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: Ara 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:
        Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1}
    + Details:
        Signatures used:
        ----------------
          s :: ["A"(1)] -(1)-> "A"(1)
          mult# :: ["A"(1) x "A"(0)] -(0)-> "A"(0)
          c_4 :: ["A"(0)] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          mult#(s(x),y) -> c_4(mult#(x,y))
        2. Weak:
          

** Step 4.b:1: Ara 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:
        Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1}
    + Details:
        Signatures used:
        ----------------
          0 :: [] -(0)-> "A"(0)
          add :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          mult :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          s :: ["A"(1)] -(1)-> "A"(1)
          s :: ["A"(0)] -(0)-> "A"(0)
          add# :: ["A"(1) x "A"(0)] -(0)-> "A"(0)
          mult# :: ["A"(0) x "A"(1)] -(0)-> "A"(0)
          c_2 :: ["A"(0)] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          add#(s(x),y) -> c_2(add#(x,y))
        2. Weak:
          

WORST_CASE(?,O(n^2))