(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__U11(tt, V2) → A__U12(a__isNat(V2))
A__U11(tt, V2) → A__ISNAT(V2)
A__U31(tt, V2) → A__U32(a__isNat(V2))
A__U31(tt, V2) → A__ISNAT(V2)
A__U41(tt, N) → MARK(N)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U51(tt, M, N) → A__ISNAT(N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__U52(tt, M, N) → MARK(N)
A__U52(tt, M, N) → MARK(M)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U71(tt, M, N) → A__ISNAT(N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__U72(tt, M, N) → MARK(N)
A__U72(tt, M, N) → MARK(M)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__U21(a__isNat(V1))
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, 0) → A__ISNAT(N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__PLUS(N, s(M)) → A__ISNAT(M)
A__X(N, 0) → A__U61(a__isNat(N))
A__X(N, 0) → A__ISNAT(N)
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
A__X(N, s(M)) → A__ISNAT(M)
MARK(U11(X1, X2)) → A__U11(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → A__U12(mark(X))
MARK(U12(X)) → MARK(X)
MARK(isNat(X)) → A__ISNAT(X)
MARK(U21(X)) → A__U21(mark(X))
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → A__U31(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → A__U32(mark(X))
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
MARK(U52(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U61(X)) → A__U61(mark(X))
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
MARK(U72(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 17 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__U11(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
A__U31(tt, V2) → A__ISNAT(V2)
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__U11(tt, V2) → A__ISNAT(V2)
A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
A__ISNAT(s(V1)) → A__ISNAT(V1)
A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
A__U31(tt, V2) → A__ISNAT(V2)
A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
The TRS R consists of the following rules:
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__isNat(X) → isNat(X)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U31(X1, X2) → U31(X1, X2)
a__U32(tt) → tt
a__U32(X) → U32(X)
a__U21(tt) → tt
a__U21(X) → U21(X)
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U11(X1, X2) → U11(X1, X2)
a__U12(tt) → tt
a__U12(X) → U12(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- A__ISNAT(plus(V1, V2)) → A__U11(a__isNat(V1), V2)
The graph contains the following edges 1 > 2
- A__ISNAT(x(V1, V2)) → A__U31(a__isNat(V1), V2)
The graph contains the following edges 1 > 2
- A__U11(tt, V2) → A__ISNAT(V2)
The graph contains the following edges 2 >= 1
- A__U31(tt, V2) → A__ISNAT(V2)
The graph contains the following edges 2 >= 1
- A__ISNAT(plus(V1, V2)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1
- A__ISNAT(s(V1)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1
- A__ISNAT(x(V1, V2)) → A__ISNAT(V1)
The graph contains the following edges 1 > 1
(9) TRUE
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U41(tt, N) → MARK(N)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
A__U52(tt, M, N) → MARK(M)
MARK(U52(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
A__U72(tt, M, N) → MARK(N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
A__U72(tt, M, N) → MARK(M)
MARK(U72(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(x(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
A__U72(tt, M, N) → MARK(M)
MARK(x(X1, X2)) → MARK(X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
POL(MARK(x1)) = | 0A | + | 0A | · | x1 |
POL(U11(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U12(x1)) = | -I | + | 0A | · | x1 |
POL(U21(x1)) = | -I | + | 0A | · | x1 |
POL(U31(x1, x2)) = | 1A | + | 0A | · | x1 | + | 1A | · | x2 |
POL(U32(x1)) = | -I | + | 0A | · | x1 |
POL(U41(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(A__U41(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(mark(x1)) = | 0A | + | 0A | · | x1 |
POL(U51(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U51(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U52(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__isNat(x1)) = | 0A | + | 0A | · | x1 |
POL(A__PLUS(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U52(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(plus(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U61(x1)) = | 0A | + | 0A | · | x1 |
POL(U71(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 1A | · | x2 | + | 0A | · | x3 |
POL(A__U71(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 1A | · | x2 | + | 0A | · | x3 |
POL(A__U72(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 1A | · | x2 | + | 0A | · | x3 |
POL(a__x(x1, x2)) = | 1A | + | 0A | · | x1 | + | 1A | · | x2 |
POL(A__X(x1, x2)) = | 0A | + | 0A | · | x1 | + | 1A | · | x2 |
