WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            d(0()) -> 0()
            d(s(X)) -> s(s(d(X)))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
            p(X,0()) -> X
            p(0(),X) -> X
            p(s(X),s(Y)) -> s(s(p(X,Y)))
            q(0()) -> 0()
            q(s(X)) -> s(p(q(X),d(X)))
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            d(0()) -> 0()
            d(s(X)) -> s(s(d(X)))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
            p(X,0()) -> X
            p(0(),X) -> X
            p(s(X),s(Y)) -> s(s(p(X,Y)))
            q(0()) -> 0()
            q(s(X)) -> s(p(q(X),d(X)))
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          a(x){x -> nf(x,y)} =
            a(nf(x,y)) ->^+ f(a(x),a(y))
              = C[a(x) = a(x){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            d(0()) -> 0()
            d(s(X)) -> s(s(d(X)))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
            p(X,0()) -> X
            p(0(),X) -> X
            p(s(X),s(Y)) -> s(s(p(X,Y)))
            q(0()) -> 0()
            q(s(X)) -> s(p(q(X),d(X)))
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a,d,f,p,q,s,t} and constructors {0,cs,nf,nil,ns,nt,r}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          a#(X) -> c_1()
          a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2))
          a#(ns(X)) -> c_3(s#(a(X)),a#(X))
          a#(nt(X)) -> c_4(t#(a(X)),a#(X))
          d#(0()) -> c_5()
          d#(s(X)) -> c_6(s#(s(d(X))),s#(d(X)),d#(X))
          f#(X1,X2) -> c_7()
          f#(0(),X) -> c_8()
          f#(s(X),cs(Y,Z)) -> c_9(a#(Z))
          p#(X,0()) -> c_10()
          p#(0(),X) -> c_11()
          p#(s(X),s(Y)) -> c_12(s#(s(p(X,Y))),s#(p(X,Y)),p#(X,Y))
          q#(0()) -> c_13()
          q#(s(X)) -> c_14(s#(p(q(X),d(X))),p#(q(X),d(X)),q#(X),d#(X))
          s#(X) -> c_15()
          t#(N) -> c_16(q#(N))
          t#(X) -> c_17()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(X) -> c_1()
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2))
            a#(ns(X)) -> c_3(s#(a(X)),a#(X))
            a#(nt(X)) -> c_4(t#(a(X)),a#(X))
            d#(0()) -> c_5()
            d#(s(X)) -> c_6(s#(s(d(X))),s#(d(X)),d#(X))
            f#(X1,X2) -> c_7()
            f#(0(),X) -> c_8()
            f#(s(X),cs(Y,Z)) -> c_9(a#(Z))
            p#(X,0()) -> c_10()
            p#(0(),X) -> c_11()
            p#(s(X),s(Y)) -> c_12(s#(s(p(X,Y))),s#(p(X,Y)),p#(X,Y))
            q#(0()) -> c_13()
            q#(s(X)) -> c_14(s#(p(q(X),d(X))),p#(q(X),d(X)),q#(X),d#(X))
            s#(X) -> c_15()
            t#(N) -> c_16(q#(N))
            t#(X) -> c_17()
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            d(0()) -> 0()
            d(s(X)) -> s(s(d(X)))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
            p(X,0()) -> X
            p(0(),X) -> X
            p(s(X),s(Y)) -> s(s(p(X,Y)))
            q(0()) -> 0()
            q(s(X)) -> s(p(q(X),d(X)))
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/3,c_3/2,c_4/2,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/3,c_13/0,c_14/4,c_15/0,c_16/1,c_17/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,5,6,7,8,9,10,11,12,13,14,15,17}
        by application of
          Pre({1,5,6,7,8,9,10,11,12,13,14,15,17}) = {2,3,4,16}.
        Here rules are labelled as follows:
          1: a#(X) -> c_1()
          2: a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2))
          3: a#(ns(X)) -> c_3(s#(a(X)),a#(X))
          4: a#(nt(X)) -> c_4(t#(a(X)),a#(X))
          5: d#(0()) -> c_5()
          6: d#(s(X)) -> c_6(s#(s(d(X))),s#(d(X)),d#(X))
          7: f#(X1,X2) -> c_7()
          8: f#(0(),X) -> c_8()
          9: f#(s(X),cs(Y,Z)) -> c_9(a#(Z))
          10: p#(X,0()) -> c_10()
          11: p#(0(),X) -> c_11()
          12: p#(s(X),s(Y)) -> c_12(s#(s(p(X,Y))),s#(p(X,Y)),p#(X,Y))
          13: q#(0()) -> c_13()
          14: q#(s(X)) -> c_14(s#(p(q(X),d(X))),p#(q(X),d(X)),q#(X),d#(X))
          15: s#(X) -> c_15()
          16: t#(N) -> c_16(q#(N))
          17: t#(X) -> c_17()
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2))
            a#(ns(X)) -> c_3(s#(a(X)),a#(X))
            a#(nt(X)) -> c_4(t#(a(X)),a#(X))
            t#(N) -> c_16(q#(N))
        - Weak DPs:
            a#(X) -> c_1()
            d#(0()) -> c_5()
            d#(s(X)) -> c_6(s#(s(d(X))),s#(d(X)),d#(X))
            f#(X1,X2) -> c_7()
            f#(0(),X) -> c_8()
            f#(s(X),cs(Y,Z)) -> c_9(a#(Z))
            p#(X,0()) -> c_10()
            p#(0(),X) -> c_11()
            p#(s(X),s(Y)) -> c_12(s#(s(p(X,Y))),s#(p(X,Y)),p#(X,Y))
            q#(0()) -> c_13()
            q#(s(X)) -> c_14(s#(p(q(X),d(X))),p#(q(X),d(X)),q#(X),d#(X))
            s#(X) -> c_15()
            t#(X) -> c_17()
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            d(0()) -> 0()
            d(s(X)) -> s(s(d(X)))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
            p(X,0()) -> X
            p(0(),X) -> X
            p(s(X),s(Y)) -> s(s(p(X,Y)))
            q(0()) -> 0()
            q(s(X)) -> s(p(q(X),d(X)))
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/3,c_3/2,c_4/2,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/3,c_13/0,c_14/4,c_15/0,c_16/1,c_17/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {4}
        by application of
          Pre({4}) = {3}.
        Here rules are labelled as follows:
          1: a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2))
          2: a#(ns(X)) -> c_3(s#(a(X)),a#(X))
          3: a#(nt(X)) -> c_4(t#(a(X)),a#(X))
          4: t#(N) -> c_16(q#(N))
          5: a#(X) -> c_1()
          6: d#(0()) -> c_5()
          7: d#(s(X)) -> c_6(s#(s(d(X))),s#(d(X)),d#(X))
          8: f#(X1,X2) -> c_7()
          9: f#(0(),X) -> c_8()
          10: f#(s(X),cs(Y,Z)) -> c_9(a#(Z))
          11: p#(X,0()) -> c_10()
          12: p#(0(),X) -> c_11()
          13: p#(s(X),s(Y)) -> c_12(s#(s(p(X,Y))),s#(p(X,Y)),p#(X,Y))
          14: q#(0()) -> c_13()
          15: q#(s(X)) -> c_14(s#(p(q(X),d(X))),p#(q(X),d(X)),q#(X),d#(X))
          16: s#(X) -> c_15()
          17: t#(X) -> c_17()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2))
            a#(ns(X)) -> c_3(s#(a(X)),a#(X))
            a#(nt(X)) -> c_4(t#(a(X)),a#(X))
        - Weak DPs:
            a#(X) -> c_1()
            d#(0()) -> c_5()
            d#(s(X)) -> c_6(s#(s(d(X))),s#(d(X)),d#(X))
            f#(X1,X2) -> c_7()
            f#(0(),X) -> c_8()
            f#(s(X),cs(Y,Z)) -> c_9(a#(Z))
            p#(X,0()) -> c_10()
            p#(0(),X) -> c_11()
            p#(s(X),s(Y)) -> c_12(s#(s(p(X,Y))),s#(p(X,Y)),p#(X,Y))
            q#(0()) -> c_13()
            q#(s(X)) -> c_14(s#(p(q(X),d(X))),p#(q(X),d(X)),q#(X),d#(X))
            s#(X) -> c_15()
            t#(N) -> c_16(q#(N))
            t#(X) -> c_17()
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            d(0()) -> 0()
            d(s(X)) -> s(s(d(X)))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
            p(X,0()) -> X
            p(0(),X) -> X
            p(s(X),s(Y)) -> s(s(p(X,Y)))
            q(0()) -> 0()
            q(s(X)) -> s(p(q(X),d(X)))
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/3,c_3/2,c_4/2,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/3,c_13/0,c_14/4,c_15/0,c_16/1,c_17/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2))
             -->_3 a#(nt(X)) -> c_4(t#(a(X)),a#(X)):3
             -->_2 a#(nt(X)) -> c_4(t#(a(X)),a#(X)):3
             -->_3 a#(ns(X)) -> c_3(s#(a(X)),a#(X)):2
             -->_2 a#(ns(X)) -> c_3(s#(a(X)),a#(X)):2
             -->_1 f#(0(),X) -> c_8():8
             -->_1 f#(X1,X2) -> c_7():7
             -->_3 a#(X) -> c_1():4
             -->_2 a#(X) -> c_1():4
             -->_3 a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2)):1
             -->_2 a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2)):1
          
