YES Problem 1: (VAR v_NonEmpty:S X:S X1:S X2:S XS:S) (RULES active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVE(head(cons(X:S,XS:S))) -> MARK(X:S) ACTIVE(incr(cons(X:S,XS:S))) -> CONS(s(X:S),incr(XS:S)) ACTIVE(incr(cons(X:S,XS:S))) -> INCR(XS:S) ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(incr(cons(X:S,XS:S))) -> S(X:S) ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) 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) HEAD(active(X:S)) -> HEAD(X:S) HEAD(mark(X:S)) -> HEAD(X:S) INCR(active(X:S)) -> INCR(X:S) INCR(mark(X:S)) -> INCR(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(head(X:S)) -> ACTIVE(head(mark(X:S))) MARK(head(X:S)) -> HEAD(mark(X:S)) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> INCR(mark(X:S)) MARK(incr(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)) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> TAIL(mark(X:S)) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) TAIL(active(X:S)) -> TAIL(X:S) TAIL(mark(X:S)) -> TAIL(X:S) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1: SCC Processor: -> Pairs: ACTIVE(head(cons(X:S,XS:S))) -> MARK(X:S) ACTIVE(incr(cons(X:S,XS:S))) -> CONS(s(X:S),incr(XS:S)) ACTIVE(incr(cons(X:S,XS:S))) -> INCR(XS:S) ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(incr(cons(X:S,XS:S))) -> S(X:S) ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) 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) HEAD(active(X:S)) -> HEAD(X:S) HEAD(mark(X:S)) -> HEAD(X:S) INCR(active(X:S)) -> INCR(X:S) INCR(mark(X:S)) -> INCR(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(head(X:S)) -> ACTIVE(head(mark(X:S))) MARK(head(X:S)) -> HEAD(mark(X:S)) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> INCR(mark(X:S)) MARK(incr(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)) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> TAIL(mark(X:S)) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) TAIL(active(X:S)) -> TAIL(X:S) TAIL(mark(X:S)) -> TAIL(X:S) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TAIL(active(X:S)) -> TAIL(X:S) TAIL(mark(X:S)) -> TAIL(X:S) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->->Cycle: ->->-> Pairs: S(active(X:S)) -> S(X:S) S(mark(X:S)) -> S(X:S) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->->Cycle: ->->-> Pairs: INCR(active(X:S)) -> INCR(X:S) INCR(mark(X:S)) -> INCR(X:S) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->->Cycle: ->->-> Pairs: HEAD(active(X:S)) -> HEAD(X:S) HEAD(mark(X:S)) -> HEAD(X:S) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(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(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->->Cycle: ->->-> Pairs: ACTIVE(head(cons(X:S,XS:S))) -> MARK(X:S) ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> ACTIVE(head(mark(X:S))) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) The problem is decomposed in 6 subproblems. Problem 1.1: Subterm Processor: -> Pairs: TAIL(active(X:S)) -> TAIL(X:S) TAIL(mark(X:S)) -> TAIL(X:S) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Projection: pi(TAIL) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X: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(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Projection: pi(S) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: INCR(active(X:S)) -> INCR(X:S) INCR(mark(X:S)) -> INCR(X:S) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Projection: pi(INCR) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: HEAD(active(X:S)) -> HEAD(X:S) HEAD(mark(X:S)) -> HEAD(X:S) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Projection: pi(HEAD) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: 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(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Projection: pi(CONS) = 1 Problem 1.5: 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(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(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(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.5: 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(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Projection: pi(CONS) = 2 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(head(cons(X:S,XS:S))) -> MARK(X:S) ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> ACTIVE(head(mark(X:S))) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) -> Usable rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [head](X) = 2.X + 2 [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X + 1 [0] = 0 [fSNonEmpty] = 0 [nats] = 0 [odds] = 0 [pairs] = 0 [ACTIVE](X) = 2.X [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [MARK](X) = 2.X [S](X) = 0 [TAIL](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> ACTIVE(head(mark(X:S))) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> ACTIVE(head(mark(X:S))) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> ACTIVE(head(mark(X:S))) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) -> Usable rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [head](X) = 2.X + 1 [incr](X) = 2.X [mark](X) = X [s](X) = X [tail](X) = 2.X + 1 [0] = 0 [fSNonEmpty] = 0 [nats] = 0 [odds] = 0 [pairs] = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 [TAIL](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> ACTIVE(head(mark(X:S))) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> ACTIVE(head(mark(X:S))) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S)) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> ACTIVE(head(mark(X:S))) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) -> Usable rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 0 [cons](X1,X2) = 1 [head](X) = 2 [incr](X) = 