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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> EQ(X:S,Y:S) 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)) -> CONS(X:S,inf(s(X:S))) ACTIVE(inf(X:S)) -> INF(s(X:S)) ACTIVE(inf(X:S)) -> MARK(cons(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))) -> MARK(s(length(L:S))) ACTIVE(length(cons(X:S,L:S))) -> S(length(L:S)) ACTIVE(length(nil)) -> MARK(0) 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))) -> MARK(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(0,X:S)) -> MARK(nil) CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) EQ(active(X1:S),X2:S) -> EQ(X1:S,X2:S) EQ(mark(X1:S),X2:S) -> EQ(X1:S,X2:S) EQ(X1:S,active(X2:S)) -> EQ(X1:S,X2:S) EQ(X1:S,mark(X2:S)) -> EQ(X1:S,X2:S) INF(active(X:S)) -> INF(X:S) INF(mark(X:S)) -> INF(X:S) LENGTH(active(X:S)) -> LENGTH(X:S) LENGTH(mark(X:S)) -> LENGTH(X:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(X1:S,X2:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> ACTIVE(inf(mark(X:S))) MARK(inf(X:S)) -> INF(mark(X:S)) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(X:S)) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(take(X1:S,X2:S)) -> TAKE(mark(X1:S),mark(X2:S)) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) TAKE(active(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(X1:S,active(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) Problem 1: SCC Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> EQ(X:S,Y:S) 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)) -> CONS(X:S,inf(s(X:S))) ACTIVE(inf(X:S)) -> INF(s(X:S)) ACTIVE(inf(X:S)) -> MARK(cons(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))) -> MARK(s(length(L:S))) ACTIVE(length(cons(X:S,L:S))) -> S(length(L:S)) ACTIVE(length(nil)) -> MARK(0) 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))) -> MARK(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(0,X:S)) -> MARK(nil) CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S) EQ(active(X1:S),X2:S) -> EQ(X1:S,X2:S) EQ(mark(X1:S),X2:S) -> EQ(X1:S,X2:S) EQ(X1:S,active(X2:S)) -> EQ(X1:S,X2:S) EQ(X1:S,mark(X2:S)) -> EQ(X1:S,X2:S) INF(active(X:S)) -> INF(X:S) INF(mark(X:S)) -> INF(X:S) LENGTH(active(X:S)) -> LENGTH(X:S) LENGTH(mark(X:S)) -> LENGTH(X:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(X1:S,X2:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> ACTIVE(inf(mark(X:S))) MARK(inf(X:S)) -> INF(mark(X:S)) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(X:S)) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(X2:S) MARK(take(X1:S,X2:S)) -> TAKE(mark(X1:S),mark(X2:S)) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) TAKE(active(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(X1:S,active(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TAKE(active(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(X1:S,active(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->->Cycle: ->->-> Pairs: S(active(X:S)) -> S(X:S) S(mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->->Cycle: ->->-> Pairs: LENGTH(active(X:S)) -> LENGTH(X:S) LENGTH(mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->->Cycle: ->->-> Pairs: INF(active(X:S)) -> INF(X:S) INF(mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->->Cycle: ->->-> Pairs: EQ(active(X1:S),X2:S) -> EQ(X1:S,X2:S) EQ(mark(X1:S),X2:S) -> EQ(X1:S,X2:S) EQ(X1:S,active(X2:S)) -> EQ(X1:S,X2:S) EQ(X1:S,mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->->Cycle: ->->-> Pairs: CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y: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(take(s(X:S),cons(Y:S,L:S))) -> MARK(cons(Y:S,take(X:S,L:S))) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> ACTIVE(inf(mark(X:S))) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) The problem is decomposed in 7 subproblems. Problem 1.1: Subterm Processor: -> Pairs: TAKE(active(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(mark(X1:S),X2:S) -> TAKE(X1:S,X2:S) TAKE(X1:S,active(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Projection: pi(TAKE) = 1 Problem 1.1: SCC Processor: -> Pairs: TAKE(X1:S,active(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TAKE(X1:S,active(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) Problem 1.1: Subterm Processor: -> Pairs: TAKE(X1:S,active(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2: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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: S(active(X:S)) -> S(X:S) S(mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2: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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: LENGTH(active(X:S)) -> LENGTH(X:S) LENGTH(mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2: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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: INF(active(X:S)) -> INF(X:S) INF(mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2: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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: EQ(active(X1:S),X2:S) -> EQ(X1:S,X2:S) EQ(mark(X1:S),X2:S) -> EQ(X1:S,X2:S) EQ(X1:S,active(X2:S)) -> EQ(X1:S,X2:S) EQ(X1:S,mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Projection: pi(EQ) = 1 Problem 1.5: SCC Processor: -> Pairs: EQ(X1:S,active(X2:S)) -> EQ(X1:S,X2:S) EQ(X1:S,mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: EQ(X1:S,active(X2:S)) -> EQ(X1:S,X2:S) EQ(X1:S,mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) Problem 1.5: Subterm Processor: -> Pairs: EQ(X1:S,active(X2:S)) -> EQ(X1:S,X2:S) EQ(X1:S,mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Projection: pi(EQ) = 2 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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Subterm Processor: -> Pairs: CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Projection: pi(CONS) = 1 Problem 1.6: SCC Processor: -> Pairs: CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) Problem 1.6: Subterm Processor: -> Pairs: CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S) CONS(X1:S,mark(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Projection: pi(CONS) = 2 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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y: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(take(s(X:S),cons(Y:S,L:S))) -> MARK(cons(Y:S,take(X:S,L:S))) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> ACTIVE(inf(mark(X:S))) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2: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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 1 [eq](X1,X2) = 2 [inf](X) = 2.X + 2 [length](X) = X + 2 [mark](X) = X [s](X) = 2 [take](X1,X2) = 2.X1 + 2.X2 + 1 [0] = 2 [fSNonEmpty] = 0 [false] = 1 [nil] = 0 [true] = 1 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [EQ](X1,X2) = 0 [INF](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 [TAKE](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) ACTIVE(length(cons(X:S,L:S))) -> MARK(s(length(L:S))) ACTIVE(take(s(X:S),cons(Y:S,L:S))) -> MARK(cons(Y:S,take(X:S,L:S))) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> ACTIVE(inf(mark(X:S))) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) ACTIVE(length(cons(X:S,L:S))) -> MARK(s(length(L:S))) ACTIVE(take(s(X:S),cons(Y:S,L:S))) -> MARK(cons(Y:S,take(X:S,L:S))) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> ACTIVE(inf(mark(X:S))) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) ACTIVE(length(cons(X:S,L:S))) -> MARK(s(length(L:S))) ACTIVE(take(s(X:S),cons(Y:S,L:S))) -> MARK(cons(Y:S,take(X:S,L:S))) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> ACTIVE(inf(mark(X:S))) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2: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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X + 1 [cons](X1,X2) = 1 [eq](X1,X2) = 0 [inf](X) = 2.X + 2 [length](X) = 2.X + 2 [mark](X) = 2.X + 1 [s](X) = 0 [take](X1,X2) = X1 + X2 + 2 [0] = 1 [fSNonEmpty] = 0 [false] = 0 [nil] = 1 [true] = 0 [ACTIVE](X) = X [CONS](X1,X2) = 0 [EQ](X1,X2) = 0 [INF](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X [S](X) = 0 [TAKE](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) ACTIVE(take(s(X:S),cons(Y:S,L:S))) -> MARK(cons(Y:S,take(X:S,L:S))) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> ACTIVE(inf(mark(X:S))) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) ACTIVE(take(s(X:S),cons(Y:S,L:S))) -> MARK(cons(Y:S,take(X:S,L:S))) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> ACTIVE(inf(mark(X:S))) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) ACTIVE(take(s(X:S),cons(Y:S,L:S))) -> MARK(cons(Y:S,take(X:S,L:S))) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> ACTIVE(inf(mark(X:S))) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2: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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X + 2 [cons](X1,X2) = X1 + 2 [eq](X1,X2) = 2 [inf](X) = 2.X + 2 [length](X) = 2.X + 2 [mark](X) = 2.X [s](X) = 2 [take](X1,X2) = 2.X1 + 2.X2 + 2 [0] = 2 [fSNonEmpty] = 0 [false] = 2 [nil] = 2 [true] = 2 [ACTIVE](X) = X + 2 [CONS](X1,X2) = 0 [EQ](X1,X2) = 0 [INF](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X [S](X) = 0 [TAKE](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> ACTIVE(inf(mark(X:S))) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> ACTIVE(inf(mark(X:S))) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> ACTIVE(inf(mark(X:S))) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2: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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X + 1 [cons](X1,X2) = X1 + 1 [eq](X1,X2) = 1 [inf](X) = 2.X + 2 [length](X) = 2.X + 2 [mark](X) = 2.X [s](X) = 1 [take](X1,X2) = X1 + 2.X2 + 2 [0] = 2 [fSNonEmpty] = 0 [false] = 1 [nil] = 1 [true] = 1 [ACTIVE](X) = 2 [CONS](X1,X2) = 0 [EQ](X1,X2) = 0 [INF](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X [S](X) = 0 [TAKE](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(inf(X:S)) -> MARK(X:S) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2: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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X2 [eq](X1,X2) = 2.X1 + X2 + 2 [inf](X) = 2.X + 1 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [take](X1,X2) = 2.X1 + 2.X2 [0] = 2 [fSNonEmpty] = 0 [false] = 1 [nil] = 1 [true] = 1 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [EQ](X1,X2) = 0 [INF](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 [TAKE](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(length(X:S)) -> ACTIVE(length(mark(X:S))) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2: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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2 [cons](X1,X2) = 0 [eq](X1,X2) = 2 [inf](X) = 0 [length](X) = 0 [mark](X) = 2 [s](X) = 0 [take](X1,X2) = 2 [0] = 2 [fSNonEmpty] = 0 [false] = 2 [nil] = 2 [true] = 1 [ACTIVE](X) = X [CONS](X1,X2) = 0 [EQ](X1,X2) = 0 [INF](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2 [S](X) = 0 [TAKE](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(length(X:S)) -> MARK(X:S) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2: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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X2 [eq](X1,X2) = 2.X1 + 2.X2 + 2 [inf](X) = 2 [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [take](X1,X2) = X1 + 2.X2 + 2 [0] = 2 [fSNonEmpty] = 0 [false] = 1 [nil] = 2 [true] = 2 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [EQ](X1,X2) = 0 [INF](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 [TAKE](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(take(X1:S,X2:S)) -> ACTIVE(take(mark(X1:S),mark(X2:S))) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2: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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2 [cons](X1,X2) = 2 [eq](X1,X2) = 2 [inf](X) = 2 [length](X) = 0 [mark](X) = 2 [s](X) = 2 [take](X1,X2) = 1 [0] = 2 [fSNonEmpty] = 0 [false] = 2 [nil] = 2 [true] = 2 [ACTIVE](X) = X [CONS](X1,X2) = 0 [EQ](X1,X2) = 0 [INF](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2 [S](X) = 0 [TAKE](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) Problem 1.7: Reduction Pairs Processor: -> Pairs: ACTIVE(eq(s(X:S),s(Y:S))) -> MARK(eq(X:S,Y:S)) MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) -> Usable rules: eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 0 [eq](X1,X2) = 2.X2 + 1 [inf](X) = 0 [length](X) = 0 [mark](X) = X [s](X) = 2.X + 2 [take](X1,X2) = 2.X1 + 2.X2 [0] = 0 [fSNonEmpty] = 0 [false] = 0 [nil] = 0 [true] = 0 [ACTIVE](X) = 2.X + 1 [CONS](X1,X2) = 0 [EQ](X1,X2) = 0 [INF](X) = 0 [LENGTH](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 [TAKE](X1,X2) = 0 Problem 1.7: SCC Processor: -> Pairs: MARK(eq(X1:S,X2:S)) -> ACTIVE(eq(X1:S,X2:S)) MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) Problem 1.7: Subterm Processor: -> Pairs: MARK(take(X1:S,X2:S)) -> MARK(X1:S) MARK(take(X1:S,X2:S)) -> MARK(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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Projection: pi(MARK) = 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)) -> 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(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) cons(active(X1:S),X2:S) -> cons(X1:S,X2:S) cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S) cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S) cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S) eq(active(X1:S),X2:S) -> eq(X1:S,X2:S) eq(mark(X1:S),X2:S) -> eq(X1:S,X2:S) eq(X1:S,active(X2:S)) -> eq(X1:S,X2:S) eq(X1:S,mark(X2:S)) -> eq(X1:S,X2:S) inf(active(X:S)) -> inf(X:S) inf(mark(X:S)) -> inf(X:S) length(active(X:S)) -> length(X:S) length(mark(X:S)) -> length(X:S) mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S)) mark(eq(X1:S,X2:S)) -> active(eq(X1:S,X2:S)) mark(inf(X:S)) -> active(inf(mark(X:S))) mark(length(X:S)) -> active(length(mark(X:S))) mark(s(X:S)) -> active(s(X:S)) mark(take(X1:S,X2:S)) -> active(take(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(ffalse) -> active(ffalse) mark(nil) -> active(nil) mark(ttrue) -> active(ttrue) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) take(active(X1:S),X2:S) -> take(X1:S,X2:S) take(mark(X1:S),X2:S) -> take(X1:S,X2:S) take(X1:S,active(X2:S)) -> take(X1:S,X2:S) take(X1:S,mark(X2:S)) -> take(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.