YES(O(1),O(n^1))
0.00/0.15	YES(O(1),O(n^1))
0.00/0.15	
0.00/0.15	We are left with following problem, upon which TcT provides the
0.00/0.15	certificate YES(O(1),O(n^1)).
0.00/0.15	
0.00/0.15	Strict Trs:
0.00/0.15	  { sum(0()) -> 0()
0.00/0.15	  , sum(s(x)) -> +(sum(x), s(x))
0.00/0.15	  , sum1(0()) -> 0()
0.00/0.15	  , sum1(s(x)) -> s(+(sum1(x), +(x, x))) }
0.00/0.15	Obligation:
0.00/0.15	  runtime complexity
0.00/0.15	Answer:
0.00/0.15	  YES(O(1),O(n^1))
0.00/0.15	
0.00/0.15	The weightgap principle applies (using the following nonconstant
0.00/0.15	growth matrix-interpretation)
0.00/0.15	
0.00/0.15	The following argument positions are usable:
0.00/0.15	  Uargs(s) = {1}, Uargs(+) = {1}
0.00/0.15	
0.00/0.15	TcT has computed the following matrix interpretation satisfying
0.00/0.15	not(EDA) and not(IDA(1)).
0.00/0.15	
0.00/0.15	    [sum](x1) = [1] x1 + [0]
0.00/0.15	                            
0.00/0.15	          [0] = [0]         
0.00/0.15	                            
0.00/0.15	      [s](x1) = [1] x1 + [1]
0.00/0.15	                            
0.00/0.15	  [+](x1, x2) = [1] x1 + [0]
0.00/0.15	                            
0.00/0.15	   [sum1](x1) = [0]         
0.00/0.15	
0.00/0.15	The order satisfies the following ordering constraints:
0.00/0.15	
0.00/0.15	    [sum(0())] =  [0]                     
0.00/0.15	               >= [0]                     
0.00/0.15	               =  [0()]                   
0.00/0.15	                                          
0.00/0.15	   [sum(s(x))] =  [1] x + [1]             
0.00/0.15	               >  [1] x + [0]             
0.00/0.15	               =  [+(sum(x), s(x))]       
0.00/0.15	                                          
0.00/0.15	   [sum1(0())] =  [0]                     
0.00/0.15	               >= [0]                     
0.00/0.15	               =  [0()]                   
0.00/0.15	                                          
0.00/0.15	  [sum1(s(x))] =  [0]                     
0.00/0.15	               ?  [1]                     
0.00/0.15	               =  [s(+(sum1(x), +(x, x)))]
0.00/0.15	                                          
0.00/0.15	
0.00/0.15	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.15	
0.00/0.15	We are left with following problem, upon which TcT provides the
0.00/0.15	certificate YES(O(1),O(n^1)).
0.00/0.15	
0.00/0.15	Strict Trs:
0.00/0.15	  { sum(0()) -> 0()
0.00/0.15	  , sum1(0()) -> 0()
0.00/0.15	  , sum1(s(x)) -> s(+(sum1(x), +(x, x))) }
0.00/0.15	Weak Trs: { sum(s(x)) -> +(sum(x), s(x)) }
0.00/0.15	Obligation:
0.00/0.15	  runtime complexity
0.00/0.15	Answer:
0.00/0.15	  YES(O(1),O(n^1))
0.00/0.15	
0.00/0.15	The weightgap principle applies (using the following nonconstant
0.00/0.15	growth matrix-interpretation)
0.00/0.15	
0.00/0.15	The following argument positions are usable:
0.00/0.15	  Uargs(s) = {1}, Uargs(+) = {1}
0.00/0.15	
0.00/0.15	TcT has computed the following matrix interpretation satisfying
0.00/0.15	not(EDA) and not(IDA(1)).
