/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1),?)
* Step 1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            gcd(x,y) -> gcd[Ite](equal0(x,y),x,y)
            monus(S(x'),S(x)) -> monus(x',x)
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            gcd[False][Ite](False(),x,y) -> gcd(y,monus(y,x))
            gcd[False][Ite](True(),x,y) -> gcd(monus(x,y),y)
            gcd[Ite](False(),x,y) -> gcd[False][Ite](<(x,y),x,y)
            gcd[Ite](True(),x,y) -> x
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2} / {0/0,False/0
            ,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,equal0,equal0[Ite],equal0[True][Ite],gcd
            ,gcd[False][Ite],gcd[Ite],monus} and constructors {0,False,S,True}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            gcd(x,y) -> gcd[Ite](equal0(x,y),x,y)
            monus(S(x'),S(x)) -> monus(x',x)
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            gcd[False][Ite](False(),x,y) -> gcd(y,monus(y,x))
            gcd[False][Ite](True(),x,y) -> gcd(monus(x,y),y)
            gcd[Ite](False(),x,y) -> gcd[False][Ite](<(x,y),x,y)
            gcd[Ite](True(),x,y) -> x
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2} / {0/0,False/0
            ,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,equal0,equal0[Ite],equal0[True][Ite],gcd
            ,gcd[False][Ite],gcd[Ite],monus} and constructors {0,False,S,True}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            gcd(x,y) -> gcd[Ite](equal0(x,y),x,y)
            monus(S(x'),S(x)) -> monus(x',x)
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            gcd[False][Ite](False(),x,y) -> gcd(y,monus(y,x))
            gcd[False][Ite](True(),x,y) -> gcd(monus(x,y),y)
            gcd[Ite](False(),x,y) -> gcd[False][Ite](<(x,y),x,y)
            gcd[Ite](True(),x,y) -> x
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2} / {0/0,False/0
            ,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,equal0,equal0[Ite],equal0[True][Ite],gcd
            ,gcd[False][Ite],gcd[Ite],monus} and constructors {0,False,S,True}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          monus(x,y){x -> S(x),y -> S(y)} =
            monus(S(x),S(y)) ->^+ monus(x,y)
              = C[monus(x,y) = monus(x,y){}]

