YES(O(1),O(1))
0.00/0.47	YES(O(1),O(1))
0.00/0.47	
0.00/0.47	We are left with following problem, upon which TcT provides the
0.00/0.47	certificate YES(O(1),O(1)).
0.00/0.47	
0.00/0.47	Strict Trs:
0.00/0.47	  { *(x, 0()) -> 0()
0.00/0.47	  , *(*(x, y), z) -> *(x, *(y, z))
0.00/0.47	  , *(i(x), x) -> 1()
0.00/0.47	  , *(1(), y) -> y }
0.00/0.47	Obligation:
0.00/0.47	  runtime complexity
0.00/0.47	Answer:
0.00/0.47	  YES(O(1),O(1))
0.00/0.47	
0.00/0.47	We add the following weak dependency pairs:
0.00/0.47	
0.00/0.47	Strict DPs:
0.00/0.47	  { *^#(x, 0()) -> c_1()
0.00/0.47	  , *^#(*(x, y), z) -> c_2(*^#(x, *(y, z)))
0.00/0.47	  , *^#(i(x), x) -> c_3()
0.00/0.47	  , *^#(1(), y) -> c_4(y) }
0.00/0.47	
0.00/0.47	and mark the set of starting terms.
0.00/0.47	
0.00/0.47	We are left with following problem, upon which TcT provides the
0.00/0.47	certificate YES(O(1),O(1)).
0.00/0.47	
0.00/0.47	Strict DPs:
0.00/0.47	  { *^#(x, 0()) -> c_1()
0.00/0.47	  , *^#(*(x, y), z) -> c_2(*^#(x, *(y, z)))
0.00/0.47	  , *^#(i(x), x) -> c_3()
0.00/0.47	  , *^#(1(), y) -> c_4(y) }
0.00/0.47	Strict Trs:
0.00/0.47	  { *(x, 0()) -> 0()
0.00/0.47	  , *(*(x, y), z) -> *(x, *(y, z))
0.00/0.47	  , *(i(x), x) -> 1()
0.00/0.47	  , *(1(), y) -> y }
0.00/0.47	Obligation:
0.00/0.47	  runtime complexity
0.00/0.47	Answer:
0.00/0.47	  YES(O(1),O(1))
0.00/0.47	
0.00/0.47	The weightgap principle applies (using the following constant
0.00/0.47	growth matrix-interpretation)
0.00/0.47	
0.00/0.47	The following argument positions are usable:
0.00/0.47	  Uargs(*) = {2}, Uargs(*^#) = {2}, Uargs(c_2) = {1},
0.00/0.47	  Uargs(c_4) = {1}
0.00/0.47	
0.00/0.47	TcT has computed the following constructor-restricted matrix
0.00/0.47	interpretation.
0.00/0.47	
0.00/0.47	    [*](x1, x2) = [0 1] x1 + [1 0] x2 + [1]
0.00/0.47	                  [1 1]      [0 1]      [2]
0.00/0.47	                                           
0.00/0.47	        [i](x1) = [2]                      
0.00/0.47	                  [0]                      
0.00/0.47	                                           
0.00/0.47	            [1] = [0]                      
0.00/0.47	                  [0]                      
0.00/0.47	                                           
0.00/0.47	            [0] = [0]                      
0.00/0.47	                  [0]                      
0.00/0.47	                                           
0.00/0.47	  [*^#](x1, x2) = [0 1] x1 + [1 0] x2 + [0]
0.00/0.47	                  [0 0]      [0 0]      [0]
0.00/0.47	                                           
0.00/0.47	          [c_1] = [1]                      
0.00/0.47	                  [0]                      
0.00/0.47	                                           
0.00/0.47	      [c_2](x1) = [1 0] x1 + [1]           
0.00/0.47	                  [0 1]      [2]           
0.00/0.47	                                           
0.00/0.47	          [c_3] = [2]                      
0.00/0.47	                  [0]                      
0.00/0.47	                                           
0.00/0.47	      [c_4](x1) = [1 0] x1 + [2]           
0.00/0.47	                  [0 1]      [0]           
0.00/0.47	
0.00/0.47	The order satisfies the following ordering constraints:
0.00/0.47	
0.00/0.47	        [*(x, 0())] = [0 1] x + [1]                    
0.00/0.47	                      [1 1]     [2]                    
0.00/0.47	                    > [0]                              
0.00/0.47	                      [0]                              
0.00/0.47	                    = [0()]                            
0.00/0.47	                                                       
0.00/0.47	    [*(*(x, y), z)] = [1 1] x + [0 1] y + [1 0] z + [3]
0.00/0.47	                      [1 2]     [1 1]     [0 1]     [5]
