Termination w.r.t. Q proof of Transformed_CSR_04_MYNAT_complete

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 QTRSToCSRProof (⇔, 0 ms)

↳2 CSR

↳3 CSDependencyPairsProof (⇔, 3 ms)

↳4 QCSDP

↳5 QCSDependencyGraphProof (⇔, 0 ms)

↳6 AND

    ↳7 QCSDP

        ↳8 QCSDPReductionPairProof (⇔, 49 ms)

        ↳9 QCSDP

        ↳10 QCSDependencyGraphProof (⇔, 0 ms)

        ↳11 AND

            ↳12 QCSDP

                ↳13 QCSDPReductionPairProof (⇔, 170 ms)

                ↳14 QCSDP

                ↳15 PIsEmptyProof (⇔, 0 ms)

                ↳16 YES

            ↳17 QCSDP

                ↳18 QCSDPReductionPairProof (⇔, 173 ms)

                ↳19 QCSDP

                ↳20 QCSDependencyGraphProof (⇔, 0 ms)

                ↳21 TRUE

    ↳22 QCSDP

        ↳23 QCSDPSubtermProof (⇔, 0 ms)

        ↳24 QCSDP

        ↳25 QCSDependencyGraphProof (⇔, 0 ms)

        ↳26 TRUE

    ↳27 QCSDP

        ↳28 QCSDPSubtermProof (⇔, 0 ms)

        ↳29 QCSDP

        ↳30 QCSDependencyGraphProof (⇔, 0 ms)

        ↳31 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) QTRSToCSRProof (EQUIVALENT transformation)

The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
Special symbols used for the transformation (see [GM04]):
top: top, active: active, mark: mark, ok: ok, proper: proper
The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U13: {1}
U21: {1}
U22: {1}
U31: {1}
U32: {1}
U33: {1}
U41: {1}
U51: {1}
s: {1}
plus: {1, 2}
U61: {1}
0: empty set
U71: {1}
x: {1, 2}
and: {1}
isNatKind: empty set
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).

(2) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The replacement map contains the following entries:

(3) CSDependencyPairsProof (EQUIVALENT transformation)

Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.

(4) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, U33, s, plus, U61, x, U13', U22', U33', PLUS, X, U61'} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U32, U41, U51, U71, and, U12', U11', U21', U32', U31', U51', U71', AND, U41'} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, ISNAT, ISNATKIND, U} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:

U11'(tt, V1, V2) → U12'(isNat(V1), V2)
U11'(tt, V1, V2) → ISNAT(V1)
U12'(tt, V2) → U13'(isNat(V2))
U12'(tt, V2) → ISNAT(V2)
U21'(tt, V1) → U22'(isNat(V1))
U21'(tt, V1) → ISNAT(V1)
U31'(tt, V1, V2) → U32'(isNat(V1), V2)
U31'(tt, V1, V2) → ISNAT(V1)
U32'(tt, V2) → U33'(isNat(V2))
U32'(tt, V2) → ISNAT(V2)
U51'(tt, M, N) → PLUS(N, M)
U71'(tt, M, N) → PLUS(x(N, M), N)
U71'(tt, M, N) → X(N, M)
ISNAT(plus(V1, V2)) → U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
ISNAT(plus(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNAT(plus(V1, V2)) → ISNATKIND(V1)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
ISNAT(s(V1)) → ISNATKIND(V1)
ISNAT(x(V1, V2)) → U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
ISNAT(x(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNAT(x(V1, V2)) → ISNATKIND(V1)
ISNATKIND(plus(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATKIND(x(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNATKIND(x(V1, V2)) → ISNATKIND(V1)
PLUS(N, 0) → U41'(and(isNat(N), isNatKind(N)), N)
PLUS(N, 0) → AND(isNat(N), isNatKind(N))
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U51'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
PLUS(N, s(M)) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
PLUS(N, s(M)) → AND(isNat(M), isNatKind(M))
PLUS(N, s(M)) → ISNAT(M)
X(N, 0) → U61'(and(isNat(N), isNatKind(N)))
X(N, 0) → AND(isNat(N), isNatKind(N))
X(N, 0) → ISNAT(N)
X(N, s(M)) → U71'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
X(N, s(M)) → AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
X(N, s(M)) → AND(isNat(M), isNatKind(M))
X(N, s(M)) → ISNAT(M)

