/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 94 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) QDP (5) QDPOrderProof [EQUIVALENT, 217 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) AND (9) QDP (10) QDPQMonotonicMRRProof [EQUIVALENT, 53 ms] (11) QDP (12) QDPQMonotonicMRRProof [EQUIVALENT, 38 ms] (13) QDP (14) QDPOrderProof [EQUIVALENT, 213 ms] (15) QDP (16) DependencyGraphProof [EQUIVALENT, 0 ms] (17) TRUE (18) QDP (19) QDPOrderProof [EQUIVALENT, 93 ms] (20) QDP (21) QDPOrderProof [EQUIVALENT, 0 ms] (22) QDP (23) QDPOrderProof [EQUIVALENT, 64 ms] (24) QDP (25) QDPOrderProof [EQUIVALENT, 82 ms] (26) QDP (27) DependencyGraphProof [EQUIVALENT, 0 ms] (28) AND (29) QDP (30) UsableRulesProof [EQUIVALENT, 1 ms] (31) QDP (32) QReductionProof [EQUIVALENT, 0 ms] (33) QDP (34) QDPSizeChangeProof [EQUIVALENT, 0 ms] (35) YES (36) QDP (37) QDPOrderProof [EQUIVALENT, 91 ms] (38) QDP (39) QDPQMonotonicMRRProof [EQUIVALENT, 86 ms] (40) QDP (41) QDPQMonotonicMRRProof [EQUIVALENT, 66 ms] (42) QDP (43) QDPOrderProof [EQUIVALENT, 218 ms] (44) QDP (45) QDPQMonotonicMRRProof [EQUIVALENT, 127 ms] (46) QDP (47) DependencyGraphProof [EQUIVALENT, 0 ms] (48) QDP (49) UsableRulesProof [EQUIVALENT, 0 ms] (50) QDP (51) QReductionProof [EQUIVALENT, 0 ms] (52) QDP (53) QDPSizeChangeProof [EQUIVALENT, 0 ms] (54) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) ---------------------------------------- (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: A__AND(tt, T) -> MARK(T) A__ISNATILIST(IL) -> A__ISNATLIST(IL) A__ISNAT(s(N)) -> A__ISNAT(N) A__ISNAT(length(L)) -> A__ISNATLIST(L) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNAT(N) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNAT(N) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATLIST(take(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATLIST(take(N, IL)) -> A__ISNAT(N) A__ISNATLIST(take(N, IL)) -> A__ISNATILIST(IL) A__TAKE(0, IL) -> A__UTAKE1(a__isNatIList(IL)) A__TAKE(0, IL) -> A__ISNATILIST(IL) A__TAKE(s(M), cons(N, IL)) -> A__UTAKE2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) A__TAKE(s(M), cons(N, IL)) -> A__AND(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))) A__TAKE(s(M), cons(N, IL)) -> A__ISNAT(M) A__TAKE(s(M), cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__TAKE(s(M), cons(N, IL)) -> A__ISNAT(N) A__TAKE(s(M), cons(N, IL)) -> A__ISNATILIST(IL) A__UTAKE2(tt, M, N, IL) -> MARK(N) A__LENGTH(cons(N, L)) -> A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L) A__LENGTH(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__LENGTH(cons(N, L)) -> A__ISNAT(N) A__LENGTH(cons(N, L)) -> A__ISNATLIST(L) A__ULENGTH(tt, L) -> A__LENGTH(mark(L)) A__ULENGTH(tt, L) -> MARK(L) MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> A__ISNATILIST(X) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(zeros) -> A__ZEROS MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) MARK(uTake1(X)) -> A__UTAKE1(mark(X)) MARK(uTake1(X)) -> MARK(X) MARK(uTake2(X1, X2, X3, X4)) -> A__UTAKE2(mark(X1), X2, X3, X4) MARK(uTake2(X1, X2, X3, X4)) -> MARK(X1) MARK(uLength(X1, X2)) -> A__ULENGTH(mark(X1), X2) MARK(uLength(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) 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 3 less nodes. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) A__AND(tt, T) -> MARK(T) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> A__ISNATILIST(X) A__ISNATILIST(IL) -> A__ISNATLIST(IL) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNAT(N) A__ISNAT(s(N)) -> A__ISNAT(N) A__ISNAT(length(L)) -> A__ISNATLIST(L) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATLIST(take(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATLIST(take(N, IL)) -> A__ISNAT(N) A__ISNATLIST(take(N, IL)) -> A__ISNATILIST(IL) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNAT(N) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(length(X)) -> A__LENGTH(mark(X)) A__LENGTH(cons(N, L)) -> A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L) A__ULENGTH(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__LENGTH(cons(N, L)) -> A__ISNAT(N) A__LENGTH(cons(N, L)) -> A__ISNATLIST(L) A__ULENGTH(tt, L) -> MARK(L) MARK(length(X)) -> MARK(X) MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