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:
  { concat(leaf(), Y) -> Y
  , concat(cons(U, V), Y) -> cons(U, concat(V, Y))
  , lessleaves(X, leaf()) -> false()
  , lessleaves(leaf(), cons(W, Z)) -> true()
  , lessleaves(cons(U, V), cons(W, Z)) ->
    lessleaves(concat(U, V), concat(W, Z)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

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

Trs:
  { lessleaves(X, leaf()) -> false()
  , lessleaves(leaf(), cons(W, Z)) -> true() }

The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^1)) . These rules are moved into the corresponding weak
component(s).

Sub-proof:
----------
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
        [concat](x1, x2) = [1] x1 + [1] x2 + [0]
                                                
                  [leaf] = [0]                  
                                                
          [cons](x1, x2) = [1] x1 + [1] x2 + [0]
                                                
    [lessleaves](x1, x2) = [1] x1 + [2] x2 + [1]
                                                
                 [false] = [0]                  
                                                
                  [true] = [0]                  
  
  This order satisfies the following ordering constraints:
  
                     [concat(leaf(), Y)] =  [1] Y + [0]                             
                                         >= [1] Y + [0]                             
                                         =  [Y]                                     
                                                                                    
                 [concat(cons(U, V), Y)] =  [1] Y + [1] U + [1] V + [0]             
                                         >= [1] Y + [1] U + [1] V + [0]             
                                         =  [cons(U, concat(V, Y))]                 
                                                                                    
                 [lessleaves(X, leaf())] =  [1] X + [1]                             
                                         >  [0]                                     
                                         =  [false()]                               
                                                                                    
        [lessleaves(leaf(), cons(W, Z))] =  [2] W + [2] Z + [1]                     
                                         >  [0]                                     
                                         =  [true()]                                
                                                                                    
    [lessleaves(cons(U, V), cons(W, Z))] =  [1] U + [1] V + [2] W + [2] Z + [1]     
                                         >= [1] U + [1] V + [2] W + [2] Z + [1]     
                                         =  [lessleaves(concat(U, V), concat(W, Z))]
                                                                                    

We return to the main proof.

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

Strict Trs:
  { concat(leaf(), Y) -> Y
  , concat(cons(U, V), Y) -> cons(U, concat(V, Y))
  , lessleaves(cons(U, V), cons(W, Z)) ->
    lessleaves(concat(U, V), concat(W, Z)) }
Weak Trs:
  { lessleaves(X, leaf()) -> false()
  , lessleaves(leaf(), cons(W, Z)) -> true() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

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

Trs:
  { concat(leaf(), Y) -> Y
  , lessleaves(cons(U, V), cons(W, Z)) ->
    lessleaves(concat(U, V), concat(W, Z)) }

The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^1)) . These rules are moved into the corresponding weak
component(s).

Sub-proof:
----------
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
        [concat](x1, x2) = [1] x1 + [1] x2 + [0]
                                                
                  [leaf] = [2]                  
                                                
          [cons](x1, x2) = [1] x1 + [1] x2 + [2]
                                                
    [lessleaves](x1, x2) = [2] x1 + [1] x2 + [0]
                                                
                 [false] = [1]                  
                                                
                  [true] = [1]                  
  
  This order satisfies the following ordering constraints:
  
                     [concat(leaf(), Y)] =  [1] Y + [2]                             
                                         >  [1] Y + [0]                             
                                         =  [Y]                                     
                                                                                    
                 [concat(cons(U, V), Y)] =  [1] Y + [1] U + [1] V + [2]             
                                         >= [1] Y + [1] U + [1] V + [2]             
                                         =  [cons(U, concat(V, Y))]                 
                                                                                    
                 [lessleaves(X, leaf())] =  [2] X + [2]                             
                                         >  [1]                                     
                                         =  [false()]                               
                                                                                    
        [lessleaves(leaf(), cons(W, Z))] =  [1] W + [1] Z + [6]                     
                                         >  [1]                                     
                                         =  [true()]                                
                                                                                    
    [lessleaves(cons(U, V), cons(W, Z))] =  [2] U + [2] V + [1] W + [1] Z + [6]     
                                         >  [2] U + [2] V + [1] W + [1] Z + [0]     
                                         =  [lessleaves(concat(U, V), concat(W, Z))]
                                                                                    

We return to the main proof.

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

Strict Trs: { concat(cons(U, V), Y) -> cons(U, concat(V, Y)) }
Weak Trs:
  { concat(leaf(), Y) -> Y
  , lessleaves(X, leaf()) -> false()
  , lessleaves(leaf(), cons(W, Z)) -> true()
  , lessleaves(cons(U, V), cons(W, Z)) ->
    lessleaves(concat(U, V), concat(W, Z)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We use the processor 'matrix interpretation of dimension 2' to
orient following rules strictly.

Trs: { concat(cons(U, V), Y) -> cons(U, concat(V, Y)) }

The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^1)) . These rules are moved into the corresponding weak
component(s).

Sub-proof:
----------
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA) and not(IDA(1)).
  
        [concat](x1, x2) = [1 1] x1 + [1 0] x2 + [0]
                           [0 1]      [0 2]      [2]
                                                    
                  [leaf] = [0]                      
                           [0]                      
                                                    
          [cons](x1, x2) = [1 2] x1 + [1 0] x2 + [2]
                           [0 0]      [0 1]      [2]
                                                    
    [lessleaves](x1, x2) = [1 0] x1 + [2 0] x2 + [0]
                           [0 0]      [0 0]      [0]
                                                    
                 [false] = [0]                      
                           [0]                      
                                                    
                  [true] = [3]                      
                           [0]                      
  
  This order satisfies the following ordering constraints:
  
                     [concat(leaf(), Y)] =  [1 0] Y + [0]                              
                                            [0 2]     [2]                              
                                         >= [1 0] Y + [0]                              
                                            [0 1]     [0]                              
                                         =  [Y]                                        
                                                                                       
                 [concat(cons(U, V), Y)] =  [1 0] Y + [1 2] U + [1 1] V + [4]          
                                            [0 2]     [0 0]     [0 1]     [4]          
                                         >  [1 0] Y + [1 2] U + [1 1] V + [2]          
                                            [0 2]     [0 0]     [0 1]     [4]          
                                         =  [cons(U, concat(V, Y))]                    
                                                                                       
                 [lessleaves(X, leaf())] =  [1 0] X + [0]                              
                                            [0 0]     [0]                              
                                         >= [0]                                        
                                            [0]                                        
                                         =  [false()]                                  
                                                                                       
        [lessleaves(leaf(), cons(W, Z))] =  [2 4] W + [2 0] Z + [4]                    
                                            [0 0]     [0 0]     [0]                    
                                         >  [3]                                        
                                            [0]                                        
                                         =  [true()]                                   
                                                                                       
    [lessleaves(cons(U, V), cons(W, Z))] =  [1 2] U + [1 0] V + [2 4] W + [2 0] Z + [6]
                                            [0 0]     [0 0]     [0 0]     [0 0]     [0]
                                         >  [1 1] U + [1 0] V + [2 2] W + [2 0] Z + [0]
                                            [0 0]     [0 0]     [0 0]     [0 0]     [0]
                                         =  [lessleaves(concat(U, V), concat(W, Z))]   
                                                                                       

We return to the main proof.

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

Weak Trs:
  { concat(leaf(), Y) -> Y
  , concat(cons(U, V), Y) -> cons(U, concat(V, Y))
  , lessleaves(X, leaf()) -> false()
  , lessleaves(leaf(), cons(W, Z)) -> true()
  , lessleaves(cons(U, V), cons(W, Z)) ->
    lessleaves(concat(U, V), concat(W, Z)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

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