodel2 :::"-wff"::: ) for ($#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 ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); cluster (Set "S" ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) "m") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "S" be ($#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"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) bbbadV2_PRE_POLY() ($#v4_fomodel2 :::"wff"::: ) 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 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); let "w" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); func "I" :::"-TruthEval"::: "w" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) means :: FOMODEL2:def 26 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool "w" "is" (Set (Var "m")) ($#v3_fomodel2 :::"-wff"::: ) )) "holds" (Bool it ($#r1_hidden :::"="::: ) (Set (Set "(" "(" "I" "," (Set (Var "m")) ")" ($#k24_fomodel2 :::"-TruthEval"::: ) ")" ) ($#k1_funct_1 :::"."::: ) "w"))); end; :: deftheorem defines :::"-TruthEval"::: FOMODEL2:def 26 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~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"))) (Bool "for" (Set (Var "w")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "w")))) "iff" (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "w")) "is" (Set (Var "m")) ($#v3_fomodel2 :::"-wff"::: ) )) "holds" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set (Set "(" "(" (Set (Var "I")) "," (Set (Var "m")) ")" ($#k24_fomodel2 :::"-TruthEval"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "w"))))) ")" )))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); func :::"AllFormulasOf"::: "S" -> ($#m1_hidden :::"set"::: ) equals :: FOMODEL2:def 27 "{" (Set (Var "w")) where w "is" ($#m2_subset_1 :::"string":::) "of" "S" : (Bool "ex" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set (Var "w")) "is" (Set (Var "m")) ($#v3_fomodel2 :::"-wff"::: ) )) "}" ; end; :: deftheorem defines :::"AllFormulasOf"::: FOMODEL2:def 27 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k27_fomodel2 :::"AllFormulasOf"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "w")) where w "is" ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) : (Bool "ex" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set (Var "w")) "is" (Set (Var "m")) ($#v3_fomodel2 :::"-wff"::: ) )) "}" )); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k27_fomodel2 :::"AllFormulasOf"::: ) "S") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: FOMODEL2:3 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "U")) (Bool "for" (Set (Var "t")) "being" ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "I")) "being" (Set (Var "b2")) "," (Set (Var "b3")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" "(" (Set (Var "I")) "," (Set (Var "u")) ")" ($#k8_fomodel2 :::"-TermEval"::: ) ")" ) ($#k10_funct_5 :::"."::: ) (Set "(" (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "I")) ($#k4_fomodel2 :::"."::: ) (Set "(" (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "t")) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set "(" "(" (Set (Var "I")) "," (Set (Var "u")) ")" ($#k8_fomodel2 :::"-TermEval"::: ) ")" ) ($#k8_nat_1 :::"."::: ) (Set (Var "m")) ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k33_fomodel1 :::"SubTerms"::: ) (Set (Var "t")) ")" ) ")" ))) & "(" (Bool (Bool (Set (Var "t")) "is" (Set ($#k6_numbers :::"0"::: ) ) ($#v14_fomodel1 :::"-termal"::: ) )) "implies" (Bool (Set (Set "(" (Set "(" "(" (Set (Var "I")) "," (Set (Var "u")) ")" ($#k8_fomodel2 :::"-TermEval"::: ) ")" ) ($#k10_funct_5 :::"."::: ) (Set "(" (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "I")) ($#k4_fomodel2 :::"."::: ) (Set "(" (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "t")) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ")" ")" ))))))) ; theorem :: FOMODEL2:4 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "u1")) "," (Set (Var "u2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "U")) (Bool "for" (Set (Var "I")) "being" (Set (Var "b3")) "," (Set (Var "b4")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "t")) "being" (Set (Var "b1")) ($#v14_fomodel1 :::"-termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set "(" (Set "(" "(" (Set (Var "I")) "," (Set (Var "u1")) ")" ($#k8_fomodel2 :::"-TermEval"::: ) ")" ) ($#k10_funct_5 :::"."::: ) (Set "(" (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" "(" (Set (Var "I")) "," (Set (Var "u2")) ")" ($#k8_fomodel2 :::"-TermEval"::: ) ")" ) ($#k10_funct_5 :::"."::: ) (Set "(" (Set "(" (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "t")))))))))) ; theorem :: FOMODEL2:5 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_funct_5 :::"curry"::: ) (Set "(" "(" (Set (Var "S")) "," (Set (Var "U")) ")" ($#k23_fomodel2 :::"-TruthEval"::: ) (Set (Var "m")) ")" )) "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k3_rfunct_3 :::"PFuncs"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ")" ))))) ; theorem :: FOMODEL2:6 (Bool "for" (Set (Var "x")) "," (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Y")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Z")))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) "or" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) "or" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z"))) ")" ) ")" )) ; theorem :: FOMODEL2:7 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set (Set (Var "S")) ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k32_fomodel1 :::"AtomicFormulasOf"::: ) (Set (Var "S"))))) ; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); redefine func "S" :::"-formulasOfMaxDepth"::: "m" means :: FOMODEL2:def 28 (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")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S") "holds" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" "(" (Set (Var "I")) "," "m" ")" ($#k24_fomodel2 :::"-TruthEval"::: ) ")" ))))); end; :: deftheorem defines :::"-formulasOfMaxDepth"::: FOMODEL2:def 28 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) (Set (Var "m")))) "iff" (Bool "for" (Set (Var "U")) "being" ($#~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"))) "holds" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" "(" (Set (Var "I")) "," (Set (Var "m")) ")" ($#k24_fomodel2 :::"-TruthEval"::: ) ")" ))))) ")" )))); theorem :: FOMODEL2:8 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set "(" (Set (Var "S")) "," (Set (Var "U")) ")" ($#k23_fomodel2 :::"-TruthEval"::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S")) ")" ) "," (Set "(" (Set (Var "S")) ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) (Set (Var "m")) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )) & (Bool (Set (Set "(" "(" (Set (Var "S")) "," (Set (Var "U")) ")" ($#k22_fomodel2 :::"-TruthEval"::: ) ")" ) ($#k8_nat_1 :::"."::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k8_mcart_1 :::"[:"::: ) (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S")) ")" ) "," (Set "(" (Set (Var "S")) ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) (Set (Var "m")) ")" ) ($#k8_mcart_1 :::":]"::: ) ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )) ")" )))) ; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); func "m" :::"-ExFormulasOf"::: "S" -> ($#m1_hidden :::"set"::: ) equals :: FOMODEL2:def 29 "{" (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "v")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "phi")) ")" ) where