We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict Trs:
  { fac(s(x)) -> *(fac(p(s(x))), s(x))
  , p(s(s(x))) -> s(p(s(x)))
  , p(s(0())) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

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

The following argument positions are usable:
  Uargs(fac) = {1}, Uargs(s) = {1}, Uargs(*) = {1}

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

    [fac](x1) = [1] x1 + [0]
                            
      [s](x1) = [1] x1 + [0]
                            
  [*](x1, x2) = [1] x1 + [0]
                            
      [p](x1) = [1] x1 + [1]
                            
          [0] = [5]         

The order satisfies the following ordering constraints:

   [fac(s(x))] =  [1] x + [0]            
               ?  [1] x + [1]            
               =  [*(fac(p(s(x))), s(x))]
                                         
  [p(s(s(x)))] =  [1] x + [1]            
               >= [1] x + [1]            
               =  [s(p(s(x)))]           
                                         
   [p(s(0()))] =  [6]                    
               >  [5]                    
               =  [0()]                  
                                         

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^2)).

Strict Trs:
  { fac(s(x)) -> *(fac(p(s(x))), s(x))
  , p(s(s(x))) -> s(p(s(x))) }
Weak Trs: { p(s(0())) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

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

The following argument positions are usable:
  Uargs(fac) = {1}, Uargs(s) = {1}, Uargs(*) = {1}

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

    [fac](x1) = [1] x1 + [0]
                            
      [s](x1) = [1] x1 + [1]
                            
  [*](x1, x2) = [1] x1 + [0]
                            
      [p](x1) = [0]         
                            
          [0] = [0]         

The order satisfies the following ordering constraints:

   [fac(s(x))] =  [1] x + [1]            
               >  [0]                    
               =  [*(fac(p(s(x))), s(x))]
                                         
  [p(s(s(x)))] =  [0]                    
               ?  [1]                    
               =  [s(p(s(x)))]           
                                         
   [p(s(0()))] =  [0]                    
               >= [0]                    
               =  [0()]                  
                                         

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^2)).

Strict Trs: { p(s(s(x))) -> s(p(s(x))) }
Weak Trs:
  { fac(s(x)) -> *(fac(p(s(x))), s(x))
  , p(s(0())) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

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

Trs: { p(s(s(x))) -> s(p(s(x))) }

The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^2)) . These rules are moved into the corresponding weak
component(s).

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(fac) = {1}, Uargs(s) = {1}, Uargs(*) = {1}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA) and not(IDA(2)).
  
                  [2 1 0]      [0]
      [fac](x1) = [0 0 0] x1 + [1]
                  [0 0 0]      [0]
                                  
                  [1 1 3]      [3]
        [s](x1) = [0 1 0] x1 + [6]
                  [0 1 0]      [1]
                                  
                  [1 2 0]      [0]
    [*](x1, x2) = [0 0 0] x1 + [0]
                  [0 0 1]      [0]
                                  
                  [1 0 0]      [0]
        [p](x1) = [0 0 1] x1 + [1]
                  [0 0 1]      [1]
                                  
                  [2]             
            [0] = [2]             
                  [0]             
  
  The order satisfies the following ordering constraints:
  
     [fac(s(x))] = [2 3 6]     [12]       
                   [0 0 0] x + [1]        
                   [0 0 0]     [0]        
                 > [2 3 6]     [10]       
                   [0 0 0] x + [0]        
                   [0 0 0]     [0]        
                 = [*(fac(p(s(x))), s(x))]
                                          
    [p(s(s(x)))] = [1 5 3]     [15]       
                   [0 1 0] x + [8]        
                   [0 1 0]     [8]        
                 > [1 5 3]     [14]       
                   [0 1 0] x + [8]        
                   [0 1 0]     [3]        
                 = [s(p(s(x)))]           
                                          
     [p(s(0()))] = [7]                    
                   [4]                    
                   [4]                    
                 > [2]                    
                   [2]                    
                   [0]                    
                 = [0()]                  
                                          

We return to the main proof.

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

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

Empty rules are trivially bounded

Hurray, we answered YES(O(1),O(n^2))