YES(O(1),O(n^2))

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict Trs:
  { min(X, 0()) -> X
  , min(s(X), s(Y)) -> min(X, Y)
  , quot(0(), s(Y)) -> 0()
  , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
  , log(s(0())) -> 0()
  , log(s(s(X))) -> s(log(s(quot(X, s(s(0())))))) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^2))

We add the following weak dependency pairs:

Strict DPs:
  { min^#(X, 0()) -> c_1(X)
  , min^#(s(X), s(Y)) -> c_2(min^#(X, Y))
  , quot^#(0(), s(Y)) -> c_3()
  , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y)))
  , log^#(s(0())) -> c_5()
  , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs:
  { min^#(X, 0()) -> c_1(X)
  , min^#(s(X), s(Y)) -> c_2(min^#(X, Y))
  , quot^#(0(), s(Y)) -> c_3()
  , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y)))
  , log^#(s(0())) -> c_5()
  , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) }
Strict Trs:
  { min(X, 0()) -> X
  , min(s(X), s(Y)) -> min(X, Y)
  , quot(0(), s(Y)) -> 0()
  , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
  , log(s(0())) -> 0()
  , log(s(s(X))) -> s(log(s(quot(X, s(s(0())))))) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^2))

We replace rewrite rules by usable rules:

  Strict Usable Rules:
    { min(X, 0()) -> X
    , min(s(X), s(Y)) -> min(X, Y)
    , quot(0(), s(Y)) -> 0()
    , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs:
  { min^#(X, 0()) -> c_1(X)
  , min^#(s(X), s(Y)) -> c_2(min^#(X, Y))
  , quot^#(0(), s(Y)) -> c_3()
  , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y)))
  , log^#(s(0())) -> c_5()
  , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) }
Strict Trs:
  { min(X, 0()) -> X
  , min(s(X), s(Y)) -> min(X, Y)
  , quot(0(), s(Y)) -> 0()
  , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^2))

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(min) = {1}, Uargs(s) = {1}, Uargs(quot) = {1},
  Uargs(c_2) = {1}, Uargs(quot^#) = {1}, Uargs(c_4) = {1},
  Uargs(log^#) = {1}, Uargs(c_6) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

     [min](x1, x2) = [1 0] x1 + [1]           
                     [0 1]      [0]           
                                              
               [0] = [0]                      
                     [0]                      
                                              
           [s](x1) = [1 2] x1 + [1]           
                     [0 1]      [1]           
                                              
    [quot](x1, x2) = [1 2] x1 + [1]           
                     [0 1]      [0]           
                                              
   [min^#](x1, x2) = [0 0] x1 + [0 0] x2 + [2]
                     [1 2]      [2 2]      [0]
                                              
         [c_1](x1) = [0 0] x1 + [1]           
                     [1 1]      [1]           
                                              
         [c_2](x1) = [1 0] x1 + [1]           
                     [0 1]      [2]           
                                              
  [quot^#](x1, x2) = [1 0] x1 + [0]           
                     [0 0]      [0]           
                                              
             [c_3] = [1]                      
                     [1]                      
                                              
         [c_4](x1) = [1 0] x1 + [1]           
                     [0 1]      [2]           
                                              
       [log^#](x1) = [1 0] x1 + [0]           
                     [0 0]      [0]           
                                              
             [c_5] = [1]                      
                     [1]                      
                                              
         [c_6](x1) = [1 0] x1 + [2]           
                     [0 1]      [2]           

The following symbols are considered usable

  {min, quot, min^#, quot^#, log^#}

The order satisfies the following ordering constraints:

         [min(X, 0())] = [1 0] X + [1]                      
                         [0 1]     [0]                      
                       > [1 0] X + [0]                      
                         [0 1]     [0]                      
                       = [X]                                
                                                            
     [min(s(X), s(Y))] = [1 2] X + [2]                      
                         [0 1]     [1]                      
                       > [1 0] X + [1]                      
                         [0 1]     [0]                      
                       = [min(X, Y)]                        
                                                            
