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:
  { :(z, +(x, f(y))) -> :(g(z, y), +(x, a()))
  , :(:(x, y), z) -> :(x, :(y, z))
  , :(+(x, y), z) -> +(:(x, z), :(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following dependency tuples:

Strict DPs:
  { :^#(z, +(x, f(y))) -> c_1(:^#(g(z, y), +(x, a())))
  , :^#(:(x, y), z) -> c_2(:^#(x, :(y, z)), :^#(y, z))
  , :^#(+(x, y), z) -> c_3(:^#(x, z), :^#(y, z)) }

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:
  { :^#(z, +(x, f(y))) -> c_1(:^#(g(z, y), +(x, a())))
  , :^#(:(x, y), z) -> c_2(:^#(x, :(y, z)), :^#(y, z))
  , :^#(+(x, y), z) -> c_3(:^#(x, z), :^#(y, z)) }
Weak Trs:
  { :(z, +(x, f(y))) -> :(g(z, y), +(x, a()))
  , :(:(x, y), z) -> :(x, :(y, z))
  , :(+(x, y), z) -> +(:(x, z), :(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

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

  DPs:
    { 1: :^#(z, +(x, f(y))) -> c_1(:^#(g(z, y), +(x, a())))
    , 2: :^#(:(x, y), z) -> c_2(:^#(x, :(y, z)), :^#(y, z))
    , 3: :^#(+(x, y), z) -> c_3(:^#(x, z), :^#(y, z)) }

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

Strict DPs:
  { :^#(:(x, y), z) -> c_2(:^#(x, :(y, z)), :^#(y, z))
  , :^#(+(x, y), z) -> c_3(:^#(x, z), :^#(y, z)) }
Weak DPs: { :^#(z, +(x, f(y))) -> c_1(:^#(g(z, y), +(x, a()))) }
Weak Trs:
  { :(z, +(x, f(y))) -> :(g(z, y), +(x, a()))
  , :(:(x, y), z) -> :(x, :(y, z))
  , :(+(x, y), z) -> +(:(x, z), :(y, z)) }
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.

{ :^#(z, +(x, f(y))) -> c_1(:^#(g(z, y), +(x, a()))) }

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

Strict DPs:
  { :^#(:(x, y), z) -> c_2(:^#(x, :(y, z)), :^#(y, z))
  , :^#(+(x, y), z) -> c_3(:^#(x, z), :^#(y, z)) }
Weak Trs:
  { :(z, +(x, f(y))) -> :(g(z, y), +(x, a()))
  , :(:(x, y), z) -> :(x, :(y, z))
  , :(+(x, y), z) -> +(:(x, z), :(y, z)) }
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:

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

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

Strict DPs:
  { :^#(:(x, y), z) -> c_1(:^#(y, z))
  , :^#(+(x, y), z) -> c_2(:^#(x, z), :^#(y, z)) }
Weak Trs:
  { :(z, +(x, f(y))) -> :(g(z, y), +(x, a()))
  , :(:(x, y), z) -> :(x, :(y, z))
  , :(+(x, y), z) -> +(:(x, z), :(y, z)) }
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:
  { :^#(:(x, y), z) -> c_1(:^#(y, z))
  , :^#(+(x, y), z) -> c_2(:^#(x, z), :^#(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: :^#(:(x, y), z) -> c_1(:^#(y, z))
  , 2: :^#(+(x, y), z) -> c_2(:^#(x, z), :^#(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).
  
      [:](x1, x2) = [7] x1 + [4] x2 + [4]
                                         
      [+](x1, x2) = [1] x1 + [1] x2 + [4]
                                         
          [f](x1) = [1] x1 + [0]         
                                         
      [g](x1, x2) = [1] x1 + [1] x2 + [0]
                                         
              [a] = [0]                  
                                         
    [:^#](x1, x2) = [2] x1 + [0]         
                                         
        [c_1](x1) = [7] x1 + [0]         
                                         
    [c_2](x1, x2) = [7] x1 + [7] x2 + [0]
                                         
    [c_3](x1, x2) = [7] x1 + [7] x2 + [0]
                                         
              [c] = [0]                  
                                         
        [c_1](x1) = [1] x1 + [1]         
                                         
    [c_2](x1, x2) = [1] x1 + [1] x2 + [3]
  
  The following symbols are considered usable
  
    {:^#}
  
  The order satisfies the following ordering constraints:
  
    [:^#(:(x, y), z)] = [14] x + [8] y + [8]       
                      > [2] y + [1]                
                      = [c_1(:^#(y, z))]           
                                                   
    [:^#(+(x, y), z)] = [2] x + [2] y + [8]        
                      > [2] x + [2] y + [3]        
                      = [c_2(:^#(x, z), :^#(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:
  { :^#(:(x, y), z) -> c_1(:^#(y, z))
  , :^#(+(x, y), z) -> c_2(:^#(x, z), :^#(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.

{ :^#(:(x, y), z) -> c_1(:^#(y, z))
, :^#(+(x, y), z) -> c_2(:^#(x, z), :^#(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))