4.88/1.94 YES 4.88/1.95 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 4.88/1.95 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.88/1.95 4.88/1.95 4.88/1.95 Termination of the given CSR could be proven: 4.88/1.95 4.88/1.95 (0) CSR 4.88/1.95 (1) CSRRRRProof [EQUIVALENT, 55 ms] 4.88/1.95 (2) CSR 4.88/1.95 (3) CSRRRRProof [EQUIVALENT, 0 ms] 4.88/1.95 (4) CSR 4.88/1.95 (5) CSRRRRProof [EQUIVALENT, 9 ms] 4.88/1.95 (6) CSR 4.88/1.95 (7) CSRRRRProof [EQUIVALENT, 3 ms] 4.88/1.95 (8) CSR 4.88/1.95 (9) CSRRRRProof [EQUIVALENT, 0 ms] 4.88/1.95 (10) CSR 4.88/1.95 (11) CSRRRRProof [EQUIVALENT, 11 ms] 4.88/1.95 (12) CSR 4.88/1.95 (13) CSRRRRProof [EQUIVALENT, 0 ms] 4.88/1.95 (14) CSR 4.88/1.95 (15) CSRRRRProof [EQUIVALENT, 0 ms] 4.88/1.95 (16) CSR 4.88/1.95 (17) CSRRRRProof [EQUIVALENT, 3 ms] 4.88/1.95 (18) CSR 4.88/1.95 (19) CSRRRRProof [EQUIVALENT, 0 ms] 4.88/1.95 (20) CSR 4.88/1.95 (21) RisEmptyProof [EQUIVALENT, 0 ms] 4.88/1.95 (22) YES 4.88/1.95 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (0) 4.88/1.95 Obligation: 4.88/1.95 Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U13(tt) -> tt 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 U22(tt) -> tt 4.88/1.95 U31(tt, N) -> N 4.88/1.95 U41(tt, M, N) -> s(plus(N, M)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(0) -> tt 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 plus(N, 0) -> U31(and(isNat(N), isNatKind(N)), N) 4.88/1.95 plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 U31: {1} 4.88/1.95 U41: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (1) CSRRRRProof (EQUIVALENT) 4.88/1.95 The following CSR is given: Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U13(tt) -> tt 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 U22(tt) -> tt 4.88/1.95 U31(tt, N) -> N 4.88/1.95 U41(tt, M, N) -> s(plus(N, M)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(0) -> tt 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 plus(N, 0) -> U31(and(isNat(N), isNatKind(N)), N) 4.88/1.95 plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 U31: {1} 4.88/1.95 U41: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 Used ordering: 4.88/1.95 Polynomial interpretation [POLO]: 4.88/1.95 4.88/1.95 POL(0) = 0 4.88/1.95 POL(U11(x_1, x_2, x_3)) = x_1 4.88/1.95 POL(U12(x_1, x_2)) = x_1 4.88/1.95 POL(U13(x_1)) = x_1 4.88/1.95 POL(U21(x_1, x_2)) = x_1 4.88/1.95 POL(U22(x_1)) = x_1 4.88/1.95 POL(U31(x_1, x_2)) = 1 + x_1 + x_2 4.88/1.95 POL(U41(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 4.88/1.95 POL(and(x_1, x_2)) = x_1 + x_2 4.88/1.95 POL(isNat(x_1)) = 0 4.88/1.95 POL(isNatKind(x_1)) = 0 4.88/1.95 POL(plus(x_1, x_2)) = 1 + x_1 + x_2 4.88/1.95 POL(s(x_1)) = x_1 4.88/1.95 POL(tt) = 0 4.88/1.95 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.88/1.95 4.88/1.95 U31(tt, N) -> N 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (2) 4.88/1.95 Obligation: 4.88/1.95 Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U13(tt) -> tt 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 U22(tt) -> tt 4.88/1.95 U41(tt, M, N) -> s(plus(N, M)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(0) -> tt 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 plus(N, 0) -> U31(and(isNat(N), isNatKind(N)), N) 4.88/1.95 plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 U31: {1} 4.88/1.95 U41: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (3) CSRRRRProof (EQUIVALENT) 4.88/1.95 The following CSR is given: Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U13(tt) -> tt 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 U22(tt) -> tt 4.88/1.95 U41(tt, M, N) -> s(plus(N, M)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(0) -> tt 