NO proof of Transformed_CSR_04_OvConsOS_nokinds_FR.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished Termination w.r.t. Q of the given QTRS could be disproven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 15 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) QDP (5) QDPOrderProof [EQUIVALENT, 180 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 71 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) AND (13) QDP (14) TransformationProof [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, 99 ms] (29) QDP (30) QDPOrderProof [EQUIVALENT, 66 ms] (31) QDP (32) QDPOrderProof [EQUIVALENT, 248 ms] (33) QDP (34) QDP (35) QDPOrderProof [EQUIVALENT, 0 ms] (36) QDP (37) DependencyGraphProof [EQUIVALENT, 0 ms] (38) QDP (39) QDPOrderProof [EQUIVALENT, 171 ms] (40) QDP (41) DependencyGraphProof [EQUIVALENT, 0 ms] (42) AND (43) QDP (44) TransformationProof [EQUIVALENT, 0 ms] (45) QDP (46) TransformationProof [EQUIVALENT, 0 ms] (47) QDP (48) DependencyGraphProof [EQUIVALENT, 0 ms] (49) QDP (50) TransformationProof [EQUIVALENT, 12 ms] (51) QDP (52) DependencyGraphProof [EQUIVALENT, 0 ms] (53) QDP (54) TransformationProof [EQUIVALENT, 21 ms] (55) QDP (56) DependencyGraphProof [EQUIVALENT, 0 ms] (57) QDP (58) TransformationProof [EQUIVALENT, 30 ms] (59) QDP (60) DependencyGraphProof [EQUIVALENT, 0 ms] (61) QDP (62) TransformationProof [EQUIVALENT, 0 ms] (63) QDP (64) DependencyGraphProof [EQUIVALENT, 0 ms] (65) QDP (66) QDPOrderProof [EQUIVALENT, 56 ms] (67) QDP (68) QDPOrderProof [EQUIVALENT, 180 ms] (69) QDP (70) QDPOrderProof [EQUIVALENT, 58 ms] (71) QDP (72) QDP (73) QDPOrderProof [EQUIVALENT, 36 ms] (74) QDP (75) QDPOrderProof [EQUIVALENT, 104 ms] (76) QDP (77) TransformationProof [EQUIVALENT, 0 ms] (78) QDP (79) DependencyGraphProof [EQUIVALENT, 0 ms] (80) QDP (81) TransformationProof [EQUIVALENT, 0 ms] (82) QDP (83) DependencyGraphProof [EQUIVALENT, 0 ms] (84) QDP (85) TransformationProof [EQUIVALENT, 0 ms] (86) QDP (87) DependencyGraphProof [EQUIVALENT, 0 ms] (88) AND (89) QDP (90) QDPOrderProof [EQUIVALENT, 114 ms] (91) QDP (92) QDPOrderProof [EQUIVALENT, 37 ms] (93) QDP (94) QDPOrderProof [EQUIVALENT, 108 ms] (95) QDP (96) QDPOrderProof [EQUIVALENT, 34 ms] (97) QDP (98) QDPOrderProof [EQUIVALENT, 66 ms] (99) QDP (100) QDP (101) QDPOrderProof [EQUIVALENT, 94 ms] (102) QDP (103) QDPOrderProof [EQUIVALENT, 36 ms] (104) QDP (105) QDPOrderProof [EQUIVALENT, 79 ms] (106) QDP (107) QDPOrderProof [EQUIVALENT, 35 ms] (108) QDP (109) NonTerminationLoopProof [COMPLETE, 5855 ms] (110) NO ---------------------------------------- (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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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(n__isNat(M), n__isNat(N))), activate(IL), M, N) TAKE(s(M), cons(N, IL)) -> AND(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))) TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL)) TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL) ACTIVATE(n__zeros) -> ZEROS ACTIVATE(n__take(X1, X2)) -> TAKE(activate(X1), activate(X2)) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2) ACTIVATE(n__0) -> 0^1 ACTIVATE(n__length(X)) -> LENGTH(activate(X)) ACTIVATE(n__length(X)) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> S(activate(X)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> CONS(activate(X1), X2) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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(activate(X1), X2) ACTIVATE(n__and(X1, X2)) -> ACTIVATE(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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(activate(X1), activate(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__take(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2) ACTIVATE(n__length(X)) -> LENGTH(activate(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__length(X)) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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(activate(X1), X2) ACTIVATE(n__and(X1, X2)) -> ACTIVATE(X1) 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(n__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(n__isNat(M), n__isNat(N))) TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL)) TAKE(s(M), cons(N, IL)) -> 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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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. ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__take(X1, X2)) -> ACTIVATE(X2) LENGTH(cons(N, L)) -> AND(isNatList(activate(L)), n__isNat(N)) ISNATLIST(n__cons(V1, V2)) -> ISNAT(activate(V1)) ISNAT(n__length(V1)) -> ISNATLIST(activate(V1)) ISNATLIST(n__cons(V1, V2)) -> ACTIVATE(V1) ACTIVATE(n__length(X)) -> ACTIVATE(X) 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) TAKE(s(M), cons(N, IL)) -> U31^1(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N) TAKE(s(M), cons(N, IL)) -> AND(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))) TAKE(s(M), cons(N, IL)) -> ACTIVATE(IL) 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) ) = 2x_1 POL( ISNATILIST_1(x_1) ) = 2x_1 + 2 POL( ISNATLIST_1(x_1) ) = x_1 + 2 POL( LENGTH_1(x_1) ) = 2x_1 + 2 POL( TAKE_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( U11^1_2(x_1, x_2) ) = 2x_2 + 2 POL( U31^1_4(x_1, ..., x_4) ) = 2x_2 + x_3 + x_4 POL( n__isNatIList_1(x_1) ) = 2x_1 + 2 POL( n__isNatList_1(x_1) ) = x_1 + 2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 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) ) = 2x_1 + x_2 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNatIList_1(x_1) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = x_1 + 2 POL( and_2(x_1, x_2) ) = x_1 + x_2 POL( isNat_1(x_1) ) = 2x_1 POL( tt ) = 0 POL( n__isNat_1(x_1) ) = 2x_1 POL( n__and_2(x_1, x_2) ) = x_1 + x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U11_2(x_1, x_2) ) = 2x_2 + 2 POL( U21_1(x_1) ) = 2 POL( U31_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 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: activate(n__zeros) -> zeros activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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) length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__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 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(activate(X1), activate(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(activate(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)) -> ISNATLIST(activate(L)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) ACTIVATE(n__isNat(X)) -> ISNAT(X) ACTIVATE(n__and(X1, X2)) -> AND(activate(X1), X2) ACTIVATE(n__and(X1, X2)) -> ACTIVATE(X1) 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))) 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)) -> 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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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 1 SCC with 3 less nodes. ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 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__take(X1, X2)) -> TAKE(activate(X1), activate(X2)) TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL)) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) ACTIVATE(n__length(X)) -> LENGTH(activate(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)) -> ISNATLIST(activate(L)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) 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(activate(X1), X2) ACTIVATE(n__and(X1, X2)) -> ACTIVATE(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) 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)) -> ISNATLIST(activate(L)) 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) ) = 2x_1 + 1 POL( ISNATLIST_1(x_1) ) = 2x_1 POL( LENGTH_1(x_1) ) = 2x_1 + 2 POL( TAKE_2(x_1, x_2) ) = 2x_2 + 1 POL( U11^1_2(x_1, x_2) ) = 2x_2 + 2 POL( n__isNatIList_1(x_1) ) = 2x_1 + 1 POL( n__isNatList_1(x_1) ) = 2x_1 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 2 POL( zeros ) = 2 POL( n__take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 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) ) = 2x_1 + x_2 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNatIList_1(x_1) ) = 2x_1 + 1 POL( isNatList_1(x_1) ) = 2x_1 POL( and_2(x_1, x_2) ) = 2x_1 + x_2 POL( isNat_1(x_1) ) = 2x_1 POL( tt ) = 0 POL( n__isNat_1(x_1) ) = 2x_1 POL( n__and_2(x_1, x_2) ) = 2x_1 + x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U11_2(x_1, x_2) ) = 2x_2 + 2 POL( U21_1(x_1) ) = 1 POL( U31_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 1 POL( ACTIVATE_1(x_1) ) = x_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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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) take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 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) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__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 0 -> n__0 nil -> n__nil ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: 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__take(X1, X2)) -> TAKE(activate(X1), activate(X2)) TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL)) ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) ACTIVATE(n__length(X)) -> LENGTH(activate(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__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) ACTIVATE(n__isNatIList(X)) -> ISNATILIST(X) ACTIVATE(n__isNatList(X)) -> ISNATLIST(X) 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(activate(X1), X2) ACTIVATE(n__and(X1, X2)) -> ACTIVATE(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) 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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) 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(activate(x0), activate(x1))),U11^1(tt, n__take(x0, x1)) -> LENGTH(take(activate(x0), activate(x1)))) (U11^1(tt, n__0) -> LENGTH(0),U11^1(tt, n__0) -> LENGTH(0)) (U11^1(tt, n__length(x0)) -> LENGTH(length(activate(x0))),U11^1(tt, n__length(x0)) -> LENGTH(length(activate(x0)))) (U11^1(tt, n__s(x0)) -> LENGTH(s(activate(x0))),U11^1(tt, n__s(x0)) -> LENGTH(s(activate(x0)))) (U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(activate(x0), x1)),U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(activate(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(activate(x0), x1)),U11^1(tt, n__and(x0, x1)) -> LENGTH(and(activate(x0), x1))) (U11^1(tt, x0) -> LENGTH(x0),U11^1(tt, x0) -> LENGTH(x0)) ---------------------------------------- (15) 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(activate(x0), activate(x1))) U11^1(tt, n__0) -> LENGTH(0) U11^1(tt, n__length(x0)) -> LENGTH(length(activate(x0))) U11^1(tt, n__s(x0)) -> LENGTH(s(activate(x0))) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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__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)) ---------------------------------------- (17) 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(activate(x0), activate(x1))) U11^1(tt, n__0) -> LENGTH(0) U11^1(tt, n__length(x0)) -> LENGTH(length(activate(x0))) U11^1(tt, n__s(x0)) -> LENGTH(s(activate(x0))) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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: U11^1(tt, n__take(x0, x1)) -> LENGTH(take(activate(x0), activate(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(activate(x0))) U11^1(tt, n__s(x0)) -> LENGTH(s(activate(x0))) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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__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)) ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(tt, n__take(x0, x1)) -> LENGTH(take(activate(x0), activate(x1))) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__length(x0)) -> LENGTH(length(activate(x0))) U11^1(tt, n__s(x0)) -> LENGTH(s(activate(x0))) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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: LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__take(x0, x1)) -> LENGTH(take(activate(x0), activate(x1))) U11^1(tt, n__length(x0)) -> LENGTH(length(activate(x0))) U11^1(tt, n__s(x0)) -> LENGTH(s(activate(x0))) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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: LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__take(x0, x1)) -> LENGTH(take(activate(x0), activate(x1))) U11^1(tt, n__length(x0)) -> LENGTH(length(activate(x0))) U11^1(tt, n__s(x0)) -> LENGTH(s(activate(x0))) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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: U11^1(tt, n__take(x0, x1)) -> LENGTH(take(activate(x0), activate(x1))) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__length(x0)) -> LENGTH(length(activate(x0))) U11^1(tt, n__s(x0)) -> LENGTH(s(activate(x0))) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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__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 + 2 POL( U11^1_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__take_2(x_1, x_2) ) = max{0, -2} POL( take_2(x_1, x_2) ) = 0 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 0 POL( length_1(x_1) ) = max{0, -1} POL( n__s_1(x_1) ) = 0 POL( s_1(x_1) ) = max{0, -1} POL( n__cons_2(x_1, x_2) ) = 2x_2 POL( cons_2(x_1, x_2) ) = 2x_2 POL( n__isNatIList_1(x_1) ) = 1 POL( isNatIList_1(x_1) ) = 1 POL( isNatList_1(x_1) ) = 1 POL( and_2(x_1, x_2) ) = x_2 POL( isNat_1(x_1) ) = 1 POL( n__isNatList_1(x_1) ) = 1 POL( tt ) = 1 