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

We add the following weak dependency pairs:

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

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] = [0]           
                [0]           
                              
  [fac^#](x1) = [2 0] x1 + [0]
                [0 0]      [0]
                              
    [c_2](x1) = [1 0] x1 + [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()]              
                                        
  [fac^#(s(x))] =  [2 0] x + [4]        
                   [0 0]     [0]        
                >= [2 0] x + [4]        
                   [0 0]     [0]        
                =  [c_2(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()
  , fac^#(s(x)) -> c_2(fac^#(p(s(x))))
  , fac^#(0()) -> c_3() }
Weak Trs: { 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: p^#(s(x)) -> c_1()
    , 2: fac^#(s(x)) -> c_2(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: { fac^#(s(x)) -> c_2(fac^#(p(s(x)))) }
Weak DPs:
  { p^#(s(x)) -> c_1()
  , fac^#(0()) -> c_3() }
Weak Trs: { 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.

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

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

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)).
  
                  [0 0 1]      [0]
        [p](x1) = [0 1 1] x1 + [0]
                  [1 1 0]      [0]
                                  
                  [1 0 0]      [1]
        [s](x1) = [0 1 4] x1 + [1]
                  [1 0 0]      [0]
                                  
                  [4 0 0]      [2]
    [fac^#](x1) = [0 1 1] x1 + [0]
                  [0 0 0]      [3]
                                  
                  [1 0 1]      [0]
      [c_2](x1) = [0 0 0] x1 + [0]
                  [0 0 0]      [0]
  
  The order satisfies the following ordering constraints:
  
        [p(s(x))] =  [1 0 0]     [0]      
                     [1 1 4] x + [1]      
                     [1 1 4]     [2]      
                  >= [1 0 0]     [0]      
                     [0 1 0] x + [0]      
                     [0 0 1]     [0]      
                  =  [x]                  
                                          
    [fac^#(s(x))] =  [4 0 0]     [6]      
                     [1 1 4] x + [1]      
                     [0 0 0]     [3]      
                  >  [4 0 0]     [5]      
                     [0 0 0] x + [0]      
                     [0 0 0]     [0]      
                  =  [c_2(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: { fac^#(s(x)) -> c_2(fac^#(p(s(x)))) }
Weak Trs: { 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.

{ fac^#(s(x)) -> c_2(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:
  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))