WORST_CASE(?,O(n^4)) * Step 1: DependencyPairs WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() nub(l) -> nub#1(l) nub#1(dd(x,xs)) -> dd(x,nub(remove(x,xs))) nub#1(nil()) -> nil() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4} / {#0/0 ,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {#and,#eq,#equal,and,eq,eq#1,eq#2,eq#3,nub,nub#1,remove ,remove#1,remove#2} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs #and#(#false(),#false()) -> c_1() #and#(#false(),#true()) -> c_2() #and#(#true(),#false()) -> c_3() #and#(#true(),#true()) -> c_4() #eq#(#0(),#0()) -> c_5() #eq#(#0(),#neg(y)) -> c_6() #eq#(#0(),#pos(y)) -> c_7() #eq#(#0(),#s(y)) -> c_8() #eq#(#neg(x),#0()) -> c_9() #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#neg(x),#pos(y)) -> c_11() #eq#(#pos(x),#0()) -> c_12() #eq#(#pos(x),#neg(y)) -> c_13() #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#0()) -> c_15() #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)),#eq#(x'1,y'1),#eq#(x'2,y'2)) #eq#(dd(x'1,x'2),nil()) -> c_18() #eq#(nil(),dd(y'1,y'2)) -> c_19() #eq#(nil(),nil()) -> c_20() #equal#(x,y) -> c_21(#eq#(x,y)) and#(x,y) -> c_22(#and#(x,y)) eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#1#(nil(),l2) -> c_25(eq#2#(l2)) eq#2#(dd(y,ys)) -> c_26() eq#2#(nil()) -> c_27() eq#3#(dd(y,ys),x,xs) -> c_28(and#(#equal(x,y),eq(xs,ys)),#equal#(x,y),eq#(xs,ys)) eq#3#(nil(),x,xs) -> c_29() nub#(l) -> c_30(nub#1#(l)) nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) nub#1#(nil()) -> c_32() remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) remove#1#(nil(),x) -> c_35() remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: #and#(#false(),#false()) -> c_1() #and#(#false(),#true()) -> c_2() #and#(#true(),#false()) -> c_3() #and#(#true(),#true()) -> c_4() #eq#(#0(),#0()) -> c_5() #eq#(#0(),#neg(y)) -> c_6() #eq#(#0(),#pos(y)) -> c_7() #eq#(#0(),#s(y)) -> c_8() #eq#(#neg(x),#0()) -> c_9() #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#neg(x),#pos(y)) -> c_11() #eq#(#pos(x),#0()) -> c_12() #eq#(#pos(x),#neg(y)) -> c_13() #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#0()) -> c_15() #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)),#eq#(x'1,y'1),#eq#(x'2,y'2)) #eq#(dd(x'1,x'2),nil()) -> c_18() #eq#(nil(),dd(y'1,y'2)) -> c_19() #eq#(nil(),nil()) -> c_20() #equal#(x,y) -> c_21(#eq#(x,y)) and#(x,y) -> c_22(#and#(x,y)) eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#1#(nil(),l2) -> c_25(eq#2#(l2)) eq#2#(dd(y,ys)) -> c_26() eq#2#(nil()) -> c_27() eq#3#(dd(y,ys),x,xs) -> c_28(and#(#equal(x,y),eq(xs,ys)),#equal#(x,y),eq#(xs,ys)) eq#3#(nil(),x,xs) -> c_29() nub#(l) -> c_30(nub#1#(l)) nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) nub#1#(nil()) -> c_32() remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) remove#1#(nil(),x) -> c_35() remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() nub(l) -> nub#1(l) nub#1(dd(x,xs)) -> dd(x,nub(remove(x,xs))) nub#1(nil()) -> nil() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/3,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/3,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,5,6,7,8,9,11,12,13,15,18,19,20,26,27,29,32,35} by application of Pre({1,2,3,4,5,6,7,8,9,11,12,13,15,18,19,20,26,27,29,32,35}) = {10,14,16,17,21,22,24,25,30,33}. Here rules are labelled as follows: 1: #and#(#false(),#false()) -> c_1() 2: #and#(#false(),#true()) -> c_2() 3: #and#(#true(),#false()) -> c_3() 4: #and#(#true(),#true()) -> c_4() 5: #eq#(#0(),#0()) -> c_5() 6: #eq#(#0(),#neg(y)) -> c_6() 7: #eq#(#0(),#pos(y)) -> c_7() 8: #eq#(#0(),#s(y)) -> c_8() 9: #eq#(#neg(x),#0()) -> c_9() 10: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) 11: #eq#(#neg(x),#pos(y)) -> c_11() 12: #eq#(#pos(x),#0()) -> c_12() 13: #eq#(#pos(x),#neg(y)) -> c_13() 14: #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) 15: #eq#(#s(x),#0()) -> c_15() 16: #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) 17: #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)),#eq#(x'1,y'1),#eq#(x'2,y'2)) 18: #eq#(dd(x'1,x'2),nil()) -> c_18() 19: #eq#(nil(),dd(y'1,y'2)) -> c_19() 20: #eq#(nil(),nil()) -> c_20() 21: #equal#(x,y) -> c_21(#eq#(x,y)) 22: and#(x,y) -> c_22(#and#(x,y)) 23: eq#(l1,l2) -> c_23(eq#1#(l1,l2)) 24: eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) 25: eq#1#(nil(),l2) -> c_25(eq#2#(l2)) 26: eq#2#(dd(y,ys)) -> c_26() 27: eq#2#(nil()) -> c_27() 28: eq#3#(dd(y,ys),x,xs) -> c_28(and#(#equal(x,y),eq(xs,ys)),#equal#(x,y),eq#(xs,ys)) 29: eq#3#(nil(),x,xs) -> c_29() 30: nub#(l) -> c_30(nub#1#(l)) 31: nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) 32: nub#1#(nil()) -> c_32() 33: remove#(x,l) -> c_33(remove#1#(l,x)) 34: remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) 35: remove#1#(nil(),x) -> c_35() 36: remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) 37: remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)),#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) and#(x,y) -> c_22(#and#(x,y)) eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#1#(nil(),l2) -> c_25(eq#2#(l2)) eq#3#(dd(y,ys),x,xs) -> c_28(and#(#equal(x,y),eq(xs,ys)),#equal#(x,y),eq#(xs,ys)) nub#(l) -> c_30(nub#1#(l)) nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) - Weak DPs: #and#(#false(),#false()) -> c_1() #and#(#false(),#true()) -> c_2() #and#(#true(),#false()) -> c_3() #and#(#true(),#true()) -> c_4() #eq#(#0(),#0()) -> c_5() #eq#(#0(),#neg(y)) -> c_6() #eq#(#0(),#pos(y)) -> c_7() #eq#(#0(),#s(y)) -> c_8() #eq#(#neg(x),#0()) -> c_9() #eq#(#neg(x),#pos(y)) -> c_11() #eq#(#pos(x),#0()) -> c_12() #eq#(#pos(x),#neg(y)) -> c_13() #eq#(#s(x),#0()) -> c_15() #eq#(dd(x'1,x'2),nil()) -> c_18() #eq#(nil(),dd(y'1,y'2)) -> c_19() #eq#(nil(),nil()) -> c_20() eq#2#(dd(y,ys)) -> c_26() eq#2#(nil()) -> c_27() eq#3#(nil(),x,xs) -> c_29() nub#1#(nil()) -> c_32() remove#1#(nil(),x) -> c_35() - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() nub(l) -> nub#1(l) nub#1(dd(x,xs)) -> dd(x,nub(remove(x,xs))) nub#1(nil()) -> nil() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/3,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/3,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {6,9} by application of Pre({6,9}) = {7,10}. Here rules are labelled as follows: 1: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) 2: #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) 3: #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) 4: #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)),#eq#(x'1,y'1),#eq#(x'2,y'2)) 5: #equal#(x,y) -> c_21(#eq#(x,y)) 6: and#(x,y) -> c_22(#and#(x,y)) 7: eq#(l1,l2) -> c_23(eq#1#(l1,l2)) 8: eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) 9: eq#1#(nil(),l2) -> c_25(eq#2#(l2)) 10: eq#3#(dd(y,ys),x,xs) -> c_28(and#(#equal(x,y),eq(xs,ys)),#equal#(x,y),eq#(xs,ys)) 11: nub#(l) -> c_30(nub#1#(l)) 12: nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) 13: remove#(x,l) -> c_33(remove#1#(l,x)) 14: remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) 15: remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) 16: remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) 17: #and#(#false(),#false()) -> c_1() 18: #and#(#false(),#true()) -> c_2() 19: #and#(#true(),#false()) -> c_3() 20: #and#(#true(),#true()) -> c_4() 21: #eq#(#0(),#0()) -> c_5() 22: #eq#(#0(),#neg(y)) -> c_6() 23: #eq#(#0(),#pos(y)) -> c_7() 24: #eq#(#0(),#s(y)) -> c_8() 25: #eq#(#neg(x),#0()) -> c_9() 26: #eq#(#neg(x),#pos(y)) -> c_11() 27: #eq#(#pos(x),#0()) -> c_12() 28: #eq#(#pos(x),#neg(y)) -> c_13() 29: #eq#(#s(x),#0()) -> c_15() 30: #eq#(dd(x'1,x'2),nil()) -> c_18() 31: #eq#(nil(),dd(y'1,y'2)) -> c_19() 32: #eq#(nil(),nil()) -> c_20() 33: eq#2#(dd(y,ys)) -> c_26() 34: eq#2#(nil()) -> c_27() 35: eq#3#(nil(),x,xs) -> c_29() 36: nub#1#(nil()) -> c_32() 37: remove#1#(nil(),x) -> c_35() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)),#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(and#(#equal(x,y),eq(xs,ys)),#equal#(x,y),eq#(xs,ys)) nub#(l) -> c_30(nub#1#(l)) nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) - Weak DPs: #and#(#false(),#false()) -> c_1() #and#(#false(),#true()) -> c_2() #and#(#true(),#false()) -> c_3() #and#(#true(),#true()) -> c_4() #eq#(#0(),#0()) -> c_5() #eq#(#0(),#neg(y)) -> c_6() #eq#(#0(),#pos(y)) -> c_7() #eq#(#0(),#s(y)) -> c_8() #eq#(#neg(x),#0()) -> c_9() #eq#(#neg(x),#pos(y)) -> c_11() #eq#(#pos(x),#0()) -> c_12() #eq#(#pos(x),#neg(y)) -> c_13() #eq#(#s(x),#0()) -> c_15() #eq#(dd(x'1,x'2),nil()) -> c_18() #eq#(nil(),dd(y'1,y'2)) -> c_19() #eq#(nil(),nil()) -> c_20() and#(x,y) -> c_22(#and#(x,y)) eq#1#(nil(),l2) -> c_25(eq#2#(l2)) eq#2#(dd(y,ys)) -> c_26() eq#2#(nil()) -> c_27() eq#3#(nil(),x,xs) -> c_29() nub#1#(nil()) -> c_32() remove#1#(nil(),x) -> c_35() - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() nub(l) -> nub#1(l) nub#1(dd(x,xs)) -> dd(x,nub(remove(x,xs))) nub#1(nil()) -> nil() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/3,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/3,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:#eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) -->_1 #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)) ,#eq#(x'1,y'1) ,#eq#(x'2,y'2)):4 -->_1 #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)):3 -->_1 #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)):2 -->_1 #eq#(nil(),nil()) -> c_20():30 -->_1 #eq#(nil(),dd(y'1,y'2)) -> c_19():29 -->_1 #eq#(dd(x'1,x'2),nil()) -> c_18():28 -->_1 #eq#(#s(x),#0()) -> c_15():27 -->_1 #eq#(#pos(x),#neg(y)) -> c_13():26 -->_1 #eq#(#pos(x),#0()) -> c_12():25 -->_1 #eq#(#neg(x),#pos(y)) -> c_11():24 -->_1 #eq#(#neg(x),#0()) -> c_9():23 -->_1 #eq#(#0(),#s(y)) -> c_8():22 -->_1 #eq#(#0(),#pos(y)) -> c_7():21 -->_1 #eq#(#0(),#neg(y)) -> c_6():20 -->_1 #eq#(#0(),#0()) -> c_5():19 -->_1 #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)):1 2:S:#eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) -->_1 #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)) ,#eq#(x'1,y'1) ,#eq#(x'2,y'2)):4 -->_1 #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)):3 -->_1 #eq#(nil(),nil()) -> c_20():30 -->_1 #eq#(nil(),dd(y'1,y'2)) -> c_19():29 -->_1 #eq#(dd(x'1,x'2),nil()) -> c_18():28 -->_1 #eq#(#s(x),#0()) -> c_15():27 -->_1 #eq#(#pos(x),#neg(y)) -> c_13():26 -->_1 #eq#(#pos(x),#0()) -> c_12():25 -->_1 #eq#(#neg(x),#pos(y)) -> c_11():24 -->_1 #eq#(#neg(x),#0()) -> c_9():23 -->_1 #eq#(#0(),#s(y)) -> c_8():22 -->_1 #eq#(#0(),#pos(y)) -> c_7():21 -->_1 #eq#(#0(),#neg(y)) -> c_6():20 -->_1 #eq#(#0(),#0()) -> c_5():19 -->_1 #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)):2 -->_1 #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)):1 3:S:#eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) -->_1 #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)) ,#eq#(x'1,y'1) ,#eq#(x'2,y'2)):4 -->_1 #eq#(nil(),nil()) -> c_20():30 -->_1 #eq#(nil(),dd(y'1,y'2)) -> c_19():29 -->_1 #eq#(dd(x'1,x'2),nil()) -> c_18():28 -->_1 #eq#(#s(x),#0()) -> c_15():27 -->_1 #eq#(#pos(x),#neg(y)) -> c_13():26 -->_1 #eq#(#pos(x),#0()) -> c_12():25 -->_1 #eq#(#neg(x),#pos(y)) -> c_11():24 -->_1 #eq#(#neg(x),#0()) -> c_9():23 -->_1 #eq#(#0(),#s(y)) -> c_8():22 -->_1 #eq#(#0(),#pos(y)) -> c_7():21 -->_1 #eq#(#0(),#neg(y)) -> c_6():20 -->_1 #eq#(#0(),#0()) -> c_5():19 -->_1 #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)):3 -->_1 #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)):2 -->_1 #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)):1 4:S:#eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)),#eq#(x'1,y'1),#eq#(x'2,y'2)) -->_3 #eq#(nil(),nil()) -> c_20():30 -->_2 #eq#(nil(),nil()) -> c_20():30 -->_3 #eq#(nil(),dd(y'1,y'2)) -> c_19():29 -->_2 #eq#(nil(),dd(y'1,y'2)) -> c_19():29 -->_3 #eq#(dd(x'1,x'2),nil()) -> c_18():28 -->_2 #eq#(dd(x'1,x'2),nil()) -> c_18():28 -->_3 #eq#(#s(x),#0()) -> c_15():27 -->_2 #eq#(#s(x),#0()) -> c_15():27 -->_3 #eq#(#pos(x),#neg(y)) -> c_13():26 -->_2 #eq#(#pos(x),#neg(y)) -> c_13():26 -->_3 #eq#(#pos(x),#0()) -> c_12():25 -->_2 #eq#(#pos(x),#0()) -> c_12():25 -->_3 #eq#(#neg(x),#pos(y)) -> c_11():24 -->_2 #eq#(#neg(x),#pos(y)) -> c_11():24 -->_3 #eq#(#neg(x),#0()) -> c_9():23 -->_2 #eq#(#neg(x),#0()) -> c_9():23 -->_3 #eq#(#0(),#s(y)) -> c_8():22 -->_2 #eq#(#0(),#s(y)) -> c_8():22 -->_3 #eq#(#0(),#pos(y)) -> c_7():21 -->_2 #eq#(#0(),#pos(y)) -> c_7():21 -->_3 #eq#(#0(),#neg(y)) -> c_6():20 -->_2 #eq#(#0(),#neg(y)) -> c_6():20 -->_3 #eq#(#0(),#0()) -> c_5():19 -->_2 #eq#(#0(),#0()) -> c_5():19 -->_1 #and#(#true(),#true()) -> c_4():18 -->_1 #and#(#true(),#false()) -> c_3():17 -->_1 #and#(#false(),#true()) -> c_2():16 -->_1 #and#(#false(),#false()) -> c_1():15 -->_3 #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)),#eq#(x'1,y'1),#eq#(x'2,y'2)):4 -->_2 #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)),#eq#(x'1,y'1),#eq#(x'2,y'2)):4 -->_3 #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)):3 -->_2 #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)):3 -->_3 #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)):2 -->_2 #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)):2 -->_3 #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)):1 -->_2 #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)):1 5:S:#equal#(x,y) -> c_21(#eq#(x,y)) -->_1 #eq#(nil(),nil()) -> c_20():30 -->_1 #eq#(nil(),dd(y'1,y'2)) -> c_19():29 -->_1 #eq#(dd(x'1,x'2),nil()) -> c_18():28 -->_1 #eq#(#s(x),#0()) -> c_15():27 -->_1 #eq#(#pos(x),#neg(y)) -> c_13():26 -->_1 #eq#(#pos(x),#0()) -> c_12():25 -->_1 #eq#(#neg(x),#pos(y)) -> c_11():24 -->_1 #eq#(#neg(x),#0()) -> c_9():23 -->_1 #eq#(#0(),#s(y)) -> c_8():22 -->_1 #eq#(#0(),#pos(y)) -> c_7():21 -->_1 #eq#(#0(),#neg(y)) -> c_6():20 -->_1 #eq#(#0(),#0()) -> c_5():19 -->_1 #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)),#eq#(x'1,y'1),#eq#(x'2,y'2)):4 -->_1 #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)):3 -->_1 #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)):2 -->_1 #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)):1 6:S:eq#(l1,l2) -> c_23(eq#1#(l1,l2)) -->_1 eq#1#(nil(),l2) -> c_25(eq#2#(l2)):32 -->_1 eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)):7 7:S:eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) -->_1 eq#3#(dd(y,ys),x,xs) -> c_28(and#(#equal(x,y),eq(xs,ys)),#equal#(x,y),eq#(xs,ys)):8 -->_1 eq#3#(nil(),x,xs) -> c_29():35 8:S:eq#3#(dd(y,ys),x,xs) -> c_28(and#(#equal(x,y),eq(xs,ys)),#equal#(x,y),eq#(xs,ys)) -->_1 and#(x,y) -> c_22(#and#(x,y)):31 -->_3 eq#(l1,l2) -> c_23(eq#1#(l1,l2)):6 -->_2 #equal#(x,y) -> c_21(#eq#(x,y)):5 9:S:nub#(l) -> c_30(nub#1#(l)) -->_1 nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)):10 -->_1 nub#1#(nil()) -> c_32():36 10:S:nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) -->_2 remove#(x,l) -> c_33(remove#1#(l,x)):11 -->_1 nub#(l) -> c_30(nub#1#(l)):9 11:S:remove#(x,l) -> c_33(remove#1#(l,x)) -->_1 remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)):12 -->_1 remove#1#(nil(),x) -> c_35():37 12:S:remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) -->_1 remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)):14 -->_1 remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)):13 -->_2 eq#(l1,l2) -> c_23(eq#1#(l1,l2)):6 13:S:remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) -->_1 remove#(x,l) -> c_33(remove#1#(l,x)):11 14:S:remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) -->_1 remove#(x,l) -> c_33(remove#1#(l,x)):11 15:W:#and#(#false(),#false()) -> c_1() 16:W:#and#(#false(),#true()) -> c_2() 17:W:#and#(#true(),#false()) -> c_3() 18:W:#and#(#true(),#true()) -> c_4() 19:W:#eq#(#0(),#0()) -> c_5() 20:W:#eq#(#0(),#neg(y)) -> c_6() 21:W:#eq#(#0(),#pos(y)) -> c_7() 22:W:#eq#(#0(),#s(y)) -> c_8() 23:W:#eq#(#neg(x),#0()) -> c_9() 24:W:#eq#(#neg(x),#pos(y)) -> c_11() 25:W:#eq#(#pos(x),#0()) -> c_12() 26:W:#eq#(#pos(x),#neg(y)) -> c_13() 27:W:#eq#(#s(x),#0()) -> c_15() 28:W:#eq#(dd(x'1,x'2),nil()) -> c_18() 29:W:#eq#(nil(),dd(y'1,y'2)) -> c_19() 30:W:#eq#(nil(),nil()) -> c_20() 31:W:and#(x,y) -> c_22(#and#(x,y)) -->_1 #and#(#true(),#true()) -> c_4():18 -->_1 #and#(#true(),#false()) -> c_3():17 -->_1 #and#(#false(),#true()) -> c_2():16 -->_1 #and#(#false(),#false()) -> c_1():15 32:W:eq#1#(nil(),l2) -> c_25(eq#2#(l2)) -->_1 eq#2#(nil()) -> c_27():34 -->_1 eq#2#(dd(y,ys)) -> c_26():33 33:W:eq#2#(dd(y,ys)) -> c_26() 34:W:eq#2#(nil()) -> c_27() 35:W:eq#3#(nil(),x,xs) -> c_29() 36:W:nub#1#(nil()) -> c_32() 37:W:remove#1#(nil(),x) -> c_35() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 36: nub#1#(nil()) -> c_32() 37: remove#1#(nil(),x) -> c_35() 35: eq#3#(nil(),x,xs) -> c_29() 31: and#(x,y) -> c_22(#and#(x,y)) 32: eq#1#(nil(),l2) -> c_25(eq#2#(l2)) 33: eq#2#(dd(y,ys)) -> c_26() 34: eq#2#(nil()) -> c_27() 15: #and#(#false(),#false()) -> c_1() 16: #and#(#false(),#true()) -> c_2() 17: #and#(#true(),#false()) -> c_3() 18: #and#(#true(),#true()) -> c_4() 19: #eq#(#0(),#0()) -> c_5() 20: #eq#(#0(),#neg(y)) -> c_6() 21: #eq#(#0(),#pos(y)) -> c_7() 22: #eq#(#0(),#s(y)) -> c_8() 23: #eq#(#neg(x),#0()) -> c_9() 24: #eq#(#neg(x),#pos(y)) -> c_11() 25: #eq#(#pos(x),#0()) -> c_12() 26: #eq#(#pos(x),#neg(y)) -> c_13() 27: #eq#(#s(x),#0()) -> c_15() 28: #eq#(dd(x'1,x'2),nil()) -> c_18() 29: #eq#(nil(),dd(y'1,y'2)) -> c_19() 30: #eq#(nil(),nil()) -> c_20() * Step 5: SimplifyRHS WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)),#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(and#(#equal(x,y),eq(xs,ys)),#equal#(x,y),eq#(xs,ys)) nub#(l) -> c_30(nub#1#(l)) nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() nub(l) -> nub#1(l) nub#1(dd(x,xs)) -> dd(x,nub(remove(x,xs))) nub#1(nil()) -> nil() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/3,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/3,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:#eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) -->_1 #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)) ,#eq#(x'1,y'1) ,#eq#(x'2,y'2)):4 -->_1 #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)):3 -->_1 #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)):2 -->_1 #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)):1 2:S:#eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) -->_1 #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)) ,#eq#(x'1,y'1) ,#eq#(x'2,y'2)):4 -->_1 #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)):3 -->_1 #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)):2 -->_1 #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)):1 3:S:#eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) -->_1 #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)) ,#eq#(x'1,y'1) ,#eq#(x'2,y'2)):4 -->_1 #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)):3 -->_1 #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)):2 -->_1 #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)):1 4:S:#eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)),#eq#(x'1,y'1),#eq#(x'2,y'2)) -->_3 #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)) ,#eq#(x'1,y'1) ,#eq#(x'2,y'2)):4 -->_2 #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)),#eq#(x'1,y'1),#eq#(x'2,y'2)):4 -->_3 #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)):3 -->_2 #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)):3 -->_3 #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)):2 -->_2 #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)):2 -->_3 #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)):1 -->_2 #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)):1 5:S:#equal#(x,y) -> c_21(#eq#(x,y)) -->_1 #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#and#(#eq(x'1,y'1),#eq(x'2,y'2)) ,#eq#(x'1,y'1) ,#eq#(x'2,y'2)):4 -->_1 #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)):3 -->_1 #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)):2 -->_1 #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)):1 6:S:eq#(l1,l2) -> c_23(eq#1#(l1,l2)) -->_1 eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)):7 7:S:eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) -->_1 eq#3#(dd(y,ys),x,xs) -> c_28(and#(#equal(x,y),eq(xs,ys)),#equal#(x,y),eq#(xs,ys)):8 8:S:eq#3#(dd(y,ys),x,xs) -> c_28(and#(#equal(x,y),eq(xs,ys)),#equal#(x,y),eq#(xs,ys)) -->_3 eq#(l1,l2) -> c_23(eq#1#(l1,l2)):6 -->_2 #equal#(x,y) -> c_21(#eq#(x,y)):5 9:S:nub#(l) -> c_30(nub#1#(l)) -->_1 nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)):10 10:S:nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) -->_2 remove#(x,l) -> c_33(remove#1#(l,x)):11 -->_1 nub#(l) -> c_30(nub#1#(l)):9 11:S:remove#(x,l) -> c_33(remove#1#(l,x)) -->_1 remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)):12 12:S:remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) -->_1 remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)):14 -->_1 remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)):13 -->_2 eq#(l1,l2) -> c_23(eq#1#(l1,l2)):6 13:S:remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) -->_1 remove#(x,l) -> c_33(remove#1#(l,x)):11 14:S:remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) -->_1 remove#(x,l) -> c_33(remove#1#(l,x)):11 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) eq#3#(dd(y,ys),x,xs) -> c_28(#equal#(x,y),eq#(xs,ys)) * Step 6: UsableRules WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(#equal#(x,y),eq#(xs,ys)) nub#(l) -> c_30(nub#1#(l)) nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() nub(l) -> nub#1(l) nub#1(dd(x,xs)) -> dd(x,nub(remove(x,xs))) nub#1(nil()) -> nil() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(#equal#(x,y),eq#(xs,ys)) nub#(l) -> c_30(nub#1#(l)) nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) * Step 7: DecomposeDG WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(#equal#(x,y),eq#(xs,ys)) nub#(l) -> c_30(nub#1#(l)) nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component nub#(l) -> c_30(nub#1#(l)) nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) and a lower component #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(#equal#(x,y),eq#(xs,ys)) remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) Further, following extension rules are added to the lower component. nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) ** Step 7.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: nub#(l) -> c_30(nub#1#(l)) nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:nub#(l) -> c_30(nub#1#(l)) -->_1 nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)):2 2:S:nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs)),remove#(x,xs)) -->_1 nub#(l) -> c_30(nub#1#(l)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs))) ** Step 7.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: nub#(l) -> c_30(nub#1#(l)) nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs))) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/1,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#and) = {1,2}, uargs(and) = {1,2}, uargs(dd) = {2}, uargs(remove#2) = {1}, uargs(nub#) = {1}, uargs(c_30) = {1}, uargs(c_31) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#and) = [1] x1 + [1] x2 + [0] p(#eq) = [0] p(#equal) = [0] p(#false) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(and) = [1] x1 + [1] x2 + [0] p(dd) = [1] x2 + [0] p(eq) = [0] p(eq#1) = [0] p(eq#2) = [0] p(eq#3) = [0] p(nil) = [0] p(nub) = [0] p(nub#1) = [0] p(remove) = [1] p(remove#1) = [1] p(remove#2) = [1] x1 + [1] p(#and#) = [0] p(#eq#) = [0] p(#equal#) = [0] p(and#) = [0] p(eq#) = [0] p(eq#1#) = [0] p(eq#2#) = [0] p(eq#3#) = [0] p(nub#) = [1] x1 + [6] p(nub#1#) = [1] x1 + [1] p(remove#) = [0] p(remove#1#) = [0] p(remove#2#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [1] x1 + [0] p(c_31) = [1] x1 + [0] p(c_32) = [0] p(c_33) = [0] p(c_34) = [1] x1 + [2] x2 + [0] p(c_35) = [1] p(c_36) = [1] p(c_37) = [4] Following rules are strictly oriented: nub#(l) = [1] l + [6] > [1] l + [1] = c_30(nub#1#(l)) Following rules are (at-least) weakly oriented: nub#1#(dd(x,xs)) = [1] xs + [1] >= [7] = c_31(nub#(remove(x,xs))) #and(#false(),#false()) = [0] >= [0] = #false() #and(#false(),#true()) = [0] >= [0] = #false() #and(#true(),#false()) = [0] >= [0] = #false() #and(#true(),#true()) = [0] >= [0] = #true() #eq(#0(),#0()) = [0] >= [0] = #true() #eq(#0(),#neg(y)) = [0] >= [0] = #false() #eq(#0(),#pos(y)) = [0] >= [0] = #false() #eq(#0(),#s(y)) = [0] >= [0] = #false() #eq(#neg(x),#0()) = [0] >= [0] = #false() #eq(#neg(x),#neg(y)) = [0] >= [0] = #eq(x,y) #eq(#neg(x),#pos(y)) = [0] >= [0] = #false() #eq(#pos(x),#0()) = [0] >= [0] = #false() #eq(#pos(x),#neg(y)) = [0] >= [0] = #false() #eq(#pos(x),#pos(y)) = [0] >= [0] = #eq(x,y) #eq(#s(x),#0()) = [0] >= [0] = #false() #eq(#s(x),#s(y)) = [0] >= [0] = #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) = [0] >= [0] = #false() #eq(nil(),dd(y'1,y'2)) = [0] >= [0] = #false() #eq(nil(),nil()) = [0] >= [0] = #true() #equal(x,y) = [0] >= [0] = #eq(x,y) and(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = #and(x,y) eq(l1,l2) = [0] >= [0] = eq#1(l1,l2) eq#1(dd(x,xs),l2) = [0] >= [0] = eq#3(l2,x,xs) eq#1(nil(),l2) = [0] >= [0] = eq#2(l2) eq#2(dd(y,ys)) = [0] >= [0] = #false() eq#2(nil()) = [0] >= [0] = #true() eq#3(dd(y,ys),x,xs) = [0] >= [0] = and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) = [0] >= [0] = #false() remove(x,l) = [1] >= [1] = remove#1(l,x) remove#1(dd(y,ys),x) = [1] >= [1] = remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) = [1] >= [0] = nil() remove#2(#false(),x,y,ys) = [1] >= [1] = dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) = [1] >= [1] = remove(x,ys) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 7.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs))) - Weak DPs: nub#(l) -> c_30(nub#1#(l)) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/1,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#and) = {1,2}, uargs(and) = {1,2}, uargs(dd) = {2}, uargs(remove#2) = {1}, uargs(nub#) = {1}, uargs(c_30) = {1}, uargs(c_31) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#and) = [1] x1 + [1] x2 + [0] p(#eq) = [0] p(#equal) = [0] p(#false) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(and) = [1] x1 + [1] x2 + [0] p(dd) = [1] x2 + [4] p(eq) = [0] p(eq#1) = [0] p(eq#2) = [0] p(eq#3) = [0] p(nil) = [0] p(nub) = [0] p(nub#1) = [0] p(remove) = [1] x2 + [2] p(remove#1) = [1] x1 + [2] p(remove#2) = [1] x1 + [1] x4 + [6] p(#and#) = [0] p(#eq#) = [0] p(#equal#) = [0] p(and#) = [0] p(eq#) = [0] p(eq#1#) = [0] p(eq#2#) = [0] p(eq#3#) = [0] p(nub#) = [1] x1 + [6] p(nub#1#) = [1] x1 + [6] p(remove#) = [0] p(remove#1#) = [0] p(remove#2#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [1] x1 + [0] p(c_31) = [1] x1 + [0] p(c_32) = [0] p(c_33) = [0] p(c_34) = [0] p(c_35) = [0] p(c_36) = [1] x1 + [2] p(c_37) = [2] x1 + [0] Following rules are strictly oriented: nub#1#(dd(x,xs)) = [1] xs + [10] > [1] xs + [8] = c_31(nub#(remove(x,xs))) Following rules are (at-least) weakly oriented: nub#(l) = [1] l + [6] >= [1] l + [6] = c_30(nub#1#(l)) #and(#false(),#false()) = [0] >= [0] = #false() #and(#false(),#true()) = [0] >= [0] = #false() #and(#true(),#false()) = [0] >= [0] = #false() #and(#true(),#true()) = [0] >= [0] = #true() #eq(#0(),#0()) = [0] >= [0] = #true() #eq(#0(),#neg(y)) = [0] >= [0] = #false() #eq(#0(),#pos(y)) = [0] >= [0] = #false() #eq(#0(),#s(y)) = [0] >= [0] = #false() #eq(#neg(x),#0()) = [0] >= [0] = #false() #eq(#neg(x),#neg(y)) = [0] >= [0] = #eq(x,y) #eq(#neg(x),#pos(y)) = [0] >= [0] = #false() #eq(#pos(x),#0()) = [0] >= [0] = #false() #eq(#pos(x),#neg(y)) = [0] >= [0] = #false() #eq(#pos(x),#pos(y)) = [0] >= [0] = #eq(x,y) #eq(#s(x),#0()) = [0] >= [0] = #false() #eq(#s(x),#s(y)) = [0] >= [0] = #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) = [0] >= [0] = #false() #eq(nil(),dd(y'1,y'2)) = [0] >= [0] = #false() #eq(nil(),nil()) = [0] >= [0] = #true() #equal(x,y) = [0] >= [0] = #eq(x,y) and(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = #and(x,y) eq(l1,l2) = [0] >= [0] = eq#1(l1,l2) eq#1(dd(x,xs),l2) = [0] >= [0] = eq#3(l2,x,xs) eq#1(nil(),l2) = [0] >= [0] = eq#2(l2) eq#2(dd(y,ys)) = [0] >= [0] = #false() eq#2(nil()) = [0] >= [0] = #true() eq#3(dd(y,ys),x,xs) = [0] >= [0] = and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) = [0] >= [0] = #false() remove(x,l) = [1] l + [2] >= [1] l + [2] = remove#1(l,x) remove#1(dd(y,ys),x) = [1] ys + [6] >= [1] ys + [6] = remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) = [2] >= [0] = nil() remove#2(#false(),x,y,ys) = [1] ys + [6] >= [1] ys + [6] = dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) = [1] ys + [6] >= [1] ys + [2] = remove(x,ys) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 7.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: nub#(l) -> c_30(nub#1#(l)) nub#1#(dd(x,xs)) -> c_31(nub#(remove(x,xs))) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/1,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 7.b:1: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(#equal#(x,y),eq#(xs,ys)) remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) - Weak DPs: nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) and a lower component #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(#equal#(x,y),eq#(xs,ys)) Further, following extension rules are added to the lower component. nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) *** Step 7.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) - Weak DPs: nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:remove#(x,l) -> c_33(remove#1#(l,x)) -->_1 remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)):2 2:S:remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys),eq#(x,y)) -->_1 remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)):4 -->_1 remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)):3 3:S:remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) -->_1 remove#(x,l) -> c_33(remove#1#(l,x)):1 4:S:remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) -->_1 remove#(x,l) -> c_33(remove#1#(l,x)):1 5:W:nub#(l) -> nub#1#(l) -->_1 nub#1#(dd(x,xs)) -> remove#(x,xs):7 -->_1 nub#1#(dd(x,xs)) -> nub#(remove(x,xs)):6 6:W:nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) -->_1 nub#(l) -> nub#1#(l):5 7:W:nub#1#(dd(x,xs)) -> remove#(x,xs) -->_1 remove#(x,l) -> c_33(remove#1#(l,x)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys)) *** Step 7.b:1.