0.00/0.15	
0.00/0.15	    [sum](x1) = [1] x1 + [1]
0.00/0.15	                            
0.00/0.15	          [0] = [0]         
0.00/0.15	                            
0.00/0.15	      [s](x1) = [1] x1 + [4]
0.00/0.15	                            
0.00/0.15	  [+](x1, x2) = [1] x1 + [0]
0.00/0.15	                            
0.00/0.15	   [sum1](x1) = [0]         
0.00/0.15	
0.00/0.15	The order satisfies the following ordering constraints:
0.00/0.15	
0.00/0.15	    [sum(0())] =  [1]                     
0.00/0.15	               >  [0]                     
0.00/0.15	               =  [0()]                   
0.00/0.15	                                          
0.00/0.15	   [sum(s(x))] =  [1] x + [5]             
0.00/0.15	               >  [1] x + [1]             
0.00/0.15	               =  [+(sum(x), s(x))]       
0.00/0.15	                                          
0.00/0.15	   [sum1(0())] =  [0]                     
0.00/0.15	               >= [0]                     
0.00/0.15	               =  [0()]                   
0.00/0.15	                                          
0.00/0.15	  [sum1(s(x))] =  [0]                     
0.00/0.15	               ?  [4]                     
0.00/0.15	               =  [s(+(sum1(x), +(x, x)))]
0.00/0.15	                                          
0.00/0.15	
0.00/0.15	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.15	
0.00/0.15	We are left with following problem, upon which TcT provides the
0.00/0.15	certificate YES(O(1),O(n^1)).
0.00/0.15	
0.00/0.15	Strict Trs:
0.00/0.15	  { sum1(0()) -> 0()
0.00/0.15	  , sum1(s(x)) -> s(+(sum1(x), +(x, x))) }
0.00/0.15	Weak Trs:
0.00/0.15	  { sum(0()) -> 0()
0.00/0.15	  , sum(s(x)) -> +(sum(x), s(x)) }
0.00/0.15	Obligation:
0.00/0.15	  runtime complexity
0.00/0.15	Answer:
0.00/0.15	  YES(O(1),O(n^1))
0.00/0.15	
0.00/0.15	The weightgap principle applies (using the following nonconstant
0.00/0.15	growth matrix-interpretation)
0.00/0.15	
0.00/0.15	The following argument positions are usable:
0.00/0.15	  Uargs(s) = {1}, Uargs(+) = {1}
0.00/0.15	
0.00/0.15	TcT has computed the following matrix interpretation satisfying
0.00/0.15	not(EDA) and not(IDA(1)).
0.00/0.15	
0.00/0.15	    [sum](x1) = [1] x1 + [0]
0.00/0.15	                            
0.00/0.15	          [0] = [4]         
0.00/0.15	                            
0.00/0.15	      [s](x1) = [1] x1 + [4]
0.00/0.15	                            
0.00/0.15	  [+](x1, x2) = [1] x1 + [4]
0.00/0.15	                            
0.00/0.15	   [sum1](x1) = [1] x1 + [4]
0.00/0.15	
0.00/0.15	The order satisfies the following ordering constraints:
0.00/0.15	
0.00/0.15	    [sum(0())] =  [4]                     
0.00/0.15	               >= [4]                     
0.00/0.15	               =  [0()]                   
0.00/0.15	                                          
0.00/0.15	   [sum(s(x))] =  [1] x + [4]             
0.00/0.15	               >= [1] x + [4]             
0.00/0.15	               =  [+(sum(x), s(x))]       
0.00/0.15	                                          
0.00/0.15	   [sum1(0())] =  [8]                     
0.00/0.15	               >  [4]                     
0.00/0.15	               =  [0()]                   
0.00/0.15	                                          
0.00/0.15	  [sum1(s(x))] =  [1] x + [8]             
0.00/0.15	               ?  [1] x + [12]            
0.00/0.15	               =  [s(+(sum1(x), +(x, x)))]
0.00/0.15	                                          
0.00/0.15	
0.00/0.15	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.15	
0.00/0.15	We are left with following problem, upon which TcT provides the
0.00/0.15	certificate YES(O(1),O(n^1)).
0.00/0.15	
0.00/0.15	Strict Trs: { sum1(s(x)) -> s(+(sum1(x), +(x, x))) }
0.00/0.15	Weak Trs:
0.00/0.15	  { sum(0()) -> 0()
0.00/0.15	  , sum(s(x)) -> +(sum(x), s(x))
0.00/0.15	  , sum1(0()) -> 0() }
0.00/0.15	Obligation:
0.00/0.15	  runtime complexity
0.00/0.15	Answer:
0.00/0.15	  YES(O(1),O(n^1))
0.00/0.15	
0.00/0.15	We use the processor 'matrix interpretation of dimension 1' to
0.00/0.15	orient following rules strictly.
0.00/0.15	
0.00/0.15	Trs: { sum1(s(x)) -> s(+(sum1(x), +(x, x))) }
0.00/0.15	
0.00/0.15	The induced complexity on above rules (modulo remaining rules) is
0.00/0.15	YES(?,O(n^1)) . These rules are moved into the corresponding weak
0.00/0.15	component(s).
0.00/0.15	
0.00/0.15	Sub-proof:
0.00/0.15	----------
0.00/0.15	  The following argument positions are usable:
0.00/0.15	    Uargs(s) = {1}, Uargs(+) = {1}
0.00/0.15	  
0.00/0.15	  TcT has computed the following constructor-based matrix
0.00/0.15	  interpretation satisfying not(EDA).
0.00/0.15	  
0.00/0.15	      [sum](x1) = [0]         
0.00/0.15	                              
0.00/0.15	            [0] = [0]         
0.00/0.15	                              
0.00/0.15	        [s](x1) = [1] x1 + [4]
0.00/0.15	                              
0.00/0.15	    [+](x1, x2) = [1] x1 + [0]
0.00/0.15	                              
0.00/0.15	     [sum1](x1) = [2] x1 + [0]
0.00/0.15	  
0.00/0.15	  The order satisfies the following ordering constraints:
0.00/0.15	  
0.00/0.15	      [sum(0())] =  [0]                     
0.00/0.15	                 >= [0]                     
0.00/0.15	                 =  [0()]                   
0.00/0.15	                                            
0.00/0.15	     [sum(s(x))] =  [0]                     
0.00/0.15	                 >= [0]                     
0.00/0.15	                 =  [+(sum(x), s(x))]       
0.00/0.15	                                            
0.00/0.15	     [sum1(0())] =  [0]                     
0.00/0.15	                 >= [0]                     
0.00/0.15	                 =  [0()]                   
0.00/0.15	                                            
0.00/0.15	    [sum1(s(x))] =  [2] x + [8]             
0.00/0.15	                 >  [2] x + [4]             
0.00/0.15	                 =  [s(+(sum1(x), +(x, x)))]
0.00/0.15	                                            
0.00/0.15	
0.00/0.15	We return to the main proof.
0.00/0.15	
0.00/0.15	We are left with following problem, upon which TcT provides the
0.00/0.15	certificate YES(O(1),O(1)).
0.00/0.15	
0.00/0.15	Weak Trs:
0.00/0.15	  { sum(0()) -> 0()
0.00/0.15	  , sum(s(x)) -> +(sum(x), s(x))
0.00/0.15	  , sum1(0()) -> 0()
0.00/0.15	  , sum1(s(x)) -> s(+(sum1(x), +(x, x))) }
0.00/0.15	Obligation:
0.00/0.15	  runtime complexity
0.00/0.15	Answer:
0.00/0.15	  YES(O(1),O(1))
0.00/0.15	
0.00/0.15	Empty rules are trivially bounded
0.00/0.15	
0.00/0.15	Hurray, we answered YES(O(1),O(n^1))
0.00/0.15	EOF