We consider the following Problem:

  Strict Trs:
    {  le(0(), y) -> true()
     , le(s(x), 0()) -> false()
     , le(s(x), s(y)) -> le(x, y)
     , minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , gcd(0(), y) -> y
     , gcd(s(x), 0()) -> s(x)
     , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
     , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
     , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))}
  StartTerms: basic terms
  Strategy: innermost

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

Proof:
  We consider the following Problem:
  
    Strict Trs:
      {  le(0(), y) -> true()
       , le(s(x), 0()) -> false()
       , le(s(x), s(y)) -> le(x, y)
       , minus(x, 0()) -> x
       , minus(s(x), s(y)) -> minus(x, y)
       , gcd(0(), y) -> y
       , gcd(s(x), 0()) -> s(x)
       , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
       , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
       , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))}
    StartTerms: basic terms
    Strategy: innermost
  
  Certificate: YES(?,O(n^2))
  
  Proof:
    The weightgap principle applies, where following rules are oriented strictly:
    
    TRS Component:
      {  le(0(), y) -> true()
       , le(s(x), 0()) -> false()}
    
    Interpretation of nonconstant growth:
    -------------------------------------
      The following argument positions are usable:
        Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
        Uargs(if_gcd) = {1}
      We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
      Interpretation Functions:
       le(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                    [0 0]      [0 0]      [1]
       0() = [0]
             [0]
       true() = [0]
                [0]
       s(x1) = [0 0] x1 + [0]
               [1 1]      [0]
       false() = [0]
                 [0]
       minus(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
                       [0 0]      [0 0]      [1]
       gcd(x1, x2) = [1 1] x1 + [1 1] x2 + [1]
                     [0 0]      [0 0]      [1]
       if_gcd(x1, x2, x3) = [1 0] x1 + [0 1] x2 + [0 1] x3 + [0]
                            [0 0]      [0 0]      [0 0]      [1]
    
    The strictly oriented rules are moved into the weak component.
    
    We consider the following Problem:
    
      Strict Trs:
        {  le(s(x), s(y)) -> le(x, y)
         , minus(x, 0()) -> x
         , minus(s(x), s(y)) -> minus(x, y)
         , gcd(0(), y) -> y
         , gcd(s(x), 0()) -> s(x)
         , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
         , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
         , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))}
      Weak Trs:
        {  le(0(), y) -> true()
         , le(s(x), 0()) -> false()}
      StartTerms: basic terms
      Strategy: innermost
    
    Certificate: YES(?,O(n^2))
    
    Proof:
      The weightgap principle applies, where following rules are oriented strictly:
      
      TRS Component:
        {if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))}
      
      Interpretation of nonconstant growth:
      -------------------------------------
        The following argument positions are usable:
          Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
          Uargs(if_gcd) = {1}
        We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
        Interpretation Functions:
         le(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                      [0 0]      [0 0]      [1]
         0() = [0]
               [0]
         true() = [1]
                  [0]
         s(x1) = [0 0] x1 + [0]
                 [1 1]      [0]
         false() = [0]
                   [0]
         minus(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
                         [0 0]      [0 0]      [1]
         gcd(x1, x2) = [1 1] x1 + [1 1] x2 + [1]
                       [0 0]      [0 0]      [1]
         if_gcd(x1, x2, x3) = [1 0] x1 + [0 1] x2 + [0 1] x3 + [3]
                              [0 0]      [0 0]      [0 0]      [1]
      
      The strictly oriented rules are moved into the weak component.
      
      We consider the following Problem:
      
        Strict Trs:
          {  le(s(x), s(y)) -> le(x, y)
           , minus(x, 0()) -> x
           , minus(s(x), s(y)) -> minus(x, y)
           , gcd(0(), y) -> y
           , gcd(s(x), 0()) -> s(x)
           , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
           , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))}
        Weak Trs:
          {  if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
           , le(0(), y) -> true()
           , le(s(x), 0()) -> false()}
        StartTerms: basic terms
        Strategy: innermost
      
      Certificate: YES(?,O(n^2))
      
      Proof:
        The weightgap principle applies, where following rules are oriented strictly:
        
        TRS Component:
          {if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))}
        
        Interpretation of nonconstant growth:
        -------------------------------------
          The following argument positions are usable:
            Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
            Uargs(if_gcd) = {1}
          We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
          Interpretation Functions:
           le(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                        [0 0]      [0 0]      [1]
           0() = [0]
                 [0]
           true() = [0]
                    [0]
           s(x1) = [0 0] x1 + [0]
                   [1 1]      [0]
           false() = [1]
                     [0]
           minus(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
                           [0 0]      [0 0]      [1]
           gcd(x1, x2) = [1 1] x1 + [1 1] x2 + [1]
                         [0 0]      [0 0]      [1]
           if_gcd(x1, x2, x3) = [1 0] x1 + [0 1] x2 + [0 1] x3 + [3]
                                [0 0]      [0 0]      [0 0]      [1]
        
        The strictly oriented rules are moved into the weak component.
        