) A__TAKE(0, IL) -> A__ISNATILIST(IL) A__TAKE(s(M), cons(N, IL)) -> A__UTAKE2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) A__UTAKE2(tt, M, N, IL) -> MARK(N) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) MARK(uTake1(X)) -> MARK(X) MARK(uTake2(X1, X2, X3, X4)) -> A__UTAKE2(mark(X1), X2, X3, X4) MARK(uTake2(X1, X2, X3, X4)) -> MARK(X1) MARK(uLength(X1, X2)) -> A__ULENGTH(mark(X1), X2) MARK(uLength(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) A__TAKE(s(M), cons(N, IL)) -> A__AND(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))) A__TAKE(s(M), cons(N, IL)) -> A__ISNAT(M) A__TAKE(s(M), cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__TAKE(s(M), cons(N, IL)) -> A__ISNAT(N) A__TAKE(s(M), cons(N, IL)) -> A__ISNATILIST(IL) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) 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. A__LENGTH(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__LENGTH(cons(N, L)) -> A__ISNAT(N) A__LENGTH(cons(N, L)) -> A__ISNATLIST(L) A__ULENGTH(tt, L) -> MARK(L) MARK(length(X)) -> MARK(X) MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) A__TAKE(0, IL) -> A__ISNATILIST(IL) A__TAKE(s(M), cons(N, IL)) -> A__UTAKE2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) MARK(uTake2(X1, X2, X3, X4)) -> A__UTAKE2(mark(X1), X2, X3, X4) MARK(uTake2(X1, X2, X3, X4)) -> MARK(X1) MARK(uLength(X1, X2)) -> MARK(X1) A__TAKE(s(M), cons(N, IL)) -> A__AND(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))) A__TAKE(s(M), cons(N, IL)) -> A__ISNAT(M) A__TAKE(s(M), cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__TAKE(s(M), cons(N, IL)) -> A__ISNAT(N) A__TAKE(s(M), cons(N, IL)) -> A__ISNATILIST(IL) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__AND_2(x_1, x_2) ) = 2x_2 POL( A__LENGTH_1(x_1) ) = x_1 + 2 POL( A__TAKE_2(x_1, x_2) ) = 2x_2 + 1 POL( A__ULENGTH_2(x_1, x_2) ) = x_2 + 2 POL( A__UTAKE2_4(x_1, ..., x_4) ) = x_3 POL( mark_1(x_1) ) = x_1 POL( and_2(x_1, x_2) ) = x_1 + 2x_2 POL( a__and_2(x_1, x_2) ) = x_1 + 2x_2 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( a__isNatIList_1(x_1) ) = 0 POL( a__isNatList_1(x_1) ) = 0 POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( a__isNat_1(x_1) ) = 0 POL( take_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = x_1 POL( length_1(x_1) ) = x_1 + 2 POL( a__length_1(x_1) ) = x_1 + 2 POL( zeros ) = 0 POL( a__zeros ) = 0 POL( a__take_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( uTake1_1(x_1) ) = x_1 POL( a__uTake1_1(x_1) ) = x_1 POL( uTake2_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2x_4 + 2 POL( a__uTake2_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2x_4 + 2 POL( uLength_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( a__uLength_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( 0 ) = 0 POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 POL( A__ISNATILIST_1(x_1) ) = 0 POL( A__ISNATLIST_1(x_1) ) = 0 POL( A__ISNAT_1(x_1) ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) a__and(tt, T) -> mark(T) mark(isNatIList(X)) -> a__isNatIList(X) a__isNatIList(IL) -> a__isNatList(IL) a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__isNat(0) -> tt a__isNat(X) -> isNat(X) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__isNatIList(zeros) -> tt a__isNatIList(X) -> isNatIList(X) a__and(X1, X2) -> and(X1, X2) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(X) -> length(X) a__take(X1, X2) -> take(X1, X2) a__uTake1(tt) -> nil a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) a__uLength(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) A__AND(tt, T) -> MARK(T) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> A__ISNATILIST(X) A__ISNATILIST(IL) -> A__ISNATLIST(IL) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNAT(N) A__ISNAT(s(N)) -> A__ISNAT(N) A__ISNAT(length(L)) -> A__ISNATLIST(L) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATLIST(take(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATLIST(take(N, IL)) -> A__ISNAT(N) A__ISNATLIST(take(N, IL)) -> A__ISNATILIST(IL) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNAT(N) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(length(X)) -> A__LENGTH(mark(X)) A__LENGTH(cons(N, L)) -> A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L) A__ULENGTH(tt, L) -> A__LENGTH(mark(L)) A__UTAKE2(tt, M, N, IL) -> MARK(N) MARK(uTake1(X)) -> MARK(X) MARK(uLength(X1, X2)) -> A__ULENGTH(mark(X1), X2) MARK(s(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: A__ULENGTH(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented rules of the TRS R: a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(A__LENGTH(x_1)) = 2*x_1 POL(A__ULENGTH(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__and(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__isNat(x_1)) = 0 POL(a__isNatIList(x_1)) = 0 POL(a__isNatList(x_1)) = 0 POL(a__length(x_1)) = x_1 POL(a__take(x_1, x_2)) = 2 + x_1 + x_2 POL(a__uLength(x_1, x_2)) = x_1 + x_2 POL(a__uTake1(x_1)) = 2*x_1 POL(a__uTake2(x_1, x_2, x_3, x_4)) = 2 + x_1 + x_2 + x_3 + x_4 POL(a__zeros) = 0 POL(and(x_1, x_2)) = 2*x_1 + 2*x_2 POL(cons(x_1, x_2)) = x_1 + x_2 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(take(x_1, x_2)) = 2 + x_1 + x_2 POL(tt) = 0 POL(uLength(x_1, x_2)) = x_1 + x_2 POL(uTake1(x_1)) = 2*x_1 POL(uTake2(x_1, x_2, x_3, x_4)) = 2 + x_1 + x_2 + x_3 + x_4 POL(zeros) = 0 ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: A__ULENGTH(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented rules of the TRS R: a__uTake1(tt) -> nil Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(A__LENGTH(x_1)) = 2*x_1 POL(A__ULENGTH(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__and(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__isNat(x_1)) = 0 POL(a__isNatIList(x_1)) = 0 POL(a__isNatList(x_1)) = 0 POL(a__length(x_1)) = x_1 POL(a__take(x_1, x_2)) = x_1 + x_2 POL(a__uLength(x_1, x_2)) = x_1 + 2*x_2 POL(a__uTake1(x_1)) = 2 + 2*x_1 POL(a__uTake2(x_1, x_2, x_3, x_4)) = 2*x_1 + 2*x_2 + x_3 + 2*x_4 POL(a__zeros) = 0 POL(and(x_1, x_2)) = 2*x_1 + 2*x_2 POL(cons(x_1, x_2)) = x_1 + 2*x_2 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = 2*x_1 POL(take(x_1, x_2)) = x_1 + x_2 POL(tt) = 0 POL(uLength(x_1, x_2)) = x_1 + 2*x_2 POL(uTake1(x_1)) = 2 + 2*x_1 POL(uTake2(x_1, x_2, x_3, x_4)) = 2*x_1 + 2*x_2 + x_3 + 2*x_4 POL(zeros) = 0 ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: A__ULENGTH(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A__LENGTH(cons(N, L)) -> A__ULENGTH(a__and(a__isNat(N), a__isNatList(L)), L) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(A__ULENGTH(x_1, x_2)) = [[-I]] + [[1A]] * x_1 + [[1A]] * x_2 >>> <<< POL(tt) = [[2A]] >>> <<< POL(A__LENGTH(x_1)) = [[3A]] + [[1A]] * x_1 >>> <<< POL(mark(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(a__and(x_1, x_2)) = [[1A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(a__isNat(x_1)) = [[1A]] + [[1A]] * x_1 >>> <<< POL(a__isNatList(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(and(x_1, x_2)) = [[-I]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(isNatIList(x_1)) = [[-I]] + [[2A]] * x_1 >>> <<< POL(a__isNatIList(x_1)) = [[0A]] + [[2A]] * x_1 >>> <<< POL(take(x_1, x_2)) = [[3A]] + [[2A]] * x_1 + [[2A]] * x_2 >>> <<< POL(isNatList(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNat(x_1)) = [[0A]] + [[1A]] * x_1 >>> <<< POL(s(x_1)) = [[2A]] + [[1A]] * x_1 >>> <<< POL(length(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(a__length(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(zeros) = [[0A]] >>> <<< POL(a__zeros) = [[1A]] >>> <<< POL(a__take(x_1, x_2)) = [[3A]] + [[2A]] * x_1 + [[2A]] * x_2 >>> <<< POL(uTake1(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(a__uTake1(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(uTake2(x_1, x_2, x_3, x_4)) = [[4A]] + [[-I]] * x_1 + [[3A]] * x_2 + [[-I]] * x_3 + [[3A]] * x_4 >>> <<< POL(a__uTake2(x_1, x_2, x_3, x_4)) = [[4A]] + [[-I]] * x_1 + [[3A]] * x_2 + [[-I]] * x_3 + [[3A]] * x_4 >>> <<< POL(uLength(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(a__uLength(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(0) = [[1A]] >>> <<< POL(nil) = [[2A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) a__and(tt, T) -> mark(T) mark(isNatIList(X)) -> a__isNatIList(X) a__isNatIList(IL) -> a__isNatList(IL) a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__isNat(0) -> tt a__isNat(X) -> isNat(X) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__and(X1, X2) -> and(X1, X2) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__isNatIList(zeros) -> tt