/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR X X1 X2 XS) (RULES active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVE(head(cons(X,XS))) -> MARK(X) ACTIVE(incr(cons(X,XS))) -> CONS(s(X),incr(XS)) ACTIVE(incr(cons(X,XS))) -> INCR(XS) ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(incr(cons(X,XS))) -> S(X) ACTIVE(tail(cons(X,XS))) -> MARK(XS) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) HEAD(active(X)) -> HEAD(X) HEAD(mark(X)) -> HEAD(X) INCR(active(X)) -> INCR(X) INCR(mark(X)) -> INCR(X) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> CONS(mark(X1),X2) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> ACTIVE(head(mark(X))) MARK(head(X)) -> HEAD(mark(X)) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> INCR(mark(X)) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(tail(X)) -> TAIL(mark(X)) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) S(active(X)) -> S(X) S(mark(X)) -> S(X) TAIL(active(X)) -> TAIL(X) TAIL(mark(X)) -> TAIL(X) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1: SCC Processor: -> Pairs: ACTIVE(head(cons(X,XS))) -> MARK(X) ACTIVE(incr(cons(X,XS))) -> CONS(s(X),incr(XS)) ACTIVE(incr(cons(X,XS))) -> INCR(XS) ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(incr(cons(X,XS))) -> S(X) ACTIVE(tail(cons(X,XS))) -> MARK(XS) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) HEAD(active(X)) -> HEAD(X) HEAD(mark(X)) -> HEAD(X) INCR(active(X)) -> INCR(X) INCR(mark(X)) -> INCR(X) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> CONS(mark(X1),X2) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> ACTIVE(head(mark(X))) MARK(head(X)) -> HEAD(mark(X)) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> INCR(mark(X)) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(s(X)) -> S(mark(X)) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(tail(X)) -> TAIL(mark(X)) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) S(active(X)) -> S(X) S(mark(X)) -> S(X) TAIL(active(X)) -> TAIL(X) TAIL(mark(X)) -> TAIL(X) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TAIL(active(X)) -> TAIL(X) TAIL(mark(X)) -> TAIL(X) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->->Cycle: ->->-> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->->Cycle: ->->-> Pairs: INCR(active(X)) -> INCR(X) INCR(mark(X)) -> INCR(X) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->->Cycle: ->->-> Pairs: HEAD(active(X)) -> HEAD(X) HEAD(mark(X)) -> HEAD(X) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->->Cycle: ->->-> Pairs: CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->->Cycle: ->->-> Pairs: ACTIVE(head(cons(X,XS))) -> MARK(X) ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(tail(cons(X,XS))) -> MARK(XS) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> ACTIVE(head(mark(X))) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) The problem is decomposed in 6 subproblems. Problem 1.1: Subterm Processor: -> Pairs: TAIL(active(X)) -> TAIL(X) TAIL(mark(X)) -> TAIL(X) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Projection: pi(TAIL) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Projection: pi(S) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: INCR(active(X)) -> INCR(X) INCR(mark(X)) -> INCR(X) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Projection: pi(INCR) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: HEAD(active(X)) -> HEAD(X) HEAD(mark(X)) -> HEAD(X) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Projection: pi(HEAD) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Projection: pi(CONS) = 1 Problem 1.5: SCC Processor: -> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.5: Subterm Processor: -> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Projection: pi(CONS) = 2 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(head(cons(X,XS))) -> MARK(X) ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(tail(cons(X,XS))) -> MARK(XS) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> ACTIVE(head(mark(X))) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 + 2.X2 [head](X) = 2.X + 2 [incr](X) = 2.X [mark](X) = X [s](X) = 2.X [tail](X) = 2.X + 2 [0] = 0 [nats] = 0 [odds] = 0 [pairs] = 0 [ACTIVE](X) = X + 2 [MARK](X) = X + 2 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(tail(cons(X,XS))) -> MARK(XS) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> ACTIVE(head(mark(X))) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(tail(cons(X,XS))) -> MARK(XS) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> ACTIVE(head(mark(X))) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(tail(cons(X,XS))) -> MARK(XS) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> ACTIVE(head(mark(X))) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + X2 [head](X) = 2.X + 2 [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = X + 2 [0] = 0 [nats] = 