YES(O(1),O(1))
0.00/0.78	YES(O(1),O(1))
0.00/0.78	
0.00/0.78	We are left with following problem, upon which TcT provides the
0.00/0.78	certificate YES(O(1),O(1)).
0.00/0.78	
0.00/0.78	Strict Trs:
0.00/0.78	  { p(f(f(x))) -> q(f(g(x)))
0.00/0.78	  , p(g(g(x))) -> q(g(f(x)))
0.00/0.78	  , q(f(f(x))) -> p(f(g(x)))
0.00/0.78	  , q(g(g(x))) -> p(g(f(x))) }
0.00/0.78	Obligation:
0.00/0.78	  innermost runtime complexity
0.00/0.78	Answer:
0.00/0.78	  YES(O(1),O(1))
0.00/0.78	
0.00/0.78	We add the following weak dependency pairs:
0.00/0.78	
0.00/0.78	Strict DPs:
0.00/0.78	  { p^#(f(f(x))) -> c_1(q^#(f(g(x))))
0.00/0.78	  , p^#(g(g(x))) -> c_2(q^#(g(f(x))))
0.00/0.78	  , q^#(f(f(x))) -> c_3(p^#(f(g(x))))
0.00/0.78	  , q^#(g(g(x))) -> c_4(p^#(g(f(x)))) }
0.00/0.78	
0.00/0.78	and mark the set of starting terms.
0.00/0.78	
0.00/0.78	We are left with following problem, upon which TcT provides the
0.00/0.78	certificate YES(O(1),O(1)).
0.00/0.78	
0.00/0.78	Strict DPs:
0.00/0.78	  { p^#(f(f(x))) -> c_1(q^#(f(g(x))))
0.00/0.78	  , p^#(g(g(x))) -> c_2(q^#(g(f(x))))
0.00/0.78	  , q^#(f(f(x))) -> c_3(p^#(f(g(x))))
0.00/0.78	  , q^#(g(g(x))) -> c_4(p^#(g(f(x)))) }
0.00/0.78	Strict Trs:
0.00/0.78	  { p(f(f(x))) -> q(f(g(x)))
0.00/0.78	  , p(g(g(x))) -> q(g(f(x)))
0.00/0.78	  , q(f(f(x))) -> p(f(g(x)))
0.00/0.78	  , q(g(g(x))) -> p(g(f(x))) }
0.00/0.78	Obligation:
0.00/0.78	  innermost runtime complexity
0.00/0.78	Answer:
0.00/0.78	  YES(O(1),O(1))
0.00/0.78	
0.00/0.78	No rule is usable, rules are removed from the input problem.
0.00/0.78	
0.00/0.78	We are left with following problem, upon which TcT provides the
0.00/0.78	certificate YES(O(1),O(1)).
0.00/0.78	
0.00/0.78	Strict DPs:
0.00/0.78	  { p^#(f(f(x))) -> c_1(q^#(f(g(x))))
0.00/0.78	  , p^#(g(g(x))) -> c_2(q^#(g(f(x))))
0.00/0.78	  , q^#(f(f(x))) -> c_3(p^#(f(g(x))))
0.00/0.78	  , q^#(g(g(x))) -> c_4(p^#(g(f(x)))) }
0.00/0.78	Obligation:
0.00/0.78	  innermost runtime complexity
0.00/0.78	Answer:
0.00/0.78	  YES(O(1),O(1))
0.00/0.78	
0.00/0.78	The weightgap principle applies (using the following constant
0.00/0.78	growth matrix-interpretation)
0.00/0.78	
0.00/0.78	The following argument positions are usable:
0.00/0.78	  none
0.00/0.78	
0.00/0.78	TcT has computed the following constructor-restricted matrix
0.00/0.78	interpretation.
0.00/0.78	
0.00/0.78	    [f](x1) = [2]           
0.00/0.78	              [2]           
0.00/0.78	                            
0.00/0.78	    [g](x1) = [0]           
0.00/0.78	              [0]           
0.00/0.78	                            
0.00/0.78	  [p^#](x1) = [0 2] x1 + [0]
0.00/0.78	              [2 0]      [0]
0.00/0.78	                            
0.00/0.78	  [c_1](x1) = [1]           
0.00/0.78	              [1]           
0.00/0.78	                            
0.00/0.78	  [q^#](x1) = [0]           
0.00/0.78	              [0]           
0.00/0.78	                            
0.00/0.78	  [c_2](x1) = [1]           
0.00/0.78	              [1]           
0.00/0.78	                            
0.00/0.78	  [c_3](x1) = [1]           
0.00/0.78	              [0]           
0.00/0.78	                            
0.00/0.78	  [c_4](x1) = [1]           
0.00/0.78	              [0]           
0.00/0.78	
0.00/0.78	The order satisfies the following ordering constraints:
0.00/0.78	
0.00/0.78	  [p^#(f(f(x)))] = [4]                
0.00/0.78	                   [4]                
0.00/0.78	                 > [1]                
0.00/0.78	                   [1]                
0.00/0.78	                 = [c_1(q^#(f(g(x))))]
0.00/0.78	                                      
0.00/0.78	  [p^#(g(g(x)))] = [0]                
0.00/0.78	                   [0]                
0.00/0.78	                 ? [1]                
0.00/0.78	                   [1]                
0.00/0.78	                 = [c_2(q^#(g(f(x))))]
0.00/0.78	                                      
0.00/0.78	  [q^#(f(f(x)))] = [0]                
0.00/0.78	                   [0]                
0.00/0.78	                 ? [1]                
0.00/0.78	                   [0]                
0.00/0.78	                 = [c_3(p^#(f(g(x))))]
0.00/0.78	                                      
0.00/0.78	  [q^#(g(g(x)))] = [0]                
0.00/0.78	                   [0]                
0.00/0.78	                 ? [1]                
0.00/0.78	                   [0]                
0.00/0.78	                 = [c_4(p^#(g(f(x))))]
0.00/0.78	                                      
0.00/0.78	
0.00/0.78	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.78	
0.00/0.78	We are left with following problem, upon which TcT provides the
0.00/0.78	certificate YES(O(1),O(1)).
0.00/0.78	
0.00/0.78	Strict DPs:
0.00/0.78	  { p^#(g(g(x))) -> c_2(q^#(g(f(x))))
0.00/0.78	  , q^#(f(f(x))) -> c_3(p^#(f(g(x))))
0.00/0.78	  , q^#(g(g(x))) -> c_4(p^#(g(f(x)))) }
0.00/0.78	Weak DPs: { p^#(f(f(x))) -> c_1(q^#(f(g(x)))) }
0.00/0.78	Obligation:
0.00/0.78	  innermost runtime complexity
0.00/0.78	Answer:
0.00/0.78	  YES(O(1),O(1))
0.00/0.78	
0.00/0.78	We estimate the number of application of {1,2,3} by applications of
0.00/0.78	Pre({1,2,3}) = {}. Here rules are labeled as follows:
0.00/0.78	
0.00/0.78	  DPs:
0.00/0.78	    { 1: p^#(g(g(x))) -> c_2(q^#(g(f(x))))
0.00/0.78	    , 2: q^#(f(f(x))) -> c_3(p^#(f(g(x))))
0.00/0.78	    , 3: q^#(g(g(x))) -> c_4(p^#(g(f(x))))
0.00/0.78	    , 4: p^#(f(f(x))) -> c_1(q^#(f(g(x)))) }
0.00/0.78	
0.00/0.78	We are left with following problem, upon which TcT provides the
0.00/0.78	certificate YES(O(1),O(1)).
0.00/0.78	
0.00/0.78	Weak DPs:
0.00/0.78	  { p^#(f(f(x))) -> c_1(q^#(f(g(x))))
0.00/0.78	  , p^#(g(g(x))) -> c_2(q^#(g(f(x))))
0.00/0.78	  , q^#(f(f(x))) -> c_3(p^#(f(g(x))))
0.00/0.78	  , q^#(g(g(x))) -> c_4(p^#(g(f(x)))) }
0.00/0.78	Obligation:
0.00/0.78	  innermost runtime complexity
0.00/0.78	Answer:
0.00/0.78	  YES(O(1),O(1))
0.00/0.78	
0.00/0.78	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.78	closed under successors. The DPs are removed.
0.00/0.78	
0.00/0.78	{ p^#(f(f(x))) -> c_1(q^#(f(g(x))))
0.00/0.78	, p^#(g(g(x))) -> c_2(q^#(g(f(x))))
0.00/0.78	, q^#(f(f(x))) -> c_3(p^#(f(g(x))))
0.00/0.78	, q^#(g(g(x))) -> c_4(p^#(g(f(x)))) }
0.00/0.78	
0.00/0.78	We are left with following problem, upon which TcT provides the
0.00/0.78	certificate YES(O(1),O(1)).
0.00/0.78	
0.00/0.78	Rules: Empty
0.00/0.78	Obligation:
0.00/0.78	  innermost runtime complexity
0.00/0.78	Answer:
0.00/0.78	  YES(O(1),O(1))
0.00/0.78	
0.00/0.78	Empty rules are trivially bounded
0.00/0.78	
0.00/0.78	Hurray, we answered YES(O(1),O(1))
0.00/0.79	EOF