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 r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22 Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: trivial reduction triple (T x T, T x T, Φ) base order 2: model-based reduction triple (=A, >=A!, >A!): finite algebra: carrier: {0,1} interpretations: app#_A(0,0) = 0 app#_A(0,1) = 0 app#_A(1,0) = 0 app#_A(1,1) = 0 forall_1_A(0) = 1 forall_1_A(1) = 1 forall_2#_A(0,0) = 0 forall_2#_A(0,1) = 0 forall_2#_A(1,0) = 0 forall_2#_A(1,1) = 0 cons_2_A(0,0) = 1 cons_2_A(0,1) = 1 cons_2_A(1,0) = 1 cons_2_A(1,1) = 1 forsome_1_A(0) = 1 forsome_1_A(1) = 1 forsome_2#_A(0,0) = 0 forsome_2#_A(0,1) = 0 forsome_2#_A(1,0) = 0 forsome_2#_A(1,1) = 0 app_A(0,0) = 1 app_A(0,1) = 1 app_A(1,0) = 0 app_A(1,1) = 0 and_0_A = 0 and_1_A(0) = 1 and_1_A(1) = 1 and_2_A(0,0) = 0 and_2_A(0,1) = 0 and_2_A(1,0) = 0 and_2_A(1,1) = 0 cons_0_A = 1 cons_1_A(0) = 0 cons_1_A(1) = 0 forall_0_A = 0 forall_2_A(0,0) = 0 forall_2_A(0,1) = 0 forall_2_A(1,0) = 0 forall_2_A(1,1) = 0 forsome_0_A = 0 forsome_2_A(0,0) = 0 forsome_2_A(0,1) = 0 forsome_2_A(1,0) = 0 forsome_2_A(1,1) = 0 or_0_A = 0 or_1_A(0) = 1 or_1_A(1) = 1 or_2_A(0,0) = 0 or_2_A(0,1) = 0 or_2_A(1,0) = 0 or_2_A(1,1) = 0 true_0_A = 0 false_0_A = 0 nil_0_A = 1 app#!_A(0,0) = 0 app#!_A(0,1) = 0 app#!_A(1,0) = 0 app#!_A(1,1) = 0 forall_1!_A(0) = 1 forall_1!_A(1) = 1 forall_2#!_A(0,0) = 0 forall_2#!_A(0,1) = 0 forall_2#!_A(1,0) = 0 forall_2#!_A(1,1) = 0 cons_2!_A(0,0) = 1 cons_2!_A(0,1) = 1 cons_2!_A(1,0) = 1 cons_2!_A(1,1) = 1 forsome_1!_A(0) = 1 forsome_1!_A(1) = 1 forsome_2#!_A(0,0) = 0 forsome_2#!_A(0,1) = 0 forsome_2#!_A(1,0) = 0 forsome_2#!_A(1,1) = 0 app!_A(0,0) = 1 app!_A(0,1) = 1 app!_A(1,0) = 1 app!_A(1,1) = 1 and_0!_A = 1 and_1!_A(0) = 1 and_1!_A(1) = 1 and_2!_A(0,0) = 0 and_2!_A(0,1) = 1 and_2!_A(1,0) = 1 and_2!_A(1,1) = 0 cons_0!_A = 1 cons_1!_A(0) = 1 cons_1!_A(1) = 0 forall_0!_A = 1 forall_2!_A(0,0) = 0 forall_2!_A(0,1) = 1 forall_2!_A(1,0) = 0 forall_2!_A(1,1) = 0 forsome_0!_A = 1 forsome_2!_A(0,0) = 1 forsome_2!_A(0,1) = 0 forsome_2!_A(1,0) = 0 forsome_2!_A(1,1) = 0 or_0!_A = 1 or_1!_A(0) = 1 or_1!_A(1) = 1 or_2!_A(0,0) = 0 or_2!_A(0,1) = 1 or_2!_A(1,0) = 0 or_2!_A(1,1) = 1 true_0!_A = 0 false_0!_A = 0 nil_0!_A = 1 partial status: pi(app#) = [1] pi(forall_1) = [1] pi(forall_2#) = [1] pi(cons_2) = [2] pi(forsome_1) = [1] pi(forsome_2#) = [1] pi(app) = [2] pi(and_0) = [] pi(and_1) = [1] pi(and_2) = [] pi(cons_0) = [] pi(cons_1) = [] pi(forall_0) = [] pi(forall_2) = [] pi(forsome_0) = [] pi(forsome_2) = [] pi(or_0) = [] pi(or_1) = [1] pi(or_2) = [] pi(true_0) = [] pi(false_0) = [] pi(nil_0) = [] The following rules are removed: p4 -- SCC decomposition. 