MAYBE

Problem:
 le(s(x),0()) -> false()
 le(0(),y) -> true()
 le(s(x),s(y)) -> le(x,y)
 plus(0(),y) -> y
 plus(s(x),y) -> s(plus(x,y))
 times(0(),y) -> 0()
 times(s(x),y) -> plus(y,times(x,y))
 log(x,0()) -> baseError()
 log(x,s(0())) -> baseError()
 log(0(),s(s(b))) -> logZeroError()
 log(s(x),s(s(b))) -> loop(s(x),s(s(b)),s(0()),0())
 loop(x,s(s(b)),s(y),z) -> if(le(x,s(y)),x,s(s(b)),s(y),z)
 if(true(),x,b,y,z) -> z
 if(false(),x,b,y,z) -> loop(x,b,times(b,y),s(z))

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   plus#(s(x),y) -> plus#(x,y)
   times#(s(x),y) -> times#(x,y)
   times#(s(x),y) -> plus#(y,times(x,y))
   log#(s(x),s(s(b))) -> loop#(s(x),s(s(b)),s(0()),0())
   loop#(x,s(s(b)),s(y),z) -> le#(x,s(y))
   loop#(x,s(s(b)),s(y),z) -> if#(le(x,s(y)),x,s(s(b)),s(y),z)
   if#(false(),x,b,y,z) -> times#(b,y)
   if#(false(),x,b,y,z) -> loop#(x,b,times(b,y),s(z))
  TRS:
   le(s(x),0()) -> false()
   le(0(),y) -> true()
   le(s(x),s(y)) -> le(x,y)
   plus(0(),y) -> y
   plus(s(x),y) -> s(plus(x,y))
   times(0(),y) -> 0()
   times(s(x),y) -> plus(y,times(x,y))
   log(x,0()) -> baseError()
   log(x,s(0())) -> baseError()
   log(0(),s(s(b))) -> logZeroError()
   log(s(x),s(s(b))) -> loop(s(x),s(s(b)),s(0()),0())
   loop(x,s(s(b)),s(y),z) -> if(le(x,s(y)),x,s(s(b)),s(y),z)
   if(true(),x,b,y,z) -> z
   if(false(),x,b,y,z) -> loop(x,b,times(b,y),s(z))
  TDG Processor:
   DPs:
    le#(s(x),s(y)) -> le#(x,y)
    plus#(s(x),y) -> plus#(x,y)
    times#(s(x),y) -> times#(x,y)
    times#(s(x),y) -> plus#(y,times(x,y))
    log#(s(x),s(s(b))) -> loop#(s(x),s(s(b)),s(0()),0())
    loop#(x,s(s(b)),s(y),z) -> le#(x,s(y))
    loop#(x,s(s(b)),s(y),z) -> if#(le(x,s(y)),x,s(s(b)),s(y),z)
    if#(false(),x,b,y,z) -> times#(b,y)
    if#(false(),x,b,y,z) -> loop#(x,b,times(b,y),s(z))
   TRS:
    le(s(x),0()) -> false()
    le(0(),y) -> true()
    le(s(x),s(y)) -> le(x,y)
    plus(0(),y) -> y
    plus(s(x),y) -> s(plus(x,y))
    times(0(),y) -> 0()
    times(s(x),y) -> plus(y,times(x,y))
    log(x,0()) -> baseError()
    log(x,s(0())) -> baseError()
    log(0(),s(s(b))) -> logZeroError()
    log(s(x),s(s(b))) -> loop(s(x),s(s(b)),s(0()),0())
    loop(x,s(s(b)),s(y),z) -> if(le(x,s(y)),x,s(s(b)),s(y),z)
    if(true(),x,b,y,z) -> z
    if(false(),x,b,y,z) -> loop(x,b,times(b,y),s(z))
   graph:
    if#(false(),x,b,y,z) -> loop#(x,b,times(b,y),s(z)) ->
    loop#(x,s(s(b)),s(y),z) -> if#(le(x,s(y)),x,s(s(b)),s(y),z)
    if#(false(),x,b,y,z) -> loop#(x,b,times(b,y),s(z)) ->
    loop#(x,s(s(b)),s(y),z) -> le#(x,s(y))
    if#(false(),x,b,y,z) -> times#(b,y) ->
    times#(s(x),y) -> plus#(y,times(x,y))
    if#(false(),x,b,y,z) -> times#(b,y) ->
    times#(s(x),y) -> times#(x,y)
    loop#(x,s(s(b)),s(y),z) -> if#(le(x,s(y)),x,s(s(b)),s(y),z) ->
    if#(false(),x,b,y,z) -> loop#(x,b,times(b,y),s(z))
    loop#(x,s(s(b)),s(y),z) -> if#(le(x,s(y)),x,s(s(b)),s(y),z) ->
    if#(false(),x,b,y,z) -> times#(b,y)
    loop#(x,s(s(b)),s(y),z) -> le#(x,s(y)) ->
    le#(s(x),s(y)) -> le#(x,y)
    log#(s(x),s(s(b))) -> loop#(s(x),s(s(b)),s(0()),0()) ->
    loop#(x,s(s(b)),s(y),z) -> if#(le(x,s(y)),x,s(s(b)),s(y),z)
    log#(s(x),s(s(b))) -> loop#(s(x),s(s(b)),s(0()),0()) ->
    loop#(x,s(s(b)),s(y),z) -> le#(x,s(y))
    times#(s(x),y) -> times#(x,y) ->
    times#(s(x),y) -> plus#(y,times(x,y))
    times#(s(x),y) -> times#(x,y) ->
    times#(s(x),y) -> times#(x,y)
    times#(s(x),y) -> plus#(y,times(x,y)) -> plus#(s(x),y) -> plus#(x,y)
    plus#(s(x),y) -> plus#(x,y) -> plus#(s(x),y) -> plus#(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: 14/81
    DPs:
     if#(false(),x,b,y,z) -> loop#(x,b,times(b,y),s(z))
     loop#(x,s(s(b)),s(y),z) -> if#(le(x,s(y)),x,s(s(b)),s(y),z)
    TRS:
     le(s(x),0()) -> false()
     le(0(),y) -> true()
     le(s(x),s(y)) -> le(x,y)
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     times(0(),y) -> 0()
     times(s(x),y) -> plus(y,times(x,y))
     log(x,0()) -> baseError()
     log(x,s(0())) -> baseError()
     log(0(),s(s(b))) -> logZeroError()
     log(s(x),s(s(b))) -> loop(s(x),s(s(b)),s(0()),0())
     loop(x,s(s(b)),s(y),z) -> if(le(x,s(y)),x,s(s(b)),s(y),z)
     if(true(),x,b,y,z) -> z
     if(false(),x,b,y,z) -> loop(x,b,times(b,y),s(z))
    Open
    
