WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) forrest(graph(N,E)) -> kruskal(sort(E),nil(),partitions(N)) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) kruskal(dd(e,E),F,P) -> kruskal#q(inBlock(e,P),e,E,F,P) kruskal(nil(),F,P) -> pair(F,P) kruskal#q(false(),e,E,F,P) -> kruskal(E,dd(e,F),join(e,P,nil())) kruskal#q(true(),e,E,F,P) -> kruskal(E,F,P) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {and,elem,eq,forrest,inBlock,insert,insert#q,join,join#q ,kruskal,kruskal#q,leq,or,partitions,pp,sort,src,trg,wt} and constructors {0,dd,edge,false,graph,nil,pair,s ,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs and#(false(),false()) -> c_1() and#(false(),true()) -> c_2() and#(true(),false()) -> c_3() and#(true(),true()) -> c_4() elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)) elem#(n,nil()) -> c_6() eq#(0(),0()) -> c_7() eq#(0(),s(m)) -> c_8() eq#(s(n),0()) -> c_9() eq#(s(n),s(m)) -> c_10(eq#(n,m)) forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(or#(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) ,and#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e) ,inBlock#(e,P)) inBlock#(e,nil()) -> c_13() insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f)),wt#(e),wt#(f)) insert#(e,nil()) -> c_15() insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) insert#q#(true(),e,f,E) -> c_17() join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)) join#(e,nil(),q) -> c_19() join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#(nil(),F,P) -> c_23() kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) leq#(0(),0()) -> c_26() leq#(0(),s(m)) -> c_27() leq#(s(n),0()) -> c_28() leq#(s(n),s(m)) -> c_29(leq#(n,m)) or#(false(),false()) -> c_30() or#(false(),true()) -> c_31() or#(true(),false()) -> c_32() or#(true(),true()) -> c_33() partitions#(dd(n,N)) -> c_34(partitions#(N)) partitions#(nil()) -> c_35() pp#(dd(n,p),q) -> c_36(pp#(p,q)) pp#(nil(),q) -> c_37() sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) sort#(nil()) -> c_39() src#(edge(n,w,m)) -> c_40() trg#(edge(n,w,m)) -> c_41() wt#(edge(n,w,m)) -> c_42() Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: and#(false(),false()) -> c_1() and#(false(),true()) -> c_2() and#(true(),false()) -> c_3() and#(true(),true()) -> c_4() elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)) elem#(n,nil()) -> c_6() eq#(0(),0()) -> c_7() eq#(0(),s(m)) -> c_8() eq#(s(n),0()) -> c_9() eq#(s(n),s(m)) -> c_10(eq#(n,m)) forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(or#(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) ,and#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e) ,inBlock#(e,P)) inBlock#(e,nil()) -> c_13() insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f)),wt#(e),wt#(f)) insert#(e,nil()) -> c_15() insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) insert#q#(true(),e,f,E) -> c_17() join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)) join#(e,nil(),q) -> c_19() join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#(nil(),F,P) -> c_23() kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) leq#(0(),0()) -> c_26() leq#(0(),s(m)) -> c_27() leq#(s(n),0()) -> c_28() leq#(s(n),s(m)) -> c_29(leq#(n,m)) or#(false(),false()) -> c_30() or#(false(),true()) -> c_31() or#(true(),false()) -> c_32() or#(true(),true()) -> c_33() partitions#(dd(n,N)) -> c_34(partitions#(N)) partitions#(nil()) -> c_35() pp#(dd(n,p),q) -> c_36(pp#(p,q)) pp#(nil(),q) -> c_37() sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) sort#(nil()) -> c_39() src#(edge(n,w,m)) -> c_40() trg#(edge(n,w,m)) -> c_41() wt#(edge(n,w,m)) -> c_42() - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) forrest(graph(N,E)) -> kruskal(sort(E),nil(),partitions(N)) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) kruskal(dd(e,E),F,P) -> kruskal#q(inBlock(e,P),e,E,F,P) kruskal(nil(),F,P) -> pair(F,P) kruskal#q(false(),e,E,F,P) -> kruskal(E,dd(e,F),join(e,P,nil())) kruskal#q(true(),e,E,F,P) -> kruskal(E,F,P) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/7,c_13/0,c_14/4,c_15/0,c_16/1,c_17/0,c_18/6,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w and#(false(),false()) -> c_1() and#(false(),true()) -> c_2() and#(true(),false()) -> c_3() and#(true(),true()) -> c_4() elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)) elem#(n,nil()) -> c_6() eq#(0(),0()) -> c_7() eq#(0(),s(m)) -> c_8() eq#(s(n),0()) -> c_9() eq#(s(n),s(m)) -> c_10(eq#(n,m)) forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(or#(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) ,and#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e) ,inBlock#(e,P)) inBlock#(e,nil()) -> c_13() insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f)),wt#(e),wt#(f)) insert#(e,nil()) -> c_15() insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) insert#q#(true(),e,f,E) -> c_17() join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)) join#(e,nil(),q) -> c_19() join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#(nil(),F,P) -> c_23() kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) leq#(0(),0()) -> c_26() leq#(0(),s(m)) -> c_27() leq#(s(n),0()) -> c_28() leq#(s(n),s(m)) -> c_29(leq#(n,m)) or#(false(),false()) -> c_30() or#(false(),true()) -> c_31() or#(true(),false()) -> c_32() or#(true(),true()) -> c_33() partitions#(dd(n,N)) -> c_34(partitions#(N)) partitions#(nil()) -> c_35() pp#(dd(n,p),q) -> c_36(pp#(p,q)) pp#(nil(),q) -> c_37() sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) sort#(nil()) -> c_39() src#(edge(n,w,m)) -> c_40() trg#(edge(n,w,m)) -> c_41() wt#(edge(n,w,m)) -> c_42() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: and#(false(),false()) -> c_1() and#(false(),true()) -> c_2() and#(true(),false()) -> c_3() and#(true(),true()) -> c_4() elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)) elem#(n,nil()) -> c_6() eq#(0(),0()) -> c_7() eq#(0(),s(m)) -> c_8() eq#(s(n),0()) -> c_9() eq#(s(n),s(m)) -> c_10(eq#(n,m)) forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(or#(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) ,and#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e) ,inBlock#(e,P)) inBlock#(e,nil()) -> c_13() insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f)),wt#(e),wt#(f)) insert#(e,nil()) -> c_15() insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) insert#q#(true(),e,f,E) -> c_17() join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)) join#(e,nil(),q) -> c_19() join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#(nil(),F,P) -> c_23() kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) leq#(0(),0()) -> c_26() leq#(0(),s(m)) -> c_27() leq#(s(n),0()) -> c_28() leq#(s(n),s(m)) -> c_29(leq#(n,m)) or#(false(),false()) -> c_30() or#(false(),true()) -> c_31() or#(true(),false()) -> c_32() or#(true(),true()) -> c_33() partitions#(dd(n,N)) -> c_34(partitions#(N)) partitions#(nil()) -> c_35() pp#(dd(n,p),q) -> c_36(pp#(p,q)) pp#(nil(),q) -> c_37() sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) sort#(nil()) -> c_39() src#(edge(n,w,m)) -> c_40() trg#(edge(n,w,m)) -> c_41() wt#(edge(n,w,m)) -> c_42() - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/7,c_13/0,c_14/4,c_15/0,c_16/1,c_17/0,c_18/6,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,6,7,8,9,13,15,17,19,23,26,27,28,30,31,32,33,35,37,39,40,41,42} by application of Pre({1,2,3,4,6,7,8,9,13,15,17,19,23,26,27,28,30,31,32,33,35,37,39,40,41,42}) = {5,10,11,12,14,16,18,20,21 ,22,24,25,29,34,36,38}. Here rules are labelled as follows: 1: and#(false(),false()) -> c_1() 2: and#(false(),true()) -> c_2() 3: and#(true(),false()) -> c_3() 4: and#(true(),true()) -> c_4() 5: elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)) 6: elem#(n,nil()) -> c_6() 7: eq#(0(),0()) -> c_7() 8: eq#(0(),s(m)) -> c_8() 9: eq#(s(n),0()) -> c_9() 10: eq#(s(n),s(m)) -> c_10(eq#(n,m)) 11: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) 12: inBlock#(e,dd(p,P)) -> c_12(or#(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) ,and#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e) ,inBlock#(e,P)) 13: inBlock#(e,nil()) -> c_13() 14: insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f)),wt#(e),wt#(f)) 15: insert#(e,nil()) -> c_15() 16: insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) 17: insert#q#(true(),e,f,E) -> c_17() 18: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)) 19: join#(e,nil(),q) -> c_19() 20: join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) 21: join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) 22: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) 23: kruskal#(nil(),F,P) -> c_23() 24: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) 25: kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) 26: leq#(0(),0()) -> c_26() 27: leq#(0(),s(m)) -> c_27() 28: leq#(s(n),0()) -> c_28() 29: leq#(s(n),s(m)) -> c_29(leq#(n,m)) 30: or#(false(),false()) -> c_30() 31: or#(false(),true()) -> c_31() 32: or#(true(),false()) -> c_32() 33: or#(true(),true()) -> c_33() 34: partitions#(dd(n,N)) -> c_34(partitions#(N)) 35: partitions#(nil()) -> c_35() 36: pp#(dd(n,p),q) -> c_36(pp#(p,q)) 37: pp#(nil(),q) -> c_37() 38: sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) 39: sort#(nil()) -> c_39() 40: src#(edge(n,w,m)) -> c_40() 41: trg#(edge(n,w,m)) -> c_41() 42: wt#(edge(n,w,m)) -> c_42() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)) eq#(s(n),s(m)) -> c_10(eq#(n,m)) forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(or#(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) ,and#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e) ,inBlock#(e,P)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f)),wt#(e),wt#(f)) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak DPs: and#(false(),false()) -> c_1() and#(false(),true()) -> c_2() and#(true(),false()) -> c_3() and#(true(),true()) -> c_4() elem#(n,nil()) -> c_6() eq#(0(),0()) -> c_7() eq#(0(),s(m)) -> c_8() eq#(s(n),0()) -> c_9() inBlock#(e,nil()) -> c_13() insert#(e,nil()) -> c_15() insert#q#(true(),e,f,E) -> c_17() join#(e,nil(),q) -> c_19() kruskal#(nil(),F,P) -> c_23() leq#(0(),0()) -> c_26() leq#(0(),s(m)) -> c_27() leq#(s(n),0()) -> c_28() or#(false(),false()) -> c_30() or#(false(),true()) -> c_31() or#(true(),false()) -> c_32() or#(true(),true()) -> c_33() partitions#(nil()) -> c_35() pp#(nil(),q) -> c_37() sort#(nil()) -> c_39() src#(edge(n,w,m)) -> c_40() trg#(edge(n,w,m)) -> c_41() wt#(edge(n,w,m)) -> c_42() - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/7,c_13/0,c_14/4,c_15/0,c_16/1,c_17/0,c_18/6,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)) -->_2 eq#(s(n),s(m)) -> c_10(eq#(n,m)):2 -->_1 or#(true(),true()) -> c_33():36 -->_1 or#(true(),false()) -> c_32():35 -->_1 or#(false(),true()) -> c_31():34 -->_1 or#(false(),false()) -> c_30():33 -->_2 eq#(s(n),0()) -> c_9():24 -->_2 eq#(0(),s(m)) -> c_8():23 -->_2 eq#(0(),0()) -> c_7():22 -->_3 elem#(n,nil()) -> c_6():21 -->_3 elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)):1 2:S:eq#(s(n),s(m)) -> c_10(eq#(n,m)) -->_1 eq#(s(n),0()) -> c_9():24 -->_1 eq#(0(),s(m)) -> c_8():23 -->_1 eq#(0(),0()) -> c_7():22 -->_1 eq#(s(n),s(m)) -> c_10(eq#(n,m)):2 3:S:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):16 -->_3 partitions#(dd(n,N)) -> c_34(partitions#(N)):14 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):10 -->_2 sort#(nil()) -> c_39():39 -->_3 partitions#(nil()) -> c_35():37 -->_1 kruskal#(nil(),F,P) -> c_23():29 4:S:inBlock#(e,dd(p,P)) -> c_12(or#(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) ,and#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e) ,inBlock#(e,P)) -->_6 trg#(edge(n,w,m)) -> c_41():41 -->_4 src#(edge(n,w,m)) -> c_40():40 -->_1 or#(true(),true()) -> c_33():36 -->_1 or#(true(),false()) -> c_32():35 -->_1 or#(false(),true()) -> c_31():34 -->_1 or#(false(),false()) -> c_30():33 -->_7 inBlock#(e,nil()) -> c_13():25 -->_5 elem#(n,nil()) -> c_6():21 -->_3 elem#(n,nil()) -> c_6():21 -->_2 and#(true(),true()) -> c_4():20 -->_2 and#(true(),false()) -> c_3():19 -->_2 and#(false(),true()) -> c_2():18 -->_2 and#(false(),false()) -> c_1():17 -->_7 inBlock#(e,dd(p,P)) -> c_12(or#(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) ,and#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e) ,inBlock#(e,P)):4 -->_5 elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)):1 -->_3 elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)):1 5:S:insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f)),wt#(e),wt#(f)) -->_2 leq#(s(n),s(m)) -> c_29(leq#(n,m)):13 -->_1 insert#q#(false(),e,f,E) -> c_16(insert#(e,E)):6 -->_4 wt#(edge(n,w,m)) -> c_42():42 -->_3 wt#(edge(n,w,m)) -> c_42():42 -->_2 leq#(s(n),0()) -> c_28():32 -->_2 leq#(0(),s(m)) -> c_27():31 -->_2 leq#(0(),0()) -> c_26():30 -->_1 insert#q#(true(),e,f,E) -> c_17():27 6:S:insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) -->_1 insert#(e,nil()) -> c_15():26 -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f)),wt#(e),wt#(f)):5 7:S:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):9 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):8 -->_6 trg#(edge(n,w,m)) -> c_41():41 -->_4 src#(edge(n,w,m)) -> c_40():40 -->_2 or#(true(),true()) -> c_33():36 -->_2 or#(true(),false()) -> c_32():35 -->_2 or#(false(),true()) -> c_31():34 -->_2 or#(false(),false()) -> c_30():33 -->_5 elem#(n,nil()) -> c_6():21 -->_3 elem#(n,nil()) -> c_6():21 -->_5 elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)):1 -->_3 elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)):1 8:S:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,nil(),q) -> c_19():28 -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)):7 9:S:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):15 -->_2 pp#(nil(),q) -> c_37():38 -->_1 join#(e,nil(),q) -> c_19():28 -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)):7 10:S:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):12 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):11 -->_2 inBlock#(e,nil()) -> c_13():25 -->_2 inBlock#(e,dd(p,P)) -> c_12(or#(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) ,and#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e) ,inBlock#(e,P)):4 11:S:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_1 kruskal#(nil(),F,P) -> c_23():29 -->_2 join#(e,nil(),q) -> c_19():28 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):10 -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)):7 12:S:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(nil(),F,P) -> c_23():29 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):10 13:S:leq#(s(n),s(m)) -> c_29(leq#(n,m)) -->_1 leq#(s(n),0()) -> c_28():32 -->_1 leq#(0(),s(m)) -> c_27():31 -->_1 leq#(0(),0()) -> c_26():30 -->_1 leq#(s(n),s(m)) -> c_29(leq#(n,m)):13 14:S:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(nil()) -> c_35():37 -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):14 15:S:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(nil(),q) -> c_37():38 -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):15 16:S:sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) -->_2 sort#(nil()) -> c_39():39 -->_1 insert#(e,nil()) -> c_15():26 -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):16 -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f)),wt#(e),wt#(f)):5 17:W:and#(false(),false()) -> c_1() 18:W:and#(false(),true()) -> c_2() 19:W:and#(true(),false()) -> c_3() 20:W:and#(true(),true()) -> c_4() 21:W:elem#(n,nil()) -> c_6() 22:W:eq#(0(),0()) -> c_7() 23:W:eq#(0(),s(m)) -> c_8() 24:W:eq#(s(n),0()) -> c_9() 25:W:inBlock#(e,nil()) -> c_13() 26:W:insert#(e,nil()) -> c_15() 27:W:insert#q#(true(),e,f,E) -> c_17() 28:W:join#(e,nil(),q) -> c_19() 29:W:kruskal#(nil(),F,P) -> c_23() 30:W:leq#(0(),0()) -> c_26() 31:W:leq#(0(),s(m)) -> c_27() 32:W:leq#(s(n),0()) -> c_28() 33:W:or#(false(),false()) -> c_30() 34:W:or#(false(),true()) -> c_31() 35:W:or#(true(),false()) -> c_32() 36:W:or#(true(),true()) -> c_33() 37:W:partitions#(nil()) -> c_35() 38:W:pp#(nil(),q) -> c_37() 39:W:sort#(nil()) -> c_39() 40:W:src#(edge(n,w,m)) -> c_40() 41:W:trg#(edge(n,w,m)) -> c_41() 42:W:wt#(edge(n,w,m)) -> c_42() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 17: and#(false(),false()) -> c_1() 18: and#(false(),true()) -> c_2() 19: and#(true(),false()) -> c_3() 20: and#(true(),true()) -> c_4() 25: inBlock#(e,nil()) -> c_13() 40: src#(edge(n,w,m)) -> c_40() 41: trg#(edge(n,w,m)) -> c_41() 38: pp#(nil(),q) -> c_37() 28: join#(e,nil(),q) -> c_19() 29: kruskal#(nil(),F,P) -> c_23() 37: partitions#(nil()) -> c_35() 27: insert#q#(true(),e,f,E) -> c_17() 42: wt#(edge(n,w,m)) -> c_42() 30: leq#(0(),0()) -> c_26() 31: leq#(0(),s(m)) -> c_27() 32: leq#(s(n),0()) -> c_28() 26: insert#(e,nil()) -> c_15() 39: sort#(nil()) -> c_39() 21: elem#(n,nil()) -> c_6() 33: or#(false(),false()) -> c_30() 34: or#(false(),true()) -> c_31() 35: or#(true(),false()) -> c_32() 36: or#(true(),true()) -> c_33() 22: eq#(0(),0()) -> c_7() 23: eq#(0(),s(m)) -> c_8() 24: eq#(s(n),0()) -> c_9() * Step 5: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)) eq#(s(n),s(m)) -> c_10(eq#(n,m)) forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(or#(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) ,and#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e) ,inBlock#(e,P)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f)),wt#(e),wt#(f)) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/3,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/7,c_13/0,c_14/4,c_15/0,c_16/1,c_17/0,c_18/6,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)) -->_2 eq#(s(n),s(m)) -> c_10(eq#(n,m)):2 -->_3 elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)):1 2:S:eq#(s(n),s(m)) -> c_10(eq#(n,m)) -->_1 eq#(s(n),s(m)) -> c_10(eq#(n,m)):2 3:S:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):16 -->_3 partitions#(dd(n,N)) -> c_34(partitions#(N)):14 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):10 4:S:inBlock#(e,dd(p,P)) -> c_12(or#(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) ,and#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e) ,inBlock#(e,P)) -->_7 inBlock#(e,dd(p,P)) -> c_12(or#(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) ,and#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e) ,inBlock#(e,P)):4 -->_5 elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)):1 -->_3 elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)):1 5:S:insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f)),wt#(e),wt#(f)) -->_2 leq#(s(n),s(m)) -> c_29(leq#(n,m)):13 -->_1 insert#q#(false(),e,f,E) -> c_16(insert#(e,E)):6 6:S:insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f)),wt#(e),wt#(f)):5 7:S:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):9 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):8 -->_5 elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)):1 -->_3 elem#(n,dd(m,p)) -> c_5(or#(eq(n,m),elem(n,p)),eq#(n,m),elem#(n,p)):1 8:S:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)):7 9:S:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):15 -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)):7 10:S:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):12 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):11 -->_2 inBlock#(e,dd(p,P)) -> c_12(or#(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) ,and#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e) ,inBlock#(e,P)):4 11:S:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):10 -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,or#(elem(src(e),p),elem(trg(e),p)) ,elem#(src(e),p) ,src#(e) ,elem#(trg(e),p) ,trg#(e)):7 12:S:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):10 13:S:leq#(s(n),s(m)) -> c_29(leq#(n,m)) -->_1 leq#(s(n),s(m)) -> c_29(leq#(n,m)):13 14:S:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):14 15:S:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):15 16:S:sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):16 -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f)),wt#(e),wt#(f)):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) * Step 6: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) eq#(s(n),s(m)) -> c_10(eq#(n,m)) forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) eq#(s(n),s(m)) -> c_10(eq#(n,m)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2 ,or/2,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2 ,insert#q#/4,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1 ,trg#/1,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1 ,c_21/2,c_22/2,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0 ,c_36/1,c_37/0,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q# ,join#,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd ,edge,false,graph,nil,pair,s,true} Problem (S) - Strict DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak DPs: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) eq#(s(n),s(m)) -> c_10(eq#(n,m)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2 ,or/2,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2 ,insert#q#/4,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1 ,trg#/1,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1 ,c_21/2,c_22/2,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0 ,c_36/1,c_37/0,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q# ,join#,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd ,edge,false,graph,nil,pair,s,true} ** Step 6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) eq#(s(n),s(m)) -> c_10(eq#(n,m)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) -->_1 eq#(s(n),s(m)) -> c_10(eq#(n,m)):2 -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 2:S:eq#(s(n),s(m)) -> c_10(eq#(n,m)) -->_1 eq#(s(n),s(m)) -> c_10(eq#(n,m)):2 3:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):16 -->_3 partitions#(dd(n,N)) -> c_34(partitions#(N)):14 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):10 4:W:inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 -->_1 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 -->_3 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):4 5:W:insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) -->_2 leq#(s(n),s(m)) -> c_29(leq#(n,m)):13 -->_1 insert#q#(false(),e,f,E) -> c_16(insert#(e,E)):6 6:W:insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):5 7:W:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) -->_3 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):9 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):8 8:W:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):7 9:W:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):7 -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):15 10:W:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_2 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):4 -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):12 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):11 11:W:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):7 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):10 12:W:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):10 13:W:leq#(s(n),s(m)) -> c_29(leq#(n,m)) -->_1 leq#(s(n),s(m)) -> c_29(leq#(n,m)):13 14:W:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):14 15:W:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):15 16:W:sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):5 -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):16 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 15: pp#(dd(n,p),q) -> c_36(pp#(p,q)) 14: partitions#(dd(n,N)) -> c_34(partitions#(N)) 16: sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) 5: insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) 6: insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) 13: leq#(s(n),s(m)) -> c_29(leq#(n,m)) ** Step 6.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) eq#(s(n),s(m)) -> c_10(eq#(n,m)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) -->_1 eq#(s(n),s(m)) -> c_10(eq#(n,m)):2 -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 2:S:eq#(s(n),s(m)) -> c_10(eq#(n,m)) -->_1 eq#(s(n),s(m)) -> c_10(eq#(n,m)):2 3:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):10 4:W:inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 -->_1 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 -->_3 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):4 7:W:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) -->_3 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):9 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):8 8:W:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):7 9:W:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):7 10:W:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_2 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):4 -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):12 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):11 11:W:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):7 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):10 12:W:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):10 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) ** Step 6.a:3: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) eq#(s(n),s(m)) -> c_10(eq#(n,m)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) and a lower component elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) eq#(s(n),s(m)) -> c_10(eq#(n,m)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) Further, following extension rules are added to the lower component. forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) kruskal#(dd(e,E),F,P) -> inBlock#(e,P) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) *** Step 6.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) Consider the set of all dependency pairs 1: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) 2: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) 3: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) 4: kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2,3,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 6.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_22) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1} Following symbols are considered usable: {insert,insert#q,sort,and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join#,join#q#,kruskal#,kruskal#q# ,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} TcT has computed the following interpretation: p(0) = [2] p(and) = [2] x2 + [1] p(dd) = [1] x2 + [1] p(edge) = [1] x1 + [1] x2 + [5] p(elem) = [1] x1 + [2] x2 + [0] p(eq) = [0] p(false) = [0] p(forrest) = [0] p(graph) = [1] x2 + [0] p(inBlock) = [3] x1 + [0] p(insert) = [1] x2 + [4] p(insert#q) = [1] x4 + [5] p(join) = [2] x1 + [3] p(join#q) = [1] x1 + [0] p(kruskal) = [0] p(kruskal#q) = [0] p(leq) = [4] x2 + [0] p(nil) = [0] p(or) = [5] x2 + [1] p(pair) = [1] x1 + [1] x2 + [0] p(partitions) = [0] p(pp) = [5] x1 + [0] p(s) = [1] x1 + [0] p(sort) = [4] x1 + [0] p(src) = [6] p(trg) = [1] x1 + [2] p(true) = [0] p(wt) = [1] x1 + [2] p(and#) = [0] p(elem#) = [0] p(eq#) = [0] p(forrest#) = [4] x1 + [5] p(inBlock#) = [0] p(insert#) = [0] p(insert#q#) = [1] x3 + [0] p(join#) = [4] x2 + [3] p(join#q#) = [1] x1 + [1] x3 + [0] p(kruskal#) = [1] x1 + [0] p(kruskal#q#) = [1] x3 + [0] p(leq#) = [4] x1 + [1] x2 + [1] p(or#) = [1] x1 + [0] p(partitions#) = [4] p(pp#) = [1] x2 + [2] p(sort#) = [1] x1 + [2] p(src#) = [1] p(trg#) = [2] p(wt#) = [2] p(c_1) = [1] p(c_2) = [1] p(c_3) = [1] p(c_4) = [0] p(c_5) = [1] x2 + [0] p(c_6) = [1] p(c_7) = [1] p(c_8) = [1] p(c_9) = [1] p(c_10) = [1] x1 + [1] p(c_11) = [1] x1 + [5] p(c_12) = [1] x1 + [1] x3 + [0] p(c_13) = [0] p(c_14) = [2] x1 + [0] p(c_15) = [1] p(c_16) = [0] p(c_17) = [0] p(c_18) = [1] x2 + [4] p(c_19) = [4] p(c_20) = [1] x1 + [2] p(c_21) = [2] p(c_22) = [1] x1 + [0] p(c_23) = [0] p(c_24) = [1] x1 + [0] p(c_25) = [1] x1 + [0] p(c_26) = [4] p(c_27) = [0] p(c_28) = [1] p(c_29) = [0] p(c_30) = [4] p(c_31) = [2] p(c_32) = [2] p(c_33) = [0] p(c_34) = [4] x1 + [0] p(c_35) = [2] p(c_36) = [4] x1 + [1] p(c_37) = [1] p(c_38) = [1] x2 + [2] p(c_39) = [2] p(c_40) = [1] p(c_41) = [1] p(c_42) = [1] Following rules are strictly oriented: kruskal#(dd(e,E),F,P) = [1] E + [1] > [1] E + [0] = c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) Following rules are (at-least) weakly oriented: forrest#(graph(N,E)) = [4] E + [5] >= [4] E + [5] = c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#q#(false(),e,E,F,P) = [1] E + [0] >= [1] E + [0] = c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) = [1] E + [0] >= [1] E + [0] = c_25(kruskal#(E,F,P)) insert(e,dd(f,E)) = [1] E + [5] >= [1] E + [5] = insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) = [4] >= [1] = dd(e,nil()) insert#q(false(),e,f,E) = [1] E + [5] >= [1] E + [5] = dd(f,insert(e,E)) insert#q(true(),e,f,E) = [1] E + [5] >= [1] E + [2] = dd(e,dd(f,E)) sort(dd(e,E)) = [4] E + [4] >= [4] E + [4] = insert(e,sort(E)) sort(nil()) = [0] >= [0] = nil() **** Step 6.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):2 2:W:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):4 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):3 3:W:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):2 4:W:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) 2: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) 4: kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) 3: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) **** Step 6.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) eq#(s(n),s(m)) -> c_10(eq#(n,m)) - Weak DPs: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) kruskal#(dd(e,E),F,P) -> inBlock#(e,P) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: eq#(s(n),s(m)) -> c_10(eq#(n,m)) Consider the set of all dependency pairs 1: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) 2: eq#(s(n),s(m)) -> c_10(eq#(n,m)) 3: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) 4: inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) 5: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) 6: join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) 7: join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) 8: kruskal#(dd(e,E),F,P) -> inBlock#(e,P) 9: kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) 10: kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) 11: kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) 12: kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 6.a:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) eq#(s(n),s(m)) -> c_10(eq#(n,m)) - Weak DPs: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) kruskal#(dd(e,E),F,P) -> inBlock#(e,P) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_5) = {1,2}, uargs(c_10) = {1}, uargs(c_12) = {1,2,3}, uargs(c_18) = {1,2,3}, uargs(c_20) = {1}, uargs(c_21) = {1} Following symbols are considered usable: {join,join#q,partitions,pp,and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join#,join#q#,kruskal# ,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} TcT has computed the following interpretation: p(0) = [2] p(and) = [6] x1 + [5] p(dd) = [1] x1 + [1] x2 + [0] p(edge) = [1] p(elem) = [1] x1 + [1] x2 + [0] p(eq) = [4] x2 + [0] p(false) = [0] p(forrest) = [1] x1 + [2] p(graph) = [1] x1 + [1] p(inBlock) = [1] x1 + [0] p(insert) = [4] x2 + [0] p(insert#q) = [1] x2 + [2] x3 + [1] x4 + [2] p(join) = [1] x2 + [1] x3 + [0] p(join#q) = [1] x3 + [1] x4 + [1] x5 + [0] p(kruskal) = [2] x3 + [0] p(kruskal#q) = [4] p(leq) = [0] p(nil) = [0] p(or) = [1] x1 + [0] p(pair) = [0] p(partitions) = [1] x1 + [0] p(pp) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [2] p(sort) = [1] x1 + [0] p(src) = [1] p(trg) = [2] p(true) = [0] p(wt) = [1] x1 + [1] p(and#) = [2] x1 + [0] p(elem#) = [1] x2 + [0] p(eq#) = [1] x2 + [0] p(forrest#) = [4] x1 + [2] p(inBlock#) = [2] x2 + [0] p(insert#) = [1] x1 + [4] x2 + [1] p(insert#q#) = [1] x2 + [1] x4 + [2] p(join#) = [4] x2 + [0] p(join#q#) = [4] x4 + [0] p(kruskal#) = [4] x3 + [0] p(kruskal#q#) = [4] x5 + [0] p(leq#) = [1] p(or#) = [2] x2 + [1] p(partitions#) = [2] p(pp#) = [1] x2 + [0] p(sort#) = [0] p(src#) = [2] x1 + [0] p(trg#) = [1] p(wt#) = [2] x1 + [0] p(c_1) = [2] p(c_2) = [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [2] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [4] p(c_12) = [1] x1 + [1] x2 + [1] x3 + [0] p(c_13) = [0] p(c_14) = [4] x1 + [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [1] p(c_18) = [1] x1 + [2] x2 + [2] x3 + [0] p(c_19) = [2] p(c_20) = [1] x1 + [0] p(c_21) = [1] x1 + [0] p(c_22) = [1] x1 + [4] x2 + [1] p(c_23) = [1] p(c_24) = [2] x1 + [1] x2 + [0] p(c_25) = [4] x1 + [1] p(c_26) = [1] p(c_27) = [0] p(c_28) = [0] p(c_29) = [1] x1 + [2] p(c_30) = [1] p(c_31) = [1] p(c_32) = [1] p(c_33) = [2] p(c_34) = [1] x1 + [0] p(c_35) = [0] p(c_36) = [2] p(c_37) = [4] p(c_38) = [2] x2 + [0] p(c_39) = [0] p(c_40) = [1] p(c_41) = [0] p(c_42) = [1] Following rules are strictly oriented: eq#(s(n),s(m)) = [1] m + [2] > [1] m + [0] = c_10(eq#(n,m)) Following rules are (at-least) weakly oriented: elem#(n,dd(m,p)) = [1] m + [1] p + [0] >= [1] m + [1] p + [0] = c_5(eq#(n,m),elem#(n,p)) forrest#(graph(N,E)) = [4] N + [6] >= [4] N + [0] = kruskal#(sort(E),nil(),partitions(N)) inBlock#(e,dd(p,P)) = [2] P + [2] p + [0] >= [2] P + [2] p + [0] = c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) = [4] P + [4] p + [0] >= [4] P + [4] p + [0] = c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q),elem#(src(e),p),elem#(trg(e),p)) join#q#(false(),e,p,P,q) = [4] P + [0] >= [4] P + [0] = c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) = [4] P + [0] >= [4] P + [0] = c_21(join#(e,P,pp(p,q))) kruskal#(dd(e,E),F,P) = [4] P + [0] >= [2] P + [0] = inBlock#(e,P) kruskal#(dd(e,E),F,P) = [4] P + [0] >= [4] P + [0] = kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) = [4] P + [0] >= [4] P + [0] = join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) = [4] P + [0] >= [4] P + [0] = kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) = [4] P + [0] >= [4] P + [0] = kruskal#(E,F,P) join(e,dd(p,P),q) = [1] P + [1] p + [1] q + [0] >= [1] P + [1] p + [1] q + [0] = join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) = [1] q + [0] >= [1] q + [0] = dd(q,nil()) join#q(false(),e,p,P,q) = [1] P + [1] p + [1] q + [0] >= [1] P + [1] p + [1] q + [0] = dd(p,join(e,P,q)) join#q(true(),e,p,P,q) = [1] P + [1] p + [1] q + [0] >= [1] P + [1] p + [1] q + [0] = join(e,P,pp(p,q)) partitions(dd(n,N)) = [1] N + [1] n + [0] >= [1] N + [1] n + [0] = dd(dd(n,nil()),partitions(N)) partitions(nil()) = [0] >= [0] = nil() pp(dd(n,p),q) = [1] n + [1] p + [1] q + [0] >= [1] n + [1] p + [1] q + [0] = dd(n,pp(p,q)) pp(nil(),q) = [1] q + [0] >= [1] q + [0] = q **** Step 6.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) - Weak DPs: eq#(s(n),s(m)) -> c_10(eq#(n,m)) forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) kruskal#(dd(e,E),F,P) -> inBlock#(e,P) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) - Weak DPs: eq#(s(n),s(m)) -> c_10(eq#(n,m)) forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) kruskal#(dd(e,E),F,P) -> inBlock#(e,P) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) -->_1 eq#(s(n),s(m)) -> c_10(eq#(n,m)):2 -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 2:W:eq#(s(n),s(m)) -> c_10(eq#(n,m)) -->_1 eq#(s(n),s(m)) -> c_10(eq#(n,m)):2 3:W:forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) -->_1 kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P):9 -->_1 kruskal#(dd(e,E),F,P) -> inBlock#(e,P):8 4:W:inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) -->_3 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):4 -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 -->_1 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 5:W:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))):7 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):6 -->_3 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 6:W:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):5 7:W:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):5 8:W:kruskal#(dd(e,E),F,P) -> inBlock#(e,P) -->_1 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):4 9:W:kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) -->_1 kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P):12 -->_1 kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())):11 -->_1 kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()):10 10:W:kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):5 11:W:kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P):9 -->_1 kruskal#(dd(e,E),F,P) -> inBlock#(e,P):8 12:W:kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) -->_1 kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P):9 -->_1 kruskal#(dd(e,E),F,P) -> inBlock#(e,P):8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: eq#(s(n),s(m)) -> c_10(eq#(n,m)) **** Step 6.a:3.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) - Weak DPs: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) kruskal#(dd(e,E),F,P) -> inBlock#(e,P) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 3:W:forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) -->_1 kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P):9 -->_1 kruskal#(dd(e,E),F,P) -> inBlock#(e,P):8 4:W:inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) -->_3 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):4 -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 -->_1 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 5:W:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))):7 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):6 -->_3 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):1 6:W:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):5 7:W:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):5 8:W:kruskal#(dd(e,E),F,P) -> inBlock#(e,P) -->_1 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):4 9:W:kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) -->_1 kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P):12 -->_1 kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())):11 -->_1 kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()):10 10:W:kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):5 11:W:kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P):9 -->_1 kruskal#(dd(e,E),F,P) -> inBlock#(e,P):8 12:W:kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) -->_1 kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P):9 -->_1 kruskal#(dd(e,E),F,P) -> inBlock#(e,P):8 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: elem#(n,dd(m,p)) -> c_5(elem#(n,p)) **** Step 6.a:3.b:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: elem#(n,dd(m,p)) -> c_5(elem#(n,p)) - Weak DPs: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) kruskal#(dd(e,E),F,P) -> inBlock#(e,P) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: elem#(n,dd(m,p)) -> c_5(elem#(n,p)) Consider the set of all dependency pairs 1: elem#(n,dd(m,p)) -> c_5(elem#(n,p)) 2: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) 3: inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) 4: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) 5: join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) 6: join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) 7: kruskal#(dd(e,E),F,P) -> inBlock#(e,P) 8: kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) 9: kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) 10: kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) 11: kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 6.a:3.b:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: elem#(n,dd(m,p)) -> c_5(elem#(n,p)) - Weak DPs: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) kruskal#(dd(e,E),F,P) -> inBlock#(e,P) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_12) = {1,2,3}, uargs(c_18) = {1,2,3}, uargs(c_20) = {1}, uargs(c_21) = {1} Following symbols are considered usable: {inBlock,insert,insert#q,join,join#q,or,partitions,pp,sort,and#,elem#,eq#,forrest#,inBlock#,insert# ,insert#q#,join#,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} TcT has computed the following interpretation: p(0) = [0] p(and) = [4] x2 + [5] p(dd) = [1] x1 + [1] x2 + [1] p(edge) = [1] x3 + [0] p(elem) = [1] x1 + [3] x2 + [0] p(eq) = [4] p(false) = [1] p(forrest) = [1] x1 + [0] p(graph) = [1] x1 + [1] x2 + [0] p(inBlock) = [2] p(insert) = [1] x1 + [1] x2 + [1] p(insert#q) = [1] x2 + [1] x3 + [1] x4 + [2] p(join) = [1] x2 + [1] x3 + [1] p(join#q) = [1] x3 + [1] x4 + [1] x5 + [2] p(kruskal) = [1] x3 + [1] p(kruskal#q) = [1] x1 + [1] x4 + [2] p(leq) = [2] x1 + [4] p(nil) = [0] p(or) = [1] x2 + [0] p(pair) = [1] x1 + [1] x2 + [1] p(partitions) = [2] x1 + [0] p(pp) = [1] x1 + [1] x2 + [1] p(s) = [1] p(sort) = [1] x1 + [0] p(src) = [4] p(trg) = [0] p(true) = [1] p(wt) = [1] p(and#) = [1] x1 + [1] p(elem#) = [1] x2 + [0] p(eq#) = [2] x2 + [0] p(forrest#) = [7] x1 + [5] p(inBlock#) = [2] x2 + [4] p(insert#) = [4] x1 + [1] x2 + [0] p(insert#q#) = [1] x2 + [4] x4 + [1] p(join#) = [2] x2 + [0] p(join#q#) = [2] x4 + [0] p(kruskal#) = [7] x1 + [1] x2 + [2] x3 + [3] p(kruskal#q#) = [4] x1 + [1] x2 + [7] x3 + [1] x4 + [2] x5 + [2] p(leq#) = [2] x1 + [0] p(or#) = [1] p(partitions#) = [1] p(pp#) = [4] x2 + [1] p(sort#) = [1] x1 + [1] p(src#) = [2] p(trg#) = [4] x1 + [1] p(wt#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] p(c_7) = [2] p(c_8) = [1] p(c_9) = [0] p(c_10) = [4] p(c_11) = [4] x1 + [1] p(c_12) = [1] x1 + [1] x2 + [1] x3 + [2] p(c_13) = [4] p(c_14) = [2] x1 + [4] x2 + [0] p(c_15) = [1] p(c_16) = [4] x1 + [1] p(c_17) = [0] p(c_18) = [1] x1 + [1] x2 + [1] x3 + [0] p(c_19) = [1] p(c_20) = [1] x1 + [0] p(c_21) = [1] x1 + [0] p(c_22) = [1] x1 + [1] x2 + [1] p(c_23) = [0] p(c_24) = [4] x1 + [1] x2 + [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [1] p(c_28) = [4] p(c_29) = [1] x1 + [1] p(c_30) = [1] p(c_31) = [4] p(c_32) = [1] p(c_33) = [0] p(c_34) = [0] p(c_35) = [0] p(c_36) = [0] p(c_37) = [4] p(c_38) = [2] x1 + [0] p(c_39) = [1] p(c_40) = [0] p(c_41) = [0] p(c_42) = [1] Following rules are strictly oriented: elem#(n,dd(m,p)) = [1] m + [1] p + [1] > [1] p + [0] = c_5(elem#(n,p)) Following rules are (at-least) weakly oriented: forrest#(graph(N,E)) = [7] E + [7] N + [5] >= [7] E + [4] N + [3] = kruskal#(sort(E),nil(),partitions(N)) inBlock#(e,dd(p,P)) = [2] P + [2] p + [6] >= [2] P + [2] p + [6] = c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) = [2] P + [2] p + [2] >= [2] P + [2] p + [0] = c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q),elem#(src(e),p),elem#(trg(e),p)) join#q#(false(),e,p,P,q) = [2] P + [0] >= [2] P + [0] = c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) = [2] P + [0] >= [2] P + [0] = c_21(join#(e,P,pp(p,q))) kruskal#(dd(e,E),F,P) = [7] E + [1] F + [2] P + [7] e + [10] >= [2] P + [4] = inBlock#(e,P) kruskal#(dd(e,E),F,P) = [7] E + [1] F + [2] P + [7] e + [10] >= [7] E + [1] F + [2] P + [1] e + [10] = kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) = [7] E + [1] F + [2] P + [1] e + [6] >= [2] P + [0] = join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) = [7] E + [1] F + [2] P + [1] e + [6] >= [7] E + [1] F + [2] P + [1] e + [6] = kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) = [7] E + [1] F + [2] P + [1] e + [6] >= [7] E + [1] F + [2] P + [3] = kruskal#(E,F,P) inBlock(e,dd(p,P)) = [2] >= [2] = or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) = [2] >= [1] = false() insert(e,dd(f,E)) = [1] E + [1] e + [1] f + [2] >= [1] E + [1] e + [1] f + [2] = insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) = [1] e + [1] >= [1] e + [1] = dd(e,nil()) insert#q(false(),e,f,E) = [1] E + [1] e + [1] f + [2] >= [1] E + [1] e + [1] f + [2] = dd(f,insert(e,E)) insert#q(true(),e,f,E) = [1] E + [1] e + [1] f + [2] >= [1] E + [1] e + [1] f + [2] = dd(e,dd(f,E)) join(e,dd(p,P),q) = [1] P + [1] p + [1] q + [2] >= [1] P + [1] p + [1] q + [2] = join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) = [1] q + [1] >= [1] q + [1] = dd(q,nil()) join#q(false(),e,p,P,q) = [1] P + [1] p + [1] q + [2] >= [1] P + [1] p + [1] q + [2] = dd(p,join(e,P,q)) join#q(true(),e,p,P,q) = [1] P + [1] p + [1] q + [2] >= [1] P + [1] p + [1] q + [2] = join(e,P,pp(p,q)) or(false(),false()) = [1] >= [1] = false() or(false(),true()) = [1] >= [1] = true() or(true(),false()) = [1] >= [1] = true() or(true(),true()) = [1] >= [1] = true() partitions(dd(n,N)) = [2] N + [2] n + [2] >= [2] N + [1] n + [2] = dd(dd(n,nil()),partitions(N)) partitions(nil()) = [0] >= [0] = nil() pp(dd(n,p),q) = [1] n + [1] p + [1] q + [2] >= [1] n + [1] p + [1] q + [2] = dd(n,pp(p,q)) pp(nil(),q) = [1] q + [1] >= [1] q + [0] = q sort(dd(e,E)) = [1] E + [1] e + [1] >= [1] E + [1] e + [1] = insert(e,sort(E)) sort(nil()) = [0] >= [0] = nil() ***** Step 6.a:3.b:1.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: elem#(n,dd(m,p)) -> c_5(elem#(n,p)) forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) kruskal#(dd(e,E),F,P) -> inBlock#(e,P) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.a:3.b:1.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: elem#(n,dd(m,p)) -> c_5(elem#(n,p)) forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) kruskal#(dd(e,E),F,P) -> inBlock#(e,P) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:elem#(n,dd(m,p)) -> c_5(elem#(n,p)) -->_1 elem#(n,dd(m,p)) -> c_5(elem#(n,p)):1 2:W:forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) -->_1 kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P):8 -->_1 kruskal#(dd(e,E),F,P) -> inBlock#(e,P):7 3:W:inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) -->_3 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):3 -->_2 elem#(n,dd(m,p)) -> c_5(elem#(n,p)):1 -->_1 elem#(n,dd(m,p)) -> c_5(elem#(n,p)):1 4:W:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))):6 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):5 -->_3 elem#(n,dd(m,p)) -> c_5(elem#(n,p)):1 -->_2 elem#(n,dd(m,p)) -> c_5(elem#(n,p)):1 5:W:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):4 6:W:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):4 7:W:kruskal#(dd(e,E),F,P) -> inBlock#(e,P) -->_1 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):3 8:W:kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) -->_1 kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P):11 -->_1 kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())):10 -->_1 kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()):9 9:W:kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):4 10:W:kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P):8 -->_1 kruskal#(dd(e,E),F,P) -> inBlock#(e,P):7 11:W:kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) -->_1 kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P):8 -->_1 kruskal#(dd(e,E),F,P) -> inBlock#(e,P):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) 8: kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) 11: kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) 10: kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) 9: kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) 4: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) 6: join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q))) 5: join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) 7: kruskal#(dd(e,E),F,P) -> inBlock#(e,P) 3: inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) 1: elem#(n,dd(m,p)) -> c_5(elem#(n,p)) ***** Step 6.a:3.b:1.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/1,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak DPs: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) eq#(s(n),s(m)) -> c_10(eq#(n,m)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):14 -->_3 partitions#(dd(n,N)) -> c_34(partitions#(N)):12 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):8 2:S:inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):15 -->_1 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):15 -->_3 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):2 3:S:insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) -->_2 leq#(s(n),s(m)) -> c_29(leq#(n,m)):11 -->_1 insert#q#(false(),e,f,E) -> c_16(insert#(e,E)):4 4:S:insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):3 5:S:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) -->_3 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):15 -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):15 -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):7 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):6 6:S:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):5 7:S:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):13 -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):5 8:S:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):10 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):9 -->_2 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):2 9:S:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):8 -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):5 10:S:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):8 11:S:leq#(s(n),s(m)) -> c_29(leq#(n,m)) -->_1 leq#(s(n),s(m)) -> c_29(leq#(n,m)):11 12:S:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):12 13:S:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):13 14:S:sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):14 -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):3 15:W:elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) -->_1 eq#(s(n),s(m)) -> c_10(eq#(n,m)):16 -->_2 elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)):15 16:W:eq#(s(n),s(m)) -> c_10(eq#(n,m)) -->_1 eq#(s(n),s(m)) -> c_10(eq#(n,m)):16 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 15: elem#(n,dd(m,p)) -> c_5(eq#(n,m),elem#(n,p)) 16: eq#(s(n),s(m)) -> c_10(eq#(n,m)) ** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/3,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/3,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):14 -->_3 partitions#(dd(n,N)) -> c_34(partitions#(N)):12 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):8 2:S:inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)) -->_3 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):2 3:S:insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) -->_2 leq#(s(n),s(m)) -> c_29(leq#(n,m)):11 -->_1 insert#q#(false(),e,f,E) -> c_16(insert#(e,E)):4 4:S:insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):3 5:S:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):7 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):6 6:S:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):5 7:S:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):13 -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):5 8:S:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):10 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):9 -->_2 inBlock#(e,dd(p,P)) -> c_12(elem#(src(e),p),elem#(trg(e),p),inBlock#(e,P)):2 9:S:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):8 -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) ,elem#(src(e),p) ,elem#(trg(e),p)):5 10:S:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):8 11:S:leq#(s(n),s(m)) -> c_29(leq#(n,m)) -->_1 leq#(s(n),s(m)) -> c_29(leq#(n,m)):11 12:S:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):12 13:S:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):13 14:S:sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):14 -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) ** Step 6.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak DPs: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2 ,or/2,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2 ,insert#q#/4,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1 ,trg#/1,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1 ,c_21/2,c_22/2,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0 ,c_36/1,c_37/0,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q# ,join#,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd ,edge,false,graph,nil,pair,s,true} Problem (S) - Strict DPs: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2 ,or/2,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2 ,insert#q#/4,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1 ,trg#/1,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/3,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1 ,c_21/2,c_22/2,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0 ,c_36/1,c_37/0,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q# ,join#,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd ,edge,false,graph,nil,pair,s,true} *** Step 6.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak DPs: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):14 -->_3 partitions#(dd(n,N)) -> c_34(partitions#(N)):12 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):8 2:W:inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) -->_1 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):2 3:S:insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) -->_2 leq#(s(n),s(m)) -> c_29(leq#(n,m)):11 -->_1 insert#q#(false(),e,f,E) -> c_16(insert#(e,E)):4 4:S:insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):3 5:W:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):7 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):6 6:W:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):5 7:W:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):5 -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):13 8:W:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_2 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):2 -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):10 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):9 9:W:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):5 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):8 10:W:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):8 11:S:leq#(s(n),s(m)) -> c_29(leq#(n,m)) -->_1 leq#(s(n),s(m)) -> c_29(leq#(n,m)):11 12:W:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):12 13:W:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):13 14:S:sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):3 -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):14 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) 10: kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) 9: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) 2: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) 5: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) 7: join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) 6: join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) 13: pp#(dd(n,p),q) -> c_36(pp#(p,q)) 12: partitions#(dd(n,N)) -> c_34(partitions#(N)) *** Step 6.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):14 3:S:insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) -->_2 leq#(s(n),s(m)) -> c_29(leq#(n,m)):11 -->_1 insert#q#(false(),e,f,E) -> c_16(insert#(e,E)):4 4:S:insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):3 11:S:leq#(s(n),s(m)) -> c_29(leq#(n,m)) -->_1 leq#(s(n),s(m)) -> c_29(leq#(n,m)):11 14:S:sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):3 -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):14 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: forrest#(graph(N,E)) -> c_11(sort#(E)) *** Step 6.b:3.a:3: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: forrest#(graph(N,E)) -> c_11(sort#(E)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() wt(edge(n,w,m)) -> w forrest#(graph(N,E)) -> c_11(sort#(E)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) *** Step 6.b:3.a:4: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: forrest#(graph(N,E)) -> c_11(sort#(E)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: forrest#(graph(N,E)) -> c_11(sort#(E)) 4: leq#(s(n),s(m)) -> c_29(leq#(n,m)) The strictly oriented rules are moved into the weak component. **** Step 6.b:3.a:4.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: forrest#(graph(N,E)) -> c_11(sort#(E)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_14) = {1,2}, uargs(c_16) = {1}, uargs(c_29) = {1}, uargs(c_38) = {1,2} Following symbols are considered usable: {insert,insert#q,leq,sort,wt,and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join#,join#q#,kruskal# ,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} TcT has computed the following interpretation: p(0) = 1 p(and) = 0 p(dd) = x1 + x2 p(edge) = x2 p(elem) = 0 p(eq) = 0 p(false) = 1 p(forrest) = 0 p(graph) = x2 p(inBlock) = 0 p(insert) = x1 + x2 p(insert#q) = x2 + x3 + x4 p(join) = 0 p(join#q) = 0 p(kruskal) = 0 p(kruskal#q) = 0 p(leq) = x2 p(nil) = 0 p(or) = 0 p(pair) = 0 p(partitions) = 0 p(pp) = 0 p(s) = 1 + x1 p(sort) = x1 p(src) = 0 p(trg) = 0 p(true) = 0 p(wt) = x1 p(and#) = 0 p(elem#) = 0 p(eq#) = 0 p(forrest#) = 1 + x1 + x1^2 p(inBlock#) = 0 p(insert#) = x1 + x1*x2 + x1^2 p(insert#q#) = x1*x2 + x2*x4 + x2^2 p(join#) = 0 p(join#q#) = 0 p(kruskal#) = 0 p(kruskal#q#) = 0 p(leq#) = x1 p(or#) = 0 p(partitions#) = 0 p(pp#) = 0 p(sort#) = x1 + x1^2 p(src#) = 0 p(trg#) = 0 p(wt#) = 0 p(c_1) = 0 p(c_2) = 0 p(c_3) = 0 p(c_4) = 0 p(c_5) = 0 p(c_6) = 0 p(c_7) = 0 p(c_8) = 0 p(c_9) = 0 p(c_10) = 0 p(c_11) = x1 p(c_12) = 0 p(c_13) = 0 p(c_14) = x1 + x2 p(c_15) = 0 p(c_16) = x1 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = 0 p(c_21) = 0 p(c_22) = 0 p(c_23) = 0 p(c_24) = 0 p(c_25) = 0 p(c_26) = 0 p(c_27) = 0 p(c_28) = 0 p(c_29) = x1 p(c_30) = 0 p(c_31) = 0 p(c_32) = 0 p(c_33) = 0 p(c_34) = 0 p(c_35) = 0 p(c_36) = 0 p(c_37) = 0 p(c_38) = x1 + x2 p(c_39) = 0 p(c_40) = 0 p(c_41) = 0 p(c_42) = 0 Following rules are strictly oriented: forrest#(graph(N,E)) = 1 + E + E^2 > E + E^2 = c_11(sort#(E)) leq#(s(n),s(m)) = 1 + n > n = c_29(leq#(n,m)) Following rules are (at-least) weakly oriented: insert#(e,dd(f,E)) = E*e + e + e*f + e^2 >= E*e + e + e*f + e^2 = c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) = E*e + e + e^2 >= E*e + e + e^2 = c_16(insert#(e,E)) sort#(dd(e,E)) = E + 2*E*e + E^2 + e + e^2 >= E + E*e + E^2 + e + e^2 = c_38(insert#(e,sort(E)),sort#(E)) insert(e,dd(f,E)) = E + e + f >= E + e + f = insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) = e >= e = dd(e,nil()) insert#q(false(),e,f,E) = E + e + f >= E + e + f = dd(f,insert(e,E)) insert#q(true(),e,f,E) = E + e + f >= E + e + f = dd(e,dd(f,E)) leq(0(),0()) = 1 >= 0 = true() leq(0(),s(m)) = 1 + m >= 0 = true() leq(s(n),0()) = 1 >= 1 = false() leq(s(n),s(m)) = 1 + m >= m = leq(n,m) sort(dd(e,E)) = E + e >= E + e = insert(e,sort(E)) sort(nil()) = 0 >= 0 = nil() wt(edge(n,w,m)) = w >= w = w **** Step 6.b:3.a:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak DPs: forrest#(graph(N,E)) -> c_11(sort#(E)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) - Weak TRS: insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.b:3.a:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak DPs: forrest#(graph(N,E)) -> c_11(sort#(E)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) - Weak TRS: insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) -->_2 leq#(s(n),s(m)) -> c_29(leq#(n,m)):5 -->_1 insert#q#(false(),e,f,E) -> c_16(insert#(e,E)):2 2:S:insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):1 3:S:sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):3 -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):1 4:W:forrest#(graph(N,E)) -> c_11(sort#(E)) -->_1 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):3 5:W:leq#(s(n),s(m)) -> c_29(leq#(n,m)) -->_1 leq#(s(n),s(m)) -> c_29(leq#(n,m)):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: leq#(s(n),s(m)) -> c_29(leq#(n,m)) **** Step 6.b:3.a:4.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak DPs: forrest#(graph(N,E)) -> c_11(sort#(E)) - Weak TRS: insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) -->_1 insert#q#(false(),e,f,E) -> c_16(insert#(e,E)):2 2:S:insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):1 3:S:sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):3 -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):1 4:W:forrest#(graph(N,E)) -> c_11(sort#(E)) -->_1 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E)) **** Step 6.b:3.a:4.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E)) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak DPs: forrest#(graph(N,E)) -> c_11(sort#(E)) - Weak TRS: insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) 3: sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) Consider the set of all dependency pairs 1: insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E)) 2: insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) 3: sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) 4: forrest#(graph(N,E)) -> c_11(sort#(E)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {2,3} These cover all (indirect) predecessors of dependency pairs {1,2,3,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 6.b:3.a:4.b:3.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E)) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak DPs: forrest#(graph(N,E)) -> c_11(sort#(E)) - Weak TRS: insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1}, uargs(c_38) = {1,2} Following symbols are considered usable: {insert,insert#q,leq,sort,and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join#,join#q#,kruskal# ,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} TcT has computed the following interpretation: p(0) = 0 p(and) = 0 p(dd) = 1 + x2 p(edge) = 0 p(elem) = 0 p(eq) = 0 p(false) = 1 p(forrest) = 0 p(graph) = x2 p(inBlock) = 0 p(insert) = 1 + x2 p(insert#q) = 1 + x1 + x1*x4 p(join) = 0 p(join#q) = 0 p(kruskal) = 0 p(kruskal#q) = 0 p(leq) = 1 p(nil) = 0 p(or) = 0 p(pair) = 0 p(partitions) = 0 p(pp) = 0 p(s) = 0 p(sort) = x1 p(src) = 0 p(trg) = 0 p(true) = 1 p(wt) = 0 p(and#) = 0 p(elem#) = 0 p(eq#) = 0 p(forrest#) = x1 + x1^2 p(inBlock#) = 0 p(insert#) = x2 p(insert#q#) = 1 + x4 p(join#) = 0 p(join#q#) = 0 p(kruskal#) = 0 p(kruskal#q#) = 0 p(leq#) = 0 p(or#) = 0 p(partitions#) = 0 p(pp#) = 0 p(sort#) = x1 + x1^2 p(src#) = 0 p(trg#) = 0 p(wt#) = 0 p(c_1) = 0 p(c_2) = 0 p(c_3) = 0 p(c_4) = 0 p(c_5) = 0 p(c_6) = 0 p(c_7) = 0 p(c_8) = 0 p(c_9) = 0 p(c_10) = 0 p(c_11) = x1 p(c_12) = 0 p(c_13) = 0 p(c_14) = x1 p(c_15) = 0 p(c_16) = x1 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = 0 p(c_21) = 0 p(c_22) = 0 p(c_23) = 0 p(c_24) = 0 p(c_25) = 0 p(c_26) = 0 p(c_27) = 0 p(c_28) = 0 p(c_29) = 0 p(c_30) = 0 p(c_31) = 0 p(c_32) = 0 p(c_33) = 0 p(c_34) = 0 p(c_35) = 0 p(c_36) = 0 p(c_37) = 0 p(c_38) = x1 + x2 p(c_39) = 0 p(c_40) = 0 p(c_41) = 0 p(c_42) = 0 Following rules are strictly oriented: insert#q#(false(),e,f,E) = 1 + E > E = c_16(insert#(e,E)) sort#(dd(e,E)) = 2 + 3*E + E^2 > 2*E + E^2 = c_38(insert#(e,sort(E)),sort#(E)) Following rules are (at-least) weakly oriented: forrest#(graph(N,E)) = E + E^2 >= E + E^2 = c_11(sort#(E)) insert#(e,dd(f,E)) = 1 + E >= 1 + E = c_14(insert#q#(leq(wt(e),wt(f)),e,f,E)) insert(e,dd(f,E)) = 2 + E >= 2 + E = insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) = 1 >= 1 = dd(e,nil()) insert#q(false(),e,f,E) = 2 + E >= 2 + E = dd(f,insert(e,E)) insert#q(true(),e,f,E) = 2 + E >= 2 + E = dd(e,dd(f,E)) leq(0(),0()) = 1 >= 1 = true() leq(0(),s(m)) = 1 >= 1 = true() leq(s(n),0()) = 1 >= 1 = false() leq(s(n),s(m)) = 1 >= 1 = leq(n,m) sort(dd(e,E)) = 1 + E >= 1 + E = insert(e,sort(E)) sort(nil()) = 0 >= 0 = nil() ***** Step 6.b:3.a:4.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E)) - Weak DPs: forrest#(graph(N,E)) -> c_11(sort#(E)) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.b:3.a:4.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: forrest#(graph(N,E)) -> c_11(sort#(E)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E)) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:forrest#(graph(N,E)) -> c_11(sort#(E)) -->_1 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):4 2:W:insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E)) -->_1 insert#q#(false(),e,f,E) -> c_16(insert#(e,E)):3 3:W:insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E)):2 4:W:sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):4 -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: forrest#(graph(N,E)) -> c_11(sort#(E)) 4: sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) 2: insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E)) 3: insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) ***** Step 6.b:3.a:4.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) leq#(s(n),s(m)) -> c_29(leq#(n,m)) sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) -->_1 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):1 2:S:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):4 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):3 3:S:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):2 4:S:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):9 -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):2 5:S:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):7 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):6 -->_2 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):1 6:S:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):5 -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):2 7:S:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):5 8:S:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):8 9:S:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):9 10:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):14 -->_3 partitions#(dd(n,N)) -> c_34(partitions#(N)):8 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):5 11:W:insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) -->_2 leq#(s(n),s(m)) -> c_29(leq#(n,m)):13 -->_1 insert#q#(false(),e,f,E) -> c_16(insert#(e,E)):12 12:W:insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):11 13:W:leq#(s(n),s(m)) -> c_29(leq#(n,m)) -->_1 leq#(s(n),s(m)) -> c_29(leq#(n,m)):13 14:W:sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) -->_2 sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)):14 -->_1 insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))):11 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: sort#(dd(e,E)) -> c_38(insert#(e,sort(E)),sort#(E)) 11: insert#(e,dd(f,E)) -> c_14(insert#q#(leq(wt(e),wt(f)),e,f,E),leq#(wt(e),wt(f))) 12: insert#q#(false(),e,f,E) -> c_16(insert#(e,E)) 13: leq#(s(n),s(m)) -> c_29(leq#(n,m)) *** Step 6.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/3,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) -->_1 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):1 2:S:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):4 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):3 3:S:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):2 4:S:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):9 -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):2 5:S:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):7 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):6 -->_2 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):1 6:S:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):5 -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):2 7:S:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):5 8:S:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):8 9:S:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):9 10:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),sort#(E),partitions#(N)) -->_3 partitions#(dd(n,N)) -> c_34(partitions#(N)):8 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) *** Step 6.b:3.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2 ,or/2,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2 ,insert#q#/4,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1 ,trg#/1,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1 ,c_21/2,c_22/2,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0 ,c_36/1,c_37/0,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q# ,join#,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd ,edge,false,graph,nil,pair,s,true} Problem (S) - Strict DPs: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2 ,or/2,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2 ,insert#q#/4,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1 ,trg#/1,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1 ,c_21/2,c_22/2,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0 ,c_36/1,c_37/0,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q# ,join#,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd ,edge,false,graph,nil,pair,s,true} **** Step 6.