v "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k16_fomodel1 :::"LettersOf"::: ) "S"), phi "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set "S" ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) "m") : (Bool verum) "}" ; func "m" :::"-NorFormulasOf"::: "S" -> ($#m1_hidden :::"set"::: ) equals :: FOMODEL2:def 30 "{" (Set "(" (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "phi2")) ")" ) where phi1, phi2 "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set "S" ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) "m") : (Bool verum) "}" ; end; :: deftheorem defines :::"-ExFormulasOf"::: FOMODEL2:def 29 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "m")) ($#k28_fomodel2 :::"-ExFormulasOf"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "v")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "phi")) ")" ) where v "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k16_fomodel1 :::"LettersOf"::: ) (Set (Var "S"))), phi "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) (Set (Var "m"))) : (Bool verum) "}" ))); :: deftheorem defines :::"-NorFormulasOf"::: FOMODEL2:def 30 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "m")) ($#k29_fomodel2 :::"-NorFormulasOf"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "phi2")) ")" ) where phi1, phi2 "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "S")) ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) (Set (Var "m"))) : (Bool verum) "}" ))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "w1", "w2" be ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); :: original: :::"^"::: redefine func "w1" :::"^"::: "w2" -> ($#m2_subset_1 :::"string":::) "of" "S"; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "s" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); :: original: :::"<*"::: redefine func :::"<*":::"s":::"*>"::: -> ($#m2_subset_1 :::"string":::) "of" "S"; end; theorem :: FOMODEL2:9 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set (Set (Var "S")) ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) (Set "(" (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "m")) ($#k28_fomodel2 :::"-ExFormulasOf"::: ) (Set (Var "S")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" (Set (Var "m")) ($#k29_fomodel2 :::"-NorFormulasOf"::: ) (Set (Var "S")) ")" ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" (Set (Var "S")) ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) (Set (Var "m")) ")" ))))) ; theorem :: FOMODEL2:10 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k32_fomodel1 :::"AtomicFormulasOf"::: ) (Set (Var "S"))) "is" (Set (Var "S")) ($#v12_fomodel1 :::"-prefix"::: ) )) ; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k31_fomodel1 :::"AtomicFormulasOf"::: ) "S") -> "S" ($#v12_fomodel1 :::"-prefix"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set "S" ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) (Set ($#k6_numbers :::"0"::: ) )) -> "S" ($#v12_fomodel1 :::"-prefix"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); func :::"Depth"::: "phi" -> ($#m1_hidden :::"Nat":::) means :: FOMODEL2:def 31 (Bool "(" (Bool "phi" "is" it ($#v3_fomodel2 :::"-wff"::: ) ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool "phi" "is" (Set (Var "n")) ($#v3_fomodel2 :::"-wff"::: ) )) "holds" (Bool it ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) ")" ) ")" ); end; :: deftheorem defines :::"Depth"::: FOMODEL2:def 31 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k32_fomodel2 :::"Depth"::: ) (Set (Var "phi")))) "iff" (Bool "(" (Bool (Set (Var "phi")) "is" (Set (Var "b3")) ($#v3_fomodel2 :::"-wff"::: ) ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "phi")) "is" (Set (Var "n")) ($#v3_fomodel2 :::"-wff"::: ) )) "holds" (Bool (Set (Var "b3")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) ")" ) ")" ) ")" )))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); let "phi1", "phi2" be (Set (Const "m")) ($#v3_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 "m" ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#v3_fomodel2 :::"-wff"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#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" ")" ) ($#k7_finseq_1 :::"^"::: ) "phi2") -> ($#v4_fomodel2 :::"wff"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); let "phi" be (Set (Const "m")) ($#v3_fomodel2 :::"-wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "v" be ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); cluster (Set (Set ($#k31_fomodel2 :::"<*"::: ) "v" ($#k31_fomodel2 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) "phi") -> (Set "m" ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#v3_fomodel2 :::"-wff"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "l" be ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set (Set ($#k31_fomodel2 :::"<*"::: ) "l" ($#k31_fomodel2 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) "phi") -> ($#v4_fomodel2 :::"wff"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "w" be ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "s" be ($#~v7_fomodel1 "non" ($#v7_fomodel1 :::"relational"::: ) ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); cluster (Set (Set ($#k31_fomodel2 :::"<*"::: ) "s" ($#k31_fomodel2 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) "w") -> ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "w1", "w2" be ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "s" be ($#~v7_fomodel1 "non" ($#v7_fomodel1 :::"relational"::: ) ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); cluster (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) "s" ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "w1" ")" ) ($#k7_finseq_1 :::"^"::: ) "w2") -> ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k8_fomodel1 :::"TheNorSymbOf"::: ) "S") -> ($#~v7_fomodel1 "non" ($#v7_fomodel1 :::"relational"::: ) ) for ($#m1_subset_1 :::"Element":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "w" be ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) "w") -> ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "l" be ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); let "w" be ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set (Set ($#k31_fomodel2 :::"<*"::: ) "l" ($#k31_fomodel2 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) "w") -> ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "w" be ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); attr "w" is :::"exal"::: means :: FOMODEL2:def 32 (Bool (Set (Set "(" "S" ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) "w") "is" ($#v4_fomodel1 :::"literal"::: ) ); end; :: deftheorem defines :::"exal"::: FOMODEL2:def 32 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "w")) "being" ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "w")) "is" ($#v5_fomodel2 :::"exal"::: ) ) "iff" (Bool (Set (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "w"))) "is" ($#v4_fomodel1 :::"literal"::: ) ) ")" ))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "w" be ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "l" be ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); cluster (Set (Set ($#k31_fomodel2 :::"<*"::: ) "l" ($#k31_fomodel2 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) "w") -> ($#v5_fomodel2 :::"exal"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "m1" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); 