YES(O(1),O(n^1))
4.44/1.38	YES(O(1),O(n^1))
4.44/1.38	
4.44/1.38	We are left with following problem, upon which TcT provides the
4.44/1.38	certificate YES(O(1),O(n^1)).
4.44/1.38	
4.44/1.38	Strict Trs:
4.44/1.38	  { :(z, +(x, f(y))) -> :(g(z, y), +(x, a()))
4.44/1.38	  , :(:(x, y), z) -> :(x, :(y, z))
4.44/1.38	  , :(+(x, y), z) -> +(:(x, z), :(y, z)) }
4.44/1.38	Obligation:
4.44/1.38	  innermost runtime complexity
4.44/1.38	Answer:
4.44/1.38	  YES(O(1),O(n^1))
4.44/1.38	
4.44/1.38	We add the following dependency tuples:
4.44/1.38	
4.44/1.38	Strict DPs:
4.44/1.38	  { :^#(z, +(x, f(y))) -> c_1(:^#(g(z, y), +(x, a())))
4.44/1.38	  , :^#(:(x, y), z) -> c_2(:^#(x, :(y, z)), :^#(y, z))
4.44/1.38	  , :^#(+(x, y), z) -> c_3(:^#(x, z), :^#(y, z)) }
4.44/1.38	
4.44/1.38	and mark the set of starting terms.
4.44/1.38	
4.44/1.38	We are left with following problem, upon which TcT provides the
4.44/1.38	certificate YES(O(1),O(n^1)).
4.44/1.38	
4.44/1.38	Strict DPs:
4.44/1.38	  { :^#(z, +(x, f(y))) -> c_1(:^#(g(z, y), +(x, a())))
4.44/1.38	  , :^#(:(x, y), z) -> c_2(:^#(x, :(y, z)), :^#(y, z))
4.44/1.38	  , :^#(+(x, y), z) -> c_3(:^#(x, z), :^#(y, z)) }
4.44/1.38	Weak Trs:
4.44/1.38	  { :(z, +(x, f(y))) -> :(g(z, y), +(x, a()))
4.44/1.38	  , :(:(x, y), z) -> :(x, :(y, z))
4.44/1.38	  , :(+(x, y), z) -> +(:(x, z), :(y, z)) }
4.44/1.38	Obligation:
4.44/1.38	  innermost runtime complexity
4.44/1.38	Answer:
4.44/1.38	  YES(O(1),O(n^1))
4.44/1.38	
4.44/1.38	We estimate the number of application of {1} by applications of
4.44/1.38	Pre({1}) = {2,3}. Here rules are labeled as follows:
4.44/1.38	
4.44/1.38	  DPs:
4.44/1.38	    { 1: :^#(z, +(x, f(y))) -> c_1(:^#(g(z, y), +(x, a())))
4.44/1.38	    , 2: :^#(:(x, y), z) -> c_2(:^#(x, :(y, z)), :^#(y, z))
4.44/1.38	    , 3: :^#(+(x, y), z) -> c_3(:^#(x, z), :^#(y, z)) }
4.44/1.38	
4.44/1.38	We are left with following problem, upon which TcT provides the
4.44/1.38	certificate YES(O(1),O(n^1)).
4.44/1.38	
4.44/1.38	Strict DPs:
4.44/1.38	  { :^#(:(x, y), z) -> c_2(:^#(x, :(y, z)), :^#(y, z))
4.44/1.38	  , :^#(+(x, y), z) -> c_3(:^#(x, z), :^#(y, z)) }
4.44/1.38	Weak DPs: { :^#(z, +(x, f(y))) -> c_1(:^#(g(z, y), +(x, a()))) }
4.44/1.38	Weak Trs:
4.44/1.38	  { :(z, +(x, f(y))) -> :(g(z, y), +(x, a()))
4.44/1.38	  , :(:(x, y), z) -> :(x, :(y, z))
4.44/1.38	  , :(+(x, y), z) -> +(:(x, z), :(y, z)) }
4.44/1.38	Obligation:
4.44/1.38	  innermost runtime complexity
4.44/1.38	Answer:
4.44/1.38	  YES(O(1),O(n^1))
4.44/1.38	
4.44/1.38	The following weak DPs constitute a sub-graph of the DG that is
4.44/1.38	closed under successors. The DPs are removed.
4.44/1.38	
4.44/1.38	{ :^#(z, +(x, f(y))) -> c_1(:^#(g(z, y), +(x, a()))) }
4.44/1.38	
4.44/1.38	We are left with following problem, upon which TcT provides the
4.44/1.38	certificate YES(O(1),O(n^1)).
4.44/1.38	
4.44/1.38	Strict DPs:
4.44/1.38	  { :^#(:(x, y), z) -> c_2(:^#(x, :(y, z)), :^#(y, z))
4.44/1.38	  , :^#(+(x, y), z) -> c_3(:^#(x, z), :^#(y, z)) }
4.44/1.38	Weak Trs:
4.44/1.38	  { :(z, +(x, f(y))) -> :(g(z, y), +(x, a()))
4.44/1.38	  , :(:(x, y), z) -> :(x, :(y, z))
4.44/1.38	  , :(+(x, y), z) -> +(:(x, z), :(y, z)) }
4.44/1.38	Obligation:
4.44/1.38	  innermost runtime complexity
4.44/1.38	Answer:
4.44/1.38	  YES(O(1),O(n^1))
4.44/1.38	
4.44/1.38	Due to missing edges in the dependency-graph, the right-hand sides
4.44/1.38	of following rules could be simplified:
4.44/1.38	
4.44/1.38	  { :^#(:(x, y), z) -> c_2(:^#(x, :(y, z)), :^#(y, z)) }
4.44/1.38	
4.44/1.38	We are left with following problem, upon which TcT provides the
4.44/1.38	certificate YES(O(1),O(n^1)).
