YES Problem 1: (VAR v_NonEmpty:S M:S N:S X:S X1:S X2:S) (RULES active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S))) ACTIVE(plus(N:S,s(M:S))) -> PLUS(N:S,M:S) ACTIVE(plus(N:S,s(M:S))) -> S(plus(N:S,M:S)) ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) ACTIVE(x(N:S,s(M:S))) -> PLUS(x(N:S,M:S),N:S) ACTIVE(x(N:S,s(M:S))) -> X(N:S,M:S) ACTIVE(x(N:S,0)) -> MARK(0) AND(active(X1:S),X2:S) -> AND(X1:S,X2:S) AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2: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)) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) MARK(x(X1:S,X2:S)) -> X(mark(X1:S),mark(X2:S)) PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) X(active(X1:S),X2:S) -> X(X1:S,X2:S) X(mark(X1:S),X2:S) -> X(X1:S,X2:S) X(X1:S,active(X2:S)) -> X(X1:S,X2:S) X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) Problem 1: SCC Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S))) ACTIVE(plus(N:S,s(M:S))) -> PLUS(N:S,M:S) ACTIVE(plus(N:S,s(M:S))) -> S(plus(N:S,M:S)) ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) ACTIVE(x(N:S,s(M:S))) -> PLUS(x(N:S,M:S),N:S) ACTIVE(x(N:S,s(M:S))) -> X(N:S,M:S) ACTIVE(x(N:S,0)) -> MARK(0) AND(active(X1:S),X2:S) -> AND(X1:S,X2:S) AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2: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)) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) MARK(x(X1:S,X2:S)) -> X(mark(X1:S),mark(X2:S)) PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) X(active(X1:S),X2:S) -> X(X1:S,X2:S) X(mark(X1:S),X2:S) -> X(X1:S,X2:S) X(X1:S,active(X2:S)) -> X(X1:S,X2:S) X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: X(active(X1:S),X2:S) -> X(X1:S,X2:S) X(mark(X1:S),X2:S) -> X(X1:S,X2:S) X(X1:S,active(X2:S)) -> X(X1:S,X2:S) X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->->Cycle: ->->-> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->->Cycle: ->->-> Pairs: PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->->Cycle: ->->-> Pairs: AND(active(X1:S),X2:S) -> AND(X1:S,X2:S) AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S))) ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) The problem is decomposed in 5 subproblems. Problem 1.1: Subterm Processor: -> Pairs: X(active(X1:S),X2:S) -> X(X1:S,X2:S) X(mark(X1:S),X2:S) -> X(X1:S,X2:S) X(X1:S,active(X2:S)) -> X(X1:S,X2:S) X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Projection: pi(X) = 1 Problem 1.1: SCC Processor: -> Pairs: X(X1:S,active(X2:S)) -> X(X1:S,X2:S) X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: X(X1:S,active(X2:S)) -> X(X1:S,X2:S) X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) Problem 1.1: Subterm Processor: -> Pairs: X(X1:S,active(X2:S)) -> X(X1:S,X2:S) X(X1:S,mark(X2:S)) -> X(X1:S,X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Projection: pi(X) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(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(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Projection: pi(S) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Projection: pi(PLUS) = 1 Problem 1.3: SCC Processor: -> Pairs: PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) Problem 1.3: Subterm Processor: -> Pairs: PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S) PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Projection: pi(PLUS) = 2 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: AND(active(X1:S),X2:S) -> AND(X1:S,X2:S) AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S) AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Projection: pi(AND) = 1 Problem 1.4: SCC Processor: -> Pairs: AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) Problem 1.4: Subterm Processor: -> Pairs: AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S) AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Projection: pi(AND) = 2 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Reduction Pairs Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S))) ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) -> Usable rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2 [and](X1,X2) = 1 [mark](X) = 2 [plus](X1,X2) = 1 [s](X) = 0 [x](X1,X2) = 1 [0] = 2 [fSNonEmpty] = 0 [tt] = 1 [ACTIVE](X) = 2.X [AND](X1,X2) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 [X](X1,X2) = 0 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S))) ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S))) ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) Problem 1.5: Reduction Pairs Processor: -> Pairs: ACTIVE(and(tt,X:S)) -> MARK(X:S) ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S))) ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) -> Usable rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [and](X1,X2) = 2.X1 + 2.X2 + 2 [mark](X) = X [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [x](X1,X2) = X1.X2 + 2.X1 + 2.X2 + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [ACTIVE](X) = 2.X.X + X + 2 [AND](X1,X2) = 0 [MARK](X) = 2.X.X + 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 [X](X1,X2) = 0 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S))) ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S))) ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) Problem 1.5: Reduction Pairs Processor: -> Pairs: ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S))) ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) -> Usable rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2 [and](X1,X2) = 0 [mark](X) = 2 [plus](X1,X2) = 1 [s](X) = 0 [x](X1,X2) = 1 [0] = 0 [fSNonEmpty] = 0 [tt] = 2 [ACTIVE](X) = 2.X [AND](X1,X2) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 [X](X1,X2) = 0 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S))) ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S))) ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) Problem 1.5: Reduction Pairs Processor: -> Pairs: ACTIVE(plus(N:S,s(M:S))) -> MARK(s(plus(N:S,M:S))) ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) -> Usable rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [and](X1,X2) = X1.X2 + X1 + X2 + 2 [mark](X) = X [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [0] = 2 [fSNonEmpty] = 0 [tt] = 0 [ACTIVE](X) = 2.X.X + 2.X + 2 [AND](X1,X2) = 0 [MARK](X) = 2.X.X + 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 [X](X1,X2) = 0 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) Problem 1.5: Reduction Pairs Processor: -> Pairs: ACTIVE(plus(N:S,0)) -> MARK(N:S) ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) -> Usable rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [and](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1 [mark](X) = X [plus](X1,X2) = X1 + 2.X2 + 2 [s](X) = X + 2 [x](X1,X2) = 2.X1.X2 + 2.X1 + X2 + 1 [0] = 0 [fSNonEmpty] = 0 [tt] = 2 [ACTIVE](X) = 2.X + 2 [AND](X1,X2) = 0 [MARK](X) = 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 [X](X1,X2) = 0 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) Problem 1.5: Reduction Pairs Processor: -> Pairs: ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S))) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) -> Usable rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2 [and](X1,X2) = 2 [mark](X) = 2 [plus](X1,X2) = 0 [s](X) = 2 [x](X1,X2) = 1 [0] = 0 [fSNonEmpty] = 0 [tt] = 2 [ACTIVE](X) = X + 1 [AND](X1,X2) = 0 [MARK](X) = 2 [PLUS](X1,X2) = 0 [S](X) = 0 [X](X1,X2) = 0 Problem 1.5: SCC Processor: -> Pairs: ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) Problem 1.5: Reduction Pairs Processor: -> Pairs: ACTIVE(x(N:S,s(M:S))) -> MARK(plus(x(N:S,M:S),N:S)) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) -> Usable rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Interpretation type: Simple mixed ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [and](X1,X2) = 2.X1 + X2 + 2 [mark](X) = X [plus](X1,X2) = X1 + 2.X2 [s](X) = X + 1 [x](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 2 [0] = 1 [fSNonEmpty] = 0 [tt] = 0 [ACTIVE](X) = 2.X.X + 2.X + 1 [AND](X1,X2) = 0 [MARK](X) = 2.X.X + 2.X + 2 [PLUS](X1,X2) = 0 [S](X) = 0 [X](X1,X2) = 0 Problem 1.5: SCC Processor: -> Pairs: MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> ACTIVE(x(mark(X1:S),mark(X2:S))) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) ->->-> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) Problem 1.5: Subterm Processor: -> Pairs: MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X1:S) MARK(plus(X1:S,X2:S)) -> MARK(X2:S) MARK(s(X:S)) -> MARK(X:S) MARK(x(X1:S,X2:S)) -> MARK(X1:S) MARK(x(X1:S,X2:S)) -> MARK(X2:S) -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Projection: pi(MARK) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(and(tt,X:S)) -> mark(X:S) active(plus(N:S,s(M:S))) -> mark(s(plus(N:S,M:S))) active(plus(N:S,0)) -> mark(N:S) active(x(N:S,s(M:S))) -> mark(plus(x(N:S,M:S),N:S)) active(x(N:S,0)) -> mark(0) and(active(X1:S),X2:S) -> and(X1:S,X2:S) and(mark(X1:S),X2:S) -> and(X1:S,X2:S) and(X1:S,active(X2:S)) -> and(X1:S,X2:S) and(X1:S,mark(X2:S)) -> and(X1:S,X2:S) mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S)) mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(x(X1:S,X2:S)) -> active(x(mark(X1:S),mark(X2:S))) mark(0) -> active(0) mark(tt) -> active(tt) plus(active(X1:S),X2:S) -> plus(X1:S,X2:S) plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S) plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S) plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) x(active(X1:S),X2:S) -> x(X1:S,X2:S) x(mark(X1:S),X2:S) -> x(X1:S,X2:S) x(X1:S,active(X2:S)) -> x(X1:S,X2:S) x(X1:S,mark(X2:S)) -> x(X1:S,X2:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.