YES(O(1),O(n^1))
11.88/5.69	YES(O(1),O(n^1))
11.88/5.69	
11.88/5.69	We are left with following problem, upon which TcT provides the
11.88/5.69	certificate YES(O(1),O(n^1)).
11.88/5.69	
11.88/5.69	Strict Trs:
11.88/5.69	  { a__f(X1, X2, X3) -> f(X1, X2, X3)
11.88/5.69	  , a__f(X, g(X), Y) -> a__f(Y, Y, Y)
11.88/5.69	  , a__g(X) -> g(X)
11.88/5.69	  , a__g(b()) -> c()
11.88/5.69	  , a__b() -> b()
11.88/5.69	  , a__b() -> c()
11.88/5.69	  , mark(g(X)) -> a__g(mark(X))
11.88/5.69	  , mark(b()) -> a__b()
11.88/5.69	  , mark(c()) -> c()
11.88/5.69	  , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) }
11.88/5.69	Obligation:
11.88/5.69	  innermost runtime complexity
11.88/5.69	Answer:
11.88/5.69	  YES(O(1),O(n^1))
11.88/5.69	
11.88/5.69	We add the following dependency tuples:
11.88/5.69	
11.88/5.69	Strict DPs:
11.88/5.69	  { a__f^#(X1, X2, X3) -> c_1()
11.88/5.69	  , a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y))
11.88/5.69	  , a__g^#(X) -> c_3()
11.88/5.69	  , a__g^#(b()) -> c_4()
11.88/5.69	  , a__b^#() -> c_5()
11.88/5.69	  , a__b^#() -> c_6()
11.88/5.69	  , mark^#(g(X)) -> c_7(a__g^#(mark(X)), mark^#(X))
11.88/5.69	  , mark^#(b()) -> c_8(a__b^#())
11.88/5.69	  , mark^#(c()) -> c_9()
11.88/5.69	  , mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3)) }
11.88/5.69	
11.88/5.69	and mark the set of starting terms.
11.88/5.69	
11.88/5.69	We are left with following problem, upon which TcT provides the
11.88/5.69	certificate YES(O(1),O(n^1)).
11.88/5.69	
11.88/5.69	Strict DPs:
11.88/5.69	  { a__f^#(X1, X2, X3) -> c_1()
11.88/5.69	  , a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y))
11.88/5.69	  , a__g^#(X) -> c_3()
11.88/5.69	  , a__g^#(b()) -> c_4()
11.88/5.69	  , a__b^#() -> c_5()
11.88/5.69	  , a__b^#() -> c_6()
11.88/5.69	  , mark^#(g(X)) -> c_7(a__g^#(mark(X)), mark^#(X))
11.88/5.69	  , mark^#(b()) -> c_8(a__b^#())
11.88/5.69	  , mark^#(c()) -> c_9()
11.88/5.69	  , mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3)) }
11.88/5.69	Weak Trs:
11.88/5.69	  { a__f(X1, X2, X3) -> f(X1, X2, X3)
11.88/5.69	  , a__f(X, g(X), Y) -> a__f(Y, Y, Y)
11.88/5.69	  , a__g(X) -> g(X)
11.88/5.69	  , a__g(b()) -> c()
11.88/5.69	  , a__b() -> b()
11.88/5.69	  , a__b() -> c()
11.88/5.69	  , mark(g(X)) -> a__g(mark(X))
11.88/5.69	  , mark(b()) -> a__b()
11.88/5.69	  , mark(c()) -> c()
11.88/5.69	  , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) }
11.88/5.69	Obligation:
11.88/5.69	  innermost runtime complexity
11.88/5.69	Answer:
11.88/5.69	  YES(O(1),O(n^1))
11.88/5.69	
11.88/5.69	We estimate the number of application of {1,3,4,5,6,9} by
11.88/5.69	applications of Pre({1,3,4,5,6,9}) = {2,7,8,10}. Here rules are
11.88/5.69	labeled as follows:
11.88/5.69	
11.88/5.69	  DPs:
11.88/5.69	    { 1: a__f^#(X1, X2, X3) -> c_1()
11.88/5.69	    , 2: a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y))
11.88/5.69	    , 3: a__g^#(X) -> c_3()
11.88/5.69	    , 4: a__g^#(b()) -> c_4()
11.88/5.69	    , 5: a__b^#() -> c_5()
11.88/5.69	    , 6: a__b^#() -> c_6()
11.88/5.70	    , 7: mark^#(g(X)) -> c_7(a__g^#(mark(X)), mark^#(X))
11.88/5.70	    , 8: mark^#(b()) -> c_8(a__b^#())
11.88/5.70	    , 9: mark^#(c()) -> c_9()
11.88/5.70	    , 10: mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3)) }
11.88/5.70	
11.88/5.70	We are left with following problem, upon which TcT provides the
11.88/5.70	certificate YES(O(1),O(n^1)).
