We consider the following Problem:

  Strict Trs:
    {  cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
     , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
     , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
     , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
     , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)
     , gr(0(), x) -> false()
     , gr(s(x), 0()) -> true()
     , gr(s(x), s(y)) -> gr(x, y)
     , or(false(), false()) -> false()
     , or(true(), x) -> true()
     , or(x, true()) -> true()
     , p(0()) -> 0()
     , p(s(x)) -> x}
  StartTerms: basic terms
  Strategy: innermost

Certificate: YES(?,O(n^1))

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
       , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
       , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
       , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
       , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)
       , gr(0(), x) -> false()
       , gr(s(x), 0()) -> true()
       , gr(s(x), s(y)) -> gr(x, y)
       , or(false(), false()) -> false()
       , or(true(), x) -> true()
       , or(x, true()) -> true()
       , p(0()) -> 0()
       , p(s(x)) -> x}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^1))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
       , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(cond1) = {1, 2, 3}, Uargs(cond2) = {1}, Uargs(gr) = {},
        Uargs(or) = {1, 2}, Uargs(p) = {}, Uargs(cond3) = {1},
        Uargs(s) = {}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       cond1(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                           [0 0]      [0 0]      [0 0]      [1]
       true() = [0]
                [0]
       cond2(x1, x2, x3) = [1 1] x1 + [1 1] x2 + [1 1] x3 + [1]
                           [0 0]      [0 0]      [0 0]      [1]
       gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                    [0 0]      [0 0]      [0]
       0() = [0]
             [0]
       or(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                    [0 0]      [0 0]      [1]
       p(x1) = [0 1] x1 + [0]
               [0 0]      [0]
       false() = [1]
                 [3]
       cond3(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                           [0 0]      [0 0]      [0 0]      [1]
       s(x1) = [0 0] x1 + [0]
               [1 0]      [0]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
         , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
         , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
         , gr(0(), x) -> false()
         , gr(s(x), 0()) -> true()
         , gr(s(x), s(y)) -> gr(x, y)
         , or(false(), false()) -> false()
         , or(true(), x) -> true()
         , or(x, true()) -> true()
         , p(0()) -> 0()
         , p(s(x)) -> x}
      Weak Trs:
        {  cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
         , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^1))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component:
        {  cond3(true(), x, y) ->
           cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
         , or(false(), false()) -> false()}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(cond1) = {1, 2, 3}, Uargs(cond2) = {1}, Uargs(gr) = {},
          Uargs(or) = {1, 2}, Uargs(p) = {}, Uargs(cond3) = {1},
          Uargs(s) = {}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         cond1(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                             [0 0]      [0 0]      [0 0]      [1]
         true() = [0]
                  [0]
         cond2(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                             [0 0]      [0 0]      [0 0]      [1]
         gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                      [1 0]      [0 0]      [0]
         0() = [1]
               [0]
         or(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                      [0 0]      [0 0]      [1]
         p(x1) = [0 1] x1 + [0]
                 [0 0]      [0]
         false() = [2]
                   [0]
         cond3(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [3]
                             [0 0]      [0 0]      [0 0]      [1]
         s(x1) = [0 0] x1 + [0]
                 [1 0]      [0]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
           , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
           , gr(0(), x) -> false()
           , gr(s(x), 0()) -> true()
           , gr(s(x), s(y)) -> gr(x, y)
           , or(true(), x) -> true()
           , or(x, true()) -> true()
           , p(0()) -> 0()
           , p(s(x)) -> x}
        Weak Trs:
          {  cond3(true(), x, y) ->
             cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
           , or(false(), false()) -> false()
           , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
           , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^1))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component: {p(0()) -> 0()}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(cond1) = {1, 2, 3}, Uargs(cond2) = {1}, Uargs(gr) = {},
            Uargs(or) = {1, 2}, Uargs(p) = {}, Uargs(cond3) = {1},
            Uargs(s) = {}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           cond1(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                               [0 0]      [0 0]      [0 0]      [1]
           true() = [0]
                    [0]
           cond2(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                               [0 0]      [0 0]      [0 0]      [1]
           gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                        [0 0]      [0 0]      [1]
           0() = [0]
                 [1]
           or(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                        [0 0]      [0 0]      [1]
           p(x1) = [0 1] x1 + [0]
                   [0 0]      [2]
           false() = [2]
                     [0]
           cond3(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [3]
                               [0 0]      [0 0]      [0 0]      [1]
           s(x1) = [0 0] x1 + [0]
                   [1 0]      [0]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs:
            {  cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
             , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
             , gr(0(), x) -> false()
             , gr(s(x), 0()) -> true()
             , gr(s(x), s(y)) -> gr(x, y)
             , or(true(), x) -> true()
             , or(x, true()) -> true()
             , p(s(x)) -> x}
          Weak Trs:
            {  p(0()) -> 0()
             , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
             , or(false(), false()) -> false()
             , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
             , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^1))
        
        Proof:
          The weightgap principle applies, where following rules are oriented strictly:
          
          TRS Component:
            {  or(true(), x) -> true()
             , or(x, true()) -> true()}
          
          Interpretation of nonconstant growth:
          -------------------------------------
            The following argument positions are usable:
              Uargs(cond1) = {1, 2, 3}, Uargs(cond2) = {1}, Uargs(gr) = {},
              Uargs(or) = {1, 2}, Uargs(p) = {}, Uargs(cond3) = {1},
              Uargs(s) = {}
            We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
            Interpretation Functions:
             cond1(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                                 [0 0]      [0 0]      [0 0]      [1]
             true() = [0]
                      [0]
             cond2(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                                 [0 0]      [0 0]      [0 0]      [1]
             gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                          [0 0]      [0 0]      [1]
             0() = [0]
                   [0]
             or(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                          [0 0]      [0 0]      [1]
             p(x1) = [0 1] x1 + [0]
                     [0 0]      [0]
             false() = [2]
                       [0]
             cond3(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [3]
                                 [0 0]      [0 0]      [0 0]      [1]
             s(x1) = [0 0] x1 + [0]
                     [1 0]      [0]
          
          The strictly oriented rules are moved into the weak component.
          
