:: FINSEQ_1 semantic presentation begin definitionlet "n" be ($#m1_hidden :::"Nat":::); func :::"Seg"::: "n" -> ($#m1_hidden :::"set"::: ) equals :: FINSEQ_1:def 1 "{" (Set (Var "k")) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) "n") ")" ) "}" ; end; :: deftheorem defines :::"Seg"::: FINSEQ_1:def 1 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k1_finseq_1 :::"Seg"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "k")) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) ")" ) "}" )); definitionlet "n" be ($#m1_hidden :::"Nat":::); :: original: :::"Seg"::: redefine func :::"Seg"::: "n" -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ); end; theorem :: FINSEQ_1:1 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "b")))) "iff" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) ")" ) ")" )) ; registrationlet "n" be ($#v1_xboole_0 :::"zero"::: ) ($#m1_hidden :::"Nat":::); cluster (Set ($#k1_finseq_1 :::"Seg"::: ) "n") -> ($#v1_xboole_0 :::"empty"::: ) ; end; theorem :: FINSEQ_1:2 (Bool "(" (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Num 1) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Num 1) "," (Num 2) ($#k2_tarski :::"}"::: ) )) ")" ) ; theorem :: FINSEQ_1:3 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "or" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "a")))) ")" )) ; registrationlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); cluster (Set ($#k1_finseq_1 :::"Seg"::: ) "n") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: FINSEQ_1:4 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "a")) ($#k1_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "a")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) ; theorem :: FINSEQ_1:5 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) "iff" (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "a"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "b")))) ")" )) ; theorem :: FINSEQ_1:6 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "b"))))) "holds" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "b")))) ; theorem :: FINSEQ_1:7 (Bool "for" (Set (Var "c")) "," (Set (Var "a")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "a")))) "holds" (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "a")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "c")) ")" )))) ; theorem :: FINSEQ_1:8 (Bool "for" (Set (Var "c")) "," (Set (Var "a")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "c")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "a")) ")" )))) "holds" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "a")))) ; theorem :: FINSEQ_1:9 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "a")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "a")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "a")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) ; theorem :: FINSEQ_1:10 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "k")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k1_tarski :::"}"::: ) )))) ; definitionlet "IT" be ($#m1_hidden :::"Relation":::); attr "IT" is :::"FinSequence-like"::: means :: FINSEQ_1:def 2 (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) "IT") ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))); end; :: deftheorem defines :::"FinSequence-like"::: FINSEQ_1:def 2 : (Bool "for" (Set (Var "IT")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_finseq_1 :::"FinSequence-like"::: ) ) "iff" (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "IT"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) ")" )); registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_xboole_0 :::"empty"::: ) -> ($#v1_finseq_1 :::"FinSequence-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionmode FinSequence is ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#m1_hidden :::"Function":::); end; registrationlet "n" be ($#m1_hidden :::"Nat":::); cluster (Set ($#k1_finseq_1 :::"Seg"::: ) "n") -> ($#v1_finset_1 :::"finite"::: ) ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) -> ($#v1_finset_1 :::"finite"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "n" be ($#m1_hidden :::"Nat":::); cluster (Set ($#k1_finseq_1 :::"Seg"::: ) "n") -> "n" ($#v3_card_1 :::"-element"::: ) ; end; notationlet "p" be ($#m1_hidden :::"FinSequence":::); synonym :::"len"::: "p" for :::"card"::: "p"; end; definitionlet "p" be ($#m1_hidden :::"FinSequence":::); :: original: :::"len"::: redefine func :::"len"::: "p" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: FINSEQ_1:def 3 (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) "p")); end; :: deftheorem defines :::"len"::: FINSEQ_1:def 3 : (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")))) "iff" (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "p")))) ")" ))); definitionlet "p" be ($#m1_hidden :::"FinSequence":::); :: original: :::"dom"::: redefine func :::"dom"::: "p" -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ); end; theorem :: FINSEQ_1:11 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool "ex" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Set (Var "f")) ($#r1_tarski :::"c="::: ) (Set (Var "p"))))) ; scheme :: FINSEQ_1:sch 1 SeqEx{ F1() -> ($#m1_hidden :::"Nat":::), P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set F1 "(" ")" ))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set F1 "(" ")" )))) "holds" (Bool P1[(Set (Var "k")) "," (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))]) ")" ) ")" )) provided (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set F1 "(" ")" )))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool P1[(Set (Var "k")) "," (Set (Var "x"))]))) proof end; scheme :: FINSEQ_1:sch 2 SeqLambda{ F1() -> ($#m1_hidden :::"Nat":::), F2( ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set F1 "(" ")" )) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set F2 "(" (Set (Var "k")) ")" )) ")" ) ")" )) proof end; theorem :: FINSEQ_1:12 (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "p")))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p")))) & (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "k")) "," (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")) ")" ) ($#k4_tarski :::"]"::: ) )) ")" )))) ; theorem :: FINSEQ_1:13 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "q")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" )) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q")))) ; theorem :: FINSEQ_1:14 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" )) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q")))) ; theorem :: FINSEQ_1:15 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "a")) ")" )) "is" ($#m1_hidden :::"FinSequence":::)))) ; theorem :: FINSEQ_1:16 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set (Var "p"))) "is" ($#m1_hidden :::"FinSequence":::)))) ; theorem :: FINSEQ_1:17 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "a")) ")" )))) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "a")))) ")" ))) ; definitionlet "D" be ($#m1_hidden :::"set"::: ) ; mode :::"FinSequence"::: "of" "D" -> ($#m1_hidden :::"FinSequence":::) means :: FINSEQ_1:def 4 (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) it) ($#r1_tarski :::"c="::: ) "D"); end; :: deftheorem defines :::"FinSequence"::: FINSEQ_1:def 4 : (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "b2")) "is" ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D"))) "iff" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "b2"))) ($#r1_tarski :::"c="::: ) (Set (Var "D"))) ")" ))); registrationlet "D" be ($#m1_hidden :::"set"::: ) ; cluster -> "D" ($#v5_relat_1 :::"-valued"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" "D"; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_xboole_0 :::"empty"::: ) -> ($#v1_finseq_1 :::"FinSequence-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "D" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) "D" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) "," "D" ($#k2_zfmisc_1 :::":]"::: ) )); end; definitionlet "D" be ($#m1_hidden :::"set"::: ) ; :: original: :::"FinSequence"::: redefine mode :::"FinSequence"::: "of" "D" -> ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," "D"; end; theorem :: FINSEQ_1:18 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set (Var "p")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "a")) ")" )) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")))))) ; theorem :: FINSEQ_1:19 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "a")))))) ; theorem :: FINSEQ_1:20 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "iff" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::">="::: ) (Num 1)) ")" )) ; definitionlet "x" be ($#m1_hidden :::"set"::: ) ; func :::"<*":::"x":::"*>"::: -> ($#m1_hidden :::"set"::: ) equals :: FINSEQ_1:def 5 (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Num 1) "," "x" ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ); end; :: deftheorem defines :::"<*"::: FINSEQ_1:def 5 : (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k5_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k5_finseq_1 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Num 1) "," (Set (Var "x")) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ))); definitionlet "D" be ($#m1_hidden :::"set"::: ) ; func :::"<*>"::: "D" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D" equals :: FINSEQ_1:def 6 (Set ($#k1_xboole_0 :::"{}"::: ) ); end; :: deftheorem defines :::"<*>"::: FINSEQ_1:def 6 : (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k6_finseq_1 :::"<*>"::: ) (Set (Var "D"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))); registrationlet "D" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k6_finseq_1 :::"<*>"::: ) "D") -> ($#v1_xboole_0 :::"empty"::: ) ; end; registrationlet "D" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) "D" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_xboole_0 :::"empty"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" "D"; end; definitionlet "p", "q" be ($#m1_hidden :::"FinSequence":::); func "p" :::"^"::: "q" -> ($#m1_hidden :::"FinSequence":::) means :: FINSEQ_1:def 7 (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) "p" ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) "q" ")" ) ")" ))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "p"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set "p" ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" ) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "q"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) "p" ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set "q" ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" ) ")" ); end; :: deftheorem defines :::"^"::: FINSEQ_1:def 7 : (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")))) "iff" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "q")) ")" ) ")" ))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" ) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "q"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" ) ")" ) ")" ))); theorem :: FINSEQ_1:21 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p")) ")" )))) ; theorem :: FINSEQ_1:22 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "q")) ")" )))) ; theorem :: FINSEQ_1:23 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "q")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ")" ))))) ; theorem :: FINSEQ_1:24 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ")" ))))) ; theorem :: FINSEQ_1:25 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" "not" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ))) "or" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p")))) "or" (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "q")))) & (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "n")))) ")" )) ")" ))) ; theorem :: FINSEQ_1:26 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" )))) ; theorem :: FINSEQ_1:27 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "q")) "," (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "q"))))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ))) ")" )))) ; theorem :: FINSEQ_1:28 (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "q")) "," (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "q"))))) "holds" (Bool (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ))))) ; theorem :: FINSEQ_1:29 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" )))) ; theorem :: FINSEQ_1:30 (Bool "for" (Set (Var "q")) "," (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "q"))) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" )))) ; theorem :: FINSEQ_1:31 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "q")) ")" )))) ; theorem :: FINSEQ_1:32 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set "(" (Set (Var "q")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "r")) ")" )))) ; theorem :: FINSEQ_1:33 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool "(" (Bool (Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "r")))) "or" (Bool (Set (Set (Var "r")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")))) ")" )) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q")))) ; theorem :: FINSEQ_1:34 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool (Set (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" )) ; theorem :: FINSEQ_1:35 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) ; definitionlet "D" be ($#m1_hidden :::"set"::: ) ; let "p", "q" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); :: original: :::"^"::: redefine func "p" :::"^"::: "q" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D"; end; definitionlet "x" be ($#m1_hidden :::"set"::: ) ; :: original: :::"<*"::: redefine func :::"<*":::"x":::"*>"::: -> ($#m1_hidden :::"Function":::) means :: FINSEQ_1:def 8 (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 1))) & (Bool (Set it ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) "x") ")" ); end; :: deftheorem defines :::"<*"::: FINSEQ_1:def 8 : (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 1))) & (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ) ")" ))); registrationlet "x" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k5_finseq_1 :::"<*"::: ) "x" ($#k5_finseq_1 :::"*>"::: ) ) -> ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ; end; registrationlet "x" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k5_finseq_1 :::"<*"::: ) "x" ($#k5_finseq_1 :::"*>"::: ) ) -> ($#v1_finseq_1 :::"FinSequence-like"::: ) ; end; theorem :: FINSEQ_1:36 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q"))) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")))) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D"))) & (Bool (Set (Var "q")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D"))) ")" ))) ; definitionlet "x", "y" be ($#m1_hidden :::"set"::: ) ; func :::"<*":::"x" "," "y":::"*>"::: -> ($#m1_hidden :::"set"::: ) equals :: FINSEQ_1:def 9 (Set (Set ($#k9_finseq_1 :::"<*"::: ) "x" ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) "y" ($#k9_finseq_1 :::"*>"::: ) )); let "z" be ($#m1_hidden :::"set"::: ) ; func :::"<*":::"x" "," "y" "," "z":::"*>"::: -> ($#m1_hidden :::"set"::: ) equals :: FINSEQ_1:def 10 (Set (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) "x" ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) "y" ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) "z" ($#k9_finseq_1 :::"*>"::: ) )); end; :: deftheorem defines :::"<*"::: FINSEQ_1:def 9 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "y")) ($#k9_finseq_1 :::"*>"::: ) )))); :: deftheorem defines :::"<*"::: FINSEQ_1:def 10 : (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "y")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "z")) ($#k9_finseq_1 :::"*>"::: ) )))); registrationlet "x", "y" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) -> ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ; let "z" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k11_finseq_1 :::"<*"::: ) "x" "," "y" "," "z" ($#k11_finseq_1 :::"*>"::: ) ) -> ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ; end; registrationlet "x", "y" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) -> ($#v1_finseq_1 :::"FinSequence-like"::: ) ; let "z" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k11_finseq_1 :::"<*"::: ) "x" "," "y" "," "z" ($#k11_finseq_1 :::"*>"::: ) ) -> ($#v1_finseq_1 :::"FinSequence-like"::: ) ; end; theorem :: FINSEQ_1:37 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Num 1) "," (Set (Var "x")) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: FINSEQ_1:38 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) )) "iff" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Num 1))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ")" ) ")" ))) ; theorem :: FINSEQ_1:39 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ")" ) ")" ))) ; theorem :: FINSEQ_1:40 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ) ")" ))) ; theorem :: FINSEQ_1:41 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set (Var "p")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x"))))) ; theorem :: FINSEQ_1:42 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x"))))) ; theorem :: FINSEQ_1:43 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k10_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "z")) ($#k9_finseq_1 :::"*>"::: ) ))) ")" )) ; theorem :: FINSEQ_1:44 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 2)) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" ) ")" ))) ; theorem :: FINSEQ_1:45 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 3)) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "z"))) ")" ) ")" ))) ; theorem :: FINSEQ_1:46 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "ex" (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::)(Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) )))))) ; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); :: original: :::"<*"::: redefine func :::"<*":::"x":::"*>"::: -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D"; end; scheme :: FINSEQ_1:sch 3 IndSeq{ P1[ ($#m1_hidden :::"FinSequence":::)] } : (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool P1[(Set (Var "p"))])) provided (Bool P1[(Set ($#k1_xboole_0 :::"{}"::: ) )]) and (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool P1[(Set (Var "p"))])) "holds" (Bool P1[(Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ))]))) proof end; theorem :: FINSEQ_1:47 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "s")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "r"))))) "holds" (Bool "ex" (Set (Var "t")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "t"))) ($#r1_hidden :::"="::: ) (Set (Var "r"))))) ; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) -> (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "D" be ($#m1_hidden :::"set"::: ) ; func "D" :::"*"::: -> ($#m1_hidden :::"set"::: ) means :: FINSEQ_1:def 11 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "x")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D") ")" )); end; :: deftheorem defines :::"*"::: FINSEQ_1:def 11 : (Bool "for" (Set (Var "D")) "," (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "D")) ($#k13_finseq_1 :::"*"::: ) )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "x")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D"))) ")" )) ")" )); registrationlet "D" be ($#m1_hidden :::"set"::: ) ; cluster (Set "D" ($#k13_finseq_1 :::"*"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: FINSEQ_1:48 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "q")))) & (Bool (Set (Var "p")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "q")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q"))))) ; theorem :: FINSEQ_1:49 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Set (Var "D")) ($#k13_finseq_1 :::"*"::: ) ))) ; scheme :: FINSEQ_1:sch 4 SepSeq{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"FinSequence":::)] } : (Bool "ex" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) "iff" (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set F1 "(" ")" ) ($#k13_finseq_1 :::"*"::: ) )) & (Bool P1[(Set (Var "p"))]) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" )) ")" ))) proof end; definitionlet "IT" be ($#m1_hidden :::"Function":::); attr "IT" is :::"FinSubsequence-like"::: means :: FINSEQ_1:def 12 (Bool "ex" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) "IT") ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))))); end; :: deftheorem defines :::"FinSubsequence-like"::: FINSEQ_1:def 12 : (Bool "for" (Set (Var "IT")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ) "iff" (Bool "ex" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "IT"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k"))))) ")" )); registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionmode FinSubsequence is ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#m1_hidden :::"Function":::); end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) -> ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_hidden :::"set"::: ) ; let "p" be ($#m1_hidden :::"FinSubsequence":::); let "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set "p" ($#k5_relat_1 :::"|"::: ) "X") -> ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ; cluster (Set "X" ($#k6_relat_1 :::"|`"::: ) "p") -> ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) -> (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; given "k" being ($#m1_hidden :::"Nat":::) such that (Bool (Set (Const "X")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Const "k")))) ; func :::"Sgm"::: "X" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: FINSEQ_1:def 13 (Bool "(" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) it) ($#r1_hidden :::"="::: ) "X") & (Bool "(" "for" (Set (Var "l")) "," (Set (Var "m")) "," (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "l"))) & (Bool (Set (Var "l")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) it)) & (Bool (Set (Var "k1")) ($#r1_hidden :::"="::: ) (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "l")))) & (Bool (Set (Var "k2")) ($#r1_hidden :::"="::: ) (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "m"))))) "holds" (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k2"))) ")" ) ")" ); end; :: deftheorem defines :::"Sgm"::: FINSEQ_1:def 13 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "ex" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))))) "holds" (Bool "for" (Set (Var "b2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "X")))) "iff" (Bool "(" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool "(" "for" (Set (Var "l")) "," (Set (Var "m")) "," (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "l"))) & (Bool (Set (Var "l")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2")))) & (Bool (Set (Var "k1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "l")))) & (Bool (Set (Var "k2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))))) "holds" (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k2"))) ")" ) ")" ) ")" ))); theorem :: FINSEQ_1:50 (Bool "for" (Set (Var "p9")) "being" ($#m1_hidden :::"FinSubsequence":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "p9")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "p9"))))) ; definitionlet "p9" be ($#m1_hidden :::"FinSubsequence":::); func :::"Seq"::: "p9" -> ($#m1_hidden :::"Function":::) equals :: FINSEQ_1:def 14 (Set "p9" ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) "p9" ")" ) ")" )); end; :: deftheorem defines :::"Seq"::: FINSEQ_1:def 14 : (Bool "for" (Set (Var "p9")) "being" ($#m1_hidden :::"FinSubsequence":::) "holds" (Bool (Set ($#k15_finseq_1 :::"Seq"::: ) (Set (Var "p9"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p9")) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k14_finseq_1 :::"Sgm"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "p9")) ")" ) ")" )))); registrationlet "p9" be ($#m1_hidden :::"FinSubsequence":::); cluster (Set ($#k15_finseq_1 :::"Seq"::: ) "p9") -> ($#v1_finseq_1 :::"FinSequence-like"::: ) ; end; theorem :: FINSEQ_1:51 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "ex" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))))) "holds" (Bool "(" (Bool (Set ($#k14_finseq_1 :::"Sgm"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "iff" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) ; begin theorem :: FINSEQ_1:52 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "D")) "is" ($#v1_finset_1 :::"finite"::: ) ) "iff" (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Set (Var "D")) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "p"))))) ")" )) ; begin theorem :: FINSEQ_1:53 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))) "," (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m"))) ($#r2_wellord2 :::"are_equipotent"::: ) )) "holds" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Var "m")))) ; theorem :: FINSEQ_1:54 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))) "," (Set (Var "n")) ($#r2_wellord2 :::"are_equipotent"::: ) )) ; theorem :: FINSEQ_1:55 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_card_1 :::"card"::: ) (Set (Var "n"))))) ; theorem :: FINSEQ_1:56 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_finset_1 :::"finite"::: ) )) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Set (Var "X")) "," (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))) ($#r2_wellord2 :::"are_equipotent"::: ) ))) ; theorem :: FINSEQ_1:57 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "n")))) ; begin registrationlet "x" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k5_finseq_1 :::"<*"::: ) "x" ($#k5_finseq_1 :::"*>"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "f1" be ($#m1_hidden :::"FinSequence":::); let "f2" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"FinSequence":::); cluster (Set "f1" ($#k7_finseq_1 :::"^"::: ) "f2") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; cluster (Set "f2" ($#k7_finseq_1 :::"^"::: ) "f1") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "x", "y" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; let "z" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k11_finseq_1 :::"<*"::: ) "x" "," "y" "," "z" ($#k11_finseq_1 :::"*>"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; scheme :: FINSEQ_1:sch 5 SeqDEx{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#m1_hidden :::"Nat":::), P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set F1 "(" ")" ) "st" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set F2 "(" ")" ))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set F2 "(" ")" )))) "holds" (Bool P1[(Set (Var "k")) "," (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))]) ")" ) ")" )) provided (Bool "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set F2 "(" ")" )))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) "st" (Bool P1[(Set (Var "k")) "," (Set (Var "x"))]))) proof end; definitionlet "m" be ($#m1_hidden :::"Nat":::); let "p" be ($#m1_hidden :::"FinSequence":::); func "p" :::"|"::: "m" -> ($#m1_hidden :::"FinSequence":::) equals :: FINSEQ_1:def 15 (Set "p" ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) "m" ")" )); end; :: deftheorem defines :::"|"::: FINSEQ_1:def 15 : (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Set (Var "p")) ($#k16_finseq_1 :::"|"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")) ")" ))))); definitionlet "D" be ($#m1_hidden :::"set"::: ) ; let "m" be ($#m1_hidden :::"Nat":::); let "p" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); :: original: :::"|"::: redefine func "p" :::"|"::: "m" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D"; end; registrationlet "f" be ($#m1_hidden :::"FinSequence":::); cluster (Set "f" ($#k16_finseq_1 :::"|"::: ) (Set ($#k6_numbers :::"0"::: ) )) -> ($#v1_xboole_0 :::"empty"::: ) ; end; theorem :: FINSEQ_1:58 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i")))) "holds" (Bool (Set (Set (Var "q")) ($#k16_finseq_1 :::"|"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "q"))))) ; theorem :: FINSEQ_1:59 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "q")) ($#k16_finseq_1 :::"|"::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "i"))))) ; theorem :: FINSEQ_1:60 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "m")) ")" )))) ; theorem :: FINSEQ_1:61 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "i")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "m")) ")" )))) "holds" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "m")) ")" ))) ")" )) ; definitionlet "R" be ($#m1_hidden :::"Relation":::); func "R" :::"[*]"::: -> ($#m1_hidden :::"Relation":::) means :: FINSEQ_1:def 16 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relat_1 :::"field"::: ) "R")) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relat_1 :::"field"::: ) "R")) & (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "R") ")" ) ")" )) ")" ) ")" )); end; :: deftheorem defines :::"[*]"::: FINSEQ_1:def 16 : (Bool "for" (Set (Var "R")) "," (Set (Var "b2")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "R")) ($#k18_finseq_1 :::"[*]"::: ) )) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relat_1 :::"field"::: ) (Set (Var "R")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relat_1 :::"field"::: ) (Set (Var "R")))) & (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) ")" ) ")" )) ")" ) ")" )) ")" )); theorem :: FINSEQ_1:62 (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "D1")) ($#r1_tarski :::"c="::: ) (Set (Var "D2")))) "holds" (Bool (Set (Set (Var "D1")) ($#k13_finseq_1 :::"*"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "D2")) ($#k13_finseq_1 :::"*"::: ) ))) ; registrationlet "D" be ($#m1_hidden :::"set"::: ) ; cluster (Set "D" ($#k13_finseq_1 :::"*"::: ) ) -> ($#v4_funct_1 :::"functional"::: ) ; end; theorem :: FINSEQ_1:63 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_tarski :::"c="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q"))))) ; theorem :: FINSEQ_1:64 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))))) ; theorem :: FINSEQ_1:65 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q"))))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))))) ; scheme :: FINSEQ_1:sch 6 FinSegRng{ F1() -> ($#m1_hidden :::"Nat":::), F2() -> ($#m1_hidden :::"Nat":::), F3( ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) , P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "{" (Set F3 "(" (Set (Var "i")) ")" ) where i "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set F1 "(" ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set F2 "(" ")" )) & (Bool P1[(Set (Var "i"))]) ")" ) "}" "is" ($#v1_finset_1 :::"finite"::: ) ) proof end; theorem :: FINSEQ_1:66 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x1")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x2")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x3")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x4")) ($#k9_finseq_1 :::"*>"::: ) )))) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 4)) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x1"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "x2"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "x3"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set (Var "x4"))) ")" ))) ; theorem :: FINSEQ_1:67 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x1")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x2")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x3")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x4")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x5")) ($#k9_finseq_1 :::"*>"::: ) )))) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 5)) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x1"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "x2"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "x3"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set (Var "x4"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set (Var "x5"))) ")" ))) ; theorem :: FINSEQ_1:68 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "," (Set (Var "x6")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x1")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x2")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x3")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x4")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x5")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x6")) ($#k9_finseq_1 :::"*>"::: ) )))) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 6)) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x1"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "x2"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "x3"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set (Var "x4"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set (Var "x5"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 6)) ($#r1_hidden :::"="::: ) (Set (Var "x6"))) ")" ))) ; theorem :: FINSEQ_1:69 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "," (Set (Var "x6")) "," (Set (Var "x7")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x1")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x2")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x3")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x4")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x5")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x6")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x7")) ($#k9_finseq_1 :::"*>"::: ) )))) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 7)) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x1"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "x2"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "x3"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set (Var "x4"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set (Var "x5"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 6)) ($#r1_hidden :::"="::: ) (Set (Var "x6"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 7)) ($#r1_hidden :::"="::: ) (Set (Var "x7"))) ")" ))) ; theorem :: FINSEQ_1:70 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "," (Set (Var "x6")) "," (Set (Var "x7")) "," (Set (Var "x8")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x1")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x2")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x3")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x4")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x5")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x6")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x7")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x8")) ($#k9_finseq_1 :::"*>"::: ) )))) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 8)) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x1"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "x2"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "x3"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set (Var "x4"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set (Var "x5"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 6)) ($#r1_hidden :::"="::: ) (Set (Var "x6"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 7)) ($#r1_hidden :::"="::: ) (Set (Var "x7"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 8)) ($#r1_hidden :::"="::: ) (Set (Var "x8"))) ")" ))) ; theorem :: FINSEQ_1:71 (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "," (Set (Var "x3")) "," (Set (Var "x4")) "," (Set (Var "x5")) "," (Set (Var "x6")) "," (Set (Var "x7")) "," (Set (Var "x8")) "," (Set (Var "x9")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x1")) ($#k9_finseq_1 :::"*>"::: ) ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x2")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x3")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x4")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x5")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x6")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x7")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x8")) ($#k9_finseq_1 :::"*>"::: ) ) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x9")) ($#k9_finseq_1 :::"*>"::: ) )))) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 9)) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x1"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Var "x2"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set (Var "x3"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Set (Var "x4"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Set (Var "x5"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 6)) ($#r1_hidden :::"="::: ) (Set (Var "x6"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 7)) ($#r1_hidden :::"="::: ) (Set (Var "x7"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 8)) ($#r1_hidden :::"="::: ) (Set (Var "x8"))) & (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Num 9)) ($#r1_hidden :::"="::: ) (Set (Var "x9"))) ")" ))) ; theorem :: FINSEQ_1:72 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Set (Var "p")) ($#k16_finseq_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: FINSEQ_1:73 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set (Set (Var "f")) ($#k16_finseq_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k16_finseq_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" )))) ; theorem :: FINSEQ_1:74 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k12_finseq_1 :::"*>"::: ) ) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D"))))) ; theorem :: FINSEQ_1:75 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set (Var "p")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "q"))) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D"))))) ; theorem :: FINSEQ_1:76 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "a")) ($#k9_finseq_1 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "b")) ($#k9_finseq_1 :::"*>"::: ) ))) "holds" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "b")))) ; theorem :: FINSEQ_1:77 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k10_finseq_1 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k10_finseq_1 :::"*>"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "d"))) ")" )) ; theorem :: FINSEQ_1:78 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) ($#k11_finseq_1 :::"*>"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "d")) "," (Set (Var "e")) "," (Set (Var "f")) ($#k11_finseq_1 :::"*>"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "e"))) & (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set (Var "f"))) ")" )) ; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v2_relat_1 :::"non-empty"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FINSEQ_1:79 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")))))) ; theorem :: FINSEQ_1:80 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )))) "holds" (Bool "ex" (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q"))))))) ; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) "D" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" "D"; end; definitionlet "p", "q" be ($#m1_hidden :::"FinSequence":::); redefine pred "p" :::"="::: "q" means :: FINSEQ_1:def 17 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) "p") ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "q")) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "p"))) "holds" (Bool (Set "p" ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set "q" ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" ) ")" ); end; :: deftheorem defines :::"="::: FINSEQ_1:def 17 : (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q"))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_funct_1 :::"."::: ) (Set (Var "k")))) ")" ) ")" ) ")" )); theorem :: FINSEQ_1:81 (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "M1"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "M2"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Var "M1")) ($#r1_hidden :::"="::: ) (Set (Var "M2")))) ; registrationlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); cluster (Set ($#k1_finseq_1 :::"Seg"::: ) "n") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: FINSEQ_1:82 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k16_finseq_1 :::"|"::: ) (Set (Var "n")) ")" ) ($#k16_finseq_1 :::"|"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k16_finseq_1 :::"|"::: ) (Set (Var "m")))))) ; theorem :: FINSEQ_1:83 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )))) ; registrationlet "n" be ($#m1_hidden :::"Nat":::); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) "n" ($#v3_card_1 :::"-element"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "x" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k5_finseq_1 :::"<*"::: ) "x" ($#k5_finseq_1 :::"*>"::: ) ) -> (Num 1) ($#v3_card_1 :::"-element"::: ) ; let "y" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k10_finseq_1 :::"<*"::: ) "x" "," "y" ($#k10_finseq_1 :::"*>"::: ) ) -> (Num 2) ($#v3_card_1 :::"-element"::: ) ; let "z" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k11_finseq_1 :::"<*"::: ) "x" "," "y" "," "z" ($#k11_finseq_1 :::"*>"::: ) ) -> (Num 3) ($#v3_card_1 :::"-element"::: ) ; end; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; attr "X" is :::"FinSequence-membered"::: means :: FINSEQ_1:def 18 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set (Var "x")) "is" ($#m1_hidden :::"FinSequence":::))); end; :: deftheorem defines :::"FinSequence-membered"::: FINSEQ_1:def 18 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v3_finseq_1 :::"FinSequence-membered"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "x")) "is" ($#m1_hidden :::"FinSequence":::))) ")" )); registration cluster ($#v1_xboole_0 :::"empty"::: ) -> ($#v3_finseq_1 :::"FinSequence-membered"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_finseq_1 :::"FinSequence-membered"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set "X" ($#k13_finseq_1 :::"*"::: ) ) -> ($#v3_finseq_1 :::"FinSequence-membered"::: ) ; end; theorem :: FINSEQ_1:84 (Bool "for" (Set (Var "x")) "," (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "D")) ($#k13_finseq_1 :::"*"::: ) )) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))))) ; registration cluster ($#v3_finseq_1 :::"FinSequence-membered"::: ) -> ($#v4_funct_1 :::"functional"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FINSEQ_1:85 (Bool "for" (Set (Var "X")) "being" ($#v3_finseq_1 :::"FinSequence-membered"::: ) ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "Y")) ($#k13_finseq_1 :::"*"::: ) )))) ; registrationlet "X" be ($#v3_finseq_1 :::"FinSequence-membered"::: ) ($#m1_hidden :::"set"::: ) ; cluster -> ($#v1_finseq_1 :::"FinSequence-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" "X"; end; registrationlet "X" be ($#v3_finseq_1 :::"FinSequence-membered"::: ) ($#m1_hidden :::"set"::: ) ; cluster -> ($#v3_finseq_1 :::"FinSequence-membered"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "X"); end; theorem :: FINSEQ_1:86 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k5_relat_1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "q"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))))) ; theorem :: FINSEQ_1:87 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q")))) "or" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "p")))) ")" )) "holds" (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: FINSEQ_1:88 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" "not" (Bool (Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) )) "or" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ) "or" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) )) ")" ) ")" ))) ; theorem :: FINSEQ_1:89 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" (Set (Var "b1")) ($#v3_card_1 :::"-element"::: ) ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))))) ; registrationlet "n" be ($#m1_hidden :::"Nat":::); let "m" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be (Set (Const "n")) ($#v3_card_1 :::"-element"::: ) ($#m1_hidden :::"FinSequence":::); let "g" be (Set (Const "m")) ($#v3_card_1 :::"-element"::: ) ($#m1_hidden :::"FinSequence":::); cluster (Set "f" ($#k7_finseq_1 :::"^"::: ) "g") -> (Set "n" ($#k1_nat_1 :::"+"::: ) "m") ($#v3_card_1 :::"-element"::: ) ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v3_valued_0 :::"real-valued"::: ) ($#v5_valued_0 :::"increasing"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) -> ($#v2_funct_1 :::"one-to-one"::: ) ($#v3_valued_0 :::"real-valued"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: FINSEQ_1:90 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "y")) ($#k9_finseq_1 :::"*>"::: ) )))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Num 1))) ; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) "X" ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "D" be ($#v3_finseq_1 :::"FinSequence-membered"::: ) ($#m1_hidden :::"set"::: ) ; let "f" be (Set (Const "D")) ($#v5_relat_1 :::"-valued"::: ) ($#m1_hidden :::"Function":::); let "x" be ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k1_funct_1 :::"."::: ) "x") -> ($#v1_finseq_1 :::"FinSequence-like"::: ) ; end;