YES We show the termination of the TRS R: app(not(),app(not(),x)) -> x app(not(),app(app(or(),x),y)) -> app(app(and(),app(not(),x)),app(not(),y)) app(not(),app(app(and(),x),y)) -> app(app(or(),app(not(),x)),app(not(),y)) app(app(and(),x),app(app(or(),y),z)) -> app(app(or(),app(app(and(),x),y)),app(app(and(),x),z)) app(app(and(),app(app(or(),y),z)),x) -> app(app(or(),app(app(and(),x),y)),app(app(and(),x),z)) 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(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(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(not_0(),y) -> not_1(y) app(or_0(),y) -> or_1(y) app(or_1(x1),y) -> or_2(x1,y) not_1(not_1(x)) -> x not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) 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#(and_1(x1),y) -> and_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: app#(not_0(),y) -> not_1#(y) p6: not_1#(or_2(x,y)) -> and_2#(not_1(x),not_1(y)) p7: not_1#(or_2(x,y)) -> not_1#(x) p8: not_1#(or_2(x,y)) -> not_1#(y) p9: not_1#(and_2(x,y)) -> not_1#(x) p10: not_1#(and_2(x,y)) -> not_1#(y) p11: and_2#(x,or_2(y,z)) -> and_2#(x,y) p12: and_2#(x,or_2(y,z)) -> and_2#(x,z) p13: and_2#(or_2(y,z),x) -> and_2#(x,y) p14: and_2#(or_2(y,z),x) -> and_2#(x,z) p15: map_2#(f,cons_2(x,xs)) -> app#(f,x) p16: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs) p17: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs) p18: filter_2#(f,cons_2(x,xs)) -> app#(f,x) p19: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs) p20: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: {p2, p3, p4, p15, p16, p17, p18, p19, p20} {p7, p8, p9, p10} {p11, p12, p13, p14} -- 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(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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r4, r5, r6, r7, r8, r9, r10, r11, r14, r15, r16, r23, r24, r25, r26 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: map_2#_A(x1,x2) = x1 + x2 + 4 cons_2_A(x1,x2) = x1 + x2 + 3 app#_A(x1,x2) = x1 + x2 + 4 map_1_A(x1) = x1 + 2 filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4 filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 5 false_0_A() = 1 filter_2#_A(x1,x2) = x1 + x2 + 3 filter_1_A(x1) = x1 + 2 app_A(x1,x2) = x1 + x2 + 4 true_0_A() = 1 and_0_A() = 1 and_1_A(x1) = x1 + 4 cons_0_A() = 1 cons_1_A(x1) = x1 + 2 filter_0_A() = 1 filter_2_A(x1,x2) = x1 + x2 + 3 filter2_0_A() = 1 filter2_1_A(x1) = x1 + 2 filter2_2_A(x1,x2) = x1 + x2 + 3 filter2_4_A(x1,x2,x3,x4) = x2 + x3 + x4 + 6 map_0_A() = 1 or_0_A() = 2 or_1_A(x1) = x1 + 5 or_2_A(x1,x2) = x1 + x2 + 6 not_1_A(x1) = x1 + 2 nil_0_A() = 1 base order 2: precedence: app# > filter2_4# = filter_2 > map_2# = filter_2# = filter2_4 > cons_2 = false_0 = app > map_1 = filter2_3 = filter_1 = true_0 = cons_1 = or_1 = nil_0 > cons_0 = filter_0 = filter2_2 = map_0 = or_0 = or_2 > filter2_1 > and_1 = filter2_0 = not_1 > and_0 partial status: pi(map_2#) = [2] pi(cons_2) = [] pi(app#) = [2] pi(map_1) = [1] pi(filter2_3) = [3] pi(filter2_4#) = [2] pi(false_0) = [] pi(filter_2#) = [] pi(filter_1) = [1] pi(app) = [2] pi(true_0) = [] pi(and_0) = [] pi(and_1) = [1] pi(cons_0) = [] pi(cons_1) = [1] pi(filter_0) = [] pi(filter_2) = [] pi(filter2_0) = [] pi(filter2_1) = [1] pi(filter2_2) = [2] pi(filter2_4) = [4] pi(map_0) = [] pi(or_0) = [] pi(or_1) = [1] pi(or_2) = [1] pi(not_1) = [1] pi(nil_0) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: app#(map_1(x1),y) -> map_2#(x1,y) p2: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs) p3: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y) p4: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs) p5: filter_2#(f,cons_2(x,xs)) -> app#(f,x) p6: app#(filter_1(x1),y) -> filter_2#(x1,y) p7: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs) p8: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: {p3, p4, p5, p6, p7, p8} {p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: filter_2#(f,cons_2(x,xs)) -> app#(f,x) p2: app#(filter_1(x1),y) -> filter_2#(x1,y) p3: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs) p4: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs) p5: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs) p6: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r4, r5, r7, r8, r9, r11, r14, r15, r23 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: filter_2#_A(x1,x2) = x1 + x2 + 4 cons_2_A(x1,x2) = x1 + x2 + 3 app#_A(x1,x2) = x1 + x2 + 4 filter_1_A(x1) = x1 + 3 filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 5 app_A(x1,x2) = x1 + x2 + 4 true_0_A() = 1 false_0_A() = 1 filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4 and_0_A() = 1 and_1_A(x1) = x1 + 3 cons_0_A() = 1 cons_1_A(x1) = x1 + 2 filter_0_A() = 1 filter2_0_A() = 1 filter2_1_A(x1) = x1 + 2 filter2_2_A(x1,x2) = x1 + x2 + 3 map_0_A() = 1 map_1_A(x1) = x1 + 2 or_0_A() = 3 or_1_A(x1) = x1 + 4 or_2_A(x1,x2) = 3 filter_2_A(x1,x2) = x1 + x2 + 2 nil_0_A() = 1 base order 2: precedence: app# > filter_2# = filter2_4# = filter_2 > app = false_0 = nil_0 > cons_2 = filter_1 = filter2_3 = and_1 = map_1 = or_1 > true_0 = and_0 = cons_1 = filter_0 = filter2_2 = map_0 = or_0 = or_2 > cons_0 = filter2_1 > filter2_0 partial status: pi(filter_2#) = [] pi(cons_2) = [2] pi(app#) = [] pi(filter_1) = [] pi(filter2_4#) = [2] pi(app) = [] pi(true_0) = [] pi(false_0) = [] pi(filter2_3) = [3] pi(and_0) = [] pi(and_1) = [1] pi(cons_0) = [] pi(cons_1) = [1] pi(filter_0) = [] pi(filter2_0) = [] pi(filter2_1) = [1] pi(filter2_2) = [1, 2] pi(map_0) = [] pi(map_1) = [1] pi(or_0) = [] pi(or_1) = [1] pi(or_2) = [] pi(filter_2) = [2] pi(nil_0) = [] The following rules are removed: p1 -- SCC decomposition. 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: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs) p5: app#(filter2_3(x1,x2,x3),y) -> filter2_4#(x1,x2,x3,y) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: {p2, p3, 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(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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r7, r15, r16, r21, r22, r23 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: filter_2#_A(x1,x2) = x2 + 2 cons_2_A(x1,x2) = x2 + 4 filter2_4#_A(x1,x2,x3,x4) = x4 + 3 app_A(x1,x2) = x1 + 2 false_0_A() = 1 true_0_A() = 1 filter2_0_A() = 1 filter2_1_A(x1) = 2 or_1_A(x1) = x1 + 5 or_2_A(x1,x2) = 4 not_1_A(x1) = x1 + 2 map_2_A(x1,x2) = x1 + x2 + 7 nil_0_A() = 1 filter_2_A(x1,x2) = x2 + 7 base order 2: precedence: filter_2 > cons_2 = app = false_0 = map_2 > filter_2# = or_1 = or_2 = nil_0 > filter2_4# = true_0 = filter2_1 = not_1 > filter2_0 partial status: pi(filter_2#) = [] pi(cons_2) = [] pi(filter2_4#) = [] pi(app) = [] pi(false_0) = [] pi(true_0) = [] pi(filter2_0) = [] pi(filter2_1) = [] pi(or_1) = [1] pi(or_2) = [] pi(not_1) = [] pi(map_2) = [2] pi(nil_0) = [] pi(filter_2) = [2] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs) p2: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: (no SCCs) -- 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(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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r5, r7, r8, r9, r11, r14, r16, r23 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: map_2#_A(x1,x2) = x2 cons_2_A(x1,x2) = x1 + x2 + 5 app_A(x1,x2) = x2 + 4 and_0_A() = 2 and_1_A(x1) = 1 cons_0_A() = 1 cons_1_A(x1) = 2 filter_0_A() = 2 filter_1_A(x1) = 1 filter2_0_A() = 4 filter2_1_A(x1) = 3 filter2_2_A(x1,x2) = 2 filter2_3_A(x1,x2,x3) = 1 map_0_A() = 2 map_1_A(x1) = 1 or_0_A() = 2 or_1_A(x1) = 1 not_1_A(x1) = x1 + 2 filter_2_A(x1,x2) = x1 + x2 + 5 nil_0_A() = 1 base order 2: precedence: filter_2 > nil_0 > cons_2 > map_2# = cons_1 = not_1 > app = and_0 = cons_0 = filter_0 = filter2_0 = map_0 = or_0 > and_1 = filter_1 = filter2_1 = map_1 = or_1 > filter2_2 > filter2_3 partial status: pi(map_2#) = [] pi(cons_2) = [1] pi(app) = [] pi(and_0) = [] pi(and_1) = [] pi(cons_0) = [] pi(cons_1) = [] pi(filter_0) = [] pi(filter_1) = [] pi(filter2_0) = [] pi(filter2_1) = [] pi(filter2_2) = [] pi(filter2_3) = [] pi(map_0) = [] pi(map_1) = [] pi(or_0) = [] pi(or_1) = [] pi(not_1) = [1] pi(filter_2) = [] 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: not_1#(and_2(x,y)) -> not_1#(y) p2: not_1#(and_2(x,y)) -> not_1#(x) p3: not_1#(or_2(x,y)) -> not_1#(y) p4: not_1#(or_2(x,y)) -> not_1#(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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r5, r7, r8, r9, r11, r14, r16, r23 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: not_1#_A(x1) = x1 and_2_A(x1,x2) = x1 + x2 + 11 or_2_A(x1,x2) = x1 + x2 + 6 app_A(x1,x2) = x2 + 5 and_0_A() = 2 and_1_A(x1) = 1 cons_0_A() = 2 cons_1_A(x1) = 1 filter_0_A() = 2 filter_1_A(x1) = 1 filter2_0_A() = 5 filter2_1_A(x1) = 4 filter2_2_A(x1,x2) = 3 filter2_3_A(x1,x2,x3) = 2 map_0_A() = 2 map_1_A(x1) = 1 or_0_A() = 2 or_1_A(x1) = 1 not_1_A(x1) = x1 + 2 filter_2_A(x1,x2) = x1 + x2 + 6 nil_0_A() = 1 base order 2: precedence: or_0 > not_1# = and_2 = or_2 = or_1 = not_1 = filter_2 > app = and_0 = cons_0 = filter_0 = map_0 > and_1 = cons_1 = filter_1 = filter2_1 = map_1 = nil_0 > filter2_0 = filter2_2 > filter2_3 partial status: pi(not_1#) = [1] pi(and_2) = [1, 2] pi(or_2) = [] pi(app) = [] pi(and_0) = [] pi(and_1) = [] pi(cons_0) = [] pi(cons_1) = [] pi(filter_0) = [] pi(filter_1) = [] pi(filter2_0) = [] pi(filter2_1) = [] pi(filter2_2) = [] pi(filter2_3) = [] pi(map_0) = [] pi(map_1) = [] pi(or_0) = [] pi(or_1) = [] pi(not_1) = [1] pi(filter_2) = [2] pi(nil_0) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: not_1#(and_2(x,y)) -> not_1#(x) p2: not_1#(or_2(x,y)) -> not_1#(y) p3: not_1#(or_2(x,y)) -> not_1#(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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,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: not_1#(and_2(x,y)) -> not_1#(x) p2: not_1#(or_2(x,y)) -> not_1#(x) p3: not_1#(or_2(x,y)) -> not_1#(y) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r4, r5, r7, r8, r9, r11, r14, r15, r16, r23 