/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR X X1 X2 X3 Y Z) (RULES active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVE(add(s(X),Y)) -> ADD(X,Y) ACTIVE(add(s(X),Y)) -> MARK(s(add(X,Y))) ACTIVE(add(s(X),Y)) -> S(add(X,Y)) ACTIVE(add(0,X)) -> MARK(X) ACTIVE(and(false,Y)) -> MARK(false) ACTIVE(and(true,X)) -> MARK(X) ACTIVE(first(s(X),cons(Y,Z))) -> CONS(Y,first(X,Z)) ACTIVE(first(s(X),cons(Y,Z))) -> FIRST(X,Z) ACTIVE(first(s(X),cons(Y,Z))) -> MARK(cons(Y,first(X,Z))) ACTIVE(first(0,X)) -> MARK(nil) ACTIVE(from(X)) -> CONS(X,from(s(X))) ACTIVE(from(X)) -> FROM(s(X)) ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(from(X)) -> S(X) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) ADD(active(X1),X2) -> ADD(X1,X2) ADD(mark(X1),X2) -> ADD(X1,X2) ADD(X1,active(X2)) -> ADD(X1,X2) ADD(X1,mark(X2)) -> ADD(X1,X2) AND(active(X1),X2) -> AND(X1,X2) AND(mark(X1),X2) -> AND(X1,X2) AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) 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) FIRST(active(X1),X2) -> FIRST(X1,X2) FIRST(mark(X1),X2) -> FIRST(X1,X2) FIRST(X1,active(X2)) -> FIRST(X1,X2) FIRST(X1,mark(X2)) -> FIRST(X1,X2) FROM(active(X)) -> FROM(X) FROM(mark(X)) -> FROM(X) IF(active(X1),X2,X3) -> IF(X1,X2,X3) IF(mark(X1),X2,X3) -> IF(X1,X2,X3) IF(X1,active(X2),X3) -> IF(X1,X2,X3) IF(X1,mark(X2),X3) -> IF(X1,X2,X3) IF(X1,X2,active(X3)) -> IF(X1,X2,X3) IF(X1,X2,mark(X3)) -> IF(X1,X2,X3) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> ADD(mark(X1),X2) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> FIRST(mark(X1),mark(X2)) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> IF(mark(X1),X2,X3) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1: SCC Processor: -> Pairs: ACTIVE(add(s(X),Y)) -> ADD(X,Y) ACTIVE(add(s(X),Y)) -> MARK(s(add(X,Y))) ACTIVE(add(s(X),Y)) -> S(add(X,Y)) ACTIVE(add(0,X)) -> MARK(X) ACTIVE(and(false,Y)) -> MARK(false) ACTIVE(and(true,X)) -> MARK(X) ACTIVE(first(s(X),cons(Y,Z))) -> CONS(Y,first(X,Z)) ACTIVE(first(s(X),cons(Y,Z))) -> FIRST(X,Z) ACTIVE(first(s(X),cons(Y,Z))) -> MARK(cons(Y,first(X,Z))) ACTIVE(first(0,X)) -> MARK(nil) ACTIVE(from(X)) -> CONS(X,from(s(X))) ACTIVE(from(X)) -> FROM(s(X)) ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(from(X)) -> S(X) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) ADD(active(X1),X2) -> ADD(X1,X2) ADD(mark(X1),X2) -> ADD(X1,X2) ADD(X1,active(X2)) -> ADD(X1,X2) ADD(X1,mark(X2)) -> ADD(X1,X2) AND(active(X1),X2) -> AND(X1,X2) AND(mark(X1),X2) -> AND(X1,X2) AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) 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) FIRST(active(X1),X2) -> FIRST(X1,X2) FIRST(mark(X1),X2) -> FIRST(X1,X2) FIRST(X1,active(X2)) -> FIRST(X1,X2) FIRST(X1,mark(X2)) -> FIRST(X1,X2) FROM(active(X)) -> FROM(X) FROM(mark(X)) -> FROM(X) IF(active(X1),X2,X3) -> IF(X1,X2,X3) IF(mark(X1),X2,X3) -> IF(X1,X2,X3) IF(X1,active(X2),X3) -> IF(X1,X2,X3) IF(X1,mark(X2),X3) -> IF(X1,X2,X3) IF(X1,X2,active(X3)) -> IF(X1,X2,X3) IF(X1,X2,mark(X3)) -> IF(X1,X2,X3) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> ADD(mark(X1),X2) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> AND(mark(X1),X2) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> FIRST(mark(X1),mark(X2)) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> IF(mark(X1),X2,X3) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->->Cycle: ->->-> Pairs: IF(active(X1),X2,X3) -> IF(X1,X2,X3) IF(mark(X1),X2,X3) -> IF(X1,X2,X3) IF(X1,active(X2),X3) -> IF(X1,X2,X3) IF(X1,mark(X2),X3) -> IF(X1,X2,X3) IF(X1,X2,active(X3)) -> IF(X1,X2,X3) IF(X1,X2,mark(X3)) -> IF(X1,X2,X3) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->->Cycle: ->->-> Pairs: FROM(active(X)) -> FROM(X) FROM(mark(X)) -> FROM(X) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->->Cycle: ->->-> Pairs: FIRST(active(X1),X2) -> FIRST(X1,X2) FIRST(mark(X1),X2) -> FIRST(X1,X2) FIRST(X1,active(X2)) -> FIRST(X1,X2) FIRST(X1,mark(X2)) -> FIRST(X1,X2) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->->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(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->->Cycle: ->->-> Pairs: AND(active(X1),X2) -> AND(X1,X2) AND(mark(X1),X2) -> AND(X1,X2) AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->->Cycle: ->->-> Pairs: ADD(active(X1),X2) -> ADD(X1,X2) ADD(mark(X1),X2) -> ADD(X1,X2) ADD(X1,active(X2)) -> ADD(X1,X2) ADD(X1,mark(X2)) -> ADD(X1,X2) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->->Cycle: ->->-> Pairs: ACTIVE(add(s(X),Y)) -> MARK(s(add(X,Y))) ACTIVE(add(0,X)) -> MARK(X) ACTIVE(and(true,X)) -> MARK(X) ACTIVE(first(s(X),cons(Y,Z))) -> MARK(cons(Y,first(X,Z))) ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) The problem is decomposed in 8 subproblems. Problem 1.1: Subterm Processor: -> Pairs: S(active(X)) -> S(X) S(mark(X)) -> S(X) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(S) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: IF(active(X1),X2,X3) -> IF(X1,X2,X3) IF(mark(X1),X2,X3) -> IF(X1,X2,X3) IF(X1,active(X2),X3) -> IF(X1,X2,X3) IF(X1,mark(X2),X3) -> IF(X1,X2,X3) IF(X1,X2,active(X3)) -> IF(X1,X2,X3) IF(X1,X2,mark(X3)) -> IF(X1,X2,X3) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(IF) = 1 Problem 1.2: SCC Processor: -> Pairs: IF(X1,active(X2),X3) -> IF(X1,X2,X3) IF(X1,mark(X2),X3) -> IF(X1,X2,X3) IF(X1,X2,active(X3)) -> IF(X1,X2,X3) IF(X1,X2,mark(X3)) -> IF(X1,X2,X3) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: IF(X1,active(X2),X3) -> IF(X1,X2,X3) IF(X1,mark(X2),X3) -> IF(X1,X2,X3) IF(X1,X2,active(X3)) -> IF(X1,X2,X3) IF(X1,X2,mark(X3)) -> IF(X1,X2,X3) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1.2: Subterm Processor: -> Pairs: IF(X1,active(X2),X3) -> IF(X1,X2,X3) IF(X1,mark(X2),X3) -> IF(X1,X2,X3) IF(X1,X2,active(X3)) -> IF(X1,X2,X3) IF(X1,X2,mark(X3)) -> IF(X1,X2,X3) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(IF) = 2 Problem 1.2: SCC Processor: -> Pairs: IF(X1,X2,active(X3)) -> IF(X1,X2,X3) IF(X1,X2,mark(X3)) -> IF(X1,X2,X3) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: IF(X1,X2,active(X3)) -> IF(X1,X2,X3) IF(X1,X2,mark(X3)) -> IF(X1,X2,X3) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1.2: Subterm Processor: -> Pairs: IF(X1,X2,active(X3)) -> IF(X1,X2,X3) IF(X1,X2,mark(X3)) -> IF(X1,X2,X3) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(IF) = 