YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given ETRS could be proven:

(0) ETRS
(1) RRRPoloETRSProof [EQUIVALENT, 221 ms]
(2) ETRS
(3) RRRPoloETRSProof [EQUIVALENT, 121 ms]
(4) ETRS
(5) RRRPoloETRSProof [EQUIVALENT, 67 ms]
(6) ETRS
(7) RRRPoloETRSProof [EQUIVALENT, 38 ms]
(8) ETRS
(9) RRRPoloETRSProof [EQUIVALENT, 24 ms]
(10) ETRS
(11) RRRPoloETRSProof [EQUIVALENT, 0 ms]
(12) ETRS
(13) RisEmptyProof [EQUIVALENT, 0 ms]
(14) YES


----------------------------------------

(0)
Obligation:
Equational rewrite system:
The TRS R consists of the following rules:

   union(X, empty) -> X
   union(empty, X) -> X
   0(z) -> z
   U11(tt, X, Y) -> U12(tt, X, Y)
   U12(tt, X, Y) -> 0(mult(X, Y))
   U21(tt, X, Y) -> U22(tt, X, Y)
   U22(tt, X, Y) -> plus(0(mult(X, Y)), Y)
   U31(tt, X, Y) -> U32(tt, X, Y)
   U32(tt, X, Y) -> 0(plus(X, Y))
   U41(tt, X, Y) -> U42(tt, X, Y)
   U42(tt, X, Y) -> 1(plus(X, Y))
   U51(tt, X, Y) -> U52(tt, X, Y)
   U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z)))
   U61(tt, A, B) -> U62(tt, A, B)
   U62(tt, A, B) -> mult(prod(A), prod(B))
   U71(tt, A, B) -> U72(tt, A, B)
   U72(tt, A, B) -> plus(sum(A), sum(B))
   mult(z, X) -> z
   mult(0(X), Y) -> U11(tt, X, Y)
   mult(1(X), Y) -> U21(tt, X, Y)
   plus(z, X) -> X
   plus(0(X), 0(Y)) -> U31(tt, X, Y)
   plus(0(X), 1(Y)) -> U41(tt, X, Y)
   plus(1(X), 1(Y)) -> U51(tt, X, Y)
   prod(empty) -> 1(z)
   prod(singl(X)) -> X
   prod(union(A, B)) -> U61(tt, A, B)
   sum(empty) -> 0(z)
   sum(singl(X)) -> X
   sum(union(A, B)) -> U71(tt, A, B)

The set E consists of the following equations:

   mult(x, y) == mult(y, x)
   plus(x, y) == plus(y, x)
   union(x, y) == union(y, x)
   mult(mult(x, y), z') == mult(x, mult(y, z'))
   plus(plus(x, y), z') == plus(x, plus(y, z'))
   union(union(x, y), z') == union(x, union(y, z'))


----------------------------------------

(1) RRRPoloETRSProof (EQUIVALENT)
The following E TRS is given: Equational rewrite system:
The TRS R consists of the following rules:

   union(X, empty) -> X
   union(empty, X) -> X
   0(z) -> z
   U11(tt, X, Y) -> U12(tt, X, Y)
   U12(tt, X, Y) -> 0(mult(X, Y))
   U21(tt, X, Y) -> U22(tt, X, Y)
   U22(tt, X, Y) -> plus(0(mult(X, Y)), Y)
   U31(tt, X, Y) -> U32(tt, X, Y)
   U32(tt, X, Y) -> 0(plus(X, Y))
   U41(tt, X, Y) -> U42(tt, X, Y)
   U42(tt, X, Y) -> 1(plus(X, Y))
   U51(tt, X, Y) -> U52(tt, X, Y)
   U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z)))
   U61(tt, A, B) -> U62(tt, A, B)
   U62(tt, A, B) -> mult(prod(A), prod(B))
   U71(tt, A, B) -> U72(tt, A, B)
   U72(tt, A, B) -> plus(sum(A), sum(B))
   mult(z, X) -> z
   mult(0(X), Y) -> U11(tt, X, Y)
   mult(1(X), Y) -> U21(tt, X, Y)
   plus(z, X) -> X
   plus(0(X), 0(Y)) -> U31(tt, X, Y)
   plus(0(X), 1(Y)) -> U41(tt, X, Y)
   plus(1(X), 1(Y)) -> U51(tt, X, Y)
   prod(empty) -> 1(z)
   prod(singl(X)) -> X
   prod(union(A, B)) -> U61(tt, A, B)
   sum(empty) -> 0(z)
   sum(singl(X)) -> X
   sum(union(A, B)) -> U71(tt, A, B)

The set E consists of the following equations:

   mult(x, y) == mult(y, x)
   plus(x, y) == plus(y, x)
   union(x, y) == union(y, x)
   mult(mult(x, y), z') == mult(x, mult(y, z'))
   plus(plus(x, y), z') == plus(x, plus(y, z'))
   union(union(x, y), z') == union(x, union(y, z'))

The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:

   union(X, empty) -> X
   union(empty, X) -> X
   U22(tt, X, Y) -> plus(0(mult(X, Y)), Y)
   U51(tt, X, Y) -> U52(tt, X, Y)
   prod(empty) -> 1(z)
   sum(empty) -> 0(z)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = x_1
   POL(1(x_1)) = 1 + x_1
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   POL(empty) = 2
   POL(mult(x_1, x_2)) = x_1 + 2*x_1*x_2 + x_2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(prod(x_1)) = x_1
   POL(singl(x_1)) = 3*x_1 + 3*x_1^2
   POL(sum(x_1)) = 2*x_1
   POL(tt) = 0
   POL(union(x_1, x_2)) = x_1 + 2*x_1*x_2 + x_2
   POL(z) = 0




----------------------------------------

(2)
Obligation:
Equational rewrite system:
The TRS R consists of the following rules:

   0(z) -> z
   U11(tt, X, Y) -> U12(tt, X, Y)
   U12(tt, X, Y) -> 0(mult(X, Y))
   U21(tt, X, Y) -> U22(tt, X, Y)
   U31(tt, X, Y) -> U32(tt, X, Y)
   U32(tt, X, Y) -> 0(plus(X, Y))
   U41(tt, X, Y) -> U42(tt, X, Y)
   U42(tt, X, Y) -> 1(plus(X, Y))
   U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z)))
   U61(tt, A, B) -> U62(tt, A, B)
   U62(tt, A, B) -> mult(prod(A), prod(B))
   U71(tt, A, B) -> U72(tt, A, B)
   U72(tt, A, B) -> plus(sum(A), sum(B))
   mult(z, X) -> z
   mult(0(X), Y) -> U11(tt, X, Y)
   mult(1(X), Y) -> U21(tt, X, Y)
   plus(z, X) -> X
   plus(0(X), 0(Y)) -> U31(tt, X, Y)
   plus(0(X), 1(Y)) -> U41(tt, X, Y)
   plus(1(X), 1(Y)) -> U51(tt, X, Y)
   prod(singl(X)) -> X
   prod(union(A, B)) -> U61(tt, A, B)
   sum(singl(X)) -> X
   sum(union(A, B)) -> U71(tt, A, B)

The set E consists of the following equations:

   mult(x, y) == mult(y, x)
   plus(x, y) == plus(y, x)
   union(x, y) == union(y, x)
   mult(mult(x, y), z') == mult(x, mult(y, z'))
   plus(plus(x, y), z') == plus(x, plus(y, z'))
   union(union(x, y), z') == union(x, union(y, z'))


----------------------------------------

(3) RRRPoloETRSProof (EQUIVALENT)
The following E TRS is given: Equational rewrite system:
The TRS R consists of the following rules:

