WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: COMP_F_G#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c3(COMP_F_G#1(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1(z2,z3,z4)) COMP_F_G#1(comp_f_g(z0,z1),cons_x(z2),z3) -> c4(COMP_F_G#1(z0,z1,Cons(z2,z3))) COMP_F_G#1(cons_x(z0),comp_f_g(z1,z2),z3) -> c5(COMP_F_G#1(z1,z2,z3)) COMP_F_G#1(cons_x(z0),cons_x(z1),z2) -> c6() MAIN(Leaf(z0)) -> c7() MAIN(Node(z0,z1)) -> c8(COMP_F_G#1(walk#1(z0),walk#1(z1),Nil()),WALK#1(z0)) MAIN(Node(z0,z1)) -> c9(COMP_F_G#1(walk#1(z0),walk#1(z1),Nil()),WALK#1(z1)) WALK#1(Leaf(z0)) -> c() WALK#1(Node(z0,z1)) -> c1(WALK#1(z0)) WALK#1(Node(z0,z1)) -> c2(WALK#1(z1)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) main(Leaf(z0)) -> Cons(z0,Nil()) main(Node(z0,z1)) -> comp_f_g#1(walk#1(z0),walk#1(z1),Nil()) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2 ,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2,comp_f_g/2,cons_x/1} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1,MAIN,WALK#1,comp_f_g#1,main ,walk#1} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: COMP_F_G#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c3(COMP_F_G#1(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1(z2,z3,z4)) COMP_F_G#1(comp_f_g(z0,z1),cons_x(z2),z3) -> c4(COMP_F_G#1(z0,z1,Cons(z2,z3))) COMP_F_G#1(cons_x(z0),comp_f_g(z1,z2),z3) -> c5(COMP_F_G#1(z1,z2,z3)) COMP_F_G#1(cons_x(z0),cons_x(z1),z2) -> c6() MAIN(Leaf(z0)) -> c7() MAIN(Node(z0,z1)) -> c8(COMP_F_G#1(walk#1(z0),walk#1(z1),Nil()),WALK#1(z0)) MAIN(Node(z0,z1)) -> c9(COMP_F_G#1(walk#1(z0),walk#1(z1),Nil()),WALK#1(z1)) WALK#1(Leaf(z0)) -> c() WALK#1(Node(z0,z1)) -> c1(WALK#1(z0)) WALK#1(Node(z0,z1)) -> c2(WALK#1(z1)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) main(Leaf(z0)) -> Cons(z0,Nil()) main(Node(z0,z1)) -> comp_f_g#1(walk#1(z0),walk#1(z1),Nil()) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2 ,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2,comp_f_g/2,cons_x/1} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1,MAIN,WALK#1,comp_f_g#1,main ,walk#1} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: COMP_F_G#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c3(COMP_F_G#1(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1(z2,z3,z4)) COMP_F_G#1(comp_f_g(z0,z1),cons_x(z2),z3) -> c4(COMP_F_G#1(z0,z1,Cons(z2,z3))) COMP_F_G#1(cons_x(z0),comp_f_g(z1,z2),z3) -> c5(COMP_F_G#1(z1,z2,z3)) COMP_F_G#1(cons_x(z0),cons_x(z1),z2) -> c6() MAIN(Leaf(z0)) -> c7() MAIN(Node(z0,z1)) -> c8(COMP_F_G#1(walk#1(z0),walk#1(z1),Nil()),WALK#1(z0)) MAIN(Node(z0,z1)) -> c9(COMP_F_G#1(walk#1(z0),walk#1(z1),Nil()),WALK#1(z1)) WALK#1(Leaf(z0)) -> c() WALK#1(Node(z0,z1)) -> c1(WALK#1(z0)) WALK#1(Node(z0,z1)) -> c2(WALK#1(z1)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) main(Leaf(z0)) -> Cons(z0,Nil()) main(Node(z0,z1)) -> comp_f_g#1(walk#1(z0),walk#1(z1),Nil()) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2 ,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2,comp_f_g/2,cons_x/1} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1,MAIN,WALK#1,comp_f_g#1,main ,walk#1} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: WALK#1(x){x -> Node(x,y)} = WALK#1(Node(x,y)) ->^+ c1(WALK#1(x)) = C[WALK#1(x) = WALK#1(x){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: COMP_F_G#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c3(COMP_F_G#1(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1(z2,z3,z4)) COMP_F_G#1(comp_f_g(z0,z1),cons_x(z2),z3) -> c4(COMP_F_G#1(z0,z1,Cons(z2,z3))) COMP_F_G#1(cons_x(z0),comp_f_g(z1,z2),z3) -> c5(COMP_F_G#1(z1,z2,z3)) COMP_F_G#1(cons_x(z0),cons_x(z1),z2) -> c6() MAIN(Leaf(z0)) -> c7() MAIN(Node(z0,z1)) -> c8(COMP_F_G#1(walk#1(z0),walk#1(z1),Nil()),WALK#1(z0)) MAIN(Node(z0,z1)) -> c9(COMP_F_G#1(walk#1(z0),walk#1(z1),Nil()),WALK#1(z1)) WALK#1(Leaf(z0)) -> c() WALK#1(Node(z0,z1)) -> c1(WALK#1(z0)) WALK#1(Node(z0,z1)) -> c2(WALK#1(z1)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) main(Leaf(z0)) -> Cons(z0,Nil()) main(Node(z0,z1)) -> comp_f_g#1(walk#1(z0),walk#1(z1),Nil()) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2 ,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2,comp_f_g/2,cons_x/1} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1,MAIN,WALK#1,comp_f_g#1,main ,walk#1} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4() MAIN#(Leaf(z0)) -> c_5() MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) WALK#1#(Leaf(z0)) -> c_8() WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) Weak DPs comp_f_g#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_11(comp_f_g#1#(z0,z1,comp_f_g#1(z2,z3,z4))) comp_f_g#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_12(comp_f_g#1#(z0,z1,Cons(z2,z3))) comp_f_g#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_13(comp_f_g#1#(z1,z2,z3)) comp_f_g#1#(cons_x(z0),cons_x(z1),z2) -> c_14() main#(Leaf(z0)) -> c_15() main#(Node(z0,z1)) -> c_16(comp_f_g#1#(walk#1(z0),walk#1(z1),Nil())) walk#1#(Leaf(z0)) -> c_17() walk#1#(Node(z0,z1)) -> c_18(walk#1#(z0),walk#1#(z1)) and mark the set of starting terms. ** Step 1.b:2: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4() MAIN#(Leaf(z0)) -> c_5() MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) WALK#1#(Leaf(z0)) -> c_8() WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Strict TRS: COMP_F_G#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c3(COMP_F_G#1(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1(z2,z3,z4)) COMP_F_G#1(comp_f_g(z0,z1),cons_x(z2),z3) -> c4(COMP_F_G#1(z0,z1,Cons(z2,z3))) COMP_F_G#1(cons_x(z0),comp_f_g(z1,z2),z3) -> c5(COMP_F_G#1(z1,z2,z3)) COMP_F_G#1(cons_x(z0),cons_x(z1),z2) -> c6() MAIN(Leaf(z0)) -> c7() MAIN(Node(z0,z1)) -> c8(COMP_F_G#1(walk#1(z0),walk#1(z1),Nil()),WALK#1(z0)) MAIN(Node(z0,z1)) -> c9(COMP_F_G#1(walk#1(z0),walk#1(z1),Nil()),WALK#1(z1)) WALK#1(Leaf(z0)) -> c() WALK#1(Node(z0,z1)) -> c1(WALK#1(z0)) WALK#1(Node(z0,z1)) -> c2(WALK#1(z1)) - Weak DPs: comp_f_g#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_11(comp_f_g#1#(z0,z1,comp_f_g#1(z2,z3,z4))) comp_f_g#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_12(comp_f_g#1#(z0,z1,Cons(z2,z3))) comp_f_g#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_13(comp_f_g#1#(z1,z2,z3)) comp_f_g#1#(cons_x(z0),cons_x(z1),z2) -> c_14() main#(Leaf(z0)) -> c_15() main#(Node(z0,z1)) -> c_16(comp_f_g#1#(walk#1(z0),walk#1(z1),Nil())) walk#1#(Leaf(z0)) -> c_17() walk#1#(Node(z0,z1)) -> c_18(walk#1#(z0),walk#1#(z1)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) main(Leaf(z0)) -> Cons(z0,Nil()) main(Node(z0,z1)) -> comp_f_g#1(walk#1(z0),walk#1(z1),Nil()) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4() MAIN#(Leaf(z0)) -> c_5() MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) WALK#1#(Leaf(z0)) -> c_8() WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) comp_f_g#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_11(comp_f_g#1#(z0,z1,comp_f_g#1(z2,z3,z4))) comp_f_g#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_12(comp_f_g#1#(z0,z1,Cons(z2,z3))) comp_f_g#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_13(comp_f_g#1#(z1,z2,z3)) comp_f_g#1#(cons_x(z0),cons_x(z1),z2) -> c_14() main#(Leaf(z0)) -> c_15() main#(Node(z0,z1)) -> c_16(comp_f_g#1#(walk#1(z0),walk#1(z1),Nil())) walk#1#(Leaf(z0)) -> c_17() walk#1#(Node(z0,z1)) -> c_18(walk#1#(z0),walk#1#(z1)) ** Step 1.b:3: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4() MAIN#(Leaf(z0)) -> c_5() MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) WALK#1#(Leaf(z0)) -> c_8() WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Weak DPs: comp_f_g#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_11(comp_f_g#1#(z0,z1,comp_f_g#1(z2,z3,z4))) comp_f_g#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_12(comp_f_g#1#(z0,z1,Cons(z2,z3))) comp_f_g#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_13(comp_f_g#1#(z1,z2,z3)) comp_f_g#1#(cons_x(z0),cons_x(z1),z2) -> c_14() main#(Leaf(z0)) -> c_15() main#(Node(z0,z1)) -> c_16(comp_f_g#1#(walk#1(z0),walk#1(z1),Nil())) walk#1#(Leaf(z0)) -> c_17() walk#1#(Node(z0,z1)) -> c_18(walk#1#(z0),walk#1#(z1)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4,5,8} by application of Pre({4,5,8}) = {1,2,3,6,7,9,10}. Here rules are labelled as follows: 1: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) 2: COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) 3: COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) 4: COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4() 5: MAIN#(Leaf(z0)) -> c_5() 6: MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) 7: MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) 8: WALK#1#(Leaf(z0)) -> c_8() 9: WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) 10: WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) 11: comp_f_g#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_11(comp_f_g#1#(z0,z1,comp_f_g#1(z2,z3,z4))) 12: comp_f_g#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_12(comp_f_g#1#(z0,z1,Cons(z2,z3))) 13: comp_f_g#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_13(comp_f_g#1#(z1,z2,z3)) 14: comp_f_g#1#(cons_x(z0),cons_x(z1),z2) -> c_14() 15: main#(Leaf(z0)) -> c_15() 16: main#(Node(z0,z1)) -> c_16(comp_f_g#1#(walk#1(z0),walk#1(z1),Nil())) 17: walk#1#(Leaf(z0)) -> c_17() 18: walk#1#(Node(z0,z1)) -> c_18(walk#1#(z0),walk#1#(z1)) ** Step 1.b:4: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Weak DPs: COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4() MAIN#(Leaf(z0)) -> c_5() WALK#1#(Leaf(z0)) -> c_8() comp_f_g#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_11(comp_f_g#1#(z0,z1,comp_f_g#1(z2,z3,z4))) comp_f_g#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_12(comp_f_g#1#(z0,z1,Cons(z2,z3))) comp_f_g#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_13(comp_f_g#1#(z1,z2,z3)) comp_f_g#1#(cons_x(z0),cons_x(z1),z2) -> c_14() main#(Leaf(z0)) -> c_15() main#(Node(z0,z1)) -> c_16(comp_f_g#1#(walk#1(z0),walk#1(z1),Nil())) walk#1#(Leaf(z0)) -> c_17() walk#1#(Node(z0,z1)) -> c_18(walk#1#(z0),walk#1#(z1)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) -->_2 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_2 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_2 COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4():8 -->_1 COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4():8 -->_2 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 2:S:COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4():8 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 3:S:COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) -->_1 COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4():8 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 4:S:MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) -->_2 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):7 -->_2 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):6 -->_2 WALK#1#(Leaf(z0)) -> c_8():10 -->_1 COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4():8 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 5:S:MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) -->_2 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):7 -->_2 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):6 -->_2 WALK#1#(Leaf(z0)) -> c_8():10 -->_1 COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4():8 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 6:S:WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):7 -->_1 WALK#1#(Leaf(z0)) -> c_8():10 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):6 7:S:WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) -->_1 WALK#1#(Leaf(z0)) -> c_8():10 -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):7 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):6 8:W:COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4() 9:W:MAIN#(Leaf(z0)) -> c_5() 10:W:WALK#1#(Leaf(z0)) -> c_8() 11:W:comp_f_g#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_11(comp_f_g#1#(z0,z1,comp_f_g#1(z2,z3,z4))) -->_1 comp_f_g#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_13(comp_f_g#1#(z1,z2,z3)):13 -->_1 comp_f_g#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_12(comp_f_g#1#(z0,z1,Cons(z2,z3))):12 -->_1 comp_f_g#1#(cons_x(z0),cons_x(z1),z2) -> c_14():14 -->_1 comp_f_g#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_11(comp_f_g#1#(z0,z1,comp_f_g#1(z2,z3,z4))):11 12:W:comp_f_g#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_12(comp_f_g#1#(z0,z1,Cons(z2,z3))) -->_1 comp_f_g#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_13(comp_f_g#1#(z1,z2,z3)):13 -->_1 comp_f_g#1#(cons_x(z0),cons_x(z1),z2) -> c_14():14 -->_1 comp_f_g#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_12(comp_f_g#1#(z0,z1,Cons(z2,z3))):12 -->_1 comp_f_g#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_11(comp_f_g#1#(z0,z1,comp_f_g#1(z2,z3,z4))):11 13:W:comp_f_g#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_13(comp_f_g#1#(z1,z2,z3)) -->_1 comp_f_g#1#(cons_x(z0),cons_x(z1),z2) -> c_14():14 -->_1 