WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mult(0(),y) -> 0()
            mult(s(x),y) -> add(y,mult(x,y))
        - Signature:
            {add/2,mult/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add,mult} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          add#(0(),y) -> c_1()
          add#(s(x),y) -> c_2(add#(x,y))
          mult#(0(),y) -> c_3()
          mult#(s(x),y) -> c_4(add#(y,mult(x,y)),mult#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add#(0(),y) -> c_1()
            add#(s(x),y) -> c_2(add#(x,y))
            mult#(0(),y) -> c_3()
            mult#(s(x),y) -> c_4(add#(y,mult(x,y)),mult#(x,y))
        - Weak TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mult(0(),y) -> 0()
            mult(s(x),y) -> add(y,mult(x,y))
        - Signature:
            {add/2,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mult#} 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: add#(0(),y) -> c_1()
          2: add#(s(x),y) -> c_2(add#(x,y))
          3: mult#(0(),y) -> c_3()
          4: mult#(s(x),y) -> c_4(add#(y,mult(x,y)),mult#(x,y))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add#(s(x),y) -> c_2(add#(x,y))
            mult#(s(x),y) -> c_4(add#(y,mult(x,y)),mult#(x,y))
        - Weak DPs:
            add#(0(),y) -> c_1()
            mult#(0(),y) -> c_3()
        - Weak TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mult(0(),y) -> 0()
            mult(s(x),y) -> add(y,mult(x,y))
        - Signature:
            {add/2,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:add#(s(x),y) -> c_2(add#(x,y))
             -->_1 add#(0(),y) -> c_1():3
             -->_1 add#(s(x),y) -> c_2(add#(x,y)):1
          
          2:S:mult#(s(x),y) -> c_4(add#(y,mult(x,y)),mult#(x,y))
             -->_2 mult#(0(),y) -> c_3():4
             -->_1 add#(0(),y) -> c_1():3
             -->_2 mult#(s(x),y) -> c_4(add#(y,mult(x,y)),mult#(x,y)):2
             -->_1 add#(s(x),y) -> c_2(add#(x,y)):1
          
          3:W:add#(0(),y) -> c_1()
             
          
          4:W:mult#(0(),y) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: mult#(0(),y) -> c_3()
          3: add#(0(),y) -> c_1()
* Step 4: DecomposeDG WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            add#(s(x),y) -> c_2(add#(x,y))
            mult#(s(x),y) -> c_4(add#(y,mult(x,y)),mult#(x,y))
        - Weak TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mult(0(),y) -> 0()
            mult(s(x),y) -> add(y,mult(x,y))
        - Signature:
            {add/2,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          mult#(s(x),y) -> c_4(add#(y,mult(x,y)),mult#(x,y))
        and a lower component
          add#(s(x),y) -> c_2(add#(x,y))
        Further, following extension rules are added to the lower component.
          mult#(s(x),y) -> add#(y,mult(x,y))
          mult#(s(x),y) -> mult#(x,y)
** Step 4.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mult#(s(x),y) -> c_4(add#(y,mult(x,y)),mult#(x,y))
        - Weak TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mult(0(),y) -> 0()
            mult(s(x),y) -> add(y,mult(x,y))
        - Signature:
            {add/2,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:mult#(s(x),y) -> c_4(add#(y,mult(x,y)),mult#(x,y))
             -->_2 mult#(s(x),y) -> c_4(add#(y,mult(x,y)),mult#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          mult#(s(x),y) -> c_4(mult#(x,y))
** Step 4.a:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mult#(s(x),y) -> c_4(mult#(x,y))
        - Weak TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mult(0(),y) -> 0()
            mult(s(x),y) -> add(y,mult(x,y))
        - Signature:
            {add/2,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          mult#(s(x),y) -> c_4(mult#(x,y))
** Step 4.a:3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mult#(s(x),y) -> c_4(mult#(x,y))
        - Signature:
            {add/2,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mult#} 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_4) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(0) = [2]                  
              p(add) = [1] x1 + [1] x2 + [1]
             p(mult) = [1] x1 + [8] x2 + [0]
                p(s) = [1] x1 + [4]         
             p(add#) = [4] x1 + [1] x2 + [1]
            p(mult#) = [1] x1 + [1] x2 + [0]
              p(c_1) = [0]                  
              p(c_2) = [4] x1 + [1]         
              p(c_3) = [2]                  
              p(c_4) = [1] x1 + [1]         
          
          Following rules are strictly oriented:
          mult#(s(x),y) = [1] x + [1] y + [4]
                        > [1] x + [1] y + [1]
                        = c_4(mult#(x,y))    
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 4.a:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mult#(s(x),y) -> c_4(mult#(x,y))
        - Signature:
            {add/2,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

** Step 4.b:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            add#(s(x),y) -> c_2(add#(x,y))
        - Weak DPs:
            mult#(s(x),y) -> add#(y,mult(x,y))
            mult#(s(x),y) -> mult#(x,y)
        - Weak TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mult(0(),y) -> 0()
            mult(s(x),y) -> add(y,mult(x,y))
        - Signature:
            {add/2,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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:
          {add#,mult#}
        TcT has computed the following interpretation:
              p(0) = [0]         
            p(add) = [0]         
           p(mult) = [2] x1 + [0]
              p(s) = [1] x1 + [4]
           p(add#) = [4] x1 + [8]
          p(mult#) = [4] x2 + [8]
            p(c_1) = [0]         
            p(c_2) = [1] x1 + [2]
            p(c_3) = [0]         
            p(c_4) = [0]         
        
        Following rules are strictly oriented:
        add#(s(x),y) = [4] x + [24]  
                     > [4] x + [10]  
                     = c_2(add#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        mult#(s(x),y) =  [4] y + [8]      
                      >= [4] y + [8]      
                      =  add#(y,mult(x,y))
        
        mult#(s(x),y) =  [4] y + [8]      
                      >= [4] y + [8]      
                      =  mult#(x,y)       
        
** Step 4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            add#(s(x),y) -> c_2(add#(x,y))
            mult#(s(x),y) -> add#(y,mult(x,y))
            mult#(s(x),y) -> mult#(x,y)
        - Weak TRS:
            add(0(),y) -> y
            add(s(x),y) -> s(add(x,y))
            mult(0(),y) -> 0()
            mult(s(x),y) -> add(y,mult(x,y))
        - Signature:
            {add/2,mult/2,add#/2,mult#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {add#,mult#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))