MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: app(x,y) -> helpa(0(),plus(length(x),length(y)),x,y) ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) greater(ys,zs) -> helpc(ge(length(ys),length(zs)),ys,zs) helpa(c,l,ys,zs) -> if(ge(c,l),c,l,ys,zs) helpb(c,l,cons(y,ys),zs) -> cons(y,helpa(s(c),l,ys,zs)) helpc(false(),ys,zs) -> zs helpc(true(),ys,zs) -> ys if(false(),c,l,ys,zs) -> helpb(c,l,greater(ys,zs),smaller(ys,zs)) if(true(),c,l,ys,zs) -> nil() length(cons(x,y)) -> s(length(y)) length(nil()) -> 0() plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) smaller(ys,zs) -> helpc(ge(length(ys),length(zs)),zs,ys) - Signature: {app/2,ge/2,greater/2,helpa/4,helpb/4,helpc/3,if/5,length/1,plus/2,smaller/2} / {0/0,cons/2,false/0,nil/0 ,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {app,ge,greater,helpa,helpb,helpc,if,length,plus ,smaller} and constructors {0,cons,false,nil,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs app#(x,y) -> c_1(helpa#(0(),plus(length(x),length(y)),x,y) ,plus#(length(x),length(y)) ,length#(x) ,length#(y)) ge#(x,0()) -> c_2() ge#(0(),s(x)) -> c_3() ge#(s(x),s(y)) -> c_4(ge#(x,y)) greater#(ys,zs) -> c_5(helpc#(ge(length(ys),length(zs)),ys,zs) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)) helpb#(c,l,cons(y,ys),zs) -> c_7(helpa#(s(c),l,ys,zs)) helpc#(false(),ys,zs) -> c_8() helpc#(true(),ys,zs) -> c_9() if#(false(),c,l,ys,zs) -> c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)),greater#(ys,zs),smaller#(ys,zs)) if#(true(),c,l,ys,zs) -> c_11() length#(cons(x,y)) -> c_12(length#(y)) length#(nil()) -> c_13() plus#(x,0()) -> c_14() plus#(x,s(y)) -> c_15(plus#(x,y)) smaller#(ys,zs) -> c_16(helpc#(ge(length(ys),length(zs)),zs,ys) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: app#(x,y) -> c_1(helpa#(0(),plus(length(x),length(y)),x,y),plus#(length(x),length(y)),length#(x),length#(y)) ge#(x,0()) -> c_2() ge#(0(),s(x)) -> c_3() ge#(s(x),s(y)) -> c_4(ge#(x,y)) greater#(ys,zs) -> c_5(helpc#(ge(length(ys),length(zs)),ys,zs) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)) helpb#(c,l,cons(y,ys),zs) -> c_7(helpa#(s(c),l,ys,zs)) helpc#(false(),ys,zs) -> c_8() helpc#(true(),ys,zs) -> c_9() if#(false(),c,l,ys,zs) -> c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)),greater#(ys,zs),smaller#(ys,zs)) if#(true(),c,l,ys,zs) -> c_11() length#(cons(x,y)) -> c_12(length#(y)) length#(nil()) -> c_13() plus#(x,0()) -> c_14() plus#(x,s(y)) -> c_15(plus#(x,y)) smaller#(ys,zs) -> c_16(helpc#(ge(length(ys),length(zs)),zs,ys) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) - Weak TRS: app(x,y) -> helpa(0(),plus(length(x),length(y)),x,y) ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) greater(ys,zs) -> helpc(ge(length(ys),length(zs)),ys,zs) helpa(c,l,ys,zs) -> if(ge(c,l),c,l,ys,zs) helpb(c,l,cons(y,ys),zs) -> cons(y,helpa(s(c),l,ys,zs)) helpc(false(),ys,zs) -> zs helpc(true(),ys,zs) -> ys if(false(),c,l,ys,zs) -> helpb(c,l,greater(ys,zs),smaller(ys,zs)) if(true(),c,l,ys,zs) -> nil() length(cons(x,y)) -> s(length(y)) length(nil()) -> 0() plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) smaller(ys,zs) -> helpc(ge(length(ys),length(zs)),zs,ys) - Signature: {app/2,ge/2,greater/2,helpa/4,helpb/4,helpc/3,if/5,length/1,plus/2,smaller/2,app#/2,ge#/2,greater#/2 ,helpa#/4,helpb#/4,helpc#/3,if#/5,length#/1,plus#/2,smaller#/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/4 ,c_2/0,c_3/0,c_4/1,c_5/4,c_6/2,c_7/1,c_8/0,c_9/0,c_10/3,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/4} - Obligation: innermost runtime complexity wrt. defined symbols {app#,ge#,greater#,helpa#,helpb#,helpc#,if#,length#,plus# ,smaller#} and constructors {0,cons,false,nil,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) greater(ys,zs) -> helpc(ge(length(ys),length(zs)),ys,zs) helpc(false(),ys,zs) -> zs helpc(true(),ys,zs) -> ys length(cons(x,y)) -> s(length(y)) length(nil()) -> 0() plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) smaller(ys,zs) -> helpc(ge(length(ys),length(zs)),zs,ys) app#(x,y) -> c_1(helpa#(0(),plus(length(x),length(y)),x,y),plus#(length(x),length(y)),length#(x),length#(y)) ge#(x,0()) -> c_2() ge#(0(),s(x)) -> c_3() ge#(s(x),s(y)) -> c_4(ge#(x,y)) greater#(ys,zs) -> c_5(helpc#(ge(length(ys),length(zs)),ys,zs) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)) helpb#(c,l,cons(y,ys),zs) -> c_7(helpa#(s(c),l,ys,zs)) helpc#(false(),ys,zs) -> c_8() helpc#(true(),ys,zs) -> c_9() if#(false(),c,l,ys,zs) -> c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)),greater#(ys,zs),smaller#(ys,zs)) if#(true(),c,l,ys,zs) -> c_11() length#(cons(x,y)) -> c_12(length#(y)) length#(nil()) -> c_13() plus#(x,0()) -> c_14() plus#(x,s(y)) -> c_15(plus#(x,y)) smaller#(ys,zs) -> c_16(helpc#(ge(length(ys),length(zs)),zs,ys) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: app#(x,y) -> c_1(helpa#(0(),plus(length(x),length(y)),x,y),plus#(length(x),length(y)),length#(x),length#(y)) ge#(x,0()) -> c_2() ge#(0(),s(x)) -> c_3() ge#(s(x),s(y)) -> c_4(ge#(x,y)) greater#(ys,zs) -> c_5(helpc#(ge(length(ys),length(zs)),ys,zs) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)) helpb#(c,l,cons(y,ys),zs) -> c_7(helpa#(s(c),l,ys,zs)) helpc#(false(),ys,zs) -> c_8() helpc#(true(),ys,zs) -> c_9() if#(false(),c,l,ys,zs) -> c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)),greater#(ys,zs),smaller#(ys,zs)) if#(true(),c,l,ys,zs) -> c_11() length#(cons(x,y)) -> c_12(length#(y)) length#(nil()) -> c_13() plus#(x,0()) -> c_14() plus#(x,s(y)) -> c_15(plus#(x,y)) smaller#(ys,zs) -> c_16(helpc#(ge(length(ys),length(zs)),zs,ys) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) greater(ys,zs) -> helpc(ge(length(ys),length(zs)),ys,zs) helpc(false(),ys,zs) -> zs helpc(true(),ys,zs) -> ys length(cons(x,y)) -> s(length(y)) length(nil()) -> 0() plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) smaller(ys,zs) -> helpc(ge(length(ys),length(zs)),zs,ys) - Signature: {app/2,ge/2,greater/2,helpa/4,helpb/4,helpc/3,if/5,length/1,plus/2,smaller/2,app#/2,ge#/2,greater#/2 ,helpa#/4,helpb#/4,helpc#/3,if#/5,length#/1,plus#/2,smaller#/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/4 ,c_2/0,c_3/0,c_4/1,c_5/4,c_6/2,c_7/1,c_8/0,c_9/0,c_10/3,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/4} - Obligation: innermost runtime complexity wrt. defined symbols {app#,ge#,greater#,helpa#,helpb#,helpc#,if#,length#,plus# ,smaller#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3,8,9,11,13,14} by application of Pre({2,3,8,9,11,13,14}) = {1,4,5,6,12,15,16}. Here rules are labelled as follows: 1: app#(x,y) -> c_1(helpa#(0(),plus(length(x),length(y)),x,y) ,plus#(length(x),length(y)) ,length#(x) ,length#(y)) 2: ge#(x,0()) -> c_2() 3: ge#(0(),s(x)) -> c_3() 4: ge#(s(x),s(y)) -> c_4(ge#(x,y)) 5: greater#(ys,zs) -> c_5(helpc#(ge(length(ys),length(zs)),ys,zs) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) 6: helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)) 7: helpb#(c,l,cons(y,ys),zs) -> c_7(helpa#(s(c),l,ys,zs)) 8: helpc#(false(),ys,zs) -> c_8() 9: helpc#(true(),ys,zs) -> c_9() 10: if#(false(),c,l,ys,zs) -> c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)) ,greater#(ys,zs) ,smaller#(ys,zs)) 11: if#(true(),c,l,ys,zs) -> c_11() 12: length#(cons(x,y)) -> c_12(length#(y)) 13: length#(nil()) -> c_13() 14: plus#(x,0()) -> c_14() 15: plus#(x,s(y)) -> c_15(plus#(x,y)) 16: smaller#(ys,zs) -> c_16(helpc#(ge(length(ys),length(zs)),zs,ys) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: app#(x,y) -> c_1(helpa#(0(),plus(length(x),length(y)),x,y),plus#(length(x),length(y)),length#(x),length#(y)) ge#(s(x),s(y)) -> c_4(ge#(x,y)) greater#(ys,zs) -> c_5(helpc#(ge(length(ys),length(zs)),ys,zs) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)) helpb#(c,l,cons(y,ys),zs) -> c_7(helpa#(s(c),l,ys,zs)) if#(false(),c,l,ys,zs) -> c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)),greater#(ys,zs),smaller#(ys,zs)) length#(cons(x,y)) -> c_12(length#(y)) plus#(x,s(y)) -> c_15(plus#(x,y)) smaller#(ys,zs) -> c_16(helpc#(ge(length(ys),length(zs)),zs,ys) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) - Weak DPs: ge#(x,0()) -> c_2() ge#(0(),s(x)) -> c_3() helpc#(false(),ys,zs) -> c_8() helpc#(true(),ys,zs) -> c_9() if#(true(),c,l,ys,zs) -> c_11() length#(nil()) -> c_13() plus#(x,0()) -> c_14() - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) greater(ys,zs) -> helpc(ge(length(ys),length(zs)),ys,zs) helpc(false(),ys,zs) -> zs helpc(true(),ys,zs) -> ys length(cons(x,y)) -> s(length(y)) length(nil()) -> 0() plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) smaller(ys,zs) -> helpc(ge(length(ys),length(zs)),zs,ys) - Signature: {app/2,ge/2,greater/2,helpa/4,helpb/4,helpc/3,if/5,length/1,plus/2,smaller/2,app#/2,ge#/2,greater#/2 ,helpa#/4,helpb#/4,helpc#/3,if#/5,length#/1,plus#/2,smaller#/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/4 ,c_2/0,c_3/0,c_4/1,c_5/4,c_6/2,c_7/1,c_8/0,c_9/0,c_10/3,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/4} - Obligation: innermost runtime complexity wrt. defined symbols {app#,ge#,greater#,helpa#,helpb#,helpc#,if#,length#,plus# ,smaller#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:app#(x,y) -> c_1(helpa#(0(),plus(length(x),length(y)),x,y) ,plus#(length(x),length(y)) ,length#(x) ,length#(y)) -->_2 plus#(x,s(y)) -> c_15(plus#(x,y)):8 -->_4 length#(cons(x,y)) -> c_12(length#(y)):7 -->_3 length#(cons(x,y)) -> c_12(length#(y)):7 -->_1 helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)):4 -->_2 plus#(x,0()) -> c_14():16 -->_4 length#(nil()) -> c_13():15 -->_3 length#(nil()) -> c_13():15 2:S:ge#(s(x),s(y)) -> c_4(ge#(x,y)) -->_1 ge#(0(),s(x)) -> c_3():11 -->_1 ge#(x,0()) -> c_2():10 -->_1 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2 3:S:greater#(ys,zs) -> c_5(helpc#(ge(length(ys),length(zs)),ys,zs) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) -->_4 length#(cons(x,y)) -> c_12(length#(y)):7 -->_3 length#(cons(x,y)) -> c_12(length#(y)):7 -->_4 length#(nil()) -> c_13():15 -->_3 length#(nil()) -> c_13():15 -->_1 helpc#(true(),ys,zs) -> c_9():13 -->_1 helpc#(false(),ys,zs) -> c_8():12 -->_2 ge#(0(),s(x)) -> c_3():11 -->_2 ge#(x,0()) -> c_2():10 -->_2 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2 4:S:helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)) -->_1 if#(false(),c,l,ys,zs) -> c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)) ,greater#(ys,zs) ,smaller#(ys,zs)):6 -->_1 if#(true(),c,l,ys,zs) -> c_11():14 -->_2 ge#(0(),s(x)) -> c_3():11 -->_2 ge#(x,0()) -> c_2():10 -->_2 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2 5:S:helpb#(c,l,cons(y,ys),zs) -> c_7(helpa#(s(c),l,ys,zs)) -->_1 helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)):4 6:S:if#(false(),c,l,ys,zs) -> c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)) ,greater#(ys,zs) ,smaller#(ys,zs)) -->_3 smaller#(ys,zs) -> c_16(helpc#(ge(length(ys),length(zs)),zs,ys) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)):9 -->_1 helpb#(c,l,cons(y,ys),zs) -> c_7(helpa#(s(c),l,ys,zs)):5 -->_2 greater#(ys,zs) -> c_5(helpc#(ge(length(ys),length(zs)),ys,zs) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)):3 7:S:length#(cons(x,y)) -> c_12(length#(y)) -->_1 length#(nil()) -> c_13():15 -->_1 length#(cons(x,y)) -> c_12(length#(y)):7 8:S:plus#(x,s(y)) -> c_15(plus#(x,y)) -->_1 plus#(x,0()) -> c_14():16 -->_1 plus#(x,s(y)) -> c_15(plus#(x,y)):8 9:S:smaller#(ys,zs) -> c_16(helpc#(ge(length(ys),length(zs)),zs,ys) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) -->_4 length#(nil()) -> c_13():15 -->_3 length#(nil()) -> c_13():15 -->_1 helpc#(true(),ys,zs) -> c_9():13 -->_1 helpc#(false(),ys,zs) -> c_8():12 -->_2 ge#(0(),s(x)) -> c_3():11 -->_2 ge#(x,0()) -> c_2():10 -->_4 length#(cons(x,y)) -> c_12(length#(y)):7 -->_3 length#(cons(x,y)) -> c_12(length#(y)):7 -->_2 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2 10:W:ge#(x,0()) -> c_2() 11:W:ge#(0(),s(x)) -> c_3() 12:W:helpc#(false(),ys,zs) -> c_8() 13:W:helpc#(true(),ys,zs) -> c_9() 