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

We add following weak dependency pairs:

Strict DPs:
  { f^#(s(X)) -> c_1(f^#(X))
  , g^#(cons(s(X), Y)) -> c_2()
  , g^#(cons(0(), Y)) -> c_3(g^#(Y))
  , h^#(cons(X, Y)) -> c_4(h^#(g(cons(X, Y)))) }

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^#(X))
  , g^#(cons(s(X), Y)) -> c_2()
  , g^#(cons(0(), Y)) -> c_3(g^#(Y))
  , h^#(cons(X, Y)) -> c_4(h^#(g(cons(X, Y)))) }
Strict Trs:
  { f(s(X)) -> f(X)
  , g(cons(s(X), Y)) -> s(X)
  , g(cons(0(), Y)) -> g(Y)
  , h(cons(X, Y)) -> h(g(cons(X, Y))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We replace rewrite rules by usable rules:

  Strict Usable Rules:
    { g(cons(s(X), Y)) -> s(X)
    , g(cons(0(), Y)) -> g(Y) }

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^#(X))
  , g^#(cons(s(X), Y)) -> c_2()
  , g^#(cons(0(), Y)) -> c_3(g^#(Y))
  , h^#(cons(X, Y)) -> c_4(h^#(g(cons(X, Y)))) }
Strict Trs:
  { g(cons(s(X), Y)) -> s(X)
  , g(cons(0(), Y)) -> g(Y) }
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_3) = {1}, Uargs(h^#) = {1},
  Uargs(c_4) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

         [s](x1) = [1] x1 + [2]         
                                        
         [g](x1) = [1] x1 + [2]         
                                        
  [cons](x1, x2) = [1] x1 + [1] x2 + [0]
                                        
             [0] = [1]                  
                                        
       [f^#](x1) = [1] x1 + [2]         
                                        
       [c_1](x1) = [1] x1 + [1]         
                                        
       [g^#](x1) = [2] x1 + [2]         
                                        
           [c_2] = [1]                  
                                        
       [c_3](x1) = [1] x1 + [1]         
                                        
       [h^#](x1) = [2] x1 + [2]         
                                        
       [c_4](x1) = [1] x1 + [1]         

This order satisfies following ordering constraints:

  [g(cons(s(X), Y))] = [1] X + [1] Y + [4]
                     > [1] X + [2]        
                     = [s(X)]             
                                          
   [g(cons(0(), Y))] = [1] Y + [3]        
                     > [1] Y + [2]        
                     = [g(Y)]             
                                          

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)).

Strict DPs: { h^#(cons(X, Y)) -> c_4(h^#(g(cons(X, Y)))) }
Weak DPs:
  { f^#(s(X)) -> c_1(f^#(X))
  , g^#(cons(s(X), Y)) -> c_2()
  , g^#(cons(0(), Y)) -> c_3(g^#(Y)) }
Weak Trs:
  { g(cons(s(X), Y)) -> s(X)
  , g(cons(0(), Y)) -> g(Y) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

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

  DPs:
    { 1: h^#(cons(X, Y)) -> c_4(h^#(g(cons(X, Y))))
    , 2: f^#(s(X)) -> c_1(f^#(X))
    , 3: g^#(cons(s(X), Y)) -> c_2()
    , 4: g^#(cons(0(), Y)) -> c_3(g^#(Y)) }

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

Weak DPs:
  { f^#(s(X)) -> c_1(f^#(X))
  , g^#(cons(s(X), Y)) -> c_2()
  , g^#(cons(0(), Y)) -> c_3(g^#(Y))
  , h^#(cons(X, Y)) -> c_4(h^#(g(cons(X, Y)))) }
Weak Trs:
  { g(cons(s(X), Y)) -> s(X)
  , g(cons(0(), Y)) -> g(Y) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,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^#(X))
, g^#(cons(s(X), Y)) -> c_2()
, g^#(cons(0(), Y)) -> c_3(g^#(Y))
, h^#(cons(X, Y)) -> c_4(h^#(g(cons(X, Y)))) }

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

Weak Trs:
  { g(cons(s(X), Y)) -> s(X)
  , g(cons(0(), Y)) -> g(Y) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,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)).

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

The input was oriented with the instance of 'Small Polynomial Path
Order (PS)' as induced by the safe mapping


and precedence

 empty .

Following symbols are considered recursive:

 {}

The recursion depth is 0.

Further, following argument filtering is employed:

 empty

Usable defined function symbols are a subset of:

 {}

For your convenience, here are the satisfied ordering constraints:


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