YES We show the termination of the TRS R: app(D(),t()) -> |1|() app(D(),constant()) -> |0|() app(D(),app(app(+(),x),y)) -> app(app(+(),app(D(),x)),app(D(),y)) app(D(),app(app(*(),x),y)) -> app(app(+(),app(app(*(),y),app(D(),x))),app(app(*(),x),app(D(),y))) app(D(),app(app(-(),x),y)) -> app(app(-(),app(D(),x)),app(D(),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(*_0(),y) -> *_1(y) app(*_1(x1),y) -> *_2(x1,y) app(+_0(),y) -> +_1(y) app(+_1(x1),y) -> +_2(x1,y) app(-_0(),y) -> -_1(y) app(-_1(x1),y) -> -_2(x1,y) app(D_0(),y) -> D_1(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) D_1(t_0()) -> |1|_0() D_1(constant_0()) -> |0|_0() D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) D_1(-_2(x,y)) -> -_2(D_1(x),D_1(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#(D_0(),y) -> D_1#(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: D_1#(+_2(x,y)) -> D_1#(x) p6: D_1#(+_2(x,y)) -> D_1#(y) p7: D_1#(*_2(x,y)) -> D_1#(x) p8: D_1#(*_2(x,y)) -> D_1#(y) p9: D_1#(-_2(x,y)) -> D_1#(x) p10: D_1#(-_2(x,y)) -> D_1#(y) p11: map_2#(f,cons_2(x,xs)) -> app#(f,x) p12: map_2#(f,cons_2(x,xs)) -> map_2#(f,xs) p13: filter_2#(f,cons_2(x,xs)) -> filter2_4#(app(f,x),f,x,xs) p14: filter_2#(f,cons_2(x,xs)) -> app#(f,x) p15: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs) p16: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: {p2, p3, p4, p11, p12, p13, p14, p15, p16} {p5, p6, p7, p8, p9, p10} -- 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(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r8, r9, r10, r12, r13, r14, r16, r18, r19, r25 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 + 3 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 *_0_A() = 1 *_1_A(x1) = x1 + 2 *_2_A(x1,x2) = x1 + x2 + 3 +_0_A() = 1 +_1_A(x1) = x1 + 2 +_2_A(x1,x2) = x1 + x2 + 3 -_0_A() = 1 -_1_A(x1) = x1 + 2 -_2_A(x1,x2) = x1 + x2 + 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 D_1_A(x1) = x1 + 2 t_0_A() = 1 |1|_0_A() = 0 constant_0_A() = 1 |0|_0_A() = 0 filter_2_A(x1,x2) = x1 + x2 + 3 nil_0_A() = 1 base order 2: precedence: app# > map_2# = filter2_4# > cons_2 = filter2_3 > filter_2# = filter_2 > false_0 = app = true_0 = +_2 = t_0 = nil_0 > map_1 = filter_1 = *_2 = +_1 = -_2 = cons_1 = filter2_2 > *_1 = +_0 = -_1 = cons_0 = filter_0 = filter2_1 = map_0 = D_1 = constant_0 > *_0 = -_0 = filter2_0 = |1|_0 = |0|_0 partial status: pi(map_2#) = [] pi(cons_2) = [2] pi(app#) = [] pi(map_1) = [1] pi(filter2_3) = [3] pi(filter2_4#) = [] pi(false_0) = [] pi(filter_2#) = [2] pi(filter_1) = [1] pi(app) = [2] pi(true_0) = [] pi(*_0) = [] pi(*_1) = [1] pi(*_2) = [2] pi(+_0) = [] pi(+_1) = [1] pi(+_2) = [] pi(-_0) = [] pi(-_1) = [1] pi(-_2) = [2] pi(cons_0) = [] pi(cons_1) = [1] pi(filter_0) = [] pi(filter2_0) = [] pi(filter2_1) = [1] pi(filter2_2) = [2] pi(map_0) = [] pi(D_1) = [] pi(t_0) = [] pi(|1|_0) = [] pi(constant_0) = [] pi(|0|_0) = [] pi(filter_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#(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(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: 