YES(O(1),O(1))
0.00/0.16	YES(O(1),O(1))
0.00/0.16	
0.00/0.16	We are left with following problem, upon which TcT provides the
0.00/0.16	certificate YES(O(1),O(1)).
0.00/0.16	
0.00/0.16	Strict Trs:
0.00/0.16	  { g(x, y) -> x
0.00/0.16	  , g(x, y) -> y
0.00/0.16	  , f(s(x), y, y) -> f(y, x, s(x)) }
0.00/0.16	Obligation:
0.00/0.16	  innermost runtime complexity
0.00/0.16	Answer:
0.00/0.16	  YES(O(1),O(1))
0.00/0.16	
0.00/0.16	We add the following weak dependency pairs:
0.00/0.16	
0.00/0.16	Strict DPs:
0.00/0.16	  { g^#(x, y) -> c_1()
0.00/0.16	  , g^#(x, y) -> c_2()
0.00/0.16	  , f^#(s(x), y, y) -> c_3(f^#(y, x, s(x))) }
0.00/0.16	
0.00/0.16	and mark the set of starting terms.
0.00/0.16	
0.00/0.16	We are left with following problem, upon which TcT provides the
0.00/0.16	certificate YES(O(1),O(1)).
0.00/0.16	
0.00/0.16	Strict DPs:
0.00/0.16	  { g^#(x, y) -> c_1()
0.00/0.16	  , g^#(x, y) -> c_2()
0.00/0.16	  , f^#(s(x), y, y) -> c_3(f^#(y, x, s(x))) }
0.00/0.16	Strict Trs:
0.00/0.16	  { g(x, y) -> x
0.00/0.16	  , g(x, y) -> y
0.00/0.16	  , f(s(x), y, y) -> f(y, x, s(x)) }
0.00/0.16	Obligation:
0.00/0.16	  innermost runtime complexity
0.00/0.16	Answer:
0.00/0.16	  YES(O(1),O(1))
0.00/0.16	
0.00/0.16	No rule is usable, rules are removed from the input problem.
0.00/0.16	
0.00/0.16	We are left with following problem, upon which TcT provides the
0.00/0.16	certificate YES(O(1),O(1)).
0.00/0.16	
0.00/0.16	Strict DPs:
0.00/0.16	  { g^#(x, y) -> c_1()
0.00/0.16	  , g^#(x, y) -> c_2()
0.00/0.16	  , f^#(s(x), y, y) -> c_3(f^#(y, x, s(x))) }
0.00/0.16	Obligation:
0.00/0.16	  innermost runtime complexity
0.00/0.16	Answer:
0.00/0.16	  YES(O(1),O(1))
0.00/0.16	
0.00/0.16	The weightgap principle applies (using the following constant
0.00/0.16	growth matrix-interpretation)
0.00/0.16	
0.00/0.16	The following argument positions are usable:
0.00/0.16	  none
0.00/0.16	
0.00/0.16	TcT has computed the following constructor-restricted matrix
0.00/0.16	interpretation.
0.00/0.16	
0.00/0.16	            [s](x1) = [0]
0.00/0.16	                      [0]
0.00/0.16	                         
0.00/0.16	      [g^#](x1, x2) = [1]
0.00/0.16	                      [1]
0.00/0.16	                         
0.00/0.16	              [c_1] = [0]
0.00/0.16	                      [1]
0.00/0.16	                         
0.00/0.16	              [c_2] = [0]
0.00/0.16	                      [1]
0.00/0.16	                         
0.00/0.16	  [f^#](x1, x2, x3) = [1]
0.00/0.16	                      [0]
0.00/0.16	                         
0.00/0.16	          [c_3](x1) = [2]
0.00/0.16	                      [2]
0.00/0.16	
0.00/0.16	The order satisfies the following ordering constraints:
0.00/0.16	
0.00/0.16	        [g^#(x, y)] = [1]                   
0.00/0.16	                      [1]                   
0.00/0.16	                    > [0]                   
0.00/0.16	                      [1]                   
0.00/0.16	                    = [c_1()]               
0.00/0.16	                                            
0.00/0.16	        [g^#(x, y)] = [1]                   
0.00/0.16	                      [1]                   
0.00/0.16	                    > [0]                   
0.00/0.16	                      [1]                   
0.00/0.16	                    = [c_2()]               
0.00/0.16	                                            
0.00/0.16	  [f^#(s(x), y, y)] = [1]                   
0.00/0.16	                      [0]                   
0.00/0.16	                    ? [2]                   
0.00/0.16	                      [2]                   
0.00/0.16	                    = [c_3(f^#(y, x, s(x)))]
0.00/0.16	                                            
0.00/0.16	
0.00/0.16	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.16	
0.00/0.16	We are left with following problem, upon which TcT provides the
0.00/0.16	certificate YES(O(1),O(1)).
0.00/0.16	
0.00/0.16	Strict DPs: { f^#(s(x), y, y) -> c_3(f^#(y, x, s(x))) }
0.00/0.16	Weak DPs:
0.00/0.16	  { g^#(x, y) -> c_1()
0.00/0.16	  , g^#(x, y) -> c_2() }
0.00/0.16	Obligation:
0.00/0.16	  innermost runtime complexity
0.00/0.16	Answer:
0.00/0.16	  YES(O(1),O(1))
0.00/0.16	
0.00/0.16	We estimate the number of application of {1} by applications of
0.00/0.16	Pre({1}) = {}. Here rules are labeled as follows:
0.00/0.16	
0.00/0.16	  DPs:
0.00/0.16	    { 1: f^#(s(x), y, y) -> c_3(f^#(y, x, s(x)))
0.00/0.16	    , 2: g^#(x, y) -> c_1()
0.00/0.16	    , 3: g^#(x, y) -> c_2() }
0.00/0.16	
0.00/0.16	We are left with following problem, upon which TcT provides the
0.00/0.16	certificate YES(O(1),O(1)).
0.00/0.16	
0.00/0.16	Weak DPs:
0.00/0.16	  { g^#(x, y) -> c_1()
0.00/0.16	  , g^#(x, y) -> c_2()
0.00/0.16	  , f^#(s(x), y, y) -> c_3(f^#(y, x, s(x))) }
0.00/0.16	Obligation:
0.00/0.16	  innermost runtime complexity
0.00/0.16	Answer:
0.00/0.16	  YES(O(1),O(1))
0.00/0.16	
0.00/0.16	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.16	closed under successors. The DPs are removed.
0.00/0.16	
0.00/0.16	{ g^#(x, y) -> c_1()
0.00/0.16	, g^#(x, y) -> c_2()
0.00/0.16	, f^#(s(x), y, y) -> c_3(f^#(y, x, s(x))) }
0.00/0.16	
0.00/0.16	We are left with following problem, upon which TcT provides the
0.00/0.16	certificate YES(O(1),O(1)).
0.00/0.16	
0.00/0.16	Rules: Empty
0.00/0.16	Obligation:
0.00/0.16	  innermost runtime complexity
0.00/0.16	Answer:
0.00/0.16	  YES(O(1),O(1))
0.00/0.16	
0.00/0.16	Empty rules are trivially bounded
0.00/0.16	
0.00/0.16	Hurray, we answered YES(O(1),O(1))
0.00/0.16	EOF