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 Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
  , sel(0(), cons(X, Y)) -> X
  , activate(X) -> X
  , activate(n__from(X)) -> from(X) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following weak dependency pairs:

Strict DPs:
  { from^#(X) -> c_1(X, X)
  , from^#(X) -> c_2(X)
  , sel^#(s(X), cons(Y, Z)) -> c_3(sel^#(X, activate(Z)))
  , sel^#(0(), cons(X, Y)) -> c_4(X)
  , activate^#(X) -> c_5(X)
  , activate^#(n__from(X)) -> c_6(from^#(X)) }

and mark the set of starting terms.

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

Strict DPs:
  { from^#(X) -> c_1(X, X)
  , from^#(X) -> c_2(X)
  , sel^#(s(X), cons(Y, Z)) -> c_3(sel^#(X, activate(Z)))
  , sel^#(0(), cons(X, Y)) -> c_4(X)
  , activate^#(X) -> c_5(X)
  , activate^#(n__from(X)) -> c_6(from^#(X)) }
Strict Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
  , sel(0(), cons(X, Y)) -> X
  , activate(X) -> X
  , activate(n__from(X)) -> from(X) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

We replace rewrite rules by usable rules:

  Strict Usable Rules:
    { from(X) -> cons(X, n__from(s(X)))
    , from(X) -> n__from(X)
    , activate(X) -> X
    , activate(n__from(X)) -> from(X) }

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

Strict DPs:
  { from^#(X) -> c_1(X, X)
  , from^#(X) -> c_2(X)
  , sel^#(s(X), cons(Y, Z)) -> c_3(sel^#(X, activate(Z)))
  , sel^#(0(), cons(X, Y)) -> c_4(X)
  , activate^#(X) -> c_5(X)
  , activate^#(n__from(X)) -> c_6(from^#(X)) }
Strict Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , activate(X) -> X
  , activate(n__from(X)) -> from(X) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

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

The following argument positions are usable:
  Uargs(sel^#) = {2}, Uargs(c_3) = {1}, Uargs(c_6) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

        [from](x1) = [1]                      
                     [0]                      
                                              
    [cons](x1, x2) = [1 0] x2 + [0]           
                     [0 0]      [0]           
                                              
     [n__from](x1) = [0]                      
                     [0]                      
                                              
           [s](x1) = [1 2] x1 + [0]           
                     [0 0]      [0]           
                                              
               [0] = [0]                      
                     [0]                      
                                              
    [activate](x1) = [1 0] x1 + [2]           
                     [0 2]      [0]           
                                              
      [from^#](x1) = [0 0] x1 + [1]           
                     [1 2]      [1]           
                                              
     [c_1](x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                     [1 2]      [2 2]      [1]
                                              
         [c_2](x1) = [0 0] x1 + [2]           
                     [1 1]      [1]           
                                              
   [sel^#](x1, x2) = [1 0] x2 + [0]           
                     [0 0]      [0]           
                                              
         [c_3](x1) = [1 0] x1 + [2]           
                     [0 1]      [2]           
                                              
         [c_4](x1) = [1]                      
                     [0]                      
                                              
  [activate^#](x1) = [1 1] x1 + [2]           
                     [1 1]      [2]           
                                              
         [c_5](x1) = [1 1] x1 + [1]           
                     [1 1]      [1]           
                                              
         [c_6](x1) = [1 0] x1 + [2]           
                     [0 1]      [2]           

The following symbols are considered usable

  {from, activate, from^#, sel^#, activate^#}

The order satisfies the following ordering constraints:

                  [from(X)] = [1]                         
                              [0]                         
                            > [0]                         
                              [0]                         
                            = [cons(X, n__from(s(X)))]    
                                                          
                  [from(X)] = [1]                         
                              [0]                         
                            > [0]                         
                              [0]                         
                            = [n__from(X)]                
                                                          
              [activate(X)] = [1 0] X + [2]               
                              [0 2]     [0]               
                            > [1 0] X + [0]               
                              [0 1]     [0]               
                            = [X]                         
                                                          
     [activate(n__from(X))] = [2]                         
                              [0]                         
                            > [1]                         
                              [0]                         
                            = [from(X)]                   
                                                          
                [from^#(X)] = [0 0] X + [1]               
                              [1 2]     [1]               
                            ? [0 0] X + [1]               
                              [3 4]     [1]               
                            = [c_1(X, X)]                 
                                                          
                [from^#(X)] = [0 0] X + [1]               
                              [1 2]     [1]               
                            ? [0 0] X + [2]               
                              [1 1]     [1]               
                            = [c_2(X)]                    
                                                          
  [sel^#(s(X), cons(Y, Z))] = [1 0] Z + [0]               
                              [0 0]     [0]               
                            ? [1 0] Z + [4]               
                              [0 0]     [2]               
                            = [c_3(sel^#(X, activate(Z)))]
                                                          
   [sel^#(0(), cons(X, Y))] = [1 0] Y + [0]               
                              [0 0]     [0]               
                            ? [1]                         
                              [0]                         
                            = [c_4(X)]                    
                                                          
            [activate^#(X)] = [1 1] X + [2]               
                              [1 1]     [2]               
                            > [1 1] X + [1]               
                              [1 1]     [1]               
                            = [c_5(X)]                    
                                                          
   [activate^#(n__from(X))] = [2]                         
                              [2]                         
                            ? [0 0] X + [3]               
                              [1 2]     [3]               
                            = [c_6(from^#(X))]            
                                                          

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(1),O(n^1)).

Strict DPs:
  { from^#(X) -> c_1(X, X)
  , from^#(X) -> c_2(X)
  , sel^#(s(X), cons(Y, Z)) -> c_3(sel^#(X, activate(Z)))
  , sel^#(0(), cons(X, Y)) -> c_4(X)
  , activate^#(n__from(X)) -> c_6(from^#(X)) }
Weak DPs: { activate^#(X) -> c_5(X) }
Weak Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , activate(X) -> X
  , activate(n__from(X)) -> from(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:
  { 5: activate^#(n__from(X)) -> c_6(from^#(X))
  , 6: activate^#(X) -> c_5(X) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_3) = {1}, Uargs(c_6) = {1}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
          [from](x1) = [1] x1 + [0]         
                                            
      [cons](x1, x2) = [1] x1 + [0]         
                                            
       [n__from](x1) = [1] x1 + [0]         
                                            
             [s](x1) = [1] x1 + [0]         
                                            
       [sel](x1, x2) = [7] x1 + [7] x2 + [0]
                                            
                 [0] = [7]                  
                                            
      [activate](x1) = [1] x1 + [0]         
                                            
        [from^#](x1) = [1] x1 + [0]         
                                            
       [c_1](x1, x2) = [1] x2 + [0]         
                                            
           [c_2](x1) = [1] x1 + [0]         
                                            
     [sel^#](x1, x2) = [0]                  
                                            
           [c_3](x1) = [4] x1 + [0]         
                                            
           [c_4](x1) = [0]                  
                                            
    [activate^#](x1) = [5] x1 + [7]         
                                            
           [c_5](x1) = [5] x1 + [6]         
                                            
           [c_6](x1) = [4] x1 + [1]         
  
  The following symbols are considered usable
  
    {from, activate, from^#, sel^#, activate^#}
  
  The order satisfies the following ordering constraints:
  
                    [from(X)] =  [1] X + [0]                 
                              >= [1] X + [0]                 
                              =  [cons(X, n__from(s(X)))]    
                                                             
                    [from(X)] =  [1] X + [0]                 
                              >= [1] X + [0]                 
                              =  [n__from(X)]                
                                                             
                [activate(X)] =  [1] X + [0]                 
                              >= [1] X + [0]                 
                              =  [X]                         
                                                             
       [activate(n__from(X))] =  [1] X + [0]                 
                              >= [1] X + [0]                 
                              =  [from(X)]                   
                                                             
                  [from^#(X)] =  [1] X + [0]                 
                              >= [1] X + [0]                 
                              =  [c_1(X, X)]                 
                                                             
                  [from^#(X)] =  [1] X + [0]                 
                              >= [1] X + [0]                 
                              =  [c_2(X)]                    
                                                             
    [sel^#(s(X), cons(Y, Z))] =  [0]                         
                              >= [0]                         
                              =  [c_3(sel^#(X, activate(Z)))]
                                                             
     [sel^#(0(), cons(X, Y))] =  [0]                         
                              >= [0]                         
                              =  [c_4(X)]                    
                                                             
              [activate^#(X)] =  [5] X + [7]                 
                              >  [5] X + [6]                 
                              =  [c_5(X)]                    
                                                             
     [activate^#(n__from(X))] =  [5] X + [7]                 
                              >  [4] X + [1]                 
                              =  [c_6(from^#(X))]            
                                                             

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:
  { from^#(X) -> c_1(X, X)
  , from^#(X) -> c_2(X)
  , sel^#(s(X), cons(Y, Z)) -> c_3(sel^#(X, activate(Z)))
  , sel^#(0(), cons(X, Y)) -> c_4(X) }
Weak DPs:
  { activate^#(X) -> c_5(X)
  , activate^#(n__from(X)) -> c_6(from^#(X)) }
Weak Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , activate(X) -> X
  , activate(n__from(X)) -> from(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: from^#(X) -> c_1(X, X)
  , 2: from^#(X) -> c_2(X)
  , 5: activate^#(X) -> c_5(X)
  , 6: activate^#(n__from(X)) -> c_6(from^#(X)) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_3) = {1}, Uargs(c_6) = {1}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
          [from](x1) = [1] x1 + [0]         
                                            
      [cons](x1, x2) = [1] x1 + [0]         
                                            
       [n__from](x1) = [1] x1 + [0]         
                                            
             [s](x1) = [1] x1 + [0]         
                                            
       [sel](x1, x2) = [7] x1 + [7] x2 + [0]
                                            
                 [0] = [7]                  
                                            
      [activate](x1) = [1] x1 + [0]         
                                            
        [from^#](x1) = [7] x1 + [1]         
                                            
       [c_1](x1, x2) = [3] x1 + [3] x2 + [0]
                                            
           [c_2](x1) = [7] x1 + [0]         
                                            
     [sel^#](x1, x2) = [0]                  
                                            
           [c_3](x1) = [4] x1 + [0]         
                                            
           [c_4](x1) = [0]                  
                                            
    [activate^#](x1) = [7] x1 + [7]         
                                            
           [c_5](x1) = [7] x1 + [6]         
                                            
           [c_6](x1) = [1] x1 + [0]         
  
  The following symbols are considered usable
  
    {from, activate, from^#, sel^#, activate^#}
  
  The order satisfies the following ordering constraints:
  
                    [from(X)] =  [1] X + [0]                 
                              >= [1] X + [0]                 
                              =  [cons(X, n__from(s(X)))]    
                                                             
                    [from(X)] =  [1] X + [0]                 
                              >= [1] X + [0]                 
                              =  [n__from(X)]                
                                                             
                [activate(X)] =  [1] X + [0]                 
                              >= [1] X + [0]                 
                              =  [X]                         
                                                             
       [activate(n__from(X))] =  [1] X + [0]                 
                              >= [1] X + [0]                 
                              =  [from(X)]                   
                                                             
                  [from^#(X)] =  [7] X + [1]                 
                              >  [6] X + [0]                 
                              =  [c_1(X, X)]                 
                                                             
                  [from^#(X)] =  [7] X + [1]                 
                              >  [7] X + [0]                 
                              =  [c_2(X)]                    
                                                             
    [sel^#(s(X), cons(Y, Z))] =  [0]                         
                              >= [0]                         
                              =  [c_3(sel^#(X, activate(Z)))]
                                                             
     [sel^#(0(), cons(X, Y))] =  [0]                         
                              >= [0]                         
                              =  [c_4(X)]                    
                                                             
              [activate^#(X)] =  [7] X + [7]                 
                              >  [7] X + [6]                 
                              =  [c_5(X)]                    
                                                             
     [activate^#(n__from(X))] =  [7] X + [7]                 
                              >  [7] X + [1]                 
                              =  [c_6(from^#(X))]            
                                                             

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:
  { sel^#(s(X), cons(Y, Z)) -> c_3(sel^#(X, activate(Z)))
  , sel^#(0(), cons(X, Y)) -> c_4(X) }
Weak DPs:
  { from^#(X) -> c_1(X, X)
  , from^#(X) -> c_2(X)
  , activate^#(X) -> c_5(X)
  , activate^#(n__from(X)) -> c_6(from^#(X)) }
Weak Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , activate(X) -> X
  , activate(n__from(X)) -> from(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: sel^#(s(X), cons(Y, Z)) -> c_3(sel^#(X, activate(Z)))
  , 5: activate^#(X) -> c_5(X)
  , 6: activate^#(n__from(X)) -> c_6(from^#(X)) }
Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , activate(X) -> X
  , activate(n__from(X)) -> from(X) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_3) = {1}, Uargs(c_6) = {1}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
          [from](x1) = [1] x1 + [4]         
                                            
      [cons](x1, x2) = [1] x1 + [0]         
                                            
       [n__from](x1) = [1] x1 + [1]         
                                            
             [s](x1) = [1] x1 + [2]         
                                            
       [sel](x1, x2) = [7] x1 + [7] x2 + [0]
                                            
                 [0] = [0]                  
                                            
      [activate](x1) = [1] x1 + [7]         
                                            
        [from^#](x1) = [0]                  
                                            
       [c_1](x1, x2) = [0]                  
                                            
           [c_2](x1) = [0]                  
                                            
     [sel^#](x1, x2) = [4] x1 + [0]         
                                            
           [c_3](x1) = [1] x1 + [1]         
                                            
           [c_4](x1) = [0]                  
                                            
    [activate^#](x1) = [7]                  
                                            
           [c_5](x1) = [6]                  
                                            
           [c_6](x1) = [4] x1 + [1]         
  
  The following symbols are considered usable
  
    {from, activate, from^#, sel^#, activate^#}
  
  The order satisfies the following ordering constraints:
  
                    [from(X)] =  [1] X + [4]                 
                              >  [1] X + [0]                 
                              =  [cons(X, n__from(s(X)))]    
                                                             
                    [from(X)] =  [1] X + [4]                 
                              >  [1] X + [1]                 
                              =  [n__from(X)]                
                                                             
                [activate(X)] =  [1] X + [7]                 
                              >  [1] X + [0]                 
                              =  [X]                         
                                                             
       [activate(n__from(X))] =  [1] X + [8]                 
                              >  [1] X + [4]                 
                              =  [from(X)]                   
                                                             
                  [from^#(X)] =  [0]                         
                              >= [0]                         
                              =  [c_1(X, X)]                 
                                                             
                  [from^#(X)] =  [0]                         
                              >= [0]                         
                              =  [c_2(X)]                    
                                                             
    [sel^#(s(X), cons(Y, Z))] =  [4] X + [8]                 
                              >  [4] X + [1]                 
                              =  [c_3(sel^#(X, activate(Z)))]
                                                             
     [sel^#(0(), cons(X, Y))] =  [0]                         
                              >= [0]                         
                              =  [c_4(X)]                    
                                                             
              [activate^#(X)] =  [7]                         
                              >  [6]                         
                              =  [c_5(X)]                    
                                                             
     [activate^#(n__from(X))] =  [7]                         
                              >  [1]                         
                              =  [c_6(from^#(X))]            
                                                             

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)).

Strict DPs: { sel^#(0(), cons(X, Y)) -> c_4(X) }
Weak DPs:
  { from^#(X) -> c_1(X, X)
  , from^#(X) -> c_2(X)
  , sel^#(s(X), cons(Y, Z)) -> c_3(sel^#(X, activate(Z)))
  , activate^#(X) -> c_5(X)
  , activate^#(n__from(X)) -> c_6(from^#(X)) }
Weak Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , activate(X) -> X
  , activate(n__from(X)) -> from(X) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(1))

We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.

