Termination w.r.t. Q proof of Transformed_CSR_04_ExSec4_2_DLMMU04

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

0 QTRS

↳1 QTRSToCSRProof (⇔, 0 ms)

↳2 CSR

↳3 CSRInnermostProof (⇔, 0 ms)

↳4 CSR

↳5 CSDependencyPairsProof (⇔, 0 ms)

↳6 QCSDP

↳7 QCSDependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 QCSDP

        ↳10 QCSDPSubtermProof (⇔, 0 ms)

        ↳11 QCSDP

        ↳12 PIsEmptyProof (⇔, 0 ms)

        ↳13 YES

    ↳14 QCSDP

        ↳15 QCSDPSubtermProof (⇔, 0 ms)

        ↳16 QCSDP

        ↳17 PIsEmptyProof (⇔, 0 ms)

        ↳18 YES

(0) Obligation:

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

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(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(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(fst(pair(XS, YS))) → mark(XS)
active(snd(pair(XS, YS))) → mark(YS)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(u(splitAt(N, XS), N, X, XS))
active(u(pair(YS, ZS), N, X, XS)) → mark(pair(cons(X, YS), ZS))
active(head(cons(N, XS))) → mark(N)
active(tail(cons(N, XS))) → mark(XS)
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(natsFrom(X)) → natsFrom(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(fst(X)) → fst(active(X))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(u(X1, X2, X3, X4)) → u(active(X1), X2, X3, X4)
active(head(X)) → head(active(X))
active(tail(X)) → tail(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
natsFrom(mark(X)) → mark(natsFrom(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
fst(mark(X)) → mark(fst(X))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
u(mark(X1), X2, X3, X4) → mark(u(X1, X2, X3, X4))
head(mark(X)) → mark(head(X))
tail(mark(X)) → mark(tail(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(fst(X)) → fst(proper(X))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(u(X1, X2, X3, X4)) → u(proper(X1), proper(X2), proper(X3), proper(X4))
proper(head(X)) → head(proper(X))
proper(tail(X)) → tail(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
natsFrom(ok(X)) → ok(natsFrom(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
fst(ok(X)) → ok(fst(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
u(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(u(X1, X2, X3, X4))
head(ok(X)) → ok(head(X))
tail(ok(X)) → ok(tail(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
take(ok(X1), ok(X2)) → ok(take(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:

natsFrom: {1}
cons: {1}
s: {1}
fst: {1}
pair: {1, 2}
snd: {1}
splitAt: {1, 2}
0: empty set
nil: empty set
u: {1}
head: {1}
tail: {1}
sel: {1, 2}
afterNth: {1, 2}
take: {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:

natsFrom(N) → cons(N, natsFrom(s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, XS), N, X, XS)
u(pair(YS, ZS), N, X, XS) → pair(cons(X, YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → XS
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))

The replacement map contains the following entries:

natsFrom: {1}
cons: {1}
s: {1}
fst: {1}
pair: {1, 2}
snd: {1}
splitAt: {1, 2}
0: empty set
nil: empty set
u: {1}
head: {1}
tail: {1}
sel: {1, 2}
afterNth: {1, 2}
take: {1, 2}

(3) CSRInnermostProof (EQUIVALENT transformation)

The CSR is orthogonal. By [CS_Inn] we can switch to innermost.

(4) Obligation:

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

natsFrom(N) → cons(N, natsFrom(s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, XS), N, X, XS)
u(pair(YS, ZS), N, X, XS) → pair(cons(X, YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → XS
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))

The replacement map contains the following entries:

natsFrom: {1}
cons: {1}
s: {1}
fst: {1}
pair: {1, 2}
snd: {1}
splitAt: {1, 2}
0: empty set
nil: empty set
u: {1}
head: {1}
tail: {1}
sel: {1, 2}
afterNth: {1, 2}
take: {1, 2}

Innermost Strategy.

(5) CSDependencyPairsProof (EQUIVALENT transformation)

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

(6) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {natsFrom, s, fst, pair, snd, splitAt, head, tail, sel, afterNth, take, SPLITAT, HEAD, SEL, AFTERNTH, FST, TAKE, SND, TAIL, NATSFROM} are replacing on all positions.
For all symbols f in {cons, u, U} we have µ(f) = {1}.
The symbols in {U'} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:

SPLITAT(s(N), cons(X, XS)) → U(splitAt(N, XS), N, X, XS)
SPLITAT(s(N), cons(X, XS)) → SPLITAT(N, XS)
SEL(N, XS) → HEAD(afterNth(N, XS))
SEL(N, XS) → AFTERNTH(N, XS)
TAKE(N, XS) → FST(splitAt(N, XS))
TAKE(N, XS) → SPLITAT(N, XS)
AFTERNTH(N, XS) → SND(splitAt(N, XS))
AFTERNTH(N, XS) → SPLITAT(N, XS)

The collapsing dependency pairs are DP_c:

SPLITAT(s(N), cons(X, XS)) → XS
U(pair(YS, ZS), N, X, XS) → X
TAIL(cons(N, XS)) → XS

The hidden terms of R are:

natsFrom(s(x0))

Every hiding context is built from:

s on positions {1}
natsFrom on positions {1}

Hence, the new unhiding pairs DP_u are :

SPLITAT(s(N), cons(X, XS)) → U'(XS)
U(pair(YS, ZS), N, X, XS) → U'(X)
TAIL(cons(N, XS)) → U'(XS)
U'(s(x_0)) → U'(x_0)
U'(natsFrom(x_0)) → U'(x_0)
U'(natsFrom(s(x0))) → NATSFROM(s(x0))

