/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR L X X1 X2) (RULES active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> ADX(L) ACTIVE(adx(cons(X,L))) -> CONS(X,adx(L)) ACTIVE(adx(cons(X,L))) -> INCR(cons(X,adx(L))) ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(adx(nil)) -> MARK(nil) ACTIVE(head(cons(X,L))) -> MARK(X) ACTIVE(incr(cons(X,L))) -> CONS(s(X),incr(L)) ACTIVE(incr(cons(X,L))) -> INCR(L) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(incr(cons(X,L))) -> S(X) ACTIVE(incr(nil)) -> MARK(nil) ACTIVE(tail(cons(X,L))) -> MARK(L) ACTIVE(nats) -> MARK(adx(zeros)) ACTIVE(zeros) -> MARK(cons(0,zeros)) ADX(active(X)) -> ADX(X) ADX(mark(X)) -> ADX(X) 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(adx(X)) -> ACTIVE(adx(mark(X))) MARK(adx(X)) -> ADX(mark(X)) MARK(adx(X)) -> MARK(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(zeros) -> ACTIVE(zeros) S(active(X)) -> S(X) S(mark(X)) -> S(X) TAIL(active(X)) -> TAIL(X) TAIL(mark(X)) -> TAIL(X) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> ADX(L) ACTIVE(adx(cons(X,L))) -> CONS(X,adx(L)) ACTIVE(adx(cons(X,L))) -> INCR(cons(X,adx(L))) ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(adx(nil)) -> MARK(nil) ACTIVE(head(cons(X,L))) -> MARK(X) ACTIVE(incr(cons(X,L))) -> CONS(s(X),incr(L)) ACTIVE(incr(cons(X,L))) -> INCR(L) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(incr(cons(X,L))) -> S(X) ACTIVE(incr(nil)) -> MARK(nil) ACTIVE(tail(cons(X,L))) -> MARK(L) ACTIVE(nats) -> MARK(adx(zeros)) ACTIVE(zeros) -> MARK(cons(0,zeros)) ADX(active(X)) -> ADX(X) ADX(mark(X)) -> ADX(X) 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(adx(X)) -> ACTIVE(adx(mark(X))) MARK(adx(X)) -> ADX(mark(X)) MARK(adx(X)) -> MARK(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(zeros) -> ACTIVE(zeros) S(active(X)) -> S(X) S(mark(X)) -> S(X) TAIL(active(X)) -> TAIL(X) TAIL(mark(X)) -> TAIL(X) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->->Cycle: ->->-> Pairs: ADX(active(X)) -> ADX(X) ADX(mark(X)) -> ADX(X) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->->Cycle: ->->-> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(head(cons(X,L))) -> MARK(X) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(tail(cons(X,L))) -> MARK(L) ACTIVE(nats) -> MARK(adx(zeros)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(adx(X)) -> MARK(X) 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(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 7 subproblems. Problem 1.1: Subterm Processor: -> Pairs: TAIL(active(X)) -> TAIL(X) TAIL(mark(X)) -> TAIL(X) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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: Subterm Processor: -> Pairs: ADX(active(X)) -> ADX(X) ADX(mark(X)) -> ADX(X) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) ->Projection: pi(ADX) = 1 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(head(cons(X,L))) -> MARK(X) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(tail(cons(X,L))) -> MARK(L) ACTIVE(nats) -> MARK(adx(zeros)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(adx(X)) -> MARK(X) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 [adx](X) = X + 1 [cons](X1,X2) = 2.X1 + X2 [head](X) = X + 2 [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X [0] = 0 [nats] = 2 [nil] = 2 [zeros] = 1 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(tail(cons(X,L))) -> MARK(L) ACTIVE(nats) -> MARK(adx(zeros)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(adx(X)) -> MARK(X) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(tail(cons(X,L))) -> MARK(L) ACTIVE(nats) -> MARK(adx(zeros)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(adx(X)) -> MARK(X) 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(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(tail(cons(X,L))) -> MARK(L) ACTIVE(nats) -> MARK(adx(zeros)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(adx(X)) -> MARK(X) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 [adx](X) = 2.X [cons](X1,X2) = 2.X1 + 2.X2 [head](X) = X + 1 [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X + 1 [0] = 0 [nats] = 2 [nil] = 2 [zeros] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(nats) -> MARK(adx(zeros)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(adx(X)) -> MARK(X) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(nats) -> MARK(adx(zeros)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(adx(X)) -> MARK(X) 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(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(nats) -> MARK(adx(zeros)) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(adx(X)) -> MARK(X) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 [adx](X) = 2.X [cons](X1,X2) = 2.X1 + 2.X2 [head](X) = 2.X [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X + 2 [0] = 0 [nats] = 2 [nil] = 2 [zeros] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(adx(X)) -> MARK(X) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(adx(X)) -> MARK(X) 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(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(adx(X)) -> MARK(X) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 [adx](X) = X + 1 [cons](X1,X2) = 2.X1 + X2 [head](X) = X [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X + 2 [0] = 0 [nats] = 2 [nil] = 2 [zeros] = 1 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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.X [adx](X) = 1 [cons](X1,X2) = 0 [head](X) = 1 [incr](X) = 1 [mark](X) = 2 [s](X) = 1 [tail](X) = 1 [0] = 1 [nats] = 1 [nil] = 1 [zeros] = 1 [ACTIVE](X) = X + 1 [MARK](X) = 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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) = 0 [adx](X) = 1 [cons](X1,X2) = 0 [head](X) = 0 [incr](X) = 1 [mark](X) = 0 [s](X) = 0 [tail](X) = 1 [0] = 0 [nats] = 0 [nil] = 2 [zeros] = 1 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 [adx](X) = 2.X [cons](X1,X2) = 2.X1 + 2.X2 [head](X) = X + 1 [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X [0] = 0 [nats] = 1 [nil] = 2 [zeros] = 0 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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.X [adx](X) = 1 [cons](X1,X2) = 1 [head](X) = 1 [incr](X) = 1 [mark](X) = 2 [s](X) = 0 [tail](X) = 1 [0] = 0 [nats] = 1 [nil] = 0 [zeros] = 1 [ACTIVE](X) = X + 1 [MARK](X) = 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 [adx](X) = 1 [cons](X1,X2) = 1 [head](X) = 0 [incr](X) = 1 [mark](X) = 2 [s](X) = 1 [tail](X) = 0 [0] = 0 [nats] = 2 [nil] = 0 [zeros] = 1 [ACTIVE](X) = 2.X [MARK](X) = 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) 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(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 [adx](X) = 2.X [cons](X1,X2) = X1 + 2.X2 [head](X) = 2.X + 2 [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = X + 2 [0] = 0 [nats] = 1 [nil] = 2 [zeros] = 0 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(adx(cons(X,L))) -> MARK(incr(cons(X,adx(L)))) ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 [adx](X) = X + 1/2 [cons](X1,X2) = X1 + 1/2.X2 [head](X) = 2.X + 1/2 [incr](X) = X [mark](X) = X [s](X) = X [tail](X) = 2.X + 1/2 [0] = 1/2 [nats] = 2 [nil] = 0 [zeros] = 1 [ACTIVE](X) = 2.X [MARK](X) = 2.X Problem 1.7: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(adx(X)) -> ACTIVE(adx(mark(X))) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 [adx](X) = 1 [cons](X1,X2) = 0 [head](X) = 0 [incr](X) = 2 [mark](X) = 1 [s](X) = 1 [tail](X) = 0 [0] = 2 [nats] = 0 [nil] = 0 [zeros] = 2 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(incr(cons(X,L))) -> MARK(cons(s(X),incr(L))) ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 [adx](X) = 2.X + 1 [cons](X1,X2) = X1 + 1/2.X2 [head](X) = 2.X + 2 [incr](X) = X + 1/2 [mark](X) = X [s](X) = X [tail](X) = 2.X + 1 [0] = 0 [nats] = 2 [nil] = 0 [zeros] = 1/2 [ACTIVE](X) = 2.X [MARK](X) = 2.X Problem 1.7: SCC Processor: -> Pairs: ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(zeros) -> MARK(cons(0,zeros)) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> ACTIVE(incr(mark(X))) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 [adx](X) = X [cons](X1,X2) = 2.X1 + X2 [head](X) = 2.X + 2 [incr](X) = 0 [mark](X) = X [s](X) = 0 [tail](X) = 2.X + 2 [0] = 0 [nats] = 2 [nil] = 0 [zeros] = 2 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(zeros) -> MARK(cons(0,zeros)) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(zeros) -> MARK(cons(0,zeros)) MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) -> Usable rules: 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) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 [incr](X) = X [mark](X) = X [s](X) = 2.X [0] = 1 [zeros] = 2 [ACTIVE](X) = X + 2 [MARK](X) = 2.X + 1 Problem 1.7: SCC Processor: -> Pairs: MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(zeros) -> ACTIVE(zeros) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) ->->-> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) s(active(X)) -> s(X) s(mark(X)) -> s(X) tail(active(X)) -> tail(X) tail(mark(X)) -> tail(X) Problem 1.7: Subterm Processor: -> Pairs: MARK(cons(X1,X2)) -> MARK(X1) MARK(incr(X)) -> MARK(X) MARK(s(X)) -> MARK(X) -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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 1.7: SCC Processor: -> Pairs: Empty -> Rules: active(adx(cons(X,L))) -> mark(incr(cons(X,adx(L)))) active(adx(nil)) -> mark(nil) active(head(cons(X,L))) -> mark(X) active(incr(cons(X,L))) -> mark(cons(s(X),incr(L))) active(incr(nil)) -> mark(nil) active(tail(cons(X,L))) -> mark(L) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(active(X)) -> adx(X) adx(mark(X)) -> adx(X) 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(adx(X)) -> active(adx(mark(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(nil) -> active(nil) mark(zeros) -> active(zeros) 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.