POL(U72(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 1A | · | x2 | + | 0A | · | x3 |
POL(x(x1, x2)) = | 1A | + | 0A | · | x1 | + | 1A | · | x2 |
POL(a__U72(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 1A | · | x2 | + | 0A | · | x3 |
POL(a__U71(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 1A | · | x2 | + | 0A | · | x3 |
POL(a__U61(x1)) = | 0A | + | 0A | · | x1 |
POL(a__plus(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U52(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U51(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U41(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U32(x1)) = | 0A | + | 0A | · | x1 |
POL(a__U31(x1, x2)) = | 1A | + | 0A | · | x1 | + | 1A | · | x2 |
POL(a__U21(x1)) = | 0A | + | 0A | · | x1 |
POL(isNat(x1)) = | -I | + | 0A | · | x1 |
POL(a__U12(x1)) = | -I | + | 0A | · | x1 |
POL(a__U11(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
The following usable rules [FROCOS05] were oriented:
a__x(X1, X2) → x(X1, X2)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U61(X) → U61(X)
a__plus(X1, X2) → plus(X1, X2)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U41(X1, X2) → U41(X1, X2)
a__U32(X) → U32(X)
a__U31(X1, X2) → U31(X1, X2)
a__U21(X) → U21(X)
a__isNat(X) → isNat(X)
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__x(N, 0) → a__U61(a__isNat(N))
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U21(tt) → tt
a__U32(tt) → tt
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U61(tt) → 0
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(tt) → tt
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__U41(tt, N) → mark(N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U41(tt, N) → MARK(N)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
A__U52(tt, M, N) → MARK(M)
MARK(U52(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
A__U72(tt, M, N) → MARK(N)
MARK(U71(X1, X2, X3)) → MARK(X1)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
MARK(U72(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
MARK(U71(X1, X2, X3)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
POL(MARK(x1)) = | -I | + | 0A | · | x1 |
POL(U11(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U12(x1)) = | 0A | + | 0A | · | x1 |
POL(U21(x1)) = | 0A | + | 0A | · | x1 |
POL(U31(x1, x2)) = | 0A | + | 0A | · | x1 | + | 1A | · | x2 |
POL(U32(x1)) = | 0A | + | 0A | · | x1 |
POL(U41(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(A__U41(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(mark(x1)) = | -I | + | 0A | · | x1 |
POL(U51(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U51(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U52(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__isNat(x1)) = | -I | + | 0A | · | x1 |
POL(A__PLUS(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U52(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(plus(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U61(x1)) = | 0A | + | 0A | · | x1 |
POL(U71(x1, x2, x3)) = | -I | + | 1A | · | x1 | + | 1A | · | x2 | + | 0A | · | x3 |
POL(A__U71(x1, x2, x3)) = | -I | + | 1A | · | x1 | + | 1A | · | x2 | + | 0A | · | x3 |
POL(A__U72(x1, x2, x3)) = | 1A | + | 0A | · | x1 | + | 1A | · | x2 | + | 0A | · | x3 |
POL(a__x(x1, x2)) = | 1A | + | 0A | · | x1 | + | 1A | · | x2 |
POL(A__X(x1, x2)) = | 1A | + | 0A | · | x1 | + | 1A | · | x2 |
POL(U72(x1, x2, x3)) = | 1A | + | 0A | · | x1 | + | 1A | · | x2 | + | 0A | · | x3 |
POL(x(x1, x2)) = | 1A | + | 0A | · | x1 | + | 1A | · | x2 |
POL(a__U72(x1, x2, x3)) = | 1A | + | 0A | · | x1 | + | 1A | · | x2 | + | 0A | · | x3 |
POL(a__U71(x1, x2, x3)) = | -I | + | 1A | · | x1 | + | 1A | · | x2 | + | 0A | · | x3 |
POL(a__U61(x1)) = | 0A | + | 0A | · | x1 |
POL(a__plus(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U52(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U51(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U41(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U32(x1)) = | 0A | + | 0A | · | x1 |
POL(a__U31(x1, x2)) = | 0A | + | 0A | · | x1 | + | 1A | · | x2 |
POL(a__U21(x1)) = | 0A | + | 0A | · | x1 |
POL(isNat(x1)) = | -I | + | 0A | · | x1 |
POL(a__U12(x1)) = | 0A | + | 0A | · | x1 |
POL(a__U11(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
The following usable rules [FROCOS05] were oriented:
a__x(X1, X2) → x(X1, X2)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U61(X) → U61(X)
a__plus(X1, X2) → plus(X1, X2)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U41(X1, X2) → U41(X1, X2)
a__U32(X) → U32(X)
a__U31(X1, X2) → U31(X1, X2)
a__U21(X) → U21(X)
a__isNat(X) → isNat(X)
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__x(N, 0) → a__U61(a__isNat(N))
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U21(tt) → tt
a__U32(tt) → tt
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U61(tt) → 0
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(tt) → tt
