:: ASYMPT_0 semantic presentation begin scheme :: ASYMPT_0:sch 1 FinSegRng1{ F1() -> ($#m1_hidden :::"Nat":::), F2() -> ($#m1_hidden :::"Nat":::), F3() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F4( ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F3 "(" ")" ) } : (Bool "{" (Set F4 "(" (Set (Var "i")) ")" ) where i "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set F1 "(" ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set F2 "(" ")" )) ")" ) "}" "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set F3 "(" ")" )) provided (Bool (Set F1 "(" ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set F2 "(" ")" )) proof end; scheme :: ASYMPT_0:sch 2 FinImInit1{ F1() -> ($#m1_hidden :::"Nat":::), F2() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F3( ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F2 "(" ")" ) } : (Bool "{" (Set F3 "(" (Set (Var "n")) ")" ) where n "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set F1 "(" ")" )) "}" "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set F2 "(" ")" )) proof end; scheme :: ASYMPT_0:sch 3 FinImInit2{ F1() -> ($#m1_hidden :::"Nat":::), F2() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F3( ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F2 "(" ")" ) } : (Bool "{" (Set F3 "(" (Set (Var "n")) ")" ) where n "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set F1 "(" ")" )) "}" "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set F2 "(" ")" )) provided (Bool (Set F1 "(" ")" ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) proof end; definitionlet "c" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; attr "c" is :::"logbase"::: means :: ASYMPT_0:def 1 (Bool "(" (Bool "c" ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "c" ($#r1_hidden :::"<>"::: ) (Num 1)) ")" ); end; :: deftheorem defines :::"logbase"::: ASYMPT_0:def 1 : (Bool "for" (Set (Var "c")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "c")) "is" ($#v1_asympt_0 :::"logbase"::: ) ) "iff" (Bool "(" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "c")) ($#r1_hidden :::"<>"::: ) (Num 1)) ")" ) ")" )); registration cluster bbbadV1_XCMPLX_0() ($#v1_xreal_0 :::"real"::: ) ($#v1_xxreal_0 :::"ext-real"::: ) ($#v2_xxreal_0 :::"positive"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); cluster bbbadV1_XCMPLX_0() ($#v1_xreal_0 :::"real"::: ) ($#v1_xxreal_0 :::"ext-real"::: ) ($#v3_xxreal_0 :::"negative"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); cluster bbbadV1_XCMPLX_0() ($#v1_xreal_0 :::"real"::: ) ($#v1_xxreal_0 :::"ext-real"::: ) ($#v1_asympt_0 :::"logbase"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); cluster bbbadV1_XCMPLX_0() ($#v1_xreal_0 :::"real"::: ) ($#v1_xxreal_0 :::"ext-real"::: ) ($#~v3_xxreal_0 "non" ($#v3_xxreal_0 :::"negative"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); cluster bbbadV1_XCMPLX_0() ($#v1_xreal_0 :::"real"::: ) ($#v1_xxreal_0 :::"ext-real"::: ) ($#~v2_xxreal_0 "non" ($#v2_xxreal_0 :::"positive"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); cluster bbbadV1_XCMPLX_0() ($#v1_xreal_0 :::"real"::: ) ($#v1_xxreal_0 :::"ext-real"::: ) ($#~v1_asympt_0 "non" ($#v1_asympt_0 :::"logbase"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ); end; definitionlet "f" be ($#m1_subset_1 :::"Real_Sequence":::); attr "f" is :::"eventually-nonnegative"::: means :: ASYMPT_0:def 2 (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )))); attr "f" is :::"positive"::: means :: ASYMPT_0:def 3 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))); attr "f" is :::"eventually-positive"::: means :: ASYMPT_0:def 4 (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )))); attr "f" is :::"eventually-nonzero"::: means :: ASYMPT_0:def 5 (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )))); attr "f" is :::"eventually-nondecreasing"::: means :: ASYMPT_0:def 6 (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set "f" ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))))); end; :: deftheorem defines :::"eventually-nonnegative"::: ASYMPT_0:def 2 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ) "iff" (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ")" )); :: deftheorem defines :::"positive"::: ASYMPT_0:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_asympt_0 :::"positive"::: ) ) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" )); :: deftheorem defines :::"eventually-positive"::: ASYMPT_0:def 4 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v4_asympt_0 :::"eventually-positive"::: ) ) "iff" (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ")" )); :: deftheorem defines :::"eventually-nonzero"::: ASYMPT_0:def 5 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_asympt_0 :::"eventually-nonzero"::: ) ) "iff" (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ")" )); :: deftheorem defines :::"eventually-nondecreasing"::: ASYMPT_0:def 6 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) ) "iff" (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))))) ")" )); registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) bbbadV1_RELAT_1() bbbadV4_RELAT_1((Set ($#k5_numbers :::"NAT"::: ) )) bbbadV5_RELAT_1((Set ($#k1_numbers :::"REAL"::: ) )) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1((Set ($#k5_numbers :::"NAT"::: ) )) ($#v1_funct_2 :::"quasi_total"::: ) bbbadV1_VALUED_0() bbbadV2_VALUED_0() bbbadV3_VALUED_0() ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#v3_asympt_0 :::"positive"::: ) ($#v4_asympt_0 :::"eventually-positive"::: ) ($#v5_asympt_0 :::"eventually-nonzero"::: ) ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK2_ZFMISC_1((Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ))))); end; registration cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v3_asympt_0 :::"positive"::: ) -> ($#v4_asympt_0 :::"eventually-positive"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK2_ZFMISC_1((Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ))))); cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v4_asympt_0 :::"eventually-positive"::: ) -> ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#v5_asympt_0 :::"eventually-nonzero"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK2_ZFMISC_1((Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ))))); cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#v5_asympt_0 :::"eventually-nonzero"::: ) -> ($#v4_asympt_0 :::"eventually-positive"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK2_ZFMISC_1((Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ))))); end; definitionlet "f", "g" be ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); :: original: :::"+"::: redefine func "f" :::"+"::: "g" -> ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); end; definitionlet "f" be ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); let "c" be ($#v2_xxreal_0 :::"positive"::: ) ($#m1_subset_1 :::"Real":::); :: original: :::"(#)"::: redefine func "c" :::"(#)"::: "f" -> ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); end; definitionlet "f" be ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); let "c" be ($#~v3_xxreal_0 "non" ($#v3_xxreal_0 :::"negative"::: ) ) ($#m1_subset_1 :::"Real":::); :: original: :::"+"::: redefine func "c" :::"+"::: "f" -> ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); end; definitionlet "f" be ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); let "c" be ($#v2_xxreal_0 :::"positive"::: ) ($#m1_subset_1 :::"Real":::); :: original: :::"+"::: redefine func "c" :::"+"::: "f" -> ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::); end; definitionlet "f", "g" be ($#m1_subset_1 :::"Real_Sequence":::); func :::"max"::: "(" "f" "," "g" ")" -> ($#m1_subset_1 :::"Real_Sequence":::) means :: ASYMPT_0:def 7 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k2_square_1 :::"max"::: ) "(" (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) "," (Set "(" "g" ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ")" ))); commutativity (Bool "for" (Set (Var "b1")) "," (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b1")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k2_square_1 :::"max"::: ) "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ")" )) ")" )) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b1")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k2_square_1 :::"max"::: ) "(" (Set "(" (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ")" )))) ; end; :: deftheorem defines :::"max"::: ASYMPT_0:def 7 : (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k5_asympt_0 :::"max"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" )) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k2_square_1 :::"max"::: ) "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ")" ))) ")" )); registrationlet "f" be ($#m1_subset_1 :::"Real_Sequence":::); let "g" be ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); cluster (Set ($#k5_asympt_0 :::"max"::: ) "(" "f" "," "g" ")" ) -> ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ; end; registrationlet "f" be ($#m1_subset_1 :::"Real_Sequence":::); let "g" be ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::); cluster (Set ($#k5_asympt_0 :::"max"::: ) "(" "f" "," "g" ")" ) -> ($#v4_asympt_0 :::"eventually-positive"::: ) ; end; definitionlet "f", "g" be ($#m1_subset_1 :::"Real_Sequence":::); pred "g" :::"majorizes"::: "f" means :: ASYMPT_0:def 8 (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set "g" ($#k1_seq_1 :::"."::: ) (Set (Var "n")))))); end; :: deftheorem defines :::"majorizes"::: ASYMPT_0:def 8 : (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r1_asympt_0 :::"majorizes"::: ) (Set (Var "f"))) "iff" (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))))) ")" )); theorem :: ASYMPT_0:1 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" )) "holds" (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "N")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "m"))))))) ; theorem :: ASYMPT_0:2 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k52_valued_1 :::"/""::: ) (Set (Var "f"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "g")) ($#k52_valued_1 :::"/""::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g")) ")" ) ")" ) ($#k2_real_1 :::"""::: ) )) ")" )) ; theorem :: ASYMPT_0:3 (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_comseq_2 :::"convergent"::: ) )) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "f"))))) ; theorem :: ASYMPT_0:4 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set (Var "g")) ($#r1_asympt_0 :::"majorizes"::: ) (Set (Var "f")))) "holds" (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "g"))))) ; theorem :: ASYMPT_0:5 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "g")) "being" ($#v5_asympt_0 :::"eventually-nonzero"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g"))) "is" ($#v1_limfunc1 :::"divergent_to+infty"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k52_valued_1 :::"/""::: ) (Set (Var "f"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "g")) ($#k52_valued_1 :::"/""::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; begin definitionlet "f" be ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); func :::"Big_Oh"::: "f" -> ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: ASYMPT_0:def 9 "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool "(" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "c")) ($#k8_real_1 :::"*"::: ) (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) & (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) ")" ))) "}" ; end; :: deftheorem defines :::"Big_Oh"::: ASYMPT_0:def 9 : (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool "(" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "c")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) & (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) ")" ))) "}" )); theorem :: ASYMPT_0:6 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "x")) "is" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::)))) ; theorem :: ASYMPT_0:7 (Bool "for" (Set (Var "f")) "being" ($#v3_asympt_0 :::"positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "t")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f")))) "iff" (Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "c")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" )) ")" ))) ; theorem :: ASYMPT_0:8 (Bool "for" (Set (Var "f")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "t")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "c")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ))))) ; theorem :: ASYMPT_0:9 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" (Set (Var "f")) ($#k1_asympt_0 :::"+"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k5_asympt_0 :::"max"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ")" )))) ; theorem :: ASYMPT_0:10 (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f"))))) ; theorem :: ASYMPT_0:11 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g"))))) ; theorem :: ASYMPT_0:12 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g")))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "h"))))) ; theorem :: ASYMPT_0:13 (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "c")) "being" ($#v2_xxreal_0 :::"positive"::: ) ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" (Set (Var "c")) ($#k2_asympt_0 :::"(#)"::: ) (Set (Var "f")) ")" ))))) ; theorem :: ASYMPT_0:14 (Bool "for" (Set (Var "c")) "being" ($#~v3_xxreal_0 "non" ($#v3_xxreal_0 :::"negative"::: ) ) ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "," (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" (Set (Var "c")) ($#k3_asympt_0 :::"+"::: ) (Set (Var "f")) ")" ))))) ; theorem :: ASYMPT_0:15 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g"))))) ; theorem :: ASYMPT_0:16 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g")))) & (Bool (Bool "not" (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f"))))) ")" )) ; theorem :: ASYMPT_0:17 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g"))) "is" ($#v1_limfunc1 :::"divergent_to+infty"::: ) )) "holds" (Bool "(" (Bool (Bool "not" (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g"))))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f")))) ")" )) ; begin definitionlet "f" be ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); func :::"Big_Omega"::: "f" -> ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: ASYMPT_0:def 10 "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool "(" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "d")) ($#k8_real_1 :::"*"::: ) (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) & (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) ")" ))) "}" ; end; :: deftheorem defines :::"Big_Omega"::: ASYMPT_0:def 10 : (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool "(" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "d")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) & (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) ")" ))) "}" )); theorem :: ASYMPT_0:18 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "x")) "is" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::)))) ; theorem :: ASYMPT_0:19 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "g")))) "iff" (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f")))) ")" )) ; theorem :: ASYMPT_0:20 (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "f"))))) ; theorem :: ASYMPT_0:21 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "g")))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "h"))))) ; theorem :: ASYMPT_0:22 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "g"))))) ; theorem :: ASYMPT_0:23 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "f")))) & (Bool (Bool "not" (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "g"))))) ")" )) ; theorem :: ASYMPT_0:24 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g"))) "is" ($#v1_limfunc1 :::"divergent_to+infty"::: ) )) "holds" (Bool "(" (Bool (Bool "not" (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "f"))))) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "g")))) ")" )) ; theorem :: ASYMPT_0:25 (Bool "for" (Set (Var "f")) "," (Set (Var "t")) "being" ($#v3_asympt_0 :::"positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "f")))) "iff" (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "d")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" ) ")" )) ")" )) ; theorem :: ASYMPT_0:26 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set "(" (Set (Var "f")) ($#k1_asympt_0 :::"+"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set "(" ($#k5_asympt_0 :::"max"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ")" )))) ; definitionlet "f" be ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); func :::"Big_Theta"::: "f" -> ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: ASYMPT_0:def 11 (Set (Set "(" ($#k6_asympt_0 :::"Big_Oh"::: ) "f" ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k7_asympt_0 :::"Big_Omega"::: ) "f" ")" )); end; :: deftheorem defines :::"Big_Theta"::: ASYMPT_0:def 11 : (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "f")) ")" )))); theorem :: ASYMPT_0:27 (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool "(" (Bool (Set (Set (Var "d")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "c")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ) ")" ))) "}" )) ; theorem :: ASYMPT_0:28 (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "f"))))) ; theorem :: ASYMPT_0:29 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "f"))))) ; theorem :: ASYMPT_0:30 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "g")))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "h"))))) ; theorem :: ASYMPT_0:31 (Bool "for" (Set (Var "f")) "," (Set (Var "t")) "being" ($#v3_asympt_0 :::"positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "f")))) "iff" (Bool "ex" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "d")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "c")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ) ")" )) ")" )) ; theorem :: ASYMPT_0:32 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set "(" (Set (Var "f")) ($#k1_asympt_0 :::"+"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set "(" ($#k5_asympt_0 :::"max"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ")" )))) ; theorem :: ASYMPT_0:33 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "g"))))) ; theorem :: ASYMPT_0:34 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g")))) & (Bool (Bool "not" (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "g"))))) ")" )) ; theorem :: ASYMPT_0:35 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set (Var "g"))) "is" ($#v1_limfunc1 :::"divergent_to+infty"::: ) )) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "g")))) & (Bool (Bool "not" (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "g"))))) ")" )) ; begin definitionlet "f" be ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); let "X" be ($#m1_hidden :::"set"::: ) ; func :::"Big_Oh"::: "(" "f" "," "X" ")" -> ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: ASYMPT_0:def 12 "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N"))) & (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool "(" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "c")) ($#k8_real_1 :::"*"::: ) (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) & (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) ")" ))) "}" ; end; :: deftheorem defines :::"Big_Oh"::: ASYMPT_0:def 12 : (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_asympt_0 :::"Big_Oh"::: ) "(" (Set (Var "f")) "," (Set (Var "X")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N"))) & (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "c")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) & (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) ")" ))) "}" ))); definitionlet "f" be ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); let "X" be ($#m1_hidden :::"set"::: ) ; func :::"Big_Omega"::: "(" "f" "," "X" ")" -> ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: ASYMPT_0:def 13 "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N"))) & (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool "(" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "d")) ($#k8_real_1 :::"*"::: ) (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) & (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) ")" ))) "}" ; end; :: deftheorem defines :::"Big_Omega"::: ASYMPT_0:def 13 : (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k10_asympt_0 :::"Big_Omega"::: ) "(" (Set (Var "f")) "," (Set (Var "X")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N"))) & (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "d")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) & (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ) ")" ))) "}" ))); definitionlet "f" be ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); let "X" be ($#m1_hidden :::"set"::: ) ; func :::"Big_Theta"::: "(" "f" "," "X" ")" -> ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: ASYMPT_0:def 14 "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N"))) & (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool "(" (Bool (Set (Set (Var "d")) ($#k8_real_1 :::"*"::: ) (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "c")) ($#k8_real_1 :::"*"::: ) (Set "(" "f" ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ) ")" ))) "}" ; end; :: deftheorem defines :::"Big_Theta"::: ASYMPT_0:def 14 : (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k11_asympt_0 :::"Big_Theta"::: ) "(" (Set (Var "f")) "," (Set (Var "X")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N"))) & (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set (Set (Var "d")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "c")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ) ")" ))) "}" ))); theorem :: ASYMPT_0:36 (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k11_asympt_0 :::"Big_Theta"::: ) "(" (Set (Var "f")) "," (Set (Var "X")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_asympt_0 :::"Big_Oh"::: ) "(" (Set (Var "f")) "," (Set (Var "X")) ")" ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k10_asympt_0 :::"Big_Omega"::: ) "(" (Set (Var "f")) "," (Set (Var "X")) ")" ")" ))))) ; definitionlet "f" be ($#m1_subset_1 :::"Real_Sequence":::); let "b" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func "f" :::"taken_every"::: "b" -> ($#m1_subset_1 :::"Real_Sequence":::) means :: ASYMPT_0:def 15 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set "f" ($#k1_seq_1 :::"."::: ) (Set "(" "b" ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" )))); end; :: deftheorem defines :::"taken_every"::: ASYMPT_0:def 15 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k12_asympt_0 :::"taken_every"::: ) (Set (Var "b")))) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "b")) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" )))) ")" )))); definitionlet "f" be ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); let "b" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); pred "f" :::"is_smooth_wrt"::: "b" means :: ASYMPT_0:def 16 (Bool "(" (Bool "f" "is" ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) ) & (Bool (Set "f" ($#k12_asympt_0 :::"taken_every"::: ) "b") ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) "f")) ")" ); end; :: deftheorem defines :::"is_smooth_wrt"::: ASYMPT_0:def 16 : (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_asympt_0 :::"is_smooth_wrt"::: ) (Set (Var "b"))) "iff" (Bool "(" (Bool (Set (Var "f")) "is" ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) ) & (Bool (Set (Set (Var "f")) ($#k12_asympt_0 :::"taken_every"::: ) (Set (Var "b"))) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f")))) ")" ) ")" ))); definitionlet "f" be ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::); attr "f" is :::"smooth"::: means :: ASYMPT_0:def 17 (Bool "for" (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool "f" ($#r2_asympt_0 :::"is_smooth_wrt"::: ) (Set (Var "b")))); end; :: deftheorem defines :::"smooth"::: ASYMPT_0:def 17 : (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v7_asympt_0 :::"smooth"::: ) ) "iff" (Bool "for" (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool (Set (Var "f")) ($#r2_asympt_0 :::"is_smooth_wrt"::: ) (Set (Var "b")))) ")" )); theorem :: ASYMPT_0:37 (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "ex" (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool (Set (Var "f")) ($#r2_asympt_0 :::"is_smooth_wrt"::: ) (Set (Var "b"))) ")" ))) "holds" (Bool (Set (Var "f")) "is" ($#v7_asympt_0 :::"smooth"::: ) )) ; theorem :: ASYMPT_0:38 (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "t")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v7_asympt_0 :::"smooth"::: ) ) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k9_asympt_0 :::"Big_Oh"::: ) "(" (Set (Var "f")) "," "{" (Set "(" (Set (Var "b")) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) where n "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ")" ))) "holds" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f"))))))) ; theorem :: ASYMPT_0:39 (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "t")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v7_asympt_0 :::"smooth"::: ) ) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k10_asympt_0 :::"Big_Omega"::: ) "(" (Set (Var "f")) "," "{" (Set "(" (Set (Var "b")) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) where n "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ")" ))) "holds" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "f"))))))) ; theorem :: ASYMPT_0:40 (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "t")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v7_asympt_0 :::"smooth"::: ) ) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k11_asympt_0 :::"Big_Theta"::: ) "(" (Set (Var "f")) "," "{" (Set "(" (Set (Var "b")) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) where n "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ")" ))) "holds" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "f"))))))) ; begin definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "F", "G" be ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set (Const "X")) "," (Set ($#k1_numbers :::"REAL"::: ) ); func "F" :::"+"::: "G" -> ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" "X" "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: ASYMPT_0:def 18 "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) "F") & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) "G") & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "X" "holds" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" )) "}" ; end; :: deftheorem defines :::"+"::: ASYMPT_0:def 18 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set (Var "X")) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set (Var "F")) ($#k13_asympt_0 :::"+"::: ) (Set (Var "G"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Var "G"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" )) "}" ))); definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "F", "G" be ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set (Const "X")) "," (Set ($#k1_numbers :::"REAL"::: ) ); func :::"max"::: "(" "F" "," "G" ")" -> ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" "X" "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: ASYMPT_0:def 19 "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" "X" "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) "F") & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) "G") & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "X" "holds" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k2_square_1 :::"max"::: ) "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ")" )) ")" ) ")" )) "}" ; end; :: deftheorem defines :::"max"::: ASYMPT_0:def 19 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set (Var "X")) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k14_asympt_0 :::"max"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ) ($#r1_hidden :::"="::: ) "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "X")) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Var "G"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k2_square_1 :::"max"::: ) "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) "," (Set "(" (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ")" )) ")" ) ")" )) "}" ))); theorem :: ASYMPT_0:41 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set (Set "(" ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f")) ")" ) ($#k13_asympt_0 :::"+"::: ) (Set "(" ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" (Set (Var "f")) ($#k1_asympt_0 :::"+"::: ) (Set (Var "g")) ")" )))) ; theorem :: ASYMPT_0:42 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool (Set ($#k14_asympt_0 :::"max"::: ) "(" (Set "(" ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k5_asympt_0 :::"max"::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ")" )))) ; definitionlet "F", "G" be ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); func "F" :::"to_power"::: "G" -> ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: ASYMPT_0:def 20 "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" )(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) "F") & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) "G") & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k4_power :::"to_power"::: ) (Set "(" (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ))) "}" ; end; :: deftheorem defines :::"to_power"::: ASYMPT_0:def 20 : (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set (Var "F")) ($#k15_asympt_0 :::"to_power"::: ) (Set (Var "G"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "t")) where t "is" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) : (Bool "ex" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_funct_2 :::"Element"::: ) "of" (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) ")" )(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set (Var "G"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k4_power :::"to_power"::: ) (Set "(" (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ))) "}" ));