YES(O(1),O(1))

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

Strict Trs:
  { f(g(i(a(), b(), b'()), c()), d()) ->
    if(e(), f(.(b(), c()), d'()), f(.(b'(), c()), d'()))
  , f(g(h(a(), b()), c()), d()) ->
    if(e(), f(.(b(), g(h(a(), b()), c())), d()), f(c(), d'())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add the following innermost weak dependency pairs:

Strict DPs:
  { f^#(g(i(a(), b(), b'()), c()), d()) ->
    c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'()))
  , f^#(g(h(a(), b()), c()), d()) ->
    c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) }

and mark the set of starting terms.

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

Strict DPs:
  { f^#(g(i(a(), b(), b'()), c()), d()) ->
    c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'()))
  , f^#(g(h(a(), b()), c()), d()) ->
    c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) }
Strict Trs:
  { f(g(i(a(), b(), b'()), c()), d()) ->
    if(e(), f(.(b(), c()), d'()), f(.(b'(), c()), d'()))
  , f(g(h(a(), b()), c()), d()) ->
    if(e(), f(.(b(), g(h(a(), b()), c())), d()), f(c(), d'())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),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),O(1)).

Strict DPs:
  { f^#(g(i(a(), b(), b'()), c()), d()) ->
    c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'()))
  , f^#(g(h(a(), b()), c()), d()) ->
    c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  none

TcT has computed the following constructor-restricted matrix
interpretation.

      [g](x1, x2) = [0]
                    [0]
                       
  [i](x1, x2, x3) = [0]
                    [0]
                       
              [a] = [0]
                    [0]
                       
              [b] = [0]
                    [0]
                       
             [b'] = [0]
                    [0]
                       
              [c] = [0]
                    [0]
                       
              [d] = [0]
                    [0]
                       
      [.](x1, x2) = [0]
                    [0]
                       
             [d'] = [0]
                    [0]
                       
      [h](x1, x2) = [0]
                    [0]
                       
    [f^#](x1, x2) = [1]
                    [0]
                       
    [c_1](x1, x2) = [1]
                    [0]
                       
    [c_2](x1, x2) = [0]
                    [0]

The following symbols are considered usable

  {f^#}

The order satisfies the following ordering constraints:

  [f^#(g(i(a(), b(), b'()), c()), d())] =  [1]                                                         
                                           [0]                                                         
                                        >= [1]                                                         
                                           [0]                                                         
                                        =  [c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'()))]      
                                                                                                       
        [f^#(g(h(a(), b()), c()), d())] =  [1]                                                         
                                           [0]                                                         
                                        >  [0]                                                         
                                           [0]                                                         
                                        =  [c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'()))]
                                                                                                       

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(1)).

Strict DPs:
  { f^#(g(i(a(), b(), b'()), c()), d()) ->
    c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'())) }
Weak DPs:
  { f^#(g(h(a(), b()), c()), d()) ->
    c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

  DPs:
    { 1: f^#(g(i(a(), b(), b'()), c()), d()) ->
         c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'()))
    , 2: f^#(g(h(a(), b()), c()), d()) ->
         c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) }

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

Weak DPs:
  { f^#(g(i(a(), b(), b'()), c()), d()) ->
    c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'()))
  , f^#(g(h(a(), b()), c()), d()) ->
    c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) }
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^#(g(i(a(), b(), b'()), c()), d()) ->
  c_1(f^#(.(b(), c()), d'()), f^#(.(b'(), c()), d'()))
, f^#(g(h(a(), b()), c()), d()) ->
  c_2(f^#(.(b(), g(h(a(), b()), c())), d()), f^#(c(), d'())) }

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(1))