/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES Problem 1: (VAR v_NonEmpty:S N:S X:S X1:S X2:S Y:S Z:S) (RULES activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ) Problem 1: Dependency Pairs Processor: -> Pairs: ACTIVATE(n__add(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__add(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__add(X1:S,X2:S)) -> ADD(activate(X1:S),activate(X2:S)) ACTIVATE(n__dbl(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> S(X:S) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) ADD(s(X:S),Y:S) -> ACTIVATE(X:S) ADD(s(X:S),Y:S) -> S(n__add(activate(X:S),Y:S)) DBL(s(X:S)) -> ACTIVATE(X:S) DBL(s(X:S)) -> S(n__s(n__dbl(activate(X:S)))) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) SQR(s(X:S)) -> S(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__add(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__add(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__add(X1:S,X2:S)) -> ADD(activate(X1:S),activate(X2:S)) ACTIVATE(n__dbl(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__s(X:S)) -> S(X:S) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) ADD(s(X:S),Y:S) -> ACTIVATE(X:S) ADD(s(X:S),Y:S) -> S(n__add(activate(X:S),Y:S)) DBL(s(X:S)) -> ACTIVATE(X:S) DBL(s(X:S)) -> S(n__s(n__dbl(activate(X:S)))) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) SQR(s(X:S)) -> S(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__add(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__add(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__add(X1:S,X2:S)) -> ADD(activate(X1:S),activate(X2:S)) ACTIVATE(n__dbl(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) ADD(s(X:S),Y:S) -> ACTIVATE(X:S) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) ->->-> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__add(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__add(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__add(X1:S,X2:S)) -> ADD(activate(X1:S),activate(X2:S)) ACTIVATE(n__dbl(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) ADD(s(X:S),Y:S) -> ACTIVATE(X:S) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) -> Usable rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Interpretation type: Simple mixed ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [activate](X) = X [add](X1,X2) = X1 + X2 + 1 [dbl](X) = 2.X + 2 [first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [s](X) = X + 2 [sqr](X) = 2.X.X + X + 2 [terms](X) = X + 2 [0] = 1 [cons](X1,X2) = 1/2.X1.X2 + 2.X1 + 1/2.X2 [n__add](X1,X2) = X1 + X2 + 1 [n__dbl](X) = 2.X + 2 [n__first](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 [n__s](X) = X + 2 [n__sqr](X) = 2.X.X + X + 2 [n__terms](X) = X + 2 [nil] = 0 [recip](X) = 0 [ACTIVATE](X) = 2.X + 2 [ADD](X1,X2) = 2.X1 + 1/2.X2 + 1 [DBL](X) = 2.X + 1/2 [FIRST](X1,X2) = 2.X1.X2 + 2.X1 + 1 [SQR](X) = 2.X [TERMS](X) = 