** Step 1.b:1: DependencyPairs.  MAYBE
    + Considered Problem:
        - Strict TRS:
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            gcd(x,y) -> gcd[Ite](equal0(x,y),x,y)
            monus(S(x'),S(x)) -> monus(x',x)
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            gcd[False][Ite](False(),x,y) -> gcd(y,monus(y,x))
            gcd[False][Ite](True(),x,y) -> gcd(monus(x,y),y)
            gcd[Ite](False(),x,y) -> gcd[False][Ite](<(x,y),x,y)
            gcd[Ite](True(),x,y) -> x
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2} / {0/0,False/0
            ,S/1,True/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<,equal0,equal0[Ite],equal0[True][Ite],gcd
            ,gcd[False][Ite],gcd[Ite],monus} and constructors {0,False,S,True}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          equal0#(a,b) -> c_1(equal0[Ite]#(<(a,b),a,b),<#(a,b))
          gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y),equal0#(x,y))
          monus#(S(x'),S(x)) -> c_3(monus#(x',x))
        Weak DPs
          <#(x,0()) -> c_4()
          <#(0(),S(y)) -> c_5()
          <#(S(x),S(y)) -> c_6(<#(x,y))
          equal0[Ite]#(False(),a,b) -> c_7()
          equal0[Ite]#(True(),a,b) -> c_8(equal0[True][Ite]#(<(b,a),a,b),<#(b,a))
          equal0[True][Ite]#(False(),a,b) -> c_9()
          equal0[True][Ite]#(True(),a,b) -> c_10()
          gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
          gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
          gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y),<#(x,y))
          gcd[Ite]#(True(),x,y) -> c_14()
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  MAYBE
    + Considered Problem:
        - Strict DPs:
            equal0#(a,b) -> c_1(equal0[Ite]#(<(a,b),a,b),<#(a,b))
            gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y),equal0#(x,y))
            monus#(S(x'),S(x)) -> c_3(monus#(x',x))
        - Weak DPs:
            <#(x,0()) -> c_4()
            <#(0(),S(y)) -> c_5()
            <#(S(x),S(y)) -> c_6(<#(x,y))
            equal0[Ite]#(False(),a,b) -> c_7()
            equal0[Ite]#(True(),a,b) -> c_8(equal0[True][Ite]#(<(b,a),a,b),<#(b,a))
            equal0[True][Ite]#(False(),a,b) -> c_9()
            equal0[True][Ite]#(True(),a,b) -> c_10()
            gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
            gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
            gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y),<#(x,y))
            gcd[Ite]#(True(),x,y) -> c_14()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            gcd(x,y) -> gcd[Ite](equal0(x,y),x,y)
            gcd[False][Ite](False(),x,y) -> gcd(y,monus(y,x))
            gcd[False][Ite](True(),x,y) -> gcd(monus(x,y),y)
            gcd[Ite](False(),x,y) -> gcd[False][Ite](<(x,y),x,y)
            gcd[Ite](True(),x,y) -> x
            monus(S(x'),S(x)) -> monus(x',x)
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2,<#/2,equal0#/2
            ,equal0[Ite]#/3,equal0[True][Ite]#/3,gcd#/2,gcd[False][Ite]#/3,gcd[Ite]#/3,monus#/2} / {0/0,False/0,S/1
            ,True/0,c_1/2,c_2/2,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/0,c_11/2,c_12/2,c_13/2,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,equal0#,equal0[Ite]#,equal0[True][Ite]#,gcd#
            ,gcd[False][Ite]#,gcd[Ite]#,monus#} and constructors {0,False,S,True}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {2}.
        Here rules are labelled as follows:
          1: equal0#(a,b) -> c_1(equal0[Ite]#(<(a,b),a,b),<#(a,b))
          2: gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y),equal0#(x,y))
          3: monus#(S(x'),S(x)) -> c_3(monus#(x',x))
          4: <#(x,0()) -> c_4()
          5: <#(0(),S(y)) -> c_5()
          6: <#(S(x),S(y)) -> c_6(<#(x,y))
          7: equal0[Ite]#(False(),a,b) -> c_7()
          8: equal0[Ite]#(True(),a,b) -> c_8(equal0[True][Ite]#(<(b,a),a,b),<#(b,a))
          9: equal0[True][Ite]#(False(),a,b) -> c_9()
          10: equal0[True][Ite]#(True(),a,b) -> c_10()
          11: gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
          12: gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
          13: gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y),<#(x,y))
          14: gcd[Ite]#(True(),x,y) -> c_14()
** Step 1.b:3: RemoveWeakSuffixes.  MAYBE
    + Considered Problem:
        - Strict DPs:
            gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y),equal0#(x,y))
            monus#(S(x'),S(x)) -> c_3(monus#(x',x))
        - Weak DPs:
            <#(x,0()) -> c_4()
            <#(0(),S(y)) -> c_5()
            <#(S(x),S(y)) -> c_6(<#(x,y))
            equal0#(a,b) -> c_1(equal0[Ite]#(<(a,b),a,b),<#(a,b))
            equal0[Ite]#(False(),a,b) -> c_7()
            equal0[Ite]#(True(),a,b) -> c_8(equal0[True][Ite]#(<(b,a),a,b),<#(b,a))
            equal0[True][Ite]#(False(),a,b) -> c_9()
            equal0[True][Ite]#(True(),a,b) -> c_10()
            gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
            gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
            gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y),<#(x,y))
            gcd[Ite]#(True(),x,y) -> c_14()
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            gcd(x,y) -> gcd[Ite](equal0(x,y),x,y)
            gcd[False][Ite](False(),x,y) -> gcd(y,monus(y,x))
            gcd[False][Ite](True(),x,y) -> gcd(monus(x,y),y)
            gcd[Ite](False(),x,y) -> gcd[False][Ite](<(x,y),x,y)
            gcd[Ite](True(),x,y) -> x
            monus(S(x'),S(x)) -> monus(x',x)
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2,<#/2,equal0#/2
            ,equal0[Ite]#/3,equal0[True][Ite]#/3,gcd#/2,gcd[False][Ite]#/3,gcd[Ite]#/3,monus#/2} / {0/0,False/0,S/1
            ,True/0,c_1/2,c_2/2,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/0,c_11/2,c_12/2,c_13/2,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,equal0#,equal0[Ite]#,equal0[True][Ite]#,gcd#
            ,gcd[False][Ite]#,gcd[Ite]#,monus#} and constructors {0,False,S,True}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y),equal0#(x,y))
             -->_1 gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y),<#(x,y)):13
             -->_2 equal0#(a,b) -> c_1(equal0[Ite]#(<(a,b),a,b),<#(a,b)):6
             -->_1 gcd[Ite]#(True(),x,y) -> c_14():14
          
          2:S:monus#(S(x'),S(x)) -> c_3(monus#(x',x))
             -->_1 monus#(S(x'),S(x)) -> c_3(monus#(x',x)):2
          
          3:W:<#(x,0()) -> c_4()
             
          
          4:W:<#(0(),S(y)) -> c_5()
             
          
          5:W:<#(S(x),S(y)) -> c_6(<#(x,y))
             -->_1 <#(S(x),S(y)) -> c_6(<#(x,y)):5
             -->_1 <#(0(),S(y)) -> c_5():4
             -->_1 <#(x,0()) -> c_4():3
          
          6:W:equal0#(a,b) -> c_1(equal0[Ite]#(<(a,b),a,b),<#(a,b))
             -->_1 equal0[Ite]#(True(),a,b) -> c_8(equal0[True][Ite]#(<(b,a),a,b),<#(b,a)):8
             -->_1 equal0[Ite]#(False(),a,b) -> c_7():7
             -->_2 <#(S(x),S(y)) -> c_6(<#(x,y)):5
             -->_2 <#(0(),S(y)) -> c_5():4
             -->_2 <#(x,0()) -> c_4():3
          
          7:W:equal0[Ite]#(False(),a,b) -> c_7()
             
          
          8:W:equal0[Ite]#(True(),a,b) -> c_8(equal0[True][Ite]#(<(b,a),a,b),<#(b,a))
             -->_1 equal0[True][Ite]#(True(),a,b) -> c_10():10
             -->_1 equal0[True][Ite]#(False(),a,b) -> c_9():9
             -->_2 <#(S(x),S(y)) -> c_6(<#(x,y)):5
             -->_2 <#(0(),S(y)) -> c_5():4
             -->_2 <#(x,0()) -> c_4():3
          
          9:W:equal0[True][Ite]#(False(),a,b) -> c_9()
             
          
          10:W:equal0[True][Ite]#(True(),a,b) -> c_10()
             
          
          11:W:gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
             -->_2 monus#(S(x'),S(x)) -> c_3(monus#(x',x)):2
             -->_1 gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y),equal0#(x,y)):1
          
          12:W:gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
             -->_2 monus#(S(x'),S(x)) -> c_3(monus#(x',x)):2
             -->_1 gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y),equal0#(x,y)):1
          
          13:W:gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y),<#(x,y))
             -->_1 gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y)):12
             -->_1 gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x)):11
             -->_2 <#(S(x),S(y)) -> c_6(<#(x,y)):5
             -->_2 <#(0(),S(y)) -> c_5():4
             -->_2 <#(x,0()) -> c_4():3
          
          14:W:gcd[Ite]#(True(),x,y) -> c_14()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          14: gcd[Ite]#(True(),x,y) -> c_14()
          6: equal0#(a,b) -> c_1(equal0[Ite]#(<(a,b),a,b),<#(a,b))
          7: equal0[Ite]#(False(),a,b) -> c_7()
          8: equal0[Ite]#(True(),a,b) -> c_8(equal0[True][Ite]#(<(b,a),a,b),<#(b,a))
          9: equal0[True][Ite]#(False(),a,b) -> c_9()
          10: equal0[True][Ite]#(True(),a,b) -> c_10()
          5: <#(S(x),S(y)) -> c_6(<#(x,y))
          3: <#(x,0()) -> c_4()
          4: <#(0(),S(y)) -> c_5()
** Step 1.b:4: SimplifyRHS.  MAYBE
    + Considered Problem:
        - Strict DPs:
            gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y),equal0#(x,y))
            monus#(S(x'),S(x)) -> c_3(monus#(x',x))
        - Weak DPs:
            gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
            gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
            gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y),<#(x,y))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            gcd(x,y) -> gcd[Ite](equal0(x,y),x,y)
            gcd[False][Ite](False(),x,y) -> gcd(y,monus(y,x))
            gcd[False][Ite](True(),x,y) -> gcd(monus(x,y),y)
            gcd[Ite](False(),x,y) -> gcd[False][Ite](<(x,y),x,y)
            gcd[Ite](True(),x,y) -> x
            monus(S(x'),S(x)) -> monus(x',x)
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2,<#/2,equal0#/2
            ,equal0[Ite]#/3,equal0[True][Ite]#/3,gcd#/2,gcd[False][Ite]#/3,gcd[Ite]#/3,monus#/2} / {0/0,False/0,S/1
            ,True/0,c_1/2,c_2/2,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/0,c_11/2,c_12/2,c_13/2,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,equal0#,equal0[Ite]#,equal0[True][Ite]#,gcd#
            ,gcd[False][Ite]#,gcd[Ite]#,monus#} and constructors {0,False,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y),equal0#(x,y))
             -->_1 gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y),<#(x,y)):13
          
          2:S:monus#(S(x'),S(x)) -> c_3(monus#(x',x))
             -->_1 monus#(S(x'),S(x)) -> c_3(monus#(x',x)):2
          
          11:W:gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
             -->_2 monus#(S(x'),S(x)) -> c_3(monus#(x',x)):2
             -->_1 gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y),equal0#(x,y)):1
          
          12:W:gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
             -->_2 monus#(S(x'),S(x)) -> c_3(monus#(x',x)):2
             -->_1 gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y),equal0#(x,y)):1
          