    DPs:
     times#(s(x),y) -> times#(x,y)
    TRS:
     le(s(x),0()) -> false()
     le(0(),y) -> true()
     le(s(x),s(y)) -> le(x,y)
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     times(0(),y) -> 0()
     times(s(x),y) -> plus(y,times(x,y))
     log(x,0()) -> baseError()
     log(x,s(0())) -> baseError()
     log(0(),s(s(b))) -> logZeroError()
     log(s(x),s(s(b))) -> loop(s(x),s(s(b)),s(0()),0())
     loop(x,s(s(b)),s(y),z) -> if(le(x,s(y)),x,s(s(b)),s(y),z)
     if(true(),x,b,y,z) -> z
     if(false(),x,b,y,z) -> loop(x,b,times(b,y),s(z))
    Subterm Criterion Processor:
     simple projection:
      pi(times#) = 0
     problem:
      DPs:
       
      TRS:
       le(s(x),0()) -> false()
       le(0(),y) -> true()
       le(s(x),s(y)) -> le(x,y)
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       times(0(),y) -> 0()
       times(s(x),y) -> plus(y,times(x,y))
       log(x,0()) -> baseError()
       log(x,s(0())) -> baseError()
       log(0(),s(s(b))) -> logZeroError()
       log(s(x),s(s(b))) -> loop(s(x),s(s(b)),s(0()),0())
       loop(x,s(s(b)),s(y),z) -> if(le(x,s(y)),x,s(s(b)),s(y),z)
       if(true(),x,b,y,z) -> z
       if(false(),x,b,y,z) -> loop(x,b,times(b,y),s(z))
     Qed
    
    DPs:
     plus#(s(x),y) -> plus#(x,y)
    TRS:
     le(s(x),0()) -> false()
     le(0(),y) -> true()
     le(s(x),s(y)) -> le(x,y)
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     times(0(),y) -> 0()
     times(s(x),y) -> plus(y,times(x,y))
     log(x,0()) -> baseError()
     log(x,s(0())) -> baseError()
     log(0(),s(s(b))) -> logZeroError()
     log(s(x),s(s(b))) -> loop(s(x),s(s(b)),s(0()),0())
     loop(x,s(s(b)),s(y),z) -> if(le(x,s(y)),x,s(s(b)),s(y),z)
     if(true(),x,b,y,z) -> z
     if(false(),x,b,y,z) -> loop(x,b,times(b,y),s(z))
    Subterm Criterion Processor:
     simple projection:
      pi(plus#) = 0
     problem:
      DPs:
       
      TRS:
       le(s(x),0()) -> false()
       le(0(),y) -> true()
       le(s(x),s(y)) -> le(x,y)
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       times(0(),y) -> 0()
       times(s(x),y) -> plus(y,times(x,y))
       log(x,0()) -> baseError()
       log(x,s(0())) -> baseError()
       log(0(),s(s(b))) -> logZeroError()
       log(s(x),s(s(b))) -> loop(s(x),s(s(b)),s(0()),0())
       loop(x,s(s(b)),s(y),z) -> if(le(x,s(y)),x,s(s(b)),s(y),z)
       if(true(),x,b,y,z) -> z
       if(false(),x,b,y,z) -> loop(x,b,times(b,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)
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     times(0(),y) -> 0()
     times(s(x),y) -> plus(y,times(x,y))
     log(x,0()) -> baseError()
     log(x,s(0())) -> baseError()
     log(0(),s(s(b))) -> logZeroError()
     log(s(x),s(s(b))) -> loop(s(x),s(s(b)),s(0()),0())
     loop(x,s(s(b)),s(y),z) -> if(le(x,s(y)),x,s(s(b)),s(y),z)
     if(true(),x,b,y,z) -> z
     if(false(),x,b,y,z) -> loop(x,b,times(b,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)
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       times(0(),y) -> 0()
       times(s(x),y) -> plus(y,times(x,y))
       log(x,0()) -> baseError()
       log(x,s(0())) -> baseError()
       log(0(),s(s(b))) -> logZeroError()
       log(s(x),s(s(b))) -> loop(s(x),s(s(b)),s(0()),0())
       loop(x,s(s(b)),s(y),z) -> if(le(x,s(y)),x,s(s(b)),s(y),z)
       if(true(),x,b,y,z) -> z
       if(false(),x,b,y,z) -> loop(x,b,times(b,y),s(z))
     Qed