0.00/0.49 YES 0.00/0.49 0.00/0.49 Problem 1: 0.00/0.49 0.00/0.49 (VAR v_NonEmpty:S X:S X1:S X2:S X3:S Y:S Z:S) 0.00/0.49 (RULES 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ) 0.00/0.49 (STRATEGY INNERMOST) 0.00/0.49 0.00/0.49 Problem 1: 0.00/0.49 0.00/0.49 Dependency Pairs Processor: 0.00/0.49 -> Pairs: 0.00/0.49 ACTIVE(add(s(X:S),Y:S)) -> ADD(X:S,Y:S) 0.00/0.49 ACTIVE(add(s(X:S),Y:S)) -> S(add(X:S,Y:S)) 0.00/0.49 ACTIVE(add(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(add(X1:S,X2:S)) -> ADD(active(X1:S),X2:S) 0.00/0.49 ACTIVE(and(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(and(X1:S,X2:S)) -> AND(active(X1:S),X2:S) 0.00/0.49 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> CONS(Y:S,first(X:S,Z:S)) 0.00/0.49 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> FIRST(X:S,Z:S) 0.00/0.49 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.49 ACTIVE(first(X1:S,X2:S)) -> FIRST(active(X1:S),X2:S) 0.00/0.49 ACTIVE(first(X1:S,X2:S)) -> FIRST(X1:S,active(X2:S)) 0.00/0.49 ACTIVE(from(X:S)) -> CONS(X:S,from(s(X:S))) 0.00/0.49 ACTIVE(from(X:S)) -> FROM(s(X:S)) 0.00/0.49 ACTIVE(from(X:S)) -> S(X:S) 0.00/0.49 ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(if(X1:S,X2:S,X3:S)) -> IF(active(X1:S),X2:S,X3:S) 0.00/0.49 ADD(mark(X1:S),X2:S) -> ADD(X1:S,X2:S) 0.00/0.49 ADD(ok(X1:S),ok(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.49 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) 0.00/0.49 AND(ok(X1:S),ok(X2:S)) -> AND(X1:S,X2:S) 0.00/0.49 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.49 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.49 FIRST(ok(X1:S),ok(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.49 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.49 FROM(ok(X:S)) -> FROM(X:S) 0.00/0.49 IF(mark(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) 0.00/0.49 IF(ok(X1:S),ok(X2:S),ok(X3:S)) -> IF(X1:S,X2:S,X3:S) 0.00/0.49 PROPER(add(X1:S,X2:S)) -> ADD(proper(X1:S),proper(X2:S)) 0.00/0.49 PROPER(add(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(add(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(and(X1:S,X2:S)) -> AND(proper(X1:S),proper(X2:S)) 0.00/0.49 PROPER(and(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(and(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 0.00/0.49 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(first(X1:S,X2:S)) -> FIRST(proper(X1:S),proper(X2:S)) 0.00/0.49 PROPER(first(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(first(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(from(X:S)) -> FROM(proper(X:S)) 0.00/0.49 PROPER(from(X:S)) -> PROPER(X:S) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> IF(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X3:S) 0.00/0.49 PROPER(s(X:S)) -> PROPER(X:S) 0.00/0.49 PROPER(s(X:S)) -> S(proper(X:S)) 0.00/0.49 S(ok(X:S)) -> S(X:S) 0.00/0.49 TOP(mark(X:S)) -> PROPER(X:S) 0.00/0.49 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.49 TOP(ok(X:S)) -> ACTIVE(X:S) 0.00/0.49 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 0.00/0.49 Problem 1: 0.00/0.49 0.00/0.49 SCC Processor: 0.00/0.49 -> Pairs: 0.00/0.49 ACTIVE(add(s(X:S),Y:S)) -> ADD(X:S,Y:S) 0.00/0.49 ACTIVE(add(s(X:S),Y:S)) -> S(add(X:S,Y:S)) 0.00/0.49 ACTIVE(add(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(add(X1:S,X2:S)) -> ADD(active(X1:S),X2:S) 0.00/0.49 ACTIVE(and(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(and(X1:S,X2:S)) -> AND(active(X1:S),X2:S) 0.00/0.49 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> CONS(Y:S,first(X:S,Z:S)) 0.00/0.49 ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> FIRST(X:S,Z:S) 0.00/0.49 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.49 ACTIVE(first(X1:S,X2:S)) -> FIRST(active(X1:S),X2:S) 0.00/0.49 ACTIVE(first(X1:S,X2:S)) -> FIRST(X1:S,active(X2:S)) 0.00/0.49 ACTIVE(from(X:S)) -> CONS(X:S,from(s(X:S))) 0.00/0.49 ACTIVE(from(X:S)) -> FROM(s(X:S)) 0.00/0.49 ACTIVE(from(X:S)) -> S(X:S) 0.00/0.49 ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(if(X1:S,X2:S,X3:S)) -> IF(active(X1:S),X2:S,X3:S) 0.00/0.49 ADD(mark(X1:S),X2:S) -> ADD(X1:S,X2:S) 0.00/0.49 ADD(ok(X1:S),ok(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.49 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) 0.00/0.49 AND(ok(X1:S),ok(X2:S)) -> AND(X1:S,X2:S) 0.00/0.49 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.49 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.49 FIRST(ok(X1:S),ok(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.49 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.49 FROM(ok(X:S)) -> FROM(X:S) 0.00/0.49 IF(mark(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) 0.00/0.49 IF(ok(X1:S),ok(X2:S),ok(X3:S)) -> IF(X1:S,X2:S,X3:S) 0.00/0.49 PROPER(add(X1:S,X2:S)) -> ADD(proper(X1:S),proper(X2:S)) 0.00/0.49 PROPER(add(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(add(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(and(X1:S,X2:S)) -> AND(proper(X1:S),proper(X2:S)) 0.00/0.49 PROPER(and(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(and(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) 0.00/0.49 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(first(X1:S,X2:S)) -> FIRST(proper(X1:S),proper(X2:S)) 0.00/0.49 PROPER(first(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(first(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(from(X:S)) -> FROM(proper(X:S)) 0.00/0.49 PROPER(from(X:S)) -> PROPER(X:S) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> IF(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X3:S) 0.00/0.49 PROPER(s(X:S)) -> PROPER(X:S) 0.00/0.49 PROPER(s(X:S)) -> S(proper(X:S)) 0.00/0.49 S(ok(X:S)) -> S(X:S) 0.00/0.49 TOP(mark(X:S)) -> PROPER(X:S) 0.00/0.49 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.49 TOP(ok(X:S)) -> ACTIVE(X:S) 0.00/0.49 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Strongly Connected Components: 0.00/0.49 ->->Cycle: 0.00/0.49 ->->-> Pairs: 0.00/0.49 S(ok(X:S)) -> S(X:S) 0.00/0.49 ->->-> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->->Cycle: 0.00/0.49 ->->-> Pairs: 0.00/0.49 IF(mark(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) 0.00/0.49 IF(ok(X1:S),ok(X2:S),ok(X3:S)) -> IF(X1:S,X2:S,X3:S) 0.00/0.49 ->->-> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->->Cycle: 0.00/0.49 ->->-> Pairs: 0.00/0.49 FROM(ok(X:S)) -> FROM(X:S) 0.00/0.49 ->->-> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->->Cycle: 0.00/0.49 ->->-> Pairs: 0.00/0.49 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.49 FIRST(ok(X1:S),ok(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.49 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.49 ->->-> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->->Cycle: 0.00/0.49 ->->-> Pairs: 0.00/0.49 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.49 ->->-> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->->Cycle: 0.00/0.49 ->->-> Pairs: 0.00/0.49 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) 0.00/0.49 AND(ok(X1:S),ok(X2:S)) -> AND(X1:S,X2:S) 0.00/0.49 ->->-> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->->Cycle: 0.00/0.49 ->->-> Pairs: 0.00/0.49 ADD(mark(X1:S),X2:S) -> ADD(X1:S,X2:S) 0.00/0.49 ADD(ok(X1:S),ok(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.49 ->->-> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->->Cycle: 0.00/0.49 ->->-> Pairs: 0.00/0.49 PROPER(add(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(add(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(and(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(and(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(first(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(first(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(from(X:S)) -> PROPER(X:S) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X3:S) 0.00/0.49 PROPER(s(X:S)) -> PROPER(X:S) 0.00/0.49 ->->-> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->->Cycle: 0.00/0.49 ->->-> Pairs: 0.00/0.49 ACTIVE(add(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(and(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.49 ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) 0.00/0.49 ->->-> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->->Cycle: 0.00/0.49 ->->-> Pairs: 0.00/0.49 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.49 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.49 ->->-> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 0.00/0.49 0.00/0.49 The problem is decomposed in 10 subproblems. 