The collapsing dependency pairs are DP_c:

U41'(tt, N) → N
U51'(tt, M, N) → N
U51'(tt, M, N) → M
U71'(tt, M, N) → N
U71'(tt, M, N) → M
AND(tt, X) → X

The hidden terms of R are:

isNatKind(x0)
and(isNat(x0), isNatKind(x0))
isNat(x0)

Every hiding context is built from:

and on positions {1}

Hence, the new unhiding pairs DP_u are :

U41'(tt, N) → U(N)
U51'(tt, M, N) → U(N)
U51'(tt, M, N) → U(M)
U71'(tt, M, N) → U(N)
U71'(tt, M, N) → U(M)
AND(tt, X) → U(X)
U(and(x_0, x_1)) → U(x_0)
U(isNatKind(x0)) → ISNATKIND(x0)
U(and(isNat(x0), isNatKind(x0))) → AND(isNat(x0), isNatKind(x0))
U(isNat(x0)) → ISNAT(x0)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

(5) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 3 SCCs with 21 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, U33, s, plus, U61, x} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U32, U41, U51, U71, and, U12', U11', AND, U21', U31', U32'} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, ISNAT, U, ISNATKIND} are not replacing on any position.

The TRS P consists of the following rules:

U12'(tt, V2) → ISNAT(V2)
ISNAT(plus(V1, V2)) → U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U11'(tt, V1, V2) → U12'(isNat(V1), V2)
U11'(tt, V1, V2) → ISNAT(V1)
ISNAT(plus(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
AND(tt, X) → U(X)
U(and(x_0, x_1)) → U(x_0)
U(isNatKind(x0)) → ISNATKIND(x0)
ISNATKIND(plus(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATKIND(x(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNATKIND(x(V1, V2)) → ISNATKIND(V1)
U(and(isNat(x0), isNatKind(x0))) → AND(isNat(x0), isNatKind(x0))
U(isNat(x0)) → ISNAT(x0)
ISNAT(plus(V1, V2)) → ISNATKIND(V1)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
U21'(tt, V1) → ISNAT(V1)
ISNAT(s(V1)) → ISNATKIND(V1)
ISNAT(x(V1, V2)) → U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U31'(tt, V1, V2) → U32'(isNat(V1), V2)
U32'(tt, V2) → ISNAT(V2)
ISNAT(x(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNAT(x(V1, V2)) → ISNATKIND(V1)
U31'(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

(8) QCSDPReductionPairProof (EQUIVALENT transformation)

Using the order
Polynomial interpretation [POLO]:

POL(0) = 0
POL(AND(x₁, x₂)) = 1 + 2·x₂
POL(ISNAT(x₁)) = 1
POL(ISNATKIND(x₁)) = 1
POL(U(x₁)) = 1 + 2·x₁
POL(U11(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 1
POL(U12(x₁, x₂)) = 0
POL(U12'(x₁, x₂)) = 1
POL(U13(x₁)) = 0
POL(U21(x₁, x₂)) = 0
POL(U21'(x₁, x₂)) = 1
POL(U22(x₁)) = 0
POL(U31(x₁, x₂, x₃)) = 1
POL(U31'(x₁, x₂, x₃)) = 1
POL(U32(x₁, x₂)) = 0
POL(U32'(x₁, x₂)) = 1
POL(U33(x₁)) = 0
POL(U41(x₁, x₂)) = x₂
POL(U51(x₁, x₂, x₃)) = x₃
POL(U61(x₁)) = 0
POL(U71(x₁, x₂, x₃)) = 1 + x₂
POL(and(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = 0
POL(plus(x₁, x₂)) = x₁
POL(s(x₁)) = x₁
POL(tt) = 0
POL(x(x₁, x₂)) = 1 + x₂

the following usable rules

plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U41(tt, N) → N
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U51(tt, M, N) → s(plus(N, M))

could all be oriented weakly.
Furthermore, the pairs

U(and(isNat(x0), isNatKind(x0))) → AND(isNat(x0), isNatKind(x0))
U(isNat(x0)) → ISNAT(x0)

could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].