     [quot(0(), s(Y))] = [1]                                
                         [0]                                
                       > [0]                                
                         [0]                                
                       = [0()]                              
                                                            
    [quot(s(X), s(Y))] = [1 4] X + [4]                      
                         [0 1]     [1]                      
                       > [1 4] X + [3]                      
                         [0 1]     [1]                      
                       = [s(quot(min(X, Y), s(Y)))]         
                                                            
       [min^#(X, 0())] = [0 0] X + [2]                      
                         [1 2]     [0]                      
                       ? [0 0] X + [1]                      
                         [1 1]     [1]                      
                       = [c_1(X)]                           
                                                            
   [min^#(s(X), s(Y))] = [0 0] X + [0 0] Y + [2]            
                         [1 4]     [2 6]     [7]            
                       ? [0 0] X + [0 0] Y + [3]            
                         [1 2]     [2 2]     [2]            
                       = [c_2(min^#(X, Y))]                 
                                                            
   [quot^#(0(), s(Y))] = [0]                                
                         [0]                                
                       ? [1]                                
                         [1]                                
                       = [c_3()]                            
                                                            
  [quot^#(s(X), s(Y))] = [1 2] X + [1]                      
                         [0 0]     [0]                      
                       ? [1 0] X + [2]                      
                         [0 0]     [2]                      
                       = [c_4(quot^#(min(X, Y), s(Y)))]     
                                                            
       [log^#(s(0()))] = [1]                                
                         [0]                                
                       ? [1]                                
                         [1]                                
                       = [c_5()]                            
                                                            
      [log^#(s(s(X)))] = [1 4] X + [4]                      
                         [0 0]     [0]                      
                       ? [1 4] X + [4]                      
                         [0 0]     [2]                      
                       = [c_6(log^#(s(quot(X, s(s(0()))))))]
                                                            

Further, it can be verified that all rules not oriented are covered by the weightgap condition.

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict DPs:
  { min^#(X, 0()) -> c_1(X)
  , min^#(s(X), s(Y)) -> c_2(min^#(X, Y))
  , quot^#(0(), s(Y)) -> c_3()
  , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y)))
  , log^#(s(0())) -> c_5()
  , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) }
Weak Trs:
  { min(X, 0()) -> X
  , min(s(X), s(Y)) -> min(X, Y)
  , quot(0(), s(Y)) -> 0()
  , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) }
Obligation:
  runtime complexity
Answer:
  YES(?,O(n^1))

We estimate the number of application of {3,5} by applications of
Pre({3,5}) = {1,4,6}. Here rules are labeled as follows:

  DPs:
    { 1: min^#(X, 0()) -> c_1(X)
    , 2: min^#(s(X), s(Y)) -> c_2(min^#(X, Y))
    , 3: quot^#(0(), s(Y)) -> c_3()
    , 4: quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y)))
    , 5: log^#(s(0())) -> c_5()
    , 6: log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict DPs:
  { min^#(X, 0()) -> c_1(X)
  , min^#(s(X), s(Y)) -> c_2(min^#(X, Y))
  , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y)))
  , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) }
Weak DPs:
  { quot^#(0(), s(Y)) -> c_3()
  , log^#(s(0())) -> c_5() }
Weak Trs:
  { min(X, 0()) -> X
  , min(s(X), s(Y)) -> min(X, Y)
  , quot(0(), s(Y)) -> 0()
  , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) }
Obligation:
  runtime complexity
Answer:
  YES(?,O(n^1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ quot^#(0(), s(Y)) -> c_3()
, log^#(s(0())) -> c_5() }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict DPs:
  { min^#(X, 0()) -> c_1(X)
  , min^#(s(X), s(Y)) -> c_2(min^#(X, Y))
  , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y)))
  , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) }
Weak Trs:
  { min(X, 0()) -> X
  , min(s(X), s(Y)) -> min(X, Y)
  , quot(0(), s(Y)) -> 0()
  , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) }
Obligation:
  runtime complexity
Answer:
  YES(?,O(n^1))

