YES We show the termination of the TRS R: app(app(ack(),|0|()),y) -> app(succ(),y) app(app(ack(),app(succ(),x)),y) -> app(app(ack(),x),app(succ(),|0|())) app(app(ack(),app(succ(),x)),app(succ(),y)) -> app(app(ack(),x),app(app(ack(),app(succ(),x)),y)) app(app(map(),f),nil()) -> nil() app(app(map(),f),app(app(cons(),x),xs)) -> app(app(cons(),app(f,x)),app(app(map(),f),xs)) app(app(filter(),f),nil()) -> nil() app(app(filter(),f),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(f,x)),f),x),xs) app(app(app(app(filter2(),true()),f),x),xs) -> app(app(cons(),x),app(app(filter(),f),xs)) app(app(app(app(filter2(),false()),f),x),xs) -> app(app(filter(),f),xs) Instead we prove termination of the uncurried system: app(ack_0(),y) -> ack_1(y) app(ack_1(x1),y) -> ack_2(x1,y) app(cons_0(),y) -> cons_1(y) app(cons_1(x1),y) -> cons_2(x1,y) app(filter_0(),y) -> filter_1(y) app(filter_1(x1),y) -> filter_2(x1,y) app(filter2_0(),y) -> filter2_1(y) app(filter2_1(x1),y) -> filter2_2(x1,y) app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) app(map_0(),y) -> map_1(y) app(map_1(x1),y) -> map_2(x1,y) app(succ_0(),y) -> succ_1(y) ack_2(|0|_0(),y) -> succ_1(y) ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0())) ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y)) map_2(f,nil_0()) -> nil_0() map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) filter_2(f,nil_0()) -> nil_0() filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: app#(ack_1(x1),y) -> ack_2#(x1,y) p2: app#(filter_1(x1),y) -> filter_2#(x1,y) p3: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y) p4: app#(map_1(x1),y) -> map_2#(x1,y) p5: ack_2#(succ_1(x),y) -> ack_2#(x,succ_1(|0|_0())) p6: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(x,ack_2(succ_1(x),y)) p7: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(succ_1(x),y) p8: map_2#(f,cons_2(x,xs)) -> app#(f,x) p9: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs) p10: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs) p11: filter_2#(f,cons_2(x,xs)) -> app#(f,x) p12: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs) p13: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs) and R consists of: r1: app(ack_0(),y) -> ack_1(y) r2: app(ack_1(x1),y) -> ack_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(filter_0(),y) -> filter_1(y) r6: app(filter_1(x1),y) -> filter_2(x1,y) r7: app(filter2_0(),y) -> filter2_1(y) r8: app(filter2_1(x1),y) -> filter2_2(x1,y) r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r11: app(map_0(),y) -> map_1(y) r12: app(map_1(x1),y) -> map_2(x1,y) r13: app(succ_0(),y) -> succ_1(y) r14: ack_2(|0|_0(),y) -> succ_1(y) r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0())) r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y)) r17: map_2(f,nil_0()) -> nil_0() r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r19: filter_2(f,nil_0()) -> nil_0() r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: {p2, p3, p4, p8, p9, p10, p11, p12, p13} {p5, p6, p7} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: map_2#(f,cons_2(x,xs)) -> app#(f,x) p2: app#(map_1(x1),y) -> map_2#(x1,y) p3: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs) p4: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y) p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs) p6: filter_2#(f,cons_2(x,xs)) -> app#(f,x) p7: app#(filter_1(x1),y) -> filter_2#(x1,y) p8: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs) p9: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs) and R consists of: r1: app(ack_0(),y) -> ack_1(y) r2: app(ack_1(x1),y) -> ack_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(filter_0(),y) -> filter_1(y) r6: app(filter_1(x1),y) -> filter_2(x1,y) r7: app(filter2_0(),y) -> filter2_1(y) r8: app(filter2_1(x1),y) -> filter2_2(x1,y) r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r11: app(map_0(),y) -> map_1(y) r12: app(map_1(x1),y) -> map_2(x1,y) r13: app(succ_0(),y) -> succ_1(y) r14: ack_2(|0|_0(),y) -> succ_1(y) r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0())) r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y)) r17: map_2(f,nil_0()) -> nil_0() r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r19: filter_2(f,nil_0()) -> nil_0() r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,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: map_2#_A(0,0) = 0 map_2#_A(0,1) = 0 map_2#_A(1,0) = 0 map_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) = 