YES(O(1),O(1))
12.22/4.86	YES(O(1),O(1))
12.22/4.86	
12.22/4.86	We are left with following problem, upon which TcT provides the
12.22/4.86	certificate YES(O(1),O(1)).
12.22/4.86	
12.22/4.86	Strict Trs:
12.22/4.86	  { g(X) -> u(h(X), h(X), X)
12.22/4.86	  , u(d(), c(Y), X) -> k(Y)
12.22/4.86	  , h(d()) -> c(a())
12.22/4.86	  , h(d()) -> c(b())
12.22/4.86	  , f(k(a()), k(b()), X) -> f(X, X, X) }
12.22/4.86	Obligation:
12.22/4.86	  innermost runtime complexity
12.22/4.86	Answer:
12.22/4.86	  YES(O(1),O(1))
12.22/4.86	
12.22/4.86	We add the following weak dependency pairs:
12.22/4.86	
12.22/4.86	Strict DPs:
12.22/4.86	  { g^#(X) -> c_1(u^#(h(X), h(X), X))
12.22/4.86	  , u^#(d(), c(Y), X) -> c_2()
12.22/4.86	  , h^#(d()) -> c_3()
12.22/4.86	  , h^#(d()) -> c_4()
12.22/4.86	  , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
12.22/4.86	
12.22/4.86	and mark the set of starting terms.
12.22/4.86	
12.22/4.86	We are left with following problem, upon which TcT provides the
12.22/4.86	certificate YES(O(1),O(1)).
12.22/4.86	
12.22/4.86	Strict DPs:
12.22/4.86	  { g^#(X) -> c_1(u^#(h(X), h(X), X))
12.22/4.86	  , u^#(d(), c(Y), X) -> c_2()
12.22/4.86	  , h^#(d()) -> c_3()
12.22/4.86	  , h^#(d()) -> c_4()
12.22/4.86	  , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
12.22/4.86	Strict Trs:
12.22/4.86	  { g(X) -> u(h(X), h(X), X)
12.22/4.86	  , u(d(), c(Y), X) -> k(Y)
12.22/4.86	  , h(d()) -> c(a())
12.22/4.86	  , h(d()) -> c(b())
12.22/4.86	  , f(k(a()), k(b()), X) -> f(X, X, X) }
12.22/4.86	Obligation:
12.22/4.86	  innermost runtime complexity
12.22/4.86	Answer:
12.22/4.86	  YES(O(1),O(1))
12.22/4.86	
12.22/4.86	We replace rewrite rules by usable rules:
12.22/4.86	
12.22/4.86	  Strict Usable Rules:
12.22/4.86	    { h(d()) -> c(a())
12.22/4.86	    , h(d()) -> c(b()) }
12.22/4.86	
12.22/4.86	We are left with following problem, upon which TcT provides the
12.22/4.86	certificate YES(O(1),O(1)).
12.22/4.86	
12.22/4.86	Strict DPs:
12.22/4.86	  { g^#(X) -> c_1(u^#(h(X), h(X), X))
12.22/4.86	  , u^#(d(), c(Y), X) -> c_2()
12.22/4.86	  , h^#(d()) -> c_3()
12.22/4.86	  , h^#(d()) -> c_4()
12.22/4.86	  , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
12.22/4.86	Strict Trs:
12.22/4.86	  { h(d()) -> c(a())
12.22/4.86	  , h(d()) -> c(b()) }
12.22/4.86	Obligation:
12.22/4.86	  innermost runtime complexity
12.22/4.86	Answer:
12.22/4.86	  YES(O(1),O(1))
12.22/4.86	
12.22/4.86	The weightgap principle applies (using the following constant
12.22/4.86	growth matrix-interpretation)
12.22/4.86	
12.22/4.86	The following argument positions are usable:
12.22/4.86	  Uargs(c_1) = {1}, Uargs(u^#) = {1, 2}
12.22/4.86	
12.22/4.86	TcT has computed the following constructor-restricted matrix
12.22/4.86	interpretation.