a__isNatIList(X) -> isNatIList(X) a__length(X) -> length(X) a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) a__uLength(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: A__ULENGTH(tt, L) -> A__LENGTH(mark(L)) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (17) TRUE ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: A__AND(tt, T) -> MARK(T) MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> A__ISNATILIST(X) A__ISNATILIST(IL) -> A__ISNATLIST(IL) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNAT(N) A__ISNAT(s(N)) -> A__ISNAT(N) A__ISNAT(length(L)) -> A__ISNATLIST(L) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATLIST(take(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATLIST(take(N, IL)) -> A__ISNAT(N) A__ISNATLIST(take(N, IL)) -> A__ISNATILIST(IL) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNAT(N) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(uTake1(X)) -> MARK(X) MARK(s(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(cons(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__AND_2(x_1, x_2) ) = x_2 POL( mark_1(x_1) ) = x_1 POL( and_2(x_1, x_2) ) = x_1 + 2x_2 POL( a__and_2(x_1, x_2) ) = x_1 + 2x_2 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( a__isNatIList_1(x_1) ) = 0 POL( a__isNatList_1(x_1) ) = 0 POL( cons_2(x_1, x_2) ) = 2x_1 + 2 POL( a__isNat_1(x_1) ) = 0 POL( take_2(x_1, x_2) ) = x_2 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = x_1 POL( length_1(x_1) ) = 2 POL( a__length_1(x_1) ) = 2 POL( zeros ) = 2 POL( a__zeros ) = 2 POL( a__take_2(x_1, x_2) ) = x_2 POL( uTake1_1(x_1) ) = x_1 POL( a__uTake1_1(x_1) ) = x_1 POL( uTake2_4(x_1, ..., x_4) ) = 2x_3 + 2 POL( a__uTake2_4(x_1, ..., x_4) ) = 2x_3 + 2 POL( uLength_2(x_1, x_2) ) = 2 POL( a__uLength_2(x_1, x_2) ) = 2 POL( 0 ) = 0 POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 POL( A__ISNATILIST_1(x_1) ) = 0 POL( A__ISNATLIST_1(x_1) ) = 0 POL( A__ISNAT_1(x_1) ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) a__and(tt, T) -> mark(T) mark(isNatIList(X)) -> a__isNatIList(X) a__isNatIList(IL) -> a__isNatList(IL) a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__isNat(0) -> tt a__isNat(X) -> isNat(X) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__isNatIList(zeros) -> tt a__isNatIList(X) -> isNatIList(X) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__and(X1, X2) -> and(X1, X2) a__length(X) -> length(X) a__take(X1, X2) -> take(X1, X2) a__uTake1(tt) -> nil a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) a__uLength(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: A__AND(tt, T) -> MARK(T) MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> A__ISNATILIST(X) A__ISNATILIST(IL) -> A__ISNATLIST(IL) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNAT(N) A__ISNAT(s(N)) -> A__ISNAT(N) A__ISNAT(length(L)) -> A__ISNATLIST(L) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATLIST(take(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATLIST(take(N, IL)) -> A__ISNAT(N) A__ISNATLIST(take(N, IL)) -> A__ISNATILIST(IL) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNAT(N) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(uTake1(X)) -> MARK(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(uTake1(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__AND_2(x_1, x_2) ) = 2x_2 + 2 POL( mark_1(x_1) ) = x_1 POL( and_2(x_1, x_2) ) = x_1 + 2x_2 POL( a__and_2(x_1, x_2) ) = x_1 + 2x_2 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( a__isNatIList_1(x_1) ) = 0 POL( a__isNatList_1(x_1) ) = 0 POL( cons_2(x_1, x_2) ) = max{0, -2} POL( a__isNat_1(x_1) ) = 0 POL( take_2(x_1, x_2) ) = 2 POL( isNatList_1(x_1) ) = 0 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = x_1 POL( length_1(x_1) ) = 0 POL( a__length_1(x_1) ) = max{0, -2} POL( zeros ) = 0 POL( a__zeros ) = 0 POL( a__take_2(x_1, x_2) ) = 2 POL( uTake1_1(x_1) ) = 2x_1 + 1 POL( a__uTake1_1(x_1) ) = 2x_1 + 1 POL( uTake2_4(x_1, ..., x_4) ) = 0 POL( a__uTake2_4(x_1, ..., x_4) ) = 0 POL( uLength_2(x_1, x_2) ) = 0 POL( a__uLength_2(x_1, x_2) ) = max{0, -2} POL( 0 ) = 0 POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 + 2 POL( A__ISNATILIST_1(x_1) ) = 2 POL( A__ISNATLIST_1(x_1) ) = 2 