YES(O(1),O(n^1))
0.00/0.57	YES(O(1),O(n^1))
0.00/0.57	
0.00/0.57	We are left with following problem, upon which TcT provides the
0.00/0.57	certificate YES(O(1),O(n^1)).
0.00/0.57	
0.00/0.57	Strict Trs:
0.00/0.57	  { g(x, y) -> x
0.00/0.57	  , g(x, y) -> y
0.00/0.57	  , f(x, y, s(z)) -> s(f(0(), 1(), z))
0.00/0.57	  , f(0(), 1(), x) -> f(s(x), x, x) }
0.00/0.57	Obligation:
0.00/0.57	  runtime complexity
0.00/0.57	Answer:
0.00/0.57	  YES(O(1),O(n^1))
0.00/0.57	
0.00/0.57	We add the following weak dependency pairs:
0.00/0.57	
0.00/0.57	Strict DPs:
0.00/0.57	  { g^#(x, y) -> c_1(x)
0.00/0.57	  , g^#(x, y) -> c_2(y)
0.00/0.57	  , f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.57	  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.57	
0.00/0.57	and mark the set of starting terms.
0.00/0.57	
0.00/0.57	We are left with following problem, upon which TcT provides the
0.00/0.57	certificate YES(O(1),O(n^1)).
0.00/0.57	
0.00/0.57	Strict DPs:
0.00/0.57	  { g^#(x, y) -> c_1(x)
0.00/0.57	  , g^#(x, y) -> c_2(y)
0.00/0.57	  , f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.57	  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.57	Strict Trs:
0.00/0.57	  { g(x, y) -> x
0.00/0.57	  , g(x, y) -> y
0.00/0.57	  , f(x, y, s(z)) -> s(f(0(), 1(), z))
0.00/0.57	  , f(0(), 1(), x) -> f(s(x), x, x) }
0.00/0.57	Obligation:
0.00/0.57	  runtime complexity
0.00/0.57	Answer:
0.00/0.57	  YES(O(1),O(n^1))
0.00/0.57	
0.00/0.57	No rule is usable, rules are removed from the input problem.
0.00/0.57	
0.00/0.57	We are left with following problem, upon which TcT provides the
0.00/0.57	certificate YES(O(1),O(n^1)).
0.00/0.57	
0.00/0.57	Strict DPs:
0.00/0.57	  { g^#(x, y) -> c_1(x)
0.00/0.57	  , g^#(x, y) -> c_2(y)
0.00/0.57	  , f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.57	  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.57	Obligation:
0.00/0.57	  runtime complexity
0.00/0.57	Answer:
0.00/0.57	  YES(O(1),O(n^1))
0.00/0.57	
0.00/0.57	The weightgap principle applies (using the following constant
0.00/0.57	growth matrix-interpretation)
0.00/0.57	
0.00/0.57	The following argument positions are usable:
0.00/0.57	  Uargs(c_3) = {1}, Uargs(c_4) = {1}
0.00/0.57	
0.00/0.57	TcT has computed the following constructor-restricted matrix
0.00/0.57	interpretation.
