MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: 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_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) -> carry#2(x2,x1) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {addNat#2/2,carry#2/2,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 {addNat#2,carry#2,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 addNat#2#(0(),x16) -> c_1() addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) carry#2#(x2,Nil()) -> c_3() carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(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_6() lt#2#(0(),0()) -> c_7() lt#2#(0(),S(x16)) -> c_8() lt#2#(S(x16),0()) -> c_9() lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) main#(x2,x1) -> c_11(carry#2#(x2,x1)) mult#2#(0(),x2) -> c_12() mult#2#(S(x4),x2) -> c_13(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: addNat#2#(0(),x16) -> c_1() addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) carry#2#(x2,Nil()) -> c_3() carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(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_6() lt#2#(0(),0()) -> c_7() lt#2#(0(),S(x16)) -> c_8() lt#2#(S(x16),0()) -> c_9() lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) main#(x2,x1) -> c_11(carry#2#(x2,x1)) mult#2#(0(),x2) -> c_12() mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: 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_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) -> carry#2(x2,x1) mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> addNat#2(mult#2(x4,x2),x2) - Signature: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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) addNat#2#(0(),x16) -> c_1() addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) carry#2#(x2,Nil()) -> c_3() carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(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_6() lt#2#(0(),0()) -> c_7() lt#2#(0(),S(x16)) -> c_8() lt#2#(S(x16),0()) -> c_9() lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) main#(x2,x1) -> c_11(carry#2#(x2,x1)) mult#2#(0(),x2) -> c_12() mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: addNat#2#(0(),x16) -> c_1() addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) carry#2#(x2,Nil()) -> c_3() carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(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_6() lt#2#(0(),0()) -> c_7() lt#2#(0(),S(x16)) -> c_8() lt#2#(S(x16),0()) -> c_9() lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) main#(x2,x1) -> c_11(carry#2#(x2,x1)) mult#2#(0(),x2) -> c_12() mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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 {1,3,6,7,8,9,12} by application of Pre({1,3,6,7,8,9,12}) = {2,4,5,10,11,13}. Here rules are labelled as follows: 1: addNat#2#(0(),x16) -> c_1() 2: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) 3: carry#2#(x2,Nil()) -> c_3() 4: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) 5: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) 6: cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6() 7: lt#2#(0(),0()) -> c_7() 8: lt#2#(0(),S(x16)) -> c_8() 9: lt#2#(S(x16),0()) -> c_9() 10: lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) 11: main#(x2,x1) -> c_11(carry#2#(x2,x1)) 12: mult#2#(0(),x2) -> c_12() 13: mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) main#(x2,x1) -> c_11(carry#2#(x2,x1)) mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak DPs: addNat#2#(0(),x16) -> c_1() carry#2#(x2,Nil()) -> c_3() cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6() lt#2#(0(),0()) -> c_7() lt#2#(0(),S(x16)) -> c_8() lt#2#(S(x16),0()) -> c_9() mult#2#(0(),x2) -> c_12() - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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:addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) -->_1 addNat#2#(0(),x16) -> c_1():7 -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1 2:S:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):4 -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1) ,mult#2#(S(S(0())),x3)):3 -->_2 lt#2#(S(x16),0()) -> c_9():12 -->_2 lt#2#(0(),S(x16)) -> c_8():11 -->_2 lt#2#(0(),0()) -> c_7():10 -->_1 cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6():9 3:S:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6 -->_1 carry#2#(x2,Nil()) -> c_3():8 -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):2 4:S:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) -->_1 lt#2#(S(x16),0()) -> c_9():12 -->_1 lt#2#(0(),S(x16)) -> c_8():11 -->_1 lt#2#(0(),0()) -> c_7():10 -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):4 5:S:main#(x2,x1) -> c_11(carry#2#(x2,x1)) -->_1 carry#2#(x2,Nil()) -> c_3():8 -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):2 6:S:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) -->_2 mult#2#(0(),x2) -> c_12():13 -->_1 addNat#2#(0(),x16) -> c_1():7 -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6 -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1 7:W:addNat#2#(0(),x16) -> c_1() 8:W:carry#2#(x2,Nil()) -> c_3() 9:W:cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6() 10:W:lt#2#(0(),0()) -> c_7() 11:W:lt#2#(0(),S(x16)) -> c_8() 12:W:lt#2#(S(x16),0()) -> c_9() 13:W:mult#2#(0(),x2) -> c_12() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: cond_carry_w_xs_1#(True(),x3,x2,x1) -> c_6() 8: carry#2#(x2,Nil()) -> c_3() 13: mult#2#(0(),x2) -> c_12() 10: lt#2#(0(),0()) -> c_7() 11: lt#2#(0(),S(x16)) -> c_8() 12: lt#2#(S(x16),0()) -> c_9() 7: addNat#2#(0(),x16) -> c_1() * Step 5: RemoveHeads MAYBE + Considered Problem: - Strict DPs: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) main#(x2,x1) -> c_11(carry#2#(x2,x1)) mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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:addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1 