6.64/2.67 YES 6.98/2.68 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 6.98/2.68 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 6.98/2.68 6.98/2.68 6.98/2.68 Termination w.r.t. Q of the given QTRS could be proven: 6.98/2.68 6.98/2.68 (0) QTRS 6.98/2.68 (1) DependencyPairsProof [EQUIVALENT, 60 ms] 6.98/2.68 (2) QDP 6.98/2.68 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 6.98/2.68 (4) AND 6.98/2.68 (5) QDP 6.98/2.68 (6) UsableRulesProof [EQUIVALENT, 0 ms] 6.98/2.68 (7) QDP 6.98/2.68 (8) QReductionProof [EQUIVALENT, 0 ms] 6.98/2.68 (9) QDP 6.98/2.68 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 6.98/2.68 (11) YES 6.98/2.68 (12) QDP 6.98/2.68 (13) QDPOrderProof [EQUIVALENT, 114 ms] 6.98/2.68 (14) QDP 6.98/2.68 (15) DependencyGraphProof [EQUIVALENT, 0 ms] 6.98/2.68 (16) QDP 6.98/2.68 (17) UsableRulesProof [EQUIVALENT, 0 ms] 6.98/2.68 (18) QDP 6.98/2.68 (19) QReductionProof [EQUIVALENT, 0 ms] 6.98/2.68 (20) QDP 6.98/2.68 (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] 6.98/2.68 (22) YES 6.98/2.68 6.98/2.68 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (0) 6.98/2.68 Obligation: 6.98/2.68 Q restricted rewrite system: 6.98/2.68 The TRS R consists of the following rules: 6.98/2.68 6.98/2.68 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 6.98/2.68 a__U12(tt) -> tt 6.98/2.68 a__U21(tt) -> tt 6.98/2.68 a__U31(tt, N) -> mark(N) 6.98/2.68 a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N) 6.98/2.68 a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M))) 6.98/2.68 a__isNat(0) -> tt 6.98/2.68 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 6.98/2.68 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 6.98/2.68 a__plus(N, 0) -> a__U31(a__isNat(N), N) 6.98/2.68 a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) 6.98/2.68 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 6.98/2.68 mark(U12(X)) -> a__U12(mark(X)) 6.98/2.68 mark(isNat(X)) -> a__isNat(X) 6.98/2.68 mark(U21(X)) -> a__U21(mark(X)) 6.98/2.68 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 6.98/2.68 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 6.98/2.68 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 6.98/2.68 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 6.98/2.68 mark(tt) -> tt 6.98/2.68 mark(s(X)) -> s(mark(X)) 6.98/2.68 mark(0) -> 0 6.98/2.68 a__U11(X1, X2) -> U11(X1, X2) 6.98/2.68 a__U12(X) -> U12(X) 6.98/2.68 a__isNat(X) -> isNat(X) 6.98/2.68 a__U21(X) -> U21(X) 6.98/2.68 a__U31(X1, X2) -> U31(X1, X2) 6.98/2.68 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 6.98/2.68 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 6.98/2.68 a__plus(X1, X2) -> plus(X1, X2) 6.98/2.68 6.98/2.68 The set Q consists of the following terms: 6.98/2.68 6.98/2.68 mark(U11(x0, x1)) 6.98/2.68 mark(U12(x0)) 6.98/2.68 mark(isNat(x0)) 6.98/2.68 mark(U21(x0)) 6.98/2.68 mark(U31(x0, x1)) 6.98/2.68 mark(U41(x0, x1, x2)) 6.98/2.68 mark(U42(x0, x1, x2)) 6.98/2.68 mark(plus(x0, x1)) 6.98/2.68 mark(tt) 6.98/2.68 mark(s(x0)) 6.98/2.68 mark(0) 6.98/2.68 a__U11(x0, x1) 6.98/2.68 a__U12(x0) 6.98/2.68 a__isNat(x0) 6.98/2.68 a__U21(x0) 6.98/2.68 a__U31(x0, x1) 6.98/2.68 a__U41(x0, x1, x2) 6.98/2.68 a__U42(x0, x1, x2) 6.98/2.68 a__plus(x0, x1) 6.98/2.68 6.98/2.68 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (1) DependencyPairsProof (EQUIVALENT) 6.98/2.68 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (2) 6.98/2.68 Obligation: 6.98/2.68 Q DP problem: 6.98/2.68 The TRS P consists of the following rules: 6.98/2.68 6.98/2.68 A__U11(tt, V2) -> A__U12(a__isNat(V2)) 6.98/2.68 A__U11(tt, V2) -> A__ISNAT(V2) 6.98/2.68 A__U31(tt, N) -> MARK(N) 6.98/2.68 A__U41(tt, M, N) -> A__U42(a__isNat(N), M, N) 