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

Strict Trs:
  { p(s(x)) -> x
  , fac(s(x)) -> times(s(x), fac(p(s(x))))
  , fac(0()) -> s(0()) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following weak dependency pairs:

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

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:
  { p^#(s(x)) -> c_1(x)
  , fac^#(s(x)) -> c_2(x, fac^#(p(s(x))))
  , fac^#(0()) -> c_3() }
Strict Trs:
  { p(s(x)) -> x
  , fac(s(x)) -> times(s(x), fac(p(s(x))))
  , fac(0()) -> s(0()) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

We replace rewrite rules by usable rules:

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

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

Strict DPs:
  { p^#(s(x)) -> c_1(x)
  , fac^#(s(x)) -> c_2(x, fac^#(p(s(x))))
  , fac^#(0()) -> c_3() }
Strict Trs: { 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(fac^#) = {1}, Uargs(c_2) = {2}

TcT has computed the following constructor-restricted matrix
interpretation.

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

The order satisfies the following ordering constraints:

      [p(s(x))] =  [1 0] x + [2]           
                   [0 1]     [0]           
                >  [1 0] x + [0]           
                   [0 1]     [0]           
                =  [x]                     
                                           
    [p^#(s(x))] =  [0]                     
                   [0]                     
                >= [0]                     
                   [0]                     
                =  [c_1(x)]                
                                           
  [fac^#(s(x))] =  [2 0] x + [4]           
                   [0 0]     [0]           
                >= [2 0] x + [4]           
                   [0 0]     [0]           
                =  [c_2(x, fac^#(p(s(x))))]
                                           
   [fac^#(0())] =  [0]                     
                   [0]                     
                >= [0]                     
                   [0]                     
                =  [c_3()]                 
                                           

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:
  { p^#(s(x)) -> c_1(x)
  , fac^#(s(x)) -> c_2(x, fac^#(p(s(x))))
  , fac^#(0()) -> c_3() }
Weak Trs: { p(s(x)) -> x }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

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

  DPs:
    { 1: p^#(s(x)) -> c_1(x)
    , 2: fac^#(s(x)) -> c_2(x, fac^#(p(s(x))))
    , 3: fac^#(0()) -> c_3() }

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

Strict DPs:
  { p^#(s(x)) -> c_1(x)
  , fac^#(s(x)) -> c_2(x, fac^#(p(s(x)))) }
Weak DPs: { fac^#(0()) -> c_3() }
Weak Trs: { 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.

{ fac^#(0()) -> c_3() }

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

Strict DPs:
  { p^#(s(x)) -> c_1(x)
  , fac^#(s(x)) -> c_2(x, fac^#(p(s(x)))) }
Weak Trs: { 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: p^#(s(x)) -> c_1(x) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_2) = {2}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
          [p](x1) = [0]         
                                
          [s](x1) = [0]         
                                
        [p^#](x1) = [1] x1 + [6]
                                
        [c_1](x1) = [1]         
                                
      [fac^#](x1) = [0]         
                                
    [c_2](x1, x2) = [4] x2 + [0]
  
  The order satisfies the following ordering constraints:
  
        [p(s(x))] =  [0]                     
                  ?  [1] x + [0]             
                  =  [x]                     
                                             
      [p^#(s(x))] =  [6]                     
                  >  [1]                     
                  =  [c_1(x)]                
                                             
    [fac^#(s(x))] =  [0]                     
                  >= [0]                     
                  =  [c_2(x, fac^#(p(s(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: { fac^#(s(x)) -> c_2(x, fac^#(p(s(x)))) }
Weak DPs: { p^#(s(x)) -> c_1(x) }
Weak Trs: { 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: fac^#(s(x)) -> c_2(x, fac^#(p(s(x)))) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_2) = {2}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA) and not(IDA(1)).
  
                    [0 0 1]      [0]
          [p](x1) = [0 1 0] x1 + [0]
                    [0 3 1]      [0]
                                    
                    [1 0 0]      [1]
          [s](x1) = [0 1 5] x1 + [1]
                    [1 0 0]      [0]
                                    
                    [0]             
        [p^#](x1) = [0]             
                    [0]             
                                    
                    [0]             
        [c_1](x1) = [0]             
                    [0]             
                                    
                    [5 0 0]      [4]
      [fac^#](x1) = [0 1 1] x1 + [0]
                    [0 0 0]      [3]
                                    
                    [1 0 1]      [0]
    [c_2](x1, x2) = [0 0 0] x2 + [0]
                    [0 0 0]      [0]
  
  The order satisfies the following ordering constraints:
  
        [p(s(x))] =  [1 0  0]     [0]        
                     [0 1  5] x + [1]        
                     [1 3 15]     [3]        
                  >= [1 0 0]     [0]         
                     [0 1 0] x + [0]         
                     [0 0 1]     [0]         
                  =  [x]                     
                                             
      [p^#(s(x))] =  [0]                     
                     [0]                     
                     [0]                     
                  >= [0]                     
                     [0]                     
                     [0]                     
                  =  [c_1(x)]                
                                             
    [fac^#(s(x))] =  [5 0 0]     [9]         
                     [1 1 5] x + [1]         
                     [0 0 0]     [3]         
                  >  [5 0 0]     [7]         
                     [0 0 0] x + [0]         
                     [0 0 0]     [0]         
                  =  [c_2(x, fac^#(p(s(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:
  { p^#(s(x)) -> c_1(x)
  , fac^#(s(x)) -> c_2(x, fac^#(p(s(x)))) }
Weak Trs: { 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.

{ p^#(s(x)) -> c_1(x)
, fac^#(s(x)) -> c_2(x, fac^#(p(s(x)))) }

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

Weak Trs: { 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))