:: MOEBIUS1 semantic presentation begin scheme :: MOEBIUS1:sch 1 LambdaNATC{ F1() -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ), F2() -> ($#m1_hidden :::"set"::: ) , F3( ($#m1_hidden :::"set"::: ) ) -> ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set F2 "(" ")" ) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_nat_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set F1 "(" ")" )) & (Bool "(" "for" (Set (Var "x")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k8_nat_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set F3 "(" (Set (Var "x")) ")" )) ")" ) ")" )) provided (Bool (Set F1 "(" ")" ) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" )) and (Bool "for" (Set (Var "x")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set F3 "(" (Set (Var "x")) ")" ) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) proof end; registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#v1_ordinal1 :::"epsilon-transitive"::: ) ($#v2_ordinal1 :::"epsilon-connected"::: ) ($#v3_ordinal1 :::"ordinal"::: ) ($#v7_ordinal1 :::"natural"::: ) bbbadV1_XCMPLX_0() ($#v1_xreal_0 :::"real"::: ) bbbadV1_INT_1() ($#~v1_int_2 "non" ($#v1_int_2 :::"prime"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#v1_card_1 :::"cardinal"::: ) ($#v1_xxreal_0 :::"ext-real"::: ) ($#~v3_xxreal_0 "non" ($#v3_xxreal_0 :::"negative"::: ) ) bbbadV1_RAT_1() bbbadV1_MEMBERED() bbbadV2_MEMBERED() bbbadV3_MEMBERED() bbbadV4_MEMBERED() bbbadV5_MEMBERED() bbbadV6_MEMBERED() bbbadV3_XXREAL_2() bbbadV4_XXREAL_2() bbbadV5_XXREAL_2() for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); end; theorem :: MOEBIUS1:1 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Num 1))) "holds" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2))) ; theorem :: MOEBIUS1:2 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "," (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))) "holds" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) "iff" (Bool (Set (Set (Var "k")) ($#k24_binop_2 :::"*"::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "k")) ($#k24_binop_2 :::"*"::: ) (Set (Var "n")) ")" ))) ")" )) ; theorem :: MOEBIUS1:3 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "not" (Bool (Set (Var "m")) "," (Set (Var "n")) ($#r1_int_2 :::"are_relative_prime"::: ) ) "or" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) "or" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ; theorem :: MOEBIUS1:4 (Bool "for" (Set (Var "n")) "being" ($#~v1_int_2 "non" ($#v1_int_2 :::"prime"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool "(" (Bool (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Var "n"))) ")" ))) ; theorem :: MOEBIUS1:5 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))))) ; theorem :: MOEBIUS1:6 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))) "iff" (Bool (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; theorem :: MOEBIUS1:7 (Bool (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ; theorem :: MOEBIUS1:8 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: MOEBIUS1:9 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set (Var "p")) ($#k1_newton :::"|^"::: ) (Set (Var "m"))) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))))) ; theorem :: MOEBIUS1:10 (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Set (Var "p")) ($#k1_newton :::"|^"::: ) (Num 2)) ($#r1_nat_d :::"divides"::: ) (Set (Var "a")))) "holds" (Bool (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "a"))))) ; theorem :: MOEBIUS1:11 (Bool "for" (Set (Var "p")) "being" ($#v1_int_2 :::"prime"::: ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) "," (Set (Var "n")) ($#r1_int_2 :::"are_relative_prime"::: ) ) & (Bool (Set (Set (Var "p")) ($#k13_newton :::"|^"::: ) (Num 2)) ($#r1_nat_d :::"divides"::: ) (Set (Set (Var "m")) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")))) & (Bool (Bool "not" (Set (Set (Var "p")) ($#k13_newton :::"|^"::: ) (Num 2)) ($#r1_nat_d :::"divides"::: ) (Set (Var "m"))))) "holds" (Bool (Set (Set (Var "p")) ($#k13_newton :::"|^"::: ) (Num 2)) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))))) ; theorem :: MOEBIUS1:12 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "N")) "being" ($#m1_hidden :::"Rbag":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k13_pre_poly :::"support"::: ) (Set (Var "N"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "n")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k3_uproots :::"Sum"::: ) (Set (Var "N"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "N")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))))) ; registration cluster (Set ($#k1_uproots :::"canFS"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) -> ($#v1_xboole_0 :::"empty"::: ) ; end; theorem :: MOEBIUS1:13 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n")))) "holds" (Bool "{" (Set (Var "d")) where d "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "d")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))) & (Bool (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "d"))) ")" ) "}" ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "p")) ($#k3_nat_1 :::"*"::: ) (Set (Var "d")) ")" ) where d "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "d")) ($#r1_nat_d :::"divides"::: ) (Set (Set (Var "n")) ($#k3_nat_d :::"div"::: ) (Set (Var "p")))) ")" ) "}" ))) ; theorem :: MOEBIUS1:14 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Set (Var "n")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))))) ; theorem :: MOEBIUS1:15 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Set (Var "n")) ")" ))))) "holds" (Bool (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: MOEBIUS1:16 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Set (Var "n")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Bool "not" (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Set (Var "n")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))))) "holds" (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) "is" ($#m1_hidden :::"Prime":::)))) ; theorem :: MOEBIUS1:17 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "n")))) ")" )) "holds" (Bool (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Set (Var "m")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Set (Var "n")) ")" )))) ; theorem :: MOEBIUS1:18 (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Set (Var "n")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) "holds" (Bool "ex" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )(Bool "ex" (Set (Var "e")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Set (Var "m")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "k")))) & (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k4_nat_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k13_newton :::"|^"::: ) (Set (Var "e")) ")" ))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Set (Var "m")) ")" )))) "implies" (Bool (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "n")))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Set (Var "m")) ")" ))))) "implies" (Bool (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "n")))) ")" ")" ) ")" ) ")" ))))) ; theorem :: MOEBIUS1:19 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "m"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "n")))) ")" )) "holds" (Bool (Set (Var "m")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n")))) ; begin definitionlet "x" be ($#m1_hidden :::"Nat":::); attr "x" is :::"square-containing"::: means :: MOEBIUS1:def 1 (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Set (Set (Var "p")) ($#k1_newton :::"|^"::: ) (Num 2)) ($#r1_nat_d :::"divides"::: ) "x")); end; :: deftheorem defines :::"square-containing"::: MOEBIUS1:def 1 : (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#v1_moebius1 :::"square-containing"::: ) ) "iff" (Bool "ex" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Set (Set (Var "p")) ($#k1_newton :::"|^"::: ) (Num 2)) ($#r1_nat_d :::"divides"::: ) (Set (Var "x")))) ")" )); theorem :: MOEBIUS1:20 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool "ex" (Set (Var "p")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Num 1)) & (Bool (Set (Set (Var "p")) ($#k1_newton :::"|^"::: ) (Num 2)) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))) ")" ))) "holds" (Bool (Set (Var "n")) "is" ($#v1_moebius1 :::"square-containing"::: ) )) ; notationlet "x" be ($#m1_hidden :::"Nat":::); antonym :::"square-free"::: "x" for :::"square-containing"::: ; end; theorem :: MOEBIUS1:21 (Bool (Set ($#k6_numbers :::"0"::: ) ) "is" ($#v1_moebius1 :::"square-containing"::: ) ) ; theorem :: MOEBIUS1:22 (Bool (Num 1) "is" ($#v1_moebius1 :::"square-free"::: ) ) ; theorem :: MOEBIUS1:23 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool (Set (Var "p")) "is" ($#v1_moebius1 :::"square-free"::: ) )) ; registration cluster ($#v1_int_2 :::"prime"::: ) -> ($#v1_moebius1 :::"square-free"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); end; definitionfunc :::"SCNAT"::: -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: MOEBIUS1:def 2 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "n")) "is" ($#v1_moebius1 :::"square-free"::: ) ) ")" )); end; :: deftheorem defines :::"SCNAT"::: MOEBIUS1:def 2 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k1_moebius1 :::"SCNAT"::: ) )) "iff" (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) "iff" (Bool (Set (Var "n")) "is" ($#v1_moebius1 :::"square-free"::: ) ) ")" )) ")" )); registration cluster ($#v1_ordinal1 :::"epsilon-transitive"::: ) ($#v2_ordinal1 :::"epsilon-connected"::: ) ($#v3_ordinal1 :::"ordinal"::: ) ($#v7_ordinal1 :::"natural"::: ) bbbadV1_XCMPLX_0() ($#v1_xreal_0 :::"real"::: ) bbbadV1_INT_1() ($#v1_finset_1 :::"finite"::: ) ($#v1_card_1 :::"cardinal"::: ) ($#v1_xxreal_0 :::"ext-real"::: ) ($#~v3_xxreal_0 "non" ($#v3_xxreal_0 :::"negative"::: ) ) ($#v1_moebius1 :::"square-free"::: ) for ($#m1_hidden :::"set"::: ) ; cluster ($#v1_ordinal1 :::"epsilon-transitive"::: ) ($#v2_ordinal1 :::"epsilon-connected"::: ) ($#v3_ordinal1 :::"ordinal"::: ) ($#v7_ordinal1 :::"natural"::: ) bbbadV1_XCMPLX_0() ($#v1_xreal_0 :::"real"::: ) bbbadV1_INT_1() ($#v1_finset_1 :::"finite"::: ) ($#v1_card_1 :::"cardinal"::: ) ($#v1_xxreal_0 :::"ext-real"::: ) ($#~v3_xxreal_0 "non" ($#v3_xxreal_0 :::"negative"::: ) ) ($#v1_moebius1 :::"square-containing"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#v7_ordinal1 :::"natural"::: ) ($#v1_pythtrip :::"square"::: ) -> ($#v1_moebius1 :::"square-containing"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: MOEBIUS1:24 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) "is" ($#v1_moebius1 :::"square-free"::: ) )) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Num 1)))) ; theorem :: MOEBIUS1:25 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set (Var "m")) ($#k24_binop_2 :::"*"::: ) (Set (Var "n"))) "is" ($#v1_moebius1 :::"square-free"::: ) )) "holds" (Bool (Set (Var "m")) "is" ($#v1_moebius1 :::"square-free"::: ) )) ; theorem :: MOEBIUS1:26 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) "is" ($#v1_moebius1 :::"square-free"::: ) ) & (Bool (Set (Var "n")) ($#r1_nat_d :::"divides"::: ) (Set (Var "m")))) "holds" (Bool (Set (Var "n")) "is" ($#v1_moebius1 :::"square-free"::: ) )) ; theorem :: MOEBIUS1:27 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "m")) "," (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) "is" ($#v1_moebius1 :::"square-free"::: ) ) & (Bool (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "m"))) & (Bool (Set (Var "d")) ($#r1_nat_d :::"divides"::: ) (Set (Set (Var "m")) ($#k3_nat_d :::"div"::: ) (Set (Var "p"))))) "holds" (Bool "(" (Bool (Set (Var "d")) ($#r1_nat_d :::"divides"::: ) (Set (Var "m"))) & (Bool (Bool "not" (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "d")))) ")" ))) ; theorem :: MOEBIUS1:28 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "m")) "," (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "m"))) & (Bool (Set (Var "d")) ($#r1_nat_d :::"divides"::: ) (Set (Var "m"))) & (Bool (Bool "not" (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "d"))))) "holds" (Bool (Set (Var "d")) ($#r1_nat_d :::"divides"::: ) (Set (Set (Var "m")) ($#k3_nat_d :::"div"::: ) (Set (Var "p")))))) ; theorem :: MOEBIUS1:29 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "m")) "is" ($#v1_moebius1 :::"square-free"::: ) ) & (Bool (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "m")))) "holds" (Bool "{" (Set (Var "d")) where d "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "d")) ($#r1_nat_d :::"divides"::: ) (Set (Var "m"))) & (Bool (Bool "not" (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "d")))) ")" ) "}" ($#r1_hidden :::"="::: ) "{" (Set (Var "d")) where d "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "d")) ($#r1_nat_d :::"divides"::: ) (Set (Set (Var "m")) ($#k3_nat_d :::"div"::: ) (Set (Var "p")))) ")" ) "}" ))) ; begin definitionlet "n" be ($#m1_hidden :::"Nat":::); func :::"Moebius"::: "n" -> ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) means :: MOEBIUS1:def 3 (Bool it ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) if (Bool "n" "is" ($#v1_moebius1 :::"square-containing"::: ) ) otherwise (Bool "ex" (Set (Var "n9")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "n9")) ($#r1_hidden :::"="::: ) "n") & (Bool it ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k2_newton :::"|^"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set "(" ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Set (Var "n9")) ")" ) ")" ) ")" ))) ")" )); end; :: deftheorem defines :::"Moebius"::: MOEBIUS1:def 3 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "n")) "is" ($#v1_moebius1 :::"square-containing"::: ) )) "implies" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_moebius1 :::"Moebius"::: ) (Set (Var "n")))) "iff" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "n")) "is" ($#v1_moebius1 :::"square-containing"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_moebius1 :::"Moebius"::: ) (Set (Var "n")))) "iff" (Bool "ex" (Set (Var "n9")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "n9")) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k2_newton :::"|^"::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set "(" ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k13_nat_3 :::"ppf"::: ) (Set (Var "n9")) ")" ) ")" ) ")" ))) ")" )) ")" ) ")" ")" ))); theorem :: MOEBIUS1:30 (Bool (Set ($#k2_moebius1 :::"Moebius"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Num 1)) ; theorem :: MOEBIUS1:31 (Bool (Set ($#k2_moebius1 :::"Moebius"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) ; theorem :: MOEBIUS1:32 (Bool (Set ($#k2_moebius1 :::"Moebius"::: ) (Num 3)) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) ; theorem :: MOEBIUS1:33 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) "is" ($#v1_moebius1 :::"square-free"::: ) )) "holds" (Bool (Set ($#k2_moebius1 :::"Moebius"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; registrationlet "n" be ($#v1_moebius1 :::"square-free"::: ) ($#m1_hidden :::"Nat":::); cluster (Set ($#k2_moebius1 :::"Moebius"::: ) "n") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#v1_xreal_0 :::"real"::: ) ; end; theorem :: MOEBIUS1:34 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool (Set ($#k2_moebius1 :::"Moebius"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1)))) ; theorem :: MOEBIUS1:35 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) "," (Set (Var "n")) ($#r1_int_2 :::"are_relative_prime"::: ) )) "holds" (Bool (Set ($#k2_moebius1 :::"Moebius"::: ) (Set "(" (Set (Var "m")) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_moebius1 :::"Moebius"::: ) (Set (Var "m")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" ($#k2_moebius1 :::"Moebius"::: ) (Set (Var "n")) ")" )))) ; theorem :: MOEBIUS1:36 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "n")) ($#k4_nat_1 :::"*"::: ) (Set (Var "p"))) "is" ($#v1_moebius1 :::"square-free"::: ) )) "holds" (Bool (Set ($#k2_moebius1 :::"Moebius"::: ) (Set "(" (Set (Var "n")) ($#k4_nat_1 :::"*"::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_binop_2 :::"-"::: ) (Set "(" ($#k2_moebius1 :::"Moebius"::: ) (Set (Var "n")) ")" ))))) ; theorem :: MOEBIUS1:37 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Bool "not" (Set (Var "m")) "," (Set (Var "n")) ($#r1_int_2 :::"are_relative_prime"::: ) ))) "holds" (Bool (Set ($#k2_moebius1 :::"Moebius"::: ) (Set "(" (Set (Var "m")) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: MOEBIUS1:38 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k1_moebius1 :::"SCNAT"::: ) )) "iff" (Bool (Set ($#k2_moebius1 :::"Moebius"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) ; begin definitionlet "n" be ($#m1_hidden :::"Nat":::); func :::"NatDivisors"::: "n" -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: MOEBIUS1:def 4 "{" (Set (Var "k")) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) "n") ")" ) "}" ; end; :: deftheorem defines :::"NatDivisors"::: MOEBIUS1:def 4 : (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k3_moebius1 :::"NatDivisors"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "k")) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set (Var "k")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))) ")" ) "}" )); theorem :: MOEBIUS1:39 (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k3_moebius1 :::"NatDivisors"::: ) (Set (Var "n")))) "iff" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))) ")" ) ")" )) ; theorem :: MOEBIUS1:40 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k3_moebius1 :::"NatDivisors"::: ) (Set (Var "n"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) ; registrationlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); cluster (Set ($#k3_moebius1 :::"NatDivisors"::: ) "n") -> ($#v1_finset_1 :::"finite"::: ) ($#v1_setfam_1 :::"with_non-empty_elements"::: ) ; end; theorem :: MOEBIUS1:41 (Bool (Set ($#k3_moebius1 :::"NatDivisors"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Num 1) ($#k6_domain_1 :::"}"::: ) )) ; begin definitionlet "X" be ($#m1_hidden :::"set"::: ) ; func :::"SMoebius"::: "X" -> ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: MOEBIUS1:def 5 (Bool "(" (Bool (Set ($#k13_pre_poly :::"support"::: ) it) ($#r1_hidden :::"="::: ) (Set "X" ($#k9_subset_1 :::"/\"::: ) (Set ($#k1_moebius1 :::"SCNAT"::: ) ))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k13_pre_poly :::"support"::: ) it))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k2_moebius1 :::"Moebius"::: ) (Set (Var "k")))) ")" ) ")" ); end; :: deftheorem defines :::"SMoebius"::: MOEBIUS1:def 5 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k4_moebius1 :::"SMoebius"::: ) (Set (Var "X")))) "iff" (Bool "(" (Bool (Set ($#k13_pre_poly :::"support"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k9_subset_1 :::"/\"::: ) (Set ($#k1_moebius1 :::"SCNAT"::: ) ))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k13_pre_poly :::"support"::: ) (Set (Var "b2"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set ($#k2_moebius1 :::"Moebius"::: ) (Set (Var "k")))) ")" ) ")" ) ")" ))); registrationlet "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k4_moebius1 :::"SMoebius"::: ) "X") -> ($#v3_valued_0 :::"real-valued"::: ) ; end; registrationlet "X" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k4_moebius1 :::"SMoebius"::: ) "X") -> ($#v2_pre_poly :::"finite-support"::: ) ; end; theorem :: MOEBIUS1:42 (Bool (Set ($#k3_uproots :::"Sum"::: ) (Set "(" ($#k4_moebius1 :::"SMoebius"::: ) (Set "(" ($#k3_moebius1 :::"NatDivisors"::: ) (Num 1) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) ; theorem :: MOEBIUS1:43 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set "(" ($#k13_pre_poly :::"support"::: ) (Set "(" ($#k4_moebius1 :::"SMoebius"::: ) (Set (Var "X")) ")" ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k13_pre_poly :::"support"::: ) (Set "(" ($#k4_moebius1 :::"SMoebius"::: ) (Set (Var "Y")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k13_pre_poly :::"support"::: ) (Set "(" (Set "(" ($#k4_moebius1 :::"SMoebius"::: ) (Set (Var "X")) ")" ) ($#k11_pre_poly :::"+"::: ) (Set "(" ($#k4_moebius1 :::"SMoebius"::: ) (Set (Var "Y")) ")" ) ")" )))) ; theorem :: MOEBIUS1:44 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "Y")))) "holds" (Bool (Set ($#k4_moebius1 :::"SMoebius"::: ) (Set "(" (Set (Var "X")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "Y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_moebius1 :::"SMoebius"::: ) (Set (Var "X")) ")" ) ($#k11_pre_poly :::"+"::: ) (Set "(" ($#k4_moebius1 :::"SMoebius"::: ) (Set (Var "Y")) ")" )))) ; begin definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); func :::"PFactors"::: "n" -> ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k10_newton :::"SetPrimes"::: ) ) means :: MOEBIUS1:def 6 (Bool "(" (Bool (Set ($#k13_pre_poly :::"support"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k12_nat_3 :::"pfexp"::: ) "n" ")" ))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k12_nat_3 :::"pfexp"::: ) "n" ")" )))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" ) ")" ); end; :: deftheorem defines :::"PFactors"::: MOEBIUS1:def 6 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set ($#k10_newton :::"SetPrimes"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k5_moebius1 :::"PFactors"::: ) (Set (Var "n")))) "iff" (Bool "(" (Bool (Set ($#k13_pre_poly :::"support"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k12_nat_3 :::"pfexp"::: ) (Set (Var "n")) ")" ))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_polynom2 :::"support"::: ) (Set "(" ($#k12_nat_3 :::"pfexp"::: ) (Set (Var "n")) ")" )))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" ) ")" ) ")" ))); registrationlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); cluster (Set ($#k5_moebius1 :::"PFactors"::: ) "n") -> ($#v4_valued_0 :::"natural-valued"::: ) ($#v2_pre_poly :::"finite-support"::: ) ; end; theorem :: MOEBIUS1:45 (Bool (Set ($#k5_moebius1 :::"PFactors"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k16_pre_poly :::"EmptyBag"::: ) (Set ($#k10_newton :::"SetPrimes"::: ) ))) ; theorem :: MOEBIUS1:46 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool (Set (Set "(" ($#k5_moebius1 :::"PFactors"::: ) (Set (Var "p")) ")" ) ($#k3_relat_1 :::"*"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k9_finseq_1 :::"*>"::: ) ))) ; theorem :: MOEBIUS1:47 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set "(" ($#k5_moebius1 :::"PFactors"::: ) (Set "(" (Set (Var "p")) ($#k1_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k3_relat_1 :::"*"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k9_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k9_finseq_1 :::"*>"::: ) )))) ; theorem :: MOEBIUS1:48 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" ($#k5_moebius1 :::"PFactors"::: ) (Set (Var "n")) ")" ) ($#k1_recdef_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: MOEBIUS1:49 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" ($#k5_moebius1 :::"PFactors"::: ) (Set (Var "n")) ")" ) ($#k1_recdef_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))))) ; theorem :: MOEBIUS1:50 