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_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        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: UsableRules 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))
        - 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,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:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          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))
* Step 3: 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))
        - 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 4: 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()
        - 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 5: 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))
        - 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 6: PredecessorEstimationCP 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:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: sum#(s(x)) -> c_3(sum#(x))
          
        The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalMI 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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        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:
          {sqr#,sum#}
        TcT has computed the following interpretation:
             p(*) = [0]         
             p(+) = [1] x2 + [0]
             p(0) = [2]         
             p(s) = [1] x1 + [4]
           p(sqr) = [1] x1 + [0]
           p(sum) = [1]         
          p(sqr#) = [4] x1 + [1]
          p(sum#) = [1] x1 + [0]
           p(c_1) = [1]         
           p(c_2) = [1]         
           p(c_3) = [1] x1 + [2]
           p(c_4) = [1] x1 + [4]
        
        Following rules are strictly oriented:
        sum#(s(x)) = [1] x + [4] 
                   > [1] x + [2] 
                   = c_3(sum#(x))
        
        
        Following rules are (at-least) weakly oriented:
        sum#(s(x)) =  [1] x + [4] 
                   >= [1] x + [4] 
                   =  c_4(sum#(x))
        
** Step 6.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_4(sum#(x))
        - Weak DPs:
            sum#(s(x)) -> c_3(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:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 6.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_4(sum#(x))
        - Weak DPs:
            sum#(s(x)) -> c_3(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:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: sum#(s(x)) -> c_4(sum#(x))
          
        The strictly oriented rules are moved into the weak component.
*** Step 6.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_4(sum#(x))
        - Weak DPs:
            sum#(s(x)) -> c_3(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:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        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:
          {sqr#,sum#}
        TcT has computed the following interpretation:
             p(*) = [1] x1 + [1]
             p(+) = [1] x1 + [1]
             p(0) = [1]         
             p(s) = [1] x1 + [8]
           p(sqr) = [2] x1 + [4]
           p(sum) = [1] x1 + [1]
          p(sqr#) = [0]         
          p(sum#) = [2] x1 + [0]
           p(c_1) = [1]         
           p(c_2) = [0]         
           p(c_3) = [1] x1 + [0]
           p(c_4) = [1] x1 + [8]
        
        Following rules are strictly oriented:
        sum#(s(x)) = [2] x + [16]
                   > [2] x + [8] 
                   = c_4(sum#(x))
        
        
        Following rules are (at-least) weakly oriented:
        sum#(s(x)) =  [2] x + [16]
                   >= [2] x + [0] 
                   =  c_3(sum#(x))
        
*** Step 6.b:1.a:2: Assumption 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:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.b:1.b:1: RemoveWeakSuffixes 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:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:sum#(s(x)) -> c_3(sum#(x))
             -->_1 sum#(s(x)) -> c_4(sum#(x)):2
             -->_1 sum#(s(x)) -> c_3(sum#(x)):1
          
          2:W:sum#(s(x)) -> c_4(sum#(x))
             -->_1 sum#(s(x)) -> c_4(sum#(x)):2
             -->_1 sum#(s(x)) -> c_3(sum#(x)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: sum#(s(x)) -> c_3(sum#(x))
          2: sum#(s(x)) -> c_4(sum#(x))
*** Step 6.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - 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))