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__U41(tt, N) → mark(N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U41(tt, N) → MARK(N)
MARK(U41(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
A__U52(tt, M, N) → MARK(M)
MARK(U52(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
A__U72(tt, M, N) → MARK(N)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
MARK(U72(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
MARK(U41(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
POL(MARK(x1)) = | 0A | + | 0A | · | x1 |
POL(U11(x1, x2)) = | 0A | + | 0A | · | x1 | + | -I | · | x2 |
POL(U12(x1)) = | 0A | + | 0A | · | x1 |
POL(U21(x1)) = | -I | + | 0A | · | x1 |
POL(U31(x1, x2)) = | -I | + | 0A | · | x1 | + | -I | · | x2 |
POL(U32(x1)) = | -I | + | 0A | · | x1 |
POL(U41(x1, x2)) = | 1A | + | 1A | · | x1 | + | 0A | · | x2 |
POL(A__U41(x1, x2)) = | 1A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(mark(x1)) = | 0A | + | 0A | · | x1 |
POL(U51(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U51(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U52(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__isNat(x1)) = | 0A | + | -I | · | x1 |
POL(A__PLUS(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U52(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(plus(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U61(x1)) = | 1A | + | 0A | · | x1 |
POL(U71(x1, x2, x3)) = | -I | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U71(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U72(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__x(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(A__X(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U72(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(x(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U72(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U71(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U61(x1)) = | 1A | + | 0A | · | x1 |
POL(a__plus(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U52(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U51(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U41(x1, x2)) = | 1A | + | 1A | · | x1 | + | 0A | · | x2 |
POL(a__U32(x1)) = | 0A | + | 0A | · | x1 |
POL(a__U31(x1, x2)) = | 0A | + | 0A | · | x1 | + | -I | · | x2 |
POL(a__U21(x1)) = | 0A | + | 0A | · | x1 |
POL(isNat(x1)) = | 0A | + | -I | · | x1 |
POL(a__U12(x1)) = | 0A | + | 0A | · | x1 |
POL(a__U11(x1, x2)) = | 0A | + | 0A | · | x1 | + | -I | · | x2 |
The following usable rules [FROCOS05] were oriented:
a__x(X1, X2) → x(X1, X2)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U61(X) → U61(X)
a__plus(X1, X2) → plus(X1, X2)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U41(X1, X2) → U41(X1, X2)
a__U32(X) → U32(X)
a__U31(X1, X2) → U31(X1, X2)
a__U21(X) → U21(X)
a__isNat(X) → isNat(X)
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__x(N, 0) → a__U61(a__isNat(N))
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U21(tt) → tt
a__U32(tt) → tt
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U61(tt) → 0
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(tt) → tt
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__U41(tt, N) → mark(N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U41(tt, N) → MARK(N)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
A__U52(tt, M, N) → MARK(M)
MARK(U52(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U61(X)) → MARK(X)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
A__U72(tt, M, N) → MARK(N)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
MARK(U72(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(17) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
MARK(U61(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
POL(MARK(x1)) = | 0A | + | 0A | · | x1 |
POL(U11(x1, x2)) = | -I | + | 0A | · | x1 | + | -I | · | x2 |
POL(U12(x1)) = | 0A | + | 0A | · | x1 |
POL(U21(x1)) = | -I | + | 0A | · | x1 |
POL(U31(x1, x2)) = | -I | + | 0A | · | x1 | + | -I | · | x2 |
POL(U32(x1)) = | 0A | + | 0A | · | x1 |
POL(U41(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(A__U41(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(mark(x1)) = | 0A | + | 0A | · | x1 |
POL(U51(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U51(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U52(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__isNat(x1)) = | 1A | + | -I | · | x1 |
POL(A__PLUS(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U52(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(plus(x1, x2)) = | 1A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U61(x1)) = | 1A | + | 1A | · | x1 |
POL(U71(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U71(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U72(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__x(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(A__X(x1, x2)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U72(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(x(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U72(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U71(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U61(x1)) = | 1A | + | 1A | · | x1 |
POL(a__plus(x1, x2)) = | 1A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U52(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U51(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U41(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U32(x1)) = | 0A | + | 0A | · | x1 |
POL(a__U31(x1, x2)) = | -I | + | 0A | · | x1 | + | -I | · | x2 |
POL(a__U21(x1)) = | -I | + | 0A | · | x1 |
POL(isNat(x1)) = | 1A | + | -I | · | x1 |
POL(a__U12(x1)) = | 0A | + | 0A | · | x1 |
POL(a__U11(x1, x2)) = | 0A | + | 0A | · | x1 | + | -I | · | x2 |
The following usable rules [FROCOS05] were oriented:
a__x(X1, X2) → x(X1, X2)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U61(X) → U61(X)
a__plus(X1, X2) → plus(X1, X2)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U41(X1, X2) → U41(X1, X2)
a__U32(X) → U32(X)
a__U31(X1, X2) → U31(X1, X2)
a__U21(X) → U21(X)
a__isNat(X) → isNat(X)
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__x(N, 0) → a__U61(a__isNat(N))
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U21(tt) → tt
a__U32(tt) → tt
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U61(tt) → 0
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(tt) → tt
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__U41(tt, N) → mark(N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U41(tt, N) → MARK(N)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
A__U52(tt, M, N) → MARK(M)
MARK(U52(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
A__U72(tt, M, N) → MARK(N)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
MARK(U72(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(x(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(19) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
MARK(U12(X)) → MARK(X)
MARK(U51(X1, X2, X3)) → MARK(X1)
A__U52(tt, M, N) → MARK(M)
MARK(plus(X1, X2)) → MARK(X2)
A__U72(tt, M, N) → MARK(N)
MARK(x(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
POL(MARK(x1)) = | -I | + | 0A | · | x1 |
POL(U11(x1, x2)) = | -I | + | 0A | · | x1 | + | 1A | · | x2 |
POL(U12(x1)) = | -I | + | 1A | · | x1 |
POL(U21(x1)) = | -I | + | 0A | · | x1 |
POL(U31(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U32(x1)) = | -I | + | 0A | · | x1 |
POL(U41(x1, x2)) = | 5A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(A__U41(x1, x2)) = | 5A | + | -I | · | x1 | + | 0A | · | x2 |
POL(mark(x1)) = | -I | + | 0A | · | x1 |
POL(U51(x1, x2, x3)) = | -I | + | 5A | · | x1 | + | 5A | · | x2 | + | 0A | · | x3 |
POL(A__U51(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 5A | · | x2 | + | 0A | · | x3 |
POL(A__U52(x1, x2, x3)) = | -I | + | -I | · | x1 | + | 5A | · | x2 | + | 0A | · | x3 |
POL(a__isNat(x1)) = | -I | + | 0A | · | x1 |
POL(A__PLUS(x1, x2)) = | -I | + | 0A | · | x1 | + | 5A | · | x2 |
POL(U52(x1, x2, x3)) = | 5A | + | 0A | · | x1 | + | 5A | · | x2 | + | 0A | · | x3 |
POL(plus(x1, x2)) = | 0A | + | 0A | · | x1 | + | 5A | · | x2 |
POL(U71(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 5A | · | x3 |
POL(A__U71(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 5A | · | x3 |
POL(A__U72(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 5A | · | x3 |
POL(a__x(x1, x2)) = | 0A | + | 5A | · | x1 | + | 0A | · | x2 |
POL(A__X(x1, x2)) = | 0A | + | 5A | · | x1 | + | 0A | · | x2 |
POL(U72(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 5A | · | x3 |
POL(x(x1, x2)) = | 0A | + | 5A | · | x1 | + | 0A | · | x2 |
POL(a__U72(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 5A | · | x3 |
POL(a__U71(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 5A | · | x3 |
POL(a__U61(x1)) = | 0A | + | -I | · | x1 |
POL(U61(x1)) = | 0A | + | -I | · | x1 |
POL(a__plus(x1, x2)) = | 0A | + | 0A | · | x1 | + | 5A | · | x2 |
POL(a__U52(x1, x2, x3)) = | 5A | + | 0A | · | x1 | + | 5A | · | x2 | + | 0A | · | x3 |
POL(a__U51(x1, x2, x3)) = | -I | + | 5A | · | x1 | + | 5A | · | x2 | + | 0A | · | x3 |
POL(a__U41(x1, x2)) = | 5A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U32(x1)) = | -I | + | 0A | · | x1 |
POL(a__U31(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U21(x1)) = | -I | + | 0A | · | x1 |
POL(isNat(x1)) = | -I | + | 0A | · | x1 |
POL(a__U12(x1)) = | -I | + | 1A | · | x1 |
POL(a__U11(x1, x2)) = | -I | + | 0A | · | x1 | + | 1A | · | x2 |
The following usable rules [FROCOS05] were oriented:
a__x(X1, X2) → x(X1, X2)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U61(X) → U61(X)
a__plus(X1, X2) → plus(X1, X2)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U41(X1, X2) → U41(X1, X2)
a__U32(X) → U32(X)
a__U31(X1, X2) → U31(X1, X2)
a__U21(X) → U21(X)
a__isNat(X) → isNat(X)
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__x(N, 0) → a__U61(a__isNat(N))
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U21(tt) → tt
a__U32(tt) → tt
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U61(tt) → 0
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(tt) → tt
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__U41(tt, N) → mark(N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
(20) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U41(tt, N) → MARK(N)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
MARK(U52(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
MARK(U72(X1, X2, X3)) → MARK(X1)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(21) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
MARK(U72(X1, X2, X3)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(A__PLUS(x1, x2)) = x1
POL(A__U41(x1, x2)) = x2
POL(A__U51(x1, x2, x3)) = x3
POL(A__U52(x1, x2, x3)) = x3
POL(A__U71(x1, x2, x3)) = 1
POL(A__U72(x1, x2, x3)) = 1
POL(A__X(x1, x2)) = 1
POL(MARK(x1)) = x1
POL(U11(x1, x2)) = x1
POL(U12(x1)) = x1
POL(U21(x1)) = x1
POL(U31(x1, x2)) = x1
POL(U32(x1)) = x1
POL(U41(x1, x2)) = x1 + x2
POL(U51(x1, x2, x3)) = x3
POL(U52(x1, x2, x3)) = x1 + x3
POL(U61(x1)) = 0
POL(U71(x1, x2, x3)) = 1
POL(U72(x1, x2, x3)) = 1 + x1
POL(a__U11(x1, x2)) = x1
POL(a__U12(x1)) = x1
POL(a__U21(x1)) = x1
POL(a__U31(x1, x2)) = x1
POL(a__U32(x1)) = x1
POL(a__U41(x1, x2)) = x1 + x2
POL(a__U51(x1, x2, x3)) = x3
POL(a__U52(x1, x2, x3)) = x1 + x3
POL(a__U61(x1)) = 0
POL(a__U71(x1, x2, x3)) = 1
POL(a__U72(x1, x2, x3)) = 1 + x1
POL(a__isNat(x1)) = 0
POL(a__plus(x1, x2)) = x1
POL(a__x(x1, x2)) = 1
POL(isNat(x1)) = 0
POL(mark(x1)) = x1
POL(plus(x1, x2)) = x1
POL(s(x1)) = x1
POL(tt) = 0
POL(x(x1, x2)) = 1
The following usable rules [FROCOS05] were oriented:
a__x(X1, X2) → x(X1, X2)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U61(X) → U61(X)
a__plus(X1, X2) → plus(X1, X2)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U41(X1, X2) → U41(X1, X2)
a__U32(X) → U32(X)
a__U31(X1, X2) → U31(X1, X2)
a__U21(X) → U21(X)
a__isNat(X) → isNat(X)
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__x(N, 0) → a__U61(a__isNat(N))
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U21(tt) → tt
a__U32(tt) → tt
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U61(tt) → 0
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(tt) → tt
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__U41(tt, N) → mark(N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
(22) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U41(tt, N) → MARK(N)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
MARK(U52(X1, X2, X3)) → MARK(X1)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(23) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
MARK(U52(X1, X2, X3)) → MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
POL(MARK(x1)) = | -I | + | 0A | · | x1 |
POL(U11(x1, x2)) = | -I | + | 0A | · | x1 | + | -I | · | x2 |
POL(U21(x1)) = | -I | + | 0A | · | x1 |
POL(U31(x1, x2)) = | -I | + | 0A | · | x1 | + | -I | · | x2 |
POL(U32(x1)) = | -I | + | 0A | · | x1 |
POL(U41(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(A__U41(x1, x2)) = | -I | + | -I | · | x1 | + | 0A | · | x2 |
POL(mark(x1)) = | 0A | + | 0A | · | x1 |
POL(U51(x1, x2, x3)) = | 1A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U51(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U52(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__isNat(x1)) = | 0A | + | -I | · | x1 |
POL(A__PLUS(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U52(x1, x2, x3)) = | 1A | + | 1A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(plus(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U71(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U71(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(A__U72(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__x(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(A__X(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(U72(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(x(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U72(x1, x2, x3)) = | 0A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U71(x1, x2, x3)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U61(x1)) = | -I | + | 0A | · | x1 |
POL(U61(x1)) = | -I | + | 0A | · | x1 |
POL(a__plus(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U52(x1, x2, x3)) = | 1A | + | 1A | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U51(x1, x2, x3)) = | 1A | + | -I | · | x1 | + | 0A | · | x2 | + | 0A | · | x3 |
POL(a__U41(x1, x2)) = | 0A | + | 0A | · | x1 | + | 0A | · | x2 |
POL(a__U32(x1)) = | -I | + | 0A | · | x1 |
POL(a__U31(x1, x2)) = | 0A | + | 0A | · | x1 | + | -I | · | x2 |