comp_f_g#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_13(comp_f_g#1#(z1,z2,z3)):13 -->_1 comp_f_g#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_12(comp_f_g#1#(z0,z1,Cons(z2,z3))):12 -->_1 comp_f_g#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_11(comp_f_g#1#(z0,z1,comp_f_g#1(z2,z3,z4))):11 14:W:comp_f_g#1#(cons_x(z0),cons_x(z1),z2) -> c_14() 15:W:main#(Leaf(z0)) -> c_15() 16:W:main#(Node(z0,z1)) -> c_16(comp_f_g#1#(walk#1(z0),walk#1(z1),Nil())) -->_1 comp_f_g#1#(cons_x(z0),cons_x(z1),z2) -> c_14():14 -->_1 comp_f_g#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_13(comp_f_g#1#(z1,z2,z3)):13 -->_1 comp_f_g#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_12(comp_f_g#1#(z0,z1,Cons(z2,z3))):12 -->_1 comp_f_g#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_11(comp_f_g#1#(z0,z1,comp_f_g#1(z2,z3,z4))):11 17:W:walk#1#(Leaf(z0)) -> c_17() 18:W:walk#1#(Node(z0,z1)) -> c_18(walk#1#(z0),walk#1#(z1)) -->_2 walk#1#(Node(z0,z1)) -> c_18(walk#1#(z0),walk#1#(z1)):18 -->_1 walk#1#(Node(z0,z1)) -> c_18(walk#1#(z0),walk#1#(z1)):18 -->_2 walk#1#(Leaf(z0)) -> c_17():17 -->_1 walk#1#(Leaf(z0)) -> c_17():17 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 18: walk#1#(Node(z0,z1)) -> c_18(walk#1#(z0),walk#1#(z1)) 17: walk#1#(Leaf(z0)) -> c_17() 16: main#(Node(z0,z1)) -> c_16(comp_f_g#1#(walk#1(z0),walk#1(z1),Nil())) 15: main#(Leaf(z0)) -> c_15() 11: comp_f_g#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_11(comp_f_g#1#(z0,z1,comp_f_g#1(z2,z3,z4))) 13: comp_f_g#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_13(comp_f_g#1#(z1,z2,z3)) 12: comp_f_g#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_12(comp_f_g#1#(z0,z1,Cons(z2,z3))) 14: comp_f_g#1#(cons_x(z0),cons_x(z1),z2) -> c_14() 9: MAIN#(Leaf(z0)) -> c_5() 10: WALK#1#(Leaf(z0)) -> c_8() 8: COMP_F_G#1#(cons_x(z0),cons_x(z1),z2) -> c_4() ** Step 1.b:5: Decompose. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + 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: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) - Weak DPs: MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} Problem (S) - Strict DPs: MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Weak DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} *** Step 1.b:5.a:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) - Weak DPs: MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) -->_2 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_2 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_2 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 2:S:COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 3:S:COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 4:W:MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_2 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):7 -->_2 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):6 5:W:MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_2 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):7 -->_2 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):6 6:W:WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):7 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):6 7:W:WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):6 -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) 6: WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) *** Step 1.b:5.a:2: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) - Weak DPs: MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) -->_2 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_2 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_2 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 2:S:COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 3:S:COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 4:W:MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 5:W:MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) *** Step 1.b:5.a:3: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) - Weak DPs: MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + 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: 2: COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) 3: COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) Consider the set of all dependency pairs 1: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) 2: COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) 3: COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) 4: MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) 5: MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) BEST_CASE TIME (?,?) SPACE(?,?)on application of the dependency pairs {2,3} These cover all (indirect) predecessors of dependency pairs {2,3,4,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 1.b:5.a:3.