/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: function(if(),false(),x,y) -> function(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) function(if(),true(),x,y) -> y function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(plus(),dummy,x,y) -> function(if(),function(iszero(),x,x,x),x,y) function(third(),x,y,z) -> z - Signature: {function/4} / {0/0,false/0,if/0,iszero/0,p/0,plus/0,s/1,third/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {function} and constructors {0,false,if,iszero,p,plus,s ,third,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: function(if(),false(),x,y) -> function(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) function(if(),true(),x,y) -> y function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(plus(),dummy,x,y) -> function(if(),function(iszero(),x,x,x),x,y) function(third(),x,y,z) -> z - Signature: {function/4} / {0/0,false/0,if/0,iszero/0,p/0,plus/0,s/1,third/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {function} and constructors {0,false,if,iszero,p,plus,s ,third,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () *** Step 1.a:1.a:1: Ara. MAYBE + Considered Problem: - Strict TRS: function(if(),false(),x,y) -> function(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) function(if(),true(),x,y) -> y function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(plus(),dummy,x,y) -> function(if(),function(iszero(),x,x,x),x,y) function(third(),x,y,z) -> z - Signature: {function/4} / {0/0,false/0,if/0,iszero/0,p/0,plus/0,s/1,third/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {function} and constructors {0,false,if,iszero,p,plus,s ,third,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(0, 0, 0) F (TrsFun "false") :: [] -(0)-> "A"(0, 0, 0) F (TrsFun "function") :: ["A"(0, 0, 1) x "A"(0, 0, 0) x "A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0) F (TrsFun "if") :: [] -(0)-> "A"(0, 0, 1) F (TrsFun "iszero") :: [] -(0)-> "A"(0, 0, 1) F (TrsFun "main") :: ["A"(0, 0, 1) x "A"(0, 0, 0) x "A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0) F (TrsFun "p") :: [] -(0)-> "A"(0, 0, 1) F (TrsFun "p") :: [] -(0)-> "A"(0, 0, 0) F (TrsFun "plus") :: [] -(0)-> "A"(0, 0, 1) F (TrsFun "s") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) F (TrsFun "third") :: [] -(0)-> "A"(0, 0, 1) F (TrsFun "true") :: [] -(0)-> "A"(0, 0, 0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: function(if(),false(),x,y) -> function(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) function(if(),true(),x,y) -> y function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(plus(),dummy,x,y) -> function(if(),function(iszero(),x,x,x),x,y) function(third(),x,y,z) -> z main(x1,x2,x3,x4) -> function(x1,x2,x3,x4) 2. Weak: *** Step 1.a:1.b:1: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: function(if(),false(),x,y) -> function(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) function(if(),true(),x,y) -> y function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(plus(),dummy,x,y) -> function(if(),function(iszero(),x,x,x),x,y) function(third(),x,y,z) -> z - Signature: {function/4} / {0/0,false/0,if/0,iszero/0,p/0,plus/0,s/1,third/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {function} and constructors {0,false,if,iszero,p,plus,s ,third,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: function(p(),s(x),y,z){x -> s(x)} = function(p(),s(s(x)),y,z) ->^+ s(function(p(),s(x),x,x)) = C[function(p(),s(x),x,x) = function(p(),s(x),y,z){y -> x,z -> x}] ** Step 1.b:1: DependencyPairs. MAYBE + Considered Problem: - Strict TRS: function(if(),false(),x,y) -> function(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) function(if(),true(),x,y) -> y function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(plus(),dummy,x,y) -> function(if(),function(iszero(),x,x,x),x,y) function(third(),x,y,z) -> z - Signature: {function/4} / {0/0,false/0,if/0,iszero/0,p/0,plus/0,s/1,third/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {function} and constructors {0,false,if,iszero,p,plus,s ,third,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(third(),x,y,y) ,function#(p(),x,x,y)) function#(if(),true(),x,y) -> c_2() function#(iszero(),0(),dummy,dummy2) -> c_3() function#(iszero(),s(x),dummy,dummy2) -> c_4() function#(p(),0(),dummy,dummy2) -> c_5() function#(p(),s(0()),dummy,dummy2) -> c_6() function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)) function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y),function#(iszero(),x,x,x)) function#(third(),x,y,z) -> c_9() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. MAYBE + Considered Problem: - Strict DPs: function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(third(),x,y,y) ,function#(p(),x,x,y)) function#(if(),true(),x,y) -> c_2() function#(iszero(),0(),dummy,dummy2) -> c_3() function#(iszero(),s(x),dummy,dummy2) -> c_4() function#(p(),0(),dummy,dummy2) -> c_5() function#(p(),s(0()),dummy,dummy2) -> c_6() function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)) function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y),function#(iszero(),x,x,x)) function#(third(),x,y,z) -> c_9() - Weak