YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given CSR could be proven: (0) CSR (1) CSRRRRProof [EQUIVALENT, 62 ms] (2) CSR (3) CSRRRRProof [EQUIVALENT, 5 ms] (4) CSR (5) CSRRRRProof [EQUIVALENT, 0 ms] (6) CSR (7) CSRRRRProof [EQUIVALENT, 0 ms] (8) CSR (9) CSRRRRProof [EQUIVALENT, 0 ms] (10) CSR (11) CSRRRRProof [EQUIVALENT, 11 ms] (12) CSR (13) CSRRRRProof [EQUIVALENT, 0 ms] (14) CSR (15) CSRRRRProof [EQUIVALENT, 0 ms] (16) CSR (17) CSRRRRProof [EQUIVALENT, 2 ms] (18) CSR (19) CSRRRRProof [EQUIVALENT, 0 ms] (20) CSR (21) CSRRRRProof [EQUIVALENT, 0 ms] (22) CSR (23) RisEmptyProof [EQUIVALENT, 0 ms] (24) YES ---------------------------------------- (0) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) __(X, nil) -> X __(nil, X) -> X U11(tt) -> tt U21(tt, V2) -> U22(isList(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt, V2) -> U42(isNeList(V2)) U42(tt) -> tt U51(tt, V2) -> U52(isList(V2)) U52(tt) -> tt U61(tt) -> tt U71(tt, P) -> U72(isPal(P)) U72(tt) -> tt U81(tt) -> tt isList(V) -> U11(isNeList(V)) isList(nil) -> tt isList(__(V1, V2)) -> U21(isList(V1), V2) isNeList(V) -> U31(isQid(V)) isNeList(__(V1, V2)) -> U41(isList(V1), V2) isNeList(__(V1, V2)) -> U51(isNeList(V1), V2) isNePal(V) -> U61(isQid(V)) isNePal(__(I, __(P, I))) -> U71(isQid(I), P) isPal(V) -> U81(isNePal(V)) isPal(nil) -> tt isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} nil: empty set U11: {1} tt: empty set U21: {1} U22: {1} isList: empty set U31: {1} U41: {1} U42: {1} isNeList: empty set U51: {1} U52: {1} U61: {1} U71: {1} U72: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set ---------------------------------------- (1) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) __(X, nil) -> X __(nil, X) -> X U11(tt) -> tt U21(tt, V2) -> U22(isList(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt, V2) -> U42(isNeList(V2)) U42(tt) -> tt U51(tt, V2) -> U52(isList(V2)) U52(tt) -> tt U61(tt) -> tt U71(tt, P) -> U72(isPal(P)) U72(tt) -> tt U81(tt) -> tt isList(V) -> U11(isNeList(V)) isList(nil) -> tt isList(__(V1, V2)) -> U21(isList(V1), V2) isNeList(V) -> U31(isQid(V)) isNeList(__(V1, V2)) -> U41(isList(V1), V2) isNeList(__(V1, V2)) -> U51(isNeList(V1), V2) isNePal(V) -> U61(isQid(V)) isNePal(__(I, __(P, I))) -> U71(isQid(I), P) isPal(V) -> U81(isNePal(V)) isPal(nil) -> tt isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} nil: empty set U11: {1} tt: empty set U21: {1} U22: {1} isList: empty set U31: {1} U41: {1} U42: {1} isNeList: empty set U51: {1} U52: {1} U61: {1} U71: {1} U72: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1)) = x_1 POL(U21(x_1, x_2)) = x_1 + x_2 POL(U22(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = x_1 + x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + x_2 POL(U52(x_1)) = x_1 POL(U61(x_1)) = x_1 POL(U71(x_1, x_2)) = x_1 POL(U72(x_1)) = x_1 POL(U81(x_1)) = x_1 POL(__(x_1, x_2)) = 1 + x_1 + x_2 POL(a) = 1 POL(e) = 1 POL(i) = 1 POL(isList(x_1)) = x_1 POL(isNeList(x_1)) = x_1 POL(isNePal(x_1)) = 0 POL(isPal(x_1)) = 0 POL(isQid(x_1)) = 0 POL(nil) = 0 POL(o) = 1 POL(tt) = 0 POL(u) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: __(X, nil) -> X __(nil, X) -> X isList(__(V1, V2)) -> U21(isList(V1), V2) isNeList(__(V1, V2)) -> U41(isList(V1), V2) isNeList(__(V1, V2)) -> U51(isNeList(V1), V2) ---------------------------------------- (2) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) U11(tt) -> tt U21(tt, V2) -> U22(isList(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt, V2) -> U42(isNeList(V2)) U42(tt) -> tt