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(x, 0()) -> x
  , f(f(x, y), z) -> f(x, f(y, z))
  , f(0(), y) -> y
  , f(i(x), y) -> i(x)
  , f(g(x, y), z) -> g(f(x, z), f(y, z))
  , f(1(), g(x, y)) -> x
  , f(2(), g(x, y)) -> y }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following dependency tuples:

Strict DPs:
  { f^#(x, 0()) -> c_1()
  , f^#(f(x, y), z) -> c_2(f^#(x, f(y, z)), f^#(y, z))
  , f^#(0(), y) -> c_3()
  , f^#(i(x), y) -> c_4()
  , f^#(g(x, y), z) -> c_5(f^#(x, z), f^#(y, z))
  , f^#(1(), g(x, y)) -> c_6()
  , f^#(2(), g(x, y)) -> c_7() }

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^#(x, 0()) -> c_1()
  , f^#(f(x, y), z) -> c_2(f^#(x, f(y, z)), f^#(y, z))
  , f^#(0(), y) -> c_3()
  , f^#(i(x), y) -> c_4()
  , f^#(g(x, y), z) -> c_5(f^#(x, z), f^#(y, z))
  , f^#(1(), g(x, y)) -> c_6()
  , f^#(2(), g(x, y)) -> c_7() }
Weak Trs:
  { f(x, 0()) -> x
  , f(f(x, y), z) -> f(x, f(y, z))
  , f(0(), y) -> y
  , f(i(x), y) -> i(x)
  , f(g(x, y), z) -> g(f(x, z), f(y, z))
  , f(1(), g(x, y)) -> x
  , f(2(), g(x, y)) -> y }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We estimate the number of application of {1,3,4,6,7} by
applications of Pre({1,3,4,6,7}) = {2,5}. Here rules are labeled as
follows:

  DPs:
    { 1: f^#(x, 0()) -> c_1()
    , 2: f^#(f(x, y), z) -> c_2(f^#(x, f(y, z)), f^#(y, z))
    , 3: f^#(0(), y) -> c_3()
    , 4: f^#(i(x), y) -> c_4()
    , 5: f^#(g(x, y), z) -> c_5(f^#(x, z), f^#(y, z))
    , 6: f^#(1(), g(x, y)) -> c_6()
    , 7: f^#(2(), g(x, y)) -> c_7() }

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

Strict DPs:
  { f^#(f(x, y), z) -> c_2(f^#(x, f(y, z)), f^#(y, z))
  , f^#(g(x, y), z) -> c_5(f^#(x, z), f^#(y, z)) }
Weak DPs:
  { f^#(x, 0()) -> c_1()
  , f^#(0(), y) -> c_3()
  , f^#(i(x), y) -> c_4()
  , f^#(1(), g(x, y)) -> c_6()
  , f^#(2(), g(x, y)) -> c_7() }
Weak Trs:
  { f(x, 0()) -> x
  , f(f(x, y), z) -> f(x, f(y, z))
  , f(0(), y) -> y
  , f(i(x), y) -> i(x)
  , f(g(x, y), z) -> g(f(x, z), f(y, z))
  , f(1(), g(x, y)) -> x
  , f(2(), g(x, y)) -> y }
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^#(x, 0()) -> c_1()
, f^#(0(), y) -> c_3()
, f^#(i(x), y) -> c_4()
, f^#(1(), g(x, y)) -> c_6()
, f^#(2(), g(x, y)) -> c_7() }

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

Strict DPs:
  { f^#(f(x, y), z) -> c_2(f^#(x, f(y, z)), f^#(y, z))
  , f^#(g(x, y), z) -> c_5(f^#(x, z), f^#(y, z)) }
Weak Trs:
  { f(x, 0()) -> x
  , f(f(x, y), z) -> f(x, f(y, z))
  , f(0(), y) -> y
  , f(i(x), y) -> i(x)
  , f(g(x, y), z) -> g(f(x, z), f(y, z))
  , f(1(), g(x, y)) -> x
  , f(2(), g(x, y)) -> y }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { f^#(f(x, y), z) -> c_2(f^#(x, f(y, z)), f^#(y, z)) }

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

Strict DPs:
  { f^#(f(x, y), z) -> c_1(f^#(y, z))
  , f^#(g(x, y), z) -> c_2(f^#(x, z), f^#(y, z)) }
Weak Trs:
  { f(x, 0()) -> x
  , f(f(x, y), z) -> f(x, f(y, z))
  , f(0(), y) -> y
  , f(i(x), y) -> i(x)
  , f(g(x, y), z) -> g(f(x, z), f(y, z))
  , f(1(), g(x, y)) -> x
  , f(2(), g(x, y)) -> y }
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^#(f(x, y), z) -> c_1(f^#(y, z))
  , f^#(g(x, y), z) -> c_2(f^#(x, z), f^#(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.

DPs:
  { 1: f^#(f(x, y), z) -> c_1(f^#(y, z))
  , 2: f^#(g(x, y), z) -> c_2(f^#(x, z), f^#(y, z)) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
      [f](x1, x2) = [7] x1 + [4] x2 + [4]
                                         
              [0] = [0]                  
                                         
          [i](x1) = [1] x1 + [0]         
                                         
      [g](x1, x2) = [1] x1 + [1] x2 + [4]
                                         
              [1] = [0]                  
                                         
              [2] = [0]                  
                                         
    [f^#](x1, x2) = [2] x1 + [0]         
                                         
            [c_1] = [0]                  
                                         
    [c_2](x1, x2) = [7] x1 + [7] x2 + [0]
                                         
            [c_3] = [0]                  
                                         
            [c_4] = [0]                  
                                         
    [c_5](x1, x2) = [7] x1 + [7] x2 + [0]
                                         
            [c_6] = [0]                  
                                         
            [c_7] = [0]                  
                                         
              [c] = [0]                  
                                         
        [c_1](x1) = [4] x1 + [3]         
                                         
    [c_2](x1, x2) = [1] x1 + [1] x2 + [1]
  
  The following symbols are considered usable
  
    {f^#}
  
  The order satisfies the following ordering constraints:
  
    [f^#(f(x, y), z)] = [8] y + [14] x + [8]       
                      > [8] y + [3]                
                      = [c_1(f^#(y, z))]           
                                                   
    [f^#(g(x, y), z)] = [2] y + [2] x + [8]        
                      > [2] y + [2] x + [1]        
                      = [c_2(f^#(x, z), f^#(y, z))]
                                                   

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^#(f(x, y), z) -> c_1(f^#(y, z))
  , f^#(g(x, y), z) -> c_2(f^#(x, z), f^#(y, z)) }
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^#(f(x, y), z) -> c_1(f^#(y, z))
, f^#(g(x, y), z) -> c_2(f^#(x, z), f^#(y, z)) }

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))