          13:W:gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y),<#(x,y))
             -->_1 gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y)):12
             -->_1 gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x)):11
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y))
          gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y))
** Step 1.b:5: UsableRules.  MAYBE
    + Considered Problem:
        - Strict DPs:
            gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y))
            monus#(S(x'),S(x)) -> c_3(monus#(x',x))
        - Weak DPs:
            gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
            gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
            gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            gcd(x,y) -> gcd[Ite](equal0(x,y),x,y)
            gcd[False][Ite](False(),x,y) -> gcd(y,monus(y,x))
            gcd[False][Ite](True(),x,y) -> gcd(monus(x,y),y)
            gcd[Ite](False(),x,y) -> gcd[False][Ite](<(x,y),x,y)
            gcd[Ite](True(),x,y) -> x
            monus(S(x'),S(x)) -> monus(x',x)
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2,<#/2,equal0#/2
            ,equal0[Ite]#/3,equal0[True][Ite]#/3,gcd#/2,gcd[False][Ite]#/3,gcd[Ite]#/3,monus#/2} / {0/0,False/0,S/1
            ,True/0,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/0,c_11/2,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,equal0#,equal0[Ite]#,equal0[True][Ite]#,gcd#
            ,gcd[False][Ite]#,gcd[Ite]#,monus#} and constructors {0,False,S,True}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          <(x,0()) -> False()
          <(0(),S(y)) -> True()
          <(S(x),S(y)) -> <(x,y)
          equal0(a,b) -> equal0[Ite](<(a,b),a,b)
          equal0[Ite](False(),a,b) -> False()
          equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
          equal0[True][Ite](False(),a,b) -> False()
          equal0[True][Ite](True(),a,b) -> True()
          monus(S(x'),S(x)) -> monus(x',x)
          gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y))
          gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
          gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
          gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y))
          monus#(S(x'),S(x)) -> c_3(monus#(x',x))
** Step 1.b:6: DecomposeDG.  MAYBE
    + Considered Problem:
        - Strict DPs:
            gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y))
            monus#(S(x'),S(x)) -> c_3(monus#(x',x))
        - Weak DPs:
            gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
            gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
            gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            monus(S(x'),S(x)) -> monus(x',x)
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2,<#/2,equal0#/2
            ,equal0[Ite]#/3,equal0[True][Ite]#/3,gcd#/2,gcd[False][Ite]#/3,gcd[Ite]#/3,monus#/2} / {0/0,False/0,S/1
            ,True/0,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/0,c_11/2,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,equal0#,equal0[Ite]#,equal0[True][Ite]#,gcd#
            ,gcd[False][Ite]#,gcd[Ite]#,monus#} and constructors {0,False,S,True}
    + 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
          gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y))
          gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
          gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
          gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y))
        and a lower component
          monus#(S(x'),S(x)) -> c_3(monus#(x',x))
        Further, following extension rules are added to the lower component.
          gcd#(x,y) -> gcd[Ite]#(equal0(x,y),x,y)
          gcd[False][Ite]#(False(),x,y) -> gcd#(y,monus(y,x))
          gcd[False][Ite]#(False(),x,y) -> monus#(y,x)
          gcd[False][Ite]#(True(),x,y) -> gcd#(monus(x,y),y)
          gcd[False][Ite]#(True(),x,y) -> monus#(x,y)
          gcd[Ite]#(False(),x,y) -> gcd[False][Ite]#(<(x,y),x,y)
*** Step 1.b:6.a:1: PredecessorEstimation.  MAYBE
    + Considered Problem:
        - Strict DPs:
            gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y))
            gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
            gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
        - Weak DPs:
            gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            monus(S(x'),S(x)) -> monus(x',x)
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2,<#/2,equal0#/2
            ,equal0[Ite]#/3,equal0[True][Ite]#/3,gcd#/2,gcd[False][Ite]#/3,gcd[Ite]#/3,monus#/2} / {0/0,False/0,S/1
            ,True/0,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/0,c_11/2,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,equal0#,equal0[Ite]#,equal0[True][Ite]#,gcd#
            ,gcd[False][Ite]#,gcd[Ite]#,monus#} and constructors {0,False,S,True}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {2,3}.
        Here rules are labelled as follows:
          1: gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y))
          2: gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
          3: gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
          4: gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y))
*** Step 1.b:6.a:2: SimplifyRHS.  MAYBE
    + Considered Problem:
        - Strict DPs:
            gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
            gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
        - Weak DPs:
            gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y))
            gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            monus(S(x'),S(x)) -> monus(x',x)
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2,<#/2,equal0#/2
            ,equal0[Ite]#/3,equal0[True][Ite]#/3,gcd#/2,gcd[False][Ite]#/3,gcd[Ite]#/3,monus#/2} / {0/0,False/0,S/1
            ,True/0,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/0,c_11/2,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,equal0#,equal0[Ite]#,equal0[True][Ite]#,gcd#
            ,gcd[False][Ite]#,gcd[Ite]#,monus#} and constructors {0,False,S,True}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x))
             -->_1 gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y)):3
          
          2:S:gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y))
             -->_1 gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y)):3
          
          3:W:gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y))
             -->_1 gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y)):4
          
          4:W:gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y))
             -->_1 gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y),monus#(x,y)):2
             -->_1 gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)),monus#(y,x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)))
          gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y))
*** Step 1.b:6.a:3: NaturalMI.  MAYBE
    + Considered Problem:
        - Strict DPs:
            gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)))
            gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y))
        - Weak DPs:
            gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y))
            gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            monus(S(x'),S(x)) -> monus(x',x)
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2,<#/2,equal0#/2
            ,equal0[Ite]#/3,equal0[True][Ite]#/3,gcd#/2,gcd[False][Ite]#/3,gcd[Ite]#/3,monus#/2} / {0/0,False/0,S/1
            ,True/0,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/0,c_11/1,c_12/1,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,equal0#,equal0[Ite]#,equal0[True][Ite]#,gcd#
            ,gcd[False][Ite]#,gcd[Ite]#,monus#} and constructors {0,False,S,True}
    + 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},
          uargs(c_11) = {1},
          uargs(c_12) = {1},
          uargs(c_13) = {1}
        
        Following symbols are considered usable:
          {<,monus,<#,equal0#,equal0[Ite]#,equal0[True][Ite]#,gcd#,gcd[False][Ite]#,gcd[Ite]#,monus#}
        TcT has computed the following interpretation:
                           p(0) = [8]                             
                           p(<) = [2] x1 + [2]                    
                       p(False) = [2]                             
                           p(S) = [1] x1 + [1]                    
                        p(True) = [6]                             
                      p(equal0) = [0]                             
                 p(equal0[Ite]) = [2] x1 + [15] x2 + [9] x3 + [11]
           p(equal0[True][Ite]) = [3] x1 + [1] x3 + [1]           
                         p(gcd) = [0]                             
             p(gcd[False][Ite]) = [1] x1 + [1] x3 + [2]           
                    p(gcd[Ite]) = [1] x1 + [8] x3 + [1]           
                       p(monus) = [0]                             
                          p(<#) = [1] x2 + [0]                    
                     p(equal0#) = [1] x1 + [2] x2 + [1]           
                p(equal0[Ite]#) = [1] x2 + [0]                    
          p(equal0[True][Ite]#) = [1] x1 + [1] x3 + [1]           
                        p(gcd#) = [12] x1 + [12] x2 + [8]         
            p(gcd[False][Ite]#) = [4] x1 + [12] x3 + [0]          
                   p(gcd[Ite]#) = [8] x2 + [12] x3 + [8]          
                      p(monus#) = [2] x2 + [8]                    
                         p(c_1) = [1] x2 + [2]                    
                         p(c_2) = [1] x1 + [0]                    
                         p(c_3) = [2] x1 + [1]                    
                         p(c_4) = [8]                             
                         p(c_5) = [0]                             
                         p(c_6) = [0]                             
                         p(c_7) = [1]                             
                         p(c_8) = [1] x1 + [1] x2 + [1]           
                         p(c_9) = [0]                             
                        p(c_10) = [8]                             
                        p(c_11) = [1] x1 + [0]                    
                        p(c_12) = [1] x1 + [12]                   
                        p(c_13) = [1] x1 + [0]                    
                        p(c_14) = [1]                             
        
        Following rules are strictly oriented:
        gcd[False][Ite]#(True(),x,y) = [12] y + [24]           
                                     > [12] y + [20]           
                                     = c_12(gcd#(monus(x,y),y))
        
        
        Following rules are (at-least) weakly oriented:
                            gcd#(x,y) =  [12] x + [12] y + [8]             
                                      >= [8] x + [12] y + [8]              
                                      =  c_2(gcd[Ite]#(equal0(x,y),x,y))   
        
        gcd[False][Ite]#(False(),x,y) =  [12] y + [8]                      
                                      >= [12] y + [8]                      
                                      =  c_11(gcd#(y,monus(y,x)))          
        
               gcd[Ite]#(False(),x,y) =  [8] x + [12] y + [8]              
                                      >= [8] x + [12] y + [8]              
                                      =  c_13(gcd[False][Ite]#(<(x,y),x,y))
        
                             <(x,0()) =  [2] x + [2]                       
                                      >= [2]                               
                                      =  False()                           
        
                          <(0(),S(y)) =  [18]                              
                                      >= [6]                               
                                      =  True()                            
        
                         <(S(x),S(y)) =  [2] x + [4]                       
                                      >= [2] x + [2]                       
                                      =  <(x,y)                            
        
                    monus(S(x'),S(x)) =  [0]                               
                                      >= [0]                               
                                      =  monus(x',x)                       
        
*** Step 1.b:6.a:4: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          gcd[False][Ite]#(False(),x,y) -> c_11(gcd#(y,monus(y,x)))
      - Weak DPs:
          gcd#(x,y) -> c_2(gcd[Ite]#(equal0(x,y),x,y))
          gcd[False][Ite]#(True(),x,y) -> c_12(gcd#(monus(x,y),y))
          gcd[Ite]#(False(),x,y) -> c_13(gcd[False][Ite]#(<(x,y),x,y))
      - Weak TRS:
          <(x,0()) -> False()
          <(0(),S(y)) -> True()
          <(S(x),S(y)) -> <(x,y)
          equal0(a,b) -> equal0[Ite](<(a,b),a,b)
          equal0[Ite](False(),a,b) -> False()
          equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
          equal0[True][Ite](False(),a,b) -> False()
          equal0[True][Ite](True(),a,b) -> True()
          monus(S(x'),S(x)) -> monus(x',x)
      - Signature:
          {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2,<#/2,equal0#/2
          ,equal0[Ite]#/3,equal0[True][Ite]#/3,gcd#/2,gcd[False][Ite]#/3,gcd[Ite]#/3,monus#/2} / {0/0,False/0,S/1
          ,True/0,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/0,c_11/1,c_12/1,c_13/1,c_14/0}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {<#,equal0#,equal0[Ite]#,equal0[True][Ite]#,gcd#
          ,gcd[False][Ite]#,gcd[Ite]#,monus#} and constructors {0,False,S,True}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
*** Step 1.b:6.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            monus#(S(x'),S(x)) -> c_3(monus#(x',x))
        - Weak DPs:
            gcd#(x,y) -> gcd[Ite]#(equal0(x,y),x,y)
            gcd[False][Ite]#(False(),x,y) -> gcd#(y,monus(y,x))
            gcd[False][Ite]#(False(),x,y) -> monus#(y,x)
            gcd[False][Ite]#(True(),x,y) -> gcd#(monus(x,y),y)
            gcd[False][Ite]#(True(),x,y) -> monus#(x,y)
            gcd[Ite]#(False(),x,y) -> gcd[False][Ite]#(<(x,y),x,y)
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            monus(S(x'),S(x)) -> monus(x',x)
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2,<#/2,equal0#/2
            ,equal0[Ite]#/3,equal0[True][Ite]#/3,gcd#/2,gcd[False][Ite]#/3,gcd[Ite]#/3,monus#/2} / {0/0,False/0,S/1
            ,True/0,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/0,c_11/2,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,equal0#,equal0[Ite]#,equal0[True][Ite]#,gcd#
            ,gcd[False][Ite]#,gcd[Ite]#,monus#} and constructors {0,False,S,True}
    + 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_3) = {1}
        
        Following symbols are considered usable:
          {monus,<#,equal0#,equal0[Ite]#,equal0[True][Ite]#,gcd#,gcd[False][Ite]#,gcd[Ite]#,monus#}
        TcT has computed the following interpretation:
                           p(0) = [12]                            
                           p(<) = [2] x1 + [2]                    
                       p(False) = [0]                             
                           p(S) = [1] x1 + [1]                    
                        p(True) = [1]                             
                      p(equal0) = [8] x2 + [0]                    
                 p(equal0[Ite]) = [8] x1 + [15] x2 + [8] x3 + [15]
           p(equal0[True][Ite]) = [5] x1 + [0]                    
                         p(gcd) = [2] x1 + [2]                    
             p(gcd[False][Ite]) = [1] x1 + [1]                    
                    p(gcd[Ite]) = [8] x1 + [1]                    
                       p(monus) = [0]                             
                          p(<#) = [0]                             
                     p(equal0#) = [2] x1 + [8] x2 + [1]           
                p(equal0[Ite]#) = [2] x1 + [1] x2 + [2]           
          p(equal0[True][Ite]#) = [2] x1 + [2] x2 + [1] x3 + [1]  
                        p(gcd#) = [1] x1 + [1] x2 + [0]           
            p(gcd[False][Ite]#) = [1] x2 + [1] x3 + [0]           
                   p(gcd[Ite]#) = [1] x2 + [1] x3 + [0]           
                      p(monus#) = [1] x2 + [0]                    
                         p(c_1) = [1] x1 + [1] x2 + [0]           
                         p(c_2) = [1] x1 + [1]                    
                         p(c_3) = [1] x1 + [0]                    
                         p(c_4) = [1]                             
                         p(c_5) = [1]                             
                         p(c_6) = [1]                             
                         p(c_7) = [8]                             
                         p(c_8) = [2] x1 + [2]                    
                         p(c_9) = [2]                             
                        p(c_10) = [1]                             
                        p(c_11) = [1] x1 + [1] x2 + [1]           
                        p(c_12) = [2] x1 + [2] x2 + [0]           
                        p(c_13) = [1] x1 + [1]                    
                        p(c_14) = [1]                             
        
        Following rules are strictly oriented:
        monus#(S(x'),S(x)) = [1] x + [1]      
                           > [1] x + [0]      
                           = c_3(monus#(x',x))
        
        
        Following rules are (at-least) weakly oriented:
                            gcd#(x,y) =  [1] x + [1] y + [0]         
                                      >= [1] x + [1] y + [0]         
                                      =  gcd[Ite]#(equal0(x,y),x,y)  
        
        gcd[False][Ite]#(False(),x,y) =  [1] x + [1] y + [0]         
                                      >= [1] y + [0]                 
                                      =  gcd#(y,monus(y,x))          
        
        gcd[False][Ite]#(False(),x,y) =  [1] x + [1] y + [0]         
                                      >= [1] x + [0]                 
                                      =  monus#(y,x)                 
        
         gcd[False][Ite]#(True(),x,y) =  [1] x + [1] y + [0]         
                                      >= [1] y + [0]                 
                                      =  gcd#(monus(x,y),y)          
        
         gcd[False][Ite]#(True(),x,y) =  [1] x + [1] y + [0]         
                                      >= [1] y + [0]                 
                                      =  monus#(x,y)                 
        
               gcd[Ite]#(False(),x,y) =  [1] x + [1] y + [0]         
                                      >= [1] x + [1] y + [0]         
                                      =  gcd[False][Ite]#(<(x,y),x,y)
        
                    monus(S(x'),S(x)) =  [0]                         
                                      >= [0]                         
                                      =  monus(x',x)                 
        
*** Step 1.b:6.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            gcd#(x,y) -> gcd[Ite]#(equal0(x,y),x,y)
            gcd[False][Ite]#(False(),x,y) -> gcd#(y,monus(y,x))
            gcd[False][Ite]#(False(),x,y) -> monus#(y,x)
            gcd[False][Ite]#(True(),x,y) -> gcd#(monus(x,y),y)
            gcd[False][Ite]#(True(),x,y) -> monus#(x,y)
            gcd[Ite]#(False(),x,y) -> gcd[False][Ite]#(<(x,y),x,y)
            monus#(S(x'),S(x)) -> c_3(monus#(x',x))
        - Weak TRS:
            <(x,0()) -> False()
            <(0(),S(y)) -> True()
            <(S(x),S(y)) -> <(x,y)
            equal0(a,b) -> equal0[Ite](<(a,b),a,b)
            equal0[Ite](False(),a,b) -> False()
            equal0[Ite](True(),a,b) -> equal0[True][Ite](<(b,a),a,b)
            equal0[True][Ite](False(),a,b) -> False()
            equal0[True][Ite](True(),a,b) -> True()
            monus(S(x'),S(x)) -> monus(x',x)
        - Signature:
            {</2,equal0/2,equal0[Ite]/3,equal0[True][Ite]/3,gcd/2,gcd[False][Ite]/3,gcd[Ite]/3,monus/2,<#/2,equal0#/2
            ,equal0[Ite]#/3,equal0[True][Ite]#/3,gcd#/2,gcd[False][Ite]#/3,gcd[Ite]#/3,monus#/2} / {0/0,False/0,S/1
            ,True/0,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/0,c_11/2,c_12/2,c_13/1,c_14/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {<#,equal0#,equal0[Ite]#,equal0[True][Ite]#,gcd#
            ,gcd[False][Ite]#,gcd[Ite]#,monus#} and constructors {0,False,S,True}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),?)