2:S:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):4 -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)):3 3:S:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6 -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):2 4:S:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):4 5:S:main#(x2,x1) -> c_11(carry#2#(x2,x1)) -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):2 6:S:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6 -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1 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). [(5,main#(x2,x1) -> c_11(carry#2#(x2,x1)))] * Step 6: Decompose MAYBE + Considered Problem: - Strict DPs: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main# ,mult#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2 ,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_2/1,c_3/0 ,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main# ,mult#2#} and constructors {0,Cons,False,Nil,S,True} Problem (S) - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak DPs: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2 ,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_2/1,c_3/0 ,c_4/2,c_5/2,c_6/0,c_7/0,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main# ,mult#2#} and constructors {0,Cons,False,Nil,S,True} ** Step 6.a:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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:addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1 2:W:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):4 -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1) ,mult#2#(S(S(0())),x3)):3 3:W:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):2 -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6 4:W:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):4 6:W:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1 -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) ** Step 6.a:2: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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:addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1 2:W:carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1) ,mult#2#(S(S(0())),x3)):3 3:W:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):2 -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6 6:W:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):1 -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):6 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_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) ** Step 6.a:3: DecomposeDG MAYBE + Considered Problem: - Strict DPs: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(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_2(addNat#2#(x4,x2)) mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) Further, following extension rules are added to the lower component. carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) 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.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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 = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) Consider the set of all dependency pairs 1: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) 2: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) Processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 6.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main# ,mult#2#} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(Cons) = [0 0 1] [0 0 1] [0] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [1] p(False) = [0] [0] [0] p(Nil) = [0] [0] [0] p(S) = [0 1 0] [0] [0 0 1] x1 + [1] [0 0 0] [1] p(True) = [0] [0] [0] p(addNat#2) = [0 0 0] [1 0 0] [1] [1 1 1] x1 + [0 0 0] x2 + [1] [0 1 1] [0 1 1] [1] p(carry#2) = [0] [0] [0] p(cond_carry_w_xs_1) = [0] [0] [0] p(lt#2) = [0 0 0] [0] [0 0 0] x1 + [0] [1 0 0] [0] p(main) = [0] [0] [0] p(mult#2) = [0 0 0] [0 1 0] [1] [0 0 0] x1 + [0 1 0] x2 + [0] [0 1 0] [0 0 0] [0] p(addNat#2#) = [0] [0] [0] p(carry#2#) = [0 0 1] [0] [1 0 1] x2 + [0] [0 1 1] [0] p(cond_carry_w_xs_1#) = [0 0 0] [0 0 1] [0 0 1] [1] [0 0 1] x2 + [0 0 0] x3 + [1 0 1] x4 + [1] [0 0 1] [0 0 0] [1 1 1] [1] p(lt#2#) = [0] [0] [0] p(main#) = [0] [0] [0] p(mult#2#) = [0 0 1] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(c_1) = [0] [0] [0] p(c_2) = [0] [0] [0] p(c_3) = [0] [0] [0] p(c_4) = [1 0 0] [0] [0 0 0] x1 + [1] [1 0 0] [0] p(c_5) = [1 0 0] [0 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [1] [0 0 1] [1 0 0] [0] p(c_6) = [0] [0] [0] p(c_7) = [0] [0] [0] p(c_8) = [0] [0] [0] p(c_9) = [0] [0] [0] p(c_10) = [0] [0] [0] p(c_11) = [0] [0] [0] p(c_12) = [0] [0] [0] p(c_13) = [0] [0] [0] Following rules are strictly oriented: cond_carry_w_xs_1#(False(),x3,x2,x1) = [0 0 1] [0 0 1] [0 0 0] [1] [1 0 1] x1 + [0 0 0] x2 + [0 0 1] x3 + [1] [1 1 1] [0 0 0] [0 0 1] [1] > [0 0 1] [0] [1 0 1] x1 + [1] [0 1 1] [1] = c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) Following rules are (at-least) weakly oriented: carry#2#(x6,Cons(x4,x2)) = [0 0 1] [0 0 1] [1] [0 0 2] x2 + [0 0 2] x4 + [1] [0 0 2] [0 0 1] [1] >= [0 0 1] [0 0 1] [1] [0 0 0] x2 + [0 0 0] x4 + [1] [0 0 1] [0 0 1] [1] = c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) **** Step 6.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1) ,mult#2#(S(S(0())),x3)):2 2:W:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2)) 2: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) **** Step 6.