(9) Obligation:

U12'(tt, V2) → ISNAT(V2)
ISNAT(plus(V1, V2)) → U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U11'(tt, V1, V2) → U12'(isNat(V1), V2)
U11'(tt, V1, V2) → ISNAT(V1)
ISNAT(plus(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
AND(tt, X) → U(X)
U(and(x_0, x_1)) → U(x_0)
U(isNatKind(x0)) → ISNATKIND(x0)
ISNATKIND(plus(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATKIND(x(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNATKIND(x(V1, V2)) → ISNATKIND(V1)
ISNAT(plus(V1, V2)) → ISNATKIND(V1)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
U21'(tt, V1) → ISNAT(V1)
ISNAT(s(V1)) → ISNATKIND(V1)
ISNAT(x(V1, V2)) → U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U31'(tt, V1, V2) → U32'(isNat(V1), V2)
U32'(tt, V2) → ISNAT(V2)
ISNAT(x(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNAT(x(V1, V2)) → ISNATKIND(V1)
U31'(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

(10) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 5 less nodes.

(11) Complex Obligation (AND)

(12) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, U33, s, plus, U61, x} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U32, U41, U51, U71, and, AND} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, U, ISNATKIND} are not replacing on any position.

The TRS P consists of the following rules:

U(and(x_0, x_1)) → U(x_0)
U(isNatKind(x0)) → ISNATKIND(x0)
ISNATKIND(plus(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
AND(tt, X) → U(X)
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATKIND(x(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNATKIND(x(V1, V2)) → ISNATKIND(V1)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

(13) QCSDPReductionPairProof (EQUIVALENT transformation)

Using the order
Combined order from the following AFS and order.
U(x1) = U(x1)
and(x1, x2) = and(x1, x2)
isNatKind(x1) = isNatKind(x1)
ISNATKIND(x1) = ISNATKIND(x1)
plus(x1, x2) = plus(x1, x2)
AND(x1, x2) = AND(x1, x2)
tt = tt
s(x1) = s(x1)
x(x1, x2) = x(x1, x2)
0 = 0
U41(x1, x2) = x2
isNat(x1) = isNat
U51(x1, x2, x3) = U51(x1, x2, x3)
U11(x1, x2, x3) = U11
U21(x1, x2) = U21
U31(x1, x2, x3) = U31
U12(x1, x2) = U12
U13(x1) = U13
U22(x1) = U22
U61(x1) = U61
U71(x1, x2, x3) = U71(x1, x2, x3)
U32(x1, x2) = U32
U33(x1) = x1

Recursive path order with status [RPO].
Quasi-Precedence:

[x₂, U71₃] > [plus₂, U51₃] > s₁ > and₂ > [tt, U13]
[x₂, U71₃] > [plus₂, U51₃] > s₁ > [ISNATKIND₁, AND₂] > U₁ > [tt, U13]
[x₂, U71₃] > [plus₂, U51₃] > [isNat, U31, U32] > isNatKind₁ > and₂ > [tt, U13]
[x₂, U71₃] > [plus₂, U51₃] > [isNat, U31, U32] > isNatKind₁ > [ISNATKIND₁, AND₂] > U₁ > [tt, U13]
[x₂, U71₃] > [plus₂, U51₃] > [isNat, U31, U32] > [U11, U12] > [tt, U13]
[x₂, U71₃] > [plus₂, U51₃] > [isNat, U31, U32] > U21 > U22 > [tt, U13]
[0, U61] > [isNat, U31, U32] > isNatKind₁ > and₂ > [tt, U13]
[0, U61] > [isNat, U31, U32] > isNatKind₁ > [ISNATKIND₁, AND₂] > U₁ > [tt, U13]
[0, U61] > [isNat, U31, U32] > [U11, U12] > [tt, U13]
[0, U61] > [isNat, U31, U32] > U21 > U22 > [tt, U13]

Status:

U₁: [1]
and₂: [2,1]
isNatKind₁: [1]
ISNATKIND₁: [1]
plus₂: [2,1]
AND₂: [1,2]
tt: multiset
s₁: multiset
x₂: [1,2]
0: multiset
isNat: multiset
U51₃: [2,3,1]
U11: multiset
U21: multiset
U31: multiset
U12: multiset
U13: multiset
U22: []
U61: multiset
U71₃: [3,2,1]
U32: multiset

the following usable rules

and(tt, X) → X
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U41(tt, N) → N
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U51(tt, M, N) → s(plus(N, M))

could all be oriented weakly.
Furthermore, the pairs

U(and(x_0, x_1)) → U(x_0)
U(isNatKind(x0)) → ISNATKIND(x0)
ISNATKIND(plus(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
AND(tt, X) → U(X)
ISNATKIND(plus(V1, V2)) → ISNATKIND(V1)
ISNATKIND(s(V1)) → ISNATKIND(V1)
ISNATKIND(x(V1, V2)) → AND(isNatKind(V1), isNatKind(V2))
ISNATKIND(x(V1, V2)) → ISNATKIND(V1)

could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].

