YES Problem 1: (VAR v_NonEmpty:S X:S X1:S X2:S Y:S Z:S) (RULES active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> CONS(Y:S,first(X:S,Z:S)) ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> FIRST(X:S,Z:S) ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S))) ACTIVE(first(0,X:S)) -> MARK(nil) ACTIVE(from(X:S)) -> CONS(X:S,from(s(X:S))) ACTIVE(from(X:S)) -> FROM(s(X:S)) ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) ACTIVE(from(X:S)) -> S(X:S) 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) FIRST(active(X1:S),X2:S) -> FIRST(X1:S,X2:S) FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) FROM(active(X:S)) -> FROM(X:S) FROM(mark(X:S)) -> FROM(X:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) MARK(first(X1:S,X2:S)) -> FIRST(mark(X1:S),mark(X2:S)) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> FROM(mark(X:S)) MARK(from(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(s(X:S)) -> S(mark(X:S)) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1: SCC Processor: -> Pairs: ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> CONS(Y:S,first(X:S,Z:S)) ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> FIRST(X:S,Z:S) ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S))) ACTIVE(first(0,X:S)) -> MARK(nil) ACTIVE(from(X:S)) -> CONS(X:S,from(s(X:S))) ACTIVE(from(X:S)) -> FROM(s(X:S)) ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) ACTIVE(from(X:S)) -> S(X:S) 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) FIRST(active(X1:S),X2:S) -> FIRST(X1:S,X2:S) FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) FROM(active(X:S)) -> FROM(X:S) FROM(mark(X:S)) -> FROM(X:S) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) MARK(first(X1:S,X2:S)) -> FIRST(mark(X1:S),mark(X2:S)) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> FROM(mark(X:S)) MARK(from(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(s(X:S)) -> S(mark(X:S)) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) ->->-> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: FROM(active(X:S)) -> FROM(X:S) FROM(mark(X:S)) -> FROM(X:S) ->->-> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: FIRST(active(X1:S),X2:S) -> FIRST(X1:S,X2:S) FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) ->->-> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X: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(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->->Cycle: ->->-> Pairs: ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S))) ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) The problem is decomposed in 5 subproblems. Problem 1.1: Subterm Processor: -> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(S) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: FROM(active(X:S)) -> FROM(X:S) FROM(mark(X:S)) -> FROM(X:S) -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(FROM) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: FIRST(active(X1:S),X2:S) -> FIRST(X1:S,X2:S) FIRST(mark(X1:S),X2:S) -> FIRST(X1:S,X2:S) FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(FIRST) = 1 Problem 1.3: SCC Processor: -> Pairs: FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) ->->-> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.3: Subterm Processor: -> Pairs: FIRST(X1:S,active(X2:S)) -> FIRST(X1:S,X2:S) FIRST(X1:S,mark(X2:S)) -> FIRST(X1:S,X2:S) -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(FIRST) = 2 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: 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(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(CONS) = 1 Problem 1.4: 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(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X: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(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.4: 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(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(CONS) = 2 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Reduction Pairs Processor: -> Pairs: ACTIVE(first(s(X:S),cons(Y:S,Z:S))) -> MARK(cons(Y:S,first(X:S,Z:S))) ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 + 1 [first](X1,X2) = 2.X1 + 2.X2 [from](X) = 2.X + 2 [mark](X) = 2.X [s](X) = 2.X + 2 [0] = 2 [fSNonEmpty] = 0 [nil] = 2 [ACTIVE](X) = X [CONS](X1,X2) = 0 [FIRST](X1,X2) = 0 [FROM](X) = 0 [MARK](X) = 2.X [S](X) = 0 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.5: Reduction Pairs Processor: -> Pairs: ACTIVE(from(X:S)) -> MARK(cons(X:S,from(s(X:S)))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) -> Usable rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 [first](X1,X2) = X1 + 2.X2 [from](X) = 2.X + 1 [mark](X) = X [s](X) = 2.X [0] = 2 [fSNonEmpty] = 0 [nil] = 2 [ACTIVE](X) = 2.X + 1 [CONS](X1,X2) = 0 [FIRST](X1,X2) = 0 [FROM](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 Problem 1.5: SCC Processor: -> Pairs: MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> ACTIVE(first(mark(X1:S),mark(X2:S))) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(from(X:S)) -> ACTIVE(from(mark(X:S))) MARK(from(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(from(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) Problem 1.5: Subterm Processor: -> Pairs: MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X1:S) MARK(first(X1:S,X2:S)) -> MARK(X2:S) MARK(from(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Projection: pi(MARK) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(first(s(X:S),cons(Y:S,Z:S))) -> mark(cons(Y:S,first(X:S,Z:S))) active(first(0,X:S)) -> mark(nil) active(from(X:S)) -> mark(cons(X:S,from(s(X:S)))) 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) first(active(X1:S),X2:S) -> first(X1:S,X2:S) first(mark(X1:S),X2:S) -> first(X1:S,X2:S) first(X1:S,active(X2:S)) -> first(X1:S,X2:S) first(X1:S,mark(X2:S)) -> first(X1:S,X2:S) from(active(X:S)) -> from(X:S) from(mark(X:S)) -> from(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(first(X1:S,X2:S)) -> active(first(mark(X1:S),mark(X2:S))) mark(from(X:S)) -> active(from(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(0) -> active(0) mark(nil) -> active(nil) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.