          We consider the following Problem:
          
            Strict Trs:
              {  cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
               , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
               , gr(0(), x) -> false()
               , gr(s(x), 0()) -> true()
               , gr(s(x), s(y)) -> gr(x, y)
               , p(s(x)) -> x}
            Weak Trs:
              {  or(true(), x) -> true()
               , or(x, true()) -> true()
               , p(0()) -> 0()
               , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
               , or(false(), false()) -> false()
               , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
               , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^1))
          
          Proof:
            The weightgap principle applies, where following rules are oriented strictly:
            
            TRS Component:
              {cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)}
            
            Interpretation of nonconstant growth:
            -------------------------------------
              The following argument positions are usable:
                Uargs(cond1) = {1, 2, 3}, Uargs(cond2) = {1}, Uargs(gr) = {},
                Uargs(or) = {1, 2}, Uargs(p) = {}, Uargs(cond3) = {1},
                Uargs(s) = {}
              We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
              Interpretation Functions:
               cond1(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                                   [0 0]      [0 0]      [0 0]      [1]
               true() = [0]
                        [3]
               cond2(x1, x2, x3) = [1 1] x1 + [1 1] x2 + [1 1] x3 + [1]
                                   [0 0]      [0 0]      [0 0]      [1]
               gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
               0() = [0]
                     [0]
               or(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                            [0 0]      [0 0]      [3]
               p(x1) = [0 1] x1 + [0]
                       [0 0]      [0]
               false() = [0]
                         [0]
               cond3(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                                   [0 0]      [0 0]      [0 0]      [1]
               s(x1) = [0 0] x1 + [0]
                       [1 0]      [0]
            
            The strictly oriented rules are moved into the weak component.
            
            We consider the following Problem:
            
              Strict Trs:
                {  cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
                 , gr(0(), x) -> false()
                 , gr(s(x), 0()) -> true()
                 , gr(s(x), s(y)) -> gr(x, y)
                 , p(s(x)) -> x}
              Weak Trs:
                {  cond2(true(), x, y) ->
                   cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
                 , or(true(), x) -> true()
                 , or(x, true()) -> true()
                 , p(0()) -> 0()
                 , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
                 , or(false(), false()) -> false()
                 , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
                 , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)}
              StartTerms: basic terms
              Strategy: innermost
            
            Certificate: YES(?,O(n^1))
            
            Proof:
              The weightgap principle applies, where following rules are oriented strictly:
              
              TRS Component: {p(s(x)) -> x}
              
              Interpretation of nonconstant growth:
              -------------------------------------
                The following argument positions are usable:
                  Uargs(cond1) = {1, 2, 3}, Uargs(cond2) = {1}, Uargs(gr) = {},
                  Uargs(or) = {1, 2}, Uargs(p) = {}, Uargs(cond3) = {1},
                  Uargs(s) = {}
                We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
                Interpretation Functions:
                 cond1(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [1]
                 true() = [0]
                          [0]
                 cond2(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [2]
                                     [0 0]      [0 0]      [0 0]      [1]
                 gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 1]      [0 0]      [0]
                 0() = [0]
                       [0]
                 or(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                              [0 0]      [0 0]      [1]
                 p(x1) = [0 1] x1 + [0]
                         [1 0]      [1]
                 false() = [0]
                           [0]
                 cond3(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                                     [0 0]      [0 0]      [0 0]      [1]
                 s(x1) = [0 1] x1 + [0]
                         [1 0]      [1]
              
              The strictly oriented rules are moved into the weak component.
              
              We consider the following Problem:
              
                Strict Trs:
                  {  cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
                   , gr(0(), x) -> false()
                   , gr(s(x), 0()) -> true()
                   , gr(s(x), s(y)) -> gr(x, y)}
                Weak Trs:
                  {  p(s(x)) -> x
                   , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
                   , or(true(), x) -> true()
                   , or(x, true()) -> true()
                   , p(0()) -> 0()
                   , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
                   , or(false(), false()) -> false()
                   , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
                   , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)}
                StartTerms: basic terms
                Strategy: innermost
              
              Certificate: YES(?,O(n^1))
              
              Proof:
                The weightgap principle applies, where following rules are oriented strictly:
                
                TRS Component: {cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)}
                
                Interpretation of nonconstant growth:
                -------------------------------------
                  The following argument positions are usable:
                    Uargs(cond1) = {1, 2, 3}, Uargs(cond2) = {1}, Uargs(gr) = {},
                    Uargs(or) = {1, 2}, Uargs(p) = {}, Uargs(cond3) = {1},
                    Uargs(s) = {}
                  We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
                  Interpretation Functions:
                   cond1(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [0]
                                       [0 1]      [0 0]      [0 0]      [3]
                   true() = [1]
                            [1]
                   cond2(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [0]
                                       [0 0]      [0 0]      [0 0]      [3]
                   gr(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                   0() = [0]
                         [0]
                   or(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                [0 1]      [0 1]      [0]
                   p(x1) = [0 1] x1 + [0]
                           [1 0]      [0]
                   false() = [0]
                             [0]
                   cond3(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [0]
                                       [0 0]      [0 0]      [0 0]      [3]
                   s(x1) = [0 1] x1 + [0]
                           [1 0]      [1]
                
                The strictly oriented rules are moved into the weak component.
                
