WORST_CASE(?,O(n^3))
* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1} / {0/0,p/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*,+,minus} and constructors {0,p,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          *#(0(),y) -> c_1()
          *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
          *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
          +#(0(),y) -> c_4()
          +#(p(x),y) -> c_5(+#(x,y))
          +#(s(x),y) -> c_6(+#(x,y))
          minus#(0()) -> c_7()
          minus#(p(x)) -> c_8(minus#(x))
          minus#(s(x)) -> c_9(minus#(x))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            *#(0(),y) -> c_1()
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
            +#(0(),y) -> c_4()
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
            minus#(0()) -> c_7()
            minus#(p(x)) -> c_8(minus#(x))
            minus#(s(x)) -> c_9(minus#(x))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,4,7}
        by application of
          Pre({1,4,7}) = {2,3,5,6,8,9}.
        Here rules are labelled as follows:
          1: *#(0(),y) -> c_1()
          2: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
          3: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
          4: +#(0(),y) -> c_4()
          5: +#(p(x),y) -> c_5(+#(x,y))
          6: +#(s(x),y) -> c_6(+#(x,y))
          7: minus#(0()) -> c_7()
          8: minus#(p(x)) -> c_8(minus#(x))
          9: minus#(s(x)) -> c_9(minus#(x))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
            minus#(p(x)) -> c_8(minus#(x))
            minus#(s(x)) -> c_9(minus#(x))
        - Weak DPs:
            *#(0(),y) -> c_1()
            +#(0(),y) -> c_4()
            minus#(0()) -> c_7()
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
             -->_3 minus#(s(x)) -> c_9(minus#(x)):6
             -->_3 minus#(p(x)) -> c_8(minus#(x)):5
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):4
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):3
             -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2
             -->_3 minus#(0()) -> c_7():9
             -->_1 +#(0(),y) -> c_4():8
             -->_2 *#(0(),y) -> c_1():7
             -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):1
          
          2:S:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):4
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):3
             -->_1 +#(0(),y) -> c_4():8
             -->_2 *#(0(),y) -> c_1():7
             -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2
             -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):1
          
          3:S:+#(p(x),y) -> c_5(+#(x,y))
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):4
             -->_1 +#(0(),y) -> c_4():8
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):3
          
          4:S:+#(s(x),y) -> c_6(+#(x,y))
             -->_1 +#(0(),y) -> c_4():8
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):4
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):3
          
          5:S:minus#(p(x)) -> c_8(minus#(x))
             -->_1 minus#(s(x)) -> c_9(minus#(x)):6
             -->_1 minus#(0()) -> c_7():9
             -->_1 minus#(p(x)) -> c_8(minus#(x)):5
          
          6:S:minus#(s(x)) -> c_9(minus#(x))
             -->_1 minus#(0()) -> c_7():9
             -->_1 minus#(s(x)) -> c_9(minus#(x)):6
             -->_1 minus#(p(x)) -> c_8(minus#(x)):5
          
          7:W:*#(0(),y) -> c_1()
             
          
          8:W:+#(0(),y) -> c_4()
             
          
          9:W:minus#(0()) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: *#(0(),y) -> c_1()
          8: +#(0(),y) -> c_4()
          9: minus#(0()) -> c_7()
* Step 4: PredecessorEstimationCP WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
            minus#(p(x)) -> c_8(minus#(x))
            minus#(s(x)) -> c_9(minus#(x))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
          2: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
          5: minus#(p(x)) -> c_8(minus#(x))
          6: minus#(s(x)) -> c_9(minus#(x))
          
        The strictly oriented rules are moved into the weak component.
** Step 4.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
            minus#(p(x)) -> c_8(minus#(x))
            minus#(s(x)) -> c_9(minus#(x))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_2) = {1,2,3},
          uargs(c_3) = {1,2},
          uargs(c_5) = {1},
          uargs(c_6) = {1},
          uargs(c_8) = {1},
          uargs(c_9) = {1}
        
