YES

Problem:
 min(X,0()) -> X
 min(s(X),s(Y)) -> min(X,Y)
 quot(0(),s(Y)) -> 0()
 quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
 log(s(0())) -> 0()
 log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))

Proof:
 DP Processor:
  DPs:
   min#(s(X),s(Y)) -> min#(X,Y)
   quot#(s(X),s(Y)) -> min#(X,Y)
   quot#(s(X),s(Y)) -> quot#(min(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:
   min(X,0()) -> X
   min(s(X),s(Y)) -> min(X,Y)
   quot(0(),s(Y)) -> 0()
   quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
   log(s(0())) -> 0()
   log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
  Usable Rule Processor:
   DPs:
    min#(s(X),s(Y)) -> min#(X,Y)
    quot#(s(X),s(Y)) -> min#(X,Y)
    quot#(s(X),s(Y)) -> quot#(min(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:
    f8(x,y) -> x
    f8(x,y) -> y
    min(X,0()) -> X
    min(s(X),s(Y)) -> min(X,Y)
    quot(0(),s(Y)) -> 0()
    quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
   CDG Processor:
    DPs:
     min#(s(X),s(Y)) -> min#(X,Y)
     quot#(s(X),s(Y)) -> min#(X,Y)
     quot#(s(X),s(Y)) -> quot#(min(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:
     f8(x,y) -> x
     f8(x,y) -> y
     min(X,0()) -> X
     min(s(X),s(Y)) -> min(X,Y)
     quot(0(),s(Y)) -> 0()
     quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
    graph:
     log#(s(s(X))) -> log#(s(quot(X,s(s(0()))))) ->
     log#(s(s(X))) -> 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()))) ->
     quot#(s(X),s(Y)) -> min#(X,Y)
     log#(s(s(X))) -> quot#(X,s(s(0()))) ->
     quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
     quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y)) ->
     quot#(s(X),s(Y)) -> min#(X,Y)
     quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y)) ->
     quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
     quot#(s(X),s(Y)) -> min#(X,Y) -> min#(s(X),s(Y)) -> min#(X,Y)
     min#(s(X),s(Y)) -> min#(X,Y) -> min#(s(X),s(Y)) -> min#(X,Y)
    Restore Modifier:
     DPs:
      min#(s(X),s(Y)) -> min#(X,Y)
      quot#(s(X),s(Y)) -> min#(X,Y)
      quot#(s(X),s(Y)) -> quot#(min(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:
      min(X,0()) -> X
      min(s(X),s(Y)) -> min(X,Y)
      quot(0(),s(Y)) -> 0()
      quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
      log(s(0())) -> 0()
      log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
     SCC Processor:
      #sccs: 3
      #rules: 3
      #arcs: 8/25
      DPs:
       log#(s(s(X))) -> log#(s(quot(X,s(s(0())))))
      TRS:
       min(X,0()) -> X
       min(s(X),s(Y)) -> min(X,Y)
       quot(0(),s(Y)) -> 0()
       quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
       log(s(0())) -> 0()
       log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [log#](x0) = x0 + 1,
        
        [log](x0) = x0,
        
        [quot](x0, x1) = x0,
        
        [s](x0) = x0 + 1,
        
        [min](x0, x1) = x0,
        
        [0] = 1
       orientation:
        log#(s(s(X))) = X + 3 >= X + 2 = log#(s(quot(X,s(s(0())))))
        
        min(X,0()) = X >= X = X
        
        min(s(X),s(Y)) = X + 1 >= X = min(X,Y)
        
        quot(0(),s(Y)) = 1 >= 1 = 0()
        
        quot(s(X),s(Y)) = X + 1 >= X + 1 = s(quot(min(X,Y),s(Y)))
        
        log(s(0())) = 2 >= 1 = 0()
        
        log(s(s(X))) = X + 2 >= X + 2 = s(log(s(quot(X,s(s(0()))))))
       problem:
        DPs:
         
        TRS:
         min(X,0()) -> X
         min(s(X),s(Y)) -> min(X,Y)
         quot(0(),s(Y)) -> 0()
         quot(s(X),s(Y)) -> s(quot(min(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#(min(X,Y),s(Y))
      TRS:
       min(X,0()) -> X
       min(s(X),s(Y)) -> min(X,Y)
       quot(0(),s(Y)) -> 0()
       quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
       log(s(0())) -> 0()
       log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [quot#](x0, x1) = x0,
        
        [log](x0) = x0,
        
        [quot](x0, x1) = x0,
        
        [s](x0) = x0 + 1,
        
        [min](x0, x1) = x0,
        
        [0] = 1
       orientation:
        quot#(s(X),s(Y)) = X + 1 >= X = quot#(min(X,Y),s(Y))
        
        min(X,0()) = X >= X = X
        
        min(s(X),s(Y)) = X + 1 >= X = min(X,Y)
        
        quot(0(),s(Y)) = 1 >= 1 = 0()
        
        quot(s(X),s(Y)) = X + 1 >= X + 1 = s(quot(min(X,Y),s(Y)))
        
        log(s(0())) = 2 >= 1 = 0()
        
        log(s(s(X))) = X + 2 >= X + 2 = s(log(s(quot(X,s(s(0()))))))
       problem:
        DPs:
         
        TRS:
         min(X,0()) -> X
         min(s(X),s(Y)) -> min(X,Y)
         quot(0(),s(Y)) -> 0()
         quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
         log(s(0())) -> 0()
         log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
       Qed
      
      DPs:
       min#(s(X),s(Y)) -> min#(X,Y)
      TRS:
       min(X,0()) -> X
       min(s(X),s(Y)) -> min(X,Y)
       quot(0(),s(Y)) -> 0()
       quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
       log(s(0())) -> 0()
       log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [min#](x0, x1) = x0 + x1 + 1,
        
        [log](x0) = x0,
        
        [quot](x0, x1) = x0,
        
        [s](x0) = x0 + 1,
        
        [min](x0, x1) = x0,
        
        [0] = 1
       orientation:
        min#(s(X),s(Y)) = X + Y + 3 >= X + Y + 1 = min#(X,Y)
        
        min(X,0()) = X >= X = X
        
        min(s(X),s(Y)) = X + 1 >= X = min(X,Y)
        
        quot(0(),s(Y)) = 1 >= 1 = 0()
        
        quot(s(X),s(Y)) = X + 1 >= X + 1 = s(quot(min(X,Y),s(Y)))
        
        log(s(0())) = 2 >= 1 = 0()
        
        log(s(s(X))) = X + 2 >= X + 2 = s(log(s(quot(X,s(s(0()))))))
       problem:
        DPs:
         
        TRS:
         min(X,0()) -> X
         min(s(X),s(Y)) -> min(X,Y)
         quot(0(),s(Y)) -> 0()
         quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
         log(s(0())) -> 0()
         log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
       Qed