MAYBE

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 inc(0()) -> 0()
 inc(s(x)) -> s(inc(x))
 minus(0(),y) -> 0()
 minus(x,0()) -> x
 minus(s(x),s(y)) -> minus(x,y)
 quot(0(),s(y)) -> 0()
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
 log(x) -> log2(x,0())
 log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
 if(true(),b,x,y) -> log_undefined()
 if(false(),b,x,y) -> if2(b,x,y)
 if2(true(),x,s(y)) -> y
 if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   inc#(s(x)) -> inc#(x)
   minus#(s(x),s(y)) -> minus#(x,y)
   quot#(s(x),s(y)) -> minus#(x,y)
   quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
   log#(x) -> log2#(x,0())
   log2#(x,y) -> inc#(y)
   log2#(x,y) -> le#(x,s(0()))
   log2#(x,y) -> le#(x,0())
   log2#(x,y) -> if#(le(x,0()),le(x,s(0())),x,inc(y))
   if#(false(),b,x,y) -> if2#(b,x,y)
   if2#(false(),x,y) -> quot#(x,s(s(0())))
   if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y)
  TRS:
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   inc(0()) -> 0()
   inc(s(x)) -> s(inc(x))
   minus(0(),y) -> 0()
   minus(x,0()) -> x
   minus(s(x),s(y)) -> minus(x,y)
   quot(0(),s(y)) -> 0()
   quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
   log(x) -> log2(x,0())
   log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
   if(true(),b,x,y) -> log_undefined()
   if(false(),b,x,y) -> if2(b,x,y)
   if2(true(),x,s(y)) -> y
   if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
  TDG Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    inc#(s(x)) -> inc#(x)
    minus#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    log#(x) -> log2#(x,0())
    log2#(x,y) -> inc#(y)
    log2#(x,y) -> le#(x,s(0()))
    log2#(x,y) -> le#(x,0())
    log2#(x,y) -> if#(le(x,0()),le(x,s(0())),x,inc(y))
    if#(false(),b,x,y) -> if2#(b,x,y)
    if2#(false(),x,y) -> quot#(x,s(s(0())))
    if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y)
   TRS:
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    inc(0()) -> 0()
    inc(s(x)) -> s(inc(x))
    minus(0(),y) -> 0()
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
    quot(0(),s(y)) -> 0()
    quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
    log(x) -> log2(x,0())
    log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
    if(true(),b,x,y) -> log_undefined()
    if(false(),b,x,y) -> if2(b,x,y)
    if2(true(),x,s(y)) -> y
    if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
   graph:
    if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y) ->
    log2#(x,y) -> if#(le(x,0()),le(x,s(0())),x,inc(y))
    if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y) ->
    log2#(x,y) -> le#(x,0())
    if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y) ->
    log2#(x,y) -> le#(x,s(0()))
    if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y) ->
    log2#(x,y) -> inc#(y)
    if2#(false(),x,y) -> quot#(x,s(s(0()))) ->
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    if2#(false(),x,y) -> quot#(x,s(s(0()))) ->
    quot#(s(x),s(y)) -> minus#(x,y)
    if#(false(),b,x,y) -> if2#(b,x,y) ->
    if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y)
    if#(false(),b,x,y) -> if2#(b,x,y) ->
    if2#(false(),x,y) -> quot#(x,s(s(0())))
    log2#(x,y) -> if#(le(x,0()),le(x,s(0())),x,inc(y)) ->
    if#(false(),b,x,y) -> if2#(b,x,y)
    log2#(x,y) -> inc#(y) -> inc#(s(x)) -> inc#(x)
    log2#(x,y) -> le#(x,s(0())) -> le#(s(x),s(y)) -> le#(x,y)
    log2#(x,y) -> le#(x,0()) -> le#(s(x),s(y)) -> le#(x,y)
    log#(x) -> log2#(x,0()) ->
    log2#(x,y) -> if#(le(x,0()),le(x,s(0())),x,inc(y))
    log#(x) -> log2#(x,0()) -> log2#(x,y) -> le#(x,0())
    log#(x) -> log2#(x,0()) -> log2#(x,y) -> le#(x,s(0()))
    log#(x) -> log2#(x,0()) -> log2#(x,y) -> inc#(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),s(y)) -> minus#(x,y)
    minus#(s(x),s(y)) -> minus#(x,y) -> minus#(s(x),s(y)) -> minus#(x,y)
    inc#(s(x)) -> inc#(x) -> inc#(s(x)) -> inc#(x)
    le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
   SCC Processor:
    #sccs: 5
    #rules: 7
    #arcs: 22/169
    DPs:
     if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y)
     log2#(x,y) -> if#(le(x,0()),le(x,s(0())),x,inc(y))
     if#(false(),b,x,y) -> if2#(b,x,y)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     minus(0(),y) -> 0()
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
     if(true(),b,x,y) -> log_undefined()
     if(false(),b,x,y) -> if2(b,x,y)
     if2(true(),x,s(y)) -> y
     if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
    Open
    