POL( n__isNat_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) ) = max{0, x_1 - 1} POL( U31_4(x_1, ..., x_4) ) = 0 POL( U11_2(x_1, x_2) ) = max{0, 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 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(n__isNat(M), n__isNat(N))), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) 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) isNat(n__0) -> tt isNat(X) -> n__isNat(X) 0 -> n__0 U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil 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: U11^1(tt, n__take(x0, x1)) -> LENGTH(take(activate(x0), activate(x1))) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__length(x0)) -> LENGTH(length(activate(x0))) U11^1(tt, n__s(x0)) -> LENGTH(s(activate(x0))) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(activate(x0), x1)) U11^1(tt, n__and(x0, x1)) -> LENGTH(and(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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__length(x0)) -> LENGTH(length(activate(x0))) U11^1(tt, n__s(x0)) -> LENGTH(s(activate(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 + 1 POL( U11^1_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__take_2(x_1, x_2) ) = max{0, -2} POL( take_2(x_1, x_2) ) = max{0, -2} POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = x_1 + 1 POL( length_1(x_1) ) = x_1 + 1 POL( n__s_1(x_1) ) = 1 POL( s_1(x_1) ) = 1 POL( n__cons_2(x_1, x_2) ) = 2x_2 POL( cons_2(x_1, x_2) ) = 2x_2 POL( n__isNatIList_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = 0 POL( and_2(x_1, x_2) ) = 2x_2 POL( isNat_1(x_1) ) = 0 POL( n__isNatList_1(x_1) ) = max{0, -2} POL( tt ) = 0 POL( n__isNat_1(x_1) ) = 0 POL( n__and_2(x_1, x_2) ) = 2x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U21_1(x_1) ) = x_1 POL( U31_4(x_1, ..., x_4) ) = max{0, -2} POL( U11_2(x_1, x_2) ) = 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 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(n__isNat(M), n__isNat(N))), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) 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) 0 -> n__0 isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) isNat(n__0) -> tt isNat(X) -> n__isNat(X) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil 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: U11^1(tt, n__take(x0, x1)) -> LENGTH(take(activate(x0), activate(x1))) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(activate(x0), x1)) U11^1(tt, n__and(x0, x1)) -> LENGTH(and(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) 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(activate(x0), x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(U11^1(x_1, x_2)) = [[5A]] + [[2A]] * x_1 + [[4A]] * x_2 >>> <<< POL(tt) = [[1A]] >>> <<< POL(n__take(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(LENGTH(x_1)) = [[5A]] + [[3A]] * x_1 >>> <<< POL(take(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(activate(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(and(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(isNatList(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(n__isNat(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(n__cons(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(n__and(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(n__zeros) = [[0A]] >>> <<< POL(0) = [[0A]] >>> <<< POL(zeros) = [[1A]] >>> <<< POL(n__0) = [[0A]] >>> <<< POL(n__length(x_1)) = [[3A]] + [[2A]] * x_1 >>> <<< POL(length(x_1)) = [[3A]] + [[2A]] * x_1 >>> <<< POL(n__s(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(n__isNatIList(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(isNat(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(n__isNatList(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(n__nil) = [[0A]] >>> <<< POL(nil) = [[0A]] >>> <<< POL(U21(x_1)) = [[0A]] + [[-I]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3, x_4)) = [[1A]] + [[-I]] * x_1 + [[-I]] * x_2 + [[-I]] * x_3 + [[-I]] * x_4 >>> <<< POL(U11(x_1, x_2)) = [[3A]] + [[0A]] * x_1 + [[2A]] * 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 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(n__isNat(M), n__isNat(N))), activate(IL), M, N) take(X1, X2) -> n__take(X1, X2) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) cons(X1, X2) -> n__cons(X1, X2) 0 -> n__0 length(nil) -> 0 length(X) -> n__length(X) s(X) -> n__s(X) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) isNat(n__0) -> tt isNat(X) -> n__isNat(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil 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 ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(tt, n__take(x0, x1)) -> LENGTH(take(activate(x0), activate(x1))) LENGTH(cons(N, L)) -> U11^1(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11^1(tt, n__cons(x0, x1)) -> LENGTH(cons(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__take(X1, X2)) -> TAKE(activate(X1), activate(X2)) TAKE(0, IL) -> ISNATILIST(IL) TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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(activate(X1), X2) ACTIVATE(n__and(X1, X2)) -> ACTIVATE(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) 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(activate(X1), activate(X2)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVATE(x_1)) = 1 + x_1 POL(AND(x_1, x_2)) = 1 + x_2 POL(ISNAT(x_1)) = 1 + x_1 POL(ISNATILIST(x_1)) = 1 + x_1 POL(ISNATLIST(x_1)) = 1 + x_1 POL(TAKE(x_1, x_2)) = 1 + x_2 POL(U11(x_1, x_2)) = x_2 POL(U21(x_1)) = 0 POL(U31(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4 POL(activate(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + x_2 POL(cons(x_1, x_2)) = x_1 + x_2 POL(isNat(x_1)) = x_1 POL(isNatIList(x_1)) = x_1 POL(isNatList(x_1)) = x_1 POL(length(x_1)) = x_1 POL(n__0) = 0 POL(n__and(x_1, x_2)) = x_1 + x_2 POL(n__cons(x_1, x_2)) = x_1 + x_2 POL(n__isNat(x_1)) = x_1 POL(n__isNatIList(x_1)) = x_1 POL(n__isNatList(x_1)) = x_1 POL(n__length(x_1)) = x_1 POL(n__nil) = 0 POL(n__s(x_1)) = x_1 POL(n__take(x_1, x_2)) = 1 + x_1 + x_2 POL(n__zeros) = 0 POL(nil) = 0 POL(s(x_1)) = x_1 POL(take(x_1, x_2)) = 1 + x_1 + x_2 POL(tt) = 0 POL(zeros) = 0 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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) length(nil) -> 0 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) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__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 0 -> n__0 nil -> n__nil ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(n__cons(V1, V2)) -> AND(isNat(activate(V1)), n__isNatIList(activate(V2))) AND(tt, X) -> ACTIVATE(X) TAKE(0, IL) -> ISNATILIST(IL) TAKE(s(M), cons(N, IL)) -> ISNATILIST(activate(IL)) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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(activate(X1), X2) ACTIVATE(n__and(X1, X2)) -> ACTIVATE(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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(activate(X1), X2) ACTIVATE(n__and(X1, X2)) -> ACTIVATE(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNAT(n__s(V1)) -> ACTIVATE(V1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(AND(x_1, x_2)) = [[5A]] + [[1A]] * x_1 + [[4A]] * x_2 >>> <<< POL(tt) = [[5A]] >>> <<< POL(ACTIVATE(x_1)) = [[4A]] + [[4A]] * x_1 >>> <<< POL(n__s(x_1)) = [[5A]] + [[0A]] * x_1 >>> <<< POL(n__cons(x_1, x_2)) = [[4A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(n__isNatIList(x_1)) = [[5A]] + [[1A]] * x_1 >>> <<< POL(ISNATILIST(x_1)) = [[5A]] + [[5A]] * x_1 >>> <<< POL(isNat(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(activate(x_1)) = [[4A]] + [[0A]] * x_1 >>> <<< POL(n__isNatList(x_1)) = [[4A]] + [[1A]] * x_1 >>> <<< POL(ISNATLIST(x_1)) = [[5A]] + [[5A]] * x_1 >>> <<< POL(n__isNat(x_1)) = [[1A]] + [[1A]] * x_1 >>> <<< POL(ISNAT(x_1)) = [[5A]] + [[5A]] * x_1 >>> <<< POL(n__and(x_1, x_2)) = [[1A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(n__zeros) = [[5A]] >>> <<< POL(zeros) = [[5A]] >>> <<< POL(n__take(x_1, x_2)) = [[4A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(take(x_1, x_2)) = [[4A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(n__0) = [[4A]] >>> <<< POL(0) = [[4A]] >>> <<< POL(n__length(x_1)) = [[5A]] + [[1A]] * x_1 >>> <<< POL(length(x_1)) = [[5A]] + [[1A]] * x_1 >>> <<< POL(s(x_1)) = [[5A]] + [[0A]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[4A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(isNatIList(x_1)) = [[5A]] + [[1A]] * x_1 >>> <<< POL(isNatList(x_1)) = [[4A]] + [[1A]] * x_1 >>> <<< POL(and(x_1, x_2)) = [[2A]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(n__nil) = [[4A]] >>> <<< POL(nil) = [[4A]] >>> <<< POL(U11(x_1, x_2)) = [[5A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(U21(x_1)) = [[4A]] + [[-I]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3, x_4)) = [[5A]] + [[-I]] * x_1 + [[0A]] * x_2 + [[0A]] * x_3 + [[0A]] * x_4 >>> 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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) length(nil) -> 0 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) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__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 0 -> n__0 nil -> n__nil ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: AND(tt, X) -> ACTIVATE(X) ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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)) ACTIVATE(n__and(X1, X2)) -> AND(activate(X1), X2) ACTIVATE(n__and(X1, X2)) -> ACTIVATE(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. ---------------------------------------- (42) Complex Obligation (AND) ---------------------------------------- (43) 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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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 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(activate(x0), activate(x1))),ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(x1)))) (ISNAT(n__s(n__0)) -> ISNAT(0),ISNAT(n__s(n__0)) -> ISNAT(0)) (ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))),ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0)))) (ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))),ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0)))) (ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(x0), x1)),ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(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(activate(x0), x1)),ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(activate(x0), x1))) (ISNAT(n__s(x0)) -> ISNAT(x0),ISNAT(n__s(x0)) -> ISNAT(x0)) ---------------------------------------- (45) 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(activate(x0), activate(x1))) ISNAT(n__s(n__0)) -> ISNAT(0) ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (46) 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)) ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(x1))) ISNAT(n__s(n__0)) -> ISNAT(0) ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(x1))) ISNAT(n__s(n__0)) -> ISNAT(0) ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (50) 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)) ---------------------------------------- (51) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(x1))) ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (52) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (53) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(x1))) ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (54) 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)) ---------------------------------------- (55) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(x1))) ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(x0), x1)) 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(activate(x0), x1)) ISNAT(n__s(x0)) -> ISNAT(x0) ISNAT(n__s(n__zeros)) -> ISNAT(cons(0, n__zeros)) 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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (56) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (57) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(x1))) ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(x0), x1)) 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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (58) 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))) ---------------------------------------- (59) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(x1))) ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(x0), x1)) 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(activate(x0), x1)) ISNAT(n__s(x0)) -> ISNAT(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (60) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (61) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(x1))) ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(x0), x1)) 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(activate(x0), x1)) ISNAT(n__s(x0)) -> ISNAT(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (62) 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))) ---------------------------------------- (63) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(x1))) ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(x0), x1)) 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(activate(x0), x1)) ISNAT(n__s(x0)) -> ISNAT(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (64) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(x1))) ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))) ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(x0), x1)) 