0.05/0.12 YES 0.05/0.12 0.05/0.12 Problem 1: 0.05/0.12 0.05/0.12 (VAR v_NonEmpty:S X:S X1:S X2:S Y:S Z:S) 0.05/0.12 (RULES 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ) 0.05/0.12 (STRATEGY INNERMOST) 0.05/0.12 0.05/0.12 Problem 1: 0.05/0.12 0.05/0.12 Dependency Pairs Processor: 0.05/0.12 -> Pairs: 0.05/0.12 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 0.05/0.12 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) 0.05/0.12 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> CONS(Y:S,first(X:S,Z:S)) 0.05/0.12 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> FIRST(X:S,Z:S) 0.05/0.12 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X1:S) 0.05/0.12 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X2:S) 0.05/0.12 ACTIVE(first(X1:S,X2:S)) -> FIRST(active(X1:S),X2:S) 0.05/0.12 ACTIVE(first(X1:S,X2:S)) -> FIRST(X1:S,active(X2:S)) 0.05/0.12 ACTIVE(from(X:S)) -> ACTIVE(X:S) 0.05/0.12 ACTIVE(from(X:S)) -> CONS(X:S,from(s(X:S))) 0.05/0.12 ACTIVE(from(X:S)) -> FROM(active(X:S)) 0.05/0.12 ACTIVE(from(X:S)) -> FROM(s(X:S)) 0.05/0.12 ACTIVE(from(X:S)) -> S(X:S) 0.05/0.12 ACTIVE(s(X:S)) -> ACTIVE(X:S) 0.05/0.12 ACTIVE(s(X:S)) -> S(active(X:S)) 0.05/0.12 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.05/0.12 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 0.05/0.12 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.05/0.12 FIRST(ok(X1:S),ok(X2:S)) -> FIRST(X1:S,X2:S) 0.05/0.12 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.05/0.12 FROM(mark(X:S)) -> FROM(X:S) 0.05/0.12 FROM(ok(X:S)) -> FROM(X:S) 0.05/0.12 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 0.05/0.12 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 0.05/0.12 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 0.05/0.12 PROPER(first(X1:S,X2:S)) -> FIRST(proper(X1:S),proper(X2:S)) 0.05/0.12 PROPER(first(X1:S,X2:S)) -> PROPER(X1:S) 0.05/0.12 PROPER(first(X1:S,X2:S)) -> PROPER(X2:S) 0.05/0.12 PROPER(from(X:S)) -> FROM(proper(X:S)) 0.05/0.12 PROPER(from(X:S)) -> PROPER(X:S) 0.05/0.12 PROPER(s(X:S)) -> PROPER(X:S) 0.05/0.12 PROPER(s(X:S)) -> S(proper(X:S)) 0.05/0.12 S(mark(X:S)) -> S(X:S) 0.05/0.12 S(ok(X:S)) -> S(X:S) 0.05/0.12 TOP(mark(X:S)) -> PROPER(X:S) 0.05/0.12 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.05/0.12 TOP(ok(X:S)) -> ACTIVE(X:S) 0.05/0.12 TOP(ok(X:S)) -> TOP(active(X:S)) 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 0.05/0.12 Problem 1: 0.05/0.12 0.05/0.12 SCC Processor: 0.05/0.12 -> Pairs: 0.05/0.12 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 0.05/0.12 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) 0.05/0.12 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> CONS(Y:S,first(X:S,Z:S)) 0.05/0.12 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> FIRST(X:S,Z:S) 0.05/0.12 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X1:S) 0.05/0.12 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X2:S) 0.05/0.12 ACTIVE(first(X1:S,X2:S)) -> FIRST(active(X1:S),X2:S) 0.05/0.12 ACTIVE(first(X1:S,X2:S)) -> FIRST(X1:S,active(X2:S)) 0.05/0.12 ACTIVE(from(X:S)) -> ACTIVE(X:S) 0.05/0.12 ACTIVE(from(X:S)) -> CONS(X:S,from(s(X:S))) 0.05/0.12 ACTIVE(from(X:S)) -> FROM(active(X:S)) 0.05/0.12 ACTIVE(from(X:S)) -> FROM(s(X:S)) 0.05/0.12 ACTIVE(from(X:S)) -> S(X:S) 0.05/0.12 ACTIVE(s(X:S)) -> ACTIVE(X:S) 0.05/0.12 ACTIVE(s(X:S)) -> S(active(X:S)) 0.05/0.12 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.05/0.12 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 0.05/0.12 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.05/0.12 FIRST(ok(X1:S),ok(X2:S)) -> FIRST(X1:S,X2:S) 0.05/0.12 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.05/0.12 