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) - Weak DPs: MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + 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_1) = {1,2}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1} Following symbols are considered usable: {walk#1,COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main#,walk#1#} TcT has computed the following interpretation: p(COMP_F_G#1) = [1] x2 + [0] p(Cons) = [1] x1 + [0] p(Leaf) = [4] p(MAIN) = [2] x1 + [1] p(Nil) = [0] p(Node) = [1] x1 + [1] x2 + [0] p(WALK#1) = [1] x1 + [0] p(c) = [1] p(c1) = [0] p(c2) = [1] p(c3) = [1] x1 + [1] x2 + [0] p(c4) = [1] x1 + [0] p(c5) = [0] p(c6) = [0] p(c7) = [0] p(c8) = [1] x1 + [1] x2 + [0] p(c9) = [1] x1 + [1] x2 + [0] p(comp_f_g) = [1] x1 + [1] x2 + [0] p(comp_f_g#1) = [1] x1 + [1] x2 + [2] p(cons_x) = [2] p(main) = [1] p(walk#1) = [1] x1 + [0] p(COMP_F_G#1#) = [4] x1 + [4] x2 + [0] p(MAIN#) = [4] x1 + [3] p(WALK#1#) = [0] p(comp_f_g#1#) = [0] p(main#) = [0] p(walk#1#) = [2] p(c_1) = [1] x1 + [1] x2 + [0] p(c_2) = [1] x1 + [4] p(c_3) = [1] x1 + [6] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [3] p(c_7) = [1] x1 + [1] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] Following rules are strictly oriented: COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) = [4] z0 + [4] z1 + [8] > [4] z0 + [4] z1 + [4] = c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) = [4] z1 + [4] z2 + [8] > [4] z1 + [4] z2 + [6] = c_3(COMP_F_G#1#(z1,z2,z3)) Following rules are (at-least) weakly oriented: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) = [4] z0 + [4] z1 + [4] z2 + [4] z3 + [0] >= [4] z0 + [4] z1 + [4] z2 + [4] z3 + [0] = c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)),COMP_F_G#1#(z2,z3,z4)) MAIN#(Node(z0,z1)) = [4] z0 + [4] z1 + [3] >= [4] z0 + [4] z1 + [3] = c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) MAIN#(Node(z0,z1)) = [4] z0 + [4] z1 + [3] >= [4] z0 + [4] z1 + [1] = c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) walk#1(Leaf(z0)) = [4] >= [2] = cons_x(z0) walk#1(Node(z0,z1)) = [1] z0 + [1] z1 + [0] >= [1] z0 + [1] z1 + [0] = comp_f_g(walk#1(z0),walk#1(z1)) **** Step 1.b:5.a:3.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) - Weak DPs: COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () **** Step 1.b:5.a:3.b:1: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) - Weak DPs: COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + 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: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) Consider the set of all dependency pairs 1: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) 2: COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) 3: COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) 4: MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) 5: MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) BEST_CASE TIME (?,?) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,4,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 1.b:5.a:3.b:1.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) - Weak DPs: COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + 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_1) = {1,2}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1} Following symbols are considered usable: {walk#1,COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main#,walk#1#} TcT has computed the following interpretation: p(COMP_F_G#1) = [0] p(Cons) = [1] x1 + [1] x2 + [0] p(Leaf) = [1] x1 + [1] p(MAIN) = [2] x1 + [2] p(Nil) = [0] p(Node) = [1] x1 + [1] x2 + [2] p(WALK#1) = [0] p(c) = [0] p(c1) = [0] p(c2) = [2] p(c3) = [0] p(c4) = [0] p(c5) = [0] p(c6) = [0] p(c7) = [0] p(c8) = [1] x2 + [0] p(c9) = [1] p(comp_f_g) = [1] x1 + [1] x2 + [1] p(comp_f_g#1) = [1] x2 + [0] p(cons_x) = [4] p(main) = [1] p(walk#1) = [2] x1 + [3] p(COMP_F_G#1#) = [2] x1 + [2] x2 + [0] p(MAIN#) = [4] x1 + [4] p(WALK#1#) = [2] x1 + [0] p(comp_f_g#1#) = [1] x2 + [0] p(main#) = [1] p(walk#1#) = [0] p(c_1) = [1] x1 + [1] x2 + [0] p(c_2) = [1] x1 + [1] p(c_3) = [1] x1 + [0] p(c_4) = [2] p(c_5) = [1] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [4] x1 + [1] p(c_10) = [1] p(c_11) = [1] x1 + [2] p(c_12) = [1] x1 + [1] p(c_13) = [0] p(c_14) = [1] p(c_15) = [1] p(c_16) = [1] x1 + [1] p(c_17) = [1] p(c_18) = [2] x1 + [1] x2 + [0] Following rules are strictly oriented: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) = [2] z0 + [2] z1 + [2] z2 + [2] z3 + [4] > [2] z0 + [2] z1 + [2] z2 + [2] z3 + [0] = c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)),COMP_F_G#1#(z2,z3,z4)) Following rules are (at-least) weakly oriented: COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) = [2] z0 + [2] z1 + [10] >= [2] z0 + [2] z1 + [1] = c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) = [2] z1 + [2] z2 + [10] >= [2] z1 + [2] z2 + [0] = c_3(COMP_F_G#1#(z1,z2,z3)) MAIN#(Node(z0,z1)) = [4] z0 + [4] z1 + [12] >= [4] z0 + [4] z1 + [12] = c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) MAIN#(Node(z0,z1)) = [4] z0 + [4] z1 + [12] >= [4] z0 + [4] z1 + [12] = c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) walk#1(Leaf(z0)) = [2] z0 + [5] >= [4] = cons_x(z0) walk#1(Node(z0,z1)) = [2] z0 + [2] z1 + [7] >= [2] z0 + [2] z1 + [7] = comp_f_g(walk#1(z0),walk#1(z1)) ***** Step 1.b:5.a:3.b:1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () ***** Step 1.b:5.a:3.b:1.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) -->_2 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_2 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_2 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 2:W:COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 3:W:COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 4:W:MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 5:W:MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):3 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):2 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) 4: MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil())) 1: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) 3: COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) 2: COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) ***** Step 1.b:5.a:3.b:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:5.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Weak DPs: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):7 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):6 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):5 -->_2 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_2 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 2:S:MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):7 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):6 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):5 -->_2 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_2 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 3:S:WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 4:S:WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 5:W:COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) -->_2 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):7 -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):7 -->_2 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):6 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):6 -->_2 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):5 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):5 6:W:COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):7 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):6 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):5 7:W:COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) -->_1 COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)):7 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))):6 -->_1 COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: COMP_F_G#1#(cons_x(z0),comp_f_g(z1,z2),z3) -> c_3(COMP_F_G#1#(z1,z2,z3)) 6: COMP_F_G#1#(comp_f_g(z0,z1),cons_x(z2),z3) -> c_2(COMP_F_G#1#(z0,z1,Cons(z2,z3))) 5: COMP_F_G#1#(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> c_1(COMP_F_G#1#(z0,z1,comp_f_g#1(z2,z3,z4)) ,COMP_F_G#1#(z2,z3,z4)) *** Step 