MAYBE

Problem:
 le(s(x),0()) -> false()
 le(0(),y) -> true()
 le(s(x),s(y)) -> le(x,y)
 double(0()) -> 0()
 double(s(x)) -> s(s(double(x)))
 log(0()) -> logError()
 log(s(x)) -> loop(s(x),s(0()),0())
 loop(x,s(y),z) -> if(le(x,s(y)),x,s(y),z)
 if(true(),x,y,z) -> z
 if(false(),x,y,z) -> loop(x,double(y),s(z))

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   double#(s(x)) -> double#(x)
   log#(s(x)) -> loop#(s(x),s(0()),0())
   loop#(x,s(y),z) -> le#(x,s(y))
   loop#(x,s(y),z) -> if#(le(x,s(y)),x,s(y),z)
   if#(false(),x,y,z) -> double#(y)
   if#(false(),x,y,z) -> loop#(x,double(y),s(z))
  TRS:
   le(s(x),0()) -> false()
   le(0(),y) -> true()
   le(s(x),s(y)) -> le(x,y)
   double(0()) -> 0()
   double(s(x)) -> s(s(double(x)))
   log(0()) -> logError()
   log(s(x)) -> loop(s(x),s(0()),0())
   loop(x,s(y),z) -> if(le(x,s(y)),x,s(y),z)
   if(true(),x,y,z) -> z
   if(false(),x,y,z) -> loop(x,double(y),s(z))
  TDG Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    double#(s(x)) -> double#(x)
    log#(s(x)) -> loop#(s(x),s(0()),0())
    loop#(x,s(y),z) -> le#(x,s(y))
    loop#(x,s(y),z) -> if#(le(x,s(y)),x,s(y),z)
    if#(false(),x,y,z) -> double#(y)
    if#(false(),x,y,z) -> loop#(x,double(y),s(z))
   TRS:
    le(s(x),0()) -> false()
    le(0(),y) -> true()
    le(s(x),s(y)) -> le(x,y)
    double(0()) -> 0()
    double(s(x)) -> s(s(double(x)))
    log(0()) -> logError()
    log(s(x)) -> loop(s(x),s(0()),0())
    loop(x,s(y),z) -> if(le(x,s(y)),x,s(y),z)
    if(true(),x,y,z) -> z
    if(false(),x,y,z) -> loop(x,double(y),s(z))
   graph:
    if#(false(),x,y,z) -> loop#(x,double(y),s(z)) ->
    loop#(x,s(y),z) -> if#(le(x,s(y)),x,s(y),z)
    if#(false(),x,y,z) -> loop#(x,double(y),s(z)) ->
    loop#(x,s(y),z) -> le#(x,s(y))
    if#(false(),x,y,z) -> double#(y) ->
    double#(s(x)) -> double#(x)
    loop#(x,s(y),z) -> if#(le(x,s(y)),x,s(y),z) ->
    if#(false(),x,y,z) -> loop#(x,double(y),s(z))
    loop#(x,s(y),z) -> if#(le(x,s(y)),x,s(y),z) ->
    if#(false(),x,y,z) -> double#(y)
    loop#(x,s(y),z) -> le#(x,s(y)) ->
    le#(s(x),s(y)) -> le#(x,y)
    log#(s(x)) -> loop#(s(x),s(0()),0()) ->
    loop#(x,s(y),z) -> if#(le(x,s(y)),x,s(y),z)
    log#(s(x)) -> loop#(s(x),s(0()),0()) ->
    loop#(x,s(y),z) -> le#(x,s(y))
    double#(s(x)) -> double#(x) -> double#(s(x)) -> double#(x)
    le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
   SCC Processor:
    #sccs: 3
    #rules: 4
    #arcs: 10/49
    DPs:
     if#(false(),x,y,z) -> loop#(x,double(y),s(z))
     loop#(x,s(y),z) -> if#(le(x,s(y)),x,s(y),z)
    TRS:
     le(s(x),0()) -> false()
     le(0(),y) -> true()
     le(s(x),s(y)) -> le(x,y)
     double(0()) -> 0()
     double(s(x)) -> s(s(double(x)))
     log(0()) -> logError()
     log(s(x)) -> loop(s(x),s(0()),0())
     loop(x,s(y),z) -> if(le(x,s(y)),x,s(y),z)
     if(true(),x,y,z) -> z
     if(false(),x,y,z) -> loop(x,double(y),s(z))
    Open
    
    DPs:
     double#(s(x)) -> double#(x)
    TRS:
     le(s(x),0()) -> false()
     le(0(),y) -> true()
     le(s(x),s(y)) -> le(x,y)
     double(0()) -> 0()
     double(s(x)) -> s(s(double(x)))
     log(0()) -> logError()
     log(s(x)) -> loop(s(x),s(0()),0())
     loop(x,s(y),z) -> if(le(x,s(y)),x,s(y),z)
     if(true(),x,y,z) -> z
     if(false(),x,y,z) -> loop(x,double(y),s(z))
    Subterm Criterion Processor:
     simple projection:
      pi(double#) = 0
     problem:
      DPs:
       
      TRS:
       le(s(x),0()) -> false()
       le(0(),y) -> true()
       le(s(x),s(y)) -> le(x,y)
       double(0()) -> 0()
       double(s(x)) -> s(s(double(x)))
       log(0()) -> logError()
       log(s(x)) -> loop(s(x),s(0()),0())
       loop(x,s(y),z) -> if(le(x,s(y)),x,s(y),z)
       if(true(),x,y,z) -> z
       if(false(),x,y,z) -> loop(x,double(y),s(z))
     Qed
    
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
    TRS:
     le(s(x),0()) -> false()
     le(0(),y) -> true()
     le(s(x),s(y)) -> le(x,y)
     double(0()) -> 0()
     double(s(x)) -> s(s(double(x)))
     log(0()) -> logError()
     log(s(x)) -> loop(s(x),s(0()),0())
     loop(x,s(y),z) -> if(le(x,s(y)),x,s(y),z)
     if(true(),x,y,z) -> z
     if(false(),x,y,z) -> loop(x,double(y),s(z))
    Subterm Criterion Processor:
     simple projection:
      pi(le#) = 1
     problem:
      DPs:
       
      TRS:
       le(s(x),0()) -> false()
       le(0(),y) -> true()
       le(s(x),s(y)) -> le(x,y)
       double(0()) -> 0()
       double(s(x)) -> s(s(double(x)))
       log(0()) -> logError()
       log(s(x)) -> loop(s(x),s(0()),0())
       loop(x,s(y),z) -> if(le(x,s(y)),x,s(y),z)
       if(true(),x,y,z) -> z
       if(false(),x,y,z) -> loop(x,double(y),s(z))
     Qed