0.00/0.14 YES 0.00/0.14 0.00/0.14 Problem 1: 0.00/0.14 0.00/0.14 (VAR v_NonEmpty:S X:S X1:S X2:S Y:S Z:S) 0.00/0.14 (RULES 0.00/0.14 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 active(first(0,X:S)) -> mark(nil) 0.00/0.14 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.14 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 from(active(X:S)) -> from(X:S) 0.00/0.14 from(mark(X:S)) -> from(X:S) 0.00/0.14 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.14 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.14 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.14 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.14 mark(0) -> active(0) 0.00/0.14 mark(nil) -> active(nil) 0.00/0.14 s(active(X:S)) -> s(X:S) 0.00/0.14 s(mark(X:S)) -> s(X:S) 0.00/0.14 ) 0.00/0.14 (STRATEGY INNERMOST) 0.00/0.14 0.00/0.14 Problem 1: 0.00/0.14 0.00/0.14 Dependency Pairs Processor: 0.00/0.14 -> Pairs: 0.00/0.14 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> CONS(Y:S,first(X:S,Z:S)) 0.00/0.14 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> FIRST(X:S,Z:S) 0.00/0.14 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 ACTIVE(first(0,X:S)) -> MARK(nil) 0.00/0.14 ACTIVE(from(X:S)) -> CONS(X:S,from(s(X:S))) 0.00/0.14 ACTIVE(from(X:S)) -> FROM(s(X:S)) 0.00/0.14 ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) 0.00/0.14 ACTIVE(from(X:S)) -> S(X:S) 0.00/0.14 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.14 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.14 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.14 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.14 FIRST(active(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.14 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.14 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.14 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.14 FROM(active(X:S)) -> FROM(X:S) 0.00/0.14 FROM(mark(X:S)) -> FROM(X:S) 0.00/0.14 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.14 MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S) 0.00/0.14 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.14 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) 0.00/0.14 MARK(first(X1:S,X2:S)) -> FIRST(mark(X1:S),mark(X2:S)) 0.00/0.14 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.14 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.14 MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) 0.00/0.14 MARK(from(X:S)) -> FROM(mark(X:S)) 0.00/0.14 MARK(from(X:S)) -> MARK(X:S) 0.00/0.14 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.14 MARK(s(X:S)) -> MARK(X:S) 0.00/0.14 MARK(s(X:S)) -> S(mark(X:S)) 0.00/0.14 S(active(X:S)) -> S(X:S) 0.00/0.14 S(mark(X:S)) -> S(X:S) 0.00/0.14 -> Rules: 0.00/0.14 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 active(first(0,X:S)) -> mark(nil) 0.00/0.14 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.14 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 from(active(X:S)) -> from(X:S) 0.00/0.14 from(mark(X:S)) -> from(X:S) 0.00/0.14 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.14 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.14 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.14 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.14 mark(0) -> active(0) 0.00/0.14 mark(nil) -> active(nil) 0.00/0.14 s(active(X:S)) -> s(X:S) 0.00/0.14 s(mark(X:S)) -> s(X:S) 0.00/0.14 0.00/0.14 Problem 1: 0.00/0.14 0.00/0.14 SCC Processor: 0.00/0.14 -> Pairs: 0.00/0.14 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> CONS(Y:S,first(X:S,Z:S)) 0.00/0.14 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> FIRST(X:S,Z:S) 0.00/0.14 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 ACTIVE(first(0,X:S)) -> MARK(nil) 0.00/0.14 ACTIVE(from(X:S)) -> CONS(X:S,from(s(X:S))) 0.00/0.14 ACTIVE(from(X:S)) -> FROM(s(X:S)) 0.00/0.14 ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) 0.00/0.14 ACTIVE(from(X:S)) -> S(X:S) 0.00/0.14 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.14 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.14 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.14 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.14 FIRST(active(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.14 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.14 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.14 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.14 FROM(active(X:S)) -> FROM(X:S) 0.00/0.14 FROM(mark(X:S)) -> FROM(X:S) 0.00/0.14 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.14 MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S) 0.00/0.14 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.14 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) 0.00/0.14 