1.b:5.b:2: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/2,c_7/2,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:MAIN#(Node(z0,z1)) -> c_6(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z0)) -->_2 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_2 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 2:S:MAIN#(Node(z0,z1)) -> c_7(COMP_F_G#1#(walk#1(z0),walk#1(z1),Nil()),WALK#1#(z1)) -->_2 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_2 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 3:S:WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 4:S:WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) *** Step 1.b:5.b:3: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Weak TRS: comp_f_g#1(comp_f_g(z0,z1),comp_f_g(z2,z3),z4) -> comp_f_g#1(z0,z1,comp_f_g#1(z2,z3,z4)) comp_f_g#1(comp_f_g(z0,z1),cons_x(z2),z3) -> comp_f_g#1(z0,z1,Cons(z2,z3)) comp_f_g#1(cons_x(z0),comp_f_g(z1,z2),z3) -> Cons(z0,comp_f_g#1(z1,z2,z3)) comp_f_g#1(cons_x(z0),cons_x(z1),z2) -> Cons(z0,Cons(z1,z2)) walk#1(Leaf(z0)) -> cons_x(z0) walk#1(Node(z0,z1)) -> comp_f_g(walk#1(z0),walk#1(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) *** Step 1.b:5.b:4: Decompose. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + 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: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Weak DPs: MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} Problem (S) - Strict DPs: MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) - Weak DPs: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} **** Step 1.b:5.b:4.a:1: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Weak DPs: MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + 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: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) 4: WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) Consider the set of all dependency pairs 1: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) 2: MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) 3: WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) 4: WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) BEST_CASE TIME (?,?) SPACE(?,?)on application of the dependency pairs {1,4} These cover all (indirect) predecessors of dependency pairs {1,2,4} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 1.b:5.b:4.a:1.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Weak DPs: MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + 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_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1} Following symbols are considered usable: {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main#,walk#1#} TcT has computed the following interpretation: p(COMP_F_G#1) = [1] x1 + [0] p(Cons) = [8] p(Leaf) = [1] p(MAIN) = [8] x1 + [8] p(Nil) = [0] p(Node) = [1] x1 + [1] x2 + [8] p(WALK#1) = [1] x1 + [4] p(c) = [0] p(c1) = [1] p(c2) = [0] p(c3) = [1] p(c4) = [4] p(c5) = [1] p(c6) = [2] p(c7) = [0] p(c8) = [1] p(c9) = [0] p(comp_f_g) = [1] p(comp_f_g#1) = [2] x2 + [1] x3 + [1] p(cons_x) = [0] p(main) = [0] p(walk#1) = [1] x1 + [0] p(COMP_F_G#1#) = [1] p(MAIN#) = [2] x1 + [8] p(WALK#1#) = [1] x1 + [0] p(comp_f_g#1#) = [8] x1 + [2] x3 + [1] p(main#) = [2] x1 + [0] p(walk#1#) = [1] x1 + [1] p(c_1) = [0] p(c_2) = [1] p(c_3) = [4] x1 + [1] p(c_4) = [1] p(c_5) = [1] p(c_6) = [2] x1 + [10] p(c_7) = [2] x1 + [0] p(c_8) = [2] p(c_9) = [1] x1 + [8] p(c_10) = [1] x1 + [4] p(c_11) = [4] p(c_12) = [1] x1 + [2] p(c_13) = [1] x1 + [1] p(c_14) = [0] p(c_15) = [1] p(c_16) = [1] x1 + [0] p(c_17) = [1] p(c_18) = [2] x1 + [0] Following rules are strictly oriented: MAIN#(Node(z0,z1)) = [2] z0 + [2] z1 + [24] > [2] z0 + [10] = c_6(WALK#1#(z0)) WALK#1#(Node(z0,z1)) = [1] z0 + [1] z1 + [8] > [1] z1 + [4] = c_10(WALK#1#(z1)) Following rules are (at-least) weakly oriented: MAIN#(Node(z0,z1)) = [2] z0 + [2] z1 + [24] >= [2] z1 + [0] = c_7(WALK#1#(z1)) WALK#1#(Node(z0,z1)) = [1] z0 + [1] z1 + [8] >= [1] z0 + [8] = c_9(WALK#1#(z0)) ***** Step 1.b:5.b:4.a:1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) - Weak DPs: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () ***** Step 1.b:5.b:4.a:1.b:1: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) - Weak DPs: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + 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: WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) Consider the set of all dependency pairs 1: WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) 2: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) 3: MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) 4: WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) BEST_CASE TIME (?,?) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ****** Step 1.b:5.b:4.a:1.b:1.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) - Weak DPs: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + 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_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1} Following symbols are considered usable: {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main#,walk#1#} TcT has computed the following interpretation: p(COMP_F_G#1) = [4] x1 + [2] x2 + [1] x3 + [8] p(Cons) = [0] p(Leaf) = [1] p(MAIN) = [2] x1 + [1] p(Nil) = [0] p(Node) = [1] x1 + [1] x2 + [7] p(WALK#1) = [1] x1 + [0] p(c) = [0] p(c1) = [0] p(c2) = [0] p(c3) = [1] x1 + [0] p(c4) = [1] x1 + [0] p(c5) = [0] p(c6) = [0] p(c7) = [0] p(c8) = [1] x1 + [1] x2 + [0] p(c9) = [1] x2 + [0] p(comp_f_g) = [0] p(comp_f_g#1) = [1] x1 + [1] x3 + [0] p(cons_x) = [1] x1 + [0] p(main) = [8] x1 + [0] p(walk#1) = [0] p(COMP_F_G#1#) = [8] x1 + [1] x3 + [0] p(MAIN#) = [2] x1 + [0] p(WALK#1#) = [2] x1 + [2] p(comp_f_g#1#) = [0] p(main#) = [0] p(walk#1#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [2] p(c_7) = [1] x1 + [12] p(c_8) = [2] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [14] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [2] x1 + [0] Following rules are strictly oriented: WALK#1#(Node(z0,z1)) = [2] z0 + [2] z1 + [16] > [2] z0 + [2] = c_9(WALK#1#(z0)) Following rules are (at-least) weakly oriented: MAIN#(Node(z0,z1)) = [2] z0 + [2] z1 + [14] >= [2] z0 + [4] = c_6(WALK#1#(z0)) MAIN#(Node(z0,z1)) = [2] z0 + [2] z1 + [14] >= [2] z1 + [14] = c_7(WALK#1#(z1)) WALK#1#(Node(z0,z1)) = [2] z0 + [2] z1 + [16] >= [2] z1 + [16] = c_10(WALK#1#(z1)) ****** Step 1.b:5.b:4.a:1.b:1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () ****** Step 1.b:5.b:4.a:1.b:1.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 2:W:MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 3:W:WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 4:W:WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) 1: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) 4: WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) 3: WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) ****** Step 1.b:5.b:4.a:1.b:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 1.b:5.b:4.b:1: PredecessorEstimation. WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) - Weak DPs: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {}. Here rules are labelled as follows: 1: MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) 2: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) 3: WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) 4: WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) **** Step 1.b:5.b:4.b:2: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 2:W:MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 3:W:WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 4:W:WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) -->_1 WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)):4 -->_1 WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: MAIN#(Node(z0,z1)) -> c_7(WALK#1#(z1)) 1: MAIN#(Node(z0,z1)) -> c_6(WALK#1#(z0)) 4: WALK#1#(Node(z0,z1)) -> c_10(WALK#1#(z1)) 3: WALK#1#(Node(z0,z1)) -> c_9(WALK#1#(z0)) **** Step 1.b:5.b:4.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Signature: {COMP_F_G#1/3,MAIN/1,WALK#1/1,comp_f_g#1/3,main/1,walk#1/1,COMP_F_G#1#/3,MAIN#/1,WALK#1#/1,comp_f_g#1#/3 ,main#/1,walk#1#/1} / {Cons/2,Leaf/1,Nil/0,Node/2,c/0,c1/1,c2/1,c3/2,c4/1,c5/1,c6/0,c7/0,c8/2,c9/2 ,comp_f_g/2,cons_x/1,c_1/2,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/1,c_13/1 ,c_14/0,c_15/0,c_16/1,c_17/0,c_18/2} - Obligation: innermost runtime complexity wrt. defined symbols {COMP_F_G#1#,MAIN#,WALK#1#,comp_f_g#1#,main# ,walk#1#} and constructors {Cons,Leaf,Nil,Node,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,comp_f_g,cons_x} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))