YES(O(1),O(n^1))
0.00/0.81	YES(O(1),O(n^1))
0.00/0.81	
0.00/0.81	We are left with following problem, upon which TcT provides the
0.00/0.81	certificate YES(O(1),O(n^1)).
0.00/0.81	
0.00/0.81	Strict Trs:
0.00/0.81	  { a__f(X) -> f(X)
0.00/0.81	  , a__f(f(a())) -> a__f(g(f(a())))
0.00/0.81	  , mark(f(X)) -> a__f(X)
0.00/0.81	  , mark(a()) -> a()
0.00/0.81	  , mark(g(X)) -> g(mark(X)) }
0.00/0.81	Obligation:
0.00/0.81	  runtime complexity
0.00/0.81	Answer:
0.00/0.81	  YES(O(1),O(n^1))
0.00/0.81	
0.00/0.81	The input is overlay and right-linear. Switching to innermost
0.00/0.81	rewriting.
0.00/0.81	
0.00/0.81	We are left with following problem, upon which TcT provides the
0.00/0.81	certificate YES(O(1),O(n^1)).
0.00/0.81	
0.00/0.81	Strict Trs:
0.00/0.81	  { a__f(X) -> f(X)
0.00/0.81	  , a__f(f(a())) -> a__f(g(f(a())))
0.00/0.81	  , mark(f(X)) -> a__f(X)
0.00/0.81	  , mark(a()) -> a()
0.00/0.81	  , mark(g(X)) -> g(mark(X)) }
0.00/0.81	Obligation:
0.00/0.81	  innermost runtime complexity
0.00/0.81	Answer:
0.00/0.81	  YES(O(1),O(n^1))
0.00/0.81	
0.00/0.81	We add the following weak dependency pairs:
0.00/0.81	
0.00/0.81	Strict DPs:
0.00/0.81	  { a__f^#(X) -> c_1()
0.00/0.81	  , a__f^#(f(a())) -> c_2(a__f^#(g(f(a()))))
0.00/0.81	  , mark^#(f(X)) -> c_3(a__f^#(X))
0.00/0.81	  , mark^#(a()) -> c_4()
0.00/0.81	  , mark^#(g(X)) -> c_5(mark^#(X)) }
0.00/0.81	
0.00/0.81	and mark the set of starting terms.
0.00/0.81	
0.00/0.81	We are left with following problem, upon which TcT provides the
0.00/0.81	certificate YES(O(1),O(n^1)).
0.00/0.81	
0.00/0.81	Strict DPs:
0.00/0.81	  { a__f^#(X) -> c_1()
0.00/0.81	  , a__f^#(f(a())) -> c_2(a__f^#(g(f(a()))))
0.00/0.81	  , mark^#(f(X)) -> c_3(a__f^#(X))
0.00/0.81	  , mark^#(a()) -> c_4()
0.00/0.81	  , mark^#(g(X)) -> c_5(mark^#(X)) }
0.00/0.81	Strict Trs:
0.00/0.81	  { a__f(X) -> f(X)
0.00/0.81	  , a__f(f(a())) -> a__f(g(f(a())))
0.00/0.81	  , mark(f(X)) -> a__f(X)
0.00/0.81	  , mark(a()) -> a()
0.00/0.81	  , mark(g(X)) -> g(mark(X)) }
0.00/0.81	Obligation:
0.00/0.81	  innermost runtime complexity
0.00/0.81	Answer:
0.00/0.81	  YES(O(1),O(n^1))
0.00/0.81	
0.00/0.81	No rule is usable, rules are removed from the input problem.
0.00/0.81	
0.00/0.81	We are left with following problem, upon which TcT provides the
0.00/0.81	certificate YES(O(1),O(n^1)).
0.00/0.81	
0.00/0.81	Strict DPs:
0.00/0.81	  { a__f^#(X) -> c_1()
0.00/0.81	  , a__f^#(f(a())) -> c_2(a__f^#(g(f(a()))))
0.00/0.81	  , mark^#(f(X)) -> c_3(a__f^#(X))
0.00/0.81	  , mark^#(a()) -> c_4()
0.00/0.81	  , mark^#(g(X)) -> c_5(mark^#(X)) }
0.00/0.81	Obligation:
0.00/0.81	  innermost runtime complexity
0.00/0.81	Answer:
0.00/0.81	  YES(O(1),O(n^1))
0.00/0.81	
0.00/0.81	The weightgap principle applies (using the following constant
0.00/0.81	growth matrix-interpretation)
0.00/0.81	
0.00/0.81	The following argument positions are usable:
0.00/0.81	  Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1}
0.00/0.81	
0.00/0.81	TcT has computed the following constructor-restricted matrix
0.00/0.81	interpretation.