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(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r8, r9, r10, r12, r13, r14, r16, r18, r19, r25 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 + 6 cons_2_A(x1,x2) = x1 + x2 + 3 app#_A(x1,x2) = x1 + x2 + 5 filter_1_A(x1) = x1 + 2 filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 7 app_A(x1,x2) = x1 + x2 + 8 true_0_A() = 1 false_0_A() = 1 filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 4 *_0_A() = 1 *_1_A(x1) = x1 + 2 *_2_A(x1,x2) = x1 + x2 + 3 +_0_A() = 1 +_1_A(x1) = x1 + 2 +_2_A(x1,x2) = x1 + x2 + 3 -_0_A() = 1 -_1_A(x1) = x1 + 2 -_2_A(x1,x2) = x1 + x2 + 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 D_1_A(x1) = x1 + 2 t_0_A() = 1 |1|_0_A() = 0 constant_0_A() = 1 |0|_0_A() = 0 filter_2_A(x1,x2) = x1 + x2 + 6 nil_0_A() = 1 base order 2: precedence: filter_2# = filter_2 > filter2_4# = app > filter2_3 = +_2 > *_1 = *_2 = -_0 = -_2 > cons_2 = filter_1 = false_0 = *_0 = +_1 = -_1 = filter2_2 = D_1 = t_0 = constant_0 > app# = true_0 = +_0 = cons_1 = filter_0 = filter2_1 = map_1 = |1|_0 = |0|_0 = nil_0 > cons_0 = filter2_0 = map_0 partial status: pi(filter_2#) = [2] pi(cons_2) = [] pi(app#) = [] pi(filter_1) = [1] pi(filter2_4#) = [] pi(app) = [] pi(true_0) = [] pi(false_0) = [] pi(filter2_3) = [3] pi(*_0) = [] pi(*_1) = [1] pi(*_2) = [] pi(+_0) = [] pi(+_1) = [1] pi(+_2) = [] pi(-_0) = [] pi(-_1) = [1] pi(-_2) = [1, 2] 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(D_1) = [1] pi(t_0) = [] pi(|1|_0) = [] pi(constant_0) = [] pi(|0|_0) = [] pi(filter_2) = [2] pi(nil_0) = [] The following rules are removed: p6 -- SCC decomposition. 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) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5} -- 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#(false_0(),f,x,xs) -> filter_2#(f,xs) p5: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r18, r19, r25 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 + 5 app#_A(x1,x2) = x1 + x2 + 6 filter_1_A(x1) = x1 + 3 filter2_4#_A(x1,x2,x3,x4) = x2 + x4 + 6 app_A(x1,x2) = x1 + 1 false_0_A() = 1 true_0_A() = 1 D_1_A(x1) = x1 + 3 t_0_A() = 1 |1|_0_A() = 0 constant_0_A() = 1 |0|_0_A() = 0 filter_2_A(x1,x2) = x1 + x2 + 5 nil_0_A() = 1 base order 2: precedence: app# = filter2_4# > cons_2 > nil_0 > filter_2# = constant_0 = filter_2 > filter_1 = false_0 = true_0 = D_1 = t_0 = |0|_0 > app = |1|_0 partial status: pi(filter_2#) = [] pi(cons_2) = [2] pi(app#) = [2] pi(filter_1) = [1] pi(filter2_4#) = [] pi(app) = [] pi(false_0) = [] pi(true_0) = [] pi(D_1) = [1] pi(t_0) = [] pi(|1|_0) = [] pi(constant_0) = [] pi(|0|_0) = [] 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#(false_0(),f,x,xs) -> filter_2#(f,xs) p4: filter2_4#(true_0(),f,x,xs) -> filter_2#(f,xs) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: 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#(true_0(),f,x,xs) -> filter_2#(f,xs) p3: filter2_4#(false_0(),f,x,xs) -> filter_2#(f,xs) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: 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, r23, r24, r25, r26, r27, r28 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) = 4 cons_2_A(x1,x2) = 0 filter2_4#_A(x1,x2,x3,x4) = x1 app_A(x1,x2) = 3 true_0_A() = 5 false_0_A() = 5 *_0_A() = 0 *_1_A(x1) = 0 *_2_A(x1,x2) = 0 +_0_A() = 0 +_1_A(x1) = 0 +_2_A(x1,x2) = 0 -_0_A() = 0 -_1_A(x1) = 0 -_2_A(x1,x2) = 0 D_0_A() = 0 D_1_A(x1) = 1 cons_0_A() = 0 cons_1_A(x1) = 0 filter_0_A() = 0 filter_1_A(x1) = 0 filter_2_A(x1,x2) = 2 filter2_0_A() = 0 filter2_1_A(x1) = 0 filter2_2_A(x1,x2) = 0 filter2_3_A(x1,x2,x3) = 0 filter2_4_A(x1,x2,x3,x4) = 2 map_0_A() = 0 map_1_A(x1) = 1 map_2_A(x1,x2) = 2 t_0_A() = 0 |1|_0_A() = 0 constant_0_A() = 0 |0|_0_A() = 0 nil_0_A() = 0 base order 2: precedence: true_0 = map_1 > cons_2 = map_2 > app = false_0 > filter_2# = +_2 = -_2 = cons_1 = filter_2 = filter2_3 = filter2_4 = map_0 > filter2_4# = *_2 = +_1 = -_1 = cons_0 = filter_1 = filter2_2 = nil_0 > *_1 = +_0 = -_0 = D_1 = filter_0 = filter2_1 = t_0 = constant_0 > *_0 = D_0 = filter2_0 = |1|_0 = |0|_0 partial status: pi(filter_2#) = [] pi(cons_2) = [] pi(filter2_4#) = [] pi(app) = [] pi(true_0) = [] pi(false_0) = [] pi(*_0) = [] pi(*_1) = [] pi(*_2) = [] pi(+_0) = [] pi(+_1) = [] pi(+_2) = [] pi(-_0) = [] pi(-_1) = [] pi(-_2) = [] pi(D_0) = [] pi(D_1) = [] 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(t_0) = [] pi(|1|_0) = [] pi(constant_0) = [] pi(|0|_0) = [] pi(nil_0) = [] The following rules are removed: p3 -- 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) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: 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: 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) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r8, r9, r10, r12, r13, r14, r16, r17, r18, r19, r23, r24, r25 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 + 6 filter2_4#_A(x1,x2,x3,x4) = x4 + 7 app_A(x1,x2) = x1 + x2 + 3 true_0_A() = 3 *_0_A() = 2 *_1_A(x1) = x1 + 1 *_2_A(x1,x2) = x1 + x2 + 3 +_0_A() = 1 +_1_A(x1) = x1 + 2 +_2_A(x1,x2) = x1 + x2 + 4 -_0_A() = 1 -_1_A(x1) = x1 + 2 -_2_A(x1,x2) = x1 + x2 + 4 cons_0_A() = 2 cons_1_A(x1) = x1 + 4 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 + 4 filter2_3_A(x1,x2,x3) = x1 + x2 + x3 + 6 map_0_A() = 4 map_1_A(x1) = x1 + 6 map_2_A(x1,x2) = x1 + x2 + 7 D_1_A(x1) = x1 + 1 t_0_A() = 1 |1|_0_A() = 0 constant_0_A() = 1 |0|_0_A() = 0 nil_0_A() = 0 filter_2_A(x1,x2) = x1 + 4 base order 2: precedence: map_2 = filter_2 > app = nil_0 > filter_2# = *_2 = -_2 = filter_1 = filter2_3 = map_1 > filter2_4# = *_1 = -_1 = filter_0 = filter2_2 = map_0 = D_1 = t_0 = constant_0 > cons_2 = *_0 = +_2 = -_0 = filter2_1 = |1|_0 = |0|_0 > true_0 = +_1 = cons_1 = filter2_0 > +_0 = cons_0 partial status: pi(filter_2#) = [] pi(cons_2) = [2] pi(filter2_4#) = [4] pi(app) = [1] pi(true_0) = [] pi(*_0) = [] pi(*_1) = [1] pi(*_2) = [1, 2] pi(+_0) = [] pi(+_1) = [1] pi(+_2) = [1] pi(-_0) = [] pi(-_1) = [1] pi(-_2) = [1, 2] pi(cons_0) = [] pi(cons_1) = [1] pi(filter_0) = [] pi(filter_1) = [1] pi(filter2_0) = [] pi(filter2_1) = [1] pi(filter2_2) = [1, 2] pi(filter2_3) = [1, 2, 3] pi(map_0) = [] pi(map_1) = [1] pi(map_2) = [1] pi(D_1) = [1] pi(t_0) = [] pi(|1|_0) = [] pi(constant_0) = [] pi(|0|_0) = [] pi(nil_0) = [] pi(filter_2) = [1] The following rules are removed: p2 -- 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) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: 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(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r5, r8, r10, r12, r13, r14, r16, r18, r19, r25 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 + 6 app_A(x1,x2) = x2 + 5 *_0_A() = 5 *_1_A(x1) = 4 +_0_A() = 2 +_1_A(x1) = 1 -_0_A() = 2 -_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() = 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 D_1_A(x1) = x1 + 2 t_0_A() = 1 |1|_0_A() = 0 constant_0_A() = 1 |0|_0_A() = 0 filter_2_A(x1,x2) = x1 + x2 + 6 nil_0_A() = 1 base order 2: precedence: cons_0 > cons_1 > cons_2 > nil_0 > filter_2 > map_2# = app = *_0 = +_0 = -_0 = filter_0 = filter2_0 = map_0 > *_1 = -_1 = filter_1 = filter2_1 = map_1 = D_1 = t_0 = |0|_0 > +_1 = filter2_2 = |1|_0 = constant_0 > filter2_3 partial status: pi(map_2#) = [] pi(cons_2) = [] pi(app) = [2] pi(*_0) = [] pi(*_1) = [] pi(+_0) = [] pi(+_1) = [] pi(-_0) = [] pi(-_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(D_1) = [1] pi(t_0) = [] pi(|1|_0) = [] pi(constant_0) = [] pi(|0|_0) = [] pi(filter_2) = [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: D_1#(+_2(x,y)) -> D_1#(x) p2: D_1#(-_2(x,y)) -> D_1#(y) p3: D_1#(-_2(x,y)) -> D_1#(x) p4: D_1#(*_2(x,y)) -> D_1#(y) p5: D_1#(*_2(x,y)) -> D_1#(x) p6: D_1#(+_2(x,y)) -> D_1#(y) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r5, r8, r10, r12, r13, r14, r16, r18, r19, r25 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: D_1#_A(x1) = x1 +_2_A(x1,x2) = x1 + x2 + 5 -_2_A(x1,x2) = x1 + x2 + 5 *_2_A(x1,x2) = x1 + x2 + 5 app_A(x1,x2) = x2 + 4 *_0_A() = 2 *_1_A(x1) = 1 +_0_A() = 2 +_1_A(x1) = 1 -_0_A() = 2 -_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() = 4 filter2_1_A(x1) = 3 filter2_2_A(x1,x2) = 2 filter2_3_A(x1,x2,x3) = 1 map_0_A() = 1 map_1_A(x1) = 2 D_1_A(x1) = x1 + 2 t_0_A() = 1 |1|_0_A() = 0 constant_0_A() = 1 |0|_0_A() = 2 filter_2_A(x1,x2) = x2 + 2 nil_0_A() = 1 base order 2: precedence: filter_2 > *_0 = cons_0 > *_1 = cons_1 > app = +_0 = -_0 = filter_0 = filter2_0 = map_0 = |0|_0 = nil_0 > +_2 = *_2 = filter_1 = filter2_1 = map_1 = D_1 = t_0 = constant_0 > -_2 = +_1 = filter2_2 = |1|_0 > D_1# = -_1 = filter2_3 partial status: pi(D_1#) = [] pi(+_2) = [] pi(-_2) = [2] pi(*_2) = [1, 