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"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) bbbadV2_PRE_POLY() "m1" ($#v3_fomodel2 :::"-wff"::: ) ($#v4_fomodel2 :::"wff"::: ) ($#v5_fomodel2 :::"exal"::: ) 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":::); cluster ($#v5_fomodel2 :::"exal"::: ) -> ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) for ($#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 ($#l1_fomodel1 :::"Language":::); let "m1" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); 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"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) bbbadV2_PRE_POLY() ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) "m1" ($#v3_fomodel2 :::"-wff"::: ) ($#v4_fomodel2 :::"wff"::: ) ($#v5_fomodel2 :::"exal"::: ) 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":::); 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"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) bbbadV2_PRE_POLY() ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) ($#v4_fomodel2 :::"wff"::: ) ($#v5_fomodel2 :::"exal"::: ) for ($#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 ($#l1_fomodel1 :::"Language":::); let "phi" be ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k32_fomodel2 :::"Depth"::: ) "phi") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) for ($#m1_hidden :::"Nat":::); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "w" be ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set (Set "(" "S" ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k1_funct_1 :::"."::: ) "w") -> ($#~v7_fomodel1 "non" ($#v7_fomodel1 :::"relational"::: ) ) for ($#m1_subset_1 :::"Element":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); cluster (Set "S" ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) "m") -> "S" ($#v12_fomodel1 :::"-prefix"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); :: original: :::"AllFormulasOf"::: redefine func :::"AllFormulasOf"::: "S" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster -> ($#v4_fomodel2 :::"wff"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S"); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k27_fomodel2 :::"AllFormulasOf"::: ) "S") -> "S" ($#v12_fomodel1 :::"-prefix"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FOMODEL2:11 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k21_fomodel2 :::"NorIterator"::: ) (Set "(" "(" (Set (Var "S")) "," (Set (Var "U")) ")" ($#k23_fomodel2 :::"-TruthEval"::: ) (Set (Var "m")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S")) ")" ) "," (Set "(" (Set (Var "m")) ($#k29_fomodel2 :::"-NorFormulasOf"::: ) (Set (Var "S")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))))) ; theorem :: FOMODEL2:12 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k19_fomodel2 :::"ExIterator"::: ) (Set "(" "(" (Set (Var "S")) "," (Set (Var "U")) ")" ($#k23_fomodel2 :::"-TruthEval"::: ) (Set (Var "m")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S")) ")" ) "," (Set "(" (Set (Var "m")) ($#k28_fomodel2 :::"-ExFormulasOf"::: ) (Set (Var "S")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))))) ; theorem :: FOMODEL2:13 (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" (Set (Var "U")) ($#k2_fomodel2 :::"-deltaInterpreter"::: ) ")" ) ($#k8_relset_1 :::"""::: ) (Set ($#k6_domain_1 :::"{"::: ) (Num 1) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) "{" (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "u")) "," (Set (Var "u")) ($#k10_finseq_1 :::"*>"::: ) ) where u "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "U")) : (Bool verum) "}" )) ; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); :: original: :::"TheEqSymbOf"::: redefine func :::"TheEqSymbOf"::: "S" -> ($#m1_subset_1 :::"Element":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 2)) -> ($#v1_xboole_0 :::"zero"::: ) for ($#m1_hidden :::"number"::: ) ; cluster (Set (Set "(" ($#k1_int_2 :::"abs"::: ) (Set "(" ($#k19_fomodel1 :::"ar"::: ) (Set "(" ($#k34_fomodel2 :::"TheEqSymbOf"::: ) "S" ")" ) ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 2)) -> ($#v1_xboole_0 :::"zero"::: ) for ($#m1_hidden :::"number"::: ) ; end; theorem :: FOMODEL2:14 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "phi")) "being" ($#v15_fomodel1 :::"0wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "I")) "being" (Set (Var "b1")) "," (Set (Var "b2")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "phi"))) ($#r1_hidden :::"<>"::: ) (Set ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S"))))) "implies" (Bool (Set (Set (Var "I")) ($#k14_fomodel2 :::"-AtomicEval"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "I")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "phi")) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "I")) ($#k10_fomodel2 :::"-TermEval"::: ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k34_fomodel1 :::"SubTerms"::: ) (Set (Var "phi")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Set ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S"))))) "implies" (Bool (Set (Set (Var "I")) ($#k14_fomodel2 :::"-AtomicEval"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "U")) ($#k2_fomodel2 :::"-deltaInterpreter"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "I")) ($#k10_fomodel2 :::"-TermEval"::: ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k34_fomodel1 :::"SubTerms"::: ) (Set (Var "phi")) ")" ) ")" ))) ")" ")" ))))) ; theorem :: FOMODEL2:15 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" (Set (Var "b1")) "," (Set (Var "b2")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "phi")) "being" ($#v15_fomodel1 :::"0wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Set ($#k34_fomodel2 :::"TheEqSymbOf"::: ) (Set (Var "S"))))) "holds" (Bool "(" (Bool (Set (Set (Var "I")) ($#k14_fomodel2 :::"-AtomicEval"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Num 1)) "iff" (Bool (Set (Set "(" (Set (Var "I")) ($#k10_fomodel2 :::"-TermEval"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k34_fomodel1 :::"SubTerms"::: ) (Set (Var "phi")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "I")) ($#k10_fomodel2 :::"-TermEval"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k34_fomodel1 :::"SubTerms"::: ) (Set (Var "phi")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 2) ")" ))) ")" ))))) ; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); cluster (Set "m" ($#k28_fomodel2 :::"-ExFormulasOf"::: ) "S") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); cluster (Set "m" ($#k29_fomodel2 :::"-NorFormulasOf"::: ) "S") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); :: original: :::"-NorFormulasOf"::: redefine func "m" :::"-NorFormulasOf"::: "S" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" (Set "(" ($#k1_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 "w" be ($#v5_fomodel2 :::"exal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set (Set "(" "S" ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k1_funct_1 :::"."::: ) "w") -> ($#v4_fomodel1 :::"literal"::: ) for ($#m1_subset_1 :::"Element":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); cluster -> ($#~v5_fomodel2 "non" ($#v5_fomodel2 :::"exal"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "m" ($#k35_fomodel2 :::"-NorFormulasOf"::: ) "S"); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); :: original: :::"-ExFormulasOf"::: redefine func "m" :::"-ExFormulasOf"::: "S" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" (Set "(" ($#k1_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 "m" be ($#m1_hidden :::"Nat":::); cluster -> ($#v5_fomodel2 :::"exal"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "m" ($#k36_fomodel2 :::"-ExFormulasOf"::: ) "S"); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#~v4_fomodel1 "non" ($#v4_fomodel1 :::"literal"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S"); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "w" be ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "s" be ($#~v4_fomodel1 "non" ($#v4_fomodel1 :::"literal"::: ) ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); cluster (Set (Set ($#k31_fomodel2 :::"<*"::: ) "s" ($#k31_fomodel2 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) "w") -> ($#~v5_fomodel2 "non" ($#v5_fomodel2 :::"exal"::: ) ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "w1", "w2" be ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "s" be ($#~v4_fomodel1 "non" ($#v4_fomodel1 :::"literal"::: ) ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); cluster (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) "s" ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "w1" ")" ) ($#k7_finseq_1 :::"^"::: ) "w2") -> ($#~v5_fomodel2 "non" ($#v5_fomodel2 :::"exal"::: ) ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k8_fomodel1 :::"TheNorSymbOf"::: ) "S") -> ($#~v4_fomodel1 "non" ($#v4_fomodel1 :::"literal"::: ) ) for ($#m1_subset_1 :::"Element":::) "of" "S"; end; theorem :: FOMODEL2:16 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool (Set (Var "phi")) ($#r2_hidden :::"in"::: ) (Set ($#k33_fomodel2 :::"AllFormulasOf"::: ) (Set (Var "S")))))) ; notationlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); let "w" be ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); antonym "m" :::"-nonwff"::: "w" for "m" :::"-wff"::: ; end; registrationlet "m" be ($#m1_hidden :::"Nat":::); let "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#~v3_fomodel2 "non" "m" ($#v3_fomodel2 :::"-wff"::: ) ) -> "m" ($#v3_fomodel2 :::"-nonwff"::: ) 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 "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" ")" ) ($#k7_finseq_1 :::"^"::: ) "phi2") -> (Set ($#k4_xxreal_0 :::"max"::: ) "(" (Set "(" ($#k32_fomodel2 :::"Depth"::: ) "phi1" ")" ) "," (Set "(" ($#k32_fomodel2 :::"Depth"::: ) "phi2" ")" ) ")" ) ($#v3_fomodel2 :::"-nonwff"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "l" be ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); cluster (Set (Set ($#k31_fomodel2 :::"<*"::: ) "l" ($#k31_fomodel2 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) "phi") -> (Set ($#k32_fomodel2 :::"Depth"::: ) "phi") ($#v3_fomodel2 :::"-nonwff"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "l" be ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); cluster (Set (Set ($#k31_fomodel2 :::"<*"::: ) "l" ($#k31_fomodel2 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) "phi") -> (Set (Num 1) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k32_fomodel2 :::"Depth"::: ) "phi" ")" )) ($#v3_fomodel2 :::"-wff"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster -> (Set ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S") ($#v4_relat_1 :::"-defined"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S"); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S") ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funcop_1 :::"Function-yielding"::: ) bbbadV2_FUNCOP_1() "S" "," "U" ($#v1_fomodel2 :::"-interpreter-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S"); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S") ($#v4_relat_1 :::"-defined"::: ) -> (Set ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S") ($#v4_relat_1 :::"-defined"::: ) ($#v1_partfun1 :::"total"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S"); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); let "x" be ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); let "u" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "U")); :: original: :::"ReassignIn"::: redefine func "(" "x" "," "u" ")" :::"ReassignIn"::: "I" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S"); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "w" be ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); func :::"xnot"::: "w" -> ($#m2_subset_1 :::"string":::) "of" "S" equals :: FOMODEL2:def 33 (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "w" ")" ) ($#k30_fomodel2 :::"^"::: ) "w"); end; :: deftheorem defines :::"xnot"::: FOMODEL2:def 33 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "w")) "being" ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool (Set ($#k38_fomodel2 :::"xnot"::: ) (Set (Var "w"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "w")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "w")))))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); let "phi" be (Set (Const "m")) ($#v3_fomodel2 :::"-wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k38_fomodel2 :::"xnot"::: ) "phi") -> (Set "m" ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#v3_fomodel2 :::"-wff"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k38_fomodel2 :::"xnot"::: ) "phi") -> ($#v4_fomodel2 :::"wff"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k7_fomodel1 :::"TheEqSymbOf"::: ) "S") -> ($#~v9_fomodel1 "non" ($#v9_fomodel1 :::"own"::: ) ) for ($#m1_subset_1 :::"Element":::) "of" "S"; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "X" be ($#m1_hidden :::"set"::: ) ; attr "X" is "S" :::"-mincover"::: means :: FOMODEL2:def 34 (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S" "holds" (Bool "(" (Bool (Set (Var "phi")) ($#r2_hidden :::"in"::: ) "X") "iff" (Bool (Bool "not" (Set ($#k38_fomodel2 :::"xnot"::: ) (Set (Var "phi"))) ($#r2_hidden :::"in"::: ) "X")) ")" )); end; :: deftheorem defines :::"-mincover"::: FOMODEL2:def 34 : (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")) ($#v6_fomodel2 :::"-mincover"::: ) ) "iff" (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "phi")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) "iff" (Bool (Bool "not" (Set ($#k38_fomodel2 :::"xnot"::: ) (Set (Var "phi"))) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) ")" )) ")" ))); theorem :: FOMODEL2:17 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "l")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "phi1")) "," (Set (Var "phi2")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set ($#k32_fomodel2 :::"Depth"::: ) (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set "(" ($#k32_fomodel2 :::"Depth"::: ) (Set (Var "phi1")) ")" ) "," (Set "(" ($#k32_fomodel2 :::"Depth"::: ) (Set (Var "phi2")) ")" ) ")" ")" ))) & (Bool (Set ($#k32_fomodel2 :::"Depth"::: ) (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set (Var "l")) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_fomodel2 :::"Depth"::: ) (Set (Var "phi1")) ")" ) ($#k1_nat_1 :::"+"::: ) (Num 1))) ")" )))) ; theorem :: FOMODEL2:18 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k32_fomodel2 :::"Depth"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1)))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "phi")) "is" ($#v5_fomodel2 :::"exal"::: ) )) "implies" (Bool (Set (Var "phi")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "m")) ($#k36_fomodel2 :::"-ExFormulasOf"::: ) (Set (Var "S")))) ")" & "(" (Bool (Bool (Set (Var "phi")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "m")) ($#k36_fomodel2 :::"-ExFormulasOf"::: ) (Set (Var "S"))))) "implies" (Bool (Set (Var "phi")) "is" ($#v5_fomodel2 :::"exal"::: ) ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "phi")) "is" ($#v5_fomodel2 :::"exal"::: ) ))) "implies" (Bool (Set (Var "phi")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "m")) ($#k35_fomodel2 :::"-NorFormulasOf"::: ) (Set (Var "S")))) ")" & "(" (Bool (Bool (Set (Var "phi")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "m")) ($#k35_fomodel2 :::"-NorFormulasOf"::: ) (Set (Var "S"))))) "implies" (Bool "not" (Bool (Set (Var "phi")) "is" ($#v5_fomodel2 :::"exal"::: ) )) ")" ")" )))) ; theorem :: FOMODEL2:19 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "l")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "phi")) "," (Set (Var "phi1")) "," (Set (Var "phi2")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set (Var "l")) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) ))) "implies" (Bool "ex" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "U")) "st" (Bool (Set (Set "(" "(" (Set (Var "l")) "," (Set (Var "u")) ")" ($#k37_fomodel2 :::"ReassignIn"::: ) (Set (Var "I")) ")" ) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Num 1))) ")" & "(" (Bool (Bool "ex" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "U")) "st" (Bool (Set (Set "(" "(" (Set (Var "l")) "," (Set (Var "u")) ")" ($#k37_fomodel2 :::"ReassignIn"::: ) (Set (Var "I")) ")" ) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Num 1)))) "implies" (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set (Var "l")) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" & "(" (Bool (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) ))) "implies" (Bool "(" (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "phi1"))) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) & (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "phi2"))) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) ")" ) ")" & "(" (Bool (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "phi1"))) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) & (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "phi2"))) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) ))) "implies" (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" ")" )))))) ; theorem :: FOMODEL2:20 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "U")) (Bool "for" (Set (Var "I")) "being" (Set (Var "b2")) "," (Set (Var "b3")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set "(" (Set "(" "(" (Set (Var "I")) "," (Set (Var "u")) ")" ($#k8_fomodel2 :::"-TermEval"::: ) ")" ) ($#k10_funct_5 :::"."::: ) (Set "(" (Set (Var "m")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set "(" (Set "(" (Set (Var "S")) ($#k39_fomodel1 :::"-termsOfMaxDepth"::: ) ")" ) ($#k8_nat_1 :::"."::: ) (Set (Var "m")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "I")) ($#k10_fomodel2 :::"-TermEval"::: ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set "(" (Set "(" (Set (Var "S")) ($#k39_fomodel1 :::"-termsOfMaxDepth"::: ) ")" ) ($#k8_nat_1 :::"."::: ) (Set (Var "m")) ")" )))))))) ; theorem :: FOMODEL2:21 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "t")) "being" ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "I")) "being" (Set (Var "b1")) "," (Set (Var "b2")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool (Set (Set "(" (Set (Var "I")) ($#k10_fomodel2 :::"-TermEval"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "I")) ($#k4_fomodel2 :::"."::: ) (Set "(" (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "t")) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "I")) ($#k10_fomodel2 :::"-TermEval"::: ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k33_fomodel1 :::"SubTerms"::: ) (Set (Var "t")) ")" ) ")" ))))))) ; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); func :::"SubWffsOf"::: "phi" -> ($#m1_hidden :::"set"::: ) means :: FOMODEL2:def 35 (Bool "ex" (Set (Var "phi1")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S"(Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set (Var "p")) "is" (Set ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S") ($#v5_relat_1 :::"-valued"::: ) ) & (Bool it ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "phi1")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) )) & (Bool "phi" ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" (Set "(" "S" ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) "phi" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "p")))) ")" ))) if (Bool (Bool "not" "phi" "is" ($#v15_fomodel1 :::"0wff"::: ) )) otherwise (Bool it ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) "phi" "," (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k4_tarski :::"]"::: ) )); end; :: deftheorem defines :::"SubWffsOf"::: FOMODEL2:def 35 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Bool "not" (Set (Var "phi")) "is" ($#v15_fomodel1 :::"0wff"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k39_fomodel2 :::"SubWffsOf"::: ) (Set (Var "phi")))) "iff" (Bool "ex" (Set (Var "phi1")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S"))(Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set (Var "p")) "is" (Set ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S"))) ($#v5_relat_1 :::"-valued"::: ) ) & (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "phi1")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) )) & (Bool (Set (Var "phi")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "phi")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "p")))) ")" ))) ")" ) ")" & "(" (Bool (Bool (Set (Var "phi")) "is" ($#v15_fomodel1 :::"0wff"::: ) )) "implies" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k39_fomodel2 :::"SubWffsOf"::: ) (Set (Var "phi")))) "iff" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "phi")) "," (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k4_tarski :::"]"::: ) )) ")" ) ")" ")" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); func :::"head"::: "phi" -> ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S" equals :: FOMODEL2:def 36 (Set (Set "(" ($#k39_fomodel2 :::"SubWffsOf"::: ) "phi" ")" ) ($#k1_xtuple_0 :::"`1"::: ) ); func :::"tail"::: "phi" -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ) equals :: FOMODEL2:def 37 (Set (Set "(" ($#k39_fomodel2 :::"SubWffsOf"::: ) "phi" ")" ) ($#k2_xtuple_0 :::"`2"::: ) ); end; :: deftheorem defines :::"head"::: FOMODEL2:def 36 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool (Set ($#k40_fomodel2 :::"head"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k39_fomodel2 :::"SubWffsOf"::: ) (Set (Var "phi")) ")" ) ($#k1_xtuple_0 :::"`1"::: ) )))); :: deftheorem defines :::"tail"::: FOMODEL2:def 37 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool (Set ($#k41_fomodel2 :::"tail"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k39_fomodel2 :::"SubWffsOf"::: ) (Set (Var "phi")) ")" ) ($#k2_xtuple_0 :::"`2"::: ) )))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "m" be ($#m1_hidden :::"Nat":::); cluster (Set (Set "(" "S" ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) "m" ")" ) ($#k4_xboole_0 :::"\"::: ) (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S" ")" )) -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set (Set "(" ($#k32_fomodel1 :::"AtomicFormulasOf"::: ) "S" ")" ) ($#k4_xboole_0 :::"\"::: ) (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S" ")" )) -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FOMODEL2:22 