YES(O(1),O(n^1))

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict Trs:
  { -(x, 0()) -> x
  , -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
  , -(0(), y) -> 0()
  , p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following innermost weak dependency pairs:

Strict DPs:
  { -^#(x, 0()) -> c_1()
  , -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
  , -^#(0(), y) -> c_3()
  , p^#(0()) -> c_4()
  , p^#(s(x)) -> c_5() }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { -^#(x, 0()) -> c_1()
  , -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
  , -^#(0(), y) -> c_3()
  , p^#(0()) -> c_4()
  , p^#(s(x)) -> c_5() }
Strict Trs:
  { -(x, 0()) -> x
  , -(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0())
  , -(0(), y) -> 0()
  , p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We replace rewrite rules by usable rules:

  Strict Usable Rules:
    { p(0()) -> 0()
    , p(s(x)) -> x }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { -^#(x, 0()) -> c_1()
  , -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
  , -^#(0(), y) -> c_3()
  , p^#(0()) -> c_4()
  , p^#(s(x)) -> c_5() }
Strict Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(-^#) = {2}, Uargs(c_2) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

            [0] = [0]                      
                  [0]                      
                                           
        [s](x1) = [1 0] x1 + [0]           
                  [0 1]      [0]           
                                           
        [p](x1) = [1 0] x1 + [2]           
                  [0 1]      [0]           
                                           
  [-^#](x1, x2) = [2 1] x1 + [2 0] x2 + [0]
                  [2 1]      [0 0]      [0]
                                           
          [c_1] = [1]                      
                  [1]                      
                                           
      [c_2](x1) = [1 0] x1 + [2]           
                  [0 1]      [2]           
                                           
          [c_3] = [1]                      
                  [1]                      
                                           
      [p^#](x1) = [1 2] x1 + [2]           
                  [1 2]      [2]           
                                           
          [c_4] = [1]                      
                  [1]                      
                                           
          [c_5] = [1]                      
                  [1]                      

The following symbols are considered usable

  {p, -^#, p^#}

The order satisfies the following ordering constraints:

        [p(0())] = [2]                    
                   [0]                    
                 > [0]                    
                   [0]                    
                 = [0()]                  
                                          
       [p(s(x))] = [1 0] x + [2]          
                   [0 1]     [0]          
                 > [1 0] x + [0]          
                   [0 1]     [0]          
                 = [x]                    
                                          
   [-^#(x, 0())] = [2 1] x + [0]          
                   [2 1]     [0]          
                 ? [1]                    
                   [1]                    
                 = [c_1()]                
                                          
  [-^#(x, s(y))] = [2 0] y + [2 1] x + [0]
                   [0 0]     [2 1]     [0]
                 ? [2 0] y + [2 1] x + [6]
                   [0 0]     [2 1]     [2]
                 = [c_2(-^#(x, p(s(y))))] 
                                          
   [-^#(0(), y)] = [2 0] y + [0]          
                   [0 0]     [0]          
                 ? [1]                    
                   [1]                    
                 = [c_3()]                
                                          
      [p^#(0())] = [2]                    
                   [2]                    
                 > [1]                    
                   [1]                    
                 = [c_4()]                
                                          
     [p^#(s(x))] = [1 2] x + [2]          
                   [1 2]     [2]          
                 > [1]                    
                   [1]                    
                 = [c_5()]                
                                          

Further, it can be verified that all rules not oriented are covered by the weightgap condition.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { -^#(x, 0()) -> c_1()
  , -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
  , -^#(0(), y) -> c_3() }
Weak DPs:
  { p^#(0()) -> c_4()
  , p^#(s(x)) -> c_5() }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We estimate the number of application of {1,3} by applications of
Pre({1,3}) = {2}. Here rules are labeled as follows:

  DPs:
    { 1: -^#(x, 0()) -> c_1()
    , 2: -^#(x, s(y)) -> c_2(-^#(x, p(s(y))))
    , 3: -^#(0(), y) -> c_3()
    , 4: p^#(0()) -> c_4()
    , 5: p^#(s(x)) -> c_5() }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs: { -^#(x, s(y)) -> c_2(-^#(x, p(s(y)))) }
Weak DPs:
  { -^#(x, 0()) -> c_1()
  , -^#(0(), y) -> c_3()
  , p^#(0()) -> c_4()
  , p^#(s(x)) -> c_5() }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ -^#(x, 0()) -> c_1()
, -^#(0(), y) -> c_3()
, p^#(0()) -> c_4()
, p^#(s(x)) -> c_5() }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs: { -^#(x, s(y)) -> c_2(-^#(x, p(s(y)))) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We use the processor 'matrix interpretation of dimension 3' to
orient following rules strictly.

DPs:
  { 1: -^#(x, s(y)) -> c_2(-^#(x, p(s(y)))) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_2) = {1}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA) and not(IDA(1)).
  
                        [7 7 0]      [7 7 0]      [0]             
          [-](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]             
                        [0 0 0]      [0 0 0]      [0]             
                                                                  
                        [0]                                       
                  [0] = [0]                                       
                        [0]                                       
                                                                  
                        [1 0 0]      [2]                          
              [s](x1) = [1 0 0] x1 + [0]                          
                        [0 1 1]      [0]                          
                                                                  
                        [1 0 0]      [1 0 0]      [1 0 0]      [0]
     [if](x1, x2, x3) = [1 0 0] x1 + [0 0 0] x2 + [1 0 0] x3 + [0]
                        [0 1 1]      [0 0 1]      [0 0 1]      [0]
                                                                  
                        [1 0 0]      [1 0 0]      [0]             
    [greater](x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]             
                        [0 1 0]      [0 1 0]      [0]             
                                                                  
                        [0 1 0]      [0]                          
              [p](x1) = [0 0 2] x1 + [0]                          
                        [0 0 2]      [0]                          
                                                                  
                        [4 0 0]      [0]                          
        [-^#](x1, x2) = [0 0 0] x2 + [4]                          
                        [4 0 0]      [0]                          
                                                                  
                        [0]                                       
                [c_1] = [0]                                       
                        [0]                                       
                                                                  
                        [1 0 0]      [5]                          
            [c_2](x1) = [0 0 0] x1 + [3]                          
                        [0 0 0]      [7]                          
                                                                  
                        [0]                                       
                [c_3] = [0]                                       
                        [0]                                       
                                                                  
                        [7 7 0]      [0]                          
            [p^#](x1) = [0 0 0] x1 + [0]                          
                        [0 0 0]      [0]                          
                                                                  
                        [0]                                       
                [c_4] = [0]                                       
                        [0]                                       
                                                                  
                        [0]                                       
                [c_5] = [0]                                       
                        [0]                                       
  
  The following symbols are considered usable
  
    {p, -^#}
  
  The order satisfies the following ordering constraints:
  
          [p(0())] =  [0]                   
                      [0]                   
                      [0]                   
                   >= [0]                   
                      [0]                   
                      [0]                   
                   =  [0()]                 
                                            
         [p(s(x))] =  [1 0 0]     [0]       
                      [0 2 2] x + [0]       
                      [0 2 2]     [0]       
                   >= [1 0 0]     [0]       
                      [0 1 0] x + [0]       
                      [0 0 1]     [0]       
                   =  [x]                   
                                            
    [-^#(x, s(y))] =  [4 0 0]     [8]       
                      [0 0 0] y + [4]       
                      [4 0 0]     [8]       
                   >  [4 0 0]     [5]       
                      [0 0 0] y + [3]       
                      [0 0 0]     [7]       
                   =  [c_2(-^#(x, p(s(y))))]
                                            

The strictly oriented rules are moved into the weak component.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Weak DPs: { -^#(x, s(y)) -> c_2(-^#(x, p(s(y)))) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ -^#(x, s(y)) -> c_2(-^#(x, p(s(y)))) }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

No rule is usable, rules are removed from the input problem.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

Hurray, we answered YES(O(1),O(n^1))