WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: #abs(#0()) -> #0() #abs(#neg(x)) -> #pos(x) #abs(#pos(x)) -> #pos(x) #abs(#s(x)) -> #pos(#s(x)) #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0 ,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs,#cklt,#compare,#less,insert,insert#1,insert#2 ,insertD,insertD#1,insertD#2,insertionsort,insertionsort#1,insertionsortD ,insertionsortD#1} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs #abs#(#0()) -> c_1() #abs#(#neg(x)) -> c_2() #abs#(#pos(x)) -> c_3() #abs#(#s(x)) -> c_4() #cklt#(#EQ()) -> c_5() #cklt#(#GT()) -> c_6() #cklt#(#LT()) -> c_7() #compare#(#0(),#0()) -> c_8() #compare#(#0(),#neg(y)) -> c_9() #compare#(#0(),#pos(y)) -> c_10() #compare#(#0(),#s(y)) -> c_11() #compare#(#neg(x),#0()) -> c_12() #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#neg(x),#pos(y)) -> c_14() #compare#(#pos(x),#0()) -> c_15() #compare#(#pos(x),#neg(y)) -> c_16() #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) #compare#(#s(x),#0()) -> c_18() #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#cklt#(#compare(x,y)),#compare#(x,y)) insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) insert#1#(nil(),x) -> c_23() insert#2#(#false(),x,y,ys) -> c_24() insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) insertD#1#(nil(),x) -> c_28() insertD#2#(#false(),x,y,ys) -> c_29() insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)) insertionsort#1#(nil()) -> c_33() insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)) insertionsortD#1#(nil()) -> c_36() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: #abs#(#0()) -> c_1() #abs#(#neg(x)) -> c_2() #abs#(#pos(x)) -> c_3() #abs#(#s(x)) -> c_4() #cklt#(#EQ()) -> c_5() #cklt#(#GT()) -> c_6() #cklt#(#LT()) -> c_7() #compare#(#0(),#0()) -> c_8() #compare#(#0(),#neg(y)) -> c_9() #compare#(#0(),#pos(y)) -> c_10() #compare#(#0(),#s(y)) -> c_11() #compare#(#neg(x),#0()) -> c_12() #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#neg(x),#pos(y)) -> c_14() #compare#(#pos(x),#0()) -> c_15() #compare#(#pos(x),#neg(y)) -> c_16() #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) #compare#(#s(x),#0()) -> c_18() #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#cklt#(#compare(x,y)),#compare#(x,y)) insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) insert#1#(nil(),x) -> c_23() insert#2#(#false(),x,y,ys) -> c_24() insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) insertD#1#(nil(),x) -> c_28() insertD#2#(#false(),x,y,ys) -> c_29() insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)) insertionsort#1#(nil()) -> c_33() insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)) insertionsortD#1#(nil()) -> c_36() - Weak TRS: #abs(#0()) -> #0() #abs(#neg(x)) -> #pos(x) #abs(#pos(x)) -> #pos(x) #abs(#s(x)) -> #pos(#s(x)) #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/2,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,18,23,24,28,29,33,36} by application of Pre({1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,18,23,24,28,29,33,36}) = {13,17,19,20,21,22,26,27,31,34}. Here rules are labelled as follows: 1: #abs#(#0()) -> c_1() 2: #abs#(#neg(x)) -> c_2() 3: #abs#(#pos(x)) -> c_3() 4: #abs#(#s(x)) -> c_4() 5: #cklt#(#EQ()) -> c_5() 6: #cklt#(#GT()) -> c_6() 7: #cklt#(#LT()) -> c_7() 8: #compare#(#0(),#0()) -> c_8() 9: #compare#(#0(),#neg(y)) -> c_9() 10: #compare#(#0(),#pos(y)) -> c_10() 11: #compare#(#0(),#s(y)) -> c_11() 12: #compare#(#neg(x),#0()) -> c_12() 13: #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) 14: #compare#(#neg(x),#pos(y)) -> c_14() 15: #compare#(#pos(x),#0()) -> c_15() 16: #compare#(#pos(x),#neg(y)) -> c_16() 17: #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) 18: #compare#(#s(x),#0()) -> c_18() 19: #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) 20: #less#(x,y) -> c_20(#cklt#(#compare(x,y)),#compare#(x,y)) 21: insert#(x,l) -> c_21(insert#1#(l,x)) 22: insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) 23: insert#1#(nil(),x) -> c_23() 24: insert#2#(#false(),x,y,ys) -> c_24() 25: insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) 26: insertD#(x,l) -> c_26(insertD#1#(l,x)) 27: insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) 28: insertD#1#(nil(),x) -> c_28() 29: insertD#2#(#false(),x,y,ys) -> c_29() 30: insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) 31: insertionsort#(l) -> c_31(insertionsort#1#(l)) 32: insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)) 33: insertionsort#1#(nil()) -> c_33() 34: insertionsortD#(l) -> c_34(insertionsortD#1#(l)) 35: insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)) 36: insertionsortD#1#(nil()) -> c_36() * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#cklt#(#compare(x,y)),#compare#(x,y)) insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)) insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)) - Weak DPs: #abs#(#0()) -> c_1() #abs#(#neg(x)) -> c_2() #abs#(#pos(x)) -> c_3() #abs#(#s(x)) -> c_4() #cklt#(#EQ()) -> c_5() #cklt#(#GT()) -> c_6() #cklt#(#LT()) -> c_7() #compare#(#0(),#0()) -> c_8() #compare#(#0(),#neg(y)) -> c_9() #compare#(#0(),#pos(y)) -> c_10() #compare#(#0(),#s(y)) -> c_11() #compare#(#neg(x),#0()) -> c_12() #compare#(#neg(x),#pos(y)) -> c_14() #compare#(#pos(x),#0()) -> c_15() #compare#(#pos(x),#neg(y)) -> c_16() #compare#(#s(x),#0()) -> c_18() insert#1#(nil(),x) -> c_23() insert#2#(#false(),x,y,ys) -> c_24() insertD#1#(nil(),x) -> c_28() insertD#2#(#false(),x,y,ys) -> c_29() insertionsort#1#(nil()) -> c_33() insertionsortD#1#(nil()) -> c_36() - Weak TRS: #abs(#0()) -> #0() #abs(#neg(x)) -> #pos(x) #abs(#pos(x)) -> #pos(x) #abs(#s(x)) -> #pos(#s(x)) #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/2,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:#compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) -->_1 #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)):3 -->_1 #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)):2 -->_1 #compare#(#s(x),#0()) -> c_18():30 -->_1 #compare#(#pos(x),#neg(y)) -> c_16():29 -->_1 #compare#(#pos(x),#0()) -> c_15():28 -->_1 #compare#(#neg(x),#pos(y)) -> c_14():27 -->_1 #compare#(#neg(x),#0()) -> c_12():26 -->_1 #compare#(#0(),#s(y)) -> c_11():25 -->_1 #compare#(#0(),#pos(y)) -> c_10():24 -->_1 #compare#(#0(),#neg(y)) -> c_9():23 -->_1 #compare#(#0(),#0()) -> c_8():22 -->_1 #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)):1 2:S:#compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) -->_1 #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)):3 -->_1 #compare#(#s(x),#0()) -> c_18():30 -->_1 #compare#(#pos(x),#neg(y)) -> c_16():29 -->_1 #compare#(#pos(x),#0()) -> c_15():28 -->_1 #compare#(#neg(x),#pos(y)) -> c_14():27 -->_1 #compare#(#neg(x),#0()) -> c_12():26 -->_1 #compare#(#0(),#s(y)) -> c_11():25 -->_1 #compare#(#0(),#pos(y)) -> c_10():24 -->_1 #compare#(#0(),#neg(y)) -> c_9():23 -->_1 #compare#(#0(),#0()) -> c_8():22 -->_1 #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)):2 -->_1 #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)):1 3:S:#compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) -->_1 #compare#(#s(x),#0()) -> c_18():30 -->_1 #compare#(#pos(x),#neg(y)) -> c_16():29 -->_1 #compare#(#pos(x),#0()) -> c_15():28 -->_1 #compare#(#neg(x),#pos(y)) -> c_14():27 -->_1 #compare#(#neg(x),#0()) -> c_12():26 -->_1 #compare#(#0(),#s(y)) -> c_11():25 -->_1 #compare#(#0(),#pos(y)) -> c_10():24 -->_1 #compare#(#0(),#neg(y)) -> c_9():23 -->_1 #compare#(#0(),#0()) -> c_8():22 -->_1 #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)):3 -->_1 #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)):2 -->_1 #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)):1 4:S:#less#(x,y) -> c_20(#cklt#(#compare(x,y)),#compare#(x,y)) -->_2 #compare#(#s(x),#0()) -> c_18():30 -->_2 #compare#(#pos(x),#neg(y)) -> c_16():29 -->_2 #compare#(#pos(x),#0()) -> c_15():28 -->_2 #compare#(#neg(x),#pos(y)) -> c_14():27 -->_2 #compare#(#neg(x),#0()) -> c_12():26 -->_2 #compare#(#0(),#s(y)) -> c_11():25 -->_2 #compare#(#0(),#pos(y)) -> c_10():24 -->_2 #compare#(#0(),#neg(y)) -> c_9():23 -->_2 #compare#(#0(),#0()) -> c_8():22 -->_1 #cklt#(#LT()) -> c_7():21 -->_1 #cklt#(#GT()) -> c_6():20 -->_1 #cklt#(#EQ()) -> c_5():19 -->_2 #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)):3 -->_2 #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)):2 -->_2 #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)):1 5:S:insert#(x,l) -> c_21(insert#1#(l,x)) -->_1 insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)):6 -->_1 insert#1#(nil(),x) -> c_23():31 6:S:insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) -->_1 insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)):7 -->_1 insert#2#(#false(),x,y,ys) -> c_24():32 -->_2 #less#(x,y) -> c_20(#cklt#(#compare(x,y)),#compare#(x,y)):4 7:S:insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) -->_1 insert#(x,l) -> c_21(insert#1#(l,x)):5 8:S:insertD#(x,l) -> c_26(insertD#1#(l,x)) -->_1 insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)):9 -->_1 insertD#1#(nil(),x) -> c_28():33 9:S:insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) -->_1 insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)):10 -->_1 insertD#2#(#false(),x,y,ys) -> c_29():34 -->_2 #less#(x,y) -> c_20(#cklt#(#compare(x,y)),#compare#(x,y)):4 10:S:insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) -->_1 insertD#(x,l) -> c_26(insertD#1#(l,x)):8 11:S:insertionsort#(l) -> c_31(insertionsort#1#(l)) -->_1 insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)):12 -->_1 insertionsort#1#(nil()) -> c_33():35 12:S:insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_2 insertionsort#(l) -> c_31(insertionsort#1#(l)):11 -->_1 insert#(x,l) -> c_21(insert#1#(l,x)):5 13:S:insertionsortD#(l) -> c_34(insertionsortD#1#(l)) -->_1 insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)):14 -->_1 insertionsortD#1#(nil()) -> c_36():36 14:S:insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)) -->_2 insertionsortD#(l) -> c_34(insertionsortD#1#(l)):13 -->_1 insertD#(x,l) -> c_26(insertD#1#(l,x)):8 15:W:#abs#(#0()) -> c_1() 16:W:#abs#(#neg(x)) -> c_2() 17:W:#abs#(#pos(x)) -> c_3() 18:W:#abs#(#s(x)) -> c_4() 19:W:#cklt#(#EQ()) -> c_5() 20:W:#cklt#(#GT()) -> c_6() 21:W:#cklt#(#LT()) -> c_7() 22:W:#compare#(#0(),#0()) -> c_8() 23:W:#compare#(#0(),#neg(y)) -> c_9() 24:W:#compare#(#0(),#pos(y)) -> c_10() 25:W:#compare#(#0(),#s(y)) -> c_11() 26:W:#compare#(#neg(x),#0()) -> c_12() 27:W:#compare#(#neg(x),#pos(y)) -> c_14() 28:W:#compare#(#pos(x),#0()) -> c_15() 29:W:#compare#(#pos(x),#neg(y)) -> c_16() 30:W:#compare#(#s(x),#0()) -> c_18() 31:W:insert#1#(nil(),x) -> c_23() 32:W:insert#2#(#false(),x,y,ys) -> c_24() 33:W:insertD#1#(nil(),x) -> c_28() 34:W:insertD#2#(#false(),x,y,ys) -> c_29() 35:W:insertionsort#1#(nil()) -> c_33() 36:W:insertionsortD#1#(nil()) -> c_36() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 18: #abs#(#s(x)) -> c_4() 17: #abs#(#pos(x)) -> c_3() 16: #abs#(#neg(x)) -> c_2() 15: #abs#(#0()) -> c_1() 36: insertionsortD#1#(nil()) -> c_36() 35: insertionsort#1#(nil()) -> c_33() 33: insertD#1#(nil(),x) -> c_28() 34: insertD#2#(#false(),x,y,ys) -> c_29() 31: insert#1#(nil(),x) -> c_23() 32: insert#2#(#false(),x,y,ys) -> c_24() 19: #cklt#(#EQ()) -> c_5() 20: #cklt#(#GT()) -> c_6() 21: #cklt#(#LT()) -> c_7() 22: #compare#(#0(),#0()) -> c_8() 23: #compare#(#0(),#neg(y)) -> c_9() 24: #compare#(#0(),#pos(y)) -> c_10() 25: #compare#(#0(),#s(y)) -> c_11() 26: #compare#(#neg(x),#0()) -> c_12() 27: #compare#(#neg(x),#pos(y)) -> c_14() 28: #compare#(#pos(x),#0()) -> c_15() 29: #compare#(#pos(x),#neg(y)) -> c_16() 30: #compare#(#s(x),#0()) -> c_18() * Step 4: SimplifyRHS WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#cklt#(#compare(x,y)),#compare#(x,y)) insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)) insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)) - Weak TRS: #abs(#0()) -> #0() #abs(#neg(x)) -> #pos(x) #abs(#pos(x)) -> #pos(x) #abs(#s(x)) -> #pos(#s(x)) #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/2,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:#compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) -->_1 #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)):3 -->_1 #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)):2 -->_1 #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)):1 2:S:#compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) -->_1 #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)):3 -->_1 #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)):2 -->_1 #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)):1 3:S:#compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) -->_1 #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)):3 -->_1 #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)):2 -->_1 #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)):1 4:S:#less#(x,y) -> c_20(#cklt#(#compare(x,y)),#compare#(x,y)) -->_2 #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)):3 -->_2 #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)):2 -->_2 #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)):1 5:S:insert#(x,l) -> c_21(insert#1#(l,x)) -->_1 insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)):6 6:S:insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) -->_1 insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)):7 -->_2 #less#(x,y) -> c_20(#cklt#(#compare(x,y)),#compare#(x,y)):4 7:S:insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) -->_1 insert#(x,l) -> c_21(insert#1#(l,x)):5 8:S:insertD#(x,l) -> c_26(insertD#1#(l,x)) -->_1 insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)):9 9:S:insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) -->_1 insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)):10 -->_2 #less#(x,y) -> c_20(#cklt#(#compare(x,y)),#compare#(x,y)):4 10:S:insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) -->_1 insertD#(x,l) -> c_26(insertD#1#(l,x)):8 11:S:insertionsort#(l) -> c_31(insertionsort#1#(l)) -->_1 insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)):12 12:S:insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_2 insertionsort#(l) -> c_31(insertionsort#1#(l)):11 -->_1 insert#(x,l) -> c_21(insert#1#(l,x)):5 13:S:insertionsortD#(l) -> c_34(insertionsortD#1#(l)) -->_1 insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)):14 14:S:insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)) -->_2 insertionsortD#(l) -> c_34(insertionsortD#1#(l)):13 -->_1 insertD#(x,l) -> c_26(insertD#1#(l,x)):8 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: #less#(x,y) -> c_20(#compare#(x,y)) * Step 5: UsableRules WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#compare#(x,y)) insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)) insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)) - Weak TRS: #abs(#0()) -> #0() #abs(#neg(x)) -> #pos(x) #abs(#pos(x)) -> #pos(x) #abs(#s(x)) -> #pos(#s(x)) #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#compare#(x,y)) insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)) insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)) * Step 6: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#compare#(x,y)) insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)) insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)) insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)) and a lower component #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#compare#(x,y)) insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) Further, following extension rules are added to the lower component. insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) ** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)) insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:insertionsort#(l) -> c_31(insertionsort#1#(l)) -->_1 insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)):2 2:S:insertionsort#1#(dd(x,xs)) -> c_32(insert#(x,insertionsort(xs)),insertionsort#(xs)) -->_2 insertionsort#(l) -> c_31(insertionsort#1#(l)):1 3:S:insertionsortD#(l) -> c_34(insertionsortD#1#(l)) -->_1 insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)):4 4:S:insertionsortD#1#(dd(x,xs)) -> c_35(insertD#(x,insertionsortD(xs)),insertionsortD#(xs)) -->_2 insertionsortD#(l) -> c_34(insertionsortD#1#(l)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: insertionsort#1#(dd(x,xs)) -> c_32(insertionsort#(xs)) insertionsortD#1#(dd(x,xs)) -> c_35(insertionsortD#(xs)) ** Step 6.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insertionsort#(xs)) insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertionsortD#(xs)) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/1,c_33/0,c_34/1,c_35/1,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insertionsort#(xs)) insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertionsortD#(xs)) ** Step 6.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insertionsort#(xs)) insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertionsortD#(xs)) - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/1,c_33/0,c_34/1,c_35/1,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_31) = {1}, uargs(c_32) = {1}, uargs(c_34) = {1}, uargs(c_35) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#EQ) = [0] p(#GT) = [0] p(#LT) = [0] p(#abs) = [0] p(#cklt) = [0] p(#compare) = [0] p(#false) = [0] p(#less) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(dd) = [1] x1 + [1] x2 + [0] p(insert) = [0] p(insert#1) = [0] p(insert#2) = [0] p(insertD) = [0] p(insertD#1) = [0] p(insertD#2) = [0] p(insertionsort) = [0] p(insertionsort#1) = [0] p(insertionsortD) = [0] p(insertionsortD#1) = [0] p(nil) = [0] p(#abs#) = [0] p(#cklt#) = [0] p(#compare#) = [0] p(#less#) = [0] p(insert#) = [0] p(insert#1#) = [0] p(insert#2#) = [0] p(insertD#) = [0] p(insertD#1#) = [0] p(insertD#2#) = [0] p(insertionsort#) = [0] p(insertionsort#1#) = [0] p(insertionsortD#) = [0] p(insertionsortD#1#) = [5] 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) = [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) = [1] x1 + [0] p(c_32) = [1] x1 + [0] p(c_33) = [0] p(c_34) = [1] x1 + [0] p(c_35) = [1] x1 + [0] p(c_36) = [0] Following rules are strictly oriented: insertionsortD#1#(dd(x,xs)) = [5] > [0] = c_35(insertionsortD#(xs)) Following rules are (at-least) weakly oriented: insertionsort#(l) = [0] >= [0] = c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) = [0] >= [0] = c_32(insertionsort#(xs)) insertionsortD#(l) = [0] >= [5] = c_34(insertionsortD#1#(l)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insertionsort#(xs)) insertionsortD#(l) -> c_34(insertionsortD#1#(l)) - Weak DPs: insertionsortD#1#(dd(x,xs)) -> c_35(insertionsortD#(xs)) - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/1,c_33/0,c_34/1,c_35/1,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_31) = {1}, uargs(c_32) = {1}, uargs(c_34) = {1}, uargs(c_35) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#EQ) = [2] p(#GT) = [0] p(#LT) = [0] p(#abs) = [0] p(#cklt) = [0] p(#compare) = [0] p(#false) = [0] p(#less) = [2] x2 + [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(dd) = [1] x1 + [1] x2 + [1] p(insert) = [0] p(insert#1) = [0] p(insert#2) = [0] p(insertD) = [0] p(insertD#1) = [0] p(insertD#2) = [0] p(insertionsort) = [0] p(insertionsort#1) = [0] p(insertionsortD) = [0] p(insertionsortD#1) = [0] p(nil) = [0] p(#abs#) = [0] p(#cklt#) = [0] p(#compare#) = [0] p(#less#) = [0] p(insert#) = [0] p(insert#1#) = [0] p(insert#2#) = [0] p(insertD#) = [0] p(insertD#1#) = [0] p(insertD#2#) = [2] x1 + [2] x2 + [2] x3 + [0] p(insertionsort#) = [4] p(insertionsort#1#) = [0] p(insertionsortD#) = [2] x1 + [7] p(insertionsortD#1#) = [2] x1 + [5] 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) = [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) = [1] p(c_26) = [1] x1 + [1] p(c_27) = [1] x1 + [8] x2 + [1] p(c_28) = [1] p(c_29) = [0] p(c_30) = [2] x1 + [1] p(c_31) = [1] x1 + [2] p(c_32) = [1] x1 + [12] p(c_33) = [0] p(c_34) = [1] x1 + [1] p(c_35) = [1] x1 + [0] p(c_36) = [0] Following rules are strictly oriented: insertionsort#(l) = [4] > [2] = c_31(insertionsort#1#(l)) insertionsortD#(l) = [2] l + [7] > [2] l + [6] = c_34(insertionsortD#1#(l)) Following rules are (at-least) weakly oriented: insertionsort#1#(dd(x,xs)) = [0] >= [16] = c_32(insertionsort#(xs)) insertionsortD#1#(dd(x,xs)) = [2] x + [2] xs + [7] >= [2] xs + [7] = c_35(insertionsortD#(xs)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: insertionsort#1#(dd(x,xs)) -> c_32(insertionsort#(xs)) - Weak DPs: insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertionsortD#(xs)) - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/1,c_33/0,c_34/1,c_35/1,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_31) = {1}, uargs(c_32) = {1}, uargs(c_34) = {1}, uargs(c_35) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#EQ) = [0] p(#GT) = [0] p(#LT) = [0] p(#abs) = [0] p(#cklt) = [0] p(#compare) = [0] p(#false) = [0] p(#less) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(dd) = [1] x1 + [1] x2 + [1] p(insert) = [0] p(insert#1) = [0] p(insert#2) = [0] p(insertD) = [0] p(insertD#1) = [0] p(insertD#2) = [0] p(insertionsort) = [1] p(insertionsort#1) = [0] p(insertionsortD) = [0] p(insertionsortD#1) = [0] p(nil) = [0] p(#abs#) = [0] p(#cklt#) = [0] p(#compare#) = [0] p(#less#) = [0] p(insert#) = [0] p(insert#1#) = [1] x1 + [0] p(insert#2#) = [0] p(insertD#) = [0] p(insertD#1#) = [0] p(insertD#2#) = [0] p(insertionsort#) = [1] x1 + [0] p(insertionsort#1#) = [1] x1 + [0] p(insertionsortD#) = [0] p(insertionsortD#1#) = [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) = [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) = [1] x1 + [0] p(c_32) = [1] x1 + [0] p(c_33) = [0] p(c_34) = [1] x1 + [0] p(c_35) = [1] x1 + [0] p(c_36) = [0] Following rules are strictly oriented: insertionsort#1#(dd(x,xs)) = [1] x + [1] xs + [1] > [1] xs + [0] = c_32(insertionsort#(xs)) Following rules are (at-least) weakly oriented: insertionsort#(l) = [1] l + [0] >= [1] l + [0] = c_31(insertionsort#1#(l)) insertionsortD#(l) = [0] >= [0] = c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) = [0] >= [0] = c_35(insertionsortD#(xs)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: insertionsort#(l) -> c_31(insertionsort#1#(l)) insertionsort#1#(dd(x,xs)) -> c_32(insertionsort#(xs)) insertionsortD#(l) -> c_34(insertionsortD#1#(l)) insertionsortD#1#(dd(x,xs)) -> c_35(insertionsortD#(xs)) - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/1,c_33/0,c_34/1,c_35/1,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#compare#(x,y)) insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) - Weak DPs: insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) and a lower component #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#compare#(x,y)) Further, following extension rules are added to the lower component. insert#(x,l) -> insert#1#(l,x) insert#1#(dd(y,ys),x) -> #less#(y,x) insert#1#(dd(y,ys),x) -> insert#2#(#less(y,x),x,y,ys) insert#2#(#true(),x,y,ys) -> insert#(x,ys) insertD#(x,l) -> insertD#1#(l,x) insertD#1#(dd(y,ys),x) -> #less#(y,x) insertD#1#(dd(y,ys),x) -> insertD#2#(#less(y,x),x,y,ys) insertD#2#(#true(),x,y,ys) -> insertD#(x,ys) insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) *** Step 6.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) - Weak DPs: insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:insert#(x,l) -> c_21(insert#1#(l,x)) -->_1 insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)):2 2:S:insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys),#less#(y,x)) -->_1 insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)):3 3:S:insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) -->_1 insert#(x,l) -> c_21(insert#1#(l,x)):1 4:S:insertD#(x,l) -> c_26(insertD#1#(l,x)) -->_1 insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)):5 5:S:insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys),#less#(y,x)) -->_1 insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)):6 6:S:insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) -->_1 insertD#(x,l) -> c_26(insertD#1#(l,x)):4 7:W:insertionsort#(l) -> insertionsort#1#(l) -->_1 insertionsort#1#(dd(x,xs)) -> insertionsort#(xs):9 -->_1 insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)):8 8:W:insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) -->_1 insert#(x,l) -> c_21(insert#1#(l,x)):1 9:W:insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) -->_1 insertionsort#(l) -> insertionsort#1#(l):7 10:W:insertionsortD#(l) -> insertionsortD#1#(l) -->_1 insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs):12 -->_1 insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)):11 11:W:insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) -->_1 insertD#(x,l) -> c_26(insertD#1#(l,x)):4 12:W:insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) -->_1 insertionsortD#(l) -> insertionsortD#1#(l):10 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys)) *** Step 6.b:1.