We consider the following Problem:

  Strict Trs:
    {  minus(n__0(), Y) -> 0()
     , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
     , geq(X, n__0()) -> true()
     , geq(n__0(), n__s(Y)) -> false()
     , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
     , div(0(), n__s(Y)) -> 0()
     , div(s(X), n__s(Y)) ->
       if(geq(X, activate(Y)),
          n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))),
          n__0())
     , if(true(), X, Y) -> activate(X)
     , if(false(), X, Y) -> activate(Y)
     , 0() -> n__0()
     , s(X) -> n__s(X)
     , div(X1, X2) -> n__div(X1, X2)
     , minus(X1, X2) -> n__minus(X1, X2)
     , activate(n__0()) -> 0()
     , activate(n__s(X)) -> s(activate(X))
     , activate(n__div(X1, X2)) -> div(activate(X1), X2)
     , activate(n__minus(X1, X2)) -> minus(X1, X2)
     , activate(X) -> X}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,O(n^2))

Proof:
  Arguments of following rules are not normal-forms:
  {  div(s(X), n__s(Y)) ->
     if(geq(X, activate(Y)),
        n__s(n__div(n__minus(X, activate(Y)), n__s(activate(Y)))),
        n__0())
   , div(0(), n__s(Y)) -> 0()}
  
  All above mentioned rules can be savely removed.
  
  We consider the following Problem:
  
