YES(O(1),O(n^1))
0.00/0.54	YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(n^1)).
0.00/0.54	
0.00/0.54	Strict Trs:
0.00/0.54	  { double(x) -> +(x, x)
0.00/0.54	  , double(0()) -> 0()
0.00/0.54	  , double(s(x)) -> s(s(double(x)))
0.00/0.54	  , +(x, 0()) -> x
0.00/0.54	  , +(x, s(y)) -> s(+(x, y))
0.00/0.54	  , +(s(x), y) -> s(+(x, y)) }
0.00/0.54	Obligation:
0.00/0.54	  innermost runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	We add the following weak dependency pairs:
0.00/0.54	
0.00/0.54	Strict DPs:
0.00/0.54	  { double^#(x) -> c_1(+^#(x, x))
0.00/0.54	  , double^#(0()) -> c_2()
0.00/0.54	  , double^#(s(x)) -> c_3(double^#(x))
0.00/0.54	  , +^#(x, 0()) -> c_4()
0.00/0.54	  , +^#(x, s(y)) -> c_5(+^#(x, y))
0.00/0.54	  , +^#(s(x), y) -> c_6(+^#(x, y)) }
0.00/0.54	
0.00/0.54	and mark the set of starting terms.
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(n^1)).
0.00/0.54	
0.00/0.54	Strict DPs:
0.00/0.54	  { double^#(x) -> c_1(+^#(x, x))
0.00/0.54	  , double^#(0()) -> c_2()
0.00/0.54	  , double^#(s(x)) -> c_3(double^#(x))
0.00/0.54	  , +^#(x, 0()) -> c_4()
0.00/0.54	  , +^#(x, s(y)) -> c_5(+^#(x, y))
0.00/0.54	  , +^#(s(x), y) -> c_6(+^#(x, y)) }
0.00/0.54	Strict Trs:
0.00/0.54	  { double(x) -> +(x, x)
0.00/0.54	  , double(0()) -> 0()
0.00/0.54	  , double(s(x)) -> s(s(double(x)))
0.00/0.54	  , +(x, 0()) -> x
0.00/0.54	  , +(x, s(y)) -> s(+(x, y))
0.00/0.54	  , +(s(x), y) -> s(+(x, y)) }
0.00/0.54	Obligation:
0.00/0.54	  innermost runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	No rule is usable, rules are removed from the input problem.
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(n^1)).
0.00/0.54	
0.00/0.54	Strict DPs:
0.00/0.54	  { double^#(x) -> c_1(+^#(x, x))
0.00/0.54	  , double^#(0()) -> c_2()
0.00/0.54	  , double^#(s(x)) -> c_3(double^#(x))
0.00/0.54	  , +^#(x, 0()) -> c_4()
0.00/0.54	  , +^#(x, s(y)) -> c_5(+^#(x, y))
0.00/0.54	  , +^#(s(x), y) -> c_6(+^#(x, y)) }
0.00/0.54	Obligation:
0.00/0.54	  innermost runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	The weightgap principle applies (using the following constant
0.00/0.54	growth matrix-interpretation)
0.00/0.54	
0.00/0.54	The following argument positions are usable:
0.00/0.54	  Uargs(c_1) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
0.00/0.54	  Uargs(c_6) = {1}
0.00/0.54	
0.00/0.54	TcT has computed the following constructor-restricted matrix
0.00/0.54	interpretation.