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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.a:3.b:1: DecomposeDG MAYBE + Considered Problem: - Strict DPs: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) 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_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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 carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) 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_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) and a lower component addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) Further, following extension rules are added to the lower component. carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) 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) -> addNat#2#(mult#2(x4,x2),x2) mult#2#(S(x4),x2) -> mult#2#(x4,x2) **** Step 6.a:3.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) 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: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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: mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) The strictly oriented rules are moved into the weak component. ***** Step 6.a:3.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) 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: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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_13) = {1,2} Following symbols are considered usable: {lt#2,addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} TcT has computed the following interpretation: p(0) = [10] p(Cons) = [1] x1 + [1] x2 + [2] p(False) = [2] p(Nil) = [1] p(S) = [1] x1 + [8] p(True) = [1] p(addNat#2) = [1] x2 + [14] p(carry#2) = [1] x1 + [1] p(cond_carry_w_xs_1) = [8] x1 + [1] x2 + [1] x3 + [1] x4 + [2] p(lt#2) = [1] x2 + [1] p(main) = [1] p(mult#2) = [6] p(addNat#2#) = [2] p(carry#2#) = [8] x2 + [4] p(cond_carry_w_xs_1#) = [8] x1 + [8] x4 + [12] p(lt#2#) = [1] x1 + [0] p(main#) = [1] p(mult#2#) = [1] x1 + [2] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] p(c_4) = [4] x1 + [4] p(c_5) = [1] x2 + [1] p(c_6) = [2] p(c_7) = [0] p(c_8) = [1] p(c_9) = [1] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [1] x2 + [4] Following rules are strictly oriented: mult#2#(S(x4),x2) = [1] x4 + [10] > [1] x4 + [8] = c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) Following rules are (at-least) weakly oriented: carry#2#(x6,Cons(x4,x2)) = [8] x2 + [8] x4 + [20] >= [8] x2 + [8] x4 + [20] = cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) cond_carry_w_xs_1#(False(),x3,x2,x1) = [8] x1 + [28] >= [8] x1 + [4] = carry#2#(mult#2(S(S(0())),x3),x1) cond_carry_w_xs_1#(False(),x3,x2,x1) = [8] x1 + [28] >= [28] = mult#2#(S(S(0())),x3) lt#2(0(),0()) = [11] >= [2] = False() lt#2(0(),S(x16)) = [1] x16 + [9] >= [1] = True() lt#2(S(x16),0()) = [11] >= [2] = False() lt#2(S(x4),S(x2)) = [1] x2 + [9] >= [1] x2 + [1] = lt#2(x4,x2) ***** Step 6.a:3.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) 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_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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:3.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) 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_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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:carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3):3 -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1):2 2:W:cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1) -->_1 carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2):1 3:W:cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3) -->_1 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4 4:W:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) 2: cond_carry_w_xs_1#(False(),x3,x2,x1) -> carry#2#(mult#2(S(S(0())),x3),x1) 3: cond_carry_w_xs_1#(False(),x3,x2,x1) -> mult#2#(S(S(0())),x3) 4: mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) ***** Step 6.a:3.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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.a:3.b:1.b:1: Failure MAYBE + Considered Problem: - Strict DPs: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2) 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) -> addNat#2#(mult#2(x4,x2),x2) mult#2#(S(x4),x2) -> mult#2#(x4,x2) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/1,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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. ** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak DPs: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3 -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4 -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1 3:S:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3 4:S:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):5 -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4 5:W:addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) -->_1 addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: addNat#2#(S(x4),x2) -> c_2(addNat#2#(x4,x2)) ** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3 -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4 -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1 3:S:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3 4:S:mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)) -->_2 mult#2#(S(x4),x2) -> c_13(addNat#2#(mult#2(x4,x2),x2),mult#2#(x4,x2)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2)) ** Step 6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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: mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2)) The strictly oriented rules are moved into the weak component. *** Step 6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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_4) = {1,2}, uargs(c_5) = {1,2}, uargs(c_10) = {1}, uargs(c_13) = {1} Following symbols are considered usable: {lt#2,addNat#2#,carry#2#,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 + [2] p(False) = [2] p(Nil) = [1] p(S) = [1] x1 + [4] p(True) = [6] p(addNat#2) = [4] x2 + [4] p(carry#2) = [1] x2 + [0] p(cond_carry_w_xs_1) = [4] x2 + [2] x4 + [0] p(lt#2) = [1] x2 + [2] p(main) = [1] x2 + [0] p(mult#2) = [0] p(addNat#2#) = [0] p(carry#2#) = [10] x2 + [8] p(cond_carry_w_xs_1#) = [10] x1 + [10] x4 + [8] p(lt#2#) = [0] p(main#) = [1] x1 + [1] x2 + [0] p(mult#2#) = [1] x1 + [2] p(c_1) = [2] p(c_2) = [1] x1 + [0] p(c_3) = [2] p(c_4) = [1] x1 + [8] x2 + [0] p(c_5) = [1] x1 + [2] x2 + [0] p(c_6) = [0] p(c_7) = [8] p(c_8) = [4] p(c_9) = [0] p(c_10) = [2] x1 + [0] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] Following rules are strictly oriented: mult#2#(S(x4),x2) = [1] x4 + [6] > [1] x4 + [2] = c_13(mult#2#(x4,x2)) Following rules are (at-least) weakly oriented: carry#2#(x6,Cons(x4,x2)) = [10] x2 + [10] x4 + [28] >= [10] x2 + [10] x4 + [28] = c_4(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) = [10] x1 + [28] >= [10] x1 + [28] = c_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) = [0] >= [0] = c_10(lt#2#(x4,x2)) lt#2(0(),0()) = [2] >= [2] = False() lt#2(0(),S(x16)) = [1] x16 + [6] >= [6] = True() lt#2(S(x16),0()) = [2] >= [2] = False() lt#2(S(x4),S(x2)) = [1] x2 + [6] >= [1] x2 + [2] = lt#2(x4,x2) *** Step 6.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) - Weak DPs: mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) - Weak DPs: mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3 -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) -->_2 mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2)):4 -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1 3:S:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3 4:W:mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2)) -->_1 mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: mult#2#(S(x4),x2) -> c_13(mult#2#(x4,x2)) *** Step 6.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/2,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3 -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(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_5(carry#2#(mult#2(S(S(0())),x3),x1),mult#2#(S(S(0())),x3)) -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1 3:S:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)) *** Step 6.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) 3: lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) Consider the set of all dependency pairs 1: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) 2: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)) 3: lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) 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 {1,3} These cover all (indirect) predecessors of dependency pairs {1,2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 6.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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_4) = {1,2}, uargs(c_5) = {1}, uargs(c_10) = {1} Following symbols are considered usable: {addNat#2#,carry#2#,cond_carry_w_xs_1#,lt#2#,main#,mult#2#} TcT has computed the following interpretation: p(0) = [8] p(Cons) = [1] x1 + [1] x2 + [2] p(False) = [0] p(Nil) = [2] p(S) = [1] x1 + [1] p(True) = [1] p(addNat#2) = [2] x1 + [8] x2 + [8] p(carry#2) = [0] p(cond_carry_w_xs_1) = [2] x1 + [1] x3 + [1] p(lt#2) = [0] p(main) = [2] x1 + [8] x2 + [1] p(mult#2) = [3] p(addNat#2#) = [1] x1 + [1] p(carry#2#) = [8] x2 + [0] p(cond_carry_w_xs_1#) = [8] x4 + [3] p(lt#2#) = [4] x2 + [2] p(main#) = [1] x1 + [1] p(mult#2#) = [2] x1 + [1] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] p(c_4) = [1] x1 + [2] x2 + [1] p(c_5) = [1] x1 + [3] p(c_6) = [2] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [2] p(c_11) = [1] x1 + [0] p(c_12) = [1] p(c_13) = [2] x1 + [0] Following rules are strictly oriented: carry#2#(x6,Cons(x4,x2)) = [8] x2 + [8] x4 + [16] > [8] x2 + [8] x4 + [8] = c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) lt#2#(S(x4),S(x2)) = [4] x2 + [6] > [4] x2 + [4] = c_10(lt#2#(x4,x2)) Following rules are (at-least) weakly oriented: cond_carry_w_xs_1#(False(),x3,x2,x1) = [8] x1 + [3] >= [8] x1 + [3] = c_5(carry#2#(mult#2(S(S(0())),x3),x1)) **** Step 6.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)) - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: carry#2#(x6,Cons(x4,x2)) -> c_4(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_5(carry#2#(mult#2(S(S(0())),x3),x1)) lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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:carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) -->_2 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3 -->_1 cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)):2 2:W:cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)) -->_1 carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)):1 3:W:lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) -->_1 lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: carry#2#(x6,Cons(x4,x2)) -> c_4(cond_carry_w_xs_1#(lt#2(x6,x4),x6,x4,x2),lt#2#(x6,x4)) 2: cond_carry_w_xs_1#(False(),x3,x2,x1) -> c_5(carry#2#(mult#2(S(S(0())),x3),x1)) 3: lt#2#(S(x4),S(x2)) -> c_10(lt#2#(x4,x2)) **** Step 6.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: addNat#2(0(),x16) -> x16 addNat#2(S(x4),x2) -> S(addNat#2(x4,x2)) 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: {addNat#2/2,carry#2/2,cond_carry_w_xs_1/4,lt#2/2,main/2,mult#2/2,addNat#2#/2,carry#2#/2,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_2/1,c_3/0,c_4/2,c_5/1,c_6/0,c_7/0 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {addNat#2#,carry#2#,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). MAYBE