:: REVROT_1 semantic presentation begin definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); :: original: :::"constant"::: redefine attr "f" is :::"constant"::: means :: REVROT_1:def 1 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "f")) & (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "f"))) "holds" (Bool (Set "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "m"))))); end; :: deftheorem defines :::"constant"::: REVROT_1:def 1 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_funct_1 :::"constant"::: ) ) "iff" (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "m")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "m"))))) ")" ))); theorem :: REVROT_1:1 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) ($#r2_finseq_4 :::"just_once_values"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))))) ; theorem :: REVROT_1:2 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; theorem :: REVROT_1:3 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))))) ; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); let "y" be ($#m1_hidden :::"set"::: ) ; :: original: :::"just_once_values"::: redefine pred "f" :::"just_once_values"::: "y" means :: REVROT_1:def 2 (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "f")) & (Bool "y" ($#r1_hidden :::"="::: ) (Set "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "f")) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "z"))) ($#r1_hidden :::"<>"::: ) "y") ")" ) ")" )); end; :: deftheorem defines :::"just_once_values"::: REVROT_1:def 2 : (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_revrot_1 :::"just_once_values"::: ) (Set (Var "y"))) "iff" (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "z"))) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) ")" ) ")" )) ")" )))); theorem :: REVROT_1:4 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) ($#r1_revrot_1 :::"just_once_values"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k1_finseq_5 :::"-:"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "f"))))) ; theorem :: REVROT_1:5 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "f")) ($#r1_revrot_1 :::"just_once_values"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )))) ; theorem :: REVROT_1:6 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )))))) ; theorem :: REVROT_1:7 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))))) ; theorem :: REVROT_1:8 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "f"))) "is" ($#v1_finseq_6 :::"circular"::: ) ))) ; begin theorem :: REVROT_1:9 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )))) "holds" (Bool (Set (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" ))))))) ; theorem :: REVROT_1:10 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k7_nat_d :::"-'"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" ))))))) ; theorem :: REVROT_1:11 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))))) ; theorem :: REVROT_1:12 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ))))))) ; theorem :: REVROT_1:13 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" ))))))) ; theorem :: REVROT_1:14 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))))) ; theorem :: REVROT_1:15 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))))) ; theorem :: REVROT_1:16 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1))) ")" )) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "f")))))) ; begin theorem :: REVROT_1:17 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ))))))) ; theorem :: REVROT_1:18 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" ))))))) ; registrationlet "D" be ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster bbbadV1_RELAT_1() bbbadV4_RELAT_1((Set ($#k5_numbers :::"NAT"::: ) )) bbbadV5_RELAT_1("D") ($#v1_funct_1 :::"Function-like"::: ) bbbadV3_FUNCT_1() bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) ($#v1_finseq_6 :::"circular"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" "D"; end; registrationlet "D" be ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"set"::: ) ; let "p" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); let "f" be bbbadV3_FUNCT_1() ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); cluster (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" "f" "," "p" ")" ) -> bbbadV3_FUNCT_1() ; end; begin theorem :: REVROT_1:19 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k16_euclid :::"0.REAL"::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set ($#k22_euclid :::"1.REAL"::: ) (Set (Var "n"))))) ; registrationlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"zero"::: ) ) ($#m1_hidden :::"Nat":::); cluster (Set ($#k15_euclid :::"TOP-REAL"::: ) "n") -> ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ; end; theorem :: REVROT_1:20 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "g")) ")" )))) ; theorem :: REVROT_1:21 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "g")) ")" )))) ; theorem :: REVROT_1:22 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "g")) ")" )))) ; theorem :: REVROT_1:23 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "g")) ")" )))) ; begin registrationlet "p" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); let "f" be ($#v1_finseq_6 :::"circular"::: ) ($#v1_topreal1 :::"special"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); cluster (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" "f" "," "p" ")" ) -> ($#v1_topreal1 :::"special"::: ) ; end; theorem :: REVROT_1:24 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" ) ")" ))))) ; theorem :: REVROT_1:25 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) "," (Set "(" (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))))) ; theorem :: REVROT_1:26 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k28_seq_4 :::"Incr"::: ) (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k28_seq_4 :::"Incr"::: ) (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) ")" ))))) ; theorem :: REVROT_1:27 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k28_seq_4 :::"Incr"::: ) (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k28_seq_4 :::"Incr"::: ) (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) ")" ))))) ; theorem :: REVROT_1:28 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k2_goboard2 :::"GoB"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")))))) ; theorem :: REVROT_1:29 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" bbbadV3_FUNCT_1() ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k4_finseq_5 :::"Rev"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set "(" ($#k1_goboard9 :::"Rev"::: ) (Set (Var "f")) ")" ) "," (Set (Var "p")) ")" )))) ; begin theorem :: REVROT_1:30 (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#v1_goboard5 :::"s.c.c."::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 4))) "holds" (Bool (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k6_domain_1 :::"}"::: ) ))) ; theorem :: REVROT_1:31 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k2_finseq_5 :::":-"::: ) (Set (Var "p")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ")" ))))) ; registrationlet "p" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); let "f" be ($#v1_finseq_6 :::"circular"::: ) ($#v1_goboard5 :::"s.c.c."::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); cluster (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" "f" "," "p" ")" ) -> ($#v1_goboard5 :::"s.c.c."::: ) ; end; registrationlet "p" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); let "f" be bbbadV3_FUNCT_1() ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::); cluster (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" "f" "," "p" ")" ) -> ($#v2_topreal1 :::"unfolded"::: ) ; end; theorem :: REVROT_1:32 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) "," (Set "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" ) ")" ))))) ; theorem :: REVROT_1:33 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: REVROT_1:34 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) "iff" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) ")" )))) ; registrationlet "p" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); let "f" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finseq_6 :::"circular"::: ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); cluster (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" "f" "," "p" ")" ) -> ($#v2_goboard5 :::"standard"::: ) ; end; theorem :: REVROT_1:35 (Bool "for" (Set (Var "f")) "being" bbbadV3_FUNCT_1() ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k5_goboard5 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k5_goboard5 :::"left_cell"::: ) "(" (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) "," (Set "(" (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Set "(" (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f")) ")" ) ")" ) ")" ))))) ; theorem :: REVROT_1:36 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" bbbadV3_FUNCT_1() ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))))) ; theorem :: REVROT_1:37 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" bbbadV3_FUNCT_1() ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k3_goboard9 :::"RightComp"::: ) (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")))))) ; begin registrationlet "p" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); let "f" be bbbadV3_FUNCT_1() ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::); cluster (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" "f" "," "p" ")" ) -> ($#v1_sprect_2 :::"clockwise_oriented"::: ) ; end; theorem :: REVROT_1:38 (Bool "for" (Set (Var "f")) "being" bbbadV3_FUNCT_1() ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_sprect_2 :::"clockwise_oriented"::: ) ) "or" (Bool (Set ($#k1_goboard9 :::"Rev"::: ) (Set (Var "f"))) "is" ($#v1_sprect_2 :::"clockwise_oriented"::: ) ) ")" )) ;