0.00/0.54	
0.00/0.54	             [0] = [0]           
0.00/0.54	                   [0]           
0.00/0.54	                                 
0.00/0.54	         [s](x1) = [1 0] x1 + [0]
0.00/0.54	                   [0 0]      [0]
0.00/0.54	                                 
0.00/0.54	  [double^#](x1) = [1]           
0.00/0.54	                   [1]           
0.00/0.54	                                 
0.00/0.54	       [c_1](x1) = [1 0] x1 + [2]
0.00/0.54	                   [0 1]      [1]
0.00/0.54	                                 
0.00/0.54	   [+^#](x1, x2) = [0]           
0.00/0.54	                   [1]           
0.00/0.54	                                 
0.00/0.54	           [c_2] = [0]           
0.00/0.54	                   [1]           
0.00/0.54	                                 
0.00/0.54	       [c_3](x1) = [1 0] x1 + [1]
0.00/0.54	                   [0 1]      [0]
0.00/0.54	                                 
0.00/0.54	           [c_4] = [1]           
0.00/0.54	                   [1]           
0.00/0.54	                                 
0.00/0.54	       [c_5](x1) = [1 0] x1 + [2]
0.00/0.54	                   [0 1]      [0]
0.00/0.54	                                 
0.00/0.54	       [c_6](x1) = [1 0] x1 + [2]
0.00/0.54	                   [0 1]      [1]
0.00/0.54	
0.00/0.54	The order satisfies the following ordering constraints:
0.00/0.54	
0.00/0.54	     [double^#(x)] = [1]               
0.00/0.54	                     [1]               
0.00/0.54	                   ? [2]               
0.00/0.54	                     [2]               
0.00/0.54	                   = [c_1(+^#(x, x))]  
0.00/0.54	                                       
0.00/0.54	   [double^#(0())] = [1]               
0.00/0.54	                     [1]               
0.00/0.54	                   > [0]               
0.00/0.54	                     [1]               
0.00/0.54	                   = [c_2()]           
0.00/0.54	                                       
0.00/0.54	  [double^#(s(x))] = [1]               
0.00/0.54	                     [1]               
0.00/0.54	                   ? [2]               
0.00/0.54	                     [1]               
0.00/0.54	                   = [c_3(double^#(x))]
0.00/0.54	                                       
0.00/0.54	     [+^#(x, 0())] = [0]               
0.00/0.54	                     [1]               
0.00/0.54	                   ? [1]               
0.00/0.54	                     [1]               
0.00/0.54	                   = [c_4()]           
0.00/0.54	                                       
0.00/0.54	    [+^#(x, s(y))] = [0]               
0.00/0.54	                     [1]               
0.00/0.54	                   ? [2]               
0.00/0.54	                     [1]               
0.00/0.54	                   = [c_5(+^#(x, y))]  
0.00/0.54	                                       
0.00/0.54	    [+^#(s(x), y)] = [0]               
0.00/0.54	                     [1]               
0.00/0.54	                   ? [2]               
0.00/0.54	                     [2]               
0.00/0.54	                   = [c_6(+^#(x, y))]  
0.00/0.54	                                       
0.00/0.54	
0.00/0.54	Further, it can be verified that all rules not oriented are covered by the weightgap condition.
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(n^1)).
0.00/0.54	
0.00/0.54	Strict DPs:
0.00/0.54	  { double^#(x) -> c_1(+^#(x, x))
0.00/0.54	  , double^#(s(x)) -> c_3(double^#(x))
0.00/0.54	  , +^#(x, 0()) -> c_4()
0.00/0.54	  , +^#(x, s(y)) -> c_5(+^#(x, y))
0.00/0.54	  , +^#(s(x), y) -> c_6(+^#(x, y)) }
0.00/0.54	Weak DPs: { double^#(0()) -> c_2() }
0.00/0.54	Obligation:
0.00/0.54	  innermost runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	We estimate the number of application of {3} by applications of
0.00/0.54	Pre({3}) = {1,4,5}. Here rules are labeled as follows:
0.00/0.54	
0.00/0.54	  DPs:
0.00/0.54	    { 1: double^#(x) -> c_1(+^#(x, x))
0.00/0.54	    , 2: double^#(s(x)) -> c_3(double^#(x))
0.00/0.54	    , 3: +^#(x, 0()) -> c_4()
0.00/0.54	    , 4: +^#(x, s(y)) -> c_5(+^#(x, y))
0.00/0.54	    , 5: +^#(s(x), y) -> c_6(+^#(x, y))
0.00/0.54	    , 6: double^#(0()) -> c_2() }
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(n^1)).
0.00/0.54	
0.00/0.54	Strict DPs:
0.00/0.54	  { double^#(x) -> c_1(+^#(x, x))
0.00/0.54	  , double^#(s(x)) -> c_3(double^#(x))
0.00/0.54	  , +^#(x, s(y)) -> c_5(+^#(x, y))
0.00/0.54	  , +^#(s(x), y) -> c_6(+^#(x, y)) }
0.00/0.54	Weak DPs:
0.00/0.54	  { double^#(0()) -> c_2()
0.00/0.54	  , +^#(x, 0()) -> c_4() }
0.00/0.54	Obligation:
0.00/0.54	  innermost runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.54	closed under successors. The DPs are removed.
0.00/0.54	
0.00/0.54	{ double^#(0()) -> c_2()
0.00/0.54	, +^#(x, 0()) -> c_4() }
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(n^1)).
0.00/0.54	
0.00/0.54	Strict DPs:
0.00/0.54	  { double^#(x) -> c_1(+^#(x, x))
0.00/0.54	  , double^#(s(x)) -> c_3(double^#(x))
0.00/0.54	  , +^#(x, s(y)) -> c_5(+^#(x, y))
0.00/0.54	  , +^#(s(x), y) -> c_6(+^#(x, y)) }
0.00/0.54	Obligation:
0.00/0.54	  innermost runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	We use the processor 'matrix interpretation of dimension 1' to
0.00/0.54	orient following rules strictly.