0.00/0.49 0.00/0.49 Problem 1.1: 0.00/0.49 0.00/0.49 Subterm Processor: 0.00/0.49 -> Pairs: 0.00/0.49 S(ok(X:S)) -> S(X:S) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Projection: 0.00/0.49 pi(S) = 1 0.00/0.49 0.00/0.49 Problem 1.1: 0.00/0.49 0.00/0.49 SCC Processor: 0.00/0.49 -> Pairs: 0.00/0.49 Empty 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Strongly Connected Components: 0.00/0.49 There is no strongly connected component 0.00/0.49 0.00/0.49 The problem is finite. 0.00/0.49 0.00/0.49 Problem 1.2: 0.00/0.49 0.00/0.49 Subterm Processor: 0.00/0.49 -> Pairs: 0.00/0.49 IF(mark(X1:S),X2:S,X3:S) -> IF(X1:S,X2:S,X3:S) 0.00/0.49 IF(ok(X1:S),ok(X2:S),ok(X3:S)) -> IF(X1:S,X2:S,X3:S) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Projection: 0.00/0.49 pi(IF) = 1 0.00/0.49 0.00/0.49 Problem 1.2: 0.00/0.49 0.00/0.49 SCC Processor: 0.00/0.49 -> Pairs: 0.00/0.49 Empty 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Strongly Connected Components: 0.00/0.49 There is no strongly connected component 0.00/0.49 0.00/0.49 The problem is finite. 0.00/0.49 0.00/0.49 Problem 1.3: 0.00/0.49 0.00/0.49 Subterm Processor: 0.00/0.49 -> Pairs: 0.00/0.49 FROM(ok(X:S)) -> FROM(X:S) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Projection: 0.00/0.49 pi(FROM) = 1 0.00/0.49 0.00/0.49 Problem 1.3: 0.00/0.49 0.00/0.49 SCC Processor: 0.00/0.49 -> Pairs: 0.00/0.49 Empty 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Strongly Connected Components: 0.00/0.49 There is no strongly connected component 0.00/0.49 0.00/0.49 The problem is finite. 0.00/0.49 0.00/0.49 Problem 1.4: 0.00/0.49 0.00/0.49 Subterm Processor: 0.00/0.49 -> Pairs: 0.00/0.49 FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) 0.00/0.49 FIRST(ok(X1:S),ok(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.49 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Projection: 0.00/0.49 pi(FIRST) = 1 0.00/0.49 0.00/0.49 Problem 1.4: 0.00/0.49 0.00/0.49 SCC Processor: 0.00/0.49 -> Pairs: 0.00/0.49 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Strongly Connected Components: 0.00/0.49 ->->Cycle: 0.00/0.49 ->->-> Pairs: 0.00/0.49 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.49 ->->-> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 0.00/0.49 Problem 1.4: 0.00/0.49 0.00/0.49 Subterm Processor: 0.00/0.49 -> Pairs: 0.00/0.49 FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Projection: 0.00/0.49 pi(FIRST) = 2 0.00/0.49 0.00/0.49 Problem 1.4: 0.00/0.49 0.00/0.49 SCC Processor: 0.00/0.49 -> Pairs: 0.00/0.49 Empty 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Strongly Connected Components: 0.00/0.49 There is no strongly connected component 0.00/0.49 0.00/0.49 The problem is finite. 0.00/0.49 0.00/0.49 Problem 1.5: 0.00/0.49 0.00/0.49 Subterm Processor: 0.00/0.49 -> Pairs: 0.00/0.49 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Projection: 0.00/0.49 pi(CONS) = 1 0.00/0.49 0.00/0.49 Problem 1.5: 0.00/0.49 0.00/0.49 SCC Processor: 0.00/0.49 -> Pairs: 0.00/0.49 Empty 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Strongly Connected Components: 0.00/0.49 There is no strongly connected component 0.00/0.49 0.00/0.49 The problem is finite. 0.00/0.49 0.00/0.49 Problem 1.6: 0.00/0.49 0.00/0.49 Subterm Processor: 0.00/0.49 -> Pairs: 0.00/0.49 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) 0.00/0.49 AND(ok(X1:S),ok(X2:S)) -> AND(X1:S,X2:S) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Projection: 0.00/0.49 pi(AND) = 1 0.00/0.49 0.00/0.49 Problem 1.6: 0.00/0.49 0.00/0.49 SCC Processor: 0.00/0.49 -> Pairs: 0.00/0.49 Empty 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Strongly Connected Components: 0.00/0.49 There is no strongly connected component 0.00/0.49 0.00/0.49 The problem is finite. 0.00/0.49 0.00/0.49 Problem 1.7: 0.00/0.49 0.00/0.49 Subterm Processor: 0.00/0.49 -> Pairs: 0.00/0.49 ADD(mark(X1:S),X2:S) -> ADD(X1:S,X2:S) 0.00/0.49 ADD(ok(X1:S),ok(X2:S)) -> ADD(X1:S,X2:S) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Projection: 0.00/0.49 pi(ADD) = 1 0.00/0.49 0.00/0.49 Problem 1.7: 0.00/0.49 0.00/0.49 SCC Processor: 0.00/0.49 -> Pairs: 0.00/0.49 Empty 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Strongly Connected Components: 0.00/0.49 There is no strongly connected component 0.00/0.49 0.00/0.49 The problem is finite. 0.00/0.49 0.00/0.49 Problem 1.8: 0.00/0.49 0.00/0.49 Subterm Processor: 0.00/0.49 -> Pairs: 0.00/0.49 PROPER(add(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(add(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(and(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(and(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(first(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(first(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(from(X:S)) -> PROPER(X:S) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X1:S) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X2:S) 0.00/0.49 PROPER(if(X1:S,X2:S,X3:S)) -> PROPER(X3:S) 0.00/0.49 PROPER(s(X:S)) -> PROPER(X:S) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Projection: 0.00/0.49 pi(PROPER) = 1 0.00/0.49 0.00/0.49 Problem 1.8: 0.00/0.49 0.00/0.49 SCC Processor: 0.00/0.49 -> Pairs: 0.00/0.49 Empty 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Strongly Connected Components: 0.00/0.49 There is no strongly connected component 0.00/0.49 0.00/0.49 The problem is finite. 0.00/0.49 0.00/0.49 Problem 1.9: 0.00/0.49 0.00/0.49 Subterm Processor: 0.00/0.49 -> Pairs: 0.00/0.49 ACTIVE(add(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(and(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.49 ACTIVE(first(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.49 ACTIVE(if(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Projection: 0.00/0.49 pi(ACTIVE) = 1 0.00/0.49 0.00/0.49 Problem 1.9: 0.00/0.49 0.00/0.49 SCC Processor: 0.00/0.49 -> Pairs: 0.00/0.49 Empty 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Strongly Connected Components: 0.00/0.49 There is no strongly connected component 0.00/0.49 0.00/0.49 The problem is finite. 0.00/0.49 0.00/0.49 Problem 1.10: 0.00/0.49 0.00/0.49 Reduction Pairs Processor: 0.00/0.49 -> Pairs: 0.00/0.49 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.49 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 -> Usable rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 ->Interpretation type: 0.00/0.49 Linear 0.00/0.49 ->Coefficients: 0.00/0.49 Natural Numbers 0.00/0.49 ->Dimension: 0.00/0.49 1 0.00/0.49 ->Bound: 0.00/0.49 2 0.00/0.49 ->Interpretation: 0.00/0.49 0.00/0.49 [active](X) = X 0.00/0.49 [add](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.49 [and](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.49 [cons](X1,X2) = 2.X1 0.00/0.49 [first](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.49 [from](X) = 2.X + 2 0.00/0.49 [if](X1,X2,X3) = X1 + 2.X2 + 2.X3 0.00/0.49 [proper](X) = X 0.00/0.49 [s](X) = X + 2 0.00/0.49 [top](X) = 0 0.00/0.49 [0] = 0 0.00/0.49 [fSNonEmpty] = 0 0.00/0.49 [false] = 2 0.00/0.49 [mark](X) = X + 2 0.00/0.49 [nil] = 0 0.00/0.49 [ok](X) = X 0.00/0.49 [true] = 2 0.00/0.49 [ACTIVE](X) = 0 0.00/0.49 [ADD](X1,X2) = 0 0.00/0.49 [AND](X1,X2) = 0 0.00/0.49 [CONS](X1,X2) = 0 0.00/0.49 [FIRST](X1,X2) = 0 0.00/0.49 [FROM](X) = 0 0.00/0.49 [IF](X1,X2,X3) = 0 0.00/0.49 [PROPER](X) = 0 0.00/0.49 [S](X) = 0 0.00/0.49 [TOP](X) = 2.X 0.00/0.49 0.00/0.49 Problem 1.10: 0.00/0.49 0.00/0.49 SCC Processor: 0.00/0.49 -> Pairs: 0.00/0.49 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 ->Strongly Connected Components: 0.00/0.49 ->->Cycle: 0.00/0.49 ->->-> Pairs: 0.00/0.49 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.49 ->->-> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.49 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.49 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.49 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.49 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.49 proper(0) -> ok(0) 0.00/0.49 proper(ffalse) -> ok(ffalse) 0.00/0.49 proper(nil) -> ok(nil) 0.00/0.49 proper(ttrue) -> ok(ttrue) 0.00/0.49 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.49 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.49 top(ok(X:S)) -> top(active(X:S)) 0.00/0.49 0.00/0.49 Problem 1.10: 0.00/0.49 0.00/0.49 Reduction Pairs Processor: 0.00/0.49 -> Pairs: 0.00/0.49 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.49 -> Rules: 0.00/0.49 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.49 active(add(0,X:S)) -> mark(X:S) 0.00/0.49 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.49 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.49 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.49 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.49 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.49 active(first(0,X:S)) -> mark(nil) 0.00/0.49 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.49 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.49 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.49 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.49 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.49 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.49 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.49 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.49 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.49 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.49 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.49 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.49 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.49 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.49 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.49 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.49 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.50 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.50 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.50 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.50 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.50 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.50 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.50 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.50 proper(0) -> ok(0) 0.00/0.50 proper(ffalse) -> ok(ffalse) 0.00/0.50 proper(nil) -> ok(nil) 0.00/0.50 proper(ttrue) -> ok(ttrue) 0.00/0.50 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.50 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.50 top(ok(X:S)) -> top(active(X:S)) 0.00/0.50 -> Usable rules: 0.00/0.50 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.50 active(add(0,X:S)) -> mark(X:S) 0.00/0.50 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.50 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.50 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.50 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.50 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.50 active(first(0,X:S)) -> mark(nil) 0.00/0.50 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.50 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.50 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.50 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.50 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.50 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.50 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.50 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.50 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.50 