0.00/0.47	                    > [0 1] x + [0 1] y + [1 0] z + [2]
0.00/0.47	                      [1 1]     [1 1]     [0 1]     [4]
0.00/0.47	                    = [*(x, *(y, z))]                  
0.00/0.47	                                                       
0.00/0.47	       [*(i(x), x)] = [1 0] x + [1]                    
0.00/0.47	                      [0 1]     [4]                    
0.00/0.47	                    > [0]                              
0.00/0.47	                      [0]                              
0.00/0.47	                    = [1()]                            
0.00/0.47	                                                       
0.00/0.47	        [*(1(), y)] = [1 0] y + [1]                    
0.00/0.47	                      [0 1]     [2]                    
0.00/0.47	                    > [1 0] y + [0]                    
0.00/0.47	                      [0 1]     [0]                    
0.00/0.47	                    = [y]                              
0.00/0.47	                                                       
0.00/0.47	      [*^#(x, 0())] = [0 1] x + [0]                    
0.00/0.47	                      [0 0]     [0]                    
0.00/0.47	                    ? [1]                              
0.00/0.47	                      [0]                              
0.00/0.47	                    = [c_1()]                          
0.00/0.47	                                                       
0.00/0.47	  [*^#(*(x, y), z)] = [1 1] x + [0 1] y + [1 0] z + [2]
0.00/0.47	                      [0 0]     [0 0]     [0 0]     [0]
0.00/0.47	                    ? [0 1] x + [0 1] y + [1 0] z + [2]
0.00/0.47	                      [0 0]     [0 0]     [0 0]     [2]
0.00/0.47	                    = [c_2(*^#(x, *(y, z)))]           
0.00/0.47	                                                       
0.00/0.47	     [*^#(i(x), x)] = [1 0] x + [0]                    
0.00/0.47	                      [0 0]     [0]                    
0.00/0.47	                    ? [2]                              
0.00/0.47	                      [0]                              
0.00/0.47	                    = [c_3()]                          
0.00/0.47	                                                       
0.00/0.47	      [*^#(1(), y)] = [1 0] y + [0]                    
0.00/0.47	                      [0 0]     [0]                    
0.00/0.47	                    ? [1 0] y + [2]                    
0.00/0.47	                      [0 1]     [0]                    
0.00/0.47	                    = [c_4(y)]                         
0.00/0.47	                                                       
0.00/0.47	
0.00/0.47	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.47	
0.00/0.47	We are left with following problem, upon which TcT provides the
0.00/0.47	certificate YES(O(1),O(1)).
0.00/0.47	
0.00/0.47	Strict DPs:
0.00/0.47	  { *^#(x, 0()) -> c_1()
0.00/0.47	  , *^#(*(x, y), z) -> c_2(*^#(x, *(y, z)))
0.00/0.47	  , *^#(i(x), x) -> c_3()
0.00/0.47	  , *^#(1(), y) -> c_4(y) }
0.00/0.47	Weak Trs:
0.00/0.47	  { *(x, 0()) -> 0()
0.00/0.47	  , *(*(x, y), z) -> *(x, *(y, z))
0.00/0.47	  , *(i(x), x) -> 1()
0.00/0.47	  , *(1(), y) -> y }
0.00/0.47	Obligation:
0.00/0.47	  runtime complexity
0.00/0.47	Answer:
0.00/0.47	  YES(O(1),O(1))
0.00/0.47	
0.00/0.47	We estimate the number of application of {1,3} by applications of
0.00/0.47	Pre({1,3}) = {2,4}. Here rules are labeled as follows:
0.00/0.47	
0.00/0.47	  DPs:
0.00/0.47	    { 1: *^#(x, 0()) -> c_1()
0.00/0.47	    , 2: *^#(*(x, y), z) -> c_2(*^#(x, *(y, z)))
0.00/0.47	    , 3: *^#(i(x), x) -> c_3()
0.00/0.47	    , 4: *^#(1(), y) -> c_4(y) }
0.00/0.47	
0.00/0.47	We are left with following problem, upon which TcT provides the
0.00/0.47	certificate YES(O(1),O(1)).
