WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
            D(+(x,y)) -> +(D(x),D(y))
            D(-(x,y)) -> -(D(x),D(y))
            D(constant()) -> 0()
            D(t()) -> 1()
        - Signature:
            {D/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D} and constructors {*,+,-,0,1,constant,t}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          D#(*(x,y)) -> c_1(D#(x),D#(y))
          D#(+(x,y)) -> c_2(D#(x),D#(y))
          D#(-(x,y)) -> c_3(D#(x),D#(y))
          D#(constant()) -> c_4()
          D#(t()) -> c_5()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
            D#(constant()) -> c_4()
            D#(t()) -> c_5()
        - Strict TRS:
            D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
            D(+(x,y)) -> +(D(x),D(y))
            D(-(x,y)) -> -(D(x),D(y))
            D(constant()) -> 0()
            D(t()) -> 1()
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          D#(*(x,y)) -> c_1(D#(x),D#(y))
          D#(+(x,y)) -> c_2(D#(x),D#(y))
          D#(-(x,y)) -> c_3(D#(x),D#(y))
          D#(constant()) -> c_4()
          D#(t()) -> c_5()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
            D#(constant()) -> c_4()
            D#(t()) -> c_5()
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {4,5}
        by application of
          Pre({4,5}) = {1,2,3}.
        Here rules are labelled as follows:
          1: D#(*(x,y)) -> c_1(D#(x),D#(y))
          2: D#(+(x,y)) -> c_2(D#(x),D#(y))
          3: D#(-(x,y)) -> c_3(D#(x),D#(y))
          4: D#(constant()) -> c_4()
          5: D#(t()) -> c_5()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
        - Weak DPs:
            D#(constant()) -> c_4()
            D#(t()) -> c_5()
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:D#(*(x,y)) -> c_1(D#(x),D#(y))
             -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_2 D#(t()) -> c_5():5
             -->_1 D#(t()) -> c_5():5
             -->_2 D#(constant()) -> c_4():4
             -->_1 D#(constant()) -> c_4():4
             -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
             -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
          
          2:S:D#(+(x,y)) -> c_2(D#(x),D#(y))
             -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_2 D#(t()) -> c_5():5
             -->_1 D#(t()) -> c_5():5
             -->_2 D#(constant()) -> c_4():4
             -->_1 D#(constant()) -> c_4():4
             -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
             -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
          
          3:S:D#(-(x,y)) -> c_3(D#(x),D#(y))
             -->_2 D#(t()) -> c_5():5
             -->_1 D#(t()) -> c_5():5
             -->_2 D#(constant()) -> c_4():4
             -->_1 D#(constant()) -> c_4():4
             -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
             -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
          
          4:W:D#(constant()) -> c_4()
             
          
          5:W:D#(t()) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: D#(constant()) -> c_4()
          5: D#(t()) -> c_5()
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + 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:
          3: D#(-(x,y)) -> c_3(D#(x),D#(y))
          
        The strictly oriented rules are moved into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + 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_1) = {1,2},
          uargs(c_2) = {1,2},
          uargs(c_3) = {1,2}
        
        Following symbols are considered usable:
          {D#}
        TcT has computed the following interpretation:
                 p(*) = [1] x1 + [1] x2 + [0]
                 p(+) = [1] x1 + [1] x2 + [0]
                 p(-) = [1] x1 + [1] x2 + [1]
                 p(0) = [0]                  
                 p(1) = [1]                  
                 p(D) = [0]                  
          p(constant) = [2]                  
                 p(t) = [0]                  
                p(D#) = [8] x1 + [0]         
               p(c_1) = [1] x1 + [1] x2 + [0]
               p(c_2) = [1] x1 + [1] x2 + [0]
               p(c_3) = [1] x1 + [1] x2 + [4]
               p(c_4) = [1]                  
               p(c_5) = [1]                  
        
        Following rules are strictly oriented:
        D#(-(x,y)) = [8] x + [8] y + [8]
                   > [8] x + [8] y + [4]
                   = c_3(D#(x),D#(y))   
        
        
        Following rules are (at-least) weakly oriented:
        D#(*(x,y)) =  [8] x + [8] y + [0]
                   >= [8] x + [8] y + [0]
                   =  c_1(D#(x),D#(y))   
        
        D#(+(x,y)) =  [8] x + [8] y + [0]
                   >= [8] x + [8] y + [0]
                   =  c_2(D#(x),D#(y))   
        
** Step 5.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
        - Weak DPs:
            D#(-(x,y)) -> c_3(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 5.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
        - Weak DPs:
            D#(-(x,y)) -> c_3(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + 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: D#(*(x,y)) -> c_1(D#(x),D#(y))
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
        - Weak DPs:
            D#(-(x,y)) -> c_3(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + 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_1) = {1,2},
          uargs(c_2) = {1,2},
          uargs(c_3) = {1,2}
        
        Following symbols are considered usable:
          {D#}
        TcT has computed the following interpretation:
                 p(*) = [1] x1 + [1] x2 + [2]
                 p(+) = [1] x1 + [1] x2 + [0]
                 p(-) = [1] x1 + [1] x2 + [0]
                 p(0) = [1]                  
                 p(1) = [1]                  
                 p(D) = [1]                  
          p(constant) = [2]                  
                 p(t) = [1]                  
                p(D#) = [8] x1 + [0]         
               p(c_1) = [1] x1 + [1] x2 + [4]
               p(c_2) = [1] x1 + [1] x2 + [0]
               p(c_3) = [1] x1 + [1] x2 + [0]
               p(c_4) = [0]                  
               p(c_5) = [2]                  
        
        Following rules are strictly oriented:
        D#(*(x,y)) = [8] x + [8] y + [16]
                   > [8] x + [8] y + [4] 
                   = c_1(D#(x),D#(y))    
        
        
        Following rules are (at-least) weakly oriented:
        D#(+(x,y)) =  [8] x + [8] y + [0]
                   >= [8] x + [8] y + [0]
                   =  c_2(D#(x),D#(y))   
        
        D#(-(x,y)) =  [8] x + [8] y + [0]
                   >= [8] x + [8] y + [0]
                   =  c_3(D#(x),D#(y))   
        
*** Step 5.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            D#(+(x,y)) -> c_2(D#(x),D#(y))
        - Weak DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 5.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(+(x,y)) -> c_2(D#(x),D#(y))
        - Weak DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + 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: D#(+(x,y)) -> c_2(D#(x),D#(y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 5.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            D#(+(x,y)) -> c_2(D#(x),D#(y))
        - Weak DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + 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_1) = {1,2},
          uargs(c_2) = {1,2},
          uargs(c_3) = {1,2}
        
        Following symbols are considered usable:
          {D#}
        TcT has computed the following interpretation:
                 p(*) = [1] x1 + [1] x2 + [2] 
                 p(+) = [1] x1 + [1] x2 + [1] 
                 p(-) = [1] x1 + [1] x2 + [0] 
                 p(0) = [1]                   
                 p(1) = [1]                   
                 p(D) = [1] x1 + [1]          
          p(constant) = [1]                   
                 p(t) = [1]                   
                p(D#) = [8] x1 + [0]          
               p(c_1) = [1] x1 + [1] x2 + [14]
               p(c_2) = [1] x1 + [1] x2 + [4] 
               p(c_3) = [1] x1 + [1] x2 + [0] 
               p(c_4) = [1]                   
               p(c_5) = [1]                   
        
        Following rules are strictly oriented:
        D#(+(x,y)) = [8] x + [8] y + [8]
                   > [8] x + [8] y + [4]
                   = c_2(D#(x),D#(y))   
        
        
        Following rules are (at-least) weakly oriented:
        D#(*(x,y)) =  [8] x + [8] y + [16]
                   >= [8] x + [8] y + [14]
                   =  c_1(D#(x),D#(y))    
        
        D#(-(x,y)) =  [8] x + [8] y + [0] 
                   >= [8] x + [8] y + [0] 
                   =  c_3(D#(x),D#(y))    
        
**** Step 5.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 5.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            D#(*(x,y)) -> c_1(D#(x),D#(y))
            D#(+(x,y)) -> c_2(D#(x),D#(y))
            D#(-(x,y)) -> c_3(D#(x),D#(y))
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:D#(*(x,y)) -> c_1(D#(x),D#(y))
             -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
             -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
          
          2:W:D#(+(x,y)) -> c_2(D#(x),D#(y))
             -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
             -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
          
          3:W:D#(-(x,y)) -> c_3(D#(x),D#(y))
             -->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
             -->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
             -->_2 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
             -->_1 D#(*(x,y)) -> c_1(D#(x),D#(y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: D#(*(x,y)) -> c_1(D#(x),D#(y))
          3: D#(-(x,y)) -> c_3(D#(x),D#(y))
          2: D#(+(x,y)) -> c_2(D#(x),D#(y))
**** Step 5.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/2,c_2/2,c_3/2,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))