YES(O(1),O(n^1))
0.00/0.84	YES(O(1),O(n^1))
0.00/0.84	
0.00/0.84	We are left with following problem, upon which TcT provides the
0.00/0.84	certificate YES(O(1),O(n^1)).
0.00/0.84	
0.00/0.84	Strict Trs:
0.00/0.84	  { D(t()) -> 1()
0.00/0.84	  , D(constant()) -> 0()
0.00/0.84	  , D(+(x, y)) -> +(D(x), D(y))
0.00/0.84	  , D(*(x, y)) -> +(*(y, D(x)), *(x, D(y)))
0.00/0.84	  , D(-(x, y)) -> -(D(x), D(y)) }
0.00/0.84	Obligation:
0.00/0.84	  innermost runtime complexity
0.00/0.84	Answer:
0.00/0.84	  YES(O(1),O(n^1))
0.00/0.84	
0.00/0.84	We add the following weak dependency pairs:
0.00/0.84	
0.00/0.84	Strict DPs:
0.00/0.84	  { D^#(t()) -> c_1()
0.00/0.84	  , D^#(constant()) -> c_2()
0.00/0.84	  , D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
0.00/0.84	  , D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
0.00/0.84	  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
0.00/0.84	
0.00/0.84	and mark the set of starting terms.
0.00/0.84	
0.00/0.84	We are left with following problem, upon which TcT provides the
0.00/0.84	certificate YES(O(1),O(n^1)).
0.00/0.84	
0.00/0.84	Strict DPs:
0.00/0.84	  { D^#(t()) -> c_1()
0.00/0.84	  , D^#(constant()) -> c_2()
0.00/0.84	  , D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
0.00/0.84	  , D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
0.00/0.84	  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
0.00/0.84	Strict Trs:
0.00/0.84	  { D(t()) -> 1()
0.00/0.84	  , D(constant()) -> 0()
0.00/0.84	  , D(+(x, y)) -> +(D(x), D(y))
0.00/0.84	  , D(*(x, y)) -> +(*(y, D(x)), *(x, D(y)))
0.00/0.84	  , D(-(x, y)) -> -(D(x), D(y)) }
0.00/0.84	Obligation:
0.00/0.84	  innermost runtime complexity
0.00/0.84	Answer:
0.00/0.84	  YES(O(1),O(n^1))
0.00/0.84	
0.00/0.84	No rule is usable, rules are removed from the input problem.
0.00/0.84	
0.00/0.84	We are left with following problem, upon which TcT provides the
0.00/0.84	certificate YES(O(1),O(n^1)).
0.00/0.84	
0.00/0.84	Strict DPs:
0.00/0.84	  { D^#(t()) -> c_1()
0.00/0.84	  , D^#(constant()) -> c_2()
0.00/0.84	  , D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
0.00/0.84	  , D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
0.00/0.84	  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
0.00/0.84	Obligation:
0.00/0.84	  innermost runtime complexity
0.00/0.84	Answer:
0.00/0.84	  YES(O(1),O(n^1))
0.00/0.84	
0.00/0.84	The weightgap principle applies (using the following constant
0.00/0.84	growth matrix-interpretation)
0.00/0.84	
0.00/0.84	The following argument positions are usable:
0.00/0.84	  Uargs(c_3) = {1, 2}, Uargs(c_4) = {1, 2}, Uargs(c_5) = {1, 2}
0.00/0.84	
0.00/0.84	TcT has computed the following constructor-restricted matrix
0.00/0.84	interpretation.
0.00/0.84	
0.00/0.84	            [t] = [0]                      
0.00/0.84	                  [0]                      
0.00/0.84	                                           
0.00/0.84	     [constant] = [0]                      
0.00/0.84	                  [0]                      
0.00/0.84	                                           
0.00/0.84	    [+](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
0.00/0.84	                  [0 0]      [0 0]      [0]
0.00/0.84	                                           
0.00/0.84	    [*](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
0.00/0.84	                  [0 0]      [0 0]      [0]
0.00/0.84	                                           
0.00/0.84	    [-](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
0.00/0.84	                  [0 0]      [0 0]      [0]
0.00/0.84	                                           
0.00/0.84	      [D^#](x1) = [1]                      
0.00/0.84	                  [0]                      
0.00/0.84	                                           
0.00/0.84	          [c_1] = [0]                      
0.00/0.84	                  [0]                      
0.00/0.84	                                           
0.00/0.84	          [c_2] = [0]                      
0.00/0.84	                  [0]                      
0.00/0.84	                                           
0.00/0.84	  [c_3](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
0.00/0.84	                  [0 1]      [0 1]      [0]
0.00/0.84	                                           
0.00/0.84	  [c_4](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
0.00/0.84	                  [0 1]      [0 1]      [0]
0.00/0.84	                                           
0.00/0.84	  [c_5](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
0.00/0.84	                  [0 1]      [0 1]      [2]
0.00/0.84	
0.00/0.84	The order satisfies the following ordering constraints:
0.00/0.84	
0.00/0.84	         [D^#(t())] = [1]                  
0.00/0.84	                      [0]                  
0.00/0.84	                    > [0]                  
0.00/0.84	                      [0]                  
0.00/0.84	                    = [c_1()]              
0.00/0.84	                                           
0.00/0.84	  [D^#(constant())] = [1]                  
0.00/0.84	                      [0]                  
0.00/0.84	                    > [0]                  
0.00/0.84	                      [0]                  
0.00/0.84	                    = [c_2()]              
0.00/0.84	                                           
0.00/0.84	     [D^#(+(x, y))] = [1]                  
0.00/0.84	                      [0]                  
0.00/0.84	                    ? [4]                  
0.00/0.84	                      [0]                  
0.00/0.84	                    = [c_3(D^#(x), D^#(y))]
0.00/0.84	                                           
0.00/0.84	     [D^#(*(x, y))] = [1]                  
0.00/0.84	                      [0]                  
0.00/0.84	                    ? [4]                  
0.00/0.84	                      [0]                  
0.00/0.84	                    = [c_4(D^#(x), D^#(y))]
0.00/0.84	                                           
0.00/0.84	     [D^#(-(x, y))] = [1]                  
0.00/0.84	                      [0]                  
0.00/0.84	                    ? [4]                  
0.00/0.84	                      [2]                  
0.00/0.84	                    = [c_5(D^#(x), D^#(y))]
0.00/0.84	                                           
0.00/0.84	
0.00/0.84	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.84	
0.00/0.84	We are left with following problem, upon which TcT provides the
0.00/0.84	certificate YES(O(1),O(n^1)).
0.00/0.84	
0.00/0.84	Strict DPs:
0.00/0.84	  { D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
0.00/0.84	  , D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
0.00/0.84	  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
0.00/0.84	Weak DPs:
0.00/0.84	  { D^#(t()) -> c_1()
0.00/0.84	  , D^#(constant()) -> c_2() }
0.00/0.84	Obligation:
0.00/0.84	  innermost runtime complexity
0.00/0.84	Answer:
0.00/0.84	  YES(O(1),O(n^1))
0.00/0.84	
0.00/0.84	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.84	closed under successors. The DPs are removed.
0.00/0.84	
0.00/0.84	{ D^#(t()) -> c_1()
0.00/0.84	, D^#(constant()) -> c_2() }
0.00/0.84	
0.00/0.84	We are left with following problem, upon which TcT provides the
0.00/0.84	certificate YES(O(1),O(n^1)).
0.00/0.84	
0.00/0.84	Strict DPs:
0.00/0.84	  { D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
0.00/0.84	  , D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
0.00/0.84	  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
0.00/0.84	Obligation:
0.00/0.84	  innermost runtime complexity
0.00/0.84	Answer:
0.00/0.84	  YES(O(1),O(n^1))
0.00/0.84	
0.00/0.84	We use the processor 'matrix interpretation of dimension 1' to
0.00/0.84	orient following rules strictly.
0.00/0.84	
0.00/0.84	DPs:
0.00/0.84	  { 1: D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
0.00/0.84	  , 3: D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
0.00/0.84	
0.00/0.84	Sub-proof:
0.00/0.84	----------
0.00/0.84	  The following argument positions are usable:
0.00/0.84	    Uargs(c_3) = {1, 2}, Uargs(c_4) = {1, 2}, Uargs(c_5) = {1, 2}
0.00/0.84	  
0.00/0.84	  TcT has computed the following constructor-based matrix
0.00/0.84	  interpretation satisfying not(EDA).
0.00/0.84	  
0.00/0.84	      [+](x1, x2) = [1] x1 + [1] x2 + [2]
0.00/0.84	                                         
0.00/0.84	      [*](x1, x2) = [1] x1 + [1] x2 + [0]
0.00/0.84	                                         
0.00/0.84	      [-](x1, x2) = [1] x1 + [1] x2 + [4]
0.00/0.84	                                         
0.00/0.84	        [D^#](x1) = [2] x1 + [0]         
0.00/0.84	                                         
0.00/0.84	    [c_3](x1, x2) = [1] x1 + [1] x2 + [1]
0.00/0.84	                                         
0.00/0.84	    [c_4](x1, x2) = [1] x1 + [1] x2 + [0]
0.00/0.84	                                         
0.00/0.84	    [c_5](x1, x2) = [1] x1 + [1] x2 + [7]
0.00/0.84	  
0.00/0.84	  The order satisfies the following ordering constraints:
0.00/0.84	  
0.00/0.84	    [D^#(+(x, y))] =  [2] x + [2] y + [4]  
0.00/0.84	                   >  [2] x + [2] y + [1]  
0.00/0.84	                   =  [c_3(D^#(x), D^#(y))]
0.00/0.84	                                           
0.00/0.84	    [D^#(*(x, y))] =  [2] x + [2] y + [0]  
0.00/0.84	                   >= [2] x + [2] y + [0]  
0.00/0.84	                   =  [c_4(D^#(x), D^#(y))]
0.00/0.84	                                           
0.00/0.84	    [D^#(-(x, y))] =  [2] x + [2] y + [8]  
0.00/0.84	                   >  [2] x + [2] y + [7]  
0.00/0.84	                   =  [c_5(D^#(x), D^#(y))]
0.00/0.84	                                           
0.00/0.84	
0.00/0.84	The strictly oriented rules are moved into the weak component.
0.00/0.84	
0.00/0.84	We are left with following problem, upon which TcT provides the
0.00/0.84	certificate YES(O(1),O(n^1)).
0.00/0.84	
0.00/0.84	Strict DPs: { D^#(*(x, y)) -> c_4(D^#(x), D^#(y)) }
0.00/0.84	Weak DPs:
0.00/0.84	  { D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
0.00/0.84	  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
0.00/0.84	Obligation:
0.00/0.84	  innermost runtime complexity
0.00/0.84	Answer:
0.00/0.84	  YES(O(1),O(n^1))
0.00/0.84	
0.00/0.84	We use the processor 'matrix interpretation of dimension 1' to
0.00/0.84	orient following rules strictly.
0.00/0.84	
0.00/0.84	DPs:
0.00/0.84	  { 1: D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
0.00/0.84	  , 3: D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
0.00/0.84	
0.00/0.84	Sub-proof:
0.00/0.84	----------
0.00/0.84	  The following argument positions are usable:
0.00/0.84	    Uargs(c_3) = {1, 2}, Uargs(c_4) = {1, 2}, Uargs(c_5) = {1, 2}
0.00/0.84	  
0.00/0.84	  TcT has computed the following constructor-based matrix
0.00/0.84	  interpretation satisfying not(EDA).
0.00/0.84	  
0.00/0.84	      [+](x1, x2) = [1] x1 + [1] x2 + [0]
0.00/0.84	                                         
0.00/0.84	      [*](x1, x2) = [1] x1 + [1] x2 + [4]
0.00/0.84	                                         
0.00/0.84	      [-](x1, x2) = [1] x1 + [1] x2 + [4]
0.00/0.84	                                         
0.00/0.84	        [D^#](x1) = [2] x1 + [0]         
0.00/0.84	                                         
0.00/0.84	    [c_3](x1, x2) = [1] x1 + [1] x2 + [0]
0.00/0.84	                                         
0.00/0.84	    [c_4](x1, x2) = [1] x1 + [1] x2 + [5]
0.00/0.84	                                         
0.00/0.84	    [c_5](x1, x2) = [1] x1 + [1] x2 + [7]
0.00/0.84	  
0.00/0.84	  The order satisfies the following ordering constraints:
0.00/0.84	  
0.00/0.84	    [D^#(+(x, y))] =  [2] x + [2] y + [0]  
0.00/0.84	                   >= [2] x + [2] y + [0]  
0.00/0.84	                   =  [c_3(D^#(x), D^#(y))]
0.00/0.84	                                           
0.00/0.84	    [D^#(*(x, y))] =  [2] x + [2] y + [8]  
0.00/0.84	                   >  [2] x + [2] y + [5]  
0.00/0.84	                   =  [c_4(D^#(x), D^#(y))]
0.00/0.84	                                           
0.00/0.84	    [D^#(-(x, y))] =  [2] x + [2] y + [8]  
0.00/0.84	                   >  [2] x + [2] y + [7]  
0.00/0.84	                   =  [c_5(D^#(x), D^#(y))]
0.00/0.84	                                           
0.00/0.84	
0.00/0.84	The strictly oriented rules are moved into the weak component.
0.00/0.84	
0.00/0.84	We are left with following problem, upon which TcT provides the
0.00/0.84	certificate YES(O(1),O(1)).
0.00/0.84	
0.00/0.84	Weak DPs:
0.00/0.84	  { D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
0.00/0.84	  , D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
0.00/0.84	  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
0.00/0.84	Obligation:
0.00/0.84	  innermost runtime complexity
0.00/0.84	Answer:
0.00/0.84	  YES(O(1),O(1))
0.00/0.84	
0.00/0.84	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.84	closed under successors. The DPs are removed.
0.00/0.84	
0.00/0.84	{ D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
0.00/0.84	, D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
0.00/0.84	, D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
0.00/0.84	
0.00/0.84	We are left with following problem, upon which TcT provides the
0.00/0.84	certificate YES(O(1),O(1)).
0.00/0.84	
0.00/0.84	Rules: Empty
0.00/0.84	Obligation:
0.00/0.84	  innermost runtime complexity
0.00/0.84	Answer:
0.00/0.84	  YES(O(1),O(1))
0.00/0.84	
0.00/0.84	Empty rules are trivially bounded
0.00/0.84	
0.00/0.84	Hurray, we answered YES(O(1),O(n^1))
0.00/0.84	EOF