YES(O(1),O(n^1))
0.00/0.66	YES(O(1),O(n^1))
0.00/0.66	
0.00/0.66	We are left with following problem, upon which TcT provides the
0.00/0.66	certificate YES(O(1),O(n^1)).
0.00/0.66	
0.00/0.66	Strict Trs:
0.00/0.66	  { g(x, y) -> x
0.00/0.66	  , g(x, y) -> y
0.00/0.66	  , f(x, y, s(z)) -> s(f(0(), 1(), z))
0.00/0.66	  , f(0(), 1(), x) -> f(s(x), x, x) }
0.00/0.66	Obligation:
0.00/0.66	  innermost runtime complexity
0.00/0.66	Answer:
0.00/0.66	  YES(O(1),O(n^1))
0.00/0.66	
0.00/0.66	We add the following weak dependency pairs:
0.00/0.66	
0.00/0.66	Strict DPs:
0.00/0.66	  { g^#(x, y) -> c_1()
0.00/0.66	  , g^#(x, y) -> c_2()
0.00/0.66	  , f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.66	  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.66	
0.00/0.66	and mark the set of starting terms.
0.00/0.66	
0.00/0.66	We are left with following problem, upon which TcT provides the
0.00/0.66	certificate YES(O(1),O(n^1)).
0.00/0.66	
0.00/0.66	Strict DPs:
0.00/0.66	  { g^#(x, y) -> c_1()
0.00/0.66	  , g^#(x, y) -> c_2()
0.00/0.66	  , f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.66	  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.66	Strict Trs:
0.00/0.66	  { g(x, y) -> x
0.00/0.66	  , g(x, y) -> y
0.00/0.66	  , f(x, y, s(z)) -> s(f(0(), 1(), z))
0.00/0.66	  , f(0(), 1(), x) -> f(s(x), x, x) }
0.00/0.66	Obligation:
0.00/0.66	  innermost runtime complexity
0.00/0.66	Answer:
0.00/0.66	  YES(O(1),O(n^1))
0.00/0.66	
0.00/0.66	No rule is usable, rules are removed from the input problem.
0.00/0.66	
0.00/0.66	We are left with following problem, upon which TcT provides the
0.00/0.66	certificate YES(O(1),O(n^1)).
0.00/0.66	
0.00/0.66	Strict DPs:
0.00/0.66	  { g^#(x, y) -> c_1()
0.00/0.66	  , g^#(x, y) -> c_2()
0.00/0.66	  , f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.66	  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.66	Obligation:
0.00/0.66	  innermost runtime complexity
0.00/0.66	Answer:
0.00/0.66	  YES(O(1),O(n^1))
0.00/0.66	
0.00/0.66	The weightgap principle applies (using the following constant
0.00/0.66	growth matrix-interpretation)
0.00/0.66	
0.00/0.66	The following argument positions are usable:
0.00/0.66	  Uargs(c_3) = {1}, Uargs(c_4) = {1}
0.00/0.66	
0.00/0.66	TcT has computed the following constructor-restricted matrix
0.00/0.66	interpretation.
0.00/0.66	
0.00/0.66	                [0] = [0]           
0.00/0.66	                      [0]           
0.00/0.66	                                    
0.00/0.66	                [1] = [0]           
0.00/0.66	                      [0]           
0.00/0.66	                                    
0.00/0.66	            [s](x1) = [1 0] x1 + [0]
0.00/0.66	                      [0 0]      [0]
0.00/0.66	                                    
0.00/0.66	      [g^#](x1, x2) = [2]           
0.00/0.66	                      [1]           
0.00/0.66	                                    
0.00/0.66	              [c_1] = [1]           
0.00/0.66	                      [1]           
0.00/0.66	                                    
0.00/0.66	              [c_2] = [1]           
0.00/0.66	                      [1]           
0.00/0.66	                                    
0.00/0.66	  [f^#](x1, x2, x3) = [0 0] x3 + [0]
0.00/0.66	                      [1 0]      [0]
0.00/0.66	                                    
0.00/0.66	          [c_3](x1) = [1 0] x1 + [2]
0.00/0.66	                      [0 1]      [2]
0.00/0.66	                                    
0.00/0.66	          [c_4](x1) = [1 0] x1 + [2]
0.00/0.66	                      [0 1]      [2]
0.00/0.66	
0.00/0.66	The order satisfies the following ordering constraints:
0.00/0.66	
0.00/0.66	         [g^#(x, y)] = [2]                    
0.00/0.66	                       [1]                    
0.00/0.66	                     > [1]                    
0.00/0.66	                       [1]                    
0.00/0.66	                     = [c_1()]                
0.00/0.66	                                              
0.00/0.66	         [g^#(x, y)] = [2]                    
0.00/0.66	                       [1]                    
0.00/0.66	                     > [1]                    
0.00/0.66	                       [1]                    
0.00/0.66	                     = [c_2()]                
0.00/0.66	                                              
0.00/0.66	   [f^#(x, y, s(z))] = [0 0] z + [0]          
0.00/0.66	                       [1 0]     [0]          
0.00/0.66	                     ? [0 0] z + [2]          
0.00/0.66	                       [1 0]     [2]          
0.00/0.66	                     = [c_3(f^#(0(), 1(), z))]
0.00/0.66	                                              
0.00/0.66	  [f^#(0(), 1(), x)] = [0 0] x + [0]          
0.00/0.66	                       [1 0]     [0]          
0.00/0.66	                     ? [0 0] x + [2]          
0.00/0.66	                       [1 0]     [2]          
0.00/0.66	                     = [c_4(f^#(s(x), x, x))] 
0.00/0.66	                                              
0.00/0.66	
0.00/0.66	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.66	
0.00/0.66	We are left with following problem, upon which TcT provides the
0.00/0.66	certificate YES(O(1),O(n^1)).