    Strict Trs:
      {  minus(n__0(), Y) -> 0()
       , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
       , geq(X, n__0()) -> true()
       , geq(n__0(), n__s(Y)) -> false()
       , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
       , if(true(), X, Y) -> activate(X)
       , if(false(), X, Y) -> activate(Y)
       , 0() -> n__0()
       , s(X) -> n__s(X)
       , div(X1, X2) -> n__div(X1, X2)
       , minus(X1, X2) -> n__minus(X1, X2)
       , activate(n__0()) -> 0()
       , activate(n__s(X)) -> s(activate(X))
       , activate(n__div(X1, X2)) -> div(activate(X1), X2)
       , activate(n__minus(X1, X2)) -> minus(X1, X2)
       , activate(X) -> X}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^2))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  minus(n__0(), Y) -> 0()
       , geq(X, n__0()) -> true()
       , geq(n__0(), n__s(Y)) -> false()
       , if(false(), X, Y) -> activate(Y)
       , s(X) -> n__s(X)
       , div(X1, X2) -> n__div(X1, X2)
       , minus(X1, X2) -> n__minus(X1, X2)}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(minus) = {1, 2}, Uargs(n__s) = {}, Uargs(activate) = {},
        Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1},
        Uargs(if) = {}, Uargs(n__div) = {}, Uargs(n__minus) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       minus(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                       [0 0]      [1 0]      [1]
       n__0() = [0]
                [0]
       0() = [0]
             [0]
       n__s(x1) = [1 0] x1 + [0]
                  [0 0]      [0]
       activate(x1) = [1 0] x1 + [0]
                      [0 0]      [1]
       geq(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                     [0 0]      [0 0]      [1]
       true() = [0]
                [0]
       false() = [0]
                 [0]
       div(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                     [0 0]      [0 0]      [1]
       s(x1) = [1 0] x1 + [1]
               [0 0]      [1]
       if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [1]
                        [0 0]      [0 0]      [0 0]      [1]
       n__div(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                        [0 0]      [0 0]      [0]
       n__minus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                          [0 0]      [0 0]      [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
         , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
         , if(true(), X, Y) -> activate(X)
         , 0() -> n__0()
         , activate(n__0()) -> 0()
         , activate(n__s(X)) -> s(activate(X))
         , activate(n__div(X1, X2)) -> div(activate(X1), X2)
         , activate(n__minus(X1, X2)) -> minus(X1, X2)
         , activate(X) -> X}
      Weak Trs:
        {  minus(n__0(), Y) -> 0()
         , geq(X, n__0()) -> true()
         , geq(n__0(), n__s(Y)) -> false()
         , if(false(), X, Y) -> activate(Y)
         , s(X) -> n__s(X)
         , div(X1, X2) -> n__div(X1, X2)
         , minus(X1, X2) -> n__minus(X1, X2)}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^2))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component: {if(true(), X, Y) -> activate(X)}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(minus) = {1, 2}, Uargs(n__s) = {}, Uargs(activate) = {},
          Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1},
          Uargs(if) = {}, Uargs(n__div) = {}, Uargs(n__minus) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         minus(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                         [0 0]      [1 0]      [1]
         n__0() = [0]
                  [0]
         0() = [0]
               [0]
         n__s(x1) = [1 0] x1 + [0]
                    [0 0]      [0]
         activate(x1) = [1 0] x1 + [0]
                        [0 0]      [1]
         geq(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                       [0 0]      [0 0]      [1]
         true() = [0]
                  [0]
         false() = [0]
                   [0]
         div(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                       [0 0]      [0 0]      [1]
         s(x1) = [1 0] x1 + [1]
                 [0 0]      [1]
         if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
                          [0 0]      [0 0]      [0 0]      [1]
         n__div(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                          [0 0]      [0 0]      [0]
         n__minus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                            [0 0]      [0 0]      [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
           , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
           , 0() -> n__0()
           , activate(n__0()) -> 0()
           , activate(n__s(X)) -> s(activate(X))
           , activate(n__div(X1, X2)) -> div(activate(X1), X2)
           , activate(n__minus(X1, X2)) -> minus(X1, X2)
           , activate(X) -> X}
        Weak Trs:
          {  if(true(), X, Y) -> activate(X)
           , minus(n__0(), Y) -> 0()
           , geq(X, n__0()) -> true()
           , geq(n__0(), n__s(Y)) -> false()
           , if(false(), X, Y) -> activate(Y)
           , s(X) -> n__s(X)
           , div(X1, X2) -> n__div(X1, X2)
           , minus(X1, X2) -> n__minus(X1, X2)}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^2))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {0() -> n__0()}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(minus) = {1, 2}, Uargs(n__s) = {}, Uargs(activate) = {},
            Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1},
            Uargs(if) = {}, Uargs(n__div) = {}, Uargs(n__minus) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           minus(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                           [0 0]      [1 0]      [1]
           n__0() = [0]
                    [0]
           0() = [1]
                 [0]
           n__s(x1) = [1 0] x1 + [0]
                      [0 0]      [0]
           activate(x1) = [1 0] x1 + [0]
                          [0 1]      [1]
           geq(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                         [0 0]      [0 0]      [1]
           true() = [0]
                    [0]
           false() = [0]
                     [0]
           div(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                         [0 0]      [0 0]      [1]
           s(x1) = [1 0] x1 + [1]
                   [0 0]      [1]
           if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
                            [0 0]      [0 1]      [0 1]      [1]
           n__div(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [1]
           n__minus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 0]      [0]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs:
            {  minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
             , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
             , activate(n__0()) -> 0()
             , activate(n__s(X)) -> s(activate(X))
             , activate(n__div(X1, X2)) -> div(activate(X1), X2)
             , activate(n__minus(X1, X2)) -> minus(X1, X2)
             , activate(X) -> X}
          Weak Trs:
            {  0() -> n__0()
             , if(true(), X, Y) -> activate(X)
             , minus(n__0(), Y) -> 0()
             , geq(X, n__0()) -> true()
             , geq(n__0(), n__s(Y)) -> false()
             , if(false(), X, Y) -> activate(Y)
             , s(X) -> n__s(X)
             , div(X1, X2) -> n__div(X1, X2)
             , minus(X1, X2) -> n__minus(X1, X2)}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^2))
        
        Proof:
          The weightgap principle applies, where following rules are oriented strictly:
          
          TRS Component:
            {  activate(n__0()) -> 0()
             , activate(X) -> X}
          
          Interpretation of nonconstant growth:
          -------------------------------------
            The following argument positions are usable:
              Uargs(minus) = {1, 2}, Uargs(n__s) = {}, Uargs(activate) = {},
              Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1},
              Uargs(if) = {}, Uargs(n__div) = {}, Uargs(n__minus) = {}
            We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
            Interpretation Functions:
             minus(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                             [0 0]      [1 0]      [1]
             n__0() = [1]
                      [0]
             0() = [1]
                   [0]
             n__s(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
             activate(x1) = [1 0] x1 + [2]
                            [0 1]      [0]
             geq(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                           [0 1]      [0 0]      [1]
             true() = [0]
                      [0]
             false() = [0]
                       [0]
             div(x1, x2) = [1 0] x1 + [0 0] x2 + [2]
                           [1 1]      [0 0]      [1]
             s(x1) = [1 0] x1 + [3]
                     [0 0]      [1]
             if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [3]
                              [0 0]      [0 1]      [0 1]      [1]
             n__div(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [1]
             n__minus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                [0 0]      [0 0]      [0]
          
          The strictly oriented rules are moved into the weak component.
          
          We consider the following Problem:
          
            Strict Trs:
              {  minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
               , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
               , activate(n__s(X)) -> s(activate(X))
               , activate(n__div(X1, X2)) -> div(activate(X1), X2)
               , activate(n__minus(X1, X2)) -> minus(X1, X2)}
            Weak Trs:
              {  activate(n__0()) -> 0()
               , activate(X) -> X
               , 0() -> n__0()
               , if(true(), X, Y) -> activate(X)
               , minus(n__0(), Y) -> 0()
               , geq(X, n__0()) -> true()
               , geq(n__0(), n__s(Y)) -> false()
               , if(false(), X, Y) -> activate(Y)
               , s(X) -> n__s(X)
               , div(X1, X2) -> n__div(X1, X2)
               , minus(X1, X2) -> n__minus(X1, X2)}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^2))
          
          Proof:
            The weightgap principle applies, where following rules are oriented strictly:
            
            TRS Component: {activate(n__minus(X1, X2)) -> minus(X1, X2)}
            
            Interpretation of nonconstant growth:
            -------------------------------------
              The following argument positions are usable:
                Uargs(minus) = {1, 2}, Uargs(n__s) = {}, Uargs(activate) = {},
                Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1},
                Uargs(if) = {}, Uargs(n__div) = {}, Uargs(n__minus) = {}
              We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
              Interpretation Functions:
               minus(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                               [0 0]      [0 0]      [1]
               n__0() = [0]
                        [0]
               0() = [0]
                     [0]
               n__s(x1) = [1 0] x1 + [0]
                          [0 0]      [1]
               activate(x1) = [1 0] x1 + [2]
                              [0 1]      [1]
               geq(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                             [0 0]      [0 1]      [0]
               true() = [0]
                        [0]
               false() = [0]
                         [0]
               div(x1, x2) = [1 0] x1 + [0 0] x2 + [3]
                             [0 1]      [0 0]      [0]
               s(x1) = [1 0] x1 + [3]
                       [0 0]      [1]
               if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [3]
                                [0 0]      [0 1]      [0 1]      [1]
               n__div(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
               n__minus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
            
            The strictly oriented rules are moved into the weak component.
            
            We consider the following Problem:
            
              Strict Trs:
                {  minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
                 , geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
                 , activate(n__s(X)) -> s(activate(X))
                 , activate(n__div(X1, X2)) -> div(activate(X1), X2)}
              Weak Trs:
                {  activate(n__minus(X1, X2)) -> minus(X1, X2)
                 , activate(n__0()) -> 0()
                 , activate(X) -> X
                 , 0() -> n__0()
                 , if(true(), X, Y) -> activate(X)
                 , minus(n__0(), Y) -> 0()
                 , geq(X, n__0()) -> true()
                 , geq(n__0(), n__s(Y)) -> false()
                 , if(false(), X, Y) -> activate(Y)
                 , s(X) -> n__s(X)
                 , div(X1, X2) -> n__div(X1, X2)
                 , minus(X1, X2) -> n__minus(X1, X2)}
              StartTerms: basic terms
              Strategy: innermost
            