                We consider the following Problem:
                
                  Strict Trs:
                    {  gr(0(), x) -> false()
                     , gr(s(x), 0()) -> true()
                     , gr(s(x), s(y)) -> gr(x, y)}
                  Weak Trs:
                    {  cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
                     , p(s(x)) -> x
                     , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
                     , or(true(), x) -> true()
                     , or(x, true()) -> true()
                     , p(0()) -> 0()
                     , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
                     , or(false(), false()) -> false()
                     , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
                     , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)}
                  StartTerms: basic terms
                  Strategy: innermost
                
                Certificate: YES(?,O(n^1))
                
                Proof:
                  The weightgap principle applies, where following rules are oriented strictly:
                  
                  TRS Component: {gr(s(x), s(y)) -> gr(x, y)}
                  
                  Interpretation of nonconstant growth:
                  -------------------------------------
                    The following argument positions are usable:
                      Uargs(cond1) = {1, 2, 3}, Uargs(cond2) = {1}, Uargs(gr) = {},
                      Uargs(or) = {1, 2}, Uargs(p) = {}, Uargs(cond3) = {1},
                      Uargs(s) = {}
                    We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
                    Interpretation Functions:
                     cond1(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                                         [0 0]      [0 0]      [0 0]      [0]
                     true() = [0]
                              [0]
                     cond2(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                                         [0 0]      [0 0]      [0 0]      [0]
                     gr(x1, x2) = [0 0] x1 + [1 1] x2 + [0]
                                  [0 0]      [0 0]      [0]
                     0() = [0]
                           [0]
                     or(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                     p(x1) = [0 1] x1 + [0]
                             [1 0]      [0]
                     false() = [0]
                               [0]
                     cond3(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 1] x3 + [1]
                                         [0 0]      [0 0]      [0 0]      [0]
                     s(x1) = [0 1] x1 + [2]
                             [1 0]      [3]
                  
                  The strictly oriented rules are moved into the weak component.
                  
                  We consider the following Problem:
                  
                    Strict Trs:
                      {  gr(0(), x) -> false()
                       , gr(s(x), 0()) -> true()}
                    Weak Trs:
                      {  gr(s(x), s(y)) -> gr(x, y)
                       , cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
                       , p(s(x)) -> x
                       , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
                       , or(true(), x) -> true()
                       , or(x, true()) -> true()
                       , p(0()) -> 0()
                       , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
                       , or(false(), false()) -> false()
                       , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
                       , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)}
                    StartTerms: basic terms
                    Strategy: innermost
                  
                  Certificate: YES(?,O(n^1))
                  
                  Proof:
                    The weightgap principle applies, where following rules are oriented strictly:
                    
                    TRS Component: {gr(0(), x) -> false()}
                    
                    Interpretation of nonconstant growth:
                    -------------------------------------
                      The following argument positions are usable:
                        Uargs(cond1) = {1, 2, 3}, Uargs(cond2) = {1}, Uargs(gr) = {},
                        Uargs(or) = {1, 2}, Uargs(p) = {}, Uargs(cond3) = {1},
                        Uargs(s) = {}
                      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
                      Interpretation Functions:
                       cond1(x1, x2, x3) = [1 1] x1 + [1 1] x2 + [1 0] x3 + [0]
                                           [0 0]      [0 0]      [0 0]      [0]
                       true() = [3]
                                [2]
                       cond2(x1, x2, x3) = [1 1] x1 + [1 1] x2 + [1 0] x3 + [3]
                                           [0 0]      [0 0]      [0 0]      [0]
                       gr(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                                    [0 0]      [0 0]      [0]
                       0() = [0]
                             [0]
                       or(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                                    [0 1]      [0 1]      [0]
                       p(x1) = [1 0] x1 + [3]
                               [0 1]      [3]
                       false() = [0]
                                 [0]
                       cond3(x1, x2, x3) = [1 0] x1 + [1 1] x2 + [1 0] x3 + [2]
                                           [0 0]      [0 0]      [0 0]      [0]
                       s(x1) = [1 0] x1 + [1]
                               [0 1]      [2]
                    
                    The strictly oriented rules are moved into the weak component.
                    
                    We consider the following Problem:
                    
                      Strict Trs: {gr(s(x), 0()) -> true()}
                      Weak Trs:
                        {  gr(0(), x) -> false()
                         , gr(s(x), s(y)) -> gr(x, y)
                         , cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
                         , p(s(x)) -> x
                         , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
                         , or(true(), x) -> true()
                         , or(x, true()) -> true()
                         , p(0()) -> 0()
                         , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
                         , or(false(), false()) -> false()
                         , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
                         , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)}
                      StartTerms: basic terms
                      Strategy: innermost
                    
                    Certificate: YES(?,O(n^1))
                    
                    Proof:
                      We consider the following Problem:
                      
                        Strict Trs: {gr(s(x), 0()) -> true()}
                        Weak Trs:
                          {  gr(0(), x) -> false()
                           , gr(s(x), s(y)) -> gr(x, y)
                           , cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
                           , p(s(x)) -> x
                           , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
                           , or(true(), x) -> true()
                           , or(x, true()) -> true()
                           , p(0()) -> 0()
                           , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
                           , or(false(), false()) -> false()
                           , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
                           , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)}
                        StartTerms: basic terms
                        Strategy: innermost
                      
