WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: *(@x,@y) -> #mult(@x,@y) dyade(@l1,@l2) -> dyade#1(@l1,@l2) dyade#1(::(@x,@xs),@l2) -> ::(mult(@x,@l2),dyade(@xs,@l2)) dyade#1(nil(),@l2) -> nil() mult(@n,@l) -> mult#1(@l,@n) mult#1(::(@x,@xs),@n) -> ::(*(@n,@x),mult(@n,@xs)) mult#1(nil(),@n) -> nil() - Weak TRS: #add(#0(),@y) -> @y #add(#neg(#s(#0())),@y) -> #pred(@y) #add(#neg(#s(#s(@x))),@y) -> #pred(#add(#pos(#s(@x)),@y)) #add(#pos(#s(#0())),@y) -> #succ(@y) #add(#pos(#s(#s(@x))),@y) -> #succ(#add(#pos(#s(@x)),@y)) #mult(#0(),#0()) -> #0() #mult(#0(),#neg(@y)) -> #0() #mult(#0(),#pos(@y)) -> #0() #mult(#neg(@x),#0()) -> #0() #mult(#neg(@x),#neg(@y)) -> #pos(#natmult(@x,@y)) #mult(#neg(@x),#pos(@y)) -> #neg(#natmult(@x,@y)) #mult(#pos(@x),#0()) -> #0() #mult(#pos(@x),#neg(@y)) -> #neg(#natmult(@x,@y)) #mult(#pos(@x),#pos(@y)) -> #pos(#natmult(@x,@y)) #natmult(#0(),@y) -> #0() #natmult(#s(@x),@y) -> #add(#pos(@y),#natmult(@x,@y)) #pred(#0()) -> #neg(#s(#0())) #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) #pred(#pos(#s(#0()))) -> #0() #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) #succ(#0()) -> #pos(#s(#0())) #succ(#neg(#s(#0()))) -> #0() #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) - Signature: {#add/2,#mult/2,#natmult/2,#pred/1,#succ/1,*/2,dyade/2,dyade#1/2,mult/2,mult#1/2} / {#0/0,#neg/1,#pos/1,#s/1 ,::/2,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {#add,#mult,#natmult,#pred,#succ,*,dyade,dyade#1,mult ,mult#1} and constructors {#0,#neg,#pos,#s,::,nil} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs *#(@x,@y) -> c_1(#mult#(@x,@y)) dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)) dyade#1#(nil(),@l2) -> c_4() mult#(@n,@l) -> c_5(mult#1#(@l,@n)) mult#1#(::(@x,@xs),@n) -> c_6(*#(@n,@x),mult#(@n,@xs)) mult#1#(nil(),@n) -> c_7() Weak DPs #add#(#0(),@y) -> c_8() #add#(#neg(#s(#0())),@y) -> c_9(#pred#(@y)) #add#(#neg(#s(#s(@x))),@y) -> c_10(#pred#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)) #add#(#pos(#s(#0())),@y) -> c_11(#succ#(@y)) #add#(#pos(#s(#s(@x))),@y) -> c_12(#succ#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)) #mult#(#0(),#0()) -> c_13() #mult#(#0(),#neg(@y)) -> c_14() #mult#(#0(),#pos(@y)) -> c_15() #mult#(#neg(@x),#0()) -> c_16() #mult#(#neg(@x),#neg(@y)) -> c_17(#natmult#(@x,@y)) #mult#(#neg(@x),#pos(@y)) -> c_18(#natmult#(@x,@y)) #mult#(#pos(@x),#0()) -> c_19() #mult#(#pos(@x),#neg(@y)) -> c_20(#natmult#(@x,@y)) #mult#(#pos(@x),#pos(@y)) -> c_21(#natmult#(@x,@y)) #natmult#(#0(),@y) -> c_22() #natmult#(#s(@x),@y) -> c_23(#add#(#pos(@y),#natmult(@x,@y)),#natmult#(@x,@y)) #pred#(#0()) -> c_24() #pred#(#neg(#s(@x))) -> c_25() #pred#(#pos(#s(#0()))) -> c_26() #pred#(#pos(#s(#s(@x)))) -> c_27() #succ#(#0()) -> c_28() #succ#(#neg(#s(#0()))) -> c_29() #succ#(#neg(#s(#s(@x)))) -> c_30() #succ#(#pos(#s(@x))) -> c_31() and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: *#(@x,@y) -> c_1(#mult#(@x,@y)) dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)) dyade#1#(nil(),@l2) -> c_4() mult#(@n,@l) -> c_5(mult#1#(@l,@n)) mult#1#(::(@x,@xs),@n) -> c_6(*#(@n,@x),mult#(@n,@xs)) mult#1#(nil(),@n) -> c_7() - Weak DPs: #add#(#0(),@y) -> c_8() #add#(#neg(#s(#0())),@y) -> c_9(#pred#(@y)) #add#(#neg(#s(#s(@x))),@y) -> c_10(#pred#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)) #add#(#pos(#s(#0())),@y) -> c_11(#succ#(@y)) #add#(#pos(#s(#s(@x))),@y) -> c_12(#succ#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)) #mult#(#0(),#0()) -> c_13() #mult#(#0(),#neg(@y)) -> c_14() #mult#(#0(),#pos(@y)) -> c_15() #mult#(#neg(@x),#0()) -> c_16() #mult#(#neg(@x),#neg(@y)) -> c_17(#natmult#(@x,@y)) #mult#(#neg(@x),#pos(@y)) -> c_18(#natmult#(@x,@y)) #mult#(#pos(@x),#0()) -> c_19() #mult#(#pos(@x),#neg(@y)) -> c_20(#natmult#(@x,@y)) #mult#(#pos(@x),#pos(@y)) -> c_21(#natmult#(@x,@y)) #natmult#(#0(),@y) -> c_22() #natmult#(#s(@x),@y) -> c_23(#add#(#pos(@y),#natmult(@x,@y)),#natmult#(@x,@y)) #pred#(#0()) -> c_24() #pred#(#neg(#s(@x))) -> c_25() #pred#(#pos(#s(#0()))) -> c_26() #pred#(#pos(#s(#s(@x)))) -> c_27() #succ#(#0()) -> c_28() #succ#(#neg(#s(#0()))) -> c_29() #succ#(#neg(#s(#s(@x)))) -> c_30() #succ#(#pos(#s(@x))) -> c_31() - Weak TRS: #add(#0(),@y) -> @y #add(#neg(#s(#0())),@y) -> #pred(@y) #add(#neg(#s(#s(@x))),@y) -> #pred(#add(#pos(#s(@x)),@y)) #add(#pos(#s(#0())),@y) -> #succ(@y) #add(#pos(#s(#s(@x))),@y) -> #succ(#add(#pos(#s(@x)),@y)) #mult(#0(),#0()) -> #0() #mult(#0(),#neg(@y)) -> #0() #mult(#0(),#pos(@y)) -> #0() #mult(#neg(@x),#0()) -> #0() #mult(#neg(@x),#neg(@y)) -> #pos(#natmult(@x,@y)) #mult(#neg(@x),#pos(@y)) -> #neg(#natmult(@x,@y)) #mult(#pos(@x),#0()) -> #0() #mult(#pos(@x),#neg(@y)) -> #neg(#natmult(@x,@y)) #mult(#pos(@x),#pos(@y)) -> #pos(#natmult(@x,@y)) #natmult(#0(),@y) -> #0() #natmult(#s(@x),@y) -> #add(#pos(@y),#natmult(@x,@y)) #pred(#0()) -> #neg(#s(#0())) #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) #pred(#pos(#s(#0()))) -> #0() #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) #succ(#0()) -> #pos(#s(#0())) #succ(#neg(#s(#0()))) -> #0() #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) *(@x,@y) -> #mult(@x,@y) dyade(@l1,@l2) -> dyade#1(@l1,@l2) dyade#1(::(@x,@xs),@l2) -> ::(mult(@x,@l2),dyade(@xs,@l2)) dyade#1(nil(),@l2) -> nil() mult(@n,@l) -> mult#1(@l,@n) mult#1(::(@x,@xs),@n) -> ::(*(@n,@x),mult(@n,@xs)) mult#1(nil(),@n) -> nil() - Signature: {#add/2,#mult/2,#natmult/2,#pred/1,#succ/1,*/2,dyade/2,dyade#1/2,mult/2,mult#1/2,#add#/2,#mult#/2 ,#natmult#/2,#pred#/1,#succ#/1,*#/2,dyade#/2,dyade#1#/2,mult#/2,mult#1#/2} / {#0/0,#neg/1,#pos/1,#s/1,::/2 ,nil/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/1,c_10/2,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/1,c_19/0,c_20/1,c_21/1,c_22/0,c_23/2,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {#add#,#mult#,#natmult#,#pred#,#succ#,*#,dyade#,dyade#1# ,mult#,mult#1#} and constructors {#0,#neg,#pos,#s,::,nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,7} by application of Pre({1,4,7}) = {2,5,6}. Here rules are labelled as follows: 1: *#(@x,@y) -> c_1(#mult#(@x,@y)) 2: dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) 3: dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)) 4: dyade#1#(nil(),@l2) -> c_4() 5: mult#(@n,@l) -> c_5(mult#1#(@l,@n)) 6: mult#1#(::(@x,@xs),@n) -> c_6(*#(@n,@x),mult#(@n,@xs)) 7: mult#1#(nil(),@n) -> c_7() 8: #add#(#0(),@y) -> c_8() 9: #add#(#neg(#s(#0())),@y) -> c_9(#pred#(@y)) 10: #add#(#neg(#s(#s(@x))),@y) -> c_10(#pred#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)) 11: #add#(#pos(#s(#0())),@y) -> c_11(#succ#(@y)) 12: #add#(#pos(#s(#s(@x))),@y) -> c_12(#succ#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)) 13: #mult#(#0(),#0()) -> c_13() 14: #mult#(#0(),#neg(@y)) -> c_14() 15: #mult#(#0(),#pos(@y)) -> c_15() 16: #mult#(#neg(@x),#0()) -> c_16() 17: #mult#(#neg(@x),#neg(@y)) -> c_17(#natmult#(@x,@y)) 18: #mult#(#neg(@x),#pos(@y)) -> c_18(#natmult#(@x,@y)) 19: #mult#(#pos(@x),#0()) -> c_19() 20: #mult#(#pos(@x),#neg(@y)) -> c_20(#natmult#(@x,@y)) 21: #mult#(#pos(@x),#pos(@y)) -> c_21(#natmult#(@x,@y)) 22: #natmult#(#0(),@y) -> c_22() 23: #natmult#(#s(@x),@y) -> c_23(#add#(#pos(@y),#natmult(@x,@y)),#natmult#(@x,@y)) 24: #pred#(#0()) -> c_24() 25: #pred#(#neg(#s(@x))) -> c_25() 26: #pred#(#pos(#s(#0()))) -> c_26() 27: #pred#(#pos(#s(#s(@x)))) -> c_27() 28: #succ#(#0()) -> c_28() 29: #succ#(#neg(#s(#0()))) -> c_29() 30: #succ#(#neg(#s(#s(@x)))) -> c_30() 31: #succ#(#pos(#s(@x))) -> c_31() * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)) mult#(@n,@l) -> c_5(mult#1#(@l,@n)) mult#1#(::(@x,@xs),@n) -> c_6(*#(@n,@x),mult#(@n,@xs)) - Weak DPs: #add#(#0(),@y) -> c_8() #add#(#neg(#s(#0())),@y) -> c_9(#pred#(@y)) #add#(#neg(#s(#s(@x))),@y) -> c_10(#pred#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)) #add#(#pos(#s(#0())),@y) -> c_11(#succ#(@y)) #add#(#pos(#s(#s(@x))),@y) -> c_12(#succ#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)) #mult#(#0(),#0()) -> c_13() #mult#(#0(),#neg(@y)) -> c_14() #mult#(#0(),#pos(@y)) -> c_15() #mult#(#neg(@x),#0()) -> c_16() #mult#(#neg(@x),#neg(@y)) -> c_17(#natmult#(@x,@y)) #mult#(#neg(@x),#pos(@y)) -> c_18(#natmult#(@x,@y)) #mult#(#pos(@x),#0()) -> c_19() #mult#(#pos(@x),#neg(@y)) -> c_20(#natmult#(@x,@y)) #mult#(#pos(@x),#pos(@y)) -> c_21(#natmult#(@x,@y)) #natmult#(#0(),@y) -> c_22() #natmult#(#s(@x),@y) -> c_23(#add#(#pos(@y),#natmult(@x,@y)),#natmult#(@x,@y)) #pred#(#0()) -> c_24() #pred#(#neg(#s(@x))) -> c_25() #pred#(#pos(#s(#0()))) -> c_26() #pred#(#pos(#s(#s(@x)))) -> c_27() #succ#(#0()) -> c_28() #succ#(#neg(#s(#0()))) -> c_29() #succ#(#neg(#s(#s(@x)))) -> c_30() #succ#(#pos(#s(@x))) -> c_31() *#(@x,@y) -> c_1(#mult#(@x,@y)) dyade#1#(nil(),@l2) -> c_4() mult#1#(nil(),@n) -> c_7() - Weak TRS: #add(#0(),@y) -> @y #add(#neg(#s(#0())),@y) -> #pred(@y) #add(#neg(#s(#s(@x))),@y) -> #pred(#add(#pos(#s(@x)),@y)) #add(#pos(#s(#0())),@y) -> #succ(@y) #add(#pos(#s(#s(@x))),@y) -> #succ(#add(#pos(#s(@x)),@y)) #mult(#0(),#0()) -> #0() #mult(#0(),#neg(@y)) -> #0() #mult(#0(),#pos(@y)) -> #0() #mult(#neg(@x),#0()) -> #0() #mult(#neg(@x),#neg(@y)) -> #pos(#natmult(@x,@y)) #mult(#neg(@x),#pos(@y)) -> #neg(#natmult(@x,@y)) #mult(#pos(@x),#0()) -> #0() #mult(#pos(@x),#neg(@y)) -> #neg(#natmult(@x,@y)) #mult(#pos(@x),#pos(@y)) -> #pos(#natmult(@x,@y)) #natmult(#0(),@y) -> #0() #natmult(#s(@x),@y) -> #add(#pos(@y),#natmult(@x,@y)) #pred(#0()) -> #neg(#s(#0())) #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) #pred(#pos(#s(#0()))) -> #0() #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) #succ(#0()) -> #pos(#s(#0())) #succ(#neg(#s(#0()))) -> #0() #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) *(@x,@y) -> #mult(@x,@y) dyade(@l1,@l2) -> dyade#1(@l1,@l2) dyade#1(::(@x,@xs),@l2) -> ::(mult(@x,@l2),dyade(@xs,@l2)) dyade#1(nil(),@l2) -> nil() mult(@n,@l) -> mult#1(@l,@n) mult#1(::(@x,@xs),@n) -> ::(*(@n,@x),mult(@n,@xs)) mult#1(nil(),@n) -> nil() - Signature: {#add/2,#mult/2,#natmult/2,#pred/1,#succ/1,*/2,dyade/2,dyade#1/2,mult/2,mult#1/2,#add#/2,#mult#/2 ,#natmult#/2,#pred#/1,#succ#/1,*#/2,dyade#/2,dyade#1#/2,mult#/2,mult#1#/2} / {#0/0,#neg/1,#pos/1,#s/1,::/2 ,nil/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/1,c_10/2,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/1,c_19/0,c_20/1,c_21/1,c_22/0,c_23/2,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {#add#,#mult#,#natmult#,#pred#,#succ#,*#,dyade#,dyade#1# ,mult#,mult#1#} and constructors {#0,#neg,#pos,#s,::,nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) -->_1 dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)):2 -->_1 dyade#1#(nil(),@l2) -> c_4():30 2:S:dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)) -->_1 mult#(@n,@l) -> c_5(mult#1#(@l,@n)):3 -->_2 dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)):1 3:S:mult#(@n,@l) -> c_5(mult#1#(@l,@n)) -->_1 mult#1#(::(@x,@xs),@n) -> c_6(*#(@n,@x),mult#(@n,@xs)):4 -->_1 mult#1#(nil(),@n) -> c_7():31 4:S:mult#1#(::(@x,@xs),@n) -> c_6(*#(@n,@x),mult#(@n,@xs)) -->_1 *#(@x,@y) -> c_1(#mult#(@x,@y)):29 -->_2 mult#(@n,@l) -> c_5(mult#1#(@l,@n)):3 5:W:#add#(#0(),@y) -> c_8() 6:W:#add#(#neg(#s(#0())),@y) -> c_9(#pred#(@y)) -->_1 #pred#(#pos(#s(#s(@x)))) -> c_27():24 -->_1 #pred#(#pos(#s(#0()))) -> c_26():23 -->_1 #pred#(#neg(#s(@x))) -> c_25():22 -->_1 #pred#(#0()) -> c_24():21 7:W:#add#(#neg(#s(#s(@x))),@y) -> c_10(#pred#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)) -->_2 #add#(#pos(#s(#s(@x))),@y) -> c_12(#succ#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)):9 -->_2 #add#(#pos(#s(#0())),@y) -> c_11(#succ#(@y)):8 -->_1 #pred#(#pos(#s(#s(@x)))) -> c_27():24 -->_1 #pred#(#pos(#s(#0()))) -> c_26():23 -->_1 #pred#(#neg(#s(@x))) -> c_25():22 -->_1 #pred#(#0()) -> c_24():21 8:W:#add#(#pos(#s(#0())),@y) -> c_11(#succ#(@y)) -->_1 #succ#(#pos(#s(@x))) -> c_31():28 -->_1 #succ#(#neg(#s(#s(@x)))) -> c_30():27 -->_1 #succ#(#neg(#s(#0()))) -> c_29():26 -->_1 #succ#(#0()) -> c_28():25 9:W:#add#(#pos(#s(#s(@x))),@y) -> c_12(#succ#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)) -->_1 #succ#(#pos(#s(@x))) -> c_31():28 -->_1 #succ#(#neg(#s(#s(@x)))) -> c_30():27 -->_1 #succ#(#neg(#s(#0()))) -> c_29():26 -->_1 #succ#(#0()) -> c_28():25 -->_2 #add#(#pos(#s(#s(@x))),@y) -> c_12(#succ#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)):9 -->_2 #add#(#pos(#s(#0())),@y) -> c_11(#succ#(@y)):8 10:W:#mult#(#0(),#0()) -> c_13() 11:W:#mult#(#0(),#neg(@y)) -> c_14() 12:W:#mult#(#0(),#pos(@y)) -> c_15() 13:W:#mult#(#neg(@x),#0()) -> c_16() 14:W:#mult#(#neg(@x),#neg(@y)) -> c_17(#natmult#(@x,@y)) -->_1 #natmult#(#s(@x),@y) -> c_23(#add#(#pos(@y),#natmult(@x,@y)),#natmult#(@x,@y)):20 -->_1 #natmult#(#0(),@y) -> c_22():19 15:W:#mult#(#neg(@x),#pos(@y)) -> c_18(#natmult#(@x,@y)) -->_1 #natmult#(#s(@x),@y) -> c_23(#add#(#pos(@y),#natmult(@x,@y)),#natmult#(@x,@y)):20 -->_1 #natmult#(#0(),@y) -> c_22():19 16:W:#mult#(#pos(@x),#0()) -> c_19() 17:W:#mult#(#pos(@x),#neg(@y)) -> c_20(#natmult#(@x,@y)) -->_1 #natmult#(#s(@x),@y) -> c_23(#add#(#pos(@y),#natmult(@x,@y)),#natmult#(@x,@y)):20 -->_1 #natmult#(#0(),@y) -> c_22():19 18:W:#mult#(#pos(@x),#pos(@y)) -> c_21(#natmult#(@x,@y)) -->_1 #natmult#(#s(@x),@y) -> c_23(#add#(#pos(@y),#natmult(@x,@y)),#natmult#(@x,@y)):20 -->_1 #natmult#(#0(),@y) -> c_22():19 19:W:#natmult#(#0(),@y) -> c_22() 20:W:#natmult#(#s(@x),@y) -> c_23(#add#(#pos(@y),#natmult(@x,@y)),#natmult#(@x,@y)) -->_2 #natmult#(#s(@x),@y) -> c_23(#add#(#pos(@y),#natmult(@x,@y)),#natmult#(@x,@y)):20 -->_2 #natmult#(#0(),@y) -> c_22():19 -->_1 #add#(#pos(#s(#s(@x))),@y) -> c_12(#succ#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)):9 -->_1 #add#(#pos(#s(#0())),@y) -> c_11(#succ#(@y)):8 21:W:#pred#(#0()) -> c_24() 22:W:#pred#(#neg(#s(@x))) -> c_25() 23:W:#pred#(#pos(#s(#0()))) -> c_26() 24:W:#pred#(#pos(#s(#s(@x)))) -> c_27() 25:W:#succ#(#0()) -> c_28() 26:W:#succ#(#neg(#s(#0()))) -> c_29() 27:W:#succ#(#neg(#s(#s(@x)))) -> c_30() 28:W:#succ#(#pos(#s(@x))) -> c_31() 29:W:*#(@x,@y) -> c_1(#mult#(@x,@y)) -->_1 #mult#(#pos(@x),#pos(@y)) -> c_21(#natmult#(@x,@y)):18 -->_1 #mult#(#pos(@x),#neg(@y)) -> c_20(#natmult#(@x,@y)):17 -->_1 #mult#(#pos(@x),#0()) -> c_19():16 -->_1 #mult#(#neg(@x),#pos(@y)) -> c_18(#natmult#(@x,@y)):15 -->_1 #mult#(#neg(@x),#neg(@y)) -> c_17(#natmult#(@x,@y)):14 -->_1 #mult#(#neg(@x),#0()) -> c_16():13 -->_1 #mult#(#0(),#pos(@y)) -> c_15():12 -->_1 #mult#(#0(),#neg(@y)) -> c_14():11 -->_1 #mult#(#0(),#0()) -> c_13():10 30:W:dyade#1#(nil(),@l2) -> c_4() 31:W:mult#1#(nil(),@n) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: #add#(#neg(#s(#s(@x))),@y) -> c_10(#pred#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)) 6: #add#(#neg(#s(#0())),@y) -> c_9(#pred#(@y)) 21: #pred#(#0()) -> c_24() 22: #pred#(#neg(#s(@x))) -> c_25() 23: #pred#(#pos(#s(#0()))) -> c_26() 24: #pred#(#pos(#s(#s(@x)))) -> c_27() 5: #add#(#0(),@y) -> c_8() 30: dyade#1#(nil(),@l2) -> c_4() 31: mult#1#(nil(),@n) -> c_7() 29: *#(@x,@y) -> c_1(#mult#(@x,@y)) 10: #mult#(#0(),#0()) -> c_13() 11: #mult#(#0(),#neg(@y)) -> c_14() 12: #mult#(#0(),#pos(@y)) -> c_15() 13: #mult#(#neg(@x),#0()) -> c_16() 14: #mult#(#neg(@x),#neg(@y)) -> c_17(#natmult#(@x,@y)) 15: #mult#(#neg(@x),#pos(@y)) -> c_18(#natmult#(@x,@y)) 16: #mult#(#pos(@x),#0()) -> c_19() 17: #mult#(#pos(@x),#neg(@y)) -> c_20(#natmult#(@x,@y)) 18: #mult#(#pos(@x),#pos(@y)) -> c_21(#natmult#(@x,@y)) 20: #natmult#(#s(@x),@y) -> c_23(#add#(#pos(@y),#natmult(@x,@y)),#natmult#(@x,@y)) 9: #add#(#pos(#s(#s(@x))),@y) -> c_12(#succ#(#add(#pos(#s(@x)),@y)),#add#(#pos(#s(@x)),@y)) 8: #add#(#pos(#s(#0())),@y) -> c_11(#succ#(@y)) 25: #succ#(#0()) -> c_28() 26: #succ#(#neg(#s(#0()))) -> c_29() 27: #succ#(#neg(#s(#s(@x)))) -> c_30() 28: #succ#(#pos(#s(@x))) -> c_31() 19: #natmult#(#0(),@y) -> c_22() * Step 4: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)) mult#(@n,@l) -> c_5(mult#1#(@l,@n)) mult#1#(::(@x,@xs),@n) -> c_6(*#(@n,@x),mult#(@n,@xs)) - Weak TRS: #add(#0(),@y) -> @y #add(#neg(#s(#0())),@y) -> #pred(@y) #add(#neg(#s(#s(@x))),@y) -> #pred(#add(#pos(#s(@x)),@y)) #add(#pos(#s(#0())),@y) -> #succ(@y) #add(#pos(#s(#s(@x))),@y) -> #succ(#add(#pos(#s(@x)),@y)) #mult(#0(),#0()) -> #0() #mult(#0(),#neg(@y)) -> #0() #mult(#0(),#pos(@y)) -> #0() #mult(#neg(@x),#0()) -> #0() #mult(#neg(@x),#neg(@y)) -> #pos(#natmult(@x,@y)) #mult(#neg(@x),#pos(@y)) -> #neg(#natmult(@x,@y)) #mult(#pos(@x),#0()) -> #0() #mult(#pos(@x),#neg(@y)) -> #neg(#natmult(@x,@y)) #mult(#pos(@x),#pos(@y)) -> #pos(#natmult(@x,@y)) #natmult(#0(),@y) -> #0() #natmult(#s(@x),@y) -> #add(#pos(@y),#natmult(@x,@y)) #pred(#0()) -> #neg(#s(#0())) #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) #pred(#pos(#s(#0()))) -> #0() #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) #succ(#0()) -> #pos(#s(#0())) #succ(#neg(#s(#0()))) -> #0() #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) *(@x,@y) -> #mult(@x,@y) dyade(@l1,@l2) -> dyade#1(@l1,@l2) dyade#1(::(@x,@xs),@l2) -> ::(mult(@x,@l2),dyade(@xs,@l2)) dyade#1(nil(),@l2) -> nil() mult(@n,@l) -> mult#1(@l,@n) mult#1(::(@x,@xs),@n) -> ::(*(@n,@x),mult(@n,@xs)) mult#1(nil(),@n) -> nil() - Signature: {#add/2,#mult/2,#natmult/2,#pred/1,#succ/1,*/2,dyade/2,dyade#1/2,mult/2,mult#1/2,#add#/2,#mult#/2 ,#natmult#/2,#pred#/1,#succ#/1,*#/2,dyade#/2,dyade#1#/2,mult#/2,mult#1#/2} / {#0/0,#neg/1,#pos/1,#s/1,::/2 ,nil/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/1,c_6/2,c_7/0,c_8/0,c_9/1,c_10/2,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/1,c_19/0,c_20/1,c_21/1,c_22/0,c_23/2,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {#add#,#mult#,#natmult#,#pred#,#succ#,*#,dyade#,dyade#1# ,mult#,mult#1#} and constructors {#0,#neg,#pos,#s,::,nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) -->_1 dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)):2 2:S:dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)) -->_1 mult#(@n,@l) -> c_5(mult#1#(@l,@n)):3 -->_2 dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)):1 3:S:mult#(@n,@l) -> c_5(mult#1#(@l,@n)) -->_1 mult#1#(::(@x,@xs),@n) -> c_6(*#(@n,@x),mult#(@n,@xs)):4 4:S:mult#1#(::(@x,@xs),@n) -> c_6(*#(@n,@x),mult#(@n,@xs)) -->_2 mult#(@n,@l) -> c_5(mult#1#(@l,@n)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mult#1#(::(@x,@xs),@n) -> c_6(mult#(@n,@xs)) * Step 5: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)) mult#(@n,@l) -> c_5(mult#1#(@l,@n)) mult#1#(::(@x,@xs),@n) -> c_6(mult#(@n,@xs)) - Weak TRS: #add(#0(),@y) -> @y #add(#neg(#s(#0())),@y) -> #pred(@y) #add(#neg(#s(#s(@x))),@y) -> #pred(#add(#pos(#s(@x)),@y)) #add(#pos(#s(#0())),@y) -> #succ(@y) #add(#pos(#s(#s(@x))),@y) -> #succ(#add(#pos(#s(@x)),@y)) #mult(#0(),#0()) -> #0() #mult(#0(),#neg(@y)) -> #0() #mult(#0(),#pos(@y)) -> #0() #mult(#neg(@x),#0()) -> #0() #mult(#neg(@x),#neg(@y)) -> #pos(#natmult(@x,@y)) #mult(#neg(@x),#pos(@y)) -> #neg(#natmult(@x,@y)) #mult(#pos(@x),#0()) -> #0() #mult(#pos(@x),#neg(@y)) -> #neg(#natmult(@x,@y)) #mult(#pos(@x),#pos(@y)) -> #pos(#natmult(@x,@y)) #natmult(#0(),@y) -> #0() #natmult(#s(@x),@y) -> #add(#pos(@y),#natmult(@x,@y)) #pred(#0()) -> #neg(#s(#0())) #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) #pred(#pos(#s(#0()))) -> #0() #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) #succ(#0()) -> #pos(#s(#0())) #succ(#neg(#s(#0()))) -> #0() #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) *(@x,@y) -> #mult(@x,@y) dyade(@l1,@l2) -> dyade#1(@l1,@l2) dyade#1(::(@x,@xs),@l2) -> ::(mult(@x,@l2),dyade(@xs,@l2)) dyade#1(nil(),@l2) -> nil() mult(@n,@l) -> mult#1(@l,@n) mult#1(::(@x,@xs),@n) -> ::(*(@n,@x),mult(@n,@xs)) mult#1(nil(),@n) -> nil() - Signature: {#add/2,#mult/2,#natmult/2,#pred/1,#succ/1,*/2,dyade/2,dyade#1/2,mult/2,mult#1/2,#add#/2,#mult#/2 ,#natmult#/2,#pred#/1,#succ#/1,*#/2,dyade#/2,dyade#1#/2,mult#/2,mult#1#/2} / {#0/0,#neg/1,#pos/1,#s/1,::/2 ,nil/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/2,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/1,c_19/0,c_20/1,c_21/1,c_22/0,c_23/2,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {#add#,#mult#,#natmult#,#pred#,#succ#,*#,dyade#,dyade#1# ,mult#,mult#1#} and constructors {#0,#neg,#pos,#s,::,nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)) mult#(@n,@l) -> c_5(mult#1#(@l,@n)) mult#1#(::(@x,@xs),@n) -> c_6(mult#(@n,@xs)) * Step 6: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)) mult#(@n,@l) -> c_5(mult#1#(@l,@n)) mult#1#(::(@x,@xs),@n) -> c_6(mult#(@n,@xs)) - Signature: {#add/2,#mult/2,#natmult/2,#pred/1,#succ/1,*/2,dyade/2,dyade#1/2,mult/2,mult#1/2,#add#/2,#mult#/2 ,#natmult#/2,#pred#/1,#succ#/1,*#/2,dyade#/2,dyade#1#/2,mult#/2,mult#1#/2} / {#0/0,#neg/1,#pos/1,#s/1,::/2 ,nil/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/2,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/1,c_19/0,c_20/1,c_21/1,c_22/0,c_23/2,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {#add#,#mult#,#natmult#,#pred#,#succ#,*#,dyade#,dyade#1# ,mult#,mult#1#} and constructors {#0,#neg,#pos,#s,::,nil} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)) and a lower component mult#(@n,@l) -> c_5(mult#1#(@l,@n)) mult#1#(::(@x,@xs),@n) -> c_6(mult#(@n,@xs)) Further, following extension rules are added to the lower component. dyade#(@l1,@l2) -> dyade#1#(@l1,@l2) dyade#1#(::(@x,@xs),@l2) -> dyade#(@xs,@l2) dyade#1#(::(@x,@xs),@l2) -> mult#(@x,@l2) ** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)) - Signature: {#add/2,#mult/2,#natmult/2,#pred/1,#succ/1,*/2,dyade/2,dyade#1/2,mult/2,mult#1/2,#add#/2,#mult#/2 ,#natmult#/2,#pred#/1,#succ#/1,*#/2,dyade#/2,dyade#1#/2,mult#/2,mult#1#/2} / {#0/0,#neg/1,#pos/1,#s/1,::/2 ,nil/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/2,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/1,c_19/0,c_20/1,c_21/1,c_22/0,c_23/2,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {#add#,#mult#,#natmult#,#pred#,#succ#,*#,dyade#,dyade#1# ,mult#,mult#1#} and constructors {#0,#neg,#pos,#s,::,nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) -->_1 dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)):2 2:S:dyade#1#(::(@x,@xs),@l2) -> c_3(mult#(@x,@l2),dyade#(@xs,@l2)) -->_2 dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: dyade#1#(::(@x,@xs),@l2) -> c_3(dyade#(@xs,@l2)) ** Step 6.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) dyade#1#(::(@x,@xs),@l2) -> c_3(dyade#(@xs,@l2)) - Signature: {#add/2,#mult/2,#natmult/2,#pred/1,#succ/1,*/2,dyade/2,dyade#1/2,mult/2,mult#1/2,#add#/2,#mult#/2 ,#natmult#/2,#pred#/1,#succ#/1,*#/2,dyade#/2,dyade#1#/2,mult#/2,mult#1#/2} / {#0/0,#neg/1,#pos/1,#s/1,::/2 ,nil/0,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/2,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/1,c_19/0,c_20/1,c_21/1,c_22/0,c_23/2,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {#add#,#mult#,#natmult#,#pred#,#succ#,*#,dyade#,dyade#1# ,mult#,mult#1#} and constructors {#0,#neg,#pos,#s,::,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#add) = [0] p(#mult) = [0] p(#natmult) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#pred) = [0] p(#s) = [1] x1 + [0] p(#succ) = [0] p(*) = [0] p(::) = [1] x1 + [1] x2 + [0] p(dyade) = [0] p(dyade#1) = [0] p(mult) = [0] p(mult#1) = [0] p(nil) = [0] p(#add#) = [0] p(#mult#) = [0] p(#natmult#) = [0] p(#pred#) = [0] p(#succ#) = [0] p(*#) = [0] p(dyade#) = [0] p(dyade#1#) = [3] p(mult#) = [0] p(mult#1#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [0] Following rules are strictly oriented: dyade#1#(::(@x,@xs),@l2) = [3] > [0] = c_3(dyade#(@xs,@l2)) Following rules are (at-least) weakly oriented: dyade#(@l1,@l2) = [0] >= [3] = c_2(dyade#1#(@l1,@l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) - Weak DPs: dyade#1#(::(@x,@xs),@l2) -> c_3(dyade#(@xs,@l2)) - Signature: {#add/2,#mult/2,#natmult/2,#pred/1,#succ/1,*/2,dyade/2,dyade#1/2,mult/2,mult#1/2,#add#/2,#mult#/2 ,#natmult#/2,#pred#/1,#succ#/1,*#/2,dyade#/2,dyade#1#/2,mult#/2,mult#1#/2} / {#0/0,#neg/1,#pos/1,#s/1,::/2 ,nil/0,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/2,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/1,c_19/0,c_20/1,c_21/1,c_22/0,c_23/2,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {#add#,#mult#,#natmult#,#pred#,#succ#,*#,dyade#,dyade#1# ,mult#,mult#1#} and constructors {#0,#neg,#pos,#s,::,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#add) = [0] p(#mult) = [0] p(#natmult) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#pred) = [0] p(#s) = [1] x1 + [0] p(#succ) = [0] p(*) = [0] p(::) = [1] x1 + [1] x2 + [11] p(dyade) = [0] p(dyade#1) = [1] p(mult) = [0] p(mult#1) = [0] p(nil) = [0] p(#add#) = [0] p(#mult#) = [0] p(#natmult#) = [0] p(#pred#) = [0] p(#succ#) = [0] p(*#) = [0] p(dyade#) = [1] x1 + [11] p(dyade#1#) = [1] x1 + [0] p(mult#) = [0] p(mult#1#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [0] Following rules are strictly oriented: dyade#(@l1,@l2) = [1] @l1 + [11] > [1] @l1 + [0] = c_2(dyade#1#(@l1,@l2)) Following rules are (at-least) weakly oriented: dyade#1#(::(@x,@xs),@l2) = [1] @x + [1] @xs + [11] >= [1] @xs + [11] = c_3(dyade#(@xs,@l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: dyade#(@l1,@l2) -> c_2(dyade#1#(@l1,@l2)) dyade#1#(::(@x,@xs),@l2) -> c_3(dyade#(@xs,@l2)) - Signature: {#add/2,#mult/2,#natmult/2,#pred/1,#succ/1,*/2,dyade/2,dyade#1/2,mult/2,mult#1/2,#add#/2,#mult#/2 ,#natmult#/2,#pred#/1,#succ#/1,*#/2,dyade#/2,dyade#1#/2,mult#/2,mult#1#/2} / {#0/0,#neg/1,#pos/1,#s/1,::/2 ,nil/0,c_1/1,c_2/1,c_3/1,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/2,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/1,c_19/0,c_20/1,c_21/1,c_22/0,c_23/2,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {#add#,#mult#,#natmult#,#pred#,#succ#,*#,dyade#,dyade#1# ,mult#,mult#1#} and constructors {#0,#neg,#pos,#s,::,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mult#(@n,@l) -> c_5(mult#1#(@l,@n)) mult#1#(::(@x,@xs),@n) -> c_6(mult#(@n,@xs)) - Weak DPs: dyade#(@l1,@l2) -> dyade#1#(@l1,@l2) dyade#1#(::(@x,@xs),@l2) -> dyade#(@xs,@l2) dyade#1#(::(@x,@xs),@l2) -> mult#(@x,@l2) - Signature: {#add/2,#mult/2,#natmult/2,#pred/1,#succ/1,*/2,dyade/2,dyade#1/2,mult/2,mult#1/2,#add#/2,#mult#/2 ,#natmult#/2,#pred#/1,#succ#/1,*#/2,dyade#/2,dyade#1#/2,mult#/2,mult#1#/2} / {#0/0,#neg/1,#pos/1,#s/1,::/2 ,nil/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/2,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/1,c_19/0,c_20/1,c_21/1,c_22/0,c_23/2,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {#add#,#mult#,#natmult#,#pred#,#succ#,*#,dyade#,dyade#1# ,mult#,mult#1#} and constructors {#0,#neg,#pos,#s,::,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#add) = [0] p(#mult) = [1] x2 + [0] p(#natmult) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#pred) = [0] p(#s) = [1] x1 + [0] p(#succ) = [0] p(*) = [0] p(::) = [1] x1 + [1] x2 + [0] p(dyade) = [0] p(dyade#1) = [0] p(mult) = [0] p(mult#1) = [0] p(nil) = [0] p(#add#) = [0] p(#mult#) = [0] p(#natmult#) = [0] p(#pred#) = [0] p(#succ#) = [0] p(*#) = [0] p(dyade#) = [0] p(dyade#1#) = [0] p(mult#) = [0] p(mult#1#) = [13] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [0] Following rules are strictly oriented: mult#1#(::(@x,@xs),@n) = [13] > [0] = c_6(mult#(@n,@xs)) Following rules are (at-least) weakly oriented: dyade#(@l1,@l2) = [0] >= [0] = dyade#1#(@l1,@l2) dyade#1#(::(@x,@xs),@l2) = [0] >= [0] = dyade#(@xs,@l2) dyade#1#(::(@x,@xs),@l2) = [0] >= [0] = mult#(@x,@l2) mult#(@n,@l) = [0] >= [13] = c_5(mult#1#(@l,@n)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.b:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mult#(@n,@l) -> c_5(mult#1#(@l,@n)) - Weak DPs: dyade#(@l1,@l2) -> dyade#1#(@l1,@l2) dyade#1#(::(@x,@xs),@l2) -> dyade#(@xs,@l2) dyade#1#(::(@x,@xs),@l2) -> mult#(@x,@l2) mult#1#(::(@x,@xs),@n) -> c_6(mult#(@n,@xs)) - Signature: {#add/2,#mult/2,#natmult/2,#pred/1,#succ/1,*/2,dyade/2,dyade#1/2,mult/2,mult#1/2,#add#/2,#mult#/2 ,#natmult#/2,#pred#/1,#succ#/1,*#/2,dyade#/2,dyade#1#/2,mult#/2,mult#1#/2} / {#0/0,#neg/1,#pos/1,#s/1,::/2 ,nil/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/2,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/1,c_19/0,c_20/1,c_21/1,c_22/0,c_23/2,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {#add#,#mult#,#natmult#,#pred#,#succ#,*#,dyade#,dyade#1# ,mult#,mult#1#} and constructors {#0,#neg,#pos,#s,::,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#add) = [0] p(#mult) = [0] p(#natmult) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#pred) = [0] p(#s) = [1] x1 + [0] p(#succ) = [0] p(*) = [0] p(::) = [1] x1 + [1] x2 + [1] p(dyade) = [0] p(dyade#1) = [0] p(mult) = [0] p(mult#1) = [0] p(nil) = [0] p(#add#) = [0] p(#mult#) = [0] p(#natmult#) = [0] p(#pred#) = [0] p(#succ#) = [0] p(*#) = [0] p(dyade#) = [1] x2 + [1] p(dyade#1#) = [1] x2 + [1] p(mult#) = [1] x2 + [1] p(mult#1#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [0] Following rules are strictly oriented: mult#(@n,@l) = [1] @l + [1] > [1] @l + [0] = c_5(mult#1#(@l,@n)) Following rules are (at-least) weakly oriented: dyade#(@l1,@l2) = [1] @l2 + [1] >= [1] @l2 + [1] = dyade#1#(@l1,@l2) dyade#1#(::(@x,@xs),@l2) = [1] @l2 + [1] >= [1] @l2 + [1] = dyade#(@xs,@l2) dyade#1#(::(@x,@xs),@l2) = [1] @l2 + [1] >= [1] @l2 + [1] = mult#(@x,@l2) mult#1#(::(@x,@xs),@n) = [1] @x + [1] @xs + [1] >= [1] @xs + [1] = c_6(mult#(@n,@xs)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: dyade#(@l1,@l2) -> dyade#1#(@l1,@l2) dyade#1#(::(@x,@xs),@l2) -> dyade#(@xs,@l2) dyade#1#(::(@x,@xs),@l2) -> mult#(@x,@l2) mult#(@n,@l) -> c_5(mult#1#(@l,@n)) mult#1#(::(@x,@xs),@n) -> c_6(mult#(@n,@xs)) - Signature: {#add/2,#mult/2,#natmult/2,#pred/1,#succ/1,*/2,dyade/2,dyade#1/2,mult/2,mult#1/2,#add#/2,#mult#/2 ,#natmult#/2,#pred#/1,#succ#/1,*#/2,dyade#/2,dyade#1#/2,mult#/2,mult#1#/2} / {#0/0,#neg/1,#pos/1,#s/1,::/2 ,nil/0,c_1/1,c_2/1,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/2,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0 ,c_16/0,c_17/1,c_18/1,c_19/0,c_20/1,c_21/1,c_22/0,c_23/2,c_24/0,c_25/0,c_26/0,c_27/0,c_28/0,c_29/0,c_30/0 ,c_31/0} - Obligation: innermost runtime complexity wrt. defined symbols {#add#,#mult#,#natmult#,#pred#,#succ#,*#,dyade#,dyade#1# ,mult#,mult#1#} and constructors {#0,#neg,#pos,#s,::,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))