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)
  Usable Rule 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:
    f21(x,y) -> x
    f21(x,y) -> y
    minus(0(),y) -> 0()
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
    inc(0()) -> 0()
    inc(s(x)) -> s(inc(x))
    le(0(),y) -> true()
    le(s(x),s(y)) -> le(x,y)
    le(s(x),0()) -> false()
    quot(0(),s(y)) -> 0()
    quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
   EDG 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:
     f21(x,y) -> x
     f21(x,y) -> y
     minus(0(),y) -> 0()
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     le(0(),y) -> true()
     le(s(x),s(y)) -> le(x,y)
     le(s(x),0()) -> false()
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
    graph:
     if2#(false(),x,y) -> log2#(quot(x,s(s(0()))),y) ->
     log2#(x,y) -> inc#(y)
     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) -> le#(x,0())
     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) -> quot#(x,s(s(0()))) ->
     quot#(s(x),s(y)) -> minus#(x,y)
     if2#(false(),x,y) -> quot#(x,s(s(0()))) ->
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
     if#(false(),b,x,y) -> if2#(b,x,y) ->
     if2#(false(),x,y) -> quot#(x,s(s(0())))
     if#(false(),b,x,y) -> if2#(b,x,y) ->
     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)
     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)
     log#(x) -> log2#(x,0()) -> log2#(x,y) -> inc#(y)
     log#(x) -> log2#(x,0()) -> log2#(x,y) -> le#(x,s(0()))
     log#(x) -> log2#(x,0()) -> log2#(x,y) -> le#(x,0())
     log#(x) -> log2#(x,0()) ->
     log2#(x,y) -> if#(le(x,0()),le(x,s(0())),x,inc(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)) -> 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) ->
     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)
    Restore Modifier:
     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)
     SCC Processor:
      #sccs: 5
      #rules: 7
      #arcs: 21/169
      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:
       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)
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [inc#](x0) = x0 + 1,
        
        [if2](x0, x1, x2) = x2,
        
        [log_undefined] = 0,
        
        [if](x0, x1, x2, x3) = x3,
        
        [log2](x0, x1) = x1,
        
        [log](x0) = 1,
        
        [quot](x0, x1) = x0 + 1,
        
        [minus](x0, x1) = x0,
        
        [inc](x0) = x0,
        
        [false] = 0,
        
        [s](x0) = x0 + 1,
        
        [true] = 0,
        
        [le](x0, x1) = 0,
        
        [0] = 0
       orientation:
        inc#(s(x)) = x + 2 >= x + 1 = inc#(x)
        
        le(0(),y) = 0 >= 0 = true()
        
        le(s(x),0()) = 0 >= 0 = false()
        
        le(s(x),s(y)) = 0 >= 0 = le(x,y)
        
        inc(0()) = 0 >= 0 = 0()
        
        inc(s(x)) = x + 1 >= x + 1 = s(inc(x))
        
        minus(0(),y) = 0 >= 0 = 0()
        
        minus(x,0()) = x >= x = x
        
        minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
        
        quot(0(),s(y)) = 1 >= 0 = 0()
        
        quot(s(x),s(y)) = x + 2 >= x + 2 = s(quot(minus(x,y),s(y)))
        
        log(x) = 1 >= 0 = log2(x,0())
        
        log2(x,y) = y >= y = if(le(x,0()),le(x,s(0())),x,inc(y))
        
        if(true(),b,x,y) = y >= 0 = log_undefined()
        
        if(false(),b,x,y) = y >= y = if2(b,x,y)
        
        if2(true(),x,s(y)) = y + 1 >= y = y
        
        if2(false(),x,y) = y >= y = 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:
       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)
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [le#](x0, x1) = x0 + x1 + 1,
        
        [if2](x0, x1, x2) = x2,
        
        [log_undefined] = 0,
        
        [if](x0, x1, x2, x3) = x3,
        
        [log2](x0, x1) = x1,
        
        [log](x0) = 0,
        
        [quot](x0, x1) = x0,
        
        [minus](x0, x1) = x0,
        
        [inc](x0) = x0,
        
        [false] = 0,
        
        [s](x0) = x0 + 1,
        
        [true] = 0,
        
        [le](x0, x1) = x1 + 1,
        
        [0] = 0
       orientation:
        le#(s(x),s(y)) = x + y + 3 >= x + y + 1 = le#(x,y)
        
        le(0(),y) = y + 1 >= 0 = true()
        
        le(s(x),0()) = 1 >= 0 = false()
        
        le(s(x),s(y)) = y + 2 >= y + 1 = le(x,y)
        
        inc(0()) = 0 >= 0 = 0()
        
        inc(s(x)) = x + 1 >= x + 1 = s(inc(x))
        
        minus(0(),y) = 0 >= 0 = 0()
        
        minus(x,0()) = x >= x = x
        
        minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
        
        quot(0(),s(y)) = 0 >= 0 = 0()
        
        quot(s(x),s(y)) = x + 1 >= x + 1 = s(quot(minus(x,y),s(y)))
        
        log(x) = 0 >= 0 = log2(x,0())
        
        log2(x,y) = y >= y = if(le(x,0()),le(x,s(0())),x,inc(y))
        
        if(true(),b,x,y) = y >= 0 = log_undefined()
        
        if(false(),b,x,y) = y >= y = if2(b,x,y)
        
        if2(true(),x,s(y)) = y + 1 >= y = y
        
        if2(false(),x,y) = y >= y = 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:
       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)
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [quot#](x0, x1) = x0 + x1,
        
        [if2](x0, x1, x2) = x2,
        
        [log_undefined] = 0,
        
        [if](x0, x1, x2, x3) = x3,
        
        [log2](x0, x1) = x1,
        
        [log](x0) = 0,
        
        [quot](x0, x1) = x0,
        
        [minus](x0, x1) = x0,
        
        [inc](x0) = x0,
        
        [false] = 0,
        
        [s](x0) = x0 + 1,
        
        [true] = 0,
        
        [le](x0, x1) = 0,
        
        [0] = 0
       orientation:
        quot#(s(x),s(y)) = x + y + 2 >= x + y + 1 = quot#(minus(x,y),s(y))
        
        le(0(),y) = 0 >= 0 = true()
        
        le(s(x),0()) = 0 >= 0 = false()
        
        le(s(x),s(y)) = 0 >= 0 = le(x,y)
        
        inc(0()) = 0 >= 0 = 0()
        
        inc(s(x)) = x + 1 >= x + 1 = s(inc(x))
        
        minus(0(),y) = 0 >= 0 = 0()
        
        minus(x,0()) = x >= x = x
        
        minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
        
        quot(0(),s(y)) = 0 >= 0 = 0()
        
        quot(s(x),s(y)) = x + 1 >= x + 1 = s(quot(minus(x,y),s(y)))
        
        log(x) = 0 >= 0 = log2(x,0())
        
        log2(x,y) = y >= y = if(le(x,0()),le(x,s(0())),x,inc(y))
        
        if(true(),b,x,y) = y >= 0 = log_undefined()
        
        if(false(),b,x,y) = y >= y = if2(b,x,y)
        
        if2(true(),x,s(y)) = y + 1 >= y = y
        
        if2(false(),x,y) = y >= y = 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)
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [minus#](x0, x1) = x0 + x1 + 1,
        
        [if2](x0, x1, x2) = x2,
        
        [log_undefined] = 0,
        
        [if](x0, x1, x2, x3) = x3,
        
        [log2](x0, x1) = x1,
        
        [log](x0) = 0,
        
        [quot](x0, x1) = x0,
        
        [minus](x0, x1) = x0,
        
        [inc](x0) = x0,
        
        [false] = 0,
        
        [s](x0) = x0 + 1,
        
        [true] = 0,
        
        [le](x0, x1) = x1 + 1,
        
        [0] = 0
       orientation:
        minus#(s(x),s(y)) = x + y + 3 >= x + y + 1 = minus#(x,y)
        
        le(0(),y) = y + 1 >= 0 = true()
        
        le(s(x),0()) = 1 >= 0 = false()
        
        le(s(x),s(y)) = y + 2 >= y + 1 = le(x,y)
        
        inc(0()) = 0 >= 0 = 0()
        
        inc(s(x)) = x + 1 >= x + 1 = s(inc(x))
        
        minus(0(),y) = 0 >= 0 = 0()
        
        minus(x,0()) = x >= x = x
        
        minus(s(x),s(y)) = x + 1 >= x = minus(x,y)
        
        quot(0(),s(y)) = 0 >= 0 = 0()
        
        quot(s(x),s(y)) = x + 1 >= x + 1 = s(quot(minus(x,y),s(y)))
        
        log(x) = 0 >= 0 = log2(x,0())
        
        log2(x,y) = y >= y = if(le(x,0()),le(x,s(0())),x,inc(y))
        
        if(true(),b,x,y) = y >= 0 = log_undefined()
        
        if(false(),b,x,y) = y >= y = if2(b,x,y)
        
        if2(true(),x,s(y)) = y + 1 >= y = y
        
        if2(false(),x,y) = y >= y = 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