WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(0(),y,0(),u) -> true()
            f(0(),y,s(z),u) -> false()
            f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u)
            f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
            perfectp(0()) -> false()
            perfectp(s(x)) -> f(x,s(0()),s(x),s(x))
        - Signature:
            {f/4,perfectp/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,perfectp} and constructors {0,false,if,le,minus,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(0(),y,0(),u) -> c_1()
          f#(0(),y,s(z),u) -> c_2()
          f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
          f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
          perfectp#(0()) -> c_5()
          perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),y,0(),u) -> c_1()
            f#(0(),y,s(z),u) -> c_2()
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
            perfectp#(0()) -> c_5()
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Weak TRS:
            f(0(),y,0(),u) -> true()
            f(0(),y,s(z),u) -> false()
            f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u)
            f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
            perfectp(0()) -> false()
            perfectp(s(x)) -> f(x,s(0()),s(x),s(x))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
            ,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,5}
        by application of
          Pre({1,2,5}) = {4,6}.
        Here rules are labelled as follows:
          1: f#(0(),y,0(),u) -> c_1()
          2: f#(0(),y,s(z),u) -> c_2()
          3: f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
          4: f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
          5: perfectp#(0()) -> c_5()
          6: perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Weak DPs:
            f#(0(),y,0(),u) -> c_1()
            f#(0(),y,s(z),u) -> c_2()
            perfectp#(0()) -> c_5()
        - Weak TRS:
            f(0(),y,0(),u) -> true()
            f(0(),y,s(z),u) -> false()
            f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u)
            f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
            perfectp(0()) -> false()
            perfectp(s(x)) -> f(x,s(0()),s(x),s(x))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
             -->_1 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
          
          2:S:f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
             -->_2 f#(0(),y,s(z),u) -> c_2():5
             -->_2 f#(0(),y,0(),u) -> c_1():4
             -->_2 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_2 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
          
          3:S:perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
             -->_1 f#(0(),y,s(z),u) -> c_2():5
             -->_1 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
          
          4:W:f#(0(),y,0(),u) -> c_1()
             
          
          5:W:f#(0(),y,s(z),u) -> c_2()
             
          
          6:W:perfectp#(0()) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: perfectp#(0()) -> c_5()
          4: f#(0(),y,0(),u) -> c_1()
          5: f#(0(),y,s(z),u) -> c_2()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Weak TRS:
            f(0(),y,0(),u) -> true()
            f(0(),y,s(z),u) -> false()
            f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u)
            f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
            perfectp(0()) -> false()
            perfectp(s(x)) -> f(x,s(0()),s(x),s(x))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
            ,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
             -->_1 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
          
          2:S:f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u))
             -->_2 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
             -->_2 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
          
          3:S:perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
             -->_1 f#(s(x),s(y),z,u) -> c_4(f#(s(x),minus(y,x),z,u),f#(x,u,z,u)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Weak TRS:
            f(0(),y,0(),u) -> true()
            f(0(),y,s(z),u) -> false()
            f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u)
            f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))
            perfectp(0()) -> false()
            perfectp(s(x)) -> f(x,s(0()),s(x),s(x))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
            ,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
          f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
          perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
* Step 6: RemoveHeads WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
            perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
            ,true}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
           -->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2
           -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
        
        2:S:f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
           -->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2
           -->_1 f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u)):1
        
        3:S:perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x)))
           -->_1 f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u)):2
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(3,perfectp#(s(x)) -> c_6(f#(x,s(0()),s(x),s(x))))]
* Step 7: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
            ,true}
    + 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_3) = {1},
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]                           
                    p(f) = [0]                           
                p(false) = [0]                           
                   p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
                   p(le) = [1] x1 + [1] x2 + [0]         
                p(minus) = [1] x1 + [0]                  
             p(perfectp) = [0]                           
                    p(s) = [1] x1 + [3]                  
                 p(true) = [0]                           
                   p(f#) = [7] x1 + [4] x3 + [0]         
            p(perfectp#) = [0]                           
                  p(c_1) = [0]                           
                  p(c_2) = [0]                           
                  p(c_3) = [1] x1 + [0]                  
                  p(c_4) = [1] x1 + [0]                  
                  p(c_5) = [0]                           
                  p(c_6) = [0]                           
          
          Following rules are strictly oriented:
           f#(s(x),0(),z,u) = [7] x + [4] z + [21]        
                            > [7] x + [4] z + [0]         
                            = c_3(f#(x,u,minus(z,s(x)),u))
          
          f#(s(x),s(y),z,u) = [7] x + [4] z + [21]        
                            > [7] x + [4] z + [0]         
                            = c_4(f#(x,u,z,u))            
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 8: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(s(x),0(),z,u) -> c_3(f#(x,u,minus(z,s(x)),u))
            f#(s(x),s(y),z,u) -> c_4(f#(x,u,z,u))
        - Signature:
            {f/4,perfectp/1,f#/4,perfectp#/1} / {0/0,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/0
            ,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,perfectp#} and constructors {0,false,if,le,minus,s
            ,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))