WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          sum#(0()) -> c_1()
          sum#(s(x)) -> c_2(sum#(x))
          sum1#(0()) -> c_3()
          sum1#(s(x)) -> c_4(sum1#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(0()) -> c_1()
            sum#(s(x)) -> c_2(sum#(x))
            sum1#(0()) -> c_3()
            sum1#(s(x)) -> c_4(sum1#(x))
        - Strict TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          sum#(0()) -> c_1()
          sum#(s(x)) -> c_2(sum#(x))
          sum1#(0()) -> c_3()
          sum1#(s(x)) -> c_4(sum1#(x))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(0()) -> c_1()
            sum#(s(x)) -> c_2(sum#(x))
            sum1#(0()) -> c_3()
            sum1#(s(x)) -> c_4(sum1#(x))
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3}
        by application of
          Pre({1,3}) = {2,4}.
        Here rules are labelled as follows:
          1: sum#(0()) -> c_1()
          2: sum#(s(x)) -> c_2(sum#(x))
          3: sum1#(0()) -> c_3()
          4: sum1#(s(x)) -> c_4(sum1#(x))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_2(sum#(x))
            sum1#(s(x)) -> c_4(sum1#(x))
        - Weak DPs:
            sum#(0()) -> c_1()
            sum1#(0()) -> c_3()
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sum#(s(x)) -> c_2(sum#(x))
             -->_1 sum#(0()) -> c_1():3
             -->_1 sum#(s(x)) -> c_2(sum#(x)):1
          
          2:S:sum1#(s(x)) -> c_4(sum1#(x))
             -->_1 sum1#(0()) -> c_3():4
             -->_1 sum1#(s(x)) -> c_4(sum1#(x)):2
          
          3:W:sum#(0()) -> c_1()
             
          
          4:W:sum1#(0()) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: sum1#(0()) -> c_3()
          3: sum#(0()) -> c_1()
* Step 5: Decompose WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_2(sum#(x))
            sum1#(s(x)) -> c_4(sum1#(x))
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              sum#(s(x)) -> c_2(sum#(x))
          - Weak DPs:
              sum1#(s(x)) -> c_4(sum1#(x))
          - Signature:
              {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
        
        Problem (S)
          - Strict DPs:
              sum1#(s(x)) -> c_4(sum1#(x))
          - Weak DPs:
              sum#(s(x)) -> c_2(sum#(x))
          - Signature:
              {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_2(sum#(x))
        - Weak DPs:
            sum1#(s(x)) -> c_4(sum1#(x))
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sum#(s(x)) -> c_2(sum#(x))
             -->_1 sum#(s(x)) -> c_2(sum#(x)):1
          
          2:W:sum1#(s(x)) -> c_4(sum1#(x))
             -->_1 sum1#(s(x)) -> c_4(sum1#(x)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: sum1#(s(x)) -> c_4(sum1#(x))
** Step 5.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_2(sum#(x))
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} 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_2(sum#(x))
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_2(sum#(x))
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} 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_2) = {1}
        
        Following symbols are considered usable:
          {sum#,sum1#}
        TcT has computed the following interpretation:
              p(+) = [1] x1 + [1] x2 + [0]
              p(0) = [2]                  
              p(s) = [1] x1 + [8]         
            p(sum) = [1]                  
           p(sum1) = [4] x1 + [4]         
           p(sum#) = [2] x1 + [0]         
          p(sum1#) = [2] x1 + [1]         
            p(c_1) = [1]                  
            p(c_2) = [1] x1 + [0]         
            p(c_3) = [0]                  
            p(c_4) = [1]                  
        
        Following rules are strictly oriented:
        sum#(s(x)) = [2] x + [16]
                   > [2] x + [0] 
                   = c_2(sum#(x))
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 5.a:2.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(s(x)) -> c_2(sum#(x))
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 5.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(s(x)) -> c_2(sum#(x))
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:sum#(s(x)) -> c_2(sum#(x))
             -->_1 sum#(s(x)) -> c_2(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_2(sum#(x))
*** Step 5.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum1#(s(x)) -> c_4(sum1#(x))
        - Weak DPs:
            sum#(s(x)) -> c_2(sum#(x))
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sum1#(s(x)) -> c_4(sum1#(x))
             -->_1 sum1#(s(x)) -> c_4(sum1#(x)):1
          
          2:W:sum#(s(x)) -> c_2(sum#(x))
             -->_1 sum#(s(x)) -> c_2(sum#(x)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: sum#(s(x)) -> c_2(sum#(x))
** Step 5.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum1#(s(x)) -> c_4(sum1#(x))
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} 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: sum1#(s(x)) -> c_4(sum1#(x))
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum1#(s(x)) -> c_4(sum1#(x))
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} 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_4) = {1}
        
        Following symbols are considered usable:
          {sum#,sum1#}
        TcT has computed the following interpretation:
              p(+) = [0]          
              p(0) = [2]          
              p(s) = [1] x1 + [4] 
            p(sum) = [8] x1 + [0] 
           p(sum1) = [1] x1 + [1] 
           p(sum#) = [0]          
          p(sum1#) = [4] x1 + [0] 
            p(c_1) = [0]          
            p(c_2) = [2] x1 + [1] 
            p(c_3) = [1]          
            p(c_4) = [1] x1 + [12]
        
        Following rules are strictly oriented:
        sum1#(s(x)) = [4] x + [16] 
                    > [4] x + [12] 
                    = c_4(sum1#(x))
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 5.b:2.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum1#(s(x)) -> c_4(sum1#(x))
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 5.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum1#(s(x)) -> c_4(sum1#(x))
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:sum1#(s(x)) -> c_4(sum1#(x))
             -->_1 sum1#(s(x)) -> c_4(sum1#(x)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: sum1#(s(x)) -> c_4(sum1#(x))
*** Step 5.b:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sum#,sum1#} and constructors {+,0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))