MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) main(x2,x1) -> add#2(x2,x1) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {add#2,addNat#2,carry#2,cond_add_ws1'_ws2'_2 ,cond_add_ws1'_ws2'_3,cond_carry_w_xs_1,lt#2,main,mult#2} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs add#2#(x2,Nil()) -> c_1() add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) add#2#(Nil(),Cons(x4,x2)) -> c_3() addNat#2#(0(),x16) -> c_4() addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) carry#2#(x2,Nil()) -> c_6() carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_13() lt#2#(0(),0()) -> c_14() lt#2#(0(),S(x16)) -> c_15() lt#2#(S(x16),0()) -> c_16() lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) main#(x2,x1) -> c_18(add#2#(x2,x1)) mult#2#(0(),x2) -> c_19() mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: add#2#(x2,Nil()) -> c_1() add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) add#2#(Nil(),Cons(x4,x2)) -> c_3() addNat#2#(0(),x16) -> c_4() addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) carry#2#(x2,Nil()) -> c_6() carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_13() lt#2#(0(),0()) -> c_14() lt#2#(0(),S(x16)) -> c_15() lt#2#(S(x16),0()) -> c_16() lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) main#(x2,x1) -> c_18(add#2#(x2,x1)) mult#2#(0(),x2) -> c_19() mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) main(x2,x1) -> add#2(x2,x1) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) add#2#(x2,Nil()) -> c_1() add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) add#2#(Nil(),Cons(x4,x2)) -> c_3() addNat#2#(0(),x16) -> c_4() addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) carry#2#(x2,Nil()) -> c_6() carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_13() lt#2#(0(),0()) -> c_14() lt#2#(0(),S(x16)) -> c_15() lt#2#(S(x16),0()) -> c_16() lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) main#(x2,x1) -> c_18(add#2#(x2,x1)) mult#2#(0(),x2) -> c_19() mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: add#2#(x2,Nil()) -> c_1() add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) add#2#(Nil(),Cons(x4,x2)) -> c_3() addNat#2#(0(),x16) -> c_4() addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) carry#2#(x2,Nil()) -> c_6() carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_13() lt#2#(0(),0()) -> c_14() lt#2#(0(),S(x16)) -> c_15() lt#2#(S(x16),0()) -> c_16() lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) main#(x2,x1) -> c_18(add#2#(x2,x1)) mult#2#(0(),x2) -> c_19() mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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 {1,3,4,6,13,14,15,16,19} by application of Pre({1,3,4,6,13,14,15,16,19}) = {2,5,7,8,9,10,11,12,17,18,20}. Here rules are labelled as follows: 1: add#2#(x2,Nil()) -> c_1() 2: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) 3: add#2#(Nil(),Cons(x4,x2)) -> c_3() 4: addNat#2#(0(),x16) -> c_4() 5: addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) 6: carry#2#(x2,Nil()) -> c_6() 7: carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) 8: cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) 9: cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) 10: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) 11: cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) 12: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) 13: cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_13() 14: lt#2#(0(),0()) -> c_14() 15: lt#2#(0(),S(x16)) -> c_15() 16: lt#2#(S(x16),0()) -> c_16() 17: lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) 18: main#(x2,x1) -> c_18(add#2#(x2,x1)) 19: mult#2#(0(),x2) -> c_19() 20: mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) main#(x2,x1) -> c_18(add#2#(x2,x1)) mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak DPs: add#2#(x2,Nil()) -> c_1() add#2#(Nil(),Cons(x4,x2)) -> c_3() addNat#2#(0(),x16) -> c_4() carry#2#(x2,Nil()) -> c_6() cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_13() lt#2#(0(),0()) -> c_14() lt#2#(0(),S(x16)) -> c_15() lt#2#(S(x16),0()) -> c_16() mult#2#(0(),x2) -> c_19() - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) -->_2 lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)):9 -->_1 cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))):5 -->_1 cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)):4 -->_2 lt#2#(S(x16),0()) -> c_16():19 -->_2 lt#2#(0(),S(x16)) -> c_15():18 -->_2 lt#2#(0(),0()) -> c_14():17 2:S:addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) -->_1 addNat#2#(0(),x16) -> c_4():14 -->_1 addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)):2 3:S:carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) -->_2 lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)):9 -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1) ,mult#2#(S(S(0())),x3)):8 -->_2 lt#2#(S(x16),0()) -> c_16():19 -->_2 lt#2#(0(),S(x16)) -> c_15():18 -->_2 lt#2#(0(),0()) -> c_14():17 -->_1 cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_13():16 4:S:cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) -->_2 lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)):9 -->_1 cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)):7 -->_1 cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)):6 -->_2 lt#2#(S(x16),0()) -> c_16():19 -->_2 lt#2#(0(),S(x16)) -> c_15():18 -->_2 lt#2#(0(),0()) -> c_14():17 5:S:cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) -->_1 add#2#(Nil(),Cons(x4,x2)) -> c_3():13 -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)):1 6:S:cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) -->_2 mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):11 -->_1 carry#2#(x2,Nil()) -> c_6():15 -->_3 add#2#(Nil(),Cons(x4,x2)) -> c_3():13 -->_3 add#2#(x2,Nil()) -> c_1():12 -->_1 carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):3 -->_3 add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)):1 7:S:cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) -->_1 add#2#(x2,Nil()) -> c_1():12 -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)):1 8:S:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) -->_2 mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):11 -->_1 carry#2#(x2,Nil()) -> c_6():15 -->_1 carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):3 9:S:lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) -->_1 lt#2#(S(x16),0()) -> c_16():19 -->_1 lt#2#(0(),S(x16)) -> c_15():18 -->_1 lt#2#(0(),0()) -> c_14():17 -->_1 lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)):9 10:S:main#(x2,x1) -> c_18(add#2#(x2,x1)) -->_1 add#2#(Nil(),Cons(x4,x2)) -> c_3():13 -->_1 add#2#(x2,Nil()) -> c_1():12 -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)):1 11:S:mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) -->_2 mult#2#(0(),x2) -> c_19():20 -->_1 addNat#2#(0(),x16) -> c_4():14 -->_2 mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):11 -->_1 addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)):2 12:W:add#2#(x2,Nil()) -> c_1() 13:W:add#2#(Nil(),Cons(x4,x2)) -> c_3() 14:W:addNat#2#(0(),x16) -> c_4() 15:W:carry#2#(x2,Nil()) -> c_6() 16:W:cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_13() 17:W:lt#2#(0(),0()) -> c_14() 18:W:lt#2#(0(),S(x16)) -> c_15() 19:W:lt#2#(S(x16),0()) -> c_16() 20:W:mult#2#(0(),x2) -> c_19() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 16: cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_13() 15: carry#2#(x2,Nil()) -> c_6() 14: addNat#2#(0(),x16) -> c_4() 20: mult#2#(0(),x2) -> c_19() 12: add#2#(x2,Nil()) -> c_1() 13: add#2#(Nil(),Cons(x4,x2)) -> c_3() 17: lt#2#(0(),0()) -> c_14() 18: lt#2#(0(),S(x16)) -> c_15() 19: lt#2#(S(x16),0()) -> c_16() * Step 5: RemoveHeads MAYBE + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) main#(x2,x1) -> c_18(add#2#(x2,x1)) mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) -->_2 lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)):9 -->_1 cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))):5 -->_1 cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)):4 2:S:addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) -->_1 addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)):2 3:S:carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) -->_2 lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)):9 -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1) ,mult#2#(S(S(0())),x3)):8 4:S:cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) -->_2 lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)):9 -->_1 cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)):7 -->_1 cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)):6 5:S:cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) ,lt#2#(x8,x4)):1 6:S:cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) -->_2 mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):11 -->_1 carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):3 -->_3 add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)):1 7:S:cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) ,lt#2#(x8,x4)):1 8:S:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) -->_2 mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):11 -->_1 carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):3 9:S:lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) -->_1 lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)):9 10:S:main#(x2,x1) -> c_18(add#2#(x2,x1)) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) ,lt#2#(x8,x4)):1 11:S:mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) -->_2 mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):11 -->_1 addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)):2 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(10,main#(x2,x1) -> c_18(add#2#(x2,x1)))] * Step 6: DecomposeDG MAYBE + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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 add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) and a lower component addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) Further, following extension rules are added to the lower component. add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) ** Step 6.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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: 4: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) The strictly oriented rules are moved into the weak component. *** Step 6.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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_2) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1,3}, uargs(c_11) = {1} Following symbols are considered usable: {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2#,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main# ,mult#2#} TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x2 + [1] p(False) = [0] p(Nil) = [2] p(S) = [0] p(True) = [0] p(add#2) = [3] x1 + [4] x2 + [5] p(addNat#2) = [2] x1 + [2] p(carry#2) = [5] x1 + [4] p(cond_add_ws1'_ws2'_2) = [1] x1 + [4] x3 + [2] x4 + [0] p(cond_add_ws1'_ws2'_3) = [1] x1 + [1] x2 + [1] x4 + [2] x5 + [2] p(cond_carry_w_xs_1) = [2] x1 + [1] x2 + [5] p(lt#2) = [0] p(main) = [2] x1 + [2] x2 + [0] p(mult#2) = [1] x1 + [1] p(add#2#) = [1] x1 + [0] p(addNat#2#) = [1] x1 + [4] x2 + [1] p(carry#2#) = [0] p(cond_add_ws1'_ws2'_2#) = [1] x5 + [1] p(cond_add_ws1'_ws2'_3#) = [1] x5 + [1] p(cond_carry_w_xs_1#) = [4] x1 + [0] p(lt#2#) = [4] x1 + [1] p(main#) = [1] x2 + [1] p(mult#2#) = [2] p(c_1) = [4] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [1] p(c_5) = [1] x1 + [4] p(c_6) = [1] p(c_7) = [1] x1 + [1] x2 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [1] p(c_10) = [2] x1 + [1] x3 + [0] p(c_11) = [1] x1 + [0] p(c_12) = [4] x1 + [2] x2 + [0] p(c_13) = [1] p(c_14) = [1] p(c_15) = [1] p(c_16) = [1] p(c_17) = [0] p(c_18) = [2] x1 + [0] p(c_19) = [1] p(c_20) = [2] x2 + [1] Following rules are strictly oriented: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) = [1] x1 + [1] > [1] x1 + [0] = c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)),mult#2#(S(S(0())),x2),add#2#(x1,x3)) Following rules are (at-least) weakly oriented: add#2#(Cons(x8,x6),Cons(x4,x2)) = [1] x6 + [1] >= [1] x6 + [1] = c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) = [1] x1 + [1] >= [1] x1 + [1] = c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1),lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) = [1] x1 + [1] >= [1] x1 + [1] = c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) = [1] x1 + [1] >= [1] x1 + [1] = c_11(add#2#(Cons(x2,x1),x3)) *** Step 6.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) - Weak DPs: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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 6.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) - Weak DPs: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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: 4: cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) The strictly oriented rules are moved into the weak component. **** Step 6.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) - Weak DPs: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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_2) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1,3}, uargs(c_11) = {1} Following symbols are considered usable: {lt#2,add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2#,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main# ,mult#2#} TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x2 + [3] p(False) = [2] p(Nil) = [2] p(S) = [0] p(True) = [2] p(add#2) = [2] x1 + [0] p(addNat#2) = [2] x1 + [3] x2 + [2] p(carry#2) = [1] x1 + [2] x2 + [5] p(cond_add_ws1'_ws2'_2) = [4] x1 + [2] x2 + [1] x3 + [2] x4 + [4] x5 + [0] p(cond_add_ws1'_ws2'_3) = [4] x1 + [5] p(cond_carry_w_xs_1) = [5] x1 + [2] x3 + [1] x4 + [4] p(lt#2) = [2] p(main) = [1] x1 + [1] x2 + [0] p(mult#2) = [6] p(add#2#) = [4] x2 + [0] p(addNat#2#) = [1] x1 + [2] x2 + [4] p(carry#2#) = [0] p(cond_add_ws1'_ws2'_2#) = [4] x1 + [4] x3 + [4] p(cond_add_ws1'_ws2'_3#) = [4] x3 + [5] p(cond_carry_w_xs_1#) = [2] x1 + [2] x2 + [1] x3 + [4] x4 + [1] p(lt#2#) = [1] p(main#) = [4] p(mult#2#) = [1] x1 + [2] x2 + [1] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [1] p(c_5) = [0] p(c_6) = [0] p(c_7) = [2] x1 + [0] p(c_8) = [1] x1 + [1] x2 + [6] p(c_9) = [1] x1 + [0] p(c_10) = [2] x1 + [1] x3 + [0] p(c_11) = [1] x1 + [1] p(c_12) = [1] x1 + [1] p(c_13) = [0] p(c_14) = [0] p(c_15) = [4] p(c_16) = [2] p(c_17) = [1] x1 + [4] p(c_18) = [4] x1 + [0] p(c_19) = [1] p(c_20) = [2] x1 + [0] Following rules are strictly oriented: cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) = [4] x3 + [5] > [4] x3 + [1] = c_11(add#2#(Cons(x2,x1),x3)) Following rules are (at-least) weakly oriented: add#2#(Cons(x8,x6),Cons(x4,x2)) = [4] x2 + [12] >= [4] x2 + [12] = c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) = [4] x3 + [12] >= [4] x3 + [12] = c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1),lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) = [4] x3 + [12] >= [4] x3 + [12] = c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) = [4] x3 + [5] >= [4] x3 + [0] = c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)),mult#2#(S(S(0())),x2),add#2#(x1,x3)) lt#2(0(),0()) = [2] >= [2] = False() lt#2(0(),S(x16)) = [2] >= [2] = True() lt#2(S(x16),0()) = [2] >= [2] = False() lt#2(S(x4),S(x2)) = [2] >= [2] = lt#2(x4,x2) **** Step 6.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) - Weak DPs: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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 6.a:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) - Weak DPs: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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: 3: cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) The strictly oriented rules are moved into the weak component. ***** Step 6.a:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) - Weak DPs: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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_2) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1,3}, uargs(c_11) = {1} Following symbols are considered usable: {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2#,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main# ,mult#2#} TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x2 + [1] p(False) = [0] p(Nil) = [1] p(S) = [3] p(True) = [0] p(add#2) = [1] x1 + [7] p(addNat#2) = [1] p(carry#2) = [3] p(cond_add_ws1'_ws2'_2) = [2] x3 + [5] x4 + [2] p(cond_add_ws1'_ws2'_3) = [1] x3 + [1] x5 + [1] p(cond_carry_w_xs_1) = [4] x2 + [4] x3 + [4] p(lt#2) = [0] p(main) = [1] x2 + [1] p(mult#2) = [1] x1 + [0] p(add#2#) = [1] x1 + [0] p(addNat#2#) = [2] x2 + [1] p(carry#2#) = [1] p(cond_add_ws1'_ws2'_2#) = [1] x5 + [1] p(cond_add_ws1'_ws2'_3#) = [1] x5 + [1] p(cond_carry_w_xs_1#) = [2] x1 + [0] p(lt#2#) = [4] x1 + [1] p(main#) = [2] p(mult#2#) = [1] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [1] x1 + [1] p(c_6) = [1] p(c_7) = [1] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [1] x3 + [0] p(c_11) = [1] x1 + [0] p(c_12) = [2] x1 + [0] p(c_13) = [1] p(c_14) = [1] p(c_15) = [4] p(c_16) = [0] p(c_17) = [2] x1 + [4] p(c_18) = [1] p(c_19) = [1] p(c_20) = [1] Following rules are strictly oriented: cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) = [1] x1 + [1] > [1] x1 + [0] = c_9(add#2#(x1,Cons(x4,x3))) Following rules are (at-least) weakly oriented: add#2#(Cons(x8,x6),Cons(x4,x2)) = [1] x6 + [1] >= [1] x6 + [1] = c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) = [1] x1 + [1] >= [1] x1 + [1] = c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1),lt#2#(x4,x2)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) = [1] x1 + [1] >= [1] x1 + [1] = c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)),mult#2#(S(S(0())),x2),add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) = [1] x1 + [1] >= [1] x1 + [1] = c_11(add#2#(Cons(x2,x1),x3)) ***** Step 6.a:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) - Weak DPs: cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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 6.a:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) - Weak DPs: cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) 2: cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) Consider the set of all dependency pairs 1: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) 2: cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) 3: cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) 4: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) 5: cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) 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 {1,2} These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ****** Step 6.a:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) - Weak DPs: cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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_2) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1,3}, uargs(c_11) = {1} Following symbols are considered usable: {addNat#2,lt#2,mult#2,add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2#,cond_add_ws1'_ws2'_3# ,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [4] p(False) = [2] p(Nil) = [0] p(S) = [0] p(True) = [2] p(add#2) = [0] p(addNat#2) = [1] x2 + [1] p(carry#2) = [0] p(cond_add_ws1'_ws2'_2) = [4] x1 + [1] x2 + [1] x4 + [4] x5 + [0] p(cond_add_ws1'_ws2'_3) = [4] x5 + [0] p(cond_carry_w_xs_1) = [7] p(lt#2) = [2] p(main) = [1] x1 + [1] x2 + [4] p(mult#2) = [1] x2 + [1] p(add#2#) = [1] x1 + [2] x2 + [2] p(addNat#2#) = [1] x2 + [4] p(carry#2#) = [1] x1 + [3] p(cond_add_ws1'_ws2'_2#) = [4] x1 + [2] x2 + [2] x3 + [1] x4 + [1] x5 + [2] p(cond_add_ws1'_ws2'_3#) = [2] x3 + [1] x4 + [1] x5 + [6] p(cond_carry_w_xs_1#) = [2] x2 + [2] x4 + [1] p(lt#2#) = [1] p(main#) = [2] p(mult#2#) = [1] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [2] p(c_3) = [0] p(c_4) = [1] p(c_5) = [4] x1 + [1] p(c_6) = [1] p(c_7) = [1] x2 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [1] x3 + [0] p(c_11) = [1] x1 + [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] p(c_14) = [2] p(c_15) = [1] p(c_16) = [1] p(c_17) = [4] x1 + [0] p(c_18) = [1] x1 + [0] p(c_19) = [0] p(c_20) = [1] x1 + [4] x2 + [0] Following rules are strictly oriented: add#2#(Cons(x8,x6),Cons(x4,x2)) = [2] x2 + [2] x4 + [1] x6 + [1] x8 + [14] > [2] x2 + [2] x4 + [1] x6 + [1] x8 + [13] = c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) = [1] x1 + [1] x2 + [2] x3 + [2] x4 + [10] > [1] x1 + [1] x2 + [2] x3 + [6] = c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1),lt#2#(x4,x2)) Following rules are (at-least) weakly oriented: cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) = [1] x1 + [1] x2 + [2] x3 + [2] x4 + [10] >= [1] x1 + [2] x3 + [2] x4 + [10] = c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) = [1] x1 + [1] x2 + [2] x3 + [6] >= [1] x1 + [1] x2 + [2] x3 + [6] = c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)),mult#2#(S(S(0())),x2),add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) = [1] x1 + [1] x2 + [2] x3 + [6] >= [1] x1 + [1] x2 + [2] x3 + [6] = c_11(add#2#(Cons(x2,x1),x3)) addNat#2(0(),x16) = [1] x16 + [1] >= [1] x16 + [0] = x16 addNat#2(S(x4),x2) = [1] x2 + [1] >= [0] = S(addNat#2(x4,x2)) lt#2(0(),0()) = [2] >= [2] = False() lt#2(0(),S(x16)) = [2] >= [2] = True() lt#2(S(x16),0()) = [2] >= [2] = False() lt#2(S(x4),S(x2)) = [2] >= [2] = lt#2(x4,x2) mult#2(0(),x2) = [1] x2 + [1] >= [0] = 0() mult#2(S(x4),x2) = [1] x2 + [1] >= [1] x2 + [1] = addNat#2(mult#2(x4,x2),x2) ****** Step 6.a:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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 6.a:1.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) -->_1 cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))):3 -->_1 cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)):2 2:W:cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) -->_1 cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)):5 -->_1 cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)):4 3:W:cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) ,lt#2#(x8,x4)):1 4:W:cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) -->_3 add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) ,lt#2#(x8,x4)):1 5:W:cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) ,lt#2#(x8,x4)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: add#2#(Cons(x8,x6),Cons(x4,x2)) -> c_2(cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6),lt#2#(x8,x4)) 5: cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> c_11(add#2#(Cons(x2,x1),x3)) 2: cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> c_8(cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) ,lt#2#(x4,x2)) 4: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> c_10(carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) ,mult#2#(S(S(0())),x2) ,add#2#(x1,x3)) 3: cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> c_9(add#2#(x1,Cons(x4,x3))) ****** Step 6.a:1.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: DecomposeDG MAYBE + Considered Problem: - Strict DPs: addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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 add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4) carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) and a lower component addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) Further, following extension rules are added to the lower component. add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4) carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) carry#2#(x6,Cons(x4,x2)) -> lt#2#(x6,x4) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3) *** Step 6.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4) carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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: add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4) 3: cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2) 4: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2) The strictly oriented rules are moved into the weak component. **** Step 6.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4) carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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_7) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2#,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main# ,mult#2#} TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [1] p(False) = [0] p(Nil) = [2] p(S) = [2] p(True) = [2] p(add#2) = [4] p(addNat#2) = [1] x1 + [2] x2 + [4] p(carry#2) = [3] x1 + [2] x2 + [0] p(cond_add_ws1'_ws2'_2) = [4] x4 + [1] x5 + [4] p(cond_add_ws1'_ws2'_3) = [1] x1 + [2] x2 + [1] x3 + [0] p(cond_carry_w_xs_1) = [2] x1 + [6] p(lt#2) = [0] p(main) = [1] p(mult#2) = [4] x2 + [0] p(add#2#) = [2] x2 + [1] p(addNat#2#) = [1] x1 + [1] p(carry#2#) = [0] p(cond_add_ws1'_ws2'_2#) = [2] x2 + [2] x3 + [3] p(cond_add_ws1'_ws2'_3#) = [2] x3 + [3] p(cond_carry_w_xs_1#) = [0] p(lt#2#) = [0] p(main#) = [2] x1 + [2] x2 + [0] p(mult#2#) = [0] p(c_1) = [4] p(c_2) = [0] p(c_3) = [0] p(c_4) = [2] p(c_5) = [1] x1 + [2] p(c_6) = [1] p(c_7) = [4] x1 + [4] x2 + [0] p(c_8) = [4] x2 + [0] p(c_9) = [1] x1 + [1] p(c_10) = [1] x1 + [1] x3 + [1] p(c_11) = [2] x1 + [0] p(c_12) = [1] x1 + [1] x2 + [0] p(c_13) = [4] p(c_14) = [1] p(c_15) = [0] p(c_16) = [4] p(c_17) = [2] x1 + [4] p(c_18) = [1] x1 + [1] p(c_19) = [0] p(c_20) = [1] x2 + [1] Following rules are strictly oriented: add#2#(Cons(x8,x6),Cons(x4,x2)) = [2] x2 + [2] x4 + [3] > [0] = lt#2#(x8,x4) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) = [2] x3 + [2] x4 + [3] > [0] = lt#2#(x4,x2) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) = [2] x3 + [3] > [0] = mult#2#(S(S(0())),x2) Following rules are (at-least) weakly oriented: add#2#(Cons(x8,x6),Cons(x4,x2)) = [2] x2 + [2] x4 + [3] >= [2] x2 + [2] x4 + [3] = cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) carry#2#(x6,Cons(x4,x2)) = [0] >= [0] = c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) = [2] x3 + [2] x4 + [3] >= [2] x3 + [3] = cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) = [2] x3 + [2] x4 + [3] >= [2] x3 + [2] x4 + [3] = add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) = [2] x3 + [3] >= [2] x3 + [1] = add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) = [2] x3 + [3] >= [0] = carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) = [2] x3 + [3] >= [2] x3 + [1] = add#2#(Cons(x2,x1),x3) cond_carry_w_xs_1#(False(),x3,x2,x1) = [0] >= [0] = c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) **** Step 6.