                      Certificate: YES(?,O(n^1))
                      
                      Proof:
                        We have computed the following dependency pairs
                        
                          Strict DPs: {gr^#(s(x), 0()) -> c_1()}
                          Weak DPs:
                            {  gr^#(0(), x) -> c_2()
                             , gr^#(s(x), s(y)) -> gr^#(x, y)
                             , cond1^#(true(), x, y) -> cond2^#(gr(x, 0()), x, y)
                             , p^#(s(x)) -> c_5()
                             , cond2^#(true(), x, y) ->
                               cond1^#(or(gr(x, 0()), gr(y, 0())), p(x), y)
                             , or^#(true(), x) -> c_7()
                             , or^#(x, true()) -> c_8()
                             , p^#(0()) -> c_9()
                             , cond3^#(true(), x, y) ->
                               cond1^#(or(gr(x, 0()), gr(y, 0())), x, p(y))
                             , or^#(false(), false()) -> c_11()
                             , cond2^#(false(), x, y) -> cond3^#(gr(y, 0()), x, y)
                             , cond3^#(false(), x, y) ->
                               cond1^#(or(gr(x, 0()), gr(y, 0())), x, y)}
                        
                        We consider the following Problem:
                        
                          Strict DPs: {gr^#(s(x), 0()) -> c_1()}
                          Strict Trs: {gr(s(x), 0()) -> true()}
                          Weak DPs:
                            {  gr^#(0(), x) -> c_2()
                             , gr^#(s(x), s(y)) -> gr^#(x, y)
                             , cond1^#(true(), x, y) -> cond2^#(gr(x, 0()), x, y)
                             , p^#(s(x)) -> c_5()
                             , cond2^#(true(), x, y) ->
                               cond1^#(or(gr(x, 0()), gr(y, 0())), p(x), y)
                             , or^#(true(), x) -> c_7()
                             , or^#(x, true()) -> c_8()
                             , p^#(0()) -> c_9()
                             , cond3^#(true(), x, y) ->
                               cond1^#(or(gr(x, 0()), gr(y, 0())), x, p(y))
                             , or^#(false(), false()) -> c_11()
                             , cond2^#(false(), x, y) -> cond3^#(gr(y, 0()), x, y)
                             , cond3^#(false(), x, y) ->
                               cond1^#(or(gr(x, 0()), gr(y, 0())), x, y)}
                          Weak Trs:
                            {  gr(0(), x) -> false()
                             , gr(s(x), s(y)) -> gr(x, y)
                             , cond1(true(), x, y) -> cond2(gr(x, 0()), x, y)
                             , p(s(x)) -> x
                             , cond2(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), p(x), y)
                             , or(true(), x) -> true()
                             , or(x, true()) -> true()
                             , p(0()) -> 0()
                             , cond3(true(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, p(y))
                             , or(false(), false()) -> false()
                             , cond2(false(), x, y) -> cond3(gr(y, 0()), x, y)
                             , cond3(false(), x, y) -> cond1(or(gr(x, 0()), gr(y, 0())), x, y)}
                          StartTerms: basic terms
                          Strategy: innermost
                        
                        Certificate: YES(?,O(n^1))
                        
                        Proof:
                          We replace strict/weak-rules by the corresponding usable rules:
                          
                            Strict Usable Rules: {gr(s(x), 0()) -> true()}
                            Weak Usable Rules:
                              {  gr(0(), x) -> false()
                               , gr(s(x), s(y)) -> gr(x, y)
                               , p(s(x)) -> x
                               , or(true(), x) -> true()
                               , or(x, true()) -> true()
                               , p(0()) -> 0()
                               , or(false(), false()) -> false()}
                          
                          We consider the following Problem:
                          
                            Strict DPs: {gr^#(s(x), 0()) -> c_1()}
                            Strict Trs: {gr(s(x), 0()) -> true()}
                            Weak DPs:
                              {  gr^#(0(), x) -> c_2()
                               , gr^#(s(x), s(y)) -> gr^#(x, y)
                               , cond1^#(true(), x, y) -> cond2^#(gr(x, 0()), x, y)
                               , p^#(s(x)) -> c_5()
                               , cond2^#(true(), x, y) ->
                                 cond1^#(or(gr(x, 0()), gr(y, 0())), p(x), y)
                               , or^#(true(), x) -> c_7()
                               , or^#(x, true()) -> c_8()
                               , p^#(0()) -> c_9()
                               , cond3^#(true(), x, y) ->
                                 cond1^#(or(gr(x, 0()), gr(y, 0())), x, p(y))
                               , or^#(false(), false()) -> c_11()
                               , cond2^#(false(), x, y) -> cond3^#(gr(y, 0()), x, y)
                               , cond3^#(false(), x, y) ->
                                 cond1^#(or(gr(x, 0()), gr(y, 0())), x, y)}
                            Weak Trs:
                              {  gr(0(), x) -> false()
                               , gr(s(x), s(y)) -> gr(x, y)
                               , p(s(x)) -> x
                               , or(true(), x) -> true()
                               , or(x, true()) -> true()
                               , p(0()) -> 0()
                               , or(false(), false()) -> false()}
                            StartTerms: basic terms
                            Strategy: innermost
                          
                          Certificate: YES(?,O(n^1))
                          
                          Proof:
                            We consider the following Problem:
                            