2.X Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__add(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__add(X1:S,X2:S)) -> ADD(activate(X1:S),activate(X2:S)) ACTIVATE(n__dbl(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) ADD(s(X:S),Y:S) -> ACTIVATE(X:S) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__add(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__add(X1:S,X2:S)) -> ADD(activate(X1:S),activate(X2:S)) ACTIVATE(n__dbl(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) ADD(s(X:S),Y:S) -> ACTIVATE(X:S) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) ->->-> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__add(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__add(X1:S,X2:S)) -> ADD(activate(X1:S),activate(X2:S)) ACTIVATE(n__dbl(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) ADD(s(X:S),Y:S) -> ACTIVATE(X:S) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) -> Usable rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Interpretation type: Simple mixed ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [activate](X) = X [add](X1,X2) = X1 + X2 + 1 [dbl](X) = 2.X [first](X1,X2) = 2.X1 + 2.X2 + 1/2 [s](X) = X + 1/2 [sqr](X) = 2.X.X + 2.X + 2 [terms](X) = X + 1/2 [0] = 0 [cons](X1,X2) = 1/2.X2 [n__add](X1,X2) = X1 + X2 + 1 [n__dbl](X) = 2.X [n__first](X1,X2) = 2.X1 + 2.X2 + 1/2 [n__s](X) = X + 1/2 [n__sqr](X) = 2.X.X + 2.X + 2 [n__terms](X) = X + 1/2 [nil] = 0 [recip](X) = 1/2.X + 1 [ACTIVATE](X) = X [ADD](X1,X2) = X1 + X2 + 1/2 [DBL](X) = 2.X [FIRST](X1,X2) = X1 + 2.X2 + 1/2 [SQR](X) = X + 1/2 [TERMS](X) = X + 1/2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__add(X1:S,X2:S)) -> ADD(activate(X1:S),activate(X2:S)) ACTIVATE(n__dbl(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) ADD(s(X:S),Y:S) -> ACTIVATE(X:S) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__add(X1:S,X2:S)) -> ADD(activate(X1:S),activate(X2:S)) ACTIVATE(n__dbl(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) ADD(s(X:S),Y:S) -> ACTIVATE(X:S) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) ->->-> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__add(X1:S,X2:S)) -> ADD(activate(X1:S),activate(X2:S)) ACTIVATE(n__dbl(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) ADD(s(X:S),Y:S) -> ACTIVATE(X:S) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) -> Usable rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Interpretation type: Simple mixed ->Coefficients: All rationals ->Dimension: 1 ->Bound: 2 ->Interpretation: [activate](X) = X [add](X1,X2) = X1 + X2 [dbl](X) = 2.X + 2 [first](X1,X2) = 2.X1.X2 + X1 + X2 + 2 [s](X) = X + 2 [sqr](X) = 1/2.X.X + 2.X + 2 [terms](X) = X + 2 [0] = 2 [cons](X1,X2) = 1/2.X2 [n__add](X1,X2) = X1 + X2 [n__dbl](X) = 2.X + 2 [n__first](X1,X2) = 2.X1.X2 + X1 + X2 + 2 [n__s](X) = X + 2 [n__sqr](X) = 1/2.X.X + 2.X + 2 [n__terms](X) = X + 2 [nil] = 2 [recip](X) = 2 [ACTIVATE](X) = 1/2.X + 1 [ADD](X1,X2) = 1/2.X1 [DBL](X) = 1/2.X + 1 [FIRST](X1,X2) = X1.X2 + 1/2.X1 + 1 [SQR](X) = 1/2.X + 2 [TERMS](X) = 1/2.X + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__dbl(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) ADD(s(X:S),Y:S) -> ACTIVATE(X:S) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__dbl(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) ->->-> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__dbl(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) -> Usable rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [activate](X) = X [add](X1,X2) = 2.X2 [dbl](X) = X + 1 [first](X1,X2) = 2.X1 + 2.X2 + 1 [s](X) = X [sqr](X) = 2.X + 2 [terms](X) = 2.X + 1 [0] = 2 [cons](X1,X2) = X1 + X2 [n__add](X1,X2) = 2.X2 [n__dbl](X) = X + 1 [n__first](X1,X2) = 2.X1 + 2.X2 + 1 [n__s](X) = X [n__sqr](X) = 2.X + 2 [n__terms](X) = 2.X + 1 [nil] = 2 [recip](X) = 0 [ACTIVATE](X) = 2.X [DBL](X) = 2.X [FIRST](X1,X2) = 2.X1 + 2.X2 + 2 [SQR](X) = 2.X + 2 [TERMS](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) ->->-> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__dbl(X:S)) -> DBL(activate(X:S)) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) -> Usable rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [activate](X) = X [add](X1,X2) = X2 [dbl](X) = 2.X + 2 [first](X1,X2) = X1 + 2.X2 + 1 [s](X) = X [sqr](X) = 2.X + 2 [terms](X) = X + 2 [0] = 2 [cons](X1,X2) = 2.X1 + X2 [n__add](X1,X2) = X2 [n__dbl](X) = 2.X + 2 [n__first](X1,X2) = X1 + 2.X2 + 1 [n__s](X) = X [n__sqr](X) = 2.X + 2 [n__terms](X) = X + 2 [nil] = 2 [recip](X) = 0 [ACTIVATE](X) = X + 2 [DBL](X) = 2.X + 2 [FIRST](X1,X2) = X1 + X2 + 2 [SQR](X) = X + 2 [TERMS](X) = X + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) DBL(s(X:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) ->->-> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X1:S) ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) -> Usable rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [activate](X) = X [add](X1,X2) = X1 + 2.X2 [dbl](X) = 0 [first](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [sqr](X) = 2.X + 2 [terms](X) = 2.X + 1 [0] = 0 [cons](X1,X2) = X2 [n__add](X1,X2) = X1 + 2.X2 [n__dbl](X) = 0 [n__first](X1,X2) = 2.X1 + 2.X2 + 2 [n__s](X) = X [n__sqr](X) = 2.X + 2 [n__terms](X) = 2.X + 1 [nil] = 2 [recip](X) = 2.X + 1 [ACTIVATE](X) = 2.X + 1 [FIRST](X1,X2) = 2.X1 + 2.X2 + 1 [SQR](X) = 2.X + 1 [TERMS](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) ->->-> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__first(X1:S,X2:S)) -> ACTIVATE(X2:S) ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) -> Usable rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [activate](X) = X [add](X1,X2) = X1 + 2.X2 [dbl](X) = 0 [first](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [sqr](X) = 2.X + 2 [terms](X) = X + 2 [0] = 0 [cons](X1,X2) = X2 [n__add](X1,X2) = X1 + 2.X2 [n__dbl](X) = 0 [n__first](X1,X2) = 2.X1 + 2.X2 + 2 [n__s](X) = X [n__sqr](X) = 2.X + 2 [n__terms](X) = X + 2 [nil] = 2 [recip](X) = 1 [ACTIVATE](X) = 2.X [FIRST](X1,X2) = 2.X1 + 2.X2 + 2 [SQR](X) = 2.X [TERMS](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) ->->-> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__first(X1:S,X2:S)) -> FIRST(activate(X1:S),activate(X2:S)) ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) -> Usable rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [activate](X) = X [add](X1,X2) = X1 + 2.X2 [dbl](X) = 0 [first](X1,X2) = 2.X1 + 2.X2 + 2 [s](X) = X [sqr](X) = X + 2 [terms](X) = 2.X + 2 [0] = 0 [cons](X1,X2) = 2.X1 + X2 [n__add](X1,X2) = X1 + 2.X2 [n__dbl](X) = 0 [n__first](X1,X2) = 2.X1 + 2.X2 + 2 [n__s](X) = X [n__sqr](X) = X + 2 [n__terms](X) = 2.X + 2 [nil] = 2 [recip](X) = 0 [ACTIVATE](X) = 2.X + 2 [FIRST](X1,X2) = 2.X1 + 2.X2 + 2 [SQR](X) = 2.X + 2 [TERMS](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(X:S) FIRST(s(X:S),cons(Y:S,Z:S)) -> ACTIVATE(Z:S) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) ->->-> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__sqr(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) -> Usable rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [activate](X) = 2.X + 2 [add](X1,X2) = 2.X2 + 2 [dbl](X) = 2 [first](X1,X2) = 2.X1 + 2 [s](X) = X [sqr](X) = 2.X + 2 [terms](X) = 2.X + 2 [0] = 2 [cons](X1,X2) = 2 [n__add](X1,X2) = 2.X2 + 2 [n__dbl](X) = 0 [n__first](X1,X2) = 2.X1 + 2 [n__s](X) = X [n__sqr](X) = 2.X + 2 [n__terms](X) = 2.X + 2 [nil] = 2 [recip](X) = 2.X + 2 [ACTIVATE](X) = 2.X + 2 [SQR](X) = 2.X + 2 [TERMS](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) ->->-> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__sqr(X:S)) -> SQR(activate(X:S)) ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) -> Usable rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [activate](X) = 2.X + 1 [add](X1,X2) = X2 + 2 [dbl](X) = 1 [first](X1,X2) = X2 + 2 [s](X) = X + 1 [sqr](X) = 2.X + 2 [terms](X) = 2.X + 2 [0] = 1 [cons](X1,X2) = 2 [n__add](X1,X2) = X2 + 1 [n__dbl](X) = 0 [n__first](X1,X2) = X2 + 1 [n__s](X) = X [n__sqr](X) = 2.X + 2 [n__terms](X) = 2.X + 2 [nil] = 1 [recip](X) = 2 [ACTIVATE](X) = X + 2 [SQR](X) = X + 2 [TERMS](X) = X + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) ->->-> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__terms(X:S)) -> ACTIVATE(X:S) ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) -> Usable rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [activate](X) = X + 2 [add](X1,X2) = X2 [dbl](X) = 2 [first](X1,X2) = 2 [s](X) = X [sqr](X) = X + 2 [terms](X) = X + 2 [0] = 1 [cons](X1,X2) = X2 [n__add](X1,X2) = X2 [n__dbl](X) = 2 [n__first](X1,X2) = 1 [n__s](X) = X [n__sqr](X) = X + 2 [n__terms](X) = X + 2 [nil] = 2 [recip](X) = 2.X + 2 [ACTIVATE](X) = 2.X + 2 [SQR](X) = 2.X + 2 [TERMS](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) ->->-> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) Problem 1: Reduction Pair Processor: -> Pairs: ACTIVATE(n__terms(X:S)) -> TERMS(activate(X:S)) SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) -> Usable rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [activate](X) = 2.X + 2 [add](X1,X2) = X2 + 2 [dbl](X) = 2 [first](X1,X2) = 2.X2 + 2 [s](X) = X + 1 [sqr](X) = 2.X + 2 [terms](X) = 2.X + 2 [0] = 2 [cons](X1,X2) = 2 [n__add](X1,X2) = X2 + 1 [n__dbl](X) = 1 [n__first](X1,X2) = 2.X2 + 2 [n__s](X) = X [n__sqr](X) = 2.X + 2 [n__terms](X) = 2.X + 2 [nil] = 0 [recip](X) = 2.X + 2 [ACTIVATE](X) = 2.X + 1 [SQR](X) = 2.X [TERMS](X) = 2.X Problem 1: SCC Processor: -> Pairs: SQR(s(X:S)) -> ACTIVATE(X:S) TERMS(N:S) -> SQR(N:S) -> Rules: activate(n__add(X1:S,X2:S)) -> add(activate(X1:S),activate(X2:S)) activate(n__dbl(X:S)) -> dbl(activate(X:S)) activate(n__first(X1:S,X2:S)) -> first(activate(X1:S),activate(X2:S)) activate(n__s(X:S)) -> s(X:S) activate(n__sqr(X:S)) -> sqr(activate(X:S)) activate(n__terms(X:S)) -> terms(activate(X:S)) activate(X:S) -> X:S add(s(X:S),Y:S) -> s(n__add(activate(X:S),Y:S)) add(0,X:S) -> X:S add(X1:S,X2:S) -> n__add(X1:S,X2:S) dbl(s(X:S)) -> s(n__s(n__dbl(activate(X:S)))) dbl(0) -> 0 dbl(X:S) -> n__dbl(X:S) first(s(X:S),cons(Y:S,Z:S)) -> cons(Y:S,n__first(activate(X:S),activate(Z:S))) first(0,X:S) -> nil first(X1:S,X2:S) -> n__first(X1:S,X2:S) s(X:S) -> n__s(X:S) sqr(s(X:S)) -> s(n__add(n__sqr(activate(X:S)),n__dbl(activate(X:S)))) sqr(0) -> 0 sqr(X:S) -> n__sqr(X:S) terms(N:S) -> cons(recip(sqr(N:S)),n__terms(n__s(N:S))) terms(X:S) -> n__terms(X:S) ->Strongly Connected Components: There is no strongly connected component The problem is finite.