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

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

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

TcT has computed the following constructor-restricted matrix
interpretation.

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

The following symbols are considered usable

  {p, f^#, p^#}

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]                
                                   
  [f^#(s(x))] = [2 0] x + [0]      
                [0 0]     [0]      
              ? [2 0] x + [6]      
                [0 0]     [2]      
              = [c_1(f^#(p(s(x))))]
                                   
   [f^#(0())] = [0]                
                [0]                
              ? [1]                
                [1]                
              = [c_2()]            
                                   
  [p^#(s(x))] = [1 1] x + [2]      
                [2 2]     [2]      
              > [1]                
                [1]                
              = [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:
  { f^#(s(x)) -> c_1(f^#(p(s(x))))
  , f^#(0()) -> c_2() }
Weak DPs: { p^#(s(x)) -> 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 {2} by applications of
Pre({2}) = {1}. Here rules are labeled as follows:

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

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

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

{ f^#(0()) -> c_2()
, p^#(s(x)) -> c_3() }

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

Strict DPs: { f^#(s(x)) -> c_1(f^#(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: f^#(s(x)) -> c_1(f^#(p(s(x)))) }

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

{ f^#(s(x)) -> c_1(f^#(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))