2 [odds] = 2 [pairs] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> ACTIVE(head(mark(X))) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> ACTIVE(head(mark(X))) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2)) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> ACTIVE(head(mark(X))) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 1 [head](X) = 2 [incr](X) = 2 [mark](X) = 2 [s](X) = 2 [tail](X) = 2 [0] = 0 [nats] = 2 [odds] = 2 [pairs] = 2 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> ACTIVE(head(mark(X))) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> ACTIVE(head(mark(X))) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> ACTIVE(head(mark(X))) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2 [cons](X1,X2) = 0 [head](X) = 0 [incr](X) = 1 [mark](X) = 2 [s](X) = 1 [tail](X) = 1 [0] = 0 [nats] = 1 [odds] = 1 [pairs] = 1 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(head(X)) -> MARK(X) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->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) = 2.X [mark](X) = X [s](X) = X [tail](X) = 2.X + 2 [0] = 0 [nats] = 0 [odds] = 0 [pairs] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2 [cons](X1,X2) = 0 [head](X) = 2 [incr](X) = 1 [mark](X) = 2 [s](X) = 0 [tail](X) = 1 [0] = 1 [nats] = 1 [odds] = 1 [pairs] = 1 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> ACTIVE(tail(mark(X))) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->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) = 0 [0] = 1 [nats] = 1 [odds] = 1 [pairs] = 1 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(tail(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->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) = 2.X [mark](X) = X [s](X) = X [tail](X) = 2.X + 1 [0] = 0 [nats] = 0 [odds] = 0 [pairs] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(nats) -> MARK(cons(0,incr(nats))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 + 1/2.X2 [head](X) = 2.X + 1 [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X + 1/2 [0] = 0 [nats] = 2 [odds] = 2 [pairs] = 1 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(nats) -> ACTIVE(nats) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(odds) -> MARK(incr(pairs)) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->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 + 1/2 [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X + 2 [0] = 0 [nats] = 2 [odds] = 2 [pairs] = 1 [ACTIVE](X) = X [MARK](X) = X Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(odds) -> ACTIVE(odds) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(pairs) -> ACTIVE(pairs) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) ACTIVE(pairs) -> MARK(cons(0,incr(odds))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->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 [0] = 0 [nats] = 1 [odds] = 2 [pairs] = 2 [ACTIVE](X) = 1/2.X + 1 [MARK](X) = 1/2.X + 1 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(pairs) -> ACTIVE(pairs) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->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) = X [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X + 2 [0] = 0 [nats] = 2 [odds] = 2 [pairs] = 2 [ACTIVE](X) = 1/2.X + 1/2 [MARK](X) = 1/2.X + 1/2 Problem 1.6: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.6: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,XS))) -> MARK(cons(s(X),incr(XS))) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [active](X) = X [cons](X1,X2) = 1/4.X1 + 1/4.X2 [head](X) = 4.X + 2 [incr](X) = 3/2.X + 1/4 [mark](X) = X [s](X) = 4/3.X [tail](X) = 4.X + 3/2 [0] = 1/3 [nats] = 3 [odds] = 2 [pairs] = 1 [ACTIVE](X) = 2.X + 1/4 [MARK](X) = 2.X + 1/4 Problem 1.6: SCC Processor: -> Pairs: MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.6: Subterm Processor: -> Pairs: MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Projection: pi(MARK) = 1 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: active(head(cons(X,XS))) -> mark(X) active(incr(cons(X,XS))) -> mark(cons(s(X),incr(XS))) active(tail(cons(X,XS))) -> mark(XS) active(nats) -> mark(cons(0,incr(nats))) active(odds) -> mark(incr(pairs)) active(pairs) -> mark(cons(0,incr(odds))) cons(active(X1),X2) -> cons(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) head(active(X)) -> head(X) head(mark(X)) -> head(X) incr(active(X)) -> incr(X) incr(mark(X)) -> incr(X) mark(cons(X1,X2)) -> active(cons(mark(X1),X2)) mark(head(X)) -> active(head(mark(X))) mark(incr(X)) -> active(incr(mark(X))) mark(s(X)) -> active(s(mark(X))) mark(tail(X)) -> active(tail(mark(X))) mark(0) -> active(0) mark(nats) -> active(nats) mark(odds) -> active(odds) mark(pairs) -> active(pairs) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Strongly Connected Components: There is no strongly connected component The problem is finite.