YES Problem 1: (VAR v_NonEmpty:S L:S X:S X1:S X2:S Y:S) (RULES active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2: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(eq(s(X:S),s(Y:S))) -> EQ(X:S,Y:S) ACTIVE(inf(X:S)) -> ACTIVE(X:S) ACTIVE(inf(X:S)) -> CONS(X:S,inf(s(X:S))) ACTIVE(inf(X:S)) -> INF(active(X:S)) ACTIVE(inf(X:S)) -> INF(s(X:S)) ACTIVE(inf(X:S)) -> S(X:S) ACTIVE(length(cons(X:S,L:S))) -> LENGTH(L:S) ACTIVE(length(cons(X:S,L:S))) -> S(length(L:S)) ACTIVE(length(X:S)) -> ACTIVE(X:S) ACTIVE(length(X:S)) -> LENGTH(active(X:S)) ACTIVE(take(s(X:S),cons(Y:S,L:S))) -> CONS(Y:S,take(X:S,L:S)) ACTIVE(take(s(X:S),cons(Y:S,L:S))) -> TAKE(X:S,L:S) ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X2:S) ACTIVE(take(X1:S,X2:S)) -> TAKE(active(X1:S),X2:S) ACTIVE(take(X1:S,X2:S)) -> TAKE(X1:S,active(X2:S)) CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) EQ(ok(X1:S),ok(X2:S)) -> EQ(X1:S,X2:S) INF(mark(X:S)) -> INF(X:S) INF(ok(X:S)) -> INF(X:S) LENGTH(mark(X:S)) -> LENGTH(X:S) LENGTH(ok(X:S)) -> LENGTH(X: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(eq(X1:S,X2:S)) -> EQ(proper(X1:S),proper(X2:S)) PROPER(eq(X1:S,X2:S)) -> PROPER(X1:S) PROPER(eq(X1:S,X2:S)) -> PROPER(X2:S) PROPER(inf(X:S)) -> INF(proper(X:S)) PROPER(inf(X:S)) -> PROPER(X:S) PROPER(length(X:S)) -> LENGTH(proper(X:S)) PROPER(length(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> S(proper(X:S)) PROPER(take(X1:S,X2:S)) -> PROPER(X1:S) PROPER(take(X1:S,X2:S)) -> PROPER(X2:S) PROPER(take(X1:S,X2:S)) -> TAKE(proper(X1:S),proper(X2:S)) S(ok(X:S)) -> S(X:S) TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(ok(X1:S),ok(X2:S)) -> TAKE(X1:S,X2:S) TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2: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(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) Problem 1: SCC Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> EQ(X:S,Y:S) ACTIVE(inf(X:S)) -> ACTIVE(X:S) ACTIVE(inf(X:S)) -> CONS(X:S,inf(s(X:S))) ACTIVE(inf(X:S)) -> INF(active(X:S)) ACTIVE(inf(X:S)) -> INF(s(X:S)) ACTIVE(inf(X:S)) -> S(X:S) ACTIVE(length(cons(X:S,L:S))) -> LENGTH(L:S) ACTIVE(length(cons(X:S,L:S))) -> S(length(L:S)) ACTIVE(length(X:S)) -> ACTIVE(X:S) ACTIVE(length(X:S)) -> LENGTH(active(X:S)) ACTIVE(take(s(X:S),cons(Y:S,L:S))) -> CONS(Y:S,take(X:S,L:S)) ACTIVE(take(s(X:S),cons(Y:S,L:S))) -> TAKE(X:S,L:S) ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X2:S) ACTIVE(take(X1:S,X2:S)) -> TAKE(active(X1:S),X2:S) ACTIVE(take(X1:S,X2:S)) -> TAKE(X1:S,active(X2:S)) CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) EQ(ok(X1:S),ok(X2:S)) -> EQ(X1:S,X2:S) INF(mark(X:S)) -> INF(X:S) INF(ok(X:S)) -> INF(X:S) LENGTH(mark(X:S)) -> LENGTH(X:S) LENGTH(ok(X:S)) -> LENGTH(X: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(eq(X1:S,X2:S)) -> EQ(proper(X1:S),proper(X2:S)) PROPER(eq(X1:S,X2:S)) -> PROPER(X1:S) PROPER(eq(X1:S,X2:S)) -> PROPER(X2:S) PROPER(inf(X:S)) -> INF(proper(X:S)) PROPER(inf(X:S)) -> PROPER(X:S) PROPER(length(X:S)) -> LENGTH(proper(X:S)) PROPER(length(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> S(proper(X:S)) PROPER(take(X1:S,X2:S)) -> PROPER(X1:S) PROPER(take(X1:S,X2:S)) -> PROPER(X2:S) PROPER(take(X1:S,X2:S)) -> TAKE(proper(X1:S),proper(X2:S)) S(ok(X:S)) -> S(X:S) TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(ok(X1:S),ok(X2:S)) -> TAKE(X1:S,X2:S) TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2: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(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(ok(X1:S),ok(X2:S)) -> TAKE(X1:S,X2:S) TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) ->->-> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: S(ok(X:S)) -> S(X:S) ->->-> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: LENGTH(mark(X:S)) -> LENGTH(X:S) LENGTH(ok(X:S)) -> LENGTH(X:S) ->->-> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: INF(mark(X:S)) -> INF(X:S) INF(ok(X:S)) -> INF(X:S) ->->-> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: EQ(ok(X1:S),ok(X2:S)) -> EQ(X1:S,X2:S) ->->-> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2: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(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) PROPER(eq(X1:S,X2:S)) -> PROPER(X1:S) PROPER(eq(X1:S,X2:S)) -> PROPER(X2:S) PROPER(inf(X:S)) -> PROPER(X:S) PROPER(length(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(take(X1:S,X2:S)) -> PROPER(X1:S) PROPER(take(X1:S,X2:S)) -> PROPER(X2:S) ->->-> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: ACTIVE(inf(X:S)) -> ACTIVE(X:S) ACTIVE(length(X:S)) -> ACTIVE(X:S) ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X2:S) ->->-> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2: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(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) The problem is decomposed in 9 subproblems. Problem 1.1: Subterm Processor: -> Pairs: TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(ok(X1:S),ok(X2:S)) -> TAKE(X1:S,X2:S) TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(TAKE) = 1 Problem 1.1: SCC Processor: -> Pairs: TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) ->->-> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) Problem 1.1: Subterm Processor: -> Pairs: TAKE(X1:S,mark(X2:S)) -> TAKE(X1:S,X2:S) -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(TAKE) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2: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: S(ok(X:S)) -> S(X:S) -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(S) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2: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: LENGTH(mark(X:S)) -> LENGTH(X:S) LENGTH(ok(X:S)) -> LENGTH(X:S) -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(LENGTH) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2: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: INF(mark(X:S)) -> INF(X:S) INF(ok(X:S)) -> INF(X:S) -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(INF) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2: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: EQ(ok(X1:S),ok(X2:S)) -> EQ(X1:S,X2:S) -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(EQ) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2: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: CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(CONS) = 1 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2: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: PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) PROPER(eq(X1:S,X2:S)) -> PROPER(X1:S) PROPER(eq(X1:S,X2:S)) -> PROPER(X2:S) PROPER(inf(X:S)) -> PROPER(X:S) PROPER(length(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(take(X1:S,X2:S)) -> PROPER(X1:S) PROPER(take(X1:S,X2:S)) -> PROPER(X2:S) -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(PROPER) = 1 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2: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: ACTIVE(inf(X:S)) -> ACTIVE(X:S) ACTIVE(length(X:S)) -> ACTIVE(X:S) ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(take(X1:S,X2:S)) -> ACTIVE(X2:S) -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(ACTIVE) = 1 Problem 1.8: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2: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: Reduction Pairs Processor: -> Pairs: TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) -> Usable rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 2 ->Bound: 1 ->Interpretation: [active](X) = [1 0;0 0].X + [0;1] [cons](X1,X2) = [1 0;0 1].X1 + [0;1] [eq](X1,X2) = [0 1;0 0].X1 + [0 1;0 0].X2 + [1;1] [inf](X) = [1 0;0 0].X + [1;1] [length](X) = [1 0;0 1].X + [1;0] [proper](X) = [1 0;0 1].X [s](X) = [0 0;0 1].X + [0;1] [take](X1,X2) = [1 0;0 0].X1 + [1 0;0 0].X2 + [1;1] [top](X) = 0 [0] = [1;1] [fSNonEmpty] = 0 [false] = [0;1] [mark](X) = [1 0;0 0].X + [1;1] [nil] = [1;0] [ok](X) = [1 0;0 1].X [true] = [1;1] [ACTIVE](X) = 0 [CONS](X1,X2) = 0 [EQ](X1,X2) = 0 [INF](X) = 0 [LENGTH](X) = 0 [PROPER](X) = 0 [S](X) = 0 [TAKE](X1,X2) = 0 [TOP](X) = [1 0;1 0].X Problem 1.9: SCC Processor: -> Pairs: TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2: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(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) Problem 1.9: Reduction Pairs Processor: -> Pairs: TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) -> Usable rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) s(ok(X:S)) -> ok(s(X:S)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2.X + 1 [cons](X1,X2) = 2.X1 + 2 [eq](X1,X2) = X1 + X2 + 1 [inf](X) = 2.X + 1 [length](X) = 2.X + 1 [proper](X) = 0 [s](X) = 2.X + 2 [take](X1,X2) = 2.X2 + 2 [top](X) = 0 [0] = 2 [fSNonEmpty] = 0 [false] = 1 [mark](X) = 2 [nil] = 2 [ok](X) = 2.X + 2 [true] = 0 [ACTIVE](X) = 0 [CONS](X1,X2) = 0 [EQ](X1,X2) = 0 [INF](X) = 0 [LENGTH](X) = 0 [PROPER](X) = 0 [S](X) = 0 [TAKE](X1,X2) = 0 [TOP](X) = X Problem 1.9: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X:S),s(Y:S))) -> mark(eq(X:S,Y:S)) active(eq(0,0)) -> mark(ttrue) active(eq(X:S,Y:S)) -> mark(ffalse) active(inf(X:S)) -> inf(active(X:S)) active(inf(X:S)) -> mark(cons(X:S,inf(s(X:S)))) active(length(cons(X:S,L:S))) -> mark(s(length(L:S))) active(length(nil)) -> mark(0) active(length(X:S)) -> length(active(X:S)) active(take(s(X:S),cons(Y:S,L:S))) -> mark(cons(Y:S,take(X:S,L:S))) active(take(0,X:S)) -> mark(nil) active(take(X1:S,X2:S)) -> take(active(X1:S),X2:S) active(take(X1:S,X2:S)) -> take(X1:S,active(X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) eq(ok(X1:S),ok(X2:S)) -> ok(eq(X1:S,X2:S)) inf(mark(X:S)) -> mark(inf(X:S)) inf(ok(X:S)) -> ok(inf(X:S)) length(mark(X:S)) -> mark(length(X:S)) length(ok(X:S)) -> ok(length(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(eq(X1:S,X2:S)) -> eq(proper(X1:S),proper(X2:S)) proper(inf(X:S)) -> inf(proper(X:S)) proper(length(X:S)) -> length(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(take(X1:S,X2:S)) -> take(proper(X1:S),proper(X2: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)) take(mark(X1:S),X2:S) -> mark(take(X1:S,X2:S)) take(ok(X1:S),ok(X2:S)) -> ok(take(X1:S,X2:S)) take(X1:S,mark(X2:S)) -> mark(take(X1:S,X2: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.