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:
  { admit(x, nil()) -> nil()
  , admit(x, .(u, .(v, .(w(), z)))) ->
    cond(=(sum(x, u, v), w()),
         .(u, .(v, .(w(), admit(carry(x, u, v), z)))))
  , cond(true(), y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add following weak dependency pairs:

Strict DPs:
  { admit^#(x, nil()) -> c_1()
  , admit^#(x, .(u, .(v, .(w(), z)))) ->
    c_2(cond^#(=(sum(x, u, v), w()),
               .(u, .(v, .(w(), admit(carry(x, u, v), z))))))
  , cond^#(true(), y) -> c_3() }

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:
  { admit^#(x, nil()) -> c_1()
  , admit^#(x, .(u, .(v, .(w(), z)))) ->
    c_2(cond^#(=(sum(x, u, v), w()),
               .(u, .(v, .(w(), admit(carry(x, u, v), z))))))
  , cond^#(true(), y) -> c_3() }
Strict Trs:
  { admit(x, nil()) -> nil()
  , admit(x, .(u, .(v, .(w(), z)))) ->
    cond(=(sum(x, u, v), w()),
         .(u, .(v, .(w(), admit(carry(x, u, v), z)))))
  , cond(true(), y) -> y }
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(.) = {2}, Uargs(cond) = {2}, Uargs(c_2) = {1},
  Uargs(cond^#) = {2}

TcT has computed following constructor-restricted matrix
interpretation.

      [admit](x1, x2) = [2] x2 + [1]                  
                                                      
                [nil] = [1]                           
                                                      
          [.](x1, x2) = [1] x1 + [1] x2 + [0]         
                                                      
                  [w] = [2]                           
                                                      
       [cond](x1, x2) = [1] x2 + [1]                  
                                                      
          [=](x1, x2) = [0]                           
                                                      
    [sum](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
                                                      
  [carry](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
                                                      
               [true] = [2]                           
                                                      
    [admit^#](x1, x2) = [1] x1 + [2] x2 + [2]         
                                                      
                [c_1] = [1]                           
                                                      
            [c_2](x1) = [1] x1 + [2]                  
                                                      
     [cond^#](x1, x2) = [1] x2 + [1]                  
                                                      
                [c_3] = [0]                           

This order satisfies following ordering constraints:

                  [admit(x, nil())] = [3]                                                 
                                    > [1]                                                 
                                    = [nil()]                                             
                                                                                          
  [admit(x, .(u, .(v, .(w(), z))))] = [2] u + [2] v + [2] z + [5]                         
                                    > [1] u + [1] v + [2] z + [4]                         
                                    = [cond(=(sum(x, u, v), w()),                         
                                            .(u, .(v, .(w(), admit(carry(x, u, v), z)))))]
                                                                                          
                  [cond(true(), y)] = [1] y + [1]                                         
                                    > [1] y + [0]                                         
                                    = [y]                                                 
                                                                                          

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)).

Strict DPs:
  { admit^#(x, .(u, .(v, .(w(), z)))) ->
    c_2(cond^#(=(sum(x, u, v), w()),
               .(u, .(v, .(w(), admit(carry(x, u, v), z)))))) }
Weak DPs:
  { admit^#(x, nil()) -> c_1()
  , cond^#(true(), y) -> c_3() }
Weak Trs:
  { admit(x, nil()) -> nil()
  , admit(x, .(u, .(v, .(w(), z)))) ->
    cond(=(sum(x, u, v), w()),
         .(u, .(v, .(w(), admit(carry(x, u, v), z)))))
  , cond(true(), y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

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

  DPs:
    { 1: admit^#(x, .(u, .(v, .(w(), z)))) ->
         c_2(cond^#(=(sum(x, u, v), w()),
                    .(u, .(v, .(w(), admit(carry(x, u, v), z))))))
    , 2: admit^#(x, nil()) -> c_1()
    , 3: cond^#(true(), y) -> c_3() }

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

Weak DPs:
  { admit^#(x, nil()) -> c_1()
  , admit^#(x, .(u, .(v, .(w(), z)))) ->
    c_2(cond^#(=(sum(x, u, v), w()),
               .(u, .(v, .(w(), admit(carry(x, u, v), z))))))
  , cond^#(true(), y) -> c_3() }
Weak Trs:
  { admit(x, nil()) -> nil()
  , admit(x, .(u, .(v, .(w(), z)))) ->
    cond(=(sum(x, u, v), w()),
         .(u, .(v, .(w(), admit(carry(x, u, v), z)))))
  , cond(true(), y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ admit^#(x, nil()) -> c_1()
, admit^#(x, .(u, .(v, .(w(), z)))) ->
  c_2(cond^#(=(sum(x, u, v), w()),
             .(u, .(v, .(w(), admit(carry(x, u, v), z))))))
, cond^#(true(), y) -> c_3() }

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

Weak Trs:
  { admit(x, nil()) -> nil()
  , admit(x, .(u, .(v, .(w(), z)))) ->
    cond(=(sum(x, u, v), w()),
         .(u, .(v, .(w(), admit(carry(x, u, v), z)))))
  , cond(true(), y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,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)).

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

The input was oriented with the instance of 'Small Polynomial Path
Order (PS)' as induced by the safe mapping


and precedence

 empty .

Following symbols are considered recursive:

 {}

The recursion depth is 0.

Further, following argument filtering is employed:

 empty

Usable defined function symbols are a subset of:

 {}

For your convenience, here are the satisfied ordering constraints:


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