4.44/1.38	
4.44/1.38	Strict DPs:
4.44/1.38	  { :^#(:(x, y), z) -> c_1(:^#(y, z))
4.44/1.38	  , :^#(+(x, y), z) -> c_2(:^#(x, z), :^#(y, z)) }
4.44/1.38	Weak Trs:
4.44/1.38	  { :(z, +(x, f(y))) -> :(g(z, y), +(x, a()))
4.44/1.38	  , :(:(x, y), z) -> :(x, :(y, z))
4.44/1.38	  , :(+(x, y), z) -> +(:(x, z), :(y, z)) }
4.44/1.38	Obligation:
4.44/1.38	  innermost runtime complexity
4.44/1.38	Answer:
4.44/1.38	  YES(O(1),O(n^1))
4.44/1.38	
4.44/1.38	No rule is usable, rules are removed from the input problem.
4.44/1.38	
4.44/1.38	We are left with following problem, upon which TcT provides the
4.44/1.38	certificate YES(O(1),O(n^1)).
4.44/1.38	
4.44/1.38	Strict DPs:
4.44/1.38	  { :^#(:(x, y), z) -> c_1(:^#(y, z))
4.44/1.38	  , :^#(+(x, y), z) -> c_2(:^#(x, z), :^#(y, z)) }
4.44/1.38	Obligation:
4.44/1.38	  innermost runtime complexity
4.44/1.38	Answer:
4.44/1.38	  YES(O(1),O(n^1))
4.44/1.38	
4.44/1.38	We use the processor 'matrix interpretation of dimension 1' to
4.44/1.38	orient following rules strictly.
4.44/1.38	
4.44/1.38	DPs:
4.44/1.38	  { 1: :^#(:(x, y), z) -> c_1(:^#(y, z))
4.44/1.38	  , 2: :^#(+(x, y), z) -> c_2(:^#(x, z), :^#(y, z)) }
4.44/1.38	
4.44/1.38	Sub-proof:
4.44/1.38	----------
4.44/1.38	  The following argument positions are usable:
4.44/1.38	    Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}
4.44/1.38	  
4.44/1.38	  TcT has computed the following constructor-based matrix
4.44/1.38	  interpretation satisfying not(EDA).
4.44/1.38	  
4.44/1.38	      [:](x1, x2) = [7] x1 + [4] x2 + [4]
4.44/1.38	                                         
4.44/1.38	      [+](x1, x2) = [1] x1 + [1] x2 + [4]
4.44/1.38	                                         
4.44/1.38	    [:^#](x1, x2) = [2] x1 + [0]         
4.44/1.38	                                         
4.44/1.38	        [c_1](x1) = [1] x1 + [3]         
4.44/1.38	                                         
4.44/1.38	    [c_2](x1, x2) = [1] x1 + [1] x2 + [5]
4.44/1.38	  
4.44/1.38	  The order satisfies the following ordering constraints:
4.44/1.38	  
4.44/1.38	    [:^#(:(x, y), z)] = [14] x + [8] y + [8]       
4.44/1.38	                      > [2] y + [3]                
4.44/1.38	                      = [c_1(:^#(y, z))]           
4.44/1.38	                                                   
4.44/1.38	    [:^#(+(x, y), z)] = [2] x + [2] y + [8]        
4.44/1.38	                      > [2] x + [2] y + [5]        
4.44/1.38	                      = [c_2(:^#(x, z), :^#(y, z))]
4.44/1.38	                                                   
4.44/1.38	
4.44/1.38	The strictly oriented rules are moved into the weak component.
4.44/1.38	
4.44/1.38	We are left with following problem, upon which TcT provides the
4.44/1.38	certificate YES(O(1),O(1)).
4.44/1.38	
4.44/1.38	Weak DPs:
4.44/1.38	  { :^#(:(x, y), z) -> c_1(:^#(y, z))
4.44/1.38	  , :^#(+(x, y), z) -> c_2(:^#(x, z), :^#(y, z)) }
4.44/1.38	Obligation:
4.44/1.38	  innermost runtime complexity
4.44/1.38	Answer:
4.44/1.38	  YES(O(1),O(1))
4.44/1.38	
4.44/1.38	The following weak DPs constitute a sub-graph of the DG that is
4.44/1.38	closed under successors. The DPs are removed.
4.44/1.38	
4.44/1.38	{ :^#(:(x, y), z) -> c_1(:^#(y, z))
4.44/1.38	, :^#(+(x, y), z) -> c_2(:^#(x, z), :^#(y, z)) }
4.44/1.38	
4.44/1.38	We are left with following problem, upon which TcT provides the
4.44/1.38	certificate YES(O(1),O(1)).
4.44/1.38	
4.44/1.38	Rules: Empty
4.44/1.38	Obligation:
4.44/1.38	  innermost runtime complexity
4.44/1.38	Answer:
4.44/1.38	  YES(O(1),O(1))
4.44/1.38	
4.44/1.38	Empty rules are trivially bounded
4.44/1.38	
4.44/1.38	Hurray, we answered YES(O(1),O(n^1))
4.44/1.38	EOF