YES(O(1),O(1))
0.00/0.43	YES(O(1),O(1))
0.00/0.43	
0.00/0.43	We are left with following problem, upon which TcT provides the
0.00/0.43	certificate YES(O(1),O(1)).
0.00/0.43	
0.00/0.43	Strict Trs:
0.00/0.43	  { gcd(x, 0()) -> x
0.00/0.43	  , gcd(0(), y) -> y
0.00/0.43	  , gcd(s(x), s(y)) ->
0.00/0.43	    if(<(x, y), gcd(s(x), -(y, x)), gcd(-(x, y), s(y))) }
0.00/0.43	Obligation:
0.00/0.43	  innermost runtime complexity
0.00/0.43	Answer:
0.00/0.43	  YES(O(1),O(1))
0.00/0.43	
0.00/0.43	We add the following weak dependency pairs:
0.00/0.43	
0.00/0.43	Strict DPs:
0.00/0.43	  { gcd^#(x, 0()) -> c_1()
0.00/0.43	  , gcd^#(0(), y) -> c_2()
0.00/0.43	  , gcd^#(s(x), s(y)) ->
0.00/0.43	    c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }
0.00/0.43	
0.00/0.43	and mark the set of starting terms.
0.00/0.43	
0.00/0.43	We are left with following problem, upon which TcT provides the
0.00/0.43	certificate YES(O(1),O(1)).
0.00/0.43	
0.00/0.43	Strict DPs:
0.00/0.43	  { gcd^#(x, 0()) -> c_1()
0.00/0.43	  , gcd^#(0(), y) -> c_2()
0.00/0.43	  , gcd^#(s(x), s(y)) ->
0.00/0.43	    c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }
0.00/0.43	Strict Trs:
0.00/0.43	  { gcd(x, 0()) -> x
0.00/0.43	  , gcd(0(), y) -> y
0.00/0.43	  , gcd(s(x), s(y)) ->
0.00/0.43	    if(<(x, y), gcd(s(x), -(y, x)), gcd(-(x, y), s(y))) }
0.00/0.43	Obligation:
0.00/0.43	  innermost runtime complexity
0.00/0.43	Answer:
0.00/0.43	  YES(O(1),O(1))
0.00/0.43	
0.00/0.43	No rule is usable, rules are removed from the input problem.
0.00/0.43	
0.00/0.43	We are left with following problem, upon which TcT provides the
0.00/0.43	certificate YES(O(1),O(1)).
0.00/0.43	
0.00/0.43	Strict DPs:
0.00/0.43	  { gcd^#(x, 0()) -> c_1()
0.00/0.43	  , gcd^#(0(), y) -> c_2()
0.00/0.43	  , gcd^#(s(x), s(y)) ->
0.00/0.43	    c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }
0.00/0.43	Obligation:
0.00/0.43	  innermost runtime complexity
0.00/0.43	Answer:
0.00/0.43	  YES(O(1),O(1))
0.00/0.43	
0.00/0.43	The weightgap principle applies (using the following constant
0.00/0.43	growth matrix-interpretation)
0.00/0.43	
0.00/0.43	The following argument positions are usable:
0.00/0.43	  none
0.00/0.43	
0.00/0.43	TcT has computed the following constructor-restricted matrix
0.00/0.43	interpretation.
