YES

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 minus(0(),y) -> 0()
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
 if_minus(true(),s(x),y) -> 0()
 if_minus(false(),s(x),y) -> s(minus(x,y))
 quot(0(),s(y)) -> 0()
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
 log(s(0())) -> 0()
 log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   minus#(s(x),y) -> le#(s(x),y)
   minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
   if_minus#(false(),s(x),y) -> minus#(x,y)
   quot#(s(x),s(y)) -> minus#(x,y)
   quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
   log#(s(s(x))) -> quot#(x,s(s(0())))
   log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
  TRS:
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   minus(0(),y) -> 0()
   minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
   if_minus(true(),s(x),y) -> 0()
   if_minus(false(),s(x),y) -> s(minus(x,y))
   quot(0(),s(y)) -> 0()
   quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
   log(s(0())) -> 0()
   log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
  TDG Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    minus#(s(x),y) -> le#(s(x),y)
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    if_minus#(false(),s(x),y) -> minus#(x,y)
    quot#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    log#(s(s(x))) -> quot#(x,s(s(0())))
    log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
   TRS:
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    minus(0(),y) -> 0()
    minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
    if_minus(true(),s(x),y) -> 0()
    if_minus(false(),s(x),y) -> s(minus(x,y))
    quot(0(),s(y)) -> 0()
    quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
    log(s(0())) -> 0()
    log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
   graph:
    log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) ->
    log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
    log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) ->
    log#(s(s(x))) -> quot#(x,s(s(0())))
    log#(s(s(x))) -> quot#(x,s(s(0()))) ->
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    log#(s(s(x))) -> quot#(x,s(s(0()))) ->
    quot#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) ->
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) ->
    quot#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    quot#(s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),y) -> le#(s(x),y)
    if_minus#(false(),s(x),y) -> minus#(x,y) ->
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
    if_minus#(false(),s(x),y) -> minus#(x,y) ->
    minus#(s(x),y) -> le#(s(x),y)
    minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y) ->
    if_minus#(false(),s(x),y) -> minus#(x,y)
    minus#(s(x),y) -> le#(s(x),y) -> le#(s(x),s(y)) -> le#(x,y)
    le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
   SCC Processor:
    #sccs: 4
    #rules: 5
    #arcs: 13/64
    DPs:
     log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(minus(x,y))
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     log(s(0())) -> 0()
     log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [log#](x0) = x0 + 5,
      
      [log](x0) = x0 + 0,
      
      [quot](x0, x1) = x0 + 2,
      
      [if_minus](x0, x1, x2) = 4x0 + x1 + 0,
      
      [minus](x0, x1) = x0 + 2,
      
      [false] = 0,
      
      [s](x0) = 2x0 + 4,
      
      [true] = 0,
      
      [le](x0, x1) = 0,
      
      [0] = 0
     orientation:
      log#(s(s(x))) = 4x + 6 >= 2x + 5 = log#(s(quot(x,s(s(0())))))
      
      le(0(),y) = 0 >= 0 = true()
      
      le(s(x),0()) = 0 >= 0 = false()
      
      le(s(x),s(y)) = 0 >= 0 = le(x,y)
      
      minus(0(),y) = 2 >= 0 = 0()
      
      minus(s(x),y) = 2x + 4 >= 2x + 4 = if_minus(le(s(x),y),s(x),y)
      
      if_minus(true(),s(x),y) = 2x + 4 >= 0 = 0()
      
      if_minus(false(),s(x),y) = 2x + 4 >= 2x + 4 = s(minus(x,y))
      
      quot(0(),s(y)) = 2 >= 0 = 0()
      
      quot(s(x),s(y)) = 2x + 4 >= 2x + 4 = s(quot(minus(x,y),s(y)))
      
      log(s(0())) = 4 >= 0 = 0()
      
      log(s(s(x))) = 4x + 6 >= 4x + 6 = s(log(s(quot(x,s(s(0()))))))
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(0(),y) -> 0()
       minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
       if_minus(true(),s(x),y) -> 0()
       if_minus(false(),s(x),y) -> s(minus(x,y))
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       log(s(0())) -> 0()
       log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
     Qed
    
    DPs:
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(minus(x,y))
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     log(s(0())) -> 0()
     log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
    KBO Processor:
     argument filtering:
      pi(0) = []
      pi(le) = [1]
      pi(true) = []
      pi(s) = [0]
      pi(false) = []
      pi(minus) = 0
      pi(if_minus) = 1
      pi(quot) = 0
      pi(log) = 0
      pi(quot#) = 0
     weight function:
      w0 = 1
      w(quot#) = w(false) = w(true) = w(le) = w(0) = 1
      w(log) = w(quot) = w(if_minus) = w(minus) = w(s) = 0
     precedence:
      s ~ true > quot# ~ log ~ quot ~ if_minus ~ minus ~ false ~ le ~ 0
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(0(),y) -> 0()
       minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
       if_minus(true(),s(x),y) -> 0()
       if_minus(false(),s(x),y) -> s(minus(x,y))
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       log(s(0())) -> 0()
       log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
     Qed
    
    DPs:
     minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
     if_minus#(false(),s(x),y) -> minus#(x,y)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(minus(x,y))
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     log(s(0())) -> 0()
     log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 0
      pi(if_minus#) = 1
     problem:
      DPs:
       minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(0(),y) -> 0()
       minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
       if_minus(true(),s(x),y) -> 0()
       if_minus(false(),s(x),y) -> s(minus(x,y))
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       log(s(0())) -> 0()
       log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 2/1
      
    
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(minus(x,y))
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     log(s(0())) -> 0()
     log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
    Subterm Criterion Processor:
     simple projection:
      pi(le#) = 1
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       minus(0(),y) -> 0()
       minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
       if_minus(true(),s(x),y) -> 0()
       if_minus(false(),s(x),y) -> s(minus(x,y))
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       log(s(0())) -> 0()
       log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
     Qed