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) -->_1 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):1 2:W:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):4 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):3 3:W:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):2 4:W:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):2 -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):9 5:W:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_2 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):1 -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):7 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):6 6:W:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):2 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):5 7:W:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):5 8:W:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):8 9:W:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):9 10:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):5 -->_2 partitions#(dd(n,N)) -> c_34(partitions#(N)):8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: partitions#(dd(n,N)) -> c_34(partitions#(N)) 2: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) 4: join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) 3: join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) 9: pp#(dd(n,p),q) -> c_36(pp#(p,q)) **** Step 6.b:3.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) -->_1 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):1 5:W:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_2 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):1 -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):7 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):6 6:W:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):5 7:W:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):5 10:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) **** Step 6.b:3.b:3.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) Consider the set of all dependency pairs 1: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) 2: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) 3: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) 4: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) 5: kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 6.b:3.b:3.a:3.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_12) = {1}, uargs(c_22) = {1,2}, uargs(c_24) = {1}, uargs(c_25) = {1} Following symbols are considered usable: {insert,insert#q,join,join#q,leq,or,partitions,sort,and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q# ,join#,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} TcT has computed the following interpretation: p(0) = 0 p(and) = x1^2 p(dd) = 1 + x2 p(edge) = x1 p(elem) = x2^2 p(eq) = 0 p(false) = 1 p(forrest) = 0 p(graph) = 1 + x1 + x2 p(inBlock) = x1 p(insert) = 1 + x2 p(insert#q) = 1 + x1 + x4 p(join) = 1 + x2 p(join#q) = 1 + x1*x4 + x1^2 p(kruskal) = 0 p(kruskal#q) = 0 p(leq) = 1 p(nil) = 1 p(or) = 1 p(pair) = 0 p(partitions) = 1 + x1 p(pp) = x1 + x1*x2 + x1^2 + x2^2 p(s) = 0 p(sort) = x1 p(src) = x1 p(trg) = x1 p(true) = 1 p(wt) = 0 p(and#) = 0 p(elem#) = 0 p(eq#) = 0 p(forrest#) = x1^2 p(inBlock#) = x2 p(insert#) = 0 p(insert#q#) = 0 p(join#) = 0 p(join#q#) = 0 p(kruskal#) = x1*x3 + x1^2 + x3 p(kruskal#q#) = 1 + x3 + x3*x5 + x3^2 + x5 p(leq#) = 0 p(or#) = 0 p(partitions#) = 0 p(pp#) = 0 p(sort#) = 0 p(src#) = 0 p(trg#) = 0 p(wt#) = 0 p(c_1) = 0 p(c_2) = 0 p(c_3) = 0 p(c_4) = 0 p(c_5) = 0 p(c_6) = 0 p(c_7) = 0 p(c_8) = 0 p(c_9) = 0 p(c_10) = 0 p(c_11) = x1 p(c_12) = x1 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = 0 p(c_21) = 0 p(c_22) = x1 + x2 p(c_23) = 0 p(c_24) = x1 p(c_25) = x1 p(c_26) = 0 p(c_27) = 0 p(c_28) = 0 p(c_29) = 0 p(c_30) = 0 p(c_31) = 0 p(c_32) = 0 p(c_33) = 0 p(c_34) = 0 p(c_35) = 0 p(c_36) = 0 p(c_37) = 0 p(c_38) = 0 p(c_39) = 0 p(c_40) = 0 p(c_41) = 0 p(c_42) = 0 Following rules are strictly oriented: inBlock#(e,dd(p,P)) = 1 + P > P = c_12(inBlock#(e,P)) Following rules are (at-least) weakly oriented: forrest#(graph(N,E)) = 1 + 2*E + 2*E*N + E^2 + 2*N + N^2 >= 1 + E + E*N + E^2 + N = c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) = 1 + 2*E + E*P + E^2 + 2*P >= 1 + E + E*P + E^2 + 2*P = c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) = 1 + E + E*P + E^2 + P >= 1 + E + E*P + E^2 + P = c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) = 1 + E + E*P + E^2 + P >= E*P + E^2 + P = c_25(kruskal#(E,F,P)) insert(e,dd(f,E)) = 2 + E >= 2 + E = insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) = 2 >= 2 = dd(e,nil()) insert#q(false(),e,f,E) = 2 + E >= 2 + E = dd(f,insert(e,E)) insert#q(true(),e,f,E) = 2 + E >= 2 + E = dd(e,dd(f,E)) join(e,dd(p,P),q) = 2 + P >= 2 + P = join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) = 2 >= 2 = dd(q,nil()) join#q(false(),e,p,P,q) = 2 + P >= 2 + P = dd(p,join(e,P,q)) join#q(true(),e,p,P,q) = 2 + P >= 1 + P = join(e,P,pp(p,q)) leq(0(),0()) = 1 >= 1 = true() leq(0(),s(m)) = 1 >= 1 = true() leq(s(n),0()) = 1 >= 1 = false() leq(s(n),s(m)) = 1 >= 1 = leq(n,m) or(false(),false()) = 1 >= 1 = false() or(false(),true()) = 1 >= 1 = true() or(true(),false()) = 1 >= 1 = true() or(true(),true()) = 1 >= 1 = true() partitions(dd(n,N)) = 2 + N >= 2 + N = dd(dd(n,nil()),partitions(N)) partitions(nil()) = 2 >= 1 = nil() sort(dd(e,E)) = 1 + E >= 1 + E = insert(e,sort(E)) sort(nil()) = 1 >= 1 = nil() ***** Step 6.b:3.b:3.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.b:3.b:3.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):3 2:W:inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) -->_1 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):2 3:W:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):5 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))):4 -->_2 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):2 4:W:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):3 5:W:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) 3: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) 5: kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) 4: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) 2: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) ***** Step 6.b:3.b:3.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 6.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):3 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):2 2:S:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):1 3:S:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):8 -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):1 4:S:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_2 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):10 -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):6 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):5 5:S:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):4 -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):1 6:S:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):4 7:S:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):7 8:S:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):8 9:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) -->_2 partitions#(dd(n,N)) -> c_34(partitions#(N)):7 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):4 10:W:inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) -->_1 inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)):10 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: inBlock#(e,dd(p,P)) -> c_12(inBlock#(e,P)) **** Step 6.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/2 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):3 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):2 2:S:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):1 3:S:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):8 -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):1 4:S:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):6 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):5 5:S:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):4 -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):1 6:S:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):4 7:S:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):7 8:S:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):8 9:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) -->_2 partitions#(dd(n,N)) -> c_34(partitions#(N)):7 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P),inBlock#(e,P)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) **** Step 6.b:3.b:3.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2 ,or/2,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2 ,insert#q#/4,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1 ,trg#/1,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1 ,c_21/2,c_22/1,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0 ,c_36/1,c_37/0,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q# ,join#,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd ,edge,false,graph,nil,pair,s,true} Problem (S) - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2 ,or/2,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2 ,insert#q#/4,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1 ,trg#/1,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1 ,c_21/2,c_22/1,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0 ,c_36/1,c_37/0,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q# ,join#,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd ,edge,false,graph,nil,pair,s,true} ***** Step 6.b:3.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):3 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):2 2:S:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):1 3:S:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):1 -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):8 4:W:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):6 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):5 5:W:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):1 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):4 6:W:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):4 7:W:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):7 8:S:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):8 9:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):4 -->_2 partitions#(dd(n,N)) -> c_34(partitions#(N)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: partitions#(dd(n,N)) -> c_34(partitions#(N)) ***** Step 6.b:3.b:3.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):3 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):2 2:S:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):1 3:S:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):1 -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):8 4:W:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):6 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):5 5:W:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):1 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):4 6:W:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):4 8:S:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):8 9:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) ***** Step 6.b:3.b:3.b:3.a:3: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) and a lower component join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) Further, following extension rules are added to the lower component. forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) ****** Step 6.b:3.b:3.b:3.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) Consider the set of all dependency pairs 1: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) 2: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) 3: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) 4: kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******* Step 6.b:3.b:3.b:3.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_22) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1} Following symbols are considered usable: {insert,insert#q,sort,and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join#,join#q#,kruskal#,kruskal#q# ,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} TcT has computed the following interpretation: p(0) = [1] p(and) = [4] x2 + [0] p(dd) = [1] x2 + [2] p(edge) = [1] x1 + [1] x3 + [2] p(elem) = [1] x1 + [2] x2 + [0] p(eq) = [4] x1 + [4] x2 + [6] p(false) = [0] p(forrest) = [1] x1 + [0] p(graph) = [1] x1 + [1] x2 + [2] p(inBlock) = [4] x2 + [0] p(insert) = [1] x2 + [2] p(insert#q) = [1] x4 + [4] p(join) = [0] p(join#q) = [2] x1 + [1] x3 + [2] x4 + [0] p(kruskal) = [1] x1 + [2] x2 + [2] x3 + [2] p(kruskal#q) = [2] x3 + [0] p(leq) = [2] x1 + [4] p(nil) = [0] p(or) = [3] x2 + [0] p(pair) = [1] x2 + [1] p(partitions) = [1] x1 + [2] p(pp) = [0] p(s) = [0] p(sort) = [1] x1 + [0] p(src) = [2] p(trg) = [1] x1 + [2] p(true) = [0] p(wt) = [2] x1 + [5] p(and#) = [1] x2 + [0] p(elem#) = [1] p(eq#) = [1] x2 + [0] p(forrest#) = [4] x1 + [7] p(inBlock#) = [1] x1 + [1] p(insert#) = [2] p(insert#q#) = [2] x3 + [4] x4 + [0] p(join#) = [2] x3 + [0] p(join#q#) = [1] x1 + [4] p(kruskal#) = [4] x1 + [0] p(kruskal#q#) = [4] x3 + [1] p(leq#) = [4] p(or#) = [1] x2 + [2] p(partitions#) = [0] p(pp#) = [2] x2 + [1] p(sort#) = [0] p(src#) = [1] p(trg#) = [1] p(wt#) = [4] x1 + [0] p(c_1) = [1] p(c_2) = [1] p(c_3) = [1] p(c_4) = [2] p(c_5) = [2] x1 + [2] x2 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] p(c_9) = [1] p(c_10) = [0] p(c_11) = [1] x1 + [4] p(c_12) = [2] x1 + [1] p(c_13) = [2] p(c_14) = [2] x1 + [0] p(c_15) = [0] p(c_16) = [2] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [1] p(c_21) = [1] x1 + [1] p(c_22) = [1] x1 + [7] p(c_23) = [0] p(c_24) = [1] x1 + [4] x2 + [0] p(c_25) = [1] x1 + [1] p(c_26) = [1] p(c_27) = [0] p(c_28) = [1] p(c_29) = [1] x1 + [0] p(c_30) = [4] p(c_31) = [4] p(c_32) = [1] p(c_33) = [0] p(c_34) = [1] x1 + [0] p(c_35) = [1] p(c_36) = [0] p(c_37) = [1] p(c_38) = [0] p(c_39) = [0] p(c_40) = [2] p(c_41) = [1] p(c_42) = [4] Following rules are strictly oriented: kruskal#q#(false(),e,E,F,P) = [4] E + [1] > [4] E + [0] = c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) Following rules are (at-least) weakly oriented: forrest#(graph(N,E)) = [4] E + [4] N + [15] >= [4] E + [4] = c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) = [4] E + [8] >= [4] E + [8] = c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(true(),e,E,F,P) = [4] E + [1] >= [4] E + [1] = c_25(kruskal#(E,F,P)) insert(e,dd(f,E)) = [1] E + [4] >= [1] E + [4] = insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) = [2] >= [2] = dd(e,nil()) insert#q(false(),e,f,E) = [1] E + [4] >= [1] E + [4] = dd(f,insert(e,E)) insert#q(true(),e,f,E) = [1] E + [4] >= [1] E + [4] = dd(e,dd(f,E)) sort(dd(e,E)) = [1] E + [2] >= [1] E + [2] = insert(e,sort(E)) sort(nil()) = [0] >= [0] = nil() ******* Step 6.b:3.b:3.b:3.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 6.b:3.b:3.b:3.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):2 2:W:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):4 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):3 3:W:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):2 4:W:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) 2: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) 4: kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) 3: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) ******* Step 6.b:3.b:3.b:3.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ****** Step 6.b:3.b:3.b:3.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) 4: pp#(dd(n,p),q) -> c_36(pp#(p,q)) Consider the set of all dependency pairs 1: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) 2: join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) 3: join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) 4: pp#(dd(n,p),q) -> c_36(pp#(p,q)) 5: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) 6: kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) 7: kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) 8: kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) 9: kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1,4} These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******* Step 6.b:3.b:3.b:3.a:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_18) = {1}, uargs(c_20) = {1}, uargs(c_21) = {1,2}, uargs(c_36) = {1} Following symbols are considered usable: {insert,insert#q,join,join#q,partitions,pp,sort,and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} TcT has computed the following interpretation: p(0) = [6] p(and) = [0] p(dd) = [1] x1 + [1] x2 + [1] p(edge) = [0] p(elem) = [2] x1 + [5] p(eq) = [2] x2 + [1] p(false) = [0] p(forrest) = [1] p(graph) = [1] x1 + [1] x2 + [4] p(inBlock) = [0] p(insert) = [1] x1 + [1] x2 + [1] p(insert#q) = [1] x2 + [1] x3 + [1] x4 + [2] p(join) = [1] x2 + [1] x3 + [1] p(join#q) = [1] x3 + [1] x4 + [1] x5 + [2] p(kruskal) = [1] x1 + [2] x2 + [4] x3 + [0] p(kruskal#q) = [1] x1 + [1] x2 + [1] x4 + [0] p(leq) = [1] x1 + [1] x2 + [2] p(nil) = [0] p(or) = [0] p(pair) = [2] p(partitions) = [2] x1 + [2] p(pp) = [1] x1 + [1] x2 + [0] p(s) = [6] p(sort) = [1] x1 + [0] p(src) = [7] x1 + [1] p(trg) = [1] x1 + [0] p(true) = [0] p(wt) = [6] x1 + [6] p(and#) = [0] p(elem#) = [0] p(eq#) = [1] x1 + [1] x2 + [2] p(forrest#) = [2] x1 + [0] p(inBlock#) = [0] p(insert#) = [1] x1 + [0] p(insert#q#) = [1] x4 + [0] p(join#) = [1] x2 + [0] p(join#q#) = [1] x3 + [1] x4 + [0] p(kruskal#) = [1] x1 + [1] x3 + [0] p(kruskal#q#) = [1] x3 + [1] x5 + [1] p(leq#) = [1] x1 + [2] x2 + [0] p(or#) = [1] p(partitions#) = [0] p(pp#) = [1] x1 + [0] p(sort#) = [2] x1 + [0] p(src#) = [0] p(trg#) = [1] p(wt#) = [1] x1 + [0] p(c_1) = [2] p(c_2) = [0] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] p(c_6) = [1] p(c_7) = [1] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [1] p(c_11) = [1] x1 + [0] p(c_12) = [4] p(c_13) = [2] p(c_14) = [4] x2 + [1] p(c_15) = [0] p(c_16) = [2] x1 + [4] p(c_17) = [0] p(c_18) = [1] x1 + [0] p(c_19) = [1] p(c_20) = [1] x1 + [0] p(c_21) = [1] x1 + [1] x2 + [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [1] x1 + [1] p(c_25) = [4] p(c_26) = [2] p(c_27) = [0] p(c_28) = [1] p(c_29) = [0] p(c_30) = [1] p(c_31) = [2] p(c_32) = [2] p(c_33) = [1] p(c_34) = [1] p(c_35) = [0] p(c_36) = [1] x1 + [0] p(c_37) = [1] p(c_38) = [2] x1 + [0] p(c_39) = [2] p(c_40) = [4] p(c_41) = [1] p(c_42) = [2] Following rules are strictly oriented: join#(e,dd(p,P),q) = [1] P + [1] p + [1] > [1] P + [1] p + [0] = c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) pp#(dd(n,p),q) = [1] n + [1] p + [1] > [1] p + [0] = c_36(pp#(p,q)) Following rules are (at-least) weakly oriented: forrest#(graph(N,E)) = [2] E + [2] N + [8] >= [1] E + [2] N + [2] = kruskal#(sort(E),nil(),partitions(N)) join#q#(false(),e,p,P,q) = [1] P + [1] p + [0] >= [1] P + [0] = c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) = [1] P + [1] p + [0] >= [1] P + [1] p + [0] = c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) = [1] E + [1] P + [1] e + [1] >= [1] E + [1] P + [1] = kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) = [1] E + [1] P + [1] >= [1] P + [0] = join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) = [1] E + [1] P + [1] >= [1] E + [1] P + [1] = kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) = [1] E + [1] P + [1] >= [1] E + [1] P + [0] = kruskal#(E,F,P) insert(e,dd(f,E)) = [1] E + [1] e + [1] f + [2] >= [1] E + [1] e + [1] f + [2] = insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) = [1] e + [1] >= [1] e + [1] = dd(e,nil()) insert#q(false(),e,f,E) = [1] E + [1] e + [1] f + [2] >= [1] E + [1] e + [1] f + [2] = dd(f,insert(e,E)) insert#q(true(),e,f,E) = [1] E + [1] e + [1] f + [2] >= [1] E + [1] e + [1] f + [2] = dd(e,dd(f,E)) join(e,dd(p,P),q) = [1] P + [1] p + [1] q + [2] >= [1] P + [1] p + [1] q + [2] = join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) = [1] q + [1] >= [1] q + [1] = dd(q,nil()) join#q(false(),e,p,P,q) = [1] P + [1] p + [1] q + [2] >= [1] P + [1] p + [1] q + [2] = dd(p,join(e,P,q)) join#q(true(),e,p,P,q) = [1] P + [1] p + [1] q + [2] >= [1] P + [1] p + [1] q + [1] = join(e,P,pp(p,q)) partitions(dd(n,N)) = [2] N + [2] n + [4] >= [2] N + [1] n + [4] = dd(dd(n,nil()),partitions(N)) partitions(nil()) = [2] >= [0] = nil() pp(dd(n,p),q) = [1] n + [1] p + [1] q + [1] >= [1] n + [1] p + [1] q + [1] = dd(n,pp(p,q)) pp(nil(),q) = [1] q + [0] >= [1] q + [0] = q sort(dd(e,E)) = [1] E + [1] e + [1] >= [1] E + [1] e + [1] = insert(e,sort(E)) sort(nil()) = [0] >= [0] = nil() ******* Step 6.b:3.b:3.b:3.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) - Weak DPs: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 6.b:3.b:3.b:3.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) -->_1 kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P):5 2:W:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):4 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):3 3:W:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):2 4:W:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):9 -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):2 5:W:kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) -->_1 kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P):8 -->_1 kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())):7 -->_1 kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()):6 6:W:kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):2 7:W:kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P):5 8:W:kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) -->_1 kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P):5 9:W:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):9 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: forrest#(graph(N,E)) -> kruskal#(sort(E),nil(),partitions(N)) 5: kruskal#(dd(e,E),F,P) -> kruskal#q#(inBlock(e,P),e,E,F,P) 8: kruskal#q#(true(),e,E,F,P) -> kruskal#(E,F,P) 7: kruskal#q#(false(),e,E,F,P) -> kruskal#(E,dd(e,F),join(e,P,nil())) 6: kruskal#q#(false(),e,E,F,P) -> join#(e,P,nil()) 2: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) 4: join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) 3: join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) 9: pp#(dd(n,p),q) -> c_36(pp#(p,q)) ******* Step 6.b:3.b:3.b:3.a:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ***** Step 6.b:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) pp#(dd(n,p),q) -> c_36(pp#(p,q)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):3 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):2 2:S:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_2 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):6 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):1 3:S:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):1 4:S:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):4 5:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) -->_2 partitions#(dd(n,N)) -> c_34(partitions#(N)):4 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):1 6:W:join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) -->_1 join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)):8 -->_1 join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)):7 7:W:join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):6 8:W:join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) -->_2 pp#(dd(n,p),q) -> c_36(pp#(p,q)):9 -->_1 join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)):6 9:W:pp#(dd(n,p),q) -> c_36(pp#(p,q)) -->_1 pp#(dd(n,p),q) -> c_36(pp#(p,q)):9 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: join#(e,dd(p,P),q) -> c_18(join#q#(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q)) 8: join#q#(true(),e,p,P,q) -> c_21(join#(e,P,pp(p,q)),pp#(p,q)) 7: join#q#(false(),e,p,P,q) -> c_20(join#(e,P,q)) 9: pp#(dd(n,p),q) -> c_36(pp#(p,q)) ***** Step 6.b:3.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/2,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):3 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())):2 2:S:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil())),join#(e,P,nil())) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):1 3:S:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):1 4:S:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):4 5:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) -->_2 partitions#(dd(n,N)) -> c_34(partitions#(N)):4 -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) ***** Step 6.b:3.b:3.b:3.b:3: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2 ,or/2,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2 ,insert#q#/4,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1 ,trg#/1,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1 ,c_21/2,c_22/1,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0 ,c_36/1,c_37/0,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q# ,join#,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd ,edge,false,graph,nil,pair,s,true} Problem (S) - Strict DPs: partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2 ,or/2,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2 ,insert#q#/4,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1 ,trg#/1,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1 ,c_21/2,c_22/1,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0 ,c_36/1,c_37/0,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q# ,join#,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd ,edge,false,graph,nil,pair,s,true} ****** Step 6.b:3.b:3.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):3 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))):2 2:S:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):1 3:S:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):1 4:W:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):4 5:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):1 -->_2 partitions#(dd(n,N)) -> c_34(partitions#(N)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: partitions#(dd(n,N)) -> c_34(partitions#(N)) ****** Step 6.b:3.b:3.b:3.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):3 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))):2 2:S:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):1 3:S:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):1 5:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) ****** Step 6.b:3.b:3.b:3.b:3.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) Consider the set of all dependency pairs 1: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) 2: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) 3: kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) 4: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {2,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******* Step 6.b:3.b:3.b:3.b:3.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_22) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1} Following symbols are considered usable: {insert,insert#q,sort,and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join#,join#q#,kruskal#,kruskal#q# ,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} TcT has computed the following interpretation: p(0) = [0] p(and) = [4] p(dd) = [1] x2 + [1] p(edge) = [1] x2 + [1] p(elem) = [1] x1 + [1] x2 + [0] p(eq) = [0] p(false) = [0] p(forrest) = [4] x1 + [1] p(graph) = [1] x1 + [1] x2 + [2] p(inBlock) = [6] x1 + [0] p(insert) = [1] x2 + [1] p(insert#q) = [1] x4 + [2] p(join) = [2] x1 + [4] x3 + [0] p(join#q) = [3] x2 + [5] x3 + [1] x4 + [0] p(kruskal) = [2] x1 + [2] x2 + [1] p(kruskal#q) = [1] x3 + [1] x4 + [2] x5 + [0] p(leq) = [2] x1 + [3] p(nil) = [0] p(or) = [2] x2 + [0] p(pair) = [4] p(partitions) = [1] p(pp) = [2] p(s) = [1] p(sort) = [1] x1 + [0] p(src) = [0] p(trg) = [6] x1 + [4] p(true) = [0] p(wt) = [4] x1 + [3] p(and#) = [1] x2 + [4] p(elem#) = [1] p(eq#) = [1] x1 + [2] x2 + [0] p(forrest#) = [4] x1 + [3] p(inBlock#) = [2] x1 + [2] x2 + [2] p(insert#) = [1] x1 + [1] p(insert#q#) = [1] x4 + [2] p(join#) = [4] x2 + [1] p(join#q#) = [2] x2 + [1] x3 + [1] x4 + [1] x5 + [1] p(kruskal#) = [1] x1 + [6] p(kruskal#q#) = [1] x3 + [7] p(leq#) = [1] x2 + [0] p(or#) = [0] p(partitions#) = [2] x1 + [1] p(pp#) = [2] p(sort#) = [1] p(src#) = [4] x1 + [1] p(trg#) = [1] x1 + [2] p(wt#) = [1] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] p(c_4) = [2] p(c_5) = [1] x1 + [0] p(c_6) = [2] p(c_7) = [2] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] p(c_11) = [1] x1 + [5] p(c_12) = [1] x1 + [2] p(c_13) = [2] p(c_14) = [1] x1 + [2] x2 + [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [1] p(c_18) = [1] x1 + [1] p(c_19) = [1] p(c_20) = [0] p(c_21) = [2] x1 + [4] x2 + [1] p(c_22) = [1] x1 + [0] p(c_23) = [1] p(c_24) = [1] x1 + [0] p(c_25) = [1] x1 + [1] p(c_26) = [1] p(c_27) = [0] p(c_28) = [4] p(c_29) = [1] p(c_30) = [0] p(c_31) = [0] p(c_32) = [1] p(c_33) = [0] p(c_34) = [1] x1 + [4] p(c_35) = [0] p(c_36) = [1] x1 + [1] p(c_37) = [1] p(c_38) = [0] p(c_39) = [1] p(c_40) = [2] p(c_41) = [0] p(c_42) = [1] Following rules are strictly oriented: kruskal#q#(false(),e,E,F,P) = [1] E + [7] > [1] E + [6] = c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) Following rules are (at-least) weakly oriented: forrest#(graph(N,E)) = [4] E + [4] N + [11] >= [1] E + [11] = c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) = [1] E + [7] >= [1] E + [7] = c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(true(),e,E,F,P) = [1] E + [7] >= [1] E + [7] = c_25(kruskal#(E,F,P)) insert(e,dd(f,E)) = [1] E + [2] >= [1] E + [2] = insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) = [1] >= [1] = dd(e,nil()) insert#q(false(),e,f,E) = [1] E + [2] >= [1] E + [2] = dd(f,insert(e,E)) insert#q(true(),e,f,E) = [1] E + [2] >= [1] E + [2] = dd(e,dd(f,E)) sort(dd(e,E)) = [1] E + [1] >= [1] E + [1] = insert(e,sort(E)) sort(nil()) = [0] >= [0] = nil() ******* Step 6.b:3.b:3.b:3.b:3.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 6.b:3.b:3.b:3.b:3.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) 2: kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) Consider the set of all dependency pairs 1: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) 2: kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) 3: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) 4: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1,2} These cover all (indirect) predecessors of dependency pairs {1,2,3,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******** Step 6.b:3.b:3.b:3.b:3.a:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_22) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1} Following symbols are considered usable: {insert,insert#q,sort,and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join#,join#q#,kruskal#,kruskal#q# ,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} TcT has computed the following interpretation: p(0) = [4] p(and) = [3] x1 + [6] x2 + [0] p(dd) = [1] x2 + [1] p(edge) = [1] x1 + [3] p(elem) = [1] x1 + [1] x2 + [0] p(eq) = [0] p(false) = [0] p(forrest) = [1] x1 + [2] p(graph) = [1] x1 + [1] x2 + [1] p(inBlock) = [0] p(insert) = [1] x2 + [1] p(insert#q) = [1] x4 + [2] p(join) = [0] p(join#q) = [0] p(kruskal) = [1] x1 + [1] x2 + [2] x3 + [1] p(kruskal#q) = [1] x3 + [1] p(leq) = [1] x1 + [1] x2 + [0] p(nil) = [0] p(or) = [1] p(pair) = [1] x2 + [1] p(partitions) = [4] p(pp) = [2] p(s) = [1] x1 + [0] p(sort) = [1] x1 + [0] p(src) = [3] p(trg) = [2] x1 + [1] p(true) = [0] p(wt) = [4] x1 + [0] p(and#) = [0] p(elem#) = [0] p(eq#) = [1] x1 + [1] x2 + [1] p(forrest#) = [5] x1 + [1] p(inBlock#) = [1] p(insert#) = [4] x1 + [0] p(insert#q#) = [2] x1 + [2] x2 + [1] x3 + [0] p(join#) = [1] p(join#q#) = [4] x3 + [1] x5 + [0] p(kruskal#) = [4] x1 + [0] p(kruskal#q#) = [4] x3 + [1] p(leq#) = [1] x1 + [1] p(or#) = [1] x2 + [1] p(partitions#) = [1] x1 + [2] p(pp#) = [2] x1 + [2] p(sort#) = [1] p(src#) = [1] p(trg#) = [1] p(wt#) = [4] p(c_1) = [4] p(c_2) = [0] p(c_3) = [2] p(c_4) = [0] p(c_5) = [1] x1 + [4] p(c_6) = [1] p(c_7) = [1] p(c_8) = [1] p(c_9) = [2] p(c_10) = [4] x1 + [1] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] p(c_16) = [1] x1 + [0] p(c_17) = [1] p(c_18) = [0] p(c_19) = [0] p(c_20) = [1] p(c_21) = [4] p(c_22) = [1] x1 + [2] p(c_23) = [4] p(c_24) = [1] x1 + [0] p(c_25) = [1] x1 + [0] p(c_26) = [0] p(c_27) = [1] p(c_28) = [0] p(c_29) = [0] p(c_30) = [1] p(c_31) = [0] p(c_32) = [2] p(c_33) = [0] p(c_34) = [4] x1 + [2] p(c_35) = [0] p(c_36) = [4] x1 + [0] p(c_37) = [1] p(c_38) = [4] x1 + [1] x2 + [1] p(c_39) = [4] p(c_40) = [4] p(c_41) = [2] p(c_42) = [0] Following rules are strictly oriented: kruskal#(dd(e,E),F,P) = [4] E + [4] > [4] E + [3] = c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(true(),e,E,F,P) = [4] E + [1] > [4] E + [0] = c_25(kruskal#(E,F,P)) Following rules are (at-least) weakly oriented: forrest#(graph(N,E)) = [5] E + [5] N + [6] >= [4] E + [0] = c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#q#(false(),e,E,F,P) = [4] E + [1] >= [4] E + [0] = c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) insert(e,dd(f,E)) = [1] E + [2] >= [1] E + [2] = insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) = [1] >= [1] = dd(e,nil()) insert#q(false(),e,f,E) = [1] E + [2] >= [1] E + [2] = dd(f,insert(e,E)) insert#q(true(),e,f,E) = [1] E + [2] >= [1] E + [2] = dd(e,dd(f,E)) sort(dd(e,E)) = [1] E + [1] >= [1] E + [1] = insert(e,sort(E)) sort(nil()) = [0] >= [0] = nil() ******** Step 6.b:3.b:3.b:3.b:3.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******** Step 6.b:3.b:3.b:3.b:3.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):2 2:W:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):4 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))):3 3:W:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):2 4:W:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N))) 2: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) 4: kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) 3: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) ******** Step 6.b:3.b:3.b:3.b:3.a:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ****** Step 6.b:3.b:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):1 2:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):3 -->_2 partitions#(dd(n,N)) -> c_34(partitions#(N)):1 3:W:kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) -->_1 kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)):5 -->_1 kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))):4 4:W:kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):3 5:W:kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) -->_1 kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: kruskal#(dd(e,E),F,P) -> c_22(kruskal#q#(inBlock(e,P),e,E,F,P)) 5: kruskal#q#(true(),e,E,F,P) -> c_25(kruskal#(E,F,P)) 4: kruskal#q#(false(),e,E,F,P) -> c_24(kruskal#(E,dd(e,F),join(e,P,nil()))) ****** Step 6.b:3.b:3.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak DPs: forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):1 2:W:forrest#(graph(N,E)) -> c_11(kruskal#(sort(E),nil(),partitions(N)),partitions#(N)) -->_2 partitions#(dd(n,N)) -> c_34(partitions#(N)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: forrest#(graph(N,E)) -> c_11(partitions#(N)) ****** Step 6.b:3.b:3.b:3.b:3.b:3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak DPs: forrest#(graph(N,E)) -> c_11(partitions#(N)) - Weak TRS: and(false(),false()) -> false() and(false(),true()) -> false() and(true(),false()) -> false() and(true(),true()) -> true() elem(n,dd(m,p)) -> or(eq(n,m),elem(n,p)) elem(n,nil()) -> false() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) inBlock(e,dd(p,P)) -> or(and(elem(src(e),p),elem(trg(e),p)),inBlock(e,P)) inBlock(e,nil()) -> false() insert(e,dd(f,E)) -> insert#q(leq(wt(e),wt(f)),e,f,E) insert(e,nil()) -> dd(e,nil()) insert#q(false(),e,f,E) -> dd(f,insert(e,E)) insert#q(true(),e,f,E) -> dd(e,dd(f,E)) join(e,dd(p,P),q) -> join#q(or(elem(src(e),p),elem(trg(e),p)),e,p,P,q) join(e,nil(),q) -> dd(q,nil()) join#q(false(),e,p,P,q) -> dd(p,join(e,P,q)) join#q(true(),e,p,P,q) -> join(e,P,pp(p,q)) leq(0(),0()) -> true() leq(0(),s(m)) -> true() leq(s(n),0()) -> false() leq(s(n),s(m)) -> leq(n,m) or(false(),false()) -> false() or(false(),true()) -> true() or(true(),false()) -> true() or(true(),true()) -> true() partitions(dd(n,N)) -> dd(dd(n,nil()),partitions(N)) partitions(nil()) -> nil() pp(dd(n,p),q) -> dd(n,pp(p,q)) pp(nil(),q) -> q sort(dd(e,E)) -> insert(e,sort(E)) sort(nil()) -> nil() src(edge(n,w,m)) -> n trg(edge(n,w,m)) -> m wt(edge(n,w,m)) -> w - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: forrest#(graph(N,E)) -> c_11(partitions#(N)) partitions#(dd(n,N)) -> c_34(partitions#(N)) ****** Step 6.b:3.b:3.b:3.b:3.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak DPs: forrest#(graph(N,E)) -> c_11(partitions#(N)) - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: partitions#(dd(n,N)) -> c_34(partitions#(N)) Consider the set of all dependency pairs 1: partitions#(dd(n,N)) -> c_34(partitions#(N)) 2: forrest#(graph(N,E)) -> c_11(partitions#(N)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******* Step 6.b:3.b:3.b:3.b:3.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: partitions#(dd(n,N)) -> c_34(partitions#(N)) - Weak DPs: forrest#(graph(N,E)) -> c_11(partitions#(N)) - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {1}, uargs(c_34) = {1} Following symbols are considered usable: {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join#,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions# ,pp#,sort#,src#,trg#,wt#} TcT has computed the following interpretation: p(0) = [0] p(and) = [4] p(dd) = [1] x1 + [1] x2 + [2] p(edge) = [1] x1 + [0] p(elem) = [1] x1 + [0] p(eq) = [0] p(false) = [0] p(forrest) = [0] p(graph) = [1] x1 + [1] x2 + [1] p(inBlock) = [0] p(insert) = [1] x2 + [0] p(insert#q) = [8] x1 + [0] p(join) = [8] x1 + [8] p(join#q) = [8] x1 + [2] x2 + [1] x3 + [8] x4 + [2] x5 + [1] p(kruskal) = [2] x1 + [2] x2 + [2] x3 + [0] p(kruskal#q) = [1] x1 + [1] x3 + [2] x5 + [1] p(leq) = [2] x1 + [8] x2 + [4] p(nil) = [0] p(or) = [1] x1 + [1] x2 + [0] p(pair) = [1] x1 + [0] p(partitions) = [0] p(pp) = [1] x1 + [0] p(s) = [0] p(sort) = [1] x1 + [0] p(src) = [0] p(trg) = [0] p(true) = [0] p(wt) = [2] x1 + [0] p(and#) = [0] p(elem#) = [0] p(eq#) = [1] x2 + [0] p(forrest#) = [9] x1 + [13] p(inBlock#) = [2] x1 + [0] p(insert#) = [1] x2 + [0] p(insert#q#) = [8] x2 + [1] x4 + [0] p(join#) = [8] x2 + [2] x3 + [0] p(join#q#) = [1] x1 + [1] x3 + [4] x5 + [0] p(kruskal#) = [1] x1 + [1] x2 + [1] x3 + [0] p(kruskal#q#) = [1] x1 + [1] x2 + [8] x3 + [1] x5 + [0] p(leq#) = [1] x1 + [1] x2 + [0] p(or#) = [1] x1 + [2] p(partitions#) = [1] x1 + [0] p(pp#) = [2] x2 + [0] p(sort#) = [0] p(src#) = [0] p(trg#) = [0] p(wt#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [2] x1 + [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [0] p(c_32) = [0] p(c_33) = [0] p(c_34) = [1] x1 + [0] p(c_35) = [0] p(c_36) = [0] p(c_37) = [0] p(c_38) = [0] p(c_39) = [0] p(c_40) = [0] p(c_41) = [0] p(c_42) = [0] Following rules are strictly oriented: partitions#(dd(n,N)) = [1] N + [1] n + [2] > [1] N + [0] = c_34(partitions#(N)) Following rules are (at-least) weakly oriented: forrest#(graph(N,E)) = [9] E + [9] N + [22] >= [2] N + [0] = c_11(partitions#(N)) ******* Step 6.b:3.b:3.b:3.b:3.b:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: forrest#(graph(N,E)) -> c_11(partitions#(N)) partitions#(dd(n,N)) -> c_34(partitions#(N)) - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 6.b:3.b:3.b:3.b:3.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: forrest#(graph(N,E)) -> c_11(partitions#(N)) partitions#(dd(n,N)) -> c_34(partitions#(N)) - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:forrest#(graph(N,E)) -> c_11(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):2 2:W:partitions#(dd(n,N)) -> c_34(partitions#(N)) -->_1 partitions#(dd(n,N)) -> c_34(partitions#(N)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: forrest#(graph(N,E)) -> c_11(partitions#(N)) 2: partitions#(dd(n,N)) -> c_34(partitions#(N)) ******* Step 6.b:3.b:3.b:3.b:3.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {and/2,elem/2,eq/2,forrest/1,inBlock/2,insert/2,insert#q/4,join/3,join#q/5,kruskal/3,kruskal#q/5,leq/2,or/2 ,partitions/1,pp/2,sort/1,src/1,trg/1,wt/1,and#/2,elem#/2,eq#/2,forrest#/1,inBlock#/2,insert#/2,insert#q#/4 ,join#/3,join#q#/5,kruskal#/3,kruskal#q#/5,leq#/2,or#/2,partitions#/1,pp#/2,sort#/1,src#/1,trg#/1 ,wt#/1} / {0/0,dd/2,edge/3,false/0,graph/2,nil/0,pair/2,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/1,c_21/2,c_22/1 ,c_23/0,c_24/1,c_25/1,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0,c_32/0,c_33/0,c_34/1,c_35/0,c_36/1,c_37/0 ,c_38/2,c_39/0,c_40/0,c_41/0,c_42/0} - Obligation: innermost runtime complexity wrt. defined symbols {and#,elem#,eq#,forrest#,inBlock#,insert#,insert#q#,join# ,join#q#,kruskal#,kruskal#q#,leq#,or#,partitions#,pp#,sort#,src#,trg#,wt#} and constructors {0,dd,edge,false ,graph,nil,pair,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))