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys)) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) - Weak DPs: nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/1,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#and) = {1,2}, uargs(and) = {1,2}, uargs(dd) = {2}, uargs(remove#2) = {1}, uargs(nub#) = {1}, uargs(remove#2#) = {1}, uargs(c_33) = {1}, uargs(c_34) = {1}, uargs(c_36) = {1}, uargs(c_37) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#and) = [1] x1 + [1] x2 + [0] p(#eq) = [0] p(#equal) = [0] p(#false) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(and) = [1] x1 + [1] x2 + [0] p(dd) = [1] x2 + [0] p(eq) = [0] p(eq#1) = [0] p(eq#2) = [0] p(eq#3) = [0] p(nil) = [0] p(nub) = [0] p(nub#1) = [0] p(remove) = [0] p(remove#1) = [0] p(remove#2) = [1] x1 + [0] p(#and#) = [0] p(#eq#) = [0] p(#equal#) = [0] p(and#) = [0] p(eq#) = [0] p(eq#1#) = [1] p(eq#2#) = [0] p(eq#3#) = [0] p(nub#) = [1] x1 + [0] p(nub#1#) = [0] p(remove#) = [0] p(remove#1#) = [0] p(remove#2#) = [1] x1 + [3] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [0] p(c_32) = [0] p(c_33) = [1] x1 + [0] p(c_34) = [1] x1 + [0] p(c_35) = [0] p(c_36) = [1] x1 + [0] p(c_37) = [1] x1 + [0] Following rules are strictly oriented: remove#2#(#false(),x,y,ys) = [3] > [0] = c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) = [3] > [0] = c_37(remove#(x,ys)) Following rules are (at-least) weakly oriented: nub#(l) = [1] l + [0] >= [0] = nub#1#(l) nub#1#(dd(x,xs)) = [0] >= [0] = nub#(remove(x,xs)) nub#1#(dd(x,xs)) = [0] >= [0] = remove#(x,xs) remove#(x,l) = [0] >= [0] = c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) = [0] >= [3] = c_34(remove#2#(eq(x,y),x,y,ys)) #and(#false(),#false()) = [0] >= [0] = #false() #and(#false(),#true()) = [0] >= [0] = #false() #and(#true(),#false()) = [0] >= [0] = #false() #and(#true(),#true()) = [0] >= [0] = #true() #eq(#0(),#0()) = [0] >= [0] = #true() #eq(#0(),#neg(y)) = [0] >= [0] = #false() #eq(#0(),#pos(y)) = [0] >= [0] = #false() #eq(#0(),#s(y)) = [0] >= [0] = #false() #eq(#neg(x),#0()) = [0] >= [0] = #false() #eq(#neg(x),#neg(y)) = [0] >= [0] = #eq(x,y) #eq(#neg(x),#pos(y)) = [0] >= [0] = #false() #eq(#pos(x),#0()) = [0] >= [0] = #false() #eq(#pos(x),#neg(y)) = [0] >= [0] = #false() #eq(#pos(x),#pos(y)) = [0] >= [0] = #eq(x,y) #eq(#s(x),#0()) = [0] >= [0] = #false() #eq(#s(x),#s(y)) = [0] >= [0] = #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) = [0] >= [0] = #false() #eq(nil(),dd(y'1,y'2)) = [0] >= [0] = #false() #eq(nil(),nil()) = [0] >= [0] = #true() #equal(x,y) = [0] >= [0] = #eq(x,y) and(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = #and(x,y) eq(l1,l2) = [0] >= [0] = eq#1(l1,l2) eq#1(dd(x,xs),l2) = [0] >= [0] = eq#3(l2,x,xs) eq#1(nil(),l2) = [0] >= [0] = eq#2(l2) eq#2(dd(y,ys)) = [0] >= [0] = #false() eq#2(nil()) = [0] >= [0] = #true() eq#3(dd(y,ys),x,xs) = [0] >= [0] = and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) = [0] >= [0] = #false() remove(x,l) = [0] >= [0] = remove#1(l,x) remove#1(dd(y,ys),x) = [0] >= [0] = remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) = [0] >= [0] = nil() remove#2(#false(),x,y,ys) = [0] >= [0] = dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) = [0] >= [0] = remove(x,ys) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 7.b:1.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys)) - Weak DPs: nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/1,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#and) = {1,2}, uargs(and) = {1,2}, uargs(dd) = {2}, uargs(remove#2) = {1}, uargs(nub#) = {1}, uargs(remove#2#) = {1}, uargs(c_33) = {1}, uargs(c_34) = {1}, uargs(c_36) = {1}, uargs(c_37) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#and) = [1] x1 + [1] x2 + [0] p(#eq) = [0] p(#equal) = [0] p(#false) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(and) = [1] x1 + [1] x2 + [0] p(dd) = [1] x2 + [0] p(eq) = [0] p(eq#1) = [0] p(eq#2) = [0] p(eq#3) = [0] p(nil) = [0] p(nub) = [0] p(nub#1) = [0] p(remove) = [0] p(remove#1) = [0] p(remove#2) = [1] x1 + [0] p(#and#) = [0] p(#eq#) = [0] p(#equal#) = [0] p(and#) = [1] x1 + [0] p(eq#) = [0] p(eq#1#) = [0] p(eq#2#) = [1] x1 + [0] p(eq#3#) = [0] p(nub#) = [1] x1 + [0] p(nub#1#) = [0] p(remove#) = [0] p(remove#1#) = [3] p(remove#2#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [2] 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) = [2] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [0] p(c_32) = [0] p(c_33) = [1] x1 + [0] p(c_34) = [1] x1 + [0] p(c_35) = [0] p(c_36) = [1] x1 + [0] p(c_37) = [1] x1 + [0] Following rules are strictly oriented: remove#1#(dd(y,ys),x) = [3] > [0] = c_34(remove#2#(eq(x,y),x,y,ys)) Following rules are (at-least) weakly oriented: nub#(l) = [1] l + [0] >= [0] = nub#1#(l) nub#1#(dd(x,xs)) = [0] >= [0] = nub#(remove(x,xs)) nub#1#(dd(x,xs)) = [0] >= [0] = remove#(x,xs) remove#(x,l) = [0] >= [3] = c_33(remove#1#(l,x)) remove#2#(#false(),x,y,ys) = [0] >= [0] = c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) = [0] >= [0] = c_37(remove#(x,ys)) #and(#false(),#false()) = [0] >= [0] = #false() #and(#false(),#true()) = [0] >= [0] = #false() #and(#true(),#false()) = [0] >= [0] = #false() #and(#true(),#true()) = [0] >= [0] = #true() #eq(#0(),#0()) = [0] >= [0] = #true() #eq(#0(),#neg(y)) = [0] >= [0] = #false() #eq(#0(),#pos(y)) = [0] >= [0] = #false() #eq(#0(),#s(y)) = [0] >= [0] = #false() #eq(#neg(x),#0()) = [0] >= [0] = #false() #eq(#neg(x),#neg(y)) = [0] >= [0] = #eq(x,y) #eq(#neg(x),#pos(y)) = [0] >= [0] = #false() #eq(#pos(x),#0()) = [0] >= [0] = #false() #eq(#pos(x),#neg(y)) = [0] >= [0] = #false() #eq(#pos(x),#pos(y)) = [0] >= [0] = #eq(x,y) #eq(#s(x),#0()) = [0] >= [0] = #false() #eq(#s(x),#s(y)) = [0] >= [0] = #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) = [0] >= [0] = #false() #eq(nil(),dd(y'1,y'2)) = [0] >= [0] = #false() #eq(nil(),nil()) = [0] >= [0] = #true() #equal(x,y) = [0] >= [0] = #eq(x,y) and(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = #and(x,y) eq(l1,l2) = [0] >= [0] = eq#1(l1,l2) eq#1(dd(x,xs),l2) = [0] >= [0] = eq#3(l2,x,xs) eq#1(nil(),l2) = [0] >= [0] = eq#2(l2) eq#2(dd(y,ys)) = [0] >= [0] = #false() eq#2(nil()) = [0] >= [0] = #true() eq#3(dd(y,ys),x,xs) = [0] >= [0] = and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) = [0] >= [0] = #false() remove(x,l) = [0] >= [0] = remove#1(l,x) remove#1(dd(y,ys),x) = [0] >= [0] = remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) = [0] >= [0] = nil() remove#2(#false(),x,y,ys) = [0] >= [0] = dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) = [0] >= [0] = remove(x,ys) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 7.b:1.a:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: remove#(x,l) -> c_33(remove#1#(l,x)) - Weak DPs: nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys)) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/1,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#and) = {1,2}, uargs(and) = {1,2}, uargs(dd) = {2}, uargs(remove#2) = {1}, uargs(nub#) = {1}, uargs(remove#2#) = {1}, uargs(c_33) = {1}, uargs(c_34) = {1}, uargs(c_36) = {1}, uargs(c_37) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#and) = [1] x1 + [1] x2 + [0] p(#eq) = [0] p(#equal) = [0] p(#false) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(and) = [1] x1 + [1] x2 + [0] p(dd) = [1] x1 + [1] x2 + [1] p(eq) = [0] p(eq#1) = [0] p(eq#2) = [0] p(eq#3) = [0] p(nil) = [0] p(nub) = [0] p(nub#1) = [1] x1 + [4] p(remove) = [1] x2 + [0] p(remove#1) = [1] x1 + [0] p(remove#2) = [1] x1 + [1] x3 + [1] x4 + [1] p(#and#) = [0] p(#eq#) = [1] x2 + [0] p(#equal#) = [0] p(and#) = [1] x1 + [1] p(eq#) = [1] x1 + [1] x2 + [0] p(eq#1#) = [1] x1 + [0] p(eq#2#) = [4] p(eq#3#) = [1] x3 + [0] p(nub#) = [1] x1 + [1] p(nub#1#) = [1] x1 + [0] p(remove#) = [1] x2 + [1] p(remove#1#) = [1] x1 + [0] p(remove#2#) = [1] x1 + [1] x3 + [1] x4 + [1] p(c_1) = [4] p(c_2) = [1] p(c_3) = [1] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] p(c_8) = [4] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [1] p(c_16) = [2] x1 + [0] p(c_17) = [4] x1 + [1] x2 + [0] p(c_18) = [1] p(c_19) = [1] p(c_20) = [0] p(c_21) = [2] x1 + [1] p(c_22) = [0] p(c_23) = [4] x1 + [0] p(c_24) = [4] x1 + [1] p(c_25) = [1] p(c_26) = [1] p(c_27) = [4] p(c_28) = [0] p(c_29) = [0] p(c_30) = [4] p(c_31) = [4] p(c_32) = [1] p(c_33) = [1] x1 + [0] p(c_34) = [1] x1 + [0] p(c_35) = [0] p(c_36) = [1] x1 + [0] p(c_37) = [1] x1 + [0] Following rules are strictly oriented: remove#(x,l) = [1] l + [1] > [1] l + [0] = c_33(remove#1#(l,x)) Following rules are (at-least) weakly oriented: nub#(l) = [1] l + [1] >= [1] l + [0] = nub#1#(l) nub#1#(dd(x,xs)) = [1] x + [1] xs + [1] >= [1] xs + [1] = nub#(remove(x,xs)) nub#1#(dd(x,xs)) = [1] x + [1] xs + [1] >= [1] xs + [1] = remove#(x,xs) remove#1#(dd(y,ys),x) = [1] y + [1] ys + [1] >= [1] y + [1] ys + [1] = c_34(remove#2#(eq(x,y),x,y,ys)) remove#2#(#false(),x,y,ys) = [1] y + [1] ys + [1] >= [1] ys + [1] = c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) = [1] y + [1] ys + [1] >= [1] ys + [1] = c_37(remove#(x,ys)) #and(#false(),#false()) = [0] >= [0] = #false() #and(#false(),#true()) = [0] >= [0] = #false() #and(#true(),#false()) = [0] >= [0] = #false() #and(#true(),#true()) = [0] >= [0] = #true() #eq(#0(),#0()) = [0] >= [0] = #true() #eq(#0(),#neg(y)) = [0] >= [0] = #false() #eq(#0(),#pos(y)) = [0] >= [0] = #false() #eq(#0(),#s(y)) = [0] >= [0] = #false() #eq(#neg(x),#0()) = [0] >= [0] = #false() #eq(#neg(x),#neg(y)) = [0] >= [0] = #eq(x,y) #eq(#neg(x),#pos(y)) = [0] >= [0] = #false() #eq(#pos(x),#0()) = [0] >= [0] = #false() #eq(#pos(x),#neg(y)) = [0] >= [0] = #false() #eq(#pos(x),#pos(y)) = [0] >= [0] = #eq(x,y) #eq(#s(x),#0()) = [0] >= [0] = #false() #eq(#s(x),#s(y)) = [0] >= [0] = #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) = [0] >= [0] = #false() #eq(nil(),dd(y'1,y'2)) = [0] >= [0] = #false() #eq(nil(),nil()) = [0] >= [0] = #true() #equal(x,y) = [0] >= [0] = #eq(x,y) and(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = #and(x,y) eq(l1,l2) = [0] >= [0] = eq#1(l1,l2) eq#1(dd(x,xs),l2) = [0] >= [0] = eq#3(l2,x,xs) eq#1(nil(),l2) = [0] >= [0] = eq#2(l2) eq#2(dd(y,ys)) = [0] >= [0] = #false() eq#2(nil()) = [0] >= [0] = #true() eq#3(dd(y,ys),x,xs) = [0] >= [0] = and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) = [0] >= [0] = #false() remove(x,l) = [1] l + [0] >= [1] l + [0] = remove#1(l,x) remove#1(dd(y,ys),x) = [1] y + [1] ys + [1] >= [1] y + [1] ys + [1] = remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) = [0] >= [0] = nil() remove#2(#false(),x,y,ys) = [1] y + [1] ys + [1] >= [1] y + [1] ys + [1] = dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) = [1] y + [1] ys + [1] >= [1] ys + [0] = remove(x,ys) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 7.b:1.a:5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> c_33(remove#1#(l,x)) remove#1#(dd(y,ys),x) -> c_34(remove#2#(eq(x,y),x,y,ys)) remove#2#(#false(),x,y,ys) -> c_36(remove#(x,ys)) remove#2#(#true(),x,y,ys) -> c_37(remove#(x,ys)) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/1,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 7.b:1.b:1: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(#equal#(x,y),eq#(xs,ys)) - Weak DPs: nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(#equal#(x,y),eq#(xs,ys)) nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) and a lower component #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) Further, following extension rules are added to the lower component. eq#(l1,l2) -> eq#1#(l1,l2) eq#1#(dd(x,xs),l2) -> eq#3#(l2,x,xs) eq#3#(dd(y,ys),x,xs) -> #equal#(x,y) eq#3#(dd(y,ys),x,xs) -> eq#(xs,ys) nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) **** Step 7.b:1.