        We consider the following Problem:
        
          Strict Trs:
            {  le(s(x), s(y)) -> le(x, y)
             , minus(x, 0()) -> x
             , minus(s(x), s(y)) -> minus(x, y)
             , gcd(0(), y) -> y
             , gcd(s(x), 0()) -> s(x)
             , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))}
          Weak Trs:
            {  if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))
             , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
             , le(0(), y) -> true()
             , le(s(x), 0()) -> false()}
          StartTerms: basic terms
          Strategy: innermost
        
        Certificate: YES(?,O(n^2))
        
        Proof:
          The weightgap principle applies, where following rules are oriented strictly:
          
          TRS Component: {minus(s(x), s(y)) -> minus(x, y)}
          
          Interpretation of nonconstant growth:
          -------------------------------------
            The following argument positions are usable:
              Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
              Uargs(if_gcd) = {1}
            We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
            Interpretation Functions:
             le(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                          [0 0]      [0 0]      [1]
             0() = [0]
                   [0]
             true() = [0]
                      [0]
             s(x1) = [0 0] x1 + [0]
                     [1 1]      [2]
             false() = [0]
                       [0]
             minus(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
                             [0 0]      [0 0]      [1]
             gcd(x1, x2) = [1 1] x1 + [1 1] x2 + [0]
                           [0 0]      [0 0]      [1]
             if_gcd(x1, x2, x3) = [1 0] x1 + [0 1] x2 + [0 1] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [1]
          
          The strictly oriented rules are moved into the weak component.
          
          We consider the following Problem:
          
            Strict Trs:
              {  le(s(x), s(y)) -> le(x, y)
               , minus(x, 0()) -> x
               , gcd(0(), y) -> y
               , gcd(s(x), 0()) -> s(x)
               , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))}
            Weak Trs:
              {  minus(s(x), s(y)) -> minus(x, y)
               , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))
               , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
               , le(0(), y) -> true()
               , le(s(x), 0()) -> false()}
            StartTerms: basic terms
            Strategy: innermost
          
          Certificate: YES(?,O(n^2))
          
          Proof:
            The weightgap principle applies, where following rules are oriented strictly:
            
            TRS Component: {gcd(s(x), 0()) -> s(x)}
            
            Interpretation of nonconstant growth:
            -------------------------------------
              The following argument positions are usable:
                Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
                Uargs(if_gcd) = {1}
              We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
              Interpretation Functions:
               le(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                            [0 0]      [0 0]      [1]
               0() = [0]
                     [0]
               true() = [0]
                        [0]
               s(x1) = [1 0] x1 + [0]
                       [0 0]      [0]
               false() = [0]
                         [0]
               minus(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                               [0 0]      [0 0]      [1]
               gcd(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                             [0 0]      [0 0]      [1]
               if_gcd(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [2]
                                    [0 0]      [0 0]      [0 0]      [1]
            
            The strictly oriented rules are moved into the weak component.
            
            We consider the following Problem:
            
              Strict Trs:
                {  le(s(x), s(y)) -> le(x, y)
                 , minus(x, 0()) -> x
                 , gcd(0(), y) -> y
                 , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))}
              Weak Trs:
                {  gcd(s(x), 0()) -> s(x)
                 , minus(s(x), s(y)) -> minus(x, y)
                 , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))
                 , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
                 , le(0(), y) -> true()
                 , le(s(x), 0()) -> false()}
              StartTerms: basic terms
              Strategy: innermost
            
            Certificate: YES(?,O(n^2))
            
            Proof:
              The weightgap principle applies, where following rules are oriented strictly:
              
              TRS Component: {gcd(0(), y) -> y}
              
              Interpretation of nonconstant growth:
              -------------------------------------
                The following argument positions are usable:
                  Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
                  Uargs(if_gcd) = {1}
                We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
                Interpretation Functions:
                 le(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                              [0 0]      [0 0]      [1]
                 0() = [0]
                       [0]
                 true() = [1]
                          [0]
                 s(x1) = [1 0] x1 + [0]
                         [0 0]      [0]
                 false() = [1]
                           [0]
                 minus(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                                 [0 0]      [0 0]      [1]
                 gcd(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                               [0 0]      [0 1]      [1]
                 if_gcd(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1]
                                      [0 0]      [0 0]      [0 0]      [1]
              
              The strictly oriented rules are moved into the weak component.
              
              We consider the following Problem:
              
                Strict Trs:
                  {  le(s(x), s(y)) -> le(x, y)
                   , minus(x, 0()) -> x
                   , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))}
                Weak Trs:
                  {  gcd(0(), y) -> y
                   , gcd(s(x), 0()) -> s(x)
                   , minus(s(x), s(y)) -> minus(x, y)
                   , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))
                   , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
                   , le(0(), y) -> true()
                   , le(s(x), 0()) -> false()}
                StartTerms: basic terms
                Strategy: innermost
              
              Certificate: YES(?,O(n^2))
              
              Proof:
                The weightgap principle applies, where following rules are oriented strictly:
                
                TRS Component: {gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))}
                
                Interpretation of nonconstant growth:
                -------------------------------------
                  The following argument positions are usable:
                    Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
                    Uargs(if_gcd) = {1}
                  We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
                  Interpretation Functions:
                   le(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                                [0 0]      [0 0]      [1]
                   0() = [0]
                         [0]
                   true() = [1]
                            [0]
                   s(x1) = [1 0] x1 + [2]
                           [1 0]      [1]
                   false() = [1]
                             [0]
                   minus(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                                   [0 0]      [0 0]      [1]
                   gcd(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                                 [1 0]      [0 1]      [3]
                   if_gcd(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [0 1] x3 + [1]
                                        [1 0]      [1 0]      [1 0]      [0]
                
                The strictly oriented rules are moved into the weak component.
                