        Following symbols are considered usable:
          {*#,+#,minus#}
        TcT has computed the following interpretation:
               p(*) = 0                        
               p(+) = 3*x1 + 2*x2 + 2*x2^2     
               p(0) = 2                        
           p(minus) = 0                        
               p(p) = 2 + x1                   
               p(s) = 2 + x1                   
              p(*#) = 2 + 2*x1 + 2*x1*x2 + x2^2
              p(+#) = 0                        
          p(minus#) = x1                       
             p(c_1) = 1                        
             p(c_2) = 3 + x1 + x2 + x3         
             p(c_3) = 2 + x1 + x2              
             p(c_4) = 1                        
             p(c_5) = x1                       
             p(c_6) = x1                       
             p(c_7) = 2                        
             p(c_8) = 1 + x1                   
             p(c_9) = x1                       
        
        Following rules are strictly oriented:
          *#(p(x),y) = 6 + 2*x + 2*x*y + 4*y + y^2               
                     > 5 + 2*x + 2*x*y + y + y^2                 
                     = c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
        
          *#(s(x),y) = 6 + 2*x + 2*x*y + 4*y + y^2               
                     > 4 + 2*x + 2*x*y + y^2                     
                     = c_3(+#(*(x,y),y),*#(x,y))                 
        
        minus#(p(x)) = 2 + x                                     
                     > 1 + x                                     
                     = c_8(minus#(x))                            
        
        minus#(s(x)) = 2 + x                                     
                     > x                                         
                     = c_9(minus#(x))                            
        
        
        Following rules are (at-least) weakly oriented:
        +#(p(x),y) =  0           
                   >= 0           
                   =  c_5(+#(x,y))
        
        +#(s(x),y) =  0           
                   >= 0           
                   =  c_6(+#(x,y))
        
** Step 4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
        - Weak DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
            minus#(p(x)) -> c_8(minus#(x))
            minus#(s(x)) -> c_9(minus#(x))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
        - Weak DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
            minus#(p(x)) -> c_8(minus#(x))
            minus#(s(x)) -> c_9(minus#(x))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:+#(p(x),y) -> c_5(+#(x,y))
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          2:S:+#(s(x),y) -> c_6(+#(x,y))
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          3:W:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
             -->_3 minus#(s(x)) -> c_9(minus#(x)):6
             -->_3 minus#(p(x)) -> c_8(minus#(x)):5
             -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4
             -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          4:W:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
             -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4
             -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          5:W:minus#(p(x)) -> c_8(minus#(x))
             -->_1 minus#(s(x)) -> c_9(minus#(x)):6
             -->_1 minus#(p(x)) -> c_8(minus#(x)):5
          
          6:W:minus#(s(x)) -> c_9(minus#(x))
             -->_1 minus#(s(x)) -> c_9(minus#(x)):6
             -->_1 minus#(p(x)) -> c_8(minus#(x)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: minus#(s(x)) -> c_9(minus#(x))
          5: minus#(p(x)) -> c_8(minus#(x))
** Step 4.b:2: SimplifyRHS WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
        - Weak DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:+#(p(x),y) -> c_5(+#(x,y))
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          2:S:+#(s(x),y) -> c_6(+#(x,y))
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          3:W:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
             -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4
             -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          4:W:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
             -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4
             -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
** Step 4.b:3: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
        - Weak DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
          *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        and a lower component
          +#(p(x),y) -> c_5(+#(x,y))
          +#(s(x),y) -> c_6(+#(x,y))
        Further, following extension rules are added to the lower component.
          *#(p(x),y) -> *#(x,y)
          *#(p(x),y) -> +#(*(x,y),minus(y))
          *#(s(x),y) -> *#(x,y)
          *#(s(x),y) -> +#(*(x,y),y)
*** Step 4.b:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 4.b:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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) = {1,2},
          uargs(c_3) = {1,2}
        
        Following symbols are considered usable:
          {*#,+#,minus#}
        TcT has computed the following interpretation:
               p(*) = [4] x1 + [0]         
               p(+) = [3] x1 + [2] x2 + [8]
               p(0) = [1]                  
           p(minus) = [8]                  
               p(p) = [1] x1 + [1]         
               p(s) = [1] x1 + [0]         
              p(*#) = [8] x1 + [0]         
              p(+#) = [0]                  
          p(minus#) = [0]                  
             p(c_1) = [2]                  
             p(c_2) = [1] x1 + [1] x2 + [0]
             p(c_3) = [2] x1 + [1] x2 + [0]
             p(c_4) = [0]                  
             p(c_5) = [0]                  
             p(c_6) = [0]                  
             p(c_7) = [0]                  
             p(c_8) = [0]                  
             p(c_9) = [0]                  
        