0.00/0.47	
0.00/0.47	Strict DPs:
0.00/0.47	  { *^#(*(x, y), z) -> c_2(*^#(x, *(y, z)))
0.00/0.47	  , *^#(1(), y) -> c_4(y) }
0.00/0.47	Weak DPs:
0.00/0.47	  { *^#(x, 0()) -> c_1()
0.00/0.47	  , *^#(i(x), x) -> c_3() }
0.00/0.47	Weak Trs:
0.00/0.47	  { *(x, 0()) -> 0()
0.00/0.47	  , *(*(x, y), z) -> *(x, *(y, z))
0.00/0.47	  , *(i(x), x) -> 1()
0.00/0.47	  , *(1(), y) -> y }
0.00/0.47	Obligation:
0.00/0.47	  runtime complexity
0.00/0.47	Answer:
0.00/0.47	  YES(O(1),O(1))
0.00/0.47	
0.00/0.47	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.47	closed under successors. The DPs are removed.
0.00/0.47	
0.00/0.47	{ *^#(x, 0()) -> c_1()
0.00/0.47	, *^#(i(x), x) -> c_3() }
0.00/0.47	
0.00/0.47	We are left with following problem, upon which TcT provides the
0.00/0.47	certificate YES(O(1),O(1)).
0.00/0.47	
0.00/0.47	Strict DPs:
0.00/0.47	  { *^#(*(x, y), z) -> c_2(*^#(x, *(y, z)))
0.00/0.47	  , *^#(1(), y) -> c_4(y) }
0.00/0.47	Weak Trs:
0.00/0.47	  { *(x, 0()) -> 0()
0.00/0.47	  , *(*(x, y), z) -> *(x, *(y, z))
0.00/0.47	  , *(i(x), x) -> 1()
0.00/0.47	  , *(1(), y) -> y }
0.00/0.47	Obligation:
0.00/0.47	  runtime complexity
0.00/0.47	Answer:
0.00/0.47	  YES(O(1),O(1))
0.00/0.47	
0.00/0.47	We use the processor 'matrix interpretation of dimension 1' to
0.00/0.47	orient following rules strictly.
0.00/0.47	
0.00/0.47	DPs:
0.00/0.47	  { 1: *^#(*(x, y), z) -> c_2(*^#(x, *(y, z)))
0.00/0.47	  , 2: *^#(1(), y) -> c_4(y) }
0.00/0.47	Trs:
0.00/0.47	  { *(x, 0()) -> 0()
0.00/0.47	  , *(i(x), x) -> 1()
0.00/0.47	  , *(1(), y) -> y }
0.00/0.47	
0.00/0.47	Sub-proof:
0.00/0.47	----------
0.00/0.47	  The following argument positions are usable:
0.00/0.47	    Uargs(c_2) = {1}, Uargs(c_4) = {1}
0.00/0.47	  
0.00/0.47	  TcT has computed the following constructor-restricted matrix
0.00/0.47	  interpretation. Note that the diagonal of the component-wise maxima
0.00/0.47	  of interpretation-entries (of constructors) contains no more than 0
0.00/0.47	  non-zero entries.
0.00/0.47	  
0.00/0.47	      [*](x1, x2) = [1] x1 + [1] x2 + [2]
0.00/0.47	                                         
0.00/0.47	          [i](x1) = [4]                  
0.00/0.47	                                         
0.00/0.47	              [1] = [2]                  
0.00/0.47	                                         
0.00/0.47	              [0] = [2]                  
0.00/0.47	                                         
0.00/0.47	    [*^#](x1, x2) = [4] x1 + [1] x2 + [0]
0.00/0.47	                                         
0.00/0.47	        [c_2](x1) = [1] x1 + [5]         
0.00/0.47	                                         
0.00/0.47	        [c_4](x1) = [1] x1 + [6]         
0.00/0.47	  
0.00/0.47	  The order satisfies the following ordering constraints:
0.00/0.47	  
0.00/0.47	          [*(x, 0())] =  [1] x + [4]                
0.00/0.47	                      >  [2]                        
0.00/0.47	                      =  [0()]                      
0.00/0.47	                                                    
0.00/0.47	      [*(*(x, y), z)] =  [1] x + [1] y + [1] z + [4]
0.00/0.47	                      >= [1] x + [1] y + [1] z + [4]
0.00/0.47	                      =  [*(x, *(y, z))]            
0.00/0.47	                                                    
0.00/0.47	         [*(i(x), x)] =  [1] x + [6]                
0.00/0.47	                      >  [2]                        
0.00/0.47	                      =  [1()]                      
0.00/0.47	                                                    
0.00/0.47	          [*(1(), y)] =  [1] y + [4]                
0.00/0.47	                      >  [1] y + [0]                
0.00/0.47	                      =  [y]                        
0.00/0.47	                                                    
0.00/0.47	    [*^#(*(x, y), z)] =  [4] x + [4] y + [1] z + [8]
0.00/0.47	                      >  [4] x + [1] y + [1] z + [7]
0.00/0.47	                      =  [c_2(*^#(x, *(y, z)))]     
0.00/0.47	                                                    
0.00/0.47	        [*^#(1(), y)] =  [1] y + [8]                
0.00/0.47	                      >  [1] y + [6]                
0.00/0.47	                      =  [c_4(y)]                   
0.00/0.47	                                                    
0.00/0.47	
0.00/0.47	The strictly oriented rules are moved into the weak component.
0.00/0.47	
0.00/0.47	We are left with following problem, upon which TcT provides the
0.00/0.47	certificate YES(O(1),O(1)).
0.00/0.47	
0.00/0.47	Weak DPs:
0.00/0.47	  { *^#(*(x, y), z) -> c_2(*^#(x, *(y, z)))
0.00/0.47	  , *^#(1(), y) -> c_4(y) }
0.00/0.47	Weak Trs:
0.00/0.47	  { *(x, 0()) -> 0()
0.00/0.47	  , *(*(x, y), z) -> *(x, *(y, z))
0.00/0.47	  , *(i(x), x) -> 1()
0.00/0.47	  , *(1(), y) -> y }
0.00/0.47	Obligation:
0.00/0.47	  runtime complexity
0.00/0.47	Answer:
0.00/0.47	  YES(O(1),O(1))
0.00/0.47	
0.00/0.47	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.47	closed under successors. The DPs are removed.
0.00/0.47	
0.00/0.47	{ *^#(*(x, y), z) -> c_2(*^#(x, *(y, z)))
0.00/0.47	, *^#(1(), y) -> c_4(y) }
0.00/0.47	
0.00/0.47	We are left with following problem, upon which TcT provides the
0.00/0.47	certificate YES(O(1),O(1)).
0.00/0.47	
0.00/0.47	Weak Trs:
0.00/0.47	  { *(x, 0()) -> 0()
0.00/0.47	  , *(*(x, y), z) -> *(x, *(y, z))
0.00/0.47	  , *(i(x), x) -> 1()
0.00/0.47	  , *(1(), y) -> y }
0.00/0.47	Obligation:
0.00/0.47	  runtime complexity
0.00/0.47	Answer:
0.00/0.47	  YES(O(1),O(1))
0.00/0.47	
0.00/0.47	No rule is usable, rules are removed from the input problem.
0.00/0.47	
0.00/0.47	We are left with following problem, upon which TcT provides the
0.00/0.47	certificate YES(O(1),O(1)).
0.00/0.47	
0.00/0.47	Rules: Empty
0.00/0.47	Obligation:
0.00/0.47	  runtime complexity
0.00/0.47	Answer:
0.00/0.47	  YES(O(1),O(1))
0.00/0.47	
0.00/0.47	Empty rules are trivially bounded
0.00/0.47	
0.00/0.47	Hurray, we answered YES(O(1),O(1))
0.00/0.47	EOF