(14) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, U33, s, plus, U61, x} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U32, U41, U51, U71, and} we have µ(f) = {1}.
The symbols in {isNat, isNatKind} are not replacing on any position.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

(15) PIsEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R,µ)-chain.

(16) YES

(17) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, U33, s, plus, U61, x} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U32, U41, U51, U71, and, U11', U12', U21', U31', U32'} we have µ(f) = {1}.
The symbols in {isNat, isNatKind, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

ISNAT(plus(V1, V2)) → U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U11'(tt, V1, V2) → U12'(isNat(V1), V2)
U12'(tt, V2) → ISNAT(V2)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
U21'(tt, V1) → ISNAT(V1)
ISNAT(x(V1, V2)) → U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U31'(tt, V1, V2) → U32'(isNat(V1), V2)
U32'(tt, V2) → ISNAT(V2)
U31'(tt, V1, V2) → ISNAT(V1)
U11'(tt, V1, V2) → ISNAT(V1)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

(18) QCSDPReductionPairProof (EQUIVALENT transformation)

Using the order
Combined order from the following AFS and order.
ISNAT(x1) = ISNAT(x1)
plus(x1, x2) = plus(x1, x2)
U11'(x1, x2, x3) = U11'(x2, x3)
and(x1, x2) = and(x1, x2)
isNatKind(x1) = isNatKind(x1)
tt = tt
U12'(x1, x2) = U12'(x1, x2)
isNat(x1) = x1
s(x1) = s(x1)
U21'(x1, x2) = U21'(x1, x2)
x(x1, x2) = x(x1, x2)
U31'(x1, x2, x3) = U31'(x2, x3)
U32'(x1, x2) = U32'(x1, x2)
0 = 0
U41(x1, x2) = x2
U51(x1, x2, x3) = U51(x1, x2, x3)
U11(x1, x2, x3) = U11(x2, x3)
U21(x1, x2) = U21(x2)
U31(x1, x2, x3) = U31
U12(x1, x2) = U12(x1, x2)
U13(x1) = U13
U61(x1) = U61
U71(x1, x2, x3) = U71(x1, x2, x3)
U22(x1) = x1
U32(x1, x2) = U32
U33(x1) = U33

Recursive path order with status [RPO].
Quasi-Precedence:

[tt, x₂, 0, U31, U13, U61, U71₃, U32, U33] > [plus₂, U11'₂, U12'₂, U51₃, U11₂] > s₁ > [ISNAT₁, U21'₂, U31'₂, U32'₂] > [and₂, isNatKind₁]
[tt, x₂, 0, U31, U13, U61, U71₃, U32, U33] > [plus₂, U11'₂, U12'₂, U51₃, U11₂] > s₁ > U21₁
[tt, x₂, 0, U31, U13, U61, U71₃, U32, U33] > [plus₂, U11'₂, U12'₂, U51₃, U11₂] > U12₂

Status:

ISNAT₁: multiset
plus₂: [2,1]
U11'₂: [2,1]
and₂: multiset
isNatKind₁: multiset
tt: multiset
U12'₂: [2,1]
s₁: [1]
U21'₂: multiset
x₂: [1,2]
U31'₂: multiset
U32'₂: multiset
0: multiset
U51₃: [2,3,1]
U11₂: [2,1]
U21₁: multiset
U31: []
U12₂: [2,1]
U13: []
U61: []
U71₃: [3,2,1]
U32: []
U33: []

the following usable rules

plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U41(tt, N) → N
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U51(tt, M, N) → s(plus(N, M))

could all be oriented weakly.
Furthermore, the pairs

ISNAT(plus(V1, V2)) → U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U12'(tt, V2) → ISNAT(V2)
ISNAT(s(V1)) → U21'(isNatKind(V1), V1)
U21'(tt, V1) → ISNAT(V1)
ISNAT(x(V1, V2)) → U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U32'(tt, V2) → ISNAT(V2)
U31'(tt, V1, V2) → ISNAT(V1)
U11'(tt, V1, V2) → ISNAT(V1)

could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].

