WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() sort(cons(n,x)) -> cons(min(cons(n,x)),sort(replace(min(cons(n,x)),n,x))) sort(nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,if_min,if_replace,le,min,replace ,sort} and constructors {0,cons,false,nil,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() sort(cons(n,x)) -> cons(min(cons(n,x)),sort(replace(min(cons(n,x)),n,x))) sort(nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,if_min,if_replace,le,min,replace ,sort} and constructors {0,cons,false,nil,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: eq(x,y){x -> s(x),y -> s(y)} = eq(s(x),s(y)) ->^+ eq(x,y) = C[eq(x,y) = eq(x,y){}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() sort(cons(n,x)) -> cons(min(cons(n,x)),sort(replace(min(cons(n,x)),n,x))) sort(nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,if_min,if_replace,le,min,replace ,sort} and constructors {0,cons,false,nil,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs eq#(0(),0()) -> c_1() eq#(0(),s(m)) -> c_2() eq#(s(n),0()) -> c_3() eq#(s(n),s(m)) -> c_4(eq#(n,m)) if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) if_replace#(true(),n,m,cons(k,x)) -> c_8() le#(0(),m) -> c_9() le#(s(n),0()) -> c_10() le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) min#(cons(0(),nil())) -> c_13() min#(cons(s(n),nil())) -> c_14() replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) replace#(n,m,nil()) -> c_16() sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) sort#(nil()) -> c_18() Weak DPs and mark the set of starting terms. ** Step 1.b:2: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(0(),0()) -> c_1() eq#(0(),s(m)) -> c_2() eq#(s(n),0()) -> c_3() eq#(s(n),s(m)) -> c_4(eq#(n,m)) if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) if_replace#(true(),n,m,cons(k,x)) -> c_8() le#(0(),m) -> c_9() le#(s(n),0()) -> c_10() le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) min#(cons(0(),nil())) -> c_13() min#(cons(s(n),nil())) -> c_14() replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) replace#(n,m,nil()) -> c_16() sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) sort#(nil()) -> c_18() - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() sort(cons(n,x)) -> cons(min(cons(n,x)),sort(replace(min(cons(n,x)),n,x))) sort(nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() eq#(0(),0()) -> c_1() eq#(0(),s(m)) -> c_2() eq#(s(n),0()) -> c_3() eq#(s(n),s(m)) -> c_4(eq#(n,m)) if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) if_replace#(true(),n,m,cons(k,x)) -> c_8() le#(0(),m) -> c_9() le#(s(n),0()) -> c_10() le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) min#(cons(0(),nil())) -> c_13() min#(cons(s(n),nil())) -> c_14() replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) replace#(n,m,nil()) -> c_16() sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) sort#(nil()) -> c_18() ** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(0(),0()) -> c_1() eq#(0(),s(m)) -> c_2() eq#(s(n),0()) -> c_3() eq#(s(n),s(m)) -> c_4(eq#(n,m)) if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) if_replace#(true(),n,m,cons(k,x)) -> c_8() le#(0(),m) -> c_9() le#(s(n),0()) -> c_10() le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) min#(cons(0(),nil())) -> c_13() min#(cons(s(n),nil())) -> c_14() replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) replace#(n,m,nil()) -> c_16() sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) sort#(nil()) -> c_18() - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,8,9,10,13,14,16,18} by application of Pre({1,2,3,8,9,10,13,14,16,18}) = {4,5,6,7,11,12,15,17}. Here rules are labelled as follows: 1: eq#(0(),0()) -> c_1() 2: eq#(0(),s(m)) -> c_2() 3: eq#(s(n),0()) -> c_3() 4: eq#(s(n),s(m)) -> c_4(eq#(n,m)) 5: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) 6: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) 7: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) 8: if_replace#(true(),n,m,cons(k,x)) -> c_8() 9: le#(0(),m) -> c_9() 10: le#(s(n),0()) -> c_10() 11: le#(s(n),s(m)) -> c_11(le#(n,m)) 12: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) 13: min#(cons(0(),nil())) -> c_13() 14: min#(cons(s(n),nil())) -> c_14() 15: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) 16: replace#(n,m,nil()) -> c_16() 17: sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) 18: sort#(nil()) -> c_18() ** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(s(n),s(m)) -> c_4(eq#(n,m)) if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak DPs: eq#(0(),0()) -> c_1() eq#(0(),s(m)) -> c_2() eq#(s(n),0()) -> c_3() if_replace#(true(),n,m,cons(k,x)) -> c_8() le#(0(),m) -> c_9() le#(s(n),0()) -> c_10() min#(cons(0(),nil())) -> c_13() min#(cons(s(n),nil())) -> c_14() replace#(n,m,nil()) -> c_16() sort#(nil()) -> c_18() - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:eq#(s(n),s(m)) -> c_4(eq#(n,m)) -->_1 eq#(s(n),0()) -> c_3():11 -->_1 eq#(0(),s(m)) -> c_2():10 -->_1 eq#(0(),0()) -> c_1():9 -->_1 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1 2:S:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6 -->_1 min#(cons(s(n),nil())) -> c_14():16 -->_1 min#(cons(0(),nil())) -> c_13():15 3:S:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6 -->_1 min#(cons(s(n),nil())) -> c_14():16 -->_1 min#(cons(0(),nil())) -> c_13():15 4:S:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):7 -->_1 replace#(n,m,nil()) -> c_16():17 5:S:le#(s(n),s(m)) -> c_11(le#(n,m)) -->_1 le#(s(n),0()) -> c_10():14 -->_1 le#(0(),m) -> c_9():13 -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):5 6:S:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) -->_2 le#(s(n),0()) -> c_10():14 -->_2 le#(0(),m) -> c_9():13 -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):5 -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):3 -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):2 7:S:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) -->_1 if_replace#(true(),n,m,cons(k,x)) -> c_8():12 -->_2 eq#(s(n),0()) -> c_3():11 -->_2 eq#(0(),s(m)) -> c_2():10 -->_2 eq#(0(),0()) -> c_1():9 -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):4 -->_2 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1 8:S:sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) -->_2 sort#(nil()) -> c_18():18 -->_3 replace#(n,m,nil()) -> c_16():17 -->_4 min#(cons(s(n),nil())) -> c_14():16 -->_1 min#(cons(s(n),nil())) -> c_14():16 -->_4 min#(cons(0(),nil())) -> c_13():15 -->_1 min#(cons(0(),nil())) -> c_13():15 -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))):8 -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):7 -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6 -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6 9:W:eq#(0(),0()) -> c_1() 10:W:eq#(0(),s(m)) -> c_2() 11:W:eq#(s(n),0()) -> c_3() 12:W:if_replace#(true(),n,m,cons(k,x)) -> c_8() 13:W:le#(0(),m) -> c_9() 14:W:le#(s(n),0()) -> c_10() 15:W:min#(cons(0(),nil())) -> c_13() 16:W:min#(cons(s(n),nil())) -> c_14() 17:W:replace#(n,m,nil()) -> c_16() 18:W:sort#(nil()) -> c_18() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 18: sort#(nil()) -> c_18() 17: replace#(n,m,nil()) -> c_16() 12: if_replace#(true(),n,m,cons(k,x)) -> c_8() 15: min#(cons(0(),nil())) -> c_13() 16: min#(cons(s(n),nil())) -> c_14() 13: le#(0(),m) -> c_9() 14: le#(s(n),0()) -> c_10() 9: eq#(0(),0()) -> c_1() 10: eq#(0(),s(m)) -> c_2() 11: eq#(s(n),0()) -> c_3() ** Step 1.b:5: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(s(n),s(m)) -> c_4(eq#(n,m)) if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: eq#(s(n),s(m)) -> c_4(eq#(n,m)) - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} Problem (S) - Strict DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak DPs: eq#(s(n),s(m)) -> c_4(eq#(n,m)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} *** Step 1.b:5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(s(n),s(m)) -> c_4(eq#(n,m)) - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:eq#(s(n),s(m)) -> c_4(eq#(n,m)) -->_1 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1 2:W:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6 3:W:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6 4:W:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):7 5:W:le#(s(n),s(m)) -> c_11(le#(n,m)) -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):5 6:W:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):2 -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):3 -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):5 7:W:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) -->_2 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1 -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):4 8:W:sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6 -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):6 -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):7 -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))):8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) 6: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) 3: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) 5: le#(s(n),s(m)) -> c_11(le#(n,m)) *** Step 1.b:5.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(s(n),s(m)) -> c_4(eq#(n,m)) - Weak DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:eq#(s(n),s(m)) -> c_4(eq#(n,m)) -->_1 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1 4:W:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):7 7:W:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) -->_2 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1 -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):4 8:W:sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):7 -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))):8 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) *** Step 1.b:5.a:3: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(s(n),s(m)) -> c_4(eq#(n,m)) - Weak DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) and a lower component eq#(s(n),s(m)) -> c_4(eq#(n,m)) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) Further, following extension rules are added to the lower component. sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) **** Step 1.b:5.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) The strictly oriented rules are moved into the weak component. ***** Step 1.b:5.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_17) = {1,2} Following symbols are considered usable: {if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#} TcT has computed the following interpretation: p(0) = [0] p(cons) = [1] x2 + [1] p(eq) = [0] p(false) = [0] p(if_min) = [4] x1 + [2] p(if_replace) = [1] x4 + [0] p(le) = [1] x1 + [0] p(min) = [5] p(nil) = [0] p(replace) = [1] x3 + [0] p(s) = [4] p(sort) = [1] x1 + [1] p(true) = [0] p(eq#) = [1] x2 + [0] p(if_min#) = [1] p(if_replace#) = [1] x1 + [4] x2 + [0] p(le#) = [1] x1 + [1] x2 + [0] p(min#) = [2] x1 + [0] p(replace#) = [0] p(sort#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] x1 + [1] p(c_5) = [4] p(c_6) = [1] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [1] p(c_10) = [4] p(c_11) = [1] x1 + [2] p(c_12) = [1] x2 + [0] p(c_13) = [1] p(c_14) = [1] p(c_15) = [2] x1 + [1] x2 + [0] p(c_16) = [1] p(c_17) = [1] x1 + [4] x2 + [0] p(c_18) = [1] Following rules are strictly oriented: sort#(cons(n,x)) = [1] x + [1] > [1] x + [0] = c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) Following rules are (at-least) weakly oriented: if_replace(false(),n,m,cons(k,x)) = [1] x + [1] >= [1] x + [1] = cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) = [1] x + [1] >= [1] x + [1] = cons(m,x) replace(n,m,cons(k,x)) = [1] x + [1] >= [1] x + [1] = if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) = [0] >= [0] = nil() ***** Step 1.b:5.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 1.b:5.