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: forsome_2#(p,cons_2(x,xs)) -> forsome_2#(p,xs) p5: 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: {p4} {p1, p2, p3} -- 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 r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22 Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: trivial reduction triple (T x T, T x T, Φ) base order 2: model-based reduction triple (=A, >=A!, >A!): finite algebra: carrier: {0,1} interpretations: forsome_2#_A(0,0) = 0 forsome_2#_A(0,1) = 0 forsome_2#_A(1,0) = 0 forsome_2#_A(1,1) = 0 cons_2_A(0,0) = 1 cons_2_A(0,1) = 1 cons_2_A(1,0) = 1 cons_2_A(1,1) = 1 app_A(0,0) = 1 app_A(0,1) = 1 app_A(1,0) = 0 app_A(1,1) = 0 and_0_A = 0 and_1_A(0) = 1 and_1_A(1) = 1 and_2_A(0,0) = 0 and_2_A(0,1) = 0 and_2_A(1,0) = 0 and_2_A(1,1) = 0 cons_0_A = 1 cons_1_A(0) = 0 cons_1_A(1) = 0 forall_0_A = 0 forall_1_A(0) = 1 forall_1_A(1) = 1 forall_2_A(0,0) = 0 forall_2_A(0,1) = 0 forall_2_A(1,0) = 0 forall_2_A(1,1) = 0 forsome_0_A = 0 forsome_1_A(0) = 1 forsome_1_A(1) = 1 forsome_2_A(0,0) = 0 forsome_2_A(0,1) = 0 forsome_2_A(1,0) = 0 forsome_2_A(1,1) = 0 or_0_A = 0 or_1_A(0) = 1 or_1_A(1) = 1 or_2_A(0,0) = 0 or_2_A(0,1) = 0 or_2_A(1,0) = 0 or_2_A(1,1) = 0 true_0_A = 0 false_0_A = 0 nil_0_A = 1 forsome_2#!_A(0,0) = 0 forsome_2#!_A(0,1) = 0 forsome_2#!_A(1,0) = 0 forsome_2#!_A(1,1) = 1 cons_2!_A(0,0) = 0 cons_2!_A(0,1) = 0 cons_2!_A(1,0) = 0 cons_2!_A(1,1) = 1 app!_A(0,0) = 0 app!_A(0,1) = 1 app!_A(1,0) = 1 app!_A(1,1) = 1 and_0!_A = 1 and_1!_A(0) = 0 and_1!_A(1) = 1 and_2!_A(0,0) = 1 and_2!_A(0,1) = 0 and_2!_A(1,0) = 0 and_2!_A(1,1) = 0 cons_0!_A = 0 cons_1!_A(0) = 0 cons_1!_A(1) = 0 forall_0!_A = 1 forall_1!_A(0) = 1 forall_1!_A(1) = 1 forall_2!_A(0,0) = 0 forall_2!_A(0,1) = 1 forall_2!_A(1,0) = 0 forall_2!_A(1,1) = 1 forsome_0!_A = 1 forsome_1!_A(0) = 1 forsome_1!_A(1) = 1 forsome_2!_A(0,0) = 0 forsome_2!_A(0,1) = 0 forsome_2!_A(1,0) = 1 forsome_2!_A(1,1) = 0 or_0!_A = 1 or_1!_A(0) = 1 or_1!_A(1) = 1 or_2!_A(0,0) = 0 or_2!_A(0,1) = 0 or_2!_A(1,0) = 0 or_2!_A(1,1) = 0 true_0!_A = 0 false_0!_A = 0 nil_0!_A = 1 partial status: pi(forsome_2#) = [2] pi(cons_2) = [1, 2] pi(app) = [1, 2] pi(and_0) = [] pi(and_1) = [] pi(and_2) = [] pi(cons_0) = [] pi(cons_1) = [1] pi(forall_0) = [] pi(forall_1) = [] pi(forall_2) = [] pi(forsome_0) = [] pi(forsome_1) = [] pi(forsome_2) = [] pi(or_0) = [] pi(or_1) = [] pi(or_2) = [] pi(true_0) = [] pi(false_0) = [] pi(nil_0) = [] 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: app#(forall_1(x1),y) -> forall_2#(x1,y) p2: forall_2#(p,cons_2(x,xs)) -> app#(p,x) p3: forall_2#(p,cons_2(x,xs)) -> forall_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 r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22 Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: trivial reduction triple (T x T, T x T, Φ) base order 2: model-based reduction triple (=A, >=A!, >A!): finite algebra: carrier: {0,1} interpretations: app#_A(0,0) = 0 app#_A(0,1) = 0 app#_A(1,0) = 0 app#_A(1,1) = 0 forall_1_A(0) = 1 forall_1_A(1) = 1 forall_2#_A(0,0) = 0 forall_2#_A(0,1) = 0 forall_2#_A(1,0) = 0 forall_2#_A(1,1) = 0 cons_2_A(0,0) = 1 cons_2_A(0,1) = 1 cons_2_A(1,0) = 1 cons_2_A(1,1) = 1 app_A(0,0) = 1 app_A(0,1) = 1 app_A(1,0) = 0 app_A(1,1) = 0 and_0_A = 0 and_1_A(0) = 1 and_1_A(1) = 1 and_2_A(0,0) = 0 and_2_A(0,1) = 0 and_2_A(1,0) = 0 and_2_A(1,1) = 0 cons_0_A = 1 cons_1_A(0) = 0 cons_1_A(1) = 0 forall_0_A = 0 forall_2_A(0,0) = 0 forall_2_A(0,1) = 0 forall_2_A(1,0) = 0 forall_2_A(1,1) = 0 forsome_0_A = 0 forsome_1_A(0) = 1 forsome_1_A(1) = 1 forsome_2_A(0,0) = 0 forsome_2_A(0,1) = 0 forsome_2_A(1,0) = 0 forsome_2_A(1,1) = 0 or_0_A = 0 or_1_A(0) = 1 or_1_A(1) = 1 or_2_A(0,0) = 0 or_2_A(0,1) = 0 or_2_A(1,0) = 0 or_2_A(1,1) = 0 true_0_A = 0 false_0_A = 0 nil_0_A = 0 app#!_A(0,0) = 0 app#!_A(0,1) = 0 app#!_A(1,0) = 0 app#!_A(1,1) = 0 forall_1!_A(0) = 0 forall_1!_A(1) = 0 forall_2#!_A(0,0) = 0 forall_2#!_A(0,1) = 0 forall_2#!_A(1,0) = 0 forall_2#!_A(1,1) = 0 cons_2!_A(0,0) = 0 cons_2!_A(0,1) = 0 cons_2!_A(1,0) = 0 cons_2!_A(1,1) = 0 app!_A(0,0) = 0 app!_A(0,1) = 0 app!_A(1,0) = 1 app!_A(1,1) = 1 and_0!_A = 1 and_1!_A(0) = 0 and_1!_A(1) = 0 and_2!_A(0,0) = 1 and_2!_A(0,1) = 1 and_2!_A(1,0) = 1 and_2!_A(1,1) = 0 cons_0!_A = 0 cons_1!_A(0) = 0 cons_1!_A(1) = 0 forall_0!_A = 1 forall_2!_A(0,0) = 1 forall_2!_A(0,1) = 1 forall_2!_A(1,0) = 1 forall_2!_A(1,1) = 1 forsome_0!_A = 1 forsome_1!_A(0) = 1 forsome_1!_A(1) = 1 forsome_2!_A(0,0) = 1 forsome_2!_A(0,1) = 1 forsome_2!_A(1,0) = 1 forsome_2!_A(1,1) = 1 or_0!_A = 1 or_1!_A(0) = 0 or_1!_A(1) = 0 or_2!_A(0,0) = 1 or_2!_A(0,1) = 1 or_2!_A(1,0) = 1 or_2!_A(1,1) = 1 true_0!_A = 1 false_0!_A = 1 nil_0!_A = 1 partial status: pi(app#) = [2] pi(forall_1) = [] pi(forall_2#) = [2] pi(cons_2) = [1, 2] pi(app) = [1, 2] pi(and_0) = [] pi(and_1) = [] pi(and_2) = [] pi(cons_0) = [] pi(cons_1) = [1] pi(forall_0) = [] pi(forall_2) = [] pi(forsome_0) = [] pi(forsome_1) = [] pi(forsome_2) = [] pi(or_0) = [] pi(or_1) = [] pi(or_2) = [] pi(true_0) = [] pi(false_0) = [] pi(nil_0) = [] The following rules are removed: p2 -- SCC decomposition. 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) 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} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: forall_2#(p,cons_2(x,xs)) -> forall_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 r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22 Take the reduction pair: generalized weighted path order (>=MG, >G) base order 1: trivial reduction triple (T x T, T x T, Φ) base order 2: model-based reduction triple (=A, >=A!, >A!): finite algebra: carrier: {0,1} interpretations: forall_2#_A(0,0) = 0 forall_2#_A(0,1) = 0 forall_2#_A(1,0) = 0 forall_2#_A(1,1) = 0 cons_2_A(0,0) = 1 cons_2_A(0,1) = 1 cons_2_A(1,0) = 1 cons_2_A(1,1) = 1 app_A(0,0) = 1 app_A(0,1) = 1 app_A(1,0) = 0 app_A(1,1) = 0 and_0_A = 0 and_1_A(0) = 1 and_1_A(1) = 1 and_2_A(0,0) = 0 and_2_A(0,1) = 0 and_2_A(1,0) = 0 and_2_A(1,1) = 0 cons_0_A = 1 cons_1_A(0) = 0 cons_1_A(1) = 0 forall_0_A = 0 forall_1_A(0) = 1 forall_1_A(1) = 1 forall_2_A(0,0) = 0 forall_2_A(0,1) = 0 forall_2_A(1,0) = 0 forall_2_A(1,1) = 0 forsome_0_A = 0 forsome_1_A(0) = 1 forsome_1_A(1) = 1 forsome_2_A(0,0) = 0 forsome_2_A(0,1) = 0 forsome_2_A(1,0) = 0 forsome_2_A(1,1) = 0 or_0_A = 0 or_1_A(0) = 1 or_1_A(1) = 1 or_2_A(0,0) = 0 or_2_A(0,1) = 0 or_2_A(1,0) = 0 or_2_A(1,1) = 0 true_0_A = 0 false_0_A = 0 nil_0_A = 1 forall_2#!_A(0,0) = 0 forall_2#!_A(0,1) = 0 forall_2#!_A(1,0) = 0 forall_2#!_A(1,1) = 1 cons_2!_A(0,0) = 0 cons_2!_A(0,1) = 0 cons_2!_A(1,0) = 0 cons_2!_A(1,1) = 1 app!_A(0,0) = 0 app!_A(0,1) = 1 app!_A(1,0) = 1 app!_A(1,1) = 1 and_0!_A = 1 and_1!_A(0) = 0 and_1!_A(1) = 1 and_2!_A(0,0) = 1 and_2!_A(0,1) = 0 and_2!_A(1,0) = 0 and_2!_A(1,1) = 0 cons_0!_A = 0 cons_1!_A(0) = 0 cons_1!_A(1) = 0 forall_0!_A = 1 forall_1!_A(0) = 1 forall_1!_A(1) = 1 forall_2!_A(0,0) = 0 forall_2!_A(0,1) = 1 forall_2!_A(1,0) = 0 forall_2!_A(1,1) = 1 forsome_0!_A = 1 forsome_1!_A(0) = 1 forsome_1!_A(1) = 1 forsome_2!_A(0,0) = 0 forsome_2!_A(0,1) = 0 forsome_2!_A(1,0) = 1 forsome_2!_A(1,1) = 0 or_0!_A = 1 or_1!_A(0) = 1 or_1!_A(1) = 1 or_2!_A(0,0) = 0 or_2!_A(0,1) = 0 or_2!_A(1,0) = 0 or_2!_A(1,1) = 0 true_0!_A = 0 false_0!_A = 0 nil_0!_A = 1 partial status: pi(forall_2#) = [2] pi(cons_2) = [1, 2] pi(app) = [1, 2] pi(and_0) = [] pi(and_1) = [] pi(and_2) = [] pi(cons_0) = [] pi(cons_1) = [1] pi(forall_0) = [] pi(forall_1) = [] pi(forall_2) = [] pi(forsome_0) = [] pi(forsome_1) = [] pi(forsome_2) = [] pi(or_0) = [] pi(or_1) = [] pi(or_2) = [] pi(true_0) = [] pi(false_0) = [] pi(nil_0) = [] The following rules are removed: p1 Then no dependency pair remains.