POL( A__ISNAT_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) a__and(tt, T) -> mark(T) mark(isNatIList(X)) -> a__isNatIList(X) a__isNatIList(IL) -> a__isNatList(IL) a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__isNat(0) -> tt a__isNat(X) -> isNat(X) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__isNatIList(zeros) -> tt a__isNatIList(X) -> isNatIList(X) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__and(X1, X2) -> and(X1, X2) a__length(X) -> length(X) a__take(X1, X2) -> take(X1, X2) a__uTake1(tt) -> nil a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) a__uLength(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: A__AND(tt, T) -> MARK(T) MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> A__ISNATILIST(X) A__ISNATILIST(IL) -> A__ISNATLIST(IL) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNAT(N) A__ISNAT(s(N)) -> A__ISNAT(N) A__ISNAT(length(L)) -> A__ISNATLIST(L) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATLIST(take(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATLIST(take(N, IL)) -> A__ISNAT(N) A__ISNATLIST(take(N, IL)) -> A__ISNATILIST(IL) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNAT(N) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A__ISNATLIST(take(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATLIST(take(N, IL)) -> A__ISNAT(N) A__ISNATLIST(take(N, IL)) -> A__ISNATILIST(IL) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__AND_2(x_1, x_2) ) = 2x_2 + 2 POL( mark_1(x_1) ) = x_1 POL( and_2(x_1, x_2) ) = 2x_1 + x_2 POL( a__and_2(x_1, x_2) ) = 2x_1 + x_2 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = x_1 POL( a__isNatIList_1(x_1) ) = x_1 POL( a__isNatList_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( a__isNat_1(x_1) ) = x_1 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 POL( isNatList_1(x_1) ) = x_1 POL( isNat_1(x_1) ) = x_1 POL( s_1(x_1) ) = x_1 POL( length_1(x_1) ) = 2x_1 POL( a__length_1(x_1) ) = 2x_1 POL( zeros ) = 0 POL( a__zeros ) = 0 POL( a__take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 POL( uTake1_1(x_1) ) = 0 POL( a__uTake1_1(x_1) ) = max{0, -2} POL( uTake2_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 1 POL( a__uTake2_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 1 POL( uLength_2(x_1, x_2) ) = 2x_2 POL( a__uLength_2(x_1, x_2) ) = 2x_2 POL( 0 ) = 0 POL( nil ) = 0 POL( MARK_1(x_1) ) = 2x_1 + 2 POL( A__ISNATILIST_1(x_1) ) = 2x_1 + 2 POL( A__ISNATLIST_1(x_1) ) = 2x_1 + 2 POL( A__ISNAT_1(x_1) ) = 2x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) a__and(tt, T) -> mark(T) mark(isNatIList(X)) -> a__isNatIList(X) a__isNatIList(IL) -> a__isNatList(IL) a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__isNat(0) -> tt a__isNat(X) -> isNat(X) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__isNatIList(zeros) -> tt a__isNatIList(X) -> isNatIList(X) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__and(X1, X2) -> and(X1, X2) a__length(X) -> length(X) a__take(X1, X2) -> take(X1, X2) a__uTake1(tt) -> nil a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) a__uLength(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: A__AND(tt, T) -> MARK(T) MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> A__ISNATILIST(X) A__ISNATILIST(IL) -> A__ISNATLIST(IL) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNAT(N) A__ISNAT(s(N)) -> A__ISNAT(N) A__ISNAT(length(L)) -> A__ISNATLIST(L) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNAT(N) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A__ISNAT(length(L)) -> A__ISNATLIST(L) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__AND_2(x_1, x_2) ) = x_2 + 2 POL( mark_1(x_1) ) = x_1 POL( and_2(x_1, x_2) ) = x_1 + x_2 POL( a__and_2(x_1, x_2) ) = x_1 + x_2 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 2x_1 POL( a__isNatIList_1(x_1) ) = 2x_1 POL( a__isNatList_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = x_1 + x_2 POL( a__isNat_1(x_1) ) = x_1 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( isNatList_1(x_1) ) = x_1 POL( isNat_1(x_1) ) = x_1 POL( s_1(x_1) ) = x_1 POL( length_1(x_1) ) = x_1 + 1 POL( a__length_1(x_1) ) = x_1 + 1 POL( zeros ) = 0 POL( a__zeros ) = 0 POL( a__take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( uTake1_1(x_1) ) = 0 