(19) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, U33, s, plus, U61, x} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U32, U41, U51, U71, and, U12', U11', U32', U31'} we have µ(f) = {1}.
The symbols in {isNat, isNatKind} are not replacing on any position.

The TRS P consists of the following rules:

U11'(tt, V1, V2) → U12'(isNat(V1), V2)
U31'(tt, V1, V2) → U32'(isNat(V1), V2)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

(20) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.

(21) TRUE

(22) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, U33, s, plus, U61, x, PLUS} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U32, U41, U51, U71, and, U51'} we have µ(f) = {1}.
The symbols in {isNat, isNatKind} are not replacing on any position.

The TRS P consists of the following rules:

PLUS(N, s(M)) → U51'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U51'(tt, M, N) → PLUS(N, M)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

(23) QCSDPSubtermProof (EQUIVALENT transformation)

We use the subterm processor [DA_EMMES].

The following pairs can be oriented strictly and are deleted.

PLUS(N, s(M)) → U51'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The remaining pairs can at least be oriented weakly.

U51'(tt, M, N) → PLUS(N, M)

Used ordering: Combined order from the following AFS and order.
U51'(x1, x2, x3) = x2
PLUS(x1, x2) = x2

Subterm Order

(24) Obligation:

U51'(tt, M, N) → PLUS(N, M)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

(25) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 1 less node.

(26) TRUE

(27) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U13, U22, U33, s, plus, U61, x, X} are replacing on all positions.
For all symbols f in {U11, U12, U21, U31, U32, U41, U51, U71, and, U71'} we have µ(f) = {1}.
The symbols in {isNat, isNatKind} are not replacing on any position.

The TRS P consists of the following rules:

U71'(tt, M, N) → X(N, M)
X(N, s(M)) → U71'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

(28) QCSDPSubtermProof (EQUIVALENT transformation)

We use the subterm processor [DA_EMMES].

The following pairs can be oriented strictly and are deleted.

X(N, s(M)) → U71'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

The remaining pairs can at least be oriented weakly.

U71'(tt, M, N) → X(N, M)

Used ordering: Combined order from the following AFS and order.
X(x1, x2) = x2
U71'(x1, x2, x3) = x2

Subterm Order

(29) Obligation:

U71'(tt, M, N) → X(N, M)

The TRS R consists of the following rules:

U11(tt, V1, V2) → U12(isNat(V1), V2)
U12(tt, V2) → U13(isNat(V2))
U13(tt) → tt
U21(tt, V1) → U22(isNat(V1))
U22(tt) → tt
U31(tt, V1, V2) → U32(isNat(V1), V2)
U32(tt, V2) → U33(isNat(V2))
U33(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNat(x(V1, V2)) → U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) → isNatKind(V1)
isNatKind(x(V1, V2)) → and(isNatKind(V1), isNatKind(V2))
plus(N, 0) → U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) → U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) → U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) → U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)

Q is empty.

(30) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 1 less node.

(0) Obligation:

(1) QTRSToCSRProof (EQUIVALENT transformation)

(2) Obligation:

(3) CSDependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) QCSDependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) QCSDPReductionPairProof (EQUIVALENT transformation)

(9) Obligation:

(10) QCSDependencyGraphProof (EQUIVALENT transformation)

(11) Complex Obligation (AND)

(12) Obligation:

(13) QCSDPReductionPairProof (EQUIVALENT transformation)

(14) Obligation:

(15) PIsEmptyProof (EQUIVALENT transformation)

(16) YES

(17) Obligation:

(18) QCSDPReductionPairProof (EQUIVALENT transformation)

(19) Obligation:

(20) QCSDependencyGraphProof (EQUIVALENT transformation)

(21) TRUE

(22) Obligation:

(23) QCSDPSubtermProof (EQUIVALENT transformation)

(24) Obligation:

(25) QCSDependencyGraphProof (EQUIVALENT transformation)

(26) TRUE

(27) Obligation:

(28) QCSDPSubtermProof (EQUIVALENT transformation)

(29) Obligation:

(30) QCSDependencyGraphProof (EQUIVALENT transformation)

(31) TRUE