6.98/2.68 A__U41(tt, M, N) -> A__ISNAT(N) 6.98/2.68 A__U42(tt, M, N) -> A__PLUS(mark(N), mark(M)) 6.98/2.68 A__U42(tt, M, N) -> MARK(N) 6.98/2.68 A__U42(tt, M, N) -> MARK(M) 6.98/2.68 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNat(V1), V2) 6.98/2.68 A__ISNAT(plus(V1, V2)) -> A__ISNAT(V1) 6.98/2.68 A__ISNAT(s(V1)) -> A__U21(a__isNat(V1)) 6.98/2.68 A__ISNAT(s(V1)) -> A__ISNAT(V1) 6.98/2.68 A__PLUS(N, 0) -> A__U31(a__isNat(N), N) 6.98/2.68 A__PLUS(N, 0) -> A__ISNAT(N) 6.98/2.68 A__PLUS(N, s(M)) -> A__U41(a__isNat(M), M, N) 6.98/2.68 A__PLUS(N, s(M)) -> A__ISNAT(M) 6.98/2.68 MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) 6.98/2.68 MARK(U11(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(U12(X)) -> A__U12(mark(X)) 6.98/2.68 MARK(U12(X)) -> MARK(X) 6.98/2.68 MARK(isNat(X)) -> A__ISNAT(X) 6.98/2.68 MARK(U21(X)) -> A__U21(mark(X)) 6.98/2.68 MARK(U21(X)) -> MARK(X) 6.98/2.68 MARK(U31(X1, X2)) -> A__U31(mark(X1), X2) 6.98/2.68 MARK(U31(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(U41(X1, X2, X3)) -> A__U41(mark(X1), X2, X3) 6.98/2.68 MARK(U41(X1, X2, X3)) -> MARK(X1) 6.98/2.68 MARK(U42(X1, X2, X3)) -> A__U42(mark(X1), X2, X3) 6.98/2.68 MARK(U42(X1, X2, X3)) -> MARK(X1) 6.98/2.68 MARK(plus(X1, X2)) -> A__PLUS(mark(X1), mark(X2)) 6.98/2.68 MARK(plus(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(plus(X1, X2)) -> MARK(X2) 6.98/2.68 MARK(s(X)) -> MARK(X) 6.98/2.68 6.98/2.68 The TRS R consists of the following rules: 6.98/2.68 6.98/2.68 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 6.98/2.68 a__U12(tt) -> tt 6.98/2.68 a__U21(tt) -> tt 6.98/2.68 a__U31(tt, N) -> mark(N) 6.98/2.68 a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N) 6.98/2.68 a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M))) 6.98/2.68 a__isNat(0) -> tt 6.98/2.68 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 6.98/2.68 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 6.98/2.68 a__plus(N, 0) -> a__U31(a__isNat(N), N) 6.98/2.68 a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) 6.98/2.68 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 6.98/2.68 mark(U12(X)) -> a__U12(mark(X)) 6.98/2.68 mark(isNat(X)) -> a__isNat(X) 6.98/2.68 mark(U21(X)) -> a__U21(mark(X)) 6.98/2.68 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 6.98/2.68 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 6.98/2.68 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 6.98/2.68 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 6.98/2.68 mark(tt) -> tt 6.98/2.68 mark(s(X)) -> s(mark(X)) 6.98/2.68 mark(0) -> 0 6.98/2.68 a__U11(X1, X2) -> U11(X1, X2) 6.98/2.68 a__U12(X) -> U12(X) 6.98/2.68 a__isNat(X) -> isNat(X) 6.98/2.68 a__U21(X) -> U21(X) 6.98/2.68 a__U31(X1, X2) -> U31(X1, X2) 6.98/2.68 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 6.98/2.68 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 6.98/2.68 a__plus(X1, X2) -> plus(X1, X2) 6.98/2.68 6.98/2.68 The set Q consists of the following terms: 6.98/2.68 6.98/2.68 mark(U11(x0, x1)) 6.98/2.68 mark(U12(x0)) 6.98/2.68 mark(isNat(x0)) 6.98/2.68 mark(U21(x0)) 6.98/2.68 mark(U31(x0, x1)) 6.98/2.68 mark(U41(x0, x1, x2)) 6.98/2.68 mark(U42(x0, x1, x2)) 6.98/2.68 mark(plus(x0, x1)) 6.98/2.68 mark(tt) 6.98/2.68 mark(s(x0)) 6.98/2.68 mark(0) 6.98/2.68 a__U11(x0, x1) 6.98/2.68 a__U12(x0) 6.98/2.68 a__isNat(x0) 6.98/2.68 a__U21(x0) 6.98/2.68 a__U31(x0, x1) 6.98/2.68 a__U41(x0, x1, x2) 6.98/2.68 a__U42(x0, x1, x2) 6.98/2.68 a__plus(x0, x1) 6.98/2.68 6.98/2.68 We have to consider all minimal (P,Q,R)-chains. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (3) DependencyGraphProof (EQUIVALENT) 6.98/2.68 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 9 less nodes. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (4) 6.98/2.68 Complex Obligation (AND) 6.98/2.68 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (5) 6.98/2.68 Obligation: 6.98/2.68 Q DP problem: 6.98/2.68 The TRS P consists of the following rules: 6.98/2.68 6.98/2.68 A__U11(tt, V2) -> A__ISNAT(V2) 6.98/2.68 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNat(V1), V2) 6.98/2.68 A__ISNAT(plus(V1, V2)) -> A__ISNAT(V1) 6.98/2.68 A__ISNAT(s(V1)) -> A__ISNAT(V1) 6.98/2.68 6.98/2.68 The TRS R consists of the following rules: 6.98/2.68 6.98/2.68 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 6.98/2.68 a__U12(tt) -> tt 6.98/2.68 a__U21(tt) -> tt 6.98/2.68 a__U31(tt, N) -> mark(N) 6.98/2.68 a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N) 6.98/2.68 a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M))) 6.98/2.68 a__isNat(0) -> tt 6.98/2.68 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 6.98/2.68 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 6.98/2.68 a__plus(N, 0) -> a__U31(a__isNat(N), N) 6.98/2.68 a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) 6.98/2.68 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 6.98/2.68 mark(U12(X)) -> a__U12(mark(X)) 6.98/2.68 mark(isNat(X)) -> a__isNat(X) 6.98/2.68 mark(U21(X)) -> a__U21(mark(X)) 6.98/2.68 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 6.98/2.68 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 6.98/2.68 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 6.98/2.68 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 6.98/2.68 mark(tt) -> tt 6.98/2.68 mark(s(X)) -> s(mark(X)) 6.98/2.68 mark(0) -> 0 6.98/2.68 a__U11(X1, X2) -> U11(X1, X2) 6.98/2.68 a__U12(X) -> U12(X) 6.98/2.68 a__isNat(X) -> isNat(X) 6.98/2.68 a__U21(X) -> U21(X) 6.98/2.68 a__U31(X1, X2) -> U31(X1, X2) 6.98/2.68 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 6.98/2.68 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 6.98/2.68 a__plus(X1, X2) -> plus(X1, X2) 6.98/2.68 6.98/2.68 The set Q consists of the following terms: 6.98/2.68 6.98/2.68 mark(U11(x0, x1)) 6.98/2.68 mark(U12(x0)) 6.98/2.68 mark(isNat(x0)) 6.98/2.68 mark(U21(x0)) 6.98/2.68 mark(U31(x0, x1)) 6.98/2.68 mark(U41(x0, x1, x2)) 6.98/2.68 mark(U42(x0, x1, x2)) 6.98/2.68 mark(plus(x0, x1)) 6.98/2.68 mark(tt) 6.98/2.68 mark(s(x0)) 6.98/2.68 mark(0) 6.98/2.68 a__U11(x0, x1) 6.98/2.68 a__U12(x0) 6.98/2.68 a__isNat(x0) 6.98/2.68 a__U21(x0) 6.98/2.68 a__U31(x0, x1) 6.98/2.68 a__U41(x0, x1, x2) 6.98/2.68 a__U42(x0, x1, x2) 6.98/2.68 a__plus(x0, x1) 6.98/2.68 6.98/2.68 We have to consider all minimal (P,Q,R)-chains. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (6) UsableRulesProof (EQUIVALENT) 6.98/2.68 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. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (7) 6.98/2.68 Obligation: 6.98/2.68 Q DP problem: 6.98/2.68 The TRS P consists of the following rules: 6.98/2.68 6.98/2.68 A__U11(tt, V2) -> A__ISNAT(V2) 6.98/2.68 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNat(V1), V2) 6.98/2.68 A__ISNAT(plus(V1, V2)) -> A__ISNAT(V1) 6.98/2.68 A__ISNAT(s(V1)) -> A__ISNAT(V1) 6.98/2.68 6.98/2.68 The TRS R consists of the following rules: 6.98/2.68 6.98/2.68 a__isNat(0) -> tt 6.98/2.68 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 6.98/2.68 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 6.98/2.68 a__isNat(X) -> isNat(X) 6.98/2.68 a__U21(tt) -> tt 6.98/2.68 a__U21(X) -> U21(X) 6.98/2.68 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 6.98/2.68 a__U11(X1, X2) -> U11(X1, X2) 6.98/2.68 a__U12(tt) -> tt 6.98/2.68 a__U12(X) -> U12(X) 6.98/2.68 6.98/2.68 The set Q consists of the following terms: 6.98/2.68 6.98/2.68 mark(U11(x0, x1)) 6.98/2.68 mark(U12(x0)) 6.98/2.68 mark(isNat(x0)) 6.98/2.68 mark(U21(x0)) 6.98/2.68 mark(U31(x0, x1)) 6.98/2.68 mark(U41(x0, x1, x2)) 6.98/2.68 mark(U42(x0, x1, x2)) 6.98/2.68 mark(plus(x0, x1)) 6.98/2.68 mark(tt) 6.98/2.68 mark(s(x0)) 6.98/2.68 mark(0) 6.98/2.68 a__U11(x0, x1) 6.98/2.68 a__U12(x0) 6.98/2.68 a__isNat(x0) 6.98/2.68 a__U21(x0) 6.98/2.68 a__U31(x0, x1) 6.98/2.68 a__U41(x0, x1, x2) 6.98/2.68 a__U42(x0, x1, x2) 6.98/2.68 a__plus(x0, x1) 6.98/2.68 6.98/2.68 We have to consider all minimal (P,Q,R)-chains. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (8) QReductionProof (EQUIVALENT) 6.98/2.68 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 6.98/2.68 6.98/2.68 mark(U11(x0, x1)) 6.98/2.68 mark(U12(x0)) 6.98/2.68 mark(isNat(x0)) 6.98/2.68 mark(U21(x0)) 6.98/2.68 mark(U31(x0, x1)) 6.98/2.68 mark(U41(x0, x1, x2)) 6.98/2.68 mark(U42(x0, x1, x2)) 6.98/2.68 mark(plus(x0, x1)) 6.98/2.68 mark(tt) 6.98/2.68 mark(s(x0)) 6.98/2.68 mark(0) 6.98/2.68 a__U31(x0, x1) 6.98/2.68 a__U41(x0, x1, x2) 6.98/2.68 a__U42(x0, x1, x2) 6.98/2.68 a__plus(x0, x1) 6.98/2.68 6.98/2.68 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (9) 6.98/2.68 Obligation: 6.98/2.68 Q DP problem: 6.98/2.68 The TRS P consists of the following rules: 6.98/2.68 6.98/2.68 A__U11(tt, V2) -> A__ISNAT(V2) 6.98/2.68 A__ISNAT(plus(V1, V2)) -> A__U11(a__isNat(V1), V2) 6.98/2.68 A__ISNAT(plus(V1, V2)) -> A__ISNAT(V1) 6.98/2.68 A__ISNAT(s(V1)) -> A__ISNAT(V1) 6.98/2.68 6.98/2.68 The TRS R consists of the following rules: 6.98/2.68 6.98/2.68 a__isNat(0) -> tt 6.98/2.68 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 6.98/2.68 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 6.98/2.68 a__isNat(X) -> isNat(X) 6.98/2.68 a__U21(tt) -> tt 6.98/2.68 a__U21(X) -> U21(X) 6.98/2.68 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 6.98/2.68 a__U11(X1, X2) -> U11(X1, X2) 6.98/2.68 a__U12(tt) -> tt 6.98/2.68 a__U12(X) -> U12(X) 6.98/2.68 6.98/2.68 The set Q consists of the following terms: 6.98/2.68 6.98/2.68 a__U11(x0, x1) 6.98/2.68 a__U12(x0) 6.98/2.68 a__isNat(x0) 6.98/2.68 a__U21(x0) 6.98/2.68 6.98/2.68 We have to consider all minimal (P,Q,R)-chains. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (10) QDPSizeChangeProof (EQUIVALENT) 6.98/2.68 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. 6.98/2.68 6.98/2.68 From the DPs we obtained the following set of size-change graphs: 6.98/2.68 *A__ISNAT(plus(V1, V2)) -> A__U11(a__isNat(V1), V2) 6.98/2.68 The graph contains the following edges 1 > 2 6.98/2.68 6.98/2.68 6.98/2.68 *A__U11(tt, V2) -> A__ISNAT(V2) 6.98/2.68 The graph contains the following edges 2 >= 1 6.98/2.68 6.98/2.68 6.98/2.68 *A__ISNAT(plus(V1, V2)) -> A__ISNAT(V1) 6.98/2.68 The graph contains the following edges 1 > 1 6.98/2.68 6.98/2.68 6.98/2.68 *A__ISNAT(s(V1)) -> A__ISNAT(V1) 6.98/2.68 The graph contains the following edges 1 > 1 6.98/2.68 6.98/2.68 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (11) 6.98/2.68 YES 6.98/2.68 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (12) 6.98/2.68 Obligation: 6.98/2.68 Q DP problem: 6.98/2.68 The TRS P consists of the following rules: 6.98/2.68 6.98/2.68 MARK(U11(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(U12(X)) -> MARK(X) 6.98/2.68 MARK(U21(X)) -> MARK(X) 6.98/2.68 MARK(U31(X1, X2)) -> A__U31(mark(X1), X2) 6.98/2.68 A__U31(tt, N) -> MARK(N) 6.98/2.68 MARK(U31(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(U41(X1, X2, X3)) -> A__U41(mark(X1), X2, X3) 6.98/2.68 A__U41(tt, M, N) -> A__U42(a__isNat(N), M, N) 6.98/2.68 A__U42(tt, M, N) -> A__PLUS(mark(N), mark(M)) 6.98/2.68 A__PLUS(N, 0) -> A__U31(a__isNat(N), N) 6.98/2.68 