DPs:
  { 1: sel^#(0(), cons(X, Y)) -> c_4(X)
  , 5: activate^#(X) -> c_5(X)
  , 6: activate^#(n__from(X)) -> c_6(from^#(X)) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_3) = {1}, Uargs(c_6) = {1}
  
  TcT has computed the following constructor-restricted matrix
  interpretation. Note that the diagonal of the component-wise maxima
  of interpretation-entries (of constructors) contains no more than 0
  non-zero entries.
  
          [from](x1) = [0]         
                                   
      [cons](x1, x2) = [0]         
                                   
       [n__from](x1) = [0]         
                                   
             [s](x1) = [7]         
                                   
       [sel](x1, x2) = [0]         
                                   
                 [0] = [7]         
                                   
      [activate](x1) = [4] x1 + [0]
                                   
        [from^#](x1) = [0]         
                                   
       [c_1](x1, x2) = [0]         
                                   
           [c_2](x1) = [0]         
                                   
     [sel^#](x1, x2) = [1]         
                                   
           [c_3](x1) = [1] x1 + [0]
                                   
           [c_4](x1) = [0]         
                                   
    [activate^#](x1) = [7] x1 + [7]
                                   
           [c_5](x1) = [7] x1 + [5]
                                   
           [c_6](x1) = [4] x1 + [1]
  
  The following symbols are considered usable
  
    {from, activate, from^#, sel^#, activate^#}
  
  The order satisfies the following ordering constraints:
  
                    [from(X)] =  [0]                         
                              >= [0]                         
                              =  [cons(X, n__from(s(X)))]    
                                                             
                    [from(X)] =  [0]                         
                              >= [0]                         
                              =  [n__from(X)]                
                                                             
                [activate(X)] =  [4] X + [0]                 
                              >= [1] X + [0]                 
                              =  [X]                         
                                                             
       [activate(n__from(X))] =  [0]                         
                              >= [0]                         
                              =  [from(X)]                   
                                                             
                  [from^#(X)] =  [0]                         
                              >= [0]                         
                              =  [c_1(X, X)]                 
                                                             
                  [from^#(X)] =  [0]                         
                              >= [0]                         
                              =  [c_2(X)]                    
                                                             
    [sel^#(s(X), cons(Y, Z))] =  [1]                         
                              >= [1]                         
                              =  [c_3(sel^#(X, activate(Z)))]
                                                             
     [sel^#(0(), cons(X, Y))] =  [1]                         
                              >  [0]                         
                              =  [c_4(X)]                    
                                                             
              [activate^#(X)] =  [7] X + [7]                 
                              >  [7] X + [5]                 
                              =  [c_5(X)]                    
                                                             
     [activate^#(n__from(X))] =  [7]                         
                              >  [1]                         
                              =  [c_6(from^#(X))]            
                                                             

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:
  { from^#(X) -> c_1(X, X)
  , from^#(X) -> c_2(X)
  , sel^#(s(X), cons(Y, Z)) -> c_3(sel^#(X, activate(Z)))
  , sel^#(0(), cons(X, Y)) -> c_4(X)
  , activate^#(X) -> c_5(X)
  , activate^#(n__from(X)) -> c_6(from^#(X)) }
Weak Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , activate(X) -> X
  , activate(n__from(X)) -> from(X) }
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.

{ from^#(X) -> c_1(X, X)
, from^#(X) -> c_2(X)
, sel^#(s(X), cons(Y, Z)) -> c_3(sel^#(X, activate(Z)))
, sel^#(0(), cons(X, Y)) -> c_4(X)
, activate^#(X) -> c_5(X)
, activate^#(n__from(X)) -> c_6(from^#(X)) }

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

Weak Trs:
  { from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X)
  , activate(X) -> X
  , activate(n__from(X)) -> from(X) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(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(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^1))