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:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following weak dependency pairs:

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

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)
  , -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y))))
  , -^#(0(), y) -> c_3()
  , p^#(0()) -> c_4()
  , p^#(s(x)) -> c_5(x) }
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:
  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)
  , -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y))))
  , -^#(0(), y) -> c_3()
  , p^#(0()) -> c_4()
  , p^#(s(x)) -> c_5(x) }
Strict Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  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) = {3}

TcT has computed the following constructor-restricted matrix
interpretation.

                [0] = [0]                                 
                      [2]                                 
                                                          
            [s](x1) = [1 2] x1 + [1]                      
                      [0 0]      [0]                      
                                                          
            [p](x1) = [1 2] x1 + [0]                      
                      [2 2]      [0]                      
                                                          
      [-^#](x1, x2) = [1 1] x1 + [2 0] x2 + [0]           
                      [2 1]      [0 0]      [0]           
                                                          
          [c_1](x1) = [1 1] x1 + [1]                      
                      [1 1]      [0]                      
                                                          
  [c_2](x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [2]
                      [1 1]      [2 2]      [0 1]      [2]
                                                          
              [c_3] = [1]                                 
                      [1]                                 
                                                          
          [p^#](x1) = [1 1] x1 + [1]                      
                      [2 1]      [2]                      
                                                          
              [c_4] = [2]                                 
                      [0]                                 
                                                          
          [c_5](x1) = [1 1] x1 + [1]                      
                      [1 0]      [0]                      

The following symbols are considered usable

  {p, -^#, p^#}

The order satisfies the following ordering constraints:

        [p(0())] = [4]                         
                   [4]                         
                 > [0]                         
                   [2]                         
                 = [0()]                       
                                               
       [p(s(x))] = [1 2] x + [1]               
                   [2 4]     [2]               
                 > [1 0] x + [0]               
                   [0 1]     [0]               
                 = [x]                         
                                               
   [-^#(x, 0())] = [1 1] x + [0]               
                   [2 1]     [0]               
                 ? [1 1] x + [1]               
                   [1 1]     [0]               
                 = [c_1(x)]                    
                                               
  [-^#(x, s(y))] = [2 4] y + [1 1] x + [2]     
                   [0 0]     [2 1]     [0]     
                 ? [2 4] y + [1 1] x + [4]     
                   [2 2]     [3 2]     [2]     
                 = [c_2(x, y, -^#(x, p(s(y))))]
                                               
   [-^#(0(), y)] = [2 0] y + [2]               
                   [0 0]     [2]               
                 > [1]                         
                   [1]                         
                 = [c_3()]                     
                                               
      [p^#(0())] = [3]                         
                   [4]                         
                 > [2]                         
                   [0]                         
                 = [c_4()]                     
                                               
     [p^#(s(x))] = [1 2] x + [2]               
                   [2 4]     [4]               
                 > [1 1] x + [1]               
                   [1 0]     [0]               
                 = [c_5(x)]                    
                                               

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)
  , -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y)))) }
Weak DPs:
  { -^#(0(), y) -> c_3()
  , p^#(0()) -> c_4()
  , p^#(s(x)) -> c_5(x) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  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.

{ -^#(0(), y) -> c_3()
, p^#(0()) -> c_4() }

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

We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.

DPs:
  { 1: -^#(x, 0()) -> c_1(x)
  , 3: p^#(s(x)) -> c_5(x) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_2) = {3}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
          [-](x1, x2) = [7] x1 + [7] x2 + [0]         
                                                      
                  [0] = [0]                           
                                                      
              [s](x1) = [1] x1 + [0]                  
                                                      
     [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
                                                      
    [greater](x1, x2) = [1] x1 + [1] x2 + [0]         
                                                      
              [p](x1) = [1] x1 + [0]                  
                                                      
        [-^#](x1, x2) = [7] x1 + [2]                  
                                                      
            [c_1](x1) = [7] x1 + [0]                  
                                                      
    [c_2](x1, x2, x3) = [1] x3 + [0]                  
                                                      
                [c_3] = [0]                           
                                                      
            [p^#](x1) = [7] x1 + [7]                  
                                                      
                [c_4] = [0]                           
                                                      
            [c_5](x1) = [7] x1 + [6]                  
  
  The following symbols are considered usable
  
    {p, -^#, p^#}
  
  The order satisfies the following ordering constraints:
  
          [p(0())] =  [0]                         
                   >= [0]                         
                   =  [0()]                       
                                                  
         [p(s(x))] =  [1] x + [0]                 
                   >= [1] x + [0]                 
                   =  [x]                         
                                                  
     [-^#(x, 0())] =  [7] x + [2]                 
                   >  [7] x + [0]                 
                   =  [c_1(x)]                    
                                                  
    [-^#(x, s(y))] =  [7] x + [2]                 
                   >= [7] x + [2]                 
                   =  [c_2(x, y, -^#(x, p(s(y))))]
                                                  
       [p^#(s(x))] =  [7] x + [7]                 
                   >  [7] x + [6]                 
                   =  [c_5(x)]                    
                                                  

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(n^1)).

Strict DPs: { -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y)))) }
Weak DPs:
  { -^#(x, 0()) -> c_1(x)
  , p^#(s(x)) -> c_5(x) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  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, y, -^#(x, p(s(y))))
  , 3: p^#(s(x)) -> c_5(x) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_2) = {3}
  
  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]      [1]             
              [s](x1) = [1 0 0] x1 + [0]             
                        [0 1 1]      [0]             
                                                     
                        [0]                          
     [if](x1, x2, x3) = [0]                          
                        [0]                          
                                                     
                        [0]                          
    [greater](x1, x2) = [0]                          
                        [0]                          
                                                     
                        [0 1 0]      [0]             
              [p](x1) = [0 0 1] x1 + [0]             
                        [0 0 4]      [0]             
                                                     
                        [4 0 0]      [0]             
        [-^#](x1, x2) = [0 0 0] x2 + [1]             
                        [0 0 0]      [4]             
                                                     
                        [0]                          
            [c_1](x1) = [1]                          
                        [4]                          
                                                     
                        [1 1 0]      [1]             
    [c_2](x1, x2, x3) = [0 0 0] x3 + [1]             
                        [0 0 0]      [3]             
                                                     
                        [0]                          
                [c_3] = [0]                          
                        [0]                          
                                                     
                        [7]                          
            [p^#](x1) = [7]                          
                        [7]                          
                                                     
                        [0]                          
                [c_4] = [0]                          
                        [0]                          
                                                     
                        [6]                          
            [c_5](x1) = [7]                          
                        [7]                          
  
  The following symbols are considered usable
  
    {p, -^#, 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 1 1] x + [0]             
                      [0 4 4]     [0]             
                   >= [1 0 0]     [0]             
                      [0 1 0] x + [0]             
                      [0 0 1]     [0]             
                   =  [x]                         
                                                  
     [-^#(x, 0())] =  [0]                         
                      [1]                         
                      [4]                         
                   >= [0]                         
                      [1]                         
                      [4]                         
                   =  [c_1(x)]                    
                                                  
    [-^#(x, s(y))] =  [4 0 0]     [4]             
                      [0 0 0] y + [1]             
                      [0 0 0]     [4]             
                   >  [4 0 0]     [2]             
                      [0 0 0] y + [1]             
                      [0 0 0]     [3]             
                   =  [c_2(x, y, -^#(x, p(s(y))))]
                                                  
       [p^#(s(x))] =  [7]                         
                      [7]                         
                      [7]                         
                   >  [6]                         
                      [7]                         
                      [7]                         
                   =  [c_5(x)]                    
                                                  

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, 0()) -> c_1(x)
  , -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y))))
  , p^#(s(x)) -> c_5(x) }
Weak Trs:
  { p(0()) -> 0()
  , p(s(x)) -> x }
Obligation:
  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, 0()) -> c_1(x)
, -^#(x, s(y)) -> c_2(x, y, -^#(x, p(s(y))))
, p^#(s(x)) -> c_5(x) }

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:
  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:
  runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

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