/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: cond1(s(x),y) -> cond2(gr(s(x),y),s(x),y) cond2(false(),x,y) -> cond1(p(x),y) cond2(true(),x,y) -> cond1(y,y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y)) -> neq(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1,cond2,gr,neq,p} 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: cond1(s(x),y) -> cond2(gr(s(x),y),s(x),y) cond2(false(),x,y) -> cond1(p(x),y) cond2(true(),x,y) -> cond1(y,y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y)) -> neq(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1,cond2,gr,neq,p} 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: cond1(s(x),y) -> cond2(gr(s(x),y),s(x),y) cond2(false(),x,y) -> cond1(p(x),y) cond2(true(),x,y) -> cond1(y,y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y)) -> neq(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1,cond2,gr,neq,p} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: gr(x,y){x -> s(x),y -> s(y)} = gr(s(x),s(y)) ->^+ gr(x,y) = C[gr(x,y) = gr(x,y){}] ** Step 1.b:1: DependencyPairs. MAYBE + Considered Problem: - Strict TRS: cond1(s(x),y) -> cond2(gr(s(x),y),s(x),y) cond2(false(),x,y) -> cond1(p(x),y) cond2(true(),x,y) -> cond1(y,y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y)) -> neq(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1,cond2,gr,neq,p} and constructors {0,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)) cond2#(false(),x,y) -> c_2(cond1#(p(x),y),p#(x)) cond2#(true(),x,y) -> c_3(cond1#(y,y)) gr#(0(),x) -> c_4() gr#(s(x),0()) -> c_5() gr#(s(x),s(y)) -> c_6(gr#(x,y)) neq#(0(),0()) -> c_7() neq#(0(),s(x)) -> c_8() neq#(s(x),0()) -> c_9() neq#(s(x),s(y)) -> c_10(neq#(x,y)) p#(0()) -> c_11() p#(s(x)) -> c_12() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. MAYBE + Considered Problem: - Strict DPs: cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)) cond2#(false(),x,y) -> c_2(cond1#(p(x),y),p#(x)) cond2#(true(),x,y) -> c_3(cond1#(y,y)) gr#(0(),x) -> c_4() gr#(s(x),0()) -> c_5() gr#(s(x),s(y)) -> c_6(gr#(x,y)) neq#(0(),0()) -> c_7() neq#(0(),s(x)) -> c_8() neq#(s(x),0()) -> c_9() neq#(s(x),s(y)) -> c_10(neq#(x,y)) p#(0()) -> c_11() p#(s(x)) -> c_12() - Weak TRS: cond1(s(x),y) -> cond2(gr(s(x),y),s(x),y) cond2(false(),x,y) -> cond1(p(x),y) cond2(true(),x,y) -> cond1(y,y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y)) -> neq(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1,cond1#/2,cond2#/3,gr#/2,neq#/2,p#/1} / {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_8/0,c_9/0,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s ,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4,5,7,8,9,11,12} by application of Pre({4,5,7,8,9,11,12}) = {1,2,6,10}. Here rules are labelled as follows: 1: cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)) 2: cond2#(false(),x,y) -> c_2(cond1#(p(x),y),p#(x)) 3: cond2#(true(),x,y) -> c_3(cond1#(y,y)) 4: gr#(0(),x) -> c_4() 5: gr#(s(x),0()) -> c_5() 6: gr#(s(x),s(y)) -> c_6(gr#(x,y)) 7: neq#(0(),0()) -> c_7() 8: neq#(0(),s(x)) -> c_8() 9: neq#(s(x),0()) -> c_9() 10: neq#(s(x),s(y)) -> c_10(neq#(x,y)) 11: p#(0()) -> c_11() 12: p#(s(x)) -> c_12() ** Step 1.b:3: RemoveWeakSuffixes. MAYBE + Considered Problem: - Strict DPs: cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)) cond2#(false(),x,y) -> c_2(cond1#(p(x),y),p#(x)) cond2#(true(),x,y) -> c_3(cond1#(y,y)) gr#(s(x),s(y)) -> c_6(gr#(x,y)) neq#(s(x),s(y)) -> c_10(neq#(x,y)) - Weak DPs: gr#(0(),x) -> c_4() gr#(s(x),0()) -> c_5() neq#(0(),0()) -> c_7() neq#(0(),s(x)) -> c_8() neq#(s(x),0()) -> c_9() p#(0()) -> c_11() p#(s(x)) -> c_12() - Weak TRS: cond1(s(x),y) -> cond2(gr(s(x),y),s(x),y) cond2(false(),x,y) -> cond1(p(x),y) cond2(true(),x,y) -> cond1(y,y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y)) -> neq(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1,cond1#/2,cond2#/3,gr#/2,neq#/2,p#/1} / {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_8/0,c_9/0,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)) -->_2 gr#(s(x),s(y)) -> c_6(gr#(x,y)):4 -->_1 cond2#(true(),x,y) -> c_3(cond1#(y,y)):3 -->_1 cond2#(false(),x,y) -> c_2(cond1#(p(x),y),p#(x)):2 -->_2 gr#(s(x),0()) -> c_5():7 2:S:cond2#(false(),x,y) -> c_2(cond1#(p(x),y),p#(x)) -->_2 p#(s(x)) -> c_12():12 -->_2 p#(0()) -> c_11():11 -->_1 cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)):1 3:S:cond2#(true(),x,y) -> c_3(cond1#(y,y)) -->_1 cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)):1 4:S:gr#(s(x),s(y)) -> c_6(gr#(x,y)) -->_1 gr#(s(x),0()) -> c_5():7 -->_1 gr#(0(),x) -> c_4():6 -->_1 gr#(s(x),s(y)) -> c_6(gr#(x,y)):4 5:S:neq#(s(x),s(y)) -> c_10(neq#(x,y)) -->_1 neq#(s(x),0()) -> c_9():10 -->_1 neq#(0(),s(x)) -> c_8():9 -->_1 neq#(0(),0()) -> c_7():8 -->_1 neq#(s(x),s(y)) -> c_10(neq#(x,y)):5 6:W:gr#(0(),x) -> c_4() 7:W:gr#(s(x),0()) -> c_5() 8:W:neq#(0(),0()) -> c_7() 9:W:neq#(0(),s(x)) -> c_8() 10:W:neq#(s(x),0()) -> c_9() 11:W:p#(0()) -> c_11() 12:W:p#(s(x)) -> c_12() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: neq#(0(),0()) -> c_7() 9: neq#(0(),s(x)) -> c_8() 10: neq#(s(x),0()) -> c_9() 11: p#(0()) -> c_11() 12: p#(s(x)) -> c_12() 6: gr#(0(),x) -> c_4() 7: gr#(s(x),0()) -> c_5() ** Step 1.b:4: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)) cond2#(false(),x,y) -> c_2(cond1#(p(x),y),p#(x)) cond2#(true(),x,y) -> c_3(cond1#(y,y)) gr#(s(x),s(y)) -> c_6(gr#(x,y)) neq#(s(x),s(y)) -> c_10(neq#(x,y)) - Weak TRS: cond1(s(x),y) -> cond2(gr(s(x),y),s(x),y) cond2(false(),x,y) -> cond1(p(x),y) cond2(true(),x,y) -> cond1(y,y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y)) -> neq(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1,cond1#/2,cond2#/3,gr#/2,neq#/2,p#/1} / {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_8/0,c_9/0,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s ,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)) -->_2 gr#(s(x),s(y)) -> c_6(gr#(x,y)):4 -->_1 cond2#(true(),x,y) -> c_3(cond1#(y,y)):3 -->_1 cond2#(false(),x,y) -> c_2(cond1#(p(x),y),p#(x)):2 2:S:cond2#(false(),x,y) -> c_2(cond1#(p(x),y),p#(x)) -->_1 cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)):1 3:S:cond2#(true(),x,y) -> c_3(cond1#(y,y)) -->_1 cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)):1 4:S:gr#(s(x),s(y)) -> c_6(gr#(x,y)) -->_1 gr#(s(x),s(y)) -> c_6(gr#(x,y)):4 5:S:neq#(s(x),s(y)) -> c_10(neq#(x,y)) -->_1 neq#(s(x),s(y)) -> c_10(neq#(x,y)):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: cond2#(false(),x,y) -> c_2(cond1#(p(x),y)) ** Step 1.b:5: UsableRules. MAYBE + Considered Problem: - Strict DPs: cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)) cond2#(false(),x,y) -> c_2(cond1#(p(x),y)) cond2#(true(),x,y) -> c_3(cond1#(y,y)) gr#(s(x),s(y)) -> c_6(gr#(x,y)) neq#(s(x),s(y)) -> c_10(neq#(x,y)) - Weak TRS: cond1(s(x),y) -> cond2(gr(s(x),y),s(x),y) cond2(false(),x,y) -> cond1(p(x),y) cond2(true(),x,y) -> cond1(y,y) gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y)) -> neq(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1,cond1#/2,cond2#/3,gr#/2,neq#/2,p#/1} / {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_8/0,c_9/0,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s ,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)) cond2#(false(),x,y) -> c_2(cond1#(p(x),y)) cond2#(true(),x,y) -> c_3(cond1#(y,y)) gr#(s(x),s(y)) -> c_6(gr#(x,y)) neq#(s(x),s(y)) -> c_10(neq#(x,y)) ** Step 1.b:6: DecomposeDG. MAYBE + Considered Problem: - Strict DPs: cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)) cond2#(false(),x,y) -> c_2(cond1#(p(x),y)) cond2#(true(),x,y) -> c_3(cond1#(y,y)) gr#(s(x),s(y)) -> c_6(gr#(x,y)) neq#(s(x),s(y)) -> c_10(neq#(x,y)) - Weak TRS: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1,cond1#/2,cond2#/3,gr#/2,neq#/2,p#/1} / {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_8/0,c_9/0,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} 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 cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)) cond2#(false(),x,y) -> c_2(cond1#(p(x),y)) cond2#(true(),x,y) -> c_3(cond1#(y,y)) neq#(s(x),s(y)) -> c_10(neq#(x,y)) and a lower component gr#(s(x),s(y)) -> c_6(gr#(x,y)) Further, following extension rules are added to the lower component. cond1#(s(x),y) -> cond2#(gr(s(x),y),s(x),y) cond1#(s(x),y) -> gr#(s(x),y) cond2#(false(),x,y) -> cond1#(p(x),y) cond2#(true(),x,y) -> cond1#(y,y) neq#(s(x),s(y)) -> neq#(x,y) *** Step 1.b:6.a:1: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)) cond2#(false(),x,y) -> c_2(cond1#(p(x),y)) cond2#(true(),x,y) -> c_3(cond1#(y,y)) neq#(s(x),s(y)) -> c_10(neq#(x,y)) - Weak TRS: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1,cond1#/2,cond2#/3,gr#/2,neq#/2,p#/1} / {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_8/0,c_9/0,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s ,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)) -->_1 cond2#(true(),x,y) -> c_3(cond1#(y,y)):3 -->_1 cond2#(false(),x,y) -> c_2(cond1#(p(x),y)):2 2:S:cond2#(false(),x,y) -> c_2(cond1#(p(x),y)) -->_1 cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)):1 3:S:cond2#(true(),x,y) -> c_3(cond1#(y,y)) -->_1 cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y),gr#(s(x),y)):1 4:S:neq#(s(x),s(y)) -> c_10(neq#(x,y)) -->_1 neq#(s(x),s(y)) -> c_10(neq#(x,y)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y)) *** Step 1.b:6.a:2: NaturalMI. MAYBE + Considered Problem: - Strict DPs: cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y)) cond2#(false(),x,y) -> c_2(cond1#(p(x),y)) cond2#(true(),x,y) -> c_3(cond1#(y,y)) neq#(s(x),s(y)) -> c_10(neq#(x,y)) - Weak TRS: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1,cond1#/2,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/1,c_2/1 ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} 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_1) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_10) = {1} Following symbols are considered usable: {cond1#,cond2#,gr#,neq#,p#} TcT has computed the following interpretation: p(0) = [0] p(cond1) = [0] p(cond2) = [1] x2 + [2] x3 + [8] p(false) = [0] p(gr) = [4] x2 + [0] p(neq) = [1] x1 + [8] p(p) = [2] p(s) = [1] x1 + [3] p(true) = [0] p(cond1#) = [0] p(cond2#) = [0] p(gr#) = [1] x1 + [8] x2 + [1] p(neq#) = [5] x2 + [8] p(p#) = [1] x1 + [0] p(c_1) = [8] x1 + [0] p(c_2) = [2] x1 + [0] p(c_3) = [8] x1 + [0] p(c_4) = [0] p(c_5) = [1] p(c_6) = [1] x1 + [0] p(c_7) = [1] p(c_8) = [0] p(c_9) = [1] p(c_10) = [1] x1 + [3] p(c_11) = [8] p(c_12) = [0] Following rules are strictly oriented: neq#(s(x),s(y)) = [5] y + [23] > [5] y + [11] = c_10(neq#(x,y)) Following rules are (at-least) weakly oriented: cond1#(s(x),y) = [0] >= [0] = c_1(cond2#(gr(s(x),y),s(x),y)) cond2#(false(),x,y) = [0] >= [0] = c_2(cond1#(p(x),y)) cond2#(true(),x,y) = [0] >= [0] = c_3(cond1#(y,y)) *** Step 1.b:6.a:3: Failure MAYBE + Considered Problem: - Strict DPs: cond1#(s(x),y) -> c_1(cond2#(gr(s(x),y),s(x),y)) cond2#(false(),x,y) -> c_2(cond1#(p(x),y)) cond2#(true(),x,y) -> c_3(cond1#(y,y)) - Weak DPs: neq#(s(x),s(y)) -> c_10(neq#(x,y)) - Weak TRS: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1,cond1#/2,cond2#/3,gr#/2,neq#/2,p#/1} / {0/0,false/0,s/1,true/0,c_1/1,c_2/1 ,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s ,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. *** Step 1.b:6.