82.55/83.00 YES 82.55/83.00 82.55/83.00 Problem 1: 82.55/83.00 82.55/83.00 (VAR v_NonEmpty:S X:S X1:S X2:S) 82.55/83.00 (RULES 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ) 82.55/83.00 (STRATEGY INNERMOST) 82.55/83.00 82.55/83.00 Problem 1: 82.55/83.00 82.55/83.00 Dependency Pairs Processor: 82.55/83.00 -> Pairs: 82.55/83.00 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 82.55/83.00 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) 82.55/83.00 ACTIVE(f(X:S)) -> ACTIVE(X:S) 82.55/83.00 ACTIVE(f(X:S)) -> F(active(X:S)) 82.55/83.00 ACTIVE(p(X:S)) -> ACTIVE(X:S) 82.55/83.00 ACTIVE(p(X:S)) -> P(active(X:S)) 82.55/83.00 ACTIVE(s(X:S)) -> ACTIVE(X:S) 82.55/83.00 ACTIVE(s(X:S)) -> S(active(X:S)) 82.55/83.00 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 82.55/83.00 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 82.55/83.00 F(mark(X:S)) -> F(X:S) 82.55/83.00 F(ok(X:S)) -> F(X:S) 82.55/83.00 P(mark(X:S)) -> P(X:S) 82.55/83.00 P(ok(X:S)) -> P(X:S) 82.55/83.00 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 82.55/83.00 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 82.55/83.00 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 82.55/83.00 PROPER(f(X:S)) -> F(proper(X:S)) 82.55/83.00 PROPER(f(X:S)) -> PROPER(X:S) 82.55/83.00 PROPER(p(X:S)) -> P(proper(X:S)) 82.55/83.00 PROPER(p(X:S)) -> PROPER(X:S) 82.55/83.00 PROPER(s(X:S)) -> PROPER(X:S) 82.55/83.00 PROPER(s(X:S)) -> S(proper(X:S)) 82.55/83.00 S(mark(X:S)) -> S(X:S) 82.55/83.00 S(ok(X:S)) -> S(X:S) 82.55/83.00 TOP(mark(X:S)) -> PROPER(X:S) 82.55/83.00 TOP(mark(X:S)) -> TOP(proper(X:S)) 82.55/83.00 TOP(ok(X:S)) -> ACTIVE(X:S) 82.55/83.00 TOP(ok(X:S)) -> TOP(active(X:S)) 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 82.55/83.00 Problem 1: 82.55/83.00 82.55/83.00 SCC Processor: 82.55/83.00 -> Pairs: 82.55/83.00 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 82.55/83.00 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) 82.55/83.00 ACTIVE(f(X:S)) -> ACTIVE(X:S) 82.55/83.00 ACTIVE(f(X:S)) -> F(active(X:S)) 82.55/83.00 ACTIVE(p(X:S)) -> ACTIVE(X:S) 82.55/83.00 ACTIVE(p(X:S)) -> P(active(X:S)) 82.55/83.00 ACTIVE(s(X:S)) -> ACTIVE(X:S) 82.55/83.00 ACTIVE(s(X:S)) -> S(active(X:S)) 82.55/83.00 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 82.55/83.00 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 82.55/83.00 F(mark(X:S)) -> F(X:S) 82.55/83.00 F(ok(X:S)) -> F(X:S) 82.55/83.00 P(mark(X:S)) -> P(X:S) 82.55/83.00 P(ok(X:S)) -> P(X:S) 82.55/83.00 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 82.55/83.00 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 82.55/83.00 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 82.55/83.00 PROPER(f(X:S)) -> F(proper(X:S)) 82.55/83.00 PROPER(f(X:S)) -> PROPER(X:S) 82.55/83.00 PROPER(p(X:S)) -> P(proper(X:S)) 82.55/83.00 PROPER(p(X:S)) -> PROPER(X:S) 82.55/83.00 PROPER(s(X:S)) -> PROPER(X:S) 82.55/83.00 PROPER(s(X:S)) -> S(proper(X:S)) 82.55/83.00 S(mark(X:S)) -> S(X:S) 82.55/83.00 S(ok(X:S)) -> S(X:S) 82.55/83.00 TOP(mark(X:S)) -> PROPER(X:S) 82.55/83.00 TOP(mark(X:S)) -> TOP(proper(X:S)) 82.55/83.00 