NO proof of Transformed_CSR_04_OvConsOS_nokinds_Z.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished Termination w.r.t. Q of the given QTRS could be disproven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 14 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) QDP (5) QDPOrderProof [EQUIVALENT, 136 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) AND (9) QDP (10) TransformationProof [EQUIVALENT, 0 ms] (11) QDP (12) TransformationProof [EQUIVALENT, 0 ms] (13) QDP (14) DependencyGraphProof [EQUIVALENT, 0 ms] (15) QDP (16) TransformationProof [EQUIVALENT, 0 ms] (17) QDP (18) DependencyGraphProof [EQUIVALENT, 0 ms] (19) QDP (20) TransformationProof [EQUIVALENT, 0 ms] (21) QDP (22) DependencyGraphProof [EQUIVALENT, 0 ms] (23) QDP (24) TransformationProof [EQUIVALENT, 0 ms] (25) QDP (26) DependencyGraphProof [EQUIVALENT, 0 ms] (27) QDP (28) QDPOrderProof [EQUIVALENT, 84 ms] (29) QDP (30) QDPOrderProof [EQUIVALENT, 286 ms] (31) QDP (32) QDP (33) QDPOrderProof [EQUIVALENT, 94 ms] (34) QDP (35) DependencyGraphProof [EQUIVALENT, 0 ms] (36) QDP (37) QDPOrderProof [EQUIVALENT, 61 ms] (38) QDP (39) DependencyGraphProof [EQUIVALENT, 0 ms] (40) AND (41) QDP (42) QDPOrderProof [EQUIVALENT, 70 ms] (43) QDP (44) TransformationProof [EQUIVALENT, 0 ms] (45) QDP (46) DependencyGraphProof [EQUIVALENT, 0 ms] (47) QDP (48) TransformationProof [EQUIVALENT, 14 ms] (49) QDP (50) DependencyGraphProof [EQUIVALENT, 0 ms] (51) QDP (52) TransformationProof [EQUIVALENT, 0 ms] (53) QDP (54) DependencyGraphProof [EQUIVALENT, 0 ms] (55) AND (56) QDP (57) QDPOrderProof [EQUIVALENT, 50 ms] (58) QDP (59) QDPOrderProof [EQUIVALENT, 160 ms] (60) QDP (61) QDPOrderProof [EQUIVALENT, 39 ms] (62) QDP (63) QDPOrderProof [EQUIVALENT, 80 ms] (64) QDP (65) QDPOrderProof [EQUIVALENT, 44 ms] (66) QDP (67) QDP (68) QDPOrderProof [EQUIVALENT, 101 ms] (69) QDP (70) QDPOrderProof [EQUIVALENT, 36 ms] (71) QDP (72) QDPOrderProof [EQUIVALENT, 34 ms] (73) QDP (74) QDPOrderProof [EQUIVALENT, 81 ms] (75) QDP (76) QDPOrderProof [EQUIVALENT, 62 ms] (77) QDP (78) QDPOrderProof [EQUIVALENT, 137 ms] (79) QDP (80) NonTerminationLoopProof [COMPLETE, 8159 ms] (81) NO (82) QDP (83) TransformationProof [EQUIVALENT, 0 ms] (84) QDP (85) TransformationProof [EQUIVALENT, 0 ms] (86) QDP (87) DependencyGraphProof [EQUIVALENT, 0 ms] (88) QDP (89) TransformationProof [EQUIVALENT, 0 ms] (90) QDP (91) DependencyGraphProof [EQUIVALENT, 0 ms] (92) QDP (93) TransformationProof [EQUIVALENT, 0 ms] (94) QDP (95) TransformationProof [EQUIVALENT, 0 ms] (96) QDP (97) DependencyGraphProof [EQUIVALENT, 0 ms] (98) QDP (99) TransformationProof [EQUIVALENT, 1 ms] (100) QDP (101) DependencyGraphProof [EQUIVALENT, 0 ms] (102) QDP (103) TransformationProof [EQUIVALENT, 0 ms] (104) QDP (105) DependencyGraphProof [EQUIVALENT, 0 ms] (106) QDP (107) TransformationProof [EQUIVALENT, 0 ms] (108) QDP (109) DependencyGraphProof [EQUIVALENT, 0 ms] (110) QDP (111) QDPOrderProof [EQUIVALENT, 33 ms] (112) QDP (113) QDPOrderProof [EQUIVALENT, 101 ms] (114) QDP (115) QDPOrderProof [EQUIVALENT, 112 ms] (116) QDP ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. ---------------------------------------- (1) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (2) Obligation: Q DP problem: The TRS P consists of the following rules: ZEROS -> CONS(0, n__zeros) ZEROS -> 0^1 U11^1(tt, L) -> S(length(activate(L))) U11^1(tt, L) -> LENGTH(activate(L)) U11^1(tt, L) -> ACTIVATE(L) U21^1(tt) -> NIL U31^1(tt, IL, M, N) -> CONS(activate(N), n__take(activate(M), activate(IL))) U31^1(tt, IL, M, N) -> ACTIVATE(N) U31^1(tt, IL, M, N) -> ACTIVATE(M) U31^1(tt, IL, M, N) -> ACTIVATE(IL) AND(tt, X) -> ACTIVATE(X) ISNAT(n__length(V1)) -> ISNATLIST(activate(V1)) ISNAT(n__length(V1)) -> ACTIVATE(V1) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNATILIST(V) -> ISNATLIST(activate(V)) ISNATILIST(V) -> ACTIVATE(V) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) ISNATILIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) ISNATLIST(n__take(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) ISNATLIST(n__take(V1, V2)) -> ISNAT(activate(V1)) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V2) LENGTH(nil) -> 0^1 LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) LENGTH(cons(N, L)) -> AND(isNatList(activate(L)), n__isNat(N)) LENGTH(cons(N, L)) -> ISNATLIST(activate(L)) LENGTH(cons(N, L)) -> ACTIVATE(L) TAKE(0, IL) -> U21^1(isNatIList(IL)) TAKE(0, IL) -> ISNATILIST(IL) TAKE(s(M), cons(N, IL)) -> U31^1(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) TAKE(s(M), cons(N, IL)) -> AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))) TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL)) TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL) TAKE(s(M), cons(N, IL)) -> ISNAT(M) ACTIVATE(n__zeros) -> ZEROS ACTIVATE(n__take(X1, X2)) -> TAKE(X1, X2) ACTIVATE(n__0) -> 0^1 ACTIVATE(n__length(X)) -> LENGTH(X) ACTIVATE(n__s(X)) -> S(X) ACTIVATE(n__cons(X1, X2)) -> CONS(X1, X2) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ACTIVATE(n__nil) -> NIL ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ACTIVATE(n__isNat(X)) -> ISNAT(X) ACTIVATE(n__and(X1, X2)) -> AND(X1, X2) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (3) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 12 less nodes. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__take(X1, X2)) -> TAKE(X1, X2) TAKE(0, IL) -> ISNATILIST(IL) ISNATILIST(V) -> ISNATLIST(activate(V)) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__length(X)) -> LENGTH(X) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, L) -> LENGTH(activate(L)) LENGTH(cons(N, L)) -> AND(isNatList(activate(L)), n__isNat(N)) LENGTH(cons(N, L)) -> ISNATLIST(activate(L)) ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNAT(n__length(V1)) -> ISNATLIST(activate(V1)) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(V) -> ACTIVATE(V) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) ACTIVATE(n__isNat(X)) -> ISNAT(X) ISNAT(n__length(V1)) -> ACTIVATE(V1) ACTIVATE(n__and(X1, X2)) -> AND(X1, X2) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNATLIST(n__take(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) ISNATLIST(n__take(V1, V2)) -> ISNAT(activate(V1)) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V2) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) ISNATILIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2) LENGTH(cons(N, L)) -> ACTIVATE(L) U11^1(tt, L) -> ACTIVATE(L) TAKE(s(M), cons(N, IL)) -> U31^1(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) U31^1(tt, IL, M, N) -> ACTIVATE(N) U31^1(tt, IL, M, N) -> ACTIVATE(M) U31^1(tt, IL, M, N) -> ACTIVATE(IL) TAKE(s(M), cons(N, IL)) -> AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))) TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL)) TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL) TAKE(s(M), cons(N, IL)) -> ISNAT(M) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(V) -> ISNATLIST(activate(V)) LENGTH(cons(N, L)) -> AND(isNatList(activate(L)), n__isNat(N)) LENGTH(cons(N, L)) -> ISNATLIST(activate(L)) ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNAT(n__length(V1)) -> ISNATLIST(activate(V1)) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATILIST(V) -> ACTIVATE(V) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V2) ISNAT(n__length(V1)) -> ACTIVATE(V1) ISNATLIST(n__take(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) ISNATLIST(n__take(V1, V2)) -> ISNAT(activate(V1)) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V1) ISNATLIST(n__take(V1, V2)) -> ACTIVATE(V2) ISNATILIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V1) ISNATILIST(n__cons(V1, V2)) -> ACTIVATE(V2) LENGTH(cons(N, L)) -> ACTIVATE(L) U11^1(tt, L) -> ACTIVATE(L) U31^1(tt, IL, M, N) -> ACTIVATE(N) U31^1(tt, IL, M, N) -> ACTIVATE(M) U31^1(tt, IL, M, N) -> ACTIVATE(IL) TAKE(s(M), cons(N, IL)) -> AND(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))) TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL) TAKE(s(M), cons(N, IL)) -> ISNAT(M) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = x_2 