11.88/5.70	
11.88/5.70	Strict DPs:
11.88/5.70	  { a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y))
11.88/5.70	  , mark^#(g(X)) -> c_7(a__g^#(mark(X)), mark^#(X))
11.88/5.70	  , mark^#(b()) -> c_8(a__b^#())
11.88/5.70	  , mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3)) }
11.88/5.70	Weak DPs:
11.88/5.70	  { a__f^#(X1, X2, X3) -> c_1()
11.88/5.70	  , a__g^#(X) -> c_3()
11.88/5.70	  , a__g^#(b()) -> c_4()
11.88/5.70	  , a__b^#() -> c_5()
11.88/5.70	  , a__b^#() -> c_6()
11.88/5.70	  , mark^#(c()) -> c_9() }
11.88/5.70	Weak Trs:
11.88/5.70	  { a__f(X1, X2, X3) -> f(X1, X2, X3)
11.88/5.70	  , a__f(X, g(X), Y) -> a__f(Y, Y, Y)
11.88/5.70	  , a__g(X) -> g(X)
11.88/5.70	  , a__g(b()) -> c()
11.88/5.70	  , a__b() -> b()
11.88/5.70	  , a__b() -> c()
11.88/5.70	  , mark(g(X)) -> a__g(mark(X))
11.88/5.70	  , mark(b()) -> a__b()
11.88/5.70	  , mark(c()) -> c()
11.88/5.70	  , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) }
11.88/5.70	Obligation:
11.88/5.70	  innermost runtime complexity
11.88/5.70	Answer:
11.88/5.70	  YES(O(1),O(n^1))
11.88/5.70	
11.88/5.70	We estimate the number of application of {1,3} by applications of
11.88/5.70	Pre({1,3}) = {2,4}. Here rules are labeled as follows:
11.88/5.70	
11.88/5.70	  DPs:
11.88/5.70	    { 1: a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y))
11.88/5.70	    , 2: mark^#(g(X)) -> c_7(a__g^#(mark(X)), mark^#(X))
11.88/5.70	    , 3: mark^#(b()) -> c_8(a__b^#())
11.88/5.70	    , 4: mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3))
11.88/5.70	    , 5: a__f^#(X1, X2, X3) -> c_1()
11.88/5.70	    , 6: a__g^#(X) -> c_3()
11.88/5.70	    , 7: a__g^#(b()) -> c_4()
11.88/5.70	    , 8: a__b^#() -> c_5()
11.88/5.70	    , 9: a__b^#() -> c_6()
11.88/5.70	    , 10: mark^#(c()) -> c_9() }
11.88/5.70	
11.88/5.70	We are left with following problem, upon which TcT provides the
11.88/5.70	certificate YES(O(1),O(n^1)).
11.88/5.70	
11.88/5.70	Strict DPs:
11.88/5.70	  { mark^#(g(X)) -> c_7(a__g^#(mark(X)), mark^#(X))
11.88/5.70	  , mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3)) }
11.88/5.70	Weak DPs:
11.88/5.70	  { a__f^#(X1, X2, X3) -> c_1()
11.88/5.70	  , a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y))
11.88/5.70	  , a__g^#(X) -> c_3()
11.88/5.70	  , a__g^#(b()) -> c_4()
11.88/5.70	  , a__b^#() -> c_5()
11.88/5.70	  , a__b^#() -> c_6()
11.88/5.70	  , mark^#(b()) -> c_8(a__b^#())
11.88/5.70	  , mark^#(c()) -> c_9() }
11.88/5.70	Weak Trs:
11.88/5.70	  { a__f(X1, X2, X3) -> f(X1, X2, X3)
11.88/5.70	  , a__f(X, g(X), Y) -> a__f(Y, Y, Y)
11.88/5.70	  , a__g(X) -> g(X)
11.88/5.70	  , a__g(b()) -> c()
11.88/5.70	  , a__b() -> b()
11.88/5.70	  , a__b() -> c()
11.88/5.70	  , mark(g(X)) -> a__g(mark(X))
11.88/5.70	  , mark(b()) -> a__b()
11.88/5.70	  , mark(c()) -> c()
11.88/5.70	  , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) }
11.88/5.70	Obligation:
11.88/5.70	  innermost runtime complexity
11.88/5.70	Answer:
11.88/5.70	  YES(O(1),O(n^1))
11.88/5.70	
11.88/5.70	We estimate the number of application of {2} by applications of
11.88/5.70	Pre({2}) = {1}. Here rules are labeled as follows:
11.88/5.70	
11.88/5.70	  DPs:
11.88/5.70	    { 1: mark^#(g(X)) -> c_7(a__g^#(mark(X)), mark^#(X))
11.88/5.70	    , 2: mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3))
11.88/5.70	    , 3: a__f^#(X1, X2, X3) -> c_1()
11.88/5.70	    , 4: a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y))
11.88/5.70	    , 5: a__g^#(X) -> c_3()
11.88/5.70	    , 6: a__g^#(b()) -> c_4()
11.88/5.70	    , 7: a__b^#() -> c_5()
11.88/5.70	    , 8: a__b^#() -> c_6()
11.88/5.70	    , 9: mark^#(b()) -> c_8(a__b^#())
11.88/5.70	    , 10: mark^#(c()) -> c_9() }
11.88/5.70	
11.88/5.70	We are left with following problem, upon which TcT provides the
11.88/5.70	certificate YES(O(1),O(n^1)).
11.88/5.70	
11.88/5.70	Strict DPs: { mark^#(g(X)) -> c_7(a__g^#(mark(X)), mark^#(X)) }
11.88/5.70	Weak DPs:
11.88/5.70	  { a__f^#(X1, X2, X3) -> c_1()
11.88/5.70	  , a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y))
11.88/5.70	  , a__g^#(X) -> c_3()
11.88/5.70	  , a__g^#(b()) -> c_4()
11.88/5.70	  , a__b^#() -> c_5()
11.88/5.70	  , a__b^#() -> c_6()
11.88/5.70	  , mark^#(b()) -> c_8(a__b^#())
11.88/5.70	  , mark^#(c()) -> c_9()
11.88/5.70	  , mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3)) }
11.88/5.70	Weak Trs:
11.88/5.70	  { a__f(X1, X2, X3) -> f(X1, X2, X3)
11.88/5.70	  , a__f(X, g(X), Y) -> a__f(Y, Y, Y)
11.88/5.70	  , a__g(X) -> g(X)
11.88/5.70	  , a__g(b()) -> c()
11.88/5.70	  , a__b() -> b()
11.88/5.70	  , a__b() -> c()
11.88/5.70	  , mark(g(X)) -> a__g(mark(X))
11.88/5.70	  , mark(b()) -> a__b()
11.88/5.70	  , mark(c()) -> c()
11.88/5.70	  , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) }
11.88/5.70	Obligation:
11.88/5.70	  innermost runtime complexity
11.88/5.70	Answer:
11.88/5.70	  YES(O(1),O(n^1))
11.88/5.70	
11.88/5.70	The following weak DPs constitute a sub-graph of the DG that is
11.88/5.70	closed under successors. The DPs are removed.
11.88/5.70	
11.88/5.70	{ a__f^#(X1, X2, X3) -> c_1()
11.88/5.70	, a__f^#(X, g(X), Y) -> c_2(a__f^#(Y, Y, Y))
11.88/5.70	, a__g^#(X) -> c_3()
11.88/5.70	, a__g^#(b()) -> c_4()
11.88/5.70	, a__b^#() -> c_5()
11.88/5.70	, a__b^#() -> c_6()
11.88/5.70	, mark^#(b()) -> c_8(a__b^#())
11.88/5.70	, mark^#(c()) -> c_9()
11.88/5.70	, mark^#(f(X1, X2, X3)) -> c_10(a__f^#(X1, X2, X3)) }
11.88/5.70	
11.88/5.70	We are left with following problem, upon which TcT provides the
11.88/5.70	certificate YES(O(1),O(n^1)).
11.88/5.70	
11.88/5.70	Strict DPs: { mark^#(g(X)) -> c_7(a__g^#(mark(X)), mark^#(X)) }
11.88/5.70	Weak Trs:
11.88/5.70	  { a__f(X1, X2, X3) -> f(X1, X2, X3)
11.88/5.70	  , a__f(X, g(X), Y) -> a__f(Y, Y, Y)
11.88/5.70	  , a__g(X) -> g(X)
11.88/5.70	  , a__g(b()) -> c()
11.88/5.70	  , a__b() -> b()
11.88/5.70	  , a__b() -> c()
11.88/5.70	  , mark(g(X)) -> a__g(mark(X))
11.88/5.70	  , mark(b()) -> a__b()
11.88/5.70	  , mark(c()) -> c()
11.88/5.70	  , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) }
11.88/5.70	Obligation:
11.88/5.70	  innermost runtime complexity
11.88/5.70	Answer:
11.88/5.70	  YES(O(1),O(n^1))
11.88/5.70	
11.88/5.70	Due to missing edges in the dependency-graph, the right-hand sides
11.88/5.70	of following rules could be simplified:
11.88/5.70	
11.88/5.70	  { mark^#(g(X)) -> c_7(a__g^#(mark(X)), mark^#(X)) }
11.88/5.70	
11.88/5.70	We are left with following problem, upon which TcT provides the
11.88/5.70	certificate YES(O(1),O(n^1)).
11.88/5.70	
11.88/5.70	Strict DPs: { mark^#(g(X)) -> c_1(mark^#(X)) }
11.88/5.70	Weak Trs:
11.88/5.70	  { a__f(X1, X2, X3) -> f(X1, X2, X3)
11.88/5.70	  , a__f(X, g(X), Y) -> a__f(Y, Y, Y)
11.88/5.70	  , a__g(X) -> g(X)
11.88/5.70	  , a__g(b()) -> c()
11.88/5.70	  , a__b() -> b()
11.88/5.70	  , a__b() -> c()
11.88/5.70	  , mark(g(X)) -> a__g(mark(X))
11.88/5.70	  , mark(b()) -> a__b()
11.88/5.70	  , mark(c()) -> c()
11.88/5.70	  , mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) }
12.07/5.70	Obligation:
12.07/5.70	  innermost runtime complexity
12.07/5.70	Answer:
12.07/5.70	  YES(O(1),O(n^1))
12.07/5.70	
12.07/5.70	No rule is usable, rules are removed from the input problem.
12.07/5.70	
12.07/5.70	We are left with following problem, upon which TcT provides the
12.07/5.70	certificate YES(O(1),O(n^1)).
12.07/5.70	
12.07/5.70	Strict DPs: { mark^#(g(X)) -> c_1(mark^#(X)) }
12.07/5.70	Obligation:
12.07/5.70	  innermost runtime complexity
12.07/5.70	Answer:
12.07/5.70	  YES(O(1),O(n^1))
12.07/5.70	
12.07/5.70	We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
12.07/5.70	to orient following rules strictly.
12.07/5.70	
12.07/5.70	DPs:
12.07/5.70	  { 1: mark^#(g(X)) -> c_1(mark^#(X)) }
12.07/5.70	
12.07/5.70	Sub-proof:
12.07/5.70	----------
12.07/5.70	  The input was oriented with the instance of 'Small Polynomial Path
12.07/5.70	  Order (PS,1-bounded)' as induced by the safe mapping
12.07/5.70	  
12.07/5.70	   safe(g) = {1}, safe(mark^#) = {}, safe(c_1) = {}
12.07/5.70	  
12.07/5.70	  and precedence
12.07/5.70	  
12.07/5.70	   empty .
12.07/5.70	  
12.07/5.70	  Following symbols are considered recursive:
12.07/5.70	  
12.07/5.70	   {mark^#}
12.07/5.70	  
12.07/5.70	  The recursion depth is 1.
12.07/5.70	  
12.07/5.70	  Further, following argument filtering is employed:
12.07/5.70	  
12.07/5.70	   pi(g) = [1], pi(mark^#) = [1], pi(c_1) = [1]
12.07/5.70	  
12.07/5.70	  Usable defined function symbols are a subset of:
12.07/5.70	  
12.07/5.70	   {mark^#}
12.07/5.70	  
12.07/5.70	  For your convenience, here are the satisfied ordering constraints:
12.07/5.70	  
12.07/5.70	    pi(mark^#(g(X))) = mark^#(g(; X);)   
12.07/5.70	                     > c_1(mark^#(X;);)  
12.07/5.70	                     = pi(c_1(mark^#(X)))
12.07/5.70	                                         
12.07/5.70	
12.07/5.70	The strictly oriented rules are moved into the weak component.
12.07/5.70	
12.07/5.70	We are left with following problem, upon which TcT provides the
12.07/5.70	certificate YES(O(1),O(1)).
12.07/5.70	
12.07/5.70	Weak DPs: { mark^#(g(X)) -> c_1(mark^#(X)) }
12.07/5.70	Obligation:
12.07/5.70	  innermost runtime complexity
12.07/5.70	Answer:
12.07/5.70	  YES(O(1),O(1))
12.07/5.70	
12.07/5.70	The following weak DPs constitute a sub-graph of the DG that is
12.07/5.70	closed under successors. The DPs are removed.
12.07/5.70	
12.07/5.70	{ mark^#(g(X)) -> c_1(mark^#(X)) }
12.07/5.70	
12.07/5.70	We are left with following problem, upon which TcT provides the
12.07/5.70	certificate YES(O(1),O(1)).
12.07/5.70	
12.07/5.70	Rules: Empty
12.07/5.70	Obligation:
12.07/5.70	  innermost runtime complexity
12.07/5.70	Answer:
12.07/5.70	  YES(O(1),O(1))
12.07/5.70	
12.07/5.70	Empty rules are trivially bounded
12.07/5.70	
12.07/5.70	Hurray, we answered YES(O(1),O(n^1))
12.07/5.71	EOF