65.45/65.81 YES 65.45/65.81 65.45/65.81 Problem 1: 65.45/65.81 65.45/65.81 (VAR v_NonEmpty:S X:S X1:S X2:S) 65.45/65.81 (RULES 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ) 65.45/65.81 (STRATEGY INNERMOST) 65.45/65.81 65.45/65.81 Problem 1: 65.45/65.81 65.45/65.81 Dependency Pairs Processor: 65.45/65.81 -> Pairs: 65.45/65.81 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 65.45/65.81 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) 65.45/65.81 ACTIVE(f(X:S)) -> ACTIVE(X:S) 65.45/65.81 ACTIVE(f(X:S)) -> F(active(X:S)) 65.45/65.81 ACTIVE(p(X:S)) -> ACTIVE(X:S) 65.45/65.81 ACTIVE(p(X:S)) -> P(active(X:S)) 65.45/65.81 ACTIVE(s(X:S)) -> ACTIVE(X:S) 65.45/65.81 ACTIVE(s(X:S)) -> S(active(X:S)) 65.45/65.81 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 65.45/65.81 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 65.45/65.81 F(mark(X:S)) -> F(X:S) 65.45/65.81 F(ok(X:S)) -> F(X:S) 65.45/65.81 P(mark(X:S)) -> P(X:S) 65.45/65.81 P(ok(X:S)) -> P(X:S) 65.45/65.81 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 65.45/65.81 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 65.45/65.81 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 65.45/65.81 PROPER(f(X:S)) -> F(proper(X:S)) 65.45/65.81 PROPER(f(X:S)) -> PROPER(X:S) 65.45/65.81 PROPER(p(X:S)) -> P(proper(X:S)) 65.45/65.81 PROPER(p(X:S)) -> PROPER(X:S) 65.45/65.81 PROPER(s(X:S)) -> PROPER(X:S) 65.45/65.81 PROPER(s(X:S)) -> S(proper(X:S)) 65.45/65.81 S(mark(X:S)) -> S(X:S) 65.45/65.81 S(ok(X:S)) -> S(X:S) 65.45/65.81 TOP(mark(X:S)) -> PROPER(X:S) 65.45/65.81 TOP(mark(X:S)) -> TOP(proper(X:S)) 65.45/65.81 TOP(ok(X:S)) -> ACTIVE(X:S) 65.45/65.81 TOP(ok(X:S)) -> TOP(active(X:S)) 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 65.45/65.81 Problem 1: 65.45/65.81 65.45/65.81 SCC Processor: 65.45/65.81 -> Pairs: 65.45/65.81 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 65.45/65.81 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) 65.45/65.81 ACTIVE(f(X:S)) -> ACTIVE(X:S) 65.45/65.81 ACTIVE(f(X:S)) -> F(active(X:S)) 65.45/65.81 ACTIVE(p(X:S)) -> ACTIVE(X:S) 65.45/65.81 ACTIVE(p(X:S)) -> P(active(X:S)) 65.45/65.81 ACTIVE(s(X:S)) -> ACTIVE(X:S) 65.45/65.81 ACTIVE(s(X:S)) -> S(active(X:S)) 65.45/65.81 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 65.45/65.81 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 65.45/65.81 F(mark(X:S)) -> F(X:S) 65.45/65.81 F(ok(X:S)) -> F(X:S) 65.45/65.81 P(mark(X:S)) -> P(X:S) 65.45/65.81 P(ok(X:S)) -> P(X:S) 65.45/65.81 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 65.45/65.81 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 65.45/65.81 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 65.45/65.81 PROPER(f(X:S)) -> F(proper(X:S)) 65.45/65.81 PROPER(f(X:S)) -> PROPER(X:S) 65.45/65.81 PROPER(p(X:S)) -> P(proper(X:S)) 65.45/65.81 PROPER(p(X:S)) -> PROPER(X:S) 65.45/65.81 PROPER(s(X:S)) -> PROPER(X:S) 65.45/65.81 PROPER(s(X:S)) -> S(proper(X:S)) 65.45/65.81 S(mark(X:S)) -> S(X:S) 65.45/65.81 S(ok(X:S)) -> S(X:S) 65.45/65.81 TOP(mark(X:S)) -> PROPER(X:S) 65.45/65.81 TOP(mark(X:S)) -> TOP(proper(X:S)) 65.45/65.81 TOP(ok(X:S)) -> ACTIVE(X:S) 65.45/65.81 TOP(ok(X:S)) -> TOP(active(X:S)) 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Strongly Connected Components: 65.45/65.81 ->->Cycle: 65.45/65.81 ->->-> Pairs: 65.45/65.81 S(mark(X:S)) -> S(X:S) 65.45/65.81 S(ok(X:S)) -> S(X:S) 65.45/65.81 ->->-> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->->Cycle: 65.45/65.81 ->->-> Pairs: 65.45/65.81 P(mark(X:S)) -> P(X:S) 65.45/65.81 P(ok(X:S)) -> P(X:S) 65.45/65.81 ->->-> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->->Cycle: 65.45/65.81 ->->-> Pairs: 65.45/65.81 F(mark(X:S)) -> F(X:S) 65.45/65.81 F(ok(X:S)) -> F(X:S) 65.45/65.81 ->->-> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->->Cycle: 65.45/65.81 ->->-> Pairs: 65.45/65.81 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 65.45/65.81 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 65.45/65.81 ->->-> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->->Cycle: 65.45/65.81 ->->-> Pairs: 65.45/65.81 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 65.45/65.81 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 65.45/65.81 PROPER(f(X:S)) -> PROPER(X:S) 65.45/65.81 PROPER(p(X:S)) -> PROPER(X:S) 65.45/65.81 PROPER(s(X:S)) -> PROPER(X:S) 65.45/65.81 ->->-> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->->Cycle: 65.45/65.81 ->->-> Pairs: 65.45/65.81 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 65.45/65.81 ACTIVE(f(X:S)) -> ACTIVE(X:S) 65.45/65.81 ACTIVE(p(X:S)) -> ACTIVE(X:S) 65.45/65.81 ACTIVE(s(X:S)) -> ACTIVE(X:S) 65.45/65.81 ->->-> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->->Cycle: 65.45/65.81 ->->-> Pairs: 65.45/65.81 TOP(mark(X:S)) -> TOP(proper(X:S)) 65.45/65.81 TOP(ok(X:S)) -> TOP(active(X:S)) 65.45/65.81 ->->-> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 65.45/65.81 65.45/65.81 The problem is decomposed in 7 subproblems. 65.45/65.81 65.45/65.81 Problem 1.1: 65.45/65.81 65.45/65.81 Subterm Processor: 65.45/65.81 -> Pairs: 65.45/65.81 S(mark(X:S)) -> S(X:S) 65.45/65.81 S(ok(X:S)) -> S(X:S) 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Projection: 65.45/65.81 pi(S) = 1 65.45/65.81 65.45/65.81 Problem 1.1: 65.45/65.81 65.45/65.81 SCC Processor: 65.45/65.81 -> Pairs: 65.45/65.81 Empty 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Strongly Connected Components: 65.45/65.81 There is no strongly connected component 65.45/65.81 65.45/65.81 The problem is finite. 65.45/65.81 65.45/65.81 Problem 1.2: 65.45/65.81 65.45/65.81 Subterm Processor: 65.45/65.81 -> Pairs: 65.45/65.81 P(mark(X:S)) -> P(X:S) 65.45/65.81 P(ok(X:S)) -> P(X:S) 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Projection: 65.45/65.81 pi(P) = 1 65.45/65.81 65.45/65.81 Problem 1.2: 65.45/65.81 65.45/65.81 SCC Processor: 65.45/65.81 -> Pairs: 65.45/65.81 Empty 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Strongly Connected Components: 65.45/65.81 There is no strongly connected component 65.45/65.81 65.45/65.81 The problem is finite. 65.45/65.81 65.45/65.81 Problem 1.3: 65.45/65.81 65.45/65.81 Subterm Processor: 65.45/65.81 -> Pairs: 65.45/65.81 F(mark(X:S)) -> F(X:S) 65.45/65.81 F(ok(X:S)) -> F(X:S) 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Projection: 65.45/65.81 pi(F) = 1 65.45/65.81 65.45/65.81 Problem 1.3: 65.45/65.81 65.45/65.81 SCC Processor: 65.45/65.81 -> Pairs: 65.45/65.81 Empty 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Strongly Connected Components: 65.45/65.81 There is no strongly connected component 65.45/65.81 65.45/65.81 The problem is finite. 65.45/65.81 65.45/65.81 Problem 1.4: 65.45/65.81 65.45/65.81 Subterm Processor: 65.45/65.81 -> Pairs: 65.45/65.81 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 65.45/65.81 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Projection: 65.45/65.81 pi(CONS) = 1 65.45/65.81 65.45/65.81 Problem 1.4: 65.45/65.81 65.45/65.81 SCC Processor: 65.45/65.81 -> Pairs: 65.45/65.81 Empty 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Strongly Connected Components: 65.45/65.81 There is no strongly connected component 65.45/65.81 65.45/65.81 The problem is finite. 65.45/65.81 65.45/65.81 Problem 1.5: 65.45/65.81 65.45/65.81 Subterm Processor: 65.45/65.81 -> Pairs: 65.45/65.81 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 65.45/65.81 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 65.45/65.81 PROPER(f(X:S)) -> PROPER(X:S) 65.45/65.81 PROPER(p(X:S)) -> PROPER(X:S) 65.45/65.81 PROPER(s(X:S)) -> PROPER(X:S) 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Projection: 65.45/65.81 pi(PROPER) = 1 65.45/65.81 65.45/65.81 Problem 1.5: 65.45/65.81 65.45/65.81 SCC Processor: 65.45/65.81 -> Pairs: 65.45/65.81 Empty 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Strongly Connected Components: 65.45/65.81 There is no strongly connected component 65.45/65.81 65.45/65.81 The problem is finite. 65.45/65.81 65.45/65.81 Problem 1.6: 65.45/65.81 65.45/65.81 Subterm Processor: 65.45/65.81 -> Pairs: 65.45/65.81 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 65.45/65.81 ACTIVE(f(X:S)) -> ACTIVE(X:S) 65.45/65.81 ACTIVE(p(X:S)) -> ACTIVE(X:S) 65.45/65.81 ACTIVE(s(X:S)) -> ACTIVE(X:S) 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Projection: 65.45/65.81 pi(ACTIVE) = 1 65.45/65.81 65.45/65.81 Problem 1.6: 65.45/65.81 65.45/65.81 SCC Processor: 65.45/65.81 -> Pairs: 65.45/65.81 Empty 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Strongly Connected Components: 65.45/65.81 There is no strongly connected component 65.45/65.81 65.45/65.81 The problem is finite. 65.45/65.81 65.45/65.81 Problem 1.7: 65.45/65.81 65.45/65.81 Reduction Pairs Processor: 65.45/65.81 -> Pairs: 65.45/65.81 TOP(mark(X:S)) -> TOP(proper(X:S)) 65.45/65.81 TOP(ok(X:S)) -> TOP(active(X:S)) 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 -> Usable rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 ->Interpretation type: 65.45/65.81 Linear 65.45/65.81 ->Coefficients: 65.45/65.81 Natural Numbers 65.45/65.81 ->Dimension: 65.45/65.81 2 65.45/65.81 ->Bound: 65.45/65.81 1 65.45/65.81 ->Interpretation: 65.45/65.81 65.45/65.81 [active](X) = [1 0;0 1].X 65.45/65.81 [cons](X1,X2) = [1 0;1 0].X1 65.45/65.81 [f](X) = [1 0;1 0].X + [1;1] 65.45/65.81 [p](X) = [0 1;1 0].X + [1;0] 65.45/65.81 [proper](X) = [1 0;0 1].X 65.45/65.81 [s](X) = [0 1;1 0].X + [1;0] 65.45/65.81 [top](X) = 0 65.45/65.81 [0] = [0;1] 65.45/65.81 [fSNonEmpty] = 0 65.45/65.81 [mark](X) = [1 0;0 1].X + [1;1] 65.45/65.81 [ok](X) = [1 0;0 1].X 65.45/65.81 [ACTIVE](X) = 0 65.45/65.81 [CONS](X1,X2) = 0 65.45/65.81 [F](X) = 0 65.45/65.81 [P](X) = 0 65.45/65.81 [PROPER](X) = 0 65.45/65.81 [S](X) = 0 65.45/65.81 [TOP](X) = [0 1;1 1].X 65.45/65.81 65.45/65.81 Problem 1.7: 65.45/65.81 65.45/65.81 SCC Processor: 65.45/65.81 -> Pairs: 65.45/65.81 TOP(ok(X:S)) -> TOP(active(X:S)) 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Strongly Connected Components: 65.45/65.81 ->->Cycle: 65.45/65.81 ->->-> Pairs: 65.45/65.81 TOP(ok(X:S)) -> TOP(active(X:S)) 65.45/65.81 ->->-> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 65.45/65.81 Problem 1.7: 65.45/65.81 65.45/65.81 Reduction Pairs Processor: 65.45/65.81 -> Pairs: 65.45/65.81 TOP(ok(X:S)) -> TOP(active(X:S)) 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 -> Usable rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 ->Interpretation type: 65.45/65.81 Linear 65.45/65.81 ->Coefficients: 65.45/65.81 Natural Numbers 65.45/65.81 ->Dimension: 65.45/65.81 1 65.45/65.81 ->Bound: 65.45/65.81 2 65.45/65.81 ->Interpretation: 65.45/65.81 65.45/65.81 [active](X) = 2.X + 1 65.45/65.81 [cons](X1,X2) = 2.X1 + 1 65.45/65.81 [f](X) = 2.X + 1 65.45/65.81 [p](X) = X 65.45/65.81 [proper](X) = 0 65.45/65.81 [s](X) = 2.X + 1 65.45/65.81 [top](X) = 0 65.45/65.81 [0] = 1 65.45/65.81 [fSNonEmpty] = 0 65.45/65.81 [mark](X) = 2.X + 1 65.45/65.81 [ok](X) = 2.X + 2 65.45/65.81 [ACTIVE](X) = 0 65.45/65.81 [CONS](X1,X2) = 0 65.45/65.81 [F](X) = 0 65.45/65.81 [P](X) = 0 65.45/65.81 [PROPER](X) = 0 65.45/65.81 [S](X) = 0 65.45/65.81 [TOP](X) = 2.X 65.45/65.81 65.45/65.81 Problem 1.7: 65.45/65.81 65.45/65.81 SCC Processor: 65.45/65.81 -> Pairs: 65.45/65.81 Empty 65.45/65.81 -> Rules: 65.45/65.81 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 65.45/65.81 active(f(s(0))) -> mark(f(p(s(0)))) 65.45/65.81 active(f(0)) -> mark(cons(0,f(s(0)))) 65.45/65.81 active(f(X:S)) -> f(active(X:S)) 65.45/65.81 active(p(s(X:S))) -> mark(X:S) 65.45/65.81 active(p(X:S)) -> p(active(X:S)) 65.45/65.81 active(s(X:S)) -> s(active(X:S)) 65.45/65.81 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 65.45/65.81 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 65.45/65.81 f(mark(X:S)) -> mark(f(X:S)) 65.45/65.81 f(ok(X:S)) -> ok(f(X:S)) 65.45/65.81 p(mark(X:S)) -> mark(p(X:S)) 65.45/65.81 p(ok(X:S)) -> ok(p(X:S)) 65.45/65.81 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 65.45/65.81 proper(f(X:S)) -> f(proper(X:S)) 65.45/65.81 proper(p(X:S)) -> p(proper(X:S)) 65.45/65.81 proper(s(X:S)) -> s(proper(X:S)) 65.45/65.81 proper(0) -> ok(0) 65.45/65.81 s(mark(X:S)) -> mark(s(X:S)) 65.45/65.81 s(ok(X:S)) -> ok(s(X:S)) 65.45/65.81 top(mark(X:S)) -> top(proper(X:S)) 65.45/65.81 top(ok(X:S)) -> top(active(X:S)) 65.45/65.81 ->Strongly Connected Components: 65.45/65.81 There is no strongly connected component 65.45/65.81 65.45/65.81 The problem is finite. 65.45/95.60 EOF