                              Strict DPs: {gr^#(s(x), 0()) -> c_1()}
                              Strict Trs: {gr(s(x), 0()) -> true()}
                              Weak DPs:
                                {  gr^#(0(), x) -> c_2()
                                 , gr^#(s(x), s(y)) -> gr^#(x, y)
                                 , cond1^#(true(), x, y) -> cond2^#(gr(x, 0()), x, y)
                                 , p^#(s(x)) -> c_5()
                                 , cond2^#(true(), x, y) ->
                                   cond1^#(or(gr(x, 0()), gr(y, 0())), p(x), y)
                                 , or^#(true(), x) -> c_7()
                                 , or^#(x, true()) -> c_8()
                                 , p^#(0()) -> c_9()
                                 , cond3^#(true(), x, y) ->
                                   cond1^#(or(gr(x, 0()), gr(y, 0())), x, p(y))
                                 , or^#(false(), false()) -> c_11()
                                 , cond2^#(false(), x, y) -> cond3^#(gr(y, 0()), x, y)
                                 , cond3^#(false(), x, y) ->
                                   cond1^#(or(gr(x, 0()), gr(y, 0())), x, y)}
                              Weak Trs:
                                {  gr(0(), x) -> false()
                                 , gr(s(x), s(y)) -> gr(x, y)
                                 , p(s(x)) -> x
                                 , or(true(), x) -> true()
                                 , or(x, true()) -> true()
                                 , p(0()) -> 0()
                                 , or(false(), false()) -> false()}
                              StartTerms: basic terms
                              Strategy: innermost
                            
                            Certificate: YES(?,O(n^1))
                            
                            Proof:
                              We use following congruence DG for path analysis
                              
                              ->7:{3}                                                     [      subsumed      ]
                                 |
                                 |->9:{1}                                                 [   YES(?,O(n^1))    ]
                                 |
                                 `->8:{2}                                                 [   YES(O(1),O(1))   ]
                              
                              ->6:{4,13,12,10,6}                                          [   YES(O(1),O(1))   ]
                              
                              ->5:{5}                                                     [   YES(O(1),O(1))   ]
                              
                              ->4:{7}                                                     [   YES(O(1),O(1))   ]
                              
                              ->3:{8}                                                     [   YES(O(1),O(1))   ]
                              
                              ->2:{9}                                                     [   YES(O(1),O(1))   ]
                              
                              ->1:{11}                                                    [   YES(O(1),O(1))   ]
                              
                              
                              Here dependency-pairs are as follows:
                              
                              Strict DPs:
                                {1: gr^#(s(x), 0()) -> c_1()}
                              WeakDPs DPs:
                                {  2: gr^#(0(), x) -> c_2()
                                 , 3: gr^#(s(x), s(y)) -> gr^#(x, y)
                                 , 4: cond1^#(true(), x, y) -> cond2^#(gr(x, 0()), x, y)
                                 , 5: p^#(s(x)) -> c_5()
                                 , 6: cond2^#(true(), x, y) ->
                                      cond1^#(or(gr(x, 0()), gr(y, 0())), p(x), y)
                                 , 7: or^#(true(), x) -> c_7()
                                 , 8: or^#(x, true()) -> c_8()
                                 , 9: p^#(0()) -> c_9()
                                 , 10: cond3^#(true(), x, y) ->
                                       cond1^#(or(gr(x, 0()), gr(y, 0())), x, p(y))
                                 , 11: or^#(false(), false()) -> c_11()
                                 , 12: cond2^#(false(), x, y) -> cond3^#(gr(y, 0()), x, y)
                                 , 13: cond3^#(false(), x, y) ->
                                       cond1^#(or(gr(x, 0()), gr(y, 0())), x, y)}
                              
                              * Path 7:{3}: subsumed
                                --------------------
                                
                                This path is subsumed by the proof of paths 7:{3}->9:{1},
                                                                            7:{3}->8:{2}.
                              
                              * Path 7:{3}->9:{1}: YES(?,O(n^1))
                                --------------------------------
                                
                                We consider the following Problem:
                                
                                  Strict DPs: {gr^#(s(x), 0()) -> c_1()}
                                  Strict Trs: {gr(s(x), 0()) -> true()}
                                  Weak DPs: {gr^#(s(x), s(y)) -> gr^#(x, y)}
                                  Weak Trs:
                                    {  gr(0(), x) -> false()
                                     , gr(s(x), s(y)) -> gr(x, y)
                                     , p(s(x)) -> x
                                     , or(true(), x) -> true()
                                     , or(x, true()) -> true()
                                     , p(0()) -> 0()
                                     , or(false(), false()) -> false()}
                                  StartTerms: basic terms
                                  Strategy: innermost
                                
                                Certificate: YES(?,O(n^1))
                                
                                Proof:
                                  We consider the following Problem:
                                  
                                    Strict DPs: {gr^#(s(x), 0()) -> c_1()}
                                    Strict Trs: {gr(s(x), 0()) -> true()}
                                    Weak DPs: {gr^#(s(x), s(y)) -> gr^#(x, y)}
                                    Weak Trs:
                                      {  gr(0(), x) -> false()
                                       , gr(s(x), s(y)) -> gr(x, y)
                                       , p(s(x)) -> x
                                       , or(true(), x) -> true()
                                       , or(x, true()) -> true()
                                       , p(0()) -> 0()
                                       , or(false(), false()) -> false()}
                                    StartTerms: basic terms
                                    Strategy: innermost
                                  
                                  Certificate: YES(?,O(n^1))
                                  
                                  Proof:
                                    We consider the following Problem:
                                    