0 app#_A(0,1) = 0 app#_A(1,0) = 0 app#_A(1,1) = 0 map_1_A(0) = 0 map_1_A(1) = 0 filter2_3_A(0,0,0) = 0 filter2_3_A(0,0,1) = 0 filter2_3_A(0,1,0) = 0 filter2_3_A(0,1,1) = 0 filter2_3_A(1,0,0) = 0 filter2_3_A(1,0,1) = 0 filter2_3_A(1,1,0) = 0 filter2_3_A(1,1,1) = 0 filter2_4#_A(0,0,0,0) = 0 filter2_4#_A(0,0,0,1) = 0 filter2_4#_A(0,0,1,0) = 0 filter2_4#_A(0,0,1,1) = 0 filter2_4#_A(0,1,0,0) = 0 filter2_4#_A(0,1,0,1) = 0 filter2_4#_A(0,1,1,0) = 0 filter2_4#_A(0,1,1,1) = 0 filter2_4#_A(1,0,0,0) = 0 filter2_4#_A(1,0,0,1) = 0 filter2_4#_A(1,0,1,0) = 0 filter2_4#_A(1,0,1,1) = 0 filter2_4#_A(1,1,0,0) = 0 filter2_4#_A(1,1,0,1) = 0 filter2_4#_A(1,1,1,0) = 0 filter2_4#_A(1,1,1,1) = 0 false_0_A = 1 filter_2#_A(0,0) = 0 filter_2#_A(0,1) = 0 filter_2#_A(1,0) = 0 filter_2#_A(1,1) = 0 filter_1_A(0) = 0 filter_1_A(1) = 0 app_A(0,0) = 1 app_A(0,1) = 1 app_A(1,0) = 0 app_A(1,1) = 0 true_0_A = 1 ack_0_A = 0 ack_1_A(0) = 1 ack_1_A(1) = 1 ack_2_A(0,0) = 0 ack_2_A(0,1) = 0 ack_2_A(1,0) = 0 ack_2_A(1,1) = 0 cons_0_A = 1 cons_1_A(0) = 0 cons_1_A(1) = 0 filter_0_A = 1 filter_2_A(0,0) = 1 filter_2_A(0,1) = 1 filter_2_A(1,0) = 1 filter_2_A(1,1) = 1 filter2_0_A = 1 filter2_1_A(0) = 0 filter2_1_A(1) = 0 filter2_2_A(0,0) = 1 filter2_2_A(0,1) = 1 filter2_2_A(1,0) = 1 filter2_2_A(1,1) = 1 filter2_4_A(0,0,0,0) = 1 filter2_4_A(0,0,0,1) = 1 filter2_4_A(0,0,1,0) = 1 filter2_4_A(0,0,1,1) = 1 filter2_4_A(0,1,0,0) = 1 filter2_4_A(0,1,0,1) = 1 filter2_4_A(0,1,1,0) = 1 filter2_4_A(0,1,1,1) = 1 filter2_4_A(1,0,0,0) = 1 filter2_4_A(1,0,0,1) = 1 filter2_4_A(1,0,1,0) = 1 filter2_4_A(1,0,1,1) = 1 filter2_4_A(1,1,0,0) = 1 filter2_4_A(1,1,0,1) = 1 filter2_4_A(1,1,1,0) = 1 filter2_4_A(1,1,1,1) = 1 map_0_A = 1 map_2_A(0,0) = 1 map_2_A(0,1) = 1 map_2_A(1,0) = 1 map_2_A(1,1) = 1 succ_0_A = 1 succ_1_A(0) = 0 succ_1_A(1) = 0 |0|_0_A = 1 nil_0_A = 1 map_2#!_A(0,0) = 0 map_2#!_A(0,1) = 0 map_2#!_A(1,0) = 0 map_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) = 0 app#!_A(1,1) = 0 map_1!_A(0) = 0 map_1!_A(1) = 0 filter2_3!_A(0,0,0) = 1 filter2_3!_A(0,0,1) = 1 filter2_3!_A(0,1,0) = 1 filter2_3!_A(0,1,1) = 1 filter2_3!_A(1,0,0) = 1 filter2_3!_A(1,0,1) = 1 filter2_3!_A(1,1,0) = 1 filter2_3!_A(1,1,1) = 1 filter2_4#!_A(0,0,0,0) = 0 filter2_4#!_A(0,0,0,1) = 0 filter2_4#!_A(0,0,1,0) = 0 filter2_4#!_A(0,0,1,1) = 0 filter2_4#!_A(0,1,0,0) = 0 filter2_4#!_A(0,1,0,1) = 0 filter2_4#!_A(0,1,1,0) = 0 filter2_4#!_A(0,1,1,1) = 0 filter2_4#!_A(1,0,0,0) = 0 filter2_4#!_A(1,0,0,1) = 0 filter2_4#!_A(1,0,1,0) = 0 filter2_4#!_A(1,0,1,1) = 0 filter2_4#!_A(1,1,0,0) = 0 filter2_4#!_A(1,1,0,1) = 0 filter2_4#!_A(1,1,1,0) = 0 filter2_4#!_A(1,1,1,1) = 0 false_0!_A = 0 filter_2#!_A(0,0) = 0 filter_2#!_A(0,1) = 0 filter_2#!_A(1,0) = 0 filter_2#!_A(1,1) = 0 filter_1!_A(0) = 0 filter_1!_A(1) = 0 app!_A(0,0) = 0 app!_A(0,1) = 1 app!_A(1,0) = 1 app!_A(1,1) = 1 true_0!_A = 0 ack_0!_A = 0 ack_1!_A(0) = 0 ack_1!_A(1) = 0 ack_2!_A(0,0) = 1 ack_2!_A(0,1) = 1 ack_2!_A(1,0) = 0 ack_2!_A(1,1) = 1 cons_0!_A = 0 cons_1!_A(0) = 0 cons_1!_A(1) = 0 filter_0!_A = 1 filter_2!_A(0,0) = 0 filter_2!_A(0,1) = 0 filter_2!_A(1,0) = 0 filter_2!_A(1,1) = 0 filter2_0!_A = 0 filter2_1!_A(0) = 0 filter2_1!_A(1) = 0 filter2_2!_A(0,0) = 0 filter2_2!_A(0,1) = 1 filter2_2!_A(1,0) = 0 filter2_2!_A(1,1) = 0 filter2_4!_A(0,0,0,0) = 0 filter2_4!_A(0,0,0,1) = 0 filter2_4!_A(0,0,1,0) = 0 filter2_4!_A(0,0,1,1) = 0 filter2_4!_A(0,1,0,0) = 0 filter2_4!_A(0,1,0,1) = 0 filter2_4!_A(0,1,1,0) = 0 filter2_4!_A(0,1,1,1) = 0 filter2_4!_A(1,0,0,0) = 0 filter2_4!_A(1,0,0,1) = 0 filter2_4!_A(1,0,1,0) = 0 filter2_4!_A(1,0,1,1) = 0 filter2_4!_A(1,1,0,0) = 0 filter2_4!_A(1,1,0,1) = 0 filter2_4!_A(1,1,1,0) = 0 filter2_4!_A(1,1,1,1) = 0 map_0!_A = 1 map_2!_A(0,0) = 0 map_2!_A(0,1) = 1 map_2!_A(1,0) = 0 map_2!_A(1,1) = 0 succ_0!_A = 0 succ_1!_A(0) = 0 succ_1!_A(1) = 1 |0|_0!_A = 0 nil_0!_A = 0 partial status: pi(map_2#) = [1] pi(cons_2) = [] pi(app#) = [1] pi(map_1) = [1] pi(filter2_3) = [1, 2] pi(filter2_4#) = [2] pi(false_0) = [] pi(filter_2#) = [1] pi(filter_1) = [1] pi(app) = [1, 2] pi(true_0) = [] pi(ack_0) = [] pi(ack_1) = [] pi(ack_2) = [] pi(cons_0) = [] pi(cons_1) = [1] pi(filter_0) = [] pi(filter_2) = [1] pi(filter2_0) = [] pi(filter2_1) = [1] pi(filter2_2) = [1, 2] pi(filter2_4) = [2] pi(map_0) = [] pi(map_2) = [1, 2] pi(succ_0) = [] pi(succ_1) = [] pi(|0|_0) = [] pi(nil_0) = [] The following rules are removed: p2, p4 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: map_2#(f,cons_2(x,xs)) -> app#(f,x) p2: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs) p3: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs) p4: filter_2#(f,cons_2(x,xs)) -> app#(f,x) p5: app#(filter_1(x1),y) -> filter_2#(x1,y) p6: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs) p7: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs) and R consists of: r1: app(ack_0(),y) -> ack_1(y) r2: app(ack_1(x1),y) -> ack_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(filter_0(),y) -> filter_1(y) r6: app(filter_1(x1),y) -> filter_2(x1,y) r7: app(filter2_0(),y) -> filter2_1(y) r8: app(filter2_1(x1),y) -> filter2_2(x1,y) r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r11: app(map_0(),y) -> map_1(y) r12: app(map_1(x1),y) -> map_2(x1,y) r13: app(succ_0(),y) -> succ_1(y) r14: ack_2(|0|_0(),y) -> succ_1(y) r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0())) r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y)) r17: map_2(f,nil_0()) -> nil_0() r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r19: filter_2(f,nil_0()) -> nil_0() r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: {p2} {p3, p4, p5, p6, p7} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs) and R consists of: r1: app(ack_0(),y) -> ack_1(y) r2: app(ack_1(x1),y) -> ack_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(filter_0(),y) -> filter_1(y) r6: app(filter_1(x1),y) -> filter_2(x1,y) r7: app(filter2_0(),y) -> filter2_1(y) r8: app(filter2_1(x1),y) -> filter2_2(x1,y) r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r11: app(map_0(),y) -> map_1(y) r12: app(map_1(x1),y) -> map_2(x1,y) r13: app(succ_0(),y) -> succ_1(y) r14: ack_2(|0|_0(),y) -> succ_1(y) r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0())) r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y)) r17: map_2(f,nil_0()) -> nil_0() r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r19: filter_2(f,nil_0()) -> nil_0() r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,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: map_2#_A(0,0) = 0 map_2#_A(0,1) = 0 map_2#_A(1,0) = 0 map_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) = 0 app_A(1,1) = 0 ack_0_A = 1 ack_1_A(0) = 0 ack_1_A(1) = 0 ack_2_A(0,0) = 0 ack_2_A(0,1) = 0 ack_2_A(1,0) = 0 ack_2_A(1,1) = 0 cons_0_A = 1 cons_1_A(0) = 0 cons_1_A(1) = 0 filter_0_A = 1 filter_1_A(0) = 0 filter_1_A(1) = 0 filter_2_A(0,0) = 0 filter_2_A(0,1) = 0 filter_2_A(1,0) = 0 filter_2_A(1,1) = 0 filter2_0_A = 1 filter2_1_A(0) = 0 filter2_1_A(1) = 0 filter2_2_A(0,0) = 0 filter2_2_A(0,1) = 0 filter2_2_A(1,0) = 0 filter2_2_A(1,1) = 0 filter2_3_A(0,0,0) = 0 filter2_3_A(0,0,1) = 0 filter2_3_A(0,1,0) = 0 filter2_3_A(0,1,1) = 0 filter2_3_A(1,0,0) = 0 filter2_3_A(1,0,1) = 0 filter2_3_A(1,1,0) = 0 filter2_3_A(1,1,1) = 0 filter2_4_A(0,0,0,0) = 0 filter2_4_A(0,0,0,1) = 0 filter2_4_A(0,0,1,0) = 0 filter2_4_A(0,0,1,1) = 0 filter2_4_A(0,1,0,0) = 0 filter2_4_A(0,1,0,1) = 0 filter2_4_A(0,1,1,0) = 0 filter2_4_A(0,1,1,1) = 0 filter2_4_A(1,0,0,0) = 0 filter2_4_A(1,0,0,1) = 0 filter2_4_A(1,0,1,0) = 0 filter2_4_A(1,0,1,1) = 0 filter2_4_A(1,1,0,0) = 0 filter2_4_A(1,1,0,1) = 0 filter2_4_A(1,1,1,0) = 0 filter2_4_A(1,1,1,1) = 0 map_0_A = 0 map_1_A(0) = 0 map_1_A(1) = 0 map_2_A(0,0) = 0 map_2_A(0,1) = 0 map_2_A(1,0) = 0 map_2_A(1,1) = 0 succ_0_A = 1 succ_1_A(0) = 0 succ_1_A(1) = 0 |0|_0_A = 1 nil_0_A = 0 true_0_A = 1 false_0_A = 0 map_2#!_A(0,0) = 0 map_2#!_A(0,1) = 0 map_2#!_A(1,0) = 0 map_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) = 1 app!_A(0,1) = 1 app!_A(1,0) = 1 app!_A(1,1) = 1 ack_0!_A = 1 ack_1!_A(0) = 0 ack_1!_A(1) = 0 ack_2!_A(0,0) = 0 ack_2!_A(0,1) = 0 ack_2!_A(1,0) = 0 ack_2!_A(1,1) = 0 cons_0!_A = 0 cons_1!_A(0) = 0 cons_1!_A(1) = 0 filter_0!_A = 1 filter_1!_A(0) = 0 filter_1!_A(1) = 0 filter_2!_A(0,0) = 0 filter_2!_A(0,1) = 0 filter_2!_A(1,0) = 0 filter_2!_A(1,1) = 0 filter2_0!_A = 1 filter2_1!_A(0) = 1 filter2_1!_A(1) = 1 filter2_2!_A(0,0) = 0 filter2_2!_A(0,1) = 0 filter2_2!_A(1,0) = 0 filter2_2!_A(1,1) = 0 filter2_3!_A(0,0,0) = 0 filter2_3!_A(0,0,1) = 0 filter2_3!_A(0,1,0) = 0 filter2_3!_A(0,1,1) = 1 filter2_3!_A(1,0,0) = 1 filter2_3!_A(1,0,1) = 1 filter2_3!_A(1,1,0) = 1 filter2_3!_A(1,1,1) = 1 filter2_4!_A(0,0,0,0) = 0 filter2_4!_A(0,0,0,1) = 0 filter2_4!_A(0,0,1,0) = 0 filter2_4!_A(0,0,1,1) = 0 filter2_4!_A(0,1,0,0) = 0 filter2_4!_A(0,1,0,1) = 0 filter2_4!_A(0,1,1,0) = 0 filter2_4!_A(0,1,1,1) = 0 filter2_4!_A(1,0,0,0) = 1 filter2_4!_A(1,0,0,1) = 1 filter2_4!_A(1,0,1,0) = 1 filter2_4!_A(1,0,1,1) = 1 filter2_4!_A(1,1,0,0) = 1 filter2_4!_A(1,1,0,1) = 1 filter2_4!_A(1,1,1,0) = 1 filter2_4!_A(1,1,1,1) = 1 map_0!_A = 1 map_1!_A(0) = 0 map_1!_A(1) = 0 map_2!_A(0,0) = 0 map_2!_A(0,1) = 0 map_2!_A(1,0) = 0 map_2!_A(1,1) = 0 succ_0!_A = 1 succ_1!_A(0) = 0 succ_1!_A(1) = 0 |0|_0!_A = 0 nil_0!_A = 0 true_0!_A = 1 false_0!_A = 0 partial status: pi(map_2#) = [2] pi(cons_2) = [2] pi(app) = [2] pi(ack_0) = [] pi(ack_1) = [1] pi(ack_2) = [] pi(cons_0) = [] pi(cons_1) = [1] pi(filter_0) = [] pi(filter_1) = [1] pi(filter_2) = [] pi(filter2_0) = [] pi(filter2_1) = [] pi(filter2_2) = [] pi(filter2_3) = [3] pi(filter2_4) = [] pi(map_0) = [] pi(map_1) = [1] pi(map_2) = [2] pi(succ_0) = [] pi(succ_1) = [] pi(|0|_0) = [] pi(nil_0) = [] pi(true_0) = [] pi(false_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#(filter_1(x1),y) -> filter_2#(x1,y) p2: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs) p3: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs) p4: filter_2#(f,cons_2(x,xs)) -> app#(f,x) p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs) and R consists of: r1: app(ack_0(),y) -> ack_1(y) r2: app(ack_1(x1),y) -> ack_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(filter_0(),y) -> filter_1(y) r6: app(filter_1(x1),y) -> filter_2(x1,y) r7: app(filter2_0(),y) -> filter2_1(y) r8: app(filter2_1(x1),y) -> filter2_2(x1,y) r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r11: app(map_0(),y) -> map_1(y) r12: app(map_1(x1),y) -> map_2(x1,y) r13: app(succ_0(),y) -> succ_1(y) r14: ack_2(|0|_0(),y) -> succ_1(y) r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0())) r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y)) r17: map_2(f,nil_0()) -> nil_0() r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r19: filter_2(f,nil_0()) -> nil_0() r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,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 filter_1_A(0) = 0 filter_1_A(1) = 0 filter_2#_A(0,0) = 0 filter_2#_A(0,1) = 0 filter_2#_A(1,0) = 0 filter_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 filter2_4#_A(0,0,0,0) = 0 filter2_4#_A(0,0,0,1) = 0 filter2_4#_A(0,0,1,0) = 0 filter2_4#_A(0,0,1,1) = 0 filter2_4#_A(0,1,0,0) = 0 filter2_4#_A(0,1,0,1) = 0 filter2_4#_A(0,1,1,0) = 0 filter2_4#_A(0,1,1,1) = 0 filter2_4#_A(1,0,0,0) = 0 filter2_4#_A(1,0,0,1) = 0 filter2_4#_A(1,0,1,0) = 0 filter2_4#_A(1,0,1,1) = 0 filter2_4#_A(1,1,0,0) = 0 filter2_4#_A(1,1,0,1) = 0 filter2_4#_A(1,1,1,0) = 0 filter2_4#_A(1,1,1,1) = 0 app_A(0,0) = 1 app_A(0,1) = 1 app_A(1,0) = 0 app_A(1,1) = 0 true_0_A = 1 false_0_A = 1 ack_0_A = 0 ack_1_A(0) = 1 ack_1_A(1) = 1 ack_2_A(0,0) = 0 ack_2_A(0,1) = 0 ack_2_A(1,0) = 0 ack_2_A(1,1) = 0 cons_0_A = 1 cons_1_A(0) = 0 cons_1_A(1) = 0 filter_0_A = 1 filter_2_A(0,0) = 1 filter_2_A(0,1) = 1 filter_2_A(1,0) = 1 filter_2_A(1,1) = 1 filter2_0_A = 1 filter2_1_A(0) = 0 filter2_1_A(1) = 0 filter2_2_A(0,0) = 1 filter2_2_A(0,1) = 1 filter2_2_A(1,0) = 1 filter2_2_A(1,1) = 1 filter2_3_A(0,0,0) = 0 filter2_3_A(0,0,1) = 0 filter2_3_A(0,1,0) = 0 filter2_3_A(0,1,1) = 0 filter2_3_A(1,0,0) = 0 filter2_3_A(1,0,1) = 0 filter2_3_A(1,1,0) = 0 filter2_3_A(1,1,1) = 0 filter2_4_A(0,0,0,0) = 1 filter2_4_A(0,0,0,1) = 1 filter2_4_A(0,0,1,0) = 1 filter2_4_A(0,0,1,1) = 1 filter2_4_A(0,1,0,0) = 1 filter2_4_A(0,1,0,1) = 1 filter2_4_A(0,1,1,0) = 1 filter2_4_A(0,1,1,1) = 1 filter2_4_A(1,0,0,0) = 1 filter2_4_A(1,0,0,1) = 1 filter2_4_A(1,0,1,0) = 1 filter2_4_A(1,0,1,1) = 1 filter2_4_A(1,1,0,0) = 1 filter2_4_A(1,1,0,1) = 1 filter2_4_A(1,1,1,0) = 1 filter2_4_A(1,1,1,1) = 1 map_0_A = 1 map_1_A(0) = 0 map_1_A(1) = 0 map_2_A(0,0) = 1 map_2_A(0,1) = 1 map_2_A(1,0) = 1 map_2_A(1,1) = 1 succ_0_A = 1 succ_1_A(0) = 0 succ_1_A(1) = 0 |0|_0_A = 1 nil_0_A = 1 app#!_A(0,0) = 0 app#!_A(0,1) = 0 app#!_A(1,0) = 0 app#!_A(1,1) = 0 filter_1!_A(0) = 0 filter_1!_A(1) = 0 filter_2#!_A(0,0) = 0 filter_2#!_A(0,1) = 0 filter_2#!_A(1,0) = 0 filter_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 filter2_4#!_A(0,0,0,0) = 0 filter2_4#!_A(0,0,0,1) = 0 filter2_4#!_A(0,0,1,0) = 0 filter2_4#!_A(0,0,1,1) = 0 filter2_4#!_A(0,1,0,0) = 0 filter2_4#!_A(0,1,0,1) = 0 filter2_4#!_A(0,1,1,0) = 0 filter2_4#!_A(0,1,1,1) = 0 filter2_4#!_A(1,0,0,0) = 0 filter2_4#!_A(1,0,0,1) = 0 filter2_4#!_A(1,0,1,0) = 0 filter2_4#!_A(1,0,1,1) = 0 filter2_4#!_A(1,1,0,0) = 1 filter2_4#!_A(1,1,0,1) = 0 filter2_4#!_A(1,1,1,0) = 1 filter2_4#!_A(1,1,1,1) = 1 app!