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) "," (Set (Var "n")) ($#r1_int_2 :::"are_relative_prime"::: ) )) "holds" (Bool (Set ($#k5_moebius1 :::"PFactors"::: ) (Set "(" (Set (Var "m")) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_moebius1 :::"PFactors"::: ) (Set (Var "m")) ")" ) ($#k11_pre_poly :::"+"::: ) (Set "(" ($#k5_moebius1 :::"PFactors"::: ) (Set (Var "n")) ")" )))) ; theorem :: MOEBIUS1:51 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) "{" (Set (Var "k")) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))) & (Bool (Set (Var "k")) "is" ($#v1_moebius1 :::"square-containing"::: ) ) ")" ) "}" )) "holds" (Bool (Set ($#k4_moebius1 :::"SMoebius"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k16_pre_poly :::"EmptyBag"::: ) (Set ($#k5_numbers :::"NAT"::: ) ))))) ; begin definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); func :::"Radical"::: "n" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: MOEBIUS1:def 7 (Set ($#k8_nat_3 :::"Product"::: ) (Set "(" ($#k5_moebius1 :::"PFactors"::: ) "n" ")" )); end; :: deftheorem defines :::"Radical"::: MOEBIUS1:def 7 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k8_nat_3 :::"Product"::: ) (Set "(" ($#k5_moebius1 :::"PFactors"::: ) (Set (Var "n")) ")" )))); theorem :: MOEBIUS1:52 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; registrationlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); cluster (Set ($#k6_moebius1 :::"Radical"::: ) "n") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ; end; theorem :: MOEBIUS1:53 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k6_moebius1 :::"Radical"::: ) (Set (Var "p"))))) ; theorem :: MOEBIUS1:54 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_moebius1 :::"Radical"::: ) (Set "(" (Set (Var "p")) ($#k1_newton :::"|^"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p"))))) ; theorem :: MOEBIUS1:55 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n"))) ($#r1_nat_d :::"divides"::: ) (Set (Var "n")))) ; theorem :: MOEBIUS1:56 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))) "iff" (Bool (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n")))) ")" ))) ; theorem :: MOEBIUS1:57 (Bool "for" (Set (Var "k")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "k")) "is" ($#v1_moebius1 :::"square-free"::: ) )) "holds" (Bool (Set ($#k6_moebius1 :::"Radical"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Var "k")))) ; theorem :: MOEBIUS1:58 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) ; theorem :: MOEBIUS1:59 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set "(" ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set (Var "n")))))) ; theorem :: MOEBIUS1:60 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Num 1)) "iff" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 1)) ")" )) ; theorem :: MOEBIUS1:61 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "p")) ($#k11_nat_3 :::"|-count"::: ) (Set "(" ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Num 1)))) ; registrationlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); cluster (Set ($#k6_moebius1 :::"Radical"::: ) "n") -> ($#v1_moebius1 :::"square-free"::: ) ; end; theorem :: MOEBIUS1:62 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k6_moebius1 :::"Radical"::: ) (Set "(" ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n"))))) ; theorem :: MOEBIUS1:63 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool "{" (Set (Var "k")) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n")))) & (Bool (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "k"))) ")" ) "}" ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))))) ; theorem :: MOEBIUS1:64 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool "{" (Set (Var "k")) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n")))) & (Bool (Bool "not" (Set (Var "p")) ($#r1_nat_d :::"divides"::: ) (Set (Var "k")))) ")" ) "}" ($#r1_tarski :::"c="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))))) ; theorem :: MOEBIUS1:65 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))) & (Bool (Set (Var "k")) "is" ($#v1_moebius1 :::"square-free"::: ) ) ")" ) "iff" (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n")))) ")" )) ; theorem :: MOEBIUS1:66 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool "{" (Set (Var "k")) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set (Var "n"))) & (Bool (Set (Var "k")) "is" ($#v1_moebius1 :::"square-free"::: ) ) ")" ) "}" ($#r1_hidden :::"="::: ) "{" (Set (Var "k")) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_nat_d :::"divides"::: ) (Set ($#k6_moebius1 :::"Radical"::: ) (Set (Var "n")))) ")" ) "}" )) ;