YES(O(1),O(n^1))
77.56/24.05	YES(O(1),O(n^1))
77.56/24.05	
77.56/24.05	We are left with following problem, upon which TcT provides the
77.56/24.05	certificate YES(O(1),O(n^1)).
77.56/24.05	
77.56/24.05	Strict Trs:
77.56/24.05	  { a__f(X, X) -> a__f(a(), b())
77.56/24.05	  , a__f(X1, X2) -> f(X1, X2)
77.56/24.05	  , a__b() -> a()
77.56/24.05	  , a__b() -> b()
77.56/24.05	  , mark(a()) -> a()
77.56/24.05	  , mark(b()) -> a__b()
77.56/24.05	  , mark(f(X1, X2)) -> a__f(mark(X1), X2) }
77.56/24.05	Obligation:
77.56/24.05	  runtime complexity
77.56/24.05	Answer:
77.56/24.05	  YES(O(1),O(n^1))
77.56/24.05	
77.56/24.05	The input is overlay and right-linear. Switching to innermost
77.56/24.05	rewriting.
77.56/24.05	
77.56/24.05	We are left with following problem, upon which TcT provides the
77.56/24.05	certificate YES(O(1),O(n^1)).
77.56/24.05	
77.56/24.05	Strict Trs:
77.56/24.05	  { a__f(X, X) -> a__f(a(), b())
77.56/24.05	  , a__f(X1, X2) -> f(X1, X2)
77.56/24.05	  , a__b() -> a()
77.56/24.05	  , a__b() -> b()
77.56/24.05	  , mark(a()) -> a()
77.56/24.05	  , mark(b()) -> a__b()
77.56/24.05	  , mark(f(X1, X2)) -> a__f(mark(X1), X2) }
77.56/24.05	Obligation:
77.56/24.05	  innermost runtime complexity
77.56/24.05	Answer:
77.56/24.05	  YES(O(1),O(n^1))
77.56/24.05	
77.56/24.05	We add the following dependency tuples:
77.56/24.05	
77.56/24.05	Strict DPs:
77.56/24.05	  { a__f^#(X, X) -> c_1(a__f^#(a(), b()))
77.56/24.05	  , a__f^#(X1, X2) -> c_2()
77.56/24.05	  , a__b^#() -> c_3()
77.56/24.05	  , a__b^#() -> c_4()
77.56/24.05	  , mark^#(a()) -> c_5()
77.56/24.05	  , mark^#(b()) -> c_6(a__b^#())
77.56/24.05	  , mark^#(f(X1, X2)) -> c_7(a__f^#(mark(X1), X2), mark^#(X1)) }
77.56/24.05	
77.56/24.05	and mark the set of starting terms.
77.56/24.05	
77.56/24.05	We are left with following problem, upon which TcT provides the
77.56/24.05	certificate YES(O(1),O(n^1)).
77.56/24.05	
77.56/24.05	Strict DPs:
77.56/24.05	  { a__f^#(X, X) -> c_1(a__f^#(a(), b()))
77.56/24.05	  , a__f^#(X1, X2) -> c_2()
77.56/24.05	  , a__b^#() -> c_3()
77.56/24.05	  , a__b^#() -> c_4()
77.56/24.05	  , mark^#(a()) -> c_5()
77.56/24.05	  , mark^#(b()) -> c_6(a__b^#())
77.56/24.05	  , mark^#(f(X1, X2)) -> c_7(a__f^#(mark(X1), X2), mark^#(X1)) }
77.56/24.05	Weak Trs:
77.56/24.05	  { a__f(X, X) -> a__f(a(), b())
77.56/24.05	  , a__f(X1, X2) -> f(X1, X2)
77.56/24.05	  , a__b() -> a()
77.56/24.05	  , a__b() -> b()
77.56/24.05	  , mark(a()) -> a()
77.56/24.05	  , mark(b()) -> a__b()
77.56/24.05	  , mark(f(X1, X2)) -> a__f(mark(X1), X2) }
77.56/24.05	Obligation:
77.56/24.05	  innermost runtime complexity
77.56/24.05	Answer:
77.56/24.05	  YES(O(1),O(n^1))
77.56/24.05	
77.56/24.05	We estimate the number of application of {2,3,4,5} by applications
77.56/24.05	of Pre({2,3,4,5}) = {1,6,7}. Here rules are labeled as follows:
77.56/24.05	
77.56/24.05	  DPs:
77.56/24.05	    { 1: a__f^#(X, X) -> c_1(a__f^#(a(), b()))
77.56/24.05	    , 2: a__f^#(X1, X2) -> c_2()
77.56/24.05	    , 3: a__b^#() -> c_3()
77.56/24.05	    , 4: a__b^#() -> c_4()
77.56/24.05	    , 5: mark^#(a()) -> c_5()
77.56/24.05	    , 6: mark^#(b()) -> c_6(a__b^#())
77.56/24.05	    , 7: mark^#(f(X1, X2)) -> c_7(a__f^#(mark(X1), X2), mark^#(X1)) }
77.56/24.05	
77.56/24.05	We are left with following problem, upon which TcT provides the
77.56/24.05	certificate YES(O(1),O(n^1)).
77.56/24.05	
77.56/24.05	Strict DPs:
77.56/24.05	  { a__f^#(X, X) -> c_1(a__f^#(a(), b()))
77.56/24.05	  , mark^#(b()) -> c_6(a__b^#())
77.56/24.05	  , mark^#(f(X1, X2)) -> c_7(a__f^#(mark(X1), X2), mark^#(X1)) }
77.56/24.05	Weak DPs:
77.56/24.05	  { a__f^#(X1, X2) -> c_2()
77.56/24.05	  , a__b^#() -> c_3()
77.56/24.05	  , a__b^#() -> c_4()
77.56/24.05	  , mark^#(a()) -> c_5() }
77.56/24.05	Weak Trs:
77.56/24.05	  { a__f(X, X) -> a__f(a(), b())
77.56/24.05	  , a__f(X1, X2) -> f(X1, X2)
77.56/24.05	  , a__b() -> a()
77.56/24.05	  , a__b() -> b()
77.56/24.05	  , mark(a()) -> a()
77.56/24.05	  , mark(b()) -> a__b()
77.56/24.05	  , mark(f(X1, X2)) -> a__f(mark(X1), X2) }
77.56/24.05	Obligation:
77.56/24.05	  innermost runtime complexity
77.56/24.05	Answer:
77.56/24.05	  YES(O(1),O(n^1))
77.56/24.05	
77.56/24.05	We estimate the number of application of {1,2} by applications of
77.56/24.05	Pre({1,2}) = {3}. Here rules are labeled as follows:
77.56/24.05	
77.56/24.05	  DPs:
77.56/24.05	    { 1: a__f^#(X, X) -> c_1(a__f^#(a(), b()))
77.56/24.05	    , 2: mark^#(b()) -> c_6(a__b^#())
77.56/24.05	    , 3: mark^#(f(X1, X2)) -> c_7(a__f^#(mark(X1), X2), mark^#(X1))
77.56/24.05	    , 4: a__f^#(X1, X2) -> c_2()
77.56/24.05	    , 5: a__b^#() -> c_3()
77.56/24.05	    , 6: a__b^#() -> c_4()
77.56/24.05	    , 7: mark^#(a()) -> c_5() }
77.56/24.05	
77.56/24.05	We are left with following problem, upon which TcT provides the
77.56/24.05	certificate YES(O(1),O(n^1)).