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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (66) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNAT(n__s(n__cons(x0, x1))) -> ISNAT(cons(activate(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 + 1 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 2 POL( zeros ) = 2 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) ) = 0 POL( length_1(x_1) ) = max{0, -1} POL( n__s_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = 2x_1 POL( n__cons_2(x_1, x_2) ) = 1 POL( cons_2(x_1, x_2) ) = 1 POL( n__isNatIList_1(x_1) ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = max{0, -2} POL( and_2(x_1, x_2) ) = 2x_2 POL( isNat_1(x_1) ) = 0 POL( n__isNatList_1(x_1) ) = max{0, -1} POL( tt ) = 0 POL( n__isNat_1(x_1) ) = 0 POL( n__and_2(x_1, x_2) ) = 2x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U21_1(x_1) ) = 0 POL( U31_4(x_1, ..., x_4) ) = 1 POL( U11_2(x_1, x_2) ) = x_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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 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(n__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) s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) 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) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (67) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(x1))) ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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. 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)) 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( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( n__take_2(x_1, x_2) ) = x_1 + x_2 POL( take_2(x_1, x_2) ) = x_1 + x_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) ) = 2x_1 POL( s_1(x_1) ) = 2x_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) ) = x_1 + 1 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( n__isNatList_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) ) = max{0, -2} POL( U31_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + x_4 POL( U11_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 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(n__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) s(X) -> n__s(X) 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) cons(X1, X2) -> n__cons(X1, X2) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (69) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(x1))) ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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. ISNAT(n__s(n__take(x0, x1))) -> ISNAT(take(activate(x0), activate(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 + 2 POL( activate_1(x_1) ) = 2x_1 POL( n__zeros ) = 2 POL( zeros ) = 2 POL( n__take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 POL( n__0 ) = 0 POL( 0 ) = 0 POL( n__length_1(x_1) ) = 0 POL( length_1(x_1) ) = 0 POL( n__s_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = 2x_1 POL( n__cons_2(x_1, x_2) ) = 2 POL( cons_2(x_1, x_2) ) = 2 POL( n__isNatIList_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = max{0, -2} POL( and_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( isNat_1(x_1) ) = 0 POL( n__isNatList_1(x_1) ) = 0 POL( tt ) = 0 POL( n__isNat_1(x_1) ) = 0 POL( n__and_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U21_1(x_1) ) = max{0, -2} POL( U31_4(x_1, ..., x_4) ) = 2 POL( U11_2(x_1, x_2) ) = 0 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) isNatIList(n__cons(V1, V2)) -> and(isNat(activate(V1)), n__isNatIList(activate(V2))) 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(n__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) s(X) -> n__s(X) and(X1, X2) -> n__and(X1, X2) cons(X1, X2) -> n__cons(X1, X2) 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) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) zeros -> cons(0, n__zeros) zeros -> n__zeros 0 -> n__0 nil -> n__nil ---------------------------------------- (71) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(n__s(n__length(x0))) -> ISNAT(length(activate(x0))) ISNAT(n__s(n__s(x0))) -> ISNAT(s(activate(x0))) ISNAT(n__s(n__and(x0, x1))) -> ISNAT(and(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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(activate(X1), X2) ACTIVATE(n__and(X1, X2)) -> ACTIVATE(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVATE(n__s(X)) -> ACTIVATE(X) ACTIVATE(n__and(X1, X2)) -> AND(activate(X1), X2) ACTIVATE(n__and(X1, X2)) -> ACTIVATE(X1) 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_2 + 2 POL( n__isNatIList_1(x_1) ) = 0 POL( n__isNatList_1(x_1) ) = 0 POL( activate_1(x_1) ) = x_1 POL( n__zeros ) = 0 POL( zeros ) = 1 POL( n__take_2(x_1, x_2) ) = 2x_2 + 2 POL( take_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( n__0 ) = 0 POL( 0 ) = 1 POL( n__length_1(x_1) ) = 2x_1 + 2 POL( length_1(x_1) ) = 0 POL( n__s_1(x_1) ) = 2x_1 + 2 POL( s_1(x_1) ) = max{0, -2} POL( n__cons_2(x_1, x_2) ) = 2x_1 POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2} POL( isNatIList_1(x_1) ) = 0 POL( isNatList_1(x_1) ) = 2 POL( and_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 POL( isNat_1(x_1) ) = max{0, -2} POL( tt ) = 1 POL( n__isNat_1(x_1) ) = x_1 + 1 POL( n__and_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 POL( n__nil ) = 0 POL( nil ) = 0 POL( U11_2(x_1, x_2) ) = 2x_2 + 2 POL( U21_1(x_1) ) = 2 POL( U31_4(x_1, ..., x_4) ) = max{0, 2x_3 - 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: none ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (75) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVATE(n__cons(X1, X2)) -> ACTIVATE(X1) 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 + 2x_2 - 2} POL( n__isNatIList_1(x_1) ) = 0 POL( n__isNatList_1(x_1) ) = max{0, -1} POL( activate_1(x_1) ) = 2x_1 + 2 POL( n__zeros ) = 0 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 ) = 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) ) = 1 POL( s_1(x_1) ) = 2 POL( n__cons_2(x_1, x_2) ) = x_1 + 1 POL( cons_2(x_1, x_2) ) = x_1 + 1 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) ) = 0 POL( n__and_2(x_1, x_2) ) = x_1 + x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U11_2(x_1, x_2) ) = 2x_1 POL( U21_1(x_1) ) = x_1 POL( U31_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_4 - 1} POL( ACTIVATE_1(x_1) ) = 2x_1 POL( ISNATILIST_1(x_1) ) = 0 POL( ISNATLIST_1(x_1) ) = 0 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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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) length(nil) -> 0 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) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__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 0 -> n__0 nil -> n__nil ---------------------------------------- (76) 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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (77) 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(activate(x0), activate(x1))), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(activate(x0), activate(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(activate(x0))), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1)))) (ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(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(activate(x0), x1)), n__isNatIList(activate(y1))),ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(activate(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)))) ---------------------------------------- (78) 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(activate(x0), activate(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(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (80) 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(activate(x0), activate(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(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) 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(activate(x0), activate(x1))), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(activate(x0), activate(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(activate(x0))), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1)))) (ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(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(activate(x0), x1)), n__isNatList(activate(y1))),ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(activate(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)))) ---------------------------------------- (82) 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(activate(x0), activate(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(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(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(activate(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(activate(x0), activate(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(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (84) 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(activate(x0), activate(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(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(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(activate(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(activate(x0), activate(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(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) 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))) ---------------------------------------- (86) 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(activate(x0), activate(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(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(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(activate(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(activate(x0), activate(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(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(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 2 SCCs. ---------------------------------------- (88) Complex Obligation (AND) ---------------------------------------- (89) 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(activate(x0), activate(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(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (90) 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(activate(x0), activate(x1))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatIList(x0), y1)) -> AND(isNat(isNatIList(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) ) = x_2 + 2 POL( n__isNatList_1(x_1) ) = x_1 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) ) = x_1 POL( n__0 ) = 0 POL( tt ) = 0 POL( activate_1(x_1) ) = x_1 POL( n__isNatIList_1(x_1) ) = x_1 + 1 POL( isNatIList_1(x_1) ) = x_1 + 1 POL( isNatList_1(x_1) ) = x_1 POL( n__cons_2(x_1, x_2) ) = x_1 + x_2 POL( and_2(x_1, x_2) ) = x_1 + x_2 POL( n__take_2(x_1, x_2) ) = x_1 + x_2 + 1 POL( n__isNat_1(x_1) ) = x_1 POL( n__length_1(x_1) ) = 2x_1 + 1 POL( n__s_1(x_1) ) = x_1 POL( n__and_2(x_1, x_2) ) = x_1 + x_2 POL( take_2(x_1, x_2) ) = x_1 + x_2 + 1 POL( length_1(x_1) ) = 2x_1 + 1 POL( s_1(x_1) ) = x_1 POL( n__nil ) = 0 POL( nil ) = 0 POL( U21_1(x_1) ) = 1 POL( U31_4(x_1, ..., x_4) ) = x_2 + x_3 + x_4 + 1 POL( U11_2(x_1, x_2) ) = 2x_2 + 1 POL( ISNATLIST_1(x_1) ) = x_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(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(n__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) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) ---------------------------------------- (91) 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__0, y1)) -> AND(isNat(0), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), 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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (92) 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))) 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) ) = 2x_1 + x_2 POL( n__isNatList_1(x_1) ) = 2x_1 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) ) = 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) ) = x_2 + 1 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) ) = x_2 + 1 POL( length_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = x_1 POL( n__nil ) = 1 POL( nil ) = 1 POL( U11_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1} POL( U21_1(x_1) ) = 1 POL( U31_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 1 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(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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 isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(X) -> n__length(X) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__isNat(M), n__isNat(N))), activate(IL), M, N) U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) ---------------------------------------- (93) 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(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (94) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATLIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(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, 2x_2 - 2} POL( n__isNatList_1(x_1) ) = x_1 + 2 POL( 0 ) = 0 POL( n__0 ) = 0 POL( isNat_1(x_1) ) = 2x_1 + 2 POL( tt ) = 1 POL( activate_1(x_1) ) = x_1 POL( n__isNatIList_1(x_1) ) = x_1 + 2 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 + x_2 POL( and_2(x_1, x_2) ) = x_2 POL( n__take_2(x_1, x_2) ) = x_2 + 2 POL( n__isNat_1(x_1) ) = 2x_1 + 2 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( n__zeros ) = 1 POL( zeros ) = 1 POL( take_2(x_1, x_2) ) = x_2 + 2 POL( length_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( n__nil ) = 2 POL( nil ) = 2 POL( U11_2(x_1, x_2) ) = 2x_2 