          2:S:a#(ns(X)) -> c_3(s#(a(X)),a#(X))
             -->_2 a#(nt(X)) -> c_4(t#(a(X)),a#(X)):3
             -->_1 s#(X) -> c_15():15
             -->_2 a#(X) -> c_1():4
             -->_2 a#(ns(X)) -> c_3(s#(a(X)),a#(X)):2
             -->_2 a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2)):1
          
          3:S:a#(nt(X)) -> c_4(t#(a(X)),a#(X))
             -->_1 t#(N) -> c_16(q#(N)):16
             -->_1 t#(X) -> c_17():17
             -->_2 a#(X) -> c_1():4
             -->_2 a#(nt(X)) -> c_4(t#(a(X)),a#(X)):3
             -->_2 a#(ns(X)) -> c_3(s#(a(X)),a#(X)):2
             -->_2 a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2)):1
          
          4:W:a#(X) -> c_1()
             
          
          5:W:d#(0()) -> c_5()
             
          
          6:W:d#(s(X)) -> c_6(s#(s(d(X))),s#(d(X)),d#(X))
             
          
          7:W:f#(X1,X2) -> c_7()
             
          
          8:W:f#(0(),X) -> c_8()
             
          
          9:W:f#(s(X),cs(Y,Z)) -> c_9(a#(Z))
             
          
          10:W:p#(X,0()) -> c_10()
             
          
          11:W:p#(0(),X) -> c_11()
             
          
          12:W:p#(s(X),s(Y)) -> c_12(s#(s(p(X,Y))),s#(p(X,Y)),p#(X,Y))
             
          
          13:W:q#(0()) -> c_13()
             
          
          14:W:q#(s(X)) -> c_14(s#(p(q(X),d(X))),p#(q(X),d(X)),q#(X),d#(X))
             
          
          15:W:s#(X) -> c_15()
             
          
          16:W:t#(N) -> c_16(q#(N))
             -->_1 q#(0()) -> c_13():13
          
          17:W:t#(X) -> c_17()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          14: q#(s(X)) -> c_14(s#(p(q(X),d(X))),p#(q(X),d(X)),q#(X),d#(X))
          12: p#(s(X),s(Y)) -> c_12(s#(s(p(X,Y))),s#(p(X,Y)),p#(X,Y))
          11: p#(0(),X) -> c_11()
          10: p#(X,0()) -> c_10()
          9: f#(s(X),cs(Y,Z)) -> c_9(a#(Z))
          6: d#(s(X)) -> c_6(s#(s(d(X))),s#(d(X)),d#(X))
          5: d#(0()) -> c_5()
          7: f#(X1,X2) -> c_7()
          8: f#(0(),X) -> c_8()
          15: s#(X) -> c_15()
          4: a#(X) -> c_1()
          17: t#(X) -> c_17()
          16: t#(N) -> c_16(q#(N))
          13: q#(0()) -> c_13()
** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2))
            a#(ns(X)) -> c_3(s#(a(X)),a#(X))
            a#(nt(X)) -> c_4(t#(a(X)),a#(X))
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            d(0()) -> 0()
            d(s(X)) -> s(s(d(X)))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
            p(X,0()) -> X
            p(0(),X) -> X
            p(s(X),s(Y)) -> s(s(p(X,Y)))
            q(0()) -> 0()
            q(s(X)) -> s(p(q(X),d(X)))
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/3,c_3/2,c_4/2,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/3,c_13/0,c_14/4,c_15/0,c_16/1,c_17/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2))
             -->_3 a#(nt(X)) -> c_4(t#(a(X)),a#(X)):3
             -->_2 a#(nt(X)) -> c_4(t#(a(X)),a#(X)):3
             -->_3 a#(ns(X)) -> c_3(s#(a(X)),a#(X)):2
             -->_2 a#(ns(X)) -> c_3(s#(a(X)),a#(X)):2
             -->_3 a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2)):1
             -->_2 a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2)):1
          
          2:S:a#(ns(X)) -> c_3(s#(a(X)),a#(X))
             -->_2 a#(nt(X)) -> c_4(t#(a(X)),a#(X)):3
             -->_2 a#(ns(X)) -> c_3(s#(a(X)),a#(X)):2
             -->_2 a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2)):1
          
          3:S:a#(nt(X)) -> c_4(t#(a(X)),a#(X))
             -->_2 a#(nt(X)) -> c_4(t#(a(X)),a#(X)):3
             -->_2 a#(ns(X)) -> c_3(s#(a(X)),a#(X)):2
             -->_2 a#(nf(X1,X2)) -> c_2(f#(a(X1),a(X2)),a#(X1),a#(X2)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          a#(nf(X1,X2)) -> c_2(a#(X1),a#(X2))
          a#(ns(X)) -> c_3(a#(X))
          a#(nt(X)) -> c_4(a#(X))
** Step 1.b:6: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(nf(X1,X2)) -> c_2(a#(X1),a#(X2))
            a#(ns(X)) -> c_3(a#(X))
            a#(nt(X)) -> c_4(a#(X))
        - Weak TRS:
            a(X) -> X
            a(nf(X1,X2)) -> f(a(X1),a(X2))
            a(ns(X)) -> s(a(X))
            a(nt(X)) -> t(a(X))
            d(0()) -> 0()
            d(s(X)) -> s(s(d(X)))
            f(X1,X2) -> nf(X1,X2)
            f(0(),X) -> nil()
            f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
            p(X,0()) -> X
            p(0(),X) -> X
            p(s(X),s(Y)) -> s(s(p(X,Y)))
            q(0()) -> 0()
            q(s(X)) -> s(p(q(X),d(X)))
            s(X) -> ns(X)
            t(N) -> cs(r(q(N)),nt(ns(N)))
            t(X) -> nt(X)
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/2,c_3/1,c_4/1,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/3,c_13/0,c_14/4,c_15/0,c_16/1,c_17/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          a#(nf(X1,X2)) -> c_2(a#(X1),a#(X2))
          a#(ns(X)) -> c_3(a#(X))
          a#(nt(X)) -> c_4(a#(X))
** Step 1.b:7: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(nf(X1,X2)) -> c_2(a#(X1),a#(X2))
            a#(ns(X)) -> c_3(a#(X))
            a#(nt(X)) -> c_4(a#(X))
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/2,c_3/1,c_4/1,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/3,c_13/0,c_14/4,c_15/0,c_16/1,c_17/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + 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) = {1,2},
            uargs(c_3) = {1},
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [0]                  
               p(a) = [0]                  
              p(cs) = [1] x1 + [1] x2 + [0]
               p(d) = [0]                  
               p(f) = [0]                  
              p(nf) = [1] x1 + [1] x2 + [0]
             p(nil) = [0]                  
              p(ns) = [1] x1 + [0]         
              p(nt) = [1] x1 + [3]         
               p(p) = [0]                  
               p(q) = [0]                  
               p(r) = [1] x1 + [0]         
               p(s) = [0]                  
               p(t) = [0]                  
              p(a#) = [5] x1 + [0]         
              p(d#) = [0]                  
              p(f#) = [0]                  
              p(p#) = [0]                  
              p(q#) = [0]                  
              p(s#) = [0]                  
              p(t#) = [0]                  
             p(c_1) = [0]                  
             p(c_2) = [1] x1 + [1] x2 + [0]
             p(c_3) = [1] x1 + [0]         
             p(c_4) = [1] x1 + [0]         
             p(c_5) = [0]                  
             p(c_6) = [0]                  
             p(c_7) = [0]                  
             p(c_8) = [0]                  
             p(c_9) = [0]                  
            p(c_10) = [0]                  
            p(c_11) = [0]                  
            p(c_12) = [0]                  
            p(c_13) = [0]                  
            p(c_14) = [0]                  
            p(c_15) = [0]                  
            p(c_16) = [0]                  
            p(c_17) = [0]                  
          