0.00/0.57	
0.00/0.57	                [0] = [0]                      
0.00/0.57	                      [0]                      
0.00/0.57	                                               
0.00/0.57	                [1] = [0]                      
0.00/0.57	                      [0]                      
0.00/0.57	                                               
0.00/0.57	            [s](x1) = [1 0] x1 + [0]           
0.00/0.57	                      [0 0]      [0]           
0.00/0.57	                                               
0.00/0.57	      [g^#](x1, x2) = [1 1] x1 + [1 1] x2 + [1]
0.00/0.57	                      [2 1]      [2 1]      [1]
0.00/0.57	                                               
0.00/0.57	          [c_1](x1) = [1 1] x1 + [0]           
0.00/0.57	                      [1 1]      [1]           
0.00/0.57	                                               
0.00/0.57	          [c_2](x1) = [1 1] x1 + [0]           
0.00/0.57	                      [1 1]      [1]           
0.00/0.57	                                               
0.00/0.57	  [f^#](x1, x2, x3) = [0 0] x3 + [0]           
0.00/0.57	                      [2 0]      [0]           
0.00/0.57	                                               
0.00/0.57	          [c_3](x1) = [1 0] x1 + [2]           
0.00/0.57	                      [0 1]      [2]           
0.00/0.57	                                               
0.00/0.57	          [c_4](x1) = [1 0] x1 + [2]           
0.00/0.57	                      [0 1]      [2]           
0.00/0.57	
0.00/0.57	The order satisfies the following ordering constraints:
0.00/0.57	
0.00/0.57	         [g^#(x, y)] = [1 1] x + [1 1] y + [1]
0.00/0.57	                       [2 1]     [2 1]     [1]
0.00/0.57	                     > [1 1] x + [0]          
0.00/0.57	                       [1 1]     [1]          
0.00/0.57	                     = [c_1(x)]               
0.00/0.57	                                              
0.00/0.57	         [g^#(x, y)] = [1 1] x + [1 1] y + [1]
0.00/0.57	                       [2 1]     [2 1]     [1]
0.00/0.57	                     > [1 1] y + [0]          
0.00/0.57	                       [1 1]     [1]          
0.00/0.57	                     = [c_2(y)]               
0.00/0.57	                                              
0.00/0.57	   [f^#(x, y, s(z))] = [0 0] z + [0]          
0.00/0.57	                       [2 0]     [0]          
0.00/0.57	                     ? [0 0] z + [2]          
0.00/0.57	                       [2 0]     [2]          
0.00/0.57	                     = [c_3(f^#(0(), 1(), z))]
0.00/0.57	                                              
0.00/0.57	  [f^#(0(), 1(), x)] = [0 0] x + [0]          
0.00/0.57	                       [2 0]     [0]          
0.00/0.57	                     ? [0 0] x + [2]          
0.00/0.57	                       [2 0]     [2]          
0.00/0.57	                     = [c_4(f^#(s(x), x, x))] 
0.00/0.57	                                              
0.00/0.57	
0.00/0.57	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.57	
0.00/0.57	We are left with following problem, upon which TcT provides the
0.00/0.57	certificate YES(O(1),O(n^1)).
0.00/0.57	
0.00/0.57	Strict DPs:
0.00/0.57	  { f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.57	  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.57	Weak DPs:
0.00/0.57	  { g^#(x, y) -> c_1(x)
0.00/0.57	  , g^#(x, y) -> c_2(y) }
0.00/0.57	Obligation:
0.00/0.57	  runtime complexity
0.00/0.57	Answer:
0.00/0.57	  YES(O(1),O(n^1))
0.00/0.57	
0.00/0.57	We use the processor 'matrix interpretation of dimension 1' to
0.00/0.57	orient following rules strictly.
0.00/0.57	
0.00/0.57	DPs:
0.00/0.57	  { 1: f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.57	  , 3: g^#(x, y) -> c_1(x)
0.00/0.57	  , 4: g^#(x, y) -> c_2(y) }
0.00/0.57	
0.00/0.57	Sub-proof:
0.00/0.57	----------
0.00/0.57	  The following argument positions are usable:
0.00/0.57	    Uargs(c_3) = {1}, Uargs(c_4) = {1}
0.00/0.57	  
0.00/0.57	  TcT has computed the following constructor-based matrix
0.00/0.57	  interpretation satisfying not(EDA).
0.00/0.57	  
0.00/0.57	                  [0] = [0]                  
0.00/0.57	                                             
0.00/0.57	                  [1] = [0]                  
0.00/0.57	                                             
0.00/0.57	              [s](x1) = [1] x1 + [2]         
0.00/0.57	                                             
0.00/0.57	        [g^#](x1, x2) = [7] x1 + [7] x2 + [7]
0.00/0.57	                                             
0.00/0.57	            [c_1](x1) = [7] x1 + [3]         
0.00/0.57	                                             
0.00/0.57	            [c_2](x1) = [7] x1 + [3]         
0.00/0.57	                                             
0.00/0.57	    [f^#](x1, x2, x3) = [4] x3 + [0]         
0.00/0.57	                                             
0.00/0.57	            [c_3](x1) = [1] x1 + [1]         
0.00/0.57	                                             
0.00/0.57	            [c_4](x1) = [1] x1 + [0]         
0.00/0.57	  
0.00/0.57	  The order satisfies the following ordering constraints:
0.00/0.57	  
0.00/0.57	           [g^#(x, y)] =  [7] x + [7] y + [7]    
0.00/0.57	                       >  [7] x + [3]            
0.00/0.57	                       =  [c_1(x)]               
0.00/0.57	                                                 
0.00/0.57	           [g^#(x, y)] =  [7] x + [7] y + [7]    
0.00/0.57	                       >  [7] y + [3]            
0.00/0.57	                       =  [c_2(y)]               
0.00/0.57	                                                 
0.00/0.57	     [f^#(x, y, s(z))] =  [4] z + [8]            
0.00/0.57	                       >  [4] z + [1]            
0.00/0.57	                       =  [c_3(f^#(0(), 1(), z))]
0.00/0.57	                                                 
0.00/0.57	    [f^#(0(), 1(), x)] =  [4] x + [0]            
0.00/0.57	                       >= [4] x + [0]            
0.00/0.57	                       =  [c_4(f^#(s(x), x, x))] 
0.00/0.57	                                                 
0.00/0.57	
0.00/0.57	We return to the main proof. Consider the set of all dependency
0.00/0.57	pairs
0.00/0.57	
0.00/0.57	:
0.00/0.57	  { 1: f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.57	  , 2: f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x))
0.00/0.57	  , 3: g^#(x, y) -> c_1(x)
0.00/0.57	  , 4: g^#(x, y) -> c_2(y) }
0.00/0.57	
0.00/0.57	Processor 'matrix interpretation of dimension 1' induces the
0.00/0.57	complexity certificate YES(?,O(n^1)) on application of dependency
0.00/0.57	pairs {1,3,4}. These cover all (indirect) predecessors of
0.00/0.57	dependency pairs {1,2,3,4}, their number of application is equally
0.00/0.57	bounded. The dependency pairs are shifted into the weak component.
0.00/0.57	
0.00/0.57	We are left with following problem, upon which TcT provides the
0.00/0.57	certificate YES(O(1),O(1)).
0.00/0.57	
0.00/0.57	Weak DPs:
0.00/0.57	  { g^#(x, y) -> c_1(x)
0.00/0.57	  , g^#(x, y) -> c_2(y)
0.00/0.57	  , f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.57	  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.57	Obligation:
0.00/0.57	  runtime complexity
0.00/0.57	Answer:
0.00/0.57	  YES(O(1),O(1))
0.00/0.57	
0.00/0.57	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.57	closed under successors. The DPs are removed.
0.00/0.57	
0.00/0.57	{ g^#(x, y) -> c_1(x)
0.00/0.57	, g^#(x, y) -> c_2(y)
0.00/0.57	, f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.57	, f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.57	
0.00/0.57	We are left with following problem, upon which TcT provides the
0.00/0.57	certificate YES(O(1),O(1)).
0.00/0.57	
0.00/0.57	Rules: Empty
0.00/0.57	Obligation:
0.00/0.57	  runtime complexity
0.00/0.57	Answer:
0.00/0.57	  YES(O(1),O(1))
0.00/0.57	
0.00/0.57	Empty rules are trivially bounded
0.00/0.57	
0.00/0.57	Hurray, we answered YES(O(1),O(n^1))
0.00/0.57	EOF