3 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: FROM(active(X)) -> FROM(X) FROM(mark(X)) -> FROM(X) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(FROM) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: FIRST(active(X1),X2) -> FIRST(X1,X2) FIRST(mark(X1),X2) -> FIRST(X1,X2) FIRST(X1,active(X2)) -> FIRST(X1,X2) FIRST(X1,mark(X2)) -> FIRST(X1,X2) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(FIRST) = 1 Problem 1.4: SCC Processor: -> Pairs: FIRST(X1,active(X2)) -> FIRST(X1,X2) FIRST(X1,mark(X2)) -> FIRST(X1,X2) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: FIRST(X1,active(X2)) -> FIRST(X1,X2) FIRST(X1,mark(X2)) -> FIRST(X1,X2) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1.4: Subterm Processor: -> Pairs: FIRST(X1,active(X2)) -> FIRST(X1,X2) FIRST(X1,mark(X2)) -> FIRST(X1,X2) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(FIRST) = 2 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: CONS(active(X1),X2) -> CONS(X1,X2) CONS(mark(X1),X2) -> CONS(X1,X2) CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(CONS) = 1 Problem 1.5: SCC Processor: -> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1.5: Subterm Processor: -> Pairs: CONS(X1,active(X2)) -> CONS(X1,X2) CONS(X1,mark(X2)) -> CONS(X1,X2) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(CONS) = 2 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Subterm Processor: -> Pairs: AND(active(X1),X2) -> AND(X1,X2) AND(mark(X1),X2) -> AND(X1,X2) AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(AND) = 1 Problem 1.6: SCC Processor: -> Pairs: AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1.6: Subterm Processor: -> Pairs: AND(X1,active(X2)) -> AND(X1,X2) AND(X1,mark(X2)) -> AND(X1,X2) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(AND) = 2 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.7: Subterm Processor: -> Pairs: ADD(active(X1),X2) -> ADD(X1,X2) ADD(mark(X1),X2) -> ADD(X1,X2) ADD(X1,active(X2)) -> ADD(X1,X2) ADD(X1,mark(X2)) -> ADD(X1,X2) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(ADD) = 1 Problem 1.7: SCC Processor: -> Pairs: ADD(X1,active(X2)) -> ADD(X1,X2) ADD(X1,mark(X2)) -> ADD(X1,X2) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ADD(X1,active(X2)) -> ADD(X1,X2) ADD(X1,mark(X2)) -> ADD(X1,X2) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1.7: Subterm Processor: -> Pairs: ADD(X1,active(X2)) -> ADD(X1,X2) ADD(X1,mark(X2)) -> ADD(X1,X2) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(ADD) = 2 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.8: Reduction Pair Processor: -> Pairs: ACTIVE(add(s(X),Y)) -> MARK(s(add(X,Y))) ACTIVE(add(0,X)) -> MARK(X) ACTIVE(and(true,X)) -> MARK(X) ACTIVE(first(s(X),cons(Y,Z))) -> MARK(cons(Y,first(X,Z))) ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) -> Usable rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X + 2 [add](X1,X2) = 2.X1 + 2.X2 + 2 [and](X1,X2) = 2.X1 + 2.X2 + 2 [cons](X1,X2) = 2 [first](X1,X2) = 2.X1 + 2.X2 + 2 [from](X) = 2.X + 2 [if](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [mark](X) = 2.X [s](X) = 2 [0] = 2 [false] = 2 [nil] = 2 [true] = 2 [ACTIVE](X) = X + 2 [MARK](X) = 2.X Problem 1.8: SCC Processor: -> Pairs: ACTIVE(add(0,X)) -> MARK(X) ACTIVE(and(true,X)) -> MARK(X) ACTIVE(first(s(X),cons(Y,Z))) -> MARK(cons(Y,first(X,Z))) ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(add(0,X)) -> MARK(X) ACTIVE(and(true,X)) -> MARK(X) ACTIVE(first(s(X),cons(Y,Z))) -> MARK(cons(Y,first(X,Z))) ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1.8: Reduction Pair Processor: -> Pairs: ACTIVE(add(0,X)) -> MARK(X) ACTIVE(and(true,X)) -> MARK(X) ACTIVE(first(s(X),cons(Y,Z))) -> MARK(cons(Y,first(X,Z))) ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) -> Usable rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X + 1 [add](X1,X2) = 2.X1 + 2.X2 + 2 [and](X1,X2) = 2.X1 + 2.X2 + 2 [cons](X1,X2) = 1 [first](X1,X2) = 2.X1 + 2.X2 + 2 [from](X) = X + 2 [if](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 [mark](X) = 2.X [s](X) = 2 [0] = 2 [false] = 2 [nil] = 2 [true] = 2 [ACTIVE](X) = X [MARK](X) = 2.X Problem 1.8: SCC Processor: -> Pairs: ACTIVE(and(true,X)) -> MARK(X) ACTIVE(first(s(X),cons(Y,Z))) -> MARK(cons(Y,first(X,Z))) ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(and(true,X)) -> MARK(X) ACTIVE(first(s(X),cons(Y,Z))) -> MARK(cons(Y,first(X,Z))) ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1.8: Reduction Pair Processor: -> Pairs: ACTIVE(and(true,X)) -> MARK(X) ACTIVE(first(s(X),cons(Y,Z))) -> MARK(cons(Y,first(X,Z))) ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) -> Usable rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [add](X1,X2) = 2.X1 + 2.X2 [and](X1,X2) = 2.X1 + 2.X2 + 2 [cons](X1,X2) = X1 + 1 [first](X1,X2) = 2.X1 + 2.X2 [from](X) = 2.X + 2 [if](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [mark](X) = 2.X [s](X) = 2 [0] = 2 [false] = 0 [nil] = 2 [true] = 2 [ACTIVE](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1.8: SCC Processor: -> Pairs: ACTIVE(first(s(X),cons(Y,Z))) -> MARK(cons(Y,first(X,Z))) ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(first(s(X),cons(Y,Z))) -> MARK(cons(Y,first(X,Z))) ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1.8: Reduction Pair Processor: -> Pairs: ACTIVE(first(s(X),cons(Y,Z))) -> MARK(cons(Y,first(X,Z))) ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) -> Usable rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X + 2 [add](X1,X2) = X1 + 2.X2 + 2 [and](X1,X2) = 2.X1 + 2.X2 + 2 [cons](X1,X2) = X1 + 2 [first](X1,X2) = 2.X1 + 2.X2 + 2 [from](X) = 2.X + 2 [if](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [mark](X) = 2.X [s](X) = 2 [0] = 2 [false] = 2 [nil] = 2 [true] = 2 [ACTIVE](X) = X + 2 [MARK](X) = 2.X Problem 1.8: SCC Processor: -> Pairs: ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1.8: Reduction Pair Processor: -> Pairs: ACTIVE(from(X)) -> MARK(cons(X,from(s(X)))) ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) -> Usable rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X + 1 [add](X1,X2) = X1 + 2.X2 + 2 [and](X1,X2) = 2.X1 + 2.X2 + 2 [cons](X1,X2) = 0 [first](X1,X2) = X1 + X2 + 2 [from](X) = 2.X + 2 [if](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [mark](X) = 2.X + 1 [s](X) = 2 [0] = 2 [false] = 0 [nil] = 