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

We add the following innermost weak dependency pairs:

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

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

We replace rewrite rules by usable rules:

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

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

Strict DPs:
  { h^#(X, Z) -> c_1(f^#(X, s(X), Z))
  , f^#(X, Y, g(X, Y)) -> c_2(h^#(0(), g(X, Y)))
  , g^#(X, s(Y)) -> c_3(g^#(X, Y))
  , g^#(0(), Y) -> c_4() }
Strict Trs:
  { g(X, s(Y)) -> g(X, Y)
  , g(0(), Y) -> 0() }
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(c_3) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

            [s](x1) = [0 0] x1 + [0]           
                      [0 1]      [1]           
                                               
        [g](x1, x2) = [0 1] x2 + [1]           
                      [0 0]      [0]           
                                               
                [0] = [0]                      
                      [0]                      
                                               
      [h^#](x1, x2) = [1 0] x1 + [2]           
                      [2 1]      [2]           
                                               
          [c_1](x1) = [1 0] x1 + [2]           
                      [0 1]      [2]           
                                               
  [f^#](x1, x2, x3) = [0 0] x1 + [2]           
                      [2 2]      [2]           
                                               
          [c_2](x1) = [1 0] x1 + [2]           
                      [0 1]      [2]           
                                               
      [g^#](x1, x2) = [1 2] x1 + [0 2] x2 + [2]
                      [2 2]      [1 2]      [2]
                                               
          [c_3](x1) = [1 0] x1 + [1]           
                      [0 1]      [1]           
                                               
              [c_4] = [1]                      
                      [1]                      

The following symbols are considered usable

  {g, h^#, f^#, g^#}

The order satisfies the following ordering constraints:

          [g(X, s(Y))] = [0 1] Y + [2]           
                         [0 0]     [0]           
                       > [0 1] Y + [1]           
                         [0 0]     [0]           
                       = [g(X, Y)]               
                                                 
           [g(0(), Y)] = [0 1] Y + [1]           
                         [0 0]     [0]           
                       > [0]                     
                         [0]                     
                       = [0()]                   
                                                 
           [h^#(X, Z)] = [1 0] X + [2]           
                         [2 1]     [2]           
                       ? [0 0] X + [4]           
                         [2 2]     [4]           
                       = [c_1(f^#(X, s(X), Z))]  
                                                 
  [f^#(X, Y, g(X, Y))] = [0 0] X + [2]           
                         [2 2]     [2]           
                       ? [4]                     
                         [4]                     
                       = [c_2(h^#(0(), g(X, Y)))]
                                                 
        [g^#(X, s(Y))] = [1 2] X + [0 2] Y + [4] 
                         [2 2]     [0 2]     [4] 
                       ? [1 2] X + [0 2] Y + [3] 
                         [2 2]     [1 2]     [3] 
                       = [c_3(g^#(X, Y))]        
                                                 
         [g^#(0(), Y)] = [0 2] Y + [2]           
                         [1 2]     [2]           
                       > [1]                     
                         [1]                     
                       = [c_4()]                 
                                                 

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

Consider the dependency graph:

  1: h^#(X, Z) -> c_1(f^#(X, s(X), Z))
  
  2: f^#(X, Y, g(X, Y)) -> c_2(h^#(0(), g(X, Y)))
     -->_1 h^#(X, Z) -> c_1(f^#(X, s(X), Z)) :1
  
  3: g^#(X, s(Y)) -> c_3(g^#(X, Y))
     -->_1 g^#(0(), Y) -> c_4() :4
     -->_1 g^#(X, s(Y)) -> c_3(g^#(X, Y)) :3
  
  4: g^#(0(), Y) -> c_4()
  

Only the nodes {1,3,4} are reachable from nodes {1,3,4} that start
derivation from marked basic terms. The nodes not reachable are
removed from the problem.

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

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

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

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

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

Strict DPs: { g^#(X, s(Y)) -> c_3(g^#(X, Y)) }
Weak DPs:
  { h^#(X, Z) -> c_1(f^#(X, s(X), Z))
  , g^#(0(), Y) -> c_4() }
Weak Trs:
  { g(X, s(Y)) -> g(X, Y)
  , g(0(), Y) -> 0() }
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.

{ h^#(X, Z) -> c_1(f^#(X, s(X), Z))
, g^#(0(), Y) -> c_4() }

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

Strict DPs: { g^#(X, s(Y)) -> c_3(g^#(X, Y)) }
Weak Trs:
  { g(X, s(Y)) -> g(X, Y)
  , g(0(), Y) -> 0() }
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: { g^#(X, s(Y)) -> c_3(g^#(X, Y)) }
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: g^#(X, s(Y)) -> c_3(g^#(X, Y)) }

Sub-proof:
----------
  The input was oriented with the instance of 'Small Polynomial Path
  Order (PS,1-bounded)' as induced by the safe mapping
  
   safe(h) = {}, safe(f) = {}, safe(s) = {1}, safe(g) = {},
   safe(0) = {}, safe(h^#) = {}, safe(c_1) = {}, safe(f^#) = {},
   safe(c_2) = {}, safe(g^#) = {1}, safe(c_3) = {}, safe(c_4) = {}
  
  and precedence
  
   empty .
  
  Following symbols are considered recursive:
  
   {g^#}
  
  The recursion depth is 1.
  
  Further, following argument filtering is employed:
  
   pi(h) = [], pi(f) = [], pi(s) = [1], pi(g) = [], pi(0) = [],
   pi(h^#) = [], pi(c_1) = [], pi(f^#) = [], pi(c_2) = [],
   pi(g^#) = [1, 2], pi(c_3) = [1], pi(c_4) = []
  
  Usable defined function symbols are a subset of:
  
   {h^#, f^#, g^#}
  
  For your convenience, here are the satisfied ordering constraints:
  
    pi(g^#(X, s(Y))) = g^#(s(; Y); X)    
                     > c_3(g^#(Y; X);)   
                     = pi(c_3(g^#(X, Y)))
                                         

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: { g^#(X, s(Y)) -> c_3(g^#(X, Y)) }
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.

{ g^#(X, s(Y)) -> c_3(g^#(X, Y)) }

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))