0.00/0.54	
0.00/0.54	DPs:
0.00/0.54	  { 2: double^#(s(x)) -> c_3(double^#(x))
0.00/0.54	  , 4: +^#(s(x), y) -> c_6(+^#(x, y)) }
0.00/0.54	
0.00/0.54	Sub-proof:
0.00/0.54	----------
0.00/0.54	  The following argument positions are usable:
0.00/0.54	    Uargs(c_1) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
0.00/0.54	    Uargs(c_6) = {1}
0.00/0.54	  
0.00/0.54	  TcT has computed the following constructor-based matrix
0.00/0.54	  interpretation satisfying not(EDA).
0.00/0.54	  
0.00/0.54	           [s](x1) = [1] x1 + [4]
0.00/0.54	                                 
0.00/0.54	    [double^#](x1) = [2] x1 + [0]
0.00/0.54	                                 
0.00/0.54	         [c_1](x1) = [1] x1 + [0]
0.00/0.54	                                 
0.00/0.54	     [+^#](x1, x2) = [1] x1 + [0]
0.00/0.54	                                 
0.00/0.54	         [c_3](x1) = [1] x1 + [1]
0.00/0.54	                                 
0.00/0.54	         [c_5](x1) = [1] x1 + [0]
0.00/0.54	                                 
0.00/0.54	         [c_6](x1) = [1] x1 + [1]
0.00/0.54	  
0.00/0.54	  The order satisfies the following ordering constraints:
0.00/0.54	  
0.00/0.54	       [double^#(x)] =  [2] x + [0]       
0.00/0.54	                     >= [1] x + [0]       
0.00/0.54	                     =  [c_1(+^#(x, x))]  
0.00/0.54	                                          
0.00/0.54	    [double^#(s(x))] =  [2] x + [8]       
0.00/0.54	                     >  [2] x + [1]       
0.00/0.54	                     =  [c_3(double^#(x))]
0.00/0.54	                                          
0.00/0.54	      [+^#(x, s(y))] =  [1] x + [0]       
0.00/0.54	                     >= [1] x + [0]       
0.00/0.54	                     =  [c_5(+^#(x, y))]  
0.00/0.54	                                          
0.00/0.54	      [+^#(s(x), y)] =  [1] x + [4]       
0.00/0.54	                     >  [1] x + [1]       
0.00/0.54	                     =  [c_6(+^#(x, y))]  
0.00/0.54	                                          
0.00/0.54	
0.00/0.54	We return to the main proof. Consider the set of all dependency
0.00/0.54	pairs
0.00/0.54	
0.00/0.54	:
0.00/0.54	  { 1: double^#(x) -> c_1(+^#(x, x))
0.00/0.54	  , 2: double^#(s(x)) -> c_3(double^#(x))
0.00/0.54	  , 3: +^#(x, s(y)) -> c_5(+^#(x, y))
0.00/0.54	  , 4: +^#(s(x), y) -> c_6(+^#(x, y)) }
0.00/0.54	
0.00/0.54	Processor 'matrix interpretation of dimension 1' induces the
0.00/0.54	complexity certificate YES(?,O(n^1)) on application of dependency
0.00/0.54	pairs {2,4}. These cover all (indirect) predecessors of dependency
0.00/0.54	pairs {1,2,4}, their number of application is equally bounded. The
0.00/0.54	dependency pairs are shifted into the weak component.
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(n^1)).
0.00/0.54	
0.00/0.54	Strict DPs: { +^#(x, s(y)) -> c_5(+^#(x, y)) }
0.00/0.54	Weak DPs:
0.00/0.54	  { double^#(x) -> c_1(+^#(x, x))
0.00/0.54	  , double^#(s(x)) -> c_3(double^#(x))
0.00/0.54	  , +^#(s(x), y) -> c_6(+^#(x, y)) }
0.00/0.54	Obligation:
0.00/0.54	  innermost runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(n^1))
0.00/0.54	
0.00/0.54	We use the processor 'matrix interpretation of dimension 1' to
0.00/0.54	orient following rules strictly.
0.00/0.54	
0.00/0.54	DPs:
0.00/0.54	  { 1: +^#(x, s(y)) -> c_5(+^#(x, y))
0.00/0.54	  , 3: double^#(s(x)) -> c_3(double^#(x)) }
0.00/0.54	
0.00/0.54	Sub-proof:
0.00/0.54	----------
0.00/0.54	  The following argument positions are usable:
0.00/0.54	    Uargs(c_1) = {1}, Uargs(c_3) = {1}, Uargs(c_5) = {1},
0.00/0.54	    Uargs(c_6) = {1}
0.00/0.54	  
0.00/0.54	  TcT has computed the following constructor-based matrix
0.00/0.54	  interpretation satisfying not(EDA).
0.00/0.54	  
0.00/0.54	           [s](x1) = [1] x1 + [4]
0.00/0.54	                                 
0.00/0.54	    [double^#](x1) = [1] x1 + [4]
0.00/0.54	                                 
0.00/0.54	         [c_1](x1) = [1] x1 + [0]
0.00/0.54	                                 
0.00/0.54	     [+^#](x1, x2) = [1] x2 + [4]
0.00/0.54	                                 
0.00/0.54	         [c_3](x1) = [1] x1 + [1]
0.00/0.54	                                 
0.00/0.54	         [c_5](x1) = [1] x1 + [1]
0.00/0.54	                                 
0.00/0.54	         [c_6](x1) = [1] x1 + [0]
0.00/0.54	  
0.00/0.54	  The order satisfies the following ordering constraints:
0.00/0.54	  
0.00/0.54	       [double^#(x)] =  [1] x + [4]       
0.00/0.54	                     >= [1] x + [4]       
0.00/0.54	                     =  [c_1(+^#(x, x))]  
0.00/0.54	                                          
0.00/0.54	    [double^#(s(x))] =  [1] x + [8]       
0.00/0.54	                     >  [1] x + [5]       
0.00/0.54	                     =  [c_3(double^#(x))]
0.00/0.54	                                          
0.00/0.54	      [+^#(x, s(y))] =  [1] y + [8]       
0.00/0.54	                     >  [1] y + [5]       
0.00/0.54	                     =  [c_5(+^#(x, y))]  
0.00/0.54	                                          
0.00/0.54	      [+^#(s(x), y)] =  [1] y + [4]       
0.00/0.54	                     >= [1] y + [4]       
0.00/0.54	                     =  [c_6(+^#(x, y))]  
0.00/0.54	                                          
0.00/0.54	
0.00/0.54	We return to the main proof. Consider the set of all dependency
0.00/0.54	pairs
0.00/0.54	
0.00/0.54	:
0.00/0.54	  { 1: +^#(x, s(y)) -> c_5(+^#(x, y))
0.00/0.54	  , 2: double^#(x) -> c_1(+^#(x, x))
0.00/0.54	  , 3: double^#(s(x)) -> c_3(double^#(x))
0.00/0.54	  , 4: +^#(s(x), y) -> c_6(+^#(x, y)) }
0.00/0.54	
0.00/0.54	Processor 'matrix interpretation of dimension 1' induces the
0.00/0.54	complexity certificate YES(?,O(n^1)) on application of dependency
0.00/0.54	pairs {1,3}. These cover all (indirect) predecessors of dependency
0.00/0.54	pairs {1,2,3}, their number of application is equally bounded. The
0.00/0.54	dependency pairs are shifted into the weak component.
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(1)).
0.00/0.54	
0.00/0.54	Weak DPs:
0.00/0.54	  { double^#(x) -> c_1(+^#(x, x))
0.00/0.54	  , double^#(s(x)) -> c_3(double^#(x))
0.00/0.54	  , +^#(x, s(y)) -> c_5(+^#(x, y))
0.00/0.54	  , +^#(s(x), y) -> c_6(+^#(x, y)) }
0.00/0.54	Obligation:
0.00/0.54	  innermost runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(1))
0.00/0.54	
0.00/0.54	The following weak DPs constitute a sub-graph of the DG that is
0.00/0.54	closed under successors. The DPs are removed.
0.00/0.54	
0.00/0.54	{ double^#(x) -> c_1(+^#(x, x))
0.00/0.54	, double^#(s(x)) -> c_3(double^#(x))
0.00/0.54	, +^#(x, s(y)) -> c_5(+^#(x, y))
0.00/0.54	, +^#(s(x), y) -> c_6(+^#(x, y)) }
0.00/0.54	
0.00/0.54	We are left with following problem, upon which TcT provides the
0.00/0.54	certificate YES(O(1),O(1)).
0.00/0.54	
0.00/0.54	Rules: Empty
0.00/0.54	Obligation:
0.00/0.54	  innermost runtime complexity
0.00/0.54	Answer:
0.00/0.54	  YES(O(1),O(1))
0.00/0.54	
0.00/0.54	Empty rules are trivially bounded
0.00/0.54	
0.00/0.54	Hurray, we answered YES(O(1),O(n^1))
0.00/0.54	EOF