FROM(mark(X:S)) -> FROM(X:S) 0.05/0.12 FROM(ok(X:S)) -> FROM(X:S) 0.05/0.12 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 0.05/0.12 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 0.05/0.12 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 0.05/0.12 PROPER(first(X1:S,X2:S)) -> FIRST(proper(X1:S),proper(X2:S)) 0.05/0.12 PROPER(first(X1:S,X2:S)) -> PROPER(X1:S) 0.05/0.12 PROPER(first(X1:S,X2:S)) -> PROPER(X2:S) 0.05/0.12 PROPER(from(X:S)) -> FROM(proper(X:S)) 0.05/0.12 PROPER(from(X:S)) -> PROPER(X:S) 0.05/0.12 PROPER(s(X:S)) -> PROPER(X:S) 0.05/0.12 PROPER(s(X:S)) -> S(proper(X:S)) 0.05/0.12 S(mark(X:S)) -> S(X:S) 0.05/0.12 S(ok(X:S)) -> S(X:S) 0.05/0.12 TOP(mark(X:S)) -> PROPER(X:S) 0.05/0.12 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.05/0.12 TOP(ok(X:S)) -> ACTIVE(X:S) 0.05/0.12 TOP(ok(X:S)) -> TOP(active(X:S)) 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Strongly Connected Components: 0.05/0.12 ->->Cycle: 0.05/0.12 ->->-> Pairs: 0.05/0.12 S(mark(X:S)) -> S(X:S) 0.05/0.12 S(ok(X:S)) -> S(X:S) 0.05/0.12 ->->-> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->->Cycle: 0.05/0.12 ->->-> Pairs: 0.05/0.12 FROM(mark(X:S)) -> FROM(X:S) 0.05/0.12 FROM(ok(X:S)) -> FROM(X:S) 0.05/0.12 ->->-> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->->Cycle: 0.05/0.12 ->->-> Pairs: 0.05/0.12 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.05/0.12 FIRST(ok(X1:S),ok(X2:S)) -> FIRST(X1:S,X2:S) 0.05/0.12 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.05/0.12 ->->-> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->->Cycle: 0.05/0.12 ->->-> Pairs: 0.05/0.12 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.05/0.12 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 0.05/0.12 ->->-> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->->Cycle: 0.05/0.12 ->->-> Pairs: 0.05/0.12 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 0.05/0.12 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 0.05/0.12 PROPER(first(X1:S,X2:S)) -> PROPER(X1:S) 0.05/0.12 PROPER(first(X1:S,X2:S)) -> PROPER(X2:S) 0.05/0.12 PROPER(from(X:S)) -> PROPER(X:S) 0.05/0.12 PROPER(s(X:S)) -> PROPER(X:S) 0.05/0.12 ->->-> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->->Cycle: 0.05/0.12 ->->-> Pairs: 0.05/0.12 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 0.05/0.12 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X1:S) 0.05/0.12 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X2:S) 0.05/0.12 ACTIVE(from(X:S)) -> ACTIVE(X:S) 0.05/0.12 ACTIVE(s(X:S)) -> ACTIVE(X:S) 0.05/0.12 ->->-> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->->Cycle: 0.05/0.12 ->->-> Pairs: 0.05/0.12 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.05/0.12 TOP(ok(X:S)) -> TOP(active(X:S)) 0.05/0.12 ->->-> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 0.05/0.12 0.05/0.12 The problem is decomposed in 7 subproblems. 0.05/0.12 0.05/0.12 Problem 1.1: 0.05/0.12 0.05/0.12 Subterm Processor: 0.05/0.12 -> Pairs: 0.05/0.12 S(mark(X:S)) -> S(X:S) 0.05/0.12 S(ok(X:S)) -> S(X:S) 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Projection: 0.05/0.12 pi(S) = 1 0.05/0.12 0.05/0.12 Problem 1.1: 0.05/0.12 0.05/0.12 SCC Processor: 0.05/0.12 -> Pairs: 0.05/0.12 Empty 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Strongly Connected Components: 0.05/0.12 There is no strongly connected component 0.05/0.12 0.05/0.12 The problem is finite. 0.05/0.12 0.05/0.12 Problem 1.2: 0.05/0.12 0.05/0.12 Subterm Processor: 0.05/0.12 -> Pairs: 0.05/0.12 FROM(mark(X:S)) -> FROM(X:S) 0.05/0.12 FROM(ok(X:S)) -> FROM(X:S) 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Projection: 0.05/0.12 pi(FROM) = 1 0.05/0.12 0.05/0.12 Problem 1.2: 0.05/0.12 0.05/0.12 SCC Processor: 0.05/0.12 -> Pairs: 0.05/0.12 Empty 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Strongly Connected Components: 0.05/0.12 There is no strongly connected component 0.05/0.12 0.05/0.12 The problem is finite. 0.05/0.12 0.05/0.12 Problem 1.3: 0.05/0.12 0.05/0.12 Subterm Processor: 0.05/0.12 -> Pairs: 0.05/0.12 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.05/0.12 FIRST(ok(X1:S),ok(X2:S)) -> FIRST(X1:S,X2:S) 0.05/0.12 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Projection: 0.05/0.12 pi(FIRST) = 1 0.05/0.12 0.05/0.12 Problem 1.3: 0.05/0.12 0.05/0.12 SCC Processor: 0.05/0.12 -> Pairs: 0.05/0.12 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Strongly Connected Components: 0.05/0.12 ->->Cycle: 0.05/0.12 ->->-> Pairs: 0.05/0.12 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.05/0.12 ->->-> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 0.05/0.12 Problem 1.3: 0.05/0.12 0.05/0.12 Subterm Processor: 0.05/0.12 -> Pairs: 0.05/0.12 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Projection: 0.05/0.12 pi(FIRST) = 2 0.05/0.12 0.05/0.12 Problem 1.3: 0.05/0.12 0.05/0.12 SCC Processor: 0.05/0.12 -> Pairs: 0.05/0.12 Empty 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Strongly Connected Components: 0.05/0.12 There is no strongly connected component 0.05/0.12 0.05/0.12 The problem is finite. 0.05/0.12 0.05/0.12 Problem 1.4: 0.05/0.12 0.05/0.12 Subterm Processor: 0.05/0.12 -> Pairs: 0.05/0.12 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 0.05/0.12 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Projection: 0.05/0.12 pi(CONS) = 1 0.05/0.12 0.05/0.12 Problem 1.4: 0.05/0.12 0.05/0.12 SCC Processor: 0.05/0.12 -> Pairs: 0.05/0.12 Empty 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Strongly Connected Components: 0.05/0.12 There is no strongly connected component 0.05/0.12 0.05/0.12 The problem is finite. 0.05/0.12 0.05/0.12 Problem 1.5: 0.05/0.12 0.05/0.12 Subterm Processor: 0.05/0.12 -> Pairs: 0.05/0.12 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 0.05/0.12 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 0.05/0.12 PROPER(first(X1:S,X2:S)) -> PROPER(X1:S) 0.05/0.12 PROPER(first(X1:S,X2:S)) -> PROPER(X2:S) 0.05/0.12 PROPER(from(X:S)) -> PROPER(X:S) 0.05/0.12 PROPER(s(X:S)) -> PROPER(X:S) 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Projection: 0.05/0.12 pi(PROPER) = 1 0.05/0.12 0.05/0.12 Problem 1.5: 0.05/0.12 0.05/0.12 SCC Processor: 0.05/0.12 -> Pairs: 0.05/0.12 Empty 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Strongly Connected Components: 0.05/0.12 There is no strongly connected component 0.05/0.12 0.05/0.12 The problem is finite. 0.05/0.12 0.05/0.12 Problem 1.6: 0.05/0.12 0.05/0.12 Subterm Processor: 0.05/0.12 -> Pairs: 0.05/0.12 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 0.05/0.12 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X1:S) 0.05/0.12 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X2:S) 0.05/0.12 ACTIVE(from(X:S)) -> ACTIVE(X:S) 0.05/0.12 ACTIVE(s(X:S)) -> ACTIVE(X:S) 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Projection: 0.05/0.12 pi(ACTIVE) = 1 0.05/0.12 0.05/0.12 Problem 1.6: 0.05/0.12 0.05/0.12 SCC Processor: 0.05/0.12 -> Pairs: 0.05/0.12 Empty 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Strongly Connected Components: 0.05/0.12 There is no strongly connected component 0.05/0.12 0.05/0.12 The problem is finite. 0.05/0.12 0.05/0.12 Problem 1.7: 0.05/0.12 0.05/0.12 Reduction Pairs Processor: 0.05/0.12 -> Pairs: 0.05/0.12 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.05/0.12 TOP(ok(X:S)) -> TOP(active(X:S)) 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 -> Usable rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 ->Interpretation type: 0.05/0.12 Linear 0.05/0.12 ->Coefficients: 0.05/0.12 Natural Numbers 0.05/0.12 ->Dimension: 0.05/0.12 1 0.05/0.12 ->Bound: 0.05/0.12 2 0.05/0.12 ->Interpretation: 0.05/0.12 0.05/0.12 [active](X) = X 0.05/0.12 [cons](X1,X2) = 2.X1 0.05/0.12 [first](X1,X2) = 2.X1 + 2.X2 + 2 0.05/0.12 [from](X) = 2.X + 2 0.05/0.12 [proper](X) = X 0.05/0.12 [s](X) = X + 2 0.05/0.12 [top](X) = 0 0.05/0.12 [0] = 2 0.05/0.12 [fSNonEmpty] = 0 0.05/0.12 [mark](X) = X + 2 0.05/0.12 [nil] = 2 0.05/0.12 [ok](X) = X 0.05/0.12 [ACTIVE](X) = 0 0.05/0.12 [CONS](X1,X2) = 0 0.05/0.12 [FIRST](X1,X2) = 0 0.05/0.12 [FROM](X) = 0 0.05/0.12 [PROPER](X) = 0 0.05/0.12 [S](X) = 0 0.05/0.12 [TOP](X) = X 0.05/0.12 0.05/0.12 Problem 1.7: 0.05/0.12 0.05/0.12 SCC Processor: 0.05/0.12 -> Pairs: 0.05/0.12 TOP(ok(X:S)) -> TOP(active(X:S)) 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Strongly Connected Components: 0.05/0.12 ->->Cycle: 0.05/0.12 ->->-> Pairs: 0.05/0.12 TOP(ok(X:S)) -> TOP(active(X:S)) 0.05/0.12 ->->-> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 0.05/0.12 Problem 1.7: 0.05/0.12 0.05/0.12 Reduction Pairs Processor: 0.05/0.12 -> Pairs: 0.05/0.12 TOP(ok(X:S)) -> TOP(active(X:S)) 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 -> Usable rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 ->Interpretation type: 0.05/0.12 Linear 0.05/0.12 ->Coefficients: 0.05/0.12 Natural Numbers 0.05/0.12 ->Dimension: 0.05/0.12 1 0.05/0.12 ->Bound: 0.05/0.12 2 0.05/0.12 ->Interpretation: 0.05/0.12 0.05/0.12 [active](X) = 2.X + 1 0.05/0.12 [cons](X1,X2) = 2.X1 + 2 0.05/0.12 [first](X1,X2) = X1 + X2 0.05/0.12 [from](X) = 2.X + 2 0.05/0.12 [proper](X) = 0 0.05/0.12 [s](X) = 2.X + 2 0.05/0.12 [top](X) = 0 0.05/0.12 [0] = 2 0.05/0.12 [fSNonEmpty] = 0 0.05/0.12 [mark](X) = X + 2 0.05/0.12 [nil] = 1 0.05/0.12 [ok](X) = 2.X + 2 0.05/0.12 [ACTIVE](X) = 0 0.05/0.12 [CONS](X1,X2) = 0 0.05/0.12 [FIRST](X1,X2) = 0 0.05/0.12 [FROM](X) = 0 0.05/0.12 [PROPER](X) = 0 0.05/0.12 [S](X) = 0 0.05/0.12 [TOP](X) = 2.X 0.05/0.12 0.05/0.12 Problem 1.7: 0.05/0.12 0.05/0.12 SCC Processor: 0.05/0.12 -> Pairs: 0.05/0.12 Empty 0.05/0.12 -> Rules: 0.05/0.12 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 0.05/0.12 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.05/0.12 active(first(0,X:S)) -> mark(nil) 0.05/0.12 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.05/0.12 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.05/0.12 active(from(X:S)) -> from(active(X:S)) 0.05/0.12 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.05/0.12 active(s(X:S)) -> s(active(X:S)) 0.05/0.12 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 0.05/0.12 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.05/0.12 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.05/0.12 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.05/0.12 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.05/0.12 from(mark(X:S)) -> mark(from(X:S)) 0.05/0.12 from(ok(X:S)) -> ok(from(X:S)) 0.05/0.12 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.05/0.12 proper(from(X:S)) -> from(proper(X:S)) 0.05/0.12 proper(s(X:S)) -> s(proper(X:S)) 0.05/0.12 proper(0) -> ok(0) 0.05/0.12 proper(nil) -> ok(nil) 0.05/0.12 s(mark(X:S)) -> mark(s(X:S)) 0.05/0.12 s(ok(X:S)) -> ok(s(X:S)) 0.05/0.12 top(mark(X:S)) -> top(proper(X:S)) 0.05/0.12 top(ok(X:S)) -> top(active(X:S)) 0.05/0.12 ->Strongly Connected Components: 0.05/0.12 There is no strongly connected component 0.05/0.12 0.05/0.12 The problem is finite. 0.05/0.12 EOF