0.00/0.81	
0.00/0.81	       [f](x1) = [0]           
0.00/0.81	                 [0]           
0.00/0.81	                               
0.00/0.81	           [a] = [0]           
0.00/0.81	                 [0]           
0.00/0.81	                               
0.00/0.81	       [g](x1) = [0]           
0.00/0.81	                 [0]           
0.00/0.81	                               
0.00/0.81	  [a__f^#](x1) = [0]           
0.00/0.81	                 [0]           
0.00/0.81	                               
0.00/0.81	         [c_1] = [1]           
0.00/0.81	                 [0]           
0.00/0.81	                               
0.00/0.81	     [c_2](x1) = [1 0] x1 + [1]
0.00/0.81	                 [0 1]      [0]
0.00/0.81	                               
0.00/0.81	  [mark^#](x1) = [1]           
0.00/0.81	                 [0]           
0.00/0.81	                               
0.00/0.81	     [c_3](x1) = [1 0] x1 + [0]
0.00/0.81	                 [0 1]      [0]
0.00/0.81	                               
0.00/0.81	         [c_4] = [0]           
0.00/0.81	                 [0]           
0.00/0.81	                               
0.00/0.81	     [c_5](x1) = [1 0] x1 + [1]
0.00/0.81	                 [0 1]      [0]
0.00/0.81	
0.00/0.81	The order satisfies the following ordering constraints:
0.00/0.81	
0.00/0.81	       [a__f^#(X)] = [0]                     
0.00/0.81	                     [0]                     
0.00/0.81	                   ? [1]                     
0.00/0.81	                     [0]                     
0.00/0.81	                   = [c_1()]                 
0.00/0.81	                                             
0.00/0.81	  [a__f^#(f(a()))] = [0]                     
0.00/0.81	                     [0]                     
0.00/0.81	                   ? [1]                     
0.00/0.81	                     [0]                     
0.00/0.81	                   = [c_2(a__f^#(g(f(a()))))]
0.00/0.81	                                             
0.00/0.81	    [mark^#(f(X))] = [1]                     
0.00/0.81	                     [0]                     
0.00/0.81	                   > [0]                     
0.00/0.81	                     [0]                     
0.00/0.81	                   = [c_3(a__f^#(X))]        
0.00/0.81	                                             
0.00/0.81	     [mark^#(a())] = [1]                     
0.00/0.81	                     [0]                     
0.00/0.81	                   > [0]                     
0.00/0.81	                     [0]                     
0.00/0.81	                   = [c_4()]                 
0.00/0.81	                                             
0.00/0.81	    [mark^#(g(X))] = [1]                     
0.00/0.81	                     [0]                     
0.00/0.81	                   ? [2]                     
0.00/0.81	                     [0]                     
0.00/0.81	                   = [c_5(mark^#(X))]        
0.00/0.81	                                             
0.00/0.81	
0.00/0.81	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.81	
0.00/0.81	We are left with following problem, upon which TcT provides the
0.00/0.81	certificate YES(O(1),O(n^1)).
0.00/0.81	
0.00/0.81	Strict DPs:
0.00/0.81	  { a__f^#(X) -> c_1()
0.00/0.81	  , a__f^#(f(a())) -> c_2(a__f^#(g(f(a()))))
0.00/0.81	  , mark^#(g(X)) -> c_5(mark^#(X)) }
0.00/0.81	Weak DPs:
0.00/0.81	  { mark^#(f(X)) -> c_3(a__f^#(X))
0.00/0.81	  , mark^#(a()) -> c_4() }
0.00/0.81	Obligation:
0.00/0.81	  innermost runtime complexity
0.00/0.81	Answer:
0.00/0.81	  YES(O(1),O(n^1))
0.00/0.81	
0.00/0.81	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.81	closed under successors. The DPs are removed.
0.00/0.81	
0.00/0.81	{ mark^#(a()) -> c_4() }
0.00/0.81	
0.00/0.81	We are left with following problem, upon which TcT provides the
0.00/0.81	certificate YES(O(1),O(n^1)).
0.00/0.81	
0.00/0.81	Strict DPs:
0.00/0.81	  { a__f^#(X) -> c_1()
0.00/0.81	  , a__f^#(f(a())) -> c_2(a__f^#(g(f(a()))))
0.00/0.81	  , mark^#(g(X)) -> c_5(mark^#(X)) }
0.00/0.81	Weak DPs: { mark^#(f(X)) -> c_3(a__f^#(X)) }
0.00/0.81	Obligation:
0.00/0.81	  innermost runtime complexity
0.00/0.81	Answer:
0.00/0.81	  YES(O(1),O(n^1))
0.00/0.81	
0.00/0.81	We use the processor 'matrix interpretation of dimension 1' to
0.00/0.81	orient following rules strictly.
0.00/0.81	
0.00/0.81	DPs:
0.00/0.81	  { 4: mark^#(f(X)) -> c_3(a__f^#(X)) }
0.00/0.81	
0.00/0.81	Sub-proof:
0.00/0.81	----------
0.00/0.81	  The following argument positions are usable:
0.00/0.81	    Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1}
0.00/0.81	  
0.00/0.81	  TcT has computed the following constructor-restricted matrix
0.00/0.81	  interpretation. Note that the diagonal of the component-wise maxima
0.00/0.81	  of interpretation-entries (of constructors) contains no more than 0
0.00/0.81	  non-zero entries.
0.00/0.81	  
0.00/0.81	         [f](x1) = [0]         
0.00/0.81	                               
0.00/0.81	             [a] = [7]         
0.00/0.81	                               
0.00/0.81	         [g](x1) = [0]         
0.00/0.81	                               
0.00/0.81	    [a__f^#](x1) = [0]         
0.00/0.81	                               
0.00/0.81	           [c_1] = [0]         
0.00/0.81	                               
0.00/0.81	       [c_2](x1) = [4] x1 + [0]
0.00/0.81	                               
0.00/0.81	    [mark^#](x1) = [1]         
0.00/0.81	                               
0.00/0.81	       [c_3](x1) = [1] x1 + [0]
0.00/0.81	                               
0.00/0.81	       [c_5](x1) = [1] x1 + [0]
0.00/0.81	  
0.00/0.81	  The order satisfies the following ordering constraints:
0.00/0.81	  
0.00/0.81	         [a__f^#(X)] =  [0]                     
0.00/0.81	                     >= [0]                     
0.00/0.81	                     =  [c_1()]                 
0.00/0.81	                                                
0.00/0.81	    [a__f^#(f(a()))] =  [0]                     
0.00/0.81	                     >= [0]                     
0.00/0.81	                     =  [c_2(a__f^#(g(f(a()))))]
0.00/0.81	                                                
0.00/0.81	      [mark^#(f(X))] =  [1]                     
0.00/0.81	                     >  [0]                     
0.00/0.81	                     =  [c_3(a__f^#(X))]        
0.00/0.81	                                                
0.00/0.81	      [mark^#(g(X))] =  [1]                     
0.00/0.81	                     >= [1]                     
0.00/0.81	                     =  [c_5(mark^#(X))]        
0.00/0.81	                                                
0.00/0.81	
0.00/0.81	We return to the main proof. Consider the set of all dependency
0.00/0.81	pairs
0.00/0.81	
0.00/0.81	:
0.00/0.81	  { 1: a__f^#(X) -> c_1()
0.00/0.81	  , 2: a__f^#(f(a())) -> c_2(a__f^#(g(f(a()))))
0.00/0.81	  , 3: mark^#(g(X)) -> c_5(mark^#(X))
0.00/0.81	  , 4: mark^#(f(X)) -> c_3(a__f^#(X)) }
0.00/0.81	
0.00/0.81	Processor 'matrix interpretation of dimension 1' induces the
0.00/0.81	complexity certificate YES(?,O(1)) on application of dependency
0.00/0.81	pairs {4}. These cover all (indirect) predecessors of dependency
0.00/0.81	pairs {1,2,4}, their number of application is equally bounded. The
0.00/0.81	dependency pairs are shifted into the weak component.
0.00/0.81	
0.00/0.81	We are left with following problem, upon which TcT provides the
0.00/0.81	certificate YES(O(1),O(n^1)).
0.00/0.81	
0.00/0.81	Strict DPs: { mark^#(g(X)) -> c_5(mark^#(X)) }
0.00/0.81	Weak DPs:
0.00/0.81	  { a__f^#(X) -> c_1()
0.00/0.81	  , a__f^#(f(a())) -> c_2(a__f^#(g(f(a()))))
0.00/0.81	  , mark^#(f(X)) -> c_3(a__f^#(X)) }
0.00/0.81	Obligation:
0.00/0.81	  innermost runtime complexity
0.00/0.81	Answer:
0.00/0.81	  YES(O(1),O(n^1))
0.00/0.81	
0.00/0.81	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.81	closed under successors. The DPs are removed.