MARK(first(X1:S,X2:S)) -> FIRST(mark(X1:S),mark(X2:S)) 0.00/0.14 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.14 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.14 MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) 0.00/0.14 MARK(from(X:S)) -> FROM(mark(X:S)) 0.00/0.14 MARK(from(X:S)) -> MARK(X:S) 0.00/0.14 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.14 MARK(s(X:S)) -> MARK(X:S) 0.00/0.14 MARK(s(X:S)) -> S(mark(X:S)) 0.00/0.14 S(active(X:S)) -> S(X:S) 0.00/0.14 S(mark(X:S)) -> S(X:S) 0.00/0.14 -> Rules: 0.00/0.14 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 active(first(0,X:S)) -> mark(nil) 0.00/0.14 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.14 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 from(active(X:S)) -> from(X:S) 0.00/0.14 from(mark(X:S)) -> from(X:S) 0.00/0.14 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.14 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.14 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.14 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.14 mark(0) -> active(0) 0.00/0.14 mark(nil) -> active(nil) 0.00/0.14 s(active(X:S)) -> s(X:S) 0.00/0.14 s(mark(X:S)) -> s(X:S) 0.00/0.14 ->Strongly Connected Components: 0.00/0.14 ->->Cycle: 0.00/0.14 ->->-> Pairs: 0.00/0.14 S(active(X:S)) -> S(X:S) 0.00/0.14 S(mark(X:S)) -> S(X:S) 0.00/0.14 ->->-> Rules: 0.00/0.14 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 active(first(0,X:S)) -> mark(nil) 0.00/0.14 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.14 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 from(active(X:S)) -> from(X:S) 0.00/0.14 from(mark(X:S)) -> from(X:S) 0.00/0.14 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.14 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.14 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.14 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.14 mark(0) -> active(0) 0.00/0.14 mark(nil) -> active(nil) 0.00/0.14 s(active(X:S)) -> s(X:S) 0.00/0.14 s(mark(X:S)) -> s(X:S) 0.00/0.14 ->->Cycle: 0.00/0.14 ->->-> Pairs: 0.00/0.14 FROM(active(X:S)) -> FROM(X:S) 0.00/0.14 FROM(mark(X:S)) -> FROM(X:S) 0.00/0.14 ->->-> Rules: 0.00/0.14 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 active(first(0,X:S)) -> mark(nil) 0.00/0.14 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.14 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 from(active(X:S)) -> from(X:S) 0.00/0.14 from(mark(X:S)) -> from(X:S) 0.00/0.14 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.14 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.14 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.14 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.14 mark(0) -> active(0) 0.00/0.14 mark(nil) -> active(nil) 0.00/0.14 s(active(X:S)) -> s(X:S) 0.00/0.14 s(mark(X:S)) -> s(X:S) 0.00/0.14 ->->Cycle: 0.00/0.14 ->->-> Pairs: 0.00/0.14 FIRST(active(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.14 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.14 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.14 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.14 ->->-> Rules: 0.00/0.14 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 active(first(0,X:S)) -> mark(nil) 0.00/0.14 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.14 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 from(active(X:S)) -> from(X:S) 0.00/0.14 from(mark(X:S)) -> from(X:S) 0.00/0.14 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.14 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.14 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.14 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.14 mark(0) -> active(0) 0.00/0.14 mark(nil) -> active(nil) 0.00/0.14 s(active(X:S)) -> s(X:S) 0.00/0.14 s(mark(X:S)) -> s(X:S) 0.00/0.14 ->->Cycle: 0.00/0.14 ->->-> Pairs: 0.00/0.14 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.14 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.14 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.14 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.14 ->->-> Rules: 0.00/0.14 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 active(first(0,X:S)) -> mark(nil) 0.00/0.14 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.14 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 from(active(X:S)) -> from(X:S) 0.00/0.14 from(mark(X:S)) -> from(X:S) 0.00/0.14 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.14 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.14 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.14 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.14 mark(0) -> active(0) 0.00/0.14 mark(nil) -> active(nil) 0.00/0.14 s(active(X:S)) -> s(X:S) 0.00/0.14 s(mark(X:S)) -> s(X:S) 0.00/0.14 ->->Cycle: 0.00/0.14 ->->-> Pairs: 0.00/0.14 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) 0.00/0.14 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.14 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.14 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) 0.00/0.14 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.14 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.14 MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) 0.00/0.14 MARK(from(X:S)) -> MARK(X:S) 0.00/0.14 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.14 MARK(s(X:S)) -> MARK(X:S) 0.00/0.14 ->->-> Rules: 0.00/0.14 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 active(first(0,X:S)) -> mark(nil) 0.00/0.14 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.14 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 from(active(X:S)) -> from(X:S) 0.00/0.14 from(mark(X:S)) -> from(X:S) 0.00/0.14 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.14 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.14 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.14 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.14 mark(0) -> active(0) 0.00/0.14 mark(nil) -> active(nil) 0.00/0.14 s(active(X:S)) -> s(X:S) 0.00/0.14 s(mark(X:S)) -> s(X:S) 0.00/0.14 0.00/0.14 0.00/0.14 The problem is decomposed in 5 subproblems. 0.00/0.14 0.00/0.14 Problem 1.1: 0.00/0.14 0.00/0.14 Subterm Processor: 0.00/0.14 -> Pairs: 0.00/0.14 S(active(X:S)) -> S(X:S) 0.00/0.14 S(mark(X:S)) -> S(X:S) 0.00/0.14 -> Rules: 0.00/0.14 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 active(first(0,X:S)) -> mark(nil) 0.00/0.14 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.14 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 from(active(X:S)) -> from(X:S) 0.00/0.14 from(mark(X:S)) -> from(X:S) 0.00/0.14 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.14 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.14 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.14 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.14 mark(0) -> active(0) 0.00/0.14 mark(nil) -> active(nil) 0.00/0.14 s(active(X:S)) -> s(X:S) 0.00/0.14 s(mark(X:S)) -> s(X:S) 0.00/0.14 ->Projection: 0.00/0.14 pi(S) = 1 0.00/0.14 0.00/0.14 Problem 1.1: 0.00/0.14 0.00/0.14 SCC Processor: 0.00/0.14 -> Pairs: 0.00/0.14 Empty 0.00/0.14 -> Rules: 0.00/0.14 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 active(first(0,X:S)) -> mark(nil) 0.00/0.14 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.14 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 from(active(X:S)) -> from(X:S) 0.00/0.14 from(mark(X:S)) -> from(X:S) 0.00/0.14 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.14 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.14 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.14 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.14 mark(0) -> active(0) 0.00/0.14 mark(nil) -> active(nil) 0.00/0.14 s(active(X:S)) -> s(X:S) 0.00/0.14 s(mark(X:S)) -> s(X:S) 0.00/0.14 ->Strongly Connected Components: 0.00/0.14 There is no strongly connected component 0.00/0.14 0.00/0.14 The problem is finite. 0.00/0.14 0.00/0.14 Problem 1.2: 0.00/0.14 0.00/0.14 Subterm Processor: 0.00/0.14 -> Pairs: 0.00/0.14 FROM(active(X:S)) -> FROM(X:S) 0.00/0.14 FROM(mark(X:S)) -> FROM(X:S) 0.00/0.14 -> Rules: 0.00/0.14 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 active(first(0,X:S)) -> mark(nil) 0.00/0.14 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.14 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 from(active(X:S)) -> from(X:S) 0.00/0.14 from(mark(X:S)) -> from(X:S) 0.00/0.14 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.14 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.14 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.14 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.14 mark(0) -> active(0) 0.00/0.14 mark(nil) -> active(nil) 0.00/0.14 s(active(X:S)) -> s(X:S) 0.00/0.14 s(mark(X:S)) -> s(X:S) 0.00/0.14 ->Projection: 0.00/0.14 pi(FROM) = 1 0.00/0.14 0.00/0.14 Problem 1.2: 0.00/0.14 0.00/0.14 SCC Processor: 0.00/0.14 -> Pairs: 0.00/0.14 Empty 0.00/0.14 -> Rules: 0.00/0.14 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 active(first(0,X:S)) -> mark(nil) 0.00/0.14 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.14 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 from(active(X:S)) -> from(X:S) 0.00/0.14 from(mark(X:S)) -> from(X:S) 0.00/0.14 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.14 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.14 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.14 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.14 mark(0) -> active(0) 0.00/0.14 mark(nil) -> active(nil) 0.00/0.14 s(active(X:S)) -> s(X:S) 0.00/0.14 s(mark(X:S)) -> s(X:S) 0.00/0.14 ->Strongly Connected Components: 0.00/0.14 There is no strongly connected component 0.00/0.14 0.00/0.14 The problem is finite. 