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, r14, r15, r16, r17, r21 Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: Term_sub#_A(x1,x2) = x1 + 4 Case_A(x1,x2,x3) = x1 + x2 + x3 + 2 Frozen#_A(x1,x2,x3,x4) = x1 + x2 + x3 + 1 Sum_sub_A(x1,x2) = 5 Sum_term_var_A(x1) = 3 Term_sub_A(x1,x2) = x1 + x2 Concat#_A(x1,x2) = x1 + 4 Cons_sum_A(x1,x2,x3) = x3 Cons_usual_A(x1,x2,x3) = x2 + x3 Concat_A(x1,x2) = x1 + x2 + 5 Term_inr_A(x1) = x1 Term_inl_A(x1) = x1 Term_pair_A(x1,x2) = x1 + x2 Term_app_A(x1,x2) = x1 + x2 Sum_constant_A(x1) = 3 Right_A() = 1 Left_A() = 1 Term_var_A(x1) = 0 Id_A() = 4 base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: Term_sub#_A(x1,x2) = 1 Case_A(x1,x2,x3) = max{56, x1 + 23, x2 + 34, x3 + 23} Frozen#_A(x1,x2,x3,x4) = 1 Sum_sub_A(x1,x2) = max{21, x1 + 1, x2 - 10} Sum_term_var_A(x1) = 22 Term_sub_A(x1,x2) = max{x1 + 20, x2 + 23} Concat#_A(x1,x2) = 1 Cons_sum_A(x1,x2,x3) = max{x1 + 11, x2 + 32, x3 + 10} Cons_usual_A(x1,x2,x3) = max{x2 + 12, x3 + 16} Concat_A(x1,x2) = max{71, x1 + 19, x2 + 18} Term_inr_A(x1) = max{21, x1} Term_inl_A(x1) = max{21, x1} Term_pair_A(x1,x2) = max{21, x1, x2} Term_app_A(x1,x2) = max{22, x1, x2} Sum_constant_A(x1) = max{22, x1 - 11} Right_A = 1 Left_A = 0 Term_var_A(x1) = max{24, x1 - 1} Id_A = 33 Term_sub#!_A(x1,x2) = max{0, x1 - 23} Case!_A(x1,x2,x3) = 1 Frozen#!_A(x1,x2,x3,x4) = max{0, x1 - 23, x3 - 23} Sum_sub!_A(x1,x2) = max{60, x1 + 38, x2 - 11} Sum_term_var!_A(x1) = 0 Term_sub!_A(x1,x2) = max{24, x1 + 3, x2 + 18} Concat#!_A(x1,x2) = x1 Cons_sum!_A(x1,x2,x3) = max{x1 + 62, x2 - 1, x3 + 1} Cons_usual!_A(x1,x2,x3) = max{51, x3 - 19} Concat!_A(x1,x2) = 0 Term_inr!_A(x1) = 4 Term_inl!_A(x1) = 2 Term_pair!_A(x1,x2) = 25 Term_app!_A(x1,x2) = 0 Sum_constant!_A(x1) = max{61, x1 - 2} Right!_A = 1 Left!_A = 1 Term_var!_A(x1) = 50 Id!_A = 51 partial status: pi(Term_sub#) = [] pi(Case) = [3] pi(Frozen#) = [2, 3] pi(Sum_sub) = [] pi(Sum_term_var) = [] pi(Term_sub) = [1, 2] pi(Concat#) = [] pi(Cons_sum) = [3] pi(Cons_usual) = [2, 3] pi(Concat) = [1] pi(Term_inr) = [] pi(Term_inl) = [] pi(Term_pair) = [] pi(Term_app) = [] pi(Sum_constant) = [] pi(Right) = [] pi(Left) = [] pi(Term_var) = [] pi(Id) = [] The following rules are removed: p1, p2, p4, p5, p6, p14, p15, p16, p17, p18 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: Term_sub#(Term_sub(m,s),t) -> Concat#(s,t) p2: Term_sub#(Term_sub(m,s),t) -> Term_sub#(m,Concat(s,t)) p3: Term_sub#(Term_inr(m),s) -> Term_sub#(m,s) p4: Term_sub#(Term_inl(m),s) -> Term_sub#(m,s) p5: Term_sub#(Term_pair(m,n),s) -> Term_sub#(n,s) p6: Term_sub#(Term_pair(m,n),s) -> Term_sub#(m,s) p7: Term_sub#(Term_app(m,n),s) -> Term_sub#(n,s) p8: Term_sub#(Term_app(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: {p2, p3, p4, p5, p6, p7, p8} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: Term_sub#(Term_sub(m,s),t) -> Term_sub#(m,Concat(s,t)) p2: Term_sub#(Term_app(m,n),s) -> Term_sub#(m,s) p3: Term_sub#(Term_app(m,n),s) -> Term_sub#(n,s) p4: Term_sub#(Term_pair(m,n),s) -> Term_sub#(m,s) p5: Term_sub#(Term_pair(m,n),s) -> Term_sub#(n,s) p6: Term_sub#(Term_inl(m),s) -> Term_sub#(m,s) p7: Term_sub#(Term_inr(m),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 Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: Term_sub#_A(x1,x2) = x1 + 6 Term_sub_A(x1,x2) = x1 + x2 + 7 Concat_A(x1,x2) = x2 + 4 Term_app_A(x1,x2) = x1 + x2 + 1 Term_pair_A(x1,x2) = x1 + x2 + 1 Term_inl_A(x1) = x1 + 1 Term_inr_A(x1) = x1 + 8 Term_var_A(x1) = 4 Id_A() = 5 base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: Term_sub#_A(x1,x2) = max{0, x1 - 54} Term_sub_A(x1,x2) = max{24, x1 + 9, x2 - 8} Concat_A(x1,x2) = max{42, x1 + 8} Term_app_A(x1,x2) = max{36, x1 + 26, x2 + 26} Term_pair_A(x1,x2) = max{x1 + 39, x2 + 53} Term_inl_A(x1) = max{49, x1 + 39} Term_inr_A(x1) = x1 + 27 Term_var_A(x1) = max{38, x1 - 10} Id_A = 9 Term_sub#!_A(x1,x2) = max{2, x1 - 50} Term_sub!_A(x1,x2) = max{45, x1 + 7} Concat!_A(x1,x2) = 0 Term_app!_A(x1,x2) = max{44, x1 - 76, x2 - 51} Term_pair!_A(x1,x2) = max{0, x1 - 50, x2 - 25} Term_inl!_A(x1) = 1 Term_inr!_A(x1) = 33 Term_var!_A(x1) = max{10, x1 - 4} Id!_A = 1 partial status: pi(Term_sub#) = [1] pi(Term_sub) = [] pi(Concat) = [] pi(Term_app) = [2] pi(Term_pair) = [2] pi(Term_inl) = [1] pi(Term_inr) = [] pi(Term_var) = [] pi(Id) = [] The following rules are removed: p1, p2, p3, p4, p5, p6, p7 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: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: Sum_sub#_A(x1,x2) = x1 + 2 Cons_sum_A(x1,x2,x3) = x3 + 10 Cons_usual_A(x1,x2,x3) = 1 base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: Sum_sub#_A(x1,x2) = 0 Cons_sum_A(x1,x2,x3) = max{x1 - 13, x2 - 13, x3 + 12} Cons_usual_A(x1,x2,x3) = max{x1 - 5, x2 - 7, x3 + 11} Sum_sub#!_A(x1,x2) = x2 + 15 Cons_sum!_A(x1,x2,x3) = max{2, x1 - 1, x2 - 2, x3 - 12} Cons_usual!_A(x1,x2,x3) = max{0, x1 - 1, x2 - 6, x3 - 12} partial status: pi(Sum_sub#) = [] pi(Cons_sum) = [3] pi(Cons_usual) = [] The following rules are removed: p1, p2 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 r21 Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: Term_sub#_A(x1,x2) = x1 + x2 + 1 Term_var_A(x1) = 4 Cons_usual_A(x1,x2,x3) = x3 + 6 Cons_sum_A(x1,x2,x3) = x3 + 6 Concat_A(x1,x2) = x2 + 7 Id_A() = 5 base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: Term_sub#_A(x1,x2) = 0 Term_var_A(x1) = max{6, x1 - 1} Cons_usual_A(x1,x2,x3) = max{9, x1 - 12, x2 - 12} Cons_sum_A(x1,x2,x3) = max{9, x1 - 6, x2 - 6, x3 - 12} Concat_A(x1,x2) = max{8, x1 - 22, x2 + 6} Id_A = 13 Term_sub#!_A(x1,x2) = max{17, x2 + 7} Term_var!_A(x1) = max{17, x1 - 1} Cons_usual!_A(x1,x2,x3) = max{0, x1 - 13} Cons_sum!_A(x1,x2,x3) = max{2, x1 - 1, x2 - 4, x3 - 11} Concat!_A(x1,x2) = 3 Id!_A = 1 partial status: pi(Term_sub#) = [1, 2] pi(Term_var) = [] pi(Cons_usual) = [3] pi(Cons_sum) = [] pi(Concat) = [] pi(Id) = [] The following rules are removed: p1, p2 Then no dependency pair remains.