WORST_CASE(?,O(n^5)) * Step 1: DependencyPairs WORST_CASE(?,O(n^5)) + Considered Problem: - Strict 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1 ,'s/1,'true/0,dd/2,nil/0,tuple'2/2} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt,'compare,'less,merge,merge'1,merge'2,merge'3 ,mergesort,mergesort'1,mergesort'2,mergesort'3,msplit,msplit'1,msplit'2,msplit'3} and constructors {'0,'EQ ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs 'cklt#('EQ()) -> c_1() 'cklt#('GT()) -> c_2() 'cklt#('LT()) -> c_3() 'compare#('0(),'0()) -> c_4() 'compare#('0(),'neg(y)) -> c_5() 'compare#('0(),'pos(y)) -> c_6() 'compare#('0(),'s(y)) -> c_7() 'compare#('neg(x),'0()) -> c_8() 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('neg(x),'pos(y)) -> c_10() 'compare#('pos(x),'0()) -> c_11() 'compare#('pos(x),'neg(y)) -> c_12() 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'0()) -> c_14() 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'1#(nil(),l2) -> c_19() merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)) merge'2#(nil(),x,xs) -> c_21() merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) mergesort'1#(nil()) -> c_26() mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))),msplit#(dd(x1,dd(x2,xs')))) mergesort'2#(nil(),x1) -> c_28() mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)),mergesort#(l1),mergesort#(l2)) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) msplit'1#(nil()) -> c_32() msplit'2#(dd(x2,xs'),x1) -> c_33(msplit'3#(msplit(xs'),x1,x2),msplit#(xs')) msplit'2#(nil(),x1) -> c_34() msplit'3#(tuple'2(l1,l2),x1,x2) -> c_35() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^5)) + Considered Problem: - Strict DPs: 'cklt#('EQ()) -> c_1() 'cklt#('GT()) -> c_2() 'cklt#('LT()) -> c_3() 'compare#('0(),'0()) -> c_4() 'compare#('0(),'neg(y)) -> c_5() 'compare#('0(),'pos(y)) -> c_6() 'compare#('0(),'s(y)) -> c_7() 'compare#('neg(x),'0()) -> c_8() 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('neg(x),'pos(y)) -> c_10() 'compare#('pos(x),'0()) -> c_11() 'compare#('pos(x),'neg(y)) -> c_12() 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'0()) -> c_14() 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'1#(nil(),l2) -> c_19() merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)) merge'2#(nil(),x,xs) -> c_21() merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) mergesort'1#(nil()) -> c_26() mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))),msplit#(dd(x1,dd(x2,xs')))) mergesort'2#(nil(),x1) -> c_28() mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)),mergesort#(l1),mergesort#(l2)) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) msplit'1#(nil()) -> c_32() msplit'2#(dd(x2,xs'),x1) -> c_33(msplit'3#(msplit(xs'),x1,x2),msplit#(xs')) msplit'2#(nil(),x1) -> c_34() msplit'3#(tuple'2(l1,l2),x1,x2) -> c_35() - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/2 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/2,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,5,6,7,8,10,11,12,14,19,21,26,28,32,34,35} by application of Pre({1,2,3,4,5,6,7,8,10,11,12,14,19,21,26,28,32,34,35}) = {9,13,15,16,17,18,24,25,30,31,33}. Here rules are labelled as follows: 1: 'cklt#('EQ()) -> c_1() 2: 'cklt#('GT()) -> c_2() 3: 'cklt#('LT()) -> c_3() 4: 'compare#('0(),'0()) -> c_4() 5: 'compare#('0(),'neg(y)) -> c_5() 6: 'compare#('0(),'pos(y)) -> c_6() 7: 'compare#('0(),'s(y)) -> c_7() 8: 'compare#('neg(x),'0()) -> c_8() 9: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 10: 'compare#('neg(x),'pos(y)) -> c_10() 11: 'compare#('pos(x),'0()) -> c_11() 12: 'compare#('pos(x),'neg(y)) -> c_12() 13: 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 14: 'compare#('s(x),'0()) -> c_14() 15: 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 16: 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) 17: merge#(l1,l2) -> c_17(merge'1#(l1,l2)) 18: merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) 19: merge'1#(nil(),l2) -> c_19() 20: merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)) 21: merge'2#(nil(),x,xs) -> c_21() 22: merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) 23: merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) 24: mergesort#(l) -> c_24(mergesort'1#(l)) 25: mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) 26: mergesort'1#(nil()) -> c_26() 27: mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))),msplit#(dd(x1,dd(x2,xs')))) 28: mergesort'2#(nil(),x1) -> c_28() 29: mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)),mergesort#(l1),mergesort#(l2)) 30: msplit#(l) -> c_30(msplit'1#(l)) 31: msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) 32: msplit'1#(nil()) -> c_32() 33: msplit'2#(dd(x2,xs'),x1) -> c_33(msplit'3#(msplit(xs'),x1,x2),msplit#(xs')) 34: msplit'2#(nil(),x1) -> c_34() 35: msplit'3#(tuple'2(l1,l2),x1,x2) -> c_35() * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^5)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)) merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))),msplit#(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)),mergesort#(l1),mergesort#(l2)) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) msplit'2#(dd(x2,xs'),x1) -> c_33(msplit'3#(msplit(xs'),x1,x2),msplit#(xs')) - Weak DPs: 'cklt#('EQ()) -> c_1() 'cklt#('GT()) -> c_2() 'cklt#('LT()) -> c_3() 'compare#('0(),'0()) -> c_4() 'compare#('0(),'neg(y)) -> c_5() 'compare#('0(),'pos(y)) -> c_6() 'compare#('0(),'s(y)) -> c_7() 'compare#('neg(x),'0()) -> c_8() 'compare#('neg(x),'pos(y)) -> c_10() 'compare#('pos(x),'0()) -> c_11() 'compare#('pos(x),'neg(y)) -> c_12() 'compare#('s(x),'0()) -> c_14() merge'1#(nil(),l2) -> c_19() merge'2#(nil(),x,xs) -> c_21() mergesort'1#(nil()) -> c_26() mergesort'2#(nil(),x1) -> c_28() msplit'1#(nil()) -> c_32() msplit'2#(nil(),x1) -> c_34() msplit'3#(tuple'2(l1,l2),x1,x2) -> c_35() - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/2 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/2,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('s(x),'0()) -> c_14():28 -->_1 'compare#('pos(x),'neg(y)) -> c_12():27 -->_1 'compare#('pos(x),'0()) -> c_11():26 -->_1 'compare#('neg(x),'pos(y)) -> c_10():25 -->_1 'compare#('neg(x),'0()) -> c_8():24 -->_1 'compare#('0(),'s(y)) -> c_7():23 -->_1 'compare#('0(),'pos(y)) -> c_6():22 -->_1 'compare#('0(),'neg(y)) -> c_5():21 -->_1 'compare#('0(),'0()) -> c_4():20 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 2:S:'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('s(x),'0()) -> c_14():28 -->_1 'compare#('pos(x),'neg(y)) -> c_12():27 -->_1 'compare#('pos(x),'0()) -> c_11():26 -->_1 'compare#('neg(x),'pos(y)) -> c_10():25 -->_1 'compare#('neg(x),'0()) -> c_8():24 -->_1 'compare#('0(),'s(y)) -> c_7():23 -->_1 'compare#('0(),'pos(y)) -> c_6():22 -->_1 'compare#('0(),'neg(y)) -> c_5():21 -->_1 'compare#('0(),'0()) -> c_4():20 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 3:S:'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) -->_1 'compare#('s(x),'0()) -> c_14():28 -->_1 'compare#('pos(x),'neg(y)) -> c_12():27 -->_1 'compare#('pos(x),'0()) -> c_11():26 -->_1 'compare#('neg(x),'pos(y)) -> c_10():25 -->_1 'compare#('neg(x),'0()) -> c_8():24 -->_1 'compare#('0(),'s(y)) -> c_7():23 -->_1 'compare#('0(),'pos(y)) -> c_6():22 -->_1 'compare#('0(),'neg(y)) -> c_5():21 -->_1 'compare#('0(),'0()) -> c_4():20 -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 4:S:'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) -->_2 'compare#('s(x),'0()) -> c_14():28 -->_2 'compare#('pos(x),'neg(y)) -> c_12():27 -->_2 'compare#('pos(x),'0()) -> c_11():26 -->_2 'compare#('neg(x),'pos(y)) -> c_10():25 -->_2 'compare#('neg(x),'0()) -> c_8():24 -->_2 'compare#('0(),'s(y)) -> c_7():23 -->_2 'compare#('0(),'pos(y)) -> c_6():22 -->_2 'compare#('0(),'neg(y)) -> c_5():21 -->_2 'compare#('0(),'0()) -> c_4():20 -->_1 'cklt#('LT()) -> c_3():19 -->_1 'cklt#('GT()) -> c_2():18 -->_1 'cklt#('EQ()) -> c_1():17 -->_2 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_2 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_2 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 5:S:merge#(l1,l2) -> c_17(merge'1#(l1,l2)) -->_1 merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)):6 -->_1 merge'1#(nil(),l2) -> c_19():29 6:S:merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) -->_1 merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)):7 -->_1 merge'2#(nil(),x,xs) -> c_21():30 7:S:merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)) -->_1 merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))):9 -->_1 merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)):8 -->_2 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)):4 8:S:merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) -->_1 merge#(l1,l2) -> c_17(merge'1#(l1,l2)):5 9:S:merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) -->_1 merge#(l1,l2) -> c_17(merge'1#(l1,l2)):5 10:S:mergesort#(l) -> c_24(mergesort'1#(l)) -->_1 mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)):11 -->_1 mergesort'1#(nil()) -> c_26():31 11:S:mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) -->_1 mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))) ,msplit#(dd(x1,dd(x2,xs')))):12 -->_1 mergesort'2#(nil(),x1) -> c_28():32 12:S:mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))),msplit#(dd(x1,dd(x2,xs')))) -->_2 msplit#(l) -> c_30(msplit'1#(l)):14 -->_1 mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)) ,mergesort#(l1) ,mergesort#(l2)):13 13:S:mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)),mergesort#(l1),mergesort#(l2)) -->_3 mergesort#(l) -> c_24(mergesort'1#(l)):10 -->_2 mergesort#(l) -> c_24(mergesort'1#(l)):10 -->_1 merge#(l1,l2) -> c_17(merge'1#(l1,l2)):5 14:S:msplit#(l) -> c_30(msplit'1#(l)) -->_1 msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)):15 -->_1 msplit'1#(nil()) -> c_32():33 15:S:msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) -->_1 msplit'2#(dd(x2,xs'),x1) -> c_33(msplit'3#(msplit(xs'),x1,x2),msplit#(xs')):16 -->_1 msplit'2#(nil(),x1) -> c_34():34 16:S:msplit'2#(dd(x2,xs'),x1) -> c_33(msplit'3#(msplit(xs'),x1,x2),msplit#(xs')) -->_1 msplit'3#(tuple'2(l1,l2),x1,x2) -> c_35():35 -->_2 msplit#(l) -> c_30(msplit'1#(l)):14 17:W:'cklt#('EQ()) -> c_1() 18:W:'cklt#('GT()) -> c_2() 19:W:'cklt#('LT()) -> c_3() 20:W:'compare#('0(),'0()) -> c_4() 21:W:'compare#('0(),'neg(y)) -> c_5() 22:W:'compare#('0(),'pos(y)) -> c_6() 23:W:'compare#('0(),'s(y)) -> c_7() 24:W:'compare#('neg(x),'0()) -> c_8() 25:W:'compare#('neg(x),'pos(y)) -> c_10() 26:W:'compare#('pos(x),'0()) -> c_11() 27:W:'compare#('pos(x),'neg(y)) -> c_12() 28:W:'compare#('s(x),'0()) -> c_14() 29:W:merge'1#(nil(),l2) -> c_19() 30:W:merge'2#(nil(),x,xs) -> c_21() 31:W:mergesort'1#(nil()) -> c_26() 32:W:mergesort'2#(nil(),x1) -> c_28() 33:W:msplit'1#(nil()) -> c_32() 34:W:msplit'2#(nil(),x1) -> c_34() 35:W:msplit'3#(tuple'2(l1,l2),x1,x2) -> c_35() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 31: mergesort'1#(nil()) -> c_26() 32: mergesort'2#(nil(),x1) -> c_28() 33: msplit'1#(nil()) -> c_32() 34: msplit'2#(nil(),x1) -> c_34() 35: msplit'3#(tuple'2(l1,l2),x1,x2) -> c_35() 29: merge'1#(nil(),l2) -> c_19() 30: merge'2#(nil(),x,xs) -> c_21() 17: 'cklt#('EQ()) -> c_1() 18: 'cklt#('GT()) -> c_2() 19: 'cklt#('LT()) -> c_3() 20: 'compare#('0(),'0()) -> c_4() 21: 'compare#('0(),'neg(y)) -> c_5() 22: 'compare#('0(),'pos(y)) -> c_6() 23: 'compare#('0(),'s(y)) -> c_7() 24: 'compare#('neg(x),'0()) -> c_8() 25: 'compare#('neg(x),'pos(y)) -> c_10() 26: 'compare#('pos(x),'0()) -> c_11() 27: 'compare#('pos(x),'neg(y)) -> c_12() 28: 'compare#('s(x),'0()) -> c_14() * Step 4: SimplifyRHS WORST_CASE(?,O(n^5)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)) merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))),msplit#(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)),mergesort#(l1),mergesort#(l2)) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) msplit'2#(dd(x2,xs'),x1) -> c_33(msplit'3#(msplit(xs'),x1,x2),msplit#(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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/2 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/2,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 2:S:'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 3:S:'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 4:S:'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)) -->_2 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_2 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_2 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 5:S:merge#(l1,l2) -> c_17(merge'1#(l1,l2)) -->_1 merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)):6 6:S:merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) -->_1 merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)):7 7:S:merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)) -->_1 merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))):9 -->_1 merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)):8 -->_2 'less#(x,y) -> c_16('cklt#('compare(x,y)),'compare#(x,y)):4 8:S:merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) -->_1 merge#(l1,l2) -> c_17(merge'1#(l1,l2)):5 9:S:merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) -->_1 merge#(l1,l2) -> c_17(merge'1#(l1,l2)):5 10:S:mergesort#(l) -> c_24(mergesort'1#(l)) -->_1 mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)):11 11:S:mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) -->_1 mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))) ,msplit#(dd(x1,dd(x2,xs')))):12 12:S:mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))),msplit#(dd(x1,dd(x2,xs')))) -->_2 msplit#(l) -> c_30(msplit'1#(l)):14 -->_1 mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)) ,mergesort#(l1) ,mergesort#(l2)):13 13:S:mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)),mergesort#(l1),mergesort#(l2)) -->_3 mergesort#(l) -> c_24(mergesort'1#(l)):10 -->_2 mergesort#(l) -> c_24(mergesort'1#(l)):10 -->_1 merge#(l1,l2) -> c_17(merge'1#(l1,l2)):5 14:S:msplit#(l) -> c_30(msplit'1#(l)) -->_1 msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)):15 15:S:msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) -->_1 msplit'2#(dd(x2,xs'),x1) -> c_33(msplit'3#(msplit(xs'),x1,x2),msplit#(xs')):16 16:S:msplit'2#(dd(x2,xs'),x1) -> c_33(msplit'3#(msplit(xs'),x1,x2),msplit#(xs')) -->_2 msplit#(l) -> c_30(msplit'1#(l)):14 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: 'less#(x,y) -> c_16('compare#(x,y)) msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(xs')) * Step 5: DecomposeDG WORST_CASE(?,O(n^5)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)) merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))),msplit#(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)),mergesort#(l1),mergesort#(l2)) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + 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 mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))),msplit#(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)),mergesort#(l1),mergesort#(l2)) and a lower component 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)) merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(xs')) Further, following extension rules are added to the lower component. mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) ** Step 5.a:1: SimplifyRHS WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))),msplit#(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)),mergesort#(l1),mergesort#(l2)) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:mergesort#(l) -> c_24(mergesort'1#(l)) -->_1 mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)):2 2:S:mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) -->_1 mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))) ,msplit#(dd(x1,dd(x2,xs')))):3 3:S:mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs')))),msplit#(dd(x1,dd(x2,xs')))) -->_1 mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)) ,mergesort#(l1) ,mergesort#(l2)):4 4:S:mergesort'3#(tuple'2(l1,l2)) -> c_29(merge#(mergesort(l1),mergesort(l2)),mergesort#(l1),mergesort#(l2)) -->_3 mergesort#(l) -> c_24(mergesort'1#(l)):1 -->_2 mergesort#(l) -> c_24(mergesort'1#(l)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs'))))) mergesort'3#(tuple'2(l1,l2)) -> c_29(mergesort#(l1),mergesort#(l2)) ** Step 5.a:2: UsableRules WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs'))))) mergesort'3#(tuple'2(l1,l2)) -> c_29(mergesort#(l1),mergesort#(l2)) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/1,c_28/0,c_29/2,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs'))))) mergesort'3#(tuple'2(l1,l2)) -> c_29(mergesort#(l1),mergesort#(l2)) ** Step 5.a:3: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs'))))) mergesort'3#(tuple'2(l1,l2)) -> c_29(mergesort#(l1),mergesort#(l2)) - Weak TRS: msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/1,c_28/0,c_29/2,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + 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(msplit'3) = {1}, uargs(mergesort'3#) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1}, uargs(c_27) = {1}, uargs(c_29) = {1,2} 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('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(merge) = [0] p(merge'1) = [0] p(merge'2) = [0] p(merge'3) = [0] p(mergesort) = [0] p(mergesort'1) = [0] p(mergesort'2) = [0] p(mergesort'3) = [0] p(msplit) = [7] p(msplit'1) = [7] p(msplit'2) = [7] p(msplit'3) = [1] x1 + [0] p(nil) = [0] p(tuple'2) = [7] p('cklt#) = [0] p('compare#) = [0] p('less#) = [0] p(merge#) = [0] p(merge'1#) = [0] p(merge'2#) = [0] p(merge'3#) = [0] p(mergesort#) = [0] p(mergesort'1#) = [0] p(mergesort'2#) = [0] p(mergesort'3#) = [1] x1 + [0] p(msplit#) = [0] p(msplit'1#) = [0] p(msplit'2#) = [0] p(msplit'3#) = [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) = [1] x1 + [0] p(c_25) = [1] x1 + [0] p(c_26) = [0] p(c_27) = [1] x1 + [0] p(c_28) = [0] p(c_29) = [1] x1 + [1] x2 + [0] p(c_30) = [0] p(c_31) = [0] p(c_32) = [0] p(c_33) = [0] p(c_34) = [0] p(c_35) = [0] Following rules are strictly oriented: mergesort'3#(tuple'2(l1,l2)) = [7] > [0] = c_29(mergesort#(l1),mergesort#(l2)) Following rules are (at-least) weakly oriented: mergesort#(l) = [0] >= [0] = c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) = [0] >= [0] = c_25(mergesort'2#(xs,x1)) mergesort'2#(dd(x2,xs'),x1) = [0] >= [7] = c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs'))))) msplit(l) = [7] >= [7] = msplit'1(l) msplit'1(dd(x1,xs)) = [7] >= [7] = msplit'2(xs,x1) msplit'1(nil()) = [7] >= [7] = tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) = [7] >= [7] = msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) = [7] >= [7] = tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) = [7] >= [7] = tuple'2(dd(x1,l1),dd(x2,l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 5.a:4: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs'))))) - Weak DPs: mergesort'3#(tuple'2(l1,l2)) -> c_29(mergesort#(l1),mergesort#(l2)) - Weak TRS: msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/1,c_28/0,c_29/2,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + 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(msplit'3) = {1}, uargs(mergesort'3#) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1}, uargs(c_27) = {1}, uargs(c_29) = {1,2} 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('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) = [0] p(merge) = [0] p(merge'1) = [0] p(merge'2) = [0] p(merge'3) = [0] p(mergesort) = [0] p(mergesort'1) = [0] p(mergesort'2) = [0] p(mergesort'3) = [0] p(msplit) = [0] p(msplit'1) = [0] p(msplit'2) = [0] p(msplit'3) = [1] x1 + [0] p(nil) = [0] p(tuple'2) = [1] x2 + [0] p('cklt#) = [0] p('compare#) = [0] p('less#) = [0] p(merge#) = [0] p(merge'1#) = [0] p(merge'2#) = [0] p(merge'3#) = [0] p(mergesort#) = [1] p(mergesort'1#) = [0] p(mergesort'2#) = [7] p(mergesort'3#) = [1] x1 + [7] p(msplit#) = [0] p(msplit'1#) = [0] p(msplit'2#) = [0] p(msplit'3#) = [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) = [2] p(c_20) = [4] p(c_21) = [1] p(c_22) = [1] p(c_23) = [4] x1 + [2] p(c_24) = [1] x1 + [0] p(c_25) = [1] x1 + [1] p(c_26) = [1] p(c_27) = [1] x1 + [4] p(c_28) = [0] p(c_29) = [1] x1 + [1] x2 + [5] p(c_30) = [2] x1 + [0] p(c_31) = [1] p(c_32) = [1] p(c_33) = [2] p(c_34) = [0] p(c_35) = [1] Following rules are strictly oriented: mergesort#(l) = [1] > [0] = c_24(mergesort'1#(l)) Following rules are (at-least) weakly oriented: mergesort'1#(dd(x1,xs)) = [0] >= [8] = c_25(mergesort'2#(xs,x1)) mergesort'2#(dd(x2,xs'),x1) = [7] >= [11] = c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs'))))) mergesort'3#(tuple'2(l1,l2)) = [1] l2 + [7] >= [7] = c_29(mergesort#(l1),mergesort#(l2)) msplit(l) = [0] >= [0] = msplit'1(l) msplit'1(dd(x1,xs)) = [0] >= [0] = msplit'2(xs,x1) msplit'1(nil()) = [0] >= [0] = tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) = [0] >= [0] = msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) = [0] >= [0] = tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) = [1] l2 + [0] >= [0] = tuple'2(dd(x1,l1),dd(x2,l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 5.a:5: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs'))))) - Weak DPs: mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'3#(tuple'2(l1,l2)) -> c_29(mergesort#(l1),mergesort#(l2)) - Weak TRS: msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/1,c_28/0,c_29/2,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + 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(msplit'3) = {1}, uargs(mergesort'3#) = {1}, uargs(c_24) = {1}, uargs(c_25) = {1}, uargs(c_27) = {1}, uargs(c_29) = {1,2} 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('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] x2 + [0] p(merge) = [0] p(merge'1) = [0] p(merge'2) = [0] p(merge'3) = [0] p(mergesort) = [0] p(mergesort'1) = [0] p(mergesort'2) = [0] p(mergesort'3) = [0] p(msplit) = [0] p(msplit'1) = [0] p(msplit'2) = [0] p(msplit'3) = [1] x1 + [0] p(nil) = [0] p(tuple'2) = [0] p('cklt#) = [0] p('compare#) = [0] p('less#) = [0] p(merge#) = [0] p(merge'1#) = [0] p(merge'2#) = [0] p(merge'3#) = [0] p(mergesort#) = [0] p(mergesort'1#) = [0] p(mergesort'2#) = [1] p(mergesort'3#) = [1] x1 + [0] p(msplit#) = [0] p(msplit'1#) = [0] p(msplit'2#) = [0] p(msplit'3#) = [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) = [1] x1 + [0] p(c_25) = [1] x1 + [0] p(c_26) = [0] p(c_27) = [1] x1 + [0] p(c_28) = [0] p(c_29) = [1] x1 + [1] x2 + [0] p(c_30) = [0] p(c_31) = [0] p(c_32) = [0] p(c_33) = [0] p(c_34) = [0] p(c_35) = [0] Following rules are strictly oriented: mergesort'2#(dd(x2,xs'),x1) = [1] > [0] = c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs'))))) Following rules are (at-least) weakly oriented: mergesort#(l) = [0] >= [0] = c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) = [0] >= [1] = c_25(mergesort'2#(xs,x1)) mergesort'3#(tuple'2(l1,l2)) = [0] >= [0] = c_29(mergesort#(l1),mergesort#(l2)) msplit(l) = [0] >= [0] = msplit'1(l) msplit'1(dd(x1,xs)) = [0] >= [0] = msplit'2(xs,x1) msplit'1(nil()) = [0] >= [0] = tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) = [0] >= [0] = msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) = [0] >= [0] = tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) = [0] >= [0] = tuple'2(dd(x1,l1),dd(x2,l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 5.a:6: MI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) - Weak DPs: mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs'))))) mergesort'3#(tuple'2(l1,l2)) -> c_29(mergesort#(l1),mergesort#(l2)) - Weak TRS: msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/1,c_28/0,c_29/2,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + Applied Processor: MI {miKind = Automaton Nothing, miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton Nothing: The following argument positions are considered usable: uargs(c_24) = {1}, uargs(c_25) = {1}, uargs(c_27) = {1}, uargs(c_29) = {1,2} Following symbols are considered usable: {msplit,msplit'1,msplit'2,msplit'3,'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3#,mergesort# ,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2#,msplit'3#} TcT has computed the following interpretation: p('0) = [0] [0] [0] p('EQ) = [0] [0] [0] p('GT) = [0] [0] [0] p('LT) = [0] [0] [0] p('cklt) = [0] [0] [0] p('compare) = [0] [0] [0] p('false) = [0] [0] [0] p('less) = [0] [0] [0] p('neg) = [0] [0] [0] p('pos) = [0] [0] [0] p('s) = [0] [0] [0] p('true) = [0] [0] [0] p(dd) = [0 0 0] [0 0 0] [1] [0 1 1] x_1 + [0 1 1] x_2 + [1] [0 0 0] [1 0 0] [0] p(merge) = [0] [0] [0] p(merge'1) = [0] [0] [0] p(merge'2) = [0] [0] [0] p(merge'3) = [0] [0] [0] p(mergesort) = [0] [0] [0] p(mergesort'1) = [0] [0] [0] p(mergesort'2) = [0] [0] [0] p(mergesort'3) = [0] [0] [0] p(msplit) = [0 1 0] [0] [1 0 1] x_1 + [0] [0 0 0] [1] p(msplit'1) = [0 1 0] [0] [1 0 1] x_1 + [0] [0 0 0] [1] p(msplit'2) = [0 1 1] [0 1 1] [1] [1 0 0] x_1 + [0 0 0] x_2 + [1] [0 0 0] [0 0 0] [1] p(msplit'3) = [1 1 1] [0 1 1] [0 1 1] [1] [0 0 1] x_1 + [0 0 0] x_2 + [0 0 0] x_3 + [1] [0 0 0] [0 0 0] [0 0 0] [1] p(nil) = [0] [0] [0] p(tuple'2) = [0 1 1] [0 1 1] [0] [1 0 0] x_1 + [1 0 0] x_2 + [0] [0 0 0] [0 0 0] [1] p('cklt#) = [0] [0] [0] p('compare#) = [0] [0] [0] p('less#) = [0] [0] [0] p(merge#) = [0] [0] [0] p(merge'1#) = [0] [0] [0] p(merge'2#) = [0] [0] [0] p(merge'3#) = [0] [0] [0] p(mergesort#) = [0 1 1] [0] [1 1 1] x_1 + [0] [0 1 1] [0] p(mergesort'1#) = [0 1 1] [0] [0 0 1] x_1 + [0] [0 0 1] [1] p(mergesort'2#) = [1 1 1] [0 1 1] [0] [1 0 0] x_1 + [0 0 0] x_2 + [1] [1 1 0] [1 0 1] [0] p(mergesort'3#) = [1 0 0] [0] [0 0 1] x_1 + [1] [1 0 0] [1] p(msplit#) = [0] [0] [0] p(msplit'1#) = [0] [0] [0] p(msplit'2#) = [0] [0] [0] p(msplit'3#) = [0] [0] [0] p(c_1) = [0] [0] [0] p(c_2) = [0] [0] [0] p(c_3) = [0] [0] [0] p(c_4) = [0] [0] [0] p(c_5) = [0] [0] [0] p(c_6) = [0] [0] [0] p(c_7) = [0] [0] [0] p(c_8) = [0] [0] [0] p(c_9) = [0] [0] [0] p(c_10) = [0] [0] [0] p(c_11) = [0] [0] [0] p(c_12) = [0] [0] [0] p(c_13) = [0] [0] [0] p(c_14) = [0] [0] [0] p(c_15) = [0] [0] [0] p(c_16) = [0] [0] [0] p(c_17) = [0] [0] [0] p(c_18) = [0] [0] [0] p(c_19) = [0] [0] [0] p(c_20) = [0] [0] [0] p(c_21) = [0] [0] [0] p(c_22) = [0] [0] [0] p(c_23) = [0] [0] [0] p(c_24) = [1 0 0] [0] [1 0 0] x_1 + [0] [0 1 0] [0] p(c_25) = [1 0 0] [0] [0 0 0] x_1 + [0] [0 1 0] [0] p(c_26) = [0] [0] [0] p(c_27) = [1 0 0] [0] [0 0 0] x_1 + [0] [0 0 0] [1] p(c_28) = [0] [0] [0] p(c_29) = [1 0 0] [1 0 0] [0] [0 0 0] x_1 + [0 0 0] x_2 + [0] [0 0 1] [0 0 0] [0] p(c_30) = [0] [0] [0] p(c_31) = [0] [0] [0] p(c_32) = [0] [0] [0] p(c_33) = [0] [0] [0] p(c_34) = [0] [0] [0] p(c_35) = [0] [0] [0] Following rules are strictly oriented: mergesort'1#(dd(x1,xs)) = [0 1 1] [1 1 1] [1] [0 0 0] x1 + [1 0 0] xs + [0] [0 0 0] [1 0 0] [1] > [0 1 1] [1 1 1] [0] [0 0 0] x1 + [0 0 0] xs + [0] [0 0 0] [1 0 0] [1] = c_25(mergesort'2#(xs,x1)) Following rules are (at-least) weakly oriented: mergesort#(l) = [0 1 1] [0] [1 1 1] l + [0] [0 1 1] [0] >= [0 1 1] [0] [0 1 1] l + [0] [0 0 1] [0] = c_24(mergesort'1#(l)) mergesort'2#(dd(x2,xs'),x1) = [0 1 1] [0 1 1] [1 1 1] [2] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] xs' + [2] [1 0 1] [0 1 1] [0 1 1] [2] >= [0 1 1] [0 1 1] [1 1 1] [2] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] xs' + [0] [0 0 0] [0 0 0] [0 0 0] [1] = c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs'))))) mergesort'3#(tuple'2(l1,l2)) = [0 1 1] [0 1 1] [0] [0 0 0] l1 + [0 0 0] l2 + [2] [0 1 1] [0 1 1] [1] >= [0 1 1] [0 1 1] [0] [0 0 0] l1 + [0 0 0] l2 + [0] [0 1 1] [0 0 0] [0] = c_29(mergesort#(l1),mergesort#(l2)) msplit(l) = [0 1 0] [0] [1 0 1] l + [0] [0 0 0] [1] >= [0 1 0] [0] [1 0 1] l + [0] [0 0 0] [1] = msplit'1(l) msplit'1(dd(x1,xs)) = [0 1 1] [0 1 1] [1] [0 0 0] x1 + [1 0 0] xs + [1] [0 0 0] [0 0 0] [1] >= [0 1 1] [0 1 1] [1] [0 0 0] x1 + [1 0 0] xs + [1] [0 0 0] [0 0 0] [1] = msplit'2(xs,x1) msplit'1(nil()) = [0] [0] [1] >= [0] [0] [1] = tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) = [0 1 1] [0 1 1] [1 1 1] [2] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] xs' + [2] [0 0 0] [0 0 0] [0 0 0] [1] >= [0 1 1] [0 1 1] [1 1 1] [2] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] xs' + [2] [0 0 0] [0 0 0] [0 0 0] [1] = msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) = [0 1 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [0 1 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] = tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) = [1 1 1] [1 1 1] [0 1 1] [0 1 1] [2] [0 0 0] l1 + [0 0 0] l2 + [0 0 0] x1 + [0 0 0] x2 + [2] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1] >= [1 1 1] [1 1 1] [0 1 1] [0 1 1] [2] [0 0 0] l1 + [0 0 0] l2 + [0 0 0] x1 + [0 0 0] x2 + [2] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1] = tuple'2(dd(x1,l1),dd(x2,l2)) ** Step 5.a:7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mergesort#(l) -> c_24(mergesort'1#(l)) mergesort'1#(dd(x1,xs)) -> c_25(mergesort'2#(xs,x1)) mergesort'2#(dd(x2,xs'),x1) -> c_27(mergesort'3#(msplit(dd(x1,dd(x2,xs'))))) mergesort'3#(tuple'2(l1,l2)) -> c_29(mergesort#(l1),mergesort#(l2)) - Weak TRS: msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/1,c_28/0,c_29/2,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)) merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(xs')) - Weak DPs: mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + 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 merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)) merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(xs')) and a lower component 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) Further, following extension rules are added to the lower component. merge#(l1,l2) -> merge'1#(l1,l2) merge'1#(dd(x,xs),l2) -> merge'2#(l2,x,xs) merge'2#(dd(y,ys),x,xs) -> 'less#(x,y) merge'2#(dd(y,ys),x,xs) -> merge'3#('less(x,y),x,xs,y,ys) merge'3#('false(),x,xs,y,ys) -> merge#(dd(x,xs),ys) merge'3#('true(),x,xs,y,ys) -> merge#(xs,dd(y,ys)) mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) msplit#(l) -> msplit'1#(l) msplit'1#(dd(x1,xs)) -> msplit'2#(xs,x1) msplit'2#(dd(x2,xs'),x1) -> msplit#(xs') *** Step 5.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)) merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(xs')) - Weak DPs: mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:merge#(l1,l2) -> c_17(merge'1#(l1,l2)) -->_1 merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)):2 2:S:merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) -->_1 merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)):3 3:S:merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys),'less#(x,y)) -->_1 merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))):5 -->_1 merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)):4 4:S:merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) -->_1 merge#(l1,l2) -> c_17(merge'1#(l1,l2)):1 5:S:merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) -->_1 merge#(l1,l2) -> c_17(merge'1#(l1,l2)):1 6:S:msplit#(l) -> c_30(msplit'1#(l)) -->_1 msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)):7 7:S:msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) -->_1 msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(xs')):8 8:S:msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(xs')) -->_1 msplit#(l) -> c_30(msplit'1#(l)):6 9:W:mergesort#(l) -> mergesort'1#(l) -->_1 mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1):10 10:W:mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) -->_1 mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))):12 -->_1 mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))):11 11:W:mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) -->_1 mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2):15 -->_1 mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1):14 -->_1 mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)):13 12:W:mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) -->_1 msplit#(l) -> c_30(msplit'1#(l)):6 13:W:mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) -->_1 merge#(l1,l2) -> c_17(merge'1#(l1,l2)):1 14:W:mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) -->_1 mergesort#(l) -> mergesort'1#(l):9 15:W:mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) -->_1 mergesort#(l) -> mergesort'1#(l):9 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys)) *** Step 5.b:1.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys)) merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(xs')) - Weak DPs: mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + 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(merge) = {1,2}, uargs(merge'3) = {1}, uargs(mergesort'3) = {1}, uargs(msplit'3) = {1}, uargs(merge#) = {1,2}, uargs(merge'3#) = {1}, uargs(mergesort'3#) = {1}, uargs(c_17) = {1}, uargs(c_18) = {1}, uargs(c_20) = {1}, uargs(c_22) = {1}, uargs(c_23) = {1}, uargs(c_30) = {1}, uargs(c_31) = {1}, uargs(c_33) = {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('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(merge) = [1] x1 + [1] x2 + [0] p(merge'1) = [1] x1 + [1] x2 + [0] p(merge'2) = [1] x1 + [1] x3 + [0] p(merge'3) = [1] x1 + [1] x3 + [1] x5 + [0] p(mergesort) = [0] p(mergesort'1) = [0] p(mergesort'2) = [0] p(mergesort'3) = [1] x1 + [0] p(msplit) = [0] p(msplit'1) = [0] p(msplit'2) = [0] p(msplit'3) = [1] x1 + [0] p(nil) = [0] p(tuple'2) = [1] x1 + [0] p('cklt#) = [0] p('compare#) = [0] p('less#) = [0] p(merge#) = [1] x1 + [1] x2 + [0] p(merge'1#) = [1] x1 + [1] x2 + [0] p(merge'2#) = [1] x1 + [1] x3 + [5] p(merge'3#) = [1] x1 + [1] x3 + [1] x5 + [0] p(mergesort#) = [0] p(mergesort'1#) = [0] p(mergesort'2#) = [0] p(mergesort'3#) = [1] x1 + [0] p(msplit#) = [0] p(msplit'1#) = [2] p(msplit'2#) = [0] p(msplit'3#) = [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) = [1] x1 + [6] p(c_18) = [1] x1 + [3] p(c_19) = [0] p(c_20) = [1] x1 + [0] p(c_21) = [0] p(c_22) = [1] x1 + [0] p(c_23) = [1] x1 + [0] p(c_24) = [0] p(c_25) = [1] x1 + [0] p(c_26) = [2] p(c_27) = [1] x1 + [4] x2 + [4] p(c_28) = [2] p(c_29) = [2] x1 + [1] x3 + [4] p(c_30) = [1] x1 + [6] p(c_31) = [1] x1 + [0] p(c_32) = [1] p(c_33) = [1] x1 + [2] p(c_34) = [1] p(c_35) = [0] Following rules are strictly oriented: merge'2#(dd(y,ys),x,xs) = [1] xs + [1] ys + [5] > [1] xs + [1] ys + [0] = c_20(merge'3#('less(x,y),x,xs,y,ys)) msplit'1#(dd(x1,xs)) = [2] > [0] = c_31(msplit'2#(xs,x1)) Following rules are (at-least) weakly oriented: merge#(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [6] = c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) = [1] l2 + [1] xs + [0] >= [1] l2 + [1] xs + [8] = c_18(merge'2#(l2,x,xs)) merge'3#('false(),x,xs,y,ys) = [1] xs + [1] ys + [0] >= [1] xs + [1] ys + [0] = c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) = [1] xs + [1] ys + [0] >= [1] xs + [1] ys + [0] = c_23(merge#(xs,dd(y,ys))) mergesort#(l) = [0] >= [0] = mergesort'1#(l) mergesort'1#(dd(x1,xs)) = [0] >= [0] = mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) = [0] >= [0] = mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) = [0] >= [0] = msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [0] >= [0] = merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [0] >= [0] = mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [0] >= [0] = mergesort#(l2) msplit#(l) = [0] >= [8] = c_30(msplit'1#(l)) msplit'2#(dd(x2,xs'),x1) = [0] >= [2] = c_33(msplit#(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)) merge(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge'1(l1,l2) merge'1(dd(x,xs),l2) = [1] l2 + [1] xs + [0] >= [1] l2 + [1] xs + [0] = merge'2(l2,x,xs) merge'1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 merge'2(dd(y,ys),x,xs) = [1] xs + [1] ys + [0] >= [1] xs + [1] ys + [0] = merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) = [1] xs + [0] >= [1] xs + [0] = dd(x,xs) merge'3('false(),x,xs,y,ys) = [1] xs + [1] ys + [0] >= [1] xs + [1] ys + [0] = dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) = [1] xs + [1] ys + [0] >= [1] xs + [1] ys + [0] = dd(x,merge(xs,dd(y,ys))) mergesort(l) = [0] >= [0] = mergesort'1(l) mergesort'1(dd(x1,xs)) = [0] >= [0] = mergesort'2(xs,x1) mergesort'1(nil()) = [0] >= [0] = nil() mergesort'2(dd(x2,xs'),x1) = [0] >= [0] = mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) = [0] >= [0] = dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) = [1] l1 + [0] >= [0] = merge(mergesort(l1),mergesort(l2)) msplit(l) = [0] >= [0] = msplit'1(l) msplit'1(dd(x1,xs)) = [0] >= [0] = msplit'2(xs,x1) msplit'1(nil()) = [0] >= [0] = tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) = [0] >= [0] = msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) = [0] >= [0] = tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) = [1] l1 + [0] >= [1] l1 + [0] = tuple'2(dd(x1,l1),dd(x2,l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 5.b:1.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) msplit#(l) -> c_30(msplit'1#(l)) msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(xs')) - Weak DPs: merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys)) mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + 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(merge) = {1,2}, uargs(merge'3) = {1}, uargs(mergesort'3) = {1}, uargs(msplit'3) = {1}, uargs(merge#) = {1,2}, uargs(merge'3#) = {1}, uargs(mergesort'3#) = {1}, uargs(c_17) = {1}, uargs(c_18) = {1}, uargs(c_20) = {1}, uargs(c_22) = {1}, uargs(c_23) = {1}, uargs(c_30) = {1}, uargs(c_31) = {1}, uargs(c_33) = {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('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 + [0] p(merge) = [1] x1 + [1] x2 + [0] p(merge'1) = [1] x1 + [1] x2 + [0] p(merge'2) = [1] x1 + [1] x3 + [0] p(merge'3) = [1] x1 + [1] x3 + [1] x5 + [0] p(mergesort) = [0] p(mergesort'1) = [0] p(mergesort'2) = [0] p(mergesort'3) = [1] x1 + [0] p(msplit) = [0] p(msplit'1) = [0] p(msplit'2) = [0] p(msplit'3) = [1] x1 + [0] p(nil) = [0] p(tuple'2) = [0] p('cklt#) = [0] p('compare#) = [0] p('less#) = [0] p(merge#) = [1] x1 + [1] x2 + [2] p(merge'1#) = [1] x1 + [1] x2 + [0] p(merge'2#) = [1] x1 + [1] x3 + [5] p(merge'3#) = [1] x1 + [1] x3 + [1] x5 + [0] p(mergesort#) = [2] p(mergesort'1#) = [2] p(mergesort'2#) = [2] p(mergesort'3#) = [1] x1 + [2] p(msplit#) = [2] p(msplit'1#) = [1] p(msplit'2#) = [1] p(msplit'3#) = [1] x1 + [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) = [1] x1 + [0] p(c_18) = [1] x1 + [0] p(c_19) = [2] p(c_20) = [1] x1 + [4] p(c_21) = [0] p(c_22) = [1] x1 + [0] p(c_23) = [1] x1 + [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) = [1] x1 + [0] p(c_31) = [1] x1 + [0] p(c_32) = [0] p(c_33) = [1] x1 + [0] p(c_34) = [0] p(c_35) = [0] Following rules are strictly oriented: merge#(l1,l2) = [1] l1 + [1] l2 + [2] > [1] l1 + [1] l2 + [0] = c_17(merge'1#(l1,l2)) msplit#(l) = [2] > [1] = c_30(msplit'1#(l)) Following rules are (at-least) weakly oriented: merge'1#(dd(x,xs),l2) = [1] l2 + [1] xs + [0] >= [1] l2 + [1] xs + [5] = c_18(merge'2#(l2,x,xs)) merge'2#(dd(y,ys),x,xs) = [1] xs + [1] ys + [5] >= [1] xs + [1] ys + [4] = c_20(merge'3#('less(x,y),x,xs,y,ys)) merge'3#('false(),x,xs,y,ys) = [1] xs + [1] ys + [0] >= [1] xs + [1] ys + [2] = c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) = [1] xs + [1] ys + [0] >= [1] xs + [1] ys + [2] = c_23(merge#(xs,dd(y,ys))) mergesort#(l) = [2] >= [2] = mergesort'1#(l) mergesort'1#(dd(x1,xs)) = [2] >= [2] = mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) = [2] >= [2] = mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) = [2] >= [2] = msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) = [2] >= [2] = merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) = [2] >= [2] = mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) = [2] >= [2] = mergesort#(l2) msplit'1#(dd(x1,xs)) = [1] >= [1] = c_31(msplit'2#(xs,x1)) msplit'2#(dd(x2,xs'),x1) = [1] >= [2] = c_33(msplit#(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)) merge(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge'1(l1,l2) merge'1(dd(x,xs),l2) = [1] l2 + [1] xs + [0] >= [1] l2 + [1] xs + [0] = merge'2(l2,x,xs) merge'1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 merge'2(dd(y,ys),x,xs) = [1] xs + [1] ys + [0] >= [1] xs + [1] ys + [0] = merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) = [1] xs + [0] >= [1] xs + [0] = dd(x,xs) merge'3('false(),x,xs,y,ys) = [1] xs + [1] ys + [0] >= [1] xs + [1] ys + [0] = dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) = [1] xs + [1] ys + [0] >= [1] xs + [1] ys + [0] = dd(x,merge(xs,dd(y,ys))) mergesort(l) = [0] >= [0] = mergesort'1(l) mergesort'1(dd(x1,xs)) = [0] >= [0] = mergesort'2(xs,x1) mergesort'1(nil()) = [0] >= [0] = nil() mergesort'2(dd(x2,xs'),x1) = [0] >= [0] = mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) = [0] >= [0] = dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) = [0] >= [0] = merge(mergesort(l1),mergesort(l2)) msplit(l) = [0] >= [0] = msplit'1(l) msplit'1(dd(x1,xs)) = [0] >= [0] = msplit'2(xs,x1) msplit'1(nil()) = [0] >= [0] = tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) = [0] >= [0] = msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) = [0] >= [0] = tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) = [0] >= [0] = tuple'2(dd(x1,l1),dd(x2,l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 5.b:1.a:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(xs')) - Weak DPs: merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys)) mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + 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(merge) = {1,2}, uargs(merge'3) = {1}, uargs(mergesort'3) = {1}, uargs(msplit'3) = {1}, uargs(merge#) = {1,2}, uargs(merge'3#) = {1}, uargs(mergesort'3#) = {1}, uargs(c_17) = {1}, uargs(c_18) = {1}, uargs(c_20) = {1}, uargs(c_22) = {1}, uargs(c_23) = {1}, uargs(c_30) = {1}, uargs(c_31) = {1}, uargs(c_33) = {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('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 + [0] p(merge) = [1] x1 + [1] x2 + [0] p(merge'1) = [1] x1 + [1] x2 + [0] p(merge'2) = [1] x1 + [1] x3 + [0] p(merge'3) = [1] x1 + [1] x3 + [1] x5 + [0] p(mergesort) = [0] p(mergesort'1) = [0] p(mergesort'2) = [0] p(mergesort'3) = [1] x1 + [0] p(msplit) = [0] p(msplit'1) = [0] p(msplit'2) = [0] p(msplit'3) = [1] x1 + [0] p(nil) = [0] p(tuple'2) = [1] x1 + [0] p('cklt#) = [0] p('compare#) = [0] p('less#) = [0] p(merge#) = [1] x1 + [1] x2 + [0] p(merge'1#) = [1] x1 + [1] x2 + [0] p(merge'2#) = [1] x1 + [1] x3 + [1] p(merge'3#) = [1] x1 + [1] x3 + [1] x5 + [1] p(mergesort#) = [7] p(mergesort'1#) = [7] p(mergesort'2#) = [7] p(mergesort'3#) = [1] x1 + [7] p(msplit#) = [7] p(msplit'1#) = [3] p(msplit'2#) = [3] p(msplit'3#) = [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) = [1] x1 + [0] p(c_18) = [1] x1 + [0] p(c_19) = [0] p(c_20) = [1] x1 + [0] p(c_21) = [0] p(c_22) = [1] x1 + [0] p(c_23) = [1] x1 + [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [2] x2 + [0] p(c_30) = [1] x1 + [4] p(c_31) = [1] x1 + [0] p(c_32) = [2] p(c_33) = [1] x1 + [0] p(c_34) = [0] p(c_35) = [0] Following rules are strictly oriented: merge'3#('false(),x,xs,y,ys) = [1] xs + [1] ys + [1] > [1] xs + [1] ys + [0] = c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) = [1] xs + [1] ys + [1] > [1] xs + [1] ys + [0] = c_23(merge#(xs,dd(y,ys))) Following rules are (at-least) weakly oriented: merge#(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) = [1] l2 + [1] xs + [0] >= [1] l2 + [1] xs + [1] = c_18(merge'2#(l2,x,xs)) merge'2#(dd(y,ys),x,xs) = [1] xs + [1] ys + [1] >= [1] xs + [1] ys + [1] = c_20(merge'3#('less(x,y),x,xs,y,ys)) mergesort#(l) = [7] >= [7] = mergesort'1#(l) mergesort'1#(dd(x1,xs)) = [7] >= [7] = mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) = [7] >= [7] = mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) = [7] >= [7] = msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [7] >= [0] = merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [7] >= [7] = mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [7] >= [7] = mergesort#(l2) msplit#(l) = [7] >= [7] = c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) = [3] >= [3] = c_31(msplit'2#(xs,x1)) msplit'2#(dd(x2,xs'),x1) = [3] >= [7] = c_33(msplit#(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)) merge(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge'1(l1,l2) merge'1(dd(x,xs),l2) = [1] l2 + [1] xs + [0] >= [1] l2 + [1] xs + [0] = merge'2(l2,x,xs) merge'1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 merge'2(dd(y,ys),x,xs) = [1] xs + [1] ys + [0] >= [1] xs + [1] ys + [0] = merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) = [1] xs + [0] >= [1] xs + [0] = dd(x,xs) merge'3('false(),x,xs,y,ys) = [1] xs + [1] ys + [0] >= [1] xs + [1] ys + [0] = dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) = [1] xs + [1] ys + [0] >= [1] xs + [1] ys + [0] = dd(x,merge(xs,dd(y,ys))) mergesort(l) = [0] >= [0] = mergesort'1(l) mergesort'1(dd(x1,xs)) = [0] >= [0] = mergesort'2(xs,x1) mergesort'1(nil()) = [0] >= [0] = nil() mergesort'2(dd(x2,xs'),x1) = [0] >= [0] = mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) = [0] >= [0] = dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) = [1] l1 + [0] >= [0] = merge(mergesort(l1),mergesort(l2)) msplit(l) = [0] >= [0] = msplit'1(l) msplit'1(dd(x1,xs)) = [0] >= [0] = msplit'2(xs,x1) msplit'1(nil()) = [0] >= [0] = tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) = [0] >= [0] = msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) = [0] >= [0] = tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) = [1] l1 + [0] >= [1] l1 + [0] = tuple'2(dd(x1,l1),dd(x2,l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 5.b:1.a:5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(xs')) - Weak DPs: merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys)) merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + 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(merge) = {1,2}, uargs(merge'3) = {1}, uargs(mergesort'3) = {1}, uargs(msplit'3) = {1}, uargs(merge#) = {1,2}, uargs(merge'3#) = {1}, uargs(mergesort'3#) = {1}, uargs(c_17) = {1}, uargs(c_18) = {1}, uargs(c_20) = {1}, uargs(c_22) = {1}, uargs(c_23) = {1}, uargs(c_30) = {1}, uargs(c_31) = {1}, uargs(c_33) = {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('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(merge) = [1] x1 + [1] x2 + [0] p(merge'1) = [1] x1 + [1] x2 + [0] p(merge'2) = [1] x1 + [1] x3 + [2] p(merge'3) = [1] x1 + [1] x3 + [1] x5 + [4] p(mergesort) = [1] x1 + [0] p(mergesort'1) = [1] x1 + [0] p(mergesort'2) = [1] x1 + [2] p(mergesort'3) = [1] x1 + [0] p(msplit) = [1] x1 + [0] p(msplit'1) = [1] x1 + [0] p(msplit'2) = [1] x1 + [2] p(msplit'3) = [1] x1 + [4] p(nil) = [0] p(tuple'2) = [1] x1 + [1] x2 + [0] p('cklt#) = [1] x1 + [2] p('compare#) = [1] x2 + [4] p('less#) = [1] p(merge#) = [1] x1 + [1] x2 + [0] p(merge'1#) = [1] x1 + [1] x2 + [0] p(merge'2#) = [1] x1 + [1] x3 + [0] p(merge'3#) = [1] x1 + [1] x3 + [1] x5 + [2] p(mergesort#) = [1] x1 + [2] p(mergesort'1#) = [1] x1 + [2] p(mergesort'2#) = [1] x1 + [4] p(mergesort'3#) = [1] x1 + [2] p(msplit#) = [0] p(msplit'1#) = [0] p(msplit'2#) = [0] p(msplit'3#) = [2] x2 + [1] x3 + [4] p(c_1) = [1] p(c_2) = [0] p(c_3) = [0] p(c_4) = [2] p(c_5) = [0] p(c_6) = [1] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] p(c_11) = [4] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [1] p(c_15) = [1] p(c_16) = [1] x1 + [0] p(c_17) = [1] x1 + [0] p(c_18) = [1] x1 + [1] p(c_19) = [2] p(c_20) = [1] x1 + [0] p(c_21) = [0] p(c_22) = [1] x1 + [0] p(c_23) = [1] x1 + [0] p(c_24) = [1] x1 + [2] p(c_25) = [1] p(c_26) = [1] p(c_27) = [4] x1 + [1] x2 + [0] p(c_28) = [1] p(c_29) = [1] x2 + [1] x3 + [0] p(c_30) = [1] x1 + [0] p(c_31) = [1] x1 + [0] p(c_32) = [0] p(c_33) = [1] x1 + [3] p(c_34) = [0] p(c_35) = [0] Following rules are strictly oriented: merge'1#(dd(x,xs),l2) = [1] l2 + [1] xs + [2] > [1] l2 + [1] xs + [1] = c_18(merge'2#(l2,x,xs)) Following rules are (at-least) weakly oriented: merge#(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = c_17(merge'1#(l1,l2)) merge'2#(dd(y,ys),x,xs) = [1] xs + [1] ys + [2] >= [1] xs + [1] ys + [2] = c_20(merge'3#('less(x,y),x,xs,y,ys)) merge'3#('false(),x,xs,y,ys) = [1] xs + [1] ys + [2] >= [1] xs + [1] ys + [2] = c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) = [1] xs + [1] ys + [2] >= [1] xs + [1] ys + [2] = c_23(merge#(xs,dd(y,ys))) mergesort#(l) = [1] l + [2] >= [1] l + [2] = mergesort'1#(l) mergesort'1#(dd(x1,xs)) = [1] xs + [4] >= [1] xs + [4] = mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) = [1] xs' + [6] >= [1] xs' + [6] = mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) = [1] xs' + [6] >= [0] = msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [2] >= [1] l1 + [1] l2 + [0] = merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [2] >= [1] l1 + [2] = mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [2] >= [1] l2 + [2] = mergesort#(l2) msplit#(l) = [0] >= [0] = c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) = [0] >= [0] = c_31(msplit'2#(xs,x1)) msplit'2#(dd(x2,xs'),x1) = [0] >= [3] = c_33(msplit#(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)) merge(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge'1(l1,l2) merge'1(dd(x,xs),l2) = [1] l2 + [1] xs + [2] >= [1] l2 + [1] xs + [2] = merge'2(l2,x,xs) merge'1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 merge'2(dd(y,ys),x,xs) = [1] xs + [1] ys + [4] >= [1] xs + [1] ys + [4] = merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) = [1] xs + [2] >= [1] xs + [2] = dd(x,xs) merge'3('false(),x,xs,y,ys) = [1] xs + [1] ys + [4] >= [1] xs + [1] ys + [4] = dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) = [1] xs + [1] ys + [4] >= [1] xs + [1] ys + [4] = dd(x,merge(xs,dd(y,ys))) mergesort(l) = [1] l + [0] >= [1] l + [0] = mergesort'1(l) mergesort'1(dd(x1,xs)) = [1] xs + [2] >= [1] xs + [2] = mergesort'2(xs,x1) mergesort'1(nil()) = [0] >= [0] = nil() mergesort'2(dd(x2,xs'),x1) = [1] xs' + [4] >= [1] xs' + [4] = mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) = [2] >= [2] = dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge(mergesort(l1),mergesort(l2)) msplit(l) = [1] l + [0] >= [1] l + [0] = msplit'1(l) msplit'1(dd(x1,xs)) = [1] xs + [2] >= [1] xs + [2] = msplit'2(xs,x1) msplit'1(nil()) = [0] >= [0] = tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) = [1] xs' + [4] >= [1] xs' + [4] = msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) = [2] >= [2] = tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) = [1] l1 + [1] l2 + [4] >= [1] l1 + [1] l2 + [4] = tuple'2(dd(x1,l1),dd(x2,l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 5.b:1.a:6: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(xs')) - Weak DPs: merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys)) merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + 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(merge) = {1,2}, uargs(merge'3) = {1}, uargs(mergesort'3) = {1}, uargs(msplit'3) = {1}, uargs(merge#) = {1,2}, uargs(merge'3#) = {1}, uargs(mergesort'3#) = {1}, uargs(c_17) = {1}, uargs(c_18) = {1}, uargs(c_20) = {1}, uargs(c_22) = {1}, uargs(c_23) = {1}, uargs(c_30) = {1}, uargs(c_31) = {1}, uargs(c_33) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p('0) = [0] p('EQ) = [1] p('GT) = [1] p('LT) = [1] p('cklt) = [1] x1 + [1] p('compare) = [1] p('false) = [2] p('less) = [2] p('neg) = [1] x1 + [1] p('pos) = [1] x1 + [0] p('s) = [1] x1 + [1] p('true) = [2] p(dd) = [1] x2 + [1] p(merge) = [1] x1 + [1] x2 + [0] p(merge'1) = [1] x1 + [1] x2 + [0] p(merge'2) = [1] x1 + [1] x3 + [1] p(merge'3) = [1] x1 + [1] x3 + [1] x5 + [0] p(mergesort) = [1] x1 + [0] p(mergesort'1) = [1] x1 + [0] p(mergesort'2) = [1] x1 + [1] p(mergesort'3) = [1] x1 + [0] p(msplit) = [1] x1 + [0] p(msplit'1) = [1] x1 + [0] p(msplit'2) = [1] x1 + [1] p(msplit'3) = [1] x1 + [2] p(nil) = [0] p(tuple'2) = [1] x1 + [1] x2 + [0] p('cklt#) = [1] x1 + [1] p('compare#) = [1] x1 + [0] p('less#) = [1] x1 + [4] x2 + [1] p(merge#) = [1] x1 + [1] x2 + [0] p(merge'1#) = [1] x1 + [1] x2 + [0] p(merge'2#) = [1] x1 + [1] x3 + [1] p(merge'3#) = [1] x1 + [1] x3 + [1] x5 + [0] p(mergesort#) = [1] x1 + [1] p(mergesort'1#) = [1] x1 + [1] p(mergesort'2#) = [1] x1 + [2] p(mergesort'3#) = [1] x1 + [1] p(msplit#) = [1] x1 + [1] p(msplit'1#) = [1] x1 + [1] p(msplit'2#) = [1] x1 + [2] p(msplit'3#) = [1] x1 + [1] x3 + [0] p(c_1) = [1] p(c_2) = [2] p(c_3) = [1] p(c_4) = [2] p(c_5) = [1] p(c_6) = [0] p(c_7) = [2] p(c_8) = [0] p(c_9) = [2] x1 + [1] p(c_10) = [1] p(c_11) = [0] p(c_12) = [1] p(c_13) = [4] p(c_14) = [1] p(c_15) = [1] x1 + [0] p(c_16) = [1] p(c_17) = [1] x1 + [0] p(c_18) = [1] x1 + [0] p(c_19) = [0] p(c_20) = [1] x1 + [0] p(c_21) = [0] p(c_22) = [1] x1 + [1] p(c_23) = [1] x1 + [0] p(c_24) = [1] x1 + [2] p(c_25) = [0] p(c_26) = [1] p(c_27) = [1] x2 + [1] p(c_28) = [1] p(c_29) = [1] x2 + [0] p(c_30) = [1] x1 + [0] p(c_31) = [1] x1 + [0] p(c_32) = [0] p(c_33) = [1] x1 + [0] p(c_34) = [2] p(c_35) = [1] Following rules are strictly oriented: msplit'2#(dd(x2,xs'),x1) = [1] xs' + [3] > [1] xs' + [1] = c_33(msplit#(xs')) Following rules are (at-least) weakly oriented: merge#(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) = [1] l2 + [1] xs + [1] >= [1] l2 + [1] xs + [1] = c_18(merge'2#(l2,x,xs)) merge'2#(dd(y,ys),x,xs) = [1] xs + [1] ys + [2] >= [1] xs + [1] ys + [2] = c_20(merge'3#('less(x,y),x,xs,y,ys)) merge'3#('false(),x,xs,y,ys) = [1] xs + [1] ys + [2] >= [1] xs + [1] ys + [2] = c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) = [1] xs + [1] ys + [2] >= [1] xs + [1] ys + [1] = c_23(merge#(xs,dd(y,ys))) mergesort#(l) = [1] l + [1] >= [1] l + [1] = mergesort'1#(l) mergesort'1#(dd(x1,xs)) = [1] xs + [2] >= [1] xs + [2] = mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) = [1] xs' + [3] >= [1] xs' + [3] = mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) = [1] xs' + [3] >= [1] xs' + [3] = msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [1] >= [1] l1 + [1] l2 + [0] = merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [1] >= [1] l1 + [1] = mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [1] >= [1] l2 + [1] = mergesort#(l2) msplit#(l) = [1] l + [1] >= [1] l + [1] = c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) = [1] xs + [2] >= [1] xs + [2] = c_31(msplit'2#(xs,x1)) 'cklt('EQ()) = [2] >= [2] = 'false() 'cklt('GT()) = [2] >= [2] = 'false() 'cklt('LT()) = [2] >= [2] = 'true() 'compare('0(),'0()) = [1] >= [1] = 'EQ() 'compare('0(),'neg(y)) = [1] >= [1] = 'GT() 'compare('0(),'pos(y)) = [1] >= [1] = 'LT() 'compare('0(),'s(y)) = [1] >= [1] = 'LT() 'compare('neg(x),'0()) = [1] >= [1] = 'LT() 'compare('neg(x),'neg(y)) = [1] >= [1] = 'compare(y,x) 'compare('neg(x),'pos(y)) = [1] >= [1] = 'LT() 'compare('pos(x),'0()) = [1] >= [1] = 'GT() 'compare('pos(x),'neg(y)) = [1] >= [1] = 'GT() 'compare('pos(x),'pos(y)) = [1] >= [1] = 'compare(x,y) 'compare('s(x),'0()) = [1] >= [1] = 'GT() 'compare('s(x),'s(y)) = [1] >= [1] = 'compare(x,y) 'less(x,y) = [2] >= [2] = 'cklt('compare(x,y)) merge(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge'1(l1,l2) merge'1(dd(x,xs),l2) = [1] l2 + [1] xs + [1] >= [1] l2 + [1] xs + [1] = merge'2(l2,x,xs) merge'1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 merge'2(dd(y,ys),x,xs) = [1] xs + [1] ys + [2] >= [1] xs + [1] ys + [2] = merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) = [1] xs + [1] >= [1] xs + [1] = dd(x,xs) merge'3('false(),x,xs,y,ys) = [1] xs + [1] ys + [2] >= [1] xs + [1] ys + [2] = dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) = [1] xs + [1] ys + [2] >= [1] xs + [1] ys + [2] = dd(x,merge(xs,dd(y,ys))) mergesort(l) = [1] l + [0] >= [1] l + [0] = mergesort'1(l) mergesort'1(dd(x1,xs)) = [1] xs + [1] >= [1] xs + [1] = mergesort'2(xs,x1) mergesort'1(nil()) = [0] >= [0] = nil() mergesort'2(dd(x2,xs'),x1) = [1] xs' + [2] >= [1] xs' + [2] = mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) = [1] >= [1] = dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge(mergesort(l1),mergesort(l2)) msplit(l) = [1] l + [0] >= [1] l + [0] = msplit'1(l) msplit'1(dd(x1,xs)) = [1] xs + [1] >= [1] xs + [1] = msplit'2(xs,x1) msplit'1(nil()) = [0] >= [0] = tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) = [1] xs' + [2] >= [1] xs' + [2] = msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) = [1] >= [1] = tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) = [1] l1 + [1] l2 + [2] >= [1] l1 + [1] l2 + [2] = tuple'2(dd(x1,l1),dd(x2,l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 5.b:1.a:7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: merge#(l1,l2) -> c_17(merge'1#(l1,l2)) merge'1#(dd(x,xs),l2) -> c_18(merge'2#(l2,x,xs)) merge'2#(dd(y,ys),x,xs) -> c_20(merge'3#('less(x,y),x,xs,y,ys)) merge'3#('false(),x,xs,y,ys) -> c_22(merge#(dd(x,xs),ys)) merge'3#('true(),x,xs,y,ys) -> c_23(merge#(xs,dd(y,ys))) mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) msplit#(l) -> c_30(msplit'1#(l)) msplit'1#(dd(x1,xs)) -> c_31(msplit'2#(xs,x1)) msplit'2#(dd(x2,xs'),x1) -> c_33(msplit#(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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/1,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 5.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) - Weak DPs: merge#(l1,l2) -> merge'1#(l1,l2) merge'1#(dd(x,xs),l2) -> merge'2#(l2,x,xs) merge'2#(dd(y,ys),x,xs) -> 'less#(x,y) merge'2#(dd(y,ys),x,xs) -> merge'3#('less(x,y),x,xs,y,ys) merge'3#('false(),x,xs,y,ys) -> merge#(dd(x,xs),ys) merge'3#('true(),x,xs,y,ys) -> merge#(xs,dd(y,ys)) mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) msplit#(l) -> msplit'1#(l) msplit'1#(dd(x1,xs)) -> msplit'2#(xs,x1) msplit'2#(dd(x2,xs'),x1) -> msplit#(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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 2:S:'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 3:S:'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 4:S:'less#(x,y) -> c_16('compare#(x,y)) -->_1 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)):3 -->_1 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)):2 -->_1 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)):1 5:W:merge#(l1,l2) -> merge'1#(l1,l2) -->_1 merge'1#(dd(x,xs),l2) -> merge'2#(l2,x,xs):6 6:W:merge'1#(dd(x,xs),l2) -> merge'2#(l2,x,xs) -->_1 merge'2#(dd(y,ys),x,xs) -> merge'3#('less(x,y),x,xs,y,ys):8 -->_1 merge'2#(dd(y,ys),x,xs) -> 'less#(x,y):7 7:W:merge'2#(dd(y,ys),x,xs) -> 'less#(x,y) -->_1 'less#(x,y) -> c_16('compare#(x,y)):4 8:W:merge'2#(dd(y,ys),x,xs) -> merge'3#('less(x,y),x,xs,y,ys) -->_1 merge'3#('true(),x,xs,y,ys) -> merge#(xs,dd(y,ys)):10 -->_1 merge'3#('false(),x,xs,y,ys) -> merge#(dd(x,xs),ys):9 9:W:merge'3#('false(),x,xs,y,ys) -> merge#(dd(x,xs),ys) -->_1 merge#(l1,l2) -> merge'1#(l1,l2):5 10:W:merge'3#('true(),x,xs,y,ys) -> merge#(xs,dd(y,ys)) -->_1 merge#(l1,l2) -> merge'1#(l1,l2):5 11:W:mergesort#(l) -> mergesort'1#(l) -->_1 mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1):12 12:W:mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) -->_1 mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))):14 -->_1 mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))):13 13:W:mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) -->_1 mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2):17 -->_1 mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1):16 -->_1 mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)):15 14:W:mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) -->_1 msplit#(l) -> msplit'1#(l):18 15:W:mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) -->_1 merge#(l1,l2) -> merge'1#(l1,l2):5 16:W:mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) -->_1 mergesort#(l) -> mergesort'1#(l):11 17:W:mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) -->_1 mergesort#(l) -> mergesort'1#(l):11 18:W:msplit#(l) -> msplit'1#(l) -->_1 msplit'1#(dd(x1,xs)) -> msplit'2#(xs,x1):19 19:W:msplit'1#(dd(x1,xs)) -> msplit'2#(xs,x1) -->_1 msplit'2#(dd(x2,xs'),x1) -> msplit#(xs'):20 20:W:msplit'2#(dd(x2,xs'),x1) -> msplit#(xs') -->_1 msplit#(l) -> msplit'1#(l):18 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: mergesort'2#(dd(x2,xs'),x1) -> msplit#(dd(x1,dd(x2,xs'))) 18: msplit#(l) -> msplit'1#(l) 20: msplit'2#(dd(x2,xs'),x1) -> msplit#(xs') 19: msplit'1#(dd(x1,xs)) -> msplit'2#(xs,x1) *** Step 5.b:1.b:2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) - Weak DPs: merge#(l1,l2) -> merge'1#(l1,l2) merge'1#(dd(x,xs),l2) -> merge'2#(l2,x,xs) merge'2#(dd(y,ys),x,xs) -> 'less#(x,y) merge'2#(dd(y,ys),x,xs) -> merge'3#('less(x,y),x,xs,y,ys) merge'3#('false(),x,xs,y,ys) -> merge#(dd(x,xs),ys) merge'3#('true(),x,xs,y,ys) -> merge#(xs,dd(y,ys)) mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_9) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1} Following symbols are considered usable: {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3#,mergesort#,mergesort'1#,mergesort'2# ,mergesort'3#,msplit#,msplit'1#,msplit'2#,msplit'3#} TcT has computed the following interpretation: p('0) = [0] p('EQ) = [0] p('GT) = [0] p('LT) = [3] p('cklt) = [1] x1 + [1] p('compare) = [1] x2 + [0] p('false) = [2] p('less) = [2] x1 + [0] p('neg) = [1] x1 + [0] p('pos) = [6] p('s) = [1] x1 + [0] p('true) = [0] p(dd) = [0] p(merge) = [0] p(merge'1) = [1] x1 + [0] p(merge'2) = [0] p(merge'3) = [3] x1 + [0] p(mergesort) = [0] p(mergesort'1) = [0] p(mergesort'2) = [0] p(mergesort'3) = [0] p(msplit) = [3] p(msplit'1) = [0] p(msplit'2) = [0] p(msplit'3) = [3] x1 + [0] p(nil) = [5] p(tuple'2) = [1] x1 + [1] x2 + [0] p('cklt#) = [0] p('compare#) = [0] p('less#) = [1] p(merge#) = [1] p(merge'1#) = [1] p(merge'2#) = [1] p(merge'3#) = [1] p(mergesort#) = [4] p(mergesort'1#) = [4] p(mergesort'2#) = [4] p(mergesort'3#) = [4] p(msplit#) = [0] p(msplit'1#) = [0] p(msplit'2#) = [0] p(msplit'3#) = [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) = [1] p(c_8) = [0] p(c_9) = [4] x1 + [0] p(c_10) = [1] p(c_11) = [1] p(c_12) = [0] p(c_13) = [4] x1 + [0] p(c_14) = [0] p(c_15) = [4] x1 + [0] p(c_16) = [4] x1 + [0] p(c_17) = [1] x1 + [1] p(c_18) = [0] p(c_19) = [1] p(c_20) = [2] x1 + [1] x2 + [1] p(c_21) = [4] p(c_22) = [1] x1 + [2] p(c_23) = [2] p(c_24) = [0] p(c_25) = [1] x1 + [1] p(c_26) = [1] p(c_27) = [1] x2 + [4] p(c_28) = [1] p(c_29) = [1] x1 + [4] x2 + [2] p(c_30) = [1] x1 + [0] p(c_31) = [1] x1 + [4] p(c_32) = [1] p(c_33) = [4] x1 + [4] p(c_34) = [1] p(c_35) = [0] Following rules are strictly oriented: 'less#(x,y) = [1] > [0] = c_16('compare#(x,y)) Following rules are (at-least) weakly oriented: 'compare#('neg(x),'neg(y)) = [0] >= [0] = c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) = [0] >= [0] = c_13('compare#(x,y)) 'compare#('s(x),'s(y)) = [0] >= [0] = c_15('compare#(x,y)) merge#(l1,l2) = [1] >= [1] = merge'1#(l1,l2) merge'1#(dd(x,xs),l2) = [1] >= [1] = merge'2#(l2,x,xs) merge'2#(dd(y,ys),x,xs) = [1] >= [1] = 'less#(x,y) merge'2#(dd(y,ys),x,xs) = [1] >= [1] = merge'3#('less(x,y),x,xs,y,ys) merge'3#('false(),x,xs,y,ys) = [1] >= [1] = merge#(dd(x,xs),ys) merge'3#('true(),x,xs,y,ys) = [1] >= [1] = merge#(xs,dd(y,ys)) mergesort#(l) = [4] >= [4] = mergesort'1#(l) mergesort'1#(dd(x1,xs)) = [4] >= [4] = mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) = [4] >= [4] = mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) = [4] >= [1] = merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) = [4] >= [4] = mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) = [4] >= [4] = mergesort#(l2) *** Step 5.b:1.b:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) - Weak DPs: 'less#(x,y) -> c_16('compare#(x,y)) merge#(l1,l2) -> merge'1#(l1,l2) merge'1#(dd(x,xs),l2) -> merge'2#(l2,x,xs) merge'2#(dd(y,ys),x,xs) -> 'less#(x,y) merge'2#(dd(y,ys),x,xs) -> merge'3#('less(x,y),x,xs,y,ys) merge'3#('false(),x,xs,y,ys) -> merge#(dd(x,xs),ys) merge'3#('true(),x,xs,y,ys) -> merge#(xs,dd(y,ys)) mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + 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(merge) = {1,2}, uargs(merge'3) = {1}, uargs(mergesort'3) = {1}, uargs(msplit'3) = {1}, uargs(merge#) = {1,2}, uargs(merge'3#) = {1}, uargs(mergesort'3#) = {1}, uargs(c_9) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p('0) = [4] p('EQ) = [0] p('GT) = [0] p('LT) = [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) = [1] x1 + [1] p('true) = [0] p(dd) = [1] x1 + [1] x2 + [0] p(merge) = [1] x1 + [1] x2 + [0] p(merge'1) = [1] x1 + [1] x2 + [0] p(merge'2) = [1] x1 + [1] x2 + [1] x3 + [0] p(merge'3) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] p(mergesort) = [1] x1 + [0] p(mergesort'1) = [1] x1 + [0] p(mergesort'2) = [1] x1 + [1] x2 + [0] p(mergesort'3) = [1] x1 + [0] p(msplit) = [1] x1 + [0] p(msplit'1) = [1] x1 + [0] p(msplit'2) = [1] x1 + [1] x2 + [0] p(msplit'3) = [1] x1 + [1] x2 + [1] x3 + [0] p(nil) = [0] p(tuple'2) = [1] x1 + [1] x2 + [0] p('cklt#) = [1] p('compare#) = [1] x1 + [1] x2 + [0] p('less#) = [1] x1 + [1] x2 + [0] p(merge#) = [1] x1 + [1] x2 + [0] p(merge'1#) = [1] x1 + [1] x2 + [0] p(merge'2#) = [1] x1 + [1] x2 + [1] x3 + [0] p(merge'3#) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] p(mergesort#) = [1] x1 + [1] p(mergesort'1#) = [1] x1 + [1] p(mergesort'2#) = [1] x1 + [1] x2 + [1] p(mergesort'3#) = [1] x1 + [1] p(msplit#) = [4] p(msplit'1#) = [1] x1 + [4] p(msplit'2#) = [1] x2 + [1] p(msplit'3#) = [2] x1 + [2] x2 + [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] p(c_4) = [1] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [4] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [0] p(c_17) = [4] x1 + [1] p(c_18) = [1] p(c_19) = [2] p(c_20) = [2] p(c_21) = [1] p(c_22) = [0] p(c_23) = [2] x1 + [0] p(c_24) = [2] p(c_25) = [1] x1 + [0] p(c_26) = [0] p(c_27) = [1] x2 + [1] p(c_28) = [2] p(c_29) = [1] x1 + [1] x2 + [2] p(c_30) = [0] p(c_31) = [1] p(c_32) = [4] p(c_33) = [1] x1 + [0] p(c_34) = [4] p(c_35) = [1] Following rules are strictly oriented: 'compare#('s(x),'s(y)) = [1] x + [1] y + [2] > [1] x + [1] y + [0] = c_15('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_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = c_13('compare#(x,y)) 'less#(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = c_16('compare#(x,y)) merge#(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge'1#(l1,l2) merge'1#(dd(x,xs),l2) = [1] l2 + [1] x + [1] xs + [0] >= [1] l2 + [1] x + [1] xs + [0] = merge'2#(l2,x,xs) merge'2#(dd(y,ys),x,xs) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] y + [0] = 'less#(x,y) merge'2#(dd(y,ys),x,xs) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] xs + [1] y + [1] ys + [0] = merge'3#('less(x,y),x,xs,y,ys) merge'3#('false(),x,xs,y,ys) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] xs + [1] ys + [0] = merge#(dd(x,xs),ys) merge'3#('true(),x,xs,y,ys) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] xs + [1] y + [1] ys + [0] = merge#(xs,dd(y,ys)) mergesort#(l) = [1] l + [1] >= [1] l + [1] = mergesort'1#(l) mergesort'1#(dd(x1,xs)) = [1] x1 + [1] xs + [1] >= [1] x1 + [1] xs + [1] = mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) = [1] x1 + [1] x2 + [1] xs' + [1] >= [1] x1 + [1] x2 + [1] xs' + [1] = mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [1] >= [1] l1 + [1] l2 + [0] = merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [1] >= [1] l1 + [1] = mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [1] >= [1] l2 + [1] = mergesort#(l2) '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)) merge(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge'1(l1,l2) merge'1(dd(x,xs),l2) = [1] l2 + [1] x + [1] xs + [0] >= [1] l2 + [1] x + [1] xs + [0] = merge'2(l2,x,xs) merge'1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 merge'2(dd(y,ys),x,xs) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] xs + [1] y + [1] ys + [0] = merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) = [1] x + [1] xs + [0] >= [1] x + [1] xs + [0] = dd(x,xs) merge'3('false(),x,xs,y,ys) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] xs + [1] y + [1] ys + [0] = dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] xs + [1] y + [1] ys + [0] = dd(x,merge(xs,dd(y,ys))) mergesort(l) = [1] l + [0] >= [1] l + [0] = mergesort'1(l) mergesort'1(dd(x1,xs)) = [1] x1 + [1] xs + [0] >= [1] x1 + [1] xs + [0] = mergesort'2(xs,x1) mergesort'1(nil()) = [0] >= [0] = nil() mergesort'2(dd(x2,xs'),x1) = [1] x1 + [1] x2 + [1] xs' + [0] >= [1] x1 + [1] x2 + [1] xs' + [0] = mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) = [1] x1 + [0] >= [1] x1 + [0] = dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge(mergesort(l1),mergesort(l2)) msplit(l) = [1] l + [0] >= [1] l + [0] = msplit'1(l) msplit'1(dd(x1,xs)) = [1] x1 + [1] xs + [0] >= [1] x1 + [1] xs + [0] = msplit'2(xs,x1) msplit'1(nil()) = [0] >= [0] = tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) = [1] x1 + [1] x2 + [1] xs' + [0] >= [1] x1 + [1] x2 + [1] xs' + [0] = msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) = [1] x1 + [0] >= [1] x1 + [0] = tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) = [1] l1 + [1] l2 + [1] x1 + [1] x2 + [0] >= [1] l1 + [1] l2 + [1] x1 + [1] x2 + [0] = tuple'2(dd(x1,l1),dd(x2,l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 5.b:1.b:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) - Weak DPs: 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) merge#(l1,l2) -> merge'1#(l1,l2) merge'1#(dd(x,xs),l2) -> merge'2#(l2,x,xs) merge'2#(dd(y,ys),x,xs) -> 'less#(x,y) merge'2#(dd(y,ys),x,xs) -> merge'3#('less(x,y),x,xs,y,ys) merge'3#('false(),x,xs,y,ys) -> merge#(dd(x,xs),ys) merge'3#('true(),x,xs,y,ys) -> merge#(xs,dd(y,ys)) mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + 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(merge) = {1,2}, uargs(merge'3) = {1}, uargs(mergesort'3) = {1}, uargs(msplit'3) = {1}, uargs(merge#) = {1,2}, uargs(merge'3#) = {1}, uargs(mergesort'3#) = {1}, uargs(c_9) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_16) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p('0) = [4] p('EQ) = [0] p('GT) = [0] p('LT) = [0] p('cklt) = [1] x1 + [0] p('compare) = [0] p('false) = [0] p('less) = [0] p('neg) = [1] x1 + [4] p('pos) = [1] x1 + [0] p('s) = [1] x1 + [6] p('true) = [0] p(dd) = [1] x1 + [1] x2 + [0] p(merge) = [1] x1 + [1] x2 + [0] p(merge'1) = [1] x1 + [1] x2 + [0] p(merge'2) = [1] x1 + [1] x2 + [1] x3 + [0] p(merge'3) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] p(mergesort) = [1] x1 + [0] p(mergesort'1) = [1] x1 + [0] p(mergesort'2) = [1] x1 + [1] x2 + [0] p(mergesort'3) = [1] x1 + [0] p(msplit) = [1] x1 + [0] p(msplit'1) = [1] x1 + [0] p(msplit'2) = [1] x1 + [1] x2 + [0] p(msplit'3) = [1] x1 + [1] x2 + [1] x3 + [0] p(nil) = [0] p(tuple'2) = [1] x1 + [1] x2 + [0] p('cklt#) = [1] x1 + [1] p('compare#) = [1] x1 + [1] x2 + [2] p('less#) = [1] x1 + [1] x2 + [4] p(merge#) = [1] x1 + [1] x2 + [4] p(merge'1#) = [1] x1 + [1] x2 + [4] p(merge'2#) = [1] x1 + [1] x2 + [1] x3 + [4] p(merge'3#) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [4] p(mergesort#) = [1] x1 + [4] p(mergesort'1#) = [1] x1 + [4] p(mergesort'2#) = [1] x1 + [1] x2 + [4] p(mergesort'3#) = [1] x1 + [4] p(msplit#) = [0] p(msplit'1#) = [4] x1 + [0] p(msplit'2#) = [1] x1 + [2] x2 + [0] p(msplit'3#) = [0] p(c_1) = [1] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] p(c_5) = [2] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] p(c_11) = [0] p(c_12) = [1] p(c_13) = [1] x1 + [1] p(c_14) = [0] p(c_15) = [1] x1 + [1] p(c_16) = [1] x1 + [2] p(c_17) = [4] x1 + [0] p(c_18) = [2] x1 + [0] p(c_19) = [1] p(c_20) = [1] x1 + [0] p(c_21) = [4] p(c_22) = [1] x1 + [0] p(c_23) = [1] x1 + [0] p(c_24) = [0] p(c_25) = [4] x1 + [1] p(c_26) = [4] p(c_27) = [1] x1 + [1] p(c_28) = [1] p(c_29) = [1] x1 + [1] p(c_30) = [1] p(c_31) = [2] x1 + [1] p(c_32) = [2] p(c_33) = [1] x1 + [4] p(c_34) = [0] p(c_35) = [1] Following rules are strictly oriented: 'compare#('neg(x),'neg(y)) = [1] x + [1] y + [10] > [1] x + [1] y + [2] = c_9('compare#(y,x)) Following rules are (at-least) weakly oriented: 'compare#('pos(x),'pos(y)) = [1] x + [1] y + [2] >= [1] x + [1] y + [3] = c_13('compare#(x,y)) 'compare#('s(x),'s(y)) = [1] x + [1] y + [14] >= [1] x + [1] y + [3] = c_15('compare#(x,y)) 'less#(x,y) = [1] x + [1] y + [4] >= [1] x + [1] y + [4] = c_16('compare#(x,y)) merge#(l1,l2) = [1] l1 + [1] l2 + [4] >= [1] l1 + [1] l2 + [4] = merge'1#(l1,l2) merge'1#(dd(x,xs),l2) = [1] l2 + [1] x + [1] xs + [4] >= [1] l2 + [1] x + [1] xs + [4] = merge'2#(l2,x,xs) merge'2#(dd(y,ys),x,xs) = [1] x + [1] xs + [1] y + [1] ys + [4] >= [1] x + [1] y + [4] = 'less#(x,y) merge'2#(dd(y,ys),x,xs) = [1] x + [1] xs + [1] y + [1] ys + [4] >= [1] x + [1] xs + [1] y + [1] ys + [4] = merge'3#('less(x,y),x,xs,y,ys) merge'3#('false(),x,xs,y,ys) = [1] x + [1] xs + [1] y + [1] ys + [4] >= [1] x + [1] xs + [1] ys + [4] = merge#(dd(x,xs),ys) merge'3#('true(),x,xs,y,ys) = [1] x + [1] xs + [1] y + [1] ys + [4] >= [1] xs + [1] y + [1] ys + [4] = merge#(xs,dd(y,ys)) mergesort#(l) = [1] l + [4] >= [1] l + [4] = mergesort'1#(l) mergesort'1#(dd(x1,xs)) = [1] x1 + [1] xs + [4] >= [1] x1 + [1] xs + [4] = mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) = [1] x1 + [1] x2 + [1] xs' + [4] >= [1] x1 + [1] x2 + [1] xs' + [4] = mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [4] >= [1] l1 + [1] l2 + [4] = merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [4] >= [1] l1 + [4] = mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [4] >= [1] l2 + [4] = mergesort#(l2) '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)) merge(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge'1(l1,l2) merge'1(dd(x,xs),l2) = [1] l2 + [1] x + [1] xs + [0] >= [1] l2 + [1] x + [1] xs + [0] = merge'2(l2,x,xs) merge'1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 merge'2(dd(y,ys),x,xs) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] xs + [1] y + [1] ys + [0] = merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) = [1] x + [1] xs + [0] >= [1] x + [1] xs + [0] = dd(x,xs) merge'3('false(),x,xs,y,ys) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] xs + [1] y + [1] ys + [0] = dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] xs + [1] y + [1] ys + [0] = dd(x,merge(xs,dd(y,ys))) mergesort(l) = [1] l + [0] >= [1] l + [0] = mergesort'1(l) mergesort'1(dd(x1,xs)) = [1] x1 + [1] xs + [0] >= [1] x1 + [1] xs + [0] = mergesort'2(xs,x1) mergesort'1(nil()) = [0] >= [0] = nil() mergesort'2(dd(x2,xs'),x1) = [1] x1 + [1] x2 + [1] xs' + [0] >= [1] x1 + [1] x2 + [1] xs' + [0] = mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) = [1] x1 + [0] >= [1] x1 + [0] = dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge(mergesort(l1),mergesort(l2)) msplit(l) = [1] l + [0] >= [1] l + [0] = msplit'1(l) msplit'1(dd(x1,xs)) = [1] x1 + [1] xs + [0] >= [1] x1 + [1] xs + [0] = msplit'2(xs,x1) msplit'1(nil()) = [0] >= [0] = tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) = [1] x1 + [1] x2 + [1] xs' + [0] >= [1] x1 + [1] x2 + [1] xs' + [0] = msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) = [1] x1 + [0] >= [1] x1 + [0] = tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) = [1] l1 + [1] l2 + [1] x1 + [1] x2 + [0] >= [1] l1 + [1] l2 + [1] x1 + [1] x2 + [0] = tuple'2(dd(x1,l1),dd(x2,l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 5.b:1.b:5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) - Weak DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) merge#(l1,l2) -> merge'1#(l1,l2) merge'1#(dd(x,xs),l2) -> merge'2#(l2,x,xs) merge'2#(dd(y,ys),x,xs) -> 'less#(x,y) merge'2#(dd(y,ys),x,xs) -> merge'3#('less(x,y),x,xs,y,ys) merge'3#('false(),x,xs,y,ys) -> merge#(dd(x,xs),ys) merge'3#('true(),x,xs,y,ys) -> merge#(xs,dd(y,ys)) mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + 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(merge) = {1,2}, uargs(merge'3) = {1}, uargs(mergesort'3) = {1}, uargs(msplit'3) = {1}, uargs(merge#) = {1,2}, uargs(merge'3#) = {1}, uargs(mergesort'3#) = {1}, uargs(c_9) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_16) = {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('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 + [4] p('true) = [0] p(dd) = [1] x1 + [1] x2 + [0] p(merge) = [1] x1 + [1] x2 + [0] p(merge'1) = [1] x1 + [1] x2 + [0] p(merge'2) = [1] x1 + [1] x2 + [1] x3 + [0] p(merge'3) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] p(mergesort) = [1] x1 + [0] p(mergesort'1) = [1] x1 + [0] p(mergesort'2) = [1] x1 + [1] x2 + [0] p(mergesort'3) = [1] x1 + [0] p(msplit) = [1] x1 + [0] p(msplit'1) = [1] x1 + [0] p(msplit'2) = [1] x1 + [1] x2 + [0] p(msplit'3) = [1] x1 + [1] x2 + [1] x3 + [0] p(nil) = [0] p(tuple'2) = [1] x1 + [1] x2 + [0] p('cklt#) = [1] x1 + [0] p('compare#) = [1] x1 + [1] x2 + [0] p('less#) = [1] x1 + [1] x2 + [0] p(merge#) = [1] x1 + [1] x2 + [0] p(merge'1#) = [1] x1 + [1] x2 + [0] p(merge'2#) = [1] x1 + [1] x2 + [1] x3 + [0] p(merge'3#) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] p(mergesort#) = [1] x1 + [0] p(mergesort'1#) = [1] x1 + [0] p(mergesort'2#) = [1] x1 + [1] x2 + [0] p(mergesort'3#) = [1] x1 + [0] p(msplit#) = [1] x1 + [0] p(msplit'1#) = [4] p(msplit'2#) = [1] x1 + [4] x2 + [0] p(msplit'3#) = [1] x1 + [1] p(c_1) = [4] p(c_2) = [2] p(c_3) = [1] p(c_4) = [0] p(c_5) = [1] p(c_6) = [0] p(c_7) = [1] p(c_8) = [4] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] p(c_12) = [2] p(c_13) = [1] x1 + [0] p(c_14) = [0] p(c_15) = [1] x1 + [3] p(c_16) = [1] x1 + [0] p(c_17) = [1] x1 + [0] p(c_18) = [4] p(c_19) = [1] p(c_20) = [2] x2 + [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [4] p(c_24) = [1] x1 + [0] p(c_25) = [1] p(c_26) = [2] p(c_27) = [1] x1 + [1] x2 + [1] p(c_28) = [1] p(c_29) = [4] x2 + [2] x3 + [1] p(c_30) = [1] x1 + [1] p(c_31) = [1] p(c_32) = [0] p(c_33) = [1] p(c_34) = [0] p(c_35) = [1] Following rules are strictly oriented: 'compare#('pos(x),'pos(y)) = [1] x + [1] y + [2] > [1] x + [1] y + [0] = c_13('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_9('compare#(y,x)) 'compare#('s(x),'s(y)) = [1] x + [1] y + [8] >= [1] x + [1] y + [3] = c_15('compare#(x,y)) 'less#(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = c_16('compare#(x,y)) merge#(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge'1#(l1,l2) merge'1#(dd(x,xs),l2) = [1] l2 + [1] x + [1] xs + [0] >= [1] l2 + [1] x + [1] xs + [0] = merge'2#(l2,x,xs) merge'2#(dd(y,ys),x,xs) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] y + [0] = 'less#(x,y) merge'2#(dd(y,ys),x,xs) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] xs + [1] y + [1] ys + [0] = merge'3#('less(x,y),x,xs,y,ys) merge'3#('false(),x,xs,y,ys) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] xs + [1] ys + [0] = merge#(dd(x,xs),ys) merge'3#('true(),x,xs,y,ys) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] xs + [1] y + [1] ys + [0] = merge#(xs,dd(y,ys)) mergesort#(l) = [1] l + [0] >= [1] l + [0] = mergesort'1#(l) mergesort'1#(dd(x1,xs)) = [1] x1 + [1] xs + [0] >= [1] x1 + [1] xs + [0] = mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) = [1] x1 + [1] x2 + [1] xs' + [0] >= [1] x1 + [1] x2 + [1] xs' + [0] = mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [0] >= [1] l1 + [0] = mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [0] >= [1] l2 + [0] = mergesort#(l2) '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)) merge(l1,l2) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge'1(l1,l2) merge'1(dd(x,xs),l2) = [1] l2 + [1] x + [1] xs + [0] >= [1] l2 + [1] x + [1] xs + [0] = merge'2(l2,x,xs) merge'1(nil(),l2) = [1] l2 + [0] >= [1] l2 + [0] = l2 merge'2(dd(y,ys),x,xs) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] xs + [1] y + [1] ys + [0] = merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) = [1] x + [1] xs + [0] >= [1] x + [1] xs + [0] = dd(x,xs) merge'3('false(),x,xs,y,ys) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] xs + [1] y + [1] ys + [0] = dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) = [1] x + [1] xs + [1] y + [1] ys + [0] >= [1] x + [1] xs + [1] y + [1] ys + [0] = dd(x,merge(xs,dd(y,ys))) mergesort(l) = [1] l + [0] >= [1] l + [0] = mergesort'1(l) mergesort'1(dd(x1,xs)) = [1] x1 + [1] xs + [0] >= [1] x1 + [1] xs + [0] = mergesort'2(xs,x1) mergesort'1(nil()) = [0] >= [0] = nil() mergesort'2(dd(x2,xs'),x1) = [1] x1 + [1] x2 + [1] xs' + [0] >= [1] x1 + [1] x2 + [1] xs' + [0] = mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) = [1] x1 + [0] >= [1] x1 + [0] = dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) = [1] l1 + [1] l2 + [0] >= [1] l1 + [1] l2 + [0] = merge(mergesort(l1),mergesort(l2)) msplit(l) = [1] l + [0] >= [1] l + [0] = msplit'1(l) msplit'1(dd(x1,xs)) = [1] x1 + [1] xs + [0] >= [1] x1 + [1] xs + [0] = msplit'2(xs,x1) msplit'1(nil()) = [0] >= [0] = tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) = [1] x1 + [1] x2 + [1] xs' + [0] >= [1] x1 + [1] x2 + [1] xs' + [0] = msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) = [1] x1 + [0] >= [1] x1 + [0] = tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) = [1] l1 + [1] l2 + [1] x1 + [1] x2 + [0] >= [1] l1 + [1] l2 + [1] x1 + [1] x2 + [0] = tuple'2(dd(x1,l1),dd(x2,l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 5.b:1.b:6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 'compare#('neg(x),'neg(y)) -> c_9('compare#(y,x)) 'compare#('pos(x),'pos(y)) -> c_13('compare#(x,y)) 'compare#('s(x),'s(y)) -> c_15('compare#(x,y)) 'less#(x,y) -> c_16('compare#(x,y)) merge#(l1,l2) -> merge'1#(l1,l2) merge'1#(dd(x,xs),l2) -> merge'2#(l2,x,xs) merge'2#(dd(y,ys),x,xs) -> 'less#(x,y) merge'2#(dd(y,ys),x,xs) -> merge'3#('less(x,y),x,xs,y,ys) merge'3#('false(),x,xs,y,ys) -> merge#(dd(x,xs),ys) merge'3#('true(),x,xs,y,ys) -> merge#(xs,dd(y,ys)) mergesort#(l) -> mergesort'1#(l) mergesort'1#(dd(x1,xs)) -> mergesort'2#(xs,x1) mergesort'2#(dd(x2,xs'),x1) -> mergesort'3#(msplit(dd(x1,dd(x2,xs')))) mergesort'3#(tuple'2(l1,l2)) -> merge#(mergesort(l1),mergesort(l2)) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l1) mergesort'3#(tuple'2(l1,l2)) -> mergesort#(l2) - 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)) merge(l1,l2) -> merge'1(l1,l2) merge'1(dd(x,xs),l2) -> merge'2(l2,x,xs) merge'1(nil(),l2) -> l2 merge'2(dd(y,ys),x,xs) -> merge'3('less(x,y),x,xs,y,ys) merge'2(nil(),x,xs) -> dd(x,xs) merge'3('false(),x,xs,y,ys) -> dd(y,merge(dd(x,xs),ys)) merge'3('true(),x,xs,y,ys) -> dd(x,merge(xs,dd(y,ys))) mergesort(l) -> mergesort'1(l) mergesort'1(dd(x1,xs)) -> mergesort'2(xs,x1) mergesort'1(nil()) -> nil() mergesort'2(dd(x2,xs'),x1) -> mergesort'3(msplit(dd(x1,dd(x2,xs')))) mergesort'2(nil(),x1) -> dd(x1,nil()) mergesort'3(tuple'2(l1,l2)) -> merge(mergesort(l1),mergesort(l2)) msplit(l) -> msplit'1(l) msplit'1(dd(x1,xs)) -> msplit'2(xs,x1) msplit'1(nil()) -> tuple'2(nil(),nil()) msplit'2(dd(x2,xs'),x1) -> msplit'3(msplit(xs'),x1,x2) msplit'2(nil(),x1) -> tuple'2(dd(x1,nil()),nil()) msplit'3(tuple'2(l1,l2),x1,x2) -> tuple'2(dd(x1,l1),dd(x2,l2)) - Signature: {'cklt/1,'compare/2,'less/2,merge/2,merge'1/2,merge'2/3,merge'3/5,mergesort/1,mergesort'1/1,mergesort'2/2 ,mergesort'3/1,msplit/1,msplit'1/1,msplit'2/2,msplit'3/3,'cklt#/1,'compare#/2,'less#/2,merge#/2,merge'1#/2 ,merge'2#/3,merge'3#/5,mergesort#/1,mergesort'1#/1,mergesort'2#/2,mergesort'3#/1,msplit#/1,msplit'1#/1 ,msplit'2#/2,msplit'3#/3} / {'0/0,'EQ/0,'GT/0,'LT/0,'false/0,'neg/1,'pos/1,'s/1,'true/0,dd/2,nil/0,tuple'2/2 ,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_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/1,c_16/1 ,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/2,c_28/0,c_29/3,c_30/1,c_31/1 ,c_32/0,c_33/1,c_34/0,c_35/0} - Obligation: innermost runtime complexity wrt. defined symbols {'cklt#,'compare#,'less#,merge#,merge'1#,merge'2#,merge'3# ,mergesort#,mergesort'1#,mergesort'2#,mergesort'3#,msplit#,msplit'1#,msplit'2# ,msplit'3#} and constructors {'0,'EQ,'GT,'LT,'false,'neg,'pos,'s,'true,dd,nil,tuple'2} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^5))