                We consider the following Problem:
                
                  Strict Trs:
                    {  le(s(x), s(y)) -> le(x, y)
                     , minus(x, 0()) -> x}
                  Weak Trs:
                    {  gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
                     , gcd(0(), y) -> y
                     , gcd(s(x), 0()) -> s(x)
                     , minus(s(x), s(y)) -> minus(x, y)
                     , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))
                     , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
                     , le(0(), y) -> true()
                     , le(s(x), 0()) -> false()}
                  StartTerms: basic terms
                  Strategy: innermost
                
                Certificate: YES(?,O(n^2))
                
                Proof:
                  The weightgap principle applies, where following rules are oriented strictly:
                  
                  TRS Component: {minus(x, 0()) -> x}
                  
                  Interpretation of nonconstant growth:
                  -------------------------------------
                    The following argument positions are usable:
                      Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
                      Uargs(if_gcd) = {1}
                    We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
                    Interpretation Functions:
                     le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [1]
                     0() = [2]
                           [0]
                     true() = [0]
                              [0]
                     s(x1) = [1 0] x1 + [2]
                             [0 1]      [2]
                     false() = [0]
                               [0]
                     minus(x1, x2) = [1 0] x1 + [0 0] x2 + [1]
                                     [0 1]      [0 0]      [1]
                     gcd(x1, x2) = [1 0] x1 + [1 0] x2 + [1]
                                   [0 1]      [0 1]      [0]
                     if_gcd(x1, x2, x3) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1]
                                          [0 0]      [0 1]      [0 1]      [0]
                  
                  The strictly oriented rules are moved into the weak component.
                  
                  We consider the following Problem:
                  
                    Strict Trs: {le(s(x), s(y)) -> le(x, y)}
                    Weak Trs:
                      {  minus(x, 0()) -> x
                       , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
                       , gcd(0(), y) -> y
                       , gcd(s(x), 0()) -> s(x)
                       , minus(s(x), s(y)) -> minus(x, y)
                       , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))
                       , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
                       , le(0(), y) -> true()
                       , le(s(x), 0()) -> false()}
                    StartTerms: basic terms
                    Strategy: innermost
                  
                  Certificate: YES(?,O(n^2))
                  
                  Proof:
                    We consider the following Problem:
                    
                      Strict Trs: {le(s(x), s(y)) -> le(x, y)}
                      Weak Trs:
                        {  minus(x, 0()) -> x
                         , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
                         , gcd(0(), y) -> y
                         , gcd(s(x), 0()) -> s(x)
                         , minus(s(x), s(y)) -> minus(x, y)
                         , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x))
                         , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y))
                         , le(0(), y) -> true()
                         , le(s(x), 0()) -> false()}
                      StartTerms: basic terms
                      Strategy: innermost
                    
                    Certificate: YES(?,O(n^2))
                    
                    Proof:
                      The following argument positions are usable:
                        Uargs(le) = {}, Uargs(s) = {}, Uargs(minus) = {}, Uargs(gcd) = {1},
                        Uargs(if_gcd) = {1}
                      We have the following constructor-based EDA-non-satisfying and IDA(2)-non-satisfying matrix interpretation:
                      Interpretation Functions:
                       le(x1, x2) = [0 0 0] x1 + [0 0 1] x2 + [0]
                                    [0 2 0]      [0 0 0]      [1]
                                    [1 0 0]      [0 0 0]      [1]
                       0() = [0]
                             [2]
                             [0]
                       true() = [0]
                                [0]
                                [0]
                       s(x1) = [1 0 0] x1 + [0]
                               [0 1 2]      [0]
                               [0 0 1]      [1]
                       false() = [0]
                                 [0]
                                 [0]
                       minus(x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [0]
                                       [0 1 0]      [0 0 0]      [0]
                                       [0 0 1]      [0 0 0]      [0]
                       gcd(x1, x2) = [1 2 1] x1 + [1 2 0] x2 + [0]
                                     [0 2 1]      [0 2 0]      [0]
                                     [0 0 2]      [0 0 2]      [1]
                       if_gcd(x1, x2, x3) = [1 0 0] x1 + [1 2 0] x2 + [1 2 0] x3 + [0]
                                            [0 0 0]      [0 2 0]      [0 2 0]      [1]
                                            [0 0 0]      [0 0 2]      [0 0 2]      [1]

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