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

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

(0) Obligation:

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

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

The replacement map contains the following entries:

U11: {1}
tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
U31: {1}
0: empty set
U41: {1}
x: {1, 2}
and: {1}
isNat: empty set

(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 {s, plus, U31, x, PLUS, X, U31'} are replacing on all positions.
For all symbols f in {U11, U21, U41, and, U21', U41', AND, U11'} 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:

U21'(tt, M, N) → PLUS(N, M)
U41'(tt, M, N) → PLUS(x(N, M), N)
U41'(tt, M, N) → X(N, M)
ISNAT(plus(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNAT(x(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(x(V1, V2)) → ISNAT(V1)
PLUS(N, 0) → U11'(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U21'(and(isNat(M), isNat(N)), M, N)
PLUS(N, s(M)) → AND(isNat(M), isNat(N))
PLUS(N, s(M)) → ISNAT(M)
X(N, 0) → U31'(isNat(N))
X(N, 0) → ISNAT(N)
X(N, s(M)) → U41'(and(isNat(M), isNat(N)), M, N)
X(N, s(M)) → AND(isNat(M), isNat(N))
X(N, s(M)) → ISNAT(M)

The collapsing dependency pairs are DPc:

U11'(tt, N) → N
U21'(tt, M, N) → N
U21'(tt, M, N) → M
U41'(tt, M, N) → N
U41'(tt, M, N) → M
AND(tt, X) → X


The hidden terms of R are:

isNat(x0)

Every hiding context is built from:none

Hence, the new unhiding pairs DPu are :

U11'(tt, N) → U(N)
U21'(tt, M, N) → U(N)
U21'(tt, M, N) → U(M)
U41'(tt, M, N) → U(N)
U41'(tt, M, N) → U(M)
AND(tt, X) → U(X)
U(isNat(x0)) → ISNAT(x0)

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)

Q is empty.

(5) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 3 SCCs with 14 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

The TRS P consists of the following rules:

AND(tt, X) → U(X)
U(isNat(x0)) → ISNAT(x0)
ISNAT(plus(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNAT(x(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(x(V1, V2)) → ISNAT(V1)

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)

Q is empty.

(8) QCSDPReductionPairProof (EQUIVALENT transformation)

Using the order
Combined order from the following AFS and order.
AND(x1, x2)  =  x2
tt  =  tt
U(x1)  =  x1
isNat(x1)  =  isNat(x1)
ISNAT(x1)  =  ISNAT(x1)
plus(x1, x2)  =  plus(x1, x2)
s(x1)  =  s(x1)
x(x1, x2)  =  x(x1, x2)
0  =  0
and(x1, x2)  =  x2
U11(x1, x2)  =  U11(x1, x2)
U21(x1, x2, x3)  =  U21(x1, x2, x3)
U31(x1)  =  x1
U41(x1, x2, x3)  =  U41(x1, x2, x3)

Recursive path order with status [RPO].
Quasi-Precedence:
[x2, U413] > [tt, isNat1, ISNAT1, plus2, U213] > s1
[x2, U413] > [tt, isNat1, ISNAT1, plus2, U213] > 0
[x2, U413] > [tt, isNat1, ISNAT1, plus2, U213] > U112

Status:
plus2: [1,2]
tt: multiset
U413: [2,3,1]
s1: multiset
x2: [2,1]
U112: [2,1]
isNat1: [1]
0: multiset
U213: [3,2,1]
ISNAT1: [1]


the following usable rules

isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)

could all be oriented weakly.
Furthermore, the pairs

ISNAT(plus(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(plus(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNAT(x(V1, V2)) → AND(isNat(V1), isNat(V2))
ISNAT(x(V1, V2)) → ISNAT(V1)

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

(9) Obligation:

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

The TRS P consists of the following rules:

AND(tt, X) → U(X)
U(isNat(x0)) → ISNAT(x0)

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), 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 {s, plus, U31, x, PLUS} are replacing on all positions.
For all symbols f in {U11, U21, U41, and, U21'} we have µ(f) = {1}.
The symbols in {isNat} are not replacing on any position.

The TRS P consists of the following rules:

PLUS(N, s(M)) → U21'(and(isNat(M), isNat(N)), M, N)
U21'(tt, M, N) → PLUS(N, M)

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), 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)) → U21'(and(isNat(M), isNat(N)), M, N)
The remaining pairs can at least be oriented weakly.

U21'(tt, M, N) → PLUS(N, M)
Used ordering: Combined order from the following AFS and order.
U21'(x1, x2, x3)  =  x2
PLUS(x1, x2)  =  x2

Subterm Order

(14) Obligation:

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

The TRS P consists of the following rules:

U21'(tt, M, N) → PLUS(N, M)

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)

Q is empty.

(15) QCSDependencyGraphProof (EQUIVALENT transformation)

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

(16) TRUE

(17) Obligation:

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

The TRS P consists of the following rules:

U41'(tt, M, N) → X(N, M)
X(N, s(M)) → U41'(and(isNat(M), isNat(N)), M, N)

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), 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)) → U41'(and(isNat(M), isNat(N)), M, N)
The remaining pairs can at least be oriented weakly.

U41'(tt, M, N) → X(N, M)
Used ordering: Combined order from the following AFS and order.
X(x1, x2)  =  x2
U41'(x1, x2, x3)  =  x2

Subterm Order

(19) Obligation:

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

The TRS P consists of the following rules:

U41'(tt, M, N) → X(N, M)

The TRS R consists of the following rules:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
U31(tt) → 0
U41(tt, M, N) → plus(x(N, M), N)
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
isNat(x(V1, V2)) → and(isNat(V1), isNat(V2))
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)
x(N, 0) → U31(isNat(N))
x(N, s(M)) → U41(and(isNat(M), isNat(N)), M, N)

Q is empty.

(20) QCSDependencyGraphProof (EQUIVALENT transformation)

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

(21) TRUE