14:W:if#(true(),c,l,ys,zs) -> c_11() 15:W:length#(nil()) -> c_13() 16:W:plus#(x,0()) -> c_14() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: if#(true(),c,l,ys,zs) -> c_11() 10: ge#(x,0()) -> c_2() 11: ge#(0(),s(x)) -> c_3() 12: helpc#(false(),ys,zs) -> c_8() 13: helpc#(true(),ys,zs) -> c_9() 15: length#(nil()) -> c_13() 16: plus#(x,0()) -> c_14() * Step 5: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: app#(x,y) -> c_1(helpa#(0(),plus(length(x),length(y)),x,y),plus#(length(x),length(y)),length#(x),length#(y)) ge#(s(x),s(y)) -> c_4(ge#(x,y)) greater#(ys,zs) -> c_5(helpc#(ge(length(ys),length(zs)),ys,zs) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)) helpb#(c,l,cons(y,ys),zs) -> c_7(helpa#(s(c),l,ys,zs)) if#(false(),c,l,ys,zs) -> c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)),greater#(ys,zs),smaller#(ys,zs)) length#(cons(x,y)) -> c_12(length#(y)) plus#(x,s(y)) -> c_15(plus#(x,y)) smaller#(ys,zs) -> c_16(helpc#(ge(length(ys),length(zs)),zs,ys) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) greater(ys,zs) -> helpc(ge(length(ys),length(zs)),ys,zs) helpc(false(),ys,zs) -> zs helpc(true(),ys,zs) -> ys length(cons(x,y)) -> s(length(y)) length(nil()) -> 0() plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) smaller(ys,zs) -> helpc(ge(length(ys),length(zs)),zs,ys) - Signature: {app/2,ge/2,greater/2,helpa/4,helpb/4,helpc/3,if/5,length/1,plus/2,smaller/2,app#/2,ge#/2,greater#/2 ,helpa#/4,helpb#/4,helpc#/3,if#/5,length#/1,plus#/2,smaller#/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/4 ,c_2/0,c_3/0,c_4/1,c_5/4,c_6/2,c_7/1,c_8/0,c_9/0,c_10/3,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/4} - Obligation: innermost runtime complexity wrt. defined symbols {app#,ge#,greater#,helpa#,helpb#,helpc#,if#,length#,plus# ,smaller#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:app#(x,y) -> c_1(helpa#(0(),plus(length(x),length(y)),x,y) ,plus#(length(x),length(y)) ,length#(x) ,length#(y)) -->_2 plus#(x,s(y)) -> c_15(plus#(x,y)):8 -->_4 length#(cons(x,y)) -> c_12(length#(y)):7 -->_3 length#(cons(x,y)) -> c_12(length#(y)):7 -->_1 helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)):4 2:S:ge#(s(x),s(y)) -> c_4(ge#(x,y)) -->_1 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2 3:S:greater#(ys,zs) -> c_5(helpc#(ge(length(ys),length(zs)),ys,zs) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) -->_4 length#(cons(x,y)) -> c_12(length#(y)):7 -->_3 length#(cons(x,y)) -> c_12(length#(y)):7 -->_2 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2 4:S:helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)) -->_1 if#(false(),c,l,ys,zs) -> c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)) ,greater#(ys,zs) ,smaller#(ys,zs)):6 -->_2 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2 5:S:helpb#(c,l,cons(y,ys),zs) -> c_7(helpa#(s(c),l,ys,zs)) -->_1 helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)):4 6:S:if#(false(),c,l,ys,zs) -> c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)) ,greater#(ys,zs) ,smaller#(ys,zs)) -->_3 smaller#(ys,zs) -> c_16(helpc#(ge(length(ys),length(zs)),zs,ys) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)):9 -->_1 helpb#(c,l,cons(y,ys),zs) -> c_7(helpa#(s(c),l,ys,zs)):5 -->_2 greater#(ys,zs) -> c_5(helpc#(ge(length(ys),length(zs)),ys,zs) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)):3 7:S:length#(cons(x,y)) -> c_12(length#(y)) -->_1 length#(cons(x,y)) -> c_12(length#(y)):7 8:S:plus#(x,s(y)) -> c_15(plus#(x,y)) -->_1 plus#(x,s(y)) -> c_15(plus#(x,y)):8 9:S:smaller#(ys,zs) -> c_16(helpc#(ge(length(ys),length(zs)),zs,ys) ,ge#(length(ys),length(zs)) ,length#(ys) ,length#(zs)) -->_4 length#(cons(x,y)) -> c_12(length#(y)):7 -->_3 length#(cons(x,y)) -> c_12(length#(y)):7 -->_2 ge#(s(x),s(y)) -> c_4(ge#(x,y)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: greater#(ys,zs) -> c_5(ge#(length(ys),length(zs)),length#(ys),length#(zs)) smaller#(ys,zs) -> c_16(ge#(length(ys),length(zs)),length#(ys),length#(zs)) * Step 6: NaturalMI MAYBE + Considered Problem: - Strict DPs: app#(x,y) -> c_1(helpa#(0(),plus(length(x),length(y)),x,y),plus#(length(x),length(y)),length#(x),length#(y)) ge#(s(x),s(y)) -> c_4(ge#(x,y)) greater#(ys,zs) -> c_5(ge#(length(ys),length(zs)),length#(ys),length#(zs)) helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)) helpb#(c,l,cons(y,ys),zs) -> c_7(helpa#(s(c),l,ys,zs)) if#(false(),c,l,ys,zs) -> c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)),greater#(ys,zs),smaller#(ys,zs)) length#(cons(x,y)) -> c_12(length#(y)) plus#(x,s(y)) -> c_15(plus#(x,y)) smaller#(ys,zs) -> c_16(ge#(length(ys),length(zs)),length#(ys),length#(zs)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) greater(ys,zs) -> helpc(ge(length(ys),length(zs)),ys,zs) helpc(false(),ys,zs) -> zs helpc(true(),ys,zs) -> ys length(cons(x,y)) -> s(length(y)) length(nil()) -> 0() plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) smaller(ys,zs) -> helpc(ge(length(ys),length(zs)),zs,ys) - Signature: {app/2,ge/2,greater/2,helpa/4,helpb/4,helpc/3,if/5,length/1,plus/2,smaller/2,app#/2,ge#/2,greater#/2 ,helpa#/4,helpb#/4,helpc#/3,if#/5,length#/1,plus#/2,smaller#/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/4 ,c_2/0,c_3/0,c_4/1,c_5/3,c_6/2,c_7/1,c_8/0,c_9/0,c_10/3,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {app#,ge#,greater#,helpa#,helpb#,helpc#,if#,length#,plus# ,smaller#} and constructors {0,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_1) = {1,2,3,4}, uargs(c_4) = {1}, uargs(c_5) = {1,2,3}, uargs(c_6) = {1,2}, uargs(c_7) = {1}, uargs(c_10) = {1,2,3}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1,2,3} Following symbols are considered usable: {app#,ge#,greater#,helpa#,helpb#,helpc#,if#,length#,plus#,smaller#} TcT has computed the following interpretation: p(0) = [0] p(app) = [0] p(cons) = [2] p(false) = [0] p(ge) = [0] p(greater) = [4] x1 + [2] p(helpa) = [4] x1 + [0] p(helpb) = [1] x1 + [1] x3 + [0] p(helpc) = [7] x2 + [1] x3 + [4] p(if) = [4] x5 + [4] p(length) = [2] x1 + [0] p(nil) = [1] p(plus) = [4] x2 + [2] p(s) = [0] p(smaller) = [4] x1 + [0] p(true) = [4] p(app#) = [7] p(ge#) = [0] p(greater#) = [0] p(helpa#) = [2] x1 + [0] p(helpb#) = [0] p(helpc#) = [1] x2 + [0] p(if#) = [0] p(length#) = [0] p(plus#) = [5] p(smaller#) = [0] p(c_1) = [4] x1 + [1] x2 + [4] x3 + [1] x4 + [0] p(c_2) = [2] p(c_3) = [0] p(c_4) = [4] x1 + [0] p(c_5) = [1] x1 + [1] x2 + [1] x3 + [0] p(c_6) = [4] x1 + [2] x2 + [0] p(c_7) = [4] x1 + [0] p(c_8) = [1] p(c_9) = [2] p(c_10) = [4] x1 + [2] x2 + [1] x3 + [0] p(c_11) = [0] p(c_12) = [2] x1 + [0] p(c_13) = [2] p(c_14) = [1] p(c_15) = [1] x1 + [0] p(c_16) = [2] x1 + [4] x2 + [4] x3 + [0] Following rules are strictly oriented: app#(x,y) = [7] > [5] = c_1(helpa#(0(),plus(length(x),length(y)),x,y),plus#(length(x),length(y)),length#(x),length#(y)) Following rules are (at-least) weakly oriented: ge#(s(x),s(y)) = [0] >= [0] = c_4(ge#(x,y)) greater#(ys,zs) = [0] >= [0] = c_5(ge#(length(ys),length(zs)),length#(ys),length#(zs)) helpa#(c,l,ys,zs) = [2] c + [0] >= [0] = c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)) helpb#(c,l,cons(y,ys),zs) = [0] >= [0] = c_7(helpa#(s(c),l,ys,zs)) if#(false(),c,l,ys,zs) = [0] >= [0] = c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)),greater#(ys,zs),smaller#(ys,zs)) length#(cons(x,y)) = [0] >= [0] = c_12(length#(y)) plus#(x,s(y)) = [5] >= [5] = c_15(plus#(x,y)) smaller#(ys,zs) = [0] >= [0] = c_16(ge#(length(ys),length(zs)),length#(ys),length#(zs)) * Step 7: Failure MAYBE + Considered Problem: - Strict DPs: ge#(s(x),s(y)) -> c_4(ge#(x,y)) greater#(ys,zs) -> c_5(ge#(length(ys),length(zs)),length#(ys),length#(zs)) helpa#(c,l,ys,zs) -> c_6(if#(ge(c,l),c,l,ys,zs),ge#(c,l)) helpb#(c,l,cons(y,ys),zs) -> c_7(helpa#(s(c),l,ys,zs)) if#(false(),c,l,ys,zs) -> c_10(helpb#(c,l,greater(ys,zs),smaller(ys,zs)),greater#(ys,zs),smaller#(ys,zs)) length#(cons(x,y)) -> c_12(length#(y)) plus#(x,s(y)) -> c_15(plus#(x,y)) smaller#(ys,zs) -> c_16(ge#(length(ys),length(zs)),length#(ys),length#(zs)) - Weak DPs: app#(x,y) -> c_1(helpa#(0(),plus(length(x),length(y)),x,y),plus#(length(x),length(y)),length#(x),length#(y)) - Weak TRS: ge(x,0()) -> true() ge(0(),s(x)) -> false() ge(s(x),s(y)) -> ge(x,y) greater(ys,zs) -> helpc(ge(length(ys),length(zs)),ys,zs) helpc(false(),ys,zs) -> zs helpc(true(),ys,zs) -> ys length(cons(x,y)) -> s(length(y)) length(nil()) -> 0() plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) smaller(ys,zs) -> helpc(ge(length(ys),length(zs)),zs,ys) - Signature: {app/2,ge/2,greater/2,helpa/4,helpb/4,helpc/3,if/5,length/1,plus/2,smaller/2,app#/2,ge#/2,greater#/2 ,helpa#/4,helpb#/4,helpc#/3,if#/5,length#/1,plus#/2,smaller#/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/4 ,c_2/0,c_3/0,c_4/1,c_5/3,c_6/2,c_7/1,c_8/0,c_9/0,c_10/3,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {app#,ge#,greater#,helpa#,helpb#,helpc#,if#,length#,plus# ,smaller#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE