WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: #less(@x,@y) -> #cklt(#compare(@x,@y)) merge(@l1,@l2) -> merge#1(@l1,@l2) merge#1(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@x2,@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) - 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,::/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,::,nil,tuple#2} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs #less#(@x,@y) -> c_1(#cklt#(#compare(@x,@y)),#compare#(@x,@y)) merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#1#(nil(),@l2) -> c_4() merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys),#less#(@x,@y)) merge#2#(nil(),@x,@xs) -> c_6() merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#1#(nil()) -> c_11() mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#2#(nil(),@x1) -> c_13() mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#1#(nil()) -> c_17() msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#3#(msplit(@xs'),@x1,@x2),msplit#(@xs')) msplit#2#(nil(),@x1) -> c_19() msplit#3#(tuple#2(@l1,@l2),@x1,@x2) -> c_20() Weak DPs #cklt#(#EQ()) -> c_21() #cklt#(#GT()) -> c_22() #cklt#(#LT()) -> c_23() #compare#(#0(),#0()) -> c_24() #compare#(#0(),#neg(@y)) -> c_25() #compare#(#0(),#pos(@y)) -> c_26() #compare#(#0(),#s(@y)) -> c_27() #compare#(#neg(@x),#0()) -> c_28() #compare#(#neg(@x),#neg(@y)) -> c_29(#compare#(@y,@x)) #compare#(#neg(@x),#pos(@y)) -> c_30() #compare#(#pos(@x),#0()) -> c_31() #compare#(#pos(@x),#neg(@y)) -> c_32() #compare#(#pos(@x),#pos(@y)) -> c_33(#compare#(@x,@y)) #compare#(#s(@x),#0()) -> c_34() #compare#(#s(@x),#s(@y)) -> c_35(#compare#(@x,@y)) and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: #less#(@x,@y) -> c_1(#cklt#(#compare(@x,@y)),#compare#(@x,@y)) merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#1#(nil(),@l2) -> c_4() merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys),#less#(@x,@y)) merge#2#(nil(),@x,@xs) -> c_6() merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#1#(nil()) -> c_11() mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#2#(nil(),@x1) -> c_13() mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#1#(nil()) -> c_17() msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#3#(msplit(@xs'),@x1,@x2),msplit#(@xs')) msplit#2#(nil(),@x1) -> c_19() msplit#3#(tuple#2(@l1,@l2),@x1,@x2) -> c_20() - Weak DPs: #cklt#(#EQ()) -> c_21() #cklt#(#GT()) -> c_22() #cklt#(#LT()) -> c_23() #compare#(#0(),#0()) -> c_24() #compare#(#0(),#neg(@y)) -> c_25() #compare#(#0(),#pos(@y)) -> c_26() #compare#(#0(),#s(@y)) -> c_27() #compare#(#neg(@x),#0()) -> c_28() #compare#(#neg(@x),#neg(@y)) -> c_29(#compare#(@y,@x)) #compare#(#neg(@x),#pos(@y)) -> c_30() #compare#(#pos(@x),#0()) -> c_31() #compare#(#pos(@x),#neg(@y)) -> c_32() #compare#(#pos(@x),#pos(@y)) -> c_33(#compare#(@x,@y)) #compare#(#s(@x),#0()) -> c_34() #compare#(#s(@x),#s(@y)) -> c_35(#compare#(@x,@y)) - 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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/2,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,6,11,13,17,19,20} by application of Pre({1,4,6,11,13,17,19,20}) = {2,3,5,9,10,15,16,18}. Here rules are labelled as follows: 1: #less#(@x,@y) -> c_1(#cklt#(#compare(@x,@y)),#compare#(@x,@y)) 2: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) 3: merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) 4: merge#1#(nil(),@l2) -> c_4() 5: merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys),#less#(@x,@y)) 6: merge#2#(nil(),@x,@xs) -> c_6() 7: merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) 8: merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) 9: mergesort#(@l) -> c_9(mergesort#1#(@l)) 10: mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) 11: mergesort#1#(nil()) -> c_11() 12: mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) 13: mergesort#2#(nil(),@x1) -> c_13() 14: mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) 15: msplit#(@l) -> c_15(msplit#1#(@l)) 16: msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) 17: msplit#1#(nil()) -> c_17() 18: msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#3#(msplit(@xs'),@x1,@x2),msplit#(@xs')) 19: msplit#2#(nil(),@x1) -> c_19() 20: msplit#3#(tuple#2(@l1,@l2),@x1,@x2) -> c_20() 21: #cklt#(#EQ()) -> c_21() 22: #cklt#(#GT()) -> c_22() 23: #cklt#(#LT()) -> c_23() 24: #compare#(#0(),#0()) -> c_24() 25: #compare#(#0(),#neg(@y)) -> c_25() 26: #compare#(#0(),#pos(@y)) -> c_26() 27: #compare#(#0(),#s(@y)) -> c_27() 28: #compare#(#neg(@x),#0()) -> c_28() 29: #compare#(#neg(@x),#neg(@y)) -> c_29(#compare#(@y,@x)) 30: #compare#(#neg(@x),#pos(@y)) -> c_30() 31: #compare#(#pos(@x),#0()) -> c_31() 32: #compare#(#pos(@x),#neg(@y)) -> c_32() 33: #compare#(#pos(@x),#pos(@y)) -> c_33(#compare#(@x,@y)) 34: #compare#(#s(@x),#0()) -> c_34() 35: #compare#(#s(@x),#s(@y)) -> c_35(#compare#(@x,@y)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys),#less#(@x,@y)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#3#(msplit(@xs'),@x1,@x2),msplit#(@xs')) - Weak DPs: #cklt#(#EQ()) -> c_21() #cklt#(#GT()) -> c_22() #cklt#(#LT()) -> c_23() #compare#(#0(),#0()) -> c_24() #compare#(#0(),#neg(@y)) -> c_25() #compare#(#0(),#pos(@y)) -> c_26() #compare#(#0(),#s(@y)) -> c_27() #compare#(#neg(@x),#0()) -> c_28() #compare#(#neg(@x),#neg(@y)) -> c_29(#compare#(@y,@x)) #compare#(#neg(@x),#pos(@y)) -> c_30() #compare#(#pos(@x),#0()) -> c_31() #compare#(#pos(@x),#neg(@y)) -> c_32() #compare#(#pos(@x),#pos(@y)) -> c_33(#compare#(@x,@y)) #compare#(#s(@x),#0()) -> c_34() #compare#(#s(@x),#s(@y)) -> c_35(#compare#(@x,@y)) #less#(@x,@y) -> c_1(#cklt#(#compare(@x,@y)),#compare#(@x,@y)) merge#1#(nil(),@l2) -> c_4() merge#2#(nil(),@x,@xs) -> c_6() mergesort#1#(nil()) -> c_11() mergesort#2#(nil(),@x1) -> c_13() msplit#1#(nil()) -> c_17() msplit#2#(nil(),@x1) -> c_19() msplit#3#(tuple#2(@l1,@l2),@x1,@x2) -> c_20() - 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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/2,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) -->_1 merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)):2 -->_1 merge#1#(nil(),@l2) -> c_4():29 2:S:merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) -->_1 merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys),#less#(@x,@y)):3 -->_1 merge#2#(nil(),@x,@xs) -> c_6():30 3:S:merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys),#less#(@x,@y)) -->_2 #less#(@x,@y) -> c_1(#cklt#(#compare(@x,@y)),#compare#(@x,@y)):28 -->_1 merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))):5 -->_1 merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)):4 4:S:merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 5:S:merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 6:S:mergesort#(@l) -> c_9(mergesort#1#(@l)) -->_1 mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)):7 -->_1 mergesort#1#(nil()) -> c_11():31 7:S:mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) -->_1 mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))):8 -->_1 mergesort#2#(nil(),@x1) -> c_13():32 8:S:mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) -->_2 msplit#(@l) -> c_15(msplit#1#(@l)):10 -->_1 mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)):9 9:S:mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) -->_3 mergesort#(@l) -> c_9(mergesort#1#(@l)):6 -->_2 mergesort#(@l) -> c_9(mergesort#1#(@l)):6 -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 10:S:msplit#(@l) -> c_15(msplit#1#(@l)) -->_1 msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)):11 -->_1 msplit#1#(nil()) -> c_17():33 11:S:msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) -->_1 msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#3#(msplit(@xs'),@x1,@x2),msplit#(@xs')):12 -->_1 msplit#2#(nil(),@x1) -> c_19():34 12:S:msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#3#(msplit(@xs'),@x1,@x2),msplit#(@xs')) -->_1 msplit#3#(tuple#2(@l1,@l2),@x1,@x2) -> c_20():35 -->_2 msplit#(@l) -> c_15(msplit#1#(@l)):10 13:W:#cklt#(#EQ()) -> c_21() 14:W:#cklt#(#GT()) -> c_22() 15:W:#cklt#(#LT()) -> c_23() 16:W:#compare#(#0(),#0()) -> c_24() 17:W:#compare#(#0(),#neg(@y)) -> c_25() 18:W:#compare#(#0(),#pos(@y)) -> c_26() 19:W:#compare#(#0(),#s(@y)) -> c_27() 20:W:#compare#(#neg(@x),#0()) -> c_28() 21:W:#compare#(#neg(@x),#neg(@y)) -> c_29(#compare#(@y,@x)) -->_1 #compare#(#s(@x),#s(@y)) -> c_35(#compare#(@x,@y)):27 -->_1 #compare#(#pos(@x),#pos(@y)) -> c_33(#compare#(@x,@y)):25 -->_1 #compare#(#s(@x),#0()) -> c_34():26 -->_1 #compare#(#pos(@x),#neg(@y)) -> c_32():24 -->_1 #compare#(#pos(@x),#0()) -> c_31():23 -->_1 #compare#(#neg(@x),#pos(@y)) -> c_30():22 -->_1 #compare#(#neg(@x),#neg(@y)) -> c_29(#compare#(@y,@x)):21 -->_1 #compare#(#neg(@x),#0()) -> c_28():20 -->_1 #compare#(#0(),#s(@y)) -> c_27():19 -->_1 #compare#(#0(),#pos(@y)) -> c_26():18 -->_1 #compare#(#0(),#neg(@y)) -> c_25():17 -->_1 #compare#(#0(),#0()) -> c_24():16 22:W:#compare#(#neg(@x),#pos(@y)) -> c_30() 23:W:#compare#(#pos(@x),#0()) -> c_31() 24:W:#compare#(#pos(@x),#neg(@y)) -> c_32() 25:W:#compare#(#pos(@x),#pos(@y)) -> c_33(#compare#(@x,@y)) -->_1 #compare#(#s(@x),#s(@y)) -> c_35(#compare#(@x,@y)):27 -->_1 #compare#(#s(@x),#0()) -> c_34():26 -->_1 #compare#(#pos(@x),#pos(@y)) -> c_33(#compare#(@x,@y)):25 -->_1 #compare#(#pos(@x),#neg(@y)) -> c_32():24 -->_1 #compare#(#pos(@x),#0()) -> c_31():23 -->_1 #compare#(#neg(@x),#pos(@y)) -> c_30():22 -->_1 #compare#(#neg(@x),#neg(@y)) -> c_29(#compare#(@y,@x)):21 -->_1 #compare#(#neg(@x),#0()) -> c_28():20 -->_1 #compare#(#0(),#s(@y)) -> c_27():19 -->_1 #compare#(#0(),#pos(@y)) -> c_26():18 -->_1 #compare#(#0(),#neg(@y)) -> c_25():17 -->_1 #compare#(#0(),#0()) -> c_24():16 26:W:#compare#(#s(@x),#0()) -> c_34() 27:W:#compare#(#s(@x),#s(@y)) -> c_35(#compare#(@x,@y)) -->_1 #compare#(#s(@x),#s(@y)) -> c_35(#compare#(@x,@y)):27 -->_1 #compare#(#s(@x),#0()) -> c_34():26 -->_1 #compare#(#pos(@x),#pos(@y)) -> c_33(#compare#(@x,@y)):25 -->_1 #compare#(#pos(@x),#neg(@y)) -> c_32():24 -->_1 #compare#(#pos(@x),#0()) -> c_31():23 -->_1 #compare#(#neg(@x),#pos(@y)) -> c_30():22 -->_1 #compare#(#neg(@x),#neg(@y)) -> c_29(#compare#(@y,@x)):21 -->_1 #compare#(#neg(@x),#0()) -> c_28():20 -->_1 #compare#(#0(),#s(@y)) -> c_27():19 -->_1 #compare#(#0(),#pos(@y)) -> c_26():18 -->_1 #compare#(#0(),#neg(@y)) -> c_25():17 -->_1 #compare#(#0(),#0()) -> c_24():16 28:W:#less#(@x,@y) -> c_1(#cklt#(#compare(@x,@y)),#compare#(@x,@y)) -->_2 #compare#(#s(@x),#s(@y)) -> c_35(#compare#(@x,@y)):27 -->_2 #compare#(#s(@x),#0()) -> c_34():26 -->_2 #compare#(#pos(@x),#pos(@y)) -> c_33(#compare#(@x,@y)):25 -->_2 #compare#(#pos(@x),#neg(@y)) -> c_32():24 -->_2 #compare#(#pos(@x),#0()) -> c_31():23 -->_2 #compare#(#neg(@x),#pos(@y)) -> c_30():22 -->_2 #compare#(#neg(@x),#neg(@y)) -> c_29(#compare#(@y,@x)):21 -->_2 #compare#(#neg(@x),#0()) -> c_28():20 -->_2 #compare#(#0(),#s(@y)) -> c_27():19 -->_2 #compare#(#0(),#pos(@y)) -> c_26():18 -->_2 #compare#(#0(),#neg(@y)) -> c_25():17 -->_2 #compare#(#0(),#0()) -> c_24():16 -->_1 #cklt#(#LT()) -> c_23():15 -->_1 #cklt#(#GT()) -> c_22():14 -->_1 #cklt#(#EQ()) -> c_21():13 29:W:merge#1#(nil(),@l2) -> c_4() 30:W:merge#2#(nil(),@x,@xs) -> c_6() 31:W:mergesort#1#(nil()) -> c_11() 32:W:mergesort#2#(nil(),@x1) -> c_13() 33:W:msplit#1#(nil()) -> c_17() 34:W:msplit#2#(nil(),@x1) -> c_19() 35:W:msplit#3#(tuple#2(@l1,@l2),@x1,@x2) -> c_20() 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_11() 32: mergesort#2#(nil(),@x1) -> c_13() 33: msplit#1#(nil()) -> c_17() 34: msplit#2#(nil(),@x1) -> c_19() 35: msplit#3#(tuple#2(@l1,@l2),@x1,@x2) -> c_20() 29: merge#1#(nil(),@l2) -> c_4() 30: merge#2#(nil(),@x,@xs) -> c_6() 28: #less#(@x,@y) -> c_1(#cklt#(#compare(@x,@y)),#compare#(@x,@y)) 13: #cklt#(#EQ()) -> c_21() 14: #cklt#(#GT()) -> c_22() 15: #cklt#(#LT()) -> c_23() 27: #compare#(#s(@x),#s(@y)) -> c_35(#compare#(@x,@y)) 25: #compare#(#pos(@x),#pos(@y)) -> c_33(#compare#(@x,@y)) 21: #compare#(#neg(@x),#neg(@y)) -> c_29(#compare#(@y,@x)) 16: #compare#(#0(),#0()) -> c_24() 17: #compare#(#0(),#neg(@y)) -> c_25() 18: #compare#(#0(),#pos(@y)) -> c_26() 19: #compare#(#0(),#s(@y)) -> c_27() 20: #compare#(#neg(@x),#0()) -> c_28() 22: #compare#(#neg(@x),#pos(@y)) -> c_30() 23: #compare#(#pos(@x),#0()) -> c_31() 24: #compare#(#pos(@x),#neg(@y)) -> c_32() 26: #compare#(#s(@x),#0()) -> c_34() * Step 4: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys),#less#(@x,@y)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/2,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/2,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) -->_1 merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)):2 2:S:merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) -->_1 merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys),#less#(@x,@y)):3 3:S:merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys),#less#(@x,@y)) -->_1 merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))):5 -->_1 merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)):4 4:S:merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 5:S:merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 6:S:mergesort#(@l) -> c_9(mergesort#1#(@l)) -->_1 mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)):7 7:S:mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) -->_1 mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))):8 8:S:mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) -->_2 msplit#(@l) -> c_15(msplit#1#(@l)):10 -->_1 mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)):9 9:S:mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) -->_3 mergesort#(@l) -> c_9(mergesort#1#(@l)):6 -->_2 mergesort#(@l) -> c_9(mergesort#1#(@l)):6 -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 10:S:msplit#(@l) -> c_15(msplit#1#(@l)) -->_1 msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)):11 11:S:msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) -->_1 msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#3#(msplit(@xs'),@x1,@x2),msplit#(@xs')):12 12:S:msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#3#(msplit(@xs'),@x1,@x2),msplit#(@xs')) -->_2 msplit#(@l) -> c_15(msplit#1#(@l)):10 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) * Step 5: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) - Weak DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0 ,tuple#2/2,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/3,c_15/1 ,c_16/1,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0 ,c_31/0,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} Problem (S) - Strict DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) - Weak DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) - 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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0 ,tuple#2/2,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/3,c_15/1 ,c_16/1,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0 ,c_31/0,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} ** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) - Weak DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) -->_1 merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)):2 2:S:merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) -->_1 merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)):3 3:S:merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) -->_1 merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))):5 -->_1 merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)):4 4:S:merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 5:S:merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 6:W:mergesort#(@l) -> c_9(mergesort#1#(@l)) -->_1 mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)):7 7:W:mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) -->_1 mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))):8 8:W:mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) -->_2 msplit#(@l) -> c_15(msplit#1#(@l)):10 -->_1 mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)):9 9:W:mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 -->_3 mergesort#(@l) -> c_9(mergesort#1#(@l)):6 -->_2 mergesort#(@l) -> c_9(mergesort#1#(@l)):6 10:W:msplit#(@l) -> c_15(msplit#1#(@l)) -->_1 msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)):11 11:W:msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) -->_1 msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')):12 12:W:msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) -->_1 msplit#(@l) -> c_15(msplit#1#(@l)):10 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: msplit#(@l) -> c_15(msplit#1#(@l)) 12: msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) 11: msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) ** Step 5.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) - Weak DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) -->_1 merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)):2 2:S:merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) -->_1 merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)):3 3:S:merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) -->_1 merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))):5 -->_1 merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)):4 4:S:merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 5:S:merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 6:W:mergesort#(@l) -> c_9(mergesort#1#(@l)) -->_1 mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)):7 7:W:mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) -->_1 mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))):8 8:W:mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) -->_1 mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)):9 9:W:mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 -->_3 mergesort#(@l) -> c_9(mergesort#1#(@l)):6 -->_2 mergesort#(@l) -> c_9(mergesort#1#(@l)):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs'))))) ** Step 5.a:3: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) - Weak DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs'))))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs'))))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) and a lower component merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) Further, following extension rules are added to the lower component. mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) - Weak DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) Consider the set of all dependency pairs 1: mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) 2: mergesort#(@l) -> c_9(mergesort#1#(@l)) 3: mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) 4: mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs'))))) Processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2,3,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 5.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) - Weak DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_14) = {1,2,3} 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] [1] p(#EQ) = [1] [1] [1] p(#GT) = [1] [0] [0] p(#LT) = [0] [0] [0] p(#cklt) = [1 1 0] [0] [1 0 1] x1 + [1] [1 0 0] [1] p(#compare) = [0 0 0] [0 1 0] [0] [1 1 1] x1 + [0 0 1] x2 + [0] [1 1 0] [1 0 0] [0] p(#false) = [0] [1] [0] p(#less) = [0 0 0] [1 0 0] [0] [0 1 0] x1 + [0 0 1] x2 + [1] [0 0 0] [1 1 1] [1] p(#neg) = [0] [1] [1] p(#pos) = [0] [0] [0] p(#s) = [0 0 1] [1] [0 0 1] x1 + [0] [0 0 0] [1] p(#true) = [0] [1] [0] p(::) = [0 1 0] [0] [0 1 1] x2 + [0] [0 0 0] [1] p(merge) = [0 0 0] [1 1 1] [0] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 0] [0] p(merge#1) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(merge#2) = [0 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(merge#3) = [0 0 0] [0 0 0] [1 1 0] [1] [1 0 1] x1 + [0 0 0] x2 + [0 0 0] x4 + [1] [0 1 1] [0 1 0] [0 0 0] [1] p(mergesort) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(mergesort#1) = [0 0 1] [0] [0 0 1] x1 + [1] [0 0 0] [1] p(mergesort#2) = [0 0 1] [0] [0 1 1] x1 + [0] [0 1 0] [0] p(mergesort#3) = [0 0 0] [0] [1 0 0] x1 + [0] [0 0 1] [0] p(msplit) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [1] p(msplit#1) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [1] p(msplit#2) = [0 1 0] [0] [0 1 1] x1 + [1] [0 0 0] [1] p(msplit#3) = [0 1 0] [0] [0 1 1] x1 + [1] [0 0 0] [1] p(nil) = [0] [0] [0] p(tuple#2) = [0 1 0] [0 1 0] [0] [0 1 1] x1 + [0 1 1] x2 + [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] [1] [0] p(merge#1#) = [0] [0] [0] p(merge#2#) = [0] [0] [0] p(merge#3#) = [0] [0] [0] p(mergesort#) = [0 1 0] [0] [1 0 0] x1 + [0] [0 0 0] [0] p(mergesort#1#) = [0 1 0] [0] [0 0 1] x1 + [1] [1 0 0] [0] p(mergesort#2#) = [0 1 1] [0 0 0] [0] [0 0 0] x1 + [1 0 1] x2 + [1] [0 0 1] [1 0 1] [1] p(mergesort#3#) = [1 0 1] [0] [0 0 1] x1 + [1] [0 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) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(c_10) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_11) = [0] [0] [0] p(c_12) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(c_13) = [0] [0] [0] p(c_14) = [1 0 0] [1 0 0] [1 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [1] [0 1 0] [0 0 0] [0 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) = [0] [0] [0] p(c_25) = [0] [0] [0] p(c_26) = [0] [0] [0] p(c_27) = [0] [0] [0] p(c_28) = [0] [0] [0] p(c_29) = [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#3#(tuple#2(@l1,@l2)) = [0 1 0] [0 1 0] [1] [0 0 0] @l1 + [0 0 0] @l2 + [2] [0 0 0] [0 0 0] [1] > [0 1 0] [0 1 0] [0] [0 0 0] @l1 + [0 0 0] @l2 + [2] [0 0 0] [0 0 0] [1] = c_14(merge#(mergesort(@l1),mergesort(@l2)),mergesort#(@l1),mergesort#(@l2)) Following rules are (at-least) weakly oriented: mergesort#(@l) = [0 1 0] [0] [1 0 0] @l + [0] [0 0 0] [0] >= [0 1 0] [0] [1 0 0] @l + [0] [0 0 0] [0] = c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) = [0 1 1] [0] [0 0 0] @xs + [2] [0 1 0] [0] >= [0 1 1] [0] [0 0 0] @xs + [0] [0 0 0] [0] = c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) = [0 0 0] [0 1 1] [1] [1 0 1] @x1 + [0 0 0] @xs' + [1] [1 0 1] [0 0 0] [2] >= [0 1 1] [1] [0 0 0] @xs' + [0] [0 0 0] [1] = c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs'))))) msplit(@l) = [1 0 0] [0] [0 1 1] @l + [0] [0 0 0] [1] >= [1 0 0] [0] [0 1 1] @l + [0] [0 0 0] [1] = msplit#1(@l) msplit#1(::(@x1,@xs)) = [0 1 0] [0] [0 1 1] @xs + [1] [0 0 0] [1] >= [0 1 0] [0] [0 1 1] @xs + [1] [0 0 0] [1] = msplit#2(@xs,@x1) msplit#1(nil()) = [0] [0] [1] >= [0] [0] [1] = tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) = [0 1 1] [0] [0 1 1] @xs' + [2] [0 0 0] [1] >= [0 1 1] [0] [0 1 1] @xs' + [2] [0 0 0] [1] = msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) = [0] [1] [1] >= [0] [1] [1] = tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) = [0 1 1] [0 1 1] [0] [0 1 1] @l1 + [0 1 1] @l2 + [2] [0 0 0] [0 0 0] [1] >= [0 1 1] [0 1 1] [0] [0 1 1] @l1 + [0 1 1] @l2 + [2] [0 0 0] [0 0 0] [1] = tuple#2(::(@x1,@l1),::(@x2,@l2)) **** Step 5.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs'))))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs'))))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:mergesort#(@l) -> c_9(mergesort#1#(@l)) -->_1 mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)):2 2:W:mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) -->_1 mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs'))))):3 3:W:mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs'))))) -->_1 mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)):4 4:W:mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) -->_3 mergesort#(@l) -> c_9(mergesort#1#(@l)):1 -->_2 mergesort#(@l) -> c_9(mergesort#1#(@l)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: mergesort#(@l) -> c_9(mergesort#1#(@l)) 4: mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) 3: mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs'))))) 2: mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) **** Step 5.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 5.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) - Weak DPs: mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 5: merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) The strictly oriented rules are moved into the weak component. **** Step 5.a:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) - Weak DPs: mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {merge,merge#1,merge#2,merge#3,mergesort,mergesort#1,mergesort#2,mergesort#3,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] p(#EQ) = [0] p(#GT) = [0] p(#LT) = [0] p(#cklt) = [7] x1 + [0] p(#compare) = [2] x1 + [2] x2 + [2] p(#false) = [0] p(#less) = [0] p(#neg) = [0] p(#pos) = [1] x1 + [0] p(#s) = [0] p(#true) = [0] p(::) = [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] x3 + [1] x5 + [2] p(mergesort) = [4] x1 + [0] p(mergesort#1) = [4] x1 + [0] p(mergesort#2) = [4] x1 + [4] p(mergesort#3) = [4] 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] p(#compare#) = [0] p(#less#) = [0] p(merge#) = [1] x1 + [0] p(merge#1#) = [1] x1 + [0] p(merge#2#) = [1] x3 + [1] p(merge#3#) = [1] x3 + [1] p(mergesort#) = [4] x1 + [0] p(mergesort#1#) = [4] x1 + [0] p(mergesort#2#) = [4] x1 + [4] p(mergesort#3#) = [4] x1 + [0] p(msplit#) = [4] p(msplit#1#) = [1] x1 + [2] p(msplit#2#) = [1] x2 + [0] p(msplit#3#) = [1] x2 + [1] x3 + [0] p(c_1) = [2] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [4] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [2] p(c_10) = [2] p(c_11) = [1] p(c_12) = [2] x1 + [0] p(c_13) = [2] p(c_14) = [2] x1 + [4] x3 + [0] p(c_15) = [1] x1 + [1] p(c_16) = [2] x1 + [4] p(c_17) = [1] p(c_18) = [1] x1 + [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [1] p(c_22) = [0] p(c_23) = [4] p(c_24) = [1] p(c_25) = [2] p(c_26) = [0] p(c_27) = [1] p(c_28) = [2] p(c_29) = [1] x1 + [1] p(c_30) = [0] p(c_31) = [1] p(c_32) = [0] p(c_33) = [2] p(c_34) = [1] p(c_35) = [1] Following rules are strictly oriented: merge#3#(#true(),@x,@xs,@y,@ys) = [1] @xs + [1] > [1] @xs + [0] = c_8(merge#(@xs,::(@y,@ys))) Following rules are (at-least) weakly oriented: merge#(@l1,@l2) = [1] @l1 + [0] >= [1] @l1 + [0] = c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) = [1] @xs + [1] >= [1] @xs + [1] = c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) = [1] @xs + [1] >= [1] @xs + [1] = c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) = [1] @xs + [1] >= [1] @xs + [1] = c_7(merge#(::(@x,@xs),@ys)) mergesort#(@l) = [4] @l + [0] >= [4] @l + [0] = mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) = [4] @xs + [4] >= [4] @xs + [4] = mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) = [4] @xs' + [8] >= [4] @xs' + [8] = mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) = [4] @l1 + [4] @l2 + [0] >= [4] @l1 + [0] = merge#(mergesort(@l1),mergesort(@l2)) mergesort#3#(tuple#2(@l1,@l2)) = [4] @l1 + [4] @l2 + [0] >= [4] @l1 + [0] = mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) = [4] @l1 + [4] @l2 + [0] >= [4] @l2 + [0] = mergesort#(@l2) merge(@l1,@l2) = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = merge#1(@l1,@l2) merge#1(::(@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(::(@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] = ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) = [4] @l + [0] >= [4] @l + [0] = mergesort#1(@l) mergesort#1(::(@x1,@xs)) = [4] @xs + [4] >= [4] @xs + [4] = mergesort#2(@xs,@x1) mergesort#1(nil()) = [0] >= [0] = nil() mergesort#2(::(@x2,@xs'),@x1) = [4] @xs' + [8] >= [4] @xs' + [8] = mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) = [4] >= [1] = ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) = [4] @l1 + [4] @l2 + [0] >= [4] @l1 + [4] @l2 + [0] = merge(mergesort(@l1),mergesort(@l2)) msplit(@l) = [1] @l + [0] >= [1] @l + [0] = msplit#1(@l) msplit#1(::(@x1,@xs)) = [1] @xs + [1] >= [1] @xs + [1] = msplit#2(@xs,@x1) msplit#1(nil()) = [0] >= [0] = tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) = [1] @xs' + [2] >= [1] @xs' + [2] = msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) = [1] >= [1] = tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) = [1] @l1 + [1] @l2 + [2] >= [1] @l1 + [1] @l2 + [2] = tuple#2(::(@x1,@l1),::(@x2,@l2)) **** Step 5.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) - Weak DPs: merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.a:3.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) - Weak DPs: merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 4: merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) The strictly oriented rules are moved into the weak component. ***** Step 5.a:3.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) - Weak DPs: merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {merge,merge#1,merge#2,merge#3,mergesort,mergesort#1,mergesort#2,mergesort#3,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] p(#EQ) = [4] p(#GT) = [5] p(#LT) = [0] p(#cklt) = [2] x1 + [0] p(#compare) = [7] x1 + [4] x2 + [7] p(#false) = [0] p(#less) = [0] p(#neg) = [0] p(#pos) = [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(::) = [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] 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#) = [2] p(#compare#) = [1] x1 + [1] p(#less#) = [4] p(merge#) = [1] x2 + [0] p(merge#1#) = [1] x2 + [0] p(merge#2#) = [1] x1 + [0] p(merge#3#) = [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#) = [1] x1 + [0] p(msplit#2#) = [1] x1 + [0] p(msplit#3#) = [1] x2 + [0] p(c_1) = [4] x1 + [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [1] x1 + [1] p(c_8) = [1] x1 + [0] p(c_9) = [1] p(c_10) = [4] p(c_11) = [2] p(c_12) = [1] p(c_13) = [0] p(c_14) = [4] x1 + [1] x2 + [1] x3 + [2] p(c_15) = [1] x1 + [0] p(c_16) = [0] p(c_17) = [2] p(c_18) = [1] x1 + [1] p(c_19) = [1] p(c_20) = [0] p(c_21) = [1] p(c_22) = [1] p(c_23) = [4] p(c_24) = [1] p(c_25) = [0] p(c_26) = [2] p(c_27) = [2] p(c_28) = [2] p(c_29) = [1] x1 + [2] p(c_30) = [0] p(c_31) = [4] p(c_32) = [0] p(c_33) = [1] p(c_34) = [0] p(c_35) = [1] Following rules are strictly oriented: merge#3#(#false(),@x,@xs,@y,@ys) = [1] @ys + [2] > [1] @ys + [1] = c_7(merge#(::(@x,@xs),@ys)) Following rules are (at-least) weakly oriented: merge#(@l1,@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) = [1] @ys + [2] >= [1] @ys + [2] = c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#true(),@x,@xs,@y,@ys) = [1] @ys + [2] >= [1] @ys + [2] = c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) = [1] @l + [2] >= [1] @l + [2] = mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) = [1] @xs + [4] >= [1] @xs + [4] = mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) = [1] @xs' + [6] >= [1] @xs' + [6] = mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) = [1] @l1 + [1] @l2 + [2] >= [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) merge(@l1,@l2) = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = merge#1(@l1,@l2) merge#1(::(@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(::(@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] = ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) = [1] @xs + [1] @ys + [4] >= [1] @xs + [1] @ys + [4] = ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) = [1] @xs + [1] @ys + [4] >= [1] @xs + [1] @ys + [4] = ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) = [1] @l + [0] >= [1] @l + [0] = mergesort#1(@l) mergesort#1(::(@x1,@xs)) = [1] @xs + [2] >= [1] @xs + [2] = mergesort#2(@xs,@x1) mergesort#1(nil()) = [0] >= [0] = nil() mergesort#2(::(@x2,@xs'),@x1) = [1] @xs' + [4] >= [1] @xs' + [4] = mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) = [2] >= [2] = ::(@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(::(@x1,@xs)) = [1] @xs + [2] >= [1] @xs + [2] = msplit#2(@xs,@x1) msplit#1(nil()) = [0] >= [0] = tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) = [1] @xs' + [4] >= [1] @xs' + [4] = msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) = [2] >= [2] = tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) = [1] @l1 + [1] @l2 + [4] >= [1] @l1 + [1] @l2 + [4] = tuple#2(::(@x1,@l1),::(@x2,@l2)) ***** Step 5.a:3.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) - Weak DPs: merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 5.a:3.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) - Weak DPs: merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) 3: merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) Consider the set of all dependency pairs 1: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) 2: merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) 3: merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) 4: merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) 5: merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) 6: mergesort#(@l) -> mergesort#1#(@l) 7: mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) 8: mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) 9: mergesort#3#(tuple#2(@l1,@l2)) -> merge#(mergesort(@l1),mergesort(@l2)) 10: mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) 11: mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {2,3} These cover all (indirect) predecessors of dependency pairs {2,3,4,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ****** Step 5.a:3.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) - Weak DPs: merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {merge,merge#1,merge#2,merge#3,mergesort,mergesort#1,mergesort#2,mergesort#3,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] p(#EQ) = [0] p(#GT) = [0] p(#LT) = [0] p(#cklt) = [7] x1 + [1] p(#compare) = [2] x1 + [2] x2 + [2] p(#false) = [0] p(#less) = [2] x1 + [0] p(#neg) = [1] p(#pos) = [0] p(#s) = [1] p(#true) = [0] p(::) = [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] x3 + [1] x5 + [2] p(mergesort) = [2] x1 + [0] p(mergesort#1) = [2] x1 + [0] p(mergesort#2) = [2] x1 + [2] p(mergesort#3) = [2] 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] p(#compare#) = [2] x1 + [1] x2 + [0] p(#less#) = [2] x1 + [2] p(merge#) = [2] x1 + [2] x2 + [0] p(merge#1#) = [2] x1 + [2] x2 + [0] p(merge#2#) = [2] x1 + [2] x3 + [1] p(merge#3#) = [2] x3 + [2] x5 + [2] p(mergesort#) = [4] x1 + [0] p(mergesort#1#) = [4] x1 + [0] p(mergesort#2#) = [4] x1 + [4] p(mergesort#3#) = [4] x1 + [0] p(msplit#) = [2] x1 + [4] p(msplit#1#) = [0] p(msplit#2#) = [4] x1 + [4] x2 + [0] p(msplit#3#) = [2] x1 + [1] p(c_1) = [1] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [2] p(c_5) = [1] x1 + [0] p(c_6) = [2] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [1] p(c_10) = [4] p(c_11) = [1] p(c_12) = [1] x1 + [0] p(c_13) = [1] p(c_14) = [2] x1 + [2] x2 + [0] p(c_15) = [1] x1 + [0] p(c_16) = [4] x1 + [4] p(c_17) = [1] p(c_18) = [4] x1 + [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [1] p(c_22) = [1] p(c_23) = [1] p(c_24) = [1] p(c_25) = [0] p(c_26) = [1] p(c_27) = [1] p(c_28) = [0] p(c_29) = [1] x1 + [0] p(c_30) = [0] p(c_31) = [1] p(c_32) = [1] p(c_33) = [2] x1 + [4] p(c_34) = [0] p(c_35) = [1] x1 + [1] Following rules are strictly oriented: merge#1#(::(@x,@xs),@l2) = [2] @l2 + [2] @xs + [2] > [2] @l2 + [2] @xs + [1] = c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) = [2] @xs + [2] @ys + [3] > [2] @xs + [2] @ys + [2] = c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) Following rules are (at-least) weakly oriented: merge#(@l1,@l2) = [2] @l1 + [2] @l2 + [0] >= [2] @l1 + [2] @l2 + [0] = c_2(merge#1#(@l1,@l2)) merge#3#(#false(),@x,@xs,@y,@ys) = [2] @xs + [2] @ys + [2] >= [2] @xs + [2] @ys + [2] = c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) = [2] @xs + [2] @ys + [2] >= [2] @xs + [2] @ys + [2] = c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) = [4] @l + [0] >= [4] @l + [0] = mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) = [4] @xs + [4] >= [4] @xs + [4] = mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) = [4] @xs' + [8] >= [4] @xs' + [8] = mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) = [4] @l1 + [4] @l2 + [0] >= [4] @l1 + [4] @l2 + [0] = merge#(mergesort(@l1),mergesort(@l2)) mergesort#3#(tuple#2(@l1,@l2)) = [4] @l1 + [4] @l2 + [0] >= [4] @l1 + [0] = mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) = [4] @l1 + [4] @l2 + [0] >= [4] @l2 + [0] = mergesort#(@l2) merge(@l1,@l2) = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = merge#1(@l1,@l2) merge#1(::(@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(::(@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] = ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) = [2] @l + [0] >= [2] @l + [0] = mergesort#1(@l) mergesort#1(::(@x1,@xs)) = [2] @xs + [2] >= [2] @xs + [2] = mergesort#2(@xs,@x1) mergesort#1(nil()) = [0] >= [0] = nil() mergesort#2(::(@x2,@xs'),@x1) = [2] @xs' + [4] >= [2] @xs' + [4] = mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) = [2] >= [1] = ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) = [2] @l1 + [2] @l2 + [0] >= [2] @l1 + [2] @l2 + [0] = merge(mergesort(@l1),mergesort(@l2)) msplit(@l) = [1] @l + [0] >= [1] @l + [0] = msplit#1(@l) msplit#1(::(@x1,@xs)) = [1] @xs + [1] >= [1] @xs + [1] = msplit#2(@xs,@x1) msplit#1(nil()) = [0] >= [0] = tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) = [1] @xs' + [2] >= [1] @xs' + [2] = msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) = [1] >= [1] = tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) = [1] @l1 + [1] @l2 + [2] >= [1] @l1 + [1] @l2 + [2] = tuple#2(::(@x1,@l1),::(@x2,@l2)) ****** Step 5.a:3.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) - Weak DPs: merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 5.a:3.b:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) - Weak DPs: merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) Consider the set of all dependency pairs 1: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) 2: merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) 3: merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) 4: merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) 5: merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) 6: mergesort#(@l) -> mergesort#1#(@l) 7: mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) 8: mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) 9: mergesort#3#(tuple#2(@l1,@l2)) -> merge#(mergesort(@l1),mergesort(@l2)) 10: mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) 11: mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******* Step 5.a:3.b:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) - Weak DPs: merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {merge,merge#1,merge#2,merge#3,mergesort,mergesort#1,mergesort#2,mergesort#3,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) = [3] p(#EQ) = [0] p(#GT) = [0] p(#LT) = [0] p(#cklt) = [5] x1 + [0] p(#compare) = [2] x1 + [3] p(#false) = [0] p(#less) = [0] p(#neg) = [2] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [3] p(#true) = [0] p(::) = [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] 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 + [0] p(#compare#) = [4] x2 + [4] p(#less#) = [4] p(merge#) = [1] x1 + [1] x2 + [1] p(merge#1#) = [1] x1 + [1] x2 + [0] p(merge#2#) = [1] x1 + [1] x3 + [2] p(merge#3#) = [1] x3 + [1] x5 + [3] p(mergesort#) = [2] x1 + [1] p(mergesort#1#) = [2] x1 + [1] p(mergesort#2#) = [2] x1 + [5] p(mergesort#3#) = [2] x1 + [1] p(msplit#) = [1] x1 + [1] p(msplit#1#) = [2] x1 + [0] p(msplit#2#) = [4] x1 + [1] x2 + [0] p(msplit#3#) = [1] x1 + [2] x2 + [2] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [1] x1 + [1] p(c_6) = [2] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [4] p(c_10) = [2] x1 + [1] p(c_11) = [1] p(c_12) = [4] p(c_13) = [0] p(c_14) = [1] x1 + [1] x3 + [0] p(c_15) = [1] p(c_16) = [1] p(c_17) = [1] p(c_18) = [2] p(c_19) = [1] p(c_20) = [1] p(c_21) = [0] p(c_22) = [0] p(c_23) = [1] p(c_24) = [1] p(c_25) = [0] p(c_26) = [0] p(c_27) = [2] p(c_28) = [4] p(c_29) = [2] x1 + [1] p(c_30) = [0] p(c_31) = [2] p(c_32) = [4] p(c_33) = [1] p(c_34) = [2] p(c_35) = [1] Following rules are strictly oriented: merge#(@l1,@l2) = [1] @l1 + [1] @l2 + [1] > [1] @l1 + [1] @l2 + [0] = c_2(merge#1#(@l1,@l2)) Following rules are (at-least) weakly oriented: merge#1#(::(@x,@xs),@l2) = [1] @l2 + [1] @xs + [2] >= [1] @l2 + [1] @xs + [2] = c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) = [1] @xs + [1] @ys + [4] >= [1] @xs + [1] @ys + [4] = c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) = [1] @xs + [1] @ys + [3] >= [1] @xs + [1] @ys + [3] = c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) = [1] @xs + [1] @ys + [3] >= [1] @xs + [1] @ys + [3] = c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) = [2] @l + [1] >= [2] @l + [1] = mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) = [2] @xs + [5] >= [2] @xs + [5] = mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) = [2] @xs' + [9] >= [2] @xs' + [9] = mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) = [2] @l1 + [2] @l2 + [1] >= [1] @l1 + [1] @l2 + [1] = merge#(mergesort(@l1),mergesort(@l2)) mergesort#3#(tuple#2(@l1,@l2)) = [2] @l1 + [2] @l2 + [1] >= [2] @l1 + [1] = mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) = [2] @l1 + [2] @l2 + [1] >= [2] @l2 + [1] = mergesort#(@l2) merge(@l1,@l2) = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = merge#1(@l1,@l2) merge#1(::(@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(::(@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] = ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) = [1] @xs + [1] @ys + [4] >= [1] @xs + [1] @ys + [4] = ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) = [1] @xs + [1] @ys + [4] >= [1] @xs + [1] @ys + [4] = ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) = [1] @l + [0] >= [1] @l + [0] = mergesort#1(@l) mergesort#1(::(@x1,@xs)) = [1] @xs + [2] >= [1] @xs + [2] = mergesort#2(@xs,@x1) mergesort#1(nil()) = [0] >= [0] = nil() mergesort#2(::(@x2,@xs'),@x1) = [1] @xs' + [4] >= [1] @xs' + [4] = mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) = [2] >= [2] = ::(@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(::(@x1,@xs)) = [1] @xs + [2] >= [1] @xs + [2] = msplit#2(@xs,@x1) msplit#1(nil()) = [0] >= [0] = tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) = [1] @xs' + [4] >= [1] @xs' + [4] = msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) = [2] >= [2] = tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) = [1] @l1 + [1] @l2 + [4] >= [1] @l1 + [1] @l2 + [4] = tuple#2(::(@x1,@l1),::(@x2,@l2)) ******* Step 5.a:3.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 5.a:3.b:1.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) -->_1 merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)):2 2:W:merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) -->_1 merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)):3 3:W:merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) -->_1 merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))):5 -->_1 merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)):4 4:W:merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 5:W:merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 6:W:mergesort#(@l) -> mergesort#1#(@l) -->_1 mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1):7 7:W:mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) -->_1 mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))):8 8:W:mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) -->_1 mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2):11 -->_1 mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1):10 -->_1 mergesort#3#(tuple#2(@l1,@l2)) -> merge#(mergesort(@l1),mergesort(@l2)):9 9:W:mergesort#3#(tuple#2(@l1,@l2)) -> merge#(mergesort(@l1),mergesort(@l2)) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):1 10:W:mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) -->_1 mergesort#(@l) -> mergesort#1#(@l):6 11:W:mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) -->_1 mergesort#(@l) -> mergesort#1#(@l):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: mergesort#(@l) -> mergesort#1#(@l) 11: mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) 8: mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) 7: mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) 10: mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) 9: mergesort#3#(tuple#2(@l1,@l2)) -> merge#(mergesort(@l1),mergesort(@l2)) 1: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) 5: merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) 3: merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) 2: merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) 4: merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) ******* Step 5.a:3.b:1.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/1,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) - Weak DPs: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) - 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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:mergesort#(@l) -> c_9(mergesort#1#(@l)) -->_1 mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)):2 2:S:mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) -->_1 mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))):3 3:S:mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) -->_2 msplit#(@l) -> c_15(msplit#1#(@l)):5 -->_1 mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)):4 4:S:mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):8 -->_3 mergesort#(@l) -> c_9(mergesort#1#(@l)):1 -->_2 mergesort#(@l) -> c_9(mergesort#1#(@l)):1 5:S:msplit#(@l) -> c_15(msplit#1#(@l)) -->_1 msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)):6 6:S:msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) -->_1 msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')):7 7:S:msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) -->_1 msplit#(@l) -> c_15(msplit#1#(@l)):5 8:W:merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) -->_1 merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)):9 9:W:merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) -->_1 merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)):10 10:W:merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) -->_1 merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))):12 -->_1 merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)):11 11:W:merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):8 12:W:merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) -->_1 merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)):8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: merge#(@l1,@l2) -> c_2(merge#1#(@l1,@l2)) 12: merge#3#(#true(),@x,@xs,@y,@ys) -> c_8(merge#(@xs,::(@y,@ys))) 10: merge#2#(::(@y,@ys),@x,@xs) -> c_5(merge#3#(#less(@x,@y),@x,@xs,@y,@ys)) 9: merge#1#(::(@x,@xs),@l2) -> c_3(merge#2#(@l2,@x,@xs)) 11: merge#3#(#false(),@x,@xs,@y,@ys) -> c_7(merge#(::(@x,@xs),@ys)) ** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/3,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:mergesort#(@l) -> c_9(mergesort#1#(@l)) -->_1 mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)):2 2:S:mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) -->_1 mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))):3 3:S:mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) -->_2 msplit#(@l) -> c_15(msplit#1#(@l)):5 -->_1 mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)):4 4:S:mergesort#3#(tuple#2(@l1,@l2)) -> c_14(merge#(mergesort(@l1),mergesort(@l2)) ,mergesort#(@l1) ,mergesort#(@l2)) -->_3 mergesort#(@l) -> c_9(mergesort#1#(@l)):1 -->_2 mergesort#(@l) -> c_9(mergesort#1#(@l)):1 5:S:msplit#(@l) -> c_15(msplit#1#(@l)) -->_1 msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)):6 6:S:msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) -->_1 msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')):7 7:S:msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) -->_1 msplit#(@l) -> c_15(msplit#1#(@l)):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mergesort#3#(tuple#2(@l1,@l2)) -> c_14(mergesort#(@l1),mergesort#(@l2)) ** Step 5.b:3: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(mergesort#(@l1),mergesort#(@l2)) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(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(::(@x,@xs),@l2) -> merge#2(@l2,@x,@xs) merge#1(nil(),@l2) -> @l2 merge#2(::(@y,@ys),@x,@xs) -> merge#3(#less(@x,@y),@x,@xs,@y,@ys) merge#2(nil(),@x,@xs) -> ::(@x,@xs) merge#3(#false(),@x,@xs,@y,@ys) -> ::(@y,merge(::(@x,@xs),@ys)) merge#3(#true(),@x,@xs,@y,@ys) -> ::(@x,merge(@xs,::(@y,@ys))) mergesort(@l) -> mergesort#1(@l) mergesort#1(::(@x1,@xs)) -> mergesort#2(@xs,@x1) mergesort#1(nil()) -> nil() mergesort#2(::(@x2,@xs'),@x1) -> mergesort#3(msplit(::(@x1,::(@x2,@xs')))) mergesort#2(nil(),@x1) -> ::(@x1,nil()) mergesort#3(tuple#2(@l1,@l2)) -> merge(mergesort(@l1),mergesort(@l2)) msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@x2,@l2)) mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(mergesort#(@l1),mergesort#(@l2)) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) ** Step 5.b:4: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(mergesort#(@l1),mergesort#(@l2)) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(mergesort#(@l1),mergesort#(@l2)) and a lower component msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) Further, following extension rules are added to the lower component. mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) *** Step 5.b:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(mergesort#(@l1),mergesort#(@l2)) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) Consider the set of all dependency pairs 1: mergesort#(@l) -> c_9(mergesort#1#(@l)) 2: mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) 3: mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) 4: mergesort#3#(tuple#2(@l1,@l2)) -> c_14(mergesort#(@l1),mergesort#(@l2)) Processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {3} These cover all (indirect) predecessors of dependency pairs {1,2,3,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 5.b:4.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(mergesort#(@l1),mergesort#(@l2)) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_14) = {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(::) = [0 0 0] [0 1 0] [0] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [1] 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) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(msplit#1) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(msplit#2) = [0 1 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 1] [1] p(msplit#3) = [0 0 1] [0 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [1] [0 1 1] [0 0 1] [0 0 1] [1] p(nil) = [0] [0] [0] p(tuple#2) = [0 1 0] [0 1 0] [0] [0 0 0] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 1] [0] 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 0] [0] [0 0 1] x1 + [0] [1 1 0] [0] p(mergesort#1#) = [0 1 0] [0] [1 1 0] x1 + [0] [1 0 1] [0] p(mergesort#2#) = [0 0 1] [0] [0 1 0] x1 + [0] [0 1 1] [0] p(mergesort#3#) = [1 0 0] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(msplit#) = [0 1 0] [0] [1 0 0] x1 + [0] [0 1 0] [1] 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) = [1 0 0] [0] [0 0 0] x1 + [0] [0 1 0] [0] p(c_10) = [1 0 0] [0] [0 1 0] x1 + [0] [1 1 0] [0] p(c_11) = [0] [0] [0] p(c_12) = [1 0 0] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [1 0 0] [1 0 0] [0] p(c_13) = [0] [0] [0] p(c_14) = [1 0 0] [1 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 1 0] [0 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) = [0] [0] [0] p(c_25) = [0] [0] [0] p(c_26) = [0] [0] [0] p(c_27) = [0] [0] [0] p(c_28) = [0] [0] [0] p(c_29) = [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#2#(::(@x2,@xs'),@x1) = [0 0 1] [0 0 1] [1] [0 0 0] @x2 + [0 0 1] @xs' + [0] [0 0 1] [0 0 2] [1] > [0 0 0] [0 0 1] [0] [0 0 0] @x2 + [0 0 0] @xs' + [0] [0 0 1] [0 0 2] [1] = c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))),msplit#(::(@x1,::(@x2,@xs')))) Following rules are (at-least) weakly oriented: mergesort#(@l) = [0 1 0] [0] [0 0 1] @l + [0] [1 1 0] [0] >= [0 1 0] [0] [0 0 0] @l + [0] [1 1 0] [0] = c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) = [0 0 0] [0 0 1] [0] [0 0 0] @x1 + [0 1 1] @xs + [0] [0 0 1] [0 1 1] [1] >= [0 0 1] [0] [0 1 0] @xs + [0] [0 1 1] [0] = c_10(mergesort#2#(@xs,@x1)) mergesort#3#(tuple#2(@l1,@l2)) = [0 1 0] [0 1 0] [0] [0 0 1] @l1 + [0 0 1] @l2 + [1] [0 0 1] [0 0 1] [1] >= [0 1 0] [0 1 0] [0] [0 0 1] @l1 + [0 0 0] @l2 + [0] [0 0 1] [0 0 0] [0] = c_14(mergesort#(@l1),mergesort#(@l2)) msplit(@l) = [1 0 0] [0] [0 0 0] @l + [1] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] @l + [1] [0 0 1] [0] = msplit#1(@l) msplit#1(::(@x1,@xs)) = [0 0 0] [0 1 0] [0] [0 0 0] @x1 + [0 0 0] @xs + [1] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 1 0] [0] [0 0 0] @x1 + [0 0 0] @xs + [1] [0 0 1] [0 0 1] [1] = msplit#2(@xs,@x1) msplit#1(nil()) = [0] [1] [0] >= [0] [1] [0] = tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) = [0 0 0] [0 0 0] [0 0 1] [0] [0 0 0] @x1 + [0 0 0] @x2 + [0 0 0] @xs' + [1] [0 0 1] [0 0 1] [0 0 1] [2] >= [0 0 0] [0 0 0] [0 0 1] [0] [0 0 0] @x1 + [0 0 0] @x2 + [0 0 0] @xs' + [1] [0 0 1] [0 0 1] [0 0 1] [2] = msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) = [0 0 0] [0] [0 0 0] @x1 + [1] [0 0 1] [1] >= [0 0 0] [0] [0 0 0] @x1 + [1] [0 0 1] [1] = tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) = [0 0 1] [0 0 1] [0 0 0] [0 0 0] [0] [0 0 0] @l1 + [0 0 0] @l2 + [0 0 0] @x1 + [0 0 0] @x2 + [1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [2] >= [0 0 1] [0 0 1] [0 0 0] [0 0 0] [0] [0 0 0] @l1 + [0 0 0] @l2 + [0 0 0] @x1 + [0 0 0] @x2 + [1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [2] = tuple#2(::(@x1,@l1),::(@x2,@l2)) **** Step 5.b:4.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(mergesort#(@l1),mergesort#(@l2)) - Weak DPs: mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.b:4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mergesort#(@l) -> c_9(mergesort#1#(@l)) mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) mergesort#3#(tuple#2(@l1,@l2)) -> c_14(mergesort#(@l1),mergesort#(@l2)) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:mergesort#(@l) -> c_9(mergesort#1#(@l)) -->_1 mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)):2 2:W:mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) -->_1 mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))):3 3:W:mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) -->_1 mergesort#3#(tuple#2(@l1,@l2)) -> c_14(mergesort#(@l1),mergesort#(@l2)):4 4:W:mergesort#3#(tuple#2(@l1,@l2)) -> c_14(mergesort#(@l1),mergesort#(@l2)) -->_2 mergesort#(@l) -> c_9(mergesort#1#(@l)):1 -->_1 mergesort#(@l) -> c_9(mergesort#1#(@l)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: mergesort#(@l) -> c_9(mergesort#1#(@l)) 4: mergesort#3#(tuple#2(@l1,@l2)) -> c_14(mergesort#(@l1),mergesort#(@l2)) 3: mergesort#2#(::(@x2,@xs'),@x1) -> c_12(mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) ,msplit#(::(@x1,::(@x2,@xs')))) 2: mergesort#1#(::(@x1,@xs)) -> c_10(mergesort#2#(@xs,@x1)) **** Step 5.b:4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 5.b:4.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) - Weak DPs: mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) The strictly oriented rules are moved into the weak component. **** Step 5.b:4.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) - Weak DPs: mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_18) = {1} 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) = [2] p(#EQ) = [1] p(#GT) = [4] p(#LT) = [0] p(#cklt) = [0] p(#compare) = [0] p(#false) = [8] p(#less) = [2] x1 + [1] x2 + [2] p(#neg) = [1] x1 + [8] p(#pos) = [8] p(#s) = [1] x1 + [2] p(#true) = [1] p(::) = [1] x1 + [1] x2 + [2] p(merge) = [1] p(merge#1) = [4] x2 + [1] p(merge#2) = [1] x3 + [1] p(merge#3) = [4] x3 + [1] p(mergesort) = [1] x1 + [0] p(mergesort#1) = [0] p(mergesort#2) = [2] x2 + [1] p(mergesort#3) = [1] p(msplit) = [1] x1 + [0] p(msplit#1) = [1] x1 + [0] p(msplit#2) = [1] x1 + [1] x2 + [2] p(msplit#3) = [1] x1 + [1] x2 + [1] x3 + [4] p(nil) = [0] p(tuple#2) = [1] x1 + [1] x2 + [0] p(#cklt#) = [8] x1 + [1] p(#compare#) = [1] x1 + [1] x2 + [1] p(#less#) = [1] p(merge#) = [1] x2 + [8] p(merge#1#) = [1] x2 + [2] p(merge#2#) = [8] x1 + [1] x2 + [8] x3 + [1] p(merge#3#) = [2] x1 + [2] x2 + [1] x5 + [0] p(mergesort#) = [4] x1 + [2] p(mergesort#1#) = [4] x1 + [2] p(mergesort#2#) = [4] x1 + [4] x2 + [10] p(mergesort#3#) = [4] x1 + [2] p(msplit#) = [4] x1 + [0] p(msplit#1#) = [4] x1 + [0] p(msplit#2#) = [4] x1 + [4] x2 + [8] p(msplit#3#) = [1] x1 + [1] x2 + [1] x3 + [0] p(c_1) = [1] x2 + [1] p(c_2) = [8] p(c_3) = [2] x1 + [0] p(c_4) = [0] p(c_5) = [1] p(c_6) = [1] p(c_7) = [2] p(c_8) = [1] x1 + [8] p(c_9) = [8] x1 + [1] p(c_10) = [1] p(c_11) = [1] p(c_12) = [1] x2 + [2] p(c_13) = [1] p(c_14) = [1] x1 + [2] x2 + [1] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [0] p(c_17) = [2] p(c_18) = [1] x1 + [8] p(c_19) = [2] p(c_20) = [1] p(c_21) = [1] p(c_22) = [1] p(c_23) = [1] p(c_24) = [1] p(c_25) = [1] p(c_26) = [1] p(c_27) = [0] p(c_28) = [2] p(c_29) = [1] x1 + [1] p(c_30) = [1] p(c_31) = [0] p(c_32) = [1] p(c_33) = [2] x1 + [1] p(c_34) = [0] p(c_35) = [1] Following rules are strictly oriented: msplit#2#(::(@x2,@xs'),@x1) = [4] @x1 + [4] @x2 + [4] @xs' + [16] > [4] @xs' + [8] = c_18(msplit#(@xs')) Following rules are (at-least) weakly oriented: mergesort#(@l) = [4] @l + [2] >= [4] @l + [2] = mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) = [4] @x1 + [4] @xs + [10] >= [4] @x1 + [4] @xs + [10] = mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) = [4] @x1 + [4] @x2 + [4] @xs' + [18] >= [4] @x1 + [4] @x2 + [4] @xs' + [18] = mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) = [4] @x1 + [4] @x2 + [4] @xs' + [18] >= [4] @x1 + [4] @x2 + [4] @xs' + [16] = msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) = [4] @l1 + [4] @l2 + [2] >= [4] @l1 + [2] = mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) = [4] @l1 + [4] @l2 + [2] >= [4] @l2 + [2] = mergesort#(@l2) msplit#(@l) = [4] @l + [0] >= [4] @l + [0] = c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) = [4] @x1 + [4] @xs + [8] >= [4] @x1 + [4] @xs + [8] = c_16(msplit#2#(@xs,@x1)) msplit(@l) = [1] @l + [0] >= [1] @l + [0] = msplit#1(@l) msplit#1(::(@x1,@xs)) = [1] @x1 + [1] @xs + [2] >= [1] @x1 + [1] @xs + [2] = msplit#2(@xs,@x1) msplit#1(nil()) = [0] >= [0] = tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) = [1] @x1 + [1] @x2 + [1] @xs' + [4] >= [1] @x1 + [1] @x2 + [1] @xs' + [4] = msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) = [1] @x1 + [2] >= [1] @x1 + [2] = tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) = [1] @l1 + [1] @l2 + [1] @x1 + [1] @x2 + [4] >= [1] @l1 + [1] @l2 + [1] @x1 + [1] @x2 + [4] = tuple#2(::(@x1,@l1),::(@x2,@l2)) **** Step 5.b:4.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) - Weak DPs: mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.b:4.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) - Weak DPs: mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) Consider the set of all dependency pairs 1: msplit#(@l) -> c_15(msplit#1#(@l)) 2: msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) 3: mergesort#(@l) -> mergesort#1#(@l) 4: mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) 5: mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) 6: mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) 7: mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) 8: mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) 9: msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {2,9} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 5.b:4.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) - Weak DPs: mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_18) = {1} 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) = [1] p(#EQ) = [2] p(#GT) = [2] p(#LT) = [0] p(#cklt) = [8] x1 + [0] p(#compare) = [1] x1 + [1] x2 + [1] p(#false) = [0] p(#less) = [1] x1 + [1] x2 + [1] p(#neg) = [1] p(#pos) = [0] p(#s) = [0] p(#true) = [2] p(::) = [1] x2 + [1] p(merge) = [1] x1 + [1] x2 + [0] p(merge#1) = [1] p(merge#2) = [1] x2 + [1] x3 + [0] p(merge#3) = [1] x4 + [1] x5 + [2] p(mergesort) = [8] p(mergesort#1) = [0] p(mergesort#2) = [1] x1 + [1] p(mergesort#3) = [1] x1 + [2] 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#) = [2] x1 + [1] p(#compare#) = [1] x2 + [8] p(#less#) = [1] x2 + [1] p(merge#) = [1] x1 + [0] p(merge#1#) = [1] x1 + [1] x2 + [0] p(merge#2#) = [4] x1 + [1] x2 + [1] x3 + [1] p(merge#3#) = [1] x1 + [2] x5 + [0] p(mergesort#) = [8] x1 + [4] p(mergesort#1#) = [8] x1 + [4] p(mergesort#2#) = [8] x1 + [12] p(mergesort#3#) = [8] x1 + [4] p(msplit#) = [8] x1 + [4] p(msplit#1#) = [8] x1 + [4] p(msplit#2#) = [8] x1 + [4] p(msplit#3#) = [1] x1 + [1] x2 + [1] x3 + [1] p(c_1) = [1] x1 + [2] x2 + [1] p(c_2) = [1] x1 + [1] p(c_3) = [2] x1 + [0] p(c_4) = [0] p(c_5) = [1] p(c_6) = [1] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [1] p(c_9) = [1] x1 + [1] p(c_10) = [1] p(c_11) = [1] p(c_12) = [4] p(c_13) = [2] p(c_14) = [2] x1 + [1] x2 + [2] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [7] p(c_17) = [1] p(c_18) = [1] x1 + [8] p(c_19) = [2] p(c_20) = [1] p(c_21) = [1] p(c_22) = [0] p(c_23) = [1] p(c_24) = [2] p(c_25) = [1] p(c_26) = [1] p(c_27) = [1] p(c_28) = [8] p(c_29) = [8] x1 + [0] p(c_30) = [0] p(c_31) = [8] p(c_32) = [1] p(c_33) = [1] p(c_34) = [1] p(c_35) = [2] x1 + [1] Following rules are strictly oriented: msplit#1#(::(@x1,@xs)) = [8] @xs + [12] > [8] @xs + [11] = c_16(msplit#2#(@xs,@x1)) Following rules are (at-least) weakly oriented: mergesort#(@l) = [8] @l + [4] >= [8] @l + [4] = mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) = [8] @xs + [12] >= [8] @xs + [12] = mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) = [8] @xs' + [20] >= [8] @xs' + [20] = mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) = [8] @xs' + [20] >= [8] @xs' + [20] = msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) = [8] @l1 + [8] @l2 + [4] >= [8] @l1 + [4] = mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) = [8] @l1 + [8] @l2 + [4] >= [8] @l2 + [4] = mergesort#(@l2) msplit#(@l) = [8] @l + [4] >= [8] @l + [4] = c_15(msplit#1#(@l)) msplit#2#(::(@x2,@xs'),@x1) = [8] @xs' + [12] >= [8] @xs' + [12] = c_18(msplit#(@xs')) msplit(@l) = [1] @l + [0] >= [1] @l + [0] = msplit#1(@l) msplit#1(::(@x1,@xs)) = [1] @xs + [1] >= [1] @xs + [1] = msplit#2(@xs,@x1) msplit#1(nil()) = [0] >= [0] = tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) = [1] @xs' + [2] >= [1] @xs' + [2] = msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) = [1] >= [1] = tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) = [1] @l1 + [1] @l2 + [2] >= [1] @l1 + [1] @l2 + [2] = tuple#2(::(@x1,@l1),::(@x2,@l2)) ***** Step 5.b:4.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: msplit#(@l) -> c_15(msplit#1#(@l)) - Weak DPs: mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 5.b:4.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: msplit#(@l) -> c_15(msplit#1#(@l)) - Weak DPs: mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: msplit#(@l) -> c_15(msplit#1#(@l)) Consider the set of all dependency pairs 1: msplit#(@l) -> c_15(msplit#1#(@l)) 2: mergesort#(@l) -> mergesort#1#(@l) 3: mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) 4: mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) 5: mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) 6: mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) 7: mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) 8: msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) 9: msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,8,9} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ****** Step 5.b:4.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: msplit#(@l) -> c_15(msplit#1#(@l)) - Weak DPs: mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_18) = {1} 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) = [1] p(#EQ) = [0] p(#GT) = [1] p(#LT) = [0] p(#cklt) = [1] p(#compare) = [0] p(#false) = [1] p(#less) = [1] x1 + [1] p(#neg) = [0] p(#pos) = [1] x1 + [4] p(#s) = [1] x1 + [4] p(#true) = [0] p(::) = [1] x2 + [1] p(merge) = [1] x1 + [4] p(merge#1) = [8] x1 + [2] x2 + [2] p(merge#2) = [1] x1 + [1] x2 + [0] p(merge#3) = [4] x3 + [8] x4 + [1] p(mergesort) = [4] x1 + [2] p(mergesort#1) = [8] x1 + [1] p(mergesort#2) = [1] x1 + [2] x2 + [0] p(mergesort#3) = [1] x1 + [8] 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 + [1] x2 + [1] p(#less#) = [2] x1 + [4] x2 + [1] p(merge#) = [2] x2 + [1] p(merge#1#) = [1] x1 + [0] p(merge#2#) = [1] x1 + [1] x2 + [1] x3 + [1] p(merge#3#) = [1] x1 + [1] p(mergesort#) = [4] x1 + [10] p(mergesort#1#) = [4] x1 + [10] p(mergesort#2#) = [4] x1 + [14] p(mergesort#3#) = [4] x1 + [10] p(msplit#) = [4] x1 + [4] p(msplit#1#) = [4] x1 + [0] p(msplit#2#) = [4] x1 + [0] p(msplit#3#) = [8] x1 + [2] x2 + [1] p(c_1) = [2] p(c_2) = [1] x1 + [1] p(c_3) = [1] p(c_4) = [0] p(c_5) = [2] p(c_6) = [1] p(c_7) = [1] p(c_8) = [2] p(c_9) = [1] x1 + [1] p(c_10) = [4] x1 + [2] p(c_11) = [2] p(c_12) = [1] x1 + [1] x2 + [2] p(c_13) = [1] p(c_14) = [1] x1 + [0] p(c_15) = [1] x1 + [3] p(c_16) = [1] x1 + [4] p(c_17) = [2] p(c_18) = [1] x1 + [0] p(c_19) = [1] p(c_20) = [2] p(c_21) = [0] p(c_22) = [1] p(c_23) = [1] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [2] p(c_29) = [0] p(c_30) = [1] p(c_31) = [0] p(c_32) = [1] p(c_33) = [1] p(c_34) = [2] p(c_35) = [0] Following rules are strictly oriented: msplit#(@l) = [4] @l + [4] > [4] @l + [3] = c_15(msplit#1#(@l)) Following rules are (at-least) weakly oriented: mergesort#(@l) = [4] @l + [10] >= [4] @l + [10] = mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) = [4] @xs + [14] >= [4] @xs + [14] = mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) = [4] @xs' + [18] >= [4] @xs' + [18] = mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) = [4] @xs' + [18] >= [4] @xs' + [12] = msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) = [4] @l1 + [4] @l2 + [10] >= [4] @l1 + [10] = mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) = [4] @l1 + [4] @l2 + [10] >= [4] @l2 + [10] = mergesort#(@l2) msplit#1#(::(@x1,@xs)) = [4] @xs + [4] >= [4] @xs + [4] = c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) = [4] @xs' + [4] >= [4] @xs' + [4] = c_18(msplit#(@xs')) msplit(@l) = [1] @l + [0] >= [1] @l + [0] = msplit#1(@l) msplit#1(::(@x1,@xs)) = [1] @xs + [1] >= [1] @xs + [1] = msplit#2(@xs,@x1) msplit#1(nil()) = [0] >= [0] = tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) = [1] @xs' + [2] >= [1] @xs' + [2] = msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) = [1] >= [1] = tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) = [1] @l1 + [1] @l2 + [2] >= [1] @l1 + [1] @l2 + [2] = tuple#2(::(@x1,@l1),::(@x2,@l2)) ****** Step 5.b:4.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 5.b:4.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mergesort#(@l) -> mergesort#1#(@l) mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) msplit#(@l) -> c_15(msplit#1#(@l)) msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:mergesort#(@l) -> mergesort#1#(@l) -->_1 mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1):2 2:W:mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) -->_1 mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))):4 -->_1 mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))):3 3:W:mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) -->_1 mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2):6 -->_1 mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1):5 4:W:mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) -->_1 msplit#(@l) -> c_15(msplit#1#(@l)):7 5:W:mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) -->_1 mergesort#(@l) -> mergesort#1#(@l):1 6:W:mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) -->_1 mergesort#(@l) -> mergesort#1#(@l):1 7:W:msplit#(@l) -> c_15(msplit#1#(@l)) -->_1 msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)):8 8:W:msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) -->_1 msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')):9 9:W:msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) -->_1 msplit#(@l) -> c_15(msplit#1#(@l)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: mergesort#(@l) -> mergesort#1#(@l) 6: mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l2) 3: mergesort#2#(::(@x2,@xs'),@x1) -> mergesort#3#(msplit(::(@x1,::(@x2,@xs')))) 2: mergesort#1#(::(@x1,@xs)) -> mergesort#2#(@xs,@x1) 5: mergesort#3#(tuple#2(@l1,@l2)) -> mergesort#(@l1) 4: mergesort#2#(::(@x2,@xs'),@x1) -> msplit#(::(@x1,::(@x2,@xs'))) 7: msplit#(@l) -> c_15(msplit#1#(@l)) 9: msplit#2#(::(@x2,@xs'),@x1) -> c_18(msplit#(@xs')) 8: msplit#1#(::(@x1,@xs)) -> c_16(msplit#2#(@xs,@x1)) ****** Step 5.b:4.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: msplit(@l) -> msplit#1(@l) msplit#1(::(@x1,@xs)) -> msplit#2(@xs,@x1) msplit#1(nil()) -> tuple#2(nil(),nil()) msplit#2(::(@x2,@xs'),@x1) -> msplit#3(msplit(@xs'),@x1,@x2) msplit#2(nil(),@x1) -> tuple#2(::(@x1,nil()),nil()) msplit#3(tuple#2(@l1,@l2),@x1,@x2) -> tuple#2(::(@x1,@l1),::(@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,::/2,nil/0,tuple#2/2 ,c_1/2,c_2/1,c_3/1,c_4/0,c_5/1,c_6/0,c_7/1,c_8/1,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0,c_14/2,c_15/1,c_16/1 ,c_17/0,c_18/1,c_19/0,c_20/0,c_21/0,c_22/0,c_23/0,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/1,c_30/0,c_31/0 ,c_32/0,c_33/1,c_34/0,c_35/1} - 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,::,nil,tuple#2} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))