0.00/0.43	
0.00/0.43	              [0] = [0]           
0.00/0.43	                    [0]           
0.00/0.43	                                  
0.00/0.43	          [s](x1) = [0]           
0.00/0.43	                    [2]           
0.00/0.43	                                  
0.00/0.43	      [-](x1, x2) = [0]           
0.00/0.43	                    [0]           
0.00/0.43	                                  
0.00/0.43	  [gcd^#](x1, x2) = [0 2] x2 + [0]
0.00/0.43	                    [0 0]      [0]
0.00/0.43	                                  
0.00/0.43	            [c_1] = [1]           
0.00/0.43	                    [0]           
0.00/0.43	                                  
0.00/0.43	            [c_2] = [1]           
0.00/0.43	                    [0]           
0.00/0.43	                                  
0.00/0.43	    [c_3](x1, x2) = [1]           
0.00/0.43	                    [0]           
0.00/0.43	
0.00/0.43	The order satisfies the following ordering constraints:
0.00/0.43	
0.00/0.43	      [gcd^#(x, 0())] = [0]                                              
0.00/0.43	                        [0]                                              
0.00/0.43	                      ? [1]                                              
0.00/0.43	                        [0]                                              
0.00/0.43	                      = [c_1()]                                          
0.00/0.43	                                                                         
0.00/0.43	      [gcd^#(0(), y)] = [0 2] y + [0]                                    
0.00/0.43	                        [0 0]     [0]                                    
0.00/0.43	                      ? [1]                                              
0.00/0.43	                        [0]                                              
0.00/0.43	                      = [c_2()]                                          
0.00/0.43	                                                                         
0.00/0.43	  [gcd^#(s(x), s(y))] = [4]                                              
0.00/0.43	                        [0]                                              
0.00/0.43	                      > [1]                                              
0.00/0.43	                        [0]                                              
0.00/0.43	                      = [c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y)))]
0.00/0.43	                                                                         
0.00/0.43	
0.00/0.43	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.43	
0.00/0.43	We are left with following problem, upon which TcT provides the
0.00/0.43	certificate YES(O(1),O(1)).
0.00/0.43	
0.00/0.43	Strict DPs:
0.00/0.43	  { gcd^#(x, 0()) -> c_1()
0.00/0.43	  , gcd^#(0(), y) -> c_2() }
0.00/0.43	Weak DPs:
0.00/0.43	  { gcd^#(s(x), s(y)) ->
0.00/0.43	    c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }
0.00/0.43	Obligation:
0.00/0.43	  innermost runtime complexity
0.00/0.43	Answer:
0.00/0.43	  YES(O(1),O(1))
0.00/0.43	
0.00/0.43	We estimate the number of application of {1,2} by applications of
0.00/0.43	Pre({1,2}) = {}. Here rules are labeled as follows:
0.00/0.43	
0.00/0.43	  DPs:
0.00/0.43	    { 1: gcd^#(x, 0()) -> c_1()
0.00/0.43	    , 2: gcd^#(0(), y) -> c_2()
0.00/0.43	    , 3: gcd^#(s(x), s(y)) ->
0.00/0.43	         c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }
0.00/0.43	
0.00/0.43	We are left with following problem, upon which TcT provides the
0.00/0.43	certificate YES(O(1),O(1)).
0.00/0.43	
0.00/0.43	Weak DPs:
0.00/0.43	  { gcd^#(x, 0()) -> c_1()
0.00/0.43	  , gcd^#(0(), y) -> c_2()
0.00/0.43	  , gcd^#(s(x), s(y)) ->
0.00/0.43	    c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }
0.00/0.43	Obligation:
0.00/0.43	  innermost runtime complexity
0.00/0.43	Answer:
0.00/0.43	  YES(O(1),O(1))
0.00/0.43	
0.00/0.43	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.43	closed under successors. The DPs are removed.
0.00/0.43	
0.00/0.43	{ gcd^#(x, 0()) -> c_1()
0.00/0.43	, gcd^#(0(), y) -> c_2()
0.00/0.43	, gcd^#(s(x), s(y)) ->
0.00/0.43	  c_3(gcd^#(s(x), -(y, x)), gcd^#(-(x, y), s(y))) }
0.00/0.43	
0.00/0.43	We are left with following problem, upon which TcT provides the
0.00/0.43	certificate YES(O(1),O(1)).
0.00/0.43	
0.00/0.43	Rules: Empty
0.00/0.43	Obligation:
0.00/0.43	  innermost runtime complexity
0.00/0.43	Answer:
0.00/0.43	  YES(O(1),O(1))
0.00/0.43	
0.00/0.43	Empty rules are trivially bounded
0.00/0.43	
0.00/0.43	Hurray, we answered YES(O(1),O(1))
0.00/0.43	EOF