0.00/0.81	
0.00/0.81	{ a__f^#(X) -> c_1()
0.00/0.81	, a__f^#(f(a())) -> c_2(a__f^#(g(f(a()))))
0.00/0.81	, mark^#(f(X)) -> c_3(a__f^#(X)) }
0.00/0.81	
0.00/0.81	We are left with following problem, upon which TcT provides the
0.00/0.81	certificate YES(O(1),O(n^1)).
0.00/0.81	
0.00/0.81	Strict DPs: { mark^#(g(X)) -> c_5(mark^#(X)) }
0.00/0.81	Obligation:
0.00/0.81	  innermost runtime complexity
0.00/0.81	Answer:
0.00/0.81	  YES(O(1),O(n^1))
0.00/0.81	
0.00/0.81	We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
0.00/0.81	to orient following rules strictly.
0.00/0.81	
0.00/0.81	DPs:
0.00/0.81	  { 1: mark^#(g(X)) -> c_5(mark^#(X)) }
0.00/0.81	
0.00/0.81	Sub-proof:
0.00/0.81	----------
0.00/0.81	  The input was oriented with the instance of 'Small Polynomial Path
0.00/0.81	  Order (PS,1-bounded)' as induced by the safe mapping
0.00/0.81	  
0.00/0.81	   safe(f) = {1}, safe(a) = {}, safe(g) = {1}, safe(a__f^#) = {},
0.00/0.81	   safe(c_1) = {}, safe(c_2) = {}, safe(mark^#) = {}, safe(c_3) = {},
0.00/0.81	   safe(c_5) = {}
0.00/0.81	  
0.00/0.81	  and precedence
0.00/0.81	  
0.00/0.81	   empty .
0.00/0.81	  
0.00/0.81	  Following symbols are considered recursive:
0.00/0.81	  
0.00/0.81	   {mark^#}
0.00/0.81	  
0.00/0.81	  The recursion depth is 1.
0.00/0.81	  
0.00/0.81	  Further, following argument filtering is employed:
0.00/0.81	  
0.00/0.81	   pi(f) = [], pi(a) = [], pi(g) = [1], pi(a__f^#) = [], pi(c_1) = [],
0.00/0.81	   pi(c_2) = [], pi(mark^#) = [1], pi(c_3) = [], pi(c_5) = [1]
0.00/0.81	  
0.00/0.81	  Usable defined function symbols are a subset of:
0.00/0.81	  
0.00/0.81	   {a__f^#, mark^#}
0.00/0.81	  
0.00/0.81	  For your convenience, here are the satisfied ordering constraints:
0.00/0.81	  
0.00/0.81	    pi(mark^#(g(X))) = mark^#(g(; X);)   
0.00/0.81	                     > c_5(mark^#(X;);)  
0.00/0.81	                     = pi(c_5(mark^#(X)))
0.00/0.81	                                         
0.00/0.81	
0.00/0.81	The strictly oriented rules are moved into the weak component.
0.00/0.81	
0.00/0.81	We are left with following problem, upon which TcT provides the
0.00/0.81	certificate YES(O(1),O(1)).
0.00/0.81	
0.00/0.81	Weak DPs: { mark^#(g(X)) -> c_5(mark^#(X)) }
0.00/0.81	Obligation:
0.00/0.81	  innermost runtime complexity
0.00/0.81	Answer:
0.00/0.81	  YES(O(1),O(1))
0.00/0.81	
0.00/0.81	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.81	closed under successors. The DPs are removed.
0.00/0.81	
0.00/0.81	{ mark^#(g(X)) -> c_5(mark^#(X)) }
0.00/0.81	
0.00/0.81	We are left with following problem, upon which TcT provides the
0.00/0.81	certificate YES(O(1),O(1)).
0.00/0.81	
0.00/0.81	Rules: Empty
0.00/0.81	Obligation:
0.00/0.81	  innermost runtime complexity
0.00/0.81	Answer:
0.00/0.81	  YES(O(1),O(1))
0.00/0.81	
0.00/0.81	Empty rules are trivially bounded
0.00/0.81	
0.00/0.81	Hurray, we answered YES(O(1),O(n^1))
0.00/0.81	EOF