The TRS R consists of the following rules:

natsFrom(N) → cons(N, natsFrom(s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, XS), N, X, XS)
u(pair(YS, ZS), N, X, XS) → pair(cons(X, YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → XS
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))

The set Q consists of the following terms:

natsFrom(x0)
fst(pair(x0, x1))
snd(pair(x0, x1))
splitAt(0, x0)
splitAt(s(x0), cons(x1, x2))
u(pair(x0, x1), x2, x3, x4)
head(cons(x0, x1))
tail(cons(x0, x1))
sel(x0, x1)
take(x0, x1)
afterNth(x0, x1)

(7) QCSDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 9 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {natsFrom, s, fst, pair, snd, splitAt, head, tail, sel, afterNth, take} are replacing on all positions.
For all symbols f in {cons, u} we have µ(f) = {1}.
The symbols in {U'} are not replacing on any position.

The TRS P consists of the following rules:

U'(s(x_0)) → U'(x_0)
U'(natsFrom(x_0)) → U'(x_0)

The TRS R consists of the following rules:

natsFrom(N) → cons(N, natsFrom(s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, XS), N, X, XS)
u(pair(YS, ZS), N, X, XS) → pair(cons(X, YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → XS
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))

The set Q consists of the following terms:

natsFrom(x0)
fst(pair(x0, x1))
snd(pair(x0, x1))
splitAt(0, x0)
splitAt(s(x0), cons(x1, x2))
u(pair(x0, x1), x2, x3, x4)
head(cons(x0, x1))
tail(cons(x0, x1))
sel(x0, x1)
take(x0, x1)
afterNth(x0, x1)

(10) QCSDPSubtermProof (EQUIVALENT transformation)

We use the subterm processor [DA_EMMES].

The following pairs can be oriented strictly and are deleted.

U'(s(x_0)) → U'(x_0)
U'(natsFrom(x_0)) → U'(x_0)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
U'(x1) = x1

Subterm Order

(11) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {natsFrom, s, fst, pair, snd, splitAt, head, tail, sel, afterNth, take} are replacing on all positions.
For all symbols f in {cons, u} we have µ(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

natsFrom(N) → cons(N, natsFrom(s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, XS), N, X, XS)
u(pair(YS, ZS), N, X, XS) → pair(cons(X, YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → XS
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))

The set Q consists of the following terms:

natsFrom(x0)
fst(pair(x0, x1))
snd(pair(x0, x1))
splitAt(0, x0)
splitAt(s(x0), cons(x1, x2))
u(pair(x0, x1), x2, x3, x4)
head(cons(x0, x1))
tail(cons(x0, x1))
sel(x0, x1)
take(x0, x1)
afterNth(x0, x1)

(12) PIsEmptyProof (EQUIVALENT transformation)

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

(13) YES

(14) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {natsFrom, s, fst, pair, snd, splitAt, head, tail, sel, afterNth, take, SPLITAT} are replacing on all positions.
For all symbols f in {cons, u} we have µ(f) = {1}.

The TRS P consists of the following rules:

SPLITAT(s(N), cons(X, XS)) → SPLITAT(N, XS)

The TRS R consists of the following rules:

natsFrom(N) → cons(N, natsFrom(s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, XS), N, X, XS)
u(pair(YS, ZS), N, X, XS) → pair(cons(X, YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → XS
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))

The set Q consists of the following terms:

natsFrom(x0)
fst(pair(x0, x1))
snd(pair(x0, x1))
splitAt(0, x0)
splitAt(s(x0), cons(x1, x2))
u(pair(x0, x1), x2, x3, x4)
head(cons(x0, x1))
tail(cons(x0, x1))
sel(x0, x1)
take(x0, x1)
afterNth(x0, x1)

(15) QCSDPSubtermProof (EQUIVALENT transformation)

We use the subterm processor [DA_EMMES].

The following pairs can be oriented strictly and are deleted.

SPLITAT(s(N), cons(X, XS)) → SPLITAT(N, XS)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
SPLITAT(x1, x2) = x1

Subterm Order

(16) Obligation:

Q-restricted context-sensitive dependency pair problem:
The symbols in {natsFrom, s, fst, pair, snd, splitAt, head, tail, sel, afterNth, take} are replacing on all positions.
For all symbols f in {cons, u} we have µ(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

natsFrom(N) → cons(N, natsFrom(s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, XS), N, X, XS)
u(pair(YS, ZS), N, X, XS) → pair(cons(X, YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → XS
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))

The set Q consists of the following terms:

natsFrom(x0)
fst(pair(x0, x1))
snd(pair(x0, x1))
splitAt(0, x0)
splitAt(s(x0), cons(x1, x2))
u(pair(x0, x1), x2, x3, x4)
head(cons(x0, x1))
tail(cons(x0, x1))
sel(x0, x1)
take(x0, x1)
afterNth(x0, x1)

(17) PIsEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) QTRSToCSRProof (EQUIVALENT transformation)

(2) Obligation:

(3) CSRInnermostProof (EQUIVALENT transformation)

(4) Obligation:

(5) CSDependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) QCSDependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) QCSDPSubtermProof (EQUIVALENT transformation)

(11) Obligation:

(12) PIsEmptyProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) QCSDPSubtermProof (EQUIVALENT transformation)

(16) Obligation:

(17) PIsEmptyProof (EQUIVALENT transformation)

(18) YES