/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),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:
             runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(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:
             runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          sum(x){x -> s(x)} =
            sum(s(x)) ->^+ +(sum(x),s(x))
              = C[sum(x) = sum(x){}]

** Step 1.b: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:
             runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          sum#(0()) -> c_1()
          sum#(s(x)) -> c_2(sum#(x),x)
          sum1#(0()) -> c_3()
          sum1#(s(x)) -> c_4(sum1#(x),x,x)
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(0()) -> c_1()
            sum#(s(x)) -> c_2(sum#(x),x)
            sum1#(0()) -> c_3()
            sum1#(s(x)) -> c_4(sum1#(x),x,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_2/2,c_3/0,c_4/3}
        - Obligation:
             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),x)
          sum1#(0()) -> c_3()
          sum1#(s(x)) -> c_4(sum1#(x),x,x)
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(0()) -> c_1()
            sum#(s(x)) -> c_2(sum#(x),x)
            sum1#(0()) -> c_3()
            sum1#(s(x)) -> c_4(sum1#(x),x,x)
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
        - Obligation:
             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),x)
          3: sum1#(0()) -> c_3()
          4: sum1#(s(x)) -> c_4(sum1#(x),x,x)
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_2(sum#(x),x)
            sum1#(s(x)) -> c_4(sum1#(x),x,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_2/2,c_3/0,c_4/3}
        - Obligation:
             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),x)
             -->_2 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
             -->_2 sum1#(0()) -> c_3():4
             -->_2 sum#(0()) -> c_1():3
             -->_1 sum#(0()) -> c_1():3
             -->_2 sum#(s(x)) -> c_2(sum#(x),x):1
             -->_1 sum#(s(x)) -> c_2(sum#(x),x):1
          
          2:S:sum1#(s(x)) -> c_4(sum1#(x),x,x)
             -->_3 sum1#(0()) -> c_3():4
             -->_2 sum1#(0()) -> c_3():4
             -->_1 sum1#(0()) -> c_3():4
             -->_3 sum#(0()) -> c_1():3
             -->_2 sum#(0()) -> c_1():3
             -->_3 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
             -->_2 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
             -->_1 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
             -->_3 sum#(s(x)) -> c_2(sum#(x),x):1
             -->_2 sum#(s(x)) -> c_2(sum#(x),x):1
          
          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.
          3: sum#(0()) -> c_1()
          4: sum1#(0()) -> c_3()
** Step 1.b:5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_2(sum#(x),x)
            sum1#(s(x)) -> c_4(sum1#(x),x,x)
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
        - Obligation:
             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),x)
          
        The strictly oriented rules are moved into the weak component.
*** Step 1.b:5.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum#(s(x)) -> c_2(sum#(x),x)
            sum1#(s(x)) -> c_4(sum1#(x),x,x)
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
        - Obligation:
             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},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
              p(+) = [1] x1 + [1] x2 + [0]
              p(0) = [2]                  
              p(s) = [1] x1 + [2]         
            p(sum) = [8] x1 + [1]         
           p(sum1) = [0]                  
           p(sum#) = [1] x1 + [0]         
          p(sum1#) = [0]                  
            p(c_1) = [0]                  
            p(c_2) = [1] x1 + [0]         
            p(c_3) = [0]                  
            p(c_4) = [8] x1 + [0]         
        
        Following rules are strictly oriented:
        sum#(s(x)) = [1] x + [2]   
                   > [1] x + [0]   
                   = c_2(sum#(x),x)
        
        
        Following rules are (at-least) weakly oriented:
        sum1#(s(x)) =  [0]              
                    >= [0]              
                    =  c_4(sum1#(x),x,x)
        
*** Step 1.b:5.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            sum1#(s(x)) -> c_4(sum1#(x),x,x)
        - Weak DPs:
            sum#(s(x)) -> c_2(sum#(x),x)
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
        - Obligation:
             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 1.b:5.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum1#(s(x)) -> c_4(sum1#(x),x,x)
        - Weak DPs:
            sum#(s(x)) -> c_2(sum#(x),x)
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
        - Obligation:
             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),x,x)
          
        The strictly oriented rules are moved into the weak component.
**** Step 1.b:5.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sum1#(s(x)) -> c_4(sum1#(x),x,x)
        - Weak DPs:
            sum#(s(x)) -> c_2(sum#(x),x)
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
        - Obligation:
             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},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
              p(+) = [1] x1 + [1] x2 + [1]
              p(0) = [0]                  
              p(s) = [1] x1 + [2]         
            p(sum) = [1] x1 + [0]         
           p(sum1) = [1] x1 + [0]         
           p(sum#) = [6] x1 + [0]         
          p(sum1#) = [8] x1 + [4]         
            p(c_1) = [1]                  
            p(c_2) = [1] x1 + [12]        
            p(c_3) = [0]                  
            p(c_4) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        sum1#(s(x)) = [8] x + [20]     
                    > [8] x + [4]      
                    = c_4(sum1#(x),x,x)
        
        
        Following rules are (at-least) weakly oriented:
        sum#(s(x)) =  [6] x + [12]  
                   >= [6] x + [12]  
                   =  c_2(sum#(x),x)
        
**** Step 1.b:5.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(s(x)) -> c_2(sum#(x),x)
            sum1#(s(x)) -> c_4(sum1#(x),x,x)
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
        - Obligation:
             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 1.b:5.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sum#(s(x)) -> c_2(sum#(x),x)
            sum1#(s(x)) -> c_4(sum1#(x),x,x)
        - Signature:
            {sum/1,sum1/1,sum#/1,sum1#/1} / {+/2,0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/3}
        - Obligation:
             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),x)
             -->_2 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
             -->_2 sum#(s(x)) -> c_2(sum#(x),x):1
             -->_1 sum#(s(x)) -> c_2(sum#(x),x):1
          
          2:W:sum1#(s(x)) -> c_4(sum1#(x),x,x)
             -->_3 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
             -->_2 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
             -->_1 sum1#(s(x)) -> c_4(sum1#(x),x,x):2
             -->_3 sum#(s(x)) -> c_2(sum#(x),x):1
             -->_2 sum#(s(x)) -> c_2(sum#(x),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),x)
          2: sum1#(s(x)) -> c_4(sum1#(x),x,x)
**** Step 1.b:5.b:1.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_2/2,c_3/0,c_4/3}
        - Obligation:
             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(Omega(n^1),O(n^1))