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys)) insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys)) insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) - Weak DPs: insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/1,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#cklt) = {1}, uargs(dd) = {2}, uargs(insert) = {2}, uargs(insert#2) = {1}, uargs(insertD) = {2}, uargs(insertD#2) = {1}, uargs(insert#) = {2}, uargs(insert#2#) = {1}, uargs(insertD#) = {2}, uargs(insertD#2#) = {1}, uargs(c_21) = {1}, uargs(c_22) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1}, uargs(c_27) = {1}, uargs(c_30) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#EQ) = [0] p(#GT) = [0] p(#LT) = [0] p(#abs) = [0] p(#cklt) = [1] x1 + [0] p(#compare) = [0] p(#false) = [0] p(#less) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [0] p(#true) = [0] p(dd) = [1] x2 + [0] p(insert) = [1] x2 + [0] p(insert#1) = [1] x1 + [0] p(insert#2) = [1] x1 + [1] x4 + [0] p(insertD) = [1] x2 + [0] p(insertD#1) = [1] x1 + [0] p(insertD#2) = [1] x1 + [1] x4 + [0] p(insertionsort) = [1] p(insertionsort#1) = [1] p(insertionsortD) = [2] p(insertionsortD#1) = [2] p(nil) = [1] p(#abs#) = [0] p(#cklt#) = [0] p(#compare#) = [0] p(#less#) = [0] p(insert#) = [1] x2 + [1] p(insert#1#) = [1] x1 + [0] p(insert#2#) = [1] x1 + [1] x4 + [0] p(insertD#) = [1] x2 + [0] p(insertD#1#) = [1] x1 + [2] p(insertD#2#) = [1] x1 + [1] x4 + [2] p(insertionsort#) = [4] p(insertionsort#1#) = [4] p(insertionsortD#) = [1] x1 + [2] p(insertionsortD#1#) = [1] x1 + [2] 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) = [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) = [1] x1 + [0] p(c_22) = [1] x1 + [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [1] x1 + [3] p(c_26) = [1] x1 + [0] p(c_27) = [1] x1 + [6] p(c_28) = [0] p(c_29) = [4] p(c_30) = [1] x1 + [0] p(c_31) = [1] p(c_32) = [1] x1 + [1] x2 + [0] p(c_33) = [0] p(c_34) = [1] x1 + [0] p(c_35) = [2] x1 + [4] x2 + [2] p(c_36) = [0] Following rules are strictly oriented: insert#(x,l) = [1] l + [1] > [1] l + [0] = c_21(insert#1#(l,x)) insertD#2#(#true(),x,y,ys) = [1] ys + [2] > [1] ys + [0] = c_30(insertD#(x,ys)) Following rules are (at-least) weakly oriented: insert#1#(dd(y,ys),x) = [1] ys + [0] >= [1] ys + [0] = c_22(insert#2#(#less(y,x),x,y,ys)) insert#2#(#true(),x,y,ys) = [1] ys + [0] >= [1] ys + [4] = c_25(insert#(x,ys)) insertD#(x,l) = [1] l + [0] >= [1] l + [2] = c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) = [1] ys + [2] >= [1] ys + [8] = c_27(insertD#2#(#less(y,x),x,y,ys)) insertionsort#(l) = [4] >= [4] = insertionsort#1#(l) insertionsort#1#(dd(x,xs)) = [4] >= [2] = insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) = [4] >= [4] = insertionsort#(xs) insertionsortD#(l) = [1] l + [2] >= [1] l + [2] = insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) = [1] xs + [2] >= [2] = insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) = [1] xs + [2] >= [1] xs + [2] = insertionsortD#(xs) #cklt(#EQ()) = [0] >= [0] = #false() #cklt(#GT()) = [0] >= [0] = #false() #cklt(#LT()) = [0] >= [0] = #true() #compare(#0(),#0()) = [0] >= [0] = #EQ() #compare(#0(),#neg(y)) = [0] >= [0] = #GT() #compare(#0(),#pos(y)) = [0] >= [0] = #LT() #compare(#0(),#s(y)) = [0] >= [0] = #LT() #compare(#neg(x),#0()) = [0] >= [0] = #LT() #compare(#neg(x),#neg(y)) = [0] >= [0] = #compare(y,x) #compare(#neg(x),#pos(y)) = [0] >= [0] = #LT() #compare(#pos(x),#0()) = [0] >= [0] = #GT() #compare(#pos(x),#neg(y)) = [0] >= [0] = #GT() #compare(#pos(x),#pos(y)) = [0] >= [0] = #compare(x,y) #compare(#s(x),#0()) = [0] >= [0] = #GT() #compare(#s(x),#s(y)) = [0] >= [0] = #compare(x,y) #less(x,y) = [0] >= [0] = #cklt(#compare(x,y)) insert(x,l) = [1] l + [0] >= [1] l + [0] = insert#1(l,x) insert#1(dd(y,ys),x) = [1] ys + [0] >= [1] ys + [0] = insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) = [1] >= [1] = dd(x,nil()) insert#2(#false(),x,y,ys) = [1] ys + [0] >= [1] ys + [0] = dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) = [1] ys + [0] >= [1] ys + [0] = dd(y,insert(x,ys)) insertD(x,l) = [1] l + [0] >= [1] l + [0] = insertD#1(l,x) insertD#1(dd(y,ys),x) = [1] ys + [0] >= [1] ys + [0] = insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) = [1] >= [1] = dd(x,nil()) insertD#2(#false(),x,y,ys) = [1] ys + [0] >= [1] ys + [0] = dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) = [1] ys + [0] >= [1] ys + [0] = dd(y,insertD(x,ys)) insertionsort(l) = [1] >= [1] = insertionsort#1(l) insertionsort#1(dd(x,xs)) = [1] >= [1] = insert(x,insertionsort(xs)) insertionsort#1(nil()) = [1] >= [1] = nil() insertionsortD(l) = [2] >= [2] = insertionsortD#1(l) insertionsortD#1(dd(x,xs)) = [2] >= [2] = insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) = [2] >= [1] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 6.b:1.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys)) insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys)) - Weak DPs: insert#(x,l) -> c_21(insert#1#(l,x)) insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/1,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#cklt) = {1}, uargs(dd) = {2}, uargs(insert) = {2}, uargs(insert#2) = {1}, uargs(insertD) = {2}, uargs(insertD#2) = {1}, uargs(insert#) = {2}, uargs(insert#2#) = {1}, uargs(insertD#) = {2}, uargs(insertD#2#) = {1}, uargs(c_21) = {1}, uargs(c_22) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1}, uargs(c_27) = {1}, uargs(c_30) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [1] p(#EQ) = [5] p(#GT) = [5] p(#LT) = [5] p(#abs) = [2] p(#cklt) = [1] x1 + [1] p(#compare) = [5] p(#false) = [6] p(#less) = [6] p(#neg) = [1] x1 + [0] p(#pos) = [2] p(#s) = [0] p(#true) = [6] p(dd) = [1] x2 + [4] p(insert) = [1] x2 + [4] p(insert#1) = [1] x1 + [4] p(insert#2) = [1] x1 + [1] x4 + [2] p(insertD) = [1] x2 + [4] p(insertD#1) = [1] x1 + [4] p(insertD#2) = [1] x1 + [1] x4 + [2] p(insertionsort) = [1] x1 + [0] p(insertionsort#1) = [1] x1 + [0] p(insertionsortD) = [1] x1 + [0] p(insertionsortD#1) = [1] x1 + [0] p(nil) = [0] p(#abs#) = [1] x1 + [1] p(#cklt#) = [0] p(#compare#) = [1] x1 + [1] p(#less#) = [1] x1 + [2] p(insert#) = [1] x2 + [4] p(insert#1#) = [1] x1 + [3] p(insert#2#) = [1] x1 + [1] x4 + [0] p(insertD#) = [1] x2 + [3] p(insertD#1#) = [1] x1 + [5] p(insertD#2#) = [1] x1 + [1] x4 + [1] p(insertionsort#) = [1] x1 + [0] p(insertionsort#1#) = [1] x1 + [0] p(insertionsortD#) = [2] x1 + [1] p(insertionsortD#1#) = [2] x1 + [1] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [2] p(c_11) = [0] p(c_12) = [0] p(c_13) = [4] x1 + [0] p(c_14) = [1] p(c_15) = [4] p(c_16) = [0] p(c_17) = [2] x1 + [0] p(c_18) = [4] p(c_19) = [1] x1 + [0] p(c_20) = [4] x1 + [1] p(c_21) = [1] x1 + [1] p(c_22) = [1] x1 + [0] p(c_23) = [1] p(c_24) = [4] p(c_25) = [1] x1 + [0] p(c_26) = [1] x1 + [3] p(c_27) = [1] x1 + [1] p(c_28) = [1] p(c_29) = [0] p(c_30) = [1] x1 + [4] p(c_31) = [1] x1 + [1] p(c_32) = [1] x1 + [1] x2 + [0] p(c_33) = [0] p(c_34) = [2] x1 + [1] p(c_35) = [1] x1 + [1] x2 + [1] p(c_36) = [0] Following rules are strictly oriented: insert#1#(dd(y,ys),x) = [1] ys + [7] > [1] ys + [6] = c_22(insert#2#(#less(y,x),x,y,ys)) insert#2#(#true(),x,y,ys) = [1] ys + [6] > [1] ys + [4] = c_25(insert#(x,ys)) insertD#1#(dd(y,ys),x) = [1] ys + [9] > [1] ys + [8] = c_27(insertD#2#(#less(y,x),x,y,ys)) Following rules are (at-least) weakly oriented: insert#(x,l) = [1] l + [4] >= [1] l + [4] = c_21(insert#1#(l,x)) insertD#(x,l) = [1] l + [3] >= [1] l + [8] = c_26(insertD#1#(l,x)) insertD#2#(#true(),x,y,ys) = [1] ys + [7] >= [1] ys + [7] = c_30(insertD#(x,ys)) insertionsort#(l) = [1] l + [0] >= [1] l + [0] = insertionsort#1#(l) insertionsort#1#(dd(x,xs)) = [1] xs + [4] >= [1] xs + [4] = insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) = [1] xs + [4] >= [1] xs + [0] = insertionsort#(xs) insertionsortD#(l) = [2] l + [1] >= [2] l + [1] = insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) = [2] xs + [9] >= [1] xs + [3] = insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) = [2] xs + [9] >= [2] xs + [1] = insertionsortD#(xs) #cklt(#EQ()) = [6] >= [6] = #false() #cklt(#GT()) = [6] >= [6] = #false() #cklt(#LT()) = [6] >= [6] = #true() #compare(#0(),#0()) = [5] >= [5] = #EQ() #compare(#0(),#neg(y)) = [5] >= [5] = #GT() #compare(#0(),#pos(y)) = [5] >= [5] = #LT() #compare(#0(),#s(y)) = [5] >= [5] = #LT() #compare(#neg(x),#0()) = [5] >= [5] = #LT() #compare(#neg(x),#neg(y)) = [5] >= [5] = #compare(y,x) #compare(#neg(x),#pos(y)) = [5] >= [5] = #LT() #compare(#pos(x),#0()) = [5] >= [5] = #GT() #compare(#pos(x),#neg(y)) = [5] >= [5] = #GT() #compare(#pos(x),#pos(y)) = [5] >= [5] = #compare(x,y) #compare(#s(x),#0()) = [5] >= [5] = #GT() #compare(#s(x),#s(y)) = [5] >= [5] = #compare(x,y) #less(x,y) = [6] >= [6] = #cklt(#compare(x,y)) insert(x,l) = [1] l + [4] >= [1] l + [4] = insert#1(l,x) insert#1(dd(y,ys),x) = [1] ys + [8] >= [1] ys + [8] = insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) = [4] >= [4] = dd(x,nil()) insert#2(#false(),x,y,ys) = [1] ys + [8] >= [1] ys + [8] = dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) = [1] ys + [8] >= [1] ys + [8] = dd(y,insert(x,ys)) insertD(x,l) = [1] l + [4] >= [1] l + [4] = insertD#1(l,x) insertD#1(dd(y,ys),x) = [1] ys + [8] >= [1] ys + [8] = insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) = [4] >= [4] = dd(x,nil()) insertD#2(#false(),x,y,ys) = [1] ys + [8] >= [1] ys + [8] = dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) = [1] ys + [8] >= [1] ys + [8] = dd(y,insertD(x,ys)) insertionsort(l) = [1] l + [0] >= [1] l + [0] = insertionsort#1(l) insertionsort#1(dd(x,xs)) = [1] xs + [4] >= [1] xs + [4] = insert(x,insertionsort(xs)) insertionsort#1(nil()) = [0] >= [0] = nil() insertionsortD(l) = [1] l + [0] >= [1] l + [0] = insertionsortD#1(l) insertionsortD#1(dd(x,xs)) = [1] xs + [4] >= [1] xs + [4] = insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) = [0] >= [0] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 6.b:1.a:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: insertD#(x,l) -> c_26(insertD#1#(l,x)) - Weak DPs: insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys)) insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys)) insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/1,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#cklt) = {1}, uargs(dd) = {2}, uargs(insert) = {2}, uargs(insert#2) = {1}, uargs(insertD) = {2}, uargs(insertD#2) = {1}, uargs(insert#) = {2}, uargs(insert#2#) = {1}, uargs(insertD#) = {2}, uargs(insertD#2#) = {1}, uargs(c_21) = {1}, uargs(c_22) = {1}, uargs(c_25) = {1}, uargs(c_26) = {1}, uargs(c_27) = {1}, uargs(c_30) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [2] p(#EQ) = [0] p(#GT) = [0] p(#LT) = [0] p(#abs) = [1] x1 + [1] p(#cklt) = [1] x1 + [0] p(#compare) = [0] p(#false) = [0] p(#less) = [0] p(#neg) = [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(dd) = [1] x2 + [2] p(insert) = [1] x2 + [2] p(insert#1) = [1] x1 + [2] p(insert#2) = [1] x1 + [1] x4 + [4] p(insertD) = [1] x2 + [2] p(insertD#1) = [1] x1 + [2] p(insertD#2) = [1] x1 + [1] x4 + [4] p(insertionsort) = [4] x1 + [1] p(insertionsort#1) = [4] x1 + [0] p(insertionsortD) = [1] x1 + [0] p(insertionsortD#1) = [1] x1 + [0] p(nil) = [0] p(#abs#) = [0] p(#cklt#) = [1] x1 + [4] p(#compare#) = [1] x2 + [1] p(#less#) = [4] x2 + [4] p(insert#) = [1] x2 + [0] p(insert#1#) = [1] x1 + [0] p(insert#2#) = [1] x1 + [1] x4 + [1] p(insertD#) = [1] x2 + [3] p(insertD#1#) = [1] x1 + [2] p(insertD#2#) = [1] x1 + [1] x4 + [4] p(insertionsort#) = [4] x1 + [4] p(insertionsort#1#) = [4] x1 + [3] p(insertionsortD#) = [1] x1 + [1] p(insertionsortD#1#) = [1] x1 + [1] p(c_1) = [1] p(c_2) = [1] p(c_3) = [1] p(c_4) = [2] p(c_5) = [0] p(c_6) = [1] p(c_7) = [0] p(c_8) = [2] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] p(c_12) = [2] p(c_13) = [4] x1 + [2] p(c_14) = [1] p(c_15) = [0] p(c_16) = [1] p(c_17) = [0] p(c_18) = [0] p(c_19) = [1] x1 + [0] p(c_20) = [0] p(c_21) = [1] x1 + [0] p(c_22) = [1] x1 + [1] p(c_23) = [2] p(c_24) = [0] p(c_25) = [1] x1 + [1] p(c_26) = [1] x1 + [0] p(c_27) = [1] x1 + [0] p(c_28) = [1] p(c_29) = [0] p(c_30) = [1] x1 + [1] p(c_31) = [1] x1 + [0] p(c_32) = [2] x2 + [2] p(c_33) = [1] p(c_34) = [1] x1 + [4] p(c_35) = [4] x1 + [2] x2 + [0] p(c_36) = [0] Following rules are strictly oriented: insertD#(x,l) = [1] l + [3] > [1] l + [2] = c_26(insertD#1#(l,x)) Following rules are (at-least) weakly oriented: insert#(x,l) = [1] l + [0] >= [1] l + [0] = c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) = [1] ys + [2] >= [1] ys + [2] = c_22(insert#2#(#less(y,x),x,y,ys)) insert#2#(#true(),x,y,ys) = [1] ys + [1] >= [1] ys + [1] = c_25(insert#(x,ys)) insertD#1#(dd(y,ys),x) = [1] ys + [4] >= [1] ys + [4] = c_27(insertD#2#(#less(y,x),x,y,ys)) insertD#2#(#true(),x,y,ys) = [1] ys + [4] >= [1] ys + [4] = c_30(insertD#(x,ys)) insertionsort#(l) = [4] l + [4] >= [4] l + [3] = insertionsort#1#(l) insertionsort#1#(dd(x,xs)) = [4] xs + [11] >= [4] xs + [1] = insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) = [4] xs + [11] >= [4] xs + [4] = insertionsort#(xs) insertionsortD#(l) = [1] l + [1] >= [1] l + [1] = insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) = [1] xs + [3] >= [1] xs + [3] = insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) = [1] xs + [3] >= [1] xs + [1] = insertionsortD#(xs) #cklt(#EQ()) = [0] >= [0] = #false() #cklt(#GT()) = [0] >= [0] = #false() #cklt(#LT()) = [0] >= [0] = #true() #compare(#0(),#0()) = [0] >= [0] = #EQ() #compare(#0(),#neg(y)) = [0] >= [0] = #GT() #compare(#0(),#pos(y)) = [0] >= [0] = #LT() #compare(#0(),#s(y)) = [0] >= [0] = #LT() #compare(#neg(x),#0()) = [0] >= [0] = #LT() #compare(#neg(x),#neg(y)) = [0] >= [0] = #compare(y,x) #compare(#neg(x),#pos(y)) = [0] >= [0] = #LT() #compare(#pos(x),#0()) = [0] >= [0] = #GT() #compare(#pos(x),#neg(y)) = [0] >= [0] = #GT() #compare(#pos(x),#pos(y)) = [0] >= [0] = #compare(x,y) #compare(#s(x),#0()) = [0] >= [0] = #GT() #compare(#s(x),#s(y)) = [0] >= [0] = #compare(x,y) #less(x,y) = [0] >= [0] = #cklt(#compare(x,y)) insert(x,l) = [1] l + [2] >= [1] l + [2] = insert#1(l,x) insert#1(dd(y,ys),x) = [1] ys + [4] >= [1] ys + [4] = insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) = [2] >= [2] = dd(x,nil()) insert#2(#false(),x,y,ys) = [1] ys + [4] >= [1] ys + [4] = dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) = [1] ys + [4] >= [1] ys + [4] = dd(y,insert(x,ys)) insertD(x,l) = [1] l + [2] >= [1] l + [2] = insertD#1(l,x) insertD#1(dd(y,ys),x) = [1] ys + [4] >= [1] ys + [4] = insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) = [2] >= [2] = dd(x,nil()) insertD#2(#false(),x,y,ys) = [1] ys + [4] >= [1] ys + [4] = dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) = [1] ys + [4] >= [1] ys + [4] = dd(y,insertD(x,ys)) insertionsort(l) = [4] l + [1] >= [4] l + [0] = insertionsort#1(l) insertionsort#1(dd(x,xs)) = [4] xs + [8] >= [4] xs + [3] = insert(x,insertionsort(xs)) insertionsort#1(nil()) = [0] >= [0] = nil() insertionsortD(l) = [1] l + [0] >= [1] l + [0] = insertionsortD#1(l) insertionsortD#1(dd(x,xs)) = [1] xs + [2] >= [1] xs + [2] = insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) = [0] >= [0] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 6.b:1.a:5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: insert#(x,l) -> c_21(insert#1#(l,x)) insert#1#(dd(y,ys),x) -> c_22(insert#2#(#less(y,x),x,y,ys)) insert#2#(#true(),x,y,ys) -> c_25(insert#(x,ys)) insertD#(x,l) -> c_26(insertD#1#(l,x)) insertD#1#(dd(y,ys),x) -> c_27(insertD#2#(#less(y,x),x,y,ys)) insertD#2#(#true(),x,y,ys) -> c_30(insertD#(x,ys)) insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/1,c_23/0,c_24/0,c_25/1,c_26/1,c_27/1,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.b:1.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#compare#(x,y)) - Weak DPs: insert#(x,l) -> insert#1#(l,x) insert#1#(dd(y,ys),x) -> #less#(y,x) insert#1#(dd(y,ys),x) -> insert#2#(#less(y,x),x,y,ys) insert#2#(#true(),x,y,ys) -> insert#(x,ys) insertD#(x,l) -> insertD#1#(l,x) insertD#1#(dd(y,ys),x) -> #less#(y,x) insertD#1#(dd(y,ys),x) -> insertD#2#(#less(y,x),x,y,ys) insertD#2#(#true(),x,y,ys) -> insertD#(x,ys) insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#cklt) = {1}, uargs(dd) = {2}, uargs(insert) = {2}, uargs(insert#2) = {1}, uargs(insertD) = {2}, uargs(insertD#2) = {1}, uargs(insert#) = {2}, uargs(insert#2#) = {1}, uargs(insertD#) = {2}, uargs(insertD#2#) = {1}, uargs(c_13) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1}, uargs(c_20) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#EQ) = [0] p(#GT) = [0] p(#LT) = [0] p(#abs) = [1] x1 + [2] p(#cklt) = [1] x1 + [0] p(#compare) = [0] p(#false) = [0] p(#less) = [0] p(#neg) = [2] p(#pos) = [1] x1 + [0] p(#s) = [0] p(#true) = [0] p(dd) = [1] x2 + [0] p(insert) = [1] x2 + [0] p(insert#1) = [1] x1 + [0] p(insert#2) = [1] x1 + [1] x4 + [0] p(insertD) = [1] x2 + [0] p(insertD#1) = [1] x1 + [0] p(insertD#2) = [1] x1 + [1] x4 + [0] p(insertionsort) = [0] p(insertionsort#1) = [0] p(insertionsortD) = [6] p(insertionsortD#1) = [6] p(nil) = [0] p(#abs#) = [4] p(#cklt#) = [2] x1 + [1] p(#compare#) = [0] p(#less#) = [1] p(insert#) = [1] x2 + [2] p(insert#1#) = [1] x1 + [2] p(insert#2#) = [1] x1 + [1] x4 + [2] p(insertD#) = [1] x2 + [1] p(insertD#1#) = [1] x1 + [1] p(insertD#2#) = [1] x1 + [1] x4 + [1] p(insertionsort#) = [1] x1 + [2] p(insertionsort#1#) = [1] x1 + [2] p(insertionsortD#) = [7] p(insertionsortD#1#) = [7] p(c_1) = [1] p(c_2) = [0] p(c_3) = [4] p(c_4) = [0] p(c_5) = [1] p(c_6) = [2] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] p(c_10) = [1] p(c_11) = [0] p(c_12) = [1] p(c_13) = [1] x1 + [2] p(c_14) = [0] p(c_15) = [0] p(c_16) = [1] p(c_17) = [1] x1 + [0] p(c_18) = [1] p(c_19) = [1] x1 + [0] p(c_20) = [1] x1 + [0] p(c_21) = [2] x1 + [4] p(c_22) = [2] x1 + [1] p(c_23) = [1] p(c_24) = [0] p(c_25) = [0] p(c_26) = [1] x1 + [0] p(c_27) = [4] x1 + [2] p(c_28) = [4] p(c_29) = [0] p(c_30) = [1] x1 + [0] p(c_31) = [1] p(c_32) = [1] p(c_33) = [0] p(c_34) = [1] x1 + [1] p(c_35) = [1] x1 + [0] p(c_36) = [1] Following rules are strictly oriented: #less#(x,y) = [1] > [0] = c_20(#compare#(x,y)) Following rules are (at-least) weakly oriented: #compare#(#neg(x),#neg(y)) = [0] >= [2] = c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) = [0] >= [0] = c_17(#compare#(x,y)) #compare#(#s(x),#s(y)) = [0] >= [0] = c_19(#compare#(x,y)) insert#(x,l) = [1] l + [2] >= [1] l + [2] = insert#1#(l,x) insert#1#(dd(y,ys),x) = [1] ys + [2] >= [1] = #less#(y,x) insert#1#(dd(y,ys),x) = [1] ys + [2] >= [1] ys + [2] = insert#2#(#less(y,x),x,y,ys) insert#2#(#true(),x,y,ys) = [1] ys + [2] >= [1] ys + [2] = insert#(x,ys) insertD#(x,l) = [1] l + [1] >= [1] l + [1] = insertD#1#(l,x) insertD#1#(dd(y,ys),x) = [1] ys + [1] >= [1] = #less#(y,x) insertD#1#(dd(y,ys),x) = [1] ys + [1] >= [1] ys + [1] = insertD#2#(#less(y,x),x,y,ys) insertD#2#(#true(),x,y,ys) = [1] ys + [1] >= [1] ys + [1] = insertD#(x,ys) insertionsort#(l) = [1] l + [2] >= [1] l + [2] = insertionsort#1#(l) insertionsort#1#(dd(x,xs)) = [1] xs + [2] >= [2] = insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) = [1] xs + [2] >= [1] xs + [2] = insertionsort#(xs) insertionsortD#(l) = [7] >= [7] = insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) = [7] >= [7] = insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) = [7] >= [7] = insertionsortD#(xs) #cklt(#EQ()) = [0] >= [0] = #false() #cklt(#GT()) = [0] >= [0] = #false() #cklt(#LT()) = [0] >= [0] = #true() #compare(#0(),#0()) = [0] >= [0] = #EQ() #compare(#0(),#neg(y)) = [0] >= [0] = #GT() #compare(#0(),#pos(y)) = [0] >= [0] = #LT() #compare(#0(),#s(y)) = [0] >= [0] = #LT() #compare(#neg(x),#0()) = [0] >= [0] = #LT() #compare(#neg(x),#neg(y)) = [0] >= [0] = #compare(y,x) #compare(#neg(x),#pos(y)) = [0] >= [0] = #LT() #compare(#pos(x),#0()) = [0] >= [0] = #GT() #compare(#pos(x),#neg(y)) = [0] >= [0] = #GT() #compare(#pos(x),#pos(y)) = [0] >= [0] = #compare(x,y) #compare(#s(x),#0()) = [0] >= [0] = #GT() #compare(#s(x),#s(y)) = [0] >= [0] = #compare(x,y) #less(x,y) = [0] >= [0] = #cklt(#compare(x,y)) insert(x,l) = [1] l + [0] >= [1] l + [0] = insert#1(l,x) insert#1(dd(y,ys),x) = [1] ys + [0] >= [1] ys + [0] = insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) = [0] >= [0] = dd(x,nil()) insert#2(#false(),x,y,ys) = [1] ys + [0] >= [1] ys + [0] = dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) = [1] ys + [0] >= [1] ys + [0] = dd(y,insert(x,ys)) insertD(x,l) = [1] l + [0] >= [1] l + [0] = insertD#1(l,x) insertD#1(dd(y,ys),x) = [1] ys + [0] >= [1] ys + [0] = insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) = [0] >= [0] = dd(x,nil()) insertD#2(#false(),x,y,ys) = [1] ys + [0] >= [1] ys + [0] = dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) = [1] ys + [0] >= [1] ys + [0] = dd(y,insertD(x,ys)) insertionsort(l) = [0] >= [0] = insertionsort#1(l) insertionsort#1(dd(x,xs)) = [0] >= [0] = insert(x,insertionsort(xs)) insertionsort#1(nil()) = [0] >= [0] = nil() insertionsortD(l) = [6] >= [6] = insertionsortD#1(l) insertionsortD#1(dd(x,xs)) = [6] >= [6] = insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) = [6] >= [0] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 6.b:1.b:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) - Weak DPs: #less#(x,y) -> c_20(#compare#(x,y)) insert#(x,l) -> insert#1#(l,x) insert#1#(dd(y,ys),x) -> #less#(y,x) insert#1#(dd(y,ys),x) -> insert#2#(#less(y,x),x,y,ys) insert#2#(#true(),x,y,ys) -> insert#(x,ys) insertD#(x,l) -> insertD#1#(l,x) insertD#1#(dd(y,ys),x) -> #less#(y,x) insertD#1#(dd(y,ys),x) -> insertD#2#(#less(y,x),x,y,ys) insertD#2#(#true(),x,y,ys) -> insertD#(x,ys) insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#cklt) = {1}, uargs(dd) = {2}, uargs(insert) = {2}, uargs(insert#2) = {1}, uargs(insertD) = {2}, uargs(insertD#2) = {1}, uargs(insert#) = {2}, uargs(insert#2#) = {1}, uargs(insertD#) = {2}, uargs(insertD#2#) = {1}, uargs(c_13) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1}, uargs(c_20) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [2] p(#EQ) = [0] p(#GT) = [0] p(#LT) = [0] p(#abs) = [1] x1 + [1] p(#cklt) = [1] x1 + [0] p(#compare) = [0] p(#false) = [0] p(#less) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [1] p(#true) = [0] p(dd) = [1] x1 + [1] x2 + [0] p(insert) = [1] x1 + [1] x2 + [0] p(insert#1) = [1] x1 + [1] x2 + [0] p(insert#2) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [0] p(insertD) = [1] x1 + [1] x2 + [0] p(insertD#1) = [1] x1 + [1] x2 + [0] p(insertD#2) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [0] p(insertionsort) = [1] x1 + [0] p(insertionsort#1) = [1] x1 + [0] p(insertionsortD) = [2] x1 + [0] p(insertionsortD#1) = [2] x1 + [0] p(nil) = [3] p(#abs#) = [4] x1 + [1] p(#cklt#) = [4] p(#compare#) = [1] x1 + [1] x2 + [0] p(#less#) = [1] x1 + [1] x2 + [0] p(insert#) = [1] x1 + [1] x2 + [0] p(insert#1#) = [1] x1 + [1] x2 + [0] p(insert#2#) = [1] x1 + [1] x2 + [1] x4 + [0] p(insertD#) = [1] x1 + [1] x2 + [0] p(insertD#1#) = [1] x1 + [1] x2 + [0] p(insertD#2#) = [1] x1 + [1] x2 + [1] x4 + [0] p(insertionsort#) = [2] x1 + [0] p(insertionsort#1#) = [2] x1 + [0] p(insertionsortD#) = [3] x1 + [3] p(insertionsortD#1#) = [3] x1 + [3] p(c_1) = [0] p(c_2) = [4] p(c_3) = [1] p(c_4) = [2] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] p(c_9) = [0] p(c_10) = [0] p(c_11) = [2] p(c_12) = [4] p(c_13) = [1] x1 + [0] p(c_14) = [1] p(c_15) = [1] p(c_16) = [0] p(c_17) = [1] x1 + [3] p(c_18) = [0] p(c_19) = [1] x1 + [0] p(c_20) = [1] x1 + [0] p(c_21) = [1] x1 + [0] p(c_22) = [1] x1 + [2] x2 + [0] p(c_23) = [2] p(c_24) = [0] p(c_25) = [2] x1 + [1] p(c_26) = [2] x1 + [0] p(c_27) = [2] x1 + [1] x2 + [1] p(c_28) = [2] p(c_29) = [1] p(c_30) = [2] x1 + [4] p(c_31) = [4] x1 + [0] p(c_32) = [1] x1 + [4] x2 + [0] p(c_33) = [4] p(c_34) = [4] x1 + [1] p(c_35) = [4] x1 + [1] x2 + [1] p(c_36) = [0] Following rules are strictly oriented: #compare#(#s(x),#s(y)) = [1] x + [1] y + [2] > [1] x + [1] y + [0] = c_19(#compare#(x,y)) Following rules are (at-least) weakly oriented: #compare#(#neg(x),#neg(y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [3] = c_17(#compare#(x,y)) #less#(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = c_20(#compare#(x,y)) insert#(x,l) = [1] l + [1] x + [0] >= [1] l + [1] x + [0] = insert#1#(l,x) insert#1#(dd(y,ys),x) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [0] = #less#(y,x) insert#1#(dd(y,ys),x) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] ys + [0] = insert#2#(#less(y,x),x,y,ys) insert#2#(#true(),x,y,ys) = [1] x + [1] ys + [0] >= [1] x + [1] ys + [0] = insert#(x,ys) insertD#(x,l) = [1] l + [1] x + [0] >= [1] l + [1] x + [0] = insertD#1#(l,x) insertD#1#(dd(y,ys),x) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [0] = #less#(y,x) insertD#1#(dd(y,ys),x) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] ys + [0] = insertD#2#(#less(y,x),x,y,ys) insertD#2#(#true(),x,y,ys) = [1] x + [1] ys + [0] >= [1] x + [1] ys + [0] = insertD#(x,ys) insertionsort#(l) = [2] l + [0] >= [2] l + [0] = insertionsort#1#(l) insertionsort#1#(dd(x,xs)) = [2] x + [2] xs + [0] >= [1] x + [1] xs + [0] = insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) = [2] x + [2] xs + [0] >= [2] xs + [0] = insertionsort#(xs) insertionsortD#(l) = [3] l + [3] >= [3] l + [3] = insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) = [3] x + [3] xs + [3] >= [1] x + [2] xs + [0] = insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) = [3] x + [3] xs + [3] >= [3] xs + [3] = insertionsortD#(xs) #cklt(#EQ()) = [0] >= [0] = #false() #cklt(#GT()) = [0] >= [0] = #false() #cklt(#LT()) = [0] >= [0] = #true() #compare(#0(),#0()) = [0] >= [0] = #EQ() #compare(#0(),#neg(y)) = [0] >= [0] = #GT() #compare(#0(),#pos(y)) = [0] >= [0] = #LT() #compare(#0(),#s(y)) = [0] >= [0] = #LT() #compare(#neg(x),#0()) = [0] >= [0] = #LT() #compare(#neg(x),#neg(y)) = [0] >= [0] = #compare(y,x) #compare(#neg(x),#pos(y)) = [0] >= [0] = #LT() #compare(#pos(x),#0()) = [0] >= [0] = #GT() #compare(#pos(x),#neg(y)) = [0] >= [0] = #GT() #compare(#pos(x),#pos(y)) = [0] >= [0] = #compare(x,y) #compare(#s(x),#0()) = [0] >= [0] = #GT() #compare(#s(x),#s(y)) = [0] >= [0] = #compare(x,y) #less(x,y) = [0] >= [0] = #cklt(#compare(x,y)) insert(x,l) = [1] l + [1] x + [0] >= [1] l + [1] x + [0] = insert#1(l,x) insert#1(dd(y,ys),x) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) = [1] x + [3] >= [1] x + [3] = dd(x,nil()) insert#2(#false(),x,y,ys) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = dd(y,insert(x,ys)) insertD(x,l) = [1] l + [1] x + [0] >= [1] l + [1] x + [0] = insertD#1(l,x) insertD#1(dd(y,ys),x) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) = [1] x + [3] >= [1] x + [3] = dd(x,nil()) insertD#2(#false(),x,y,ys) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = dd(y,insertD(x,ys)) insertionsort(l) = [1] l + [0] >= [1] l + [0] = insertionsort#1(l) insertionsort#1(dd(x,xs)) = [1] x + [1] xs + [0] >= [1] x + [1] xs + [0] = insert(x,insertionsort(xs)) insertionsort#1(nil()) = [3] >= [3] = nil() insertionsortD(l) = [2] l + [0] >= [2] l + [0] = insertionsortD#1(l) insertionsortD#1(dd(x,xs)) = [2] x + [2] xs + [0] >= [1] x + [2] xs + [0] = insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) = [6] >= [3] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 6.b:1.b:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) - Weak DPs: #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#compare#(x,y)) insert#(x,l) -> insert#1#(l,x) insert#1#(dd(y,ys),x) -> #less#(y,x) insert#1#(dd(y,ys),x) -> insert#2#(#less(y,x),x,y,ys) insert#2#(#true(),x,y,ys) -> insert#(x,ys) insertD#(x,l) -> insertD#1#(l,x) insertD#1#(dd(y,ys),x) -> #less#(y,x) insertD#1#(dd(y,ys),x) -> insertD#2#(#less(y,x),x,y,ys) insertD#2#(#true(),x,y,ys) -> insertD#(x,ys) insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#cklt) = {1}, uargs(dd) = {2}, uargs(insert) = {2}, uargs(insert#2) = {1}, uargs(insertD) = {2}, uargs(insertD#2) = {1}, uargs(insert#) = {2}, uargs(insert#2#) = {1}, uargs(insertD#) = {2}, uargs(insertD#2#) = {1}, uargs(c_13) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1}, uargs(c_20) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [2] p(#EQ) = [0] p(#GT) = [0] p(#LT) = [0] p(#abs) = [2] x1 + [1] p(#cklt) = [1] x1 + [0] p(#compare) = [0] p(#false) = [0] p(#less) = [0] p(#neg) = [1] x1 + [1] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [4] p(#true) = [0] p(dd) = [1] x1 + [1] x2 + [0] p(insert) = [1] x1 + [1] x2 + [0] p(insert#1) = [1] x1 + [1] x2 + [0] p(insert#2) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [0] p(insertD) = [1] x1 + [1] x2 + [0] p(insertD#1) = [1] x1 + [1] x2 + [0] p(insertD#2) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [0] p(insertionsort) = [1] x1 + [0] p(insertionsort#1) = [1] x1 + [0] p(insertionsortD) = [1] x1 + [0] p(insertionsortD#1) = [1] x1 + [0] p(nil) = [0] p(#abs#) = [1] p(#cklt#) = [0] p(#compare#) = [1] x1 + [1] x2 + [0] p(#less#) = [1] x1 + [1] x2 + [0] p(insert#) = [1] x1 + [1] x2 + [0] p(insert#1#) = [1] x1 + [1] x2 + [0] p(insert#2#) = [1] x1 + [1] x2 + [1] x4 + [0] p(insertD#) = [2] x1 + [1] x2 + [0] p(insertD#1#) = [1] x1 + [2] x2 + [0] p(insertD#2#) = [1] x1 + [2] x2 + [1] x3 + [1] x4 + [0] p(insertionsort#) = [1] x1 + [0] p(insertionsort#1#) = [1] x1 + [0] p(insertionsortD#) = [4] x1 + [0] p(insertionsortD#1#) = [4] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] p(c_5) = [1] p(c_6) = [1] p(c_7) = [1] p(c_8) = [0] p(c_9) = [2] p(c_10) = [1] p(c_11) = [0] p(c_12) = [1] p(c_13) = [1] x1 + [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [2] p(c_17) = [1] x1 + [1] p(c_18) = [1] p(c_19) = [1] x1 + [2] p(c_20) = [1] x1 + [0] p(c_21) = [1] p(c_22) = [1] x1 + [2] x2 + [0] p(c_23) = [0] p(c_24) = [1] p(c_25) = [0] p(c_26) = [1] p(c_27) = [2] x1 + [1] x2 + [0] p(c_28) = [4] p(c_29) = [0] p(c_30) = [1] x1 + [0] p(c_31) = [0] p(c_32) = [1] p(c_33) = [1] p(c_34) = [1] p(c_35) = [1] x2 + [1] p(c_36) = [4] Following rules are strictly oriented: #compare#(#neg(x),#neg(y)) = [1] x + [1] y + [2] > [1] x + [1] y + [0] = c_13(#compare#(y,x)) Following rules are (at-least) weakly oriented: #compare#(#pos(x),#pos(y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [1] = c_17(#compare#(x,y)) #compare#(#s(x),#s(y)) = [1] x + [1] y + [8] >= [1] x + [1] y + [2] = c_19(#compare#(x,y)) #less#(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = c_20(#compare#(x,y)) insert#(x,l) = [1] l + [1] x + [0] >= [1] l + [1] x + [0] = insert#1#(l,x) insert#1#(dd(y,ys),x) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [0] = #less#(y,x) insert#1#(dd(y,ys),x) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] ys + [0] = insert#2#(#less(y,x),x,y,ys) insert#2#(#true(),x,y,ys) = [1] x + [1] ys + [0] >= [1] x + [1] ys + [0] = insert#(x,ys) insertD#(x,l) = [1] l + [2] x + [0] >= [1] l + [2] x + [0] = insertD#1#(l,x) insertD#1#(dd(y,ys),x) = [2] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [0] = #less#(y,x) insertD#1#(dd(y,ys),x) = [2] x + [1] y + [1] ys + [0] >= [2] x + [1] y + [1] ys + [0] = insertD#2#(#less(y,x),x,y,ys) insertD#2#(#true(),x,y,ys) = [2] x + [1] y + [1] ys + [0] >= [2] x + [1] ys + [0] = insertD#(x,ys) insertionsort#(l) = [1] l + [0] >= [1] l + [0] = insertionsort#1#(l) insertionsort#1#(dd(x,xs)) = [1] x + [1] xs + [0] >= [1] x + [1] xs + [0] = insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) = [1] x + [1] xs + [0] >= [1] xs + [0] = insertionsort#(xs) insertionsortD#(l) = [4] l + [0] >= [4] l + [0] = insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) = [4] x + [4] xs + [0] >= [2] x + [1] xs + [0] = insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) = [4] x + [4] xs + [0] >= [4] xs + [0] = insertionsortD#(xs) #cklt(#EQ()) = [0] >= [0] = #false() #cklt(#GT()) = [0] >= [0] = #false() #cklt(#LT()) = [0] >= [0] = #true() #compare(#0(),#0()) = [0] >= [0] = #EQ() #compare(#0(),#neg(y)) = [0] >= [0] = #GT() #compare(#0(),#pos(y)) = [0] >= [0] = #LT() #compare(#0(),#s(y)) = [0] >= [0] = #LT() #compare(#neg(x),#0()) = [0] >= [0] = #LT() #compare(#neg(x),#neg(y)) = [0] >= [0] = #compare(y,x) #compare(#neg(x),#pos(y)) = [0] >= [0] = #LT() #compare(#pos(x),#0()) = [0] >= [0] = #GT() #compare(#pos(x),#neg(y)) = [0] >= [0] = #GT() #compare(#pos(x),#pos(y)) = [0] >= [0] = #compare(x,y) #compare(#s(x),#0()) = [0] >= [0] = #GT() #compare(#s(x),#s(y)) = [0] >= [0] = #compare(x,y) #less(x,y) = [0] >= [0] = #cklt(#compare(x,y)) insert(x,l) = [1] l + [1] x + [0] >= [1] l + [1] x + [0] = insert#1(l,x) insert#1(dd(y,ys),x) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) = [1] x + [0] >= [1] x + [0] = dd(x,nil()) insert#2(#false(),x,y,ys) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = dd(y,insert(x,ys)) insertD(x,l) = [1] l + [1] x + [0] >= [1] l + [1] x + [0] = insertD#1(l,x) insertD#1(dd(y,ys),x) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) = [1] x + [0] >= [1] x + [0] = dd(x,nil()) insertD#2(#false(),x,y,ys) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = dd(y,insertD(x,ys)) insertionsort(l) = [1] l + [0] >= [1] l + [0] = insertionsort#1(l) insertionsort#1(dd(x,xs)) = [1] x + [1] xs + [0] >= [1] x + [1] xs + [0] = insert(x,insertionsort(xs)) insertionsort#1(nil()) = [0] >= [0] = nil() insertionsortD(l) = [1] l + [0] >= [1] l + [0] = insertionsortD#1(l) insertionsortD#1(dd(x,xs)) = [1] x + [1] xs + [0] >= [1] x + [1] xs + [0] = insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) = [0] >= [0] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 6.b:1.b:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) - Weak DPs: #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#compare#(x,y)) insert#(x,l) -> insert#1#(l,x) insert#1#(dd(y,ys),x) -> #less#(y,x) insert#1#(dd(y,ys),x) -> insert#2#(#less(y,x),x,y,ys) insert#2#(#true(),x,y,ys) -> insert#(x,ys) insertD#(x,l) -> insertD#1#(l,x) insertD#1#(dd(y,ys),x) -> #less#(y,x) insertD#1#(dd(y,ys),x) -> insertD#2#(#less(y,x),x,y,ys) insertD#2#(#true(),x,y,ys) -> insertD#(x,ys) insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#cklt) = {1}, uargs(dd) = {2}, uargs(insert) = {2}, uargs(insert#2) = {1}, uargs(insertD) = {2}, uargs(insertD#2) = {1}, uargs(insert#) = {2}, uargs(insert#2#) = {1}, uargs(insertD#) = {2}, uargs(insertD#2#) = {1}, uargs(c_13) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1}, uargs(c_20) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#EQ) = [0] p(#GT) = [0] p(#LT) = [0] p(#abs) = [1] x1 + [1] p(#cklt) = [1] x1 + [0] p(#compare) = [0] p(#false) = [0] p(#less) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [1] p(#s) = [1] x1 + [0] p(#true) = [0] p(dd) = [1] x1 + [1] x2 + [0] p(insert) = [1] x1 + [1] x2 + [0] p(insert#1) = [1] x1 + [1] x2 + [0] p(insert#2) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [0] p(insertD) = [4] x1 + [1] x2 + [0] p(insertD#1) = [1] x1 + [4] x2 + [0] p(insertD#2) = [1] x1 + [4] x2 + [1] x3 + [1] x4 + [0] p(insertionsort) = [1] x1 + [0] p(insertionsort#1) = [1] x1 + [0] p(insertionsortD) = [6] x1 + [0] p(insertionsortD#1) = [6] x1 + [0] p(nil) = [0] p(#abs#) = [4] p(#cklt#) = [4] x1 + [0] p(#compare#) = [1] x1 + [1] x2 + [0] p(#less#) = [1] x1 + [1] x2 + [0] p(insert#) = [1] x1 + [1] x2 + [2] p(insert#1#) = [1] x1 + [1] x2 + [2] p(insert#2#) = [1] x1 + [1] x2 + [1] x4 + [2] p(insertD#) = [1] x1 + [1] x2 + [0] p(insertD#1#) = [1] x1 + [1] x2 + [0] p(insertD#2#) = [1] x1 + [1] x2 + [1] x4 + [0] p(insertionsort#) = [2] x1 + [2] p(insertionsort#1#) = [2] x1 + [2] p(insertionsortD#) = [6] x1 + [0] p(insertionsortD#1#) = [6] x1 + [0] p(c_1) = [0] p(c_2) = [2] p(c_3) = [2] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] p(c_7) = [0] p(c_8) = [1] p(c_9) = [0] p(c_10) = [4] p(c_11) = [0] p(c_12) = [1] p(c_13) = [1] x1 + [0] p(c_14) = [1] p(c_15) = [2] p(c_16) = [1] p(c_17) = [1] x1 + [0] p(c_18) = [0] p(c_19) = [1] x1 + [0] p(c_20) = [1] x1 + [0] p(c_21) = [4] x1 + [0] p(c_22) = [2] x1 + [2] x2 + [1] p(c_23) = [2] p(c_24) = [0] p(c_25) = [1] x1 + [1] p(c_26) = [1] p(c_27) = [1] x1 + [1] x2 + [2] p(c_28) = [0] p(c_29) = [4] p(c_30) = [1] x1 + [1] p(c_31) = [0] p(c_32) = [2] x1 + [4] x2 + [2] p(c_33) = [0] p(c_34) = [1] x1 + [0] p(c_35) = [1] p(c_36) = [0] Following rules are strictly oriented: #compare#(#pos(x),#pos(y)) = [1] x + [1] y + [2] > [1] x + [1] y + [0] = c_17(#compare#(x,y)) Following rules are (at-least) weakly oriented: #compare#(#neg(x),#neg(y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = c_13(#compare#(y,x)) #compare#(#s(x),#s(y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = c_19(#compare#(x,y)) #less#(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = c_20(#compare#(x,y)) insert#(x,l) = [1] l + [1] x + [2] >= [1] l + [1] x + [2] = insert#1#(l,x) insert#1#(dd(y,ys),x) = [1] x + [1] y + [1] ys + [2] >= [1] x + [1] y + [0] = #less#(y,x) insert#1#(dd(y,ys),x) = [1] x + [1] y + [1] ys + [2] >= [1] x + [1] ys + [2] = insert#2#(#less(y,x),x,y,ys) insert#2#(#true(),x,y,ys) = [1] x + [1] ys + [2] >= [1] x + [1] ys + [2] = insert#(x,ys) insertD#(x,l) = [1] l + [1] x + [0] >= [1] l + [1] x + [0] = insertD#1#(l,x) insertD#1#(dd(y,ys),x) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [0] = #less#(y,x) insertD#1#(dd(y,ys),x) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] ys + [0] = insertD#2#(#less(y,x),x,y,ys) insertD#2#(#true(),x,y,ys) = [1] x + [1] ys + [0] >= [1] x + [1] ys + [0] = insertD#(x,ys) insertionsort#(l) = [2] l + [2] >= [2] l + [2] = insertionsort#1#(l) insertionsort#1#(dd(x,xs)) = [2] x + [2] xs + [2] >= [1] x + [1] xs + [2] = insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) = [2] x + [2] xs + [2] >= [2] xs + [2] = insertionsort#(xs) insertionsortD#(l) = [6] l + [0] >= [6] l + [0] = insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) = [6] x + [6] xs + [0] >= [1] x + [6] xs + [0] = insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) = [6] x + [6] xs + [0] >= [6] xs + [0] = insertionsortD#(xs) #cklt(#EQ()) = [0] >= [0] = #false() #cklt(#GT()) = [0] >= [0] = #false() #cklt(#LT()) = [0] >= [0] = #true() #compare(#0(),#0()) = [0] >= [0] = #EQ() #compare(#0(),#neg(y)) = [0] >= [0] = #GT() #compare(#0(),#pos(y)) = [0] >= [0] = #LT() #compare(#0(),#s(y)) = [0] >= [0] = #LT() #compare(#neg(x),#0()) = [0] >= [0] = #LT() #compare(#neg(x),#neg(y)) = [0] >= [0] = #compare(y,x) #compare(#neg(x),#pos(y)) = [0] >= [0] = #LT() #compare(#pos(x),#0()) = [0] >= [0] = #GT() #compare(#pos(x),#neg(y)) = [0] >= [0] = #GT() #compare(#pos(x),#pos(y)) = [0] >= [0] = #compare(x,y) #compare(#s(x),#0()) = [0] >= [0] = #GT() #compare(#s(x),#s(y)) = [0] >= [0] = #compare(x,y) #less(x,y) = [0] >= [0] = #cklt(#compare(x,y)) insert(x,l) = [1] l + [1] x + [0] >= [1] l + [1] x + [0] = insert#1(l,x) insert#1(dd(y,ys),x) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) = [1] x + [0] >= [1] x + [0] = dd(x,nil()) insert#2(#false(),x,y,ys) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) = [1] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = dd(y,insert(x,ys)) insertD(x,l) = [1] l + [4] x + [0] >= [1] l + [4] x + [0] = insertD#1(l,x) insertD#1(dd(y,ys),x) = [4] x + [1] y + [1] ys + [0] >= [4] x + [1] y + [1] ys + [0] = insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) = [4] x + [0] >= [1] x + [0] = dd(x,nil()) insertD#2(#false(),x,y,ys) = [4] x + [1] y + [1] ys + [0] >= [1] x + [1] y + [1] ys + [0] = dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) = [4] x + [1] y + [1] ys + [0] >= [4] x + [1] y + [1] ys + [0] = dd(y,insertD(x,ys)) insertionsort(l) = [1] l + [0] >= [1] l + [0] = insertionsort#1(l) insertionsort#1(dd(x,xs)) = [1] x + [1] xs + [0] >= [1] x + [1] xs + [0] = insert(x,insertionsort(xs)) insertionsort#1(nil()) = [0] >= [0] = nil() insertionsortD(l) = [6] l + [0] >= [6] l + [0] = insertionsortD#1(l) insertionsortD#1(dd(x,xs)) = [6] x + [6] xs + [0] >= [4] x + [6] xs + [0] = insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) = [0] >= [0] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 6.b:1.b:5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: #compare#(#neg(x),#neg(y)) -> c_13(#compare#(y,x)) #compare#(#pos(x),#pos(y)) -> c_17(#compare#(x,y)) #compare#(#s(x),#s(y)) -> c_19(#compare#(x,y)) #less#(x,y) -> c_20(#compare#(x,y)) insert#(x,l) -> insert#1#(l,x) insert#1#(dd(y,ys),x) -> #less#(y,x) insert#1#(dd(y,ys),x) -> insert#2#(#less(y,x),x,y,ys) insert#2#(#true(),x,y,ys) -> insert#(x,ys) insertD#(x,l) -> insertD#1#(l,x) insertD#1#(dd(y,ys),x) -> #less#(y,x) insertD#1#(dd(y,ys),x) -> insertD#2#(#less(y,x),x,y,ys) insertD#2#(#true(),x,y,ys) -> insertD#(x,ys) insertionsort#(l) -> insertionsort#1#(l) insertionsort#1#(dd(x,xs)) -> insert#(x,insertionsort(xs)) insertionsort#1#(dd(x,xs)) -> insertionsort#(xs) insertionsortD#(l) -> insertionsortD#1#(l) insertionsortD#1#(dd(x,xs)) -> insertD#(x,insertionsortD(xs)) insertionsortD#1#(dd(x,xs)) -> insertionsortD#(xs) - Weak TRS: #cklt(#EQ()) -> #false() #cklt(#GT()) -> #false() #cklt(#LT()) -> #true() #compare(#0(),#0()) -> #EQ() #compare(#0(),#neg(y)) -> #GT() #compare(#0(),#pos(y)) -> #LT() #compare(#0(),#s(y)) -> #LT() #compare(#neg(x),#0()) -> #LT() #compare(#neg(x),#neg(y)) -> #compare(y,x) #compare(#neg(x),#pos(y)) -> #LT() #compare(#pos(x),#0()) -> #GT() #compare(#pos(x),#neg(y)) -> #GT() #compare(#pos(x),#pos(y)) -> #compare(x,y) #compare(#s(x),#0()) -> #GT() #compare(#s(x),#s(y)) -> #compare(x,y) #less(x,y) -> #cklt(#compare(x,y)) insert(x,l) -> insert#1(l,x) insert#1(dd(y,ys),x) -> insert#2(#less(y,x),x,y,ys) insert#1(nil(),x) -> dd(x,nil()) insert#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insert#2(#true(),x,y,ys) -> dd(y,insert(x,ys)) insertD(x,l) -> insertD#1(l,x) insertD#1(dd(y,ys),x) -> insertD#2(#less(y,x),x,y,ys) insertD#1(nil(),x) -> dd(x,nil()) insertD#2(#false(),x,y,ys) -> dd(x,dd(y,ys)) insertD#2(#true(),x,y,ys) -> dd(y,insertD(x,ys)) insertionsort(l) -> insertionsort#1(l) insertionsort#1(dd(x,xs)) -> insert(x,insertionsort(xs)) insertionsort#1(nil()) -> nil() insertionsortD(l) -> insertionsortD#1(l) insertionsortD#1(dd(x,xs)) -> insertD(x,insertionsortD(xs)) insertionsortD#1(nil()) -> nil() - Signature: {#abs/1,#cklt/1,#compare/2,#less/2,insert/2,insert#1/2,insert#2/4,insertD/2,insertD#1/2,insertD#2/4 ,insertionsort/1,insertionsort#1/1,insertionsortD/1,insertionsortD#1/1,#abs#/1,#cklt#/1,#compare#/2,#less#/2 ,insert#/2,insert#1#/2,insert#2#/4,insertD#/2,insertD#1#/2,insertD#2#/4,insertionsort#/1,insertionsort#1#/1 ,insertionsortD#/1,insertionsortD#1#/1} / {#0/0,#EQ/0,#GT/0,#LT/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2 ,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0,c_7/0,c_8/0,c_9/0,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/0,c_19/1,c_20/1,c_21/1,c_22/2,c_23/0,c_24/0,c_25/1,c_26/1,c_27/2,c_28/0,c_29/0,c_30/1 ,c_31/1,c_32/2,c_33/0,c_34/1,c_35/2,c_36/0} - Obligation: innermost runtime complexity wrt. defined symbols {#abs#,#cklt#,#compare#,#less#,insert#,insert#1#,insert#2# ,insertD#,insertD#1#,insertD#2#,insertionsort#,insertionsort#1#,insertionsortD# ,insertionsortD#1#} and constructors {#0,#EQ,#GT,#LT,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))