        Following rules are strictly oriented:
        *#(p(x),y) = [8] x + [8]                     
                   > [8] x + [0]                     
                   = c_2(+#(*(x,y),minus(y)),*#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        *#(s(x),y) =  [8] x + [0]              
                   >= [8] x + [0]              
                   =  c_3(+#(*(x,y),y),*#(x,y))
        
**** Step 4.b:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        - Weak DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 4.b:3.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        - Weak DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
          
        The strictly oriented rules are moved into the weak component.
***** Step 4.b:3.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        - Weak DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,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) = {1,2},
          uargs(c_3) = {1,2}
        
        Following symbols are considered usable:
          {*#,+#,minus#}
        TcT has computed the following interpretation:
               p(*) = [12] x1 + [2] x2 + [0]
               p(+) = [2] x1 + [1]          
               p(0) = [0]                   
           p(minus) = [10] x1 + [9]         
               p(p) = [1] x1 + [1]          
               p(s) = [1] x1 + [1]          
              p(*#) = [4] x1 + [2] x2 + [0] 
              p(+#) = [0]                   
          p(minus#) = [0]                   
             p(c_1) = [0]                   
             p(c_2) = [4] x1 + [1] x2 + [2] 
             p(c_3) = [4] x1 + [1] x2 + [0] 
             p(c_4) = [8]                   
             p(c_5) = [1] x1 + [0]          
             p(c_6) = [2] x1 + [1]          
             p(c_7) = [0]                   
             p(c_8) = [2] x1 + [0]          
             p(c_9) = [1]                   
        
        Following rules are strictly oriented:
        *#(s(x),y) = [4] x + [2] y + [4]      
                   > [4] x + [2] y + [0]      
                   = c_3(+#(*(x,y),y),*#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        *#(p(x),y) =  [4] x + [2] y + [4]             
                   >= [4] x + [2] y + [2]             
                   =  c_2(+#(*(x,y),minus(y)),*#(x,y))
        
***** Step 4.b:3.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 4.b:3.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
             -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2
             -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)):1
          
          2:W:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
             -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2
             -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
          2: *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
***** Step 4.b:3.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 4.b:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
        - Weak DPs:
            *#(p(x),y) -> *#(x,y)
            *#(p(x),y) -> +#(*(x,y),minus(y))
            *#(s(x),y) -> *#(x,y)
            *#(s(x),y) -> +#(*(x,y),y)
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          2: +#(s(x),y) -> c_6(+#(x,y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 4.b:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
        - Weak DPs:
            *#(p(x),y) -> *#(x,y)
            *#(p(x),y) -> +#(*(x,y),minus(y))
            *#(s(x),y) -> *#(x,y)
            *#(s(x),y) -> +#(*(x,y),y)
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_5) = {1},
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          {*,+,minus,*#,+#,minus#}
        TcT has computed the following interpretation:
               p(*) = 2*x1*x2 + x1^2                         
               p(+) = x1 + 2*x2                              
               p(0) = 0                                      
           p(minus) = 2 + x1                                 
               p(p) = 2 + x1                                 
               p(s) = 2 + x1                                 
              p(*#) = 1 + x1 + 2*x1*x2 + x1^2 + 2*x2 + 2*x2^2
              p(+#) = x1 + x2 + x2^2                         
          p(minus#) = x1 + 2*x1^2                            
             p(c_1) = 2                                      
             p(c_2) = 0                                      
             p(c_3) = 0                                      
             p(c_4) = 1                                      
             p(c_5) = 2 + x1                                 
             p(c_6) = x1                                     
             p(c_7) = 0                                      
             p(c_8) = 0                                      
             p(c_9) = 1                                      
        
        Following rules are strictly oriented:
        +#(s(x),y) = 2 + x + y + y^2
                   > x + y + y^2    
                   = c_6(+#(x,y))   
        
        
        Following rules are (at-least) weakly oriented:
         *#(p(x),y) =  7 + 5*x + 2*x*y + x^2 + 6*y + 2*y^2
                    >= 1 + x + 2*x*y + x^2 + 2*y + 2*y^2  
                    =  *#(x,y)                            
        
         *#(p(x),y) =  7 + 5*x + 2*x*y + x^2 + 6*y + 2*y^2
                    >= 6 + 2*x*y + x^2 + 5*y + y^2        
                    =  +#(*(x,y),minus(y))                
        
         *#(s(x),y) =  7 + 5*x + 2*x*y + x^2 + 6*y + 2*y^2
                    >= 1 + x + 2*x*y + x^2 + 2*y + 2*y^2  
                    =  *#(x,y)                            
        
         *#(s(x),y) =  7 + 5*x + 2*x*y + x^2 + 6*y + 2*y^2
                    >= 2*x*y + x^2 + y + y^2              
                    =  +#(*(x,y),y)                       
        
         +#(p(x),y) =  2 + x + y + y^2                    
                    >= 2 + x + y + y^2                    
                    =  c_5(+#(x,y))                       
        
           *(0(),y) =  0                                  
                    >= 0                                  
                    =  0()                                
        
          *(p(x),y) =  4 + 4*x + 2*x*y + x^2 + 4*y        
                    >= 4 + 2*x*y + x^2 + 2*y              
                    =  +(*(x,y),minus(y))                 
        
          *(s(x),y) =  4 + 4*x + 2*x*y + x^2 + 4*y        
                    >= 2*x*y + x^2 + 2*y                  
                    =  +(*(x,y),y)                        
        
           +(0(),y) =  2*y                                
                    >= y                                  
                    =  y                                  
        
          +(p(x),y) =  2 + x + 2*y                        
                    >= 2 + x + 2*y                        
                    =  p(+(x,y))                          
        
          +(s(x),y) =  2 + x + 2*y                        
                    >= 2 + x + 2*y                        
                    =  s(+(x,y))                          
        
         minus(0()) =  2                                  
                    >= 0                                  
                    =  0()                                
        
        minus(p(x)) =  4 + x                              
                    >= 4 + x                              
                    =  s(minus(x))                        
        
        minus(s(x)) =  4 + x                              
                    >= 4 + x                              
                    =  p(minus(x))                        
        
**** Step 4.b:3.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            +#(p(x),y) -> c_5(+#(x,y))
        - Weak DPs:
            *#(p(x),y) -> *#(x,y)
            *#(p(x),y) -> +#(*(x,y),minus(y))
            *#(s(x),y) -> *#(x,y)
            *#(s(x),y) -> +#(*(x,y),y)
            +#(s(x),y) -> c_6(+#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 4.b:3.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(p(x),y) -> c_5(+#(x,y))
        - Weak DPs:
            *#(p(x),y) -> *#(x,y)
            *#(p(x),y) -> +#(*(x,y),minus(y))
            *#(s(x),y) -> *#(x,y)
            *#(s(x),y) -> +#(*(x,y),y)
            +#(s(x),y) -> c_6(+#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: +#(p(x),y) -> c_5(+#(x,y))
          
        The strictly oriented rules are moved into the weak component.
***** Step 4.b:3.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(p(x),y) -> c_5(+#(x,y))
        - Weak DPs:
            *#(p(x),y) -> *#(x,y)
            *#(p(x),y) -> +#(*(x,y),minus(y))
            *#(s(x),y) -> *#(x,y)
            *#(s(x),y) -> +#(*(x,y),y)
            +#(s(x),y) -> c_6(+#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_5) = {1},
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          {*,+,minus,*#,+#,minus#}
        TcT has computed the following interpretation:
               p(*) = 2*x1 + x1*x2               
               p(+) = 2 + x1 + x2                
               p(0) = 0                          
           p(minus) = 1 + x1                     
               p(p) = 2 + x1                     
               p(s) = 2 + x1                     
              p(*#) = 3 + 3*x1*x2 + x1^2 + 3*x2^2
              p(+#) = 2*x1 + 3*x2^2              
          p(minus#) = 1                          
             p(c_1) = 1                          
             p(c_2) = 0                          
             p(c_3) = 0                          
             p(c_4) = 0                          
             p(c_5) = x1                         
             p(c_6) = 1 + x1                     
             p(c_7) = 1                          
             p(c_8) = 2 + x1                     
             p(c_9) = 2                          
        
        Following rules are strictly oriented:
        +#(p(x),y) = 4 + 2*x + 3*y^2
                   > 2*x + 3*y^2    
                   = c_5(+#(x,y))   
        
        
        Following rules are (at-least) weakly oriented:
         *#(p(x),y) =  7 + 4*x + 3*x*y + x^2 + 6*y + 3*y^2
                    >= 3 + 3*x*y + x^2 + 3*y^2            
                    =  *#(x,y)                            
        
         *#(p(x),y) =  7 + 4*x + 3*x*y + x^2 + 6*y + 3*y^2
                    >= 3 + 4*x + 2*x*y + 6*y + 3*y^2      
                    =  +#(*(x,y),minus(y))                
        
         *#(s(x),y) =  7 + 4*x + 3*x*y + x^2 + 6*y + 3*y^2
                    >= 3 + 3*x*y + x^2 + 3*y^2            
                    =  *#(x,y)                            
        
         *#(s(x),y) =  7 + 4*x + 3*x*y + x^2 + 6*y + 3*y^2
                    >= 4*x + 2*x*y + 3*y^2                
                    =  +#(*(x,y),y)                       
        
         +#(s(x),y) =  4 + 2*x + 3*y^2                    
                    >= 1 + 2*x + 3*y^2                    
                    =  c_6(+#(x,y))                       
        
           *(0(),y) =  0                                  
                    >= 0                                  
                    =  0()                                
        
          *(p(x),y) =  4 + 2*x + x*y + 2*y                
                    >= 3 + 2*x + x*y + y                  
                    =  +(*(x,y),minus(y))                 
        
          *(s(x),y) =  4 + 2*x + x*y + 2*y                
                    >= 2 + 2*x + x*y + y                  
                    =  +(*(x,y),y)                        
        
           +(0(),y) =  2 + y                              
                    >= y                                  
                    =  y                                  
        
          +(p(x),y) =  4 + x + y                          
                    >= 4 + x + y                          
                    =  p(+(x,y))                          
        
          +(s(x),y) =  4 + x + y                          
                    >= 4 + x + y                          
                    =  s(+(x,y))                          
        
         minus(0()) =  1                                  
                    >= 0                                  
                    =  0()                                
        
        minus(p(x)) =  3 + x                              
                    >= 3 + x                              
                    =  s(minus(x))                        
        
        minus(s(x)) =  3 + x                              
                    >= 3 + x                              
                    =  p(minus(x))                        
        
***** Step 4.b:3.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            *#(p(x),y) -> *#(x,y)
            *#(p(x),y) -> +#(*(x,y),minus(y))
            *#(s(x),y) -> *#(x,y)
            *#(s(x),y) -> +#(*(x,y),y)
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

***** Step 4.b:3.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            *#(p(x),y) -> *#(x,y)
            *#(p(x),y) -> +#(*(x,y),minus(y))
            *#(s(x),y) -> *#(x,y)
            *#(s(x),y) -> +#(*(x,y),y)
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:*#(p(x),y) -> *#(x,y)
             -->_1 *#(s(x),y) -> +#(*(x,y),y):4
             -->_1 *#(s(x),y) -> *#(x,y):3
             -->_1 *#(p(x),y) -> +#(*(x,y),minus(y)):2
             -->_1 *#(p(x),y) -> *#(x,y):1
          
          2:W:*#(p(x),y) -> +#(*(x,y),minus(y))
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):6
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):5
          
          3:W:*#(s(x),y) -> *#(x,y)
             -->_1 *#(s(x),y) -> +#(*(x,y),y):4
             -->_1 *#(s(x),y) -> *#(x,y):3
             -->_1 *#(p(x),y) -> +#(*(x,y),minus(y)):2
             -->_1 *#(p(x),y) -> *#(x,y):1
          
          4:W:*#(s(x),y) -> +#(*(x,y),y)
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):6
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):5
          
          5:W:+#(p(x),y) -> c_5(+#(x,y))
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):6
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):5
          
          6:W:+#(s(x),y) -> c_6(+#(x,y))
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):6
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: *#(p(x),y) -> *#(x,y)
          3: *#(s(x),y) -> *#(x,y)
          2: *#(p(x),y) -> +#(*(x,y),minus(y))
          4: *#(s(x),y) -> +#(*(x,y),y)
          6: +#(s(x),y) -> c_6(+#(x,y))
          5: +#(p(x),y) -> c_5(+#(x,y))
***** Step 4.b:3.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {*#,+#,minus#} and constructors {0,p,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^3))