   0(z) -> z
   U11(tt, X, Y) -> U12(tt, X, Y)
   U12(tt, X, Y) -> 0(mult(X, Y))
   U21(tt, X, Y) -> U22(tt, X, Y)
   U31(tt, X, Y) -> U32(tt, X, Y)
   U32(tt, X, Y) -> 0(plus(X, Y))
   U41(tt, X, Y) -> U42(tt, X, Y)
   U42(tt, X, Y) -> 1(plus(X, Y))
   U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z)))
   U61(tt, A, B) -> U62(tt, A, B)
   U62(tt, A, B) -> mult(prod(A), prod(B))
   U71(tt, A, B) -> U72(tt, A, B)
   U72(tt, A, B) -> plus(sum(A), sum(B))
   mult(z, X) -> z
   mult(0(X), Y) -> U11(tt, X, Y)
   mult(1(X), Y) -> U21(tt, X, Y)
   plus(z, X) -> X
   plus(0(X), 0(Y)) -> U31(tt, X, Y)
   plus(0(X), 1(Y)) -> U41(tt, X, Y)
   plus(1(X), 1(Y)) -> U51(tt, X, Y)
   prod(singl(X)) -> X
   prod(union(A, B)) -> U61(tt, A, B)
   sum(singl(X)) -> X
   sum(union(A, B)) -> U71(tt, A, B)

The set E consists of the following equations:

   mult(x, y) == mult(y, x)
   plus(x, y) == plus(y, x)
   union(x, y) == union(y, x)
   mult(mult(x, y), z') == mult(x, mult(y, z'))
   plus(plus(x, y), z') == plus(x, plus(y, z'))
   union(union(x, y), z') == union(x, union(y, z'))

The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:

   prod(union(A, B)) -> U61(tt, A, B)
   sum(singl(X)) -> X
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = x_1
   POL(1(x_1)) = x_1
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   POL(mult(x_1, x_2)) = x_1 + x_2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(prod(x_1)) = 2*x_1
   POL(singl(x_1)) = 3*x_1 + 3*x_1^2
   POL(sum(x_1)) = 1 + x_1
   POL(tt) = 0
   POL(union(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2
   POL(z) = 0




----------------------------------------

(4)
Obligation:
Equational rewrite system:
The TRS R consists of the following rules:

   0(z) -> z
   U11(tt, X, Y) -> U12(tt, X, Y)
   U12(tt, X, Y) -> 0(mult(X, Y))
   U21(tt, X, Y) -> U22(tt, X, Y)
   U31(tt, X, Y) -> U32(tt, X, Y)
   U32(tt, X, Y) -> 0(plus(X, Y))
   U41(tt, X, Y) -> U42(tt, X, Y)
   U42(tt, X, Y) -> 1(plus(X, Y))
   U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z)))
   U61(tt, A, B) -> U62(tt, A, B)
   U62(tt, A, B) -> mult(prod(A), prod(B))
   U71(tt, A, B) -> U72(tt, A, B)
   U72(tt, A, B) -> plus(sum(A), sum(B))
   mult(z, X) -> z
   mult(0(X), Y) -> U11(tt, X, Y)
   mult(1(X), Y) -> U21(tt, X, Y)
   plus(z, X) -> X
   plus(0(X), 0(Y)) -> U31(tt, X, Y)
   plus(0(X), 1(Y)) -> U41(tt, X, Y)
   plus(1(X), 1(Y)) -> U51(tt, X, Y)
   prod(singl(X)) -> X
   sum(union(A, B)) -> U71(tt, A, B)

The set E consists of the following equations:

   mult(x, y) == mult(y, x)
   plus(x, y) == plus(y, x)
   union(x, y) == union(y, x)
   mult(mult(x, y), z') == mult(x, mult(y, z'))
   plus(plus(x, y), z') == plus(x, plus(y, z'))
   union(union(x, y), z') == union(x, union(y, z'))


----------------------------------------

(5) RRRPoloETRSProof (EQUIVALENT)
The following E TRS is given: Equational rewrite system:
The TRS R consists of the following rules:

   0(z) -> z
   U11(tt, X, Y) -> U12(tt, X, Y)
   U12(tt, X, Y) -> 0(mult(X, Y))
   U21(tt, X, Y) -> U22(tt, X, Y)
   U31(tt, X, Y) -> U32(tt, X, Y)
   U32(tt, X, Y) -> 0(plus(X, Y))
   U41(tt, X, Y) -> U42(tt, X, Y)
   U42(tt, X, Y) -> 1(plus(X, Y))
   U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z)))
   U61(tt, A, B) -> U62(tt, A, B)
   U62(tt, A, B) -> mult(prod(A), prod(B))
   U71(tt, A, B) -> U72(tt, A, B)
   U72(tt, A, B) -> plus(sum(A), sum(B))
   mult(z, X) -> z
   mult(0(X), Y) -> U11(tt, X, Y)
   mult(1(X), Y) -> U21(tt, X, Y)
   plus(z, X) -> X
   plus(0(X), 0(Y)) -> U31(tt, X, Y)
   plus(0(X), 1(Y)) -> U41(tt, X, Y)
   plus(1(X), 1(Y)) -> U51(tt, X, Y)
   prod(singl(X)) -> X
   sum(union(A, B)) -> U71(tt, A, B)

The set E consists of the following equations:

   mult(x, y) == mult(y, x)
   plus(x, y) == plus(y, x)
   union(x, y) == union(y, x)
   mult(mult(x, y), z') == mult(x, mult(y, z'))
   plus(plus(x, y), z') == plus(x, plus(y, z'))
   union(union(x, y), z') == union(x, union(y, z'))

The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:

   0(z) -> z
   U31(tt, X, Y) -> U32(tt, X, Y)
   U41(tt, X, Y) -> U42(tt, X, Y)
   U61(tt, A, B) -> U62(tt, A, B)
   plus(0(X), 0(Y)) -> U31(tt, X, Y)
   plus(0(X), 1(Y)) -> U41(tt, X, Y)
   prod(singl(X)) -> X
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = 2 + 2*x_1
   POL(1(x_1)) = 2*x_1
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   POL(mult(x_1, x_2)) = x_1 + x_1*x_2 + x_2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(prod(x_1)) = x_1
   POL(singl(x_1)) = 3 + 3*x_1 + 3*x_1^2
   POL(sum(x_1)) = 2*x_1
   POL(tt) = 0
   POL(union(x_1, x_2)) = x_1 + x_1*x_2 + x_2
   POL(z) = 0




----------------------------------------

(6)
Obligation:
Equational rewrite system:
The TRS R consists of the following rules:

   U11(tt, X, Y) -> U12(tt, X, Y)
   U12(tt, X, Y) -> 0(mult(X, Y))
   U21(tt, X, Y) -> U22(tt, X, Y)
   U32(tt, X, Y) -> 0(plus(X, Y))
   U42(tt, X, Y) -> 1(plus(X, Y))
   U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z)))
   U62(tt, A, B) -> mult(prod(A), prod(B))
   U71(tt, A, B) -> U72(tt, A, B)
   U72(tt, A, B) -> plus(sum(A), sum(B))
   mult(z, X) -> z
   mult(0(X), Y) -> U11(tt, X, Y)
   mult(1(X), Y) -> U21(tt, X, Y)
   plus(z, X) -> X
   plus(1(X), 1(Y)) -> U51(tt, X, Y)
   sum(union(A, B)) -> U71(tt, A, B)

The set E consists of the following equations:

   mult(x, y) == mult(y, x)
   plus(x, y) == plus(y, x)
   union(x, y) == union(y, x)
   mult(mult(x, y), z') == mult(x, mult(y, z'))
   plus(plus(x, y), z') == plus(x, plus(y, z'))
   union(union(x, y), z') == union(x, union(y, z'))


----------------------------------------

(7) RRRPoloETRSProof (EQUIVALENT)
The following E TRS is given: Equational rewrite system:
The TRS R consists of the following rules:

   U11(tt, X, Y) -> U12(tt, X, Y)
   U12(tt, X, Y) -> 0(mult(X, Y))
   U21(tt, X, Y) -> U22(tt, X, Y)
   U32(tt, X, Y) -> 0(plus(X, Y))
   U42(tt, X, Y) -> 1(plus(X, Y))
   U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z)))
   U62(tt, A, B) -> mult(prod(A), prod(B))
   U71(tt, A, B) -> U72(tt, A, B)
   U72(tt, A, B) -> plus(sum(A), sum(B))
   mult(z, X) -> z
   mult(0(X), Y) -> U11(tt, X, Y)
   mult(1(X), Y) -> U21(tt, X, Y)
   plus(z, X) -> X
   plus(1(X), 1(Y)) -> U51(tt, X, Y)
   sum(union(A, B)) -> U71(tt, A, B)

The set E consists of the following equations:

   mult(x, y) == mult(y, x)
   plus(x, y) == plus(y, x)
   union(x, y) == union(y, x)
   mult(mult(x, y), z') == mult(x, mult(y, z'))
   plus(plus(x, y), z') == plus(x, plus(y, z'))
   union(union(x, y), z') == union(x, union(y, z'))

The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:

   U21(tt, X, Y) -> U22(tt, X, Y)
   U32(tt, X, Y) -> 0(plus(X, Y))
   U42(tt, X, Y) -> 1(plus(X, Y))
   U52(tt, X, Y) -> 0(plus(plus(X, Y), 1(z)))
   mult(z, X) -> z
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = x_1
   POL(1(x_1)) = 2*x_1
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   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
   POL(mult(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(prod(x_1)) = x_1
   POL(sum(x_1)) = 3*x_1
   POL(tt) = 0
   POL(union(x_1, x_2)) = x_1 + 3*x_1*x_2 + x_2
   POL(z) = 0




----------------------------------------

(8)
Obligation:
Equational rewrite system:
The TRS R consists of the following rules:

   U11(tt, X, Y) -> U12(tt, X, Y)
   U12(tt, X, Y) -> 0(mult(X, Y))
   U62(tt, A, B) -> mult(prod(A), prod(B))
   U71(tt, A, B) -> U72(tt, A, B)
   U72(tt, A, B) -> plus(sum(A), sum(B))
   mult(0(X), Y) -> U11(tt, X, Y)
   mult(1(X), Y) -> U21(tt, X, Y)
   plus(z, X) -> X
   plus(1(X), 1(Y)) -> U51(tt, X, Y)
   sum(union(A, B)) -> U71(tt, A, B)

The set E consists of the following equations:

   mult(x, y) == mult(y, x)
   plus(x, y) == plus(y, x)
   union(x, y) == union(y, x)
   mult(mult(x, y), z') == mult(x, mult(y, z'))
   plus(plus(x, y), z') == plus(x, plus(y, z'))
   union(union(x, y), z') == union(x, union(y, z'))


----------------------------------------

(9) RRRPoloETRSProof (EQUIVALENT)
The following E TRS is given: Equational rewrite system:
The TRS R consists of the following rules:

   U11(tt, X, Y) -> U12(tt, X, Y)
   U12(tt, X, Y) -> 0(mult(X, Y))
   U62(tt, A, B) -> mult(prod(A), prod(B))
   U71(tt, A, B) -> U72(tt, A, B)
   U72(tt, A, B) -> plus(sum(A), sum(B))
   mult(0(X), Y) -> U11(tt, X, Y)
   mult(1(X), Y) -> U21(tt, X, Y)
   plus(z, X) -> X
   plus(1(X), 1(Y)) -> U51(tt, X, Y)
   sum(union(A, B)) -> U71(tt, A, B)

The set E consists of the following equations:

   mult(x, y) == mult(y, x)
   plus(x, y) == plus(y, x)
   union(x, y) == union(y, x)
   mult(mult(x, y), z') == mult(x, mult(y, z'))
   plus(plus(x, y), z') == plus(x, plus(y, z'))
   union(union(x, y), z') == union(x, union(y, z'))

The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:

   mult(0(X), Y) -> U11(tt, X, Y)
   mult(1(X), Y) -> U21(tt, X, Y)
   plus(z, X) -> X
   plus(1(X), 1(Y)) -> U51(tt, X, Y)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = 1 + x_1
   POL(1(x_1)) = 3 + 2*x_1 + 3*x_1^2
   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
   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
   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
   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
   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
   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
   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
   POL(mult(x_1, x_2)) = 1 + 2*x_1 + 2*x_1*x_2 + 2*x_2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(prod(x_1)) = x_1
   POL(sum(x_1)) = x_1
   POL(tt) = 0
   POL(union(x_1, x_2)) = x_1 + 2*x_1*x_2 + x_2
   POL(z) = 1




----------------------------------------

(10)
Obligation:
Equational rewrite system:
The TRS R consists of the following rules:

   U11(tt, X, Y) -> U12(tt, X, Y)
   U12(tt, X, Y) -> 0(mult(X, Y))
   U62(tt, A, B) -> mult(prod(A), prod(B))
   U71(tt, A, B) -> U72(tt, A, B)
   U72(tt, A, B) -> plus(sum(A), sum(B))
   sum(union(A, B)) -> U71(tt, A, B)

The set E consists of the following equations:

   mult(x, y) == mult(y, x)
   plus(x, y) == plus(y, x)
   union(x, y) == union(y, x)
   mult(mult(x, y), z') == mult(x, mult(y, z'))
   plus(plus(x, y), z') == plus(x, plus(y, z'))
   union(union(x, y), z') == union(x, union(y, z'))


----------------------------------------

(11) RRRPoloETRSProof (EQUIVALENT)
The following E TRS is given: Equational rewrite system:
The TRS R consists of the following rules:

   U11(tt, X, Y) -> U12(tt, X, Y)
   U12(tt, X, Y) -> 0(mult(X, Y))
   U62(tt, A, B) -> mult(prod(A), prod(B))
   U71(tt, A, B) -> U72(tt, A, B)
   U72(tt, A, B) -> plus(sum(A), sum(B))
   sum(union(A, B)) -> U71(tt, A, B)

The set E consists of the following equations:

   mult(x, y) == mult(y, x)
   plus(x, y) == plus(y, x)
   union(x, y) == union(y, x)
   mult(mult(x, y), z') == mult(x, mult(y, z'))
   plus(plus(x, y), z') == plus(x, plus(y, z'))
   union(union(x, y), z') == union(x, union(y, z'))

The following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:

   U11(tt, X, Y) -> U12(tt, X, Y)
   U12(tt, X, Y) -> 0(mult(X, Y))
   U62(tt, A, B) -> mult(prod(A), prod(B))
   U71(tt, A, B) -> U72(tt, A, B)
   U72(tt, A, B) -> plus(sum(A), sum(B))
   sum(union(A, B)) -> U71(tt, A, B)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = 1 + x_1
   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
   POL(U12(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_2*x_3 + 2*x_3
   POL(U62(x_1, x_2, x_3)) = x_1 + 2*x_1*x_3 + x_2 + x_2*x_3 + x_3
   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
   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
   POL(mult(x_1, x_2)) = x_1 + x_2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(prod(x_1)) = x_1
   POL(sum(x_1)) = 3*x_1
   POL(tt) = 2
   POL(union(x_1, x_2)) = 3 + 3*x_1 + 2*x_1*x_2 + 3*x_2




----------------------------------------

(12)
Obligation:
Equational rewrite system:
R is empty.
The set E consists of the following equations:

   mult(x, y) == mult(y, x)
   plus(x, y) == plus(y, x)
   union(x, y) == union(y, x)
   mult(mult(x, y), z') == mult(x, mult(y, z'))
   plus(plus(x, y), z') == plus(x, plus(y, z'))
   union(union(x, y), z') == union(x, union(y, z'))


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(13) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(14)
YES