0.00/0.66	
0.00/0.66	Strict DPs:
0.00/0.66	  { f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.66	  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.66	Weak DPs:
0.00/0.66	  { g^#(x, y) -> c_1()
0.00/0.66	  , g^#(x, y) -> c_2() }
0.00/0.66	Obligation:
0.00/0.66	  innermost runtime complexity
0.00/0.66	Answer:
0.00/0.66	  YES(O(1),O(n^1))
0.00/0.66	
0.00/0.66	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.66	closed under successors. The DPs are removed.
0.00/0.66	
0.00/0.66	{ g^#(x, y) -> c_1()
0.00/0.66	, g^#(x, y) -> c_2() }
0.00/0.66	
0.00/0.66	We are left with following problem, upon which TcT provides the
0.00/0.66	certificate YES(O(1),O(n^1)).
0.00/0.66	
0.00/0.66	Strict DPs:
0.00/0.66	  { f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.66	  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.66	Obligation:
0.00/0.66	  innermost runtime complexity
0.00/0.66	Answer:
0.00/0.66	  YES(O(1),O(n^1))
0.00/0.66	
0.00/0.66	We use the processor 'matrix interpretation of dimension 1' to
0.00/0.66	orient following rules strictly.
0.00/0.66	
0.00/0.66	DPs:
0.00/0.66	  { 1: f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z)) }
0.00/0.66	
0.00/0.66	Sub-proof:
0.00/0.66	----------
0.00/0.66	  The following argument positions are usable:
0.00/0.66	    Uargs(c_3) = {1}, Uargs(c_4) = {1}
0.00/0.66	  
0.00/0.66	  TcT has computed the following constructor-based matrix
0.00/0.66	  interpretation satisfying not(EDA).
0.00/0.66	  
0.00/0.66	                  [0] = [0]         
0.00/0.66	                                    
0.00/0.66	                  [1] = [0]         
0.00/0.66	                                    
0.00/0.66	              [s](x1) = [1] x1 + [2]
0.00/0.66	                                    
0.00/0.66	    [f^#](x1, x2, x3) = [4] x3 + [0]
0.00/0.66	                                    
0.00/0.66	            [c_3](x1) = [1] x1 + [1]
0.00/0.66	                                    
0.00/0.66	            [c_4](x1) = [1] x1 + [0]
0.00/0.66	  
0.00/0.66	  The order satisfies the following ordering constraints:
0.00/0.66	  
0.00/0.66	     [f^#(x, y, s(z))] =  [4] z + [8]            
0.00/0.66	                       >  [4] z + [1]            
0.00/0.66	                       =  [c_3(f^#(0(), 1(), z))]
0.00/0.66	                                                 
0.00/0.66	    [f^#(0(), 1(), x)] =  [4] x + [0]            
0.00/0.66	                       >= [4] x + [0]            
0.00/0.66	                       =  [c_4(f^#(s(x), x, x))] 
0.00/0.66	                                                 
0.00/0.66	
0.00/0.66	We return to the main proof. Consider the set of all dependency
0.00/0.66	pairs
0.00/0.66	
0.00/0.66	:
0.00/0.66	  { 1: f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.66	  , 2: f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.66	
0.00/0.66	Processor 'matrix interpretation of dimension 1' induces the
0.00/0.66	complexity certificate YES(?,O(n^1)) on application of dependency
0.00/0.66	pairs {1}. These cover all (indirect) predecessors of dependency
0.00/0.66	pairs {1,2}, their number of application is equally bounded. The
0.00/0.66	dependency pairs are shifted into the weak component.
0.00/0.66	
0.00/0.66	We are left with following problem, upon which TcT provides the
0.00/0.66	certificate YES(O(1),O(1)).
0.00/0.66	
0.00/0.66	Weak DPs:
0.00/0.66	  { f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.66	  , f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.66	Obligation:
0.00/0.66	  innermost runtime complexity
0.00/0.66	Answer:
0.00/0.66	  YES(O(1),O(1))
0.00/0.66	
0.00/0.66	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.66	closed under successors. The DPs are removed.
0.00/0.66	
0.00/0.66	{ f^#(x, y, s(z)) -> c_3(f^#(0(), 1(), z))
0.00/0.66	, f^#(0(), 1(), x) -> c_4(f^#(s(x), x, x)) }
0.00/0.66	
0.00/0.66	We are left with following problem, upon which TcT provides the
0.00/0.66	certificate YES(O(1),O(1)).
0.00/0.66	
0.00/0.66	Rules: Empty
0.00/0.66	Obligation:
0.00/0.66	  innermost runtime complexity
0.00/0.66	Answer:
0.00/0.66	  YES(O(1),O(1))
0.00/0.66	
0.00/0.66	Empty rules are trivially bounded
0.00/0.66	
0.00/0.66	Hurray, we answered YES(O(1),O(n^1))
0.00/0.66	EOF