We employ 'linear path analysis' using the following approximated
dependency graph:
->{1,2}                                      [         ?          ]
   |
   |->{3}                                    [  YES(O(1),O(n^1))  ]
   |
   `->{4}                                    [  YES(O(1),O(n^1))  ]


Here dependency-pairs are as follows:

Strict DPs:
  { 1: min^#(X, 0()) -> c_1(X)
  , 2: min^#(s(X), s(Y)) -> c_2(min^#(X, Y))
  , 3: quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y)))
  , 4: log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) }

* Path {1,2}->{3}: YES(O(1),O(n^1))
  ---------------------------------
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(n^1)).
  
  Strict DPs:
    { min^#(X, 0()) -> c_1(X)
    , min^#(s(X), s(Y)) -> c_2(min^#(X, Y))
    , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) }
  Weak Trs:
    { min(X, 0()) -> X
    , min(s(X), s(Y)) -> min(X, Y)
    , quot(0(), s(Y)) -> 0()
    , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) }
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(n^1))
  
  We replace rewrite rules by usable rules:
  
    Weak Usable Rules:
      { min(X, 0()) -> X
      , min(s(X), s(Y)) -> min(X, Y) }
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(n^1)).
  
  Strict DPs:
    { min^#(X, 0()) -> c_1(X)
    , min^#(s(X), s(Y)) -> c_2(min^#(X, Y))
    , quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) }
  Weak Trs:
    { min(X, 0()) -> X
    , min(s(X), s(Y)) -> min(X, Y) }
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(n^1))
  
  We use the processor 'matrix interpretation of dimension 1' to
  orient following rules strictly.
  
  DPs:
    { 3: quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) }
  Trs: { min(s(X), s(Y)) -> min(X, Y) }
  
  Sub-proof:
  ----------
    The following argument positions are usable:
      Uargs(c_2) = {1}, Uargs(c_4) = {1}
    
    TcT has computed the following constructor-based matrix
    interpretation satisfying not(EDA).
    
         [min](x1, x2) = [1] x1 + [0]         
                                              
                   [0] = [0]                  
                                              
               [s](x1) = [1] x1 + [1]         
                                              
        [quot](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
             [log](x1) = [7] x1 + [0]         
                                              
       [min^#](x1, x2) = [0]                  
                                              
             [c_1](x1) = [0]                  
                                              
             [c_2](x1) = [4] x1 + [0]         
                                              
      [quot^#](x1, x2) = [3] x1 + [1] x2 + [4]
                                              
                 [c_3] = [0]                  
                                              
             [c_4](x1) = [1] x1 + [1]         
                                              
           [log^#](x1) = [7] x1 + [0]         
                                              
                 [c_5] = [0]                  
                                              
             [c_6](x1) = [7] x1 + [0]         
    
    The following symbols are considered usable
    
      {min, min^#, quot^#}
    
    The order satisfies the following ordering constraints:
    
             [min(X, 0())] =  [1] X + [0]                   
                           >= [1] X + [0]                   
                           =  [X]                           
                                                            
         [min(s(X), s(Y))] =  [1] X + [1]                   
                           >  [1] X + [0]                   
                           =  [min(X, Y)]                   
                                                            
           [min^#(X, 0())] =  [0]                           
                           >= [0]                           
                           =  [c_1(X)]                      
                                                            
       [min^#(s(X), s(Y))] =  [0]                           
                           >= [0]                           
                           =  [c_2(min^#(X, Y))]            
                                                            
      [quot^#(s(X), s(Y))] =  [3] X + [1] Y + [8]           
                           >  [3] X + [1] Y + [6]           
                           =  [c_4(quot^#(min(X, Y), s(Y)))]
                                                            
  
  The strictly oriented rules are moved into the weak component.
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(n^1)).
  
  Strict DPs:
    { min^#(X, 0()) -> c_1(X)
    , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  Weak DPs: { quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) }
  Weak Trs:
    { min(X, 0()) -> X
    , min(s(X), s(Y)) -> min(X, Y) }
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(n^1))
  
  The following weak DPs constitute a sub-graph of the DG that is
  closed under successors. The DPs are removed.
  
  { quot^#(s(X), s(Y)) -> c_4(quot^#(min(X, Y), s(Y))) }
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(n^1)).
  
  Strict DPs:
    { min^#(X, 0()) -> c_1(X)
    , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  Weak Trs:
    { min(X, 0()) -> X
    , min(s(X), s(Y)) -> min(X, Y) }
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(n^1))
  
  No rule is usable, rules are removed from the input problem.
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(n^1)).
  
  Strict DPs:
    { min^#(X, 0()) -> c_1(X)
    , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(n^1))
  
  We use the processor 'matrix interpretation of dimension 1' to
  orient following rules strictly.
  
  DPs:
    { 1: min^#(X, 0()) -> c_1(X) }
  
  Sub-proof:
  ----------
    The following argument positions are usable:
      Uargs(c_2) = {1}
    
    TcT has computed the following constructor-based matrix
    interpretation satisfying not(EDA).
    
         [min](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
                   [0] = [1]                  
                                              
               [s](x1) = [1] x1 + [0]         
                                              
        [quot](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
             [log](x1) = [7] x1 + [0]         
                                              
       [min^#](x1, x2) = [2] x2 + [0]         
                                              
             [c_1](x1) = [0]                  
                                              
             [c_2](x1) = [1] x1 + [0]         
                                              
      [quot^#](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
                 [c_3] = [0]                  
                                              
             [c_4](x1) = [7] x1 + [0]         
                                              
           [log^#](x1) = [7] x1 + [0]         
                                              
                 [c_5] = [0]                  
                                              
             [c_6](x1) = [7] x1 + [0]         
    
    The following symbols are considered usable
    
      {min^#}
    
    The order satisfies the following ordering constraints:
    
          [min^#(X, 0())] =  [2]               
                          >  [0]               
                          =  [c_1(X)]          
                                               
      [min^#(s(X), s(Y))] =  [2] Y + [0]       
                          >= [2] Y + [0]       
                          =  [c_2(min^#(X, Y))]
                                               
  
  The strictly oriented rules are moved into the weak component.
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(n^1)).
  
  Strict DPs: { min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  Weak DPs: { min^#(X, 0()) -> c_1(X) }
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(n^1))
  
  We use the processor 'matrix interpretation of dimension 1' to
  orient following rules strictly.
  
  DPs:
    { 1: min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  
  Sub-proof:
  ----------
    The following argument positions are usable:
      Uargs(c_2) = {1}
    
    TcT has computed the following constructor-based matrix
    interpretation satisfying not(EDA).
    
         [min](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
                   [0] = [7]                  
                                              
               [s](x1) = [1] x1 + [4]         
                                              
        [quot](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
             [log](x1) = [7] x1 + [0]         
                                              
       [min^#](x1, x2) = [2] x1 + [0]         
                                              
             [c_1](x1) = [1] x1 + [0]         
                                              
             [c_2](x1) = [1] x1 + [5]         
                                              
      [quot^#](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
                 [c_3] = [0]                  
                                              
             [c_4](x1) = [7] x1 + [0]         
                                              
           [log^#](x1) = [7] x1 + [0]         
                                              
                 [c_5] = [0]                  
                                              
             [c_6](x1) = [7] x1 + [0]         
    
    The following symbols are considered usable
    
      {min^#}
    
    The order satisfies the following ordering constraints:
    
          [min^#(X, 0())] =  [2] X + [0]       
                          >= [1] X + [0]       
                          =  [c_1(X)]          
                                               
      [min^#(s(X), s(Y))] =  [2] X + [8]       
                          >  [2] X + [5]       
                          =  [c_2(min^#(X, Y))]
                                               
  
  The strictly oriented rules are moved into the weak component.
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(1)).
  
  Weak DPs:
    { min^#(X, 0()) -> c_1(X)
    , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(1))
  
  The following weak DPs constitute a sub-graph of the DG that is
  closed under successors. The DPs are removed.
  
  { min^#(X, 0()) -> c_1(X)
  , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(1)).
  
  Rules: Empty
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(1))
  
  Empty rules are trivially bounded

* Path {1,2}->{4}: YES(O(1),O(n^1))
  ---------------------------------
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(n^1)).
  
  Strict DPs:
    { min^#(X, 0()) -> c_1(X)
    , min^#(s(X), s(Y)) -> c_2(min^#(X, Y))
    , log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) }
  Weak Trs:
    { min(X, 0()) -> X
    , min(s(X), s(Y)) -> min(X, Y)
    , quot(0(), s(Y)) -> 0()
    , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) }
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(n^1))
  
  We use the processor 'matrix interpretation of dimension 1' to
  orient following rules strictly.
  
  DPs:
    { 3: log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) }
  Trs: { min(s(X), s(Y)) -> min(X, Y) }
  
  Sub-proof:
  ----------
    The following argument positions are usable:
      Uargs(c_2) = {1}, Uargs(c_6) = {1}
    
    TcT has computed the following constructor-based matrix
    interpretation satisfying not(EDA).
    
         [min](x1, x2) = [1] x1 + [0]         
                                              
                   [0] = [0]                  
                                              
               [s](x1) = [1] x1 + [1]         
                                              
        [quot](x1, x2) = [1] x1 + [0]         
                                              
             [log](x1) = [7] x1 + [0]         
                                              
       [min^#](x1, x2) = [0]                  
                                              
             [c_1](x1) = [0]                  
                                              
             [c_2](x1) = [1] x1 + [0]         
                                              
      [quot^#](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
                 [c_3] = [0]                  
                                              
             [c_4](x1) = [7] x1 + [0]         
                                              
           [log^#](x1) = [4] x1 + [0]         
                                              
                 [c_5] = [0]                  
                                              
             [c_6](x1) = [1] x1 + [1]         
    
    The following symbols are considered usable
    
      {min, quot, min^#, log^#}
    
    The order satisfies the following ordering constraints:
    
            [min(X, 0())] =  [1] X + [0]                        
                          >= [1] X + [0]                        
                          =  [X]                                
                                                                
        [min(s(X), s(Y))] =  [1] X + [1]                        
                          >  [1] X + [0]                        
                          =  [min(X, Y)]                        
                                                                
        [quot(0(), s(Y))] =  [0]                                
                          >= [0]                                
                          =  [0()]                              
                                                                
       [quot(s(X), s(Y))] =  [1] X + [1]                        
                          >= [1] X + [1]                        
                          =  [s(quot(min(X, Y), s(Y)))]         
                                                                
          [min^#(X, 0())] =  [0]                                
                          >= [0]                                
                          =  [c_1(X)]                           
                                                                
      [min^#(s(X), s(Y))] =  [0]                                
                          >= [0]                                
                          =  [c_2(min^#(X, Y))]                 
                                                                
         [log^#(s(s(X)))] =  [4] X + [8]                        
                          >  [4] X + [5]                        
                          =  [c_6(log^#(s(quot(X, s(s(0()))))))]
                                                                
  
  The strictly oriented rules are moved into the weak component.
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(n^1)).
  
  Strict DPs:
    { min^#(X, 0()) -> c_1(X)
    , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  Weak DPs: { log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) }
  Weak Trs:
    { min(X, 0()) -> X
    , min(s(X), s(Y)) -> min(X, Y)
    , quot(0(), s(Y)) -> 0()
    , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) }
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(n^1))
  
  The following weak DPs constitute a sub-graph of the DG that is
  closed under successors. The DPs are removed.
  
  { log^#(s(s(X))) -> c_6(log^#(s(quot(X, s(s(0())))))) }
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(n^1)).
  
  Strict DPs:
    { min^#(X, 0()) -> c_1(X)
    , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  Weak Trs:
    { min(X, 0()) -> X
    , min(s(X), s(Y)) -> min(X, Y)
    , quot(0(), s(Y)) -> 0()
    , quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y))) }
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(n^1))
  
  No rule is usable, rules are removed from the input problem.
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(n^1)).
  
  Strict DPs:
    { min^#(X, 0()) -> c_1(X)
    , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(n^1))
  
  We use the processor 'matrix interpretation of dimension 1' to
  orient following rules strictly.
  
  DPs:
    { 1: min^#(X, 0()) -> c_1(X) }
  
  Sub-proof:
  ----------
    The following argument positions are usable:
      Uargs(c_2) = {1}
    
    TcT has computed the following constructor-based matrix
    interpretation satisfying not(EDA).
    
         [min](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
                   [0] = [1]                  
                                              
               [s](x1) = [1] x1 + [0]         
                                              
        [quot](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
             [log](x1) = [7] x1 + [0]         
                                              
       [min^#](x1, x2) = [2] x2 + [0]         
                                              
             [c_1](x1) = [0]                  
                                              
             [c_2](x1) = [1] x1 + [0]         
                                              
      [quot^#](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
                 [c_3] = [0]                  
                                              
             [c_4](x1) = [7] x1 + [0]         
                                              
           [log^#](x1) = [7] x1 + [0]         
                                              
                 [c_5] = [0]                  
                                              
             [c_6](x1) = [7] x1 + [0]         
    
    The following symbols are considered usable
    
      {min^#}
    
    The order satisfies the following ordering constraints:
    
          [min^#(X, 0())] =  [2]               
                          >  [0]               
                          =  [c_1(X)]          
                                               
      [min^#(s(X), s(Y))] =  [2] Y + [0]       
                          >= [2] Y + [0]       
                          =  [c_2(min^#(X, Y))]
                                               
  
  The strictly oriented rules are moved into the weak component.
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(n^1)).
  
  Strict DPs: { min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  Weak DPs: { min^#(X, 0()) -> c_1(X) }
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(n^1))
  
  We use the processor 'matrix interpretation of dimension 1' to
  orient following rules strictly.
  
  DPs:
    { 1: min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  
  Sub-proof:
  ----------
    The following argument positions are usable:
      Uargs(c_2) = {1}
    
    TcT has computed the following constructor-based matrix
    interpretation satisfying not(EDA).
    
         [min](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
                   [0] = [7]                  
                                              
               [s](x1) = [1] x1 + [4]         
                                              
        [quot](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
             [log](x1) = [7] x1 + [0]         
                                              
       [min^#](x1, x2) = [2] x1 + [0]         
                                              
             [c_1](x1) = [1] x1 + [0]         
                                              
             [c_2](x1) = [1] x1 + [5]         
                                              
      [quot^#](x1, x2) = [7] x1 + [7] x2 + [0]
                                              
                 [c_3] = [0]                  
                                              
             [c_4](x1) = [7] x1 + [0]         
                                              
           [log^#](x1) = [7] x1 + [0]         
                                              
                 [c_5] = [0]                  
                                              
             [c_6](x1) = [7] x1 + [0]         
    
    The following symbols are considered usable
    
      {min^#}
    
    The order satisfies the following ordering constraints:
    
          [min^#(X, 0())] =  [2] X + [0]       
                          >= [1] X + [0]       
                          =  [c_1(X)]          
                                               
      [min^#(s(X), s(Y))] =  [2] X + [8]       
                          >  [2] X + [5]       
                          =  [c_2(min^#(X, Y))]
                                               
  
  The strictly oriented rules are moved into the weak component.
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(1)).
  
  Weak DPs:
    { min^#(X, 0()) -> c_1(X)
    , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(1))
  
  The following weak DPs constitute a sub-graph of the DG that is
  closed under successors. The DPs are removed.
  
  { min^#(X, 0()) -> c_1(X)
  , min^#(s(X), s(Y)) -> c_2(min^#(X, Y)) }
  
  We are left with following problem, upon which TcT provides the
  certificate YES(O(1),O(1)).
  
  Rules: Empty
  Obligation:
    runtime complexity
  Answer:
    YES(O(1),O(1))
  
  Empty rules are trivially bounded

Hurray, we answered YES(O(1),O(n^2))