YES We show the termination of the TRS R: app(app(and(),true()),true()) -> true() app(app(and(),true()),false()) -> false() app(app(and(),false()),true()) -> false() app(app(and(),false()),false()) -> false() app(app(or(),true()),true()) -> true() app(app(or(),true()),false()) -> true() app(app(or(),false()),true()) -> true() app(app(or(),false()),false()) -> false() app(app(forall(),p),nil()) -> true() app(app(forall(),p),app(app(cons(),x),xs)) -> app(app(and(),app(p,x)),app(app(forall(),p),xs)) app(app(forsome(),p),nil()) -> false() app(app(forsome(),p),app(app(cons(),x),xs)) -> app(app(or(),app(p,x)),app(app(forsome(),p),xs)) Instead we prove termination of the uncurried system: app(and_0(),y) -> and_1(y) app(and_1(x1),y) -> and_2(x1,y) app(cons_0(),y) -> cons_1(y) app(cons_1(x1),y) -> cons_2(x1,y) app(forall_0(),y) -> forall_1(y) app(forall_1(x1),y) -> forall_2(x1,y) app(forsome_0(),y) -> forsome_1(y) app(forsome_1(x1),y) -> forsome_2(x1,y) app(or_0(),y) -> or_1(y) app(or_1(x1),y) -> or_2(x1,y) and_2(true_0(),true_0()) -> true_0() and_2(true_0(),false_0()) -> false_0() and_2(false_0(),true_0()) -> false_0() and_2(false_0(),false_0()) -> false_0() or_2(true_0(),true_0()) -> true_0() or_2(true_0(),false_0()) -> true_0() or_2(false_0(),true_0()) -> true_0() or_2(false_0(),false_0()) -> false_0() forall_2(p,nil_0()) -> true_0() forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs)) forsome_2(p,nil_0()) -> false_0() forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs)) -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: app#(and_1(x1),y) -> and_2#(x1,y) p2: app#(forall_1(x1),y) -> forall_2#(x1,y) p3: app#(forsome_1(x1),y) -> forsome_2#(x1,y) p4: app#(or_1(x1),y) -> or_2#(x1,y) p5: forall_2#(p,cons_2(x,xs)) -> and_2#(app(p,x),forall_2(p,xs)) p6: forall_2#(p,cons_2(x,xs)) -> app#(p,x) p7: forall_2#(p,cons_2(x,xs)) -> forall_2#(p,xs) p8: forsome_2#(p,cons_2(x,xs)) -> or_2#(app(p,x),forsome_2(p,xs)) p9: forsome_2#(p,cons_2(x,xs)) -> app#(p,x) p10: forsome_2#(p,cons_2(x,xs)) -> forsome_2#(p,xs) and R consists of: r1: app(and_0(),y) -> and_1(y) r2: app(and_1(x1),y) -> and_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(forall_0(),y) -> forall_1(y) r6: app(forall_1(x1),y) -> forall_2(x1,y) r7: app(forsome_0(),y) -> forsome_1(y) r8: app(forsome_1(x1),y) -> forsome_2(x1,y) r9: app(or_0(),y) -> or_1(y) r10: app(or_1(x1),y) -> or_2(x1,y) r11: and_2(true_0(),true_0()) -> true_0() r12: and_2(true_0(),false_0()) -> false_0() r13: and_2(false_0(),true_0()) -> false_0() r14: and_2(false_0(),false_0()) -> false_0() r15: or_2(true_0(),true_0()) -> true_0() r16: or_2(true_0(),false_0()) -> true_0() r17: or_2(false_0(),true_0()) -> true_0() r18: or_2(false_0(),false_0()) -> false_0() r19: forall_2(p,nil_0()) -> true_0() r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs)) r21: forsome_2(p,nil_0()) -> false_0() r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs)) The estimated dependency graph contains the following SCCs: {p2, p3, p6, p7, p9, p10} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: app#(forall_1(x1),y) -> forall_2#(x1,y) p2: forall_2#(p,cons_2(x,xs)) -> forall_2#(p,xs) p3: forall_2#(p,cons_2(x,xs)) -> app#(p,x) p4: app#(forsome_1(x1),y) -> forsome_2#(x1,y) p5: forsome_2#(p,cons_2(x,xs)) -> forsome_2#(p,xs) p6: forsome_2#(p,cons_2(x,xs)) -> app#(p,x) and R consists of: r1: app(and_0(),y) -> and_1(y) r2: app(and_1(x1),y) -> and_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(forall_0(),y) -> forall_1(y) r6: app(forall_1(x1),y) -> forall_2(x1,y) r7: app(forsome_0(),y) -> forsome_1(y) r8: app(forsome_1(x1),y) -> forsome_2(x1,y) r9: app(or_0(),y) -> or_1(y) r10: app(or_1(x1),y) -> or_2(x1,y) r11: and_2(true_0(),true_0()) -> true_0() r12: and_2(true_0(),false_0()) -> false_0() r13: and_2(false_0(),true_0()) -> false_0() r14: and_2(false_0(),false_0()) -> false_0() r15: or_2(true_0(),true_0()) -> true_0() r16: or_2(true_0(),false_0()) -> true_0() r17: or_2(false_0(),true_0()) -> true_0() r18: or_2(false_0(),false_0()) -> false_0() r19: forall_2(p,nil_0()) -> true_0() r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs)) r21: forsome_2(p,nil_0()) -> false_0() r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs)) 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: app#_A(x1,x2) = max{4, x2 - 1} forall_1_A(x1) = max{2, x1 - 14} forall_2#_A(x1,x2) = max{3, x2 - 6} cons_2_A(x1,x2) = max{x1 + 10, x2 + 1} forsome_1_A(x1) = max{3, x1 - 14} forsome_2#_A(x1,x2) = max{4, x2 - 11} base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: app#_A(x1,x2) = max{0, x1 - 3, x2 - 16} forall_1_A(x1) = x1 + 2 forall_2#_A(x1,x2) = max{0, x1 - 3, x2 - 16} cons_2_A(x1,x2) = max{15, x1 + 7, x2 + 7} forsome_1_A(x1) = x1 + 2 forsome_2#_A(x1,x2) = max{0, x1 - 3, x2 - 16} app#!_A(x1,x2) = max{4, x1 - 33, x2 - 16} forall_1!_A(x1) = max{1, x1 - 30} forall_2#!_A(x1,x2) = max{5, x1 - 30, x2 - 17} cons_2!_A(x1,x2) = max{3, x1 - 15, x2 - 18} forsome_1!_A(x1) = max{2, x1 - 34} forsome_2#!_A(x1,x2) = max{4, x1 - 33, x2 - 16} partial status: pi(app#) = [] pi(forall_1) = [] pi(forall_2#) = [] pi(cons_2) = [2] pi(forsome_1) = [] pi(forsome_2#) = [] The following rules are removed: p1, p2, p3 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: app#(forsome_1(x1),y) -> forsome_2#(x1,y) p2: forsome_2#(p,cons_2(x,xs)) -> forsome_2#(p,xs) p3: forsome_2#(p,cons_2(x,xs)) -> app#(p,x) and R consists of: r1: app(and_0(),y) -> and_1(y) r2: app(and_1(x1),y) -> and_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(forall_0(),y) -> forall_1(y) r6: app(forall_1(x1),y) -> forall_2(x1,y) r7: app(forsome_0(),y) -> forsome_1(y) r8: app(forsome_1(x1),y) -> forsome_2(x1,y) r9: app(or_0(),y) -> or_1(y) r10: app(or_1(x1),y) -> or_2(x1,y) r11: and_2(true_0(),true_0()) -> true_0() r12: and_2(true_0(),false_0()) -> false_0() r13: and_2(false_0(),true_0()) -> false_0() r14: and_2(false_0(),false_0()) -> false_0() r15: or_2(true_0(),true_0()) -> true_0() r16: or_2(true_0(),false_0()) -> true_0() r17: or_2(false_0(),true_0()) -> true_0() r18: or_2(false_0(),false_0()) -> false_0() r19: forall_2(p,nil_0()) -> true_0() r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs)) r21: forsome_2(p,nil_0()) -> false_0() r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs)) The estimated dependency graph contains the following SCCs: {p1, p2, p3} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: app#(forsome_1(x1),y) -> forsome_2#(x1,y) p2: forsome_2#(p,cons_2(x,xs)) -> app#(p,x) p3: forsome_2#(p,cons_2(x,xs)) -> forsome_2#(p,xs) and R consists of: r1: app(and_0(),y) -> and_1(y) r2: app(and_1(x1),y) -> and_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(forall_0(),y) -> forall_1(y) r6: app(forall_1(x1),y) -> forall_2(x1,y) r7: app(forsome_0(),y) -> forsome_1(y) r8: app(forsome_1(x1),y) -> forsome_2(x1,y) r9: app(or_0(),y) -> or_1(y) r10: app(or_1(x1),y) -> or_2(x1,y) r11: and_2(true_0(),true_0()) -> true_0() r12: and_2(true_0(),false_0()) -> false_0() r13: and_2(false_0(),true_0()) -> false_0() r14: and_2(false_0(),false_0()) -> false_0() r15: or_2(true_0(),true_0()) -> true_0() r16: or_2(true_0(),false_0()) -> true_0() r17: or_2(false_0(),true_0()) -> true_0() r18: or_2(false_0(),false_0()) -> false_0() r19: forall_2(p,nil_0()) -> true_0() r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs)) r21: forsome_2(p,nil_0()) -> false_0() r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs)) 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: app#_A(x1,x2) = max{4, x1 - 7} forsome_1_A(x1) = max{13, x1 + 6} forsome_2#_A(x1,x2) = max{5, x1 - 2} cons_2_A(x1,x2) = max{3, x1 - 7, x2 - 7} base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: app#_A(x1,x2) = max{0, x1 - 14, x2 - 2} forsome_1_A(x1) = max{13, x1 + 1} forsome_2#_A(x1,x2) = max{0, x1 - 14, x2 - 2} cons_2_A(x1,x2) = max{1, x1, x2} app#!_A(x1,x2) = max{6, x1 - 14, x2 - 3} forsome_1!_A(x1) = max{2, x1 - 15} forsome_2#!_A(x1,x2) = max{6, x1 - 14, x2 - 2} cons_2!_A(x1,x2) = max{5, x1 - 4, x2 - 3} partial status: pi(app#) = [] pi(forsome_1) = [1] pi(forsome_2#) = [] pi(cons_2) = [] The following rules are removed: p1, p2 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: forsome_2#(p,cons_2(x,xs)) -> forsome_2#(p,xs) and R consists of: r1: app(and_0(),y) -> and_1(y) r2: app(and_1(x1),y) -> and_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(forall_0(),y) -> forall_1(y) r6: app(forall_1(x1),y) -> forall_2(x1,y) r7: app(forsome_0(),y) -> forsome_1(y) r8: app(forsome_1(x1),y) -> forsome_2(x1,y) r9: app(or_0(),y) -> or_1(y) r10: app(or_1(x1),y) -> or_2(x1,y) r11: and_2(true_0(),true_0()) -> true_0() r12: and_2(true_0(),false_0()) -> false_0() r13: and_2(false_0(),true_0()) -> false_0() r14: and_2(false_0(),false_0()) -> false_0() r15: or_2(true_0(),true_0()) -> true_0() r16: or_2(true_0(),false_0()) -> true_0() r17: or_2(false_0(),true_0()) -> true_0() r18: or_2(false_0(),false_0()) -> false_0() r19: forall_2(p,nil_0()) -> true_0() r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs)) r21: forsome_2(p,nil_0()) -> false_0() r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs)) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: forsome_2#(p,cons_2(x,xs)) -> forsome_2#(p,xs) and R consists of: r1: app(and_0(),y) -> and_1(y) r2: app(and_1(x1),y) -> and_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(forall_0(),y) -> forall_1(y) r6: app(forall_1(x1),y) -> forall_2(x1,y) r7: app(forsome_0(),y) -> forsome_1(y) r8: app(forsome_1(x1),y) -> forsome_2(x1,y) r9: app(or_0(),y) -> or_1(y) r10: app(or_1(x1),y) -> or_2(x1,y) r11: and_2(true_0(),true_0()) -> true_0() r12: and_2(true_0(),false_0()) -> false_0() r13: and_2(false_0(),true_0()) -> false_0() r14: and_2(false_0(),false_0()) -> false_0() r15: or_2(true_0(),true_0()) -> true_0() r16: or_2(true_0(),false_0()) -> true_0() r17: or_2(false_0(),true_0()) -> true_0() r18: or_2(false_0(),false_0()) -> false_0() r19: forall_2(p,nil_0()) -> true_0() r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs)) r21: forsome_2(p,nil_0()) -> false_0() r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs)) 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: forsome_2#_A(x1,x2) = max{6, x1 + 1} cons_2_A(x1,x2) = max{5, x1 - 11, x2 - 11} base order 2: marked-algebra reduction triple (>=A, >=A!, >A!): max/plus interpretation on natural numbers: forsome_2#_A(x1,x2) = x2 cons_2_A(x1,x2) = max{11, x1 + 9, x2 + 3} forsome_2#!_A(x1,x2) = max{4, x2 - 4} cons_2!_A(x1,x2) = max{3, x1 - 27, x2 - 34} partial status: pi(forsome_2#) = [1] pi(cons_2) = [] The following rules are removed: p1 Then no dependency pair remains.