MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: cond_fst_p#1(P(rpm_t_m_2(x2),x1),x3) -> rpm_t_m_2#1(x2,x3) cond_fst_p#1(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> rpm_t_m_9#1(x2,x3,x4,x5) fst#2(P(rpm_t_m_2(x2),x1),x3) -> rpm_t_m_2#1(x2,x3) fst#2(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> rpm_t_m_9#1(x2,x3,x4,x5) main(x1) -> cond_fst_p#1(x#1(f(x1),bot[0]()),bot[1]()) min#2(0(),x12) -> 0() min#2(S(x16),0()) -> 0() min#2(S(x4),S(x2)) -> S(min#2(x4,x2)) rpm#3(Leaf(x8),f_p(x12,x16),x4) -> P(rpm_t_m_2(f_p(x12,x16)),rpm_t_m_3(x8)) rpm#3(Node(x8,x6),f_p(x2,x4),x10) -> rpm_t_m_8#1(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) rpm_t_m_2#1(f_p(snd(),x(x2)),x3) -> Leaf(snd#2(x#1(x2,bot[2]()),bot[3]())) rpm_t_m_8#1(x3,x2,x1) -> P(rpm_t_m_9(fst(),x3,x2),rpm_t_m_10(x3,x2)) rpm_t_m_9#1(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> Node(rpm_t_m_2#1(x4,bot[5]()),fst#2(x5,bot[6]())) rpm_t_m_9#1(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> Node(rpm_t_m_9#1(x4,x6,x8,bot[5]()) ,fst#2(x5,bot[6]())) snd#2(P(x18,rpm_t_m_3(x14)),x18) -> x14 snd#2(P(x2,rpm_t_m_10(x6,x4)),x2) -> min#2(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) x#1(f(x4),x3) -> rpm#3(x4,f_p(snd(),x(f(x4))),bot[11]()) - Signature: {cond_fst_p#1/2,fst#2/2,main/1,min#2/2,rpm#3/3,rpm_t_m_2#1/2,rpm_t_m_8#1/3,rpm_t_m_9#1/4,snd#2/2 ,x#1/2} / {0/0,Leaf/1,Node/2,P/2,S/1,bot[0]/0,bot[10]/0,bot[11]/0,bot[1]/0,bot[2]/0,bot[3]/0,bot[5]/0 ,bot[6]/0,bot[7]/0,bot[8]/0,bot[9]/0,f/1,f_p/2,fst/0,rpm_t_m_10/2,rpm_t_m_2/1,rpm_t_m_3/1,rpm_t_m_9/3,snd/0 ,x/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_fst_p#1,fst#2,main,min#2,rpm#3,rpm_t_m_2#1 ,rpm_t_m_8#1,rpm_t_m_9#1,snd#2,x#1} and constructors {0,Leaf,Node,P,S,bot[0],bot[10],bot[11],bot[1],bot[2] ,bot[3],bot[5],bot[6],bot[7],bot[8],bot[9],f,f_p,fst,rpm_t_m_10,rpm_t_m_2,rpm_t_m_3,rpm_t_m_9,snd,x} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)) cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) main#(x1) -> c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) min#2#(0(),x12) -> c_6() min#2#(S(x16),0()) -> c_7() min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)) rpm#3#(Leaf(x8),f_p(x12,x16),x4) -> c_9() rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())) rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) rpm_t_m_8#1#(x3,x2,x1) -> c_12() rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())) snd#2#(P(x18,rpm_t_m_3(x14)),x18) -> c_15() snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())) x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)) cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) main#(x1) -> c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) min#2#(0(),x12) -> c_6() min#2#(S(x16),0()) -> c_7() min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)) rpm#3#(Leaf(x8),f_p(x12,x16),x4) -> c_9() rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())) rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) rpm_t_m_8#1#(x3,x2,x1) -> c_12() rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())) snd#2#(P(x18,rpm_t_m_3(x14)),x18) -> c_15() snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())) x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) - Weak TRS: cond_fst_p#1(P(rpm_t_m_2(x2),x1),x3) -> rpm_t_m_2#1(x2,x3) cond_fst_p#1(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> rpm_t_m_9#1(x2,x3,x4,x5) fst#2(P(rpm_t_m_2(x2),x1),x3) -> rpm_t_m_2#1(x2,x3) fst#2(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> rpm_t_m_9#1(x2,x3,x4,x5) main(x1) -> cond_fst_p#1(x#1(f(x1),bot[0]()),bot[1]()) min#2(0(),x12) -> 0() min#2(S(x16),0()) -> 0() min#2(S(x4),S(x2)) -> S(min#2(x4,x2)) rpm#3(Leaf(x8),f_p(x12,x16),x4) -> P(rpm_t_m_2(f_p(x12,x16)),rpm_t_m_3(x8)) rpm#3(Node(x8,x6),f_p(x2,x4),x10) -> rpm_t_m_8#1(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) rpm_t_m_2#1(f_p(snd(),x(x2)),x3) -> Leaf(snd#2(x#1(x2,bot[2]()),bot[3]())) rpm_t_m_8#1(x3,x2,x1) -> P(rpm_t_m_9(fst(),x3,x2),rpm_t_m_10(x3,x2)) rpm_t_m_9#1(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> Node(rpm_t_m_2#1(x4,bot[5]()),fst#2(x5,bot[6]())) rpm_t_m_9#1(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> Node(rpm_t_m_9#1(x4,x6,x8,bot[5]()) ,fst#2(x5,bot[6]())) snd#2(P(x18,rpm_t_m_3(x14)),x18) -> x14 snd#2(P(x2,rpm_t_m_10(x6,x4)),x2) -> min#2(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) x#1(f(x4),x3) -> rpm#3(x4,f_p(snd(),x(f(x4))),bot[11]()) - Signature: {cond_fst_p#1/2,fst#2/2,main/1,min#2/2,rpm#3/3,rpm_t_m_2#1/2,rpm_t_m_8#1/3,rpm_t_m_9#1/4,snd#2/2,x#1/2 ,cond_fst_p#1#/2,fst#2#/2,main#/1,min#2#/2,rpm#3#/3,rpm_t_m_2#1#/2,rpm_t_m_8#1#/3,rpm_t_m_9#1#/4,snd#2#/2 ,x#1#/2} / {0/0,Leaf/1,Node/2,P/2,S/1,bot[0]/0,bot[10]/0,bot[11]/0,bot[1]/0,bot[2]/0,bot[3]/0,bot[5]/0 ,bot[6]/0,bot[7]/0,bot[8]/0,bot[9]/0,f/1,f_p/2,fst/0,rpm_t_m_10/2,rpm_t_m_2/1,rpm_t_m_3/1,rpm_t_m_9/3,snd/0 ,x/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/3,c_11/2,c_12/0,c_13/2,c_14/2,c_15/0,c_16/3 ,c_17/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_fst_p#1#,fst#2#,main#,min#2#,rpm#3#,rpm_t_m_2#1# ,rpm_t_m_8#1#,rpm_t_m_9#1#,snd#2#,x#1#} and constructors {0,Leaf,Node,P,S,bot[0],bot[10],bot[11],bot[1] ,bot[2],bot[3],bot[5],bot[6],bot[7],bot[8],bot[9],f,f_p,fst,rpm_t_m_10,rpm_t_m_2,rpm_t_m_3,rpm_t_m_9,snd,x} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: min#2(0(),x12) -> 0() min#2(S(x16),0()) -> 0() min#2(S(x4),S(x2)) -> S(min#2(x4,x2)) rpm#3(Leaf(x8),f_p(x12,x16),x4) -> P(rpm_t_m_2(f_p(x12,x16)),rpm_t_m_3(x8)) rpm#3(Node(x8,x6),f_p(x2,x4),x10) -> rpm_t_m_8#1(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) rpm_t_m_8#1(x3,x2,x1) -> P(rpm_t_m_9(fst(),x3,x2),rpm_t_m_10(x3,x2)) snd#2(P(x18,rpm_t_m_3(x14)),x18) -> x14 snd#2(P(x2,rpm_t_m_10(x6,x4)),x2) -> min#2(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) x#1(f(x4),x3) -> rpm#3(x4,f_p(snd(),x(f(x4))),bot[11]()) cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)) cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) main#(x1) -> c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) min#2#(0(),x12) -> c_6() min#2#(S(x16),0()) -> c_7() min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)) rpm#3#(Leaf(x8),f_p(x12,x16),x4) -> c_9() rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())) rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) rpm_t_m_8#1#(x3,x2,x1) -> c_12() rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())) snd#2#(P(x18,rpm_t_m_3(x14)),x18) -> c_15() snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())) x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)) cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) main#(x1) -> c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) min#2#(0(),x12) -> c_6() min#2#(S(x16),0()) -> c_7() min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)) rpm#3#(Leaf(x8),f_p(x12,x16),x4) -> c_9() rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())) rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) rpm_t_m_8#1#(x3,x2,x1) -> c_12() rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())) snd#2#(P(x18,rpm_t_m_3(x14)),x18) -> c_15() snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())) x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) - Weak TRS: min#2(0(),x12) -> 0() min#2(S(x16),0()) -> 0() min#2(S(x4),S(x2)) -> S(min#2(x4,x2)) rpm#3(Leaf(x8),f_p(x12,x16),x4) -> P(rpm_t_m_2(f_p(x12,x16)),rpm_t_m_3(x8)) rpm#3(Node(x8,x6),f_p(x2,x4),x10) -> rpm_t_m_8#1(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) rpm_t_m_8#1(x3,x2,x1) -> P(rpm_t_m_9(fst(),x3,x2),rpm_t_m_10(x3,x2)) snd#2(P(x18,rpm_t_m_3(x14)),x18) -> x14 snd#2(P(x2,rpm_t_m_10(x6,x4)),x2) -> min#2(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) x#1(f(x4),x3) -> rpm#3(x4,f_p(snd(),x(f(x4))),bot[11]()) - Signature: {cond_fst_p#1/2,fst#2/2,main/1,min#2/2,rpm#3/3,rpm_t_m_2#1/2,rpm_t_m_8#1/3,rpm_t_m_9#1/4,snd#2/2,x#1/2 ,cond_fst_p#1#/2,fst#2#/2,main#/1,min#2#/2,rpm#3#/3,rpm_t_m_2#1#/2,rpm_t_m_8#1#/3,rpm_t_m_9#1#/4,snd#2#/2 ,x#1#/2} / {0/0,Leaf/1,Node/2,P/2,S/1,bot[0]/0,bot[10]/0,bot[11]/0,bot[1]/0,bot[2]/0,bot[3]/0,bot[5]/0 ,bot[6]/0,bot[7]/0,bot[8]/0,bot[9]/0,f/1,f_p/2,fst/0,rpm_t_m_10/2,rpm_t_m_2/1,rpm_t_m_3/1,rpm_t_m_9/3,snd/0 ,x/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/3,c_11/2,c_12/0,c_13/2,c_14/2,c_15/0,c_16/3 ,c_17/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_fst_p#1#,fst#2#,main#,min#2#,rpm#3#,rpm_t_m_2#1# ,rpm_t_m_8#1#,rpm_t_m_9#1#,snd#2#,x#1#} and constructors {0,Leaf,Node,P,S,bot[0],bot[10],bot[11],bot[1] ,bot[2],bot[3],bot[5],bot[6],bot[7],bot[8],bot[9],f,f_p,fst,rpm_t_m_10,rpm_t_m_2,rpm_t_m_3,rpm_t_m_9,snd,x} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {6,7,9,12,15} by application of Pre({6,7,9,12,15}) = {8,10,11,16,17}. Here rules are labelled as follows: 1: cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)) 2: cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) 3: fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)) 4: fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) 5: main#(x1) -> c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) 6: min#2#(0(),x12) -> c_6() 7: min#2#(S(x16),0()) -> c_7() 8: min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)) 9: rpm#3#(Leaf(x8),f_p(x12,x16),x4) -> c_9() 10: rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())) 11: rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) 12: rpm_t_m_8#1#(x3,x2,x1) -> c_12() 13: rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) 14: rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())) 15: snd#2#(P(x18,rpm_t_m_3(x14)),x18) -> c_15() 16: snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())) 17: x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)) cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) main#(x1) -> c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)) rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())) rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())) snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())) x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) - Weak DPs: min#2#(0(),x12) -> c_6() min#2#(S(x16),0()) -> c_7() rpm#3#(Leaf(x8),f_p(x12,x16),x4) -> c_9() rpm_t_m_8#1#(x3,x2,x1) -> c_12() snd#2#(P(x18,rpm_t_m_3(x14)),x18) -> c_15() - Weak TRS: min#2(0(),x12) -> 0() min#2(S(x16),0()) -> 0() min#2(S(x4),S(x2)) -> S(min#2(x4,x2)) rpm#3(Leaf(x8),f_p(x12,x16),x4) -> P(rpm_t_m_2(f_p(x12,x16)),rpm_t_m_3(x8)) rpm#3(Node(x8,x6),f_p(x2,x4),x10) -> rpm_t_m_8#1(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) rpm_t_m_8#1(x3,x2,x1) -> P(rpm_t_m_9(fst(),x3,x2),rpm_t_m_10(x3,x2)) snd#2(P(x18,rpm_t_m_3(x14)),x18) -> x14 snd#2(P(x2,rpm_t_m_10(x6,x4)),x2) -> min#2(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) x#1(f(x4),x3) -> rpm#3(x4,f_p(snd(),x(f(x4))),bot[11]()) - Signature: {cond_fst_p#1/2,fst#2/2,main/1,min#2/2,rpm#3/3,rpm_t_m_2#1/2,rpm_t_m_8#1/3,rpm_t_m_9#1/4,snd#2/2,x#1/2 ,cond_fst_p#1#/2,fst#2#/2,main#/1,min#2#/2,rpm#3#/3,rpm_t_m_2#1#/2,rpm_t_m_8#1#/3,rpm_t_m_9#1#/4,snd#2#/2 ,x#1#/2} / {0/0,Leaf/1,Node/2,P/2,S/1,bot[0]/0,bot[10]/0,bot[11]/0,bot[1]/0,bot[2]/0,bot[3]/0,bot[5]/0 ,bot[6]/0,bot[7]/0,bot[8]/0,bot[9]/0,f/1,f_p/2,fst/0,rpm_t_m_10/2,rpm_t_m_2/1,rpm_t_m_3/1,rpm_t_m_9/3,snd/0 ,x/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/3,c_11/2,c_12/0,c_13/2,c_14/2,c_15/0,c_16/3 ,c_17/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_fst_p#1#,fst#2#,main#,min#2#,rpm#3#,rpm_t_m_2#1# ,rpm_t_m_8#1#,rpm_t_m_9#1#,snd#2#,x#1#} and constructors {0,Leaf,Node,P,S,bot[0],bot[10],bot[11],bot[1] ,bot[2],bot[3],bot[5],bot[6],bot[7],bot[8],bot[9],f,f_p,fst,rpm_t_m_10,rpm_t_m_2,rpm_t_m_3,rpm_t_m_9,snd,x} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)) -->_1 rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())):8 2:S:cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) -->_1 rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())):10 -->_1 rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()) ,fst#2#(x5,bot[6]())):9 3:S:fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)) -->_1 rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())):8 4:S:fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) -->_1 rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())):10 -->_1 rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()) ,fst#2#(x5,bot[6]())):9 5:S:main#(x1) -> c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) -->_2 x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())):12 -->_1 cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)):2 -->_1 cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)):1 6:S:min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)) -->_1 min#2#(S(x16),0()) -> c_7():14 -->_1 min#2#(0(),x12) -> c_6():13 -->_1 min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)):6 7:S:rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())) -->_1 rpm_t_m_8#1#(x3,x2,x1) -> c_12():16 -->_3 rpm#3#(Leaf(x8),f_p(x12,x16),x4) -> c_9():15 -->_2 rpm#3#(Leaf(x8),f_p(x12,x16),x4) -> c_9():15 -->_3 rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())):7 -->_2 rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())):7 8:S:rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) -->_2 x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())):12 -->_1 snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())):11 -->_1 snd#2#(P(x18,rpm_t_m_3(x14)),x18) -> c_15():17 9:S:rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) -->_1 rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())):8 -->_2 fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)):4 -->_2 fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)):3 10:S:rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())) -->_1 rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())):10 -->_1 rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()) ,fst#2#(x5,bot[6]())):9 -->_2 fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)):4 -->_2 fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)):3 11:S:snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())) -->_3 snd#2#(P(x18,rpm_t_m_3(x14)),x18) -> c_15():17 -->_2 snd#2#(P(x18,rpm_t_m_3(x14)),x18) -> c_15():17 -->_1 min#2#(S(x16),0()) -> c_7():14 -->_1 min#2#(0(),x12) -> c_6():13 -->_3 snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())):11 -->_2 snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())):11 -->_1 min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)):6 12:S:x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) -->_1 rpm#3#(Leaf(x8),f_p(x12,x16),x4) -> c_9():15 -->_1 rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())):7 13:W:min#2#(0(),x12) -> c_6() 14:W:min#2#(S(x16),0()) -> c_7() 15:W:rpm#3#(Leaf(x8),f_p(x12,x16),x4) -> c_9() 16:W:rpm_t_m_8#1#(x3,x2,x1) -> c_12() 17:W:snd#2#(P(x18,rpm_t_m_3(x14)),x18) -> c_15() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: min#2#(0(),x12) -> c_6() 14: min#2#(S(x16),0()) -> c_7() 17: snd#2#(P(x18,rpm_t_m_3(x14)),x18) -> c_15() 16: rpm_t_m_8#1#(x3,x2,x1) -> c_12() 15: rpm#3#(Leaf(x8),f_p(x12,x16),x4) -> c_9() * Step 5: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)) cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) main#(x1) -> c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)) rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())) rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())) snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())) x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) - Weak TRS: min#2(0(),x12) -> 0() min#2(S(x16),0()) -> 0() min#2(S(x4),S(x2)) -> S(min#2(x4,x2)) rpm#3(Leaf(x8),f_p(x12,x16),x4) -> P(rpm_t_m_2(f_p(x12,x16)),rpm_t_m_3(x8)) rpm#3(Node(x8,x6),f_p(x2,x4),x10) -> rpm_t_m_8#1(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) rpm_t_m_8#1(x3,x2,x1) -> P(rpm_t_m_9(fst(),x3,x2),rpm_t_m_10(x3,x2)) snd#2(P(x18,rpm_t_m_3(x14)),x18) -> x14 snd#2(P(x2,rpm_t_m_10(x6,x4)),x2) -> min#2(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) x#1(f(x4),x3) -> rpm#3(x4,f_p(snd(),x(f(x4))),bot[11]()) - Signature: {cond_fst_p#1/2,fst#2/2,main/1,min#2/2,rpm#3/3,rpm_t_m_2#1/2,rpm_t_m_8#1/3,rpm_t_m_9#1/4,snd#2/2,x#1/2 ,cond_fst_p#1#/2,fst#2#/2,main#/1,min#2#/2,rpm#3#/3,rpm_t_m_2#1#/2,rpm_t_m_8#1#/3,rpm_t_m_9#1#/4,snd#2#/2 ,x#1#/2} / {0/0,Leaf/1,Node/2,P/2,S/1,bot[0]/0,bot[10]/0,bot[11]/0,bot[1]/0,bot[2]/0,bot[3]/0,bot[5]/0 ,bot[6]/0,bot[7]/0,bot[8]/0,bot[9]/0,f/1,f_p/2,fst/0,rpm_t_m_10/2,rpm_t_m_2/1,rpm_t_m_3/1,rpm_t_m_9/3,snd/0 ,x/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/3,c_11/2,c_12/0,c_13/2,c_14/2,c_15/0,c_16/3 ,c_17/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_fst_p#1#,fst#2#,main#,min#2#,rpm#3#,rpm_t_m_2#1# ,rpm_t_m_8#1#,rpm_t_m_9#1#,snd#2#,x#1#} and constructors {0,Leaf,Node,P,S,bot[0],bot[10],bot[11],bot[1] ,bot[2],bot[3],bot[5],bot[6],bot[7],bot[8],bot[9],f,f_p,fst,rpm_t_m_10,rpm_t_m_2,rpm_t_m_3,rpm_t_m_9,snd,x} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)) -->_1 rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())):8 2:S:cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) -->_1 rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())):10 -->_1 rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()) ,fst#2#(x5,bot[6]())):9 3:S:fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)) -->_1 rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())):8 4:S:fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) -->_1 rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())):10 -->_1 rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()) ,fst#2#(x5,bot[6]())):9 5:S:main#(x1) -> c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) -->_2 x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())):12 -->_1 cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)):2 -->_1 cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)):1 6:S:min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)) -->_1 min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)):6 7:S:rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())) -->_3 rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())):7 -->_2 rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())):7 8:S:rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) -->_2 x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())):12 -->_1 snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())):11 9:S:rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) -->_1 rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())):8 -->_2 fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)):4 -->_2 fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)):3 10:S:rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())) -->_1 rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())):10 -->_1 rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()) ,fst#2#(x5,bot[6]())):9 -->_2 fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)):4 -->_2 fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)):3 11:S:snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())) -->_3 snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())):11 -->_2 snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())):11 -->_1 min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)):6 12:S:x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) -->_1 rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm_t_m_8#1#(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) ,rpm#3#(x8,f_p(x2,x4),bot[9]()) ,rpm#3#(x6,f_p(x2,x4),bot[10]())):7 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm#3#(x8,f_p(x2,x4),bot[9]()),rpm#3#(x6,f_p(x2,x4),bot[10]())) * Step 6: NaturalMI MAYBE + Considered Problem: - Strict DPs: cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)) cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) main#(x1) -> c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)) rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm#3#(x8,f_p(x2,x4),bot[9]()),rpm#3#(x6,f_p(x2,x4),bot[10]())) rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())) snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())) x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) - Weak TRS: min#2(0(),x12) -> 0() min#2(S(x16),0()) -> 0() min#2(S(x4),S(x2)) -> S(min#2(x4,x2)) rpm#3(Leaf(x8),f_p(x12,x16),x4) -> P(rpm_t_m_2(f_p(x12,x16)),rpm_t_m_3(x8)) rpm#3(Node(x8,x6),f_p(x2,x4),x10) -> rpm_t_m_8#1(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) rpm_t_m_8#1(x3,x2,x1) -> P(rpm_t_m_9(fst(),x3,x2),rpm_t_m_10(x3,x2)) snd#2(P(x18,rpm_t_m_3(x14)),x18) -> x14 snd#2(P(x2,rpm_t_m_10(x6,x4)),x2) -> min#2(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) x#1(f(x4),x3) -> rpm#3(x4,f_p(snd(),x(f(x4))),bot[11]()) - Signature: {cond_fst_p#1/2,fst#2/2,main/1,min#2/2,rpm#3/3,rpm_t_m_2#1/2,rpm_t_m_8#1/3,rpm_t_m_9#1/4,snd#2/2,x#1/2 ,cond_fst_p#1#/2,fst#2#/2,main#/1,min#2#/2,rpm#3#/3,rpm_t_m_2#1#/2,rpm_t_m_8#1#/3,rpm_t_m_9#1#/4,snd#2#/2 ,x#1#/2} / {0/0,Leaf/1,Node/2,P/2,S/1,bot[0]/0,bot[10]/0,bot[11]/0,bot[1]/0,bot[2]/0,bot[3]/0,bot[5]/0 ,bot[6]/0,bot[7]/0,bot[8]/0,bot[9]/0,f/1,f_p/2,fst/0,rpm_t_m_10/2,rpm_t_m_2/1,rpm_t_m_3/1,rpm_t_m_9/3,snd/0 ,x/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/2,c_12/0,c_13/2,c_14/2,c_15/0,c_16/3 ,c_17/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_fst_p#1#,fst#2#,main#,min#2#,rpm#3#,rpm_t_m_2#1# ,rpm_t_m_8#1#,rpm_t_m_9#1#,snd#2#,x#1#} and constructors {0,Leaf,Node,P,S,bot[0],bot[10],bot[11],bot[1] ,bot[2],bot[3],bot[5],bot[6],bot[7],bot[8],bot[9],f,f_p,fst,rpm_t_m_10,rpm_t_m_2,rpm_t_m_3,rpm_t_m_9,snd,x} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_8) = {1}, uargs(c_10) = {1,2}, uargs(c_11) = {1,2}, uargs(c_13) = {1,2}, uargs(c_14) = {1,2}, uargs(c_16) = {1,2,3}, uargs(c_17) = {1} Following symbols are considered usable: {cond_fst_p#1#,fst#2#,main#,min#2#,rpm#3#,rpm_t_m_2#1#,rpm_t_m_8#1#,rpm_t_m_9#1#,snd#2#,x#1#} TcT has computed the following interpretation: p(0) = [4] p(Leaf) = [0] p(Node) = [2] p(P) = [0] p(S) = [2] p(bot[0]) = [0] p(bot[10]) = [2] p(bot[11]) = [0] p(bot[1]) = [0] p(bot[2]) = [0] p(bot[3]) = [4] p(bot[5]) = [4] p(bot[6]) = [0] p(bot[7]) = [4] p(bot[8]) = [0] p(bot[9]) = [0] p(cond_fst_p#1) = [4] p(f) = [0] p(f_p) = [0] p(fst) = [0] p(fst#2) = [1] x2 + [1] p(main) = [2] p(min#2) = [2] x1 + [0] p(rpm#3) = [4] x1 + [4] x3 + [1] p(rpm_t_m_10) = [0] p(rpm_t_m_2) = [0] p(rpm_t_m_2#1) = [1] x2 + [0] p(rpm_t_m_3) = [0] p(rpm_t_m_8#1) = [2] x1 + [1] x3 + [0] p(rpm_t_m_9) = [2] p(rpm_t_m_9#1) = [1] x3 + [0] p(snd) = [1] p(snd#2) = [4] x1 + [4] p(x) = [3] p(x#1) = [2] x1 + [1] p(cond_fst_p#1#) = [1] x2 + [0] p(fst#2#) = [2] x2 + [0] p(main#) = [4] p(min#2#) = [0] p(rpm#3#) = [0] p(rpm_t_m_2#1#) = [0] p(rpm_t_m_8#1#) = [1] x2 + [0] p(rpm_t_m_9#1#) = [0] p(snd#2#) = [0] p(x#1#) = [0] p(c_1) = [1] x1 + [0] p(c_2) = [2] x1 + [0] p(c_3) = [4] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [4] x1 + [2] x2 + [2] p(c_6) = [1] p(c_7) = [1] p(c_8) = [4] x1 + [0] p(c_9) = [0] p(c_10) = [1] x1 + [1] x2 + [0] p(c_11) = [1] x1 + [2] x2 + [0] p(c_12) = [1] p(c_13) = [4] x1 + [2] x2 + [0] p(c_14) = [1] x1 + [1] x2 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [4] x2 + [4] x3 + [0] p(c_17) = [1] x1 + [0] Following rules are strictly oriented: main#(x1) = [4] > [2] = c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) Following rules are (at-least) weakly oriented: cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) = [1] x3 + [0] >= [0] = c_1(rpm_t_m_2#1#(x2,x3)) cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) = [1] x5 + [0] >= [0] = c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) fst#2#(P(rpm_t_m_2(x2),x1),x3) = [2] x3 + [0] >= [0] = c_3(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) = [2] x5 + [0] >= [0] = c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) min#2#(S(x4),S(x2)) = [0] >= [0] = c_8(min#2#(x4,x2)) rpm#3#(Node(x8,x6),f_p(x2,x4),x10) = [0] >= [0] = c_10(rpm#3#(x8,f_p(x2,x4),bot[9]()),rpm#3#(x6,f_p(x2,x4),bot[10]())) rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) = [0] >= [0] = c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) = [0] >= [0] = c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) = [0] >= [0] = c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()),fst#2#(x5,bot[6]())) snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) = [0] >= [0] = c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())),snd#2#(x6,bot[7]()),snd#2#(x4,bot[8]())) x#1#(f(x4),x3) = [0] >= [0] = c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) * Step 7: NaturalMI MAYBE + Considered Problem: - Strict DPs: cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)) cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)) rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm#3#(x8,f_p(x2,x4),bot[9]()),rpm#3#(x6,f_p(x2,x4),bot[10]())) rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())) snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())) x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) - Weak DPs: main#(x1) -> c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) - Weak TRS: min#2(0(),x12) -> 0() min#2(S(x16),0()) -> 0() min#2(S(x4),S(x2)) -> S(min#2(x4,x2)) rpm#3(Leaf(x8),f_p(x12,x16),x4) -> P(rpm_t_m_2(f_p(x12,x16)),rpm_t_m_3(x8)) rpm#3(Node(x8,x6),f_p(x2,x4),x10) -> rpm_t_m_8#1(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) rpm_t_m_8#1(x3,x2,x1) -> P(rpm_t_m_9(fst(),x3,x2),rpm_t_m_10(x3,x2)) snd#2(P(x18,rpm_t_m_3(x14)),x18) -> x14 snd#2(P(x2,rpm_t_m_10(x6,x4)),x2) -> min#2(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) x#1(f(x4),x3) -> rpm#3(x4,f_p(snd(),x(f(x4))),bot[11]()) - Signature: {cond_fst_p#1/2,fst#2/2,main/1,min#2/2,rpm#3/3,rpm_t_m_2#1/2,rpm_t_m_8#1/3,rpm_t_m_9#1/4,snd#2/2,x#1/2 ,cond_fst_p#1#/2,fst#2#/2,main#/1,min#2#/2,rpm#3#/3,rpm_t_m_2#1#/2,rpm_t_m_8#1#/3,rpm_t_m_9#1#/4,snd#2#/2 ,x#1#/2} / {0/0,Leaf/1,Node/2,P/2,S/1,bot[0]/0,bot[10]/0,bot[11]/0,bot[1]/0,bot[2]/0,bot[3]/0,bot[5]/0 ,bot[6]/0,bot[7]/0,bot[8]/0,bot[9]/0,f/1,f_p/2,fst/0,rpm_t_m_10/2,rpm_t_m_2/1,rpm_t_m_3/1,rpm_t_m_9/3,snd/0 ,x/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/2,c_12/0,c_13/2,c_14/2,c_15/0,c_16/3 ,c_17/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_fst_p#1#,fst#2#,main#,min#2#,rpm#3#,rpm_t_m_2#1# ,rpm_t_m_8#1#,rpm_t_m_9#1#,snd#2#,x#1#} and constructors {0,Leaf,Node,P,S,bot[0],bot[10],bot[11],bot[1] ,bot[2],bot[3],bot[5],bot[6],bot[7],bot[8],bot[9],f,f_p,fst,rpm_t_m_10,rpm_t_m_2,rpm_t_m_3,rpm_t_m_9,snd,x} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_8) = {1}, uargs(c_10) = {1,2}, uargs(c_11) = {1,2}, uargs(c_13) = {1,2}, uargs(c_14) = {1,2}, uargs(c_16) = {1,2,3}, uargs(c_17) = {1} Following symbols are considered usable: {cond_fst_p#1#,fst#2#,main#,min#2#,rpm#3#,rpm_t_m_2#1#,rpm_t_m_8#1#,rpm_t_m_9#1#,snd#2#,x#1#} TcT has computed the following interpretation: p(0) = [0] p(Leaf) = [0] p(Node) = [4] p(P) = [0] p(S) = [2] p(bot[0]) = [0] p(bot[10]) = [2] p(bot[11]) = [0] p(bot[1]) = [1] p(bot[2]) = [2] p(bot[3]) = [4] p(bot[5]) = [1] p(bot[6]) = [0] p(bot[7]) = [1] p(bot[8]) = [0] p(bot[9]) = [1] p(cond_fst_p#1) = [1] x2 + [1] p(f) = [1] p(f_p) = [0] p(fst) = [0] p(fst#2) = [1] p(main) = [4] p(min#2) = [1] x2 + [2] p(rpm#3) = [1] x1 + [2] x3 + [0] p(rpm_t_m_10) = [1] p(rpm_t_m_2) = [0] p(rpm_t_m_2#1) = [1] x1 + [0] p(rpm_t_m_3) = [1] p(rpm_t_m_8#1) = [1] x1 + [1] x3 + [6] p(rpm_t_m_9) = [1] p(rpm_t_m_9#1) = [1] x1 + [0] p(snd) = [1] p(snd#2) = [4] x2 + [0] p(x) = [0] p(x#1) = [5] x1 + [2] x2 + [0] p(cond_fst_p#1#) = [3] p(fst#2#) = [0] p(main#) = [1] x1 + [3] p(min#2#) = [0] p(rpm#3#) = [0] p(rpm_t_m_2#1#) = [0] p(rpm_t_m_8#1#) = [1] x1 + [1] p(rpm_t_m_9#1#) = [0] p(snd#2#) = [0] p(x#1#) = [0] p(c_1) = [4] x1 + [1] p(c_2) = [2] x1 + [3] p(c_3) = [2] x1 + [0] p(c_4) = [4] x1 + [0] p(c_5) = [1] x1 + [1] x2 + [0] p(c_6) = [0] p(c_7) = [1] p(c_8) = [2] x1 + [0] p(c_9) = [1] p(c_10) = [4] x1 + [2] x2 + [0] p(c_11) = [2] x1 + [2] x2 + [0] p(c_12) = [0] p(c_13) = [2] x1 + [4] x2 + [0] p(c_14) = [4] x1 + [1] x2 + [0] p(c_15) = [4] p(c_16) = [1] x1 + [1] x2 + [2] x3 + [0] p(c_17) = [4] x1 + [0] Following rules are strictly oriented: cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) = [3] > [1] = c_1(rpm_t_m_2#1#(x2,x3)) Following rules are (at-least) weakly oriented: cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) = [3] >= [3] = c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) fst#2#(P(rpm_t_m_2(x2),x1),x3) = [0] >= [0] = c_3(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) = [0] >= [0] = c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) main#(x1) = [1] x1 + [3] >= [3] = c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) min#2#(S(x4),S(x2)) = [0] >= [0] = c_8(min#2#(x4,x2)) rpm#3#(Node(x8,x6),f_p(x2,x4),x10) = [0] >= [0] = c_10(rpm#3#(x8,f_p(x2,x4),bot[9]()),rpm#3#(x6,f_p(x2,x4),bot[10]())) rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) = [0] >= [0] = c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) = [0] >= [0] = c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) = [0] >= [0] = c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()),fst#2#(x5,bot[6]())) snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) = [0] >= [0] = c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())),snd#2#(x6,bot[7]()),snd#2#(x4,bot[8]())) x#1#(f(x4),x3) = [0] >= [0] = c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) * Step 8: NaturalMI MAYBE + Considered Problem: - Strict DPs: cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)) rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm#3#(x8,f_p(x2,x4),bot[9]()),rpm#3#(x6,f_p(x2,x4),bot[10]())) rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())) snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())) x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) - Weak DPs: cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)) main#(x1) -> c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) - Weak TRS: min#2(0(),x12) -> 0() min#2(S(x16),0()) -> 0() min#2(S(x4),S(x2)) -> S(min#2(x4,x2)) rpm#3(Leaf(x8),f_p(x12,x16),x4) -> P(rpm_t_m_2(f_p(x12,x16)),rpm_t_m_3(x8)) rpm#3(Node(x8,x6),f_p(x2,x4),x10) -> rpm_t_m_8#1(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) rpm_t_m_8#1(x3,x2,x1) -> P(rpm_t_m_9(fst(),x3,x2),rpm_t_m_10(x3,x2)) snd#2(P(x18,rpm_t_m_3(x14)),x18) -> x14 snd#2(P(x2,rpm_t_m_10(x6,x4)),x2) -> min#2(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) x#1(f(x4),x3) -> rpm#3(x4,f_p(snd(),x(f(x4))),bot[11]()) - Signature: {cond_fst_p#1/2,fst#2/2,main/1,min#2/2,rpm#3/3,rpm_t_m_2#1/2,rpm_t_m_8#1/3,rpm_t_m_9#1/4,snd#2/2,x#1/2 ,cond_fst_p#1#/2,fst#2#/2,main#/1,min#2#/2,rpm#3#/3,rpm_t_m_2#1#/2,rpm_t_m_8#1#/3,rpm_t_m_9#1#/4,snd#2#/2 ,x#1#/2} / {0/0,Leaf/1,Node/2,P/2,S/1,bot[0]/0,bot[10]/0,bot[11]/0,bot[1]/0,bot[2]/0,bot[3]/0,bot[5]/0 ,bot[6]/0,bot[7]/0,bot[8]/0,bot[9]/0,f/1,f_p/2,fst/0,rpm_t_m_10/2,rpm_t_m_2/1,rpm_t_m_3/1,rpm_t_m_9/3,snd/0 ,x/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/2,c_12/0,c_13/2,c_14/2,c_15/0,c_16/3 ,c_17/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_fst_p#1#,fst#2#,main#,min#2#,rpm#3#,rpm_t_m_2#1# ,rpm_t_m_8#1#,rpm_t_m_9#1#,snd#2#,x#1#} and constructors {0,Leaf,Node,P,S,bot[0],bot[10],bot[11],bot[1] ,bot[2],bot[3],bot[5],bot[6],bot[7],bot[8],bot[9],f,f_p,fst,rpm_t_m_10,rpm_t_m_2,rpm_t_m_3,rpm_t_m_9,snd,x} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1,2}, uargs(c_8) = {1}, uargs(c_10) = {1,2}, uargs(c_11) = {1,2}, uargs(c_13) = {1,2}, uargs(c_14) = {1,2}, uargs(c_16) = {1,2,3}, uargs(c_17) = {1} Following symbols are considered usable: {cond_fst_p#1#,fst#2#,main#,min#2#,rpm#3#,rpm_t_m_2#1#,rpm_t_m_8#1#,rpm_t_m_9#1#,snd#2#,x#1#} TcT has computed the following interpretation: p(0) = [1] p(Leaf) = [0] p(Node) = [1] p(P) = [0] p(S) = [3] p(bot[0]) = [1] p(bot[10]) = [4] p(bot[11]) = [4] p(bot[1]) = [0] p(bot[2]) = [1] p(bot[3]) = [1] p(bot[5]) = [0] p(bot[6]) = [0] p(bot[7]) = [0] p(bot[8]) = [5] p(bot[9]) = [1] p(cond_fst_p#1) = [1] x1 + [1] x2 + [0] p(f) = [1] p(f_p) = [0] p(fst) = [4] p(fst#2) = [1] x2 + [4] p(main) = [2] p(min#2) = [4] x1 + [0] p(rpm#3) = [2] x1 + [3] x3 + [2] p(rpm_t_m_10) = [0] p(rpm_t_m_2) = [1] p(rpm_t_m_2#1) = [1] x1 + [1] x2 + [0] p(rpm_t_m_3) = [1] p(rpm_t_m_8#1) = [1] x1 + [1] x3 + [1] p(rpm_t_m_9) = [0] p(rpm_t_m_9#1) = [1] x1 + [1] x4 + [2] p(snd) = [0] p(snd#2) = [1] x2 + [1] p(x) = [0] p(x#1) = [5] x2 + [0] p(cond_fst_p#1#) = [5] x2 + [2] p(fst#2#) = [4] x2 + [0] p(main#) = [4] x1 + [4] p(min#2#) = [0] p(rpm#3#) = [4] x2 + [0] p(rpm_t_m_2#1#) = [0] p(rpm_t_m_8#1#) = [1] x3 + [0] p(rpm_t_m_9#1#) = [4] x4 + [0] p(snd#2#) = [0] p(x#1#) = [0] p(c_1) = [1] x1 + [2] p(c_2) = [1] x1 + [0] p(c_3) = [2] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [2] x1 + [4] x2 + [0] p(c_6) = [1] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] p(c_10) = [4] x1 + [4] x2 + [0] p(c_11) = [4] x1 + [4] x2 + [0] p(c_12) = [0] p(c_13) = [1] x1 + [4] x2 + [0] p(c_14) = [4] x1 + [4] x2 + [0] p(c_15) = [2] p(c_16) = [1] x1 + [1] x2 + [4] x3 + [0] p(c_17) = [4] x1 + [0] Following rules are strictly oriented: cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) = [5] x5 + [2] > [4] x5 + [0] = c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) Following rules are (at-least) weakly oriented: cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) = [5] x3 + [2] >= [2] = c_1(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_2(x2),x1),x3) = [4] x3 + [0] >= [0] = c_3(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) = [4] x5 + [0] >= [4] x5 + [0] = c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) main#(x1) = [4] x1 + [4] >= [4] = c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) min#2#(S(x4),S(x2)) = [0] >= [0] = c_8(min#2#(x4,x2)) rpm#3#(Node(x8,x6),f_p(x2,x4),x10) = [0] >= [0] = c_10(rpm#3#(x8,f_p(x2,x4),bot[9]()),rpm#3#(x6,f_p(x2,x4),bot[10]())) rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) = [0] >= [0] = c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) = [4] x3 + [0] >= [0] = c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) = [4] x3 + [0] >= [0] = c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()),fst#2#(x5,bot[6]())) snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) = [0] >= [0] = c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())),snd#2#(x6,bot[7]()),snd#2#(x4,bot[8]())) x#1#(f(x4),x3) = [0] >= [0] = c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) * Step 9: Failure MAYBE + Considered Problem: - Strict DPs: fst#2#(P(rpm_t_m_2(x2),x1),x3) -> c_3(rpm_t_m_2#1#(x2,x3)) fst#2#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_4(rpm_t_m_9#1#(x2,x3,x4,x5)) min#2#(S(x4),S(x2)) -> c_8(min#2#(x4,x2)) rpm#3#(Node(x8,x6),f_p(x2,x4),x10) -> c_10(rpm#3#(x8,f_p(x2,x4),bot[9]()),rpm#3#(x6,f_p(x2,x4),bot[10]())) rpm_t_m_2#1#(f_p(snd(),x(x2)),x3) -> c_11(snd#2#(x#1(x2,bot[2]()),bot[3]()),x#1#(x2,bot[2]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_2(x4),x2),x5,x3) -> c_13(rpm_t_m_2#1#(x4,bot[5]()),fst#2#(x5,bot[6]())) rpm_t_m_9#1#(fst(),P(rpm_t_m_9(x4,x6,x8),x2),x5,x3) -> c_14(rpm_t_m_9#1#(x4,x6,x8,bot[5]()) ,fst#2#(x5,bot[6]())) snd#2#(P(x2,rpm_t_m_10(x6,x4)),x2) -> c_16(min#2#(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) ,snd#2#(x6,bot[7]()) ,snd#2#(x4,bot[8]())) x#1#(f(x4),x3) -> c_17(rpm#3#(x4,f_p(snd(),x(f(x4))),bot[11]())) - Weak DPs: cond_fst_p#1#(P(rpm_t_m_2(x2),x1),x3) -> c_1(rpm_t_m_2#1#(x2,x3)) cond_fst_p#1#(P(rpm_t_m_9(x2,x3,x4),x1),x5) -> c_2(rpm_t_m_9#1#(x2,x3,x4,x5)) main#(x1) -> c_5(cond_fst_p#1#(x#1(f(x1),bot[0]()),bot[1]()),x#1#(f(x1),bot[0]())) - Weak TRS: min#2(0(),x12) -> 0() min#2(S(x16),0()) -> 0() min#2(S(x4),S(x2)) -> S(min#2(x4,x2)) rpm#3(Leaf(x8),f_p(x12,x16),x4) -> P(rpm_t_m_2(f_p(x12,x16)),rpm_t_m_3(x8)) rpm#3(Node(x8,x6),f_p(x2,x4),x10) -> rpm_t_m_8#1(rpm#3(x8,f_p(x2,x4),bot[9]()) ,rpm#3(x6,f_p(x2,x4),bot[10]()) ,x10) rpm_t_m_8#1(x3,x2,x1) -> P(rpm_t_m_9(fst(),x3,x2),rpm_t_m_10(x3,x2)) snd#2(P(x18,rpm_t_m_3(x14)),x18) -> x14 snd#2(P(x2,rpm_t_m_10(x6,x4)),x2) -> min#2(snd#2(x6,bot[7]()),snd#2(x4,bot[8]())) x#1(f(x4),x3) -> rpm#3(x4,f_p(snd(),x(f(x4))),bot[11]()) - Signature: {cond_fst_p#1/2,fst#2/2,main/1,min#2/2,rpm#3/3,rpm_t_m_2#1/2,rpm_t_m_8#1/3,rpm_t_m_9#1/4,snd#2/2,x#1/2 ,cond_fst_p#1#/2,fst#2#/2,main#/1,min#2#/2,rpm#3#/3,rpm_t_m_2#1#/2,rpm_t_m_8#1#/3,rpm_t_m_9#1#/4,snd#2#/2 ,x#1#/2} / {0/0,Leaf/1,Node/2,P/2,S/1,bot[0]/0,bot[10]/0,bot[11]/0,bot[1]/0,bot[2]/0,bot[3]/0,bot[5]/0 ,bot[6]/0,bot[7]/0,bot[8]/0,bot[9]/0,f/1,f_p/2,fst/0,rpm_t_m_10/2,rpm_t_m_2/1,rpm_t_m_3/1,rpm_t_m_9/3,snd/0 ,x/1,c_1/1,c_2/1,c_3/1,c_4/1,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/2,c_12/0,c_13/2,c_14/2,c_15/0,c_16/3 ,c_17/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_fst_p#1#,fst#2#,main#,min#2#,rpm#3#,rpm_t_m_2#1# ,rpm_t_m_8#1#,rpm_t_m_9#1#,snd#2#,x#1#} and constructors {0,Leaf,Node,P,S,bot[0],bot[10],bot[11],bot[1] ,bot[2],bot[3],bot[5],bot[6],bot[7],bot[8],bot[9],f,f_p,fst,rpm_t_m_10,rpm_t_m_2,rpm_t_m_3,rpm_t_m_9,snd,x} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE