:: GOBOARD8 semantic presentation begin theorem :: GOBOARD8:1 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:2 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:3 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:4 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:5 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:6 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:7 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Num 1) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Num 1) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Num 2) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Num 1) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Num 2) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Num 1) ")" ")" ) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Num 2) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:8 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Num 1) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Num 2) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Num 2) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Num 1) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Num 1) ")" ")" ) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Num 1) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Num 2) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:9 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:10 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:11 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:12 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "j")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:13 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "j")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:14 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "j")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set (Var "j")) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set (Var "j")) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:15 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "j")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:16 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "j")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:17 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:18 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:19 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:20 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "j")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ")" ) ")" )) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: GOBOARD8:21 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) ")" )) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: GOBOARD8:22 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Num 1) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) ")" )) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: GOBOARD8:23 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) ")" )) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: GOBOARD8:24 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) ")" )) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: GOBOARD8:25 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Num 1) ")" ")" ) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Num 1) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Num 1) ")" ")" ) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: GOBOARD8:26 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Num 1) ")" ")" ) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) ; theorem :: GOBOARD8:27 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) "," (Num 1) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Num 1) ")" ")" ) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Num 1) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) ; theorem :: GOBOARD8:28 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: GOBOARD8:29 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) ; theorem :: GOBOARD8:30 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) ; theorem :: GOBOARD8:31 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: GOBOARD8:32 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 2) ")" ")" ) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) ; theorem :: GOBOARD8:33 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ")" ) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) ; theorem :: GOBOARD8:34 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ")" ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: GOBOARD8:35 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Num 1) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set "(" ($#k4_xcmplx_0 :::"-"::: ) (Num 1) ")" ) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Num 1) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Num 2) ")" ")" ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) ; theorem :: GOBOARD8:36 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Set ($#k6_numbers :::"0"::: ) ) ($#k19_euclid :::"]|"::: ) ) ")" ) "," (Set "(" (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) ; theorem :: GOBOARD8:37 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k5_goboard5 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: GOBOARD8:38 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k4_goboard5 :::"right_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ;