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(f(x)) -> f(x)
  , f(s(x)) -> f(x)
  , g(s(0())) -> g(f(s(0()))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following innermost weak dependency pairs:

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

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

We replace rewrite rules by usable rules:

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

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

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

TcT has computed the following constructor-restricted matrix
interpretation.

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

The following symbols are considered usable

  {f, f^#, g^#}

The order satisfies the following ordering constraints:

      [f(f(x))] = [0 1] x + [2]        
                  [0 1]     [4]        
                > [0 1] x + [0]        
                  [0 1]     [2]        
                = [f(x)]               
                                       
      [f(s(x))] = [0 1] x + [2]        
                  [0 1]     [4]        
                > [0 1] x + [0]        
                  [0 1]     [2]        
                = [f(x)]               
                                       
    [f^#(f(x))] = [0 0] x + [2]        
                  [0 3]     [6]        
                ? [0 0] x + [3]        
                  [1 2]     [3]        
                = [c_1(f^#(x))]        
                                       
    [f^#(s(x))] = [0 0] x + [2]        
                  [0 2]     [6]        
                ? [0 0] x + [3]        
                  [1 2]     [3]        
                = [c_2(f^#(x))]        
                                       
  [g^#(s(0()))] = [2]                  
                  [2]                  
                ? [7]                  
                  [4]                  
                = [c_3(g^#(f(s(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^1)).

Strict DPs:
  { f^#(f(x)) -> c_1(f^#(x))
  , f^#(s(x)) -> c_2(f^#(x))
  , g^#(s(0())) -> c_3(g^#(f(s(0())))) }
Weak Trs:
  { f(f(x)) -> f(x)
  , f(s(x)) -> f(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: f^#(f(x)) -> c_1(f^#(x))
    , 2: f^#(s(x)) -> c_2(f^#(x))
    , 3: g^#(s(0())) -> c_3(g^#(f(s(0())))) }

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

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

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

Strict DPs: { f^#(s(x)) -> c_2(f^#(x)) }
Weak Trs:
  { f(f(x)) -> f(x)
  , f(s(x)) -> f(x) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^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(n^1)).

Strict DPs: { f^#(s(x)) -> c_2(f^#(x)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
to orient following rules strictly.

DPs:
  { 1: f^#(s(x)) -> c_2(f^#(x)) }

Sub-proof:
----------
  The input was oriented with the instance of 'Small Polynomial Path
  Order (PS,1-bounded)' as induced by the safe mapping
  
   safe(f) = {}, safe(s) = {1}, safe(g) = {}, safe(0) = {},
   safe(f^#) = {}, safe(c_1) = {}, safe(c_2) = {}, safe(g^#) = {},
   safe(c_3) = {}
  
  and precedence
  
   empty .
  
  Following symbols are considered recursive:
  
   {f^#}
  
  The recursion depth is 1.
  
  Further, following argument filtering is employed:
  
   pi(f) = [], pi(s) = [1], pi(g) = [], pi(0) = [], pi(f^#) = [1],
   pi(c_1) = [], pi(c_2) = [1], pi(g^#) = [], pi(c_3) = []
  
  Usable defined function symbols are a subset of:
  
   {f^#, g^#}
  
  For your convenience, here are the satisfied ordering constraints:
  
    pi(f^#(s(x))) = f^#(s(; x);)   
                  > c_2(f^#(x;);)  
                  = pi(c_2(f^#(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_2(f^#(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_2(f^#(x)) }

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))