YES(O(1),O(n^1))
5.63/1.61	YES(O(1),O(n^1))
5.63/1.61	
5.63/1.61	We are left with following problem, upon which TcT provides the
5.63/1.61	certificate YES(O(1),O(n^1)).
5.63/1.61	
5.63/1.61	Strict Trs:
5.63/1.61	  { f(a()) -> b()
5.63/1.61	  , f(c()) -> d()
5.63/1.61	  , f(g(x, y)) -> g(f(x), f(y))
5.63/1.61	  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
5.63/1.61	  , g(x, x) -> h(e(), x) }
5.63/1.61	Obligation:
5.63/1.61	  innermost runtime complexity
5.63/1.61	Answer:
5.63/1.61	  YES(O(1),O(n^1))
5.63/1.61	
5.63/1.61	We add the following dependency tuples:
5.63/1.61	
5.63/1.61	Strict DPs:
5.63/1.61	  { f^#(a()) -> c_1()
5.63/1.61	  , f^#(c()) -> c_2()
5.63/1.61	  , f^#(g(x, y)) -> c_3(g^#(f(x), f(y)), f^#(x), f^#(y))
5.63/1.61	  , f^#(h(x, y)) -> c_4(g^#(h(y, f(x)), h(x, f(y))), f^#(x), f^#(y))
5.63/1.61	  , g^#(x, x) -> c_5() }
5.63/1.61	
5.63/1.61	and mark the set of starting terms.
5.63/1.61	
5.63/1.61	We are left with following problem, upon which TcT provides the
5.63/1.61	certificate YES(O(1),O(n^1)).
5.63/1.61	
5.63/1.61	Strict DPs:
5.63/1.61	  { f^#(a()) -> c_1()
5.63/1.61	  , f^#(c()) -> c_2()
5.63/1.61	  , f^#(g(x, y)) -> c_3(g^#(f(x), f(y)), f^#(x), f^#(y))
5.63/1.61	  , f^#(h(x, y)) -> c_4(g^#(h(y, f(x)), h(x, f(y))), f^#(x), f^#(y))
5.63/1.61	  , g^#(x, x) -> c_5() }
5.63/1.61	Weak Trs:
5.63/1.61	  { f(a()) -> b()
5.63/1.61	  , f(c()) -> d()
5.63/1.61	  , f(g(x, y)) -> g(f(x), f(y))
5.63/1.61	  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
5.63/1.61	  , g(x, x) -> h(e(), x) }
5.63/1.61	Obligation:
5.63/1.61	  innermost runtime complexity
5.63/1.61	Answer:
5.63/1.61	  YES(O(1),O(n^1))
5.63/1.61	
5.63/1.61	We estimate the number of application of {1,2,5} by applications of
5.63/1.61	Pre({1,2,5}) = {3,4}. Here rules are labeled as follows:
5.63/1.61	
5.63/1.61	  DPs:
5.63/1.61	    { 1: f^#(a()) -> c_1()
5.63/1.61	    , 2: f^#(c()) -> c_2()
5.63/1.61	    , 3: f^#(g(x, y)) -> c_3(g^#(f(x), f(y)), f^#(x), f^#(y))
5.63/1.61	    , 4: f^#(h(x, y)) ->
5.63/1.61	         c_4(g^#(h(y, f(x)), h(x, f(y))), f^#(x), f^#(y))
5.63/1.61	    , 5: g^#(x, x) -> c_5() }
5.63/1.61	
5.63/1.61	We are left with following problem, upon which TcT provides the
5.63/1.61	certificate YES(O(1),O(n^1)).
5.63/1.61	
5.63/1.61	Strict DPs:
5.63/1.61	  { f^#(g(x, y)) -> c_3(g^#(f(x), f(y)), f^#(x), f^#(y))
5.63/1.61	  , f^#(h(x, y)) ->
5.63/1.61	    c_4(g^#(h(y, f(x)), h(x, f(y))), f^#(x), f^#(y)) }
5.63/1.61	Weak DPs:
5.63/1.61	  { f^#(a()) -> c_1()
5.63/1.61	  , f^#(c()) -> c_2()
5.63/1.61	  , g^#(x, x) -> c_5() }
5.63/1.61	Weak Trs:
5.63/1.61	  { f(a()) -> b()
5.63/1.61	  , f(c()) -> d()
5.63/1.61	  , f(g(x, y)) -> g(f(x), f(y))
5.63/1.61	  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
5.63/1.61	  , g(x, x) -> h(e(), x) }
5.63/1.61	Obligation:
5.63/1.61	  innermost runtime complexity
5.63/1.61	Answer:
5.63/1.61	  YES(O(1),O(n^1))
5.63/1.61	
5.63/1.61	The following weak DPs constitute a sub-graph of the DG that is
5.63/1.61	closed under successors. The DPs are removed.
5.63/1.61	
5.63/1.61	{ f^#(a()) -> c_1()
5.63/1.61	, f^#(c()) -> c_2()
5.63/1.61	, g^#(x, x) -> c_5() }
5.63/1.61	
5.63/1.61	We are left with following problem, upon which TcT provides the
5.63/1.61	certificate YES(O(1),O(n^1)).
5.63/1.61	
5.63/1.61	Strict DPs:
5.63/1.61	  { f^#(g(x, y)) -> c_3(g^#(f(x), f(y)), f^#(x), f^#(y))
5.63/1.61	  , f^#(h(x, y)) ->
5.63/1.61	    c_4(g^#(h(y, f(x)), h(x, f(y))), f^#(x), f^#(y)) }
5.63/1.61	Weak Trs:
5.63/1.61	  { f(a()) -> b()
5.63/1.61	  , f(c()) -> d()
5.63/1.61	  , f(g(x, y)) -> g(f(x), f(y))
5.63/1.61	  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
5.63/1.61	  , g(x, x) -> h(e(), x) }
5.63/1.61	Obligation:
5.63/1.61	  innermost runtime complexity
5.63/1.61	Answer:
5.63/1.61	  YES(O(1),O(n^1))
5.63/1.61	
5.63/1.61	Due to missing edges in the dependency-graph, the right-hand sides
5.63/1.61	of following rules could be simplified:
5.63/1.61	
5.63/1.61	  { f^#(g(x, y)) -> c_3(g^#(f(x), f(y)), f^#(x), f^#(y))
5.63/1.61	  , f^#(h(x, y)) ->
5.63/1.61	    c_4(g^#(h(y, f(x)), h(x, f(y))), f^#(x), f^#(y)) }
5.63/1.61	
5.63/1.61	We are left with following problem, upon which TcT provides the
5.63/1.61	certificate YES(O(1),O(n^1)).
5.63/1.61	
5.63/1.61	Strict DPs:
5.63/1.61	  { f^#(g(x, y)) -> c_1(f^#(x), f^#(y))
5.63/1.61	  , f^#(h(x, y)) -> c_2(f^#(x), f^#(y)) }
5.63/1.61	Weak Trs:
5.63/1.61	  { f(a()) -> b()
5.63/1.61	  , f(c()) -> d()
5.63/1.61	  , f(g(x, y)) -> g(f(x), f(y))
5.63/1.61	  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
5.63/1.61	  , g(x, x) -> h(e(), x) }
5.63/1.61	Obligation:
5.63/1.61	  innermost runtime complexity
5.63/1.61	Answer:
5.63/1.61	  YES(O(1),O(n^1))
5.63/1.61	
5.63/1.61	No rule is usable, rules are removed from the input problem.
5.63/1.61	
5.63/1.61	We are left with following problem, upon which TcT provides the
5.63/1.61	certificate YES(O(1),O(n^1)).
