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:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sqr(s(x)), sum(x))
  , sum(s(x)) -> +(*(s(x), s(x)), sum(x))
  , sqr(x) -> *(x, x) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following innermost weak dependency pairs:

Strict DPs:
  { sum^#(0()) -> c_1()
  , sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
  , sum^#(s(x)) -> c_3(sum^#(x))
  , sqr^#(x) -> c_4() }

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:
  { sum^#(0()) -> c_1()
  , sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
  , sum^#(s(x)) -> c_3(sum^#(x))
  , sqr^#(x) -> c_4() }
Strict Trs:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sqr(s(x)), sum(x))
  , sum(s(x)) -> +(*(s(x), s(x)), sum(x))
  , sqr(x) -> *(x, x) }
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:
  { sum^#(0()) -> c_1()
  , sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
  , sum^#(s(x)) -> c_3(sum^#(x))
  , sqr^#(x) -> c_4() }
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(c_2) = {1, 2}, Uargs(c_3) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

            [0] = [0]                      
                  [0]                      
                                           
        [s](x1) = [0]                      
                  [0]                      
                                           
    [sum^#](x1) = [0]                      
                  [0]                      
                                           
          [c_1] = [1]                      
                  [0]                      
                                           
  [c_2](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                  [0 1]      [0 1]      [2]
                                           
    [sqr^#](x1) = [1]                      
                  [0]                      
                                           
      [c_3](x1) = [1 0] x1 + [2]           
                  [0 1]      [0]           
                                           
          [c_4] = [0]                      
                  [0]                      

The following symbols are considered usable

  {sum^#, sqr^#}

The order satisfies the following ordering constraints:

   [sum^#(0())] = [0]                         
                  [0]                         
                ? [1]                         
                  [0]                         
                = [c_1()]                     
                                              
  [sum^#(s(x))] = [0]                         
                  [0]                         
                ? [1]                         
                  [2]                         
                = [c_2(sqr^#(s(x)), sum^#(x))]
                                              
  [sum^#(s(x))] = [0]                         
                  [0]                         
                ? [2]                         
                  [0]                         
                = [c_3(sum^#(x))]             
                                              
     [sqr^#(x)] = [1]                         
                  [0]                         
                > [0]                         
                  [0]                         
                = [c_4()]                     
                                              

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(n^1)).

Strict DPs:
  { sum^#(0()) -> c_1()
  , sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
  , sum^#(s(x)) -> c_3(sum^#(x)) }
Weak DPs: { sqr^#(x) -> c_4() }
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: sum^#(0()) -> c_1()
    , 2: sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
    , 3: sum^#(s(x)) -> c_3(sum^#(x))
    , 4: sqr^#(x) -> c_4() }

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

Strict DPs:
  { sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
  , sum^#(s(x)) -> c_3(sum^#(x)) }
Weak DPs:
  { sum^#(0()) -> c_1()
  , sqr^#(x) -> c_4() }
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.

{ sum^#(0()) -> c_1()
, sqr^#(x) -> c_4() }

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

Strict DPs:
  { sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
  , sum^#(s(x)) -> c_3(sum^#(x)) }
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:

  { sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x)) }

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

Strict DPs:
  { sum^#(s(x)) -> c_1(sum^#(x))
  , sum^#(s(x)) -> c_2(sum^#(x)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
to orient following rules strictly.

DPs:
  { 1: sum^#(s(x)) -> c_1(sum^#(x))
  , 2: sum^#(s(x)) -> c_2(sum^#(x)) }

Sub-proof:
----------
  The input was oriented with the instance of 'Small Polynomial Path
  Order (PS,1-bounded)' as induced by the safe mapping
  
   safe(sum) = {}, safe(0) = {}, safe(s) = {1}, safe(+) = {1, 2},
   safe(sqr) = {}, safe(*) = {1, 2}, safe(sum^#) = {}, safe(c_1) = {},
   safe(c_2) = {}, safe(sqr^#) = {}, safe(c_3) = {}, safe(c_4) = {},
   safe(c) = {}, safe(c_1) = {}, safe(c_2) = {}
  
  and precedence
  
   empty .
  
  Following symbols are considered recursive:
  
   {sum^#}
  
  The recursion depth is 1.
  
  Further, following argument filtering is employed:
  
   pi(sum) = [], pi(0) = [], pi(s) = [1], pi(+) = [], pi(sqr) = [],
   pi(*) = [], pi(sum^#) = [1], pi(c_1) = [], pi(c_2) = [],
   pi(sqr^#) = [], pi(c_3) = [], pi(c_4) = [], pi(c) = [],
   pi(c_1) = [1], pi(c_2) = [1]
  
  Usable defined function symbols are a subset of:
  
   {sum^#, sqr^#}
  
  For your convenience, here are the satisfied ordering constraints:
  
    pi(sum^#(s(x))) = sum^#(s(; x);)   
                    > c_1(sum^#(x;);)  
                    = pi(c_1(sum^#(x)))
                                       
    pi(sum^#(s(x))) = sum^#(s(; x);)   
                    > c_2(sum^#(x;);)  
                    = pi(c_2(sum^#(x)))
                                       

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:
  { sum^#(s(x)) -> c_1(sum^#(x))
  , sum^#(s(x)) -> c_2(sum^#(x)) }
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.

{ sum^#(s(x)) -> c_1(sum^#(x))
, sum^#(s(x)) -> c_2(sum^#(x)) }

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))