POL( ISNAT_1(x_1) ) = x_1 POL( ISNATILIST_1(x_1) ) = x_1 + 2 POL( ISNATLIST_1(x_1) ) = x_1 + 1 POL( LENGTH_1(x_1) ) = x_1 + 2 POL( U11^1_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( U31^1_4(x_1, ..., x_4) ) = 2x_2 + x_3 + x_4 + 2 POL( n__and_2(x_1, x_2) ) = x_2 POL( n__isNatIList_1(x_1) ) = x_1 + 2 POL( n__isNatList_1(x_1) ) = x_1 + 1 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__take_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( take_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = x_1 + 2 POL( length_1(x_1) ) = x_1 + 2 POL( n__s_1(x_1) ) = x_1 POL( s_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNatIList_1(x_1) ) = x_1 + 2 POL( isNatList_1(x_1) ) = x_1 + 1 POL( and_2(x_1, x_2) ) = x_2 POL( isNat_1(x_1) ) = x_1 POL( tt ) = 0 POL( n__isNat_1(x_1) ) = x_1 POL( n__nil ) = 0 POL( nil ) = 0 POL( U21_1(x_1) ) = 2 POL( U11_2(x_1, x_2) ) = x_2 + 2 POL( U31_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2x_4 + 2 POL( ACTIVATE_1(x_1) ) = x_1 POL( TAKE_2(x_1, x_2) ) = x_1 + 2x_2 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(X) -> n__isNat(X) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) take(X1, X2) -> n__take(X1, X2) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) cons(X1, X2) -> n__cons(X1, X2) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__take(X1, X2)) -> TAKE(X1, X2) TAKE(0, IL) -> ISNATILIST(IL) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__length(X)) -> LENGTH(X) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, L) -> LENGTH(activate(L)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ACTIVATE(n__isNat(X)) -> ISNAT(X) ACTIVATE(n__and(X1, X2)) -> AND(X1, X2) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) TAKE(s(M), cons(N, IL)) -> U31^1(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(tt, L) -> LENGTH(activate(L)) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U11^1(tt, L) -> LENGTH(activate(L)) at position [0] we obtained the following new rules [LPAR04]: (U11^1(tt, n__zeros) -> LENGTH(zeros),U11^1(tt, n__zeros) -> LENGTH(zeros)) (U11^1(tt, n__take(x0, x1)) -> LENGTH(take(x0, x1)),U11^1(tt, n__take(x0, x1)) -> LENGTH(take(x0, x1))) (U11^1(tt, n__0) -> LENGTH(0),U11^1(tt, n__0) -> LENGTH(0)) (U11^1(tt, n__length(x0)) -> LENGTH(length(x0)),U11^1(tt, n__length(x0)) -> LENGTH(length(x0))) (U11^1(tt, n__s(x0)) -> LENGTH(s(x0)),U11^1(tt, n__s(x0)) -> LENGTH(s(x0))) (U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(x0, x1)),U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(x0, x1))) (U11^1(tt, n__isNatIList(x0)) -> LENGTH(isNatIList(x0)),U11^1(tt, n__isNatIList(x0)) -> LENGTH(isNatIList(x0))) (U11^1(tt, n__nil) -> LENGTH(nil),U11^1(tt, n__nil) -> LENGTH(nil)) (U11^1(tt, n__isNatList(x0)) -> LENGTH(isNatList(x0)),U11^1(tt, n__isNatList(x0)) -> LENGTH(isNatList(x0))) (U11^1(tt, n__isNat(x0)) -> LENGTH(isNat(x0)),U11^1(tt, n__isNat(x0)) -> LENGTH(isNat(x0))) (U11^1(tt, n__and(x0, x1)) -> LENGTH(and(x0, x1)),U11^1(tt, n__and(x0, x1)) -> LENGTH(and(x0, x1))) (U11^1(tt, x0) -> LENGTH(x0),U11^1(tt, x0) -> LENGTH(x0)) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__zeros) -> LENGTH(zeros) U11^1(tt, n__take(x0, x1)) -> LENGTH(take(x0, x1)) U11^1(tt, n__0) -> LENGTH(0) U11^1(tt, n__length(x0)) -> LENGTH(length(x0)) U11^1(tt, n__s(x0)) -> LENGTH(s(x0)) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(x0, x1)) U11^1(tt, n__isNatIList(x0)) -> LENGTH(isNatIList(x0)) U11^1(tt, n__nil) -> LENGTH(nil) U11^1(tt, n__isNatList(x0)) -> LENGTH(isNatList(x0)) U11^1(tt, n__isNat(x0)) -> LENGTH(isNat(x0)) U11^1(tt, n__and(x0, x1)) -> LENGTH(and(x0, x1)) U11^1(tt, x0) -> LENGTH(x0) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U11^1(tt, n__zeros) -> LENGTH(zeros) at position [0] we obtained the following new rules [LPAR04]: (U11^1(tt, n__zeros) -> LENGTH(cons(0, n__zeros)),U11^1(tt, n__zeros) -> LENGTH(cons(0, n__zeros))) (U11^1(tt, n__zeros) -> LENGTH(n__zeros),U11^1(tt, n__zeros) -> LENGTH(n__zeros)) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__take(x0, x1)) -> LENGTH(take(x0, x1)) U11^1(tt, n__0) -> LENGTH(0) U11^1(tt, n__length(x0)) -> LENGTH(length(x0)) U11^1(tt, n__s(x0)) -> LENGTH(s(x0)) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(x0, x1)) U11^1(tt, n__isNatIList(x0)) -> LENGTH(isNatIList(x0)) U11^1(tt, n__nil) -> LENGTH(nil) U11^1(tt, n__isNatList(x0)) -> LENGTH(isNatList(x0)) U11^1(tt, n__isNat(x0)) -> LENGTH(isNat(x0)) U11^1(tt, n__and(x0, x1)) -> LENGTH(and(x0, x1)) U11^1(tt, x0) -> LENGTH(x0) U11^1(tt, n__zeros) -> LENGTH(cons(0, n__zeros)) U11^1(tt, n__zeros) -> LENGTH(n__zeros) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(tt, n__take(x0, x1)) -> LENGTH(take(x0, x1)) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__0) -> LENGTH(0) U11^1(tt, n__length(x0)) -> LENGTH(length(x0)) U11^1(tt, n__s(x0)) -> LENGTH(s(x0)) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(x0, x1)) U11^1(tt, n__isNatIList(x0)) -> LENGTH(isNatIList(x0)) U11^1(tt, n__nil) -> LENGTH(nil) U11^1(tt, n__isNatList(x0)) -> LENGTH(isNatList(x0)) U11^1(tt, n__isNat(x0)) -> LENGTH(isNat(x0)) U11^1(tt, n__and(x0, x1)) -> LENGTH(and(x0, x1)) U11^1(tt, x0) -> LENGTH(x0) U11^1(tt, n__zeros) -> LENGTH(cons(0, n__zeros)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U11^1(tt, n__0) -> LENGTH(0) at position [0] we obtained the following new rules [LPAR04]: (U11^1(tt, n__0) -> LENGTH(n__0),U11^1(tt, n__0) -> LENGTH(n__0)) ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(tt, n__take(x0, x1)) -> LENGTH(take(x0, x1)) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__length(x0)) -> LENGTH(length(x0)) U11^1(tt, n__s(x0)) -> LENGTH(s(x0)) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(x0, x1)) U11^1(tt, n__isNatIList(x0)) -> LENGTH(isNatIList(x0)) U11^1(tt, n__nil) -> LENGTH(nil) U11^1(tt, n__isNatList(x0)) -> LENGTH(isNatList(x0)) U11^1(tt, n__isNat(x0)) -> LENGTH(isNat(x0)) U11^1(tt, n__and(x0, x1)) -> LENGTH(and(x0, x1)) U11^1(tt, x0) -> LENGTH(x0) U11^1(tt, n__zeros) -> LENGTH(cons(0, n__zeros)) U11^1(tt, n__0) -> LENGTH(n__0) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__take(x0, x1)) -> LENGTH(take(x0, x1)) U11^1(tt, n__length(x0)) -> LENGTH(length(x0)) U11^1(tt, n__s(x0)) -> LENGTH(s(x0)) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(x0, x1)) U11^1(tt, n__isNatIList(x0)) -> LENGTH(isNatIList(x0)) U11^1(tt, n__nil) -> LENGTH(nil) U11^1(tt, n__isNatList(x0)) -> LENGTH(isNatList(x0)) U11^1(tt, n__isNat(x0)) -> LENGTH(isNat(x0)) U11^1(tt, n__and(x0, x1)) -> LENGTH(and(x0, x1)) U11^1(tt, x0) -> LENGTH(x0) U11^1(tt, n__zeros) -> LENGTH(cons(0, n__zeros)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U11^1(tt, n__s(x0)) -> LENGTH(s(x0)) at position [0] we obtained the following new rules [LPAR04]: (U11^1(tt, n__s(x0)) -> LENGTH(n__s(x0)),U11^1(tt, n__s(x0)) -> LENGTH(n__s(x0))) ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__take(x0, x1)) -> LENGTH(take(x0, x1)) U11^1(tt, n__length(x0)) -> LENGTH(length(x0)) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(x0, x1)) U11^1(tt, n__isNatIList(x0)) -> LENGTH(isNatIList(x0)) U11^1(tt, n__nil) -> LENGTH(nil) U11^1(tt, n__isNatList(x0)) -> LENGTH(isNatList(x0)) U11^1(tt, n__isNat(x0)) -> LENGTH(isNat(x0)) U11^1(tt, n__and(x0, x1)) -> LENGTH(and(x0, x1)) U11^1(tt, x0) -> LENGTH(x0) U11^1(tt, n__zeros) -> LENGTH(cons(0, n__zeros)) U11^1(tt, n__s(x0)) -> LENGTH(n__s(x0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(tt, n__take(x0, x1)) -> LENGTH(take(x0, x1)) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__length(x0)) -> LENGTH(length(x0)) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(x0, x1)) U11^1(tt, n__isNatIList(x0)) -> LENGTH(isNatIList(x0)) U11^1(tt, n__nil) -> LENGTH(nil) U11^1(tt, n__isNatList(x0)) -> LENGTH(isNatList(x0)) U11^1(tt, n__isNat(x0)) -> LENGTH(isNat(x0)) U11^1(tt, n__and(x0, x1)) -> LENGTH(and(x0, x1)) U11^1(tt, x0) -> LENGTH(x0) U11^1(tt, n__zeros) -> LENGTH(cons(0, n__zeros)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule U11^1(tt, n__nil) -> LENGTH(nil) at position [0] we obtained the following new rules [LPAR04]: (U11^1(tt, n__nil) -> LENGTH(n__nil),U11^1(tt, n__nil) -> LENGTH(n__nil)) ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(tt, n__take(x0, x1)) -> LENGTH(take(x0, x1)) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__length(x0)) -> LENGTH(length(x0)) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(x0, x1)) U11^1(tt, n__isNatIList(x0)) -> LENGTH(isNatIList(x0)) U11^1(tt, n__isNatList(x0)) -> LENGTH(isNatList(x0)) U11^1(tt, n__isNat(x0)) -> LENGTH(isNat(x0)) U11^1(tt, n__and(x0, x1)) -> LENGTH(and(x0, x1)) U11^1(tt, x0) -> LENGTH(x0) U11^1(tt, n__zeros) -> LENGTH(cons(0, n__zeros)) U11^1(tt, n__nil) -> LENGTH(n__nil) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__take(x0, x1)) -> LENGTH(take(x0, x1)) U11^1(tt, n__length(x0)) -> LENGTH(length(x0)) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(x0, x1)) U11^1(tt, n__isNatIList(x0)) -> LENGTH(isNatIList(x0)) U11^1(tt, n__isNatList(x0)) -> LENGTH(isNatList(x0)) U11^1(tt, n__isNat(x0)) -> LENGTH(isNat(x0)) U11^1(tt, n__and(x0, x1)) -> LENGTH(and(x0, x1)) U11^1(tt, x0) -> LENGTH(x0) U11^1(tt, n__zeros) -> LENGTH(cons(0, n__zeros)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U11^1(tt, n__length(x0)) -> LENGTH(length(x0)) U11^1(tt, n__isNatIList(x0)) -> LENGTH(isNatIList(x0)) U11^1(tt, n__isNatList(x0)) -> LENGTH(isNatList(x0)) U11^1(tt, n__isNat(x0)) -> LENGTH(isNat(x0)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( LENGTH_1(x_1) ) = x_1 POL( U11^1_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__take_2(x_1, x_2) ) = x_2 POL( take_2(x_1, x_2) ) = x_2 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2x_1 + 2 POL( length_1(x_1) ) = 2x_1 + 2 POL( n__s_1(x_1) ) = x_1 POL( s_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( n__isNatIList_1(x_1) ) = 2 POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2 POL( and_2(x_1, x_2) ) = x_2 POL( isNat_1(x_1) ) = x_1 + 2 POL( n__isNatList_1(x_1) ) = 2 POL( tt ) = 2 POL( n__isNat_1(x_1) ) = x_1 + 2 POL( n__and_2(x_1, x_2) ) = x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U21_1(x_1) ) = 0 POL( U31_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U11_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__nil) -> nil activate(X) -> X isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) length(X) -> n__length(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) isNat(n__0) -> tt isNat(X) -> n__isNat(X) 0 -> n__0 U21(tt) -> nil U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__take(x0, x1)) -> LENGTH(take(x0, x1)) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(x0, x1)) U11^1(tt, n__and(x0, x1)) -> LENGTH(and(x0, x1)) U11^1(tt, x0) -> LENGTH(x0) U11^1(tt, n__zeros) -> LENGTH(cons(0, n__zeros)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. U11^1(tt, n__and(x0, x1)) -> LENGTH(and(x0, x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(LENGTH(x_1)) = [[3A]] + [[0A]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(U11^1(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(and(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(isNatList(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(activate(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(n__isNat(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(tt) = [[3A]] >>> <<< POL(n__take(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(take(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(n__cons(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__and(x_1, x_2)) = [[3A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(n__zeros) = [[1A]] >>> <<< POL(0) = [[0A]] >>> <<< POL(zeros) = [[2A]] >>> <<< POL(n__0) = [[0A]] >>> <<< POL(n__length(x_1)) = [[4A]] + [[0A]] * x_1 >>> <<< POL(length(x_1)) = [[4A]] + [[0A]] * x_1 >>> <<< POL(n__s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(n__isNatIList(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(isNat(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(n__isNatList(x_1)) = [[5A]] + [[-I]] * x_1 >>> <<< POL(n__nil) = [[0A]] >>> <<< POL(nil) = [[2A]] >>> <<< POL(U21(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3, x_4)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 + [[-I]] * x_4 >>> <<< POL(U11(x_1, x_2)) = [[4A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__nil) -> nil activate(X) -> X isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) cons(X1, X2) -> n__cons(X1, X2) 0 -> n__0 isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil isNat(n__0) -> tt isNat(X) -> n__isNat(X) length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__take(x0, x1)) -> LENGTH(take(x0, x1)) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(x0, x1)) U11^1(tt, x0) -> LENGTH(x0) U11^1(tt, n__zeros) -> LENGTH(cons(0, n__zeros)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: TAKE(0, IL) -> ISNATILIST(IL) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__take(X1, X2)) -> TAKE(X1, X2) TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ACTIVATE(n__isNat(X)) -> ISNAT(X) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ACTIVATE(n__and(X1, X2)) -> AND(X1, X2) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVATE(n__take(X1, X2)) -> TAKE(X1, X2) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = x_1 + 2x_2 POL( ISNAT_1(x_1) ) = 2x_1 POL( ISNATILIST_1(x_1) ) = 2x_1 + 2 POL( n__isNatIList_1(x_1) ) = x_1 + 1 POL( n__isNatList_1(x_1) ) = x_1 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 2 POL( zeros ) = 2 POL( n__take_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( take_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = x_1 POL( length_1(x_1) ) = x_1 POL( n__s_1(x_1) ) = x_1 POL( s_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = x_1 + x_2 POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( isNatIList_1(x_1) ) = x_1 + 1 POL( isNatList_1(x_1) ) = x_1 POL( and_2(x_1, x_2) ) = x_1 + x_2 POL( isNat_1(x_1) ) = x_1 POL( tt ) = 0 POL( n__isNat_1(x_1) ) = x_1 POL( n__and_2(x_1, x_2) ) = x_1 + x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U21_1(x_1) ) = max{0, -2} POL( U11_2(x_1, x_2) ) = x_2 POL( U31_4(x_1, ..., x_4) ) = x_2 + x_3 + x_4 + 2 POL( TAKE_2(x_1, x_2) ) = 2x_2 + 2 POL( ACTIVATE_1(x_1) ) = 2x_1 POL( ISNATLIST_1(x_1) ) = 2x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(X) -> n__isNat(X) take(X1, X2) -> n__take(X1, X2) take(0, IL) -> U21(isNatIList(IL)) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) cons(X1, X2) -> n__cons(X1, X2) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: TAKE(0, IL) -> ISNATILIST(IL) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ACTIVATE(n__isNat(X)) -> ISNAT(X) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ACTIVATE(n__and(X1, X2)) -> AND(X1, X2) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ACTIVATE(n__isNat(X)) -> ISNAT(X) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ACTIVATE(n__and(X1, X2)) -> AND(X1, X2) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVATE(n__isNat(X)) -> ISNAT(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = x_2 POL( ISNAT_1(x_1) ) = x_1 POL( n__isNatIList_1(x_1) ) = x_1 + 1 POL( n__isNatList_1(x_1) ) = x_1 + 1 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__take_2(x_1, x_2) ) = x_2 POL( take_2(x_1, x_2) ) = x_2 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 2x_1 + 1 POL( length_1(x_1) ) = 2x_1 + 1 POL( n__s_1(x_1) ) = x_1 POL( s_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( isNatIList_1(x_1) ) = x_1 + 1 POL( isNatList_1(x_1) ) = x_1 + 1 POL( and_2(x_1, x_2) ) = x_2 POL( isNat_1(x_1) ) = x_1 + 1 POL( tt ) = 1 POL( n__isNat_1(x_1) ) = x_1 + 1 POL( n__and_2(x_1, x_2) ) = x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U21_1(x_1) ) = 0 POL( U11_2(x_1, x_2) ) = x_1 + 2x_2 POL( U31_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( ACTIVATE_1(x_1) ) = x_1 POL( ISNATILIST_1(x_1) ) = x_1 + 1 POL( ISNATLIST_1(x_1) ) = x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(X) -> n__isNat(X) take(X1, X2) -> n__take(X1, X2) take(0, IL) -> U21(isNatIList(IL)) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) cons(X1, X2) -> n__cons(X1, X2) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ISNAT(n__s(V1)) -> ISNAT(activate(V1)) ISNAT(n__s(V1)) -> ACTIVATE(V1) ACTIVATE(n__and(X1, X2)) -> AND(X1, X2) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. ---------------------------------------- (40) Complex Obligation (AND) ---------------------------------------- (41) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ACTIVATE(n__and(X1, X2)) -> AND(X1, X2) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (42) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVATE(n__and(X1, X2)) -> AND(X1, X2) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = x_1 + x_2 POL( n__isNatIList_1(x_1) ) = max{0, -2} POL( n__isNatList_1(x_1) ) = 0 POL( activate_1(x_1) ) = 2x_1 + 2 POL( n__zeros ) = 2 POL( zeros ) = 2 POL( n__take_2(x_1, x_2) ) = 2x_2 + 2 POL( take_2(x_1, x_2) ) = 2x_2 + 2 POL( n__0 ) = 1 POL( 0 ) = 1 POL( n__length_1(x_1) ) = x_1 POL( length_1(x_1) ) = x_1 + 2 POL( n__s_1(x_1) ) = 0 POL( s_1(x_1) ) = 0 POL( n__cons_2(x_1, x_2) ) = 2 POL( cons_2(x_1, x_2) ) = 2 POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2 POL( and_2(x_1, x_2) ) = x_1 + 2x_2 POL( isNat_1(x_1) ) = 2 POL( tt ) = 2 POL( n__isNat_1(x_1) ) = 2 POL( n__and_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1} POL( n__nil ) = 0 POL( nil ) = 0 POL( U21_1(x_1) ) = max{0, -2} POL( U11_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U31_4(x_1, ..., x_4) ) = 2 POL( ACTIVATE_1(x_1) ) = x_1 + 2 POL( ISNATILIST_1(x_1) ) = 2 POL( ISNATLIST_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(X) -> n__isNat(X) take(X1, X2) -> n__take(X1, X2) take(0, IL) -> U21(isNatIList(IL)) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) cons(X1, X2) -> n__cons(X1, X2) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (43) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (44) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) at position [0] we obtained the following new rules [LPAR04]: (ISNATILIST(n__cons(y0, y1)) -> AND(n__isNat(activate(y0)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(y0, y1)) -> AND(n__isNat(activate(y0)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))),ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1)))) ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) ISNATILIST(n__cons(y0, y1)) -> AND(n__isNat(activate(y0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (46) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNATLIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatList(activate(V2))) at position [0] we obtained the following new rules [LPAR04]: (ISNATLIST(n__cons(y0, y1)) -> AND(n__isNat(activate(y0)), n__isNatList(activate(y1))),ISNATLIST(n__cons(y0, y1)) -> AND(n__isNat(activate(y0)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))),ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1)))) ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(y0, y1)) -> AND(n__isNat(activate(y0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (50) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (51) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (52) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule AND(tt, X) -> ACTIVATE(X) we obtained the following new rules [LPAR04]: (AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)),AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3))) (AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)),AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3))) ---------------------------------------- (53) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (54) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (55) Complex Obligation (AND) ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = max{0, x_2 - 2} POL( n__isNatList_1(x_1) ) = 2x_1 + 2 POL( zeros ) = 0 POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( 0 ) = 0 POL( n__zeros ) = 0 POL( isNat_1(x_1) ) = 2x_1 + 1 POL( n__0 ) = 0 POL( tt ) = 1 POL( activate_1(x_1) ) = x_1 POL( n__isNatIList_1(x_1) ) = 2x_1 + 2 POL( isNatIList_1(x_1) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = 2x_1 + 2 POL( n__cons_2(x_1, x_2) ) = x_1 + x_2 POL( and_2(x_1, x_2) ) = x_2 POL( n__take_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( n__isNat_1(x_1) ) = 2x_1 + 1 POL( n__length_1(x_1) ) = x_1 + 2 POL( n__s_1(x_1) ) = x_1 POL( n__and_2(x_1, x_2) ) = x_2 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( length_1(x_1) ) = x_1 + 2 POL( s_1(x_1) ) = x_1 POL( n__nil ) = 0 POL( nil ) = 0 POL( U21_1(x_1) ) = max{0, x_1 - 2} POL( U31_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 POL( U11_2(x_1, x_2) ) = x_2 + 2 POL( ISNATLIST_1(x_1) ) = 2x_1 POL( ACTIVATE_1(x_1) ) = max{0, x_1 - 2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: zeros -> cons(0, n__zeros) zeros -> n__zeros isNat(n__0) -> tt activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) U21(tt) -> nil U11(tt, L) -> s(length(activate(L))) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATLIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = 2x_1 + x_2 POL( n__isNatList_1(x_1) ) = 2x_1 POL( zeros ) = 1 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( 0 ) = 0 POL( n__zeros ) = 1 POL( isNat_1(x_1) ) = x_1 + 1 POL( n__0 ) = 0 POL( tt ) = 1 POL( activate_1(x_1) ) = x_1 POL( n__isNatIList_1(x_1) ) = 2x_1 POL( isNatIList_1(x_1) ) = 2x_1 POL( isNatList_1(x_1) ) = 2x_1 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( and_2(x_1, x_2) ) = x_2 POL( n__take_2(x_1, x_2) ) = 2x_2 POL( n__isNat_1(x_1) ) = x_1 + 1 POL( n__length_1(x_1) ) = 2x_1 POL( n__s_1(x_1) ) = x_1 POL( n__and_2(x_1, x_2) ) = x_2 POL( take_2(x_1, x_2) ) = 2x_2 POL( length_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = x_1 POL( n__nil ) = 1 POL( nil ) = 1 POL( U21_1(x_1) ) = x_1 POL( U31_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( U11_2(x_1, x_2) ) = 2x_2 POL( ISNATLIST_1(x_1) ) = 2x_1 + 2 POL( ACTIVATE_1(x_1) ) = x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: zeros -> cons(0, n__zeros) zeros -> n__zeros isNat(n__0) -> tt activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) 0 -> n__0 s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil and(X1, X2) -> n__and(X1, X2) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATLIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = max{0, x_2 - 1} POL( n__isNatList_1(x_1) ) = 2x_1 + 2 POL( take_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( 0 ) = 0 POL( U21_1(x_1) ) = 2 POL( isNatIList_1(x_1) ) = 2x_1 + 2 POL( s_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( U31_4(x_1, ..., x_4) ) = x_2 + x_3 + x_4 + 2 POL( and_2(x_1, x_2) ) = x_2 POL( activate_1(x_1) ) = x_1 POL( n__and_2(x_1, x_2) ) = x_2 POL( isNat_1(x_1) ) = x_1 POL( n__isNat_1(x_1) ) = x_1 POL( n__take_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( n__0 ) = 0 POL( tt ) = 0 POL( n__isNatIList_1(x_1) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = 2x_1 + 2 POL( n__cons_2(x_1, x_2) ) = x_1 + x_2 POL( n__length_1(x_1) ) = 2x_1 + 2 POL( n__s_1(x_1) ) = x_1 POL( n__zeros ) = 1 POL( zeros ) = 1 POL( length_1(x_1) ) = 2x_1 + 2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U11_2(x_1, x_2) ) = 2x_2 + 2 POL( ACTIVATE_1(x_1) ) = max{0, x_1 - 1} POL( ISNATLIST_1(x_1) ) = 2x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) isNat(n__0) -> tt activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X 0 -> n__0 s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) and(X1, X2) -> n__and(X1, X2) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(AND(x_1, x_2)) = [[1]] + [[0, 0]] * x_1 + [[1, 0]] * x_2 >>> <<< POL(tt) = [[0], [0]] >>> <<< POL(n__isNatList(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(ACTIVATE(x_1)) = [[1]] + [[1, 0]] * x_1 >>> <<< POL(ISNATLIST(x_1)) = [[1]] + [[1, 0]] * x_1 >>> <<< POL(n__cons(x_1, x_2)) = [[0], [1]] + [[1, 1], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 >>> <<< POL(n__0) = [[0], [0]] >>> <<< POL(isNat(x_1)) = [[1], [0]] + [[1, 1], [1, 0]] * x_1 >>> <<< POL(0) = [[0], [0]] >>> <<< POL(activate(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(n__s(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(s(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[0], [1]] + [[1, 1], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 >>> <<< POL(n__and(x_1, x_2)) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 + [[1, 1], [0, 1]] * x_2 >>> <<< POL(and(x_1, x_2)) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 + [[1, 1], [0, 1]] * x_2 >>> <<< POL(n__isNatIList(x_1)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(isNatList(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(n__take(x_1, x_2)) = [[1], [1]] + [[0, 1], [1, 0]] * x_1 + [[1, 0], [1, 1]] * x_2 >>> <<< POL(n__isNat(x_1)) = [[1], [0]] + [[1, 1], [1, 0]] * x_1 >>> <<< POL(n__length(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(n__zeros) = [[1], [1]] >>> <<< POL(zeros) = [[1], [1]] >>> <<< POL(take(x_1, x_2)) = [[1], [1]] + [[0, 1], [1, 0]] * x_1 + [[1, 0], [1, 1]] * x_2 >>> <<< POL(length(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(n__nil) = [[0], [0]] >>> <<< POL(nil) = [[0], [0]] >>> <<< POL(U21(x_1)) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U11(x_1, x_2)) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 >>> <<< POL(U31(x_1, x_2, x_3, x_4)) = [[1], [1]] + [[0, 0], [0, 0]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[0, 1], [0, 0]] * x_3 + [[1, 1], [0, 0]] * x_4 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0 -> n__0 isNat(n__0) -> tt activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) and(X1, X2) -> n__and(X1, X2) take(X1, X2) -> n__take(X1, X2) take(0, IL) -> U21(isNatIList(IL)) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(AND(x_1, x_2)) = [[0]] + [[0, 0]] * x_1 + [[0, 1]] * x_2 >>> <<< POL(tt) = [[1], [1]] >>> <<< POL(n__isNatList(x_1)) = [[1], [1]] + [[0, 0], [1, 1]] * x_1 >>> <<< POL(ACTIVATE(x_1)) = [[0]] + [[0, 1]] * x_1 >>> <<< POL(ISNATLIST(x_1)) = [[1]] + [[1, 1]] * x_1 >>> <<< POL(n__cons(x_1, x_2)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[0, 1], [1, 0]] * x_2 >>> <<< POL(n__0) = [[0], [1]] >>> <<< POL(isNat(x_1)) = [[1], [1]] + [[0, 0], [0, 1]] * x_1 >>> <<< POL(0) = [[0], [1]] >>> <<< POL(activate(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(n__s(x_1)) = [[1], [0]] + [[0, 0], [0, 1]] * x_1 >>> <<< POL(s(x_1)) = [[1], [0]] + [[0, 0], [0, 1]] * x_1 >>> <<< POL(n__and(x_1, x_2)) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(and(x_1, x_2)) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(n__isNatIList(x_1)) = [[1], [1]] + [[0, 0], [1, 1]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[1], [1]] + [[0, 0], [1, 1]] * x_1 >>> <<< POL(isNatList(x_1)) = [[1], [1]] + [[0, 0], [1, 1]] * x_1 >>> <<< POL(n__take(x_1, x_2)) = [[1], [1]] + [[0, 1], [0, 1]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(n__isNat(x_1)) = [[1], [1]] + [[0, 0], [0, 1]] * x_1 >>> <<< POL(n__length(x_1)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(n__zeros) = [[1], [1]] >>> <<< POL(zeros) = [[1], [1]] >>> <<< POL(take(x_1, x_2)) = [[1], [1]] + [[0, 1], [0, 1]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(length(x_1)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[0, 1], [1, 0]] * x_2 >>> <<< POL(n__nil) = [[1], [0]] >>> <<< POL(nil) = [[1], [0]] >>> <<< POL(U21(x_1)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U11(x_1, x_2)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [1, 1]] * x_2 >>> <<< POL(U31(x_1, x_2, x_3, x_4)) = [[1], [1]] + [[0, 0], [0, 0]] * x_1 + [[0, 1], [1, 0]] * x_2 + [[0, 1], [0, 1]] * x_3 + [[1, 0], [0, 0]] * x_4 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0 -> n__0 isNat(n__0) -> tt activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X s(X) -> n__s(X) and(X1, X2) -> n__and(X1, X2) take(X1, X2) -> n__take(X1, X2) take(0, IL) -> U21(isNatIList(IL)) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) cons(X1, X2) -> n__cons(X1, X2) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, n__isNatList(y_3)) -> ACTIVATE(n__isNatList(y_3)) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ISNATLIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (68) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNat(x0), y1)) -> AND(isNat(isNat(x0)), n__isNatIList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( n__isNatIList_1(x_1) ) = x_1 + 2 POL( zeros ) = 0 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( 0 ) = 0 POL( n__zeros ) = 0 POL( isNat_1(x_1) ) = x_1 + 1 POL( n__0 ) = 0 POL( tt ) = 0 POL( activate_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = x_1 + 2 POL( isNatList_1(x_1) ) = x_1 + 2 POL( n__cons_2(x_1, x_2) ) = x_1 + 2x_2 POL( and_2(x_1, x_2) ) = x_2 POL( n__isNatList_1(x_1) ) = x_1 + 2 POL( n__take_2(x_1, x_2) ) = x_2 POL( n__isNat_1(x_1) ) = x_1 + 1 POL( n__length_1(x_1) ) = x_1 + 1 POL( n__s_1(x_1) ) = x_1 POL( n__and_2(x_1, x_2) ) = x_2 POL( take_2(x_1, x_2) ) = x_2 POL( length_1(x_1) ) = x_1 + 1 POL( s_1(x_1) ) = x_1 POL( n__nil ) = 0 POL( nil ) = 0 POL( U21_1(x_1) ) = max{0, -1} POL( U31_4(x_1, ..., x_4) ) = 2x_2 + x_4 POL( U11_2(x_1, x_2) ) = 2x_2 + 1 POL( ISNATILIST_1(x_1) ) = 2x_1 + 2 POL( ACTIVATE_1(x_1) ) = max{0, 2x_1 - 2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: zeros -> cons(0, n__zeros) zeros -> n__zeros isNat(n__0) -> tt activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) U21(tt) -> nil U11(tt, L) -> s(length(activate(L))) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) ---------------------------------------- (69) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (70) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(n__cons(n__zeros, y1)) -> AND(isNat(zeros), n__isNatIList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2} POL( n__isNatIList_1(x_1) ) = 2x_1 + 2 POL( zeros ) = 2 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( 0 ) = 0 POL( n__zeros ) = 2 POL( isNat_1(x_1) ) = 2x_1 + 2 POL( n__0 ) = 0 POL( tt ) = 2 POL( activate_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = 2x_1 + 2 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( and_2(x_1, x_2) ) = x_2 POL( n__isNatList_1(x_1) ) = 2x_1 + 2 POL( n__take_2(x_1, x_2) ) = x_2 POL( n__isNat_1(x_1) ) = 2x_1 + 2 POL( n__length_1(x_1) ) = 2x_1 + 2 POL( n__s_1(x_1) ) = x_1 POL( n__and_2(x_1, x_2) ) = x_2 POL( take_2(x_1, x_2) ) = x_2 POL( length_1(x_1) ) = 2x_1 + 2 POL( s_1(x_1) ) = x_1 POL( n__nil ) = 0 POL( nil ) = 0 POL( U21_1(x_1) ) = max{0, -2} POL( U31_4(x_1, ..., x_4) ) = x_2 + 2x_4 POL( U11_2(x_1, x_2) ) = x_1 + 2x_2 POL( ISNATILIST_1(x_1) ) = 2x_1 + 2 POL( ACTIVATE_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: zeros -> cons(0, n__zeros) zeros -> n__zeros