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