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 plus(N, 0) -> U31(and(isNat(N), isNatKind(N)), N) 4.88/1.95 plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 U31: {1} 4.88/1.95 U41: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 Used ordering: 4.88/1.95 Polynomial interpretation [POLO]: 4.88/1.95 4.88/1.95 POL(0) = 2 4.88/1.95 POL(U11(x_1, x_2, x_3)) = x_1 4.88/1.95 POL(U12(x_1, x_2)) = 2*x_1 4.88/1.95 POL(U13(x_1)) = 2*x_1 4.88/1.95 POL(U21(x_1, x_2)) = x_1 4.88/1.95 POL(U22(x_1)) = x_1 4.88/1.95 POL(U31(x_1, x_2)) = 2*x_1 4.88/1.95 POL(U41(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 4.88/1.95 POL(and(x_1, x_2)) = 2*x_1 + 2*x_2 4.88/1.95 POL(isNat(x_1)) = 0 4.88/1.95 POL(isNatKind(x_1)) = 0 4.88/1.95 POL(plus(x_1, x_2)) = x_1 + 2*x_2 4.88/1.95 POL(s(x_1)) = x_1 4.88/1.95 POL(tt) = 0 4.88/1.95 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.88/1.95 4.88/1.95 plus(N, 0) -> U31(and(isNat(N), isNatKind(N)), N) 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (4) 4.88/1.95 Obligation: 4.88/1.95 Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U13(tt) -> tt 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 U22(tt) -> tt 4.88/1.95 U41(tt, M, N) -> s(plus(N, M)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(0) -> tt 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 U41: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (5) CSRRRRProof (EQUIVALENT) 4.88/1.95 The following CSR is given: Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U13(tt) -> tt 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 U22(tt) -> tt 4.88/1.95 U41(tt, M, N) -> s(plus(N, M)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(0) -> tt 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 U41: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 Used ordering: 4.88/1.95 Polynomial interpretation [POLO]: 4.88/1.95 4.88/1.95 POL(0) = 0 4.88/1.95 POL(U11(x_1, x_2, x_3)) = 2*x_1 4.88/1.95 POL(U12(x_1, x_2)) = x_1 4.88/1.95 POL(U13(x_1)) = x_1 4.88/1.95 POL(U21(x_1, x_2)) = x_1 4.88/1.95 POL(U22(x_1)) = x_1 4.88/1.95 POL(U41(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3 4.88/1.95 POL(and(x_1, x_2)) = 2*x_1 + 2*x_2 4.88/1.95 POL(isNat(x_1)) = 0 4.88/1.95 POL(isNatKind(x_1)) = 0 4.88/1.95 POL(plus(x_1, x_2)) = x_1 + 2*x_2 4.88/1.95 POL(s(x_1)) = 1 + x_1 4.88/1.95 POL(tt) = 0 4.88/1.95 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.88/1.95 4.88/1.95 plus(N, s(M)) -> U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (6) 4.88/1.95 Obligation: 4.88/1.95 Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U13(tt) -> tt 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 U22(tt) -> tt 4.88/1.95 U41(tt, M, N) -> s(plus(N, M)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(0) -> tt 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 U41: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (7) CSRRRRProof (EQUIVALENT) 4.88/1.95 The following CSR is given: Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U13(tt) -> tt 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 U22(tt) -> tt 4.88/1.95 U41(tt, M, N) -> s(plus(N, M)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(0) -> tt 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 U41: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 Used ordering: 4.88/1.95 Polynomial interpretation [POLO]: 4.88/1.95 4.88/1.95 POL(0) = 1 4.88/1.95 POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 4.88/1.95 POL(U12(x_1, x_2)) = 1 + x_1 + x_2 4.88/1.95 POL(U13(x_1)) = 1 + x_1 4.88/1.95 POL(U21(x_1, x_2)) = x_1 + x_2 4.88/1.95 POL(U22(x_1)) = x_1 4.88/1.95 POL(U41(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 4.88/1.95 POL(and(x_1, x_2)) = x_1 + x_2 4.88/1.95 POL(isNat(x_1)) = x_1 4.88/1.95 POL(isNatKind(x_1)) = 0 4.88/1.95 POL(plus(x_1, x_2)) = 1 + x_1 + x_2 4.88/1.95 POL(s(x_1)) = x_1 4.88/1.95 POL(tt) = 0 4.88/1.95 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.88/1.95 4.88/1.95 U13(tt) -> tt 4.88/1.95 isNat(0) -> tt 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (8) 4.88/1.95 Obligation: 4.88/1.95 Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 U22(tt) -> tt 4.88/1.95 U41(tt, M, N) -> s(plus(N, M)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 U41: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (9) CSRRRRProof (EQUIVALENT) 4.88/1.95 The following CSR is given: Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 U22(tt) -> tt 4.88/1.95 U41(tt, M, N) -> s(plus(N, M)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 U41: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 Used ordering: 4.88/1.95 Polynomial interpretation [POLO]: 4.88/1.95 4.88/1.95 POL(0) = 1 4.88/1.95 POL(U11(x_1, x_2, x_3)) = x_1 + x_2 + x_3 4.88/1.95 POL(U12(x_1, x_2)) = x_1 + x_2 4.88/1.95 POL(U13(x_1)) = x_1 4.88/1.95 POL(U21(x_1, x_2)) = 1 + x_1 + x_2 4.88/1.95 POL(U22(x_1)) = 1 + x_1 4.88/1.95 POL(U41(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 4.88/1.95 POL(and(x_1, x_2)) = x_1 + x_2 4.88/1.95 POL(isNat(x_1)) = x_1 4.88/1.95 POL(isNatKind(x_1)) = 0 4.88/1.95 POL(plus(x_1, x_2)) = x_1 + x_2 4.88/1.95 POL(s(x_1)) = 1 + x_1 4.88/1.95 POL(tt) = 0 4.88/1.95 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.88/1.95 4.88/1.95 U22(tt) -> tt 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (10) 4.88/1.95 Obligation: 4.88/1.95 Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 U41(tt, M, N) -> s(plus(N, M)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 U41: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (11) CSRRRRProof (EQUIVALENT) 4.88/1.95 The following CSR is given: Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 U41(tt, M, N) -> s(plus(N, M)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 U41: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 Used ordering: 4.88/1.95 Polynomial interpretation [POLO]: 4.88/1.95 4.88/1.95 POL(0) = 2 4.88/1.95 POL(U11(x_1, x_2, x_3)) = 2*x_1 4.88/1.95 POL(U12(x_1, x_2)) = x_1 4.88/1.95 POL(U13(x_1)) = x_1 4.88/1.95 POL(U21(x_1, x_2)) = 2*x_1 4.88/1.95 POL(U22(x_1)) = 2*x_1 4.88/1.95 POL(U41(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 4.88/1.95 POL(and(x_1, x_2)) = 2*x_1 + 2*x_2 4.88/1.95 POL(isNat(x_1)) = 0 4.88/1.95 POL(isNatKind(x_1)) = 0 4.88/1.95 POL(plus(x_1, x_2)) = x_1 + 2*x_2 4.88/1.95 POL(s(x_1)) = x_1 4.88/1.95 POL(tt) = 0 4.88/1.95 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.88/1.95 4.88/1.95 U41(tt, M, N) -> s(plus(N, M)) 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (12) 4.88/1.95 Obligation: 4.88/1.95 Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (13) CSRRRRProof (EQUIVALENT) 4.88/1.95 The following CSR is given: Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U13: {1} 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 Used ordering: 4.88/1.95 Polynomial interpretation [POLO]: 4.88/1.95 4.88/1.95 POL(0) = 1 4.88/1.95 POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 4.88/1.95 POL(U12(x_1, x_2)) = 1 + x_1 + x_2 4.88/1.95 POL(U13(x_1)) = x_1 4.88/1.95 POL(U21(x_1, x_2)) = 1 + x_1 + x_2 4.88/1.95 POL(U22(x_1)) = 1 + x_1 4.88/1.95 POL(and(x_1, x_2)) = x_1 + x_2 4.88/1.95 POL(isNat(x_1)) = x_1 4.88/1.95 POL(isNatKind(x_1)) = 0 4.88/1.95 POL(plus(x_1, x_2)) = 1 + x_1 + x_2 4.88/1.95 POL(s(x_1)) = 1 + x_1 4.88/1.95 POL(tt) = 0 4.88/1.95 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.88/1.95 4.88/1.95 U12(tt, V2) -> U13(isNat(V2)) 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (14) 4.88/1.95 Obligation: 4.88/1.95 Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (15) CSRRRRProof (EQUIVALENT) 4.88/1.95 The following CSR is given: Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 Used ordering: 4.88/1.95 Polynomial interpretation [POLO]: 4.88/1.95 4.88/1.95 POL(0) = 1 4.88/1.95 POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 4.88/1.95 POL(U12(x_1, x_2)) = x_1 + x_2 4.88/1.95 POL(U21(x_1, x_2)) = 1 + x_1 + x_2 4.88/1.95 POL(U22(x_1)) = x_1 4.88/1.95 POL(and(x_1, x_2)) = x_1 + x_2 4.88/1.95 POL(isNat(x_1)) = 1 + x_1 4.88/1.95 POL(isNatKind(x_1)) = 0 4.88/1.95 POL(plus(x_1, x_2)) = 1 + x_1 + x_2 4.88/1.95 POL(s(x_1)) = x_1 4.88/1.95 POL(tt) = 0 4.88/1.95 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.88/1.95 4.88/1.95 isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2) 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (16) 4.88/1.95 Obligation: 4.88/1.95 Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (17) CSRRRRProof (EQUIVALENT) 4.88/1.95 The following CSR is given: Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 Used ordering: 4.88/1.95 Polynomial interpretation [POLO]: 4.88/1.95 4.88/1.95 POL(0) = 1 4.88/1.95 POL(U11(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 4.88/1.95 POL(U12(x_1, x_2)) = x_1 + x_2 4.88/1.95 POL(U21(x_1, x_2)) = 1 + x_1 + x_2 4.88/1.95 POL(U22(x_1)) = x_1 4.88/1.95 POL(and(x_1, x_2)) = x_1 + x_2 4.88/1.95 POL(isNat(x_1)) = 1 + x_1 4.88/1.95 POL(isNatKind(x_1)) = 0 4.88/1.95 POL(plus(x_1, x_2)) = 1 + x_1 + x_2 4.88/1.95 POL(s(x_1)) = 1 + x_1 4.88/1.95 POL(tt) = 0 4.88/1.95 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.88/1.95 4.88/1.95 isNat(s(V1)) -> U21(isNatKind(V1), V1) 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (18) 4.88/1.95 Obligation: 4.88/1.95 Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (19) CSRRRRProof (EQUIVALENT) 4.88/1.95 The following CSR is given: Context-sensitive rewrite system: 4.88/1.95 The TRS R consists of the following rules: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 The replacement map contains the following entries: 4.88/1.95 4.88/1.95 U11: {1} 4.88/1.95 tt: empty set 4.88/1.95 U12: {1} 4.88/1.95 isNat: empty set 4.88/1.95 U21: {1} 4.88/1.95 U22: {1} 4.88/1.95 s: {1} 4.88/1.95 plus: {1, 2} 4.88/1.95 and: {1} 4.88/1.95 0: empty set 4.88/1.95 isNatKind: empty set 4.88/1.95 Used ordering: 4.88/1.95 Polynomial interpretation [POLO]: 4.88/1.95 4.88/1.95 POL(0) = 2 4.88/1.95 POL(U11(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 4.88/1.95 POL(U12(x_1, x_2)) = 2*x_1 + 2*x_2 4.88/1.95 POL(U21(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 4.88/1.95 POL(U22(x_1)) = 2*x_1 4.88/1.95 POL(and(x_1, x_2)) = x_1 + x_2 4.88/1.95 POL(isNat(x_1)) = x_1 4.88/1.95 POL(isNatKind(x_1)) = x_1 4.88/1.95 POL(plus(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 4.88/1.95 POL(s(x_1)) = 2 + 2*x_1 4.88/1.95 POL(tt) = 1 4.88/1.95 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.88/1.95 4.88/1.95 U11(tt, V1, V2) -> U12(isNat(V1), V2) 4.88/1.95 U21(tt, V1) -> U22(isNat(V1)) 4.88/1.95 and(tt, X) -> X 4.88/1.95 isNatKind(0) -> tt 4.88/1.95 isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2)) 4.88/1.95 isNatKind(s(V1)) -> isNatKind(V1) 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (20) 4.88/1.95 Obligation: 4.88/1.95 Context-sensitive rewrite system: 4.88/1.95 R is empty. 4.88/1.95 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (21) RisEmptyProof (EQUIVALENT) 4.88/1.95 The CSR R is empty. Hence, termination is trivially proven. 4.88/1.95 ---------------------------------------- 4.88/1.95 4.88/1.95 (22) 4.88/1.95 YES 5.22/2.08 EOF