YES We show the termination of the TRS R: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) Term_sub(Term_var(x),Id()) -> Term_var(x) Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) Sum_sub(xi,Id()) -> Sum_term_var(xi) Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) Concat(Id(),s) -> s -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: Term_sub#(Case(m,xi,n),s) -> Frozen#(m,Sum_sub(xi,s),n,s) p2: Term_sub#(Case(m,xi,n),s) -> Sum_sub#(xi,s) p3: Frozen#(m,Sum_constant(Left()),n,s) -> Term_sub#(m,s) p4: Frozen#(m,Sum_constant(Right()),n,s) -> Term_sub#(n,s) p5: Frozen#(m,Sum_term_var(xi),n,s) -> Term_sub#(m,s) p6: Frozen#(m,Sum_term_var(xi),n,s) -> Term_sub#(n,s) p7: Term_sub#(Term_app(m,n),s) -> Term_sub#(m,s) p8: Term_sub#(Term_app(m,n),s) -> Term_sub#(n,s) p9: Term_sub#(Term_pair(m,n),s) -> Term_sub#(m,s) p10: Term_sub#(Term_pair(m,n),s) -> Term_sub#(n,s) p11: Term_sub#(Term_inl(m),s) -> Term_sub#(m,s) p12: Term_sub#(Term_inr(m),s) -> Term_sub#(m,s) p13: Term_sub#(Term_var(x),Cons_usual(y,m,s)) -> Term_sub#(Term_var(x),s) p14: Term_sub#(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub#(Term_var(x),s) p15: Term_sub#(Term_sub(m,s),t) -> Term_sub#(m,Concat(s,t)) p16: Term_sub#(Term_sub(m,s),t) -> Concat#(s,t) p17: Sum_sub#(xi,Cons_sum(psi,k,s)) -> Sum_sub#(xi,s) p18: Sum_sub#(xi,Cons_usual(y,m,s)) -> Sum_sub#(xi,s) p19: Concat#(Concat(s,t),u) -> Concat#(s,Concat(t,u)) p20: Concat#(Concat(s,t),u) -> Concat#(t,u) p21: Concat#(Cons_usual(x,m,s),t) -> Term_sub#(m,t) p22: Concat#(Cons_usual(x,m,s),t) -> Concat#(s,t) p23: Concat#(Cons_sum(xi,k,s),t) -> Concat#(s,t) and R consists of: r1: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) r2: Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) r3: Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) r4: Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) r5: Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) r6: Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) r7: Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) r8: Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) r9: Term_sub(Term_var(x),Id()) -> Term_var(x) r10: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m r11: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) r12: Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) r13: Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) r14: Sum_sub(xi,Id()) -> Sum_term_var(xi) r15: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) r16: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) r17: Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) r18: Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) r19: Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) r20: Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) r21: Concat(Id(),s) -> s The estimated dependency graph contains the following SCCs: {p1, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p15, p16, p19, p20, p21, p22, p23} {p17, p18} {p13, p14} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: Term_sub#(Case(m,xi,n),s) -> Frozen#(m,Sum_sub(xi,s),n,s) p2: Frozen#(m,Sum_term_var(xi),n,s) -> Term_sub#(n,s) p3: Term_sub#(Term_sub(m,s),t) -> Concat#(s,t) p4: Concat#(Cons_sum(xi,k,s),t) -> Concat#(s,t) p5: Concat#(Cons_usual(x,m,s),t) -> Concat#(s,t) p6: Concat#(Cons_usual(x,m,s),t) -> Term_sub#(m,t) p7: Term_sub#(Term_sub(m,s),t) -> Term_sub#(m,Concat(s,t)) p8: Term_sub#(Term_inr(m),s) -> Term_sub#(m,s) p9: Term_sub#(Term_inl(m),s) -> Term_sub#(m,s) p10: Term_sub#(Term_pair(m,n),s) -> Term_sub#(n,s) p11: Term_sub#(Term_pair(m,n),s) -> Term_sub#(m,s) p12: Term_sub#(Term_app(m,n),s) -> Term_sub#(n,s) p13: Term_sub#(Term_app(m,n),s) -> Term_sub#(m,s) p14: Concat#(Concat(s,t),u) -> Concat#(t,u) p15: Concat#(Concat(s,t),u) -> Concat#(s,Concat(t,u)) p16: Frozen#(m,Sum_term_var(xi),n,s) -> Term_sub#(m,s) p17: Frozen#(m,Sum_constant(Right()),n,s) -> Term_sub#(n,s) p18: Frozen#(m,Sum_constant(Left()),n,s) -> Term_sub#(m,s) and