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: not_1#_A(x1) = x1 and_2_A(x1,x2) = x1 + x2 + 8 or_2_A(x1,x2) = x1 + x2 + 3 app_A(x1,x2) = x1 + x2 + 4 and_0_A() = 2 and_1_A(x1) = x1 + 5 cons_0_A() = 1 cons_1_A(x1) = x1 + 2 cons_2_A(x1,x2) = x1 + x2 + 3 filter_0_A() = 1 filter_1_A(x1) = x1 + 2 filter2_0_A() = 1 filter2_1_A(x1) = x1 + 2 filter2_2_A(x1,x2) = x1 + x2 + 3 filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4 map_0_A() = 1 map_1_A(x1) = x1 + 2 or_0_A() = 1 or_1_A(x1) = x1 + 2 not_1_A(x1) = x1 + 2 filter_2_A(x1,x2) = x1 + x2 + 3 nil_0_A() = 1 base order 2: precedence: filter_2 > filter2_3 = nil_0 > and_1 = filter2_2 > app > and_2 = or_2 = and_0 = cons_2 = filter_1 = filter2_1 = map_1 > not_1# = cons_1 = filter_0 = filter2_0 = map_0 = or_1 = not_1 > cons_0 = or_0 partial status: pi(not_1#) = [1] pi(and_2) = [] pi(or_2) = [1, 2] pi(app) = [1, 2] pi(and_0) = [] pi(and_1) = [1] pi(cons_0) = [] pi(cons_1) = [1] pi(cons_2) = [] pi(filter_0) = [] pi(filter_1) = [1] pi(filter2_0) = [] pi(filter2_1) = [1] pi(filter2_2) = [2] pi(filter2_3) = [] pi(map_0) = [] pi(map_1) = [1] pi(or_0) = [] pi(or_1) = [1] pi(not_1) = [1] pi(filter_2) = [2] pi(nil_0) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: not_1#(or_2(x,y)) -> not_1#(x) p2: not_1#(or_2(x,y)) -> not_1#(y) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: not_1#(or_2(x,y)) -> not_1#(x) p2: not_1#(or_2(x,y)) -> not_1#(y) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r4, r5, r7, r8, r9, r11, r14, r15, r16, r23 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: not_1#_A(x1) = x1 or_2_A(x1,x2) = x1 + x2 + 6 app_A(x1,x2) = x1 + x2 + 8 and_0_A() = 1 and_1_A(x1) = x1 + 4 cons_0_A() = 1 cons_1_A(x1) = x1 + 2 cons_2_A(x1,x2) = x1 + x2 + 3 filter_0_A() = 1 filter_1_A(x1) = x1 + 2 filter2_0_A() = 1 filter2_1_A(x1) = x1 + 2 filter2_2_A(x1,x2) = x1 + x2 + 3 filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4 map_0_A() = 1 map_1_A(x1) = x1 + 2 or_0_A() = 2 or_1_A(x1) = x1 + 1 not_1_A(x1) = x1 + 2 filter_2_A(x1,x2) = x1 + x2 + 6 nil_0_A() = 1 base order 2: precedence: app = filter_2 > cons_2 = filter_1 = filter2_3 > or_2 = and_1 = cons_1 = filter_0 = filter2_2 > not_1# = and_0 = cons_0 = filter2_1 = map_1 = or_1 = not_1 = nil_0 > filter2_0 = map_0 = or_0 partial status: pi(not_1#) = [1] pi(or_2) = [2] pi(app) = [2] pi(and_0) = [] pi(and_1) = [1] pi(cons_0) = [] pi(cons_1) = [1] pi(cons_2) = [] pi(filter_0) = [] pi(filter_1) = [1] pi(filter2_0) = [] pi(filter2_1) = [1] pi(filter2_2) = [2] pi(filter2_3) = [] pi(map_0) = [] pi(map_1) = [1] pi(or_0) = [] pi(or_1) = [1] pi(not_1) = [1] pi(filter_2) = [2] pi(nil_0) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: not_1#(or_2(x,y)) -> not_1#(y) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: 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: not_1#(or_2(x,y)) -> not_1#(y) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r5, r7, r8, r9, r11, r14, r16, r23 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: not_1#_A(x1) = x1 or_2_A(x1,x2) = x1 + x2 + 11 app_A(x1,x2) = x2 + 10 and_0_A() = 1 and_1_A(x1) = x1 + 2 cons_0_A() = 1 cons_1_A(x1) = 2 filter_0_A() = 2 filter_1_A(x1) = 1 filter2_0_A() = 4 filter2_1_A(x1) = 3 filter2_2_A(x1,x2) = 2 filter2_3_A(x1,x2,x3) = 1 