TRS: function(if(),false(),x,y) -> function(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) function(if(),true(),x,y) -> y function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(plus(),dummy,x,y) -> function(if(),function(iszero(),x,x,x),x,y) function(third(),x,y,z) -> z - Signature: {function/4,function#/4} / {0/0,false/0,if/0,iszero/0,p/0,plus/0,s/1,third/0,true/0,c_1/3,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {function#} and constructors {0,false,if,iszero,p,plus,s ,third,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3,4,5,6,9} by application of Pre({2,3,4,5,6,9}) = {1,7,8}. Here rules are labelled as follows: 1: function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(third(),x,y,y) ,function#(p(),x,x,y)) 2: function#(if(),true(),x,y) -> c_2() 3: function#(iszero(),0(),dummy,dummy2) -> c_3() 4: function#(iszero(),s(x),dummy,dummy2) -> c_4() 5: function#(p(),0(),dummy,dummy2) -> c_5() 6: function#(p(),s(0()),dummy,dummy2) -> c_6() 7: function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)) 8: function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y) ,function#(iszero(),x,x,x)) 9: function#(third(),x,y,z) -> c_9() ** Step 1.b:3: RemoveWeakSuffixes. MAYBE + Considered Problem: - Strict DPs: function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(third(),x,y,y) ,function#(p(),x,x,y)) function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)) function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y),function#(iszero(),x,x,x)) - Weak DPs: function#(if(),true(),x,y) -> c_2() function#(iszero(),0(),dummy,dummy2) -> c_3() function#(iszero(),s(x),dummy,dummy2) -> c_4() function#(p(),0(),dummy,dummy2) -> c_5() function#(p(),s(0()),dummy,dummy2) -> c_6() function#(third(),x,y,z) -> c_9() - Weak TRS: function(if(),false(),x,y) -> function(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) function(if(),true(),x,y) -> y function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(plus(),dummy,x,y) -> function(if(),function(iszero(),x,x,x),x,y) function(third(),x,y,z) -> z - Signature: {function/4,function#/4} / {0/0,false/0,if/0,iszero/0,p/0,plus/0,s/1,third/0,true/0,c_1/3,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {function#} and constructors {0,false,if,iszero,p,plus,s ,third,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(third(),x,y,y) ,function#(p(),x,x,y)) -->_1 function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y) ,function#(iszero(),x,x,x)):3 -->_3 function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)):2 -->_2 function#(third(),x,y,z) -> c_9():9 -->_3 function#(p(),s(0()),dummy,dummy2) -> c_6():8 -->_3 function#(p(),0(),dummy,dummy2) -> c_5():7 2:S:function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)) -->_1 function#(p(),s(0()),dummy,dummy2) -> c_6():8 -->_1 function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)):2 3:S:function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y) ,function#(iszero(),x,x,x)) -->_2 function#(iszero(),s(x),dummy,dummy2) -> c_4():6 -->_2 function#(iszero(),0(),dummy,dummy2) -> c_3():5 -->_1 function#(if(),true(),x,y) -> c_2():4 -->_1 function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(third(),x,y,y) ,function#(p(),x,x,y)):1 4:W:function#(if(),true(),x,y) -> c_2() 5:W:function#(iszero(),0(),dummy,dummy2) -> c_3() 6:W:function#(iszero(),s(x),dummy,dummy2) -> c_4() 7:W:function#(p(),0(),dummy,dummy2) -> c_5() 8:W:function#(p(),s(0()),dummy,dummy2) -> c_6() 9:W:function#(third(),x,y,z) -> c_9() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: function#(p(),0(),dummy,dummy2) -> c_5() 9: function#(third(),x,y,z) -> c_9() 8: function#(p(),s(0()),dummy,dummy2) -> c_6() 4: function#(if(),true(),x,y) -> c_2() 5: function#(iszero(),0(),dummy,dummy2) -> c_3() 6: function#(iszero(),s(x),dummy,dummy2) -> c_4() ** Step 1.b:4: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(third(),x,y,y) ,function#(p(),x,x,y)) function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)) function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y),function#(iszero(),x,x,x)) - Weak TRS: function(if(),false(),x,y) -> function(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) function(if(),true(),x,y) -> y function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(plus(),dummy,x,y) -> function(if(),function(iszero(),x,x,x),x,y) function(third(),x,y,z) -> z - Signature: {function/4,function#/4} / {0/0,false/0,if/0,iszero/0,p/0,plus/0,s/1,third/0,true/0,c_1/3,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {function#} and constructors {0,false,if,iszero,p,plus,s ,third,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(third(),x,y,y) ,function#(p(),x,x,y)) -->_1 function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y) ,function#(iszero(),x,x,x)):3 -->_3 function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)):2 2:S:function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)) -->_1 function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)):2 