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.50 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.50 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.50 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.50 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.50 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.50 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.50 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.50 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.50 ->Interpretation type: 0.00/0.50 Linear 0.00/0.50 ->Coefficients: 0.00/0.50 Natural Numbers 0.00/0.50 ->Dimension: 0.00/0.50 1 0.00/0.50 ->Bound: 0.00/0.50 2 0.00/0.50 ->Interpretation: 0.00/0.50 0.00/0.50 [active](X) = 2.X + 1 0.00/0.50 [add](X1,X2) = 2.X1 + 2 0.00/0.50 [and](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.50 [cons](X1,X2) = 2.X1 + 2 0.00/0.50 [first](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.50 [from](X) = 2.X + 2 0.00/0.50 [if](X1,X2,X3) = 2.X2 + X3 + 2 0.00/0.50 [proper](X) = 0 0.00/0.50 [s](X) = 2.X 0.00/0.50 [top](X) = 0 0.00/0.50 [0] = 2 0.00/0.50 [fSNonEmpty] = 0 0.00/0.50 [false] = 2 0.00/0.50 [mark](X) = 2 0.00/0.50 [nil] = 0 0.00/0.50 [ok](X) = 2.X + 2 0.00/0.50 [true] = 1 0.00/0.50 [ACTIVE](X) = 0 0.00/0.50 [ADD](X1,X2) = 0 0.00/0.50 [AND](X1,X2) = 0 0.00/0.50 [CONS](X1,X2) = 0 0.00/0.50 [FIRST](X1,X2) = 0 0.00/0.50 [FROM](X) = 0 0.00/0.50 [IF](X1,X2,X3) = 0 0.00/0.50 [PROPER](X) = 0 0.00/0.50 [S](X) = 0 0.00/0.50 [TOP](X) = 2.X 0.00/0.50 0.00/0.50 Problem 1.10: 0.00/0.50 0.00/0.50 SCC Processor: 0.00/0.50 -> Pairs: 0.00/0.50 Empty 0.00/0.50 -> Rules: 0.00/0.50 active(add(s(X:S),Y:S)) -> mark(s(add(X:S,Y:S))) 0.00/0.50 active(add(0,X:S)) -> mark(X:S) 0.00/0.50 active(add(X1:S,X2:S)) -> add(active(X1:S),X2:S) 0.00/0.50 active(and(ffalse,Y:S)) -> mark(ffalse) 0.00/0.50 active(and(ttrue,X:S)) -> mark(X:S) 0.00/0.50 active(and(X1:S,X2:S)) -> and(active(X1:S),X2:S) 0.00/0.50 active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) 0.00/0.50 active(first(0,X:S)) -> mark(nil) 0.00/0.50 active(first(X1:S,X2:S)) -> first(active(X1:S),X2:S) 0.00/0.50 active(first(X1:S,X2:S)) -> first(X1:S,active(X2:S)) 0.00/0.50 active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 0.00/0.50 active(if(ffalse,X:S,Y:S)) -> mark(Y:S) 0.00/0.50 active(if(ttrue,X:S,Y:S)) -> mark(X:S) 0.00/0.50 active(if(X1:S,X2:S,X3:S)) -> if(active(X1:S),X2:S,X3:S) 0.00/0.50 add(mark(X1:S),X2:S) -> mark(add(X1:S,X2:S)) 0.00/0.50 add(ok(X1:S),ok(X2:S)) -> ok(add(X1:S,X2:S)) 0.00/0.50 and(mark(X1:S),X2:S) -> mark(and(X1:S,X2:S)) 0.00/0.50 and(ok(X1:S),ok(X2:S)) -> ok(and(X1:S,X2:S)) 0.00/0.50 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) 0.00/0.50 first(mark(X1:S),X2:S) -> mark(first(X1:S,X2:S)) 0.00/0.50 first(ok(X1:S),ok(X2:S)) -> ok(first(X1:S,X2:S)) 0.00/0.50 first(X1:S,mark(X2:S)) -> mark(first(X1:S,X2:S)) 0.00/0.50 from(ok(X:S)) -> ok(from(X:S)) 0.00/0.50 if(mark(X1:S),X2:S,X3:S) -> mark(if(X1:S,X2:S,X3:S)) 0.00/0.50 if(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(if(X1:S,X2:S,X3:S)) 0.00/0.50 proper(add(X1:S,X2:S)) -> add(proper(X1:S),proper(X2:S)) 0.00/0.50 proper(and(X1:S,X2:S)) -> and(proper(X1:S),proper(X2:S)) 0.00/0.50 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) 0.00/0.50 proper(first(X1:S,X2:S)) -> first(proper(X1:S),proper(X2:S)) 0.00/0.50 proper(from(X:S)) -> from(proper(X:S)) 0.00/0.50 proper(if(X1:S,X2:S,X3:S)) -> if(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.50 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.50 proper(0) -> ok(0) 0.00/0.50 proper(ffalse) -> ok(ffalse) 0.00/0.50 proper(nil) -> ok(nil) 0.00/0.50 proper(ttrue) -> ok(ttrue) 0.00/0.50 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.50 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.50 top(ok(X:S)) -> top(active(X:S)) 0.00/0.50 ->Strongly Connected Components: 0.00/0.50 There is no strongly connected component 0.00/0.50 0.00/0.50 The problem is finite. 0.00/0.50 EOF