R consists of: r1: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) r2: Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) r3: Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) r4: Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) r5: Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) r6: Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) r7: Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) r8: Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) r9: Term_sub(Term_var(x),Id()) -> Term_var(x) r10: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m r11: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) r12: Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) r13: Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) r14: Sum_sub(xi,Id()) -> Sum_term_var(xi) r15: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) r16: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) r17: Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) r18: Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) r19: Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) r20: Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) r21: Concat(Id(),s) -> s The set of usable rules consists of r9, r10, r11, r12, r21 Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: max/plus interpretation on natural numbers: Term_sub#_A(x1,x2) = 12 Case_A(x1,x2,x3) = max{13, x1, x2 - 36, x3} Frozen#_A(x1,x2,x3,x4) = 12 Sum_sub_A(x1,x2) = max{14, x2 - 39} Sum_term_var_A(x1) = x1 + 37 Term_sub_A(x1,x2) = max{x1 + 34, x2} Concat#_A(x1,x2) = 12 Cons_sum_A(x1,x2,x3) = max{x1 + 20, x2 - 13, x3} Cons_usual_A(x1,x2,x3) = max{38, x1 + 20, x2 + 20, x3 + 11} Concat_A(x1,x2) = max{24, x1 + 14, x2 + 20} Term_inr_A(x1) = 0 Term_inl_A(x1) = 13 Term_pair_A(x1,x2) = max{18, x1, x2 - 36} Term_app_A(x1,x2) = 13 Sum_constant_A(x1) = max{13, x1 - 52} Right_A = 70 Left_A = 13 Term_var_A(x1) = max{39, x1 - 35} Id_A = 1 base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: Term_sub#_A(x1,x2) = max{56, x1 + 16} Case_A(x1,x2,x3) = max{38, x1, x3} Frozen#_A(x1,x2,x3,x4) = max{56, x1 + 16, x3 + 16} Sum_sub_A(x1,x2) = 7 Sum_term_var_A(x1) = 65 Term_sub_A(x1,x2) = max{x1 + 16, x2 + 23} Concat#_A(x1,x2) = max{56, x1 + 39} Cons_sum_A(x1,x2,x3) = max{11, x1 + 2, x2 - 8, x3} Cons_usual_A(x1,x2,x3) = max{16, x1 + 2, x2 - 23, x3 + 1} Concat_A(x1,x2) = max{x1 + 13, x2} Term_inr_A(x1) = max{40, x1} Term_inl_A(x1) = max{40, x1} Term_pair_A(x1,x2) = max{27, x1, x2 + 10} Term_app_A(x1,x2) = max{40, x1, x2} Sum_constant_A(x1) = 64 Right_A = 75 Left_A = 75 Term_var_A(x1) = max{35, x1 - 17} Id_A = 11 Term_sub#!_A(x1,x2) = max{55, x1 + 16} Case!_A(x1,x2,x3) = 0 Frozen#!_A(x1,x2,x3,x4) = max{55, x1 + 16, x3 + 16} Sum_sub!_A(x1,x2) = max{56, x1 - 1, x2 - 12} Sum_term_var!_A(x1) = max{53, x1 - 2} Term_sub!_A(x1,x2) = 55 Concat#!_A(x1,x2) = x1 + 39 Cons_sum!_A(x1,x2,x3) = 1 Cons_usual!_A(x1,x2,x3) = 30 Concat!_A(x1,x2) = 55 Term_inr!_A(x1) = max{56, x1 + 17} Term_inl!_A(x1) = max{0, x1 - 24} Term_pair!_A(x1,x2) = 0 Term_app!_A(x1,x2) = max{55, x1 - 39, x2 - 24} Sum_constant!_A(x1) = max{55, x1 - 21} Right!_A = 56 Left!_A = 55 Term_var!_A(x1) = max{55, x1 - 3} Id!_A = 54 partial status: pi(Term_sub#) = [] pi(Case) = [3] pi(Frozen#) = [] pi(Sum_sub) = [] pi(Sum_term_var) = [] pi(Term_sub) = [1, 2] pi(Concat#) = [] pi(Cons_sum) = [1, 3] pi(Cons_usual) = [1, 2, 3] pi(Concat) = [1, 2] pi(Term_inr) = [] pi(Term_inl) = [] pi(Term_pair) = [1] pi(Term_app) = [] pi(Sum_constant) = [] pi(Right) = [] pi(Left) = [] pi(Term_var) = [] pi(Id) = [] The following rules are removed: p5, p15 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: Term_sub#(Case(m,xi,n),s) -> Frozen#(m,Sum_sub(xi,s),n,s) p2: Frozen#(m,Sum_term_var(xi),n,s) -> Term_sub#(n,s) p3: Term_sub#(Term_sub(m,s),t) -> Concat#(s,t) p4: Concat#(Cons_sum(xi,k,s),t) -> Concat#(s,t) p5: Concat#(Cons_usual(x,m,s),t) -> Term_sub#(m,t) p6: Term_sub#(Term_sub(m,s),t) -> Term_sub#(m,Concat(s,t)) p7: Term_sub#(Term_inr(m),s) -> Term_sub#(m,s) p8: Term_sub#(Term_inl(m),s) -> Term_sub#(m,s) p9: Term_sub#(Term_pair(m,n),s) -> Term_sub#(n,s) p10: Term_sub#(Term_pair(m,n),s) -> Term_sub#(m,s) p11: Term_sub#(Term_app(m,n),s) -> Term_sub#(n,s) p12: Term_sub#(Term_app(m,n),s) -> Term_sub#(m,s) p13: Concat#(Concat(s,t),u) -> Concat#(t,u) p14: Frozen#(m,Sum_term_var(xi),n,s) -> Term_sub#(m,s) p15: Frozen#(m,Sum_constant(Right()),n,s) -> Term_sub#(n,s) p16: Frozen#(m,Sum_constant(Left()),n,s) -> Term_sub#(m,s) and R consists of: r1: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) r2: Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) r3: Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) r4: Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) r5: Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) r6: Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) r7: Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) r8: Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) r9: Term_sub(Term_var(x),Id()) -> Term_var(x) r10: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m r11: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) r12: Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) r13: Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) r14: Sum_sub(xi,Id()) -> Sum_term_var(xi) r15: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) r16: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) r17: Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) r18: Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) r19: Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) r20: Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) r21: Concat(Id(),s) -> s The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: Term_sub#(Case(m,xi,n),s) -> Frozen#(m,Sum_sub(xi,s),n,s) p2: Frozen#(m,Sum_constant(Left()),n,s) -> Term_sub#(m,s) p3: Term_sub#(Term_app(m,n),s) -> Term_sub#(m,s) p4: Term_sub#(Term_app(m,n),s) -> Term_sub#(n,s) p5: Term_sub#(Term_pair(m,n),s) -> Term_sub#(m,s) p6: Term_sub#(Term_pair(m,n),s) -> Term_sub#(n,s) p7: Term_sub#(Term_inl(m),s) -> Term_sub#(m,s) p8: Term_sub#(Term_inr(m),s) -> Term_sub#(m,s) p9: Term_sub#(Term_sub(m,s),t) -> Term_sub#(m,Concat(s,t)) p10: Term_sub#(Term_sub(m,s),t) -> Concat#(s,t) p11: Concat#(Concat(s,t),u) -> Concat#(t,u) p12: Concat#(Cons_usual(x,m,s),t) -> Term_sub#(m,t) p13: Concat#(Cons_sum(xi,k,s),t) -> Concat#(s,t) p14: Frozen#(m,Sum_constant(Right()),n,s) -> Term_sub#(n,s) p15: Frozen#(m,Sum_term_var(xi),n,s) -> Term_sub#(m,s) p16: Frozen#(m,Sum_term_var(xi),n,s) -> Term_sub#(n,s) and R consists of: r1: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) r2: Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) r3: Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) r4: Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) r5: Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) r6: Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) r7: Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) r8: Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) r9: Term_sub(Term_var(x),Id()) -> Term_var(x) r10: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m r11: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) r12: Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) r13: Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) r14: Sum_sub(xi,Id()) -> Sum_term_var(xi) r15: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) r16: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) r17: Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) r18: Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) r19: Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) r20: Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) r21: Concat(Id(),s) -> s The set of usable rules consists of r9, r14, r15, r16, r17 Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: max/plus interpretation on natural numbers: Term_sub#_A(x1,x2) = 8 Case_A(x1,x2,x3) = max{23, x1 + 22, x2 - 1, x3 + 22} Frozen#_A(x1,x2,x3,x4) = max{1, x2 - 11} Sum_sub_A(x1,x2) = 19 Sum_constant_A(x1) = 19 Left_A = 8 Term_app_A(x1,x2) = max{23, x1 - 22, x2 - 22} Term_pair_A(x1,x2) = max{23, x2 - 22} Term_inl_A(x1) = max{23, x1 - 22} Term_inr_A(x1) = max{23, x1 - 22} Term_sub_A(x1,x2) = 0 Concat_A(x1,x2) = x1 + 13 Concat#_A(x1,x2) = 8 Cons_usual_A(x1,x2,x3) = 0 Cons_sum_A(x1,x2,x3) = max{18, x1 + 2, x2 + 1, x3 + 4} Right_A = 9 Sum_term_var_A(x1) = 19 Term_var_A(x1) = 0 Id_A = 20 base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: Term_sub#_A(x1,x2) = max{1, x1 - 18} Case_A(x1,x2,x3) = max{x1 + 14, x2 - 32, x3 + 14} Frozen#_A(x1,x2,x3,x4) = max{1, x1 - 4, x2 - 63, x3 - 4} Sum_sub_A(x1,x2) = x1 + 13 Sum_constant_A(x1) = 10 Left_A = 1 Term_app_A(x1,x2) = max{35, x1 + 17, x2 + 17} Term_pair_A(x1,x2) = max{35, x1, x2 + 17} Term_inl_A(x1) = max{35, x1 + 17} Term_inr_A(x1) = max{35, x1 + 17} Term_sub_A(x1,x2) = max{x1 + 17, x2 + 14} Concat_A(x1,x2) = max{33, x1 + 14, x2} Concat#_A(x1,x2) = max{1, x1 - 11} Cons_usual_A(x1,x2,x3) = max{10, x1 - 53, x2 - 5} Cons_sum_A(x1,x2,x3) = max{34, x1 - 15, x2 - 53, x3 + 9} Right_A = 3 Sum_term_var_A(x1) = x1 + 13 Term_var_A(x1) = max{91, x1 - 18} Id_A = 76 Term_sub#!_A(x1,x2) = max{0, x1 - 15} Case!_A(x1,x2,x3) = max{3, x1 - 18} Frozen#!_A(x1,x2,x3,x4) = max{0, x1 - 14, x3 - 15} Sum_sub!_A(x1,x2) = x1 + 2 Sum_constant!_A(x1) = 0 Left!_A = 1 Term_app!_A(x1,x2) = max{0, x1 - 18, x2 - 23} Term_pair!_A(x1,x2) = max{0, x1 - 23, x2 - 18} Term_inl!_A(x1) = max{0, x1 - 18} Term_inr!_A(x1) = max{0, x1 - 18} Term_sub!_A(x1,x2) = max{2, x2 - 10} Concat!_A(x1,x2) = max{65, x1 + 52, x2 + 34} Concat#!_A(x1,x2) = max{1, x1 - 7} Cons_usual!_A(x1,x2,x3) = max{65, x2 - 40, x3 - 35} Cons_sum!_A(x1,x2,x3) = max{x1 + 36, x3} Right!_A = 1 Sum_term_var!_A(x1) = x1 + 1 Term_var!_A(x1) = 66 Id!_A = 1 partial status: pi(Term_sub#) = [] pi(Case) = [1] pi(Frozen#) = [] pi(Sum_sub) = [] pi(Sum_constant) = [] pi(Left) = [] pi(Term_app) = [] pi(Term_pair) = [] pi(Term_inl) = [] pi(Term_inr) = [] pi(Term_sub) = [] pi(Concat) = [1] pi(Concat#) = [] pi(Cons_usual) = [] pi(Cons_sum) = [1, 2, 3] pi(Right) = [] pi(Sum_term_var) = [] pi(Term_var) = [] pi(Id) = [] The following rules are removed: p3, p4, p6, p7, p8, p9, p10, p12, p13 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: Term_sub#(Case(m,xi,n),s) -> Frozen#(m,Sum_sub(xi,s),n,s) p2: Frozen#(m,Sum_constant(Left()),n,s) -> Term_sub#(m,s) p3: Term_sub#(Term_pair(m,n),s) -> Term_sub#(m,s) p4: Concat#(Concat(s,t),u) -> Concat#(t,u) p5: Frozen#(m,Sum_constant(Right()),n,s) -> Term_sub#(n,s) p6: Frozen#(m,Sum_term_var(xi),n,s) -> Term_sub#(m,s) p7: Frozen#(m,Sum_term_var(xi),n,s) -> Term_sub#(n,s) and R consists of: r1: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) r2: Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) r3: Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) r4: Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) r5: Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) r6: Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) r7: Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) r8: Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) r9: Term_sub(Term_var(x),Id()) -> Term_var(x) r10: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m r11: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) r12: Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) r13: Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) r14: Sum_sub(xi,Id()) -> Sum_term_var(xi) r15: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) r16: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) r17: Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) r18: Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) r19: Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) r20: Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) r21: Concat(Id(),s) -> s The estimated dependency graph contains the following