0.00/0.14 0.00/0.14 Problem 1.3: 0.00/0.14 0.00/0.14 Subterm Processor: 0.00/0.14 -> Pairs: 0.00/0.14 FIRST(active(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.14 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.14 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.14 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.14 -> Rules: 0.00/0.14 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.14 active(first(0,X:S)) -> mark(nil) 0.00/0.14 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.14 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.14 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.14 from(active(X:S)) -> from(X:S) 0.00/0.14 from(mark(X:S)) -> from(X:S) 0.00/0.14 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.14 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.14 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Projection: 0.00/0.15 pi(FIRST) = 1 0.00/0.15 0.00/0.15 Problem 1.3: 0.00/0.15 0.00/0.15 SCC Processor: 0.00/0.15 -> Pairs: 0.00/0.15 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.15 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.15 -> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Strongly Connected Components: 0.00/0.15 ->->Cycle: 0.00/0.15 ->->-> Pairs: 0.00/0.15 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.15 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.15 ->->-> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 0.00/0.15 Problem 1.3: 0.00/0.15 0.00/0.15 Subterm Processor: 0.00/0.15 -> Pairs: 0.00/0.15 FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.15 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.15 -> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Projection: 0.00/0.15 pi(FIRST) = 2 0.00/0.15 0.00/0.15 Problem 1.3: 0.00/0.15 0.00/0.15 SCC Processor: 0.00/0.15 -> Pairs: 0.00/0.15 Empty 0.00/0.15 -> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Strongly Connected Components: 0.00/0.15 There is no strongly connected component 0.00/0.15 0.00/0.15 The problem is finite. 0.00/0.15 0.00/0.15 Problem 1.4: 0.00/0.15 0.00/0.15 Subterm Processor: 0.00/0.15 -> Pairs: 0.00/0.15 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.15 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.00/0.15 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.15 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.15 -> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Projection: 0.00/0.15 pi(CONS) = 1 0.00/0.15 0.00/0.15 Problem 1.4: 0.00/0.15 0.00/0.15 SCC Processor: 0.00/0.15 -> Pairs: 0.00/0.15 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.15 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.15 -> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Strongly Connected Components: 0.00/0.15 ->->Cycle: 0.00/0.15 ->->-> Pairs: 0.00/0.15 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.15 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.15 ->->-> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 0.00/0.15 Problem 1.4: 0.00/0.15 0.00/0.15 Subterm Processor: 0.00/0.15 -> Pairs: 0.00/0.15 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.15 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.15 -> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Projection: 0.00/0.15 pi(CONS) = 2 0.00/0.15 0.00/0.15 Problem 1.4: 0.00/0.15 0.00/0.15 SCC Processor: 0.00/0.15 -> Pairs: 0.00/0.15 Empty 0.00/0.15 -> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Strongly Connected Components: 0.00/0.15 There is no strongly connected component 0.00/0.15 0.00/0.15 The problem is finite. 0.00/0.15 0.00/0.15 Problem 1.5: 0.00/0.15 0.00/0.15 Reduction Pairs Processor: 0.00/0.15 -> Pairs: 0.00/0.15 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) 0.00/0.15 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.15 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.15 MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) 0.00/0.15 MARK(from(X:S)) -> MARK(X:S) 0.00/0.15 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.15 MARK(s(X:S)) -> MARK(X:S) 0.00/0.15 -> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 -> Usable rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Interpretation type: 0.00/0.15 Linear 0.00/0.15 ->Coefficients: 0.00/0.15 Natural Numbers 0.00/0.15 ->Dimension: 0.00/0.15 1 0.00/0.15 ->Bound: 0.00/0.15 2 0.00/0.15 ->Interpretation: 0.00/0.15 0.00/0.15 [active](X) = X 0.00/0.15 [cons](X1,X2) = X1 + 1 0.00/0.15 [first](X1,X2) = 2.X1 + 2.X2 0.00/0.15 [from](X) = 2.X + 2 0.00/0.15 [mark](X) = 2.X 0.00/0.15 [s](X) = 2.X + 2 0.00/0.15 [0] = 2 0.00/0.15 [fSNonEmpty] = 0 0.00/0.15 [nil] = 2 0.00/0.15 [ACTIVE](X) = X 0.00/0.15 [CONS](X1,X2) = 0 0.00/0.15 [FIRST](X1,X2) = 0 0.00/0.15 [FROM](X) = 0 0.00/0.15 [MARK](X) = 2.X 0.00/0.15 [S](X) = 0 0.00/0.15 0.00/0.15 Problem 1.5: 0.00/0.15 0.00/0.15 SCC Processor: 0.00/0.15 -> Pairs: 0.00/0.15 ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) 0.00/0.15 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.15 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.15 MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) 0.00/0.15 MARK(from(X:S)) -> MARK(X:S) 0.00/0.15 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.15 MARK(s(X:S)) -> MARK(X:S) 0.00/0.15 -> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Strongly Connected Components: 0.00/0.15 ->->Cycle: 0.00/0.15 ->->-> Pairs: 0.00/0.15 ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) 0.00/0.15 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.15 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.15 MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) 0.00/0.15 MARK(from(X:S)) -> MARK(X:S) 0.00/0.15 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.15 MARK(s(X:S)) -> MARK(X:S) 0.00/0.15 ->->-> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 0.00/0.15 Problem 1.5: 0.00/0.15 0.00/0.15 Reduction Pairs Processor: 0.00/0.15 -> Pairs: 0.00/0.15 ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) 0.00/0.15 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.15 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.15 MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) 0.00/0.15 MARK(from(X:S)) -> MARK(X:S) 0.00/0.15 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.15 MARK(s(X:S)) -> MARK(X:S) 0.00/0.15 -> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 -> Usable rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Interpretation type: 0.00/0.15 Linear 0.00/0.15 ->Coefficients: 0.00/0.15 Natural Numbers 0.00/0.15 ->Dimension: 0.00/0.15 1 0.00/0.15 ->Bound: 0.00/0.15 2 0.00/0.15 ->Interpretation: 0.00/0.15 0.00/0.15 [active](X) = X 0.00/0.15 [cons](X1,X2) = 2.X1 0.00/0.15 [first](X1,X2) = X1 + 2.X2 0.00/0.15 [from](X) = 2.X + 1 0.00/0.15 [mark](X) = X 0.00/0.15 [s](X) = 2.X 0.00/0.15 [0] = 2 0.00/0.15 [fSNonEmpty] = 0 0.00/0.15 [nil] = 2 0.00/0.15 [ACTIVE](X) = 2.X + 1 0.00/0.15 [CONS](X1,X2) = 0 0.00/0.15 [FIRST](X1,X2) = 0 0.00/0.15 [FROM](X) = 0 0.00/0.15 [MARK](X) = 2.X + 2 0.00/0.15 [S](X) = 0 0.00/0.15 0.00/0.15 Problem 1.5: 0.00/0.15 0.00/0.15 SCC Processor: 0.00/0.15 -> Pairs: 0.00/0.15 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) 0.00/0.15 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.15 MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) 0.00/0.15 MARK(from(X:S)) -> MARK(X:S) 0.00/0.15 MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) 0.00/0.15 MARK(s(X:S)) -> MARK(X:S) 0.00/0.15 -> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Strongly Connected Components: 0.00/0.15 ->->Cycle: 0.00/0.15 ->->-> Pairs: 0.00/0.15 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.15 MARK(from(X:S)) -> MARK(X:S) 0.00/0.15 MARK(s(X:S)) -> MARK(X:S) 0.00/0.15 ->->-> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 0.00/0.15 Problem 1.5: 0.00/0.15 0.00/0.15 Subterm Processor: 0.00/0.15 -> Pairs: 0.00/0.15 MARK(cons(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X1:S) 0.00/0.15 MARK(first(X1:S,X2:S)) -> MARK(X2:S) 0.00/0.15 MARK(from(X:S)) -> MARK(X:S) 0.00/0.15 MARK(s(X:S)) -> MARK(X:S) 0.00/0.15 -> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Projection: 0.00/0.15 pi(MARK) = 1 0.00/0.15 0.00/0.15 Problem 1.5: 0.00/0.15 0.00/0.15 SCC Processor: 0.00/0.15 -> Pairs: 0.00/0.15 Empty 0.00/0.15 -> Rules: 0.00/0.15 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.15 active(first(0,X:S)) -> mark(nil) 0.00/0.15 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.15 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) 0.00/0.15 first(active(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(mark(X1:S),X2:S) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,active(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) 0.00/0.15 from(active(X:S)) -> from(X:S) 0.00/0.15 from(mark(X:S)) -> from(X:S) 0.00/0.15 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) 0.00/0.15 mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) 0.00/0.15 mark(from(X:S)) -> active(from(mark(X:S))) 0.00/0.15 mark(s(X:S)) -> active(s(mark(X:S))) 0.00/0.15 mark(0) -> active(0) 0.00/0.15 mark(nil) -> active(nil) 0.00/0.15 s(active(X:S)) -> s(X:S) 0.00/0.15 s(mark(X:S)) -> s(X:S) 0.00/0.15 ->Strongly Connected Components: 0.00/0.15 There is no strongly connected component 0.00/0.15 0.00/0.15 The problem is finite. 0.00/0.15 EOF