TOP(ok(X:S)) -> ACTIVE(X:S) 82.55/83.00 TOP(ok(X:S)) -> TOP(active(X:S)) 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Strongly Connected Components: 82.55/83.00 ->->Cycle: 82.55/83.00 ->->-> Pairs: 82.55/83.00 S(mark(X:S)) -> S(X:S) 82.55/83.00 S(ok(X:S)) -> S(X:S) 82.55/83.00 ->->-> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->->Cycle: 82.55/83.00 ->->-> Pairs: 82.55/83.00 P(mark(X:S)) -> P(X:S) 82.55/83.00 P(ok(X:S)) -> P(X:S) 82.55/83.00 ->->-> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->->Cycle: 82.55/83.00 ->->-> Pairs: 82.55/83.00 F(mark(X:S)) -> F(X:S) 82.55/83.00 F(ok(X:S)) -> F(X:S) 82.55/83.00 ->->-> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->->Cycle: 82.55/83.00 ->->-> Pairs: 82.55/83.00 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 82.55/83.00 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 82.55/83.00 ->->-> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->->Cycle: 82.55/83.00 ->->-> Pairs: 82.55/83.00 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 82.55/83.00 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 82.55/83.00 PROPER(f(X:S)) -> PROPER(X:S) 82.55/83.00 PROPER(p(X:S)) -> PROPER(X:S) 82.55/83.00 PROPER(s(X:S)) -> PROPER(X:S) 82.55/83.00 ->->-> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->->Cycle: 82.55/83.00 ->->-> Pairs: 82.55/83.00 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 82.55/83.00 ACTIVE(f(X:S)) -> ACTIVE(X:S) 82.55/83.00 ACTIVE(p(X:S)) -> ACTIVE(X:S) 82.55/83.00 ACTIVE(s(X:S)) -> ACTIVE(X:S) 82.55/83.00 ->->-> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->->Cycle: 82.55/83.00 ->->-> Pairs: 82.55/83.00 TOP(mark(X:S)) -> TOP(proper(X:S)) 82.55/83.00 TOP(ok(X:S)) -> TOP(active(X:S)) 82.55/83.00 ->->-> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 82.55/83.00 82.55/83.00 The problem is decomposed in 7 subproblems. 82.55/83.00 82.55/83.00 Problem 1.1: 82.55/83.00 82.55/83.00 Subterm Processor: 82.55/83.00 -> Pairs: 82.55/83.00 S(mark(X:S)) -> S(X:S) 82.55/83.00 S(ok(X:S)) -> S(X:S) 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Projection: 82.55/83.00 pi(S) = 1 82.55/83.00 82.55/83.00 Problem 1.1: 82.55/83.00 82.55/83.00 SCC Processor: 82.55/83.00 -> Pairs: 82.55/83.00 Empty 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Strongly Connected Components: 82.55/83.00 There is no strongly connected component 82.55/83.00 82.55/83.00 The problem is finite. 82.55/83.00 82.55/83.00 Problem 1.2: 82.55/83.00 82.55/83.00 Subterm Processor: 82.55/83.00 -> Pairs: 82.55/83.00 P(mark(X:S)) -> P(X:S) 82.55/83.00 P(ok(X:S)) -> P(X:S) 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Projection: 82.55/83.00 pi(P) = 1 82.55/83.00 82.55/83.00 Problem 1.2: 82.55/83.00 82.55/83.00 SCC Processor: 82.55/83.00 -> Pairs: 82.55/83.00 Empty 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Strongly Connected Components: 82.55/83.00 There is no strongly connected component 82.55/83.00 82.55/83.00 The problem is finite. 