A__PLUS(N, s(M)) -> A__U41(a__isNat(M), M, N) 6.98/2.68 A__U42(tt, M, N) -> MARK(N) 6.98/2.68 MARK(U41(X1, X2, X3)) -> MARK(X1) 6.98/2.68 MARK(U42(X1, X2, X3)) -> A__U42(mark(X1), X2, X3) 6.98/2.68 A__U42(tt, M, N) -> MARK(M) 6.98/2.68 MARK(U42(X1, X2, X3)) -> MARK(X1) 6.98/2.68 MARK(plus(X1, X2)) -> A__PLUS(mark(X1), mark(X2)) 6.98/2.68 MARK(plus(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(plus(X1, X2)) -> MARK(X2) 6.98/2.68 MARK(s(X)) -> MARK(X) 6.98/2.68 6.98/2.68 The TRS R consists of the following rules: 6.98/2.68 6.98/2.68 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 6.98/2.68 a__U12(tt) -> tt 6.98/2.68 a__U21(tt) -> tt 6.98/2.68 a__U31(tt, N) -> mark(N) 6.98/2.68 a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N) 6.98/2.68 a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M))) 6.98/2.68 a__isNat(0) -> tt 6.98/2.68 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 6.98/2.68 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 6.98/2.68 a__plus(N, 0) -> a__U31(a__isNat(N), N) 6.98/2.68 a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) 6.98/2.68 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 6.98/2.68 mark(U12(X)) -> a__U12(mark(X)) 6.98/2.68 mark(isNat(X)) -> a__isNat(X) 6.98/2.68 mark(U21(X)) -> a__U21(mark(X)) 6.98/2.68 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 6.98/2.68 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 6.98/2.68 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 6.98/2.68 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 6.98/2.68 mark(tt) -> tt 6.98/2.68 mark(s(X)) -> s(mark(X)) 6.98/2.68 mark(0) -> 0 6.98/2.68 a__U11(X1, X2) -> U11(X1, X2) 6.98/2.68 a__U12(X) -> U12(X) 6.98/2.68 a__isNat(X) -> isNat(X) 6.98/2.68 a__U21(X) -> U21(X) 6.98/2.68 a__U31(X1, X2) -> U31(X1, X2) 6.98/2.68 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 6.98/2.68 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 6.98/2.68 a__plus(X1, X2) -> plus(X1, X2) 6.98/2.68 6.98/2.68 The set Q consists of the following terms: 6.98/2.68 6.98/2.68 mark(U11(x0, x1)) 6.98/2.68 mark(U12(x0)) 6.98/2.68 mark(isNat(x0)) 6.98/2.68 mark(U21(x0)) 6.98/2.68 mark(U31(x0, x1)) 6.98/2.68 mark(U41(x0, x1, x2)) 6.98/2.68 mark(U42(x0, x1, x2)) 6.98/2.68 mark(plus(x0, x1)) 6.98/2.68 mark(tt) 6.98/2.68 mark(s(x0)) 6.98/2.68 mark(0) 6.98/2.68 a__U11(x0, x1) 6.98/2.68 a__U12(x0) 6.98/2.68 a__isNat(x0) 6.98/2.68 a__U21(x0) 6.98/2.68 a__U31(x0, x1) 6.98/2.68 a__U41(x0, x1, x2) 6.98/2.68 a__U42(x0, x1, x2) 6.98/2.68 a__plus(x0, x1) 6.98/2.68 6.98/2.68 We have to consider all minimal (P,Q,R)-chains. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (13) QDPOrderProof (EQUIVALENT) 6.98/2.68 We use the reduction pair processor [LPAR04,JAR06]. 6.98/2.68 6.98/2.68 6.98/2.68 The following pairs can be oriented strictly and are deleted. 6.98/2.68 6.98/2.68 MARK(U31(X1, X2)) -> A__U31(mark(X1), X2) 6.98/2.68 A__U31(tt, N) -> MARK(N) 6.98/2.68 MARK(U31(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(U41(X1, X2, X3)) -> A__U41(mark(X1), X2, X3) 6.98/2.68 A__PLUS(N, s(M)) -> A__U41(a__isNat(M), M, N) 6.98/2.68 MARK(U41(X1, X2, X3)) -> MARK(X1) 6.98/2.68 MARK(U42(X1, X2, X3)) -> A__U42(mark(X1), X2, X3) 6.98/2.68 MARK(U42(X1, X2, X3)) -> MARK(X1) 6.98/2.68 MARK(s(X)) -> MARK(X) 6.98/2.68 The remaining pairs can at least be oriented weakly. 