YES(O(1),O(1))
0.00/0.24	YES(O(1),O(1))
0.00/0.24	
0.00/0.24	We are left with following problem, upon which TcT provides the
0.00/0.24	certificate YES(O(1),O(1)).
0.00/0.24	
0.00/0.24	Strict Trs:
0.00/0.24	  { f(X, X) -> c(X)
0.00/0.24	  , f(X, c(X)) -> f(s(X), X)
0.00/0.24	  , f(s(X), X) -> f(X, a(X)) }
0.00/0.24	Obligation:
0.00/0.24	  innermost runtime complexity
0.00/0.24	Answer:
0.00/0.24	  YES(O(1),O(1))
0.00/0.24	
0.00/0.24	We add the following weak dependency pairs:
0.00/0.24	
0.00/0.24	Strict DPs:
0.00/0.24	  { f^#(X, X) -> c_1()
0.00/0.24	  , f^#(X, c(X)) -> c_2(f^#(s(X), X))
0.00/0.24	  , f^#(s(X), X) -> c_3(f^#(X, a(X))) }
0.00/0.24	
0.00/0.24	and mark the set of starting terms.
0.00/0.24	
0.00/0.24	We are left with following problem, upon which TcT provides the
0.00/0.24	certificate YES(O(1),O(1)).
0.00/0.24	
0.00/0.24	Strict DPs:
0.00/0.24	  { f^#(X, X) -> c_1()
0.00/0.24	  , f^#(X, c(X)) -> c_2(f^#(s(X), X))
0.00/0.24	  , f^#(s(X), X) -> c_3(f^#(X, a(X))) }
0.00/0.24	Strict Trs:
0.00/0.24	  { f(X, X) -> c(X)
0.00/0.24	  , f(X, c(X)) -> f(s(X), X)
0.00/0.24	  , f(s(X), X) -> f(X, a(X)) }
0.00/0.24	Obligation:
0.00/0.24	  innermost runtime complexity
0.00/0.24	Answer:
0.00/0.24	  YES(O(1),O(1))
0.00/0.24	
0.00/0.24	No rule is usable, rules are removed from the input problem.
0.00/0.24	
0.00/0.24	We are left with following problem, upon which TcT provides the
0.00/0.24	certificate YES(O(1),O(1)).
0.00/0.24	
0.00/0.24	Strict DPs:
0.00/0.24	  { f^#(X, X) -> c_1()
0.00/0.24	  , f^#(X, c(X)) -> c_2(f^#(s(X), X))
0.00/0.24	  , f^#(s(X), X) -> c_3(f^#(X, a(X))) }
0.00/0.24	Obligation:
0.00/0.24	  innermost runtime complexity
0.00/0.24	Answer:
0.00/0.24	  YES(O(1),O(1))
0.00/0.24	
0.00/0.24	The weightgap principle applies (using the following constant
0.00/0.24	growth matrix-interpretation)
0.00/0.24	
0.00/0.24	The following argument positions are usable:
0.00/0.24	  Uargs(c_2) = {1}
0.00/0.24	
0.00/0.24	TcT has computed the following constructor-restricted matrix
0.00/0.24	interpretation.
0.00/0.24	
0.00/0.24	        [s](x1) = [0]           
0.00/0.24	                  [0]           
0.00/0.24	                                
0.00/0.24	        [a](x1) = [0]           
0.00/0.24	                  [0]           
0.00/0.24	                                
0.00/0.24	        [c](x1) = [0]           
0.00/0.24	                  [0]           
0.00/0.24	                                
0.00/0.24	  [f^#](x1, x2) = [1]           
0.00/0.24	                  [0]           
0.00/0.24	                                
0.00/0.24	          [c_1] = [0]           
0.00/0.24	                  [0]           
0.00/0.24	                                
0.00/0.24	      [c_2](x1) = [1 0] x1 + [0]
0.00/0.24	                  [0 1]      [1]
0.00/0.24	                                
0.00/0.24	      [c_3](x1) = [1]           
0.00/0.24	                  [0]           
0.00/0.24	
0.00/0.24	The order satisfies the following ordering constraints:
0.00/0.24	
0.00/0.24	     [f^#(X, X)] =  [1]                
0.00/0.24	                    [0]                
0.00/0.24	                 >  [0]                
0.00/0.24	                    [0]                
0.00/0.24	                 =  [c_1()]            
0.00/0.24	                                       
0.00/0.24	  [f^#(X, c(X))] =  [1]                
0.00/0.24	                    [0]                
0.00/0.24	                 ?  [1]                
0.00/0.24	                    [1]                
0.00/0.24	                 =  [c_2(f^#(s(X), X))]
0.00/0.24	                                       
0.00/0.24	  [f^#(s(X), X)] =  [1]                
0.00/0.24	                    [0]                
0.00/0.24	                 >= [1]                
0.00/0.24	                    [0]                
0.00/0.24	                 =  [c_3(f^#(X, a(X)))]
0.00/0.24	                                       
0.00/0.24	
0.00/0.24	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.24	
0.00/0.24	We are left with following problem, upon which TcT provides the
0.00/0.24	certificate YES(O(1),O(1)).
