YES(O(1),O(1))
0.00/0.20	YES(O(1),O(1))
0.00/0.20	
0.00/0.20	We are left with following problem, upon which TcT provides the
0.00/0.20	certificate YES(O(1),O(1)).
0.00/0.20	
0.00/0.20	Strict Trs:
0.00/0.20	  { g(x, y) -> x
0.00/0.20	  , g(x, y) -> y
0.00/0.20	  , f(s(x), y, y) -> f(y, x, s(x)) }
0.00/0.20	Obligation:
0.00/0.20	  runtime complexity
0.00/0.20	Answer:
0.00/0.20	  YES(O(1),O(1))
0.00/0.20	
0.00/0.20	We add the following weak dependency pairs:
0.00/0.20	
0.00/0.20	Strict DPs:
0.00/0.20	  { g^#(x, y) -> c_1(x)
0.00/0.20	  , g^#(x, y) -> c_2(y)
0.00/0.20	  , f^#(s(x), y, y) -> c_3(f^#(y, x, s(x))) }
0.00/0.20	
0.00/0.20	and mark the set of starting terms.
0.00/0.20	
0.00/0.20	We are left with following problem, upon which TcT provides the
0.00/0.20	certificate YES(O(1),O(1)).
0.00/0.20	
0.00/0.20	Strict DPs:
0.00/0.20	  { g^#(x, y) -> c_1(x)
0.00/0.20	  , g^#(x, y) -> c_2(y)
0.00/0.20	  , f^#(s(x), y, y) -> c_3(f^#(y, x, s(x))) }
0.00/0.20	Strict Trs:
0.00/0.20	  { g(x, y) -> x
0.00/0.20	  , g(x, y) -> y
0.00/0.20	  , f(s(x), y, y) -> f(y, x, s(x)) }
0.00/0.20	Obligation:
0.00/0.20	  runtime complexity
0.00/0.20	Answer:
0.00/0.20	  YES(O(1),O(1))
0.00/0.20	
0.00/0.20	No rule is usable, rules are removed from the input problem.
0.00/0.20	
0.00/0.20	We are left with following problem, upon which TcT provides the
0.00/0.20	certificate YES(O(1),O(1)).
0.00/0.20	
0.00/0.20	Strict DPs:
0.00/0.20	  { g^#(x, y) -> c_1(x)
0.00/0.20	  , g^#(x, y) -> c_2(y)
0.00/0.20	  , f^#(s(x), y, y) -> c_3(f^#(y, x, s(x))) }
0.00/0.20	Obligation:
0.00/0.20	  runtime complexity
0.00/0.20	Answer:
0.00/0.20	  YES(O(1),O(1))
0.00/0.20	
0.00/0.20	The weightgap principle applies (using the following constant
0.00/0.20	growth matrix-interpretation)
0.00/0.20	
0.00/0.20	The following argument positions are usable:
0.00/0.20	  none
0.00/0.20	
0.00/0.20	TcT has computed the following constructor-restricted matrix
0.00/0.20	interpretation.
0.00/0.20	
0.00/0.20	            [s](x1) = [0]                      
0.00/0.20	                      [0]                      
0.00/0.20	                                               
0.00/0.20	      [g^#](x1, x2) = [1 1] x1 + [1 1] x2 + [1]
0.00/0.20	                      [1 2]      [1 1]      [1]
0.00/0.20	                                               
0.00/0.20	          [c_1](x1) = [1 1] x1 + [0]           
0.00/0.20	                      [1 1]      [1]           
0.00/0.20	                                               
0.00/0.20	          [c_2](x1) = [1 1] x1 + [0]           
0.00/0.20	                      [1 1]      [1]           
0.00/0.20	                                               
0.00/0.20	  [f^#](x1, x2, x3) = [1]                      
0.00/0.20	                      [0]                      
0.00/0.20	                                               
0.00/0.20	          [c_3](x1) = [2]                      
0.00/0.20	                      [2]                      
0.00/0.20	
0.00/0.20	The order satisfies the following ordering constraints:
0.00/0.20	
0.00/0.20	        [g^#(x, y)] = [1 1] x + [1 1] y + [1]
0.00/0.20	                      [1 2]     [1 1]     [1]
0.00/0.20	                    > [1 1] x + [0]          
0.00/0.20	                      [1 1]     [1]          
0.00/0.20	                    = [c_1(x)]               
0.00/0.20	                                             
0.00/0.20	        [g^#(x, y)] = [1 1] x + [1 1] y + [1]
0.00/0.20	                      [1 2]     [1 1]     [1]
0.00/0.20	                    > [1 1] y + [0]          
0.00/0.20	                      [1 1]     [1]          
0.00/0.20	                    = [c_2(y)]               
0.00/0.20	                                             
0.00/0.20	  [f^#(s(x), y, y)] = [1]                    
0.00/0.20	                      [0]                    
0.00/0.20	                    ? [2]                    
0.00/0.20	                      [2]                    
0.00/0.20	                    = [c_3(f^#(y, x, s(x)))] 
0.00/0.20	                                             
0.00/0.20	
0.00/0.20	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.20	
0.00/0.20	We are left with following problem, upon which TcT provides the
0.00/0.20	certificate YES(O(1),O(1)).
0.00/0.20	
0.00/0.20	Strict DPs: { f^#(s(x), y, y) -> c_3(f^#(y, x, s(x))) }
0.00/0.20	Weak DPs:
0.00/0.20	  { g^#(x, y) -> c_1(x)
0.00/0.20	  , g^#(x, y) -> c_2(y) }
0.00/0.20	Obligation:
0.00/0.20	  runtime complexity
0.00/0.20	Answer:
0.00/0.20	  YES(O(1),O(1))
0.00/0.20	
0.00/0.20	We use the processor 'matrix interpretation of dimension 1' to
0.00/0.20	orient following rules strictly.
0.00/0.20	
0.00/0.20	DPs:
0.00/0.20	  { 2: g^#(x, y) -> c_1(x)
0.00/0.20	  , 3: g^#(x, y) -> c_2(y) }
0.00/0.20	
0.00/0.20	Sub-proof:
0.00/0.20	----------
0.00/0.20	  The following argument positions are usable:
0.00/0.20	    none
0.00/0.20	  
0.00/0.20	  TcT has computed the following constructor-restricted matrix
0.00/0.20	  interpretation. Note that the diagonal of the component-wise maxima
0.00/0.20	  of interpretation-entries (of constructors) contains no more than 0
0.00/0.20	  non-zero entries.
0.00/0.20	  
0.00/0.20	              [s](x1) = [0]                  
0.00/0.20	                                             
0.00/0.20	        [g^#](x1, x2) = [7] x1 + [7] x2 + [7]
0.00/0.20	                                             
0.00/0.20	            [c_1](x1) = [7] x1 + [3]         
0.00/0.20	                                             
0.00/0.20	            [c_2](x1) = [7] x1 + [3]         
0.00/0.20	                                             
0.00/0.20	    [f^#](x1, x2, x3) = [1] x2 + [0]         
0.00/0.20	                                             
0.00/0.20	            [c_3](x1) = [0]                  
0.00/0.20	  
0.00/0.20	  The order satisfies the following ordering constraints:
0.00/0.20	  
0.00/0.20	          [g^#(x, y)] =  [7] x + [7] y + [7]   
0.00/0.20	                      >  [7] x + [3]           
0.00/0.20	                      =  [c_1(x)]              
0.00/0.20	                                               
0.00/0.20	          [g^#(x, y)] =  [7] x + [7] y + [7]   
0.00/0.20	                      >  [7] y + [3]           
0.00/0.20	                      =  [c_2(y)]              
0.00/0.20	                                               
0.00/0.20	    [f^#(s(x), y, y)] =  [1] y + [0]           
0.00/0.20	                      >= [0]                   
0.00/0.20	                      =  [c_3(f^#(y, x, s(x)))]
0.00/0.20	                                               
0.00/0.20	
0.00/0.20	We return to the main proof. Consider the set of all dependency
0.00/0.20	pairs
0.00/0.20	
0.00/0.20	:
0.00/0.20	  { 1: f^#(s(x), y, y) -> c_3(f^#(y, x, s(x)))
0.00/0.20	  , 2: g^#(x, y) -> c_1(x)
0.00/0.20	  , 3: g^#(x, y) -> c_2(y) }
0.00/0.20	
0.00/0.20	Processor 'matrix interpretation of dimension 1' induces the
0.00/0.20	complexity certificate YES(?,O(1)) on application of dependency
0.00/0.20	pairs {2,3}. These cover all (indirect) predecessors of dependency
0.00/0.20	pairs {1,2,3}, their number of application is equally bounded. The
0.00/0.20	dependency pairs are shifted into the weak component.
0.00/0.20	
0.00/0.20	We are left with following problem, upon which TcT provides the
0.00/0.20	certificate YES(O(1),O(1)).
0.00/0.20	
0.00/0.20	Weak DPs:
0.00/0.20	  { g^#(x, y) -> c_1(x)
0.00/0.20	  , g^#(x, y) -> c_2(y)
0.00/0.20	  , f^#(s(x), y, y) -> c_3(f^#(y, x, s(x))) }
0.00/0.20	Obligation:
0.00/0.20	  runtime complexity
0.00/0.20	Answer:
0.00/0.20	  YES(O(1),O(1))
0.00/0.20	
0.00/0.20	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.20	closed under successors. The DPs are removed.
0.00/0.20	
0.00/0.20	{ g^#(x, y) -> c_1(x)
0.00/0.20	, g^#(x, y) -> c_2(y)
0.00/0.20	, f^#(s(x), y, y) -> c_3(f^#(y, x, s(x))) }
0.00/0.20	
0.00/0.20	We are left with following problem, upon which TcT provides the
0.00/0.20	certificate YES(O(1),O(1)).
0.00/0.20	
0.00/0.20	Rules: Empty
0.00/0.20	Obligation:
0.00/0.20	  runtime complexity
0.00/0.20	Answer:
0.00/0.20	  YES(O(1),O(1))
0.00/0.20	
0.00/0.20	Empty rules are trivially bounded
0.00/0.20	
0.00/0.20	Hurray, we answered YES(O(1),O(1))
0.00/0.21	EOF