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()))))))
  TDG 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()))))))
   graph:
    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))) -> log#(s(quot(X,s(s(0()))))) ->
    log#(s(s(X))) -> quot#(X,s(s(0())))
    log#(s(s(X))) -> quot#(X,s(s(0()))) ->
    quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
    log#(s(s(X))) -> quot#(X,s(s(0()))) ->
    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)) -> quot#(min(X,Y),s(Y)) ->
    quot#(s(X),s(Y)) -> min#(X,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)
   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()))))))
    Usable Rule Processor:
     DPs:
      log#(s(s(X))) -> log#(s(quot(X,s(s(0())))))
     TRS:
      quot(0(),s(Y)) -> 0()
      quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
      min(X,0()) -> X
      min(s(X),s(Y)) -> min(X,Y)
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       quot(0(),s(Y)) -> 0()
       quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
       min(X,0()) -> X
       min(s(X),s(Y)) -> min(X,Y)
      interpretation:
       [log#](x0) = x0,
       
       [quot](x0, x1) = x0,
       
       [s](x0) = 2x0,
       
       [min](x0, x1) = x0,
       
       [0] = 0
      orientation:
       log#(s(s(X))) = 4X >= 2X = log#(s(quot(X,s(s(0())))))
       
       quot(0(),s(Y)) = 0 >= 0 = 0()
       
       quot(s(X),s(Y)) = 2X >= 2X = s(quot(min(X,Y),s(Y)))
       
       min(X,0()) = X >= X = X
       
       min(s(X),s(Y)) = 2X >= X = min(X,Y)
      problem:
       DPs:
        
       TRS:
        quot(0(),s(Y)) -> 0()
        quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
        min(X,0()) -> X
        min(s(X),s(Y)) -> min(X,Y)
      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()))))))
    Usable Rule Processor:
     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)
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       min(X,0()) -> X
       min(s(X),s(Y)) -> min(X,Y)
      interpretation:
       [quot#](x0, x1) = 3x0 + -16,
       
       [s](x0) = 1x0 + -3,
       
       [min](x0, x1) = x0 + -11,
       
       [0] = 8
      orientation:
       quot#(s(X),s(Y)) = 4X + 0 >= 3X + -8 = quot#(min(X,Y),s(Y))
       
       min(X,0()) = X + -11 >= X = X
       
       min(s(X),s(Y)) = 1X + -3 >= X + -11 = min(X,Y)
      problem:
       DPs:
        
       TRS:
        min(X,0()) -> X
        min(s(X),s(Y)) -> min(X,Y)
      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()))))))
    Subterm Criterion Processor:
     simple projection:
      pi(min#) = 1
     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