2 [mark](X) = 0 [s](X) = 2 [tail](X) = 1 [0] = 2 [fSNonEmpty] = 0 [nats] = 2 [odds] = 2 [pairs] = 2 [ACTIVE](X) = X [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [MARK](X) = 2 [S](X) = 0 [TAIL](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> ACTIVE(head(mark(X:S))) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> ACTIVE(head(mark(X:S))) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> ACTIVE(head(mark(X:S))) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) -> Usable rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 1 [cons](X1,X2) = 0 [head](X) = 0 [incr](X) = 1 [mark](X) = 1 [s](X) = 0 [tail](X) = 1 [0] = 2 [fSNonEmpty] = 0 [nats] = 1 [odds] = 1 [pairs] = 1 [ACTIVE](X) = 2.X [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [MARK](X) = 2 [S](X) = 0 [TAIL](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(head(X:S)) -> MARK(X:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) -> Usable rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + X2 [head](X) = 2.X + 1 [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X + 1 [0] = 0 [fSNonEmpty] = 0 [nats] = 1 [odds] = 0 [pairs] = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 [TAIL](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> ACTIVE(s(mark(X:S))) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) -> Usable rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2 [cons](X1,X2) = 0 [head](X) = 2 [incr](X) = 2 [mark](X) = 2 [s](X) = 1 [tail](X) = 1 [0] = 0 [fSNonEmpty] = 0 [nats] = 2 [odds] = 2 [pairs] = 2 [ACTIVE](X) = X [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [MARK](X) = 2 [S](X) = 0 [TAIL](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S))) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) -> Usable rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 0 [cons](X1,X2) = 0 [head](X) = 2 [incr](X) = 1 [mark](X) = 0 [s](X) = 2 [tail](X) = 0 [0] = 1 [fSNonEmpty] = 0 [nats] = 1 [odds] = 1 [pairs] = 1 [ACTIVE](X) = 2.X [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [MARK](X) = 2 [S](X) = 0 [TAIL](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(tail(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) -> Usable rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 2.X2 [head](X) = X + 2 [incr](X) = 2.X [mark](X) = X [s](X) = X [tail](X) = 2.X + 1 [0] = 0 [fSNonEmpty] = 0 [nats] = 0 [odds] = 0 [pairs] = 0 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 [TAIL](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) -> Usable rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 1/2.X2 [head](X) = 2.X + 2 [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X [0] = 0 [fSNonEmpty] = 0 [nats] = 1 [odds] = 2 [pairs] = 2 [ACTIVE](X) = 1/2.X + 1 [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [MARK](X) = 1/2.X + 1 [S](X) = 0 [TAIL](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) -> Usable rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 1/2.X2 [head](X) = 1/2.X + 1 [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X [0] = 0 [fSNonEmpty] = 0 [nats] = 2 [odds] = 1 [pairs] = 1/2 [ACTIVE](X) = 1/2.X + 2 [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [MARK](X) = 1/2.X + 2 [S](X) = 0 [TAIL](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) -> Usable rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 1/2.X2 [head](X) = X + 2 [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X [0] = 0 [fSNonEmpty] = 0 [nats] = 0 [odds] = 1/2 [pairs] = 1/2 [ACTIVE](X) = 2.X [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [MARK](X) = 2.X [S](X) = 0 [TAIL](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) -> Usable rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 + 1/2.X2 + 1/2 [head](X) = 2.X [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X + 1/2 [0] = 1/2 [fSNonEmpty] = 0 [nats] = 2 [odds] = 2 [pairs] = 2 [ACTIVE](X) = 2.X + 2 [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [MARK](X) = 2.X + 2 [S](X) = 0 [TAIL](X) = 0 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.6: Reduction Pairs Processor: -> Pairs: ACTIVE(incr(cons(X:S,XS:S))) -> MARK(cons(s(X:S),incr(XS:S))) MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) -> Usable rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [active](X) = X [cons](X1,X2) = 3/4.X1 + 1/3.X2 [head](X) = 3/2.X + 3/2 [incr](X) = X + 1/4 [mark](X) = X [s](X) = X [tail](X) = 3.X + 1 [0] = 0 [fSNonEmpty] = 0 [nats] = 4 [odds] = 1/2 [pairs] = 1/4 [ACTIVE](X) = 1/3.X + 2 [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [MARK](X) = 1/3.X + 2 [S](X) = 0 [TAIL](X) = 0 Problem 1.6: SCC Processor: -> Pairs: MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S))) MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) Problem 1.6: Subterm Processor: -> Pairs: MARK(incr(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Projection: pi(MARK) = 1 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: active(head(cons(X:S,XS:S))) -> mark(X:S) active(incr(cons(X:S,XS:S))) -> mark(cons(s(X:S),incr(XS:S))) active(tail(cons(X:S,XS:S))) -> mark(XS:S) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) 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) head(active(X:S)) -> head(X:S) head(mark(X:S)) -> head(X:S) incr(active(X:S)) -> incr(X:S) incr(mark(X:S)) -> incr(X:S) mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S)) mark(head(X:S)) -> active(head(mark(X:S))) mark(incr(X:S)) -> active(incr(mark(X:S))) mark(s(X:S)) -> active(s(mark(X:S))) mark(tail(X:S)) -> active(tail(mark(X:S))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X:S)) -> s(X:S) s(mark(X:S)) -> s(X:S) tail(active(X:S)) -> tail(X:S) tail(mark(X:S)) -> tail(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.