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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 6.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1) ,mult#2#(S(S(0())),x3)):2 2:S:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) -->_1 carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1 3:W:add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) -->_1 cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)):7 -->_1 cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1):5 -->_1 cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2):6 4:W:add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4) 5:W:cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) -->_1 cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3):11 -->_1 cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)):9 -->_1 cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3):8 -->_1 cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2):10 6:W:cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2) 7:W:cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4):4 -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6):3 8:W:cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4):4 -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6):3 9:W:cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) -->_1 carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1 10:W:cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2) 11:W:cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4):4 -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2) 10: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2) 4: add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4) **** Step 6.b:1.a:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1) ,mult#2#(S(S(0())),x3)):2 2:S:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) -->_1 carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1 3:W:add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) -->_1 cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)):7 -->_1 cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1):5 5:W:cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) -->_1 cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3):11 -->_1 cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)):9 -->_1 cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3):8 7:W:cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6):3 8:W:cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6):3 9:W:cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) -->_1 carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1 11:W:cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1)) **** Step 6.b:1.a:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1)) - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/1,c_8/2,c_9/1,c_10/3,c_11/1,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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: carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) Consider the set of all dependency pairs 1: carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) 2: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1)) 3: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) 4: cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) 5: cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) 6: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) 7: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) 8: cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) 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} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 6.b:1.a:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1)) - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/1,c_8/2,c_9/1,c_10/3,c_11/1,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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_7) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {add#2,carry#2,cond_add_ws1'_ws2'_2,cond_add_ws1'_ws2'_3,cond_carry_w_xs_1,add#2#,addNat#2#,carry#2# ,cond_add_ws1'_ws2'_2#,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} TcT has computed the following interpretation: p(0) = [1] p(Cons) = [1] x2 + [2] p(False) = [0] p(Nil) = [0] p(S) = [4] p(True) = [0] p(add#2) = [1] x1 + [1] x2 + [2] p(addNat#2) = [1] p(carry#2) = [1] x2 + [2] p(cond_add_ws1'_ws2'_2) = [1] x3 + [1] x5 + [6] p(cond_add_ws1'_ws2'_3) = [1] x3 + [1] x5 + [6] p(cond_carry_w_xs_1) = [1] x4 + [4] p(lt#2) = [0] p(main) = [1] x1 + [0] p(mult#2) = [0] p(add#2#) = [2] x1 + [2] x2 + [2] p(addNat#2#) = [2] x1 + [2] x2 + [2] p(carry#2#) = [2] x2 + [0] p(cond_add_ws1'_ws2'_2#) = [2] x3 + [2] x5 + [6] p(cond_add_ws1'_ws2'_3#) = [2] x3 + [2] x5 + [6] p(cond_carry_w_xs_1#) = [2] x4 + [0] p(lt#2#) = [0] p(main#) = [1] x2 + [0] p(mult#2#) = [2] p(c_1) = [1] p(c_2) = [4] x1 + [1] x2 + [0] p(c_3) = [4] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [1] x1 + [3] p(c_8) = [1] p(c_9) = [1] p(c_10) = [1] x1 + [1] x2 + [2] p(c_11) = [1] p(c_12) = [1] x1 + [0] p(c_13) = [2] p(c_14) = [1] p(c_15) = [1] p(c_16) = [4] p(c_17) = [2] x1 + [4] p(c_18) = [1] p(c_19) = [1] p(c_20) = [1] Following rules are strictly oriented: carry#2#(x6,Cons(x4,x2)) = [2] x2 + [4] > [2] x2 + [3] = c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) Following rules are (at-least) weakly oriented: add#2#(Cons(x8,x6),Cons(x4,x2)) = [2] x2 + [2] x6 + [10] >= [2] x2 + [2] x6 + [6] = cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) = [2] x1 + [2] x3 + [6] >= [2] x1 + [2] x3 + [6] = cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) = [2] x1 + [2] x3 + [6] >= [2] x1 + [2] x3 + [6] = add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) = [2] x1 + [2] x3 + [6] >= [2] x1 + [2] x3 + [2] = add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) = [2] x1 + [2] x3 + [6] >= [2] x1 + [2] x3 + [4] = carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) = [2] x1 + [2] x3 + [6] >= [2] x1 + [2] x3 + [6] = add#2#(Cons(x2,x1),x3) cond_carry_w_xs_1#(False(),x3,x2,x1) = [2] x1 + [0] >= [2] x1 + [0] = c_12(carry#2#(mult#2(S(S(0())),x3),x1)) add#2(x2,Nil()) = [1] x2 + [2] >= [1] x2 + [0] = x2 add#2(Cons(x8,x6),Cons(x4,x2)) = [1] x2 + [1] x6 + [6] >= [1] x2 + [1] x6 + [6] = cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) = [1] x2 + [4] >= [1] x2 + [2] = Cons(x4,x2) carry#2(x2,Nil()) = [2] >= [2] = Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) = [1] x2 + [4] >= [1] x2 + [4] = cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) = [1] x1 + [1] x3 + [6] >= [1] x1 + [1] x3 + [6] = cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) = [1] x1 + [1] x3 + [6] >= [1] x1 + [1] x3 + [6] = Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) = [1] x1 + [1] x3 + [6] >= [1] x1 + [1] x3 + [4] = carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) = [1] x1 + [1] x3 + [6] >= [1] x1 + [1] x3 + [6] = Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) = [1] x1 + [4] >= [1] x1 + [2] = carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) = [1] x1 + [4] >= [1] x1 + [4] = Cons(x3,Cons(x2,x1)) ***** Step 6.b:1.a:1.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1)) - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/1,c_8/2,c_9/1,c_10/3,c_11/1,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} 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 6.b:1.a:1.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/1,c_8/2,c_9/1,c_10/3,c_11/1,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) -->_1 cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)):4 -->_1 cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1):3 2:W:carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1)):8 3:W:cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) -->_1 cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3):7 -->_1 cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)):6 -->_1 cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3):5 4:W:cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6):1 5:W:cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6):1 6:W:cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) -->_1 carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)):2 7:W:cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) -->_1 add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6):1 8:W:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1)) -->_1 carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) 7: cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) 3: cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) 5: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) 4: cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) 6: cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) 2: carry#2#(x6,Cons(x4,x2)) -> c_7(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) 8: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_12(carry#2#(mult#2(S(S(0())),x3),x1)) ***** Step 6.b:1.a:1.