5.63/1.61	
5.63/1.61	Strict DPs:
5.63/1.61	  { f^#(g(x, y)) -> c_1(f^#(x), f^#(y))
5.63/1.61	  , f^#(h(x, y)) -> c_2(f^#(x), f^#(y)) }
5.63/1.61	Obligation:
5.63/1.61	  innermost runtime complexity
5.63/1.61	Answer:
5.63/1.61	  YES(O(1),O(n^1))
5.63/1.61	
5.63/1.61	We use the processor 'matrix interpretation of dimension 1' to
5.63/1.61	orient following rules strictly.
5.63/1.61	
5.63/1.61	DPs:
5.63/1.61	  { 1: f^#(g(x, y)) -> c_1(f^#(x), f^#(y))
5.63/1.61	  , 2: f^#(h(x, y)) -> c_2(f^#(x), f^#(y)) }
5.63/1.61	
5.63/1.61	Sub-proof:
5.63/1.61	----------
5.63/1.61	  The following argument positions are usable:
5.63/1.61	    Uargs(c_1) = {1, 2}, Uargs(c_2) = {1, 2}
5.63/1.61	  
5.63/1.61	  TcT has computed the following constructor-based matrix
5.63/1.61	  interpretation satisfying not(EDA).
5.63/1.61	  
5.63/1.61	      [g](x1, x2) = [4] x1 + [4] x2 + [4]
5.63/1.61	                                         
5.63/1.61	      [h](x1, x2) = [1] x1 + [1] x2 + [4]
5.63/1.61	                                         
5.63/1.61	        [f^#](x1) = [2] x1 + [0]         
5.63/1.61	                                         
5.63/1.61	    [c_1](x1, x2) = [4] x1 + [1] x2 + [1]
5.63/1.61	                                         
5.63/1.61	    [c_2](x1, x2) = [1] x1 + [1] x2 + [3]
5.63/1.61	  
5.63/1.61	  The order satisfies the following ordering constraints:
5.63/1.61	  
5.63/1.61	    [f^#(g(x, y))] = [8] x + [8] y + [8]  
5.63/1.61	                   > [8] x + [2] y + [1]  
5.63/1.61	                   = [c_1(f^#(x), f^#(y))]
5.63/1.61	                                          
5.63/1.61	    [f^#(h(x, y))] = [2] x + [2] y + [8]  
5.63/1.61	                   > [2] x + [2] y + [3]  
5.63/1.61	                   = [c_2(f^#(x), f^#(y))]
5.63/1.61	                                          
5.63/1.61	
5.63/1.61	The strictly oriented rules are moved into the weak component.
5.63/1.61	
5.63/1.61	We are left with following problem, upon which TcT provides the
5.63/1.61	certificate YES(O(1),O(1)).
5.63/1.61	
5.63/1.61	Weak DPs:
5.63/1.61	  { f^#(g(x, y)) -> c_1(f^#(x), f^#(y))
5.63/1.61	  , f^#(h(x, y)) -> c_2(f^#(x), f^#(y)) }
5.63/1.61	Obligation:
5.63/1.61	  innermost runtime complexity
5.63/1.61	Answer:
5.63/1.61	  YES(O(1),O(1))
5.63/1.61	
5.63/1.61	The following weak DPs constitute a sub-graph of the DG that is
5.63/1.61	closed under successors. The DPs are removed.
5.63/1.61	
5.63/1.61	{ f^#(g(x, y)) -> c_1(f^#(x), f^#(y))
5.63/1.61	, f^#(h(x, y)) -> c_2(f^#(x), f^#(y)) }
5.63/1.61	
5.63/1.61	We are left with following problem, upon which TcT provides the
5.63/1.61	certificate YES(O(1),O(1)).
5.63/1.61	
5.63/1.61	Rules: Empty
5.63/1.61	Obligation:
5.63/1.61	  innermost runtime complexity
5.63/1.61	Answer:
5.63/1.61	  YES(O(1),O(1))
5.63/1.61	
5.63/1.61	Empty rules are trivially bounded
5.63/1.61	
5.63/1.61	Hurray, we answered YES(O(1),O(n^1))
5.63/1.62	EOF