map_0_A() = 2 map_1_A(x1) = 1 or_0_A() = 2 or_1_A(x1) = 1 not_1_A(x1) = x1 + 4 filter_2_A(x1,x2) = x2 + 2 nil_0_A() = 1 base order 2: precedence: not_1 > or_2 = filter_0 = filter2_0 = map_0 > not_1# = app = or_0 > and_1 = filter2_2 = or_1 > and_0 = cons_1 = filter2_1 = map_1 > cons_0 = filter2_3 = filter_2 > filter_1 = nil_0 partial status: pi(not_1#) = [] pi(or_2) = [] pi(app) = [2] pi(and_0) = [] pi(and_1) = [1] pi(cons_0) = [] pi(cons_1) = [] pi(filter_0) = [] pi(filter_1) = [] pi(filter2_0) = [] pi(filter2_1) = [] pi(filter2_2) = [] pi(filter2_3) = [] pi(map_0) = [] pi(map_1) = [] pi(or_0) = [] pi(or_1) = [] pi(not_1) = [1] pi(filter_2) = [] 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: and_2#(x,or_2(y,z)) -> and_2#(x,y) p2: and_2#(or_2(y,z),x) -> and_2#(x,z) p3: and_2#(or_2(y,z),x) -> and_2#(x,y) p4: and_2#(x,or_2(y,z)) -> and_2#(x,z) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r5, r7, r8, r9, r11, r14, r23 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: and_2#_A(x1,x2) = x1 + x2 + 5 or_2_A(x1,x2) = x1 + x2 + 4 app_A(x1,x2) = x2 + 3 and_0_A() = 2 and_1_A(x1) = 1 cons_0_A() = 2 cons_1_A(x1) = 1 filter_0_A() = 2 filter_1_A(x1) = 1 filter2_0_A() = 3 filter2_1_A(x1) = 2 filter2_2_A(x1,x2) = 1 filter2_3_A(x1,x2,x3) = x3 + 2 map_0_A() = 2 map_1_A(x1) = 1 or_0_A() = 2 or_1_A(x1) = 1 filter_2_A(x1,x2) = x1 + x2 + 4 nil_0_A() = 1 base order 2: precedence: filter2_0 > filter2_1 = filter2_3 > filter2_2 = filter_2 > and_2# = app = and_0 = cons_0 = filter_0 = map_0 = or_0 = nil_0 > cons_1 = filter_1 = map_1 > or_2 > or_1 > and_1 partial status: pi(and_2#) = [] pi(or_2) = [] pi(app) = [] pi(and_0) = [] pi(and_1) = [] pi(cons_0) = [] pi(cons_1) = [] pi(filter_0) = [] pi(filter_1) = [] pi(filter2_0) = [] pi(filter2_1) = [] pi(filter2_2) = [] pi(filter2_3) = [3] pi(map_0) = [] pi(map_1) = [] pi(or_0) = [] pi(or_1) = [] pi(filter_2) = [2] pi(nil_0) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: and_2#(or_2(y,z),x) -> and_2#(x,z) p2: and_2#(or_2(y,z),x) -> and_2#(x,y) p3: and_2#(x,or_2(y,z)) -> and_2#(x,z) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,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: and_2#(or_2(y,z),x) -> and_2#(x,z) p2: and_2#(x,or_2(y,z)) -> and_2#(x,z) p3: and_2#(or_2(y,z),x) -> and_2#(x,y) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r5, r7, r8, r9, r11, r14, r16, r23 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: and_2#_A(x1,x2) = x1 + x2 + 4 or_2_A(x1,x2) = x1 + x2 + 3 app_A(x1,x2) = x2 + 7 and_0_A() = 2 and_1_A(x1) = 1 cons_0_A() = 2 cons_1_A(x1) = 1 filter_0_A() = 2 filter_1_A(x1) = 1 filter2_0_A() = 1 filter2_1_A(x1) = x1 + 2 filter2_2_A(x1,x2) = 3 filter2_3_A(x1,x2,x3) = 1 map_0_A() = 2 map_1_A(x1) = 1 or_0_A() = 2 or_1_A(x1) = 1 not_1_A(x1) = x1 + 2 filter_2_A(x1,x2) = x1 + x2 + 8 nil_0_A() = 1 base order 2: precedence: and_2# > or_2 = filter2_2 > app = cons_0 = filter_0 = map_0 = or_0 = filter_2 > and_0 = cons_1 = filter_1 = filter2_1 = map_1 = or_1 = nil_0 > and_1 = filter2_0 = filter2_3 = not_1 partial status: pi(and_2#) = [] pi(or_2) = [] pi(app) = [2] pi(and_0) = [] pi(and_1) = [] pi(cons_0) = [] pi(cons_1) = [] pi(filter_0) = [] pi(filter_1) = [] pi(filter2_0) = [] pi(filter2_1) = [1] pi(filter2_2) = [] pi(filter2_3) = [] pi(map_0) = [] pi(map_1) = [] pi(or_0) = [] pi(or_1) = [] pi(not_1) = [] pi(filter_2) = [1, 2] pi(nil_0) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: and_2#(x,or_2(y,z)) -> and_2#(x,z) p2: and_2#(or_2(y,z),x) -> and_2#(x,y) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: and_2#(x,or_2(y,z)) -> and_2#(x,z) p2: and_2#(or_2(y,z),x) -> and_2#(x,y) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r4, r5, r7, r8, r9, r11, r14, r15, r16, r23 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: and_2#_A(x1,x2) = x1 + x2 + 7 or_2_A(x1,x2) = x1 + x2 + 6 app_A(x1,x2) = x1 + x2 + 7 and_0_A() = 1 and_1_A(x1) = x1 + 2 cons_0_A() = 1 cons_1_A(x1) = x1 + 2 cons_2_A(x1,x2) = x1 + x2 + 3 filter_0_A() = 1 filter_1_A(x1) = x1 + 2 filter2_0_A() = 1 filter2_1_A(x1) = x1 + 2 filter2_2_A(x1,x2) = x1 + x2 + 3 filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4 map_0_A() = 1 map_1_A(x1) = x1 + 2 or_0_A() = 2 or_1_A(x1) = x1 + 1 not_1_A(x1) = x1 + 2 filter_2_A(x1,x2) = x2 + 3 nil_0_A() = 1 base order 2: precedence: cons_2 > app > and_2# = or_2 = cons_1 = map_1 = filter_2 > and_1 = cons_0 = filter_1 = map_0 = or_1 = nil_0 > and_0 = filter_0 = filter2_3 = or_0 = not_1 > filter2_2 > filter2_1 > filter2_0 partial status: pi(and_2#) = [] pi(or_2) = [2] pi(app) = [2] pi(and_0) = [] pi(and_1) = [1] pi(cons_0) = [] pi(cons_1) = [1] pi(cons_2) = [] pi(filter_0) = [] pi(filter_1) = [1] pi(filter2_0) = [] pi(filter2_1) = [1] pi(filter2_2) = [2] pi(filter2_3) = [3] pi(map_0) = [] pi(map_1) = [1] pi(or_0) = [] pi(or_1) = [1] pi(not_1) = [1] pi(filter_2) = [] pi(nil_0) = [] The following rules are removed: p2 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: and_2#(x,or_2(y,z)) -> and_2#(x,z) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: 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: and_2#(x,or_2(y,z)) -> and_2#(x,z) 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(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(not_0(),y) -> not_1(y) r14: app(or_0(),y) -> or_1(y) r15: app(or_1(x1),y) -> or_2(x1,y) r16: not_1(not_1(x)) -> x r17: not_1(or_2(x,y)) -> and_2(not_1(x),not_1(y)) r18: not_1(and_2(x,y)) -> or_2(not_1(x),not_1(y)) r19: and_2(x,or_2(y,z)) -> or_2(and_2(x,y),and_2(x,z)) r20: and_2(or_2(y,z),x) -> or_2(and_2(x,y),and_2(x,z)) r21: map_2(f,nil_0()) -> nil_0() r22: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r23: filter_2(f,nil_0()) -> nil_0() r24: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r25: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r26: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r16, r23, r24, r25, r26 Take the reduction pair: generalized weighted path order (>=MG, >MG) base order 1: matrix interpretation with standard point-wise order: carrier: N^1 order: standard order interpretations: and_2#_A(x1,x2) = x2 or_2_A(x1,x2) = x2 + 3 not_1_A(x1) = x1 + 2 filter_2_A(x1,x2) = x1 + x2 + 2 nil_0_A() = 1 cons_2_A(x1,x2) = x2 + 2 filter2_4_A(x1,x2,x3,x4) = x2 + x4 + 4 app_A(x1,x2) = x2 true_0_A() = 1 false_0_A() = 1 base order 2: precedence: or_2 > and_2# = not_1 > app > filter_2 > filter2_4 = false_0 > cons_2 > nil_0 = true_0 partial status: pi(and_2#) = [] pi(or_2) = [] pi(not_1) = [1] pi(filter_2) = [1] pi(nil_0) = [] pi(cons_2) = [] pi(filter2_4) = [2, 4] pi(app) = [] pi(true_0) = [] pi(false_0) = [] The following rules are removed: p1 Then no dependency pair remains.