5.99/2.39 YES 5.99/2.40 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 5.99/2.40 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.99/2.40 5.99/2.40 5.99/2.40 Termination of the given ETRS could be proven: 5.99/2.40 5.99/2.40 (0) ETRS 5.99/2.40 (1) RRRPoloETRSProof [EQUIVALENT, 220 ms] 5.99/2.40 (2) ETRS 5.99/2.40 (3) RRRPoloETRSProof [EQUIVALENT, 105 ms] 5.99/2.40 (4) ETRS 5.99/2.40 (5) RRRPoloETRSProof [EQUIVALENT, 46 ms] 5.99/2.40 (6) ETRS 5.99/2.40 (7) RRRPoloETRSProof [EQUIVALENT, 8 ms] 5.99/2.40 (8) ETRS 5.99/2.40 (9) RRRPoloETRSProof [EQUIVALENT, 13 ms] 5.99/2.40 (10) ETRS 5.99/2.40 (11) RRRPoloETRSProof [EQUIVALENT, 0 ms] 5.99/2.40 (12) ETRS 5.99/2.40 (13) RisEmptyProof [EQUIVALENT, 0 ms] 5.99/2.40 (14) YES 5.99/2.40 5.99/2.40 5.99/2.40 ---------------------------------------- 5.99/2.40 5.99/2.40 (0) 5.99/2.40 Obligation: 5.99/2.40 Equational rewrite system: 5.99/2.40 The TRS R consists of the following rules: 5.99/2.40 5.99/2.40 union(X, empty) -> X 5.99/2.40 union(empty, X) -> X 5.99/2.40 0(z) -> z 5.99/2.40 U11(tt, X, Y) -> U12(tt, X, Y) 5.99/2.40 U12(tt, X, Y) -> 0(mult(X, Y)) 5.99/2.40 U21(tt, X, Y) -> U22(tt, X, Y) 5.99/2.40 U22(tt, X, Y) -> plus(0(mult(X, Y)), Y) 5.99/2.40 U31(tt, X, Y) -> U32(tt, X, Y) 5.99/2.40 U32(tt, X, Y) -> 0(plus(X, Y)) 5.99/2.40 U41(tt, X, Y) -> U42(tt, X, Y) 5.99/2.40 U42(tt, X, Y) -> 1(plus(X, Y)) 5.99/2.40 U51(tt, X, Y) -> U52(tt, X, Y) 5.99/2.40 U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) 5.99/2.40 U61(tt, A, B) -> U62(tt, A, B) 5.99/2.40 U62(tt, A, B) -> mult(prod(A), prod(B)) 5.99/2.40 U71(tt, A, B) -> U72(tt, A, B) 5.99/2.40 U72(tt, A, B) -> plus(sum(A), sum(B)) 5.99/2.40 mult(z, X) -> z 5.99/2.40 mult(0(X), Y) -> U11(tt, X, Y) 5.99/2.40 mult(1(X), Y) -> U21(tt, X, Y) 5.99/2.40 plus(z, X) -> X 5.99/2.40 plus(0(X), 0(Y)) -> U31(tt, X, Y) 5.99/2.40 plus(0(X), 1(Y)) -> U41(tt, X, Y) 5.99/2.40 plus(1(X), 1(Y)) -> U51(tt, X, Y) 5.99/2.40 prod(empty) -> 1(z) 5.99/2.40 prod(singl(X)) -> X 5.99/2.40 prod(union(A, B)) -> U61(tt, A, B) 5.99/2.40 sum(empty) -> 0(z) 5.99/2.40 sum(singl(X)) -> X 5.99/2.40 sum(union(A, B)) -> U71(tt, A, B) 5.99/2.40 5.99/2.40 The set E consists of the following equations: 5.99/2.40 5.99/2.40 mult(x, y) == mult(y, x) 5.99/2.40 plus(x, y) == plus(y, x) 5.99/2.40 union(x, y) == union(y, x) 5.99/2.40 mult(mult(x, y), z') == mult(x, mult(y, z')) 5.99/2.40 plus(plus(x, y), z') == plus(x, plus(y, z')) 5.99/2.40 union(union(x, y), z') == union(x, union(y, z')) 5.99/2.40 5.99/2.40 5.99/2.40 ---------------------------------------- 5.99/2.40 5.99/2.40 (1) RRRPoloETRSProof (EQUIVALENT) 5.99/2.40 The following E TRS is given: Equational rewrite system: 5.99/2.40 The TRS R consists of the following rules: 5.99/2.40 5.99/2.40 union(X, empty) -> X 5.99/2.40 union(empty, X) -> X 5.99/2.40 0(z) -> z 5.99/2.40 U11(tt, X, Y) -> U12(tt, X, Y) 5.99/2.40 U12(tt, X, Y) -> 0(mult(X, Y)) 5.99/2.40 U21(tt, X, Y) -> U22(tt, X, Y) 5.99/2.40 U22(tt, X, Y) -> plus(0(mult(X, Y)), Y) 5.99/2.40 U31(tt, X, Y) -> U32(tt, X, Y) 5.99/2.40 U32(tt, X, Y) -> 0(plus(X, Y)) 5.99/2.40 U41(tt, X, Y) -> U42(tt, X, Y) 5.99/2.40 U42(tt, X, Y) -> 1(plus(X, Y)) 5.99/2.40 U51(tt, X, Y) -> U52(tt, X, Y) 5.99/2.40 U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) 5.99/2.40 U61(tt, A, B) -> U62(tt, A, B) 5.99/2.40 U62(tt, A, B) -> mult(prod(A), prod(B)) 5.99/2.40 U71(tt, A, B) -> U72(tt, A, B) 5.99/2.40 U72(tt, A, B) -> plus(sum(A), sum(B)) 5.99/2.40 mult(z, X) -> z 5.99/2.40 mult(0(X), Y) -> U11(tt, X, Y) 5.99/2.40 mult(1(X), Y) -> U21(tt, X, Y) 5.99/2.40 plus(z, X) -> X 5.99/2.40 plus(0(X), 0(Y)) -> U31(tt, X, Y) 5.99/2.40 plus(0(X), 1(Y)) -> U41(tt, X, Y) 5.99/2.40 plus(1(X), 1(Y)) -> U51(tt, X, Y) 5.99/2.40 prod(empty) -> 1(z) 5.99/2.40 prod(singl(X)) -> X 5.99/2.40 prod(union(A, B)) -> U61(tt, A, B) 5.99/2.40 sum(empty) -> 0(z) 5.99/2.40 sum(singl(X)) -> X 5.99/2.40 sum(union(A, B)) -> U71(tt, A, B) 5.99/2.40 5.99/2.40 The set E consists of the following equations: 5.99/2.40 5.99/2.40 mult(x, y) == mult(y, x) 5.99/2.40 plus(x, y) == plus(y, x) 5.99/2.40 union(x, y) == union(y, x) 5.99/2.40 mult(mult(x, y), z') == mult(x, mult(y, z')) 5.99/2.40 plus(plus(x, y), z') == plus(x, plus(y, z')) 5.99/2.40 union(union(x, y), z') == union(x, union(y, z')) 5.99/2.40 5.99/2.40 The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering: 5.99/2.40 5.99/2.40 union(X, empty) -> X 5.99/2.40 union(empty, X) -> X 5.99/2.40 U22(tt, X, Y) -> plus(0(mult(X, Y)), Y) 5.99/2.40 U51(tt, X, Y) -> U52(tt, X, Y) 5.99/2.40 prod(empty) -> 1(z) 5.99/2.40 sum(empty) -> 0(z) 5.99/2.40 Used ordering: 5.99/2.40 Polynomial interpretation [POLO]: 5.99/2.40 5.99/2.40 POL(0(x_1)) = x_1 5.99/2.40 POL(1(x_1)) = 1 + x_1 5.99/2.40 POL(U11(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_2*x_3 + x_3 5.99/2.40 POL(U12(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_2*x_3 + x_3 5.99/2.40 POL(U21(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_2*x_3 + 3*x_3 5.99/2.40 POL(U22(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_2*x_3 + 2*x_3 5.99/2.40 POL(U31(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 3*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U32(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 3*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U41(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_1*x_2 + 3*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U42(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_1*x_2 + 3*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U51(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 3*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U52(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_1*x_2 + 3*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_2*x_3 + x_3 5.99/2.40 POL(U62(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_2*x_3 + x_3 5.99/2.40 POL(U71(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_3 5.99/2.40 POL(U72(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_3 5.99/2.40 POL(empty) = 2 5.99/2.40 POL(mult(x_1, x_2)) = x_1 + 2*x_1*x_2 + x_2 5.99/2.40 POL(plus(x_1, x_2)) = x_1 + x_2 5.99/2.40 POL(prod(x_1)) = x_1 5.99/2.40 POL(singl(x_1)) = 3*x_1 + 3*x_1^2 5.99/2.40 POL(sum(x_1)) = 2*x_1 5.99/2.40 POL(tt) = 0 5.99/2.40 POL(union(x_1, x_2)) = x_1 + 2*x_1*x_2 + x_2 5.99/2.40 POL(z) = 0 5.99/2.40 5.99/2.40 5.99/2.40 5.99/2.40 5.99/2.40 ---------------------------------------- 5.99/2.40 5.99/2.40 (2) 5.99/2.40 Obligation: 5.99/2.40 Equational rewrite system: 5.99/2.40 The TRS R consists of the following rules: 5.99/2.40 5.99/2.40 0(z) -> z 5.99/2.40 U11(tt, X, Y) -> U12(tt, X, Y) 5.99/2.40 U12(tt, X, Y) -> 0(mult(X, Y)) 5.99/2.40 U21(tt, X, Y) -> U22(tt, X, Y) 5.99/2.40 U31(tt, X, Y) -> U32(tt, X, Y) 5.99/2.40 U32(tt, X, Y) -> 0(plus(X, Y)) 5.99/2.40 U41(tt, X, Y) -> U42(tt, X, Y) 5.99/2.40 U42(tt, X, Y) -> 1(plus(X, Y)) 5.99/2.40 U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) 5.99/2.40 U61(tt, A, B) -> U62(tt, A, B) 5.99/2.40 U62(tt, A, B) -> mult(prod(A), prod(B)) 5.99/2.40 U71(tt, A, B) -> U72(tt, A, B) 5.99/2.40 U72(tt, A, B) -> plus(sum(A), sum(B)) 5.99/2.40 mult(z, X) -> z 5.99/2.40 mult(0(X), Y) -> U11(tt, X, Y) 5.99/2.40 mult(1(X), Y) -> U21(tt, X, Y) 5.99/2.40 plus(z, X) -> X 5.99/2.40 plus(0(X), 0(Y)) -> U31(tt, X, Y) 5.99/2.40 plus(0(X), 1(Y)) -> U41(tt, X, Y) 5.99/2.40 plus(1(X), 1(Y)) -> U51(tt, X, Y) 5.99/2.40 prod(singl(X)) -> X 5.99/2.40 prod(union(A, B)) -> U61(tt, A, B) 5.99/2.40 sum(singl(X)) -> X 5.99/2.40 sum(union(A, B)) -> U71(tt, A, B) 5.99/2.40 5.99/2.40 The set E consists of the following equations: 5.99/2.40 5.99/2.40 mult(x, y) == mult(y, x) 5.99/2.40 plus(x, y) == plus(y, x) 5.99/2.40 union(x, y) == union(y, x) 5.99/2.40 mult(mult(x, y), z') == mult(x, mult(y, z')) 5.99/2.40 plus(plus(x, y), z') == plus(x, plus(y, z')) 5.99/2.40 union(union(x, y), z') == union(x, union(y, z')) 5.99/2.40 5.99/2.40 5.99/2.40 ---------------------------------------- 5.99/2.40 5.99/2.40 (3) RRRPoloETRSProof (EQUIVALENT) 5.99/2.40 The following E TRS is given: Equational rewrite system: 5.99/2.40 The TRS R consists of the following rules: 5.99/2.40 5.99/2.40 0(z) -> z 5.99/2.40 U11(tt, X, Y) -> U12(tt, X, Y) 5.99/2.40 U12(tt, X, Y) -> 0(mult(X, Y)) 5.99/2.40 U21(tt, X, Y) -> U22(tt, X, Y) 5.99/2.40 U31(tt, X, Y) -> U32(tt, X, Y) 5.99/2.40 U32(tt, X, Y) -> 0(plus(X, Y)) 5.99/2.40 U41(tt, X, Y) -> U42(tt, X, Y) 5.99/2.40 U42(tt, X, Y) -> 1(plus(X, Y)) 5.99/2.40 U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) 5.99/2.40 U61(tt, A, B) -> U62(tt, A, B) 5.99/2.40 U62(tt, A, B) -> mult(prod(A), prod(B)) 5.99/2.40 U71(tt, A, B) -> U72(tt, A, B) 5.99/2.40 U72(tt, A, B) -> plus(sum(A), sum(B)) 5.99/2.40 mult(z, X) -> z 5.99/2.40 mult(0(X), Y) -> U11(tt, X, Y) 5.99/2.40 mult(1(X), Y) -> U21(tt, X, Y) 5.99/2.40 plus(z, X) -> X 5.99/2.40 plus(0(X), 0(Y)) -> U31(tt, X, Y) 5.99/2.40 plus(0(X), 1(Y)) -> U41(tt, X, Y) 5.99/2.40 plus(1(X), 1(Y)) -> U51(tt, X, Y) 5.99/2.40 prod(singl(X)) -> X 5.99/2.40 prod(union(A, B)) -> U61(tt, A, B) 5.99/2.40 sum(singl(X)) -> X 5.99/2.40 sum(union(A, B)) -> U71(tt, A, B) 5.99/2.40 5.99/2.40 The set E consists of the following equations: 5.99/2.40 5.99/2.40 mult(x, y) == mult(y, x) 5.99/2.40 plus(x, y) == plus(y, x) 5.99/2.40 union(x, y) == union(y, x) 5.99/2.40 mult(mult(x, y), z') == mult(x, mult(y, z')) 5.99/2.40 plus(plus(x, y), z') == plus(x, plus(y, z')) 5.99/2.40 union(union(x, y), z') == union(x, union(y, z')) 5.99/2.40 5.99/2.40 The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering: 5.99/2.40 5.99/2.40 prod(union(A, B)) -> U61(tt, A, B) 5.99/2.40 sum(singl(X)) -> X 5.99/2.40 Used ordering: 5.99/2.40 Polynomial interpretation [POLO]: 5.99/2.40 5.99/2.40 POL(0(x_1)) = x_1 5.99/2.40 POL(1(x_1)) = x_1 5.99/2.40 POL(U11(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U12(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U21(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U22(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U31(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U32(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U41(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U42(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U51(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U52(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_2*x_3 + x_3 5.99/2.40 POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 3*x_2 + x_2*x_3 + 2*x_3 5.99/2.40 POL(U62(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_2*x_3 + 2*x_3 5.99/2.40 POL(U71(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U72(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(mult(x_1, x_2)) = x_1 + x_2 5.99/2.40 POL(plus(x_1, x_2)) = x_1 + x_2 5.99/2.40 POL(prod(x_1)) = 2*x_1 5.99/2.40 POL(singl(x_1)) = 3*x_1 + 3*x_1^2 5.99/2.40 POL(sum(x_1)) = 1 + x_1 5.99/2.40 POL(tt) = 0 5.99/2.40 POL(union(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2 5.99/2.40 POL(z) = 0 5.99/2.40 5.99/2.40 5.99/2.40 5.99/2.40 5.99/2.40 ---------------------------------------- 5.99/2.40 5.99/2.40 (4) 5.99/2.40 Obligation: 5.99/2.40 Equational rewrite system: 5.99/2.40 The TRS R consists of the following rules: 5.99/2.40 5.99/2.40 0(z) -> z 5.99/2.40 U11(tt, X, Y) -> U12(tt, X, Y) 5.99/2.40 U12(tt, X, Y) -> 0(mult(X, Y)) 5.99/2.40 U21(tt, X, Y) -> U22(tt, X, Y) 5.99/2.40 U31(tt, X, Y) -> U32(tt, X, Y) 5.99/2.40 U32(tt, X, Y) -> 0(plus(X, Y)) 5.99/2.40 U41(tt, X, Y) -> U42(tt, X, Y) 5.99/2.40 U42(tt, X, Y) -> 1(plus(X, Y)) 5.99/2.40 U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) 5.99/2.40 U61(tt, A, B) -> U62(tt, A, B) 5.99/2.40 U62(tt, A, B) -> mult(prod(A), prod(B)) 5.99/2.40 U71(tt, A, B) -> U72(tt, A, B) 5.99/2.40 U72(tt, A, B) -> plus(sum(A), sum(B)) 5.99/2.40 mult(z, X) -> z 5.99/2.40 mult(0(X), Y) -> U11(tt, X, Y) 5.99/2.40 mult(1(X), Y) -> U21(tt, X, Y) 5.99/2.40 plus(z, X) -> X 5.99/2.40 plus(0(X), 0(Y)) -> U31(tt, X, Y) 5.99/2.40 plus(0(X), 1(Y)) -> U41(tt, X, Y) 5.99/2.40 plus(1(X), 1(Y)) -> U51(tt, X, Y) 5.99/2.40 prod(singl(X)) -> X 5.99/2.40 sum(union(A, B)) -> U71(tt, A, B) 5.99/2.40 5.99/2.40 The set E consists of the following equations: 5.99/2.40 5.99/2.40 mult(x, y) == mult(y, x) 5.99/2.40 plus(x, y) == plus(y, x) 5.99/2.40 union(x, y) == union(y, x) 5.99/2.40 mult(mult(x, y), z') == mult(x, mult(y, z')) 5.99/2.40 plus(plus(x, y), z') == plus(x, plus(y, z')) 5.99/2.40 union(union(x, y), z') == union(x, union(y, z')) 5.99/2.40 5.99/2.40 5.99/2.40 ---------------------------------------- 5.99/2.40 5.99/2.40 (5) RRRPoloETRSProof (EQUIVALENT) 5.99/2.40 The following E TRS is given: Equational rewrite system: 5.99/2.40 The TRS R consists of the following rules: 5.99/2.40 5.99/2.40 0(z) -> z 5.99/2.40 U11(tt, X, Y) -> U12(tt, X, Y) 5.99/2.40 U12(tt, X, Y) -> 0(mult(X, Y)) 5.99/2.40 U21(tt, X, Y) -> U22(tt, X, Y) 5.99/2.40 U31(tt, X, Y) -> U32(tt, X, Y) 5.99/2.40 U32(tt, X, Y) -> 0(plus(X, Y)) 5.99/2.40 U41(tt, X, Y) -> U42(tt, X, Y) 5.99/2.40 U42(tt, X, Y) -> 1(plus(X, Y)) 5.99/2.40 U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) 5.99/2.40 U61(tt, A, B) -> U62(tt, A, B) 5.99/2.40 U62(tt, A, B) -> mult(prod(A), prod(B)) 5.99/2.40 U71(tt, A, B) -> U72(tt, A, B) 5.99/2.40 U72(tt, A, B) -> plus(sum(A), sum(B)) 5.99/2.40 mult(z, X) -> z 5.99/2.40 mult(0(X), Y) -> U11(tt, X, Y) 5.99/2.40 mult(1(X), Y) -> U21(tt, X, Y) 5.99/2.40 plus(z, X) -> X 5.99/2.40 plus(0(X), 0(Y)) -> U31(tt, X, Y) 5.99/2.40 plus(0(X), 1(Y)) -> U41(tt, X, Y) 5.99/2.40 plus(1(X), 1(Y)) -> U51(tt, X, Y) 5.99/2.40 prod(singl(X)) -> X 5.99/2.40 sum(union(A, B)) -> U71(tt, A, B) 5.99/2.40 5.99/2.40 The set E consists of the following equations: 5.99/2.40 5.99/2.40 mult(x, y) == mult(y, x) 5.99/2.40 plus(x, y) == plus(y, x) 5.99/2.40 union(x, y) == union(y, x) 5.99/2.40 mult(mult(x, y), z') == mult(x, mult(y, z')) 5.99/2.40 plus(plus(x, y), z') == plus(x, plus(y, z')) 5.99/2.40 union(union(x, y), z') == union(x, union(y, z')) 5.99/2.40 5.99/2.40 The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering: 5.99/2.40 5.99/2.40 0(z) -> z 5.99/2.40 U31(tt, X, Y) -> U32(tt, X, Y) 5.99/2.40 U41(tt, X, Y) -> U42(tt, X, Y) 5.99/2.40 U61(tt, A, B) -> U62(tt, A, B) 5.99/2.40 plus(0(X), 0(Y)) -> U31(tt, X, Y) 5.99/2.40 plus(0(X), 1(Y)) -> U41(tt, X, Y) 5.99/2.40 prod(singl(X)) -> X 5.99/2.40 Used ordering: 5.99/2.40 Polynomial interpretation [POLO]: 5.99/2.40 5.99/2.40 POL(0(x_1)) = 2 + 2*x_1 5.99/2.40 POL(1(x_1)) = 2*x_1 5.99/2.40 POL(U11(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_2*x_3 + 2*x_3 5.99/2.40 POL(U12(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_2*x_3 + 2*x_3 5.99/2.40 POL(U21(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U22(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U31(x_1, x_2, x_3)) = 3 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_3 5.99/2.40 POL(U32(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_3 5.99/2.40 POL(U41(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_3 5.99/2.40 POL(U42(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_3 5.99/2.40 POL(U51(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.40 POL(U52(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_3 5.99/2.41 POL(U61(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_2*x_3 + x_3 5.99/2.41 POL(U62(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_2*x_3 + x_3 5.99/2.41 POL(U71(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_3 5.99/2.41 POL(U72(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_3 5.99/2.41 POL(mult(x_1, x_2)) = x_1 + x_1*x_2 + x_2 5.99/2.41 POL(plus(x_1, x_2)) = x_1 + x_2 5.99/2.41 POL(prod(x_1)) = x_1 5.99/2.41 POL(singl(x_1)) = 3 + 3*x_1 + 3*x_1^2 5.99/2.41 POL(sum(x_1)) = 2*x_1 5.99/2.41 POL(tt) = 0 5.99/2.41 POL(union(x_1, x_2)) = x_1 + x_1*x_2 + x_2 5.99/2.41 POL(z) = 0 5.99/2.41 5.99/2.41 5.99/2.41 5.99/2.41 5.99/2.41 ---------------------------------------- 5.99/2.41 5.99/2.41 (6) 5.99/2.41 Obligation: 5.99/2.41 Equational rewrite system: 5.99/2.41 The TRS R consists of the following rules: 5.99/2.41 5.99/2.41 U11(tt, X, Y) -> U12(tt, X, Y) 5.99/2.41 U12(tt, X, Y) -> 0(mult(X, Y)) 5.99/2.41 U21(tt, X, Y) -> U22(tt, X, Y) 5.99/2.41 U32(tt, X, Y) -> 0(plus(X, Y)) 5.99/2.41 U42(tt, X, Y) -> 1(plus(X, Y)) 5.99/2.41 U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) 5.99/2.41 U62(tt, A, B) -> mult(prod(A), prod(B)) 5.99/2.41 U71(tt, A, B) -> U72(tt, A, B) 5.99/2.41 U72(tt, A, B) -> plus(sum(A), sum(B)) 5.99/2.41 mult(z, X) -> z 5.99/2.41 mult(0(X), Y) -> U11(tt, X, Y) 5.99/2.41 mult(1(X), Y) -> U21(tt, X, Y) 5.99/2.41 plus(z, X) -> X 5.99/2.41 plus(1(X), 1(Y)) -> U51(tt, X, Y) 5.99/2.41 sum(union(A, B)) -> U71(tt, A, B) 5.99/2.41 5.99/2.41 The set E consists of the following equations: 5.99/2.41 5.99/2.41 mult(x, y) == mult(y, x) 5.99/2.41 plus(x, y) == plus(y, x) 5.99/2.41 union(x, y) == union(y, x) 5.99/2.41 mult(mult(x, y), z') == mult(x, mult(y, z')) 5.99/2.41 plus(plus(x, y), z') == plus(x, plus(y, z')) 5.99/2.41 union(union(x, y), z') == union(x, union(y, z')) 5.99/2.41 5.99/2.41 5.99/2.41 ---------------------------------------- 5.99/2.41 5.99/2.41 (7) RRRPoloETRSProof (EQUIVALENT) 5.99/2.41 The following E TRS is given: Equational rewrite system: 5.99/2.41 The TRS R consists of the following rules: 5.99/2.41 5.99/2.41 U11(tt, X, Y) -> U12(tt, X, Y) 5.99/2.41 U12(tt, X, Y) -> 0(mult(X, Y)) 5.99/2.41 U21(tt, X, Y) -> U22(tt, X, Y) 5.99/2.41 U32(tt, X, Y) -> 0(plus(X, Y)) 5.99/2.41 U42(tt, X, Y) -> 1(plus(X, Y)) 5.99/2.41 U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) 5.99/2.41 U62(tt, A, B) -> mult(prod(A), prod(B)) 5.99/2.41 U71(tt, A, B) -> U72(tt, A, B) 5.99/2.41 U72(tt, A, B) -> plus(sum(A), sum(B)) 5.99/2.41 mult(z, X) -> z 5.99/2.41 mult(0(X), Y) -> U11(tt, X, Y) 5.99/2.41 mult(1(X), Y) -> U21(tt, X, Y) 5.99/2.41 plus(z, X) -> X 5.99/2.41 plus(1(X), 1(Y)) -> U51(tt, X, Y) 5.99/2.41 sum(union(A, B)) -> U71(tt, A, B) 5.99/2.41 5.99/2.41 The set E consists of the following equations: 5.99/2.41 5.99/2.41 mult(x, y) == mult(y, x) 5.99/2.41 plus(x, y) == plus(y, x) 5.99/2.41 union(x, y) == union(y, x) 5.99/2.41 mult(mult(x, y), z') == mult(x, mult(y, z')) 5.99/2.41 plus(plus(x, y), z') == plus(x, plus(y, z')) 5.99/2.41 union(union(x, y), z') == union(x, union(y, z')) 5.99/2.41 5.99/2.41 The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering: 5.99/2.41 5.99/2.41 U21(tt, X, Y) -> U22(tt, X, Y) 5.99/2.41 U32(tt, X, Y) -> 0(plus(X, Y)) 5.99/2.41 U42(tt, X, Y) -> 1(plus(X, Y)) 5.99/2.41 U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) 5.99/2.41 mult(z, X) -> z 5.99/2.41 Used ordering: 5.99/2.41 Polynomial interpretation [POLO]: 5.99/2.41 5.99/2.41 POL(0(x_1)) = x_1 5.99/2.41 POL(1(x_1)) = 2*x_1 5.99/2.41 POL(U11(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_2*x_3 + 2*x_3 5.99/2.41 POL(U12(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_2*x_3 + 2*x_3 5.99/2.41 POL(U21(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_2*x_3 + x_3 5.99/2.41 POL(U22(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.41 POL(U32(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_2*x_3 + 2*x_3 5.99/2.41 POL(U42(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 3*x_2*x_3 + 2*x_3 5.99/2.41 POL(U51(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.41 POL(U52(x_1, x_2, x_3)) = 3 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_3 5.99/2.41 POL(U62(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 3*x_2 + 2*x_2*x_3 + 2*x_3 5.99/2.41 POL(U71(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 3*x_2 + 2*x_2*x_3 + 3*x_3 5.99/2.41 POL(U72(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 3*x_2 + 3*x_3 5.99/2.41 POL(mult(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2 5.99/2.41 POL(plus(x_1, x_2)) = x_1 + x_2 5.99/2.41 POL(prod(x_1)) = x_1 5.99/2.41 POL(sum(x_1)) = 3*x_1 5.99/2.41 POL(tt) = 0 5.99/2.41 POL(union(x_1, x_2)) = x_1 + 3*x_1*x_2 + x_2 5.99/2.41 POL(z) = 0 5.99/2.41 5.99/2.41 5.99/2.41 5.99/2.41 5.99/2.41 ---------------------------------------- 5.99/2.41 5.99/2.41 (8) 5.99/2.41 Obligation: 5.99/2.41 Equational rewrite system: 5.99/2.41 The TRS R consists of the following rules: 5.99/2.41 5.99/2.41 U11(tt, X, Y) -> U12(tt, X, Y) 5.99/2.41 U12(tt, X, Y) -> 0(mult(X, Y)) 5.99/2.41 U62(tt, A, B) -> mult(prod(A), prod(B)) 5.99/2.41 U71(tt, A, B) -> U72(tt, A, B) 5.99/2.41 U72(tt, A, B) -> plus(sum(A), sum(B)) 5.99/2.41 mult(0(X), Y) -> U11(tt, X, Y) 5.99/2.41 mult(1(X), Y) -> U21(tt, X, Y) 5.99/2.41 plus(z, X) -> X 5.99/2.41 plus(1(X), 1(Y)) -> U51(tt, X, Y) 5.99/2.41 sum(union(A, B)) -> U71(tt, A, B) 5.99/2.41 5.99/2.41 The set E consists of the following equations: 5.99/2.41 5.99/2.41 mult(x, y) == mult(y, x) 5.99/2.41 plus(x, y) == plus(y, x) 5.99/2.41 union(x, y) == union(y, x) 5.99/2.41 mult(mult(x, y), z') == mult(x, mult(y, z')) 5.99/2.41 plus(plus(x, y), z') == plus(x, plus(y, z')) 5.99/2.41 union(union(x, y), z') == union(x, union(y, z')) 5.99/2.41 5.99/2.41 5.99/2.41 ---------------------------------------- 5.99/2.41 5.99/2.41 (9) RRRPoloETRSProof (EQUIVALENT) 5.99/2.41 The following E TRS is given: Equational rewrite system: 5.99/2.41 The TRS R consists of the following rules: 5.99/2.41 5.99/2.41 U11(tt, X, Y) -> U12(tt, X, Y) 5.99/2.41 U12(tt, X, Y) -> 0(mult(X, Y)) 5.99/2.41 U62(tt, A, B) -> mult(prod(A), prod(B)) 5.99/2.41 U71(tt, A, B) -> U72(tt, A, B) 5.99/2.41 U72(tt, A, B) -> plus(sum(A), sum(B)) 5.99/2.41 mult(0(X), Y) -> U11(tt, X, Y) 5.99/2.41 mult(1(X), Y) -> U21(tt, X, Y) 5.99/2.41 plus(z, X) -> X 5.99/2.41 plus(1(X), 1(Y)) -> U51(tt, X, Y) 5.99/2.41 sum(union(A, B)) -> U71(tt, A, B) 5.99/2.41 5.99/2.41 The set E consists of the following equations: 5.99/2.41 5.99/2.41 mult(x, y) == mult(y, x) 5.99/2.41 plus(x, y) == plus(y, x) 5.99/2.41 union(x, y) == union(y, x) 5.99/2.41 mult(mult(x, y), z') == mult(x, mult(y, z')) 5.99/2.41 plus(plus(x, y), z') == plus(x, plus(y, z')) 5.99/2.41 union(union(x, y), z') == union(x, union(y, z')) 5.99/2.41 5.99/2.41 The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering: 5.99/2.41 5.99/2.41 mult(0(X), Y) -> U11(tt, X, Y) 5.99/2.41 mult(1(X), Y) -> U21(tt, X, Y) 5.99/2.41 plus(z, X) -> X 5.99/2.41 plus(1(X), 1(Y)) -> U51(tt, X, Y) 5.99/2.41 Used ordering: 5.99/2.41 Polynomial interpretation [POLO]: 5.99/2.41 5.99/2.41 POL(0(x_1)) = 1 + x_1 5.99/2.41 POL(1(x_1)) = 3 + 2*x_1 + 3*x_1^2 5.99/2.41 POL(U11(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_2*x_3 + 2*x_3 5.99/2.41 POL(U12(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 2*x_2*x_3 + 2*x_3 5.99/2.41 POL(U21(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + 2*x_2*x_3 + 2*x_3 5.99/2.41 POL(U51(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.41 POL(U62(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + 3*x_2*x_3 + 2*x_3 5.99/2.41 POL(U71(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.41 POL(U72(x_1, x_2, x_3)) = 2*x_1 + 2*x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + x_2 + x_3 5.99/2.41 POL(mult(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2 5.99/2.41 POL(plus(x_1, x_2)) = x_1 + x_2 5.99/2.41 POL(prod(x_1)) = x_1 5.99/2.41 POL(sum(x_1)) = x_1 5.99/2.41 POL(tt) = 0 5.99/2.41 POL(union(x_1, x_2)) = x_1 + 2*x_1*x_2 + x_2 5.99/2.41 POL(z) = 1 5.99/2.41 5.99/2.41 5.99/2.41 5.99/2.41 5.99/2.41 ---------------------------------------- 5.99/2.41 5.99/2.41 (10) 5.99/2.41 Obligation: 5.99/2.41 Equational rewrite system: 5.99/2.41 The TRS R consists of the following rules: 5.99/2.41 5.99/2.41 U11(tt, X, Y) -> U12(tt, X, Y) 5.99/2.41 U12(tt, X, Y) -> 0(mult(X, Y)) 5.99/2.41 U62(tt, A, B) -> mult(prod(A), prod(B)) 5.99/2.41 U71(tt, A, B) -> U72(tt, A, B) 5.99/2.41 U72(tt, A, B) -> plus(sum(A), sum(B)) 5.99/2.41 sum(union(A, B)) -> U71(tt, A, B) 5.99/2.41 5.99/2.41 The set E consists of the following equations: 5.99/2.41 5.99/2.41 mult(x, y) == mult(y, x) 5.99/2.41 plus(x, y) == plus(y, x) 5.99/2.41 union(x, y) == union(y, x) 5.99/2.41 mult(mult(x, y), z') == mult(x, mult(y, z')) 5.99/2.41 plus(plus(x, y), z') == plus(x, plus(y, z')) 5.99/2.41 union(union(x, y), z') == union(x, union(y, z')) 5.99/2.41 5.99/2.41 5.99/2.41 ---------------------------------------- 5.99/2.41 5.99/2.41 (11) RRRPoloETRSProof (EQUIVALENT) 5.99/2.41 The following E TRS is given: Equational rewrite system: 5.99/2.41 The TRS R consists of the following rules: 5.99/2.41 5.99/2.41 U11(tt, X, Y) -> U12(tt, X, Y) 5.99/2.41 U12(tt, X, Y) -> 0(mult(X, Y)) 5.99/2.41 U62(tt, A, B) -> mult(prod(A), prod(B)) 5.99/2.41 U71(tt, A, B) -> U72(tt, A, B) 5.99/2.41 U72(tt, A, B) -> plus(sum(A), sum(B)) 5.99/2.41 sum(union(A, B)) -> U71(tt, A, B) 5.99/2.41 5.99/2.41 The set E consists of the following equations: 5.99/2.41 5.99/2.41 mult(x, y) == mult(y, x) 5.99/2.41 plus(x, y) == plus(y, x) 5.99/2.41 union(x, y) == union(y, x) 5.99/2.41 mult(mult(x, y), z') == mult(x, mult(y, z')) 5.99/2.41 plus(plus(x, y), z') == plus(x, plus(y, z')) 5.99/2.41 union(union(x, y), z') == union(x, union(y, z')) 5.99/2.41 5.99/2.41 The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering: 5.99/2.41 5.99/2.41 U11(tt, X, Y) -> U12(tt, X, Y) 5.99/2.41 U12(tt, X, Y) -> 0(mult(X, Y)) 5.99/2.41 U62(tt, A, B) -> mult(prod(A), prod(B)) 5.99/2.41 U71(tt, A, B) -> U72(tt, A, B) 5.99/2.41 U72(tt, A, B) -> plus(sum(A), sum(B)) 5.99/2.41 sum(union(A, B)) -> U71(tt, A, B) 5.99/2.41 Used ordering: 5.99/2.41 Polynomial interpretation [POLO]: 5.99/2.41 5.99/2.41 POL(0(x_1)) = 1 + x_1 5.99/2.41 POL(U11(x_1, x_2, x_3)) = 3 + x_1 + x_1*x_2 + 2*x_1*x_2*x_3 + 2*x_1*x_3 + 2*x_2 + x_3 5.99/2.41 POL(U12(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_2*x_3 + 2*x_3 5.99/2.41 POL(U62(x_1, x_2, x_3)) = x_1 + 2*x_1*x_3 + x_2 + x_2*x_3 + x_3 5.99/2.41 POL(U71(x_1, x_2, x_3)) = 2 + 3*x_1 + 3*x_1*x_2 + 2*x_1*x_2*x_3 + 3*x_1*x_3 + 2*x_2 + 2*x_3 5.99/2.41 POL(U72(x_1, x_2, x_3)) = 3 + 2*x_1 + 2*x_1*x_2 + x_2 + 2*x_2*x_3 + 3*x_3 5.99/2.41 POL(mult(x_1, x_2)) = x_1 + x_2 5.99/2.41 POL(plus(x_1, x_2)) = x_1 + x_2 5.99/2.41 POL(prod(x_1)) = x_1 5.99/2.41 POL(sum(x_1)) = 3*x_1 5.99/2.41 POL(tt) = 2 5.99/2.41 POL(union(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2 5.99/2.41 5.99/2.41 5.99/2.41 5.99/2.41 5.99/2.41 ---------------------------------------- 5.99/2.41 5.99/2.41 (12) 5.99/2.41 Obligation: 5.99/2.41 Equational rewrite system: 5.99/2.41 R is empty. 5.99/2.41 The set E consists of the following equations: 5.99/2.41 5.99/2.41 mult(x, y) == mult(y, x) 5.99/2.41 plus(x, y) == plus(y, x) 5.99/2.41 union(x, y) == union(y, x) 5.99/2.41 mult(mult(x, y), z') == mult(x, mult(y, z')) 5.99/2.41 plus(plus(x, y), z') == plus(x, plus(y, z')) 5.99/2.41 union(union(x, y), z') == union(x, union(y, z')) 5.99/2.41 5.99/2.41 5.99/2.41 ---------------------------------------- 5.99/2.41 5.99/2.41 (13) RisEmptyProof (EQUIVALENT) 5.99/2.41 The TRS R is empty. Hence, termination is trivially proven. 5.99/2.41 ---------------------------------------- 5.99/2.41 5.99/2.41 (14) 5.99/2.41 YES 6.19/2.48 EOF