2] pi(app) = [2] pi(*_0) = [] pi(*_1) = [] pi(+_0) = [] pi(+_1) = [] pi(-_0) = [] pi(-_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(D_1) = [1] pi(t_0) = [] pi(|1|_0) = [] pi(constant_0) = [] pi(|0|_0) = [] 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: D_1#(-_2(x,y)) -> D_1#(y) p2: D_1#(-_2(x,y)) -> D_1#(x) p3: D_1#(*_2(x,y)) -> D_1#(y) p4: D_1#(*_2(x,y)) -> D_1#(x) p5: D_1#(+_2(x,y)) -> D_1#(y) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: D_1#(-_2(x,y)) -> D_1#(y) p2: D_1#(+_2(x,y)) -> D_1#(y) p3: D_1#(*_2(x,y)) -> D_1#(x) p4: D_1#(*_2(x,y)) -> D_1#(y) p5: D_1#(-_2(x,y)) -> D_1#(x) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r5, r8, r10, r12, r13, r14, r16, r18, r19, r25 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: D_1#_A(x1) = x1 + 3 -_2_A(x1,x2) = x1 + x2 + 2 +_2_A(x1,x2) = x1 + x2 + 6 *_2_A(x1,x2) = x1 + x2 + 4 app_A(x1,x2) = x2 + 5 *_0_A() = 2 *_1_A(x1) = 1 +_0_A() = 2 +_1_A(x1) = 1 -_0_A() = 2 -_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) = 1 map_0_A() = 1 map_1_A(x1) = 2 D_1_A(x1) = x1 + 1 t_0_A() = 1 |1|_0_A() = 0 constant_0_A() = 1 |0|_0_A() = 0 filter_2_A(x1,x2) = x2 + 2 nil_0_A() = 1 base order 2: precedence: +_2 > D_1# > -_2 > app = +_0 = -_0 = filter_0 = filter2_0 = map_0 = t_0 = constant_0 = filter_2 > *_2 = +_1 = -_1 = cons_1 = filter_1 = filter2_1 = D_1 = |1|_0 > *_1 = cons_0 = filter2_2 = map_1 = |0|_0 = nil_0 > *_0 = filter2_3 partial status: pi(D_1#) = [] pi(-_2) = [1, 2] pi(+_2) = [] pi(*_2) = [1, 2] pi(app) = [2] pi(*_0) = [] pi(*_1) = [] pi(+_0) = [] pi(+_1) = [] pi(-_0) = [] pi(-_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(D_1) = [1] pi(t_0) = [] pi(|1|_0) = [] pi(constant_0) = [] pi(|0|_0) = [] pi(filter_2) = [2] pi(nil_0) = [] The following rules are removed: p5 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: D_1#(-_2(x,y)) -> D_1#(y) p2: D_1#(+_2(x,y)) -> D_1#(y) p3: D_1#(*_2(x,y)) -> D_1#(x) p4: D_1#(*_2(x,y)) -> D_1#(y) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: D_1#(-_2(x,y)) -> D_1#(y) p2: D_1#(*_2(x,y)) -> D_1#(y) p3: D_1#(*_2(x,y)) -> D_1#(x) p4: D_1#(+_2(x,y)) -> D_1#(y) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r5, r8, r10, r12, r13, r14, r16, r18, r19, r25 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: D_1#_A(x1) = x1 -_2_A(x1,x2) = x1 + x2 + 8 *_2_A(x1,x2) = x1 + x2 + 6 +_2_A(x1,x2) = x1 + x2 + 4 app_A(x1,x2) = x2 + 7 *_0_A() = 2 *_1_A(x1) = 1 +_0_A() = 2 +_1_A(x1) = 1 -_0_A() = 2 -_1_A(x1) = 1 cons_0_A() = 2 cons_1_A(x1) = 1 filter_0_A() = 4 filter_1_A(x1) = 3 filter2_0_A() = 7 filter2_1_A(x1) = 6 filter2_2_A(x1,x2) = 5 filter2_3_A(x1,x2,x3) = 4 map_0_A() = 2 map_1_A(x1) = 1 D_1_A(x1) = x1 + 5 t_0_A() = 1 |1|_0_A() = 0 constant_0_A() = 1 |0|_0_A() = 0 filter_2_A(x1,x2) = 2 nil_0_A() = 1 base order 2: precedence: filter_0 = map_0 > +_2 = +_0 = filter_1 = map_1 > -_2 = *_2 = +_1 = D_1 = filter_2 > app = filter2_0 > filter2_1 > *_0 = -_0 = filter2_2 = constant_0 > D_1# = *_1 = -_1 = cons_0 = filter2_3 = |1|_0 = |0|_0 = nil_0 > t_0 > cons_1 partial status: pi(D_1#) = [] pi(-_2) = [1, 2] pi(*_2) = [1, 2] pi(+_2) = [1, 2] pi(app) = [] pi(*_0) = [] pi(*_1) = [] pi(+_0) = [] pi(+_1) = [] pi(-_0) = [] pi(-_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(D_1) = [1] pi(t_0) = [] pi(|1|_0) = [] pi(constant_0) = [] pi(|0|_0) = [] pi(filter_2) = [] pi(nil_0) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: D_1#(*_2(x,y)) -> D_1#(y) p2: D_1#(*_2(x,y)) -> D_1#(x) p3: D_1#(+_2(x,y)) -> D_1#(y) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: 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: D_1#(*_2(x,y)) -> D_1#(y) p2: D_1#(+_2(x,y)) -> D_1#(y) p3: D_1#(*_2(x,y)) -> D_1#(x) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r5, r10, r12, r13, r14, r16, r18, r19, r24, r25, r27 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: D_1#_A(x1) = x1 *_2_A(x1,x2) = x1 + x2 + 3 +_2_A(x1,x2) = x1 + x2 + 6 app_A(x1,x2) = 5 *_0_A() = 5 *_1_A(x1) = 4 +_0_A() = 2 +_1_A(x1) = 1 -_0_A() = 5 -_1_A(x1) = 4 filter_0_A() = 5 filter_1_A(x1) = 4 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() = 5 map_1_A(x1) = 4 D_1_A(x1) = x1 + 2 t_0_A() = 1 |1|_0_A() = 0 constant_0_A() = 1 |0|_0_A() = 0 map_2_A(x1,x2) = 7 cons_2_A(x1,x2) = 6 filter_2_A(x1,x2) = 8 nil_0_A() = 1 filter2_4_A(x1,x2,x3,x4) = 7 true_0_A() = 9 base order 2: precedence: filter2_0 > *_0 = filter_0 = filter2_1 > +_2 = *_1 = filter_1 = filter2_2 = true_0 > D_1# = *_2 = filter2_3 = map_0 = D_1 = t_0 = constant_0 = filter_2 > map_1 = |1|_0 = |0|_0 = cons_2 = filter2_4 > map_2 > app = +_0 = -_0 = nil_0 > +_1 = -_1 partial status: pi(D_1#) = [] pi(*_2) = [2] pi(+_2) = [1, 2] pi(app) = [] pi(*_0) = [] pi(*_1) = [] pi(+_0) = [] pi(+_1) = [] pi(-_0) = [] pi(-_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(D_1) = [1] pi(t_0) = [] pi(|1|_0) = [] pi(constant_0) = [] pi(|0|_0) = [] pi(map_2) = [] pi(cons_2) = [] pi(filter_2) = [] pi(nil_0) = [] pi(filter2_4) = [] pi(true_0) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: D_1#(+_2(x,y)) -> D_1#(y) p2: D_1#(*_2(x,y)) -> D_1#(x) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: 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: D_1#(+_2(x,y)) -> D_1#(y) p2: D_1#(*_2(x,y)) -> D_1#(x) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r5, r8, r10, r12, r13, r14, r16, r18, r19, r25 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: D_1#_A(x1) = x1 +_2_A(x1,x2) = x1 + x2 + 4 *_2_A(x1,x2) = x1 + x2 + 7 app_A(x1,x2) = 5 -_0_A() = 5 -_1_A(x1) = 4 cons_0_A() = 5 cons_1_A(x1) = 4 filter_0_A() = 5 filter_1_A(x1) = 4 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() = 4 map_1_A(x1) = 3 D_1_A(x1) = x1 + 2 t_0_A() = 1 |1|_0_A() = 0 constant_0_A() = 1 |0|_0_A() = 0 filter_2_A(x1,x2) = 6 nil_0_A() = 1 base order 2: precedence: D_1# > +_2 = cons_0 = filter_0 = map_0 = D_1 = t_0 = constant_0 > *_2 = cons_1 = map_1 = |1|_0 = |0|_0 > filter2_2 = filter_2 = nil_0 > app = filter2_0 > -_1 = filter_1 = filter2_1 = filter2_3 > -_0 partial status: pi(D_1#) = [] pi(+_2) = [1] pi(*_2) = [1, 2] pi(app) = [] pi(-_0) = [] pi(-_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(D_1) = [1] pi(t_0) = [] pi(|1|_0) = [] pi(constant_0) = [] pi(|0|_0) = [] pi(filter_2) = [] pi(nil_0) = [] The following rules are removed: p1 -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: D_1#(*_2(x,y)) -> D_1#(x) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: 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: D_1#(*_2(x,y)) -> D_1#(x) and R consists of: r1: app(*_0(),y) -> *_1(y) r2: app(*_1(x1),y) -> *_2(x1,y) r3: app(+_0(),y) -> +_1(y) r4: app(+_1(x1),y) -> +_2(x1,y) r5: app(-_0(),y) -> -_1(y) r6: app(-_1(x1),y) -> -_2(x1,y) r7: app(D_0(),y) -> D_1(y) r8: app(cons_0(),y) -> cons_1(y) r9: app(cons_1(x1),y) -> cons_2(x1,y) r10: app(filter_0(),y) -> filter_1(y) r11: app(filter_1(x1),y) -> filter_2(x1,y) r12: app(filter2_0(),y) -> filter2_1(y) r13: app(filter2_1(x1),y) -> filter2_2(x1,y) r14: app(filter2_2(x1,x2),y) -> filter2_3(x1,x2,y) r15: app(filter2_3(x1,x2,x3),y) -> filter2_4(x1,x2,x3,y) r16: app(map_0(),y) -> map_1(y) r17: app(map_1(x1),y) -> map_2(x1,y) r18: D_1(t_0()) -> |1|_0() r19: D_1(constant_0()) -> |0|_0() r20: D_1(+_2(x,y)) -> +_2(D_1(x),D_1(y)) r21: D_1(*_2(x,y)) -> +_2(*_2(y,D_1(x)),*_2(x,D_1(y))) r22: D_1(-_2(x,y)) -> -_2(D_1(x),D_1(y)) r23: map_2(f,nil_0()) -> nil_0() r24: map_2(f,cons_2(x,xs)) -> cons_2(app(f,x),map_2(f,xs)) r25: filter_2(f,nil_0()) -> nil_0() r26: filter_2(f,cons_2(x,xs)) -> filter2_4(app(f,x),f,x,xs) r27: filter2_4(true_0(),f,x,xs) -> cons_2(x,filter_2(f,xs)) r28: filter2_4(false_0(),f,x,xs) -> filter_2(f,xs) The set of usable rules consists of r1, r3, r4, r5, r8, r10, r12, r13, r14, r16, r19, r20, r21, r25 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: D_1#_A(x1) = x1 + 7 *_2_A(x1,x2) = x1 + x2 + 6 app_A(x1,x2) = x2 + 5 *_0_A() = 2 *_1_A(x1) = 1 +_0_A() = 3 +_1_A(x1) = 2 +_2_A(x1,x2) = 1 -_0_A() = 2 -_1_A(x1) = 1 cons_0_A() = 2 cons_1_A(x1) = 1 filter_0_A() = 1 filter_1_A(x1) = x1 + 2 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 D_1_A(x1) = x1 + 2 constant_0_A() = 1 |0|_0_A() = 0 filter_2_A(x1,x2) = x1 + x2 + 6 nil_0_A() = 1 base order 2: precedence: *_2 > app = *_0 = +_2 = -_0 = cons_0 = filter2_0 = map_0 = D_1 = constant_0 > D_1# = *_1 = +_1 = -_1 = cons_1 = filter2_1 = map_1 = |0|_0 > +_0 = filter2_2 = filter_2 = nil_0 > filter_1 = filter2_3 > filter_0 partial status: pi(D_1#) = [] pi(*_2) = [2] pi(app) = [] pi(*_0) = [] pi(*_1) = [] pi(+_0) = [] pi(+_1) = [] pi(+_2) = [] pi(-_0) = [] pi(-_1) = [] pi(cons_0) = [] pi(cons_1) = [] pi(filter_0) = [] pi(filter_1) = [1] pi(filter2_0) = [] pi(filter2_1) = [] pi(filter2_2) = [] pi(filter2_3) = [] pi(map_0) = [] pi(map_1) = [] pi(D_1) = [] pi(constant_0) = [] pi(|0|_0) = [] pi(filter_2) = [] pi(nil_0) = [] The following rules are removed: p1 Then no dependency pair remains.