YES(O(1),O(n^1))
8.37/3.46	YES(O(1),O(n^1))
8.37/3.46	
8.37/3.46	We are left with following problem, upon which TcT provides the
8.37/3.46	certificate YES(O(1),O(n^1)).
8.37/3.46	
8.37/3.46	Strict Trs: { h(f(x, y)) -> f(f(a(), h(h(y))), x) }
8.37/3.46	Obligation:
8.37/3.46	  innermost runtime complexity
8.37/3.46	Answer:
8.37/3.46	  YES(O(1),O(n^1))
8.37/3.46	
8.37/3.46	We add the following dependency tuples:
8.37/3.46	
8.37/3.46	Strict DPs: { h^#(f(x, y)) -> c_1(h^#(h(y)), h^#(y)) }
8.37/3.46	
8.37/3.46	and mark the set of starting terms.
8.37/3.46	
8.37/3.46	We are left with following problem, upon which TcT provides the
8.37/3.46	certificate YES(O(1),O(n^1)).
8.37/3.46	
8.37/3.46	Strict DPs: { h^#(f(x, y)) -> c_1(h^#(h(y)), h^#(y)) }
8.37/3.46	Weak Trs: { h(f(x, y)) -> f(f(a(), h(h(y))), x) }
8.37/3.46	Obligation:
8.37/3.46	  innermost runtime complexity
8.37/3.46	Answer:
8.37/3.46	  YES(O(1),O(n^1))
8.37/3.46	
8.37/3.46	We use the processor 'matrix interpretation of dimension 2' to
8.37/3.46	orient following rules strictly.
8.37/3.46	
8.37/3.46	DPs:
8.37/3.46	  { 1: h^#(f(x, y)) -> c_1(h^#(h(y)), h^#(y)) }
8.37/3.46	
8.37/3.46	Sub-proof:
8.37/3.46	----------
8.37/3.46	  The following argument positions are usable:
8.37/3.46	    Uargs(c_1) = {1, 2}
8.37/3.46	  
8.37/3.46	  TcT has computed the following constructor-based matrix
8.37/3.46	  interpretation satisfying not(EDA) and not(IDA(1)).
8.37/3.46	  
8.37/3.46	          [h](x1) = [0 3] x1 + [0]           
8.37/3.46	                    [1 0]      [0]           
8.37/3.46	                                             
8.37/3.46	      [f](x1, x2) = [1 1] x1 + [0 0] x2 + [4]
8.37/3.46	                    [0 0]      [1 1]      [4]
8.37/3.46	                                             
8.37/3.46	              [a] = [0]                      
8.37/3.46	                    [0]                      
8.37/3.46	                                             
8.37/3.46	        [h^#](x1) = [0 1] x1 + [0]           
8.37/3.46	                    [0 0]      [0]           
8.37/3.46	                                             
8.37/3.46	    [c_1](x1, x2) = [1 0] x1 + [1 0] x2 + [1]
8.37/3.46	                    [0 0]      [0 0]      [0]
8.37/3.46	  
8.37/3.46	  The order satisfies the following ordering constraints:
8.37/3.46	  
8.37/3.46	      [h(f(x, y))] =  [0 0] x + [3 3] y + [12]
8.37/3.46	                      [1 1]     [0 0]     [4] 
8.37/3.46	                   >= [0 0] x + [3 3] y + [12]
8.37/3.46	                      [1 1]     [0 0]     [4] 
8.37/3.46	                   =  [f(f(a(), h(h(y))), x)] 
8.37/3.46	                                              
8.37/3.46	    [h^#(f(x, y))] =  [1 1] y + [4]           
8.37/3.46	                      [0 0]     [0]           
8.37/3.46	                   >  [1 1] y + [1]           
8.37/3.46	                      [0 0]     [0]           
8.37/3.46	                   =  [c_1(h^#(h(y)), h^#(y))]
8.37/3.46	                                              
8.37/3.46	
8.37/3.46	The strictly oriented rules are moved into the weak component.
8.37/3.46	
8.37/3.46	We are left with following problem, upon which TcT provides the
8.37/3.46	certificate YES(O(1),O(1)).
8.37/3.46	
8.37/3.46	Weak DPs: { h^#(f(x, y)) -> c_1(h^#(h(y)), h^#(y)) }
8.37/3.46	Weak Trs: { h(f(x, y)) -> f(f(a(), h(h(y))), x) }
8.37/3.46	Obligation:
8.37/3.46	  innermost runtime complexity
8.37/3.46	Answer:
8.37/3.46	  YES(O(1),O(1))
8.37/3.46	
8.37/3.46	The following weak DPs constitute a sub-graph of the DG that is
8.37/3.46	closed under successors. The DPs are removed.
8.37/3.46	
8.37/3.46	{ h^#(f(x, y)) -> c_1(h^#(h(y)), h^#(y)) }
8.37/3.46	
8.37/3.46	We are left with following problem, upon which TcT provides the
8.37/3.46	certificate YES(O(1),O(1)).
8.37/3.46	
8.37/3.46	Weak Trs: { h(f(x, y)) -> f(f(a(), h(h(y))), x) }
8.37/3.46	Obligation:
8.37/3.46	  innermost runtime complexity
8.37/3.46	Answer:
8.37/3.46	  YES(O(1),O(1))
8.37/3.46	
8.37/3.46	No rule is usable, rules are removed from the input problem.
8.37/3.46	
8.37/3.46	We are left with following problem, upon which TcT provides the
8.37/3.46	certificate YES(O(1),O(1)).
8.37/3.46	
8.37/3.46	Rules: Empty
8.37/3.46	Obligation:
8.37/3.46	  innermost runtime complexity
8.37/3.46	Answer:
8.37/3.46	  YES(O(1),O(1))
8.37/3.46	
8.37/3.46	Empty rules are trivially bounded
8.37/3.46	
8.37/3.46	Hurray, we answered YES(O(1),O(n^1))
8.37/3.47	EOF