U51(tt, V2) -> U52(isList(V2)) U52(tt) -> tt U61(tt) -> tt U71(tt, P) -> U72(isPal(P)) U72(tt) -> tt U81(tt) -> tt isList(V) -> U11(isNeList(V)) isList(nil) -> tt isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isNePal(__(I, __(P, I))) -> U71(isQid(I), P) isPal(V) -> U81(isNePal(V)) isPal(nil) -> tt isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} nil: empty set U11: {1} tt: empty set U21: {1} U22: {1} isList: empty set U31: {1} U41: {1} U42: {1} isNeList: empty set U51: {1} U52: {1} U61: {1} U71: {1} U72: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set ---------------------------------------- (3) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) U11(tt) -> tt U21(tt, V2) -> U22(isList(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt, V2) -> U42(isNeList(V2)) U42(tt) -> tt U51(tt, V2) -> U52(isList(V2)) U52(tt) -> tt U61(tt) -> tt U71(tt, P) -> U72(isPal(P)) U72(tt) -> tt U81(tt) -> tt isList(V) -> U11(isNeList(V)) isList(nil) -> tt isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isNePal(__(I, __(P, I))) -> U71(isQid(I), P) isPal(V) -> U81(isNePal(V)) isPal(nil) -> tt isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} nil: empty set U11: {1} tt: empty set U21: {1} U22: {1} isList: empty set U31: {1} U41: {1} U42: {1} isNeList: empty set U51: {1} U52: {1} U61: {1} U71: {1} U72: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1)) = x_1 POL(U21(x_1, x_2)) = 1 + x_1 + x_2 POL(U22(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = 1 + x_1 + x_2 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = 1 + x_1 + x_2 POL(U52(x_1)) = x_1 POL(U61(x_1)) = 1 + x_1 POL(U71(x_1, x_2)) = 1 + x_1 POL(U72(x_1)) = x_1 POL(U81(x_1)) = x_1 POL(__(x_1, x_2)) = 1 + x_1 + x_2 POL(a) = 1 POL(e) = 1 POL(i) = 1 POL(isList(x_1)) = x_1 POL(isNeList(x_1)) = x_1 POL(isNePal(x_1)) = 1 POL(isPal(x_1)) = 1 POL(isQid(x_1)) = 0 POL(nil) = 1 POL(o) = 1 POL(tt) = 0 POL(u) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U21(tt, V2) -> U22(isList(V2)) U41(tt, V2) -> U42(isNeList(V2)) U51(tt, V2) -> U52(isList(V2)) U61(tt) -> tt isList(nil) -> tt isPal(nil) -> tt ---------------------------------------- (4) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) U11(tt) -> tt U22(tt) -> tt U31(tt) -> tt U42(tt) -> tt U52(tt) -> tt U71(tt, P) -> U72(isPal(P)) U72(tt) -> tt U81(tt) -> tt isList(V) -> U11(isNeList(V)) isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isNePal(__(I, __(P, I))) -> U71(isQid(I), P) isPal(V) -> U81(isNePal(V)) isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} U11: {1} tt: empty set U22: {1} isList: empty set U31: {1} U42: {1} isNeList: empty set U52: {1} U61: {1} U71: {1} U72: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set ---------------------------------------- (5) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) U11(tt) -> tt U22(tt) -> tt U31(tt) -> tt U42(tt) -> tt U52(tt) -> tt U71(tt, P) -> U72(isPal(P)) U72(tt) -> tt U81(tt) -> tt isList(V) -> U11(isNeList(V)) isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isNePal(__(I, __(P, I))) -> U71(isQid(I), P) isPal(V) -> U81(isNePal(V)) isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} U11: {1} tt: empty set U22: {1} isList: empty set U31: {1} U42: {1} isNeList: empty set U52: {1} U61: {1} U71: {1} U72: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1)) = x_1 POL(U22(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U42(x_1)) = x_1 POL(U52(x_1)) = x_1 POL(U61(x_1)) = x_1 POL(U71(x_1, x_2)) = 1 + x_1 + x_2 POL(U72(x_1)) = x_1 POL(U81(x_1)) = x_1 POL(__(x_1, x_2)) = 1 + x_1 + x_2 POL(a) = 0 POL(e) = 0 POL(i) = 0 POL(isList(x_1)) = 1 + x_1 POL(isNeList(x_1)) = 1 + x_1 POL(isNePal(x_1)) = 1 + x_1 POL(isPal(x_1)) = 1 + x_1 POL(isQid(x_1)) = 1 + x_1 POL(o) = 0 POL(tt) = 1 POL(u) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U71(tt, P) -> U72(isPal(P)) isNePal(__(I, __(P, I))) -> U71(isQid(I), P) ---------------------------------------- (6) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) U11(tt) -> tt U22(tt) -> tt U31(tt) -> tt U42(tt) -> tt U52(tt) -> tt U72(tt) -> tt U81(tt) -> tt isList(V) -> U11(isNeList(V)) isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isPal(V) -> U81(isNePal(V)) isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} U11: {1} tt: empty set U22: {1} isList: empty set U31: {1} U42: {1} isNeList: empty set U52: {1} U61: {1} U72: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set ---------------------------------------- (7) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) U11(tt) -> tt U22(tt) -> tt U31(tt) -> tt U42(tt) -> tt U52(tt) -> tt U72(tt) -> tt U81(tt) -> tt isList(V) -> U11(isNeList(V)) isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isPal(V) -> U81(isNePal(V)) isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} U11: {1} tt: empty set U22: {1} isList: empty set U31: {1} U42: {1} isNeList: empty set U52: {1} U61: {1} U72: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1)) = x_1 POL(U22(x_1)) = 1 + x_1 POL(U31(x_1)) = 1 + x_1 POL(U42(x_1)) = x_1 POL(U52(x_1)) = x_1 POL(U61(x_1)) = 1 + x_1 POL(U72(x_1)) = x_1 POL(U81(x_1)) = x_1 POL(__(x_1, x_2)) = x_1 + x_2 POL(a) = 1 POL(e) = 1 POL(i) = 1 POL(isList(x_1)) = 1 + x_1 POL(isNeList(x_1)) = 1 + x_1 POL(isNePal(x_1)) = 1 + x_1 POL(isPal(x_1)) = 1 + x_1 POL(isQid(x_1)) = x_1 POL(o) = 1 POL(tt) = 1 POL(u) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U22(tt) -> tt U31(tt) -> tt ---------------------------------------- (8) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) U11(tt) -> tt U42(tt) -> tt U52(tt) -> tt U72(tt) -> tt U81(tt) -> tt isList(V) -> U11(isNeList(V)) isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isPal(V) -> U81(isNePal(V)) isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} U11: {1} tt: empty set isList: empty set U31: {1} U42: {1} isNeList: empty set U52: {1} U61: {1} U72: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set ---------------------------------------- (9) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) U11(tt) -> tt U42(tt) -> tt U52(tt) -> tt U72(tt) -> tt U81(tt) -> tt isList(V) -> U11(isNeList(V)) isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isPal(V) -> U81(isNePal(V)) isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} U11: {1} tt: empty set isList: empty set U31: {1} U42: {1} isNeList: empty set U52: {1} U61: {1} U72: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1)) = x_1 POL(U31(x_1)) = 1 + x_1 POL(U42(x_1)) = 1 + x_1 POL(U52(x_1)) = x_1 POL(U61(x_1)) = 1 + x_1 POL(U72(x_1)) = x_1 POL(U81(x_1)) = x_1 POL(__(x_1, x_2)) = x_1 + x_2 POL(a) = 1 POL(e) = 1 POL(i) = 1 POL(isList(x_1)) = 1 + x_1 POL(isNeList(x_1)) = 1 + x_1 POL(isNePal(x_1)) = 1 + x_1 POL(isPal(x_1)) = 1 + x_1 POL(isQid(x_1)) = x_1 POL(o) = 1 POL(tt) = 1 POL(u) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U42(tt) -> tt ---------------------------------------- (10) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) U11(tt) -> tt U52(tt) -> tt U72(tt) -> tt U81(tt) -> tt isList(V) -> U11(isNeList(V)) isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isPal(V) -> U81(isNePal(V)) isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} U11: {1} tt: empty set isList: empty set U31: {1} isNeList: empty set U52: {1} U61: {1} U72: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set ---------------------------------------- (11) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) U11(tt) -> tt U52(tt) -> tt U72(tt) -> tt U81(tt) -> tt isList(V) -> U11(isNeList(V)) isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isPal(V) -> U81(isNePal(V)) isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} U11: {1} tt: empty set isList: empty set U31: {1} isNeList: empty set U52: {1} U61: {1} U72: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(__(x_1, x_2)) = [[1], [0]] + [[1, 0], [0, 1]] * x_1 + [[1, 1], [0, 0]] * x_2 >>> <<< POL(U11(x_1)) = [[1], [1]] + [[1, 0], [1, 0]] * x_1 >>> <<< POL(tt) = [[1], [1]] >>> <<< POL(U52(x_1)) = [[1], [1]] + [[1, 0], [1, 1]] * x_1 >>> <<< POL(U72(x_1)) = [[1], [1]] + [[1, 0], [1, 1]] * x_1 >>> <<< POL(U81(x_1)) = [[1], [1]] + [[1, 0], [1, 0]] * x_1 >>> <<< POL(isList(x_1)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(isNeList(x_1)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(U31(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(isQid(x_1)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(isNePal(x_1)) = [[0], [1]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(U61(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1 >>> <<< POL(isPal(x_1)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 >>> <<< POL(a) = [[0], [1]] >>> <<< POL(e) = [[1], [0]] >>> <<< POL(i) = [[0], [1]] >>> <<< POL(o) = [[1], [0]] >>> <<< POL(u) = [[0], [1]] >>> With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U11(tt) -> tt U52(tt) -> tt U72(tt) -> tt U81(tt) -> tt ---------------------------------------- (12) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) isList(V) -> U11(isNeList(V)) isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isPal(V) -> U81(isNePal(V)) isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} U11: {1} tt: empty set isList: empty set U31: {1} isNeList: empty set U61: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set ---------------------------------------- (13) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: __(__(X, Y), Z) -> __(X, __(Y, Z)) isList(V) -> U11(isNeList(V)) isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isPal(V) -> U81(isNePal(V)) isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: __: {1, 2} U11: {1} tt: empty set isList: empty set U31: {1} isNeList: empty set U61: {1} isPal: empty set U81: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set Used ordering: Polynomial interpretation [POLO]: POL(U11(x_1)) = 2*x_1 POL(U31(x_1)) = 2*x_1 POL(U61(x_1)) = 2*x_1 POL(U81(x_1)) = x_1 POL(__(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(a) = 2 POL(e) = 2 POL(i) = 2 POL(isList(x_1)) = 2 + 2*x_1 POL(isNeList(x_1)) = x_1 POL(isNePal(x_1)) = 2*x_1 POL(isPal(x_1)) = 2 + 2*x_1 POL(isQid(x_1)) = 0 POL(o) = 2 POL(tt) = 0 POL(u) = 2 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: __(__(X, Y), Z) -> __(X, __(Y, Z)) isList(V) -> U11(isNeList(V)) isPal(V) -> U81(isNePal(V)) ---------------------------------------- (14) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: tt: empty set U31: {1} isNeList: empty set U61: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set ---------------------------------------- (15) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isQid(a) -> tt isQid(e) -> tt isQid(i) -> tt isQid(o) -> tt isQid(u) -> tt The replacement map contains the following entries: tt: empty set U31: {1} isNeList: empty set U61: {1} isQid: empty set isNePal: empty set a: empty set e: empty set i: empty set o: empty set u: empty set Used ordering: Polynomial interpretation [POLO]: POL(U31(x_1)) = x_1 POL(U61(x_1)) = x_1 POL(a) = 1 POL(e) = 1 POL(i) = 0 POL(isNeList(x_1)) = 1 + x_1 POL(isNePal(x_1)) = 1 + x_1 POL(isQid(x_1)) = 1 + x_1 POL(o) = 1 POL(tt) = 1 POL(u) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: isQid(a) -> tt isQid(e) -> tt isQid(o) -> tt isQid(u) -> tt ---------------------------------------- (16) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isQid(i) -> tt The replacement map contains the following entries: tt: empty set U31: {1} isNeList: empty set U61: {1} isQid: empty set isNePal: empty set i: empty set ---------------------------------------- (17) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) isQid(i) -> tt The replacement map contains the following entries: tt: empty set U31: {1} isNeList: empty set U61: {1} isQid: empty set isNePal: empty set i: empty set Used ordering: Polynomial interpretation [POLO]: POL(U31(x_1)) = 2*x_1 POL(U61(x_1)) = 2*x_1 POL(i) = 2 POL(isNeList(x_1)) = 2 + 2*x_1 POL(isNePal(x_1)) = 2 + 2*x_1 POL(isQid(x_1)) = 1 + x_1 POL(tt) = 2 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: isQid(i) -> tt ---------------------------------------- (18) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) The replacement map contains the following entries: U31: {1} isNeList: empty set U61: {1} isQid: empty set isNePal: empty set ---------------------------------------- (19) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: isNeList(V) -> U31(isQid(V)) isNePal(V) -> U61(isQid(V)) The replacement map contains the following entries: U31: {1} isNeList: empty set U61: {1} isQid: empty set isNePal: empty set Used ordering: Polynomial interpretation [POLO]: POL(U31(x_1)) = 2*x_1 POL(U61(x_1)) = x_1 POL(isNeList(x_1)) = 2 + 2*x_1 POL(isNePal(x_1)) = 2 + 2*x_1 POL(isQid(x_1)) = 1 + x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: isNePal(V) -> U61(isQid(V)) ---------------------------------------- (20) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: isNeList(V) -> U31(isQid(V)) The replacement map contains the following entries: U31: {1} isNeList: empty set isQid: empty set ---------------------------------------- (21) CSRRRRProof (EQUIVALENT) The following CSR is given: Context-sensitive rewrite system: The TRS R consists of the following rules: isNeList(V) -> U31(isQid(V)) The replacement map contains the following entries: U31: {1} isNeList: empty set isQid: empty set Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) : <<< POL(isNeList(x_1)) = [[1]] + [[1, 1]] * x_1 >>> <<< POL(U31(x_1)) = [[0]] + [[1, 1]] * x_1 >>> <<< POL(isQid(x_1)) = [[0], [0]] + [[1, 1], [0, 0]] * x_1 >>> With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: isNeList(V) -> U31(isQid(V)) ---------------------------------------- (22) Obligation: Context-sensitive rewrite system: R is empty. ---------------------------------------- (23) RisEmptyProof (EQUIVALENT) The CSR R is empty. Hence, termination is trivially proven. ---------------------------------------- (24) YES