WORST_CASE(Omega(n^1),O(n^3))
* Step 1: Sum WORST_CASE(Omega(n^1),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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + 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:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          *(x,y){x -> p(x)} =
            *(p(x),y) ->^+ +(*(x,y),minus(y))
              = C[*(x,y) = *(x,y){}]

** Step 1.b: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 1.b: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 1.b: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 1.b:4: DecomposeDG 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:
        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
          *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(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))
          minus#(p(x)) -> c_8(minus#(x))
          minus#(s(x)) -> c_9(minus#(x))
        Further, following extension rules are added to the lower component.
          *#(p(x),y) -> *#(x,y)
          *#(p(x),y) -> +#(*(x,y),minus(y))
          *#(p(x),y) -> minus#(y)
          *#(s(x),y) -> *#(x,y)
          *#(s(x),y) -> +#(*(x,y),y)
*** Step 1.b:4.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + 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))
        - 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_2(+#(*(x,y),minus(y)),*#(x,y),minus#(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),minus#(y)):1
          
          2:S:*#(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),minus#(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))
          *#(s(x),y) -> c_3(*#(x,y))
*** Step 1.b:4.a:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(p(x),y) -> c_2(*#(x,y))
            *#(s(x),y) -> c_3(*#(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_2/1,c_3/1,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:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          *#(p(x),y) -> c_2(*#(x,y))
          *#(s(x),y) -> c_3(*#(x,y))
*** Step 1.b:4.a:3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            *#(p(x),y) -> c_2(*#(x,y))
            *#(s(x),y) -> c_3(*#(x,y))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/1,c_3/1,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:
        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_2) = {1},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(*) = [0]         
                 p(+) = [0]         
                 p(0) = [0]         
             p(minus) = [1] x1 + [0]
                 p(p) = [1] x1 + [3]
                 p(s) = [1] x1 + [1]
                p(*#) = [7] x1 + [0]
                p(+#) = [0]         
            p(minus#) = [0]         
               p(c_1) = [0]         
               p(c_2) = [1] x1 + [0]
               p(c_3) = [1] x1 + [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) = [7] x + [21]
                     > [7] x + [0] 
                     = c_2(*#(x,y))
          
          *#(s(x),y) = [7] x + [7] 
                     > [7] x + [0] 
                     = c_3(*#(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 1.b:4.a:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            *#(p(x),y) -> c_2(*#(x,y))
            *#(s(x),y) -> c_3(*#(x,y))
        - Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/1,c_3/1,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 1.b:4.b: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))
            minus#(p(x)) -> c_8(minus#(x))
            minus#(s(x)) -> c_9(minus#(x))
        - Weak DPs:
            *#(p(x),y) -> *#(x,y)
            *#(p(x),y) -> +#(*(x,y),minus(y))
            *#(p(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_2/3,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 = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          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(+) = 0            
               p(0) = 4            
           p(minus) = 2            
               p(p) = x1           
               p(s) = 4 + x1       
              p(*#) = 4 + x1 + 4*x2
              p(+#) = 0            
          p(minus#) = 2*x1         
             p(c_1) = 2            
             p(c_2) = x1           
             p(c_3) = x1           
             p(c_4) = 2            
             p(c_5) = x1           
             p(c_6) = x1           
             p(c_7) = 0            
             p(c_8) = x1           
             p(c_9) = x1           
        
        Following rules are strictly oriented:
        minus#(s(x)) = 8 + 2*x       
                     > 2*x           
                     = c_9(minus#(x))
        
        
        Following rules are (at-least) weakly oriented:
          *#(p(x),y) =  4 + x + 4*y        
                     >= 4 + x + 4*y        
                     =  *#(x,y)            
        
          *#(p(x),y) =  4 + x + 4*y        
                     >= 0                  
                     =  +#(*(x,y),minus(y))
        
          *#(p(x),y) =  4 + x + 4*y        
                     >= 2*y                
                     =  minus#(y)          
        
          *#(s(x),y) =  8 + x + 4*y        
                     >= 4 + x + 4*y        
                     =  *#(x,y)            
        
          *#(s(x),y) =  8 + x + 4*y        
                     >= 0                  
                     =  +#(*(x,y),y)       
        
          +#(p(x),y) =  0                  
                     >= 0                  
                     =  c_5(+#(x,y))       
        
          +#(s(x),y) =  0                  
                     >= 0                  
                     =  c_6(+#(x,y))       
        
        minus#(p(x)) =  2*x                
                     >= 2*x                
                     =  c_8(minus#(x))     
        
*** Step 1.b:4.b:2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
            minus#(p(x)) -> c_8(minus#(x))
        - Weak DPs:
            *#(p(x),y) -> *#(x,y)
            *#(p(x),y) -> +#(*(x,y),minus(y))
            *#(p(x),y) -> minus#(y)
            *#(s(x),y) -> *#(x,y)
            *#(s(x),y) -> +#(*(x,y),y)
            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 = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          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(*) = 1 + 7*x1 + x2
               p(+) = 2*x1         
               p(0) = 1            
           p(minus) = 4*x1         
               p(p) = 1 + x1       
               p(s) = x1           
              p(*#) = 2*x1 + 7*x2  
              p(+#) = 0            
          p(minus#) = 1 + 7*x1     
             p(c_1) = 0            
             p(c_2) = 1            
             p(c_3) = 4            
             p(c_4) = 1            
             p(c_5) = x1           
             p(c_6) = x1           
             p(c_7) = 0            
             p(c_8) = x1           
             p(c_9) = x1           
        
        Following rules are strictly oriented:
        minus#(p(x)) = 8 + 7*x       
                     > 1 + 7*x       
                     = c_8(minus#(x))
        
        
        Following rules are (at-least) weakly oriented:
          *#(p(x),y) =  2 + 2*x + 7*y      
                     >= 2*x + 7*y          
                     =  *#(x,y)            
        
          *#(p(x),y) =  2 + 2*x + 7*y      
                     >= 0                  
                     =  +#(*(x,y),minus(y))
        
          *#(p(x),y) =  2 + 2*x + 7*y      
                     >= 1 + 7*y            
                     =  minus#(y)          
        
          *#(s(x),y) =  2*x + 7*y          
                     >= 2*x + 7*y          
                     =  *#(x,y)            
        
          *#(s(x),y) =  2*x + 7*y          
                     >= 0                  
                     =  +#(*(x,y),y)       
        
          +#(p(x),y) =  0                  
                     >= 0                  
                     =  c_5(+#(x,y))       
        
          +#(s(x),y) =  0                  
                     >= 0                  
                     =  c_6(+#(x,y))       
        
        minus#(s(x)) =  1 + 7*x            
                     >= 1 + 7*x            
                     =  c_9(minus#(x))     
        
*** Step 1.b:4.b:3: 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))
            *#(p(x),y) -> minus#(y)
            *#(s(x),y) -> *#(x,y)
            *#(s(x),y) -> +#(*(x,y),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 any 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},
          uargs(c_8) = {1},
          uargs(c_9) = {1}
        
        Following symbols are considered usable:
          {*,+,minus,*#,+#,minus#}
        TcT has computed the following interpretation:
               p(*) = x1*x2      
               p(+) = x1 + x2    
               p(0) = 0          
           p(minus) = 2*x1       
               p(p) = 2 + x1     
               p(s) = 2 + x1     
              p(*#) = 2 + 3*x1*x2
              p(+#) = 2 + 2*x1   
          p(minus#) = 1 + 2*x1   
             p(c_1) = 0          
             p(c_2) = x1         
             p(c_3) = 1          
             p(c_4) = 0          
             p(c_5) = 3 + x1     
             p(c_6) = 3 + x1     
             p(c_7) = 0          
             p(c_8) = 3 + x1     
             p(c_9) = x1         
        
        Following rules are strictly oriented:
        +#(p(x),y) = 6 + 2*x     
                   > 5 + 2*x     
                   = c_5(+#(x,y))
        
        +#(s(x),y) = 6 + 2*x     
                   > 5 + 2*x     
                   = c_6(+#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
          *#(p(x),y) =  2 + 3*x*y + 6*y    
                     >= 2 + 3*x*y          
                     =  *#(x,y)            
        
          *#(p(x),y) =  2 + 3*x*y + 6*y    
                     >= 2 + 2*x*y          
                     =  +#(*(x,y),minus(y))
        
          *#(p(x),y) =  2 + 3*x*y + 6*y    
                     >= 1 + 2*y            
                     =  minus#(y)          
        
          *#(s(x),y) =  2 + 3*x*y + 6*y    
                     >= 2 + 3*x*y          
                     =  *#(x,y)            
        
          *#(s(x),y) =  2 + 3*x*y + 6*y    
                     >= 2 + 2*x*y          
                     =  +#(*(x,y),y)       
        
        minus#(p(x)) =  5 + 2*x            
                     >= 4 + 2*x            
                     =  c_8(minus#(x))     
        
        minus#(s(x)) =  5 + 2*x            
                     >= 1 + 2*x            
                     =  c_9(minus#(x))     
        
            *(0(),y) =  0                  
                     >= 0                  
                     =  0()                
        
           *(p(x),y) =  x*y + 2*y          
                     >= x*y + 2*y          
                     =  +(*(x,y),minus(y)) 
        
           *(s(x),y) =  x*y + 2*y          
                     >= x*y + y            
                     =  +(*(x,y),y)        
        
            +(0(),y) =  y                  
                     >= y                  
                     =  y                  
        
           +(p(x),y) =  2 + x + y          
                     >= 2 + x + y          
                     =  p(+(x,y))          
        
           +(s(x),y) =  2 + x + y          
                     >= 2 + x + y          
                     =  s(+(x,y))          
        
          minus(0()) =  0                  
                     >= 0                  
                     =  0()                
        
         minus(p(x)) =  4 + 2*x            
                     >= 2 + 2*x            
                     =  s(minus(x))        
        
         minus(s(x)) =  4 + 2*x            
                     >= 2 + 2*x            
                     =  p(minus(x))        
        
*** Step 1.b:4.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            *#(p(x),y) -> *#(x,y)
            *#(p(x),y) -> +#(*(x,y),minus(y))
            *#(p(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))
            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:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^3))