:: MODELC_1 semantic presentation begin definitionlet "x", "S" be ($#m1_hidden :::"set"::: ) ; let "a" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "S")); func :::"k_id"::: "(" "x" "," "S" "," "a" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" "S" equals :: MODELC_1:def 1 "x" if (Bool "x" ($#r2_hidden :::"in"::: ) "S") otherwise "a"; end; :: deftheorem defines :::"k_id"::: MODELC_1:def 1 : (Bool "for" (Set (Var "x")) "," (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "implies" (Bool (Set ($#k1_modelc_1 :::"k_id"::: ) "(" (Set (Var "x")) "," (Set (Var "S")) "," (Set (Var "a")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))))) "implies" (Bool (Set ($#k1_modelc_1 :::"k_id"::: ) "(" (Set (Var "x")) "," (Set (Var "S")) "," (Set (Var "a")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) ")" ")" ))); definitionlet "x" be ($#m1_hidden :::"set"::: ) ; func :::"k_nat"::: "x" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: MODELC_1:def 2 "x" if (Bool "x" ($#r2_hidden :::"in"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) otherwise (Set ($#k6_numbers :::"0"::: ) ); end; :: deftheorem defines :::"k_nat"::: MODELC_1:def 2 : (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k5_numbers :::"NAT"::: ) ))) "implies" (Bool (Set ($#k2_modelc_1 :::"k_nat"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k5_numbers :::"NAT"::: ) )))) "implies" (Bool (Set ($#k2_modelc_1 :::"k_nat"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ")" )); definitionlet "f" be ($#m1_hidden :::"Function":::); let "x", "a" be ($#m1_hidden :::"set"::: ) ; func :::"UnivF"::: "(" "x" "," "f" "," "a" ")" -> ($#m1_hidden :::"set"::: ) equals :: MODELC_1:def 3 (Set "f" ($#k1_funct_1 :::"."::: ) "x") if (Bool "x" ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "f")) otherwise "a"; end; :: deftheorem defines :::"UnivF"::: MODELC_1:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "x")) "," (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "implies" (Bool (Set ($#k3_modelc_1 :::"UnivF"::: ) "(" (Set (Var "x")) "," (Set (Var "f")) "," (Set (Var "a")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")))))) "implies" (Bool (Set ($#k3_modelc_1 :::"UnivF"::: ) "(" (Set (Var "x")) "," (Set (Var "f")) "," (Set (Var "a")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) ")" ")" ))); definitionlet "a" be ($#m1_hidden :::"set"::: ) ; func :::"Castboolean"::: "a" -> ($#v1_xboolean :::"boolean"::: ) ($#m1_hidden :::"set"::: ) equals :: MODELC_1:def 4 "a" if (Bool "a" "is" ($#v1_xboolean :::"boolean"::: ) ($#m1_hidden :::"set"::: ) ) otherwise (Set ($#k7_margrel1 :::"FALSE"::: ) ); end; :: deftheorem defines :::"Castboolean"::: MODELC_1:def 4 : (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "a")) "is" ($#v1_xboolean :::"boolean"::: ) ($#m1_hidden :::"set"::: ) )) "implies" (Bool (Set ($#k4_modelc_1 :::"Castboolean"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))) ")" & "(" (Bool (Bool (Set (Var "a")) "is" (Bool "not" ($#v1_xboolean :::"boolean"::: ) ($#m1_hidden :::"set"::: ) ))) "implies" (Bool (Set ($#k4_modelc_1 :::"Castboolean"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) ")" ")" )); definitionlet "X", "a" be ($#m1_hidden :::"set"::: ) ; func :::"CastBool"::: "(" "a" "," "X" ")" -> ($#m1_subset_1 :::"Subset":::) "of" "X" equals :: MODELC_1:def 5 "a" if (Bool "a" ($#r1_tarski :::"c="::: ) "X") otherwise (Set ($#k1_xboole_0 :::"{}"::: ) ); end; :: deftheorem defines :::"CastBool"::: MODELC_1:def 5 : (Bool "for" (Set (Var "X")) "," (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "a")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "implies" (Bool (Set ($#k5_modelc_1 :::"CastBool"::: ) "(" (Set (Var "a")) "," (Set (Var "X")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "a")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))))) "implies" (Bool (Set ($#k5_modelc_1 :::"CastBool"::: ) "(" (Set (Var "a")) "," (Set (Var "X")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ")" )); definitionlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"atom."::: "n" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: MODELC_1:def 6 (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Num 5) ($#k2_nat_1 :::"+"::: ) "n" ")" ) ($#k12_finseq_1 :::"*>"::: ) ); end; :: deftheorem defines :::"atom."::: MODELC_1:def 6 : (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_modelc_1 :::"atom."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Num 5) ($#k2_nat_1 :::"+"::: ) (Set (Var "n")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ))); definitionlet "p" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"'not'"::: "p" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: MODELC_1:def 7 (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) "p"); let "q" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func "p" :::"'&'"::: "q" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: MODELC_1:def 8 (Set (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Num 1) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) "p" ")" ) ($#k8_finseq_1 :::"^"::: ) "q"); end; :: deftheorem defines :::"'not'"::: MODELC_1:def 7 : (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k7_modelc_1 :::"'not'"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "p"))))); :: deftheorem defines :::"'&'"::: MODELC_1:def 8 : (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "p")) ($#k8_modelc_1 :::"'&'"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Num 1) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "p")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "q"))))); definitionlet "p" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"EX"::: "p" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: MODELC_1:def 9 (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Num 2) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) "p"); func :::"EG"::: "p" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: MODELC_1:def 10 (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Num 3) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) "p"); let "q" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func "p" :::"EU"::: "q" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: MODELC_1:def 11 (Set (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Num 4) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) "p" ")" ) ($#k8_finseq_1 :::"^"::: ) "q"); end; :: deftheorem defines :::"EX"::: MODELC_1:def 9 : (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k9_modelc_1 :::"EX"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Num 2) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "p"))))); :: deftheorem defines :::"EG"::: MODELC_1:def 10 : (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k10_modelc_1 :::"EG"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Num 3) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "p"))))); :: deftheorem defines :::"EU"::: MODELC_1:def 11 : (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "p")) ($#k11_modelc_1 :::"EU"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k12_finseq_1 :::"<*"::: ) (Num 4) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "p")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "q"))))); definitionfunc :::"CTL_WFF"::: -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) means :: MODELC_1:def 12 (Bool "(" (Bool "(" "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) it)) "holds" (Bool (Set (Var "a")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_modelc_1 :::"atom."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) it) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) it)) "holds" (Bool (Set ($#k7_modelc_1 :::"'not'"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) it) ")" ) & (Bool "(" "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) it) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) it)) "holds" (Bool (Set (Set (Var "p")) ($#k8_modelc_1 :::"'&'"::: ) (Set (Var "q"))) ($#r2_hidden :::"in"::: ) it) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) it)) "holds" (Bool (Set ($#k9_modelc_1 :::"EX"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) it) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) it)) "holds" (Bool (Set ($#k10_modelc_1 :::"EG"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) it) ")" ) & (Bool "(" "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) it) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) it)) "holds" (Bool (Set (Set (Var "p")) ($#k11_modelc_1 :::"EU"::: ) (Set (Var "q"))) ($#r2_hidden :::"in"::: ) it) ")" ) & (Bool "(" "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "(" "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Var "a")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_modelc_1 :::"atom."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set ($#k7_modelc_1 :::"'not'"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "p")) ($#k8_modelc_1 :::"'&'"::: ) (Set (Var "q"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set ($#k9_modelc_1 :::"EX"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set ($#k10_modelc_1 :::"EG"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "p")) ($#k11_modelc_1 :::"EU"::: ) (Set (Var "q"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" )) "holds" (Bool it ($#r1_tarski :::"c="::: ) (Set (Var "D"))) ")" ) ")" ); end; :: deftheorem defines :::"CTL_WFF"::: MODELC_1:def 12 : (Bool "for" (Set (Var "b1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) )) "iff" (Bool "(" (Bool "(" "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "b1")))) "holds" (Bool (Set (Var "a")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_modelc_1 :::"atom."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "b1")))) "holds" (Bool (Set ($#k7_modelc_1 :::"'not'"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set (Var "b1")))) "holds" (Bool (Set (Set (Var "p")) ($#k8_modelc_1 :::"'&'"::: ) (Set (Var "q"))) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "b1")))) "holds" (Bool (Set ($#k9_modelc_1 :::"EX"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "b1")))) "holds" (Bool (Set ($#k10_modelc_1 :::"EG"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set (Var "b1")))) "holds" (Bool (Set (Set (Var "p")) ($#k11_modelc_1 :::"EU"::: ) (Set (Var "q"))) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) ")" ) & (Bool "(" "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "(" "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Var "a")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_modelc_1 :::"atom."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set ($#k7_modelc_1 :::"'not'"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "p")) ($#k8_modelc_1 :::"'&'"::: ) (Set (Var "q"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set ($#k9_modelc_1 :::"EX"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set ($#k10_modelc_1 :::"EG"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" ) & (Bool "(" "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "p")) ($#k11_modelc_1 :::"EU"::: ) (Set (Var "q"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) ")" )) "holds" (Bool (Set (Var "b1")) ($#r1_tarski :::"c="::: ) (Set (Var "D"))) ")" ) ")" ) ")" )); definitionlet "IT" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); attr "IT" is :::"CTL-formula-like"::: means :: MODELC_1:def 13 (Bool "IT" "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) )); end; :: deftheorem defines :::"CTL-formula-like"::: MODELC_1:def 13 : (Bool "for" (Set (Var "IT")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_modelc_1 :::"CTL-formula-like"::: ) ) "iff" (Bool (Set (Var "IT")) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) )) ")" )); registration cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v1_modelc_1 :::"CTL-formula-like"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); end; definitionmode CTL-formula is ($#v1_modelc_1 :::"CTL-formula-like"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); end; theorem :: MODELC_1:1 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "a")) "is" ($#m2_finseq_1 :::"CTL-formula":::)) "iff" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) )) ")" )) ; registrationlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k6_modelc_1 :::"atom."::: ) "n") -> ($#v1_modelc_1 :::"CTL-formula-like"::: ) ; end; registrationlet "H" be ($#m2_finseq_1 :::"CTL-formula":::); cluster (Set ($#k7_modelc_1 :::"'not'"::: ) "H") -> ($#v1_modelc_1 :::"CTL-formula-like"::: ) ; cluster (Set ($#k9_modelc_1 :::"EX"::: ) "H") -> ($#v1_modelc_1 :::"CTL-formula-like"::: ) ; cluster (Set ($#k10_modelc_1 :::"EG"::: ) "H") -> ($#v1_modelc_1 :::"CTL-formula-like"::: ) ; let "G" be ($#m2_finseq_1 :::"CTL-formula":::); cluster (Set "H" ($#k8_modelc_1 :::"'&'"::: ) "G") -> ($#v1_modelc_1 :::"CTL-formula-like"::: ) ; cluster (Set "H" ($#k11_modelc_1 :::"EU"::: ) "G") -> ($#v1_modelc_1 :::"CTL-formula-like"::: ) ; end; definitionlet "H" be ($#m2_finseq_1 :::"CTL-formula":::); attr "H" is :::"atomic"::: means :: MODELC_1:def 14 (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "H" ($#r1_hidden :::"="::: ) (Set ($#k6_modelc_1 :::"atom."::: ) (Set (Var "n"))))); attr "H" is :::"negative"::: means :: MODELC_1:def 15 (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool "H" ($#r1_hidden :::"="::: ) (Set ($#k7_modelc_1 :::"'not'"::: ) (Set (Var "H1"))))); attr "H" is :::"conjunctive"::: means :: MODELC_1:def 16 (Bool "ex" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool "H" ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k8_modelc_1 :::"'&'"::: ) (Set (Var "G"))))); attr "H" is :::"ExistNext"::: means :: MODELC_1:def 17 (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool "H" ($#r1_hidden :::"="::: ) (Set ($#k9_modelc_1 :::"EX"::: ) (Set (Var "H1"))))); attr "H" is :::"ExistGlobal"::: means :: MODELC_1:def 18 (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool "H" ($#r1_hidden :::"="::: ) (Set ($#k10_modelc_1 :::"EG"::: ) (Set (Var "H1"))))); attr "H" is :::"ExistUntill"::: means :: MODELC_1:def 19 (Bool "ex" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool "H" ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k11_modelc_1 :::"EU"::: ) (Set (Var "G"))))); end; :: deftheorem defines :::"atomic"::: MODELC_1:def 14 : (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" (Bool (Set (Var "H")) "is" ($#v2_modelc_1 :::"atomic"::: ) ) "iff" (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Var "H")) ($#r1_hidden :::"="::: ) (Set ($#k6_modelc_1 :::"atom."::: ) (Set (Var "n"))))) ")" )); :: deftheorem defines :::"negative"::: MODELC_1:def 15 : (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" (Bool (Set (Var "H")) "is" ($#v3_modelc_1 :::"negative"::: ) ) "iff" (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Set (Var "H")) ($#r1_hidden :::"="::: ) (Set ($#k7_modelc_1 :::"'not'"::: ) (Set (Var "H1"))))) ")" )); :: deftheorem defines :::"conjunctive"::: MODELC_1:def 16 : (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" (Bool (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) ) "iff" (Bool "ex" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Set (Var "H")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k8_modelc_1 :::"'&'"::: ) (Set (Var "G"))))) ")" )); :: deftheorem defines :::"ExistNext"::: MODELC_1:def 17 : (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" (Bool (Set (Var "H")) "is" ($#v5_modelc_1 :::"ExistNext"::: ) ) "iff" (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Set (Var "H")) ($#r1_hidden :::"="::: ) (Set ($#k9_modelc_1 :::"EX"::: ) (Set (Var "H1"))))) ")" )); :: deftheorem defines :::"ExistGlobal"::: MODELC_1:def 18 : (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" (Bool (Set (Var "H")) "is" ($#v6_modelc_1 :::"ExistGlobal"::: ) ) "iff" (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Set (Var "H")) ($#r1_hidden :::"="::: ) (Set ($#k10_modelc_1 :::"EG"::: ) (Set (Var "H1"))))) ")" )); :: deftheorem defines :::"ExistUntill"::: MODELC_1:def 19 : (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" (Bool (Set (Var "H")) "is" ($#v7_modelc_1 :::"ExistUntill"::: ) ) "iff" (Bool "ex" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Set (Var "H")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k11_modelc_1 :::"EU"::: ) (Set (Var "G"))))) ")" )); definitionlet "F", "G" be ($#m2_finseq_1 :::"CTL-formula":::); func "F" :::"'or'"::: "G" -> ($#m2_finseq_1 :::"CTL-formula":::) equals :: MODELC_1:def 20 (Set ($#k7_modelc_1 :::"'not'"::: ) (Set "(" (Set "(" ($#k7_modelc_1 :::"'not'"::: ) "F" ")" ) ($#k8_modelc_1 :::"'&'"::: ) (Set "(" ($#k7_modelc_1 :::"'not'"::: ) "G" ")" ) ")" )); end; :: deftheorem defines :::"'or'"::: MODELC_1:def 20 : (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool (Set (Set (Var "F")) ($#k13_modelc_1 :::"'or'"::: ) (Set (Var "G"))) ($#r1_hidden :::"="::: ) (Set ($#k7_modelc_1 :::"'not'"::: ) (Set "(" (Set "(" ($#k7_modelc_1 :::"'not'"::: ) (Set (Var "F")) ")" ) ($#k8_modelc_1 :::"'&'"::: ) (Set "(" ($#k7_modelc_1 :::"'not'"::: ) (Set (Var "G")) ")" ) ")" )))); theorem :: MODELC_1:2 (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" (Bool (Set (Var "H")) "is" ($#v2_modelc_1 :::"atomic"::: ) ) "or" (Bool (Set (Var "H")) "is" ($#v3_modelc_1 :::"negative"::: ) ) "or" (Bool (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) ) "or" (Bool (Set (Var "H")) "is" ($#v5_modelc_1 :::"ExistNext"::: ) ) "or" (Bool (Set (Var "H")) "is" ($#v6_modelc_1 :::"ExistGlobal"::: ) ) "or" (Bool (Set (Var "H")) "is" ($#v7_modelc_1 :::"ExistUntill"::: ) ) ")" )) ; definitionlet "H" be ($#m2_finseq_1 :::"CTL-formula":::); assume (Bool "(" (Bool (Set (Const "H")) "is" ($#v3_modelc_1 :::"negative"::: ) ) "or" (Bool (Set (Const "H")) "is" ($#v5_modelc_1 :::"ExistNext"::: ) ) "or" (Bool (Set (Const "H")) "is" ($#v6_modelc_1 :::"ExistGlobal"::: ) ) ")" ) ; func :::"the_argument_of"::: "H" -> ($#m2_finseq_1 :::"CTL-formula":::) means :: MODELC_1:def 21 (Bool (Set ($#k7_modelc_1 :::"'not'"::: ) it) ($#r1_hidden :::"="::: ) "H") if (Bool "H" "is" ($#v3_modelc_1 :::"negative"::: ) ) (Bool (Set ($#k9_modelc_1 :::"EX"::: ) it) ($#r1_hidden :::"="::: ) "H") if (Bool "H" "is" ($#v5_modelc_1 :::"ExistNext"::: ) ) otherwise (Bool (Set ($#k10_modelc_1 :::"EG"::: ) it) ($#r1_hidden :::"="::: ) "H"); end; :: deftheorem defines :::"the_argument_of"::: MODELC_1:def 21 : (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Bool "(" (Bool (Set (Var "H")) "is" ($#v3_modelc_1 :::"negative"::: ) ) "or" (Bool (Set (Var "H")) "is" ($#v5_modelc_1 :::"ExistNext"::: ) ) "or" (Bool (Set (Var "H")) "is" ($#v6_modelc_1 :::"ExistGlobal"::: ) ) ")" )) "holds" (Bool "for" (Set (Var "b2")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "H")) "is" ($#v3_modelc_1 :::"negative"::: ) )) "implies" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")))) "iff" (Bool (Set ($#k7_modelc_1 :::"'not'"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Var "H"))) ")" ) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v5_modelc_1 :::"ExistNext"::: ) )) "implies" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")))) "iff" (Bool (Set ($#k9_modelc_1 :::"EX"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Var "H"))) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "H")) "is" ($#v3_modelc_1 :::"negative"::: ) )) & (Bool (Bool "not" (Set (Var "H")) "is" ($#v5_modelc_1 :::"ExistNext"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")))) "iff" (Bool (Set ($#k10_modelc_1 :::"EG"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Var "H"))) ")" ) ")" ")" ))); definitionlet "H" be ($#m2_finseq_1 :::"CTL-formula":::); assume (Bool "(" (Bool (Set (Const "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) ) "or" (Bool (Set (Const "H")) "is" ($#v7_modelc_1 :::"ExistUntill"::: ) ) ")" ) ; func :::"the_left_argument_of"::: "H" -> ($#m2_finseq_1 :::"CTL-formula":::) means :: MODELC_1:def 22 (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Set it ($#k8_modelc_1 :::"'&'"::: ) (Set (Var "H1"))) ($#r1_hidden :::"="::: ) "H")) if (Bool "H" "is" ($#v4_modelc_1 :::"conjunctive"::: ) ) otherwise (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Set it ($#k11_modelc_1 :::"EU"::: ) (Set (Var "H1"))) ($#r1_hidden :::"="::: ) "H")); func :::"the_right_argument_of"::: "H" -> ($#m2_finseq_1 :::"CTL-formula":::) means :: MODELC_1:def 23 (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Set (Set (Var "H1")) ($#k8_modelc_1 :::"'&'"::: ) it) ($#r1_hidden :::"="::: ) "H")) if (Bool "H" "is" ($#v4_modelc_1 :::"conjunctive"::: ) ) otherwise (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Set (Set (Var "H1")) ($#k11_modelc_1 :::"EU"::: ) it) ($#r1_hidden :::"="::: ) "H")); end; :: deftheorem defines :::"the_left_argument_of"::: MODELC_1:def 22 : (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Bool "(" (Bool (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) ) "or" (Bool (Set (Var "H")) "is" ($#v7_modelc_1 :::"ExistUntill"::: ) ) ")" )) "holds" (Bool "for" (Set (Var "b2")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) )) "implies" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k15_modelc_1 :::"the_left_argument_of"::: ) (Set (Var "H")))) "iff" (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Set (Set (Var "b2")) ($#k8_modelc_1 :::"'&'"::: ) (Set (Var "H1"))) ($#r1_hidden :::"="::: ) (Set (Var "H")))) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k15_modelc_1 :::"the_left_argument_of"::: ) (Set (Var "H")))) "iff" (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Set (Set (Var "b2")) ($#k11_modelc_1 :::"EU"::: ) (Set (Var "H1"))) ($#r1_hidden :::"="::: ) (Set (Var "H")))) ")" ) ")" ")" ))); :: deftheorem defines :::"the_right_argument_of"::: MODELC_1:def 23 : (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Bool "(" (Bool (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) ) "or" (Bool (Set (Var "H")) "is" ($#v7_modelc_1 :::"ExistUntill"::: ) ) ")" )) "holds" (Bool "for" (Set (Var "b2")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) )) "implies" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k16_modelc_1 :::"the_right_argument_of"::: ) (Set (Var "H")))) "iff" (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Set (Set (Var "H1")) ($#k8_modelc_1 :::"'&'"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Var "H")))) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k16_modelc_1 :::"the_right_argument_of"::: ) (Set (Var "H")))) "iff" (Bool "ex" (Set (Var "H1")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Set (Set (Var "H1")) ($#k11_modelc_1 :::"EU"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Var "H")))) ")" ) ")" ")" ))); definitionlet "x" be ($#m1_hidden :::"set"::: ) ; func :::"CastCTLformula"::: "x" -> ($#m2_finseq_1 :::"CTL-formula":::) equals :: MODELC_1:def 24 "x" if (Bool "x" ($#r2_hidden :::"in"::: ) (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) )) otherwise (Set ($#k6_modelc_1 :::"atom."::: ) (Set ($#k6_numbers :::"0"::: ) )); end; :: deftheorem defines :::"CastCTLformula"::: MODELC_1:def 24 : (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ))) "implies" (Bool (Set ($#k17_modelc_1 :::"CastCTLformula"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) )))) "implies" (Bool (Set ($#k17_modelc_1 :::"CastCTLformula"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_modelc_1 :::"atom."::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" ")" )); definitionlet "Prop" be ($#m1_hidden :::"set"::: ) ; attr "c2" is :::"strict"::: ; struct :::"KripkeStr"::: "over" "Prop" -> ($#l1_orders_2 :::"RelStr"::: ) ; aggr :::"KripkeStr":::(# :::"carrier":::, :::"Starts":::, :::"InternalRel":::, :::"Label"::: #) -> ($#l1_modelc_1 :::"KripkeStr"::: ) "over" "Prop"; sel :::"Starts"::: "c2" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c2"); sel :::"Label"::: "c2" -> ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c2") "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "Prop" ")" ); end; definitionattr "c1" is :::"strict"::: ; struct :::"CTLModelStr"::: -> ($#l3_robbins1 :::"ComplULattStr"::: ) ; aggr :::"CTLModelStr":::(# :::"carrier":::, :::"BasicAssign":::, :::"L_meet":::, :::"Compl":::, :::"EneXt":::, :::"EGlobal":::, :::"EUntill"::: #) -> ($#l2_modelc_1 :::"CTLModelStr"::: ) ; sel :::"BasicAssign"::: "c1" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c1"); sel :::"EneXt"::: "c1" -> ($#m1_subset_1 :::"UnOp":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c1"); sel :::"EGlobal"::: "c1" -> ($#m1_subset_1 :::"UnOp":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c1"); sel :::"EUntill"::: "c1" -> ($#m1_subset_1 :::"BinOp":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "c1"); end; definitionlet "V" be ($#l2_modelc_1 :::"CTLModelStr"::: ) ; mode Assign of "V" is ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V"); end; definitionfunc :::"atomic_WFF"::: -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) equals :: MODELC_1:def 25 "{" (Set (Var "x")) where x "is" ($#m2_finseq_1 :::"CTL-formula":::) : (Bool (Set (Var "x")) "is" ($#v2_modelc_1 :::"atomic"::: ) ) "}" ; end; :: deftheorem defines :::"atomic_WFF"::: MODELC_1:def 25 : (Bool (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set (Var "x")) where x "is" ($#m2_finseq_1 :::"CTL-formula":::) : (Bool (Set (Var "x")) "is" ($#v2_modelc_1 :::"atomic"::: ) ) "}" ); definitionlet "V" be ($#l2_modelc_1 :::"CTLModelStr"::: ) ; let "Kai" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Const "V"))); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "V"))); pred "f" :::"is-Evaluation-for"::: "Kai" means :: MODELC_1:def 26 (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "H")) "is" ($#v2_modelc_1 :::"atomic"::: ) )) "implies" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set "Kai" ($#k1_funct_1 :::"."::: ) (Set (Var "H")))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v3_modelc_1 :::"negative"::: ) )) "implies" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_robbins1 :::"Compl"::: ) "of" "V") ($#k1_funct_1 :::"."::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) )) "implies" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_lattices :::"L_meet"::: ) "of" "V") ($#k1_binop_1 :::"."::: ) "(" (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k15_modelc_1 :::"the_left_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) "," (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k16_modelc_1 :::"the_right_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) ")" )) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v5_modelc_1 :::"ExistNext"::: ) )) "implies" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u4_modelc_1 :::"EneXt"::: ) "of" "V") ($#k1_funct_1 :::"."::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v6_modelc_1 :::"ExistGlobal"::: ) )) "implies" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u5_modelc_1 :::"EGlobal"::: ) "of" "V") ($#k1_funct_1 :::"."::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v7_modelc_1 :::"ExistUntill"::: ) )) "implies" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u6_modelc_1 :::"EUntill"::: ) "of" "V") ($#k1_binop_1 :::"."::: ) "(" (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k15_modelc_1 :::"the_left_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) "," (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k16_modelc_1 :::"the_right_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) ")" )) ")" ")" )); end; :: deftheorem defines :::"is-Evaluation-for"::: MODELC_1:def 26 : (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModelStr"::: ) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_modelc_1 :::"is-Evaluation-for"::: ) (Set (Var "Kai"))) "iff" (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "H")) "is" ($#v2_modelc_1 :::"atomic"::: ) )) "implies" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "Kai")) ($#k1_funct_1 :::"."::: ) (Set (Var "H")))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v3_modelc_1 :::"negative"::: ) )) "implies" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_robbins1 :::"Compl"::: ) "of" (Set (Var "V"))) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) )) "implies" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_lattices :::"L_meet"::: ) "of" (Set (Var "V"))) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k15_modelc_1 :::"the_left_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) "," (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k16_modelc_1 :::"the_right_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) ")" )) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v5_modelc_1 :::"ExistNext"::: ) )) "implies" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u4_modelc_1 :::"EneXt"::: ) "of" (Set (Var "V"))) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v6_modelc_1 :::"ExistGlobal"::: ) )) "implies" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u5_modelc_1 :::"EGlobal"::: ) "of" (Set (Var "V"))) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v7_modelc_1 :::"ExistUntill"::: ) )) "implies" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u6_modelc_1 :::"EUntill"::: ) "of" (Set (Var "V"))) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k15_modelc_1 :::"the_left_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) "," (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k16_modelc_1 :::"the_right_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) ")" )) ")" ")" )) ")" )))); definitionlet "V" be ($#l2_modelc_1 :::"CTLModelStr"::: ) ; let "Kai" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Const "V"))); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "V"))); let "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); pred "f" :::"is-PreEvaluation-for"::: "n" "," "Kai" means :: MODELC_1:def 27 (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_xxreal_0 :::"<="::: ) "n")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "H")) "is" ($#v2_modelc_1 :::"atomic"::: ) )) "implies" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set "Kai" ($#k1_funct_1 :::"."::: ) (Set (Var "H")))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v3_modelc_1 :::"negative"::: ) )) "implies" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_robbins1 :::"Compl"::: ) "of" "V") ($#k1_funct_1 :::"."::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) )) "implies" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_lattices :::"L_meet"::: ) "of" "V") ($#k1_binop_1 :::"."::: ) "(" (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k15_modelc_1 :::"the_left_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) "," (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k16_modelc_1 :::"the_right_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) ")" )) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v5_modelc_1 :::"ExistNext"::: ) )) "implies" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u4_modelc_1 :::"EneXt"::: ) "of" "V") ($#k1_funct_1 :::"."::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v6_modelc_1 :::"ExistGlobal"::: ) )) "implies" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u5_modelc_1 :::"EGlobal"::: ) "of" "V") ($#k1_funct_1 :::"."::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v7_modelc_1 :::"ExistUntill"::: ) )) "implies" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u6_modelc_1 :::"EUntill"::: ) "of" "V") ($#k1_binop_1 :::"."::: ) "(" (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k15_modelc_1 :::"the_left_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) "," (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k16_modelc_1 :::"the_right_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) ")" )) ")" ")" )); end; :: deftheorem defines :::"is-PreEvaluation-for"::: MODELC_1:def 27 : (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModelStr"::: ) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_modelc_1 :::"is-PreEvaluation-for"::: ) (Set (Var "n")) "," (Set (Var "Kai"))) "iff" (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "H")) "is" ($#v2_modelc_1 :::"atomic"::: ) )) "implies" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "Kai")) ($#k1_funct_1 :::"."::: ) (Set (Var "H")))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v3_modelc_1 :::"negative"::: ) )) "implies" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_robbins1 :::"Compl"::: ) "of" (Set (Var "V"))) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) )) "implies" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_lattices :::"L_meet"::: ) "of" (Set (Var "V"))) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k15_modelc_1 :::"the_left_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) "," (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k16_modelc_1 :::"the_right_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) ")" )) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v5_modelc_1 :::"ExistNext"::: ) )) "implies" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u4_modelc_1 :::"EneXt"::: ) "of" (Set (Var "V"))) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v6_modelc_1 :::"ExistGlobal"::: ) )) "implies" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u5_modelc_1 :::"EGlobal"::: ) "of" (Set (Var "V"))) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "H")) "is" ($#v7_modelc_1 :::"ExistUntill"::: ) )) "implies" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u6_modelc_1 :::"EUntill"::: ) "of" (Set (Var "V"))) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k15_modelc_1 :::"the_left_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) "," (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k16_modelc_1 :::"the_right_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) ")" )) ")" ")" )) ")" ))))); definitionlet "V" be ($#l2_modelc_1 :::"CTLModelStr"::: ) ; let "Kai" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Const "V"))); let "f", "h" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "V"))); let "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "H" be ($#m2_finseq_1 :::"CTL-formula":::); func :::"GraftEval"::: "(" "V" "," "Kai" "," "f" "," "h" "," "n" "," "H" ")" -> ($#m1_hidden :::"set"::: ) equals :: MODELC_1:def 28 (Set "f" ($#k1_funct_1 :::"."::: ) "H") if (Bool (Set ($#k3_finseq_1 :::"len"::: ) "H") ($#r1_xxreal_0 :::">"::: ) (Set "n" ($#k2_nat_1 :::"+"::: ) (Num 1))) (Set "Kai" ($#k1_funct_1 :::"."::: ) "H") if (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) "H") ($#r1_hidden :::"="::: ) (Set "n" ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool "H" "is" ($#v2_modelc_1 :::"atomic"::: ) ) ")" ) (Set (Set "the" ($#u1_robbins1 :::"Compl"::: ) "of" "V") ($#k1_funct_1 :::"."::: ) (Set "(" "h" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) "H" ")" ) ")" )) if (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) "H") ($#r1_hidden :::"="::: ) (Set "n" ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool "H" "is" ($#v3_modelc_1 :::"negative"::: ) ) ")" ) (Set (Set "the" ($#u1_lattices :::"L_meet"::: ) "of" "V") ($#k1_binop_1 :::"."::: ) "(" (Set "(" "h" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k15_modelc_1 :::"the_left_argument_of"::: ) "H" ")" ) ")" ) "," (Set "(" "h" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k16_modelc_1 :::"the_right_argument_of"::: ) "H" ")" ) ")" ) ")" ) if (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) "H") ($#r1_hidden :::"="::: ) (Set "n" ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool "H" "is" ($#v4_modelc_1 :::"conjunctive"::: ) ) ")" ) (Set (Set "the" ($#u4_modelc_1 :::"EneXt"::: ) "of" "V") ($#k1_funct_1 :::"."::: ) (Set "(" "h" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) "H" ")" ) ")" )) if (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) "H") ($#r1_hidden :::"="::: ) (Set "n" ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool "H" "is" ($#v5_modelc_1 :::"ExistNext"::: ) ) ")" ) (Set (Set "the" ($#u5_modelc_1 :::"EGlobal"::: ) "of" "V") ($#k1_funct_1 :::"."::: ) (Set "(" "h" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) "H" ")" ) ")" )) if (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) "H") ($#r1_hidden :::"="::: ) (Set "n" ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool "H" "is" ($#v6_modelc_1 :::"ExistGlobal"::: ) ) ")" ) (Set (Set "the" ($#u6_modelc_1 :::"EUntill"::: ) "of" "V") ($#k1_binop_1 :::"."::: ) "(" (Set "(" "h" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k15_modelc_1 :::"the_left_argument_of"::: ) "H" ")" ) ")" ) "," (Set "(" "h" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k16_modelc_1 :::"the_right_argument_of"::: ) "H" ")" ) ")" ) ")" ) if (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) "H") ($#r1_hidden :::"="::: ) (Set "n" ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool "H" "is" ($#v7_modelc_1 :::"ExistUntill"::: ) ) ")" ) (Set "h" ($#k1_funct_1 :::"."::: ) "H") if (Bool (Set ($#k3_finseq_1 :::"len"::: ) "H") ($#r1_xxreal_0 :::"<"::: ) (Set "n" ($#k2_nat_1 :::"+"::: ) (Num 1))) otherwise (Set ($#k1_xboole_0 :::"{}"::: ) ); end; :: deftheorem defines :::"GraftEval"::: MODELC_1:def 28 : (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModelStr"::: ) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "f")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1)))) "implies" (Bool (Set ($#k19_modelc_1 :::"GraftEval"::: ) "(" (Set (Var "V")) "," (Set (Var "Kai")) "," (Set (Var "f")) "," (Set (Var "h")) "," (Set (Var "n")) "," (Set (Var "H")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H")))) ")" & "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "H")) "is" ($#v2_modelc_1 :::"atomic"::: ) )) "implies" (Bool (Set ($#k19_modelc_1 :::"GraftEval"::: ) "(" (Set (Var "V")) "," (Set (Var "Kai")) "," (Set (Var "f")) "," (Set (Var "h")) "," (Set (Var "n")) "," (Set (Var "H")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "Kai")) ($#k1_funct_1 :::"."::: ) (Set (Var "H")))) ")" & "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "H")) "is" ($#v3_modelc_1 :::"negative"::: ) )) "implies" (Bool (Set ($#k19_modelc_1 :::"GraftEval"::: ) "(" (Set (Var "V")) "," (Set (Var "Kai")) "," (Set (Var "f")) "," (Set (Var "h")) "," (Set (Var "n")) "," (Set (Var "H")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_robbins1 :::"Compl"::: ) "of" (Set (Var "V"))) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) )) "implies" (Bool (Set ($#k19_modelc_1 :::"GraftEval"::: ) "(" (Set (Var "V")) "," (Set (Var "Kai")) "," (Set (Var "f")) "," (Set (Var "h")) "," (Set (Var "n")) "," (Set (Var "H")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_lattices :::"L_meet"::: ) "of" (Set (Var "V"))) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k15_modelc_1 :::"the_left_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) "," (Set "(" (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k16_modelc_1 :::"the_right_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) ")" )) ")" & "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "H")) "is" ($#v5_modelc_1 :::"ExistNext"::: ) )) "implies" (Bool (Set ($#k19_modelc_1 :::"GraftEval"::: ) "(" (Set (Var "V")) "," (Set (Var "Kai")) "," (Set (Var "f")) "," (Set (Var "h")) "," (Set (Var "n")) "," (Set (Var "H")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u4_modelc_1 :::"EneXt"::: ) "of" (Set (Var "V"))) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "H")) "is" ($#v6_modelc_1 :::"ExistGlobal"::: ) )) "implies" (Bool (Set ($#k19_modelc_1 :::"GraftEval"::: ) "(" (Set (Var "V")) "," (Set (Var "Kai")) "," (Set (Var "f")) "," (Set (Var "h")) "," (Set (Var "n")) "," (Set (Var "H")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u5_modelc_1 :::"EGlobal"::: ) "of" (Set (Var "V"))) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k14_modelc_1 :::"the_argument_of"::: ) (Set (Var "H")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "H")) "is" ($#v7_modelc_1 :::"ExistUntill"::: ) )) "implies" (Bool (Set ($#k19_modelc_1 :::"GraftEval"::: ) "(" (Set (Var "V")) "," (Set (Var "Kai")) "," (Set (Var "f")) "," (Set (Var "h")) "," (Set (Var "n")) "," (Set (Var "H")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u6_modelc_1 :::"EUntill"::: ) "of" (Set (Var "V"))) ($#k1_binop_1 :::"."::: ) "(" (Set "(" (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k15_modelc_1 :::"the_left_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) "," (Set "(" (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k16_modelc_1 :::"the_right_argument_of"::: ) (Set (Var "H")) ")" ) ")" ) ")" )) ")" & "(" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1)))) "implies" (Bool (Set ($#k19_modelc_1 :::"GraftEval"::: ) "(" (Set (Var "V")) "," (Set (Var "Kai")) "," (Set (Var "f")) "," (Set (Var "h")) "," (Set (Var "n")) "," (Set (Var "H")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set (Var "H")))) ")" & "(" (Bool (Bool (Bool "not" (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1)))) & (Bool "(" "not" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))) "or" "not" (Bool (Set (Var "H")) "is" ($#v2_modelc_1 :::"atomic"::: ) ) ")" ) & (Bool "(" "not" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))) "or" "not" (Bool (Set (Var "H")) "is" ($#v3_modelc_1 :::"negative"::: ) ) ")" ) & (Bool "(" "not" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))) "or" "not" (Bool (Set (Var "H")) "is" ($#v4_modelc_1 :::"conjunctive"::: ) ) ")" ) & (Bool "(" "not" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))) "or" "not" (Bool (Set (Var "H")) "is" ($#v5_modelc_1 :::"ExistNext"::: ) ) ")" ) & (Bool "(" "not" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))) "or" "not" (Bool (Set (Var "H")) "is" ($#v6_modelc_1 :::"ExistGlobal"::: ) ) ")" ) & (Bool "(" "not" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))) "or" "not" (Bool (Set (Var "H")) "is" ($#v7_modelc_1 :::"ExistUntill"::: ) ) ")" ) & (Bool (Bool "not" (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "H"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1))))) "implies" (Bool (Set ($#k19_modelc_1 :::"GraftEval"::: ) "(" (Set (Var "V")) "," (Set (Var "Kai")) "," (Set (Var "f")) "," (Set (Var "h")) "," (Set (Var "n")) "," (Set (Var "H")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ")" )))))); definitionlet "C" be ($#l2_modelc_1 :::"CTLModelStr"::: ) ; attr "C" is :::"with_basic"::: means :: MODELC_1:def 29 (Bool (Bool "not" (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" "C") "is" ($#v1_xboole_0 :::"empty"::: ) )); end; :: deftheorem defines :::"with_basic"::: MODELC_1:def 29 : (Bool "for" (Set (Var "C")) "being" ($#l2_modelc_1 :::"CTLModelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v10_modelc_1 :::"with_basic"::: ) ) "iff" (Bool (Bool "not" (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "C"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) ")" )); definitionfunc :::"TrivialCTLModel"::: -> ($#l2_modelc_1 :::"CTLModelStr"::: ) equals :: MODELC_1:def 30 (Set ($#g2_modelc_1 :::"CTLModelStr"::: ) "(#" (Num 1) "," (Set "(" ($#k2_subset_1 :::"[#]"::: ) (Num 1) ")" ) "," (Set ($#k9_funct_5 :::"op2"::: ) ) "," (Set ($#k8_funct_5 :::"op1"::: ) ) "," (Set ($#k8_funct_5 :::"op1"::: ) ) "," (Set ($#k8_funct_5 :::"op1"::: ) ) "," (Set ($#k9_funct_5 :::"op2"::: ) ) "#)" ); end; :: deftheorem defines :::"TrivialCTLModel"::: MODELC_1:def 30 : (Bool (Set ($#k20_modelc_1 :::"TrivialCTLModel"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#g2_modelc_1 :::"CTLModelStr"::: ) "(#" (Num 1) "," (Set "(" ($#k2_subset_1 :::"[#]"::: ) (Num 1) ")" ) "," (Set ($#k9_funct_5 :::"op2"::: ) ) "," (Set ($#k8_funct_5 :::"op1"::: ) ) "," (Set ($#k8_funct_5 :::"op1"::: ) ) "," (Set ($#k8_funct_5 :::"op1"::: ) ) "," (Set ($#k9_funct_5 :::"op2"::: ) ) "#)" )); registration cluster (Set ($#k20_modelc_1 :::"TrivialCTLModel"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v9_modelc_1 :::"strict"::: ) ($#v10_modelc_1 :::"with_basic"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) for ($#l2_modelc_1 :::"CTLModelStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v10_modelc_1 :::"with_basic"::: ) for ($#l2_modelc_1 :::"CTLModelStr"::: ) ; end; definitionmode CTLModel is ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v10_modelc_1 :::"with_basic"::: ) ($#l2_modelc_1 :::"CTLModelStr"::: ) ; end; registrationlet "C" be ($#l2_modelc_1 :::"CTLModel":::); cluster (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" "C") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionlet "V" be ($#l2_modelc_1 :::"CTLModel":::); let "Kai" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Const "V"))); let "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"EvalSet"::: "(" "V" "," "Kai" "," "n" ")" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) equals :: MODELC_1:def 31 "{" (Set (Var "h")) where h "is" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") : (Bool (Set (Var "h")) ($#r2_modelc_1 :::"is-PreEvaluation-for"::: ) "n" "," "Kai") "}" ; end; :: deftheorem defines :::"EvalSet"::: MODELC_1:def 31 : (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k21_modelc_1 :::"EvalSet"::: ) "(" (Set (Var "V")) "," (Set (Var "Kai")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "h")) where h "is" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) : (Bool (Set (Var "h")) ($#r2_modelc_1 :::"is-PreEvaluation-for"::: ) (Set (Var "n")) "," (Set (Var "Kai"))) "}" )))); definitionlet "V" be ($#l2_modelc_1 :::"CTLModel":::); let "v0" be ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "V"))); let "x" be ($#m1_hidden :::"set"::: ) ; func :::"CastEval"::: "(" "V" "," "x" "," "v0" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") equals :: MODELC_1:def 32 "x" if (Bool "x" ($#r2_hidden :::"in"::: ) (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") ")" )) otherwise (Set (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) ($#k1_margrel1 :::"-->"::: ) "v0"); end; :: deftheorem defines :::"CastEval"::: MODELC_1:def 32 : (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "v0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) ")" ))) "implies" (Bool (Set ($#k22_modelc_1 :::"CastEval"::: ) "(" (Set (Var "V")) "," (Set (Var "x")) "," (Set (Var "v0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) ")" )))) "implies" (Bool (Set ($#k22_modelc_1 :::"CastEval"::: ) "(" (Set (Var "V")) "," (Set (Var "x")) "," (Set (Var "v0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) ($#k1_margrel1 :::"-->"::: ) (Set (Var "v0")))) ")" ")" )))); definitionlet "V" be ($#l2_modelc_1 :::"CTLModel":::); let "Kai" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Const "V"))); func :::"EvalFamily"::: "(" "V" "," "Kai" ")" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) means :: MODELC_1:def 33 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") ")" ")" ))) & (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k21_modelc_1 :::"EvalSet"::: ) "(" "V" "," "Kai" "," (Set (Var "n")) ")" ))) ")" ) ")" )); end; :: deftheorem defines :::"EvalFamily"::: MODELC_1:def 33 : (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "b3")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k23_modelc_1 :::"EvalFamily"::: ) "(" (Set (Var "V")) "," (Set (Var "Kai")) ")" )) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) ")" ")" ))) & (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k21_modelc_1 :::"EvalSet"::: ) "(" (Set (Var "V")) "," (Set (Var "Kai")) "," (Set (Var "n")) ")" ))) ")" ) ")" )) ")" )))); theorem :: MODELC_1:3 (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool (Set (Var "f")) ($#r1_modelc_1 :::"is-Evaluation-for"::: ) (Set (Var "Kai")))))) ; theorem :: MODELC_1:4 (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool (Bool (Set (Var "f1")) ($#r1_modelc_1 :::"is-Evaluation-for"::: ) (Set (Var "Kai"))) & (Bool (Set (Var "f2")) ($#r1_modelc_1 :::"is-Evaluation-for"::: ) (Set (Var "Kai")))) "holds" (Bool (Set (Var "f1")) ($#r2_funct_2 :::"="::: ) (Set (Var "f2")))))) ; definitionlet "V" be ($#l2_modelc_1 :::"CTLModel":::); let "Kai" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Const "V"))); let "H" be ($#m2_finseq_1 :::"CTL-formula":::); func :::"Evaluate"::: "(" "H" "," "Kai" ")" -> ($#m1_subset_1 :::"Assign":::) "of" "V" means :: MODELC_1:def 34 (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_modelc_1 :::"is-Evaluation-for"::: ) "Kai") & (Bool it ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) "H")) ")" )); end; :: deftheorem defines :::"Evaluate"::: MODELC_1:def 34 : (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set (Var "H")) "," (Set (Var "Kai")) ")" )) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k12_modelc_1 :::"CTL_WFF"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_modelc_1 :::"is-Evaluation-for"::: ) (Set (Var "Kai"))) & (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "H")))) ")" )) ")" ))))); notationlet "V" be ($#l2_modelc_1 :::"CTLModel":::); let "f" be ($#m1_subset_1 :::"Assign":::) "of" (Set (Const "V")); synonym :::"'not'"::: "f" for "f" :::"`"::: ; let "g" be ($#m1_subset_1 :::"Assign":::) "of" (Set (Const "V")); synonym "f" :::"'&'"::: "g" for "f" :::""/\""::: "g"; end; definitionlet "V" be ($#l2_modelc_1 :::"CTLModel":::); let "f" be ($#m1_subset_1 :::"Assign":::) "of" (Set (Const "V")); func :::"EX"::: "f" -> ($#m1_subset_1 :::"Assign":::) "of" "V" equals :: MODELC_1:def 35 (Set (Set "the" ($#u4_modelc_1 :::"EneXt"::: ) "of" "V") ($#k3_funct_2 :::"."::: ) "f"); func :::"EG"::: "f" -> ($#m1_subset_1 :::"Assign":::) "of" "V" equals :: MODELC_1:def 36 (Set (Set "the" ($#u5_modelc_1 :::"EGlobal"::: ) "of" "V") ($#k3_funct_2 :::"."::: ) "f"); end; :: deftheorem defines :::"EX"::: MODELC_1:def 35 : (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k25_modelc_1 :::"EX"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u4_modelc_1 :::"EneXt"::: ) "of" (Set (Var "V"))) ($#k3_funct_2 :::"."::: ) (Set (Var "f")))))); :: deftheorem defines :::"EG"::: MODELC_1:def 36 : (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k26_modelc_1 :::"EG"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u5_modelc_1 :::"EGlobal"::: ) "of" (Set (Var "V"))) ($#k3_funct_2 :::"."::: ) (Set (Var "f")))))); definitionlet "V" be ($#l2_modelc_1 :::"CTLModel":::); let "f", "g" be ($#m1_subset_1 :::"Assign":::) "of" (Set (Const "V")); func "f" :::"EU"::: "g" -> ($#m1_subset_1 :::"Assign":::) "of" "V" equals :: MODELC_1:def 37 (Set (Set "the" ($#u6_modelc_1 :::"EUntill"::: ) "of" "V") ($#k5_binop_1 :::"."::: ) "(" "f" "," "g" ")" ); func "f" :::"'or'"::: "g" -> ($#m1_subset_1 :::"Assign":::) "of" "V" equals :: MODELC_1:def 38 (Set ($#k3_robbins1 :::"'not'"::: ) (Set "(" (Set "(" ($#k3_robbins1 :::"'not'"::: ) "f" ")" ) ($#k2_lattices :::"'&'"::: ) (Set "(" ($#k3_robbins1 :::"'not'"::: ) "g" ")" ) ")" )); end; :: deftheorem defines :::"EU"::: MODELC_1:def 37 : (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "f")) ($#k27_modelc_1 :::"EU"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u6_modelc_1 :::"EUntill"::: ) "of" (Set (Var "V"))) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" )))); :: deftheorem defines :::"'or'"::: MODELC_1:def 38 : (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "f")) ($#k28_modelc_1 :::"'or'"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set ($#k3_robbins1 :::"'not'"::: ) (Set "(" (Set "(" ($#k3_robbins1 :::"'not'"::: ) (Set (Var "f")) ")" ) ($#k2_lattices :::"'&'"::: ) (Set "(" ($#k3_robbins1 :::"'not'"::: ) (Set (Var "g")) ")" ) ")" ))))); theorem :: MODELC_1:5 (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) "holds" (Bool (Set ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set "(" ($#k7_modelc_1 :::"'not'"::: ) (Set (Var "H")) ")" ) "," (Set (Var "Kai")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_robbins1 :::"'not'"::: ) (Set "(" ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set (Var "H")) "," (Set (Var "Kai")) ")" ")" )))))) ; theorem :: MODELC_1:6 (Bool "for" (Set (Var "H1")) "," (Set (Var "H2")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) "holds" (Bool (Set ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set "(" (Set (Var "H1")) ($#k8_modelc_1 :::"'&'"::: ) (Set (Var "H2")) ")" ) "," (Set (Var "Kai")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set (Var "H1")) "," (Set (Var "Kai")) ")" ")" ) ($#k2_lattices :::"'&'"::: ) (Set "(" ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set (Var "H2")) "," (Set (Var "Kai")) ")" ")" )))))) ; theorem :: MODELC_1:7 (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) "holds" (Bool (Set ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set "(" ($#k9_modelc_1 :::"EX"::: ) (Set (Var "H")) ")" ) "," (Set (Var "Kai")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k25_modelc_1 :::"EX"::: ) (Set "(" ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set (Var "H")) "," (Set (Var "Kai")) ")" ")" )))))) ; theorem :: MODELC_1:8 (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) "holds" (Bool (Set ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set "(" ($#k10_modelc_1 :::"EG"::: ) (Set (Var "H")) ")" ) "," (Set (Var "Kai")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k26_modelc_1 :::"EG"::: ) (Set "(" ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set (Var "H")) "," (Set (Var "Kai")) ")" ")" )))))) ; theorem :: MODELC_1:9 (Bool "for" (Set (Var "H1")) "," (Set (Var "H2")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) "holds" (Bool (Set ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set "(" (Set (Var "H1")) ($#k11_modelc_1 :::"EU"::: ) (Set (Var "H2")) ")" ) "," (Set (Var "Kai")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set (Var "H1")) "," (Set (Var "Kai")) ")" ")" ) ($#k27_modelc_1 :::"EU"::: ) (Set "(" ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set (Var "H2")) "," (Set (Var "Kai")) ")" ")" )))))) ; theorem :: MODELC_1:10 (Bool "for" (Set (Var "H1")) "," (Set (Var "H2")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "V")) "being" ($#l2_modelc_1 :::"CTLModel":::) (Bool "for" (Set (Var "Kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set (Var "V"))) "holds" (Bool (Set ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set "(" (Set (Var "H1")) ($#k13_modelc_1 :::"'or'"::: ) (Set (Var "H2")) ")" ) "," (Set (Var "Kai")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set (Var "H1")) "," (Set (Var "Kai")) ")" ")" ) ($#k28_modelc_1 :::"'or'"::: ) (Set "(" ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set (Var "H2")) "," (Set (Var "Kai")) ")" ")" )))))) ; notationlet "f" be ($#m1_hidden :::"Function":::); let "n" be ($#m1_hidden :::"Nat":::); synonym "f" :::"|**"::: "n" for :::"iter"::: "(" "f" "," "n" ")" ; end; definitionlet "S" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "n" be ($#m1_hidden :::"Nat":::); :: original: :::"|**"::: redefine func "f" :::"|**"::: "n" -> ($#m1_subset_1 :::"Function":::) "of" "S" "," "S"; end; scheme :: MODELC_1:sch 1 ExistPath{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set F1 "(" ")" ) "," (Set F1 "(" ")" ), F3() -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ), F4( ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" )) -> ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set F1 "(" ")" ) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set F3 "(" ")" )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" )) & (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set F4 "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ) ")" )) ")" ) ")" ) ")" )) provided (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) "holds" (Bool (Set (Set "(" ($#k9_relat_1 :::"Im"::: ) "(" (Set F2 "(" ")" ) "," (Set (Var "s")) ")" ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set F4 "(" (Set (Var "s")) ")" )) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set F1 "(" ")" ))) proof end; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); mode :::"inf_path"::: "of" "R" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," "S" means :: MODELC_1:def 39 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" it ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ) "," (Set "(" it ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "R")); end; :: deftheorem defines :::"inf_path"::: MODELC_1:def 39 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R"))) "iff" (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R")))) ")" )))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; func :::"ModelSP"::: "S" -> ($#m1_hidden :::"set"::: ) equals :: MODELC_1:def 40 (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "S" "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ); end; :: deftheorem defines :::"ModelSP"::: MODELC_1:def 40 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "S")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ))); registrationlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_hidden :::"set"::: ) ; func :::"Fid"::: "(" "f" "," "S" ")" -> ($#m1_subset_1 :::"Function":::) "of" "S" "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) equals :: MODELC_1:def 41 "f" if (Bool "f" ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S")) otherwise (Set "S" ($#k1_margrel1 :::"-->"::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )); end; :: deftheorem defines :::"Fid"::: MODELC_1:def 41 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S"))))) "implies" (Bool (Set ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f"))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")))))) "implies" (Bool (Set ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k1_margrel1 :::"-->"::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) ))) ")" ")" ))); scheme :: MODELC_1:sch 2 Func1EX{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ), F3( ($#m1_hidden :::"set"::: ) "," ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) )) -> ($#v1_xboolean :::"boolean"::: ) ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "g")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set F1 "(" ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool "(" (Bool (Set F3 "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "g")) "," (Set F1 "(" ")" ) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" ) ")" ) ")" )) proof end; scheme :: MODELC_1:sch 3 Func1Unique{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ), F3( ($#m1_hidden :::"set"::: ) "," ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) )) -> ($#v1_xboolean :::"boolean"::: ) ($#m1_hidden :::"set"::: ) } : (Bool "for" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "g1")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set F1 "(" ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool "(" (Bool (Set F3 "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "g1")) "," (Set F1 "(" ")" ) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" ) ")" ) & (Bool (Set (Var "g2")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set F1 "(" ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool "(" (Bool (Set F3 "(" (Set (Var "s")) "," (Set F2 "(" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "g2")) "," (Set F1 "(" ")" ) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" ) ")" )) "holds" (Bool (Set (Var "g1")) ($#r1_hidden :::"="::: ) (Set (Var "g2")))) proof end; scheme :: MODELC_1:sch 4 UnOpEX{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2( ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) } : (Bool "ex" (Set (Var "o")) "being" ($#m1_subset_1 :::"UnOp":::) "of" (Set F1 "(" ")" ) "st" (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool (Set (Set (Var "o")) ($#k1_funct_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set F2 "(" (Set (Var "f")) ")" )))) proof end; scheme :: MODELC_1:sch 5 UnOpUnique{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F3( ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F2 "(" ")" ) } : (Bool "for" (Set (Var "o1")) "," (Set (Var "o2")) "being" ($#m1_subset_1 :::"UnOp":::) "of" (Set F2 "(" ")" ) "st" (Bool (Bool "(" "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool (Set (Set (Var "o1")) ($#k1_funct_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set F3 "(" (Set (Var "f")) ")" )) ")" ) & (Bool "(" "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool (Set (Set (Var "o2")) ($#k1_funct_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set F3 "(" (Set (Var "f")) ")" )) ")" )) "holds" (Bool (Set (Var "o1")) ($#r2_funct_2 :::"="::: ) (Set (Var "o2")))) proof end; scheme :: MODELC_1:sch 6 Func2EX{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ), F3() -> ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ), F4( ($#m1_hidden :::"set"::: ) "," ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "," ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) )) -> ($#v1_xboolean :::"boolean"::: ) ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "h")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set F1 "(" ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool "(" (Bool (Set F4 "(" (Set (Var "s")) "," (Set F2 "(" ")" ) "," (Set F3 "(" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "h")) "," (Set F1 "(" ")" ) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" ) ")" ) ")" )) proof end; scheme :: MODELC_1:sch 7 Func2Unique{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ), F3() -> ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ), F4( ($#m1_hidden :::"set"::: ) "," ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "," ($#m1_subset_1 :::"Function":::) "of" (Set F1 "(" ")" ) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) )) -> ($#v1_xboolean :::"boolean"::: ) ($#m1_hidden :::"set"::: ) } : (Bool "for" (Set (Var "h1")) "," (Set (Var "h2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "h1")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set F1 "(" ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool "(" (Bool (Set F4 "(" (Set (Var "s")) "," (Set F2 "(" ")" ) "," (Set F3 "(" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "h1")) "," (Set F1 "(" ")" ) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" ) ")" ) & (Bool (Set (Var "h2")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set F1 "(" ")" ))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set F1 "(" ")" ))) "holds" (Bool "(" (Bool (Set F4 "(" (Set (Var "s")) "," (Set F2 "(" ")" ) "," (Set F3 "(" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "h2")) "," (Set F1 "(" ")" ) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" ) ")" )) "holds" (Bool (Set (Var "h1")) ($#r1_hidden :::"="::: ) (Set (Var "h2")))) proof end; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_hidden :::"set"::: ) ; func :::"Not_0"::: "(" "f" "," "S" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S") means :: MODELC_1:def 42 (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) "S")) "holds" (Bool "(" (Bool (Set ($#k3_xboolean :::"'not'"::: ) (Set "(" ($#k4_modelc_1 :::"Castboolean"::: ) (Set "(" (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" "f" "," "S" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" it "," "S" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )); end; :: deftheorem defines :::"Not_0"::: MODELC_1:def 42 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k32_modelc_1 :::"Not_0"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" )) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool "(" (Bool (Set ($#k3_xboolean :::"'not'"::: ) (Set "(" ($#k4_modelc_1 :::"Castboolean"::: ) (Set "(" (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "b3")) "," (Set (Var "S")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )) ")" )))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; func :::"Not_"::: "S" -> ($#m1_subset_1 :::"UnOp":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) "S" ")" ) means :: MODELC_1:def 43 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k32_modelc_1 :::"Not_0"::: ) "(" (Set (Var "f")) "," "S" ")" ))); end; :: deftheorem defines :::"Not_"::: MODELC_1:def 43 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"UnOp":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k33_modelc_1 :::"Not_"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k32_modelc_1 :::"Not_0"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ))) ")" ))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "S")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ); let "x" be ($#m1_hidden :::"set"::: ) ; func :::"EneXt_univ"::: "(" "x" "," "f" "," "R" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) equals :: MODELC_1:def 44 (Set ($#k8_margrel1 :::"TRUE"::: ) ) if (Bool "(" (Bool "x" ($#r2_hidden :::"in"::: ) "S") & (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" "R" "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "x") & (Bool (Set "f" ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )) ")" ) otherwise (Set ($#k7_margrel1 :::"FALSE"::: ) ); end; :: deftheorem defines :::"EneXt_univ"::: MODELC_1:def 44 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" ))) "implies" (Bool (Set ($#k34_modelc_1 :::"EneXt_univ"::: ) "(" (Set (Var "x")) "," (Set (Var "f")) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" & "(" (Bool (Bool "(" "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) "or" (Bool "for" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "holds" (Bool "(" "not" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) "or" "not" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )) ")" )) "implies" (Bool (Set ($#k34_modelc_1 :::"EneXt_univ"::: ) "(" (Set (Var "x")) "," (Set (Var "f")) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) ")" ")" ))))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "f" be ($#m1_hidden :::"set"::: ) ; func :::"EneXt_0"::: "(" "f" "," "R" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S") means :: MODELC_1:def 45 (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) "S")) "holds" (Bool "(" (Bool (Set ($#k34_modelc_1 :::"EneXt_univ"::: ) "(" (Set (Var "s")) "," (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" "f" "," "S" ")" ")" ) "," "R" ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" it "," "S" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )); end; :: deftheorem defines :::"EneXt_0"::: MODELC_1:def 45 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k35_modelc_1 :::"EneXt_0"::: ) "(" (Set (Var "f")) "," (Set (Var "R")) ")" )) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool "(" (Bool (Set ($#k34_modelc_1 :::"EneXt_univ"::: ) "(" (Set (Var "s")) "," (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" ) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "b4")) "," (Set (Var "S")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )) ")" ))))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); func :::"EneXt_"::: "R" -> ($#m1_subset_1 :::"UnOp":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) "S" ")" ) means :: MODELC_1:def 46 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k35_modelc_1 :::"EneXt_0"::: ) "(" (Set (Var "f")) "," "R" ")" ))); end; :: deftheorem defines :::"EneXt_"::: MODELC_1:def 46 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"UnOp":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k36_modelc_1 :::"EneXt_"::: ) (Set (Var "R")))) "iff" (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k35_modelc_1 :::"EneXt_0"::: ) "(" (Set (Var "f")) "," (Set (Var "R")) ")" ))) ")" )))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "S")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ); let "x" be ($#m1_hidden :::"set"::: ) ; func :::"EGlobal_univ"::: "(" "x" "," "f" "," "R" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) equals :: MODELC_1:def 47 (Set ($#k8_margrel1 :::"TRUE"::: ) ) if (Bool "(" (Bool "x" ($#r2_hidden :::"in"::: ) "S") & (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" "R" "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "x") & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set "f" ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" ) ")" )) ")" ) otherwise (Set ($#k7_margrel1 :::"FALSE"::: ) ); end; :: deftheorem defines :::"EGlobal_univ"::: MODELC_1:def 47 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" ) ")" ))) "implies" (Bool (Set ($#k37_modelc_1 :::"EGlobal_univ"::: ) "(" (Set (Var "x")) "," (Set (Var "f")) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" & "(" (Bool (Bool "(" "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) "or" (Bool "for" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "holds" (Bool "(" "not" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) "or" (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool "not" (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )))) ")" )) ")" )) "implies" (Bool (Set ($#k37_modelc_1 :::"EGlobal_univ"::: ) "(" (Set (Var "x")) "," (Set (Var "f")) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) ")" ")" ))))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "f" be ($#m1_hidden :::"set"::: ) ; func :::"EGlobal_0"::: "(" "f" "," "R" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S") means :: MODELC_1:def 48 (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) "S")) "holds" (Bool "(" (Bool (Set ($#k37_modelc_1 :::"EGlobal_univ"::: ) "(" (Set (Var "s")) "," (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" "f" "," "S" ")" ")" ) "," "R" ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" it "," "S" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )); end; :: deftheorem defines :::"EGlobal_0"::: MODELC_1:def 48 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k38_modelc_1 :::"EGlobal_0"::: ) "(" (Set (Var "f")) "," (Set (Var "R")) ")" )) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool "(" (Bool (Set ($#k37_modelc_1 :::"EGlobal_univ"::: ) "(" (Set (Var "s")) "," (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" ) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "b4")) "," (Set (Var "S")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )) ")" ))))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); func :::"EGlobal_"::: "R" -> ($#m1_subset_1 :::"UnOp":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) "S" ")" ) means :: MODELC_1:def 49 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k38_modelc_1 :::"EGlobal_0"::: ) "(" (Set (Var "f")) "," "R" ")" ))); end; :: deftheorem defines :::"EGlobal_"::: MODELC_1:def 49 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"UnOp":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k39_modelc_1 :::"EGlobal_"::: ) (Set (Var "R")))) "iff" (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k38_modelc_1 :::"EGlobal_0"::: ) "(" (Set (Var "f")) "," (Set (Var "R")) ")" ))) ")" )))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f", "g" be ($#m1_hidden :::"set"::: ) ; func :::"And_0"::: "(" "f" "," "g" "," "S" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S") means :: MODELC_1:def 50 (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) "S")) "holds" (Bool "(" (Bool (Set (Set "(" ($#k4_modelc_1 :::"Castboolean"::: ) (Set "(" (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" "f" "," "S" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) ")" ) ($#k4_xboolean :::"'&'"::: ) (Set "(" ($#k4_modelc_1 :::"Castboolean"::: ) (Set "(" (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" "g" "," "S" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" it "," "S" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )); end; :: deftheorem defines :::"And_0"::: MODELC_1:def 50 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k40_modelc_1 :::"And_0"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "S")) ")" )) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k4_modelc_1 :::"Castboolean"::: ) (Set "(" (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) ")" ) ($#k4_xboolean :::"'&'"::: ) (Set "(" ($#k4_modelc_1 :::"Castboolean"::: ) (Set "(" (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "g")) "," (Set (Var "S")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "b4")) "," (Set (Var "S")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )) ")" )))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; func :::"And_"::: "S" -> ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) "S" ")" ) means :: MODELC_1:def 51 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S")) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S"))) "holds" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k40_modelc_1 :::"And_0"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) "," "S" ")" ))); end; :: deftheorem defines :::"And_"::: MODELC_1:def 51 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k41_modelc_1 :::"And_"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k40_modelc_1 :::"And_0"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "S")) ")" ))) ")" ))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "f", "g" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "S")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ); let "x" be ($#m1_hidden :::"set"::: ) ; func :::"EUntill_univ"::: "(" "x" "," "f" "," "g" "," "R" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) equals :: MODELC_1:def 52 (Set ($#k8_margrel1 :::"TRUE"::: ) ) if (Bool "(" (Bool "x" ($#r2_hidden :::"in"::: ) "S") & (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" "R" "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) "x") & (Bool "ex" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m")))) "holds" (Bool (Set "f" ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" ) & (Bool (Set "g" ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "m")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )) ")" )) ")" ) otherwise (Set ($#k7_margrel1 :::"FALSE"::: ) ); end; :: deftheorem defines :::"EUntill_univ"::: MODELC_1:def 52 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool "ex" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m")))) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" ) & (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "m")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )) ")" ))) "implies" (Bool (Set ($#k42_modelc_1 :::"EUntill_univ"::: ) "(" (Set (Var "x")) "," (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" & "(" (Bool (Bool "(" "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) "or" (Bool "for" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "holds" (Bool "(" "not" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) "or" (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool "ex" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m"))) & (Bool (Bool "not" (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) ))) ")" )) "or" "not" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "m")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )) ")" )) ")" )) "implies" (Bool (Set ($#k42_modelc_1 :::"EUntill_univ"::: ) "(" (Set (Var "x")) "," (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_margrel1 :::"FALSE"::: ) )) ")" ")" ))))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "f", "g" be ($#m1_hidden :::"set"::: ) ; func :::"EUntill_0"::: "(" "f" "," "g" "," "R" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S") means :: MODELC_1:def 53 (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) "S")) "holds" (Bool "(" (Bool (Set ($#k42_modelc_1 :::"EUntill_univ"::: ) "(" (Set (Var "s")) "," (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" "f" "," "S" ")" ")" ) "," (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" "g" "," "S" ")" ")" ) "," "R" ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" it "," "S" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )); end; :: deftheorem defines :::"EUntill_0"::: MODELC_1:def 53 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k43_modelc_1 :::"EUntill_0"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "R")) ")" )) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool "(" (Bool (Set ($#k42_modelc_1 :::"EUntill_univ"::: ) "(" (Set (Var "s")) "," (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" ) "," (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "g")) "," (Set (Var "S")) ")" ")" ) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "b5")) "," (Set (Var "S")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )) ")" ))))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); func :::"EUntill_"::: "R" -> ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) "S" ")" ) means :: MODELC_1:def 54 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S")) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) "S"))) "holds" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k43_modelc_1 :::"EUntill_0"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) "," "R" ")" ))); end; :: deftheorem defines :::"EUntill_"::: MODELC_1:def 54 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k44_modelc_1 :::"EUntill_"::: ) (Set (Var "R")))) "iff" (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k43_modelc_1 :::"EUntill_0"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "R")) ")" ))) ")" )))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); let "s" be ($#m1_hidden :::"set"::: ) ; func :::"F_LABEL"::: "(" "s" "," "X" ")" -> ($#m1_subset_1 :::"Subset":::) "of" "X" means :: MODELC_1:def 55 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X") & (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" "S" "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) "s") ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )) ")" ) ")" )); end; :: deftheorem defines :::"F_LABEL"::: MODELC_1:def 55 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k45_modelc_1 :::"F_LABEL"::: ) "(" (Set (Var "s")) "," (Set (Var "X")) ")" )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b4"))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )) ")" ) ")" )) ")" ))))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); func :::"Label_"::: "X" -> ($#m1_subset_1 :::"Function":::) "of" "S" "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "X" ")" ) means :: MODELC_1:def 56 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "S")) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k45_modelc_1 :::"F_LABEL"::: ) "(" (Set (Var "x")) "," "X" ")" ))); end; :: deftheorem defines :::"Label_"::: MODELC_1:def 56 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "X")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k46_modelc_1 :::"Label_"::: ) (Set (Var "X")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k45_modelc_1 :::"F_LABEL"::: ) "(" (Set (Var "x")) "," (Set (Var "X")) ")" ))) ")" )))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S0" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "Prop" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); func :::"KModel"::: "(" "R" "," "S0" "," "Prop" ")" -> ($#l1_modelc_1 :::"KripkeStr"::: ) "over" "Prop" equals :: MODELC_1:def 57 (Set ($#g1_modelc_1 :::"KripkeStr"::: ) "(#" "S" "," "S0" "," "R" "," (Set "(" ($#k46_modelc_1 :::"Label_"::: ) "Prop" ")" ) "#)" ); end; :: deftheorem defines :::"KModel"::: MODELC_1:def 57 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "Prop")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) "holds" (Bool (Set ($#k47_modelc_1 :::"KModel"::: ) "(" (Set (Var "R")) "," (Set (Var "S0")) "," (Set (Var "Prop")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g1_modelc_1 :::"KripkeStr"::: ) "(#" (Set (Var "S")) "," (Set (Var "S0")) "," (Set (Var "R")) "," (Set "(" ($#k46_modelc_1 :::"Label_"::: ) (Set (Var "Prop")) ")" ) "#)" )))))); registrationlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S0" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "Prop" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); cluster (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k47_modelc_1 :::"KModel"::: ) "(" "R" "," "S0" "," "Prop" ")" ")" )) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S0" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "Prop" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); cluster (Set ($#k30_modelc_1 :::"ModelSP"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k47_modelc_1 :::"KModel"::: ) "(" "R" "," "S0" "," "Prop" ")" ")" ))) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k47_modelc_1 :::"KModel"::: ) "(" "R" "," "S0" "," "Prop" ")" ")" )) "," (Set ($#k6_margrel1 :::"BOOLEAN"::: ) ) ")" ")" ); end; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); func :::"BASSModel"::: "(" "R" "," "BASSIGN" ")" -> ($#l2_modelc_1 :::"CTLModelStr"::: ) equals :: MODELC_1:def 58 (Set ($#g2_modelc_1 :::"CTLModelStr"::: ) "(#" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) "S" ")" ) "," "BASSIGN" "," (Set "(" ($#k41_modelc_1 :::"And_"::: ) "S" ")" ) "," (Set "(" ($#k33_modelc_1 :::"Not_"::: ) "S" ")" ) "," (Set "(" ($#k36_modelc_1 :::"EneXt_"::: ) "R" ")" ) "," (Set "(" ($#k39_modelc_1 :::"EGlobal_"::: ) "R" ")" ) "," (Set "(" ($#k44_modelc_1 :::"EUntill_"::: ) "R" ")" ) "#)" ); end; :: deftheorem defines :::"BASSModel"::: MODELC_1:def 58 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) "holds" (Bool (Set ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g2_modelc_1 :::"CTLModelStr"::: ) "(#" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) "," (Set (Var "BASSIGN")) "," (Set "(" ($#k41_modelc_1 :::"And_"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k33_modelc_1 :::"Not_"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k36_modelc_1 :::"EneXt_"::: ) (Set (Var "R")) ")" ) "," (Set "(" ($#k39_modelc_1 :::"EGlobal_"::: ) (Set (Var "R")) ")" ) "," (Set "(" ($#k44_modelc_1 :::"EUntill_"::: ) (Set (Var "R")) ")" ) "#)" ))))); registrationlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); cluster (Set ($#k48_modelc_1 :::"BASSModel"::: ) "(" "R" "," "BASSIGN" ")" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v10_modelc_1 :::"with_basic"::: ) ; end; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); let "s" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "S")); let "f" be ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Const "R")) "," (Set (Const "BASSIGN")) ")" ")" ); pred "s" :::"|="::: "f" means :: MODELC_1:def 59 (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" "f" "," "S" ")" ")" ) ($#k3_funct_2 :::"."::: ) "s") ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )); end; :: deftheorem defines :::"|="::: MODELC_1:def 59 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "f"))) "iff" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "f")) "," (Set (Var "S")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k8_margrel1 :::"TRUE"::: ) )) ")" )))))); notationlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); let "s" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "S")); let "f" be ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Const "R")) "," (Set (Const "BASSIGN")) ")" ")" ); antonym "s" :::"|/="::: "f" for "s" :::"|="::: "f"; end; theorem :: MODELC_1:11 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "BASSIGN")))) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "a"))) "iff" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k46_modelc_1 :::"Label_"::: ) (Set (Var "BASSIGN")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "s")))) ")" )))))) ; theorem :: MODELC_1:12 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k3_robbins1 :::"'not'"::: ) (Set (Var "f")))) "iff" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|/="::: ) (Set (Var "f"))) ")" )))))) ; theorem :: MODELC_1:13 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Set (Var "f")) ($#k2_lattices :::"'&'"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "f"))) & (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "g"))) ")" ) ")" )))))) ; theorem :: MODELC_1:14 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k25_modelc_1 :::"EX"::: ) (Set (Var "f")))) "iff" (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "s"))) & (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Num 1)) ($#r3_modelc_1 :::"|="::: ) (Set (Var "f"))) ")" )) ")" )))))) ; theorem :: MODELC_1:15 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k26_modelc_1 :::"EG"::: ) (Set (Var "f")))) "iff" (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r3_modelc_1 :::"|="::: ) (Set (Var "f"))) ")" ) ")" )) ")" )))))) ; theorem :: MODELC_1:16 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Set (Var "f")) ($#k27_modelc_1 :::"EU"::: ) (Set (Var "g")))) "iff" (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "s"))) & (Bool "ex" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m")))) "holds" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "j"))) ($#r3_modelc_1 :::"|="::: ) (Set (Var "f"))) ")" ) & (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r3_modelc_1 :::"|="::: ) (Set (Var "g"))) ")" )) ")" )) ")" )))))) ; theorem :: MODELC_1:17 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Set (Var "f")) ($#k28_modelc_1 :::"'or'"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "f"))) "or" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "g"))) ")" ) ")" )))))) ; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); let "kai" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Const "R")) "," (Set (Const "BASSIGN")) ")" ")" )); let "s" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "S")); let "H" be ($#m2_finseq_1 :::"CTL-formula":::); pred "s" "," "kai" :::"|="::: "H" means :: MODELC_1:def 60 (Bool "s" ($#r3_modelc_1 :::"|="::: ) (Set ($#k24_modelc_1 :::"Evaluate"::: ) "(" "H" "," "kai" ")" )); end; :: deftheorem defines :::"|="::: MODELC_1:def 60 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" )) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) "holds" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set (Var "H"))) "iff" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k24_modelc_1 :::"Evaluate"::: ) "(" (Set (Var "H")) "," (Set (Var "kai")) ")" )) ")" ))))))); notationlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); let "kai" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Const "R")) "," (Set (Const "BASSIGN")) ")" ")" )); let "s" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "S")); let "H" be ($#m2_finseq_1 :::"CTL-formula":::); antonym "s" "," "kai" :::"|/="::: "H" for "s" "," "kai" :::"|="::: "H"; end; theorem :: MODELC_1:18 (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" )) "st" (Bool (Bool (Set (Var "H")) "is" ($#v2_modelc_1 :::"atomic"::: ) )) "holds" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set (Var "H"))) "iff" (Bool (Set (Set (Var "kai")) ($#k1_funct_1 :::"."::: ) (Set (Var "H"))) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k46_modelc_1 :::"Label_"::: ) (Set (Var "BASSIGN")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "s")))) ")" ))))))) ; theorem :: MODELC_1:19 (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" )) "holds" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set ($#k7_modelc_1 :::"'not'"::: ) (Set (Var "H")))) "iff" (Bool (Set (Var "s")) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|/="::: ) (Set (Var "H"))) ")" ))))))) ; theorem :: MODELC_1:20 (Bool "for" (Set (Var "H1")) "," (Set (Var "H2")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" )) "holds" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set (Set (Var "H1")) ($#k8_modelc_1 :::"'&'"::: ) (Set (Var "H2")))) "iff" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set (Var "H1"))) & (Bool (Set (Var "s")) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set (Var "H2"))) ")" ) ")" ))))))) ; theorem :: MODELC_1:21 (Bool "for" (Set (Var "H1")) "," (Set (Var "H2")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" )) "holds" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set (Set (Var "H1")) ($#k13_modelc_1 :::"'or'"::: ) (Set (Var "H2")))) "iff" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set (Var "H1"))) "or" (Bool (Set (Var "s")) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set (Var "H2"))) ")" ) ")" ))))))) ; theorem :: MODELC_1:22 (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" )) "holds" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set ($#k9_modelc_1 :::"EX"::: ) (Set (Var "H")))) "iff" (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "s"))) & (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Num 1)) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set (Var "H"))) ")" )) ")" ))))))) ; theorem :: MODELC_1:23 (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" )) "holds" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set ($#k10_modelc_1 :::"EG"::: ) (Set (Var "H")))) "iff" (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set (Var "H"))) ")" ) ")" )) ")" ))))))) ; theorem :: MODELC_1:24 (Bool "for" (Set (Var "H1")) "," (Set (Var "H2")) "being" ($#m2_finseq_1 :::"CTL-formula":::) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "kai")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k18_modelc_1 :::"atomic_WFF"::: ) ) "," (Set "the" ($#u3_modelc_1 :::"BasicAssign"::: ) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" )) "holds" (Bool "(" (Bool (Set (Var "s")) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set (Set (Var "H1")) ($#k11_modelc_1 :::"EU"::: ) (Set (Var "H2")))) "iff" (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "s"))) & (Bool "ex" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m")))) "holds" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "j"))) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set (Var "H1"))) ")" ) & (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) "," (Set (Var "kai")) ($#r4_modelc_1 :::"|="::: ) (Set (Var "H2"))) ")" )) ")" )) ")" ))))))) ; theorem :: MODELC_1:25 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "st" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "s0"))))))) ; theorem :: MODELC_1:26 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v1_partfun1 :::"total"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) ")" ))) ")" ))) ; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "s0" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "S")); let "pai" be ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Const "R")); let "n" be ($#m1_hidden :::"set"::: ) ; func :::"PrePath"::: "(" "n" "," "s0" "," "pai" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" "S" equals :: MODELC_1:def 61 "s0" if (Bool "n" ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) otherwise (Set "pai" ($#k3_funct_2 :::"."::: ) (Set "(" ($#k2_modelc_1 :::"k_nat"::: ) (Set "(" (Set "(" ($#k2_modelc_1 :::"k_nat"::: ) "n" ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ")" )); end; :: deftheorem defines :::"PrePath"::: MODELC_1:def 61 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k49_modelc_1 :::"PrePath"::: ) "(" (Set (Var "n")) "," (Set (Var "s0")) "," (Set (Var "pai")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "s0"))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) "implies" (Bool (Set ($#k49_modelc_1 :::"PrePath"::: ) "(" (Set (Var "n")) "," (Set (Var "s0")) "," (Set (Var "pai")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k2_modelc_1 :::"k_nat"::: ) (Set "(" (Set "(" ($#k2_modelc_1 :::"k_nat"::: ) (Set (Var "n")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ")" ))) ")" ")" )))))); theorem :: MODELC_1:27 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s0")) "," (Set (Var "s1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s0")) "," (Set (Var "s1")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R")))) "holds" (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "s0"))) & (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "s1"))) ")" ))))) ; theorem :: MODELC_1:28 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k25_modelc_1 :::"EX"::: ) (Set (Var "f")))) "iff" (Bool "ex" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "s1")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool (Set (Var "s1")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "f"))) ")" )) ")" )))))) ; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "H" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); func :::"Pred"::: "(" "H" "," "R" ")" -> ($#m1_subset_1 :::"Subset":::) "of" "S" equals :: MODELC_1:def 62 "{" (Set (Var "s")) where s "is" ($#m1_subset_1 :::"Element"::: ) "of" "S" : (Bool "ex" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "S" "st" (Bool "(" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) "H") & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "R") ")" )) "}" ; end; :: deftheorem defines :::"Pred"::: MODELC_1:def 62 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "H")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool (Set ($#k50_modelc_1 :::"Pred"::: ) "(" (Set (Var "H")) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "s")) where s "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) : (Bool "ex" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set (Var "H"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) ")" )) "}" )))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); let "f" be ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Const "R")) "," (Set (Const "BASSIGN")) ")" ")" ); func :::"SIGMA"::: "f" -> ($#m1_subset_1 :::"Subset":::) "of" "S" equals :: MODELC_1:def 63 "{" (Set (Var "s")) where s "is" ($#m1_subset_1 :::"Element"::: ) "of" "S" : (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) "f") "}" ; end; :: deftheorem defines :::"SIGMA"::: MODELC_1:def 63 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "s")) where s "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) : (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "f"))) "}" ))))); theorem :: MODELC_1:29 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "st" (Bool (Bool (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))))))) ; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); func :::"Tau"::: "(" "T" "," "R" "," "BASSIGN" ")" -> ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" "R" "," "BASSIGN" ")" ")" ) means :: MODELC_1:def 64 (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) "S")) "holds" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" it "," "S" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_funct_3 :::"chi"::: ) "(" "T" "," "S" ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))))); end; :: deftheorem defines :::"Tau"::: MODELC_1:def 64 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k52_modelc_1 :::"Tau"::: ) "(" (Set (Var "T")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" )) "iff" (Bool "for" (Set (Var "s")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool (Set (Set "(" ($#k31_modelc_1 :::"Fid"::: ) "(" (Set (Var "b5")) "," (Set (Var "S")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_funct_3 :::"chi"::: ) "(" (Set (Var "T")) "," (Set (Var "S")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "s"))))) ")" )))))); theorem :: MODELC_1:30 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "T1")) "," (Set (Var "T2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k52_modelc_1 :::"Tau"::: ) "(" (Set (Var "T1")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k52_modelc_1 :::"Tau"::: ) "(" (Set (Var "T2")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ))) "holds" (Bool (Set (Var "T1")) ($#r1_hidden :::"="::: ) (Set (Var "T2"))))))) ; theorem :: MODELC_1:31 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool (Set ($#k52_modelc_1 :::"Tau"::: ) "(" (Set "(" ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "f")) ")" ) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f"))))))) ; theorem :: MODELC_1:32 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set "(" ($#k52_modelc_1 :::"Tau"::: ) "(" (Set (Var "T")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "T"))))))) ; theorem :: MODELC_1:33 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool "(" (Bool (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set "(" ($#k3_robbins1 :::"'not'"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set "(" (Set (Var "f")) ($#k2_lattices :::"'&'"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "g")) ")" ))) & (Bool (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set "(" (Set (Var "f")) ($#k28_modelc_1 :::"'or'"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "f")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "g")) ")" ))) ")" ))))) ; theorem :: MODELC_1:34 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "G1")) "," (Set (Var "G2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "G1")) ($#r1_tarski :::"c="::: ) (Set (Var "G2")))) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k52_modelc_1 :::"Tau"::: ) "(" (Set (Var "G1")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ))) "holds" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k52_modelc_1 :::"Tau"::: ) "(" (Set (Var "G2")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ))))))) ; theorem :: MODELC_1:35 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "st" (Bool (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "f1")))) "holds" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "f2"))) ")" )) "holds" (Bool (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "f1"))) ($#r1_tarski :::"c="::: ) (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "f2")))))))) ; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); let "f", "g" be ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Const "R")) "," (Set (Const "BASSIGN")) ")" ")" ); func :::"Fax"::: "(" "f" "," "g" ")" -> ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" "R" "," "BASSIGN" ")" ")" ) equals :: MODELC_1:def 65 (Set "f" ($#k2_lattices :::"'&'"::: ) (Set "(" ($#k25_modelc_1 :::"EX"::: ) "g" ")" )); end; :: deftheorem defines :::"Fax"::: MODELC_1:def 65 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool (Set ($#k53_modelc_1 :::"Fax"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_lattices :::"'&'"::: ) (Set "(" ($#k25_modelc_1 :::"EX"::: ) (Set (Var "g")) ")" ))))))); theorem :: MODELC_1:36 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "st" (Bool (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "g1")))) "holds" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "g2"))) ")" )) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k53_modelc_1 :::"Fax"::: ) "(" (Set (Var "f")) "," (Set (Var "g1")) ")" ))) "holds" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k53_modelc_1 :::"Fax"::: ) "(" (Set (Var "f")) "," (Set (Var "g2")) ")" ))))))) ; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); let "f" be ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Const "R")) "," (Set (Const "BASSIGN")) ")" ")" ); let "G" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); func :::"SigFaxTau"::: "(" "f" "," "G" "," "R" "," "BASSIGN" ")" -> ($#m1_subset_1 :::"Subset":::) "of" "S" equals :: MODELC_1:def 66 (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set "(" ($#k53_modelc_1 :::"Fax"::: ) "(" "f" "," (Set "(" ($#k52_modelc_1 :::"Tau"::: ) "(" "G" "," "R" "," "BASSIGN" ")" ")" ) ")" ")" )); end; :: deftheorem defines :::"SigFaxTau"::: MODELC_1:def 66 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) (Bool "for" (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool (Set ($#k54_modelc_1 :::"SigFaxTau"::: ) "(" (Set (Var "f")) "," (Set (Var "G")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set "(" ($#k53_modelc_1 :::"Fax"::: ) "(" (Set (Var "f")) "," (Set "(" ($#k52_modelc_1 :::"Tau"::: ) "(" (Set (Var "G")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) ")" ")" )))))))); theorem :: MODELC_1:37 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) (Bool "for" (Set (Var "G1")) "," (Set (Var "G2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "G1")) ($#r1_tarski :::"c="::: ) (Set (Var "G2")))) "holds" (Bool (Set ($#k54_modelc_1 :::"SigFaxTau"::: ) "(" (Set (Var "f")) "," (Set (Var "G1")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k54_modelc_1 :::"SigFaxTau"::: ) "(" (Set (Var "f")) "," (Set (Var "G2")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ))))))) ; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "pai" be ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Const "R")); let "k" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"PathShift"::: "(" "pai" "," "k" ")" -> ($#m1_modelc_1 :::"inf_path"::: ) "of" "R" means :: MODELC_1:def 67 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set "pai" ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) "k" ")" )))); end; :: deftheorem defines :::"PathShift"::: MODELC_1:def 67 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b5")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k55_modelc_1 :::"PathShift"::: ) "(" (Set (Var "pai")) "," (Set (Var "k")) ")" )) "iff" (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b5")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )))) ")" )))))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "pai1", "pai2" be ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Const "R")); let "n", "k" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"PathChange"::: "(" "pai1" "," "pai2" "," "k" "," "n" ")" -> ($#m1_hidden :::"set"::: ) equals :: MODELC_1:def 68 (Set "pai1" ($#k3_funct_2 :::"."::: ) "n") if (Bool "n" ($#r1_xxreal_0 :::"<"::: ) "k") otherwise (Set "pai2" ($#k1_funct_1 :::"."::: ) (Set "(" "n" ($#k6_xcmplx_0 :::"-"::: ) "k" ")" )); end; :: deftheorem defines :::"PathChange"::: MODELC_1:def 68 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "pai1")) "," (Set (Var "pai2")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k")))) "implies" (Bool (Set ($#k56_modelc_1 :::"PathChange"::: ) "(" (Set (Var "pai1")) "," (Set (Var "pai2")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "pai1")) ($#k3_funct_2 :::"."::: ) (Set (Var "n")))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))))) "implies" (Bool (Set ($#k56_modelc_1 :::"PathChange"::: ) "(" (Set (Var "pai1")) "," (Set (Var "pai2")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "pai2")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "k")) ")" ))) ")" ")" ))))); definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "pai1", "pai2" be ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Const "R")); let "k" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"PathConc"::: "(" "pai1" "," "pai2" "," "k" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," "S" means :: MODELC_1:def 69 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k56_modelc_1 :::"PathChange"::: ) "(" "pai1" "," "pai2" "," "k" "," (Set (Var "n")) ")" ))); end; :: deftheorem defines :::"PathConc"::: MODELC_1:def 69 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "pai1")) "," (Set (Var "pai2")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b6")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "b6")) ($#r1_hidden :::"="::: ) (Set ($#k57_modelc_1 :::"PathConc"::: ) "(" (Set (Var "pai1")) "," (Set (Var "pai2")) "," (Set (Var "k")) ")" )) "iff" (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b6")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k56_modelc_1 :::"PathChange"::: ) "(" (Set (Var "pai1")) "," (Set (Var "pai2")) "," (Set (Var "k")) "," (Set (Var "n")) ")" ))) ")" )))))); theorem :: MODELC_1:38 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "pai1")) "," (Set (Var "pai2")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "pai1")) ($#k3_funct_2 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "pai2")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )))) "holds" (Bool (Set ($#k57_modelc_1 :::"PathConc"::: ) "(" (Set (Var "pai1")) "," (Set (Var "pai2")) "," (Set (Var "k")) ")" ) "is" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R"))))))) ; theorem :: MODELC_1:39 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k26_modelc_1 :::"EG"::: ) (Set (Var "f")))) "iff" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k53_modelc_1 :::"Fax"::: ) "(" (Set (Var "f")) "," (Set "(" ($#k26_modelc_1 :::"EG"::: ) (Set (Var "f")) ")" ) ")" )) ")" )))))) ; theorem :: MODELC_1:40 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) (Bool "for" (Set (Var "s0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s0")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "g"))) & (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "g")))) "holds" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k25_modelc_1 :::"EX"::: ) (Set (Var "g")))) ")" )) "holds" (Bool "ex" (Set (Var "pai")) "being" ($#m1_modelc_1 :::"inf_path"::: ) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "s0"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "pai")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r3_modelc_1 :::"|="::: ) (Set (Var "g"))) ")" ) ")" ))))))) ; theorem :: MODELC_1:41 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "st" (Bool (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "g"))) "iff" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k53_modelc_1 :::"Fax"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" )) ")" ) ")" )) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "g")))) "holds" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k26_modelc_1 :::"EG"::: ) (Set (Var "f"))))))))) ; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); let "f" be ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Const "R")) "," (Set (Const "BASSIGN")) ")" ")" ); func :::"TransEG"::: "f" -> bbbadV6_COHSP_1() ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "S" ")" ) "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "S" ")" ) means :: MODELC_1:def 70 (Bool "for" (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset":::) "of" "S" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "G"))) ($#r1_hidden :::"="::: ) (Set ($#k54_modelc_1 :::"SigFaxTau"::: ) "(" "f" "," (Set (Var "G")) "," "R" "," "BASSIGN" ")" ))); end; :: deftheorem defines :::"TransEG"::: MODELC_1:def 70 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) (Bool "for" (Set (Var "b5")) "being" bbbadV6_COHSP_1() ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "S")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set ($#k58_modelc_1 :::"TransEG"::: ) (Set (Var "f")))) "iff" (Bool "for" (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "b5")) ($#k3_funct_2 :::"."::: ) (Set (Var "G"))) ($#r1_hidden :::"="::: ) (Set ($#k54_modelc_1 :::"SigFaxTau"::: ) "(" (Set (Var "f")) "," (Set (Var "G")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ))) ")" )))))); theorem :: MODELC_1:42 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool "(" (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "g"))) "iff" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k53_modelc_1 :::"Fax"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" )) ")" ) ")" ) "iff" (Bool (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "g"))) ($#r2_abian :::"is_a_fixpoint_of"::: ) (Set ($#k58_modelc_1 :::"TransEG"::: ) (Set (Var "f")))) ")" ))))) ; theorem :: MODELC_1:43 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set "(" ($#k26_modelc_1 :::"EG"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_knaster :::"gfp"::: ) "(" (Set (Var "S")) "," (Set "(" ($#k58_modelc_1 :::"TransEG"::: ) (Set (Var "f")) ")" ) ")" )))))) ; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); let "f", "g", "h" be ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Const "R")) "," (Set (Const "BASSIGN")) ")" ")" ); func :::"Foax"::: "(" "g" "," "f" "," "h" ")" -> ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" "R" "," "BASSIGN" ")" ")" ) equals :: MODELC_1:def 71 (Set "g" ($#k28_modelc_1 :::"'or'"::: ) (Set "(" ($#k53_modelc_1 :::"Fax"::: ) "(" "f" "," "h" ")" ")" )); end; :: deftheorem defines :::"Foax"::: MODELC_1:def 71 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool (Set ($#k59_modelc_1 :::"Foax"::: ) "(" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "h")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k28_modelc_1 :::"'or'"::: ) (Set "(" ($#k53_modelc_1 :::"Fax"::: ) "(" (Set (Var "f")) "," (Set (Var "h")) ")" ")" ))))))); theorem :: MODELC_1:44 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h1")) "," (Set (Var "h2")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "st" (Bool (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "h1")))) "holds" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "h2"))) ")" )) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k59_modelc_1 :::"Foax"::: ) "(" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "h1")) ")" ))) "holds" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k59_modelc_1 :::"Foax"::: ) "(" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "h2")) ")" ))))))) ; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); let "f", "g" be ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Const "R")) "," (Set (Const "BASSIGN")) ")" ")" ); let "H" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); func :::"SigFoaxTau"::: "(" "g" "," "f" "," "H" "," "R" "," "BASSIGN" ")" -> ($#m1_subset_1 :::"Subset":::) "of" "S" equals :: MODELC_1:def 72 (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set "(" ($#k59_modelc_1 :::"Foax"::: ) "(" "g" "," "f" "," (Set "(" ($#k52_modelc_1 :::"Tau"::: ) "(" "H" "," "R" "," "BASSIGN" ")" ")" ) ")" ")" )); end; :: deftheorem defines :::"SigFoaxTau"::: MODELC_1:def 72 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) (Bool "for" (Set (Var "H")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool (Set ($#k60_modelc_1 :::"SigFoaxTau"::: ) "(" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "H")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set "(" ($#k59_modelc_1 :::"Foax"::: ) "(" (Set (Var "g")) "," (Set (Var "f")) "," (Set "(" ($#k52_modelc_1 :::"Tau"::: ) "(" (Set (Var "H")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) ")" ")" )))))))); theorem :: MODELC_1:45 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) (Bool "for" (Set (Var "H1")) "," (Set (Var "H2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "H1")) ($#r1_tarski :::"c="::: ) (Set (Var "H2")))) "holds" (Bool (Set ($#k60_modelc_1 :::"SigFoaxTau"::: ) "(" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "H1")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k60_modelc_1 :::"SigFoaxTau"::: ) "(" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "H2")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ))))))) ; theorem :: MODELC_1:46 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Set (Var "f")) ($#k27_modelc_1 :::"EU"::: ) (Set (Var "g")))) "iff" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k59_modelc_1 :::"Foax"::: ) "(" (Set (Var "g")) "," (Set (Var "f")) "," (Set "(" (Set (Var "f")) ($#k27_modelc_1 :::"EU"::: ) (Set (Var "g")) ")" ) ")" )) ")" )))))) ; theorem :: MODELC_1:47 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "st" (Bool (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "h"))) "iff" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k59_modelc_1 :::"Foax"::: ) "(" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "h")) ")" )) ")" ) ")" )) "holds" (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Set (Var "f")) ($#k27_modelc_1 :::"EU"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "h")))))))) ; definitionlet "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "R" be ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "S")) "," (Set (Const "S")); let "BASSIGN" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Const "S")) ")" ); let "f", "g" be ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Const "R")) "," (Set (Const "BASSIGN")) ")" ")" ); func :::"TransEU"::: "(" "f" "," "g" ")" -> bbbadV6_COHSP_1() ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "S" ")" ) "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "S" ")" ) means :: MODELC_1:def 73 (Bool "for" (Set (Var "H")) "being" ($#m1_subset_1 :::"Subset":::) "of" "S" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set ($#k60_modelc_1 :::"SigFoaxTau"::: ) "(" "g" "," "f" "," (Set (Var "H")) "," "R" "," "BASSIGN" ")" ))); end; :: deftheorem defines :::"TransEU"::: MODELC_1:def 73 : (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) (Bool "for" (Set (Var "b6")) "being" bbbadV6_COHSP_1() ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "S")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b6")) ($#r1_hidden :::"="::: ) (Set ($#k61_modelc_1 :::"TransEU"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" )) "iff" (Bool "for" (Set (Var "H")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "b6")) ($#k3_funct_2 :::"."::: ) (Set (Var "H"))) ($#r1_hidden :::"="::: ) (Set ($#k60_modelc_1 :::"SigFoaxTau"::: ) "(" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "H")) "," (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ))) ")" )))))); theorem :: MODELC_1:48 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool "(" (Bool "(" "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set (Var "h"))) "iff" (Bool (Set (Var "s")) ($#r3_modelc_1 :::"|="::: ) (Set ($#k59_modelc_1 :::"Foax"::: ) "(" (Set (Var "g")) "," (Set (Var "f")) "," (Set (Var "h")) ")" )) ")" ) ")" ) "iff" (Bool (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "h"))) ($#r2_abian :::"is_a_fixpoint_of"::: ) (Set ($#k61_modelc_1 :::"TransEU"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" )) ")" ))))) ; theorem :: MODELC_1:49 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set "(" (Set (Var "f")) ($#k27_modelc_1 :::"EU"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_knaster :::"lfp"::: ) "(" (Set (Var "S")) "," (Set "(" ($#k61_modelc_1 :::"TransEU"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ")" ) ")" )))))) ; theorem :: MODELC_1:50 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) "holds" (Bool (Set ($#k51_modelc_1 :::"SIGMA"::: ) (Set "(" ($#k25_modelc_1 :::"EX"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k50_modelc_1 :::"Pred"::: ) "(" (Set "(" ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "f")) ")" ) "," (Set (Var "R")) ")" )))))) ; theorem :: MODELC_1:51 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set "(" ($#k58_modelc_1 :::"TransEG"::: ) (Set (Var "f")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k50_modelc_1 :::"Pred"::: ) "(" (Set (Var "X")) "," (Set (Var "R")) ")" ")" )))))))) ; theorem :: MODELC_1:52 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_partfun1 :::"total"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "BASSIGN")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k30_modelc_1 :::"ModelSP"::: ) (Set (Var "S")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Assign":::) "of" (Set "(" ($#k48_modelc_1 :::"BASSModel"::: ) "(" (Set (Var "R")) "," (Set (Var "BASSIGN")) ")" ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set "(" ($#k61_modelc_1 :::"TransEU"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "g")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set "(" ($#k51_modelc_1 :::"SIGMA"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k50_modelc_1 :::"Pred"::: ) "(" (Set (Var "X")) "," (Set (Var "R")) ")" ")" ) ")" )))))))) ;