            Certificate: YES(?,O(n^2))
            
            Proof:
              The weightgap principle applies, where following rules are oriented strictly:
              
              TRS Component:
                {minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))}
              
              Interpretation of nonconstant growth:
              -------------------------------------
                The following argument positions are usable:
                  Uargs(minus) = {1, 2}, Uargs(n__s) = {}, Uargs(activate) = {},
                  Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1},
                  Uargs(if) = {}, Uargs(n__div) = {}, Uargs(n__minus) = {}
                We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
                Interpretation Functions:
                 minus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                 [0 0]      [0 0]      [1]
                 n__0() = [0]
                          [0]
                 0() = [0]
                       [0]
                 n__s(x1) = [1 0] x1 + [2]
                            [0 0]      [3]
                 activate(x1) = [1 0] x1 + [0]
                                [0 1]      [1]
                 geq(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                               [0 0]      [0 1]      [1]
                 true() = [0]
                          [0]
                 false() = [0]
                           [0]
                 div(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                 s(x1) = [1 0] x1 + [2]
                         [0 1]      [3]
                 if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
                                  [0 0]      [0 1]      [0 1]      [1]
                 n__div(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                 n__minus(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
              
              The strictly oriented rules are moved into the weak component.
              
              We consider the following Problem:
              
                Strict Trs:
                  {  geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
                   , activate(n__s(X)) -> s(activate(X))
                   , activate(n__div(X1, X2)) -> div(activate(X1), X2)}
                Weak Trs:
                  {  minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
                   , activate(n__minus(X1, X2)) -> minus(X1, X2)
                   , activate(n__0()) -> 0()
                   , activate(X) -> X
                   , 0() -> n__0()
                   , if(true(), X, Y) -> activate(X)
                   , minus(n__0(), Y) -> 0()
                   , geq(X, n__0()) -> true()
                   , geq(n__0(), n__s(Y)) -> false()
                   , if(false(), X, Y) -> activate(Y)
                   , s(X) -> n__s(X)
                   , div(X1, X2) -> n__div(X1, X2)
                   , minus(X1, X2) -> n__minus(X1, X2)}
                StartTerms: basic terms
                Strategy: innermost
              
              Certificate: YES(?,O(n^2))
              
              Proof:
                The weightgap principle applies, where following rules are oriented strictly:
                
                TRS Component:
                  {geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))}
                
                Interpretation of nonconstant growth:
                -------------------------------------
                  The following argument positions are usable:
                    Uargs(minus) = {1, 2}, Uargs(n__s) = {}, Uargs(activate) = {},
                    Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1},
                    Uargs(if) = {}, Uargs(n__div) = {}, Uargs(n__minus) = {}
                  We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
                  Interpretation Functions:
                   minus(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                                   [0 0]      [0 0]      [1]
                   n__0() = [0]
                            [0]
                   0() = [0]
                         [0]
                   n__s(x1) = [1 0] x1 + [2]
                              [0 0]      [0]
                   activate(x1) = [1 0] x1 + [0]
                                  [0 1]      [1]
                   geq(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                 [0 0]      [1 0]      [1]
                   true() = [0]
                            [0]
                   false() = [0]
                             [0]
                   div(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                                 [0 1]      [0 0]      [0]
                   s(x1) = [1 0] x1 + [2]
                           [0 1]      [0]
                   if(x1, x2, x3) = [0 0] x1 + [1 0] x2 + [1 0] x3 + [1]
                                    [0 0]      [0 1]      [0 1]      [1]
                   n__div(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                                    [0 0]      [0 0]      [0]
                   n__minus(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                                      [0 0]      [0 0]      [0]
                
                The strictly oriented rules are moved into the weak component.
                
                We consider the following Problem:
                
                  Strict Trs:
                    {  activate(n__s(X)) -> s(activate(X))
                     , activate(n__div(X1, X2)) -> div(activate(X1), X2)}
                  Weak Trs:
                    {  geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
                     , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
                     , activate(n__minus(X1, X2)) -> minus(X1, X2)
                     , activate(n__0()) -> 0()
                     , activate(X) -> X
                     , 0() -> n__0()
                     , if(true(), X, Y) -> activate(X)
                     , minus(n__0(), Y) -> 0()
                     , geq(X, n__0()) -> true()
                     , geq(n__0(), n__s(Y)) -> false()
                     , if(false(), X, Y) -> activate(Y)
                     , s(X) -> n__s(X)
                     , div(X1, X2) -> n__div(X1, X2)
                     , minus(X1, X2) -> n__minus(X1, X2)}
                  StartTerms: basic terms
                  Strategy: innermost
                
                Certificate: YES(?,O(n^2))
                
                Proof:
                  We consider the following Problem:
                  
                    Strict Trs:
                      {  activate(n__s(X)) -> s(activate(X))
                       , activate(n__div(X1, X2)) -> div(activate(X1), X2)}
                    Weak Trs:
                      {  geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y))
                       , minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y))
                       , activate(n__minus(X1, X2)) -> minus(X1, X2)
                       , activate(n__0()) -> 0()
                       , activate(X) -> X
                       , 0() -> n__0()
                       , if(true(), X, Y) -> activate(X)
                       , minus(n__0(), Y) -> 0()
                       , geq(X, n__0()) -> true()
                       , geq(n__0(), n__s(Y)) -> false()
                       , if(false(), X, Y) -> activate(Y)
                       , s(X) -> n__s(X)
                       , div(X1, X2) -> n__div(X1, X2)
                       , minus(X1, X2) -> n__minus(X1, X2)}
                    StartTerms: basic terms
                    Strategy: innermost
                  
                  Certificate: YES(?,O(n^2))
                  
                  Proof:
                    The following argument positions are usable:
                      Uargs(minus) = {1, 2}, Uargs(n__s) = {}, Uargs(activate) = {},
                      Uargs(geq) = {1, 2}, Uargs(div) = {1}, Uargs(s) = {1},
                      Uargs(if) = {}, Uargs(n__div) = {}, Uargs(n__minus) = {}
                    We have the following constructor-based EDA-non-satisfying and IDA(2)-non-satisfying matrix interpretation:
                    Interpretation Functions:
                     minus(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [1]
                                     [0 0 0]      [0 0 0]      [1]
                                     [0 0 0]      [0 0 0]      [1]
                     n__0() = [0]
                              [0]
                              [0]
                     0() = [0]
                           [0]
                           [0]
                     n__s(x1) = [1 0 2] x1 + [1]
                                [0 0 0]      [2]
                                [0 0 1]      [2]
                     activate(x1) = [1 0 2] x1 + [0]
                                    [0 2 0]      [1]
                                    [0 0 1]      [0]
                     geq(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [1]
                                   [0 0 0]      [0 0 0]      [1]
                                   [0 0 0]      [0 0 0]      [1]
                     true() = [0]
                              [0]
                              [0]
                     false() = [0]
                               [0]
                               [0]
                     div(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [1]
                                   [0 0 0]      [0 0 0]      [1]
                                   [0 0 1]      [0 0 0]      [2]
                     s(x1) = [1 0 2] x1 + [1]
                             [0 0 0]      [2]
                             [0 0 1]      [2]
                     if(x1, x2, x3) = [0 0 0] x1 + [2 0 2] x2 + [2 0 2] x3 + [1]
                                      [0 0 0]      [0 2 0]      [0 2 0]      [1]
                                      [0 0 0]      [0 0 2]      [0 0 2]      [1]
                     n__div(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [0]
                                      [0 0 0]      [0 0 0]      [0]
                                      [0 0 1]      [0 0 0]      [2]
                     n__minus(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
                                        [0 0 0]      [0 0 0]      [0]
                                        [0 0 0]      [0 0 0]      [1]

Hurray, we answered YES(?,O(n^2))