YES(O(1),O(n^1))
0.00/0.54	YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(n^1)).
0.00/0.54	
0.00/0.54	Strict Trs:
0.00/0.54	  { f(empty(), l) -> l
0.00/0.54	  , f(cons(x, k), l) -> g(k, l, cons(x, k))
0.00/0.54	  , g(a, b, c) -> f(a, cons(b, c)) }
0.00/0.54	Obligation:
0.00/0.54	  runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	We add the following weak dependency pairs:
0.00/0.54	
0.00/0.54	Strict DPs:
0.00/0.54	  { f^#(empty(), l) -> c_1(l)
0.00/0.54	  , f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k)))
0.00/0.54	  , g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
0.00/0.54	
0.00/0.54	and mark the set of starting terms.
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(n^1)).
0.00/0.54	
0.00/0.54	Strict DPs:
0.00/0.54	  { f^#(empty(), l) -> c_1(l)
0.00/0.54	  , f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k)))
0.00/0.54	  , g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
0.00/0.54	Strict Trs:
0.00/0.54	  { f(empty(), l) -> l
0.00/0.54	  , f(cons(x, k), l) -> g(k, l, cons(x, k))
0.00/0.54	  , g(a, b, c) -> f(a, cons(b, c)) }
0.00/0.54	Obligation:
0.00/0.54	  runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	No rule is usable, rules are removed from the input problem.
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(n^1)).
0.00/0.54	
0.00/0.54	Strict DPs:
0.00/0.54	  { f^#(empty(), l) -> c_1(l)
0.00/0.54	  , f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k)))
0.00/0.54	  , g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
0.00/0.54	Obligation:
0.00/0.54	  runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	The weightgap principle applies (using the following constant
0.00/0.54	growth matrix-interpretation)
0.00/0.54	
0.00/0.54	The following argument positions are usable:
0.00/0.54	  Uargs(c_2) = {1}, Uargs(c_3) = {1}
0.00/0.54	
0.00/0.54	TcT has computed the following constructor-restricted matrix
0.00/0.54	interpretation.
0.00/0.54	
0.00/0.54	            [empty] = [0]           
0.00/0.54	                      [0]           
0.00/0.54	                                    
0.00/0.54	     [cons](x1, x2) = [1 0] x1 + [0]
0.00/0.54	                      [0 0]      [0]
0.00/0.54	                                    
0.00/0.54	      [f^#](x1, x2) = [0]           
0.00/0.54	                      [0]           
0.00/0.54	                                    
0.00/0.54	          [c_1](x1) = [1]           
0.00/0.54	                      [0]           
0.00/0.54	                                    
0.00/0.54	          [c_2](x1) = [1 0] x1 + [0]
0.00/0.54	                      [0 1]      [0]
0.00/0.54	                                    
0.00/0.54	  [g^#](x1, x2, x3) = [0 0] x2 + [1]
0.00/0.54	                      [1 1]      [0]
0.00/0.54	                                    
0.00/0.54	          [c_3](x1) = [1 0] x1 + [0]
0.00/0.54	                      [0 1]      [0]
0.00/0.54	
0.00/0.54	The order satisfies the following ordering constraints:
0.00/0.54	
0.00/0.54	     [f^#(empty(), l)] = [0]                         
0.00/0.54	                         [0]                         
0.00/0.54	                       ? [1]                         
0.00/0.54	                         [0]                         
0.00/0.54	                       = [c_1(l)]                    
0.00/0.54	                                                     
0.00/0.54	  [f^#(cons(x, k), l)] = [0]                         
0.00/0.54	                         [0]                         
0.00/0.54	                       ? [0 0] l + [1]               
0.00/0.54	                         [1 1]     [0]               
0.00/0.54	                       = [c_2(g^#(k, l, cons(x, k)))]
0.00/0.54	                                                     
0.00/0.54	        [g^#(a, b, c)] = [0 0] b + [1]               
0.00/0.54	                         [1 1]     [0]               
0.00/0.54	                       > [0]                         
0.00/0.54	                         [0]                         
0.00/0.54	                       = [c_3(f^#(a, cons(b, c)))]   
0.00/0.54	                                                     
0.00/0.54	
0.00/0.54	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(n^1)).
0.00/0.54	
0.00/0.54	Strict DPs:
0.00/0.54	  { f^#(empty(), l) -> c_1(l)
0.00/0.54	  , f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) }
0.00/0.54	Weak DPs: { g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
0.00/0.54	Obligation:
0.00/0.54	  runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	We use the processor 'matrix interpretation of dimension 1' to
0.00/0.54	orient following rules strictly.
0.00/0.54	
0.00/0.54	DPs:
0.00/0.54	  { 1: f^#(empty(), l) -> c_1(l) }
0.00/0.54	
0.00/0.54	Sub-proof:
0.00/0.54	----------
0.00/0.54	  The following argument positions are usable:
0.00/0.54	    Uargs(c_2) = {1}, Uargs(c_3) = {1}
0.00/0.54	  
0.00/0.54	  TcT has computed the following constructor-based matrix
0.00/0.54	  interpretation satisfying not(EDA).
0.00/0.54	  
0.00/0.54	              [empty] = [1]                  
0.00/0.54	                                             
0.00/0.54	       [cons](x1, x2) = [1] x1 + [1] x2 + [0]
0.00/0.54	                                             
0.00/0.54	        [f^#](x1, x2) = [1] x1 + [0]         
0.00/0.54	                                             
0.00/0.54	            [c_1](x1) = [0]                  
0.00/0.54	                                             
0.00/0.54	            [c_2](x1) = [1] x1 + [0]         
0.00/0.54	                                             
0.00/0.54	    [g^#](x1, x2, x3) = [1] x1 + [0]         
0.00/0.54	                                             
0.00/0.54	            [c_3](x1) = [1] x1 + [0]         
0.00/0.54	  
0.00/0.54	  The order satisfies the following ordering constraints:
0.00/0.54	  
0.00/0.54	       [f^#(empty(), l)] =  [1]                         
0.00/0.54	                         >  [0]                         
0.00/0.54	                         =  [c_1(l)]                    
0.00/0.54	                                                        
0.00/0.54	    [f^#(cons(x, k), l)] =  [1] x + [1] k + [0]         
0.00/0.54	                         >= [1] k + [0]                 
0.00/0.54	                         =  [c_2(g^#(k, l, cons(x, k)))]
0.00/0.54	                                                        
0.00/0.54	          [g^#(a, b, c)] =  [1] a + [0]                 
0.00/0.54	                         >= [1] a + [0]                 
0.00/0.54	                         =  [c_3(f^#(a, cons(b, c)))]   
0.00/0.54	                                                        
0.00/0.54	
0.00/0.54	The strictly oriented rules are moved into the weak component.
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(n^1)).
0.00/0.54	
0.00/0.54	Strict DPs: { f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) }
0.00/0.54	Weak DPs:
0.00/0.54	  { f^#(empty(), l) -> c_1(l)
0.00/0.54	  , g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
0.00/0.54	Obligation:
0.00/0.54	  runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	We use the processor 'matrix interpretation of dimension 1' to
0.00/0.54	orient following rules strictly.
0.00/0.54	
0.00/0.54	DPs:
0.00/0.54	  { 1: f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) }
0.00/0.54	
0.00/0.54	Sub-proof:
0.00/0.54	----------
0.00/0.54	  The following argument positions are usable:
0.00/0.54	    Uargs(c_2) = {1}, Uargs(c_3) = {1}
0.00/0.54	  
0.00/0.54	  TcT has computed the following constructor-based matrix
0.00/0.54	  interpretation satisfying not(EDA).
0.00/0.54	  
0.00/0.54	              [empty] = [0]                  
0.00/0.54	                                             
0.00/0.54	       [cons](x1, x2) = [1] x1 + [1] x2 + [2]
0.00/0.54	                                             
0.00/0.54	        [f^#](x1, x2) = [4] x1 + [0]         
0.00/0.54	                                             
0.00/0.54	            [c_1](x1) = [0]                  
0.00/0.54	                                             
0.00/0.54	            [c_2](x1) = [1] x1 + [1]         
0.00/0.54	                                             
0.00/0.54	    [g^#](x1, x2, x3) = [4] x1 + [0]         
0.00/0.54	                                             
0.00/0.54	            [c_3](x1) = [1] x1 + [0]         
0.00/0.54	  
0.00/0.54	  The order satisfies the following ordering constraints:
0.00/0.54	  
0.00/0.54	       [f^#(empty(), l)] =  [0]                         
0.00/0.54	                         >= [0]                         
0.00/0.54	                         =  [c_1(l)]                    
0.00/0.54	                                                        
0.00/0.54	    [f^#(cons(x, k), l)] =  [4] x + [4] k + [8]         
0.00/0.54	                         >  [4] k + [1]                 
0.00/0.54	                         =  [c_2(g^#(k, l, cons(x, k)))]
0.00/0.54	                                                        
0.00/0.54	          [g^#(a, b, c)] =  [4] a + [0]                 
0.00/0.54	                         >= [4] a + [0]                 
0.00/0.54	                         =  [c_3(f^#(a, cons(b, c)))]   
0.00/0.54	                                                        
0.00/0.54	
0.00/0.54	The strictly oriented rules are moved into the weak component.
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(1)).
0.00/0.54	
0.00/0.54	Weak DPs:
0.00/0.54	  { f^#(empty(), l) -> c_1(l)
0.00/0.54	  , f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k)))
0.00/0.54	  , g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
0.00/0.54	Obligation:
0.00/0.54	  runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(1))
0.00/0.54	
0.00/0.54	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.54	closed under successors. The DPs are removed.
0.00/0.54	
0.00/0.54	{ f^#(empty(), l) -> c_1(l)
0.00/0.54	, f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k)))
0.00/0.54	, g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(1)).
0.00/0.54	
0.00/0.54	Rules: Empty
0.00/0.54	Obligation:
0.00/0.54	  runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(1))
0.00/0.54	
0.00/0.54	Empty rules are trivially bounded
0.00/0.54	
0.00/0.54	Hurray, we answered YES(O(1),O(n^1))
0.00/0.54	EOF