_A(0,0) = 1 app!_A(0,1) = 1 app!_A(1,0) = 1 app!_A(1,1) = 1 true_0!_A = 1 false_0!_A = 0 ack_0!_A = 0 ack_1!_A(0) = 0 ack_1!_A(1) = 0 ack_2!_A(0,0) = 1 ack_2!_A(0,1) = 1 ack_2!_A(1,0) = 1 ack_2!_A(1,1) = 1 cons_0!_A = 1 cons_1!_A(0) = 0 cons_1!_A(1) = 0 filter_0!_A = 1 filter_2!_A(0,0) = 0 filter_2!_A(0,1) = 0 filter_2!_A(1,0) = 0 filter_2!_A(1,1) = 0 filter2_0!_A = 0 filter2_1!_A(0) = 0 filter2_1!_A(1) = 0 filter2_2!_A(0,0) = 1 filter2_2!_A(0,1) = 1 filter2_2!_A(1,0) = 1 filter2_2!_A(1,1) = 1 filter2_3!_A(0,0,0) = 0 filter2_3!_A(0,0,1) = 0 filter2_3!_A(0,1,0) = 0 filter2_3!_A(0,1,1) = 0 filter2_3!_A(1,0,0) = 0 filter2_3!_A(1,0,1) = 0 filter2_3!_A(1,1,0) = 0 filter2_3!_A(1,1,1) = 0 filter2_4!_A(0,0,0,0) = 0 filter2_4!_A(0,0,0,1) = 0 filter2_4!_A(0,0,1,0) = 0 filter2_4!_A(0,0,1,1) = 0 filter2_4!_A(0,1,0,0) = 0 filter2_4!_A(0,1,0,1) = 0 filter2_4!_A(0,1,1,0) = 0 filter2_4!_A(0,1,1,1) = 0 filter2_4!_A(1,0,0,0) = 0 filter2_4!_A(1,0,0,1) = 0 filter2_4!_A(1,0,1,0) = 0 filter2_4!_A(1,0,1,1) = 0 filter2_4!_A(1,1,0,0) = 0 filter2_4!_A(1,1,0,1) = 0 filter2_4!_A(1,1,1,0) = 0 filter2_4!_A(1,1,1,1) = 0 map_0!_A = 1 map_1!_A(0) = 1 map_1!_A(1) = 1 map_2!_A(0,0) = 1 map_2!_A(0,1) = 0 map_2!_A(1,0) = 1 map_2!_A(1,1) = 1 succ_0!_A = 1 succ_1!_A(0) = 1 succ_1!_A(1) = 1 |0|_0!_A = 1 nil_0!_A = 0 partial status: pi(app#) = [1] pi(filter_1) = [1] pi(filter_2#) = [1] pi(cons_2) = [] pi(filter2_4#) = [2] pi(app) = [1, 2] pi(true_0) = [] pi(false_0) = [] pi(ack_0) = [] pi(ack_1) = [1] pi(ack_2) = [] pi(cons_0) = [] pi(cons_1) = [1] pi(filter_0) = [] pi(filter_2) = [1] pi(filter2_0) = [] pi(filter2_1) = [1] pi(filter2_2) = [1, 2] pi(filter2_3) = [2] pi(filter2_4) = [2] pi(map_0) = [] pi(map_1) = [] pi(map_2) = [] pi(succ_0) = [] pi(succ_1) = [] pi(|0|_0) = [] pi(nil_0) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs) p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs) p3: filter_2#(f,cons_2(x,xs)) -> app#(f,x) p4: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs) and R consists of: r1: app(ack_0(),y) -> ack_1(y) r2: app(ack_1(x1),y) -> ack_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(filter_0(),y) -> filter_1(y) r6: app(filter_1(x1),y) -> filter_2(x1,y) r7: app(filter2_0(),y) -> filter2_1(y) r8: app(filter2_1(x1),y) -> filter2_2(x1,y) r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r11: app(map_0(),y) -> map_1(y) r12: app(map_1(x1),y) -> map_2(x1,y) r13: app(succ_0(),y) -> succ_1(y) r14: ack_2(|0|_0(),y) -> succ_1(y) r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0())) r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y)) r17: map_2(f,nil_0()) -> nil_0() r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r19: filter_2(f,nil_0()) -> nil_0() r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: {p1, p2, p4} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs) p2: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs) p3: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs) and R consists of: r1: app(ack_0(),y) -> ack_1(y) r2: app(ack_1(x1),y) -> ack_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(filter_0(),y) -> filter_1(y) r6: app(filter_1(x1),y) -> filter_2(x1,y) r7: app(filter2_0(),y) -> filter2_1(y) r8: app(filter2_1(x1),y) -> filter2_2(x1,y) r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r11: app(map_0(),y) -> map_1(y) r12: app(map_1(x1),y) -> map_2(x1,y) r13: app(succ_0(),y) -> succ_1(y) r14: ack_2(|0|_0(),y) -> succ_1(y) r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0())) r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y)) r17: map_2(f,nil_0()) -> nil_0() r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r19: filter_2(f,nil_0()) -> nil_0() r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,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: filter_2#_A(0,0) = 0 filter_2#_A(0,1) = 0 filter_2#_A(1,0) = 0 filter_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 filter2_4#_A(0,0,0,0) = 0 filter2_4#_A(0,0,0,1) = 0 filter2_4#_A(0,0,1,0) = 0 filter2_4#_A(0,0,1,1) = 0 filter2_4#_A(0,1,0,0) = 0 filter2_4#_A(0,1,0,1) = 0 filter2_4#_A(0,1,1,0) = 0 filter2_4#_A(0,1,1,1) = 0 filter2_4#_A(1,0,0,0) = 0 filter2_4#_A(1,0,0,1) = 0 filter2_4#_A(1,0,1,0) = 0 filter2_4#_A(1,0,1,1) = 0 filter2_4#_A(1,1,0,0) = 0 filter2_4#_A(1,1,0,1) = 0 filter2_4#_A(1,1,1,0) = 0 filter2_4#_A(1,1,1,1) = 0 app_A(0,0) = 1 app_A(0,1) = 1 app_A(1,0) = 0 app_A(1,1) = 0 false_0_A = 1 true_0_A = 1 ack_0_A = 0 