82.55/83.00 82.55/83.00 Problem 1.3: 82.55/83.00 82.55/83.00 Subterm Processor: 82.55/83.00 -> Pairs: 82.55/83.00 F(mark(X:S)) -> F(X:S) 82.55/83.00 F(ok(X:S)) -> F(X:S) 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Projection: 82.55/83.00 pi(F) = 1 82.55/83.00 82.55/83.00 Problem 1.3: 82.55/83.00 82.55/83.00 SCC Processor: 82.55/83.00 -> Pairs: 82.55/83.00 Empty 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Strongly Connected Components: 82.55/83.00 There is no strongly connected component 82.55/83.00 82.55/83.00 The problem is finite. 82.55/83.00 82.55/83.00 Problem 1.4: 82.55/83.00 82.55/83.00 Subterm Processor: 82.55/83.00 -> Pairs: 82.55/83.00 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) 82.55/83.00 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Projection: 82.55/83.00 pi(CONS) = 1 82.55/83.00 82.55/83.00 Problem 1.4: 82.55/83.00 82.55/83.00 SCC Processor: 82.55/83.00 -> Pairs: 82.55/83.00 Empty 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Strongly Connected Components: 82.55/83.00 There is no strongly connected component 82.55/83.00 82.55/83.00 The problem is finite. 82.55/83.00 82.55/83.00 Problem 1.5: 82.55/83.00 82.55/83.00 Subterm Processor: 82.55/83.00 -> Pairs: 82.55/83.00 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 82.55/83.00 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 82.55/83.00 PROPER(f(X:S)) -> PROPER(X:S) 82.55/83.00 PROPER(p(X:S)) -> PROPER(X:S) 82.55/83.00 PROPER(s(X:S)) -> PROPER(X:S) 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Projection: 82.55/83.00 pi(PROPER) = 1 82.55/83.00 82.55/83.00 Problem 1.5: 82.55/83.00 82.55/83.00 SCC Processor: 82.55/83.00 -> Pairs: 82.55/83.00 Empty 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Strongly Connected Components: 82.55/83.00 There is no strongly connected component 82.55/83.00 82.55/83.00 The problem is finite. 82.55/83.00 82.55/83.00 Problem 1.6: 82.55/83.00 82.55/83.00 Subterm Processor: 82.55/83.00 -> Pairs: 82.55/83.00 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) 82.55/83.00 ACTIVE(f(X:S)) -> ACTIVE(X:S) 82.55/83.00 ACTIVE(p(X:S)) -> ACTIVE(X:S) 82.55/83.00 ACTIVE(s(X:S)) -> ACTIVE(X:S) 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Projection: 82.55/83.00 pi(ACTIVE) = 1 82.55/83.00 82.55/83.00 Problem 1.6: 82.55/83.00 82.55/83.00 SCC Processor: 82.55/83.00 -> Pairs: 82.55/83.00 Empty 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Strongly Connected Components: 82.55/83.00 There is no strongly connected component 82.55/83.00 82.55/83.00 The problem is finite. 82.55/83.00 82.55/83.00 Problem 1.7: 82.55/83.00 82.55/83.00 Reduction Pairs Processor: 82.55/83.00 -> Pairs: 82.55/83.00 TOP(mark(X:S)) -> TOP(proper(X:S)) 82.55/83.00 TOP(ok(X:S)) -> TOP(active(X:S)) 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 -> Usable rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 ->Interpretation type: 82.55/83.00 Linear 82.55/83.00 ->Coefficients: 82.55/83.00 Natural Numbers 82.55/83.00 ->Dimension: 82.55/83.00 2 82.55/83.00 ->Bound: 82.55/83.00 1 82.55/83.00 ->Interpretation: 82.55/83.00 82.55/83.00 [active](X) = [1 0;0 1].X 82.55/83.00 [cons](X1,X2) = [0 0;0 1].X1 82.55/83.00 [f](X) = [0 0;1 1].X + [0;1] 82.55/83.00 [p](X) = [0 0;0 1].X 