POL( U21_1(x_1) ) = 2 POL( U31_4(x_1, ..., x_4) ) = x_2 + x_4 + 2 POL( ACTIVATE_1(x_1) ) = max{0, 2x_1 - 2} POL( ISNATLIST_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(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(X) -> n__length(X) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__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 ---------------------------------------- (95) 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(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (96) 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(activate(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)) = [[0]] + [[0, 0]] * x_1 + [[1, 1]] * x_2 >>> <<< POL(tt) = [[0], [1]] >>> <<< POL(n__isNatList(x_1)) = [[0], [0]] + [[0, 0], [1, 0]] * x_1 >>> <<< POL(ACTIVATE(x_1)) = [[0]] + [[0, 1]] * x_1 >>> <<< POL(ISNATLIST(x_1)) = [[0]] + [[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)) = [[0], [1]] + [[0, 0], [1, 1]] * 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, 1], [0, 1]] * x_1 >>> <<< POL(s(x_1)) = [[0], [0]] + [[1, 1], [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]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(and(x_1, x_2)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(n__isNatIList(x_1)) = [[0], [1]] + [[0, 0], [1, 0]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[0], [1]] + [[0, 0], [1, 0]] * x_1 >>> <<< POL(isNatList(x_1)) = [[0], [0]] + [[0, 0], [1, 0]] * x_1 >>> <<< POL(n__take(x_1, x_2)) = [[1], [0]] + [[1, 0], [0, 1]] * x_1 + [[1, 0], [1, 1]] * x_2 >>> <<< POL(n__isNat(x_1)) = [[0], [1]] + [[0, 0], [1, 1]] * x_1 >>> <<< POL(n__length(x_1)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(n__zeros) = [[0], [1]] >>> <<< POL(zeros) = [[0], [1]] >>> <<< POL(take(x_1, x_2)) = [[1], [0]] + [[1, 0], [0, 1]] * x_1 + [[1, 0], [1, 1]] * x_2 >>> <<< POL(length(x_1)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(n__nil) = [[1], [0]] >>> <<< POL(nil) = [[1], [0]] >>> <<< POL(U11(x_1, x_2)) = [[0], [0]] + [[0, 1], [0, 0]] * x_1 + [[1, 0], [0, 0]] * x_2 >>> <<< POL(U21(x_1)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3, x_4)) = [[1], [0]] + [[0, 0], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [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(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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) length(nil) -> 0 length(X) -> n__length(X) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__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 ---------------------------------------- (97) 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(activate(x0))), n__isNatList(activate(y1))) ISNATLIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (98) 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(activate(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]] + [[1, 0]] * x_1 + [[1, 0]] * x_2 >>> <<< POL(tt) = [[1], [0]] >>> <<< POL(n__isNatList(x_1)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 >>> <<< POL(ACTIVATE(x_1)) = [[1]] + [[1, 0]] * x_1 >>> <<< POL(ISNATLIST(x_1)) = [[1]] + [[1, 1]] * x_1 >>> <<< POL(n__cons(x_1, x_2)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [0, 0]] * x_2 >>> <<< POL(n__0) = [[0], [0]] >>> <<< POL(isNat(x_1)) = [[1], [1]] + [[1, 0], [0, 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(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)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 >>> <<< POL(isNatList(x_1)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 >>> <<< POL(n__take(x_1, x_2)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(n__isNat(x_1)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(n__length(x_1)) = [[0], [1]] + [[1, 1], [0, 0]] * x_1 >>> <<< POL(n__zeros) = [[1], [0]] >>> <<< POL(zeros) = [[1], [0]] >>> <<< POL(take(x_1, x_2)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(length(x_1)) = [[0], [1]] + [[1, 1], [0, 0]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [0, 0]] * x_2 >>> <<< POL(n__nil) = [[1], [0]] >>> <<< POL(nil) = [[1], [0]] >>> <<< POL(U11(x_1, x_2)) = [[0], [1]] + [[0, 0], [0, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 >>> <<< POL(U21(x_1)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3, x_4)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[0, 0], [0, 0]] * x_3 + [[1, 1], [1, 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(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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) length(nil) -> 0 length(X) -> n__length(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__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 ---------------------------------------- (99) 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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (100) 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(activate(x0), activate(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(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__s(x0), y1)) -> AND(isNat(s(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) 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))) ISNATILIST(n__cons(n__take(x0, x1), y1)) -> AND(isNat(take(activate(x0), activate(x1))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__length(x0), y1)) -> AND(isNat(length(activate(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__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) ) = x_1 + x_2 POL( n__isNatIList_1(x_1) ) = 2x_1 + 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) ) = 2x_1 + 1 POL( n__0 ) = 0 POL( tt ) = 1 POL( activate_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = 2x_1 + 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__isNatList_1(x_1) ) = 2x_1 POL( n__take_2(x_1, x_2) ) = x_2 + 2 POL( n__isNat_1(x_1) ) = 2x_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 + 2 POL( length_1(x_1) ) = x_1 + 1 POL( s_1(x_1) ) = x_1 POL( n__nil ) = 1 POL( nil ) = 1 POL( U21_1(x_1) ) = 2 POL( U31_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( U11_2(x_1, x_2) ) = x_2 + 1 POL( ISNATILIST_1(x_1) ) = 2x_1 + 2 POL( ACTIVATE_1(x_1) ) = x_1 + 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(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(n__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) U11(tt, L) -> s(length(activate(L))) U21(tt) -> nil U31(tt, IL, M, N) -> cons(activate(N), n__take(activate(M), activate(IL))) ---------------------------------------- (102) 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(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(x0)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ISNATILIST(n__cons(n__isNatList(x0), y1)) -> AND(isNat(isNatList(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_1 + 2x_2 - 2} POL( n__isNatIList_1(x_1) ) = 2x_1 + 1 POL( 0 ) = 0 POL( n__0 ) = 0 