POL(a__U21(x1)) = | 0A | + | 0A | · | x1 |
POL(isNat(x1)) = | 0A | + | -I | · | x1 |
POL(a__U12(x1)) = | 0A | + | -I | · | x1 |
POL(U12(x1)) = | 0A | + | -I | · | x1 |
POL(a__U11(x1, x2)) = | -I | + | 0A | · | x1 | + | -I | · | x2 |
The following usable rules [FROCOS05] were oriented:
a__x(X1, X2) → x(X1, X2)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U61(X) → U61(X)
a__plus(X1, X2) → plus(X1, X2)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U41(X1, X2) → U41(X1, X2)
a__U32(X) → U32(X)
a__U31(X1, X2) → U31(X1, X2)
a__U21(X) → U21(X)
a__isNat(X) → isNat(X)
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__x(N, 0) → a__U61(a__isNat(N))
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U21(tt) → tt
a__U32(tt) → tt
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U61(tt) → 0
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(tt) → tt
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__U41(tt, N) → mark(N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
(24) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U21(X)) → MARK(X)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U41(tt, N) → MARK(N)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(25) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(0) = 0
POL(A__PLUS(x1, x2)) = x1
POL(A__U41(x1, x2)) = x2
POL(A__U51(x1, x2, x3)) = x3
POL(A__U52(x1, x2, x3)) = x3
POL(A__U71(x1, x2, x3)) = 1 + x2 + x3
POL(A__U72(x1, x2, x3)) = 1 + x2 + x3
POL(A__X(x1, x2)) = 1 + x1 + x2
POL(MARK(x1)) = x1
POL(U11(x1, x2)) = x1
POL(U12(x1)) = 0
POL(U21(x1)) = x1
POL(U31(x1, x2)) = 1 + x1 + x2
POL(U32(x1)) = 1 + x1
POL(U41(x1, x2)) = x2
POL(U51(x1, x2, x3)) = x3
POL(U52(x1, x2, x3)) = x3
POL(U61(x1)) = 0
POL(U71(x1, x2, x3)) = 1 + x2 + x3
POL(U72(x1, x2, x3)) = 1 + x2 + x3
POL(a__U11(x1, x2)) = x1
POL(a__U12(x1)) = 0
POL(a__U21(x1)) = x1
POL(a__U31(x1, x2)) = 1 + x1 + x2
POL(a__U32(x1)) = 1 + x1
POL(a__U41(x1, x2)) = x2
POL(a__U51(x1, x2, x3)) = x3
POL(a__U52(x1, x2, x3)) = x3
POL(a__U61(x1)) = 0
POL(a__U71(x1, x2, x3)) = 1 + x2 + x3
POL(a__U72(x1, x2, x3)) = 1 + x2 + x3
POL(a__isNat(x1)) = x1
POL(a__plus(x1, x2)) = x1
POL(a__x(x1, x2)) = 1 + x1 + x2
POL(isNat(x1)) = x1
POL(mark(x1)) = x1
POL(plus(x1, x2)) = x1
POL(s(x1)) = x1
POL(tt) = 0
POL(x(x1, x2)) = 1 + x1 + x2
The following usable rules [FROCOS05] were oriented:
a__x(X1, X2) → x(X1, X2)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U61(X) → U61(X)
a__plus(X1, X2) → plus(X1, X2)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U41(X1, X2) → U41(X1, X2)
a__U32(X) → U32(X)
a__U31(X1, X2) → U31(X1, X2)
a__U21(X) → U21(X)
a__isNat(X) → isNat(X)
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__x(N, 0) → a__U61(a__isNat(N))
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U21(tt) → tt
a__U32(tt) → tt
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U61(tt) → 0
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(tt) → tt
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__U41(tt, N) → mark(N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
(26) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U21(X)) → MARK(X)
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
A__U41(tt, N) → MARK(N)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
MARK(s(X)) → MARK(X)
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(27) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
MARK(U41(X1, X2)) → A__U41(mark(X1), X2)
MARK(U51(X1, X2, X3)) → A__U51(mark(X1), X2, X3)
MARK(U52(X1, X2, X3)) → A__U52(mark(X1), X2, X3)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(U71(X1, X2, X3)) → A__U71(mark(X1), X2, X3)
A__X(N, s(M)) → A__U71(a__isNat(M), M, N)
MARK(U72(X1, X2, X3)) → A__U72(mark(X1), X2, X3)
MARK(s(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(
x1) =
x1
U11(
x1,
x2) =
x1
U21(
x1) =
x1
U41(
x1,
x2) =
U41(
x1,
x2)
A__U41(
x1,
x2) =
x2
mark(
x1) =
x1
tt =
tt
U51(
x1,
x2,
x3) =
U51(
x1,
x2,
x3)
A__U51(
x1,
x2,
x3) =
x3
A__U52(
x1,
x2,
x3) =
x3
a__isNat(
x1) =
a__isNat
A__PLUS(
x1,
x2) =
x1
0 =
0
s(
x1) =
s(
x1)
U52(
x1,
x2,
x3) =
U52(
x1,
x2,
x3)
plus(
x1,
x2) =
plus(
x1,
x2)
U71(
x1,
x2,
x3) =
U71(
x1,
x2,
x3)
A__U71(
x1,
x2,
x3) =
A__U71(
x2,
x3)
A__U72(
x1,
x2,
x3) =
A__U72(
x2,
x3)
a__x(
x1,
x2) =
a__x(
x1,
x2)
A__X(
x1,
x2) =
A__X(
x1,
x2)
U72(
x1,
x2,
x3) =
U72(
x1,
x2,
x3)
x(
x1,
x2) =
x(
x1,
x2)
a__U72(
x1,
x2,
x3) =
a__U72(
x1,
x2,
x3)
a__U71(
x1,
x2,
x3) =
a__U71(
x1,
x2,
x3)
a__U61(
x1) =
a__U61(
x1)
U61(
x1) =
U61(
x1)
a__plus(
x1,
x2) =
a__plus(
x1,
x2)
a__U52(
x1,
x2,
x3) =
a__U52(
x1,
x2,
x3)
a__U51(
x1,
x2,
x3) =
a__U51(
x1,
x2,
x3)
a__U41(
x1,
x2) =
a__U41(
x1,
x2)
a__U32(
x1) =
x1
U32(
x1) =
x1
a__U31(
x1,
x2) =
a__U31
U31(
x1,
x2) =
U31
a__U21(
x1) =
x1
isNat(
x1) =
isNat
a__U12(
x1) =
x1
U12(
x1) =
x1
a__U11(
x1,
x2) =
x1
Recursive path order with status [RPO].
Quasi-Precedence:
[U713, AU712, AU722, ax2, AX2, U723, x2, aU723, aU713] > [U412, U513, U523, plus2, aplus2, aU523, aU513, aU412] > [tt, aisNat, 0, s1, aU611, U611, aU31, U31, isNat]
Status:
U31: multiset
x2: multiset
AU722: multiset
aplus2: [2,1]
isNat: multiset
aU31: multiset
aU611: [1]
tt: multiset
s1: multiset
U513: [2,3,1]
AU712: multiset
plus2: [2,1]
aU412: [1,2]
U611: [1]
aisNat: multiset
aU523: [2,3,1]
0: multiset
aU513: [2,3,1]
ax2: multiset
U523: [2,3,1]
U412: [1,2]
AX2: multiset
U723: multiset
aU713: multiset
aU723: multiset
U713: multiset
The following usable rules [FROCOS05] were oriented:
a__x(X1, X2) → x(X1, X2)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U61(X) → U61(X)
a__plus(X1, X2) → plus(X1, X2)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U41(X1, X2) → U41(X1, X2)
a__U32(X) → U32(X)
a__U31(X1, X2) → U31(X1, X2)
a__U21(X) → U21(X)
a__isNat(X) → isNat(X)
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__x(N, 0) → a__U61(a__isNat(N))
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U21(tt) → tt
a__U32(tt) → tt
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U61(tt) → 0
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(tt) → tt
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__U41(tt, N) → mark(N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
(28) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U11(X1, X2)) → MARK(X1)
MARK(U21(X)) → MARK(X)
A__U41(tt, N) → MARK(N)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
A__PLUS(N, 0) → A__U41(a__isNat(N), N)
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → MARK(N)
A__U71(tt, M, N) → A__U72(a__isNat(N), M, N)
A__U72(tt, M, N) → A__PLUS(a__x(mark(N), mark(M)), mark(N))
A__U72(tt, M, N) → A__X(mark(N), mark(M))
MARK(x(X1, X2)) → A__X(mark(X1), mark(X2))
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(29) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 7 less nodes.
(30) Complex Obligation (AND)
(31) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U21(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(32) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(33) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(U21(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(34) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- MARK(U21(X)) → MARK(X)
The graph contains the following edges 1 > 1
- MARK(U11(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1
(35) TRUE
(36) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(37) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
A__PLUS(N, s(M)) → A__U51(a__isNat(M), M, N)
A__U52(tt, M, N) → A__PLUS(mark(N), mark(M))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__PLUS(
x1,
x2) =
A__PLUS(
x2)
s(
x1) =
s(
x1)
A__U51(
x1,
x2,
x3) =
A__U51(
x1,
x2)
a__isNat(
x1) =
a__isNat
tt =
tt
A__U52(
x1,
x2,
x3) =
A__U52(
x1,
x2)
mark(
x1) =
x1
a__x(
x1,
x2) =
a__x(
x1,
x2)
x(
x1,
x2) =
x(
x1,
x2)
a__U72(
x1,
x2,
x3) =
a__U72(
x1,
x2,
x3)
U72(
x1,
x2,
x3) =
U72(
x1,
x2,
x3)
a__U71(
x1,
x2,
x3) =
a__U71(
x1,
x2,
x3)
U71(
x1,
x2,
x3) =
U71(
x1,
x2,
x3)
a__U61(
x1) =
a__U61
U61(
x1) =
U61
a__plus(
x1,
x2) =
a__plus(
x1,
x2)
plus(
x1,
x2) =
plus(
x1,
x2)
a__U52(
x1,
x2,
x3) =
a__U52(
x1,
x2,
x3)
U52(
x1,
x2,
x3) =
U52(
x1,
x2,
x3)
a__U51(
x1,
x2,
x3) =
a__U51(
x1,
x2,
x3)
U51(
x1,
x2,
x3) =
U51(
x1,
x2,
x3)
a__U41(
x1,
x2) =
a__U41(
x1,
x2)
U41(
x1,
x2) =
U41(
x1,
x2)
a__U32(
x1) =
x1
U32(
x1) =
x1
a__U31(
x1,
x2) =
x1
U31(
x1,
x2) =
x1
a__U21(
x1) =
x1
U21(
x1) =
x1
isNat(
x1) =
isNat
a__U12(
x1) =
a__U12
U12(
x1) =
U12
a__U11(
x1,
x2) =
x1
U11(
x1,
x2) =
x1
0 =
0
Recursive path order with status [RPO].
Quasi-Precedence:
[APLUS1, AU512, AU522] > [aisNat, tt, isNat, aU12, U12] > [s1, aU412, U412]
[ax2, x2, aU723, U723, aU713, U713, aU61, U61] > [aplus2, plus2, aU523, U523, aU513, U513] > [aisNat, tt, isNat, aU12, U12] > [s1, aU412, U412]
[ax2, x2, aU723, U723, aU713, U713, aU61, U61] > 0 > [aisNat, tt, isNat, aU12, U12] > [s1, aU412, U412]
Status:
aU61: []
x2: [1,2]
isNat: multiset
aplus2: [1,2]
U61: []
tt: multiset
s1: multiset
AU522: [2,1]
aU12: multiset
U513: [3,2,1]
plus2: [1,2]
aU412: multiset
U12: multiset
aisNat: multiset
aU523: [3,2,1]
0: multiset
aU513: [3,2,1]
U523: [3,2,1]
ax2: [1,2]
APLUS1: [1]
AU512: [2,1]
U412: multiset
U723: [3,2,1]
aU713: [3,2,1]
aU723: [3,2,1]
U713: [3,2,1]
The following usable rules [FROCOS05] were oriented:
a__x(X1, X2) → x(X1, X2)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U61(X) → U61(X)
a__plus(X1, X2) → plus(X1, X2)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U41(X1, X2) → U41(X1, X2)
a__U32(X) → U32(X)
a__U31(X1, X2) → U31(X1, X2)
a__U21(X) → U21(X)
a__isNat(X) → isNat(X)
a__U12(X) → U12(X)
a__U11(X1, X2) → U11(X1, X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__isNat(0) → tt
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__x(N, 0) → a__U61(a__isNat(N))
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U21(tt) → tt
a__U32(tt) → tt
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U61(tt) → 0
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(U61(X)) → a__U61(mark(X))
mark(tt) → tt
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__U41(tt, N) → mark(N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
(38) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__U51(tt, M, N) → A__U52(a__isNat(N), M, N)
The TRS R consists of the following rules:
a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, V2) → a__U32(a__isNat(V2))
a__U32(tt) → tt
a__U41(tt, N) → mark(N)
a__U51(tt, M, N) → a__U52(a__isNat(N), M, N)
a__U52(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__U61(tt) → 0
a__U71(tt, M, N) → a__U72(a__isNat(N), M, N)
a__U72(tt, M, N) → a__plus(a__x(mark(N), mark(M)), mark(N))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__isNat(x(V1, V2)) → a__U31(a__isNat(V1), V2)
a__plus(N, 0) → a__U41(a__isNat(N), N)
a__plus(N, s(M)) → a__U51(a__isNat(M), M, N)
a__x(N, 0) → a__U61(a__isNat(N))
a__x(N, s(M)) → a__U71(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U32(X)) → a__U32(mark(X))
mark(U41(X1, X2)) → a__U41(mark(X1), X2)
mark(U51(X1, X2, X3)) → a__U51(mark(X1), X2, X3)
mark(U52(X1, X2, X3)) → a__U52(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(U61(X)) → a__U61(mark(X))
mark(U71(X1, X2, X3)) → a__U71(mark(X1), X2, X3)
mark(U72(X1, X2, X3)) → a__U72(mark(X1), X2, X3)
mark(x(X1, X2)) → a__x(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U32(X) → U32(X)
a__U41(X1, X2) → U41(X1, X2)
a__U51(X1, X2, X3) → U51(X1, X2, X3)
a__U52(X1, X2, X3) → U52(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)
a__U61(X) → U61(X)
a__U71(X1, X2, X3) → U71(X1, X2, X3)
a__U72(X1, X2, X3) → U72(X1, X2, X3)
a__x(X1, X2) → x(X1, X2)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(39) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(40) TRUE