2 [true] = 2 [ACTIVE](X) = X + 1 [MARK](X) = 2.X + 2 Problem 1.8: SCC Processor: -> Pairs: ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1.8: Reduction Pair Processor: -> Pairs: ACTIVE(if(false,X,Y)) -> MARK(Y) ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) -> Usable rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X + 2 [add](X1,X2) = 2.X1 + 2.X2 + 2 [and](X1,X2) = X1 + 2.X2 + 2 [cons](X1,X2) = 2 [first](X1,X2) = 2.X1 + 2.X2 + 2 [from](X) = 2 [if](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 [mark](X) = 2.X [s](X) = 2 [0] = 2 [false] = 2 [nil] = 2 [true] = 2 [ACTIVE](X) = X + 2 [MARK](X) = 2.X + 2 Problem 1.8: SCC Processor: -> Pairs: ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1.8: Reduction Pair Processor: -> Pairs: ACTIVE(if(true,X,Y)) -> MARK(X) MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) -> Usable rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = X [add](X1,X2) = 2.X1 + X2 [and](X1,X2) = 2.X1 + 2.X2 [cons](X1,X2) = 2.X1 + 2 [first](X1,X2) = X1 + 2.X2 + 1 [from](X) = 2.X + 2 [if](X1,X2,X3) = 2.X1 + X2 + 2.X3 + 2 [mark](X) = X [s](X) = 0 [0] = 0 [false] = 2 [nil] = 1 [true] = 2 [ACTIVE](X) = 2.X + 2 [MARK](X) = 2.X + 2 Problem 1.8: SCC Processor: -> Pairs: MARK(add(X1,X2)) -> ACTIVE(add(mark(X1),X2)) MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> ACTIVE(and(mark(X1),X2)) MARK(and(X1,X2)) -> MARK(X1) MARK(cons(X1,X2)) -> ACTIVE(cons(X1,X2)) MARK(first(X1,X2)) -> ACTIVE(first(mark(X1),mark(X2))) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(from(X)) -> ACTIVE(from(X)) MARK(if(X1,X2,X3)) -> ACTIVE(if(mark(X1),X2,X3)) MARK(if(X1,X2,X3)) -> MARK(X1) MARK(s(X)) -> ACTIVE(s(X)) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(if(X1,X2,X3)) -> MARK(X1) ->->-> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) Problem 1.8: Subterm Processor: -> Pairs: MARK(add(X1,X2)) -> MARK(X1) MARK(and(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X1) MARK(first(X1,X2)) -> MARK(X2) MARK(if(X1,X2,X3)) -> MARK(X1) -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Projection: pi(MARK) = 1 Problem 1.8: SCC Processor: -> Pairs: Empty -> Rules: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(add(0,X)) -> mark(X) active(and(false,Y)) -> mark(false) active(and(true,X)) -> mark(X) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(first(0,X)) -> mark(nil) active(from(X)) -> mark(cons(X,from(s(X)))) active(if(false,X,Y)) -> mark(Y) active(if(true,X,Y)) -> mark(X) add(active(X1),X2) -> add(X1,X2) add(mark(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) and(active(X1),X2) -> and(X1,X2) and(mark(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) 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) first(active(X1),X2) -> first(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) from(active(X)) -> from(X) from(mark(X)) -> from(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(from(X)) -> active(from(X)) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(s(X)) -> active(s(X)) 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) ->Strongly Connected Components: There is no strongly connected component The problem is finite.