0.00/0.24	
0.00/0.24	Strict DPs:
0.00/0.24	  { f^#(X, c(X)) -> c_2(f^#(s(X), X))
0.00/0.24	  , f^#(s(X), X) -> c_3(f^#(X, a(X))) }
0.00/0.24	Weak DPs: { f^#(X, X) -> c_1() }
0.00/0.24	Obligation:
0.00/0.24	  innermost runtime complexity
0.00/0.24	Answer:
0.00/0.24	  YES(O(1),O(1))
0.00/0.24	
0.00/0.24	We estimate the number of application of {2} by applications of
0.00/0.24	Pre({2}) = {1}. Here rules are labeled as follows:
0.00/0.24	
0.00/0.24	  DPs:
0.00/0.24	    { 1: f^#(X, c(X)) -> c_2(f^#(s(X), X))
0.00/0.24	    , 2: f^#(s(X), X) -> c_3(f^#(X, a(X)))
0.00/0.24	    , 3: f^#(X, X) -> c_1() }
0.00/0.24	
0.00/0.24	We are left with following problem, upon which TcT provides the
0.00/0.24	certificate YES(O(1),O(1)).
0.00/0.24	
0.00/0.24	Strict DPs: { f^#(X, c(X)) -> c_2(f^#(s(X), X)) }
0.00/0.24	Weak DPs:
0.00/0.24	  { f^#(X, X) -> c_1()
0.00/0.24	  , f^#(s(X), X) -> c_3(f^#(X, a(X))) }
0.00/0.24	Obligation:
0.00/0.24	  innermost runtime complexity
0.00/0.24	Answer:
0.00/0.24	  YES(O(1),O(1))
0.00/0.24	
0.00/0.24	We estimate the number of application of {1} by applications of
0.00/0.24	Pre({1}) = {}. Here rules are labeled as follows:
0.00/0.24	
0.00/0.24	  DPs:
0.00/0.24	    { 1: f^#(X, c(X)) -> c_2(f^#(s(X), X))
0.00/0.24	    , 2: f^#(X, X) -> c_1()
0.00/0.24	    , 3: f^#(s(X), X) -> c_3(f^#(X, a(X))) }
0.00/0.24	
0.00/0.24	We are left with following problem, upon which TcT provides the
0.00/0.24	certificate YES(O(1),O(1)).
0.00/0.24	
0.00/0.24	Weak DPs:
0.00/0.24	  { f^#(X, X) -> c_1()
0.00/0.24	  , f^#(X, c(X)) -> c_2(f^#(s(X), X))
0.00/0.24	  , f^#(s(X), X) -> c_3(f^#(X, a(X))) }
0.00/0.24	Obligation:
0.00/0.24	  innermost runtime complexity
0.00/0.24	Answer:
0.00/0.24	  YES(O(1),O(1))
0.00/0.24	
0.00/0.24	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.24	closed under successors. The DPs are removed.
0.00/0.24	
0.00/0.24	{ f^#(X, X) -> c_1()
0.00/0.24	, f^#(X, c(X)) -> c_2(f^#(s(X), X))
0.00/0.24	, f^#(s(X), X) -> c_3(f^#(X, a(X))) }
0.00/0.24	
0.00/0.24	We are left with following problem, upon which TcT provides the
0.00/0.24	certificate YES(O(1),O(1)).
0.00/0.24	
0.00/0.24	Rules: Empty
0.00/0.24	Obligation:
0.00/0.24	  innermost runtime complexity
0.00/0.24	Answer:
0.00/0.24	  YES(O(1),O(1))
0.00/0.24	
0.00/0.24	Empty rules are trivially bounded
0.00/0.24	
0.00/0.24	Hurray, we answered YES(O(1),O(1))
0.00/0.24	EOF