WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          sqr#(x) -> c_1()
          sum#(0()) -> c_2()
          sum#(s(x)) -> c_3(sum#(x))
          sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sqr#(x) -> c_1()
            sum#(0()) -> c_2()
            sum#(s(x)) -> c_3(sum#(x))
            sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
        - Weak TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1,sqr#/1,sum#/1} / {*/2,+/2,0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr#,sum#} and constructors {*,+,0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2}
        by application of
          Pre({1,2}) = {3,4}.
        Here rules are labelled as follows:
          1: sqr#(x) -> c_1()
          2: sum#(0()) -> c_2()
          3: sum#(s(x)) -> c_3(sum#(x))
          4: sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_3(sum#(x))
            sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
        - Weak DPs:
            sqr#(x) -> c_1()
            sum#(0()) -> c_2()
        - Weak TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1,sqr#/1,sum#/1} / {*/2,+/2,0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr#,sum#} and constructors {*,+,0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sum#(s(x)) -> c_3(sum#(x))
             -->_1 sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x)):2
             -->_1 sum#(0()) -> c_2():4
             -->_1 sum#(s(x)) -> c_3(sum#(x)):1
          
          2:S:sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
             -->_2 sum#(0()) -> c_2():4
             -->_1 sqr#(x) -> c_1():3
             -->_2 sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x)):2
             -->_2 sum#(s(x)) -> c_3(sum#(x)):1
          
          3:W:sqr#(x) -> c_1()
             
          
          4:W:sum#(0()) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: sqr#(x) -> c_1()
          4: sum#(0()) -> c_2()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_3(sum#(x))
            sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
        - Weak TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1,sqr#/1,sum#/1} / {*/2,+/2,0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr#,sum#} and constructors {*,+,0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:sum#(s(x)) -> c_3(sum#(x))
             -->_1 sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x)):2
             -->_1 sum#(s(x)) -> c_3(sum#(x)):1
          
          2:S:sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x))
             -->_2 sum#(s(x)) -> c_4(sqr#(s(x)),sum#(x)):2
             -->_2 sum#(s(x)) -> c_3(sum#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sum#(s(x)) -> c_4(sum#(x))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_3(sum#(x))
            sum#(s(x)) -> c_4(sum#(x))
        - Weak TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1,sqr#/1,sum#/1} / {*/2,+/2,0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr#,sum#} and constructors {*,+,0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          sum#(s(x)) -> c_3(sum#(x))
          sum#(s(x)) -> c_4(sum#(x))
* Step 6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_3(sum#(x))
            sum#(s(x)) -> c_4(sum#(x))
        - Signature:
            {sqr/1,sum/1,sqr#/1,sum#/1} / {*/2,+/2,0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr#,sum#} and constructors {*,+,0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_3) = {1},
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(*) = [1] x1 + [1] x2 + [0]
               p(+) = [1] x1 + [1] x2 + [0]
               p(0) = [0]                  
               p(s) = [1] x1 + [3]         
             p(sqr) = [0]                  
             p(sum) = [0]                  
            p(sqr#) = [0]                  
            p(sum#) = [9] x1 + [0]         
             p(c_1) = [0]                  
             p(c_2) = [0]                  
             p(c_3) = [1] x1 + [0]         
             p(c_4) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          sum#(s(x)) = [9] x + [27]
                     > [9] x + [0] 
                     = c_3(sum#(x))
          
          sum#(s(x)) = [9] x + [27]
                     > [9] x + [0] 
                     = c_4(sum#(x))
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(s(x)) -> c_3(sum#(x))
            sum#(s(x)) -> c_4(sum#(x))
        - Signature:
            {sqr/1,sum/1,sqr#/1,sum#/1} / {*/2,+/2,0/0,s/1,c_1/0,c_2/0,c_3/1,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr#,sum#} and constructors {*,+,0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))