:: INT_5 semantic presentation begin theorem :: INT_5:1 (Bool "for" (Set (Var "i1")) "," (Set (Var "i2")) "," (Set (Var "i3")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "i1")) ($#r1_int_1 :::"divides"::: ) (Set (Var "i2"))) & (Bool (Set (Var "i1")) ($#r1_int_1 :::"divides"::: ) (Set (Var "i3")))) "holds" (Bool (Set (Var "i1")) ($#r1_int_1 :::"divides"::: ) (Set (Set (Var "i2")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "i3"))))) ; theorem :: INT_5:2 (Bool "for" (Set (Var "i")) "," (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_int_1 :::"divides"::: ) (Set (Var "a"))) & (Bool (Set (Var "i")) ($#r1_int_1 :::"divides"::: ) (Set (Set (Var "a")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "b"))))) "holds" (Bool (Set (Var "i")) ($#r1_int_1 :::"divides"::: ) (Set (Var "b")))) ; definitionlet "fp" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ); func :::"Poly-INT"::: "fp" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k4_numbers :::"INT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) means :: INT_5:def 1 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) (Bool "ex" (Set (Var "fr")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fr"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "fp")) & (Bool "(" "for" (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "fr"))))) "holds" (Bool (Set (Set (Var "fr")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "fp" ($#k1_funct_1 :::"."::: ) (Set (Var "d")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "x")) ($#k1_newton :::"|^"::: ) (Set "(" (Set (Var "d")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ))) ")" ) & (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_gr_cy_1 :::"Sum"::: ) (Set (Var "fr")))) ")" ))); end; :: deftheorem defines :::"Poly-INT"::: INT_5:def 1 : (Bool "for" (Set (Var "fp")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k4_numbers :::"INT"::: ) ) "," (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_int_5 :::"Poly-INT"::: ) (Set (Var "fp")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) (Bool "ex" (Set (Var "fr")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fr"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fp")))) & (Bool "(" "for" (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "fr"))))) "holds" (Bool (Set (Set (Var "fr")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "fp")) ($#k1_funct_1 :::"."::: ) (Set (Var "d")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "x")) ($#k1_newton :::"|^"::: ) (Set "(" (Set (Var "d")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ))) ")" ) & (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_gr_cy_1 :::"Sum"::: ) (Set (Var "fr")))) ")" ))) ")" ))); theorem :: INT_5:3 (Bool "for" (Set (Var "fp")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fp"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k1_int_5 :::"Poly-INT"::: ) (Set (Var "fp"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_numbers :::"INT"::: ) ) ($#k7_funcop_1 :::"-->"::: ) (Set "(" (Set (Var "fp")) ($#k1_funct_1 :::"."::: ) (Num 1) ")" )))) ; theorem :: INT_5:4 (Bool "for" (Set (Var "fp")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fp"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool (Set (Set "(" ($#k1_int_5 :::"Poly-INT"::: ) (Set (Var "fp")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "fp")) ($#k1_funct_1 :::"."::: ) (Num 1))))) ; theorem :: INT_5:5 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f2"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k1_funct_1 :::"."::: ) (Set (Var "d")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k1_funct_1 :::"."::: ) (Set (Var "d")) ")" ))) ")" )) "holds" (Bool "ex" (Set (Var "fr")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fr"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "fr"))))) "holds" (Bool (Set (Set (Var "fr")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k1_funct_1 :::"."::: ) (Set (Var "d")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "d")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" ) & (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "fr")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "f1")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k1_funct_1 :::"."::: ) (Num 1) ")" ))) ")" )))) ; theorem :: INT_5:6 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "fp")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fp"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2)))) "holds" (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "ex" (Set (Var "fr")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) )(Bool "ex" (Set (Var "r")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fr"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool (Set (Set "(" ($#k1_int_5 :::"Poly-INT"::: ) (Set (Var "fp")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "x")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set "(" ($#k1_int_5 :::"Poly-INT"::: ) (Set (Var "fr")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r")))) ")" ) & (Bool (Set (Set (Var "fp")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "fr")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" )))))) ; theorem :: INT_5:7 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool "(" "not" (Bool (Set (Var "p")) ($#r1_int_1 :::"divides"::: ) (Set (Set (Var "i")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "j")))) "or" (Bool (Set (Var "p")) ($#r1_int_1 :::"divides"::: ) (Set (Var "i"))) "or" (Bool (Set (Var "p")) ($#r1_int_1 :::"divides"::: ) (Set (Var "j"))) ")" ))) ; theorem :: INT_5:8 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "fp")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fp"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Bool "not" (Set (Var "p")) ($#r1_int_1 :::"divides"::: ) (Set (Set (Var "fp")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "n")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ))))) "holds" (Bool "for" (Set (Var "fr")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "fr"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_int_5 :::"Poly-INT"::: ) (Set (Var "fp")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "fr")) ($#k1_funct_1 :::"."::: ) (Set (Var "d")) ")" ) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool "(" "for" (Set (Var "d")) "," (Set (Var "e")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "fr")))) & (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "fr")))) & (Bool (Set (Var "d")) ($#r1_hidden :::"<>"::: ) (Set (Var "e")))) "holds" (Bool "not" (Bool (Set (Set (Var "fr")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) "," (Set (Set (Var "fr")) ($#k1_funct_1 :::"."::: ) (Set (Var "e"))) ($#r2_int_1 :::"are_congruent_mod"::: ) (Set (Var "p")))) ")" )) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fr"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))))))) ; definitionlet "a" be ($#m1_hidden :::"Integer":::); let "m" be ($#m1_hidden :::"Nat":::); pred "a" :::"is_quadratic_residue_mod"::: "m" means :: INT_5:def 2 (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Set (Set "(" (Set "(" (Set (Var "x")) ($#k1_pepin :::"^2"::: ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) "a" ")" ) ($#k6_int_1 :::"mod"::: ) "m") ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))); end; :: deftheorem defines :::"is_quadratic_residue_mod"::: INT_5:def 2 : (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "m"))) "iff" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Set (Set "(" (Set "(" (Set (Var "x")) ($#k1_pepin :::"^2"::: ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" ))); theorem :: INT_5:9 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "a")) ($#k1_pepin :::"^2"::: ) ) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "m"))))) ; theorem :: INT_5:10 (Bool (Num 1) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Num 2)) ; theorem :: INT_5:11 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "m"))) & (Bool (Set (Var "i")) "," (Set (Var "j")) ($#r2_int_1 :::"are_congruent_mod"::: ) (Set (Var "m")))) "holds" (Bool (Set (Var "j")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "m"))))) ; theorem :: INT_5:12 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_int_1 :::"divides"::: ) (Set (Var "j")))) "holds" (Bool (Set (Set (Var "i")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set ($#k1_int_2 :::"abs"::: ) (Set (Var "i"))))) ; theorem :: INT_5:13 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Set (Var "i")) ($#k6_int_1 :::"mod"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "j")) ($#k6_int_1 :::"mod"::: ) (Set (Var "m"))))) "holds" (Bool (Set (Set "(" (Set (Var "i")) ($#k1_newton :::"|^"::: ) (Set (Var "n")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "j")) ($#k1_newton :::"|^"::: ) (Set (Var "n")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "m")))))) ; theorem :: INT_5:14 (Bool "for" (Set (Var "a")) "," (Set (Var "x")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Set (Var "a")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set "(" (Set "(" (Set (Var "x")) ($#k1_pepin :::"^2"::: ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Var "x")) "," (Set (Var "p")) ($#r1_int_2 :::"are_relative_prime"::: ) ))) ; theorem :: INT_5:15 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "a")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "a")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p")))) "holds" (Bool "ex" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool "(" (Bool (Set (Set "(" (Set "(" (Set (Var "x")) ($#k1_pepin :::"^2"::: ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set "(" (Set (Var "y")) ($#k1_pepin :::"^2"::: ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "a")) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Bool "not" (Set (Var "x")) "," (Set (Var "y")) ($#r2_int_1 :::"are_congruent_mod"::: ) (Set (Var "p")))) ")" )))) ; theorem :: INT_5:16 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2))) "holds" (Bool "ex" (Set (Var "fp")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fp"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k3_nat_d :::"div"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "fp"))))) "holds" (Bool (Set (Set "(" (Set (Var "fp")) ($#k1_recdef_1 :::"."::: ) (Set (Var "d")) ")" ) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) & (Bool "(" "for" (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "fp"))))) "holds" (Bool (Set (Set (Var "fp")) ($#k1_recdef_1 :::"."::: ) (Set (Var "d"))) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))) ")" ) & (Bool "(" "for" (Set (Var "d")) "," (Set (Var "e")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "fp")))) & (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "fp")))) & (Bool (Set (Var "d")) ($#r1_hidden :::"<>"::: ) (Set (Var "e")))) "holds" (Bool "not" (Bool (Set (Set (Var "fp")) ($#k1_recdef_1 :::"."::: ) (Set (Var "d"))) "," (Set (Set (Var "fp")) ($#k1_recdef_1 :::"."::: ) (Set (Var "e"))) ($#r2_int_1 :::"are_congruent_mod"::: ) (Set (Var "p")))) ")" ) ")" ))) ; theorem :: INT_5:17 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "a")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "a")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p")))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k1_newton :::"|^"::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k3_nat_d :::"div"::: ) (Num 2) ")" ) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)))) ; theorem :: INT_5:18 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "b")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "b")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Bool "not" (Set (Var "b")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set "(" (Set (Var "b")) ($#k1_newton :::"|^"::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k3_nat_d :::"div"::: ) (Num 2) ")" ) ")" ) ($#k4_nat_d :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))))) ; theorem :: INT_5:19 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "a")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Bool "not" (Set (Var "a")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k1_newton :::"|^"::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k3_nat_d :::"div"::: ) (Num 2) ")" ) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k6_xcmplx_0 :::"-"::: ) (Num 1))))) ; theorem :: INT_5:20 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "a")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "a")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "a")) ($#k1_newton :::"|^"::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k3_nat_d :::"div"::: ) (Num 2) ")" ) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: INT_5:21 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "a")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Bool "not" (Set (Var "a")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "a")) ($#k1_newton :::"|^"::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k3_nat_d :::"div"::: ) (Num 2) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Num 1) ")" ) ($#k6_int_1 :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: INT_5:22 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "b")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))) & (Bool (Set (Var "b")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p")))) "holds" (Bool (Set (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "b"))) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p")))))) ; theorem :: INT_5:23 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "b")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "a")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "b")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "a")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))) & (Bool (Bool "not" (Set (Var "b")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))))) "holds" (Bool "not" (Bool (Set (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "b"))) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))))))) ; theorem :: INT_5:24 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "b")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "a")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "b")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Bool "not" (Set (Var "a")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p")))) & (Bool (Bool "not" (Set (Var "b")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "b"))) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p")))))) ; definitionlet "a" be ($#m1_hidden :::"Integer":::); let "p" be ($#m1_hidden :::"Prime":::); func :::"Lege"::: "(" "a" "," "p" ")" -> ($#m1_hidden :::"Integer":::) equals :: INT_5:def 3 (Num 1) if (Bool "(" (Bool "a" ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) "p") & (Bool (Set "a" ($#k6_int_1 :::"mod"::: ) "p") ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) (Set ($#k6_numbers :::"0"::: ) ) if (Bool "(" (Bool "a" ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) "p") & (Bool (Set "a" ($#k6_int_1 :::"mod"::: ) "p") ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) otherwise (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)); end; :: deftheorem defines :::"Lege"::: INT_5:def 3 : (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "a")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))) & (Bool (Set (Set (Var "a")) ($#k6_int_1 :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "a")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Num 1)) ")" & "(" (Bool (Bool (Set (Var "a")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))) & (Bool (Set (Set (Var "a")) ($#k6_int_1 :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "a")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & "(" (Bool (Bool "(" "not" (Bool (Set (Var "a")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))) "or" "not" (Bool (Set (Set (Var "a")) ($#k6_int_1 :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool "(" "not" (Bool (Set (Var "a")) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))) "or" "not" (Bool (Set (Set (Var "a")) ($#k6_int_1 :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "implies" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "a")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1))) ")" ")" ))); theorem :: INT_5:25 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool "(" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "a")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Num 1)) "or" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "a")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "or" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "a")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1))) ")" ))) ; theorem :: INT_5:26 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Set (Var "a")) ($#k6_int_1 :::"mod"::: ) (Set (Var "p"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set "(" (Set (Var "a")) ($#k1_pepin :::"^2"::: ) ")" ) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Num 1)))) ; theorem :: INT_5:27 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "holds" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Num 1) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: INT_5:28 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "a")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "a")) "," (Set (Var "p")) ")" ) "," (Set (Set (Var "a")) ($#k1_newton :::"|^"::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k3_nat_d :::"div"::: ) (Num 2) ")" )) ($#r2_int_1 :::"are_congruent_mod"::: ) (Set (Var "p"))))) ; theorem :: INT_5:29 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "b")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "a")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "a")) "," (Set (Var "b")) ($#r2_int_1 :::"are_congruent_mod"::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "a")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "b")) "," (Set (Var "p")) ")" ))))) ; theorem :: INT_5:30 (Bool "for" (Set (Var "a")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "b")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "a")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "b")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set "(" (Set (Var "a")) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "b")) ")" ) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "a")) "," (Set (Var "p")) ")" ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "b")) "," (Set (Var "p")) ")" ")" )))))) ; theorem :: INT_5:31 (Bool "for" (Set (Var "fr")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "holds" (Bool "(" (Bool "ex" (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "fr")))) & (Bool (Bool "not" (Set (Set (Var "fr")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) ($#r1_hidden :::"="::: ) (Num 1))) & (Bool (Bool "not" (Set (Set (Var "fr")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) & (Bool (Bool "not" (Set (Set (Var "fr")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)))) ")" )) "or" (Bool (Set ($#k19_rvsum_1 :::"Product"::: ) (Set (Var "fr"))) ($#r1_hidden :::"="::: ) (Num 1)) "or" (Bool (Set ($#k19_rvsum_1 :::"Product"::: ) (Set (Var "fr"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "or" (Bool (Set ($#k19_rvsum_1 :::"Product"::: ) (Set (Var "fr"))) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1))) ")" )) ; theorem :: INT_5:32 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "f")) "," (Set (Var "fr")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fr")))) & (Bool "(" "for" (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) "," (Set (Set (Var "fr")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) ($#r2_int_1 :::"are_congruent_mod"::: ) (Set (Var "m"))) ")" )) "holds" (Bool (Set ($#k19_rvsum_1 :::"Product"::: ) (Set (Var "f"))) "," (Set ($#k19_rvsum_1 :::"Product"::: ) (Set (Var "fr"))) ($#r2_int_1 :::"are_congruent_mod"::: ) (Set (Var "m"))))) ; theorem :: INT_5:33 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "f")) "," (Set (Var "fr")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fr")))) & (Bool "(" "for" (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) "," (Set ($#k4_xcmplx_0 :::"-"::: ) (Set "(" (Set (Var "fr")) ($#k1_funct_1 :::"."::: ) (Set (Var "d")) ")" )) ($#r2_int_1 :::"are_congruent_mod"::: ) (Set (Var "m"))) ")" )) "holds" (Bool (Set ($#k19_rvsum_1 :::"Product"::: ) (Set (Var "f"))) "," (Set (Set "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k1_newton :::"|^"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" ($#k19_rvsum_1 :::"Product"::: ) (Set (Var "fr")) ")" )) ($#r2_int_1 :::"are_congruent_mod"::: ) (Set (Var "m"))))) ; theorem :: INT_5:34 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "fp")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool "(" "for" (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "fp"))))) "holds" (Bool (Set (Set "(" (Set (Var "fp")) ($#k1_recdef_1 :::"."::: ) (Set (Var "d")) ")" ) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" )) "holds" (Bool "ex" (Set (Var "fr")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k4_numbers :::"INT"::: ) ) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fr"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "fp")))) & (Bool "(" "for" (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "fr"))))) "holds" (Bool (Set (Set (Var "fr")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) ($#r1_hidden :::"="::: ) (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set "(" (Set (Var "fp")) ($#k1_recdef_1 :::"."::: ) (Set (Var "d")) ")" ) "," (Set (Var "p")) ")" )) ")" ) & (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set "(" ($#k3_wsierp_1 :::"Product"::: ) (Set (Var "fp")) ")" ) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k19_rvsum_1 :::"Product"::: ) (Set (Var "fr")))) ")" )))) ; theorem :: INT_5:35 (Bool "for" (Set (Var "d")) "," (Set (Var "e")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "d")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Set (Var "e")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set "(" (Set "(" (Set (Var "d")) ($#k1_pepin :::"^2"::: ) ")" ) ($#k4_nat_1 :::"*"::: ) (Set (Var "e")) ")" ) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "e")) "," (Set (Var "p")) ")" )))) ; theorem :: INT_5:36 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2))) "holds" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k1_newton :::"|^"::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k3_nat_d :::"div"::: ) (Num 2) ")" )))) ; theorem :: INT_5:37 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "p")) ($#k4_nat_d :::"mod"::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p")))) ; theorem :: INT_5:38 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "p")) ($#k4_nat_d :::"mod"::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Num 3))) "holds" (Bool "not" (Bool (Set ($#k4_xcmplx_0 :::"-"::: ) (Num 1)) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))))) ; begin theorem :: INT_5:39 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) "iff" (Bool (Set ($#k2_finseq_7 :::"Swap"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )))) ; theorem :: INT_5:40 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "d")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_recdef_1 :::"."::: ) (Set (Var "d"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_recdef_1 :::"."::: ) (Set (Var "d"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) ")" ) ")" ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))))) ; theorem :: INT_5:41 