          Following rules are strictly oriented:
          a#(nt(X)) = [5] X + [15]
                    > [5] X + [0] 
                    = c_4(a#(X))  
          
          
          Following rules are (at-least) weakly oriented:
          a#(nf(X1,X2)) =  [5] X1 + [5] X2 + [0]
                        >= [5] X1 + [5] X2 + [0]
                        =  c_2(a#(X1),a#(X2))   
          
              a#(ns(X)) =  [5] X + [0]          
                        >= [5] X + [0]          
                        =  c_3(a#(X))           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:8: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(nf(X1,X2)) -> c_2(a#(X1),a#(X2))
            a#(ns(X)) -> c_3(a#(X))
        - Weak DPs:
            a#(nt(X)) -> c_4(a#(X))
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/2,c_3/1,c_4/1,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/3,c_13/0,c_14/4,c_15/0,c_16/1,c_17/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + 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) = {1,2},
            uargs(c_3) = {1},
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
               p(0) = [0]                  
               p(a) = [0]                  
              p(cs) = [1] x1 + [1] x2 + [0]
               p(d) = [0]                  
               p(f) = [0]                  
              p(nf) = [1] x1 + [1] x2 + [0]
             p(nil) = [0]                  
              p(ns) = [1] x1 + [5]         
              p(nt) = [1] x1 + [0]         
               p(p) = [0]                  
               p(q) = [0]                  
               p(r) = [1] x1 + [0]         
               p(s) = [0]                  
               p(t) = [0]                  
              p(a#) = [1] x1 + [0]         
              p(d#) = [0]                  
              p(f#) = [0]                  
              p(p#) = [0]                  
              p(q#) = [0]                  
              p(s#) = [0]                  
              p(t#) = [0]                  
             p(c_1) = [0]                  
             p(c_2) = [1] x1 + [1] x2 + [0]
             p(c_3) = [1] x1 + [0]         
             p(c_4) = [1] x1 + [0]         
             p(c_5) = [0]                  
             p(c_6) = [0]                  
             p(c_7) = [0]                  
             p(c_8) = [0]                  
             p(c_9) = [0]                  
            p(c_10) = [0]                  
            p(c_11) = [0]                  
            p(c_12) = [0]                  
            p(c_13) = [0]                  
            p(c_14) = [0]                  
            p(c_15) = [0]                  
            p(c_16) = [0]                  
            p(c_17) = [0]                  
          
          Following rules are strictly oriented:
          a#(ns(X)) = [1] X + [5]
                    > [1] X + [0]
                    = c_3(a#(X)) 
          
          
          Following rules are (at-least) weakly oriented:
          a#(nf(X1,X2)) =  [1] X1 + [1] X2 + [0]
                        >= [1] X1 + [1] X2 + [0]
                        =  c_2(a#(X1),a#(X2))   
          
              a#(nt(X)) =  [1] X + [0]          
                        >= [1] X + [0]          
                        =  c_4(a#(X))           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:9: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            a#(nf(X1,X2)) -> c_2(a#(X1),a#(X2))
        - Weak DPs:
            a#(ns(X)) -> c_3(a#(X))
            a#(nt(X)) -> c_4(a#(X))
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/2,c_3/1,c_4/1,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/3,c_13/0,c_14/4,c_15/0,c_16/1,c_17/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {a#,d#,f#,p#,q#,s#,t#}
        TcT has computed the following interpretation:
             p(0) = [1]                           
             p(a) = [1] x1 + [2]                  
            p(cs) = [0]                           
             p(d) = [2]                           
             p(f) = [1] x2 + [2]                  
            p(nf) = [1] x1 + [1] x2 + [1]         
           p(nil) = [1]                           
            p(ns) = [1] x1 + [0]                  
            p(nt) = [1] x1 + [0]                  
             p(p) = [2] x1 + [1] x2 + [1]         
             p(q) = [1] x1 + [1]                  
             p(r) = [1]                           
             p(s) = [0]                           
             p(t) = [0]                           
            p(a#) = [4] x1 + [0]                  
            p(d#) = [0]                           
            p(f#) = [2] x1 + [1] x2 + [0]         
            p(p#) = [1] x1 + [4] x2 + [2]         
            p(q#) = [1] x1 + [8]                  
            p(s#) = [1] x1 + [1]                  
            p(t#) = [4] x1 + [0]                  
           p(c_1) = [2]                           
           p(c_2) = [1] x1 + [1] x2 + [0]         
           p(c_3) = [1] x1 + [0]                  
           p(c_4) = [1] x1 + [0]                  
           p(c_5) = [2]                           
           p(c_6) = [1] x3 + [2]                  
           p(c_7) = [1]                           
           p(c_8) = [8]                           
           p(c_9) = [1] x1 + [0]                  
          p(c_10) = [1]                           
          p(c_11) = [4]                           
          p(c_12) = [1] x1 + [1] x2 + [2] x3 + [1]
          p(c_13) = [0]                           
          p(c_14) = [1] x1 + [0]                  
          p(c_15) = [0]                           
          p(c_16) = [1]                           
          p(c_17) = [4]                           
        
        Following rules are strictly oriented:
        a#(nf(X1,X2)) = [4] X1 + [4] X2 + [4]
                      > [4] X1 + [4] X2 + [0]
                      = c_2(a#(X1),a#(X2))   
        
        
        Following rules are (at-least) weakly oriented:
        a#(ns(X)) =  [4] X + [0]
                  >= [4] X + [0]
                  =  c_3(a#(X)) 
        
        a#(nt(X)) =  [4] X + [0]
                  >= [4] X + [0]
                  =  c_4(a#(X)) 
        
** Step 1.b:10: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            a#(nf(X1,X2)) -> c_2(a#(X1),a#(X2))
            a#(ns(X)) -> c_3(a#(X))
            a#(nt(X)) -> c_4(a#(X))
        - Signature:
            {a/1,d/1,f/2,p/2,q/1,s/1,t/1,a#/1,d#/1,f#/2,p#/2,q#/1,s#/1,t#/1} / {0/0,cs/2,nf/2,nil/0,ns/1,nt/1,r/1,c_1/0
            ,c_2/2,c_3/1,c_4/1,c_5/0,c_6/3,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/3,c_13/0,c_14/4,c_15/0,c_16/1,c_17/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a#,d#,f#,p#,q#,s#,t#} and constructors {0,cs,nf,nil,ns,nt
            ,r}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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