SCCs: {p1, p2, p3, p5, p6, p7} {p4} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: Term_sub#(Case(m,xi,n),s) -> Frozen#(m,Sum_sub(xi,s),n,s) p2: Frozen#(m,Sum_term_var(xi),n,s) -> Term_sub#(n,s) p3: Term_sub#(Term_pair(m,n),s) -> Term_sub#(m,s) p4: Frozen#(m,Sum_term_var(xi),n,s) -> Term_sub#(m,s) p5: Frozen#(m,Sum_constant(Right()),n,s) -> Term_sub#(n,s) p6: Frozen#(m,Sum_constant(Left()),n,s) -> Term_sub#(m,s) and R consists of: r1: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) r2: Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) r3: Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) r4: Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) r5: Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) r6: Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) r7: Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) r8: Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) r9: Term_sub(Term_var(x),Id()) -> Term_var(x) r10: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m r11: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) r12: Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) r13: Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) r14: Sum_sub(xi,Id()) -> Sum_term_var(xi) r15: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) r16: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) r17: Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) r18: Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) r19: Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) r20: Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) r21: Concat(Id(),s) -> s The set of usable rules consists of r14, r15, r16, r17 Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: max/plus interpretation on natural numbers: Term_sub#_A(x1,x2) = 1 Case_A(x1,x2,x3) = max{10, x1 - 26, x2 - 11, x3 - 11} Frozen#_A(x1,x2,x3,x4) = 1 Sum_sub_A(x1,x2) = x1 + 8 Sum_term_var_A(x1) = max{7, x1 - 12} Term_pair_A(x1,x2) = max{2, x1 - 11, x2 - 11} Sum_constant_A(x1) = 3 Right_A = 18 Left_A = 18 Id_A = 6 Cons_sum_A(x1,x2,x3) = max{11, x1 - 9, x2 - 9, x3 - 9} Cons_usual_A(x1,x2,x3) = max{10, x1 - 9, x2 - 9, x3 - 34} base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: Term_sub#_A(x1,x2) = max{5, x1 - 54, x2 - 2} Case_A(x1,x2,x3) = max{x1 + 52, x2 - 11, x3 + 42} Frozen#_A(x1,x2,x3,x4) = max{5, x1 - 54, x2 - 1, x3 - 54, x4 - 2} Sum_sub_A(x1,x2) = max{0, x1 - 75, x2 - 54} Sum_term_var_A(x1) = max{0, x1 - 75} Term_pair_A(x1,x2) = max{54, x1, x2 - 11} Sum_constant_A(x1) = max{0, x1 - 60} Right_A = 65 Left_A = 64 Id_A = 0 Cons_sum_A(x1,x2,x3) = max{x1 - 28, x2 - 1, x3 + 29} Cons_usual_A(x1,x2,x3) = max{x1 - 1, x2 - 82, x3 + 29} Term_sub#!_A(x1,x2) = x1 + 12 Case!_A(x1,x2,x3) = max{9, x1 - 89, x2 - 2, x3 - 168} Frozen#!_A(x1,x2,x3,x4) = max{55, x1 + 13, x3 + 13} Sum_sub!_A(x1,x2) = max{3, x1 - 1, x2 - 57} Sum_term_var!_A(x1) = max{2, x1 - 159} Term_pair!_A(x1,x2) = max{8, x1 - 66, x2 - 1} Sum_constant!_A(x1) = max{2, x1 - 59} Right!_A = 8 Left!_A = 1 Id!_A = 1 Cons_sum!_A(x1,x2,x3) = max{4, x1 - 30, x2 - 32, x3 - 58} Cons_usual!_A(x1,x2,x3) = max{1, x1 - 30, x2 - 44, x3 - 33} partial status: pi(Term_sub#) = [] pi(Case) = [] pi(Frozen#) = [] pi(Sum_sub) = [1] pi(Sum_term_var) = [] pi(Term_pair) = [] pi(Sum_constant) = [] pi(Right) = [] pi(Left) = [] pi(Id) = [] pi(Cons_sum) = [] pi(Cons_usual) = [] The following rules are removed: p1, p2, p4, p5, p6 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: Term_sub#(Term_pair(m,n),s) -> Term_sub#(m,s) and R consists of: r1: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) r2: Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) r3: Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) r4: Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) r5: Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) r6: Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) r7: Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) r8: Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) r9: Term_sub(Term_var(x),Id()) -> Term_var(x) r10: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m r11: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) r12: Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) r13: Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) r14: Sum_sub(xi,Id()) -> Sum_term_var(xi) r15: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) r16: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) r17: Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) r18: Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) r19: Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) r20: Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) r21: Concat(Id(),s) -> s The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: Term_sub#(Term_pair(m,n),s) -> Term_sub#(m,s) and R consists of: r1: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) r2: Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) r3: Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) r4: Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) r5: Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) r6: Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) r7: Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) r8: Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) r9: Term_sub(Term_var(x),Id()) -> Term_var(x) r10: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m r11: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) r12: Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) r13: Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) r14: Sum_sub(xi,Id()) -> Sum_term_var(xi) r15: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) r16: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) r17: Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) r18: Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) r19: Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) r20: Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) r21: Concat(Id(),s) -> s The set of usable rules consists of (no rules) Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: max/plus interpretation on natural numbers: Term_sub#_A(x1,x2) = max{12, x2 - 1} Term_pair_A(x1,x2) = max{11, x1 - 16, x2 - 16} base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: Term_sub#_A(x1,x2) = max{0, x1 - 2} Term_pair_A(x1,x2) = max{4, x1 + 1, x2 - 2} Term_sub#!_A(x1,x2) = max{1, x1 - 2} Term_pair!_A(x1,x2) = max{0, x2 - 58} partial status: pi(Term_sub#) = [] pi(Term_pair) = [] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: Concat#(Concat(s,t),u) -> Concat#(t,u) and R consists of: r1: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) r2: Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) r3: Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) r4: Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) r5: Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) r6: Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) r7: Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) r8: Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) r9: Term_sub(Term_var(x),Id()) -> Term_var(x) r10: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m r11: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) r12: Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) r13: Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) r14: Sum_sub(xi,Id()) -> Sum_term_var(xi) r15: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) r16: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) r17: Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) r18: Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) r19: Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) r20: Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) r21: Concat(Id(),s) -> s The set of usable rules consists of r21 Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: max/plus interpretation on natural numbers: Concat#_A(x1,x2) = x2 + 9 Concat_A(x1,x2) = max{7, x1 - 19, x2 + 1} Id_A = 1 base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: Concat#_A(x1,x2) = max{0, x1 - 2} Concat_A(x1,x2) = max{7, x1 - 39, x2 + 1} Id_A = 13 Concat#!_A(x1,x2) = max{4, x1 - 2} Concat!_A(x1,x2) = max{3, x1 - 14, x2 - 61} Id!_A = 1 partial status: pi(Concat#) = [] pi(Concat) = [2] pi(Id) = [] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: Sum_sub#(xi,Cons_sum(psi,k,s)) -> Sum_sub#(xi,s) p2: Sum_sub#(xi,Cons_usual(y,m,s)) -> Sum_sub#(xi,s) and R consists of: r1: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) r2: Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) r3: Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) r4: Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) r5: Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) r6: Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) r7: Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) r8: Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) r9: Term_sub(Term_var(x),Id()) -> Term_var(x) r10: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m r11: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) r12: Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) r13: Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) r14: Sum_sub(xi,Id()) -> Sum_term_var(xi) r15: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) r16: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) r17: Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) r18: Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) r19: Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) r20: Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) r21: Concat(Id(),s) -> s The set of usable rules consists of (no rules) Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: max/plus interpretation on natural numbers: Sum_sub#_A(x1,x2) = max{2, x1 + 1} Cons_sum_A(x1,x2,x3) = max{1, x1 - 9, x2 - 10, x3 - 15} Cons_usual_A(x1,x2,x3) = max{23, x1 + 15, x3 - 15} base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: Sum_sub#_A(x1,x2) = max{0, x2 - 28} Cons_sum_A(x1,x2,x3) = max{x1 - 1, x2 - 1, x3 + 27} Cons_usual_A(x1,x2,x3) = max{26, x1 - 8, x2 - 8, x3 + 7} Sum_sub#!_A(x1,x2) = max{33, x2 + 7} Cons_sum!_A(x1,x2,x3) = max{28, x1 - 1, x2 - 1, x3 - 31} Cons_usual!_A(x1,x2,x3) = max{x1 + 32, x3 + 30} partial status: pi(Sum_sub#) = [1] pi(Cons_sum) = [] pi(Cons_usual) = [1] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: Sum_sub#(xi,Cons_usual(y,m,s)) -> Sum_sub#(xi,s) and R consists of: r1: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) r2: Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) r3: Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) r4: Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) r5: Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) r6: Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) r7: Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) r8: Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) r9: Term_sub(Term_var(x),Id()) -> Term_var(x) r10: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m r11: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) r12: Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) r13: Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) r14: Sum_sub(xi,Id()) -> Sum_term_var(xi) r15: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) r16: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) r17: Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) r18: Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) r19: Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) r20: Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) r21: Concat(Id(),s) -> s The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: Sum_sub#(xi,Cons_usual(y,m,s)) -> Sum_sub#(xi,s) and R consists of: r1: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) r2: Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) r3: Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) r4: Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) r5: Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) r6: Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) r7: Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) r8: Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) r9: Term_sub(Term_var(x),Id()) -> Term_var(x) r10: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m r11: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) r12: Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) r13: Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) r14: Sum_sub(xi,Id()) -> Sum_term_var(xi) r15: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) r16: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) r17: Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) r18: Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) r19: Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) r20: Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) r21: Concat(Id(),s) -> s The set of usable rules consists of (no rules) Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: max/plus interpretation on natural numbers: Sum_sub#_A(x1,x2) = max{2, x1 + 1} Cons_usual_A(x1,x2,x3) = max{x1 + 1, x2 - 19, x3 - 20} base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: Sum_sub#_A(x1,x2) = max{0, x2 - 2} Cons_usual_A(x1,x2,x3) = max{8, x1 - 1, x2 - 22, x3 + 1} Sum_sub#!_A(x1,x2) = max{5, x2 - 2} Cons_usual!_A(x1,x2,x3) = max{4, x1 - 1, x2 - 42, x3 - 40} partial status: pi(Sum_sub#) = [1] pi(Cons_usual) = [1] The following rules are removed: p1 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: Term_sub#(Term_var(x),Cons_usual(y,m,s)) -> Term_sub#(Term_var(x),s) p2: Term_sub#(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub#(Term_var(x),s) and R consists of: r1: Term_sub(Case(m,xi,n),s) -> Frozen(m,Sum_sub(xi,s),n,s) r2: Frozen(m,Sum_constant(Left()),n,s) -> Term_sub(m,s) r3: Frozen(m,Sum_constant(Right()),n,s) -> Term_sub(n,s) r4: Frozen(m,Sum_term_var(xi),n,s) -> Case(Term_sub(m,s),xi,Term_sub(n,s)) r5: Term_sub(Term_app(m,n),s) -> Term_app(Term_sub(m,s),Term_sub(n,s)) r6: Term_sub(Term_pair(m,n),s) -> Term_pair(Term_sub(m,s),Term_sub(n,s)) r7: Term_sub(Term_inl(m),s) -> Term_inl(Term_sub(m,s)) r8: Term_sub(Term_inr(m),s) -> Term_inr(Term_sub(m,s)) r9: Term_sub(Term_var(x),Id()) -> Term_var(x) r10: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> m r11: Term_sub(Term_var(x),Cons_usual(y,m,s)) -> Term_sub(Term_var(x),s) r12: Term_sub(Term_var(x),Cons_sum(xi,k,s)) -> Term_sub(Term_var(x),s) r13: Term_sub(Term_sub(m,s),t) -> Term_sub(m,Concat(s,t)) r14: Sum_sub(xi,Id()) -> Sum_term_var(xi) r15: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_constant(k) r16: Sum_sub(xi,Cons_sum(psi,k,s)) -> Sum_sub(xi,s) r17: Sum_sub(xi,Cons_usual(y,m,s)) -> Sum_sub(xi,s) r18: Concat(Concat(s,t),u) -> Concat(s,Concat(t,u)) r19: Concat(Cons_usual(x,m,s),t) -> Cons_usual(x,Term_sub(m,t),Concat(s,t)) r20: Concat(Cons_sum(xi,k,s),t) -> Cons_sum(xi,k,Concat(s,t)) r21: Concat(Id(),s) -> s The set of usable rules consists of (no rules) Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: max/plus interpretation on natural numbers: Term_sub#_A(x1,x2) = max{3, x1 - 25} Term_var_A(x1) = max{24, x1 - 5} Cons_usual_A(x1,x2,x3) = max{0, x1 - 20, x2 - 34, x3 - 31} Cons_sum_A(x1,x2,x3) = max{x1 - 30, x2 + 2, x3 - 30} base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: Term_sub#_A(x1,x2) = max{0, x2 - 7} Term_var_A(x1) = max{4, x1 - 7} Cons_usual_A(x1,x2,x3) = max{x1 - 6, x2 - 3, x3 + 6} Cons_sum_A(x1,x2,x3) = max{6, x1 - 1, x2 - 1, x3 + 1} Term_sub#!_A(x1,x2) = max{3, x2 - 2} Term_var!_A(x1) = max{5, x1 - 1} Cons_usual!_A(x1,x2,x3) = max{0, x1 - 22, x2 - 23, x3 - 52} Cons_sum!_A(x1,x2,x3) = max{2, x1 - 1, x2 - 1, x3 - 52} partial status: pi(Term_sub#) = [] pi(Term_var) = [] pi(Cons_usual) = [] pi(Cons_sum) = [2] The following rules are removed: p1, p2 Then no dependency pair remains.