ack_1_A(0) = 1 ack_1_A(1) = 1 ack_2_A(0,0) = 0 ack_2_A(0,1) = 0 ack_2_A(1,0) = 0 ack_2_A(1,1) = 0 cons_0_A = 1 cons_1_A(0) = 0 cons_1_A(1) = 0 filter_0_A = 1 filter_1_A(0) = 0 filter_1_A(1) = 0 filter_2_A(0,0) = 1 filter_2_A(0,1) = 1 filter_2_A(1,0) = 1 filter_2_A(1,1) = 1 filter2_0_A = 1 filter2_1_A(0) = 0 filter2_1_A(1) = 0 filter2_2_A(0,0) = 1 filter2_2_A(0,1) = 1 filter2_2_A(1,0) = 1 filter2_2_A(1,1) = 1 filter2_3_A(0,0,0) = 0 filter2_3_A(0,0,1) = 0 filter2_3_A(0,1,0) = 0 filter2_3_A(0,1,1) = 0 filter2_3_A(1,0,0) = 0 filter2_3_A(1,0,1) = 0 filter2_3_A(1,1,0) = 0 filter2_3_A(1,1,1) = 0 filter2_4_A(0,0,0,0) = 1 filter2_4_A(0,0,0,1) = 1 filter2_4_A(0,0,1,0) = 1 filter2_4_A(0,0,1,1) = 1 filter2_4_A(0,1,0,0) = 1 filter2_4_A(0,1,0,1) = 1 filter2_4_A(0,1,1,0) = 1 filter2_4_A(0,1,1,1) = 1 filter2_4_A(1,0,0,0) = 1 filter2_4_A(1,0,0,1) = 1 filter2_4_A(1,0,1,0) = 1 filter2_4_A(1,0,1,1) = 1 filter2_4_A(1,1,0,0) = 1 filter2_4_A(1,1,0,1) = 1 filter2_4_A(1,1,1,0) = 1 filter2_4_A(1,1,1,1) = 1 map_0_A = 1 map_1_A(0) = 0 map_1_A(1) = 0 map_2_A(0,0) = 1 map_2_A(0,1) = 1 map_2_A(1,0) = 1 map_2_A(1,1) = 1 succ_0_A = 1 succ_1_A(0) = 0 succ_1_A(1) = 0 |0|_0_A = 1 nil_0_A = 1 filter_2#!_A(0,0) = 1 filter_2#!_A(0,1) = 1 filter_2#!_A(1,0) = 0 filter_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) = 0 filter2_4#!_A(0,0,0,0) = 0 filter2_4#!_A(0,0,0,1) = 0 filter2_4#!_A(0,0,1,0) = 0 filter2_4#!_A(0,0,1,1) = 0 filter2_4#!_A(0,1,0,0) = 0 filter2_4#!_A(0,1,0,1) = 0 filter2_4#!_A(0,1,1,0) = 0 filter2_4#!_A(0,1,1,1) = 0 filter2_4#!_A(1,0,0,0) = 0 filter2_4#!_A(1,0,0,1) = 0 filter2_4#!_A(1,0,1,0) = 0 filter2_4#!_A(1,0,1,1) = 0 filter2_4#!_A(1,1,0,0) = 0 filter2_4#!_A(1,1,0,1) = 1 filter2_4#!_A(1,1,1,0) = 0 filter2_4#!_A(1,1,1,1) = 1 app!_A(0,0) = 0 app!_A(0,1) = 0 app!_A(1,0) = 0 app!_A(1,1) = 0 false_0!_A = 1 true_0!_A = 1 ack_0!_A = 1 ack_1!_A(0) = 0 ack_1!_A(1) = 0 ack_2!_A(0,0) = 0 ack_2!_A(0,1) = 0 ack_2!_A(1,0) = 0 ack_2!_A(1,1) = 0 cons_0!_A = 0 cons_1!_A(0) = 0 cons_1!_A(1) = 0 filter_0!_A = 0 filter_1!_A(0) = 0 filter_1!_A(1) = 0 filter_2!_A(0,0) = 0 filter_2!_A(0,1) = 0 filter_2!_A(1,0) = 0 filter_2!_A(1,1) = 0 filter2_0!_A = 0 filter2_1!_A(0) = 0 filter2_1!_A(1) = 0 filter2_2!_A(0,0) = 0 filter2_2!_A(0,1) = 0 filter2_2!_A(1,0) = 0 filter2_2!_A(1,1) = 0 filter2_3!_A(0,0,0) = 0 filter2_3!_A(0,0,1) = 0 filter2_3!_A(0,1,0) = 0 filter2_3!_A(0,1,1) = 0 filter2_3!_A(1,0,0) = 0 filter2_3!_A(1,0,1) = 0 filter2_3!_A(1,1,0) = 0 filter2_3!_A(1,1,1) = 0 filter2_4!_A(0,0,0,0) = 0 filter2_4!_A(0,0,0,1) = 0 filter2_4!_A(0,0,1,0) = 0 filter2_4!_A(0,0,1,1) = 0 filter2_4!_A(0,1,0,0) = 0 filter2_4!_A(0,1,0,1) = 0 filter2_4!_A(0,1,1,0) = 0 filter2_4!_A(0,1,1,1) = 0 filter2_4!_A(1,0,0,0) = 0 filter2_4!_A(1,0,0,1) = 0 filter2_4!_A(1,0,1,0) = 0 filter2_4!_A(1,0,1,1) = 0 filter2_4!_A(1,1,0,0) = 0 filter2_4!_A(1,1,0,1) = 0 filter2_4!_A(1,1,1,0) = 0 filter2_4!_A(1,1,1,1) = 0 map_0!_A = 0 map_1!_A(0) = 0 map_1!_A(1) = 0 map_2!_A(0,0) = 0 map_2!_A(0,1) = 0 map_2!_A(1,0) = 0 map_2!_A(1,1) = 0 succ_0!_A = 1 succ_1!_A(0) = 0 succ_1!_A(1) = 0 |0|_0!_A = 0 nil_0!_A = 0 partial status: pi(filter_2#) = [] pi(cons_2) = [] pi(filter2_4#) = [1] pi(app) = [] pi(false_0) = [] pi(true_0) = [] pi(ack_0) = [] pi(ack_1) = [] pi(ack_2) = [] pi(cons_0) = [] pi(cons_1) = [] pi(filter_0) = [] pi(filter_1) = [] pi(filter_2) = [] pi(filter2_0) = [] pi(filter2_1) = [] pi(filter2_2) = [] pi(filter2_3) = [] pi(filter2_4) = [] pi(map_0) = [] pi(map_1) = [] pi(map_2) = [] pi(succ_0) = [] pi(succ_1) = [] pi(|0|_0) = [] pi(nil_0) = [] The following rules are removed: p1, p2, p3 Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: ack_2#(succ_1(x),y) -> ack_2#(x,succ_1(|0|_0())) p2: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(succ_1(x),y) p3: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(x,ack_2(succ_1(x),y)) and R consists of: r1: app(ack_0(),y) -> ack_1(y) r2: app(ack_1(x1),y) -> ack_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(filter_0(),y) -> filter_1(y) r6: app(filter_1(x1),y) -> filter_2(x1,y) r7: app(filter2_0(),y) -> filter2_1(y) r8: app(filter2_1(x1),y) -> filter2_2(x1,y) r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r11: app(map_0(),y) -> map_1(y) r12: app(map_1(x1),y) -> map_2(x1,y) r13: app(succ_0(),y) -> succ_1(y) r14: ack_2(|0|_0(),y) -> succ_1(y) r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0())) r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y)) r17: map_2(f,nil_0()) -> nil_0() r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r19: filter_2(f,nil_0()) -> nil_0() r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,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: ack_2#_A(0,0) = 0 ack_2#_A(0,1) = 0 ack_2#_A(1,0) = 0 ack_2#_A(1,1) = 0 succ_1_A(0) = 0 succ_1_A(1) = 0 |0|_0_A = 0 ack_2_A(0,0) = 0 ack_2_A(0,1) = 0 ack_2_A(1,0) = 0 ack_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 ack_0_A = 0 ack_1_A(0) = 1 ack_1_A(1) = 1 cons_0_A = 0 cons_1_A(0) = 1 cons_1_A(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) = 0 filter_0_A = 0 filter_1_A(0) = 1 filter_1_A(1) = 1 filter_2_A(0,0) = 0 filter_2_A(0,1) = 0 filter_2_A(1,0) = 0 filter_2_A(1,1) = 0 filter2_0_A = 0 filter2_1_A(0) = 1 filter2_1_A(1) = 1 filter2_2_A(0,0) = 0 filter2_2_A(0,1) = 0 filter2_2_A(1,0) = 0 filter2_2_A(1,1) = 0 filter2_3_A(0,0,0) = 1 filter2_3_A(0,0,1) = 1 filter2_3_A(0,1,0) = 1 filter2_3_A(0,1,1) = 1 filter2_3_A(1,0,0) = 1 filter2_3_A(1,0,1) = 1 filter2_3_A(1,1,0) = 1 filter2_3_A(1,1,1) = 1 filter2_4_A(0,0,0,0) = 0 filter2_4_A(0,0,0,1) = 0 filter2_4_A(0,0,1,0) = 0 filter2_4_A(0,0,1,1) = 0 filter2_4_A(0,1,0,0) = 0 filter2_4_A(0,1,0,1) = 0 filter2_4_A(0,1,1,0) = 0 filter2_4_A(0,1,1,1) = 0 filter2_4_A(1,0,0,0) = 0 filter2_4_A(1,0,0,1) = 0 filter2_4_A(1,0,1,0) = 0 filter2_4_A(1,0,1,1) = 0 filter2_4_A(1,1,0,0) = 0 filter2_4_A(1,1,0,1) = 0 filter2_4_A(1,1,1,0) = 0 filter2_4_A(1,1,1,1) = 0 map_0_A = 0 map_1_A(0) = 1 map_1_A(1) = 1 map_2_A(0,0) = 0 map_2_A(0,1) = 0 map_2_A(1,0) = 0 map_2_A(1,1) = 0 succ_0_A = 1 nil_0_A = 0 true_0_A = 0 false_0_A = 1 ack_2#!_A(0,0) = 0 ack_2#!_A(0,1) = 0 ack_2#!_A(1,0) = 0 ack_2#!_A(1,1) = 1 succ_1!_A(0) = 0 succ_1!_A(1) = 0 |0|_0!_A = 0 ack_2!_A(0,0) = 1 ack_2!_A(0,1) = 1 ack_2!_A(1,0) = 0 ack_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 ack_0!_A = 0 ack_1!_A(0) = 0 ack_1!_A(1) = 0 cons_0!_A = 1 cons_1!_A(0) = 1 cons_1!_A(1) = 1 cons_2!_A(0,0) = 1 cons_2!_A(0,1) = 1 cons_2!_A(1,0) = 1 cons_2!_A(1,1) = 1 filter_0!_A = 1 filter_1!_A(0) = 0 filter_1!_A(1) = 1 filter_2!_A(0,0) = 0 filter_2!_A(0,1) = 0 filter_2!_A(1,0) = 1 filter_2!_A(1,1) = 1 filter2_0!_A = 1 filter2_1!_A(0) = 1 filter2_1!_A(1) = 1 filter2_2!_A(0,0) = 0 filter2_2!_A(0,1) = 0 filter2_2!_A(1,0) = 0 filter2_2!_A(1,1) = 0 filter2_3!_A(0,0,0) = 0 filter2_3!_A(0,0,1) = 0 filter2_3!_A(0,1,0) = 0 filter2_3!_A(0,1,1) = 0 filter2_3!_A(1,0,0) = 1 filter2_3!_A(1,0,1) = 1 filter2_3!_A(1,1,0) = 1 filter2_3!_A(1,1,1) = 0 filter2_4!_A(0,0,0,0) = 1 filter2_4!_A(0,0,0,1) = 1 filter2_4!_A(0,0,1,0) = 1 filter2_4!_A(0,0,1,1) = 1 filter2_4!_A(0,1,0,0) = 1 filter2_4!_A(0,1,0,1) = 1 filter2_4!_A(0,1,1,0) = 1 filter2_4!_A(0,1,1,1) = 1 filter2_4!_A(1,0,0,0) = 0 filter2_4!_A(1,0,0,1) = 0 filter2_4!_A(1,0,1,0) = 0 filter2_4!_A(1,0,1,1) = 0 filter2_4!_A(1,1,0,0) = 1 filter2_4!_A(1,1,0,1) = 1 filter2_4!_A(1,1,1,0) = 1 filter2_4!_A(1,1,1,1) = 1 map_0!_A = 1 map_1!_A(0) = 0 map_1!_A(1) = 1 map_2!_A(0,0) = 0 map_2!_A(0,1) = 0 map_2!_A(1,0) = 0 map_2!_A(1,1) = 0 succ_0!_A = 1 nil_0!_A = 0 true_0!_A = 1 false_0!_A = 1 partial status: pi(ack_2#) = [1] pi(succ_1) = [1] pi(|0|_0) = [] pi(ack_2) = [1, 2] pi(app) = [1, 2] pi(ack_0) = [] pi(ack_1) = [1] pi(cons_0) = [] pi(cons_1) = [] pi(cons_2) = [] pi(filter_0) = [] pi(filter_1) = [] pi(filter_2) = [] pi(filter2_0) = [] pi(filter2_1) = [] pi(filter2_2) = [] pi(filter2_3) = [] pi(filter2_4) = [] pi(map_0) = [] pi(map_1) = [] pi(map_2) = [2] pi(succ_0) = [] pi(nil_0) = [] pi(true_0) = [] pi(false_0) = [] The following rules are removed: p1, p3 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(succ_1(x),y) and R consists of: r1: app(ack_0(),y) -> ack_1(y) r2: app(ack_1(x1),y) -> ack_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(filter_0(),y) -> filter_1(y) r6: app(filter_1(x1),y) -> filter_2(x1,y) r7: app(filter2_0(),y) -> filter2_1(y) r8: app(filter2_1(x1),y) -> filter2_2(x1,y) r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r11: app(map_0(),y) -> map_1(y) r12: app(map_1(x1),y) -> map_2(x1,y) r13: app(succ_0(),y) -> succ_1(y) r14: ack_2(|0|_0(),y) -> succ_1(y) r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0())) r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y)) r17: map_2(f,nil_0()) -> nil_0() r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r19: filter_2(f,nil_0()) -> nil_0() r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: ack_2#(succ_1(x),succ_1(y)) -> ack_2#(succ_1(x),y) and R consists of: r1: app(ack_0(),y) -> ack_1(y) r2: app(ack_1(x1),y) -> ack_2(x1,y) r3: app(cons_0(),y) -> cons_1(y) r4: app(cons_1(x1),y) -> cons_2(x1,y) r5: app(filter_0(),y) -> filter_1(y) r6: app(filter_1(x1),y) -> filter_2(x1,y) r7: app(filter2_0(),y) -> filter2_1(y) r8: app(filter2_1(x1),y) -> filter2_2(x1,y) r9: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r10: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r11: app(map_0(),y) -> map_1(y) r12: app(map_1(x1),y) -> map_2(x1,y) r13: app(succ_0(),y) -> succ_1(y) r14: ack_2(|0|_0(),y) -> succ_1(y) r15: ack_2(succ_1(x),y) -> ack_2(x,succ_1(|0|_0())) r16: ack_2(succ_1(x),succ_1(y)) -> ack_2(x,ack_2(succ_1(x),y)) r17: map_2(f,nil_0()) -> nil_0() r18: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r19: filter_2(f,nil_0()) -> nil_0() r20: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r21: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r22: filter2_4(false_0(),f,x,xs) -> filter_2(f,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: ack_2#_A(0,0) = 0 ack_2#_A(0,1) = 0 ack_2#_A(1,0) = 0 ack_2#_A(1,1) = 0 succ_1_A(0) = 0 succ_1_A(1) = 0 app_A(0,0) = 1 app_A(0,1) = 1 app_A(1,0) = 0 app_A(1,1) = 0 ack_0_A = 0 ack_1_A(0) = 1 ack_1_A(1) = 1 ack_2_A(0,0) = 0 ack_2_A(0,1) = 0 ack_2_A(1,0) = 0 ack_2_A(1,1) = 0 cons_0_A = 1 cons_1_A(0) = 0 cons_1_A(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 filter_0_A = 1 filter_1_A(0) = 0 filter_1_A(1) = 0 filter_2_A(0,0) = 1 filter_2_A(0,1) = 1 filter_2_A(1,0) = 1 filter_2_A(1,1) = 1 filter2_0_A = 1 filter2_1_A(0) = 0 filter2_1_A(1) = 0 filter2_2_A(0,0) = 1 filter2_2_A(0,1) = 1 filter2_2_A(1,0) = 1 filter2_2_A(1,1) = 1 filter2_3_A(0,0,0) = 0 filter2_3_A(0,0,1) = 0 filter2_3_A(0,1,0) = 0 filter2_3_A(0,1,1) = 0 filter2_3_A(1,0,0) = 0 filter2_3_A(1,0,1) = 0 filter2_3_A(1,1,0) = 0 filter2_3_A(1,1,1) = 0 filter2_4_A(0,0,0,0) = 1 filter2_4_A(0,0,0,1) = 1 filter2_4_A(0,0,1,0) = 1 filter2_4_A(0,0,1,1) = 1 filter2_4_A(0,1,0,0) = 1 filter2_4_A(0,1,0,1) = 1 filter2_4_A(0,1,1,0) = 1 filter2_4_A(0,1,1,1) = 1 filter2_4_A(1,0,0,0) = 1 filter2_4_A(1,0,0,1) = 1 filter2_4_A(1,0,1,0) = 1 filter2_4_A(1,0,1,1) = 1 filter2_4_A(1,1,0,0) = 1 filter2_4_A(1,1,0,1) = 1 filter2_4_A(1,1,1,0) = 1 filter2_4_A(1,1,1,1) = 1 map_0_A = 1 map_1_A(0) = 0 map_1_A(1) = 0 map_2_A(0,0) = 1 map_2_A(0,1) = 1 map_2_A(1,0) = 1 map_2_A(1,1) = 1 succ_0_A = 1 |0|_0_A = 1 nil_0_A = 1 true_0_A = 1 false_0_A = 0 ack_2#!_A(0,0) = 0 ack_2#!_A(0,1) = 0 ack_2#!_A(1,0) = 1 ack_2#!_A(1,1) = 1 succ_1!_A(0) = 0 succ_1!_A(1) = 0 app!_A(0,0) = 1 app!_A(0,1) = 1 app!_A(1,0) = 1 app!_A(1,1) = 1 ack_0!_A = 1 ack_1!_A(0) = 0 ack_1!_A(1) = 0 ack_2!_A(0,0) = 1 ack_2!_A(0,1) = 1 ack_2!_A(1,0) = 1 ack_2!_A(1,1) = 1 cons_0!_A = 1 cons_1!_A(0) = 0 cons_1!_A(1) = 0 cons_2!_A(0,0) = 1 cons_2!_A(0,1) = 0 cons_2!_A(1,0) = 0 cons_2!_A(1,1) = 0 filter_0!_A = 0 filter_1!_A(0) = 0 filter_1!_A(1) = 0 filter_2!_A(0,0) = 0 filter_2!_A(0,1) = 0 filter_2!_A(1,0) = 0 filter_2!_A(1,1) = 0 filter2_0!_A = 0 filter2_1!_A(0) = 0 filter2_1!_A(1) = 0 filter2_2!_A(0,0) = 0 filter2_2!_A(0,1) = 0 filter2_2!_A(1,0) = 0 filter2_2!_A(1,1) = 0 filter2_3!_A(0,0,0) = 1 filter2_3!_A(0,0,1) = 1 filter2_3!_A(0,1,0) = 0 filter2_3!_A(0,1,1) = 0 filter2_3!_A(1,0,0) = 0 filter2_3!_A(1,0,1) = 1 filter2_3!_A(1,1,0) = 0 filter2_3!_A(1,1,1) = 1 filter2_4!_A(0,0,0,0) = 1 filter2_4!_A(0,0,0,1) = 0 filter2_4!_A(0,0,1,0) = 0 filter2_4!_A(0,0,1,1) = 0 filter2_4!_A(0,1,0,0) = 0 filter2_4!_A(0,1,0,1) = 0 filter2_4!_A(0,1,1,0) = 0 filter2_4!_A(0,1,1,1) = 0 filter2_4!_A(1,0,0,0) = 0 filter2_4!_A(1,0,0,1) = 0 filter2_4!_A(1,0,1,0) = 0 filter2_4!_A(1,0,1,1) = 0 filter2_4!_A(1,1,0,0) = 0 filter2_4!_A(1,1,0,1) = 0 filter2_4!_A(1,1,1,0) = 0 filter2_4!_A(1,1,1,1) = 0 map_0!_A = 0 map_1!_A(0) = 0 map_1!_A(1) = 0 map_2!_A(0,0) = 1 map_2!_A(0,1) = 1 map_2!_A(1,0) = 0 map_2!_A(1,1) = 0 succ_0!_A = 0 |0|_0!_A = 1 nil_0!_A = 0 true_0!_A = 1 false_0!_A = 1 partial status: pi(ack_2#) = [2] pi(succ_1) = [1] pi(app) = [1, 2] pi(ack_0) = [] pi(ack_1) = [1] pi(ack_2) = [1, 2] pi(cons_0) = [] pi(cons_1) = [] pi(cons_2) = [] pi(filter_0) = [] pi(filter_1) = [] pi(filter_2) = [] pi(filter2_0) = [] pi(filter2_1) = [1] pi(filter2_2) = [1] pi(filter2_3) = [1, 3] pi(filter2_4) = [] pi(map_0) = [] pi(map_1) = [1] pi(map_2) = [] pi(succ_0) = [] pi(|0|_0) = [] pi(nil_0) = [] pi(true_0) = [] pi(false_0) = [] The following rules are removed: p1 Then no dependency pair remains.