                                      Strict DPs: {gr^#(s(x), 0()) -> c_1()}
                                      Strict Trs: {gr(s(x), 0()) -> true()}
                                      Weak DPs: {gr^#(s(x), s(y)) -> gr^#(x, y)}
                                      Weak Trs:
                                        {  gr(0(), x) -> false()
                                         , gr(s(x), s(y)) -> gr(x, y)
                                         , p(s(x)) -> x
                                         , or(true(), x) -> true()
                                         , or(x, true()) -> true()
                                         , p(0()) -> 0()
                                         , or(false(), false()) -> false()}
                                      StartTerms: basic terms
                                      Strategy: innermost
                                    
                                    Certificate: YES(?,O(n^1))
                                    
                                    Proof:
                                      No rule is usable.
                                      
                                      We consider the following Problem:
                                      
                                        Strict DPs: {gr^#(s(x), 0()) -> c_1()}
                                        Weak DPs: {gr^#(s(x), s(y)) -> gr^#(x, y)}
                                        StartTerms: basic terms
                                        Strategy: innermost
                                      
                                      Certificate: YES(?,O(n^1))
                                      
                                      Proof:
                                        The problem is match-bounded by 1.
                                        The enriched problem is compatible with the following automaton:
                                        {  0_0() -> 2
                                         , s_0(2) -> 2
                                         , gr^#_0(2, 2) -> 1
                                         , c_1_1() -> 1}
                              
                              * Path 7:{3}->8:{2}: YES(O(1),O(1))
                                ---------------------------------
                                
                                We consider the following Problem:
                                
                                  Strict Trs: {gr(s(x), 0()) -> true()}
                                  Weak DPs: {gr^#(s(x), s(y)) -> gr^#(x, y)}
                                  Weak Trs:
                                    {  gr(0(), x) -> false()
                                     , gr(s(x), s(y)) -> gr(x, y)
                                     , p(s(x)) -> x
                                     , or(true(), x) -> true()
                                     , or(x, true()) -> true()
                                     , p(0()) -> 0()
                                     , or(false(), false()) -> false()}
                                  StartTerms: basic terms
                                  Strategy: innermost
                                
                                Certificate: YES(O(1),O(1))
                                
                                Proof:
                                  We consider the following Problem:
                                  
                                    Strict Trs: {gr(s(x), 0()) -> true()}
                                    Weak DPs: {gr^#(s(x), s(y)) -> gr^#(x, y)}
                                    Weak Trs:
                                      {  gr(0(), x) -> false()
                                       , gr(s(x), s(y)) -> gr(x, y)
                                       , p(s(x)) -> x
                                       , or(true(), x) -> true()
                                       , or(x, true()) -> true()
                                       , p(0()) -> 0()
                                       , or(false(), false()) -> false()}
                                    StartTerms: basic terms
                                    Strategy: innermost
                                  
                                  Certificate: YES(O(1),O(1))
                                  
                                  Proof:
                                    We consider the following Problem:
                                    
                                      Strict Trs: {gr(s(x), 0()) -> true()}
                                      Weak DPs: {gr^#(s(x), s(y)) -> gr^#(x, y)}
                                      Weak Trs:
                                        {  gr(0(), x) -> false()
                                         , gr(s(x), s(y)) -> gr(x, y)
                                         , p(s(x)) -> x
                                         , or(true(), x) -> true()
                                         , or(x, true()) -> true()
                                         , p(0()) -> 0()
                                         , or(false(), false()) -> false()}
                                      StartTerms: basic terms
                                      Strategy: innermost
                                    
                                    Certificate: YES(O(1),O(1))
                                    
                                    Proof:
                                      No rule is usable.
                                      
                                      We consider the following Problem:
                                      
                                        Weak DPs: {gr^#(s(x), s(y)) -> gr^#(x, y)}
                                        StartTerms: basic terms
                                        Strategy: innermost
                                      
                                      Certificate: YES(O(1),O(1))
                                      
                                      Proof:
                                        Empty rules are trivially bounded
                              
                              * Path 6:{4,13,12,10,6}: YES(O(1),O(1))
                                -------------------------------------
                                
                                We consider the following Problem:
                                
                                  Strict Trs: {gr(s(x), 0()) -> true()}
                                  Weak Trs:
                                    {  gr(0(), x) -> false()
                                     , gr(s(x), s(y)) -> gr(x, y)
                                     , p(s(x)) -> x
                                     , or(true(), x) -> true()
                                     , or(x, true()) -> true()
                                     , p(0()) -> 0()
                                     , or(false(), false()) -> false()}
                                  StartTerms: basic terms
                                  Strategy: innermost
                                
                                Certificate: YES(O(1),O(1))
                                
                                Proof:
                                  We consider the following Problem:
                                  
                                    Strict Trs: {gr(s(x), 0()) -> true()}
                                    Weak Trs:
                                      {  gr(0(), x) -> false()
                                       , gr(s(x), s(y)) -> gr(x, y)
                                       , p(s(x)) -> x
                                       , or(true(), x) -> true()
                                       , or(x, true()) -> true()
                                       , p(0()) -> 0()
                                       , or(false(), false()) -> false()}
                                    StartTerms: basic terms
                                    Strategy: innermost
                                  
                                  Certificate: YES(O(1),O(1))
                                  
                                  Proof:
                                    We consider the following Problem:
                                    
                                      Strict Trs: {gr(s(x), 0()) -> true()}
                                      Weak Trs:
                                        {  gr(0(), x) -> false()
                                         , gr(s(x), s(y)) -> gr(x, y)
                                         , p(s(x)) -> x
                                         , or(true(), x) -> true()
                                         , or(x, true()) -> true()
                                         , p(0()) -> 0()
                                         , or(false(), false()) -> false()}
                                      StartTerms: basic terms
                                      Strategy: innermost
                                    
                                    Certificate: YES(O(1),O(1))
                                    
                                    Proof:
                                      No rule is usable.
                                      
                                      We consider the following Problem:
                                      
                                        StartTerms: basic terms
                                        Strategy: innermost
                                      
                                      Certificate: YES(O(1),O(1))
                                      
                                      Proof:
                                        Empty rules are trivially bounded
                              
                              * Path 5:{5}: YES(O(1),O(1))
                                --------------------------
                                
                                We consider the following Problem:
                                
                                  Strict Trs: {gr(s(x), 0()) -> true()}
                                  Weak Trs:
                                    {  gr(0(), x) -> false()
                                     , gr(s(x), s(y)) -> gr(x, y)
                                     , p(s(x)) -> x
                                     , or(true(), x) -> true()
                                     , or(x, true()) -> true()
                                     , p(0()) -> 0()
                                     , or(false(), false()) -> false()}
                                  StartTerms: basic terms
                                  Strategy: innermost
                                
                                Certificate: YES(O(1),O(1))
                                
                                Proof:
                                  We consider the following Problem:
                                  
                                    Strict Trs: {gr(s(x), 0()) -> true()}
                                    Weak Trs:
                                      {  gr(0(), x) -> false()
                                       , gr(s(x), s(y)) -> gr(x, y)
                                       , p(s(x)) -> x
                                       , or(true(), x) -> true()
                                       , or(x, true()) -> true()
                                       , p(0()) -> 0()
                                       , or(false(), false()) -> false()}
                                    StartTerms: basic terms
                                    Strategy: innermost
                                  
                                  Certificate: YES(O(1),O(1))
                                  
                                  Proof:
                                    We consider the following Problem:
                                    
                                      Strict Trs: {gr(s(x), 0()) -> true()}
                                      Weak Trs:
                                        {  gr(0(), x) -> false()
                                         , gr(s(x), s(y)) -> gr(x, y)
                                         , p(s(x)) -> x
                                         , or(true(), x) -> true()
                                         , or(x, true()) -> true()
                                         , p(0()) -> 0()
                                         , or(false(), false()) -> false()}
                                      StartTerms: basic terms
                                      Strategy: innermost
                                    
                                    Certificate: YES(O(1),O(1))
                                    
                                    Proof:
                                      No rule is usable.
                                      
                                      We consider the following Problem:
                                      
                                        StartTerms: basic terms
                                        Strategy: innermost
                                      
                                      Certificate: YES(O(1),O(1))
                                      
                                      Proof:
                                        Empty rules are trivially bounded
                              
                              * Path 4:{7}: YES(O(1),O(1))
                                --------------------------
                                
                                We consider the following Problem:
                                
                                  Strict Trs: {gr(s(x), 0()) -> true()}
                                  Weak Trs:
                                    {  gr(0(), x) -> false()
                                     , gr(s(x), s(y)) -> gr(x, y)
                                     , p(s(x)) -> x
                                     , or(true(), x) -> true()
                                     , or(x, true()) -> true()
                                     , p(0()) -> 0()
                                     , or(false(), false()) -> false()}
                                  StartTerms: basic terms
                                  Strategy: innermost
                                
                                Certificate: YES(O(1),O(1))
                                
                                Proof:
                                  We consider the following Problem:
                                  
                                    Strict Trs: {gr(s(x), 0()) -> true()}
                                    Weak Trs:
                                      {  gr(0(), x) -> false()
                                       , gr(s(x), s(y)) -> gr(x, y)
                                       , p(s(x)) -> x
                                       , or(true(), x) -> true()
                                       , or(x, true()) -> true()
                                       , p(0()) -> 0()
                                       , or(false(), false()) -> false()}
                                    StartTerms: basic terms
                                    Strategy: innermost
                                  
                                  Certificate: YES(O(1),O(1))
                                  
                                  Proof:
                                    We consider the following Problem:
                                    
                                      Strict Trs: {gr(s(x), 0()) -> true()}
                                      Weak Trs:
                                        {  gr(0(), x) -> false()
                                         , gr(s(x), s(y)) -> gr(x, y)
                                         , p(s(x)) -> x
                                         , or(true(), x) -> true()
                                         , or(x, true()) -> true()
                                         , p(0()) -> 0()
                                         , or(false(), false()) -> false()}
                                      StartTerms: basic terms
                                      Strategy: innermost
                                    
                                    Certificate: YES(O(1),O(1))
                                    
                                    Proof:
                                      No rule is usable.
                                      
                                      We consider the following Problem:
                                      
                                        StartTerms: basic terms
                                        Strategy: innermost
                                      
                                      Certificate: YES(O(1),O(1))
                                      
                                      Proof:
                                        Empty rules are trivially bounded
                              
                              * Path 3:{8}: YES(O(1),O(1))
                                --------------------------
                                
                                We consider the following Problem:
                                
                                  Strict Trs: {gr(s(x), 0()) -> true()}
                                  Weak Trs:
                                    {  gr(0(), x) -> false()
                                     , gr(s(x), s(y)) -> gr(x, y)
                                     , p(s(x)) -> x
                                     , or(true(), x) -> true()
                                     , or(x, true()) -> true()
                                     , p(0()) -> 0()
                                     , or(false(), false()) -> false()}
                                  StartTerms: basic terms
                                  Strategy: innermost
                                
                                Certificate: YES(O(1),O(1))
                                
                                Proof:
                                  We consider the following Problem:
                                  