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) -->_1 sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) ***** Step 1.b:5.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 1.b:5.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: eq#(s(n),s(m)) -> c_4(eq#(n,m)) - Weak DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: eq#(s(n),s(m)) -> c_4(eq#(n,m)) The strictly oriented rules are moved into the weak component. ***** Step 1.b:5.a:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: eq#(s(n),s(m)) -> c_4(eq#(n,m)) - Weak DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_4) = {1}, uargs(c_7) = {1}, uargs(c_15) = {1,2} Following symbols are considered usable: {eq,if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(cons) = [0 0 0] [0 1 0] [1] [0 1 0] x1 + [0 1 0] x2 + [1] [0 0 0] [0 0 0] [0] p(eq) = [0 1 0] [1] [0 0 0] x2 + [1] [0 0 0] [0] p(false) = [0] [0] [0] p(if_min) = [0 0 1] [0] [1 1 1] x1 + [1] [0 0 1] [0] p(if_replace) = [0 0 0] [0 1 0] [1 0 0] [1] [0 0 0] x1 + [0 1 0] x3 + [0 1 0] x4 + [1] [1 1 0] [0 0 0] [0 0 0] [1] p(le) = [1 1 1] [1 1 0] [0] [0 1 1] x1 + [1 1 0] x2 + [1] [1 0 0] [1 0 1] [0] p(min) = [0] [1] [0] p(nil) = [1] [0] [0] p(replace) = [0 1 0] [1 0 0] [1] [0 1 0] x2 + [0 1 0] x3 + [1] [0 0 0] [1 1 0] [1] p(s) = [0 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(sort) = [0] [0] [0] p(true) = [0] [1] [0] p(eq#) = [0 1 0] [0] [0 1 0] x2 + [0] [0 0 0] [0] p(if_min#) = [0] [0] [0] p(if_replace#) = [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 1] x2 + [0 0 1] x3 + [1 0 0] x4 + [0] [0 1 0] [0 0 0] [0 1 0] [0 0 0] [1] p(le#) = [0] [0] [0] p(min#) = [0] [0] [0] p(replace#) = [0 0 0] [0 1 0] [1] [0 1 0] x2 + [0 1 0] x3 + [1] [0 0 0] [0 0 0] [0] p(sort#) = [0 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(c_1) = [0] [0] [0] p(c_2) = [0] [0] [0] p(c_3) = [0] [0] [0] p(c_4) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(c_5) = [0] [0] [0] p(c_6) = [0] [0] [0] p(c_7) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] p(c_8) = [0] [0] [0] p(c_9) = [0] [0] [0] p(c_10) = [0] [0] [0] p(c_11) = [0] [0] [0] p(c_12) = [0] [0] [0] p(c_13) = [0] [0] [0] p(c_14) = [0] [0] [0] p(c_15) = [1 0 0] [1 0 0] [0] [0 0 1] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(c_16) = [0] [0] [0] p(c_17) = [0] [0] [0] p(c_18) = [0] [0] [0] Following rules are strictly oriented: eq#(s(n),s(m)) = [0 1 0] [1] [0 1 0] m + [1] [0 0 0] [0] > [0 1 0] [0] [0 1 0] m + [0] [0 0 0] [0] = c_4(eq#(n,m)) Following rules are (at-least) weakly oriented: if_replace#(false(),n,m,cons(k,x)) = [0 0 0] [0 0 0] [0 1 0] [1] [0 0 1] m + [0 0 1] n + [0 1 0] x + [1] [0 1 0] [0 0 0] [0 0 0] [1] >= [0 1 0] [1] [0 0 0] x + [1] [0 0 0] [1] = c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) = [0 1 0] [0 0 0] [0 1 0] [2] [0 1 0] k + [0 1 0] m + [0 1 0] x + [2] [0 0 0] [0 0 0] [0 0 0] [0] >= [0 1 0] [0 0 0] [0 1 0] [1] [0 0 0] k + [0 1 0] m + [0 0 0] x + [2] [0 0 0] [0 0 0] [0 0 0] [0] = c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) = [0 1 0] [0 1 0] [1] [0 1 0] n + [0 1 0] x + [1] [0 0 0] [0 0 0] [0] >= [0 0 0] [0 1 0] [1] [0 1 0] n + [0 1 0] x + [1] [0 0 0] [0 0 0] [0] = replace#(min(cons(n,x)),n,x) sort#(cons(n,x)) = [0 1 0] [0 1 0] [1] [0 1 0] n + [0 1 0] x + [1] [0 0 0] [0 0 0] [0] >= [0 1 0] [0 1 0] [1] [0 1 0] n + [0 1 0] x + [1] [0 0 0] [0 0 0] [0] = sort#(replace(min(cons(n,x)),n,x)) eq(0(),0()) = [1] [1] [0] >= [0] [1] [0] = true() eq(0(),s(m)) = [0 1 0] [2] [0 0 0] m + [1] [0 0 0] [0] >= [0] [0] [0] = false() eq(s(n),0()) = [1] [1] [0] >= [0] [0] [0] = false() eq(s(n),s(m)) = [0 1 0] [2] [0 0 0] m + [1] [0 0 0] [0] >= [0 1 0] [1] [0 0 0] m + [1] [0 0 0] [0] = eq(n,m) if_replace(false(),n,m,cons(k,x)) = [0 0 0] [0 1 0] [0 1 0] [2] [0 1 0] k + [0 1 0] m + [0 1 0] x + [2] [0 0 0] [0 0 0] [0 0 0] [1] >= [0 0 0] [0 1 0] [0 1 0] [2] [0 1 0] k + [0 1 0] m + [0 1 0] x + [2] [0 0 0] [0 0 0] [0 0 0] [0] = cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) = [0 0 0] [0 1 0] [0 1 0] [2] [0 1 0] k + [0 1 0] m + [0 1 0] x + [2] [0 0 0] [0 0 0] [0 0 0] [2] >= [0 0 0] [0 1 0] [1] [0 1 0] m + [0 1 0] x + [1] [0 0 0] [0 0 0] [0] = cons(m,x) replace(n,m,cons(k,x)) = [0 0 0] [0 1 0] [0 1 0] [2] [0 1 0] k + [0 1 0] m + [0 1 0] x + [2] [0 1 0] [0 0 0] [0 2 0] [3] >= [0 0 0] [0 1 0] [0 1 0] [2] [0 1 0] k + [0 1 0] m + [0 1 0] x + [2] [0 1 0] [0 0 0] [0 0 0] [3] = if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) = [0 1 0] [2] [0 1 0] m + [1] [0 0 0] [2] >= [1] [0] [0] = nil() ***** Step 1.b:5.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: eq#(s(n),s(m)) -> c_4(eq#(n,m)) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 1.b:5.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: eq#(s(n),s(m)) -> c_4(eq#(n,m)) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:eq#(s(n),s(m)) -> c_4(eq#(n,m)) -->_1 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1 2:W:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):3 3:W:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):2 -->_2 eq#(s(n),s(m)) -> c_4(eq#(n,m)):1 4:W:sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x) -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):3 5:W:sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) -->_1 sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)):5 -->_1 sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) 4: sort#(cons(n,x)) -> replace#(min(cons(n,x)),n,x) 2: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) 3: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) 1: eq#(s(n),s(m)) -> c_4(eq#(n,m)) ***** Step 1.b:5.a:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak DPs: eq#(s(n),s(m)) -> c_4(eq#(n,m)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 2:S:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 3:S:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):6 4:S:le#(s(n),s(m)) -> c_11(le#(n,m)) -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):4 5:S:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):4 -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):2 -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):1 6:S:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) -->_2 eq#(s(n),s(m)) -> c_4(eq#(n,m)):8 -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):3 7:S:sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))):7 -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):6 -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 8:W:eq#(s(n),s(m)) -> c_4(eq#(n,m)) -->_1 eq#(s(n),s(m)) -> c_4(eq#(n,m)):8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: eq#(s(n),s(m)) -> c_4(eq#(n,m)) *** Step 1.b:5.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/2,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 2:S:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 3:S:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):6 4:S:le#(s(n),s(m)) -> c_11(le#(n,m)) -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):4 5:S:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):4 -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):2 -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):1 6:S:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)) -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):3 7:S:sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))):7 -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x)),eq#(n,k)):6 -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) *** Step 1.b:5.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) - Weak DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} Problem (S) - Strict DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} **** Step 1.b:5.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) - Weak DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 2:S:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 3:W:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):6 4:S:le#(s(n),s(m)) -> c_11(le#(n,m)) -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):4 5:S:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):1 -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):2 -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):4 6:W:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):3 7:W:sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):6 -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) 6: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) **** Step 1.b:5.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) - Weak DPs: sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 2:S:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 4:S:le#(s(n),s(m)) -> c_11(le#(n,m)) -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):4 5:S:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):1 -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):2 -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):4 7:W:sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):5 -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))):7 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x))) **** Step 1.b:5.b:3.a:3: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) - Weak DPs: sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x))) and a lower component if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) Further, following extension rules are added to the lower component. sort#(cons(n,x)) -> min#(cons(n,x)) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) ***** Step 1.b:5.b:3.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x))) The strictly oriented rules are moved into the weak component. ****** Step 1.b:5.b:3.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_17) = {2} Following symbols are considered usable: {if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#} TcT has computed the following interpretation: p(0) = [0] p(cons) = [1] x2 + [1] p(eq) = [0] p(false) = [0] p(if_min) = [0] p(if_replace) = [1] x4 + [0] p(le) = [1] x2 + [1] p(min) = [0] p(nil) = [1] p(replace) = [1] x3 + [0] p(s) = [0] p(sort) = [0] p(true) = [1] p(eq#) = [1] x2 + [0] p(if_min#) = [1] x1 + [1] p(if_replace#) = [4] x1 + [1] x3 + [0] p(le#) = [1] p(min#) = [1] p(replace#) = [4] x1 + [4] x2 + [1] p(sort#) = [4] x1 + [0] p(c_1) = [0] p(c_2) = [2] p(c_3) = [1] p(c_4) = [1] p(c_5) = [2] x1 + [0] p(c_6) = [0] p(c_7) = [4] p(c_8) = [0] p(c_9) = [1] p(c_10) = [1] p(c_11) = [1] p(c_12) = [1] x1 + [0] p(c_13) = [1] p(c_14) = [0] p(c_15) = [1] p(c_16) = [1] p(c_17) = [1] x1 + [1] x2 + [1] p(c_18) = [2] Following rules are strictly oriented: sort#(cons(n,x)) = [4] x + [4] > [4] x + [2] = c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x))) Following rules are (at-least) weakly oriented: if_replace(false(),n,m,cons(k,x)) = [1] x + [1] >= [1] x + [1] = cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) = [1] x + [1] >= [1] x + [1] = cons(m,x) replace(n,m,cons(k,x)) = [1] x + [1] >= [1] x + [1] = if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) = [1] >= [1] = nil() ****** Step 1.b:5.b:3.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 1.b:5.b:3.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x))) -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: sort#(cons(n,x)) -> c_17(min#(cons(n,x)),sort#(replace(min(cons(n,x)),n,x)),min#(cons(n,x))) ****** Step 1.b:5.b:3.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ***** Step 1.b:5.b:3.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) - Weak DPs: sort#(cons(n,x)) -> min#(cons(n,x)) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) 3: le#(s(n),s(m)) -> c_11(le#(n,m)) The strictly oriented rules are moved into the weak component. ****** Step 1.b:5.b:3.a:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) - Weak DPs: sort#(cons(n,x)) -> min#(cons(n,x)) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_11) = {1}, uargs(c_12) = {1,2} Following symbols are considered usable: {eq,if_min,if_replace,min,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#} TcT has computed the following interpretation: p(0) = [1] [1] [1] p(cons) = [0 0 0] [0 1 0] [1] [0 0 0] x1 + [0 0 1] x2 + [1] [0 0 1] [0 0 1] [1] p(eq) = [0] [1] [1] p(false) = [0] [1] [1] p(if_min) = [0 0 0] [1] [0 1 0] x2 + [0] [0 0 1] [0] p(if_replace) = [0 0 0] [0 1 0] [0 0 1] [1 1 0] [0] [0 0 1] x1 + [0 1 0] x2 + [0 0 1] x3 + [0 1 0] x4 + [0] [0 1 0] [0 0 0] [0 0 1] [0 0 1] [0] p(le) = [0 0 0] [0 0 1] [0] [1 1 0] x1 + [1 0 0] x2 + [0] [0 0 0] [1 0 0] [1] p(min) = [0 0 0] [1] [0 1 0] x1 + [1] [0 0 1] [1] p(nil) = [0] [0] [0] p(replace) = [0 1 0] [0 0 1] [1 0 1] [1] [0 1 0] x1 + [0 0 1] x2 + [0 1 0] x3 + [1] [0 0 0] [0 0 1] [0 0 1] [1] p(s) = [0 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(sort) = [0] [0] [0] p(true) = [0] [0] [0] p(eq#) = [0] [0] [0] p(if_min#) = [1 0 0] [1] [0 0 0] x2 + [0] [0 0 0] [0] p(if_replace#) = [0] [0] [0] p(le#) = [0 0 0] [0 0 1] [0] [0 0 0] x1 + [1 0 0] x2 + [0] [0 0 1] [0 0 0] [0] p(min#) = [0 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] p(replace#) = [0] [0] [0] p(sort#) = [0 0 1] [1] [0 0 0] x1 + [1] [0 0 1] [1] p(c_1) = [0] [0] [0] p(c_2) = [0] [0] [0] p(c_3) = [0] [0] [0] p(c_4) = [0] [0] [0] p(c_5) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_6) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_7) = [0] [0] [0] p(c_8) = [0] [0] [0] p(c_9) = [0] [0] [0] p(c_10) = [0] [0] [0] p(c_11) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_12) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(c_13) = [0] [0] [0] p(c_14) = [0] [0] [0] p(c_15) = [0] [0] [0] p(c_16) = [0] [0] [0] p(c_17) = [0] [0] [0] p(c_18) = [0] [0] [0] Following rules are strictly oriented: if_min#(false(),cons(n,cons(m,x))) = [0 0 1] [3] [0 0 0] x + [0] [0 0 0] [0] > [0 0 1] [2] [0 0 0] x + [0] [0 0 0] [0] = c_5(min#(cons(m,x))) le#(s(n),s(m)) = [0 0 1] [0 0 0] [1] [0 0 0] m + [0 0 0] n + [1] [0 0 0] [0 0 1] [1] > [0 0 1] [0] [0 0 0] m + [0] [0 0 0] [0] = c_11(le#(n,m)) Following rules are (at-least) weakly oriented: if_min#(true(),cons(n,cons(m,x))) = [0 0 1] [3] [0 0 0] x + [0] [0 0 0] [0] >= [0 0 1] [3] [0 0 0] x + [0] [0 0 0] [0] = c_6(min#(cons(n,x))) min#(cons(n,cons(m,x))) = [0 0 1] [0 0 1] [3] [0 0 0] m + [0 0 0] x + [1] [0 0 0] [0 0 0] [0] >= [0 0 1] [0 0 1] [3] [0 0 0] m + [0 0 0] x + [0] [0 0 0] [0 0 0] [0] = c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) sort#(cons(n,x)) = [0 0 1] [0 0 1] [2] [0 0 0] n + [0 0 0] x + [1] [0 0 1] [0 0 1] [2] >= [0 0 1] [2] [0 0 0] x + [1] [0 0 0] [0] = min#(cons(n,x)) sort#(cons(n,x)) = [0 0 1] [0 0 1] [2] [0 0 0] n + [0 0 0] x + [1] [0 0 1] [0 0 1] [2] >= [0 0 1] [0 0 1] [2] [0 0 0] n + [0 0 0] x + [1] [0 0 1] [0 0 1] [2] = sort#(replace(min(cons(n,x)),n,x)) eq(0(),0()) = [0] [1] [1] >= [0] [0] [0] = true() eq(0(),s(m)) = [0] [1] [1] >= [0] [1] [1] = false() eq(s(n),0()) = [0] [1] [1] >= [0] [1] [1] = false() eq(s(n),s(m)) = [0] [1] [1] >= [0] [1] [1] = eq(n,m) if_min(false(),cons(n,cons(m,x))) = [0 0 0] [0 0 0] [0 0 0] [1] [0 0 1] m + [0 0 0] n + [0 0 1] x + [2] [0 0 1] [0 0 1] [0 0 1] [2] >= [0 0 0] [0 0 0] [1] [0 0 0] m + [0 0 1] x + [2] [0 0 1] [0 0 1] [2] = min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) = [0 0 0] [0 0 0] [0 0 0] [1] [0 0 1] m + [0 0 0] n + [0 0 1] x + [2] [0 0 1] [0 0 1] [0 0 1] [2] >= [0 0 0] [0 0 0] [1] [0 0 0] n + [0 0 1] x + [2] [0 0 1] [0 0 1] [2] = min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) = [0 0 0] [0 0 1] [0 1 0] [0 1 1] [2] [0 0 0] k + [0 0 1] m + [0 1 0] n + [0 0 1] x + [2] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [2] >= [0 0 0] [0 0 1] [0 1 0] [0 1 0] [2] [0 0 0] k + [0 0 1] m + [0 0 0] n + [0 0 1] x + [2] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [2] = cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) = [0 0 0] [0 0 1] [0 1 0] [0 1 1] [2] [0 0 0] k + [0 0 1] m + [0 1 0] n + [0 0 1] x + [1] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [1] >= [0 0 0] [0 1 0] [1] [0 0 0] m + [0 0 1] x + [1] [0 0 1] [0 0 1] [1] = cons(m,x) min(cons(n,cons(m,x))) = [0 0 0] [0 0 0] [0 0 0] [1] [0 0 1] m + [0 0 0] n + [0 0 1] x + [3] [0 0 1] [0 0 1] [0 0 1] [3] >= [0 0 0] [0 0 0] [0 0 0] [1] [0 0 1] m + [0 0 0] n + [0 0 1] x + [2] [0 0 1] [0 0 1] [0 0 1] [2] = if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) = [1] [2] [3] >= [1] [1] [1] = 0() min(cons(s(n),nil())) = [0 0 0] [1] [0 0 0] n + [2] [0 0 1] [3] >= [0 0 0] [1] [0 0 0] n + [0] [0 0 1] [1] = s(n) replace(n,m,cons(k,x)) = [0 0 1] [0 0 1] [0 1 0] [0 1 1] [3] [0 0 0] k + [0 0 1] m + [0 1 0] n + [0 0 1] x + [2] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [2] >= [0 0 0] [0 0 1] [0 1 0] [0 1 1] [2] [0 0 0] k + [0 0 1] m + [0 1 0] n + [0 0 1] x + [2] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [2] = if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) = [0 0 1] [0 1 0] [1] [0 0 1] m + [0 1 0] n + [1] [0 0 1] [0 0 0] [1] >= [0] [0] [0] = nil() ****** Step 1.b:5.b:3.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) le#(s(n),s(m)) -> c_11(le#(n,m)) sort#(cons(n,x)) -> min#(cons(n,x)) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 1.b:5.b:3.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) le#(s(n),s(m)) -> c_11(le#(n,m)) sort#(cons(n,x)) -> min#(cons(n,x)) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):2 2:S:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):4 -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):3 -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):1 3:W:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):2 4:W:le#(s(n),s(m)) -> c_11(le#(n,m)) -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):4 5:W:sort#(cons(n,x)) -> min#(cons(n,x)) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):2 6:W:sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) -->_1 sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)):6 -->_1 sort#(cons(n,x)) -> min#(cons(n,x)):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: le#(s(n),s(m)) -> c_11(le#(n,m)) ****** Step 1.b:5.b:3.a:3.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) sort#(cons(n,x)) -> min#(cons(n,x)) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):2 2:S:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):3 -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):1 3:W:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):2 5:W:sort#(cons(n,x)) -> min#(cons(n,x)) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):2 6:W:sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) -->_1 sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)):6 -->_1 sort#(cons(n,x)) -> min#(cons(n,x)):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))) ****** Step 1.b:5.b:3.a:3.b:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))) - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) sort#(cons(n,x)) -> min#(cons(n,x)) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) The strictly oriented rules are moved into the weak component. ******* Step 1.b:5.b:3.a:3.b:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))) - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) sort#(cons(n,x)) -> min#(cons(n,x)) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {if_replace,le,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#} TcT has computed the following interpretation: p(0) = [1] p(cons) = [1] x2 + [1] p(eq) = [2] x1 + [1] x2 + [3] p(false) = [2] p(if_min) = [1] x2 + [4] p(if_replace) = [1] x4 + [0] p(le) = [2] p(min) = [1] x1 + [2] p(nil) = [2] p(replace) = [1] x3 + [0] p(s) = [1] x1 + [1] p(sort) = [4] x1 + [0] p(true) = [1] p(eq#) = [1] x1 + [1] p(if_min#) = [2] x1 + [4] x2 + [0] p(if_replace#) = [1] x1 + [1] x3 + [1] x4 + [1] p(le#) = [1] p(min#) = [4] x1 + [4] p(replace#) = [2] x1 + [2] x3 + [4] p(sort#) = [4] x1 + [4] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [4] x1 + [0] p(c_5) = [1] x1 + [1] p(c_6) = [1] x1 + [0] p(c_7) = [4] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [0] p(c_14) = [1] p(c_15) = [1] p(c_16) = [0] p(c_17) = [1] x1 + [1] x2 + [1] x3 + [1] p(c_18) = [0] Following rules are strictly oriented: if_min#(true(),cons(n,cons(m,x))) = [4] x + [10] > [4] x + [8] = c_6(min#(cons(n,x))) Following rules are (at-least) weakly oriented: if_min#(false(),cons(n,cons(m,x))) = [4] x + [12] >= [4] x + [9] = c_5(min#(cons(m,x))) min#(cons(n,cons(m,x))) = [4] x + [12] >= [4] x + [12] = c_12(if_min#(le(n,m),cons(n,cons(m,x)))) sort#(cons(n,x)) = [4] x + [8] >= [4] x + [8] = min#(cons(n,x)) sort#(cons(n,x)) = [4] x + [8] >= [4] x + [4] = sort#(replace(min(cons(n,x)),n,x)) if_replace(false(),n,m,cons(k,x)) = [1] x + [1] >= [1] x + [1] = cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) = [1] x + [1] >= [1] x + [1] = cons(m,x) le(0(),m) = [2] >= [1] = true() le(s(n),0()) = [2] >= [2] = false() le(s(n),s(m)) = [2] >= [2] = le(n,m) replace(n,m,cons(k,x)) = [1] x + [1] >= [1] x + [1] = if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) = [2] >= [2] = nil() ******* Step 1.b:5.b:3.a:3.b:1.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))) - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) sort#(cons(n,x)) -> min#(cons(n,x)) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 1.b:5.b:3.a:3.b:1.b:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))) - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) sort#(cons(n,x)) -> min#(cons(n,x)) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))) Consider the set of all dependency pairs 1: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))) 2: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) 3: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) 4: sort#(cons(n,x)) -> min#(cons(n,x)) 5: sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******** Step 1.b:5.b:3.a:3.b:1.b:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))) - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) sort#(cons(n,x)) -> min#(cons(n,x)) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#} TcT has computed the following interpretation: p(0) = [0] p(cons) = [1] x2 + [2] p(eq) = [4] x2 + [4] p(false) = [0] p(if_min) = [1] x2 + [0] p(if_replace) = [1] x4 + [2] p(le) = [0] p(min) = [0] p(nil) = [2] p(replace) = [1] x3 + [2] p(s) = [0] p(sort) = [1] x1 + [4] p(true) = [0] p(eq#) = [4] x2 + [2] p(if_min#) = [2] x2 + [2] p(if_replace#) = [4] x1 + [2] p(le#) = [4] x1 + [4] x2 + [0] p(min#) = [2] x1 + [3] p(replace#) = [2] p(sort#) = [4] x1 + [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [4] p(c_8) = [0] p(c_9) = [1] p(c_10) = [2] p(c_11) = [1] x1 + [4] p(c_12) = [1] x1 + [0] p(c_13) = [1] p(c_14) = [0] p(c_15) = [2] x1 + [1] p(c_16) = [1] p(c_17) = [1] x1 + [1] x2 + [1] p(c_18) = [0] Following rules are strictly oriented: min#(cons(n,cons(m,x))) = [2] x + [11] > [2] x + [10] = c_12(if_min#(le(n,m),cons(n,cons(m,x)))) Following rules are (at-least) weakly oriented: if_min#(false(),cons(n,cons(m,x))) = [2] x + [10] >= [2] x + [7] = c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) = [2] x + [10] >= [2] x + [7] = c_6(min#(cons(n,x))) sort#(cons(n,x)) = [4] x + [9] >= [2] x + [7] = min#(cons(n,x)) sort#(cons(n,x)) = [4] x + [9] >= [4] x + [9] = sort#(replace(min(cons(n,x)),n,x)) if_replace(false(),n,m,cons(k,x)) = [1] x + [4] >= [1] x + [4] = cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) = [1] x + [4] >= [1] x + [2] = cons(m,x) replace(n,m,cons(k,x)) = [1] x + [4] >= [1] x + [4] = if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) = [4] >= [2] = nil() ******** Step 1.b:5.b:3.a:3.b:1.b:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))) sort#(cons(n,x)) -> min#(cons(n,x)) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******** Step 1.b:5.b:3.a:3.b:1.b:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))) sort#(cons(n,x)) -> min#(cons(n,x)) sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))):3 2:W:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))):3 3:W:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))) -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):2 -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):1 4:W:sort#(cons(n,x)) -> min#(cons(n,x)) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))):3 5:W:sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) -->_1 sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)):5 -->_1 sort#(cons(n,x)) -> min#(cons(n,x)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: sort#(cons(n,x)) -> sort#(replace(min(cons(n,x)),n,x)) 4: sort#(cons(n,x)) -> min#(cons(n,x)) 1: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) 3: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x)))) 2: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) ******** Step 1.b:5.b:3.a:3.b:1.b:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/3,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 1.b:5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak DPs: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) le#(s(n),s(m)) -> c_11(le#(n,m)) min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):2 2:S:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):1 3:S:sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) -->_4 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):7 -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):7 -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))):3 -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):2 4:W:if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):7 5:W:if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) -->_1 min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)):7 6:W:le#(s(n),s(m)) -> c_11(le#(n,m)) -->_1 le#(s(n),s(m)) -> c_11(le#(n,m)):6 7:W:min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) -->_2 le#(s(n),s(m)) -> c_11(le#(n,m)):6 -->_1 if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))):5 -->_1 if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: min#(cons(n,cons(m,x))) -> c_12(if_min#(le(n,m),cons(n,cons(m,x))),le#(n,m)) 5: if_min#(true(),cons(n,cons(m,x))) -> c_6(min#(cons(n,x))) 4: if_min#(false(),cons(n,cons(m,x))) -> c_5(min#(cons(m,x))) 6: le#(s(n),s(m)) -> c_11(le#(n,m)) **** Step 1.b:5.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/4,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):2 2:S:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):1 3:S:sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))) -->_2 sort#(cons(n,x)) -> c_17(min#(cons(n,x)) ,sort#(replace(min(cons(n,x)),n,x)) ,replace#(min(cons(n,x)),n,x) ,min#(cons(n,x))):3 -->_3 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) **** Step 1.b:5.b:3.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) - Weak DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} Problem (S) - Strict DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} ***** Step 1.b:5.b:3.b:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) - Weak DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) The strictly oriented rules are moved into the weak component. ****** Step 1.b:5.b:3.b:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) - Weak DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_7) = {1}, uargs(c_15) = {1}, uargs(c_17) = {1,2} Following symbols are considered usable: {if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#} TcT has computed the following interpretation: p(0) = [1] [0] p(cons) = [1 2] x2 + [1] [0 1] [2] p(eq) = [0] [0] p(false) = [0] [0] p(if_min) = [0 1] x1 + [0 1] x2 + [0] [1 1] [0 1] [1] p(if_replace) = [1 0] x4 + [1] [0 1] [0] p(le) = [1 1] x1 + [0 0] x2 + [0] [1 3] [0 2] [2] p(min) = [0] [2] p(nil) = [1] [1] p(replace) = [1 0] x3 + [1] [0 1] [0] p(s) = [2] [0] p(sort) = [2 1] x1 + [1] [0 1] [0] p(true) = [0] [0] p(eq#) = [0 0] x1 + [1 0] x2 + [0] [0 2] [0 0] [2] p(if_min#) = [0 0] x1 + [0] [2 0] [1] p(if_replace#) = [0 0] x3 + [0 1] x4 + [0] [2 0] [1 0] [1] p(le#) = [0 0] x2 + [1] [0 2] [0] p(min#) = [2] [0] p(replace#) = [0 1] x3 + [0] [0 0] [1] p(sort#) = [1 0] x1 + [0] [0 0] [2] p(c_1) = [1] [0] p(c_2) = [1] [1] p(c_3) = [2] [0] p(c_4) = [1] [1] p(c_5) = [1] [0] p(c_6) = [2 0] x1 + [0] [2 1] [1] p(c_7) = [1 1] x1 + [0] [1 0] [0] p(c_8) = [0] [0] p(c_9) = [0] [0] p(c_10) = [0] [0] p(c_11) = [1] [0] p(c_12) = [2 2] x2 + [2] [0 0] [2] p(c_13) = [0] [0] p(c_14) = [0] [2] p(c_15) = [1 0] x1 + [0] [0 0] [1] p(c_16) = [2] [2] p(c_17) = [1 0] x1 + [2 0] x2 + [0] [0 0] [0 0] [0] p(c_18) = [1] [1] Following rules are strictly oriented: if_replace#(false(),n,m,cons(k,x)) = [0 0] m + [0 1] x + [2] [2 0] [1 2] [2] > [0 1] x + [1] [0 1] [0] = c_7(replace#(n,m,x)) Following rules are (at-least) weakly oriented: replace#(n,m,cons(k,x)) = [0 1] x + [2] [0 0] [1] >= [0 1] x + [2] [0 0] [1] = c_15(if_replace#(eq(n,k),n,m,cons(k,x))) sort#(cons(n,x)) = [1 2] x + [1] [0 0] [2] >= [1 2] x + [1] [0 0] [0] = c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) if_replace(false(),n,m,cons(k,x)) = [1 2] x + [2] [0 1] [2] >= [1 2] x + [2] [0 1] [2] = cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) = [1 2] x + [2] [0 1] [2] >= [1 2] x + [1] [0 1] [2] = cons(m,x) replace(n,m,cons(k,x)) = [1 2] x + [2] [0 1] [2] >= [1 2] x + [2] [0 1] [2] = if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) = [2] [1] >= [1] [1] = nil() ****** Step 1.b:5.b:3.b:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) - Weak DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 1.b:5.b:3.b:3.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) - Weak DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) Consider the set of all dependency pairs 1: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) 2: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) 3: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) Processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******* Step 1.b:5.b:3.b:3.