POL( isNat_1(x_1) ) = x_1 POL( tt ) = 0 POL( activate_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = 2x_1 + 1 POL( isNatList_1(x_1) ) = 2x_1 + 1 POL( n__cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( and_2(x_1, x_2) ) = x_2 POL( n__isNatList_1(x_1) ) = 2x_1 + 1 POL( n__take_2(x_1, x_2) ) = x_2 POL( n__isNat_1(x_1) ) = x_1 POL( n__length_1(x_1) ) = 2x_1 + 1 POL( n__s_1(x_1) ) = x_1 POL( n__and_2(x_1, x_2) ) = x_2 POL( n__zeros ) = 0 POL( zeros ) = 0 POL( take_2(x_1, x_2) ) = x_2 POL( length_1(x_1) ) = 2x_1 + 1 POL( s_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( n__nil ) = 0 POL( nil ) = 0 POL( U11_2(x_1, x_2) ) = 2x_2 + 1 POL( U21_1(x_1) ) = max{0, -1} POL( U31_4(x_1, ..., x_4) ) = 2x_2 + 2x_4 POL( ACTIVATE_1(x_1) ) = max{0, x_1 - 1} POL( ISNATILIST_1(x_1) ) = 2x_1 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(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__nil) -> nil activate(X) -> X s(X) -> n__s(X) cons(X1, X2) -> n__cons(X1, X2) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) and(X1, X2) -> n__and(X1, X2) take(X1, X2) -> n__take(X1, X2) length(nil) -> 0 length(X) -> n__length(X) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__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 ---------------------------------------- (104) 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(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__cons(x0, x1), y1)) -> AND(isNat(cons(activate(x0), x1)), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) 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(activate(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, 1]] * x_1 + [[1, 0]] * x_2 >>> <<< POL(tt) = [[0], [0]] >>> <<< POL(n__isNatIList(x_1)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(ACTIVATE(x_1)) = [[0]] + [[1, 0]] * x_1 >>> <<< POL(ISNATILIST(x_1)) = [[1]] + [[1, 0]] * x_1 >>> <<< POL(n__cons(x_1, x_2)) = [[0], [1]] + [[0, 1], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 >>> <<< POL(n__0) = [[0], [0]] >>> <<< POL(isNat(x_1)) = [[1], [0]] + [[0, 1], [0, 0]] * x_1 >>> <<< POL(0) = [[0], [0]] >>> <<< 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(cons(x_1, x_2)) = [[0], [1]] + [[0, 1], [1, 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)) = [[1], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(isNatList(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(n__isNatList(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 >>> <<< POL(n__take(x_1, x_2)) = [[1], [1]] + [[0, 0], [1, 1]] * x_1 + [[1, 0], [1, 1]] * x_2 >>> <<< POL(n__isNat(x_1)) = [[1], [0]] + [[0, 1], [0, 0]] * x_1 >>> <<< POL(n__length(x_1)) = [[0], [1]] + [[0, 1], [1, 0]] * x_1 >>> <<< POL(n__zeros) = [[1], [1]] >>> <<< POL(zeros) = [[1], [1]] >>> <<< POL(take(x_1, x_2)) = [[1], [1]] + [[0, 0], [1, 1]] * x_1 + [[1, 0], [1, 1]] * x_2 >>> <<< POL(length(x_1)) = [[0], [1]] + [[0, 1], [1, 0]] * x_1 >>> <<< POL(n__nil) = [[0], [0]] >>> <<< POL(nil) = [[0], [0]] >>> <<< POL(U11(x_1, x_2)) = [[1], [1]] + [[0, 1], [0, 1]] * x_1 + [[0, 0], [1, 0]] * x_2 >>> <<< POL(U21(x_1)) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3, x_4)) = [[1], [1]] + [[0, 0], [1, 1]] * x_1 + [[1, 0], [0, 0]] * x_2 + [[0, 0], [0, 0]] * x_3 + [[0, 1], [1, 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(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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) length(nil) -> 0 length(X) -> n__length(X) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__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 ---------------------------------------- (106) 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(activate(x0))), n__isNatIList(activate(y1))) ISNATILIST(n__cons(n__and(x0, x1), y1)) -> AND(isNat(and(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) 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(activate(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)) = [[0]] + [[0, 0]] * x_1 + [[0, 1]] * x_2 >>> <<< POL(tt) = [[1], [0]] >>> <<< POL(n__isNatIList(x_1)) = [[1], [1]] + [[0, 0], [1, 1]] * x_1 >>> <<< POL(ACTIVATE(x_1)) = [[0]] + [[0, 1]] * x_1 >>> <<< POL(ISNATILIST(x_1)) = [[1]] + [[1, 0]] * x_1 >>> <<< POL(n__cons(x_1, x_2)) = [[0], [0]] + [[0, 1], [1, 1]] * x_1 + [[1, 1], [0, 0]] * x_2 >>> <<< POL(n__0) = [[0], [0]] >>> <<< POL(isNat(x_1)) = [[1], [0]] + [[0, 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(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)) = [[1], [1]] + [[0, 0], [1, 1]] * x_1 >>> <<< POL(isNatList(x_1)) = [[1], [0]] + [[0, 0], [1, 1]] * x_1 >>> <<< POL(n__isNatList(x_1)) = [[1], [0]] + [[0, 0], [1, 1]] * x_1 >>> <<< POL(n__take(x_1, x_2)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(n__isNat(x_1)) = [[1], [0]] + [[0, 0], [1, 0]] * x_1 >>> <<< POL(n__length(x_1)) = [[1], [1]] + [[1, 1], [1, 0]] * x_1 >>> <<< POL(n__zeros) = [[1], [0]] >>> <<< POL(zeros) = [[1], [0]] >>> <<< POL(take(x_1, x_2)) = [[1], [0]] + [[0, 0], [0, 0]] * x_1 + [[1, 0], [0, 1]] * x_2 >>> <<< POL(length(x_1)) = [[1], [1]] + [[1, 1], [1, 0]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[0], [0]] + [[0, 1], [1, 1]] * x_1 + [[1, 1], [0, 0]] * x_2 >>> <<< POL(n__nil) = [[0], [0]] >>> <<< POL(nil) = [[0], [0]] >>> <<< POL(U11(x_1, x_2)) = [[1], [0]] + [[0, 1], [1, 0]] * x_1 + [[1, 1], [1, 1]] * x_2 >>> <<< POL(U21(x_1)) = [[0], [0]] + [[0, 0], [0, 0]] * x_1 >>> <<< POL(U31(x_1, x_2, x_3, x_4)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[0, 0], [0, 0]] * x_3 + [[0, 1], [1, 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(activate(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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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) length(nil) -> 0 length(X) -> n__length(X) cons(X1, X2) -> n__cons(X1, X2) isNatIList(n__zeros) -> tt isNatIList(X) -> n__isNatIList(X) isNatList(n__nil) -> tt isNatList(X) -> n__isNatList(X) length(cons(N, L)) -> U11(and(isNatList(activate(L)), n__isNat(N)), activate(L)) U11(tt, L) -> s(length(activate(L))) take(0, IL) -> U21(isNatIList(IL)) U21(tt) -> nil take(s(M), cons(N, IL)) -> U31(and(isNatIList(activate(IL)), n__and(n__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 ---------------------------------------- (108) 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(activate(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(n__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(activate(X1), activate(X2)) activate(n__0) -> 0 activate(n__length(X)) -> length(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__cons(X1, X2)) -> cons(activate(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(activate(X1), X2) activate(X) -> X Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) 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. ---------------------------------------- (110) NO