POL( a__uTake1_1(x_1) ) = max{0, -2} POL( uTake2_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( a__uTake2_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( uLength_2(x_1, x_2) ) = x_2 + 1 POL( a__uLength_2(x_1, x_2) ) = x_2 + 1 POL( 0 ) = 0 POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 + 2 POL( A__ISNATILIST_1(x_1) ) = 2x_1 + 2 POL( A__ISNATLIST_1(x_1) ) = x_1 + 2 POL( A__ISNAT_1(x_1) ) = x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) a__and(tt, T) -> mark(T) mark(isNatIList(X)) -> a__isNatIList(X) a__isNatIList(IL) -> a__isNatList(IL) a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__isNat(0) -> tt a__isNat(X) -> isNat(X) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__isNatIList(zeros) -> tt a__isNatIList(X) -> isNatIList(X) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__and(X1, X2) -> and(X1, X2) a__length(X) -> length(X) a__take(X1, X2) -> take(X1, X2) a__uTake1(tt) -> nil a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) a__uLength(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: A__AND(tt, T) -> MARK(T) MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> A__ISNATILIST(X) A__ISNATILIST(IL) -> A__ISNATLIST(IL) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNAT(N) A__ISNAT(s(N)) -> A__ISNAT(N) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNAT(N) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(isNat(X)) -> A__ISNAT(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (28) Complex Obligation (AND) ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: A__ISNAT(s(N)) -> A__ISNAT(N) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: A__ISNAT(s(N)) -> A__ISNAT(N) R is empty. The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: A__ISNAT(s(N)) -> A__ISNAT(N) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *A__ISNAT(s(N)) -> A__ISNAT(N) The graph contains the following edges 1 > 1 ---------------------------------------- (35) YES ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) A__AND(tt, T) -> MARK(T) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> A__ISNATILIST(X) A__ISNATILIST(IL) -> A__ISNATLIST(IL) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A__ISNATILIST(IL) -> A__ISNATLIST(IL) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__AND_2(x_1, x_2) ) = x_2 POL( mark_1(x_1) ) = x_1 POL( and_2(x_1, x_2) ) = x_1 + x_2 POL( a__and_2(x_1, x_2) ) = x_1 + x_2 POL( tt ) = 0 POL( isNatIList_1(x_1) ) = 2x_1 + 2 POL( a__isNatIList_1(x_1) ) = 2x_1 + 2 POL( a__isNatList_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( a__isNat_1(x_1) ) = 2x_1 POL( take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( isNatList_1(x_1) ) = x_1 POL( isNat_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = x_1 POL( length_1(x_1) ) = 2x_1 POL( a__length_1(x_1) ) = 2x_1 POL( zeros ) = 0 POL( a__zeros ) = 0 POL( a__take_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( uTake1_1(x_1) ) = 0 POL( a__uTake1_1(x_1) ) = max{0, -2} POL( uTake2_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( a__uTake2_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 2x_4 + 2 POL( uLength_2(x_1, x_2) ) = 2x_2 POL( a__uLength_2(x_1, x_2) ) = 2x_2 POL( 0 ) = 0 POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 POL( A__ISNATILIST_1(x_1) ) = 2x_1 + 2 POL( A__ISNATLIST_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) a__and(tt, T) -> mark(T) mark(isNatIList(X)) -> a__isNatIList(X) a__isNatIList(IL) -> a__isNatList(IL) a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__isNat(0) -> tt a__isNat(X) -> isNat(X) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__isNatIList(zeros) -> tt a__isNatIList(X) -> isNatIList(X) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__and(X1, X2) -> and(X1, X2) a__length(X) -> length(X) a__take(X1, X2) -> take(X1, X2) a__uTake1(tt) -> nil a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) a__uLength(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) A__AND(tt, T) -> MARK(T) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> A__ISNATILIST(X) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__uTake1(tt) -> nil a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented rules of the TRS R: a__uTake1(tt) -> nil Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(A__AND(x_1, x_2)) = 2*x_1 + 2*x_2 POL(A__ISNATILIST(x_1)) = 0 POL(A__ISNATLIST(x_1)) = 0 POL(MARK(x_1)) = 2*x_1 POL(a__and(x_1, x_2)) = 2*x_1 + x_2 POL(a__isNat(x_1)) = 0 POL(a__isNatIList(x_1)) = 0 POL(a__isNatList(x_1)) = 0 POL(a__length(x_1)) = 2*x_1 POL(a__take(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(a__uLength(x_1, x_2)) = 2*x_1 + 2*x_2 POL(a__uTake1(x_1)) = 2 + 2*x_1 POL(a__uTake2(x_1, x_2, x_3, x_4)) = 2 + x_1 + 2*x_2 + x_3 + x_4 POL(a__zeros) = 0 POL(and(x_1, x_2)) = 2*x_1 + x_2 POL(cons(x_1, x_2)) = x_1 + x_2 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = 2*x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(take(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(tt) = 0 POL(uLength(x_1, x_2)) = 2*x_1 + 2*x_2 POL(uTake1(x_1)) = 2 + 2*x_1 POL(uTake2(x_1, x_2, x_3, x_4)) = 2 + x_1 + 2*x_2 + x_3 + x_4 POL(zeros) = 0 ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) A__AND(tt, T) -> MARK(T) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> A__ISNATILIST(X) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented rules of the TRS R: a__take(0, IL) -> a__uTake1(a__isNatIList(IL)) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(A__AND(x_1, x_2)) = x_1 + x_2 POL(A__ISNATILIST(x_1)) = 0 POL(A__ISNATLIST(x_1)) = 0 POL(MARK(x_1)) = x_1 POL(a__and(x_1, x_2)) = x_1 + 2*x_2 POL(a__isNat(x_1)) = 0 POL(a__isNatIList(x_1)) = 0 POL(a__isNatList(x_1)) = 0 POL(a__length(x_1)) = x_1 POL(a__take(x_1, x_2)) = 2 + x_1 + x_2 POL(a__uLength(x_1, x_2)) = x_1 + x_2 POL(a__uTake1(x_1)) = 2*x_1 POL(a__uTake2(x_1, x_2, x_3, x_4)) = 2 + 2*x_1 + x_2 + x_3 + x_4 POL(a__zeros) = 0 POL(and(x_1, x_2)) = x_1 + 2*x_2 POL(cons(x_1, x_2)) = x_1 + x_2 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(take(x_1, x_2)) = 2 + x_1 + x_2 POL(tt) = 0 POL(uLength(x_1, x_2)) = x_1 + x_2 POL(uTake1(x_1)) = 2*x_1 POL(uTake2(x_1, x_2, x_3, x_4)) = 2 + 2*x_1 + x_2 + x_3 + x_4 POL(zeros) = 0 ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) A__AND(tt, T) -> MARK(T) MARK(and(X1, X2)) -> MARK(X1) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> A__ISNATILIST(X) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(and(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(MARK(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(and(x_1, x_2)) = [[-I]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(A__AND(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[0A]] * x_2 >>> <<< POL(mark(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(tt) = [[0A]] >>> <<< POL(isNatIList(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(A__ISNATILIST(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(A__ISNATLIST(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[-I]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(a__isNat(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(a__isNatList(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(a__isNatIList(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(isNatList(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(a__and(x_1, x_2)) = [[-I]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(take(x_1, x_2)) = [[0A]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(isNat(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(length(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(a__length(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(zeros) = [[2A]] >>> <<< POL(a__zeros) = [[2A]] >>> <<< POL(a__take(x_1, x_2)) = [[0A]] + [[1A]] * x_1 + [[0A]] * x_2 >>> <<< POL(uTake1(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(a__uTake1(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(uTake2(x_1, x_2, x_3, x_4)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 + [[0A]] * x_4 >>> <<< POL(a__uTake2(x_1, x_2, x_3, x_4)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[1A]] * x_3 + [[0A]] * x_4 >>> <<< POL(uLength(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(a__uLength(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[0A]] * x_2 >>> <<< POL(0) = [[1A]] >>> <<< POL(nil) = [[0A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) a__and(tt, T) -> mark(T) mark(isNatIList(X)) -> a__isNatIList(X) a__isNatIList(IL) -> a__isNatList(IL) a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__isNat(0) -> tt a__isNat(X) -> isNat(X) a__isNatList(nil) -> tt a__isNatList(X) -> isNatList(X) a__isNatIList(zeros) -> tt a__isNatIList(X) -> isNatIList(X) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__and(X1, X2) -> and(X1, X2) a__length(X) -> length(X) a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) a__uLength(tt, L) -> s(a__length(mark(L))) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) A__AND(tt, T) -> MARK(T) MARK(and(X1, X2)) -> MARK(X2) MARK(isNatIList(X)) -> A__ISNATILIST(X) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: A__AND(tt, T) -> MARK(T) MARK(and(X1, X2)) -> MARK(X2) A__ISNATLIST(cons(N, L)) -> A__AND(a__isNat(N), a__isNatList(L)) A__ISNATLIST(cons(N, L)) -> A__ISNATLIST(L) A__ISNATILIST(cons(N, IL)) -> A__AND(a__isNat(N), a__isNatIList(IL)) A__ISNATILIST(cons(N, IL)) -> A__ISNATILIST(IL) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(A__AND(x_1, x_2)) = 1 + x_2 POL(A__ISNATILIST(x_1)) = 2*x_1 POL(A__ISNATLIST(x_1)) = 2*x_1 POL(MARK(x_1)) = x_1 POL(a__and(x_1, x_2)) = 2 + x_2 POL(a__isNat(x_1)) = 2*x_1 POL(a__isNatIList(x_1)) = 2*x_1 POL(a__isNatList(x_1)) = 2*x_1 POL(a__length(x_1)) = x_1 POL(a__take(x_1, x_2)) = 1 + x_2 POL(a__uLength(x_1, x_2)) = 1 + x_2 POL(a__uTake1(x_1)) = 1 POL(a__uTake2(x_1, x_2, x_3, x_4)) = 2 + x_4 POL(a__zeros) = 1 POL(and(x_1, x_2)) = 2 + x_2 POL(cons(x_1, x_2)) = 1 + x_2 POL(isNat(x_1)) = 2*x_1 POL(isNatIList(x_1)) = 2*x_1 POL(isNatList(x_1)) = 2*x_1 POL(length(x_1)) = x_1 POL(mark(x_1)) = 1 + x_1 POL(nil) = 2 POL(s(x_1)) = x_1 POL(take(x_1, x_2)) = 1 + x_2 POL(tt) = 0 POL(uLength(x_1, x_2)) = x_2 POL(uTake1(x_1)) = 0 POL(uTake2(x_1, x_2, x_3, x_4)) = 1 + x_4 POL(zeros) = 0 ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(and(X1, X2)) -> A__AND(mark(X1), mark(X2)) MARK(isNatIList(X)) -> A__ISNATILIST(X) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__and(tt, T) -> mark(T) a__isNatIList(IL) -> a__isNatList(IL) a__isNat(0) -> tt a__isNat(s(N)) -> a__isNat(N) a__isNat(length(L)) -> a__isNatList(L) a__isNatIList(zeros) -> tt a__isNatIList(cons(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__isNatList(nil) -> tt a__isNatList(cons(N, L)) -> a__and(a__isNat(N), a__isNatList(L)) a__isNatList(take(N, IL)) -> a__and(a__isNat(N), a__isNatIList(IL)) a__zeros -> cons(0, zeros) a__take(s(M), cons(N, IL)) -> a__uTake2(a__and(a__isNat(M), a__and(a__isNat(N), a__isNatIList(IL))), M, N, IL) a__uTake2(tt, M, N, IL) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__uLength(a__and(a__isNat(N), a__isNatList(L)), L) a__uLength(tt, L) -> s(a__length(mark(L))) mark(and(X1, X2)) -> a__and(mark(X1), mark(X2)) mark(isNatIList(X)) -> a__isNatIList(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNat(X)) -> a__isNat(X) mark(length(X)) -> a__length(mark(X)) mark(zeros) -> a__zeros mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(uTake1(X)) -> a__uTake1(mark(X)) mark(uTake2(X1, X2, X3, X4)) -> a__uTake2(mark(X1), X2, X3, X4) mark(uLength(X1, X2)) -> a__uLength(mark(X1), X2) mark(tt) -> tt mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(nil) -> nil a__and(X1, X2) -> and(X1, X2) a__isNatIList(X) -> isNatIList(X) a__isNatList(X) -> isNatList(X) a__isNat(X) -> isNat(X) a__length(X) -> length(X) a__zeros -> zeros a__take(X1, X2) -> take(X1, X2) a__uTake1(X) -> uTake1(X) a__uTake2(X1, X2, X3, X4) -> uTake2(X1, X2, X3, X4) a__uLength(X1, X2) -> uLength(X1, X2) The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(s(X)) -> MARK(X) R is empty. The set Q consists of the following terms: a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. a__isNatIList(x0) a__zeros mark(and(x0, x1)) mark(isNatIList(x0)) mark(isNatList(x0)) mark(isNat(x0)) mark(length(x0)) mark(zeros) mark(take(x0, x1)) mark(uTake1(x0)) mark(uTake2(x0, x1, x2, x3)) mark(uLength(x0, x1)) mark(tt) mark(0) mark(s(x0)) mark(cons(x0, x1)) mark(nil) a__and(x0, x1) a__isNatList(x0) a__isNat(x0) a__length(x0) a__take(x0, x1) a__uTake1(x0) a__uTake2(x0, x1, x2, x3) a__uLength(x0, x1) ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(s(X)) -> MARK(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MARK(s(X)) -> MARK(X) The graph contains the following edges 1 > 1 ---------------------------------------- (54) YES