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) - Weak DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_7) = {1}, uargs(c_15) = {1}, uargs(c_17) = {1,2} Following symbols are considered usable: {if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#} TcT has computed the following interpretation: p(0) = [0] [3] p(cons) = [1 1] x2 + [2] [0 1] [1] p(eq) = [0 2] x1 + [0] [0 0] [0] p(false) = [0] [0] p(if_min) = [0 0] x1 + [0] [3 2] [1] p(if_replace) = [1 0] x4 + [1] [0 1] [0] p(le) = [1 0] x2 + [0] [1 0] [0] p(min) = [0] [3] p(nil) = [0] [0] p(replace) = [1 0] x3 + [1] [0 1] [0] p(s) = [2] [0] p(sort) = [0] [0] p(true) = [1] [0] p(eq#) = [0 0] x1 + [1 0] x2 + [0] [1 0] [0 0] [1] p(if_min#) = [1 0] x2 + [0] [0 0] [2] p(if_replace#) = [0 0] x2 + [0 2] x4 + [0] [1 2] [0 1] [3] p(le#) = [0 0] x1 + [0] [0 1] [0] p(min#) = [2 1] x1 + [0] [1 0] [0] p(replace#) = [0 2] x3 + [1] [2 2] [0] p(sort#) = [2 0] x1 + [1] [2 0] [2] p(c_1) = [0] [1] p(c_2) = [0] [0] p(c_3) = [0] [0] p(c_4) = [2 1] x1 + [0] [1 1] [0] p(c_5) = [0 0] x1 + [0] [0 1] [0] p(c_6) = [0] [0] p(c_7) = [1 0] x1 + [1] [0 0] [1] p(c_8) = [1] [0] p(c_9) = [0] [0] p(c_10) = [0] [0] p(c_11) = [1] [0] p(c_12) = [1 0] x1 + [0 1] x2 + [2] [0 0] [0 2] [0] p(c_13) = [0] [0] p(c_14) = [2] [0] p(c_15) = [1 0] x1 + [0] [0 0] [0] p(c_16) = [0] [1] p(c_17) = [1 0] x1 + [1 0] x2 + [1] [0 0] [1 0] [1] p(c_18) = [2] [0] Following rules are strictly oriented: replace#(n,m,cons(k,x)) = [0 2] x + [3] [2 4] [6] > [0 2] x + [2] [0 0] [0] = c_15(if_replace#(eq(n,k),n,m,cons(k,x))) Following rules are (at-least) weakly oriented: if_replace#(false(),n,m,cons(k,x)) = [0 0] n + [0 2] x + [2] [1 2] [0 1] [4] >= [0 2] x + [2] [0 0] [1] = c_7(replace#(n,m,x)) sort#(cons(n,x)) = [2 2] x + [5] [2 2] [6] >= [2 2] x + [5] [0 2] [2] = c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) if_replace(false(),n,m,cons(k,x)) = [1 1] x + [3] [0 1] [1] >= [1 1] x + [3] [0 1] [1] = cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) = [1 1] x + [3] [0 1] [1] >= [1 1] x + [2] [0 1] [1] = cons(m,x) replace(n,m,cons(k,x)) = [1 1] x + [3] [0 1] [1] >= [1 1] x + [3] [0 1] [1] = if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) = [1] [0] >= [0] [0] = nil() ******* Step 1.b:5.b:3.b:3.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 1.b:5.b:3.b:3.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):2 2:W:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):1 3:W:sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) -->_1 sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)):3 -->_2 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) 1: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) 2: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) ******* Step 1.b:5.b:3.b:3.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ***** Step 1.b:5.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak DPs: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) -->_2 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):3 -->_1 sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)):1 2:W:if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) -->_1 replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))):3 3:W:replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) -->_1 if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: replace#(n,m,cons(k,x)) -> c_15(if_replace#(eq(n,k),n,m,cons(k,x))) 2: if_replace#(false(),n,m,cons(k,x)) -> c_7(replace#(n,m,x)) ***** Step 1.b:5.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/2,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)) -->_1 sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x)),replace#(min(cons(n,x)),n,x)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x))) ***** Step 1.b:5.b:3.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x))) The strictly oriented rules are moved into the weak component. ****** Step 1.b:5.b:3.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_17) = {1} Following symbols are considered usable: {if_replace,replace,eq#,if_min#,if_replace#,le#,min#,replace#,sort#} TcT has computed the following interpretation: p(0) = [0] p(cons) = [1] x2 + [1] p(eq) = [0] p(false) = [0] p(if_min) = [0] p(if_replace) = [1] x4 + [0] p(le) = [4] x2 + [5] p(min) = [1] p(nil) = [7] p(replace) = [1] x3 + [0] p(s) = [2] p(sort) = [4] p(true) = [0] p(eq#) = [1] x1 + [4] p(if_min#) = [1] x2 + [0] p(if_replace#) = [1] x3 + [2] x4 + [0] p(le#) = [2] x1 + [4] p(min#) = [1] x1 + [4] p(replace#) = [1] x1 + [1] x2 + [4] x3 + [4] p(sort#) = [4] x1 + [0] p(c_1) = [1] p(c_2) = [1] p(c_3) = [0] p(c_4) = [4] x1 + [4] p(c_5) = [1] p(c_6) = [1] p(c_7) = [1] x1 + [2] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [2] p(c_12) = [0] p(c_13) = [2] p(c_14) = [1] p(c_15) = [1] x1 + [0] p(c_16) = [2] p(c_17) = [1] x1 + [0] p(c_18) = [0] Following rules are strictly oriented: sort#(cons(n,x)) = [4] x + [4] > [4] x + [0] = c_17(sort#(replace(min(cons(n,x)),n,x))) Following rules are (at-least) weakly oriented: if_replace(false(),n,m,cons(k,x)) = [1] x + [1] >= [1] x + [1] = cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) = [1] x + [1] >= [1] x + [1] = cons(m,x) replace(n,m,cons(k,x)) = [1] x + [1] >= [1] x + [1] = if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) = [7] >= [7] = nil() ****** Step 1.b:5.b:3.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 1.b:5.b:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x))) -->_1 sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: sort#(cons(n,x)) -> c_17(sort#(replace(min(cons(n,x)),n,x))) ****** Step 1.b:5.b:3.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(0(),nil())) -> 0() min(cons(s(n),nil())) -> s(n) replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() - Signature: {eq/2,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,eq#/2,if_min#/2,if_replace#/4,le#/2,min#/1 ,replace#/3,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0 ,c_9/0,c_10/0,c_11/1,c_12/2,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,if_min#,if_replace#,le#,min#,replace# ,sort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^2))