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(#equal#(x,y),eq#(xs,ys)) - Weak DPs: nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:eq#(l1,l2) -> c_23(eq#1#(l1,l2)) -->_1 eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)):2 2:S:eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) -->_1 eq#3#(dd(y,ys),x,xs) -> c_28(#equal#(x,y),eq#(xs,ys)):3 3:S:eq#3#(dd(y,ys),x,xs) -> c_28(#equal#(x,y),eq#(xs,ys)) -->_2 eq#(l1,l2) -> c_23(eq#1#(l1,l2)):1 4:W:nub#(l) -> nub#1#(l) -->_1 nub#1#(dd(x,xs)) -> remove#(x,xs):6 -->_1 nub#1#(dd(x,xs)) -> nub#(remove(x,xs)):5 5:W:nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) -->_1 nub#(l) -> nub#1#(l):4 6:W:nub#1#(dd(x,xs)) -> remove#(x,xs) -->_1 remove#(x,l) -> remove#1#(l,x):7 7:W:remove#(x,l) -> remove#1#(l,x) -->_1 remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys):9 -->_1 remove#1#(dd(y,ys),x) -> eq#(x,y):8 8:W:remove#1#(dd(y,ys),x) -> eq#(x,y) -->_1 eq#(l1,l2) -> c_23(eq#1#(l1,l2)):1 9:W:remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) -->_1 remove#2#(#true(),x,y,ys) -> remove#(x,ys):11 -->_1 remove#2#(#false(),x,y,ys) -> remove#(x,ys):10 10:W:remove#2#(#false(),x,y,ys) -> remove#(x,ys) -->_1 remove#(x,l) -> remove#1#(l,x):7 11:W:remove#2#(#true(),x,y,ys) -> remove#(x,ys) -->_1 remove#(x,l) -> remove#1#(l,x):7 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: eq#3#(dd(y,ys),x,xs) -> c_28(eq#(xs,ys)) **** Step 7.b:1.b:1.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(eq#(xs,ys)) - Weak DPs: nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#and) = {1,2}, uargs(and) = {1,2}, uargs(dd) = {2}, uargs(remove#2) = {1}, uargs(nub#) = {1}, uargs(remove#2#) = {1}, uargs(c_23) = {1}, uargs(c_24) = {1}, uargs(c_28) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#and) = [1] x1 + [1] x2 + [0] p(#eq) = [0] p(#equal) = [0] p(#false) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(and) = [1] x1 + [1] x2 + [0] p(dd) = [1] x2 + [0] p(eq) = [0] p(eq#1) = [0] p(eq#2) = [0] p(eq#3) = [0] p(nil) = [0] p(nub) = [0] p(nub#1) = [0] p(remove) = [0] p(remove#1) = [0] p(remove#2) = [1] x1 + [0] p(#and#) = [0] p(#eq#) = [0] p(#equal#) = [0] p(and#) = [0] p(eq#) = [0] p(eq#1#) = [0] p(eq#2#) = [0] p(eq#3#) = [3] p(nub#) = [1] x1 + [0] p(nub#1#) = [0] p(remove#) = [0] p(remove#1#) = [0] p(remove#2#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [1] x1 + [0] p(c_24) = [1] x1 + [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [1] x1 + [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [0] p(c_32) = [0] p(c_33) = [0] p(c_34) = [0] p(c_35) = [0] p(c_36) = [0] p(c_37) = [0] Following rules are strictly oriented: eq#3#(dd(y,ys),x,xs) = [3] > [0] = c_28(eq#(xs,ys)) Following rules are (at-least) weakly oriented: eq#(l1,l2) = [0] >= [0] = c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) = [0] >= [3] = c_24(eq#3#(l2,x,xs)) nub#(l) = [1] l + [0] >= [0] = nub#1#(l) nub#1#(dd(x,xs)) = [0] >= [0] = nub#(remove(x,xs)) nub#1#(dd(x,xs)) = [0] >= [0] = remove#(x,xs) remove#(x,l) = [0] >= [0] = remove#1#(l,x) remove#1#(dd(y,ys),x) = [0] >= [0] = eq#(x,y) remove#1#(dd(y,ys),x) = [0] >= [0] = remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) = [0] >= [0] = remove#(x,ys) remove#2#(#true(),x,y,ys) = [0] >= [0] = remove#(x,ys) #and(#false(),#false()) = [0] >= [0] = #false() #and(#false(),#true()) = [0] >= [0] = #false() #and(#true(),#false()) = [0] >= [0] = #false() #and(#true(),#true()) = [0] >= [0] = #true() #eq(#0(),#0()) = [0] >= [0] = #true() #eq(#0(),#neg(y)) = [0] >= [0] = #false() #eq(#0(),#pos(y)) = [0] >= [0] = #false() #eq(#0(),#s(y)) = [0] >= [0] = #false() #eq(#neg(x),#0()) = [0] >= [0] = #false() #eq(#neg(x),#neg(y)) = [0] >= [0] = #eq(x,y) #eq(#neg(x),#pos(y)) = [0] >= [0] = #false() #eq(#pos(x),#0()) = [0] >= [0] = #false() #eq(#pos(x),#neg(y)) = [0] >= [0] = #false() #eq(#pos(x),#pos(y)) = [0] >= [0] = #eq(x,y) #eq(#s(x),#0()) = [0] >= [0] = #false() #eq(#s(x),#s(y)) = [0] >= [0] = #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) = [0] >= [0] = #false() #eq(nil(),dd(y'1,y'2)) = [0] >= [0] = #false() #eq(nil(),nil()) = [0] >= [0] = #true() #equal(x,y) = [0] >= [0] = #eq(x,y) and(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = #and(x,y) eq(l1,l2) = [0] >= [0] = eq#1(l1,l2) eq#1(dd(x,xs),l2) = [0] >= [0] = eq#3(l2,x,xs) eq#1(nil(),l2) = [0] >= [0] = eq#2(l2) eq#2(dd(y,ys)) = [0] >= [0] = #false() eq#2(nil()) = [0] >= [0] = #true() eq#3(dd(y,ys),x,xs) = [0] >= [0] = and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) = [0] >= [0] = #false() remove(x,l) = [0] >= [0] = remove#1(l,x) remove#1(dd(y,ys),x) = [0] >= [0] = remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) = [0] >= [0] = nil() remove#2(#false(),x,y,ys) = [0] >= [0] = dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) = [0] >= [0] = remove(x,ys) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 7.b:1.b:1.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) - Weak DPs: eq#3#(dd(y,ys),x,xs) -> c_28(eq#(xs,ys)) nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#and) = {1,2}, uargs(and) = {1,2}, uargs(dd) = {2}, uargs(remove#2) = {1}, uargs(nub#) = {1}, uargs(remove#2#) = {1}, uargs(c_23) = {1}, uargs(c_24) = {1}, uargs(c_28) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#and) = [1] x1 + [1] x2 + [0] p(#eq) = [0] p(#equal) = [0] p(#false) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(and) = [1] x1 + [1] x2 + [0] p(dd) = [1] x2 + [0] p(eq) = [0] p(eq#1) = [0] p(eq#2) = [0] p(eq#3) = [0] p(nil) = [0] p(nub) = [0] p(nub#1) = [0] p(remove) = [0] p(remove#1) = [0] p(remove#2) = [1] x1 + [0] p(#and#) = [0] p(#eq#) = [0] p(#equal#) = [0] p(and#) = [0] p(eq#) = [0] p(eq#1#) = [7] p(eq#2#) = [0] p(eq#3#) = [0] p(nub#) = [1] x1 + [0] p(nub#1#) = [0] p(remove#) = [0] p(remove#1#) = [0] p(remove#2#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [1] x1 + [0] p(c_24) = [1] x1 + [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [1] x1 + [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [0] p(c_32) = [0] p(c_33) = [0] p(c_34) = [0] p(c_35) = [0] p(c_36) = [0] p(c_37) = [0] Following rules are strictly oriented: eq#1#(dd(x,xs),l2) = [7] > [0] = c_24(eq#3#(l2,x,xs)) Following rules are (at-least) weakly oriented: eq#(l1,l2) = [0] >= [7] = c_23(eq#1#(l1,l2)) eq#3#(dd(y,ys),x,xs) = [0] >= [0] = c_28(eq#(xs,ys)) nub#(l) = [1] l + [0] >= [0] = nub#1#(l) nub#1#(dd(x,xs)) = [0] >= [0] = nub#(remove(x,xs)) nub#1#(dd(x,xs)) = [0] >= [0] = remove#(x,xs) remove#(x,l) = [0] >= [0] = remove#1#(l,x) remove#1#(dd(y,ys),x) = [0] >= [0] = eq#(x,y) remove#1#(dd(y,ys),x) = [0] >= [0] = remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) = [0] >= [0] = remove#(x,ys) remove#2#(#true(),x,y,ys) = [0] >= [0] = remove#(x,ys) #and(#false(),#false()) = [0] >= [0] = #false() #and(#false(),#true()) = [0] >= [0] = #false() #and(#true(),#false()) = [0] >= [0] = #false() #and(#true(),#true()) = [0] >= [0] = #true() #eq(#0(),#0()) = [0] >= [0] = #true() #eq(#0(),#neg(y)) = [0] >= [0] = #false() #eq(#0(),#pos(y)) = [0] >= [0] = #false() #eq(#0(),#s(y)) = [0] >= [0] = #false() #eq(#neg(x),#0()) = [0] >= [0] = #false() #eq(#neg(x),#neg(y)) = [0] >= [0] = #eq(x,y) #eq(#neg(x),#pos(y)) = [0] >= [0] = #false() #eq(#pos(x),#0()) = [0] >= [0] = #false() #eq(#pos(x),#neg(y)) = [0] >= [0] = #false() #eq(#pos(x),#pos(y)) = [0] >= [0] = #eq(x,y) #eq(#s(x),#0()) = [0] >= [0] = #false() #eq(#s(x),#s(y)) = [0] >= [0] = #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) = [0] >= [0] = #false() #eq(nil(),dd(y'1,y'2)) = [0] >= [0] = #false() #eq(nil(),nil()) = [0] >= [0] = #true() #equal(x,y) = [0] >= [0] = #eq(x,y) and(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = #and(x,y) eq(l1,l2) = [0] >= [0] = eq#1(l1,l2) eq#1(dd(x,xs),l2) = [0] >= [0] = eq#3(l2,x,xs) eq#1(nil(),l2) = [0] >= [0] = eq#2(l2) eq#2(dd(y,ys)) = [0] >= [0] = #false() eq#2(nil()) = [0] >= [0] = #true() eq#3(dd(y,ys),x,xs) = [0] >= [0] = and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) = [0] >= [0] = #false() remove(x,l) = [0] >= [0] = remove#1(l,x) remove#1(dd(y,ys),x) = [0] >= [0] = remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) = [0] >= [0] = nil() remove#2(#false(),x,y,ys) = [0] >= [0] = dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) = [0] >= [0] = remove(x,ys) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 7.b:1.b:1.a:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: eq#(l1,l2) -> c_23(eq#1#(l1,l2)) - Weak DPs: eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(eq#(xs,ys)) nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#and) = {1,2}, uargs(and) = {1,2}, uargs(dd) = {2}, uargs(remove#2) = {1}, uargs(nub#) = {1}, uargs(remove#2#) = {1}, uargs(c_23) = {1}, uargs(c_24) = {1}, uargs(c_28) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#and) = [1] x1 + [1] x2 + [0] p(#eq) = [0] p(#equal) = [0] p(#false) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(and) = [1] x1 + [1] x2 + [0] p(dd) = [1] x1 + [1] x2 + [1] p(eq) = [0] p(eq#1) = [0] p(eq#2) = [0] p(eq#3) = [0] p(nil) = [0] p(nub) = [0] p(nub#1) = [0] p(remove) = [1] x2 + [1] p(remove#1) = [1] x1 + [1] p(remove#2) = [1] x1 + [1] x3 + [1] x4 + [2] p(#and#) = [1] x2 + [1] p(#eq#) = [1] x2 + [0] p(#equal#) = [0] p(and#) = [1] x1 + [1] x2 + [0] p(eq#) = [1] x1 + [1] p(eq#1#) = [1] x1 + [0] p(eq#2#) = [4] p(eq#3#) = [1] x2 + [1] x3 + [1] p(nub#) = [1] x1 + [3] p(nub#1#) = [1] x1 + [3] p(remove#) = [1] x1 + [2] p(remove#1#) = [1] x2 + [2] p(remove#2#) = [1] x1 + [1] x2 + [2] p(c_1) = [0] p(c_2) = [1] p(c_3) = [4] p(c_4) = [0] p(c_5) = [1] p(c_6) = [2] p(c_7) = [1] p(c_8) = [1] p(c_9) = [4] p(c_10) = [1] p(c_11) = [0] p(c_12) = [0] p(c_13) = [2] p(c_14) = [1] x1 + [1] p(c_15) = [0] p(c_16) = [4] x1 + [1] p(c_17) = [1] x1 + [0] p(c_18) = [4] p(c_19) = [1] p(c_20) = [0] p(c_21) = [2] x1 + [1] p(c_22) = [2] p(c_23) = [1] x1 + [0] p(c_24) = [1] x1 + [0] p(c_25) = [4] x1 + [1] p(c_26) = [2] p(c_27) = [4] p(c_28) = [1] x1 + [0] p(c_29) = [0] p(c_30) = [1] x1 + [0] p(c_31) = [0] p(c_32) = [4] p(c_33) = [4] x1 + [0] p(c_34) = [1] x1 + [1] x2 + [0] p(c_35) = [1] p(c_36) = [1] x1 + [1] p(c_37) = [1] x1 + [0] Following rules are strictly oriented: eq#(l1,l2) = [1] l1 + [1] > [1] l1 + [0] = c_23(eq#1#(l1,l2)) Following rules are (at-least) weakly oriented: eq#1#(dd(x,xs),l2) = [1] x + [1] xs + [1] >= [1] x + [1] xs + [1] = c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) = [1] x + [1] xs + [1] >= [1] xs + [1] = c_28(eq#(xs,ys)) nub#(l) = [1] l + [3] >= [1] l + [3] = nub#1#(l) nub#1#(dd(x,xs)) = [1] x + [1] xs + [4] >= [1] xs + [4] = nub#(remove(x,xs)) nub#1#(dd(x,xs)) = [1] x + [1] xs + [4] >= [1] x + [2] = remove#(x,xs) remove#(x,l) = [1] x + [2] >= [1] x + [2] = remove#1#(l,x) remove#1#(dd(y,ys),x) = [1] x + [2] >= [1] x + [1] = eq#(x,y) remove#1#(dd(y,ys),x) = [1] x + [2] >= [1] x + [2] = remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) = [1] x + [2] >= [1] x + [2] = remove#(x,ys) remove#2#(#true(),x,y,ys) = [1] x + [2] >= [1] x + [2] = remove#(x,ys) #and(#false(),#false()) = [0] >= [0] = #false() #and(#false(),#true()) = [0] >= [0] = #false() #and(#true(),#false()) = [0] >= [0] = #false() #and(#true(),#true()) = [0] >= [0] = #true() #eq(#0(),#0()) = [0] >= [0] = #true() #eq(#0(),#neg(y)) = [0] >= [0] = #false() #eq(#0(),#pos(y)) = [0] >= [0] = #false() #eq(#0(),#s(y)) = [0] >= [0] = #false() #eq(#neg(x),#0()) = [0] >= [0] = #false() #eq(#neg(x),#neg(y)) = [0] >= [0] = #eq(x,y) #eq(#neg(x),#pos(y)) = [0] >= [0] = #false() #eq(#pos(x),#0()) = [0] >= [0] = #false() #eq(#pos(x),#neg(y)) = [0] >= [0] = #false() #eq(#pos(x),#pos(y)) = [0] >= [0] = #eq(x,y) #eq(#s(x),#0()) = [0] >= [0] = #false() #eq(#s(x),#s(y)) = [0] >= [0] = #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) = [0] >= [0] = #false() #eq(nil(),dd(y'1,y'2)) = [0] >= [0] = #false() #eq(nil(),nil()) = [0] >= [0] = #true() #equal(x,y) = [0] >= [0] = #eq(x,y) and(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = #and(x,y) eq(l1,l2) = [0] >= [0] = eq#1(l1,l2) eq#1(dd(x,xs),l2) = [0] >= [0] = eq#3(l2,x,xs) eq#1(nil(),l2) = [0] >= [0] = eq#2(l2) eq#2(dd(y,ys)) = [0] >= [0] = #false() eq#2(nil()) = [0] >= [0] = #true() eq#3(dd(y,ys),x,xs) = [0] >= [0] = and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) = [0] >= [0] = #false() remove(x,l) = [1] l + [1] >= [1] l + [1] = remove#1(l,x) remove#1(dd(y,ys),x) = [1] y + [1] ys + [2] >= [1] y + [1] ys + [2] = remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) = [1] >= [0] = nil() remove#2(#false(),x,y,ys) = [1] y + [1] ys + [2] >= [1] y + [1] ys + [2] = dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) = [1] y + [1] ys + [2] >= [1] ys + [1] = remove(x,ys) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 7.b:1.b:1.a:5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: eq#(l1,l2) -> c_23(eq#1#(l1,l2)) eq#1#(dd(x,xs),l2) -> c_24(eq#3#(l2,x,xs)) eq#3#(dd(y,ys),x,xs) -> c_28(eq#(xs,ys)) nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/1,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 7.b:1.b:1.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) - Weak DPs: eq#(l1,l2) -> eq#1#(l1,l2) eq#1#(dd(x,xs),l2) -> eq#3#(l2,x,xs) eq#3#(dd(y,ys),x,xs) -> #equal#(x,y) eq#3#(dd(y,ys),x,xs) -> eq#(xs,ys) nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#and) = {1,2}, uargs(and) = {1,2}, uargs(dd) = {2}, uargs(remove#2) = {1}, uargs(nub#) = {1}, uargs(remove#2#) = {1}, uargs(c_10) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1,2}, uargs(c_21) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#and) = [1] x1 + [1] x2 + [0] p(#eq) = [0] p(#equal) = [0] p(#false) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(and) = [1] x1 + [1] x2 + [0] p(dd) = [1] x2 + [0] p(eq) = [0] p(eq#1) = [0] p(eq#2) = [0] p(eq#3) = [0] p(nil) = [0] p(nub) = [0] p(nub#1) = [0] p(remove) = [0] p(remove#1) = [0] p(remove#2) = [1] x1 + [0] p(#and#) = [0] p(#eq#) = [0] p(#equal#) = [1] p(and#) = [0] p(eq#) = [1] p(eq#1#) = [1] p(eq#2#) = [0] p(eq#3#) = [1] p(nub#) = [1] x1 + [4] p(nub#1#) = [4] p(remove#) = [1] p(remove#1#) = [1] p(remove#2#) = [1] x1 + [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [1] x1 + [2] p(c_15) = [0] p(c_16) = [1] x1 + [0] p(c_17) = [1] x1 + [1] x2 + [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [1] x1 + [0] p(c_22) = [0] p(c_23) = [0] p(c_24) = [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [0] p(c_28) = [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [0] p(c_32) = [0] p(c_33) = [0] p(c_34) = [0] p(c_35) = [0] p(c_36) = [0] p(c_37) = [0] Following rules are strictly oriented: #equal#(x,y) = [1] > [0] = c_21(#eq#(x,y)) Following rules are (at-least) weakly oriented: #eq#(#neg(x),#neg(y)) = [0] >= [0] = c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) = [0] >= [2] = c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) = [0] >= [0] = c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) eq#(l1,l2) = [1] >= [1] = eq#1#(l1,l2) eq#1#(dd(x,xs),l2) = [1] >= [1] = eq#3#(l2,x,xs) eq#3#(dd(y,ys),x,xs) = [1] >= [1] = #equal#(x,y) eq#3#(dd(y,ys),x,xs) = [1] >= [1] = eq#(xs,ys) nub#(l) = [1] l + [4] >= [4] = nub#1#(l) nub#1#(dd(x,xs)) = [4] >= [4] = nub#(remove(x,xs)) nub#1#(dd(x,xs)) = [4] >= [1] = remove#(x,xs) remove#(x,l) = [1] >= [1] = remove#1#(l,x) remove#1#(dd(y,ys),x) = [1] >= [1] = eq#(x,y) remove#1#(dd(y,ys),x) = [1] >= [1] = remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) = [1] >= [1] = remove#(x,ys) remove#2#(#true(),x,y,ys) = [1] >= [1] = remove#(x,ys) #and(#false(),#false()) = [0] >= [0] = #false() #and(#false(),#true()) = [0] >= [0] = #false() #and(#true(),#false()) = [0] >= [0] = #false() #and(#true(),#true()) = [0] >= [0] = #true() #eq(#0(),#0()) = [0] >= [0] = #true() #eq(#0(),#neg(y)) = [0] >= [0] = #false() #eq(#0(),#pos(y)) = [0] >= [0] = #false() #eq(#0(),#s(y)) = [0] >= [0] = #false() #eq(#neg(x),#0()) = [0] >= [0] = #false() #eq(#neg(x),#neg(y)) = [0] >= [0] = #eq(x,y) #eq(#neg(x),#pos(y)) = [0] >= [0] = #false() #eq(#pos(x),#0()) = [0] >= [0] = #false() #eq(#pos(x),#neg(y)) = [0] >= [0] = #false() #eq(#pos(x),#pos(y)) = [0] >= [0] = #eq(x,y) #eq(#s(x),#0()) = [0] >= [0] = #false() #eq(#s(x),#s(y)) = [0] >= [0] = #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) = [0] >= [0] = #false() #eq(nil(),dd(y'1,y'2)) = [0] >= [0] = #false() #eq(nil(),nil()) = [0] >= [0] = #true() #equal(x,y) = [0] >= [0] = #eq(x,y) and(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = #and(x,y) eq(l1,l2) = [0] >= [0] = eq#1(l1,l2) eq#1(dd(x,xs),l2) = [0] >= [0] = eq#3(l2,x,xs) eq#1(nil(),l2) = [0] >= [0] = eq#2(l2) eq#2(dd(y,ys)) = [0] >= [0] = #false() eq#2(nil()) = [0] >= [0] = #true() eq#3(dd(y,ys),x,xs) = [0] >= [0] = and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) = [0] >= [0] = #false() remove(x,l) = [0] >= [0] = remove#1(l,x) remove#1(dd(y,ys),x) = [0] >= [0] = remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) = [0] >= [0] = nil() remove#2(#false(),x,y,ys) = [0] >= [0] = dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) = [0] >= [0] = remove(x,ys) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 7.b:1.b:1.b:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) - Weak DPs: #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> eq#1#(l1,l2) eq#1#(dd(x,xs),l2) -> eq#3#(l2,x,xs) eq#3#(dd(y,ys),x,xs) -> #equal#(x,y) eq#3#(dd(y,ys),x,xs) -> eq#(xs,ys) nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#and) = {1,2}, uargs(and) = {1,2}, uargs(dd) = {2}, uargs(remove#2) = {1}, uargs(nub#) = {1}, uargs(remove#2#) = {1}, uargs(c_10) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1,2}, uargs(c_21) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#and) = [1] x1 + [1] x2 + [0] p(#eq) = [0] p(#equal) = [0] p(#false) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(and) = [1] x1 + [1] x2 + [0] p(dd) = [1] x1 + [1] x2 + [4] p(eq) = [0] p(eq#1) = [0] p(eq#2) = [0] p(eq#3) = [0] p(nil) = [0] p(nub) = [1] p(nub#1) = [1] x1 + [0] p(remove) = [1] x2 + [0] p(remove#1) = [1] x1 + [0] p(remove#2) = [1] x1 + [1] x3 + [1] x4 + [4] p(#and#) = [4] x1 + [2] p(#eq#) = [1] x1 + [0] p(#equal#) = [1] x1 + [0] p(and#) = [1] x1 + [1] x2 + [1] p(eq#) = [1] x1 + [0] p(eq#1#) = [1] x1 + [0] p(eq#2#) = [0] p(eq#3#) = [1] x2 + [1] x3 + [0] p(nub#) = [1] x1 + [5] p(nub#1#) = [1] x1 + [4] p(remove#) = [1] x1 + [1] x2 + [0] p(remove#1#) = [1] x1 + [1] x2 + [0] p(remove#2#) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] p(c_1) = [0] p(c_2) = [1] p(c_3) = [4] 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) = [1] x1 + [2] p(c_11) = [1] p(c_12) = [2] p(c_13) = [0] p(c_14) = [1] x1 + [1] p(c_15) = [4] p(c_16) = [1] x1 + [1] p(c_17) = [1] x1 + [1] x2 + [2] p(c_18) = [0] p(c_19) = [1] p(c_20) = [1] p(c_21) = [1] x1 + [0] p(c_22) = [0] p(c_23) = [1] x1 + [2] p(c_24) = [2] p(c_25) = [4] x1 + [1] p(c_26) = [1] p(c_27) = [2] p(c_28) = [1] x1 + [4] x2 + [0] p(c_29) = [0] p(c_30) = [0] p(c_31) = [1] x2 + [4] p(c_32) = [0] p(c_33) = [0] p(c_34) = [2] x1 + [2] x2 + [0] p(c_35) = [0] p(c_36) = [0] p(c_37) = [2] x1 + [1] Following rules are strictly oriented: #eq#(dd(x'1,x'2),dd(y'1,y'2)) = [1] x'1 + [1] x'2 + [4] > [1] x'1 + [1] x'2 + [2] = c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) Following rules are (at-least) weakly oriented: #eq#(#neg(x),#neg(y)) = [1] x + [0] >= [1] x + [2] = c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) = [1] x + [0] >= [1] x + [1] = c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) = [1] x + [0] >= [1] x + [1] = c_16(#eq#(x,y)) #equal#(x,y) = [1] x + [0] >= [1] x + [0] = c_21(#eq#(x,y)) eq#(l1,l2) = [1] l1 + [0] >= [1] l1 + [0] = eq#1#(l1,l2) eq#1#(dd(x,xs),l2) = [1] x + [1] xs + [4] >= [1] x + [1] xs + [0] = eq#3#(l2,x,xs) eq#3#(dd(y,ys),x,xs) = [1] x + [1] xs + [0] >= [1] x + [0] = #equal#(x,y) eq#3#(dd(y,ys),x,xs) = [1] x + [1] xs + [0] >= [1] xs + [0] = eq#(xs,ys) nub#(l) = [1] l + [5] >= [1] l + [4] = nub#1#(l) nub#1#(dd(x,xs)) = [1] x + [1] xs + [8] >= [1] xs + [5] = nub#(remove(x,xs)) nub#1#(dd(x,xs)) = [1] x + [1] xs + [8] >= [1] x + [1] xs + [0] = remove#(x,xs) remove#(x,l) = [1] l + [1] x + [0] >= [1] l + [1] x + [0] = remove#1#(l,x) remove#1#(dd(y,ys),x) = [1] x + [1] y + [1] ys + [4] >= [1] x + [0] = eq#(x,y) remove#1#(dd(y,ys),x) = [1] x + [1] y + [1] ys + [4] >= [1] x + [1] y + [1] ys + [1] = remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) = [1] x + [1] y + [1] ys + [1] >= [1] x + [1] ys + [0] = remove#(x,ys) remove#2#(#true(),x,y,ys) = [1] x + [1] y + [1] ys + [1] >= [1] x + [1] ys + [0] = remove#(x,ys) #and(#false(),#false()) = [0] >= [0] = #false() #and(#false(),#true()) = [0] >= [0] = #false() #and(#true(),#false()) = [0] >= [0] = #false() #and(#true(),#true()) = [0] >= [0] = #true() #eq(#0(),#0()) = [0] >= [0] = #true() #eq(#0(),#neg(y)) = [0] >= [0] = #false() #eq(#0(),#pos(y)) = [0] >= [0] = #false() #eq(#0(),#s(y)) = [0] >= [0] = #false() #eq(#neg(x),#0()) = [0] >= [0] = #false() #eq(#neg(x),#neg(y)) = [0] >= [0] = #eq(x,y) #eq(#neg(x),#pos(y)) = [0] >= [0] = #false() #eq(#pos(x),#0()) = [0] >= [0] = #false() #eq(#pos(x),#neg(y)) = [0] >= [0] = #false() #eq(#pos(x),#pos(y)) = [0] >= [0] = #eq(x,y) #eq(#s(x),#0()) = [0] >= [0] = #false() #eq(#s(x),#s(y)) = [0] >= [0] = #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) = [0] >= [0] = #false() #eq(nil(),dd(y'1,y'2)) = [0] >= [0] = #false() #eq(nil(),nil()) = [0] >= [0] = #true() #equal(x,y) = [0] >= [0] = #eq(x,y) and(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = #and(x,y) eq(l1,l2) = [0] >= [0] = eq#1(l1,l2) eq#1(dd(x,xs),l2) = [0] >= [0] = eq#3(l2,x,xs) eq#1(nil(),l2) = [0] >= [0] = eq#2(l2) eq#2(dd(y,ys)) = [0] >= [0] = #false() eq#2(nil()) = [0] >= [0] = #true() eq#3(dd(y,ys),x,xs) = [0] >= [0] = and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) = [0] >= [0] = #false() remove(x,l) = [1] l + [0] >= [1] l + [0] = remove#1(l,x) remove#1(dd(y,ys),x) = [1] y + [1] ys + [4] >= [1] y + [1] ys + [4] = remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) = [0] >= [0] = nil() remove#2(#false(),x,y,ys) = [1] y + [1] ys + [4] >= [1] y + [1] ys + [4] = dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) = [1] y + [1] ys + [4] >= [1] ys + [0] = remove(x,ys) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 7.b:1.b:1.b:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) - Weak DPs: #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> eq#1#(l1,l2) eq#1#(dd(x,xs),l2) -> eq#3#(l2,x,xs) eq#3#(dd(y,ys),x,xs) -> #equal#(x,y) eq#3#(dd(y,ys),x,xs) -> eq#(xs,ys) nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#and) = {1,2}, uargs(and) = {1,2}, uargs(dd) = {2}, uargs(remove#2) = {1}, uargs(nub#) = {1}, uargs(remove#2#) = {1}, uargs(c_10) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1,2}, uargs(c_21) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#and) = [1] x1 + [1] x2 + [0] p(#eq) = [0] p(#equal) = [0] p(#false) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [5] p(#true) = [0] p(and) = [1] x1 + [1] x2 + [0] p(dd) = [1] x1 + [1] x2 + [0] p(eq) = [0] p(eq#1) = [0] p(eq#2) = [0] p(eq#3) = [0] p(nil) = [0] p(nub) = [0] p(nub#1) = [0] p(remove) = [1] x2 + [0] p(remove#1) = [1] x1 + [0] p(remove#2) = [1] x1 + [1] x3 + [1] x4 + [0] p(#and#) = [0] p(#eq#) = [1] x1 + [0] p(#equal#) = [1] x1 + [0] p(and#) = [4] x2 + [0] p(eq#) = [1] x1 + [0] p(eq#1#) = [1] x1 + [0] p(eq#2#) = [0] p(eq#3#) = [1] x2 + [1] x3 + [0] p(nub#) = [1] x1 + [2] p(nub#1#) = [1] x1 + [2] p(remove#) = [1] x1 + [0] p(remove#1#) = [1] x2 + [0] p(remove#2#) = [1] x1 + [1] x2 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [2] p(c_10) = [1] x1 + [2] p(c_11) = [1] p(c_12) = [1] p(c_13) = [1] p(c_14) = [1] x1 + [1] p(c_15) = [0] p(c_16) = [1] x1 + [0] p(c_17) = [1] x1 + [1] x2 + [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [1] p(c_21) = [1] x1 + [0] p(c_22) = [1] x1 + [0] p(c_23) = [2] p(c_24) = [4] x1 + [0] p(c_25) = [0] p(c_26) = [0] p(c_27) = [1] p(c_28) = [1] x1 + [1] x2 + [0] p(c_29) = [1] p(c_30) = [2] x1 + [1] p(c_31) = [1] x1 + [4] x2 + [0] p(c_32) = [1] p(c_33) = [4] x1 + [4] p(c_34) = [4] x1 + [0] p(c_35) = [0] p(c_36) = [4] p(c_37) = [2] x1 + [0] Following rules are strictly oriented: #eq#(#s(x),#s(y)) = [1] x + [5] > [1] x + [0] = c_16(#eq#(x,y)) Following rules are (at-least) weakly oriented: #eq#(#neg(x),#neg(y)) = [1] x + [0] >= [1] x + [2] = c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) = [1] x + [0] >= [1] x + [1] = c_14(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) = [1] x'1 + [1] x'2 + [0] >= [1] x'1 + [1] x'2 + [0] = c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) = [1] x + [0] >= [1] x + [0] = c_21(#eq#(x,y)) eq#(l1,l2) = [1] l1 + [0] >= [1] l1 + [0] = eq#1#(l1,l2) eq#1#(dd(x,xs),l2) = [1] x + [1] xs + [0] >= [1] x + [1] xs + [0] = eq#3#(l2,x,xs) eq#3#(dd(y,ys),x,xs) = [1] x + [1] xs + [0] >= [1] x + [0] = #equal#(x,y) eq#3#(dd(y,ys),x,xs) = [1] x + [1] xs + [0] >= [1] xs + [0] = eq#(xs,ys) nub#(l) = [1] l + [2] >= [1] l + [2] = nub#1#(l) nub#1#(dd(x,xs)) = [1] x + [1] xs + [2] >= [1] xs + [2] = nub#(remove(x,xs)) nub#1#(dd(x,xs)) = [1] x + [1] xs + [2] >= [1] x + [0] = remove#(x,xs) remove#(x,l) = [1] x + [0] >= [1] x + [0] = remove#1#(l,x) remove#1#(dd(y,ys),x) = [1] x + [0] >= [1] x + [0] = eq#(x,y) remove#1#(dd(y,ys),x) = [1] x + [0] >= [1] x + [0] = remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) = [1] x + [0] >= [1] x + [0] = remove#(x,ys) remove#2#(#true(),x,y,ys) = [1] x + [0] >= [1] x + [0] = remove#(x,ys) #and(#false(),#false()) = [0] >= [0] = #false() #and(#false(),#true()) = [0] >= [0] = #false() #and(#true(),#false()) = [0] >= [0] = #false() #and(#true(),#true()) = [0] >= [0] = #true() #eq(#0(),#0()) = [0] >= [0] = #true() #eq(#0(),#neg(y)) = [0] >= [0] = #false() #eq(#0(),#pos(y)) = [0] >= [0] = #false() #eq(#0(),#s(y)) = [0] >= [0] = #false() #eq(#neg(x),#0()) = [0] >= [0] = #false() #eq(#neg(x),#neg(y)) = [0] >= [0] = #eq(x,y) #eq(#neg(x),#pos(y)) = [0] >= [0] = #false() #eq(#pos(x),#0()) = [0] >= [0] = #false() #eq(#pos(x),#neg(y)) = [0] >= [0] = #false() #eq(#pos(x),#pos(y)) = [0] >= [0] = #eq(x,y) #eq(#s(x),#0()) = [0] >= [0] = #false() #eq(#s(x),#s(y)) = [0] >= [0] = #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) = [0] >= [0] = #false() #eq(nil(),dd(y'1,y'2)) = [0] >= [0] = #false() #eq(nil(),nil()) = [0] >= [0] = #true() #equal(x,y) = [0] >= [0] = #eq(x,y) and(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = #and(x,y) eq(l1,l2) = [0] >= [0] = eq#1(l1,l2) eq#1(dd(x,xs),l2) = [0] >= [0] = eq#3(l2,x,xs) eq#1(nil(),l2) = [0] >= [0] = eq#2(l2) eq#2(dd(y,ys)) = [0] >= [0] = #false() eq#2(nil()) = [0] >= [0] = #true() eq#3(dd(y,ys),x,xs) = [0] >= [0] = and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) = [0] >= [0] = #false() remove(x,l) = [1] l + [0] >= [1] l + [0] = remove#1(l,x) remove#1(dd(y,ys),x) = [1] y + [1] ys + [0] >= [1] y + [1] ys + [0] = remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) = [0] >= [0] = nil() remove#2(#false(),x,y,ys) = [1] y + [1] ys + [0] >= [1] y + [1] ys + [0] = dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) = [1] y + [1] ys + [0] >= [1] ys + [0] = remove(x,ys) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 7.b:1.b:1.b:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) - Weak DPs: #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> eq#1#(l1,l2) eq#1#(dd(x,xs),l2) -> eq#3#(l2,x,xs) eq#3#(dd(y,ys),x,xs) -> #equal#(x,y) eq#3#(dd(y,ys),x,xs) -> eq#(xs,ys) nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#and) = {1,2}, uargs(and) = {1,2}, uargs(dd) = {2}, uargs(remove#2) = {1}, uargs(nub#) = {1}, uargs(remove#2#) = {1}, uargs(c_10) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1,2}, uargs(c_21) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#and) = [1] x1 + [1] x2 + [0] p(#eq) = [0] p(#equal) = [0] p(#false) = [0] p(#neg) = [1] x1 + [0] p(#pos) = [1] x1 + [1] p(#s) = [1] x1 + [2] p(#true) = [0] p(and) = [1] x1 + [1] x2 + [0] p(dd) = [1] x1 + [1] x2 + [0] p(eq) = [0] p(eq#1) = [0] p(eq#2) = [0] p(eq#3) = [0] p(nil) = [0] p(nub) = [4] x1 + [0] p(nub#1) = [1] x1 + [1] p(remove) = [1] x2 + [0] p(remove#1) = [1] x1 + [0] p(remove#2) = [1] x1 + [1] x3 + [1] x4 + [0] p(#and#) = [2] x2 + [0] p(#eq#) = [1] x1 + [0] p(#equal#) = [1] x1 + [0] p(and#) = [1] x1 + [2] x2 + [1] p(eq#) = [1] x1 + [0] p(eq#1#) = [1] x1 + [0] p(eq#2#) = [2] x1 + [0] p(eq#3#) = [1] x2 + [1] x3 + [0] p(nub#) = [1] x1 + [1] p(nub#1#) = [1] x1 + [1] p(remove#) = [1] x1 + [1] p(remove#1#) = [1] x2 + [1] p(remove#2#) = [1] x1 + [1] x2 + [1] p(c_1) = [1] p(c_2) = [1] p(c_3) = [0] p(c_4) = [1] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] p(c_8) = [1] p(c_9) = [0] p(c_10) = [1] x1 + [3] p(c_11) = [2] p(c_12) = [1] p(c_13) = [1] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [1] p(c_17) = [1] x1 + [1] x2 + [0] p(c_18) = [0] p(c_19) = [2] p(c_20) = [0] p(c_21) = [1] x1 + [0] p(c_22) = [4] p(c_23) = [1] x1 + [0] p(c_24) = [1] x1 + [1] p(c_25) = [1] x1 + [2] p(c_26) = [4] p(c_27) = [1] p(c_28) = [1] x2 + [1] p(c_29) = [0] p(c_30) = [1] x1 + [1] p(c_31) = [4] x1 + [2] x2 + [4] p(c_32) = [2] p(c_33) = [4] x1 + [0] p(c_34) = [1] x2 + [0] p(c_35) = [1] p(c_36) = [1] p(c_37) = [4] x1 + [1] Following rules are strictly oriented: #eq#(#pos(x),#pos(y)) = [1] x + [1] > [1] x + [0] = c_14(#eq#(x,y)) Following rules are (at-least) weakly oriented: #eq#(#neg(x),#neg(y)) = [1] x + [0] >= [1] x + [3] = c_10(#eq#(x,y)) #eq#(#s(x),#s(y)) = [1] x + [2] >= [1] x + [1] = c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) = [1] x'1 + [1] x'2 + [0] >= [1] x'1 + [1] x'2 + [0] = c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) = [1] x + [0] >= [1] x + [0] = c_21(#eq#(x,y)) eq#(l1,l2) = [1] l1 + [0] >= [1] l1 + [0] = eq#1#(l1,l2) eq#1#(dd(x,xs),l2) = [1] x + [1] xs + [0] >= [1] x + [1] xs + [0] = eq#3#(l2,x,xs) eq#3#(dd(y,ys),x,xs) = [1] x + [1] xs + [0] >= [1] x + [0] = #equal#(x,y) eq#3#(dd(y,ys),x,xs) = [1] x + [1] xs + [0] >= [1] xs + [0] = eq#(xs,ys) nub#(l) = [1] l + [1] >= [1] l + [1] = nub#1#(l) nub#1#(dd(x,xs)) = [1] x + [1] xs + [1] >= [1] xs + [1] = nub#(remove(x,xs)) nub#1#(dd(x,xs)) = [1] x + [1] xs + [1] >= [1] x + [1] = remove#(x,xs) remove#(x,l) = [1] x + [1] >= [1] x + [1] = remove#1#(l,x) remove#1#(dd(y,ys),x) = [1] x + [1] >= [1] x + [0] = eq#(x,y) remove#1#(dd(y,ys),x) = [1] x + [1] >= [1] x + [1] = remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) = [1] x + [1] >= [1] x + [1] = remove#(x,ys) remove#2#(#true(),x,y,ys) = [1] x + [1] >= [1] x + [1] = remove#(x,ys) #and(#false(),#false()) = [0] >= [0] = #false() #and(#false(),#true()) = [0] >= [0] = #false() #and(#true(),#false()) = [0] >= [0] = #false() #and(#true(),#true()) = [0] >= [0] = #true() #eq(#0(),#0()) = [0] >= [0] = #true() #eq(#0(),#neg(y)) = [0] >= [0] = #false() #eq(#0(),#pos(y)) = [0] >= [0] = #false() #eq(#0(),#s(y)) = [0] >= [0] = #false() #eq(#neg(x),#0()) = [0] >= [0] = #false() #eq(#neg(x),#neg(y)) = [0] >= [0] = #eq(x,y) #eq(#neg(x),#pos(y)) = [0] >= [0] = #false() #eq(#pos(x),#0()) = [0] >= [0] = #false() #eq(#pos(x),#neg(y)) = [0] >= [0] = #false() #eq(#pos(x),#pos(y)) = [0] >= [0] = #eq(x,y) #eq(#s(x),#0()) = [0] >= [0] = #false() #eq(#s(x),#s(y)) = [0] >= [0] = #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) = [0] >= [0] = #false() #eq(nil(),dd(y'1,y'2)) = [0] >= [0] = #false() #eq(nil(),nil()) = [0] >= [0] = #true() #equal(x,y) = [0] >= [0] = #eq(x,y) and(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = #and(x,y) eq(l1,l2) = [0] >= [0] = eq#1(l1,l2) eq#1(dd(x,xs),l2) = [0] >= [0] = eq#3(l2,x,xs) eq#1(nil(),l2) = [0] >= [0] = eq#2(l2) eq#2(dd(y,ys)) = [0] >= [0] = #false() eq#2(nil()) = [0] >= [0] = #true() eq#3(dd(y,ys),x,xs) = [0] >= [0] = and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) = [0] >= [0] = #false() remove(x,l) = [1] l + [0] >= [1] l + [0] = remove#1(l,x) remove#1(dd(y,ys),x) = [1] y + [1] ys + [0] >= [1] y + [1] ys + [0] = remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) = [0] >= [0] = nil() remove#2(#false(),x,y,ys) = [1] y + [1] ys + [0] >= [1] y + [1] ys + [0] = dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) = [1] y + [1] ys + [0] >= [1] ys + [0] = remove(x,ys) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 7.b:1.b:1.b:5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) - Weak DPs: #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> eq#1#(l1,l2) eq#1#(dd(x,xs),l2) -> eq#3#(l2,x,xs) eq#3#(dd(y,ys),x,xs) -> #equal#(x,y) eq#3#(dd(y,ys),x,xs) -> eq#(xs,ys) nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(#and) = {1,2}, uargs(and) = {1,2}, uargs(dd) = {2}, uargs(remove#2) = {1}, uargs(nub#) = {1}, uargs(remove#2#) = {1}, uargs(c_10) = {1}, uargs(c_14) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1,2}, uargs(c_21) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(#0) = [0] p(#and) = [1] x1 + [1] x2 + [0] p(#eq) = [0] p(#equal) = [0] p(#false) = [0] p(#neg) = [1] x1 + [1] p(#pos) = [1] x1 + [0] p(#s) = [1] x1 + [0] p(#true) = [0] p(and) = [1] x1 + [1] x2 + [0] p(dd) = [1] x1 + [1] x2 + [0] p(eq) = [0] p(eq#1) = [0] p(eq#2) = [0] p(eq#3) = [0] p(nil) = [0] p(nub) = [0] p(nub#1) = [0] p(remove) = [1] x2 + [0] p(remove#1) = [1] x1 + [0] p(remove#2) = [1] x1 + [1] x3 + [1] x4 + [0] p(#and#) = [0] p(#eq#) = [1] x2 + [0] p(#equal#) = [1] x2 + [0] p(and#) = [0] p(eq#) = [1] x2 + [0] p(eq#1#) = [1] x2 + [0] p(eq#2#) = [0] p(eq#3#) = [1] x1 + [0] p(nub#) = [1] x1 + [0] p(nub#1#) = [1] x1 + [0] p(remove#) = [1] x2 + [0] p(remove#1#) = [1] x1 + [0] p(remove#2#) = [1] x1 + [1] x3 + [1] x4 + [0] p(c_1) = [0] p(c_2) = [4] p(c_3) = [1] p(c_4) = [2] p(c_5) = [0] p(c_6) = [1] p(c_7) = [2] p(c_8) = [1] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [2] p(c_12) = [1] p(c_13) = [2] p(c_14) = [1] x1 + [0] p(c_15) = [1] p(c_16) = [1] x1 + [0] p(c_17) = [1] x1 + [1] x2 + [0] p(c_18) = [2] p(c_19) = [1] p(c_20) = [0] p(c_21) = [1] x1 + [0] p(c_22) = [1] p(c_23) = [0] p(c_24) = [1] p(c_25) = [1] x1 + [0] p(c_26) = [1] p(c_27) = [0] p(c_28) = [1] x2 + [0] p(c_29) = [4] p(c_30) = [1] x1 + [1] p(c_31) = [4] x2 + [0] p(c_32) = [1] p(c_33) = [4] x1 + [0] p(c_34) = [2] p(c_35) = [4] p(c_36) = [4] x1 + [0] p(c_37) = [1] Following rules are strictly oriented: #eq#(#neg(x),#neg(y)) = [1] y + [1] > [1] y + [0] = c_10(#eq#(x,y)) Following rules are (at-least) weakly oriented: #eq#(#pos(x),#pos(y)) = [1] y + [0] >= [1] y + [0] = c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) = [1] y + [0] >= [1] y + [0] = c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) = [1] y'1 + [1] y'2 + [0] >= [1] y'1 + [1] y'2 + [0] = c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) = [1] y + [0] >= [1] y + [0] = c_21(#eq#(x,y)) eq#(l1,l2) = [1] l2 + [0] >= [1] l2 + [0] = eq#1#(l1,l2) eq#1#(dd(x,xs),l2) = [1] l2 + [0] >= [1] l2 + [0] = eq#3#(l2,x,xs) eq#3#(dd(y,ys),x,xs) = [1] y + [1] ys + [0] >= [1] y + [0] = #equal#(x,y) eq#3#(dd(y,ys),x,xs) = [1] y + [1] ys + [0] >= [1] ys + [0] = eq#(xs,ys) nub#(l) = [1] l + [0] >= [1] l + [0] = nub#1#(l) nub#1#(dd(x,xs)) = [1] x + [1] xs + [0] >= [1] xs + [0] = nub#(remove(x,xs)) nub#1#(dd(x,xs)) = [1] x + [1] xs + [0] >= [1] xs + [0] = remove#(x,xs) remove#(x,l) = [1] l + [0] >= [1] l + [0] = remove#1#(l,x) remove#1#(dd(y,ys),x) = [1] y + [1] ys + [0] >= [1] y + [0] = eq#(x,y) remove#1#(dd(y,ys),x) = [1] y + [1] ys + [0] >= [1] y + [1] ys + [0] = remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) = [1] y + [1] ys + [0] >= [1] ys + [0] = remove#(x,ys) remove#2#(#true(),x,y,ys) = [1] y + [1] ys + [0] >= [1] ys + [0] = remove#(x,ys) #and(#false(),#false()) = [0] >= [0] = #false() #and(#false(),#true()) = [0] >= [0] = #false() #and(#true(),#false()) = [0] >= [0] = #false() #and(#true(),#true()) = [0] >= [0] = #true() #eq(#0(),#0()) = [0] >= [0] = #true() #eq(#0(),#neg(y)) = [0] >= [0] = #false() #eq(#0(),#pos(y)) = [0] >= [0] = #false() #eq(#0(),#s(y)) = [0] >= [0] = #false() #eq(#neg(x),#0()) = [0] >= [0] = #false() #eq(#neg(x),#neg(y)) = [0] >= [0] = #eq(x,y) #eq(#neg(x),#pos(y)) = [0] >= [0] = #false() #eq(#pos(x),#0()) = [0] >= [0] = #false() #eq(#pos(x),#neg(y)) = [0] >= [0] = #false() #eq(#pos(x),#pos(y)) = [0] >= [0] = #eq(x,y) #eq(#s(x),#0()) = [0] >= [0] = #false() #eq(#s(x),#s(y)) = [0] >= [0] = #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) = [0] >= [0] = #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) = [0] >= [0] = #false() #eq(nil(),dd(y'1,y'2)) = [0] >= [0] = #false() #eq(nil(),nil()) = [0] >= [0] = #true() #equal(x,y) = [0] >= [0] = #eq(x,y) and(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = #and(x,y) eq(l1,l2) = [0] >= [0] = eq#1(l1,l2) eq#1(dd(x,xs),l2) = [0] >= [0] = eq#3(l2,x,xs) eq#1(nil(),l2) = [0] >= [0] = eq#2(l2) eq#2(dd(y,ys)) = [0] >= [0] = #false() eq#2(nil()) = [0] >= [0] = #true() eq#3(dd(y,ys),x,xs) = [0] >= [0] = and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) = [0] >= [0] = #false() remove(x,l) = [1] l + [0] >= [1] l + [0] = remove#1(l,x) remove#1(dd(y,ys),x) = [1] y + [1] ys + [0] >= [1] y + [1] ys + [0] = remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) = [0] >= [0] = nil() remove#2(#false(),x,y,ys) = [1] y + [1] ys + [0] >= [1] y + [1] ys + [0] = dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) = [1] y + [1] ys + [0] >= [1] ys + [0] = remove(x,ys) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 7.b:1.b:1.b:6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: #eq#(#neg(x),#neg(y)) -> c_10(#eq#(x,y)) #eq#(#pos(x),#pos(y)) -> c_14(#eq#(x,y)) #eq#(#s(x),#s(y)) -> c_16(#eq#(x,y)) #eq#(dd(x'1,x'2),dd(y'1,y'2)) -> c_17(#eq#(x'1,y'1),#eq#(x'2,y'2)) #equal#(x,y) -> c_21(#eq#(x,y)) eq#(l1,l2) -> eq#1#(l1,l2) eq#1#(dd(x,xs),l2) -> eq#3#(l2,x,xs) eq#3#(dd(y,ys),x,xs) -> #equal#(x,y) eq#3#(dd(y,ys),x,xs) -> eq#(xs,ys) nub#(l) -> nub#1#(l) nub#1#(dd(x,xs)) -> nub#(remove(x,xs)) nub#1#(dd(x,xs)) -> remove#(x,xs) remove#(x,l) -> remove#1#(l,x) remove#1#(dd(y,ys),x) -> eq#(x,y) remove#1#(dd(y,ys),x) -> remove#2#(eq(x,y),x,y,ys) remove#2#(#false(),x,y,ys) -> remove#(x,ys) remove#2#(#true(),x,y,ys) -> remove#(x,ys) - Weak TRS: #and(#false(),#false()) -> #false() #and(#false(),#true()) -> #false() #and(#true(),#false()) -> #false() #and(#true(),#true()) -> #true() #eq(#0(),#0()) -> #true() #eq(#0(),#neg(y)) -> #false() #eq(#0(),#pos(y)) -> #false() #eq(#0(),#s(y)) -> #false() #eq(#neg(x),#0()) -> #false() #eq(#neg(x),#neg(y)) -> #eq(x,y) #eq(#neg(x),#pos(y)) -> #false() #eq(#pos(x),#0()) -> #false() #eq(#pos(x),#neg(y)) -> #false() #eq(#pos(x),#pos(y)) -> #eq(x,y) #eq(#s(x),#0()) -> #false() #eq(#s(x),#s(y)) -> #eq(x,y) #eq(dd(x'1,x'2),dd(y'1,y'2)) -> #and(#eq(x'1,y'1),#eq(x'2,y'2)) #eq(dd(x'1,x'2),nil()) -> #false() #eq(nil(),dd(y'1,y'2)) -> #false() #eq(nil(),nil()) -> #true() #equal(x,y) -> #eq(x,y) and(x,y) -> #and(x,y) eq(l1,l2) -> eq#1(l1,l2) eq#1(dd(x,xs),l2) -> eq#3(l2,x,xs) eq#1(nil(),l2) -> eq#2(l2) eq#2(dd(y,ys)) -> #false() eq#2(nil()) -> #true() eq#3(dd(y,ys),x,xs) -> and(#equal(x,y),eq(xs,ys)) eq#3(nil(),x,xs) -> #false() remove(x,l) -> remove#1(l,x) remove#1(dd(y,ys),x) -> remove#2(eq(x,y),x,y,ys) remove#1(nil(),x) -> nil() remove#2(#false(),x,y,ys) -> dd(y,remove(x,ys)) remove#2(#true(),x,y,ys) -> remove(x,ys) - Signature: {#and/2,#eq/2,#equal/2,and/2,eq/2,eq#1/2,eq#2/1,eq#3/3,nub/1,nub#1/1,remove/2,remove#1/2,remove#2/4,#and#/2 ,#eq#/2,#equal#/2,and#/2,eq#/2,eq#1#/2,eq#2#/1,eq#3#/3,nub#/1,nub#1#/1,remove#/2,remove#1#/2 ,remove#2#/4} / {#0/0,#false/0,#neg/1,#pos/1,#s/1,#true/0,dd/2,nil/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0,c_6/0 ,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1,c_15/0,c_16/1,c_17/2,c_18/0,c_19/0,c_20/0,c_21/1 ,c_22/1,c_23/1,c_24/1,c_25/1,c_26/0,c_27/0,c_28/2,c_29/0,c_30/1,c_31/2,c_32/0,c_33/1,c_34/2,c_35/0,c_36/1 ,c_37/1} - Obligation: innermost runtime complexity wrt. defined symbols {#and#,#eq#,#equal#,and#,eq#,eq#1#,eq#2#,eq#3#,nub#,nub#1# ,remove#,remove#1#,remove#2#} and constructors {#0,#false,#neg,#pos,#s,#true,dd,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^4))