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gr#(s(x),s(y)) -> c_6(gr#(x,y)) - Weak DPs: cond1#(s(x),y) -> cond2#(gr(s(x),y),s(x),y) cond1#(s(x),y) -> gr#(s(x),y) cond2#(false(),x,y) -> cond1#(p(x),y) cond2#(true(),x,y) -> cond1#(y,y) neq#(s(x),s(y)) -> neq#(x,y) - Weak TRS: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1,cond1#/2,cond2#/3,gr#/2,neq#/2,p#/1} / {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_8/0,c_9/0,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:gr#(s(x),s(y)) -> c_6(gr#(x,y)) -->_1 gr#(s(x),s(y)) -> c_6(gr#(x,y)):1 2:W:cond1#(s(x),y) -> cond2#(gr(s(x),y),s(x),y) -->_1 cond2#(true(),x,y) -> cond1#(y,y):5 -->_1 cond2#(false(),x,y) -> cond1#(p(x),y):4 3:W:cond1#(s(x),y) -> gr#(s(x),y) -->_1 gr#(s(x),s(y)) -> c_6(gr#(x,y)):1 4:W:cond2#(false(),x,y) -> cond1#(p(x),y) -->_1 cond1#(s(x),y) -> gr#(s(x),y):3 -->_1 cond1#(s(x),y) -> cond2#(gr(s(x),y),s(x),y):2 5:W:cond2#(true(),x,y) -> cond1#(y,y) -->_1 cond1#(s(x),y) -> gr#(s(x),y):3 -->_1 cond1#(s(x),y) -> cond2#(gr(s(x),y),s(x),y):2 6:W:neq#(s(x),s(y)) -> neq#(x,y) -->_1 neq#(s(x),s(y)) -> neq#(x,y):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: neq#(s(x),s(y)) -> neq#(x,y) *** Step 1.b:6.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gr#(s(x),s(y)) -> c_6(gr#(x,y)) - Weak DPs: cond1#(s(x),y) -> cond2#(gr(s(x),y),s(x),y) cond1#(s(x),y) -> gr#(s(x),y) cond2#(false(),x,y) -> cond1#(p(x),y) cond2#(true(),x,y) -> cond1#(y,y) - Weak TRS: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1,cond1#/2,cond2#/3,gr#/2,neq#/2,p#/1} / {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_8/0,c_9/0,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} 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_6) = {1} Following symbols are considered usable: {cond1#,cond2#,gr#,neq#,p#} TcT has computed the following interpretation: p(0) = [8] p(cond1) = [1] p(cond2) = [0] p(false) = [0] p(gr) = [2] x2 + [2] p(neq) = [0] p(p) = [10] p(s) = [1] x1 + [8] p(true) = [1] p(cond1#) = [3] x2 + [0] p(cond2#) = [3] x3 + [0] p(gr#) = [2] x2 + [0] p(neq#) = [8] x2 + [1] p(p#) = [1] p(c_1) = [1] x1 + [2] x2 + [0] p(c_2) = [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [2] p(c_6) = [1] x1 + [15] p(c_7) = [4] p(c_8) = [8] p(c_9) = [1] p(c_10) = [2] x1 + [1] p(c_11) = [2] p(c_12) = [1] Following rules are strictly oriented: gr#(s(x),s(y)) = [2] y + [16] > [2] y + [15] = c_6(gr#(x,y)) Following rules are (at-least) weakly oriented: cond1#(s(x),y) = [3] y + [0] >= [3] y + [0] = cond2#(gr(s(x),y),s(x),y) cond1#(s(x),y) = [3] y + [0] >= [2] y + [0] = gr#(s(x),y) cond2#(false(),x,y) = [3] y + [0] >= [3] y + [0] = cond1#(p(x),y) cond2#(true(),x,y) = [3] y + [0] >= [3] y + [0] = cond1#(y,y) *** Step 1.b:6.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: cond1#(s(x),y) -> cond2#(gr(s(x),y),s(x),y) cond1#(s(x),y) -> gr#(s(x),y) cond2#(false(),x,y) -> cond1#(p(x),y) cond2#(true(),x,y) -> cond1#(y,y) gr#(s(x),s(y)) -> c_6(gr#(x,y)) - Weak TRS: gr(0(),x) -> false() gr(s(x),0()) -> true() gr(s(x),s(y)) -> gr(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/3,gr/2,neq/2,p/1,cond1#/2,cond2#/3,gr#/2,neq#/2,p#/1} / {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_8/0,c_9/0,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,gr#,neq#,p#} 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),?)