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/1,c_8/2,c_9/1,c_10/3,c_11/1,c_12/1,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.b:1.b:1: NaturalMI MAYBE + Considered Problem: - Strict DPs: addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4) carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) carry#2#(x6,Cons(x4,x2)) -> lt#2#(x6,x4) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_17) = {1}, uargs(c_20) = {1,2} Following symbols are considered usable: {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2#,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main# ,mult#2#} TcT has computed the following interpretation: p(0) = [0] [0] p(Cons) = [0] [2] p(False) = [0] [0] p(Nil) = [0] [2] p(S) = [0 0] x1 + [0] [0 1] [1] p(True) = [0] [0] p(add#2) = [0 0] x1 + [0] [0 2] [0] p(addNat#2) = [2 0] x1 + [0 0] x2 + [0] [2 0] [1 0] [2] p(carry#2) = [0] [1] p(cond_add_ws1'_ws2'_2) = [0 2] x1 + [1 2] x4 + [3] [1 2] [0 0] [2] p(cond_add_ws1'_ws2'_3) = [0 3] x2 + [2] [0 0] [2] p(cond_carry_w_xs_1) = [0 3] x2 + [1 2] x3 + [2] [0 0] [0 0] [0] p(lt#2) = [0] [0] p(main) = [1] [2] p(mult#2) = [0 3] x1 + [1 0] x2 + [1] [0 0] [0 0] [0] p(add#2#) = [2] [2] p(addNat#2#) = [0 0] x2 + [0] [2 0] [0] p(carry#2#) = [2] [2] p(cond_add_ws1'_ws2'_2#) = [2] [2] p(cond_add_ws1'_ws2'_3#) = [2] [2] p(cond_carry_w_xs_1#) = [2] [2] p(lt#2#) = [0] [0] p(main#) = [2 2] x2 + [2] [1 0] [0] p(mult#2#) = [0 1] x1 + [0] [0 0] [2] p(c_1) = [0] [2] p(c_2) = [2] [1] p(c_3) = [0] [0] p(c_4) = [0] [0] p(c_5) = [2 0] x1 + [0] [1 1] [0] p(c_6) = [0] [0] p(c_7) = [1 1] x2 + [0] [1 1] [1] p(c_8) = [1] [1] p(c_9) = [0 2] x1 + [0] [0 2] [2] p(c_10) = [0 0] x1 + [1] [0 2] [1] p(c_11) = [0 2] x1 + [0] [1 0] [2] p(c_12) = [1] [0] p(c_13) = [0] [0] p(c_14) = [0] [2] p(c_15) = [2] [0] p(c_16) = [0] [0] p(c_17) = [2 1] x1 + [0] [1 0] [0] p(c_18) = [0 0] x1 + [1] [0 1] [1] p(c_19) = [2] [0] p(c_20) = [2 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] Following rules are strictly oriented: mult#2#(S(x4),x2) = [0 1] x4 + [1] [0 0] [2] > [0 1] x4 + [0] [0 0] [0] = c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) Following rules are (at-least) weakly oriented: add#2#(Cons(x8,x6),Cons(x4,x2)) = [2] [2] >= [2] [2] = cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) add#2#(Cons(x8,x6),Cons(x4,x2)) = [2] [2] >= [0] [0] = lt#2#(x8,x4) addNat#2#(S(x4),x2) = [0 0] x2 + [0] [2 0] [0] >= [0 0] x2 + [0] [2 0] [0] = c_5(addNat#2#(x4,x2)) carry#2#(x6,Cons(x4,x2)) = [2] [2] >= [2] [2] = cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) carry#2#(x6,Cons(x4,x2)) = [2] [2] >= [0] [0] = lt#2#(x6,x4) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) = [2] [2] >= [2] [2] = cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) = [2] [2] >= [0] [0] = lt#2#(x4,x2) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) = [2] [2] >= [2] [2] = add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) = [2] [2] >= [2] [2] = add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) = [2] [2] >= [2] [2] = carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) = [2] [2] >= [2] [2] = mult#2#(S(S(0())),x2) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) = [2] [2] >= [2] [2] = add#2#(Cons(x2,x1),x3) cond_carry_w_xs_1#(False(),x3,x2,x1) = [2] [2] >= [2] [2] = carry#2#(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1#(False(),x3,x2,x1) = [2] [2] >= [2] [2] = mult#2#(S(S(0())),x3) lt#2#(S(x4),S(x2)) = [0] [0] >= [0] [0] = c_17(lt#2#(x4,x2)) *** Step 6.b:1.b:2: Failure MAYBE + Considered Problem: - Strict DPs: addNat#2#(S(x4),x2) -> c_5(addNat#2#(x4,x2)) lt#2#(S(x4),S(x2)) -> c_17(lt#2#(x4,x2)) - Weak DPs: add#2#(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2#(lt#2(x8,x4),x4,x2,x8,x6) add#2#(Cons(x8,x6),Cons(x4,x2)) -> lt#2#(x8,x4) carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) carry#2#(x6,Cons(x4,x2)) -> lt#2#(x6,x4) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3#(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2#(False(),x4,x3,x2,x1) -> lt#2#(x4,x2) cond_add_ws1'_ws2'_2#(True(),x4,x3,x2,x1) -> add#2#(x1,Cons(x4,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> add#2#(x1,x3) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3#(False(),x4,x3,x2,x1) -> mult#2#(S(S(0())),x2) cond_add_ws1'_ws2'_3#(True(),x4,x3,x2,x1) -> add#2#(Cons(x2,x1),x3) cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3) mult#2#(S(x4),x2) -> c_20(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x8,x6),Cons(x4,x2)) -> cond_add_ws1'_ws2'_2(lt#2(x8,x4),x4,x2,x8,x6) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) carry#2(x2,Nil()) -> Cons(x2,Nil()) carry#2(x6,Cons(x4,x2)) -> cond_carry_w_xs_1(lt#2(x6,x4),x6,x4,x2) cond_add_ws1'_ws2'_2(False(),x4,x3,x2,x1) -> cond_add_ws1'_ws2'_3(lt#2(x4,x2),x4,x3,x2,x1) cond_add_ws1'_ws2'_2(True(),x4,x3,x2,x1) -> Cons(x2,add#2(x1,Cons(x4,x3))) cond_add_ws1'_ws2'_3(False(),x4,x3,x2,x1) -> carry#2(mult#2(S(S(0())),x2),add#2(x1,x3)) cond_add_ws1'_ws2'_3(True(),x4,x3,x2,x1) -> Cons(x4,add#2(Cons(x2,x1),x3)) cond_carry_w_xs_1(False(),x3,x2,x1) -> carry#2(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) lt#2(0(),0()) -> False() lt#2(0(),S(x16)) -> True() lt#2(S(x16),0()) -> False() lt#2(S(x4),S(x2)) -> lt#2(x4,x2) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {add#2/2,addNat#2/2,carry#2/2,cond_add_ws1'_ws2'_2/5,cond_add_ws1'_ws2'_3/5,cond_carry_w_xs_1/4,lt#2/2 ,main/2,mult#2/2,add#2#/2,addNat#2#/2,carry#2#/2,cond_add_ws1'_ws2'_2#/5,cond_add_ws1'_ws2'_3#/5 ,cond_carry_w_xs_1#/4,lt#2#/2,main#/2,mult#2#/2} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/0,c_2/2,c_3/0 ,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/3,c_11/1,c_12/2,c_13/0,c_14/0,c_15/0,c_16/0,c_17/1,c_18/1,c_19/0 ,c_20/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,addNat#2#,carry#2#,cond_add_ws1'_ws2'_2# ,cond_add_ws1'_ws2'_3#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE