YES(O(1),O(n^1))
8.34/3.43	YES(O(1),O(n^1))
8.34/3.43	
8.34/3.43	We are left with following problem, upon which TcT provides the
8.34/3.43	certificate YES(O(1),O(n^1)).
8.34/3.43	
8.34/3.43	Strict Trs: { h(f(x, y)) -> f(y, f(h(h(x)), a())) }
8.34/3.43	Obligation:
8.34/3.43	  runtime complexity
8.34/3.43	Answer:
8.34/3.43	  YES(O(1),O(n^1))
8.34/3.43	
8.34/3.43	The input is overlay and right-linear. Switching to innermost
8.34/3.43	rewriting.
8.34/3.43	
8.34/3.43	We are left with following problem, upon which TcT provides the
8.34/3.43	certificate YES(O(1),O(n^1)).
8.34/3.43	
8.34/3.43	Strict Trs: { h(f(x, y)) -> f(y, f(h(h(x)), a())) }
8.34/3.43	Obligation:
8.34/3.43	  innermost runtime complexity
8.34/3.43	Answer:
8.34/3.43	  YES(O(1),O(n^1))
8.34/3.43	
8.34/3.43	We add the following dependency tuples:
8.34/3.43	
8.34/3.43	Strict DPs: { h^#(f(x, y)) -> c_1(h^#(h(x)), h^#(x)) }
8.34/3.43	
8.34/3.43	and mark the set of starting terms.
8.34/3.43	
8.34/3.43	We are left with following problem, upon which TcT provides the
8.34/3.43	certificate YES(O(1),O(n^1)).
8.34/3.43	
8.34/3.43	Strict DPs: { h^#(f(x, y)) -> c_1(h^#(h(x)), h^#(x)) }
8.34/3.43	Weak Trs: { h(f(x, y)) -> f(y, f(h(h(x)), a())) }
8.34/3.43	Obligation:
8.34/3.43	  innermost runtime complexity
8.34/3.43	Answer:
8.34/3.43	  YES(O(1),O(n^1))
8.34/3.43	
8.34/3.43	We use the processor 'matrix interpretation of dimension 2' to
8.34/3.43	orient following rules strictly.
8.34/3.43	
8.34/3.43	DPs:
8.34/3.43	  { 1: h^#(f(x, y)) -> c_1(h^#(h(x)), h^#(x)) }
8.34/3.43	
8.34/3.43	Sub-proof:
8.34/3.43	----------
8.34/3.43	  The following argument positions are usable:
8.34/3.43	    Uargs(c_1) = {1, 2}
8.34/3.43	  
8.34/3.43	  TcT has computed the following constructor-based matrix
8.34/3.43	  interpretation satisfying not(EDA) and not(IDA(1)).
8.34/3.43	  
8.34/3.43	          [h](x1) = [0 3] x1 + [0]           
8.34/3.43	                    [1 0]      [0]           
8.34/3.43	                                             
8.34/3.43	      [f](x1, x2) = [0 0] x1 + [1 1] x2 + [2]
8.34/3.43	                    [1 1]      [0 0]      [2]
8.34/3.43	                                             
8.34/3.43	              [a] = [0]                      
8.34/3.43	                    [0]                      
8.34/3.43	                                             
8.34/3.43	        [h^#](x1) = [0 4] x1 + [0]           
8.34/3.43	                    [0 0]      [0]           
8.34/3.43	                                             
8.34/3.43	    [c_1](x1, x2) = [1 0] x1 + [1 0] x2 + [1]
8.34/3.43	                    [0 0]      [0 0]      [0]
8.34/3.43	  
8.34/3.43	  The order satisfies the following ordering constraints:
8.34/3.43	  
8.34/3.43	      [h(f(x, y))] =  [3 3] x + [0 0] y + [6] 
8.34/3.43	                      [0 0]     [1 1]     [2] 
8.34/3.43	                   >= [3 3] x + [0 0] y + [6] 
8.34/3.43	                      [0 0]     [1 1]     [2] 
8.34/3.43	                   =  [f(y, f(h(h(x)), a()))] 
8.34/3.43	                                              
8.34/3.43	    [h^#(f(x, y))] =  [4 4] x + [8]           
8.34/3.43	                      [0 0]     [0]           
8.34/3.43	                   >  [4 4] x + [1]           
8.34/3.43	                      [0 0]     [0]           
8.34/3.43	                   =  [c_1(h^#(h(x)), h^#(x))]
8.34/3.43	                                              
8.34/3.43	
8.34/3.43	The strictly oriented rules are moved into the weak component.
8.34/3.43	
8.34/3.43	We are left with following problem, upon which TcT provides the
8.34/3.43	certificate YES(O(1),O(1)).
8.34/3.43	
8.34/3.43	Weak DPs: { h^#(f(x, y)) -> c_1(h^#(h(x)), h^#(x)) }
8.34/3.43	Weak Trs: { h(f(x, y)) -> f(y, f(h(h(x)), a())) }
8.34/3.43	Obligation:
8.34/3.43	  innermost runtime complexity
8.34/3.43	Answer:
8.34/3.43	  YES(O(1),O(1))
8.34/3.43	
8.34/3.43	The following weak DPs constitute a sub-graph of the DG that is
8.34/3.43	closed under successors. The DPs are removed.
8.34/3.43	
8.34/3.43	{ h^#(f(x, y)) -> c_1(h^#(h(x)), h^#(x)) }
8.34/3.43	
8.34/3.43	We are left with following problem, upon which TcT provides the
8.34/3.43	certificate YES(O(1),O(1)).
8.34/3.43	
8.34/3.43	Weak Trs: { h(f(x, y)) -> f(y, f(h(h(x)), a())) }
8.34/3.43	Obligation:
8.34/3.43	  innermost runtime complexity
8.34/3.43	Answer:
8.34/3.43	  YES(O(1),O(1))
8.34/3.43	
8.34/3.43	No rule is usable, rules are removed from the input problem.
8.34/3.43	
8.34/3.43	We are left with following problem, upon which TcT provides the
8.34/3.43	certificate YES(O(1),O(1)).
8.34/3.43	
8.34/3.43	Rules: Empty
8.34/3.43	Obligation:
8.34/3.43	  innermost runtime complexity
8.34/3.43	Answer:
8.34/3.43	  YES(O(1),O(1))
8.34/3.43	
8.34/3.43	Empty rules are trivially bounded
8.34/3.43	
8.34/3.43	Hurray, we answered YES(O(1),O(n^1))
8.34/3.45	EOF