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:
  { g(x, h(y, z)) -> h(g(x, y), z)
  , g(f(x, y), z) -> f(x, g(y, z))
  , g(h(x, y), z) -> g(x, 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.

Trs:
  { g(x, h(y, z)) -> h(g(x, y), z)
  , g(h(x, y), z) -> g(x, f(y, z)) }

The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^1)) . These rules are moved into the corresponding weak
component(s).

Sub-proof:
----------
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
    [g](x1, x2) = [2] x1 + [2] x2 + [1]
                                       
    [f](x1, x2) = [1] x1 + [1] x2 + [0]
                                       
    [h](x1, x2) = [1] x1 + [1] x2 + [2]
  
  This order satisfies the following ordering constraints:
  
    [g(x, h(y, z))] =  [2] x + [2] y + [2] z + [5]
                    >  [2] x + [2] y + [1] z + [3]
                    =  [h(g(x, y), z)]            
                                                  
    [g(f(x, y), z)] =  [2] x + [2] y + [2] z + [1]
                    >= [1] x + [2] y + [2] z + [1]
                    =  [f(x, g(y, z))]            
                                                  
    [g(h(x, y), z)] =  [2] x + [2] y + [2] z + [5]
                    >  [2] x + [2] y + [2] z + [1]
                    =  [g(x, f(y, z))]            
                                                  

We return to the main proof.

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

Strict Trs: { g(f(x, y), z) -> f(x, g(y, z)) }
Weak Trs:
  { g(x, h(y, z)) -> h(g(x, y), z)
  , g(h(x, y), z) -> g(x, 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.

Trs: { g(f(x, y), z) -> f(x, g(y, z)) }

The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^1)) . These rules are moved into the corresponding weak
component(s).

Sub-proof:
----------
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
    [g](x1, x2) = [2] x1 + [2] x2 + [1]
                                       
    [f](x1, x2) = [1] x1 + [1] x2 + [2]
                                       
    [h](x1, x2) = [1] x1 + [1] x2 + [2]
  
  This order satisfies the following ordering constraints:
  
    [g(x, h(y, z))] =  [2] x + [2] y + [2] z + [5]
                    >  [2] x + [2] y + [1] z + [3]
                    =  [h(g(x, y), z)]            
                                                  
    [g(f(x, y), z)] =  [2] x + [2] y + [2] z + [5]
                    >  [1] x + [2] y + [2] z + [3]
                    =  [f(x, g(y, z))]            
                                                  
    [g(h(x, y), z)] =  [2] x + [2] y + [2] z + [5]
                    >= [2] x + [2] y + [2] z + [5]
                    =  [g(x, f(y, z))]            
                                                  

We return to the main proof.

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

Weak Trs:
  { g(x, h(y, z)) -> h(g(x, y), z)
  , g(f(x, y), z) -> f(x, g(y, z))
  , g(h(x, y), z) -> g(x, f(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

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