77.56/24.05	
77.56/24.05	Strict DPs:
77.56/24.05	  { mark^#(f(X1, X2)) -> c_7(a__f^#(mark(X1), X2), mark^#(X1)) }
77.56/24.05	Weak DPs:
77.56/24.05	  { a__f^#(X, X) -> c_1(a__f^#(a(), b()))
77.56/24.05	  , a__f^#(X1, X2) -> c_2()
77.56/24.05	  , a__b^#() -> c_3()
77.56/24.05	  , a__b^#() -> c_4()
77.56/24.05	  , mark^#(a()) -> c_5()
77.56/24.05	  , mark^#(b()) -> c_6(a__b^#()) }
77.56/24.05	Weak Trs:
77.56/24.05	  { a__f(X, X) -> a__f(a(), b())
77.56/24.05	  , a__f(X1, X2) -> f(X1, X2)
77.56/24.05	  , a__b() -> a()
77.56/24.05	  , a__b() -> b()
77.56/24.05	  , mark(a()) -> a()
77.56/24.05	  , mark(b()) -> a__b()
77.56/24.05	  , mark(f(X1, X2)) -> a__f(mark(X1), X2) }
77.56/24.05	Obligation:
77.56/24.05	  innermost runtime complexity
77.56/24.05	Answer:
77.56/24.05	  YES(O(1),O(n^1))
77.56/24.05	
77.56/24.05	The following weak DPs constitute a sub-graph of the DG that is
77.56/24.05	closed under successors. The DPs are removed.
77.56/24.05	
77.56/24.05	{ a__f^#(X, X) -> c_1(a__f^#(a(), b()))
77.56/24.05	, a__f^#(X1, X2) -> c_2()
77.56/24.05	, a__b^#() -> c_3()
77.56/24.05	, a__b^#() -> c_4()
77.56/24.05	, mark^#(a()) -> c_5()
77.56/24.05	, mark^#(b()) -> c_6(a__b^#()) }
77.56/24.05	
77.56/24.05	We are left with following problem, upon which TcT provides the
77.56/24.05	certificate YES(O(1),O(n^1)).
77.56/24.05	
77.56/24.05	Strict DPs:
77.56/24.05	  { mark^#(f(X1, X2)) -> c_7(a__f^#(mark(X1), X2), mark^#(X1)) }
77.56/24.05	Weak Trs:
77.56/24.05	  { a__f(X, X) -> a__f(a(), b())
77.56/24.05	  , a__f(X1, X2) -> f(X1, X2)
77.56/24.05	  , a__b() -> a()
77.56/24.05	  , a__b() -> b()
77.56/24.05	  , mark(a()) -> a()
77.56/24.05	  , mark(b()) -> a__b()
77.56/24.05	  , mark(f(X1, X2)) -> a__f(mark(X1), X2) }
77.56/24.05	Obligation:
77.56/24.05	  innermost runtime complexity
77.56/24.05	Answer:
77.56/24.05	  YES(O(1),O(n^1))
77.56/24.05	
77.56/24.05	Due to missing edges in the dependency-graph, the right-hand sides
77.56/24.05	of following rules could be simplified:
77.56/24.05	
77.56/24.05	  { mark^#(f(X1, X2)) -> c_7(a__f^#(mark(X1), X2), mark^#(X1)) }
77.56/24.05	
77.56/24.05	We are left with following problem, upon which TcT provides the
77.56/24.05	certificate YES(O(1),O(n^1)).
77.56/24.05	
77.56/24.05	Strict DPs: { mark^#(f(X1, X2)) -> c_1(mark^#(X1)) }
77.56/24.05	Weak Trs:
77.56/24.05	  { a__f(X, X) -> a__f(a(), b())
77.56/24.05	  , a__f(X1, X2) -> f(X1, X2)
77.56/24.05	  , a__b() -> a()
77.56/24.05	  , a__b() -> b()
77.56/24.05	  , mark(a()) -> a()
77.56/24.05	  , mark(b()) -> a__b()
77.56/24.05	  , mark(f(X1, X2)) -> a__f(mark(X1), X2) }
77.56/24.05	Obligation:
77.56/24.05	  innermost runtime complexity
77.56/24.05	Answer:
77.56/24.05	  YES(O(1),O(n^1))
77.56/24.05	
77.56/24.05	No rule is usable, rules are removed from the input problem.