    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)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     minus(0(),y) -> 0()
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
     if(true(),b,x,y) -> log_undefined()
     if(false(),b,x,y) -> if2(b,x,y)
     if2(true(),x,s(y)) -> y
     if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [quot#](x0, x1) = x0,
      
      [if2](x0, x1, x2) = 1x0 + 4x1 + 4x2 + 0,
      
      [log_undefined] = 1,
      
      [if](x0, x1, x2, x3) = 2x0 + 2x1 + 4x2 + 4x3,
      
      [log2](x0, x1) = 4x0 + 4x1,
      
      [log](x0) = 13x0 + 8,
      
      [quot](x0, x1) = x0,
      
      [minus](x0, x1) = x0,
      
      [inc](x0) = x0,
      
      [false] = 0,
      
      [s](x0) = 2x0 + 0,
      
      [true] = 3,
      
      [le](x0, x1) = 1x0,
      
      [0] = 3
     orientation:
      quot#(s(x),s(y)) = 2x + 0 >= x = quot#(minus(x,y),s(y))
      
      le(0(),y) = 4 >= 3 = true()
      
      le(s(x),0()) = 3x + 1 >= 0 = false()
      
      le(s(x),s(y)) = 3x + 1 >= 1x = le(x,y)
      
      inc(0()) = 3 >= 3 = 0()
      
      inc(s(x)) = 2x + 0 >= 2x + 0 = s(inc(x))
      
      minus(0(),y) = 3 >= 3 = 0()
      
      minus(x,0()) = x >= x = x
      
      minus(s(x),s(y)) = 2x + 0 >= x = minus(x,y)
      
      quot(0(),s(y)) = 3 >= 3 = 0()
      
      quot(s(x),s(y)) = 2x + 0 >= 2x + 0 = s(quot(minus(x,y),s(y)))
      
      log(x) = 13x + 8 >= 4x + 7 = log2(x,0())
      
      log2(x,y) = 4x + 4y >= 4x + 4y = if(le(x,0()),le(x,s(0())),x,inc(y))
      
      if(true(),b,x,y) = 2b + 4x + 4y + 5 >= 1 = log_undefined()
      
      if(false(),b,x,y) = 2b + 4x + 4y + 2 >= 1b + 4x + 4y + 0 = if2(b,x,y)
      
      if2(true(),x,s(y)) = 4x + 6y + 4 >= y = y
      
      if2(false(),x,y) = 4x + 4y + 1 >= 4x + 4y = log2(quot(x,s(s(0()))),y)
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       inc(0()) -> 0()
       inc(s(x)) -> s(inc(x))
       minus(0(),y) -> 0()
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       log(x) -> log2(x,0())
       log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
       if(true(),b,x,y) -> log_undefined()
       if(false(),b,x,y) -> if2(b,x,y)
       if2(true(),x,s(y)) -> y
       if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
     Qed
    
    DPs:
     minus#(s(x),s(y)) -> minus#(x,y)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     minus(0(),y) -> 0()
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
     if(true(),b,x,y) -> log_undefined()
     if(false(),b,x,y) -> if2(b,x,y)
     if2(true(),x,s(y)) -> y
     if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 1
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       inc(0()) -> 0()
       inc(s(x)) -> s(inc(x))
       minus(0(),y) -> 0()
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       log(x) -> log2(x,0())
       log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
       if(true(),b,x,y) -> log_undefined()
       if(false(),b,x,y) -> if2(b,x,y)
       if2(true(),x,s(y)) -> y
       if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
     Qed
    
    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)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     minus(0(),y) -> 0()
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
     if(true(),b,x,y) -> log_undefined()
     if(false(),b,x,y) -> if2(b,x,y)
     if2(true(),x,s(y)) -> y
     if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
    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)
       inc(0()) -> 0()
       inc(s(x)) -> s(inc(x))
       minus(0(),y) -> 0()
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       log(x) -> log2(x,0())
       log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
       if(true(),b,x,y) -> log_undefined()
       if(false(),b,x,y) -> if2(b,x,y)
       if2(true(),x,s(y)) -> y
       if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
     Qed
    
    DPs:
     inc#(s(x)) -> inc#(x)
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     minus(0(),y) -> 0()
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
     if(true(),b,x,y) -> log_undefined()
     if(false(),b,x,y) -> if2(b,x,y)
     if2(true(),x,s(y)) -> y
     if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
    Subterm Criterion Processor:
     simple projection:
      pi(inc#) = 0
     problem:
      DPs:
       
      TRS:
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       inc(0()) -> 0()
       inc(s(x)) -> s(inc(x))
       minus(0(),y) -> 0()
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       log(x) -> log2(x,0())
       log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y))
       if(true(),b,x,y) -> log_undefined()
       if(false(),b,x,y) -> if2(b,x,y)
       if2(true(),x,s(y)) -> y
       if2(false(),x,y) -> log2(quot(x,s(s(0()))),y)
     Qed