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) (Bool "for" (Set (Var "a")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Set (Var "a")) ($#k3_int_2 :::"gcd"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k10_rvsum_1 :::"*"::: ) (Set "(" ($#k1_finseq_2 :::"idseq"::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k3_nat_d :::"div"::: ) (Num 2) ")" ) ")" ))) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) "{" (Set (Var "k")) where k "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_euler_2 :::"mod"::: ) (Set (Var "p")) ")" ))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p")) ($#k7_xcmplx_0 :::"/"::: ) (Num 2))) ")" ) "}" ))) "holds" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "a")) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k1_newton :::"|^"::: ) (Set (Var "m"))))))) ; theorem :: INT_5:42 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2))) "holds" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Num 2) "," (Set (Var "p")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k1_newton :::"|^"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k1_pepin :::"^2"::: ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k3_nat_d :::"div"::: ) (Num 8) ")" )))) ; theorem :: INT_5:43 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool "(" (Bool (Set (Set (Var "p")) ($#k4_nat_d :::"mod"::: ) (Num 8)) ($#r1_hidden :::"="::: ) (Num 1)) "or" (Bool (Set (Set (Var "p")) ($#k4_nat_d :::"mod"::: ) (Num 8)) ($#r1_hidden :::"="::: ) (Num 7)) ")" )) "holds" (Bool (Num 2) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p")))) ; theorem :: INT_5:44 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool "(" (Bool (Set (Set (Var "p")) ($#k4_nat_d :::"mod"::: ) (Num 8)) ($#r1_hidden :::"="::: ) (Num 3)) "or" (Bool (Set (Set (Var "p")) ($#k4_nat_d :::"mod"::: ) (Num 8)) ($#r1_hidden :::"="::: ) (Num 5)) ")" )) "holds" (Bool "not" (Bool (Num 2) ($#r1_int_5 :::"is_quadratic_residue_mod"::: ) (Set (Var "p"))))) ; theorem :: INT_5:45 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Set (Var "a")) ($#k4_nat_d :::"mod"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k4_nat_d :::"mod"::: ) (Num 2)))) "holds" (Bool (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k1_newton :::"|^"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k1_newton :::"|^"::: ) (Set (Var "b"))))) ; theorem :: INT_5:46 (Bool "for" (Set (Var "f")) "," (Set (Var "h")) "," (Set (Var "g")) "," (Set (Var "k")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "h")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "k"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g")) ")" ) ($#k8_rvsum_1 :::"-"::: ) (Set "(" (Set (Var "h")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "h")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set "(" (Set (Var "g")) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "k")) ")" )))) ; theorem :: INT_5:47 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k5_finseq_2 :::"|->"::: ) (Set (Var "m")) ")" ) ($#k8_rvsum_1 :::"-"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set (Var "m")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set "(" ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "f")) ")" ))))) ; definitionlet "X" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; let "F" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Const "X"))); :: original: :::"Card"::: redefine func :::"Card"::: "F" -> ($#v1_card_3 :::"Cardinal-yielding"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); end; theorem :: INT_5:48 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "X")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "X"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool "(" "for" (Set (Var "d")) "," (Set (Var "e")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "d")) ($#r1_hidden :::"<>"::: ) (Set (Var "e")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "d"))) ($#r1_xboole_0 :::"misses"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "e")))) ")" )) "holds" (Bool (Set ($#k5_card_1 :::"card"::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_wsierp_1 :::"Sum"::: ) (Set "(" ($#k3_int_5 :::"Card"::: ) (Set (Var "f")) ")" )))))) ; theorem :: INT_5:49 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Var "q")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set "(" ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ")" ) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "q")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k1_newton :::"|^"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k3_nat_d :::"div"::: ) (Num 2) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "q")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k3_nat_d :::"div"::: ) (Num 2) ")" ) ")" )))) ; theorem :: INT_5:50 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Var "q")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Var "q"))) & (Bool (Set (Set (Var "p")) ($#k4_nat_d :::"mod"::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Num 3)) & (Bool (Set (Set (Var "q")) ($#k4_nat_d :::"mod"::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Num 3))) "holds" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_xcmplx_0 :::"-"::: ) (Set "(" ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "q")) "," (Set (Var "p")) ")" ")" )))) ; theorem :: INT_5:51 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"Prime":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Var "q")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Var "q"))) & (Bool "(" (Bool (Set (Set (Var "p")) ($#k4_nat_d :::"mod"::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Num 1)) "or" (Bool (Set (Set (Var "q")) ($#k4_nat_d :::"mod"::: ) (Num 4)) ($#r1_hidden :::"="::: ) (Num 1)) ")" )) "holds" (Bool (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_int_5 :::"Lege"::: ) "(" (Set (Var "q")) "," (Set (Var "p")) ")" ))) ;