isNat(n__0) -> tt activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) 0 -> n__0 s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil and(X1, X2) -> n__and(X1, X2) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) ---------------------------------------- (71) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (72) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(x0, x1)), n__isNatIList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = x_1 + x_2 + 1 POL( n__isNatIList_1(x_1) ) = 2x_1 POL( take_2(x_1, x_2) ) = x_1 + 2x_2 + 1 POL( 0 ) = 0 POL( U21_1(x_1) ) = 0 POL( isNatIList_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( U31_4(x_1, ..., x_4) ) = x_1 + 2x_2 + x_3 + x_4 + 1 POL( and_2(x_1, x_2) ) = x_2 POL( activate_1(x_1) ) = x_1 POL( n__and_2(x_1, x_2) ) = x_2 POL( isNat_1(x_1) ) = x_1 POL( n__isNat_1(x_1) ) = x_1 POL( n__take_2(x_1, x_2) ) = x_1 + 2x_2 + 1 POL( n__0 ) = 0 POL( tt ) = 0 POL( isNatList_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = x_1 + x_2 POL( n__isNatList_1(x_1) ) = x_1 POL( n__length_1(x_1) ) = x_1 POL( n__s_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( length_1(x_1) ) = x_1 POL( n__nil ) = 0 POL( nil ) = 0 POL( U11_2(x_1, x_2) ) = x_2 POL( ACTIVATE_1(x_1) ) = x_1 + 1 POL( ISNATILIST_1(x_1) ) = 2x_1 + 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) isNat(n__0) -> tt activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X 0 -> n__0 s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil and(X1, X2) -> n__and(X1, X2) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) zeros -> cons(0, n__zeros) zeros -> n__zeros ---------------------------------------- (73) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (74) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(n__cons(n__nil, y1)) -> AND(isNat(nil), n__isNatIList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( AND_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( n__isNatIList_1(x_1) ) = x_1 POL( 0 ) = 0 POL( n__0 ) = 0 POL( isNat_1(x_1) ) = x_1 + 1 POL( tt ) = 1 POL( activate_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( and_2(x_1, x_2) ) = x_2 POL( n__isNatList_1(x_1) ) = x_1 POL( n__take_2(x_1, x_2) ) = x_2 + 2 POL( n__isNat_1(x_1) ) = x_1 + 1 POL( n__length_1(x_1) ) = x_1 POL( n__s_1(x_1) ) = x_1 POL( n__and_2(x_1, x_2) ) = x_2 POL( n__zeros ) = 1 POL( zeros ) = 1 POL( take_2(x_1, x_2) ) = x_2 + 2 POL( length_1(x_1) ) = x_1 POL( s_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( n__nil ) = 1 POL( nil ) = 1 POL( U21_1(x_1) ) = 2 POL( U11_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( U31_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( ACTIVATE_1(x_1) ) = 2x_1 + 2 POL( ISNATILIST_1(x_1) ) = 2x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0 -> n__0 isNat(n__0) -> tt activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) nil -> n__nil and(X1, X2) -> n__and(X1, X2) take(X1, X2) -> n__take(X1, X2) take(0, IL) -> U21(isNatIList(IL)) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) zeros -> cons(0, n__zeros) zeros -> n__zeros ---------------------------------------- (75) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (76) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(x0)), n__isNatIList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(AND(x_1, x_2)) = [[1]] + [[0, 0]] * x_1 + [[0, 1]] * x_2 >>> <<< POL(tt) = [[0], [0]] >>> <<< POL(n__isNatIList(x_1)) = [[1], [0]] + [[0, 1], [1, 1]] * x_1 >>> <<< POL(ACTIVATE(x_1)) = [[0]] + [[1, 0]] * x_1 >>> <<< POL(ISNATILIST(x_1)) = [[1]] + [[0, 1]] * x_1 >>> <<< POL(n__cons(x_1, x_2)) = [[0], [0]] + [[1, 1], [0, 1]] * x_1 + [[0, 0], [1, 1]] * x_2 >>> <<< POL(n__0) = [[0], [0]] >>> <<< POL(isNat(x_1)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 >>> <<< POL(0) = [[0], [0]] >>> <<< POL(activate(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(n__s(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(s(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[0], [0]] + [[1, 1], [0, 1]] * x_1 + [[0, 0], [1, 1]] * x_2 >>> <<< POL(n__and(x_1, x_2)) = [[0], [0]] + [[0, 0], [1, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(and(x_1, x_2)) = [[0], [0]] + [[0, 0], [1, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(isNatIList(x_1)) = [[1], [0]] + [[0, 1], [1, 1]] * x_1 >>> <<< POL(isNatList(x_1)) = [[0], [0]] + [[0, 1], [1, 1]] * x_1 >>> <<< POL(n__isNatList(x_1)) = [[0], [0]] + [[0, 1], [1, 1]] * x_1 >>> <<< POL(n__take(x_1, x_2)) = [[0], [1]] + [[0, 0], [1, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(n__isNat(x_1)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 >>> <<< POL(n__length(x_1)) = [[1], [1]] + [[1, 1], [1, 0]] * x_1 >>> <<< POL(n__zeros) = [[0], [1]] >>> <<< POL(zeros) = [[0], [1]] >>> <<< POL(take(x_1, x_2)) = [[0], [1]] + [[0, 0], [1, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(length(x_1)) = [[1], [1]] + [[1, 1], [1, 0]] * x_1 >>> <<< POL(n__nil) = [[0], [1]] >>> <<< POL(nil) = [[0], [1]] >>> <<< POL(U21(x_1)) = [[0], [1]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U11(x_1, x_2)) = [[1], [1]] + [[0, 0], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 >>> <<< POL(U31(x_1, x_2, x_3, x_4)) = [[0], [1]] + [[0, 0], [0, 0]] * x_1 + [[0, 0], [1, 1]] * x_2 + [[0, 0], [1, 0]] * x_3 + [[1, 1], [0, 1]] * x_4 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0 -> n__0 isNat(n__0) -> tt activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) and(X1, X2) -> n__and(X1, X2) take(X1, X2) -> n__take(X1, X2) take(0, IL) -> U21(isNatIList(IL)) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (77) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (78) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(x0, x1)), n__isNatIList(activate(y1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(AND(x_1, x_2)) = [[0]] + [[0, 0]] * x_1 + [[1, 0]] * x_2 >>> <<< POL(tt) = [[0], [1]] >>> <<< POL(n__isNatIList(x_1)) = [[0], [1]] + [[1, 0], [1, 0]] * x_1 >>> <<< POL(ACTIVATE(x_1)) = [[0]] + [[1, 0]] * x_1 >>> <<< POL(ISNATILIST(x_1)) = [[0]] + [[1, 0]] * x_1 >>> <<< POL(n__cons(x_1, x_2)) = [[0], [1]] + [[1, 1], [0, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 >>> <<< POL(n__0) = [[0], [0]] >>> <<< POL(isNat(x_1)) = [[0], [1]] + [[1, 0], [1, 0]] * x_1 >>> <<< POL(0) = [[0], [0]] >>> <<< POL(activate(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[0], [1]] + [[1, 1], [0, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 >>> <<< POL(n__and(x_1, x_2)) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(and(x_1, x_2)) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(isNatIList(x_1)) = [[0], [1]] + [[1, 0], [1, 0]] * x_1 >>> <<< POL(isNatList(x_1)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 >>> <<< POL(n__isNatList(x_1)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 >>> <<< POL(n__take(x_1, x_2)) = [[1], [1]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(n__isNat(x_1)) = [[0], [1]] + [[1, 0], [1, 0]] * x_1 >>> <<< POL(n__length(x_1)) = [[1], [1]] + [[1, 0], [1, 1]] * x_1 >>> <<< POL(n__s(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(n__zeros) = [[1], [1]] >>> <<< POL(zeros) = [[1], [1]] >>> <<< POL(take(x_1, x_2)) = [[1], [1]] + [[1, 0], [1, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(length(x_1)) = [[1], [1]] + [[1, 0], [1, 1]] * x_1 >>> <<< POL(s(x_1)) = [[0], [1]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(n__nil) = [[1], [0]] >>> <<< POL(nil) = [[1], [0]] >>> <<< POL(U21(x_1)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U11(x_1, x_2)) = [[0], [1]] + [[0, 1], [0, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 >>> <<< POL(U31(x_1, x_2, x_3, x_4)) = [[1], [1]] + [[0, 0], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[1, 0], [1, 0]] * x_3 + [[1, 1], [0, 1]] * x_4 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0 -> n__0 isNat(n__0) -> tt activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNat(X) -> n__isNat(X) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X cons(X1, X2) -> n__cons(X1, X2) and(X1, X2) -> n__and(X1, X2) take(X1, X2) -> n__take(X1, X2) take(0, IL) -> U21(isNatIList(IL)) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) length(nil) -> 0 length(X) -> n__length(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) zeros -> cons(0, n__zeros) zeros -> n__zeros nil -> n__nil ---------------------------------------- (79) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ISNATILIST(n__cons(n__0, y1)) -> AND(isNat(0), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(x0, x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (80) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = ACTIVATE(n__isNatIList(activate(n__zeros))) evaluates to t =ACTIVATE(n__isNatIList(activate(n__zeros))) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence ACTIVATE(n__isNatIList(activate(n__zeros))) -> ACTIVATE(n__isNatIList(zeros)) with rule activate(n__zeros) -> zeros at position [0,0] and matcher [ ] ACTIVATE(n__isNatIList(zeros)) -> ACTIVATE(n__isNatIList(cons(0, n__zeros))) with rule zeros -> cons(0, n__zeros) at position [0,0] and matcher [ ] ACTIVATE(n__isNatIList(cons(0, n__zeros))) -> ACTIVATE(n__isNatIList(cons(n__0, n__zeros))) with rule 0 -> n__0 at position [0,0,0] and matcher [ ] ACTIVATE(n__isNatIList(cons(n__0, n__zeros))) -> ACTIVATE(n__isNatIList(n__cons(n__0, n__zeros))) with rule cons(X1, X2) -> n__cons(X1, X2) at position [0,0] and matcher [X1 / n__0, X2 / n__zeros] ACTIVATE(n__isNatIList(n__cons(n__0, n__zeros))) -> ISNATILIST(n__cons(n__0, n__zeros)) with rule ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) at position [] and matcher [X / n__cons(n__0, n__zeros)] ISNATILIST(n__cons(n__0, n__zeros)) -> AND(isNat(n__0), n__isNatIList(activate(n__zeros))) with rule ISNATILIST(n__cons(x0, y1)) -> AND(isNat(x0), n__isNatIList(activate(y1))) at position [] and matcher [x0 / n__0, y1 / n__zeros] AND(isNat(n__0), n__isNatIList(activate(n__zeros))) -> AND(tt, n__isNatIList(activate(n__zeros))) with rule isNat(n__0) -> tt at position [0] and matcher [ ] AND(tt, n__isNatIList(activate(n__zeros))) -> ACTIVATE(n__isNatIList(activate(n__zeros))) with rule AND(tt, n__isNatIList(y_3)) -> ACTIVATE(n__isNatIList(y_3)) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (81) NO ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(V1)) -> ISNAT(activate(V1)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNAT(n__s(V1)) -> ISNAT(activate(V1)) at position [0] we obtained the following new rules [LPAR04]: (ISNAT(n__s(n__zeros)) -> ISNAT(zeros),ISNAT(n__s(n__zeros)) -> ISNAT(zeros)) (ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)),ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1))) (ISNAT(n__s(n__0)) -> ISNAT(0),ISNAT(n__s(n__0)) -> ISNAT(0)) (ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)),ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0))) (ISNAT(n__s(n__s(x0))) -> ISNAT(s(x0)),ISNAT(n__s(n__s(x0))) -> ISNAT(s(x0))) (ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(x0, x1)),ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(x0, x1))) (ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)),ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0))) (ISNAT(n__s(n__nil)) -> ISNAT(nil),ISNAT(n__s(n__nil)) -> ISNAT(nil)) (ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)),ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0))) (ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)),ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0))) (ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)),ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1))) (ISNAT(n__s(x0)) -> ISNAT(x0),ISNAT(n__s(x0)) -> ISNAT(x0)) ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__zeros)) -> ISNAT(zeros) ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__0)) -> ISNAT(0) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__s(x0))) -> ISNAT(s(x0)) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(x0, x1)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__nil)) -> ISNAT(nil) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNAT(n__s(n__zeros)) -> ISNAT(zeros) at position [0] we obtained the following new rules [LPAR04]: (ISNAT(n__s(n__zeros)) -> ISNAT(cons(0, n__zeros)),ISNAT(n__s(n__zeros)) -> ISNAT(cons(0, n__zeros))) (ISNAT(n__s(n__zeros)) -> ISNAT(n__zeros),ISNAT(n__s(n__zeros)) -> ISNAT(n__zeros)) ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__0)) -> ISNAT(0) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__s(x0))) -> ISNAT(s(x0)) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(x0, x1)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__nil)) -> ISNAT(nil) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__zeros)) -> ISNAT(cons(0, n__zeros)) ISNAT(n__s(n__zeros)) -> ISNAT(n__zeros) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (87) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__0)) -> ISNAT(0) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__s(x0))) -> ISNAT(s(x0)) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(x0, x1)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__nil)) -> ISNAT(nil) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__zeros)) -> ISNAT(cons(0, n__zeros)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNAT(n__s(n__0)) -> ISNAT(0) at position [0] we obtained the following new rules [LPAR04]: (ISNAT(n__s(n__0)) -> ISNAT(n__0),ISNAT(n__s(n__0)) -> ISNAT(n__0)) ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__s(x0))) -> ISNAT(s(x0)) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(x0, x1)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__nil)) -> ISNAT(nil) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__zeros)) -> ISNAT(cons(0, n__zeros)) ISNAT(n__s(n__0)) -> ISNAT(n__0) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__s(x0))) -> ISNAT(s(x0)) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(x0, x1)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__nil)) -> ISNAT(nil) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__zeros)) -> ISNAT(cons(0, n__zeros)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNAT(n__s(n__s(x0))) -> ISNAT(s(x0)) at position [0] we obtained the following new rules [LPAR04]: (ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0)),ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0))) ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(x0, x1)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__nil)) -> ISNAT(nil) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__zeros)) -> ISNAT(cons(0, n__zeros)) ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(x0, x1)) at position [0] we obtained the following new rules [LPAR04]: (ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(n__cons(x0, x1)),ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(n__cons(x0, x1))) ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__nil)) -> ISNAT(nil) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__zeros)) -> ISNAT(cons(0, n__zeros)) ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0)) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(n__cons(x0, x1)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (97) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__nil)) -> ISNAT(nil) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__zeros)) -> ISNAT(cons(0, n__zeros)) ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (99) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNAT(n__s(n__nil)) -> ISNAT(nil) at position [0] we obtained the following new rules [LPAR04]: (ISNAT(n__s(n__nil)) -> ISNAT(n__nil),ISNAT(n__s(n__nil)) -> ISNAT(n__nil)) ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__zeros)) -> ISNAT(cons(0, n__zeros)) ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0)) ISNAT(n__s(n__nil)) -> ISNAT(n__nil) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__zeros)) -> ISNAT(cons(0, n__zeros)) ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNAT(n__s(n__zeros)) -> ISNAT(cons(0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]: (ISNAT(n__s(n__zeros)) -> ISNAT(n__cons(0, n__zeros)),ISNAT(n__s(n__zeros)) -> ISNAT(n__cons(0, n__zeros))) (ISNAT(n__s(n__zeros)) -> ISNAT(cons(n__0, n__zeros)),ISNAT(n__s(n__zeros)) -> ISNAT(cons(n__0, n__zeros))) ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0)) ISNAT(n__s(n__zeros)) -> ISNAT(n__cons(0, n__zeros)) ISNAT(n__s(n__zeros)) -> ISNAT(cons(n__0, n__zeros)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0)) ISNAT(n__s(n__zeros)) -> ISNAT(cons(n__0, n__zeros)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule ISNAT(n__s(n__zeros)) -> ISNAT(cons(n__0, n__zeros)) at position [0] we obtained the following new rules [LPAR04]: (ISNAT(n__s(n__zeros)) -> ISNAT(n__cons(n__0, n__zeros)),ISNAT(n__s(n__zeros)) -> ISNAT(n__cons(n__0, n__zeros))) ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0)) ISNAT(n__s(n__zeros)) -> ISNAT(n__cons(n__0, n__zeros)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(x0, x1)) ISNAT(n__s(n__isNat(x0))) -> ISNAT(isNat(x0)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ISNAT_1(x_1) ) = x_1 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( 0 ) = 0 POL( U21_1(x_1) ) = max{0, x_1 - 2} POL( isNatIList_1(x_1) ) = 2 POL( s_1(x_1) ) = 2x_1 POL( cons_2(x_1, x_2) ) = 2 POL( U31_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_3 - 2} POL( and_2(x_1, x_2) ) = 2x_2 POL( activate_1(x_1) ) = 2x_1 POL( n__and_2(x_1, x_2) ) = x_2 POL( isNat_1(x_1) ) = 2 POL( n__isNat_1(x_1) ) = 2 POL( n__take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( length_1(x_1) ) = 0 POL( nil ) = 0 POL( U11_2(x_1, x_2) ) = max{0, -2} POL( isNatList_1(x_1) ) = 2 POL( n__length_1(x_1) ) = 0 POL( n__isNatIList_1(x_1) ) = 1 POL( n__cons_2(x_1, x_2) ) = 2 POL( n__isNatList_1(x_1) ) = 1 POL( tt ) = 2 POL( n__s_1(x_1) ) = 2x_1 POL( n__zeros ) = 2 POL( n__nil ) = 0 POL( n__0 ) = 0 POL( zeros ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) length(X) -> n__length(X) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) isNat(n__0) -> tt isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__take(X1, X2)) -> take(X1, X2) U21(tt) -> nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X activate(n__length(X)) -> length(X) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) cons(X1, X2) -> n__cons(X1, X2) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (113) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNAT(n__s(n__isNatIList(x0))) -> ISNAT(isNatIList(x0)) ISNAT(n__s(n__isNatList(x0))) -> ISNAT(isNatList(x0)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ISNAT_1(x_1) ) = x_1 + 2 POL( length_1(x_1) ) = max{0, -2} POL( nil ) = 0 POL( 0 ) = 0 POL( cons_2(x_1, x_2) ) = x_1 POL( U11_2(x_1, x_2) ) = max{0, x_1 - 2} POL( and_2(x_1, x_2) ) = x_2 POL( isNatList_1(x_1) ) = 1 POL( activate_1(x_1) ) = x_1 POL( n__isNat_1(x_1) ) = 2 POL( n__length_1(x_1) ) = 0 POL( n__isNatIList_1(x_1) ) = 1 POL( isNatIList_1(x_1) ) = 1 POL( n__cons_2(x_1, x_2) ) = x_1 POL( isNat_1(x_1) ) = 2 POL( n__isNatList_1(x_1) ) = 1 POL( tt ) = 0 POL( n__take_2(x_1, x_2) ) = x_2 + 2 POL( n__s_1(x_1) ) = 2x_1 POL( n__and_2(x_1, x_2) ) = x_2 POL( n__zeros ) = 1 POL( n__nil ) = 0 POL( take_2(x_1, x_2) ) = x_2 + 2 POL( U21_1(x_1) ) = 1 POL( zeros ) = 1 POL( n__0 ) = 0 POL( s_1(x_1) ) = 2x_1 POL( U31_4(x_1, ..., x_4) ) = x_4 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) length(X) -> n__length(X) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) activate(n__take(X1, X2)) -> take(X1, X2) take(X1, X2) -> n__take(X1, X2) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNat(n__0) -> tt isNat(X) -> n__isNat(X) activate(n__length(X)) -> length(X) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) cons(X1, X2) -> n__cons(X1, X2) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (115) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(x0, x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ISNAT_1(x_1) ) = x_1 POL( length_1(x_1) ) = 0 POL( nil ) = 2 POL( 0 ) = 0 POL( cons_2(x_1, x_2) ) = 2x_2 POL( U11_2(x_1, x_2) ) = max{0, x_1 - 2} POL( and_2(x_1, x_2) ) = x_2 + 2 POL( isNatList_1(x_1) ) = 2 POL( activate_1(x_1) ) = x_1 + 2 POL( n__isNat_1(x_1) ) = 0 POL( n__length_1(x_1) ) = 0 POL( n__isNatIList_1(x_1) ) = 0 POL( isNatIList_1(x_1) ) = 2 POL( n__cons_2(x_1, x_2) ) = 2x_2 POL( isNat_1(x_1) ) = 2 POL( n__isNatList_1(x_1) ) = 0 POL( tt ) = 2 POL( n__take_2(x_1, x_2) ) = 0 POL( n__s_1(x_1) ) = 2x_1 POL( n__and_2(x_1, x_2) ) = x_2 + 2 POL( take_2(x_1, x_2) ) = 2 POL( U21_1(x_1) ) = 2 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__0 ) = 0 POL( s_1(x_1) ) = 2x_1 POL( n__nil ) = 2 POL( U31_4(x_1, ..., x_4) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) length(X) -> n__length(X) activate(n__isNatIList(X)) -> isNatIList(X) isNatIList(V) -> isNatList(activate(V)) isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) and(tt, X) -> activate(X) activate(n__isNatList(X)) -> isNatList(X) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) activate(n__isNat(X)) -> isNat(X) isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) activate(n__and(X1, X2)) -> and(X1, X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) and(X1, X2) -> n__and(X1, X2) activate(n__take(X1, X2)) -> take(X1, X2) take(X1, X2) -> n__take(X1, X2) take(0, IL) -> U21(isNatIList(IL)) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) U21(tt) -> nil activate(n__zeros) -> zeros activate(n__0) -> 0 activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__nil) -> nil activate(X) -> X isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) isNat(n__0) -> tt isNat(X) -> n__isNat(X) activate(n__length(X)) -> length(X) U11(tt, L) -> s(length(activate(L))) s(X) -> n__s(X) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) cons(X1, X2) -> n__cons(X1, X2) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (116) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__length(x0))) -> ISNAT(length(x0)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__s(x0))) -> ISNAT(n__s(x0)) The TRS R consists of the following rules: zeros -> cons(0, n__zeros) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__length(V1)) -> isNatList(activate(V1)) isNat(n__s(V1)) -> isNat(activate(V1)) isNatIList(V) -> isNatList(activate(V)) isNatIList(n__zeros) -> tt isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) isNatList(n__nil) -> tt isNatList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatList(activate(V2))) isNatList(n__take(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) length(nil) -> 0 length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) take(0, IL) -> U21(isNatIList(IL)) take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(isNat(M), n__isNat(N))), activate(IL), M, N) zeros -> n__zeros take(X1, X2) -> n__take(X1, X2) 0 -> n__0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(X) -> n__isNatIList(X) nil -> n__nil isNatList(X) -> n__isNatList(X) isNat(X) -> n__isNat(X) and(X1, X2) -> n__and(X1, X2) activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(X1, X2) activate(n__0) -> 0 activate(n__length(X)) -> length(X) activate(n__s(X)) -> s(X) activate(n__cons(X1, X2)) -> cons(X1, X2) activate(n__isNatIList(X)) -> isNatIList(X) activate(n__nil) -> nil activate(n__isNatList(X)) -> isNatList(X) activate(n__isNat(X)) -> isNat(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains.