YES Termination w.r.t. Q proof of Transformed_CSR_04_MYNAT_nokinds_noand_C.ari

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

0 QTRS
↳1 QTRSToCSRProof (⇔, 0 ms)
↳2 CSR
↳3 CSDependencyPairsProof (⇔, 0 ms)
↳4 QCSDP
↳5 QCSDependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 QCSDP
        ↳8 QCSDPSubtermProof (⇔, 0 ms)
        ↳9 QCSDP
        ↳10 QCSDependencyGraphProof (⇔, 0 ms)
        ↳11 TRUE
    ↳12 QCSDP
        ↳13 QCSDPSubtermProof (⇔, 0 ms)
        ↳14 QCSDP
        ↳15 QCSDependencyGraphProof (⇔, 0 ms)
        ↳16 TRUE
    ↳17 QCSDP
        ↳18 QCSDPSubtermProof (⇔, 0 ms)
        ↳19 QCSDP
        ↳20 QCSDependencyGraphProof (⇔, 0 ms)
        ↳21 TRUE

(0) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(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(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(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))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
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
U21: {1}
U31: {1}
U32: {1}
U41: {1}
U51: {1}
U52: {1}
s: {1}
plus: {1, 2}
U61: {1}
0: empty set
U71: {1}
U72: {1}
x: {1, 2}
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, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U31: {1}
U32: {1}
U41: {1}
U51: {1}
U52: {1}
s: {1}
plus: {1, 2}
U61: {1}
0: empty set
U71: {1}
U72: {1}
x: {1, 2}

(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 {U12, U21, U32, s, plus, U61, x, U12', U32', PLUS, X, U21', U61'} are replacing on all positions.
For all symbols f in {U11, U31, U41, U51, U52, U71, U72, U11', U31', U52', U51', U72', U71', U41'} we have µ(f) = {1}.
The symbols in {isNat, ISNAT, U} are not replacing on any position.

The ordinary context-sensitive dependency pairs DPo are:

U11'(tt, V2) → U12'(isNat(V2))
U11'(tt, V2) → ISNAT(V2)
U31'(tt, V2) → U32'(isNat(V2))
U31'(tt, V2) → ISNAT(V2)
U51'(tt, M, N) → U52'(isNat(N), M, N)
U51'(tt, M, N) → ISNAT(N)
U52'(tt, M, N) → PLUS(N, M)
U71'(tt, M, N) → U72'(isNat(N), M, N)
U71'(tt, M, N) → ISNAT(N)
U72'(tt, M, N) → PLUS(x(N, M), N)
U72'(tt, M, N) → X(N, M)
ISNAT(plus(V1, V2)) → U11'(isNat(V1), V2)
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → U21'(isNat(V1))
ISNAT(s(V1)) → ISNAT(V1)
ISNAT(x(V1, V2)) → U31'(isNat(V1), V2)
ISNAT(x(V1, V2)) → ISNAT(V1)
PLUS(N, 0) → U41'(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U51'(isNat(M), M, N)
PLUS(N, s(M)) → ISNAT(M)
X(N, 0) → U61'(isNat(N))
X(N, 0) → ISNAT(N)
X(N, s(M)) → U71'(isNat(M), M, N)
X(N, s(M)) → ISNAT(M)

The collapsing dependency pairs are DPc:

U41'(tt, N) → N
U52'(tt, M, N) → N
U52'(tt, M, N) → M
U72'(tt, M, N) → N
U72'(tt, M, N) → M


The hidden terms of R are:
none

Every hiding context is built from:none

Hence, the new unhiding pairs DPu are :

U41'(tt, N) → U(N)
U52'(tt, M, N) → U(N)
U52'(tt, M, N) → U(M)
U72'(tt, M, N) → U(N)
U72'(tt, M, N) → U(M)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)

Q is empty.

(5) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 3 SCCs with 17 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U21, U32, s, plus, U61, x} are replacing on all positions.
For all symbols f in {U11, U31, U41, U51, U52, U71, U72, U11', U31'} we have µ(f) = {1}.
The symbols in {isNat, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

ISNAT(plus(V1, V2)) → U11'(isNat(V1), V2)
U11'(tt, V2) → ISNAT(V2)
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNAT(x(V1, V2)) → U31'(isNat(V1), V2)
U31'(tt, V2) → ISNAT(V2)
ISNAT(x(V1, V2)) → ISNAT(V1)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)

Q is empty.

(8) QCSDPSubtermProof (EQUIVALENT transformation)

We use the subterm processor [DA_EMMES].


The following pairs can be oriented strictly and are deleted.


ISNAT(plus(V1, V2)) → U11'(isNat(V1), V2)
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNAT(x(V1, V2)) → U31'(isNat(V1), V2)
ISNAT(x(V1, V2)) → ISNAT(V1)
The remaining pairs can at least be oriented weakly.

U11'(tt, V2) → ISNAT(V2)
U31'(tt, V2) → ISNAT(V2)
Used ordering: Combined order from the following AFS and order.
U11'(x1, x2)  =  x2
ISNAT(x1)  =  x1
U31'(x1, x2)  =  x2

Subterm Order

(9) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U21, U32, s, plus, U61, x} are replacing on all positions.
For all symbols f in {U11, U31, U41, U51, U52, U71, U72, U11', U31'} we have µ(f) = {1}.
The symbols in {isNat, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

U11'(tt, V2) → ISNAT(V2)
U31'(tt, V2) → ISNAT(V2)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)

Q is empty.

(10) QCSDependencyGraphProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U21, U32, s, plus, U61, x, PLUS} are replacing on all positions.
For all symbols f in {U11, U31, U41, U51, U52, U71, U72, U52', U51'} we have µ(f) = {1}.
The symbols in {isNat} are not replacing on any position.

The TRS P consists of the following rules:

U51'(tt, M, N) → U52'(isNat(N), M, N)
U52'(tt, M, N) → PLUS(N, M)
PLUS(N, s(M)) → U51'(isNat(M), M, N)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)

Q is empty.

(13) 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'(isNat(M), M, N)
The remaining pairs can at least be oriented weakly.

U51'(tt, M, N) → U52'(isNat(N), M, N)
U52'(tt, M, N) → PLUS(N, M)
Used ordering: Combined order from the following AFS and order.
U52'(x1, x2, x3)  =  x2
U51'(x1, x2, x3)  =  x2
PLUS(x1, x2)  =  x2

Subterm Order

(14) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U21, U32, s, plus, U61, x, PLUS} are replacing on all positions.
For all symbols f in {U11, U31, U41, U51, U52, U71, U72, U52', U51'} we have µ(f) = {1}.
The symbols in {isNat} are not replacing on any position.

The TRS P consists of the following rules:

U51'(tt, M, N) → U52'(isNat(N), M, N)
U52'(tt, M, N) → PLUS(N, M)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)

Q is empty.

(15) QCSDependencyGraphProof (EQUIVALENT transformation)

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

(16) TRUE

(17) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U21, U32, s, plus, U61, x, X} are replacing on all positions.
For all symbols f in {U11, U31, U41, U51, U52, U71, U72, U72', U71'} we have µ(f) = {1}.
The symbols in {isNat} are not replacing on any position.

The TRS P consists of the following rules:

U72'(tt, M, N) → X(N, M)
X(N, s(M)) → U71'(isNat(M), M, N)
U71'(tt, M, N) → U72'(isNat(N), M, N)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)

Q is empty.

(18) 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'(isNat(M), M, N)
The remaining pairs can at least be oriented weakly.

U72'(tt, M, N) → X(N, M)
U71'(tt, M, N) → U72'(isNat(N), M, N)
Used ordering: Combined order from the following AFS and order.
X(x1, x2)  =  x2
U72'(x1, x2, x3)  =  x2
U71'(x1, x2, x3)  =  x2

Subterm Order

(19) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {U12, U21, U32, s, plus, U61, x, X} are replacing on all positions.
For all symbols f in {U11, U31, U41, U51, U52, U71, U72, U72', U71'} we have µ(f) = {1}.
The symbols in {isNat} are not replacing on any position.

The TRS P consists of the following rules:

U72'(tt, M, N) → X(N, M)
U71'(tt, M, N) → U72'(isNat(N), M, N)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(V2))
U32(tt) → tt
U41(tt, N) → N
U51(tt, M, N) → U52(isNat(N), M, N)
U52(tt, M, N) → s(plus(N, M))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(N), M, N)
U72(tt, M, N) → plus(x(N, M), N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
isNat(x(V1, V2)) → U31(isNat(V1), V2)
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), 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