6.98/2.68 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 6.98/2.68 6.98/2.68 POL( A__PLUS_2(x_1, x_2) ) = 2x_1 + 2x_2 6.98/2.68 POL( A__U31_2(x_1, x_2) ) = 2x_2 + 2 6.98/2.68 POL( A__U41_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 6.98/2.68 POL( A__U42_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 6.98/2.68 POL( mark_1(x_1) ) = x_1 6.98/2.68 POL( U11_2(x_1, x_2) ) = 2x_1 6.98/2.68 POL( a__U11_2(x_1, x_2) ) = 2x_1 6.98/2.68 POL( U12_1(x_1) ) = 2x_1 6.98/2.68 POL( a__U12_1(x_1) ) = 2x_1 6.98/2.68 POL( isNat_1(x_1) ) = 0 6.98/2.68 POL( a__isNat_1(x_1) ) = 0 6.98/2.68 POL( U21_1(x_1) ) = 2x_1 6.98/2.68 POL( a__U21_1(x_1) ) = 2x_1 6.98/2.68 POL( U31_2(x_1, x_2) ) = x_1 + x_2 + 2 6.98/2.68 POL( a__U31_2(x_1, x_2) ) = x_1 + x_2 + 2 6.98/2.68 POL( tt ) = 0 6.98/2.68 POL( plus_2(x_1, x_2) ) = x_1 + 2x_2 6.98/2.68 POL( a__plus_2(x_1, x_2) ) = x_1 + 2x_2 6.98/2.68 POL( 0 ) = 1 6.98/2.68 POL( U41_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 2 6.98/2.68 POL( a__U41_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 2 6.98/2.68 POL( U42_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 2 6.98/2.68 POL( a__U42_3(x_1, ..., x_3) ) = x_1 + 2x_2 + x_3 + 2 6.98/2.68 POL( s_1(x_1) ) = x_1 + 2 6.98/2.68 POL( MARK_1(x_1) ) = 2x_1 6.98/2.68 6.98/2.68 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 6.98/2.68 6.98/2.68 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 6.98/2.68 mark(U12(X)) -> a__U12(mark(X)) 6.98/2.68 mark(isNat(X)) -> a__isNat(X) 6.98/2.68 mark(U21(X)) -> a__U21(mark(X)) 6.98/2.68 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 6.98/2.68 a__U31(tt, N) -> mark(N) 6.98/2.68 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 6.98/2.68 a__plus(N, 0) -> a__U31(a__isNat(N), N) 6.98/2.68 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 6.98/2.68 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 6.98/2.68 mark(tt) -> tt 6.98/2.68 mark(s(X)) -> s(mark(X)) 6.98/2.68 mark(0) -> 0 6.98/2.68 a__isNat(0) -> tt 6.98/2.68 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 6.98/2.68 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 6.98/2.68 a__isNat(X) -> isNat(X) 6.98/2.68 a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M))) 6.98/2.68 a__plus(X1, X2) -> plus(X1, X2) 6.98/2.68 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 6.98/2.68 a__U11(X1, X2) -> U11(X1, X2) 6.98/2.68 a__U12(tt) -> tt 6.98/2.68 a__U12(X) -> U12(X) 6.98/2.68 a__U21(tt) -> tt 6.98/2.68 a__U21(X) -> U21(X) 6.98/2.68 a__U31(X1, X2) -> U31(X1, X2) 6.98/2.68 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 6.98/2.68 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 6.98/2.68 a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N) 6.98/2.68 a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) 6.98/2.68 6.98/2.68 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (14) 6.98/2.68 Obligation: 6.98/2.68 Q DP problem: 6.98/2.68 The TRS P consists of the following rules: 6.98/2.68 6.98/2.68 MARK(U11(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(U12(X)) -> MARK(X) 6.98/2.68 MARK(U21(X)) -> MARK(X) 6.98/2.68 A__U41(tt, M, N) -> A__U42(a__isNat(N), M, N) 6.98/2.68 A__U42(tt, M, N) -> A__PLUS(mark(N), mark(M)) 6.98/2.68 A__PLUS(N, 0) -> A__U31(a__isNat(N), N) 6.98/2.68 A__U42(tt, M, N) -> MARK(N) 6.98/2.68 A__U42(tt, M, N) -> MARK(M) 6.98/2.68 MARK(plus(X1, X2)) -> A__PLUS(mark(X1), mark(X2)) 6.98/2.68 MARK(plus(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(plus(X1, X2)) -> MARK(X2) 6.98/2.68 6.98/2.68 The TRS R consists of the following rules: 6.98/2.68 6.98/2.68 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 6.98/2.68 a__U12(tt) -> tt 6.98/2.68 a__U21(tt) -> tt 6.98/2.68 a__U31(tt, N) -> mark(N) 6.98/2.68 a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N) 6.98/2.68 a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M))) 6.98/2.68 a__isNat(0) -> tt 6.98/2.68 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 6.98/2.68 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 6.98/2.68 a__plus(N, 0) -> a__U31(a__isNat(N), N) 6.98/2.68 a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) 6.98/2.68 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 6.98/2.68 mark(U12(X)) -> a__U12(mark(X)) 6.98/2.68 mark(isNat(X)) -> a__isNat(X) 6.98/2.68 mark(U21(X)) -> a__U21(mark(X)) 6.98/2.68 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 6.98/2.68 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 6.98/2.68 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 6.98/2.68 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 6.98/2.68 mark(tt) -> tt 6.98/2.68 mark(s(X)) -> s(mark(X)) 6.98/2.68 mark(0) -> 0 6.98/2.68 a__U11(X1, X2) -> U11(X1, X2) 6.98/2.68 a__U12(X) -> U12(X) 6.98/2.68 a__isNat(X) -> isNat(X) 6.98/2.68 a__U21(X) -> U21(X) 6.98/2.68 a__U31(X1, X2) -> U31(X1, X2) 6.98/2.68 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 6.98/2.68 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 6.98/2.68 a__plus(X1, X2) -> plus(X1, X2) 6.98/2.68 6.98/2.68 The set Q consists of the following terms: 6.98/2.68 6.98/2.68 mark(U11(x0, x1)) 6.98/2.68 mark(U12(x0)) 6.98/2.68 mark(isNat(x0)) 6.98/2.68 mark(U21(x0)) 6.98/2.68 mark(U31(x0, x1)) 6.98/2.68 mark(U41(x0, x1, x2)) 6.98/2.68 mark(U42(x0, x1, x2)) 6.98/2.68 mark(plus(x0, x1)) 6.98/2.68 mark(tt) 6.98/2.68 mark(s(x0)) 6.98/2.68 mark(0) 6.98/2.68 a__U11(x0, x1) 6.98/2.68 a__U12(x0) 6.98/2.68 a__isNat(x0) 6.98/2.68 a__U21(x0) 6.98/2.68 a__U31(x0, x1) 6.98/2.68 a__U41(x0, x1, x2) 6.98/2.68 a__U42(x0, x1, x2) 6.98/2.68 a__plus(x0, x1) 6.98/2.68 6.98/2.68 We have to consider all minimal (P,Q,R)-chains. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (15) DependencyGraphProof (EQUIVALENT) 6.98/2.68 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (16) 6.98/2.68 Obligation: 6.98/2.68 Q DP problem: 6.98/2.68 The TRS P consists of the following rules: 6.98/2.68 6.98/2.68 MARK(U12(X)) -> MARK(X) 6.98/2.68 MARK(U11(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(U21(X)) -> MARK(X) 6.98/2.68 MARK(plus(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(plus(X1, X2)) -> MARK(X2) 6.98/2.68 6.98/2.68 The TRS R consists of the following rules: 6.98/2.68 6.98/2.68 a__U11(tt, V2) -> a__U12(a__isNat(V2)) 6.98/2.68 a__U12(tt) -> tt 6.98/2.68 a__U21(tt) -> tt 6.98/2.68 a__U31(tt, N) -> mark(N) 6.98/2.68 a__U41(tt, M, N) -> a__U42(a__isNat(N), M, N) 6.98/2.68 a__U42(tt, M, N) -> s(a__plus(mark(N), mark(M))) 6.98/2.68 a__isNat(0) -> tt 6.98/2.68 a__isNat(plus(V1, V2)) -> a__U11(a__isNat(V1), V2) 6.98/2.68 a__isNat(s(V1)) -> a__U21(a__isNat(V1)) 6.98/2.68 a__plus(N, 0) -> a__U31(a__isNat(N), N) 6.98/2.68 a__plus(N, s(M)) -> a__U41(a__isNat(M), M, N) 6.98/2.68 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 6.98/2.68 mark(U12(X)) -> a__U12(mark(X)) 6.98/2.68 mark(isNat(X)) -> a__isNat(X) 6.98/2.68 mark(U21(X)) -> a__U21(mark(X)) 6.98/2.68 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 6.98/2.68 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 6.98/2.68 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 6.98/2.68 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 6.98/2.68 mark(tt) -> tt 6.98/2.68 mark(s(X)) -> s(mark(X)) 6.98/2.68 mark(0) -> 0 6.98/2.68 a__U11(X1, X2) -> U11(X1, X2) 6.98/2.68 a__U12(X) -> U12(X) 6.98/2.68 a__isNat(X) -> isNat(X) 6.98/2.68 a__U21(X) -> U21(X) 6.98/2.68 a__U31(X1, X2) -> U31(X1, X2) 6.98/2.68 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 6.98/2.68 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 6.98/2.68 a__plus(X1, X2) -> plus(X1, X2) 6.98/2.68 6.98/2.68 The set Q consists of the following terms: 6.98/2.68 6.98/2.68 mark(U11(x0, x1)) 6.98/2.68 mark(U12(x0)) 6.98/2.68 mark(isNat(x0)) 6.98/2.68 mark(U21(x0)) 6.98/2.68 mark(U31(x0, x1)) 6.98/2.68 mark(U41(x0, x1, x2)) 6.98/2.68 mark(U42(x0, x1, x2)) 6.98/2.68 mark(plus(x0, x1)) 6.98/2.68 mark(tt) 6.98/2.68 mark(s(x0)) 6.98/2.68 mark(0) 6.98/2.68 a__U11(x0, x1) 6.98/2.68 a__U12(x0) 6.98/2.68 a__isNat(x0) 6.98/2.68 a__U21(x0) 6.98/2.68 a__U31(x0, x1) 6.98/2.68 a__U41(x0, x1, x2) 6.98/2.68 a__U42(x0, x1, x2) 6.98/2.68 a__plus(x0, x1) 6.98/2.68 6.98/2.68 We have to consider all minimal (P,Q,R)-chains. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (17) UsableRulesProof (EQUIVALENT) 6.98/2.68 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. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (18) 6.98/2.68 Obligation: 6.98/2.68 Q DP problem: 6.98/2.68 The TRS P consists of the following rules: 6.98/2.68 6.98/2.68 MARK(U12(X)) -> MARK(X) 6.98/2.68 MARK(U11(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(U21(X)) -> MARK(X) 6.98/2.68 MARK(plus(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(plus(X1, X2)) -> MARK(X2) 6.98/2.68 6.98/2.68 R is empty. 6.98/2.68 The set Q consists of the following terms: 6.98/2.68 6.98/2.68 mark(U11(x0, x1)) 6.98/2.68 mark(U12(x0)) 6.98/2.68 mark(isNat(x0)) 6.98/2.68 mark(U21(x0)) 6.98/2.68 mark(U31(x0, x1)) 6.98/2.68 mark(U41(x0, x1, x2)) 6.98/2.68 mark(U42(x0, x1, x2)) 6.98/2.68 mark(plus(x0, x1)) 6.98/2.68 mark(tt) 6.98/2.68 mark(s(x0)) 6.98/2.68 mark(0) 6.98/2.68 a__U11(x0, x1) 6.98/2.68 a__U12(x0) 6.98/2.68 a__isNat(x0) 6.98/2.68 a__U21(x0) 6.98/2.68 a__U31(x0, x1) 6.98/2.68 a__U41(x0, x1, x2) 6.98/2.68 a__U42(x0, x1, x2) 6.98/2.68 a__plus(x0, x1) 6.98/2.68 6.98/2.68 We have to consider all minimal (P,Q,R)-chains. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (19) QReductionProof (EQUIVALENT) 6.98/2.68 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 6.98/2.68 6.98/2.68 mark(U11(x0, x1)) 6.98/2.68 mark(U12(x0)) 6.98/2.68 mark(isNat(x0)) 6.98/2.68 mark(U21(x0)) 6.98/2.68 mark(U31(x0, x1)) 6.98/2.68 mark(U41(x0, x1, x2)) 6.98/2.68 mark(U42(x0, x1, x2)) 6.98/2.68 mark(plus(x0, x1)) 6.98/2.68 mark(tt) 6.98/2.68 mark(s(x0)) 6.98/2.68 mark(0) 6.98/2.68 a__U11(x0, x1) 6.98/2.68 a__U12(x0) 6.98/2.68 a__isNat(x0) 6.98/2.68 a__U21(x0) 6.98/2.68 a__U31(x0, x1) 6.98/2.68 a__U41(x0, x1, x2) 6.98/2.68 a__U42(x0, x1, x2) 6.98/2.68 a__plus(x0, x1) 6.98/2.68 6.98/2.68 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (20) 6.98/2.68 Obligation: 6.98/2.68 Q DP problem: 6.98/2.68 The TRS P consists of the following rules: 6.98/2.68 6.98/2.68 MARK(U12(X)) -> MARK(X) 6.98/2.68 MARK(U11(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(U21(X)) -> MARK(X) 6.98/2.68 MARK(plus(X1, X2)) -> MARK(X1) 6.98/2.68 MARK(plus(X1, X2)) -> MARK(X2) 6.98/2.68 6.98/2.68 R is empty. 6.98/2.68 Q is empty. 6.98/2.68 We have to consider all minimal (P,Q,R)-chains. 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (21) QDPSizeChangeProof (EQUIVALENT) 6.98/2.68 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. 6.98/2.68 6.98/2.68 From the DPs we obtained the following set of size-change graphs: 6.98/2.68 *MARK(U12(X)) -> MARK(X) 6.98/2.68 The graph contains the following edges 1 > 1 6.98/2.68 6.98/2.68 6.98/2.68 *MARK(U11(X1, X2)) -> MARK(X1) 6.98/2.68 The graph contains the following edges 1 > 1 6.98/2.68 6.98/2.68 6.98/2.68 *MARK(U21(X)) -> MARK(X) 6.98/2.68 The graph contains the following edges 1 > 1 6.98/2.68 6.98/2.68 6.98/2.68 *MARK(plus(X1, X2)) -> MARK(X1) 6.98/2.68 The graph contains the following edges 1 > 1 6.98/2.68 6.98/2.68 6.98/2.68 *MARK(plus(X1, X2)) -> MARK(X2) 6.98/2.68 The graph contains the following edges 1 > 1 6.98/2.68 6.98/2.68 6.98/2.68 ---------------------------------------- 6.98/2.68 6.98/2.68 (22) 6.98/2.68 YES 6.98/2.73 EOF