/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 Y) (RULES active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> EQ(X,Y) ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) ACTIVE(eq(0,0)) -> MARK(true) ACTIVE(eq(X,Y)) -> MARK(false) ACTIVE(inf(X)) -> CONS(X,inf(s(X))) ACTIVE(inf(X)) -> INF(s(X)) ACTIVE(inf(X)) -> MARK(cons(X,inf(s(X)))) ACTIVE(inf(X)) -> S(X) ACTIVE(length(cons(X,L))) -> LENGTH(L) ACTIVE(length(cons(X,L))) -> MARK(s(length(L))) ACTIVE(length(cons(X,L))) -> S(length(L)) ACTIVE(length(nil)) -> MARK(0) ACTIVE(take(s(X),cons(Y,L))) -> CONS(Y,take(X,L)) ACTIVE(take(s(X),cons(Y,L))) -> MARK(cons(Y,take(X,L))) ACTIVE(take(s(X),cons(Y,L))) -> TAKE(X,L) ACTIVE(take(0,X)) -> MARK(nil) 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) EQ(active(X1),X2) -> EQ(X1,X2) EQ(mark(X1),X2) -> EQ(X1,X2) EQ(X1,active(X2)) -> EQ(X1,X2) EQ(X1,mark(X2)) -> EQ(X1,X2) INF(active(X)) -> INF(X) INF(mark(X)) -> INF(X) LENGTH(active(X)) -> LENGTH(X) LENGTH(mark(X)) -> LENGTH(X) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> INF(mark(X)) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) MARK(take(X1,X2)) -> TAKE(mark(X1),mark(X2)) S(active(X)) -> S(X) S(mark(X)) -> S(X) TAKE(active(X1),X2) -> TAKE(X1,X2) TAKE(mark(X1),X2) -> TAKE(X1,X2) TAKE(X1,active(X2)) -> TAKE(X1,X2) TAKE(X1,mark(X2)) -> TAKE(X1,X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1: SCC Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> EQ(X,Y) ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) ACTIVE(eq(0,0)) -> MARK(true) ACTIVE(eq(X,Y)) -> MARK(false) ACTIVE(inf(X)) -> CONS(X,inf(s(X))) ACTIVE(inf(X)) -> INF(s(X)) ACTIVE(inf(X)) -> MARK(cons(X,inf(s(X)))) ACTIVE(inf(X)) -> S(X) ACTIVE(length(cons(X,L))) -> LENGTH(L) ACTIVE(length(cons(X,L))) -> MARK(s(length(L))) ACTIVE(length(cons(X,L))) -> S(length(L)) ACTIVE(length(nil)) -> MARK(0) ACTIVE(take(s(X),cons(Y,L))) -> CONS(Y,take(X,L)) ACTIVE(take(s(X),cons(Y,L))) -> MARK(cons(Y,take(X,L))) ACTIVE(take(s(X),cons(Y,L))) -> TAKE(X,L) ACTIVE(take(0,X)) -> MARK(nil) 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) EQ(active(X1),X2) -> EQ(X1,X2) EQ(mark(X1),X2) -> EQ(X1,X2) EQ(X1,active(X2)) -> EQ(X1,X2) EQ(X1,mark(X2)) -> EQ(X1,X2) INF(active(X)) -> INF(X) INF(mark(X)) -> INF(X) LENGTH(active(X)) -> LENGTH(X) LENGTH(mark(X)) -> LENGTH(X) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> INF(mark(X)) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) MARK(take(X1,X2)) -> TAKE(mark(X1),mark(X2)) S(active(X)) -> S(X) S(mark(X)) -> S(X) TAKE(active(X1),X2) -> TAKE(X1,X2) TAKE(mark(X1),X2) -> TAKE(X1,X2) TAKE(X1,active(X2)) -> TAKE(X1,X2) TAKE(X1,mark(X2)) -> TAKE(X1,X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TAKE(active(X1),X2) -> TAKE(X1,X2) TAKE(mark(X1),X2) -> TAKE(X1,X2) TAKE(X1,active(X2)) -> TAKE(X1,X2) TAKE(X1,mark(X2)) -> TAKE(X1,X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->->Cycle: ->->-> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->->Cycle: ->->-> Pairs: LENGTH(active(X)) -> LENGTH(X) LENGTH(mark(X)) -> LENGTH(X) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->->Cycle: ->->-> Pairs: INF(active(X)) -> INF(X) INF(mark(X)) -> INF(X) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->->Cycle: ->->-> Pairs: EQ(active(X1),X2) -> EQ(X1,X2) EQ(mark(X1),X2) -> EQ(X1,X2) EQ(X1,active(X2)) -> EQ(X1,X2) EQ(X1,mark(X2)) -> EQ(X1,X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->->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(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) ACTIVE(inf(X)) -> MARK(cons(X,inf(s(X)))) ACTIVE(length(cons(X,L))) -> MARK(s(length(L))) ACTIVE(take(s(X),cons(Y,L))) -> MARK(cons(Y,take(X,L))) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) The problem is decomposed in 7 subproblems. Problem 1.1: Subterm Processor: -> Pairs: TAKE(active(X1),X2) -> TAKE(X1,X2) TAKE(mark(X1),X2) -> TAKE(X1,X2) TAKE(X1,active(X2)) -> TAKE(X1,X2) TAKE(X1,mark(X2)) -> TAKE(X1,X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Projection: pi(TAKE) = 1 Problem 1.1: SCC Processor: -> Pairs: TAKE(X1,active(X2)) -> TAKE(X1,X2) TAKE(X1,mark(X2)) -> TAKE(X1,X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TAKE(X1,active(X2)) -> TAKE(X1,X2) TAKE(X1,mark(X2)) -> TAKE(X1,X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.1: Subterm Processor: -> Pairs: TAKE(X1,active(X2)) -> TAKE(X1,X2) TAKE(X1,mark(X2)) -> TAKE(X1,X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Projection: pi(TAKE) = 2 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->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(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Projection: pi(S) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: LENGTH(active(X)) -> LENGTH(X) LENGTH(mark(X)) -> LENGTH(X) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Projection: pi(LENGTH) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: INF(active(X)) -> INF(X) INF(mark(X)) -> INF(X) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Projection: pi(INF) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: EQ(active(X1),X2) -> EQ(X1,X2) EQ(mark(X1),X2) -> EQ(X1,X2) EQ(X1,active(X2)) -> EQ(X1,X2) EQ(X1,mark(X2)) -> EQ(X1,X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Projection: pi(EQ) = 1 Problem 1.5: SCC Processor: -> Pairs: EQ(X1,active(X2)) -> EQ(X1,X2) EQ(X1,mark(X2)) -> EQ(X1,X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: EQ(X1,active(X2)) -> EQ(X1,X2) EQ(X1,mark(X2)) -> EQ(X1,X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.5: Subterm Processor: -> Pairs: EQ(X1,active(X2)) -> EQ(X1,X2) EQ(X1,mark(X2)) -> EQ(X1,X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Projection: pi(EQ) = 2 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: 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(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Projection: pi(CONS) = 1 Problem 1.6: SCC Processor: -> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.6: Subterm Processor: -> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Projection: pi(CONS) = 2 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) ACTIVE(inf(X)) -> MARK(cons(X,inf(s(X)))) ACTIVE(length(cons(X,L))) -> MARK(s(length(L))) ACTIVE(take(s(X),cons(Y,L))) -> MARK(cons(Y,take(X,L))) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) -> Usable rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X + 1 [cons](X1,X2) = X1 + 1 [eq](X1,X2) = 1 [inf](X) = 2.X + 2 [length](X) = 2.X + 2 [mark](X) = 2.X [s](X) = 2 [take](X1,X2) = 2.X1 + 2.X2 + 2 [0] = 2 [false] = 1 [nil] = 2 [true] = 1 [ACTIVE](X) = X + 2 [MARK](X) = 2.X + 1 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) ACTIVE(length(cons(X,L))) -> MARK(s(length(L))) ACTIVE(take(s(X),cons(Y,L))) -> MARK(cons(Y,take(X,L))) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) ACTIVE(length(cons(X,L))) -> MARK(s(length(L))) ACTIVE(take(s(X),cons(Y,L))) -> MARK(cons(Y,take(X,L))) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) ACTIVE(length(cons(X,L))) -> MARK(s(length(L))) ACTIVE(take(s(X),cons(Y,L))) -> MARK(cons(Y,take(X,L))) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) -> Usable rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 + 1 [eq](X1,X2) = 0 [inf](X) = 2.X + 2 [length](X) = X + 2 [mark](X) = 2.X [s](X) = 0 [take](X1,X2) = X1 + 2.X2 + 2 [0] = 2 [false] = 0 [nil] = 2 [true] = 0 [ACTIVE](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) ACTIVE(take(s(X),cons(Y,L))) -> MARK(cons(Y,take(X,L))) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) ACTIVE(take(s(X),cons(Y,L))) -> MARK(cons(Y,take(X,L))) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) ACTIVE(take(s(X),cons(Y,L))) -> MARK(cons(Y,take(X,L))) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) -> Usable rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = 2.X1 + 2 [eq](X1,X2) = 2 [inf](X) = 2.X + 2 [length](X) = 2.X + 2 [mark](X) = X [s](X) = 2 [take](X1,X2) = 2.X1 + 2.X2 + 2 [0] = 2 [false] = 1 [nil] = 0 [true] = 1 [ACTIVE](X) = X + 2 [MARK](X) = X + 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) -> Usable rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X1 + 1 [eq](X1,X2) = 0 [inf](X) = 2.X + 2 [length](X) = 2.X + 2 [mark](X) = 2.X [s](X) = 2 [take](X1,X2) = X1 + 2.X2 + 2 [0] = 2 [false] = 0 [nil] = 2 [true] = 0 [ACTIVE](X) = 2 [MARK](X) = X + 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> ACTIVE(inf(mark(X))) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) -> Usable rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X + 1 [cons](X1,X2) = 1 [eq](X1,X2) = 0 [inf](X) = X + 2 [length](X) = 2.X + 2 [mark](X) = 2.X + 1 [s](X) = 0 [take](X1,X2) = X1 + X2 + 2 [0] = 2 [false] = 0 [nil] = 2 [true] = 0 [ACTIVE](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(inf(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) -> Usable rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X + 2 [cons](X1,X2) = X1 + 2 [eq](X1,X2) = 2 [inf](X) = 2.X + 2 [length](X) = 2.X + 2 [mark](X) = 2.X [s](X) = 2 [take](X1,X2) = 2.X1 + 2.X2 + 2 [0] = 2 [false] = 2 [nil] = 2 [true] = 2 [ACTIVE](X) = X + 2 [MARK](X) = 2.X Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) -> Usable rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2.X + 2 [cons](X1,X2) = 1 [eq](X1,X2) = 2 [inf](X) = 2 [length](X) = 1 [mark](X) = 2.X + 2 [s](X) = 1 [take](X1,X2) = 2 [0] = 1 [false] = 2 [nil] = 2 [true] = 2 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(length(X)) -> MARK(X) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) -> Usable rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [cons](X1,X2) = X2 [eq](X1,X2) = 2.X1 + X2 + 2 [inf](X) = 2.X [length](X) = 2.X + 2 [mark](X) = X [s](X) = X [take](X1,X2) = 2.X1 + 2.X2 + 2 [0] = 0 [false] = 2 [nil] = 2 [true] = 2 [ACTIVE](X) = 2.X + 1 [MARK](X) = 2.X + 1 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(s(X)) -> ACTIVE(s(X)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) -> Usable rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 0 [cons](X1,X2) = 0 [eq](X1,X2) = 2 [inf](X) = 0 [length](X) = 1 [mark](X) = 0 [s](X) = 0 [take](X1,X2) = 2 [0] = 2 [false] = 2 [nil] = 2 [true] = 2 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(take(X1,X2)) -> ACTIVE(take(mark(X1),mark(X2))) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) -> Usable rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2 [cons](X1,X2) = 2 [eq](X1,X2) = 2 [inf](X) = 2 [length](X) = 0 [mark](X) = 2 [s](X) = 0 [take](X1,X2) = 1 [0] = 2 [false] = 1 [nil] = 2 [true] = 0 [ACTIVE](X) = X [MARK](X) = 2 Problem 1.7: SCC Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.7: Reduction Pair Processor: -> Pairs: ACTIVE(eq(s(X),s(Y))) -> MARK(eq(X,Y)) MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) -> Usable rules: eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2.X [eq](X1,X2) = 2.X1 + 1 [mark](X) = 2.X [s](X) = 2.X + 1 [take](X1,X2) = X1 + 2.X2 [ACTIVE](X) = X + 2 [MARK](X) = X + 2 Problem 1.7: SCC Processor: -> Pairs: MARK(eq(X1,X2)) -> ACTIVE(eq(X1,X2)) MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) ->->-> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) Problem 1.7: Subterm Processor: -> Pairs: MARK(take(X1,X2)) -> MARK(X1) MARK(take(X1,X2)) -> MARK(X2) -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Projection: pi(MARK) = 1 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(0,0)) -> mark(true) active(eq(X,Y)) -> mark(false) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) active(length(nil)) -> mark(0) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(take(0,X)) -> mark(nil) 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) eq(active(X1),X2) -> eq(X1,X2) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) inf(active(X)) -> inf(X) inf(mark(X)) -> inf(X) length(active(X)) -> length(X) length(mark(X)) -> length(X) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(inf(X)) -> active(inf(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(s(X)) -> active(s(X)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(0) -> active(0) mark(false) -> active(false) mark(nil) -> active(nil) mark(true) -> active(true) s(active(X)) -> s(X) s(mark(X)) -> s(X) take(active(X1),X2) -> take(X1,X2) take(mark(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) ->Strongly Connected Components: There is no strongly connected component The problem is finite.