77.56/24.05	
77.56/24.05	We are left with following problem, upon which TcT provides the
77.56/24.05	certificate YES(O(1),O(n^1)).
77.56/24.05	
77.56/24.05	Strict DPs: { mark^#(f(X1, X2)) -> c_1(mark^#(X1)) }
77.56/24.05	Obligation:
77.56/24.05	  innermost runtime complexity
77.56/24.05	Answer:
77.56/24.05	  YES(O(1),O(n^1))
77.56/24.05	
77.56/24.05	We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
77.56/24.05	to orient following rules strictly.
77.56/24.05	
77.56/24.05	DPs:
77.56/24.05	  { 1: mark^#(f(X1, X2)) -> c_1(mark^#(X1)) }
77.56/24.05	
77.56/24.05	Sub-proof:
77.56/24.05	----------
77.56/24.05	  The input was oriented with the instance of 'Small Polynomial Path
77.56/24.05	  Order (PS,1-bounded)' as induced by the safe mapping
77.56/24.05	  
77.56/24.05	   safe(f) = {1, 2}, safe(mark^#) = {}, safe(c_1) = {}
77.56/24.05	  
77.56/24.05	  and precedence
77.56/24.05	  
77.56/24.05	   empty .
77.56/24.05	  
77.56/24.05	  Following symbols are considered recursive:
77.56/24.05	  
77.56/24.05	   {mark^#}
77.56/24.05	  
77.56/24.05	  The recursion depth is 1.
77.56/24.05	  
77.56/24.05	  Further, following argument filtering is employed:
77.56/24.05	  
77.56/24.05	   pi(f) = [1, 2], pi(mark^#) = [1], pi(c_1) = [1]
77.56/24.05	  
77.56/24.05	  Usable defined function symbols are a subset of:
77.56/24.05	  
77.56/24.05	   {mark^#}
77.56/24.05	  
77.56/24.05	  For your convenience, here are the satisfied ordering constraints:
77.56/24.05	  
77.56/24.05	    pi(mark^#(f(X1, X2))) = mark^#(f(; X1,  X2);)
77.56/24.05	                          > c_1(mark^#(X1;);)    
77.56/24.05	                          = pi(c_1(mark^#(X1)))  
77.56/24.05	                                                 
77.56/24.05	
77.56/24.05	The strictly oriented rules are moved into the weak component.
77.56/24.05	
77.56/24.05	We are left with following problem, upon which TcT provides the
77.56/24.05	certificate YES(O(1),O(1)).
77.56/24.05	
77.56/24.05	Weak DPs: { mark^#(f(X1, X2)) -> c_1(mark^#(X1)) }
77.56/24.05	Obligation:
77.56/24.05	  innermost runtime complexity
77.56/24.05	Answer:
77.56/24.05	  YES(O(1),O(1))
77.56/24.05	
77.56/24.05	The following weak DPs constitute a sub-graph of the DG that is
77.56/24.05	closed under successors. The DPs are removed.
77.56/24.05	
77.56/24.05	{ mark^#(f(X1, X2)) -> c_1(mark^#(X1)) }
77.56/24.05	
77.56/24.05	We are left with following problem, upon which TcT provides the
77.56/24.05	certificate YES(O(1),O(1)).
77.56/24.05	
77.56/24.05	Rules: Empty
77.56/24.05	Obligation:
77.56/24.05	  innermost runtime complexity
77.56/24.05	Answer:
77.56/24.05	  YES(O(1),O(1))
77.56/24.05	
77.56/24.05	Empty rules are trivially bounded
77.56/24.05	
77.56/24.05	Hurray, we answered YES(O(1),O(n^1))
77.56/24.05	EOF