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "l")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "phi1")) "," (Set (Var "phi2")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set ($#k32_fomodel2 :::"Depth"::: ) (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set (Var "l")) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set ($#k32_fomodel2 :::"Depth"::: ) (Set (Var "phi1")))) & (Bool (Set ($#k32_fomodel2 :::"Depth"::: ) (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set ($#k32_fomodel2 :::"Depth"::: ) (Set (Var "phi1")))) & (Bool (Set ($#k32_fomodel2 :::"Depth"::: ) (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) (Set (Var "S")) ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi1")) ")" ) ($#k30_fomodel2 :::"^"::: ) (Set (Var "phi2")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set ($#k32_fomodel2 :::"Depth"::: ) (Set (Var "phi2")))) ")" )))) ; theorem :: FOMODEL2:23 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "p2")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "phi")) "," (Set (Var "phi2")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool (Bool (Bool "not" (Set (Var "phi")) "is" ($#v15_fomodel1 :::"0wff"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "phi")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "phi2")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "p2")))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "S")) ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "phi")))) & (Bool (Set (Var "phi2")) ($#r1_hidden :::"="::: ) (Set ($#k40_fomodel2 :::"head"::: ) (Set (Var "phi")))) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set ($#k41_fomodel2 :::"tail"::: ) (Set (Var "phi")))) ")" ) ")" ))))) ; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "m1" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); 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"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) bbbadV2_PRE_POLY() ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) "m1" ($#v3_fomodel2 :::"-wff"::: ) ($#v4_fomodel2 :::"wff"::: ) ($#~v5_fomodel2 "non" ($#v5_fomodel2 :::"exal"::: ) ) for ($#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 ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#v5_fomodel2 :::"exal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k41_fomodel2 :::"tail"::: ) "phi") -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) ($#v4_fomodel2 :::"wff"::: ) ($#~v5_fomodel2 "non" ($#v5_fomodel2 :::"exal"::: ) ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); :: original: :::"tail"::: redefine func :::"tail"::: "phi" -> ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) ($#v4_fomodel2 :::"wff"::: ) ($#~v5_fomodel2 "non" ($#v5_fomodel2 :::"exal"::: ) ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k41_fomodel2 :::"tail"::: ) "phi") -> ($#v4_fomodel2 :::"wff"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi0" be ($#v15_fomodel1 :::"0wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); identify ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) ($#v4_fomodel2 :::"wff"::: ) ($#~v5_fomodel2 "non" ($#v5_fomodel2 :::"exal"::: ) ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set (Set "(" (Set "(" "S" ($#k22_fomodel1 :::"-firstChar"::: ) ")" ) ($#k3_funct_2 :::"."::: ) "phi" ")" ) ($#k5_xboole_0 :::"\+\"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" )) -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "m" be ($#m1_hidden :::"Nat":::); let "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be (Set (Set (Const "m")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#v3_fomodel2 :::"-wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k40_fomodel2 :::"head"::: ) "phi") -> "m" ($#v3_fomodel2 :::"-wff"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; registrationlet "m" be ($#m1_hidden :::"Nat":::); let "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) (Set (Set (Const "m")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#v3_fomodel2 :::"-wff"::: ) ($#~v5_fomodel2 "non" ($#v5_fomodel2 :::"exal"::: ) ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set ($#k41_fomodel2 :::"tail"::: ) "phi") -> "m" ($#v3_fomodel2 :::"-wff"::: ) for ($#m2_subset_1 :::"string":::) "of" "S"; end; theorem :: FOMODEL2:24 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (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")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) "holds" (Bool (Set "(" (Set (Var "I")) "," (Set (Var "m")) ")" ($#k24_fomodel2 :::"-TruthEval"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "(" (Set (Var "S")) ($#k25_fomodel2 :::"-formulasOfMaxDepth"::: ) (Set (Var "m")) ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" )))))) ; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); let "phi0" be ($#v15_fomodel1 :::"0wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); identify ; identify ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#~v4_fomodel1 "non" ($#v4_fomodel1 :::"literal"::: ) ) ($#v10_fomodel1 :::"ofAtomicFormula"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S"); end; theorem :: FOMODEL2:25 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "U")) (Bool "for" (Set (Var "l2")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "I")) "being" (Set (Var "b1")) "," (Set (Var "b2")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Bool "not" (Set (Var "l2")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")))))) "holds" (Bool (Set (Set "(" (Set "(" "(" (Set (Var "l2")) "," (Set (Var "u")) ")" ($#k5_fomodel2 :::"ReassignIn"::: ) (Set (Var "I")) ")" ) ($#k10_fomodel2 :::"-TermEval"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "I")) ($#k10_fomodel2 :::"-TermEval"::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "p")))))))))) ; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "s" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); attr "s" is "X" :::"-occurring"::: means :: FOMODEL2:def 38 (Bool "s" ($#r2_hidden :::"in"::: ) (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 :::"/\"::: ) "X" ")" ))); end; :: deftheorem defines :::"-occurring"::: FOMODEL2:def 38 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) "is" (Set (Var "X")) ($#v7_fomodel2 :::"-occurring"::: ) ) "iff" (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#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 :::"}"::: ) ) ")" ) ($#k8_subset_1 :::"/\"::: ) (Set (Var "X")) ")" ))) ")" )))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "s" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); let "X" be ($#m1_hidden :::"set"::: ) ; attr "X" is "s" :::"-containing"::: means :: FOMODEL2:def 39 (Bool "s" ($#r2_hidden :::"in"::: ) (Set ($#k21_fomodel0 :::"SymbolsOf"::: ) (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k3_xboole_0 :::"/\"::: ) "X" ")" ) ")" ))); end; :: deftheorem defines :::"-containing"::: FOMODEL2:def 39 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" (Set (Var "s")) ($#v8_fomodel2 :::"-containing"::: ) ) "iff" (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k21_fomodel0 :::"SymbolsOf"::: ) (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) (Set (Var "S")) ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X")) ")" ) ")" ))) ")" )))); notationlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "s" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); antonym "X" :::"-absent"::: "s" for "X" :::"-occurring"::: ; end; notationlet "S" be ($#l1_fomodel1 :::"Language":::); let "s" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "S")); let "X" be ($#m1_hidden :::"set"::: ) ; antonym "s" :::"-free"::: "X" for "s" :::"-containing"::: ; end; registrationlet "X" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); cluster ($#v4_fomodel1 :::"literal"::: ) ($#~v6_fomodel1 "non" ($#v6_fomodel1 :::"operational"::: ) ) ($#~v7_fomodel1 "non" ($#v7_fomodel1 :::"relational"::: ) ) ($#v8_fomodel1 :::"termal"::: ) ($#v9_fomodel1 :::"own"::: ) ($#v10_fomodel1 :::"ofAtomicFormula"::: ) "X" ($#v7_fomodel2 :::"-absent"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S"); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "t" be ($#v13_fomodel1 :::"termal"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); cluster (Set (Set "(" ($#k2_relset_1 :::"rng"::: ) "t" ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k16_fomodel1 :::"LettersOf"::: ) "S" ")" )) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) 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 (Set "(" ($#k10_xtuple_0 :::"rng"::: ) "phi" ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k16_fomodel1 :::"LettersOf"::: ) "S" ")" )) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "B" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "B")); cluster "A" ($#v7_fomodel2 :::"-occurring"::: ) -> "B" ($#v7_fomodel2 :::"-occurring"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S"); end; registrationlet "A", "B" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set "A" ($#k13_fomodel0 :::"null"::: ) "B") ($#v7_fomodel2 :::"-absent"::: ) -> (Set "A" ($#k3_xboole_0 :::"/\"::: ) "B") ($#v7_fomodel2 :::"-absent"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S"); end; registrationlet "F" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); cluster "F" ($#v7_fomodel2 :::"-absent"::: ) "A" ($#v7_fomodel2 :::"-absent"::: ) -> "F" ($#v7_fomodel2 :::"-absent"::: ) (Set "A" ($#k2_xboole_0 :::"\/"::: ) "F") ($#v7_fomodel2 :::"-absent"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S"); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be (Set (Const "S")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); cluster (Set (Set "(" ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S" ")" ) ($#k4_xboole_0 :::"\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "I" ")" )) -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FOMODEL2:26 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "l")) "being" ($#v4_fomodel1 :::"literal"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "I")) "being" (Set (Var "b1")) "," (Set (Var "b2")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::) (Bool "ex" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "U")) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "I")) ($#k4_fomodel2 :::"."::: ) (Set (Var "l")) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) & (Bool (Set "(" (Set (Var "l")) "," (Set (Var "u")) ")" ($#k5_fomodel2 :::"ReassignIn"::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set (Var "I"))) ")" )))))) ; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "X" be ($#m1_hidden :::"set"::: ) ; attr "X" is "S" :::"-covering"::: means :: FOMODEL2:def 40 (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S" "holds" (Bool "(" (Bool (Set (Var "phi")) ($#r2_hidden :::"in"::: ) "X") "or" (Bool (Set ($#k38_fomodel2 :::"xnot"::: ) (Set (Var "phi"))) ($#r2_hidden :::"in"::: ) "X") ")" )); end; :: deftheorem defines :::"-covering"::: FOMODEL2:def 40 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" (Set (Var "S")) ($#v9_fomodel2 :::"-covering"::: ) ) "iff" (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "phi")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) "or" (Bool (Set ($#k38_fomodel2 :::"xnot"::: ) (Set (Var "phi"))) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" )) ")" ))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster "S" ($#v6_fomodel2 :::"-mincover"::: ) -> "S" ($#v9_fomodel2 :::"-covering"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#~v15_fomodel1 "non" ($#v15_fomodel1 :::"0wff"::: ) ) ($#v4_fomodel2 :::"wff"::: ) ($#~v5_fomodel2 "non" ($#v5_fomodel2 :::"exal"::: ) ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); cluster (Set (Set "(" "I" ($#k26_fomodel2 :::"-TruthEval"::: ) "phi" ")" ) ($#k5_xboole_0 :::"\+\"::: ) (Set "(" (Set "(" "I" ($#k26_fomodel2 :::"-TruthEval"::: ) (Set "(" ($#k40_fomodel2 :::"head"::: ) "phi" ")" ) ")" ) ($#k9_xboolean :::"'nor'"::: ) (Set "(" "I" ($#k26_fomodel2 :::"-TruthEval"::: ) (Set "(" ($#k42_fomodel2 :::"tail"::: ) "phi" ")" ) ")" ) ")" )) -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); func :::"ExFormulasOf"::: "S" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" (Set "(" ($#k1_fomodel1 :::"AllSymbolsOf"::: ) "S" ")" ) ($#k3_finseq_2 :::"*"::: ) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) equals :: FOMODEL2:def 41 "{" (Set (Var "phi")) where phi "is" ($#m2_subset_1 :::"string":::) "of" "S" : (Bool "(" (Bool (Set (Var "phi")) "is" ($#v4_fomodel2 :::"wff"::: ) ) & (Bool (Set (Var "phi")) "is" ($#v5_fomodel2 :::"exal"::: ) ) ")" ) "}" ; end; :: deftheorem defines :::"ExFormulasOf"::: FOMODEL2:def 41 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) "holds" (Bool (Set ($#k43_fomodel2 :::"ExFormulasOf"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "phi")) where phi "is" ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) : (Bool "(" (Bool (Set (Var "phi")) "is" ($#v4_fomodel2 :::"wff"::: ) ) & (Bool (Set (Var "phi")) "is" ($#v5_fomodel2 :::"exal"::: ) ) ")" ) "}" )); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set ($#k43_fomodel2 :::"ExFormulasOf"::: ) "S") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster -> ($#v4_fomodel2 :::"wff"::: ) ($#v5_fomodel2 :::"exal"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k43_fomodel2 :::"ExFormulasOf"::: ) "S"); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster -> ($#v4_fomodel2 :::"wff"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k43_fomodel2 :::"ExFormulasOf"::: ) "S"); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster -> ($#v5_fomodel2 :::"exal"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k43_fomodel2 :::"ExFormulasOf"::: ) "S"); end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set (Set "(" ($#k43_fomodel2 :::"ExFormulasOf"::: ) "S" ")" ) ($#k4_xboole_0 :::"\"::: ) (Set "(" ($#k33_fomodel2 :::"AllFormulasOf"::: ) "S" ")" )) -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S1" be ($#l1_fomodel1 :::"Language":::); let "S2" be (Set (Const "S1")) ($#v16_fomodel1 :::"-extending"::: ) ($#l1_fomodel1 :::"Language":::); cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) "S2" "," "U" ($#v1_fomodel2 :::"-interpreter-like"::: ) -> "S1" "," "U" ($#v1_fomodel2 :::"-interpreter-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S1" be ($#l1_fomodel1 :::"Language":::); let "S2" be (Set (Const "S1")) ($#v16_fomodel1 :::"-extending"::: ) ($#l1_fomodel1 :::"Language":::); let "I" be (Set (Const "S2")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); cluster (Set "I" ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k37_fomodel1 :::"OwnSymbolsOf"::: ) "S1" ")" )) -> "S1" "," "U" ($#v1_fomodel2 :::"-interpreter-like"::: ) for ($#m1_hidden :::"Function":::); end; registrationlet "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S1" be ($#l1_fomodel1 :::"Language":::); let "S2" be (Set (Const "S1")) ($#v16_fomodel1 :::"-extending"::: ) ($#l1_fomodel1 :::"Language":::); let "I1" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S1"))); let "I2" be (Set (Const "S2")) "," (Set (Const "U")) ($#v1_fomodel2 :::"-interpreter-like"::: ) ($#m1_hidden :::"Function":::); cluster (Set "I2" ($#k1_funct_4 :::"+*"::: ) "I1") -> "S2" "," "U" ($#v1_fomodel2 :::"-interpreter-like"::: ) ; end; definitionlet "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); let "X" be ($#m1_hidden :::"set"::: ) ; attr "X" is "I" :::"-satisfied"::: means :: FOMODEL2:def 42 (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S" "st" (Bool (Bool (Set (Var "phi")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set "I" ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Num 1))); end; :: deftheorem defines :::"-satisfied"::: FOMODEL2:def 42 : (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" (Set (Var "I")) ($#v10_fomodel2 :::"-satisfied"::: ) ) "iff" (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "phi")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Num 1))) ")" ))))); definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "X" be ($#m1_hidden :::"set"::: ) ; let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); attr "I" is "X" :::"-satisfying"::: means :: FOMODEL2:def 43 (Bool "X" "is" "I" ($#v10_fomodel2 :::"-satisfied"::: ) ); end; :: deftheorem defines :::"-satisfying"::: FOMODEL2:def 43 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "I")) "is" (Set (Var "X")) ($#v11_fomodel2 :::"-satisfying"::: ) ) "iff" (Bool (Set (Var "X")) "is" (Set (Var "I")) ($#v10_fomodel2 :::"-satisfied"::: ) ) ")" ))))); registrationlet "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "e" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_hidden :::"set"::: ) ; let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); cluster (Set "e" ($#k13_fomodel0 :::"null"::: ) "I") -> "I" ($#v10_fomodel2 :::"-satisfied"::: ) ; end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); cluster "I" ($#v10_fomodel2 :::"-satisfied"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "X"); end; registrationlet "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); cluster "I" ($#v10_fomodel2 :::"-satisfied"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); let "X" be (Set (Const "I")) ($#v10_fomodel2 :::"-satisfied"::: ) ($#m1_hidden :::"set"::: ) ; cluster -> "I" ($#v10_fomodel2 :::"-satisfied"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "X"); end; registrationlet "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); let "X", "Y" be (Set (Const "I")) ($#v10_fomodel2 :::"-satisfied"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "X" ($#k2_xboole_0 :::"\/"::: ) "Y") -> "I" ($#v10_fomodel2 :::"-satisfied"::: ) ; end; registrationlet "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); let "X" be (Set (Const "I")) ($#v10_fomodel2 :::"-satisfied"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "I" ($#k13_fomodel0 :::"null"::: ) "X") -> "X" ($#v11_fomodel2 :::"-satisfying"::: ) for ($#m2_subset_1 :::"Element"::: ) "of" (Set "U" ($#k16_fomodel2 :::"-InterpretersOf"::: ) "S"); end; definitionlet "S" be ($#l1_fomodel1 :::"Language":::); let "X" be ($#m1_hidden :::"set"::: ) ; attr "X" is "S" :::"-correct"::: means :: FOMODEL2:def 44 (Bool "for" (Set (Var "U")) "being" ($#~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") (Bool "for" (Set (Var "x")) "being" (Set (Var "b2")) ($#v10_fomodel2 :::"-satisfied"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" "S" "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "phi")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Num 1)))))); end; :: deftheorem defines :::"-correct"::: FOMODEL2:def 44 : (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" (Set (Var "S")) ($#v12_fomodel2 :::"-correct"::: ) ) "iff" (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")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) (Bool "for" (Set (Var "x")) "being" (Set (Var "b4")) ($#v10_fomodel2 :::"-satisfied"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "phi")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Num 1)))))) ")" ))); registrationlet "S" be ($#l1_fomodel1 :::"Language":::); cluster (Set (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k13_fomodel0 :::"null"::: ) "S") -> "S" ($#v12_fomodel2 :::"-correct"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "X" be ($#m1_hidden :::"set"::: ) ; cluster "S" ($#v12_fomodel2 :::"-correct"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "X"); end; theorem :: FOMODEL2:27 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "U")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Var "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) (Set (Var "phi"))) ($#r1_hidden :::"="::: ) (Num 1)) "iff" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "phi")) ($#k6_domain_1 :::"}"::: ) ) "is" (Set (Var "I")) ($#v10_fomodel2 :::"-satisfied"::: ) ) ")" ))))) ; theorem :: FOMODEL2:28 (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "w")) "being" ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) "is" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "w")) ($#k6_domain_1 :::"}"::: ) ) ($#v7_fomodel2 :::"-occurring"::: ) ) "iff" (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "w")))) ")" )))) ; registrationlet "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "phi1", "phi2" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); cluster (Set (Set "(" "I" ($#k26_fomodel2 :::"-TruthEval"::: ) (Set "(" (Set "(" (Set ($#k31_fomodel2 :::"<*"::: ) (Set "(" ($#k1_fomodel2 :::"TheNorSymbOf"::: ) "S" ")" ) ($#k31_fomodel2 :::"*>"::: ) ) ($#k30_fomodel2 :::"^"::: ) "phi1" ")" ) ($#k30_fomodel2 :::"^"::: ) "phi2" ")" ) ")" ) ($#k5_xboole_0 :::"\+\"::: ) (Set "(" (Set "(" "I" ($#k26_fomodel2 :::"-TruthEval"::: ) "phi1" ")" ) ($#k9_xboolean :::"'nor'"::: ) (Set "(" "I" ($#k26_fomodel2 :::"-TruthEval"::: ) "phi2" ")" ) ")" )) -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); let "U" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "I" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Set (Const "U")) ($#k16_fomodel2 :::"-InterpretersOf"::: ) (Set (Const "S"))); cluster (Set (Set "(" "I" ($#k26_fomodel2 :::"-TruthEval"::: ) (Set "(" ($#k38_fomodel2 :::"xnot"::: ) "phi" ")" ) ")" ) ($#k5_xboole_0 :::"\+\"::: ) (Set "(" ($#k9_margrel1 :::"'not'"::: ) (Set "(" "I" ($#k26_fomodel2 :::"-TruthEval"::: ) "phi" ")" ) ")" )) -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "S" be ($#l1_fomodel1 :::"Language":::); let "phi" be ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Const "S")); attr "phi" is "X" :::"-implied"::: means :: FOMODEL2:def 45 (Bool "for" (Set (Var "U")) "being" ($#~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 "X" "is" (Set (Var "I")) ($#v10_fomodel2 :::"-satisfied"::: ) )) "holds" (Bool (Set (Set (Var "I")) ($#k26_fomodel2 :::"-TruthEval"::: ) "phi") ($#r1_hidden :::"="::: ) (Num 1)))); end; :: deftheorem defines :::"-implied"::: FOMODEL2:def 45 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_fomodel1 :::"Language":::) (Bool "for" (Set (Var "phi")) "being" ($#v4_fomodel2 :::"wff"::: ) ($#m2_subset_1 :::"string":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "phi")) "is" (Set (Var "X")) ($#v13_fomodel2 :::"-implied"::: ) ) "iff" (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)))) ")" ))));