82.55/83.00 [proper](X) = [1 0;0 1].X 82.55/83.00 [s](X) = [0 0;0 1].X + [1;1] 82.55/83.00 [top](X) = 0 82.55/83.00 [0] = 0 82.55/83.00 [fSNonEmpty] = 0 82.55/83.00 [mark](X) = [1 0;0 1].X + [0;1] 82.55/83.00 [ok](X) = [1 0;0 1].X 82.55/83.00 [ACTIVE](X) = 0 82.55/83.00 [CONS](X1,X2) = 0 82.55/83.00 [F](X) = 0 82.55/83.00 [P](X) = 0 82.55/83.00 [PROPER](X) = 0 82.55/83.00 [S](X) = 0 82.55/83.00 [TOP](X) = [0 1;0 1].X 82.55/83.00 82.55/83.00 Problem 1.7: 82.55/83.00 82.55/83.00 SCC Processor: 82.55/83.00 -> Pairs: 82.55/83.00 TOP(ok(X:S)) -> TOP(active(X:S)) 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Strongly Connected Components: 82.55/83.00 ->->Cycle: 82.55/83.00 ->->-> Pairs: 82.55/83.00 TOP(ok(X:S)) -> TOP(active(X:S)) 82.55/83.00 ->->-> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 82.55/83.00 Problem 1.7: 82.55/83.00 82.55/83.00 Reduction Pairs Processor: 82.55/83.00 -> Pairs: 82.55/83.00 TOP(ok(X:S)) -> TOP(active(X:S)) 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 -> Usable rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 ->Interpretation type: 82.55/83.00 Linear 82.55/83.00 ->Coefficients: 82.55/83.00 Natural Numbers 82.55/83.00 ->Dimension: 82.55/83.00 1 82.55/83.00 ->Bound: 82.55/83.00 2 82.55/83.00 ->Interpretation: 82.55/83.00 82.55/83.00 [active](X) = 2.X + 1 82.55/83.00 [cons](X1,X2) = 2.X1 + 2.X2 + 1 82.55/83.00 [f](X) = 2.X + 2 82.55/83.00 [p](X) = 2.X + 2 82.55/83.00 [proper](X) = 0 82.55/83.00 [s](X) = 2.X + 2 82.55/83.00 [top](X) = 0 82.55/83.00 [0] = 1 82.55/83.00 [fSNonEmpty] = 0 82.55/83.00 [mark](X) = 2 82.55/83.00 [ok](X) = 2.X + 2 82.55/83.00 [ACTIVE](X) = 0 82.55/83.00 [CONS](X1,X2) = 0 82.55/83.00 [F](X) = 0 82.55/83.00 [P](X) = 0 82.55/83.00 [PROPER](X) = 0 82.55/83.00 [S](X) = 0 82.55/83.00 [TOP](X) = X 82.55/83.00 82.55/83.00 Problem 1.7: 82.55/83.00 82.55/83.00 SCC Processor: 82.55/83.00 -> Pairs: 82.55/83.00 Empty 82.55/83.00 -> Rules: 82.55/83.00 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) 82.55/83.00 active(f(s(0))) -> mark(f(p(s(0)))) 82.55/83.00 active(f(0)) -> mark(cons(0,f(s(0)))) 82.55/83.00 active(f(X:S)) -> f(active(X:S)) 82.55/83.00 active(p(s(0))) -> mark(0) 82.55/83.00 active(p(X:S)) -> p(active(X:S)) 82.55/83.00 active(s(X:S)) -> s(active(X:S)) 82.55/83.00 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) 82.55/83.00 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 82.55/83.00 f(mark(X:S)) -> mark(f(X:S)) 82.55/83.00 f(ok(X:S)) -> ok(f(X:S)) 82.55/83.00 p(mark(X:S)) -> mark(p(X:S)) 82.55/83.00 p(ok(X:S)) -> ok(p(X:S)) 82.55/83.00 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 82.55/83.00 proper(f(X:S)) -> f(proper(X:S)) 82.55/83.00 proper(p(X:S)) -> p(proper(X:S)) 82.55/83.00 proper(s(X:S)) -> s(proper(X:S)) 82.55/83.00 proper(0) -> ok(0) 82.55/83.00 s(mark(X:S)) -> mark(s(X:S)) 82.55/83.00 s(ok(X:S)) -> ok(s(X:S)) 82.55/83.00 top(mark(X:S)) -> top(proper(X:S)) 82.55/83.00 top(ok(X:S)) -> top(active(X:S)) 82.55/83.00 ->Strongly Connected Components: 82.55/83.00 There is no strongly connected component 82.55/83.00 82.55/83.00 The problem is finite. 82.55/113.29 EOF