3:S:function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y) ,function#(iszero(),x,x,x)) -->_1 function#(if(),false(),x,y) -> c_1(function#(plus() ,function(third(),x,y,y) ,function(p(),x,x,y) ,s(y)) ,function#(third(),x,y,y) ,function#(p(),x,x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(p(),x,x,y)) function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y)) ** Step 1.b:5: UsableRules. MAYBE + Considered Problem: - Strict DPs: function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(p(),x,x,y)) function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)) function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y)) - Weak TRS: function(if(),false(),x,y) -> function(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) function(if(),true(),x,y) -> y function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(plus(),dummy,x,y) -> function(if(),function(iszero(),x,x,x),x,y) function(third(),x,y,z) -> z - Signature: {function/4,function#/4} / {0/0,false/0,if/0,iszero/0,p/0,plus/0,s/1,third/0,true/0,c_1/2,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {function#} and constructors {0,false,if,iszero,p,plus,s ,third,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(third(),x,y,z) -> z function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(p(),x,x,y)) function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)) function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y)) ** Step 1.b:6: DecomposeDG. MAYBE + Considered Problem: - Strict DPs: function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(p(),x,x,y)) function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)) function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y)) - Weak TRS: function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(third(),x,y,z) -> z - Signature: {function/4,function#/4} / {0/0,false/0,if/0,iszero/0,p/0,plus/0,s/1,third/0,true/0,c_1/2,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {function#} and constructors {0,false,if,iszero,p,plus,s ,third,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 function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(p(),x,x,y)) function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y)) and a lower component function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)) Further, following extension rules are added to the lower component. function#(if(),false(),x,y) -> function#(p(),x,x,y) function#(if(),false(),x,y) -> function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) function#(plus(),dummy,x,y) -> function#(if(),function(iszero(),x,x,x),x,y) *** Step 1.b:6.a:1: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(p(),x,x,y)) function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y)) - Weak TRS: function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(third(),x,y,z) -> z - Signature: {function/4,function#/4} / {0/0,false/0,if/0,iszero/0,p/0,plus/0,s/1,third/0,true/0,c_1/2,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {function#} and constructors {0,false,if,iszero,p,plus,s ,third,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) ,function#(p(),x,x,y)) -->_1 function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y)):2 2:S:function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y)) -->_1 function#(if(),false(),x,y) -> c_1(function#(plus() ,function(third(),x,y,y) ,function(p(),x,x,y) ,s(y)) ,function#(p(),x,x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y))) *** Step 1.b:6.a:2: Failure MAYBE + Considered Problem: - Strict DPs: function#(if(),false(),x,y) -> c_1(function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y))) function#(plus(),dummy,x,y) -> c_8(function#(if(),function(iszero(),x,x,x),x,y)) - Weak TRS: function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(third(),x,y,z) -> z - Signature: {function/4,function#/4} / {0/0,false/0,if/0,iszero/0,p/0,plus/0,s/1,third/0,true/0,c_1/1,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {function#} and constructors {0,false,if,iszero,p,plus,s ,third,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. *** Step 1.b:6.b:1: Failure MAYBE + Considered Problem: - Strict DPs: function#(p(),s(s(x)),dummy,dummy2) -> c_7(function#(p(),s(x),x,x)) - Weak DPs: function#(if(),false(),x,y) -> function#(p(),x,x,y) function#(if(),false(),x,y) -> function#(plus(),function(third(),x,y,y),function(p(),x,x,y),s(y)) function#(plus(),dummy,x,y) -> function#(if(),function(iszero(),x,x,x),x,y) - Weak TRS: function(iszero(),0(),dummy,dummy2) -> true() function(iszero(),s(x),dummy,dummy2) -> false() function(p(),0(),dummy,dummy2) -> 0() function(p(),s(0()),dummy,dummy2) -> 0() function(p(),s(s(x)),dummy,dummy2) -> s(function(p(),s(x),x,x)) function(third(),x,y,z) -> z - Signature: {function/4,function#/4} / {0/0,false/0,if/0,iszero/0,p/0,plus/0,s/1,third/0,true/0,c_1/2,c_2/0,c_3/0,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {function#} and constructors {0,false,if,iszero,p,plus,s ,third,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. WORST_CASE(Omega(n^1),?)