12.22/4.86	
12.22/4.86	            [h](x1) = [2]                                 
12.22/4.86	                      [0]                                 
12.22/4.86	                                                          
12.22/4.86	                [d] = [0]                                 
12.22/4.86	                      [0]                                 
12.22/4.86	                                                          
12.22/4.86	            [c](x1) = [0]                                 
12.22/4.86	                      [0]                                 
12.22/4.86	                                                          
12.22/4.86	            [k](x1) = [0]                                 
12.22/4.86	                      [0]                                 
12.22/4.86	                                                          
12.22/4.86	                [a] = [0]                                 
12.22/4.86	                      [0]                                 
12.22/4.86	                                                          
12.22/4.86	                [b] = [0]                                 
12.22/4.86	                      [0]                                 
12.22/4.86	                                                          
12.22/4.86	          [g^#](x1) = [2 1] x1 + [1]                      
12.22/4.86	                      [1 1]      [1]                      
12.22/4.86	                                                          
12.22/4.86	          [c_1](x1) = [1 0] x1 + [0]                      
12.22/4.86	                      [0 1]      [2]                      
12.22/4.86	                                                          
12.22/4.86	  [u^#](x1, x2, x3) = [1 0] x1 + [2 0] x2 + [0 0] x3 + [1]
12.22/4.86	                      [1 1]      [0 0]      [1 1]      [2]
12.22/4.86	                                                          
12.22/4.86	              [c_2] = [1]                                 
12.22/4.86	                      [1]                                 
12.22/4.86	                                                          
12.22/4.86	          [h^#](x1) = [1 1] x1 + [2]                      
12.22/4.86	                      [1 1]      [2]                      
12.22/4.86	                                                          
12.22/4.86	              [c_3] = [1]                                 
12.22/4.86	                      [1]                                 
12.22/4.86	                                                          
12.22/4.86	              [c_4] = [1]                                 
12.22/4.86	                      [2]                                 
12.22/4.86	                                                          
12.22/4.86	  [f^#](x1, x2, x3) = [1]                                 
12.22/4.86	                      [1]                                 
12.22/4.86	                                                          
12.22/4.86	          [c_5](x1) = [2]                                 
12.22/4.86	                      [2]                                 
12.22/4.86	
12.22/4.86	The order satisfies the following ordering constraints:
12.22/4.86	
12.22/4.86	                  [h(d())] =  [2]                      
12.22/4.86	                              [0]                      
12.22/4.86	                           >  [0]                      
12.22/4.86	                              [0]                      
12.22/4.86	                           =  [c(a())]                 
12.22/4.86	                                                       
12.22/4.86	                  [h(d())] =  [2]                      
12.22/4.86	                              [0]                      
12.22/4.86	                           >  [0]                      
12.22/4.86	                              [0]                      
12.22/4.86	                           =  [c(b())]                 
12.22/4.86	                                                       
12.22/4.86	                  [g^#(X)] =  [2 1] X + [1]            
12.22/4.86	                              [1 1]     [1]            
12.22/4.86	                           ?  [0 0] X + [7]            
12.22/4.86	                              [1 1]     [6]            
12.22/4.86	                           =  [c_1(u^#(h(X), h(X), X))]
12.22/4.86	                                                       
12.22/4.86	       [u^#(d(), c(Y), X)] =  [0 0] X + [1]            
12.22/4.86	                              [1 1]     [2]            
12.22/4.86	                           >= [1]                      
12.22/4.86	                              [1]                      
12.22/4.86	                           =  [c_2()]                  
12.22/4.86	                                                       
12.22/4.86	                [h^#(d())] =  [2]                      
12.22/4.86	                              [2]                      
12.22/4.86	                           >  [1]                      
12.22/4.86	                              [1]                      
12.22/4.86	                           =  [c_3()]                  
12.22/4.86	                                                       
12.22/4.86	                [h^#(d())] =  [2]                      
12.22/4.86	                              [2]                      
12.22/4.86	                           >  [1]                      
12.22/4.86	                              [2]                      
12.22/4.86	                           =  [c_4()]                  
12.22/4.86	                                                       
12.22/4.86	  [f^#(k(a()), k(b()), X)] =  [1]                      
12.22/4.86	                              [1]                      
12.22/4.86	                           ?  [2]                      
12.22/4.86	                              [2]                      
12.22/4.86	                           =  [c_5(f^#(X, X, X))]      
12.22/4.86	                                                       
12.22/4.86	
12.22/4.86	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
12.22/4.86	
12.22/4.86	We are left with following problem, upon which TcT provides the
12.22/4.86	certificate YES(O(1),O(1)).
12.22/4.86	
12.22/4.86	Strict DPs:
12.22/4.86	  { g^#(X) -> c_1(u^#(h(X), h(X), X))
12.22/4.86	  , u^#(d(), c(Y), X) -> c_2()
12.22/4.86	  , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
12.22/4.86	Weak DPs:
12.22/4.86	  { h^#(d()) -> c_3()
12.22/4.86	  , h^#(d()) -> c_4() }
12.22/4.86	Weak Trs:
12.22/4.86	  { h(d()) -> c(a())
12.22/4.86	  , h(d()) -> c(b()) }
12.22/4.86	Obligation:
12.22/4.86	  innermost runtime complexity
12.22/4.86	Answer:
12.22/4.86	  YES(O(1),O(1))
12.22/4.86	
12.22/4.86	We estimate the number of application of {1,2,3} by applications of
12.22/4.86	Pre({1,2,3}) = {}. Here rules are labeled as follows:
12.22/4.86	
12.22/4.86	  DPs:
12.22/4.86	    { 1: g^#(X) -> c_1(u^#(h(X), h(X), X))
12.22/4.86	    , 2: u^#(d(), c(Y), X) -> c_2()
12.22/4.86	    , 3: f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X))
12.22/4.86	    , 4: h^#(d()) -> c_3()
12.22/4.86	    , 5: h^#(d()) -> c_4() }
12.22/4.86	
12.22/4.86	We are left with following problem, upon which TcT provides the
12.22/4.86	certificate YES(O(1),O(1)).
12.22/4.86	
12.22/4.86	Weak DPs:
12.22/4.86	  { g^#(X) -> c_1(u^#(h(X), h(X), X))
12.22/4.86	  , u^#(d(), c(Y), X) -> c_2()
12.22/4.86	  , h^#(d()) -> c_3()
12.22/4.86	  , h^#(d()) -> c_4()
12.22/4.86	  , f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
12.22/4.86	Weak Trs:
12.22/4.86	  { h(d()) -> c(a())
12.22/4.86	  , h(d()) -> c(b()) }
12.22/4.86	Obligation:
12.22/4.86	  innermost runtime complexity
12.22/4.86	Answer:
12.22/4.86	  YES(O(1),O(1))
12.22/4.86	
12.22/4.86	The following weak DPs constitute a sub-graph of the DG that is
12.22/4.86	closed under successors. The DPs are removed.
12.22/4.86	
12.22/4.86	{ g^#(X) -> c_1(u^#(h(X), h(X), X))
12.22/4.86	, u^#(d(), c(Y), X) -> c_2()
12.22/4.86	, h^#(d()) -> c_3()
12.22/4.86	, h^#(d()) -> c_4()
12.22/4.86	, f^#(k(a()), k(b()), X) -> c_5(f^#(X, X, X)) }
12.22/4.86	
12.22/4.86	We are left with following problem, upon which TcT provides the
12.22/4.86	certificate YES(O(1),O(1)).
12.22/4.86	
12.22/4.86	Weak Trs:
12.22/4.86	  { h(d()) -> c(a())
12.22/4.86	  , h(d()) -> c(b()) }
12.22/4.86	Obligation:
12.22/4.86	  innermost runtime complexity
12.22/4.86	Answer:
12.22/4.86	  YES(O(1),O(1))
12.22/4.86	
12.22/4.86	No rule is usable, rules are removed from the input problem.
12.22/4.86	
12.22/4.86	We are left with following problem, upon which TcT provides the
12.22/4.86	certificate YES(O(1),O(1)).
12.22/4.86	
12.22/4.86	Rules: Empty
12.22/4.86	Obligation:
12.22/4.86	  innermost runtime complexity
12.22/4.86	Answer:
12.22/4.86	  YES(O(1),O(1))
12.22/4.86	
12.22/4.86	Empty rules are trivially bounded
12.22/4.86	
12.22/4.86	Hurray, we answered YES(O(1),O(1))
12.22/4.86	EOF