WORST_CASE(?,O(1))
* Step 1: DependencyPairs WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict TRS:
            f(x,y,z) -> g(<=(x,y),x,y,z)
            g(false(),x,y,z) -> f(f(p(x),y,z),f(p(y),z,x),f(p(z),x,y))
            g(true(),x,y,z) -> z
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {f/3,g/4,p/1} / {0/0,<=/2,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,p} and constructors {0,<=,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(x,y,z) -> c_1(g#(<=(x,y),x,y,z))
          g#(false(),x,y,z) -> c_2(f#(f(p(x),y,z),f(p(y),z,x),f(p(z),x,y))
                                  ,f#(p(x),y,z)
                                  ,p#(x)
                                  ,f#(p(y),z,x)
                                  ,p#(y)
                                  ,f#(p(z),x,y)
                                  ,p#(z))
          g#(true(),x,y,z) -> c_3()
          p#(0()) -> c_4()
          p#(s(x)) -> c_5()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(x,y,z) -> c_1(g#(<=(x,y),x,y,z))
            g#(false(),x,y,z) -> c_2(f#(f(p(x),y,z),f(p(y),z,x),f(p(z),x,y))
                                    ,f#(p(x),y,z)
                                    ,p#(x)
                                    ,f#(p(y),z,x)
                                    ,p#(y)
                                    ,f#(p(z),x,y)
                                    ,p#(z))
            g#(true(),x,y,z) -> c_3()
            p#(0()) -> c_4()
            p#(s(x)) -> c_5()
        - Weak TRS:
            f(x,y,z) -> g(<=(x,y),x,y,z)
            g(false(),x,y,z) -> f(f(p(x),y,z),f(p(y),z,x),f(p(z),x,y))
            g(true(),x,y,z) -> z
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {f/3,g/4,p/1,f#/3,g#/4,p#/1} / {0/0,<=/2,false/0,s/1,true/0,c_1/1,c_2/7,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,p#} and constructors {0,<=,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f(x,y,z) -> g(<=(x,y),x,y,z)
          p(0()) -> 0()
          p(s(x)) -> x
          f#(x,y,z) -> c_1(g#(<=(x,y),x,y,z))
          g#(false(),x,y,z) -> c_2(f#(f(p(x),y,z),f(p(y),z,x),f(p(z),x,y))
                                  ,f#(p(x),y,z)
                                  ,p#(x)
                                  ,f#(p(y),z,x)
                                  ,p#(y)
                                  ,f#(p(z),x,y)
                                  ,p#(z))
          g#(true(),x,y,z) -> c_3()
          p#(0()) -> c_4()
          p#(s(x)) -> c_5()
* Step 3: Trivial WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(x,y,z) -> c_1(g#(<=(x,y),x,y,z))
            g#(false(),x,y,z) -> c_2(f#(f(p(x),y,z),f(p(y),z,x),f(p(z),x,y))
                                    ,f#(p(x),y,z)
                                    ,p#(x)
                                    ,f#(p(y),z,x)
                                    ,p#(y)
                                    ,f#(p(z),x,y)
                                    ,p#(z))
            g#(true(),x,y,z) -> c_3()
            p#(0()) -> c_4()
            p#(s(x)) -> c_5()
        - Weak TRS:
            f(x,y,z) -> g(<=(x,y),x,y,z)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {f/3,g/4,p/1,f#/3,g#/4,p#/1} / {0/0,<=/2,false/0,s/1,true/0,c_1/1,c_2/7,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,p#} and constructors {0,<=,false,s,true}
    + Applied Processor:
        Trivial
    + Details:
        Consider the dependency graph
          1:S:f#(x,y,z) -> c_1(g#(<=(x,y),x,y,z))
             
          
          2:S:g#(false(),x,y,z) -> c_2(f#(f(p(x),y,z),f(p(y),z,x),f(p(z),x,y))
                                      ,f#(p(x),y,z)
                                      ,p#(x)
                                      ,f#(p(y),z,x)
                                      ,p#(y)
                                      ,f#(p(z),x,y)
                                      ,p#(z))
             -->_7 p#(s(x)) -> c_5():5
             -->_5 p#(s(x)) -> c_5():5
             -->_3 p#(s(x)) -> c_5():5
             -->_7 p#(0()) -> c_4():4
             -->_5 p#(0()) -> c_4():4
             -->_3 p#(0()) -> c_4():4
             -->_6 f#(x,y,z) -> c_1(g#(<=(x,y),x,y,z)):1
             -->_4 f#(x,y,z) -> c_1(g#(<=(x,y),x,y,z)):1
             -->_2 f#(x,y,z) -> c_1(g#(<=(x,y),x,y,z)):1
             -->_1 f#(x,y,z) -> c_1(g#(<=(x,y),x,y,z)):1
          
          3:S:g#(true(),x,y,z) -> c_3()
             
          
          4:S:p#(0()) -> c_4()
             
          
          5:S:p#(s(x)) -> c_5()
             
          
        The dependency graph contains no loops, we remove all dependency pairs.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x,y,z) -> g(<=(x,y),x,y,z)
            p(0()) -> 0()
            p(s(x)) -> x
        - Signature:
            {f/3,g/4,p/1,f#/3,g#/4,p#/1} / {0/0,<=/2,false/0,s/1,true/0,c_1/1,c_2/7,c_3/0,c_4/0,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,p#} and constructors {0,<=,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))