                                    Strict Trs: {gr(s(x), 0()) -> true()}
                                    Weak Trs:
                                      {  gr(0(), x) -> false()
                                       , gr(s(x), s(y)) -> gr(x, y)
                                       , p(s(x)) -> x
                                       , or(true(), x) -> true()
                                       , or(x, true()) -> true()
                                       , p(0()) -> 0()
                                       , or(false(), false()) -> false()}
                                    StartTerms: basic terms
                                    Strategy: innermost
                                  
                                  Certificate: YES(O(1),O(1))
                                  
                                  Proof:
                                    We consider the following Problem:
                                    
                                      Strict Trs: {gr(s(x), 0()) -> true()}
                                      Weak Trs:
                                        {  gr(0(), x) -> false()
                                         , gr(s(x), s(y)) -> gr(x, y)
                                         , p(s(x)) -> x
                                         , or(true(), x) -> true()
                                         , or(x, true()) -> true()
                                         , p(0()) -> 0()
                                         , or(false(), false()) -> false()}
                                      StartTerms: basic terms
                                      Strategy: innermost
                                    
                                    Certificate: YES(O(1),O(1))
                                    
                                    Proof:
                                      No rule is usable.
                                      
                                      We consider the following Problem:
                                      
                                        StartTerms: basic terms
                                        Strategy: innermost
                                      
                                      Certificate: YES(O(1),O(1))
                                      
                                      Proof:
                                        Empty rules are trivially bounded
                              
                              * Path 2:{9}: YES(O(1),O(1))
                                --------------------------
                                
                                We consider the following Problem:
                                
                                  Strict Trs: {gr(s(x), 0()) -> true()}
                                  Weak Trs:
                                    {  gr(0(), x) -> false()
                                     , gr(s(x), s(y)) -> gr(x, y)
                                     , p(s(x)) -> x
                                     , or(true(), x) -> true()
                                     , or(x, true()) -> true()
                                     , p(0()) -> 0()
                                     , or(false(), false()) -> false()}
                                  StartTerms: basic terms
                                  Strategy: innermost
                                
                                Certificate: YES(O(1),O(1))
                                
                                Proof:
                                  We consider the following Problem:
                                  
                                    Strict Trs: {gr(s(x), 0()) -> true()}
                                    Weak Trs:
                                      {  gr(0(), x) -> false()
                                       , gr(s(x), s(y)) -> gr(x, y)
                                       , p(s(x)) -> x
                                       , or(true(), x) -> true()
                                       , or(x, true()) -> true()
                                       , p(0()) -> 0()
                                       , or(false(), false()) -> false()}
                                    StartTerms: basic terms
                                    Strategy: innermost
                                  
                                  Certificate: YES(O(1),O(1))
                                  
                                  Proof:
                                    We consider the following Problem:
                                    
                                      Strict Trs: {gr(s(x), 0()) -> true()}
                                      Weak Trs:
                                        {  gr(0(), x) -> false()
                                         , gr(s(x), s(y)) -> gr(x, y)
                                         , p(s(x)) -> x
                                         , or(true(), x) -> true()
                                         , or(x, true()) -> true()
                                         , p(0()) -> 0()
                                         , or(false(), false()) -> false()}
                                      StartTerms: basic terms
                                      Strategy: innermost
                                    
                                    Certificate: YES(O(1),O(1))
                                    
                                    Proof:
                                      No rule is usable.
                                      
                                      We consider the following Problem:
                                      
                                        StartTerms: basic terms
                                        Strategy: innermost
                                      
                                      Certificate: YES(O(1),O(1))
                                      
                                      Proof:
                                        Empty rules are trivially bounded
                              
                              * Path 1:{11}: YES(O(1),O(1))
                                ---------------------------
                                
                                We consider the following Problem:
                                
                                  Strict Trs: {gr(s(x), 0()) -> true()}
                                  Weak Trs:
                                    {  gr(0(), x) -> false()
                                     , gr(s(x), s(y)) -> gr(x, y)
                                     , p(s(x)) -> x
                                     , or(true(), x) -> true()
                                     , or(x, true()) -> true()
                                     , p(0()) -> 0()
                                     , or(false(), false()) -> false()}
                                  StartTerms: basic terms
                                  Strategy: innermost
                                
                                Certificate: YES(O(1),O(1))
                                
                                Proof:
                                  We consider the following Problem:
                                  
                                    Strict Trs: {gr(s(x), 0()) -> true()}
                                    Weak Trs:
                                      {  gr(0(), x) -> false()
                                       , gr(s(x), s(y)) -> gr(x, y)
                                       , p(s(x)) -> x
                                       , or(true(), x) -> true()
                                       , or(x, true()) -> true()
                                       , p(0()) -> 0()
                                       , or(false(), false()) -> false()}
                                    StartTerms: basic terms
                                    Strategy: innermost
                                  
                                  Certificate: YES(O(1),O(1))
                                  
                                  Proof:
                                    We consider the following Problem:
                                    
                                      Strict Trs: {gr(s(x), 0()) -> true()}
                                      Weak Trs:
                                        {  gr(0(), x) -> false()
                                         , gr(s(x), s(y)) -> gr(x, y)
                                         , p(s(x)) -> x
                                         , or(true(), x) -> true()
                                         , or(x, true()) -> true()
                                         , p(0()) -> 0()
                                         , or(false(), false()) -> false()}
                                      StartTerms: basic terms
                                      Strategy: innermost
                                    
                                    Certificate: YES(O(1),O(1))
                                    
                                    Proof:
                                      No rule is usable.
                                      
                                      We consider the following Problem:
                                      
                                        StartTerms: basic terms
                                        Strategy: innermost
                                      
                                      Certificate: YES(O(1),O(1))
                                      
                                      Proof:
                                        Empty rules are trivially bounded

Hurray, we answered YES(?,O(n^1))