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))
  Open