:: ASYMPT_1 semantic presentation begin theorem :: ASYMPT_1:1 (Bool "for" (Set (Var "t")) "," (Set (Var "t1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (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 ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 12) ($#k4_nat_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k5_series_1 :::"to_power"::: ) (Num 3) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 5) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 36))) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "t1")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k5_series_1 :::"to_power"::: ) (Num 3) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" ))) ")" )) "holds" (Bool "ex" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "t"))) & (Bool (Set (Var "s1")) ($#r2_funct_2 :::"="::: ) (Set (Var "t1"))) & (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s1")))) ")" ))) ; theorem :: ASYMPT_1:2 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_asympt_0 :::"logbase"::: ) ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_power :::"log"::: ) "(" (Set (Var "a")) "," (Set (Var "n")) ")" )) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_power :::"log"::: ) "(" (Set (Var "b")) "," (Set (Var "n")) ")" )) ")" )) "holds" (Bool "ex" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "s1")) ($#r2_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s1")))) ")" )))) ; definitionlet "a", "b", "c" be ($#m1_subset_1 :::"Real":::); func :::"seq_a^"::: "(" "a" "," "b" "," "c" ")" -> ($#m1_subset_1 :::"Real_Sequence":::) means :: ASYMPT_1:def 1 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set "a" ($#k4_power :::"to_power"::: ) (Set "(" (Set "(" "b" ($#k8_real_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) "c" ")" )))); end; :: deftheorem defines :::"seq_a^"::: ASYMPT_1:def 1 : (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) ")" )) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b4")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k4_power :::"to_power"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k8_real_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) (Set (Var "c")) ")" )))) ")" ))); registrationlet "a" be ($#v2_xxreal_0 :::"positive"::: ) ($#m1_subset_1 :::"Real":::); let "b", "c" be ($#m1_subset_1 :::"Real":::); cluster (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" "a" "," "b" "," "c" ")" ) -> ($#v4_asympt_0 :::"eventually-positive"::: ) ; end; theorem :: ASYMPT_1:3 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v2_xxreal_0 :::"positive"::: ) ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "b")))) "holds" (Bool "not" (Bool (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Set (Var "b")) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Set (Var "a")) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ))))) ; definitionfunc :::"seq_logn"::: -> ($#m1_subset_1 :::"Real_Sequence":::) means :: ASYMPT_1:def 2 (Bool "(" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"seq_logn"::: ASYMPT_1:def 2 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k2_asympt_1 :::"seq_logn"::: ) )) "iff" (Bool "(" (Bool (Set (Set (Var "b1")) ($#k1_seq_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "b1")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" )) ")" ) ")" ) ")" )); definitionlet "a" be ($#m1_subset_1 :::"Real":::); func :::"seq_n^"::: "a" -> ($#m1_subset_1 :::"Real_Sequence":::) means :: ASYMPT_1:def 3 (Bool "(" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (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 ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k4_power :::"to_power"::: ) "a")) ")" ) ")" ); end; :: deftheorem defines :::"seq_n^"::: ASYMPT_1:def 3 : (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_asympt_1 :::"seq_n^"::: ) (Set (Var "a")))) "iff" (Bool "(" (Bool (Set (Set (Var "b2")) ($#k1_seq_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k4_power :::"to_power"::: ) (Set (Var "a")))) ")" ) ")" ) ")" ))); registration cluster (Set ($#k2_asympt_1 :::"seq_logn"::: ) ) -> ($#v4_asympt_0 :::"eventually-positive"::: ) ; end; registrationlet "a" be ($#m1_subset_1 :::"Real":::); cluster (Set ($#k3_asympt_1 :::"seq_n^"::: ) "a") -> ($#v4_asympt_0 :::"eventually-positive"::: ) ; end; theorem :: ASYMPT_1:4 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g")))) & (Bool (Bool "not" (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g"))))) "iff" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g")))) & (Bool (Bool "not" (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set (Var "g"))))) ")" ) ")" )) ; theorem :: ASYMPT_1:5 (Bool "(" (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set ($#k2_asympt_1 :::"seq_logn"::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ))) & (Bool (Bool "not" (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set ($#k2_asympt_1 :::"seq_logn"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" )))) ")" ) ; theorem :: ASYMPT_1:6 (Bool "(" (Bool (Set ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set ($#k2_asympt_1 :::"seq_logn"::: ) ))) & (Bool (Bool "not" (Set ($#k2_asympt_1 :::"seq_logn"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" )))) ")" ) ; theorem :: ASYMPT_1:7 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_series_1 :::"Sum"::: ) "(" (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set (Var "k")) ")" ) "," (Set (Var "n")) ")" )) ")" )) "holds" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))))) ; theorem :: ASYMPT_1:8 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k4_power :::"to_power"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" ))) ")" )) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "f"))) & (Bool (Bool "not" (Set (Var "s")) "is" ($#v7_asympt_0 :::"smooth"::: ) )) ")" ))) ; definitionlet "b" be ($#m1_subset_1 :::"Real":::); func :::"seq_const"::: "b" -> ($#m1_subset_1 :::"Real_Sequence":::) equals :: ASYMPT_1:def 4 (Set (Set ($#k5_numbers :::"NAT"::: ) ) ($#k8_funcop_1 :::"-->"::: ) "b"); end; :: deftheorem defines :::"seq_const"::: ASYMPT_1:def 4 : (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k4_asympt_1 :::"seq_const"::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k5_numbers :::"NAT"::: ) ) ($#k8_funcop_1 :::"-->"::: ) (Set (Var "b"))))); registration cluster (Set ($#k4_asympt_1 :::"seq_const"::: ) (Num 1)) -> ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ; end; theorem :: ASYMPT_1:9 (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "ex" (Set (Var "F")) "being" ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1) ")" ) ($#k1_tarski :::"}"::: ) )) & "(" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "F")) ($#k15_asympt_0 :::"to_power"::: ) (Set "(" ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k4_asympt_1 :::"seq_const"::: ) (Num 1) ")" ) ")" )))) "implies" (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "k")) "being" ($#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 (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "c")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set (Var "k")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ) ")" )))) ")" & "(" (Bool (Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "k")) "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")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "c")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set (Var "k")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")) ")" ))) ")" ) ")" ) ")" ))))) "implies" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "F")) ($#k15_asympt_0 :::"to_power"::: ) (Set "(" ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k4_asympt_1 :::"seq_const"::: ) (Num 1) ")" ) ")" ))) ")" ")" ))) ; begin theorem :: ASYMPT_1:10 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 3) ($#k4_nat_1 :::"*"::: ) (Set "(" (Num 10) ($#k5_series_1 :::"to_power"::: ) (Num 6) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 18) ($#k4_nat_1 :::"*"::: ) (Set "(" (Num 10) ($#k5_series_1 :::"to_power"::: ) (Num 3) ")" ) ")" ) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 27) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" )) "holds" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 2) ")" )))) ; begin theorem :: ASYMPT_1:11 (Bool (Set ($#k3_asympt_1 :::"seq_n^"::: ) (Num 2)) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 3) ")" ))) ; theorem :: ASYMPT_1:12 (Bool (Bool "not" (Set ($#k3_asympt_1 :::"seq_n^"::: ) (Num 2)) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 3) ")" )))) ; theorem :: ASYMPT_1:13 (Bool "ex" (Set (Var "s")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Num 2) "," (Num 1) "," (Num 1) ")" )) & (Bool (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Num 2) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "s")))) ")" )) ; definitionlet "a" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"seq_n!"::: "a" -> ($#m1_subset_1 :::"Real_Sequence":::) means :: ASYMPT_1:def 5 (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 (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) "a" ")" ) ($#k9_newton :::"!"::: ) ))); end; :: deftheorem defines :::"seq_n!"::: ASYMPT_1:def 5 : (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k5_asympt_1 :::"seq_n!"::: ) (Set (Var "a")))) "iff" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b2")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Set (Var "a")) ")" ) ($#k9_newton :::"!"::: ) ))) ")" ))); registrationlet "a" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k5_asympt_1 :::"seq_n!"::: ) "a") -> ($#v4_asympt_0 :::"eventually-positive"::: ) ; end; theorem :: ASYMPT_1:14 (Bool (Bool "not" (Set ($#k5_asympt_1 :::"seq_n!"::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set "(" ($#k5_asympt_1 :::"seq_n!"::: ) (Num 1) ")" )))) ; begin theorem :: ASYMPT_1:15 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 2) ")" )))) ; begin theorem :: ASYMPT_1:16 (Bool "ex" (Set (Var "s")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Num 2) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) & (Bool (Set (Num 2) ($#k2_asympt_0 :::"(#)"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1) ")" ))) & (Bool (Bool "not" (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Num 2) "," (Num 2) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))))) ")" )) ; begin theorem :: ASYMPT_1:17 "(" (Bool (Bool (Set ($#k6_power :::"log"::: ) "(" (Num 2) "," (Num 3) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Num 159) ($#k10_real_1 :::"/"::: ) (Num 100)))) "implies" (Bool "(" (Bool (Set ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Num 3) ")" ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" (Num 159) ($#k10_real_1 :::"/"::: ) (Num 100) ")" ) ")" ))) & (Bool (Bool "not" (Set ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Num 3) ")" ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k7_asympt_0 :::"Big_Omega"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" (Num 159) ($#k10_real_1 :::"/"::: ) (Num 100) ")" ) ")" )))) & (Bool (Bool "not" (Set ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Num 3) ")" ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" (Num 159) ($#k10_real_1 :::"/"::: ) (Num 100) ")" ) ")" )))) ")" ) ")" ; begin theorem :: ASYMPT_1:18 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#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 "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k6_int_1 :::"mod"::: ) (Num 2))) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k6_int_1 :::"mod"::: ) (Num 2))) ")" )) "holds" (Bool "ex" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "s1")) ($#r2_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Bool "not" (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s1"))))) & (Bool (Bool "not" (Set (Var "s1")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))))) ")" ))) ; begin theorem :: ASYMPT_1:19 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g")))) "iff" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "g")))) ")" )) ; theorem :: ASYMPT_1:20 (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")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "g")))) "iff" (Bool (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "g")))) ")" )) ; begin theorem :: ASYMPT_1:21 (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "e"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" ))) ")" )) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "f"))) & (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "e")) ")" ) ")" ))) & (Bool (Bool "not" (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "e")) ")" ) ")" )))) ")" )))) ; theorem :: ASYMPT_1:22 (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "e")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Num 1))) "holds" (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k5_series_1 :::"to_power"::: ) (Num 2) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" ))) ")" )) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "e")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s")))) & (Bool (Bool "not" (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "e")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))))) ")" )))) ; theorem :: ASYMPT_1:23 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Num 1))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k5_series_1 :::"to_power"::: ) (Num 2) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" ))) ")" )) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "f"))) & (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 8) ")" ))) & (Bool (Bool "not" (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 8) ")" )))) ")" ))) ; theorem :: ASYMPT_1:24 (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ($#k4_power :::"to_power"::: ) (Num 4))) ")" )) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 8) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s")))) ")" ))) ; theorem :: ASYMPT_1:25 (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "e"))) & (Bool (Set (Var "e")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "e")) ")" ) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) & (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 8) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s")))) & (Bool (Bool "not" (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 8) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))))) ")" ))) ; begin theorem :: ASYMPT_1:26 (Bool "for" (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 ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k4_power :::"to_power"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" ))) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k4_power :::"to_power"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set (Var "n")) ")" ))) ")" )) "holds" (Bool "ex" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "s1")) ($#r2_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s1")))) & (Bool (Bool "not" (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s1"))))) ")" ))) ; theorem :: ASYMPT_1:27 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k4_power :::"to_power"::: ) (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set (Var "n")) ")" ))) ")" )) "holds" (Bool "ex" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "s1")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Num 2) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) & (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s1")))) & (Bool (Bool "not" (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s1"))))) ")" ))) ; theorem :: ASYMPT_1:28 (Bool "ex" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Num 2) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) & (Bool (Set (Var "s1")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Num 2) "," (Num 1) "," (Num 1) ")" )) & (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s1")))) ")" )) ; theorem :: ASYMPT_1:29 (Bool "ex" (Set (Var "s")) "," (Set (Var "s1")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Num 2) "," (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) & (Bool (Set (Var "s1")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Num 2) "," (Num 2) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) & (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s1")))) & (Bool (Bool "not" (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s1"))))) ")" )) ; theorem :: ASYMPT_1:30 (Bool "ex" (Set (Var "s")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_asympt_1 :::"seq_a^"::: ) "(" (Num 2) "," (Num 2) "," (Set ($#k6_numbers :::"0"::: ) ) ")" )) & (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k5_asympt_1 :::"seq_n!"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" ))) & (Bool (Bool "not" (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k5_asympt_1 :::"seq_n!"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" )))) ")" )) ; theorem :: ASYMPT_1:31 (Bool "(" (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k5_asympt_1 :::"seq_n!"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k5_asympt_1 :::"seq_n!"::: ) (Num 1) ")" ))) & (Bool (Bool "not" (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k5_asympt_1 :::"seq_n!"::: ) (Set ($#k6_numbers :::"0"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k5_asympt_1 :::"seq_n!"::: ) (Num 1) ")" )))) ")" ) ; theorem :: ASYMPT_1:32 (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")))) ")" )) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k5_asympt_1 :::"seq_n!"::: ) (Num 1) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s")))) & (Bool (Bool "not" (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k5_asympt_1 :::"seq_n!"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s"))))) ")" ))) ; begin theorem :: ASYMPT_1:33 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "k")) "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"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_series_1 :::"Sum"::: ) "(" (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set (Var "k")) ")" ) "," (Set (Var "n")) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" (Set (Var "n")) ($#k5_series_1 :::"to_power"::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))))) ; begin theorem :: ASYMPT_1:34 (Bool "for" (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 ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" ))) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set "(" (Set (Var "n")) ($#k9_newton :::"!"::: ) ")" ) ")" )) ")" )) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "s")))) ")" ))) ; begin theorem :: ASYMPT_1:35 (Bool "for" (Set (Var "f")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "(" (Bool (Bool (Set (Set (Var "n")) ($#k6_int_1 :::"mod"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" & "(" (Bool (Bool (Set (Set (Var "n")) ($#k6_int_1 :::"mod"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Num 1))) "implies" (Bool (Set (Set (Var "t")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" ")" ) ")" )) "holds" (Bool "not" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "f"))))))) ; begin begin definitionfunc :::"POWEROF2SET"::: -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: ASYMPT_1:def 6 "{" (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) where n "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ; end; :: deftheorem defines :::"POWEROF2SET"::: ASYMPT_1:def 6 : (Bool (Set ($#k6_asympt_1 :::"POWEROF2SET"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) where n "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool verum) "}" ); theorem :: ASYMPT_1:36 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_1 :::"POWEROF2SET"::: ) ))) "implies" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_1 :::"POWEROF2SET"::: ) )))) "implies" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")))) ")" ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k11_asympt_0 :::"Big_Theta"::: ) "(" (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1) ")" ) "," (Set ($#k6_asympt_1 :::"POWEROF2SET"::: ) ) ")" )) & (Bool (Bool "not" (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1) ")" )))) & (Bool (Set ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1)) "is" ($#v7_asympt_0 :::"smooth"::: ) ) & (Bool (Bool "not" (Set (Var "f")) "is" ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) )) ")" )) ; theorem :: ASYMPT_1:37 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k3_power :::"to_power"::: ) (Set "(" (Num 2) ($#k3_power :::"to_power"::: ) (Set ($#k1_int_1 :::"[\"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" ) ($#k1_int_1 :::"/]"::: ) ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")))) ")" )) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k11_asympt_0 :::"Big_Theta"::: ) "(" (Set (Var "s")) "," (Set ($#k6_asympt_1 :::"POWEROF2SET"::: ) ) ")" )) & (Bool (Bool "not" (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "s"))))) & (Bool (Set (Var "f")) "is" ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) ) & (Bool (Set (Var "s")) "is" ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) ) & (Bool (Bool "not" (Set (Var "s")) ($#r2_asympt_0 :::"is_smooth_wrt"::: ) (Num 2))) ")" ))) ; theorem :: ASYMPT_1:38 (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_1 :::"POWEROF2SET"::: ) ))) "implies" (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_1 :::"POWEROF2SET"::: ) )))) "implies" (Bool (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k5_series_1 :::"to_power"::: ) (Num 2))) ")" ")" ) ")" )) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "g"))) & (Bool (Set ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k11_asympt_0 :::"Big_Theta"::: ) "(" (Set (Var "s")) "," (Set ($#k6_asympt_1 :::"POWEROF2SET"::: ) ) ")" )) & (Bool (Bool "not" (Set ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "s"))))) & (Bool (Set (Set (Var "s")) ($#k12_asympt_0 :::"taken_every"::: ) (Num 2)) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "s")))) & (Bool (Set ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1)) "is" ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) ) & (Bool (Bool "not" (Set (Var "s")) "is" ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) )) ")" ))) ; begin definitionlet "x" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"Step1"::: "x" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: ASYMPT_1:def 7 (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Set (Var "n")) ($#k9_newton :::"!"::: ) ) ($#r1_xxreal_0 :::"<="::: ) "x") & (Bool "x" ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k9_newton :::"!"::: ) )) & (Bool it ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k9_newton :::"!"::: ) )) ")" )) if (Bool "x" ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) otherwise (Bool it ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )); end; :: deftheorem defines :::"Step1"::: ASYMPT_1:def 7 : (Bool "for" (Set (Var "x")) "," (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k7_asympt_1 :::"Step1"::: ) (Set (Var "x")))) "iff" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Set (Var "n")) ($#k9_newton :::"!"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "x"))) & (Bool (Set (Var "x")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k9_newton :::"!"::: ) )) & (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k9_newton :::"!"::: ) )) ")" )) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )))) "implies" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k7_asympt_1 :::"Step1"::: ) (Set (Var "x")))) "iff" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ")" )); theorem :: ASYMPT_1:39 (Bool "for" (Set (Var "f")) "being" ($#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 "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k7_asympt_1 :::"Step1"::: ) (Set (Var "n")))) ")" )) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v4_asympt_0 :::"eventually-positive"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set (Var "s")) ($#r2_funct_2 :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "f")) "is" ($#v6_asympt_0 :::"eventually-nondecreasing"::: ) ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" ) & (Bool (Bool "not" (Set (Var "s")) "is" ($#v7_asympt_0 :::"smooth"::: ) )) ")" ))) ; begin theorem :: ASYMPT_1:40 (Bool "for" (Set (Var "F")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "F")) ($#r2_funct_2 :::"="::: ) (Set (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k4_asympt_1 :::"seq_const"::: ) (Num 1) ")" )))) "holds" (Bool (Set (Set "(" ($#k8_asympt_0 :::"Big_Theta"::: ) (Set (Var "F")) ")" ) ($#k13_asympt_0 :::"+"::: ) (Set "(" ($#k8_asympt_0 :::"Big_Theta"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_asympt_0 :::"Big_Theta"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1) ")" )))) ; begin theorem :: ASYMPT_1:41 (Bool "ex" (Set (Var "F")) "being" ($#m1_funct_2 :::"FUNCTION_DOMAIN"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" ) & (Bool (Bool "not" (Set ($#k3_asympt_1 :::"seq_n^"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set (Set (Var "F")) ($#k15_asympt_0 :::"to_power"::: ) (Set "(" ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" ($#k4_asympt_1 :::"seq_const"::: ) (Num 1) ")" ) ")" )))) ")" )) ; begin theorem :: ASYMPT_1:42 (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"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "ex" (Set (Var "e")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "e")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set (Var "e"))) ")" ) ")" ))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set "(" (Set (Var "c")) ($#k3_asympt_0 :::"+"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f")))))) ; begin theorem :: ASYMPT_1:43 (Bool (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) (Num 12)) ($#r1_hidden :::"="::: ) (Num 4096)) ; theorem :: ASYMPT_1:44 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 3))) "holds" (Bool (Set (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)))) ; theorem :: ASYMPT_1:45 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 10))) "holds" (Bool (Set (Num 2) ($#k4_power :::"to_power"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k5_square_1 :::"^2"::: ) ))) ; theorem :: ASYMPT_1:46 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 9))) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k5_series_1 :::"to_power"::: ) (Num 6)) ($#r1_xxreal_0 :::"<"::: ) (Set (Num 2) ($#k4_nat_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k5_series_1 :::"to_power"::: ) (Num 6) ")" )))) ; theorem :: ASYMPT_1:47 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 30))) "holds" (Bool (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "n")) ($#k5_series_1 :::"to_power"::: ) (Num 6)))) ; theorem :: ASYMPT_1:48 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r1_xxreal_0 :::">"::: ) (Num 9))) "holds" (Bool (Set (Num 2) ($#k4_power :::"to_power"::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ))) ; theorem :: ASYMPT_1:49 (Bool "ex" (Set (Var "N")) "being" ($#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 "(" ($#k7_square_1 :::"sqrt"::: ) (Set (Var "n")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" )) ($#r1_xxreal_0 :::">"::: ) (Num 1)))) ; theorem :: ASYMPT_1:50 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "c")) ($#r1_hidden :::"<>"::: ) (Num 1))) "holds" (Bool (Set (Set (Var "a")) ($#k4_power :::"to_power"::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k4_power :::"to_power"::: ) (Set "(" (Set (Var "b")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Set (Var "c")) "," (Set (Var "a")) ")" ")" ) ")" )))) ; theorem :: ASYMPT_1:51 (Bool (Set (Num 5) ($#k9_newton :::"!"::: ) ) ($#r1_hidden :::"="::: ) (Num 120)) ; theorem :: ASYMPT_1:52 (Bool (Set (Num 5) ($#k5_series_1 :::"to_power"::: ) (Num 5)) ($#r1_hidden :::"="::: ) (Num 3125)) ; theorem :: ASYMPT_1:53 (Bool (Set (Num 4) ($#k5_series_1 :::"to_power"::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Num 256)) ; theorem :: ASYMPT_1:54 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: ASYMPT_1:55 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool (Set (Set (Var "n")) ($#k9_newton :::"!"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Num 1))) ; theorem :: ASYMPT_1:56 (Bool "for" (Set (Var "n1")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n1")))) "holds" (Bool (Set (Set (Var "n")) ($#k9_newton :::"!"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "n1")) ($#k9_newton :::"!"::: ) ))) ; theorem :: ASYMPT_1:57 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Set (Var "n")) ($#k9_newton :::"!"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k9_newton :::"!"::: ) )) & (Bool "(" "for" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "m")) ($#k9_newton :::"!"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "m")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k9_newton :::"!"::: ) ))) "holds" (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" ) ")" ))) ; theorem :: ASYMPT_1:58 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool (Set ($#k2_int_1 :::"[/"::: ) (Set "(" (Set (Var "n")) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k2_int_1 :::"\]"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) ; theorem :: ASYMPT_1:59 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 3))) "holds" (Bool (Set (Set (Var "n")) ($#k9_newton :::"!"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Var "n")))) ; theorem :: ASYMPT_1:60 (Bool (Set (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Num 1) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k4_asympt_1 :::"seq_const"::: ) (Num 1) ")" )) "is" ($#v4_asympt_0 :::"eventually-positive"::: ) ) ; theorem :: ASYMPT_1:61 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1)))) ; theorem :: ASYMPT_1:62 (Bool "for" (Set (Var "a")) "being" ($#v1_asympt_0 :::"logbase"::: ) ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_power :::"log"::: ) "(" (Set (Var "a")) "," (Set (Var "n")) ")" )) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v4_asympt_0 :::"eventually-positive"::: ) ))) ; theorem :: ASYMPT_1:63 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_asympt_0 :::"eventually-nonnegative"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "holds" (Bool "(" (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 "f")))) ")" ) "iff" (Bool (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k6_asympt_0 :::"Big_Oh"::: ) (Set (Var "g")))) ")" )) ; theorem :: ASYMPT_1:64 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "a")) ($#k4_power :::"to_power"::: ) (Set (Var "c"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "b")) ($#k4_power :::"to_power"::: ) (Set (Var "c"))))) ; theorem :: ASYMPT_1:65 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 4))) "holds" (Bool (Set (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 3)) ($#r1_xxreal_0 :::"<"::: ) (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n"))))) ; theorem :: ASYMPT_1:66 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 6))) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k5_square_1 :::"^2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n"))))) ; theorem :: ASYMPT_1:67 (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">"::: ) (Num 6))) "holds" (Bool (Set (Set (Var "c")) ($#k5_square_1 :::"^2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Num 2) ($#k4_power :::"to_power"::: ) (Set (Var "c"))))) ; theorem :: ASYMPT_1:68 (Bool "for" (Set (Var "e")) "being" ($#v2_xxreal_0 :::"positive"::: ) ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set "(" (Set (Var "n")) ($#k4_power :::"to_power"::: ) (Set (Var "e")) ")" ) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set (Var "e")) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k52_valued_1 :::"/""::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set (Var "e")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; theorem :: ASYMPT_1:69 (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "e")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k2_asympt_1 :::"seq_logn"::: ) ) ($#k52_valued_1 :::"/""::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set (Var "e")) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set ($#k2_asympt_1 :::"seq_logn"::: ) ) ($#k52_valued_1 :::"/""::: ) (Set "(" ($#k3_asympt_1 :::"seq_n^"::: ) (Set (Var "e")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ; theorem :: ASYMPT_1:70 (Bool "for" (Set (Var "f")) "being" ($#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"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set ($#k6_series_1 :::"Sum"::: ) "(" (Set (Var "f")) "," (Set (Var "N")) ")" ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: ASYMPT_1:71 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#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"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" )) "holds" (Bool (Set ($#k6_series_1 :::"Sum"::: ) "(" (Set (Var "f")) "," (Set (Var "N")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_series_1 :::"Sum"::: ) "(" (Set (Var "g")) "," (Set (Var "N")) ")" )))) ; theorem :: ASYMPT_1:72 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (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 ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "b"))) ")" )) "holds" (Bool "for" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_series_1 :::"Sum"::: ) "(" (Set (Var "f")) "," (Set (Var "N")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k8_real_1 :::"*"::: ) (Set (Var "N"))))))) ; theorem :: ASYMPT_1:73 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "N")) "," (Set (Var "M")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k7_series_1 :::"Sum"::: ) "(" (Set (Var "f")) "," (Set (Var "N")) "," (Set (Var "M")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "N")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_series_1 :::"Sum"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "N")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "M")) ")" )))) ; theorem :: ASYMPT_1:74 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "M")) "," (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "N")) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "M")) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "M")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#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")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" )) "holds" (Bool (Set ($#k7_series_1 :::"Sum"::: ) "(" (Set (Var "f")) "," (Set (Var "N")) "," (Set (Var "M")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_series_1 :::"Sum"::: ) "(" (Set (Var "g")) "," (Set (Var "N")) "," (Set (Var "M")) ")" )))) ; theorem :: ASYMPT_1:75 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k2_int_1 :::"[/"::: ) (Set "(" (Set (Var "n")) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k2_int_1 :::"\]"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) ; theorem :: ASYMPT_1:76 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (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 ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Var "b"))) ")" )) "holds" (Bool "for" (Set (Var "M")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k7_series_1 :::"Sum"::: ) "(" (Set (Var "f")) "," (Set (Var "N")) "," (Set (Var "M")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "N")) ($#k9_real_1 :::"-"::: ) (Set (Var "M")) ")" ))))))) ; theorem :: ASYMPT_1:77 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "N")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (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"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" )) "holds" (Bool "(" (Bool (Set (Var "g")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "c"))) ")" )))) ; theorem :: ASYMPT_1:78 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ))) ; theorem :: ASYMPT_1:79 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ($#k7_real_1 :::"+"::: ) (Num 1) ")" )))) ; theorem :: ASYMPT_1:80 (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "e"))) & (Bool (Set (Var "e")) ($#r1_xxreal_0 :::"<"::: ) (Num 1))) "holds" (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")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set (Var "e")) ")" ) ")" ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 8) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set (Num 8) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" )))))) ; theorem :: ASYMPT_1:81 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 10))) "holds" (Bool (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k9_newton :::"!"::: ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k4_power :::"to_power"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 9) ")" ) ")" )))) ; theorem :: ASYMPT_1:82 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 3))) "holds" (Bool (Set (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 2) ")" )) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 1)))) ; theorem :: ASYMPT_1:83 (Bool "for" (Set (Var "c")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "c")) ($#k3_power :::"to_power"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_square_1 :::"sqrt"::: ) (Set (Var "c"))))) ; theorem :: ASYMPT_1:84 (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")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k7_square_1 :::"sqrt"::: ) (Set (Var "n")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "n")) ($#k10_real_1 :::"/"::: ) (Num 2))))) ; theorem :: ASYMPT_1:85 (Bool "for" (Set (Var "s")) "being" ($#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 "s")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k4_power :::"to_power"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" )) "holds" (Bool (Set (Var "s")) "is" bbbadV7_VALUED_0())) ; theorem :: ASYMPT_1:86 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ($#k4_power :::"to_power"::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k4_power :::"to_power"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) ; theorem :: ASYMPT_1:87 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Set (Var "k"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k6_newton :::"choose"::: ) (Set (Var "k")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) ; theorem :: ASYMPT_1:88 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set "(" (Set (Var "n")) ($#k9_newton :::"!"::: ) ")" ) ")" )) ")" )) "holds" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_series_1 :::"Sum"::: ) "(" (Set ($#k2_asympt_1 :::"seq_logn"::: ) ) "," (Set (Var "n")) ")" )))) ; theorem :: ASYMPT_1:89 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 4))) "holds" (Bool (Set (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k6_power :::"log"::: ) "(" (Num 2) "," (Set (Var "n")) ")" ")" )) ($#r1_xxreal_0 :::">="::: ) (Set (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n"))))) ; theorem :: ASYMPT_1:90 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool (Set (Set (Var "n")) ($#k5_square_1 :::"^2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1)))) ; theorem :: ASYMPT_1:91 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Num 1))) ; theorem :: ASYMPT_1:92 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool "not" (Bool (Set (Set "(" (Num 2) ($#k5_series_1 :::"to_power"::: ) (Set (Var "n")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k6_asympt_1 :::"POWEROF2SET"::: ) )))) ; theorem :: ASYMPT_1:93 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool (Set (Set (Var "n")) ($#k9_newton :::"!"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k9_newton :::"!"::: ) ))) "holds" (Bool (Set ($#k7_asympt_1 :::"Step1"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k9_newton :::"!"::: ) ))) ; theorem :: ASYMPT_1:94 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::">"::: ) (Num 1)) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "a"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set ($#k6_power :::"log"::: ) "(" (Set (Var "a")) "," (Set (Var "c")) ")" ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_power :::"log"::: ) "(" (Set (Var "b")) "," (Set (Var "c")) ")" ))) ;