/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination of the given CSR could be proven:

(0) CSR
(1) CSRInnermostProof [EQUIVALENT, 0 ms]
(2) CSR
(3) CSDependencyPairsProof [EQUIVALENT, 0 ms]
(4) QCSDP
(5) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) QCSDP
        (8) QCSDPSubtermProof [EQUIVALENT, 0 ms]
        (9) QCSDP
        (10) PIsEmptyProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QCSDP
        (13) QCSDPSubtermProof [EQUIVALENT, 0 ms]
        (14) QCSDP
        (15) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
        (16) TRUE


----------------------------------------

(0)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
   afterNth(N, XS) -> snd(splitAt(N, XS))
   and(tt, X) -> X
   fst(pair(X, Y)) -> X
   head(cons(N, XS)) -> N
   natsFrom(N) -> cons(N, natsFrom(s(N)))
   sel(N, XS) -> head(afterNth(N, XS))
   snd(pair(X, Y)) -> Y
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
   tail(cons(N, XS)) -> XS
   take(N, XS) -> fst(splitAt(N, XS))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
splitAt: {1, 2}
pair: {1, 2}
cons: {1}
afterNth: {1, 2}
snd: {1}
and: {1}
fst: {1}
head: {1}
natsFrom: {1}
s: {1}
sel: {1, 2}
0: empty set
nil: empty set
tail: {1}
take: {1, 2}

----------------------------------------

(1) CSRInnermostProof (EQUIVALENT)
The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
----------------------------------------

(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
   afterNth(N, XS) -> snd(splitAt(N, XS))
   and(tt, X) -> X
   fst(pair(X, Y)) -> X
   head(cons(N, XS)) -> N
   natsFrom(N) -> cons(N, natsFrom(s(N)))
   sel(N, XS) -> head(afterNth(N, XS))
   snd(pair(X, Y)) -> Y
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
   tail(cons(N, XS)) -> XS
   take(N, XS) -> fst(splitAt(N, XS))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
splitAt: {1, 2}
pair: {1, 2}
cons: {1}
afterNth: {1, 2}
snd: {1}
and: {1}
fst: {1}
head: {1}
natsFrom: {1}
s: {1}
sel: {1, 2}
0: empty set
nil: empty set
tail: {1}
take: {1, 2}


Innermost Strategy.

----------------------------------------

(3) CSDependencyPairsProof (EQUIVALENT)
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 {splitAt_2, pair_2, afterNth_2, snd_1, fst_1, head_1, natsFrom_1, s_1, sel_2, tail_1, take_2, SPLITAT_2, SND_1, AFTERNTH_2, HEAD_1, SEL_2, FST_1, TAKE_2, TAIL_1, NATSFROM_1} are replacing on all positions.
For all symbols f in {U11_4, U12_2, cons_2, and_2, U12'_2, U11'_4, AND_2} we have mu(f) = {1}.
The symbols in {U_1} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:
   U11'(tt, N, X, XS) -> U12'(splitAt(N, XS), X)
   U11'(tt, N, X, XS) -> SPLITAT(N, XS)
   AFTERNTH(N, XS) -> SND(splitAt(N, XS))
   AFTERNTH(N, XS) -> SPLITAT(N, XS)
   SEL(N, XS) -> HEAD(afterNth(N, XS))
   SEL(N, XS) -> AFTERNTH(N, XS)
   SPLITAT(s(N), cons(X, XS)) -> U11'(tt, N, X, XS)
   TAKE(N, XS) -> FST(splitAt(N, XS))
   TAKE(N, XS) -> SPLITAT(N, XS)

The collapsing dependency pairs are DP_c:
   U11'(tt, N, X, XS) -> N
   U11'(tt, N, X, XS) -> XS
   U12'(pair(YS, ZS), X) -> X
   AND(tt, X) -> X
   TAIL(cons(N, XS)) -> XS


The hidden terms of R are:

   natsFrom(s(x0))

Every hiding context is built from:
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@70ff9956
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@aa5c03d

Hence, the new unhiding pairs DP_u are :
   U11'(tt, N, X, XS) -> U(N)
   U11'(tt, N, X, XS) -> U(XS)
   U12'(pair(YS, ZS), X) -> U(X)
   AND(tt, X) -> 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:

   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
   afterNth(N, XS) -> snd(splitAt(N, XS))
   and(tt, X) -> X
   fst(pair(X, Y)) -> X
   head(cons(N, XS)) -> N
   natsFrom(N) -> cons(N, natsFrom(s(N)))
   sel(N, XS) -> head(afterNth(N, XS))
   snd(pair(X, Y)) -> Y
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
   tail(cons(N, XS)) -> XS
   take(N, XS) -> fst(splitAt(N, XS))

The set Q consists of the following terms:

   U11(tt, x0, x1, x2)
   U12(pair(x0, x1), x2)
   afterNth(x0, x1)
   and(tt, x0)
   fst(pair(x0, x1))
   head(cons(x0, x1))
   natsFrom(x0)
   sel(x0, x1)
   snd(pair(x0, x1))
   splitAt(0, x0)
   splitAt(s(x0), cons(x1, x2))
   tail(cons(x0, x1))
   take(x0, x1)


----------------------------------------

(5) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 11 less nodes.

----------------------------------------

(6)
Complex Obligation (AND)

----------------------------------------

(7)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {splitAt_2, pair_2, afterNth_2, snd_1, fst_1, head_1, natsFrom_1, s_1, sel_2, tail_1, take_2} are replacing on all positions.
For all symbols f in {U11_4, U12_2, cons_2, and_2} we have mu(f) = {1}.
The symbols in {U_1} 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:

   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
   afterNth(N, XS) -> snd(splitAt(N, XS))
   and(tt, X) -> X
   fst(pair(X, Y)) -> X
   head(cons(N, XS)) -> N
   natsFrom(N) -> cons(N, natsFrom(s(N)))
   sel(N, XS) -> head(afterNth(N, XS))
   snd(pair(X, Y)) -> Y
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
   tail(cons(N, XS)) -> XS
   take(N, XS) -> fst(splitAt(N, XS))

The set Q consists of the following terms:

   U11(tt, x0, x1, x2)
   U12(pair(x0, x1), x2)
   afterNth(x0, x1)
   and(tt, x0)
   fst(pair(x0, x1))
   head(cons(x0, x1))
   natsFrom(x0)
   sel(x0, x1)
   snd(pair(x0, x1))
   splitAt(0, x0)
   splitAt(s(x0), cons(x1, x2))
   tail(cons(x0, x1))
   take(x0, x1)


----------------------------------------

(8) QCSDPSubtermProof (EQUIVALENT)
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

----------------------------------------

(9)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {splitAt_2, pair_2, afterNth_2, snd_1, fst_1, head_1, natsFrom_1, s_1, sel_2, tail_1, take_2} are replacing on all positions.
For all symbols f in {U11_4, U12_2, cons_2, and_2} we have mu(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
   afterNth(N, XS) -> snd(splitAt(N, XS))
   and(tt, X) -> X
   fst(pair(X, Y)) -> X
   head(cons(N, XS)) -> N
   natsFrom(N) -> cons(N, natsFrom(s(N)))
   sel(N, XS) -> head(afterNth(N, XS))
   snd(pair(X, Y)) -> Y
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
   tail(cons(N, XS)) -> XS
   take(N, XS) -> fst(splitAt(N, XS))

The set Q consists of the following terms:

   U11(tt, x0, x1, x2)
   U12(pair(x0, x1), x2)
   afterNth(x0, x1)
   and(tt, x0)
   fst(pair(x0, x1))
   head(cons(x0, x1))
   natsFrom(x0)
   sel(x0, x1)
   snd(pair(x0, x1))
   splitAt(0, x0)
   splitAt(s(x0), cons(x1, x2))
   tail(cons(x0, x1))
   take(x0, x1)


----------------------------------------

(10) PIsEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
----------------------------------------

(11)
YES

----------------------------------------

(12)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {splitAt_2, pair_2, afterNth_2, snd_1, fst_1, head_1, natsFrom_1, s_1, sel_2, tail_1, take_2, SPLITAT_2} are replacing on all positions.
For all symbols f in {U11_4, U12_2, cons_2, and_2, U11'_4} we have mu(f) = {1}.

The TRS P consists of the following rules:

   U11'(tt, N, X, XS) -> SPLITAT(N, XS)
   SPLITAT(s(N), cons(X, XS)) -> U11'(tt, N, X, XS)

The TRS R consists of the following rules:

   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
   afterNth(N, XS) -> snd(splitAt(N, XS))
   and(tt, X) -> X
   fst(pair(X, Y)) -> X
   head(cons(N, XS)) -> N
   natsFrom(N) -> cons(N, natsFrom(s(N)))
   sel(N, XS) -> head(afterNth(N, XS))
   snd(pair(X, Y)) -> Y
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
   tail(cons(N, XS)) -> XS
   take(N, XS) -> fst(splitAt(N, XS))

The set Q consists of the following terms:

   U11(tt, x0, x1, x2)
   U12(pair(x0, x1), x2)
   afterNth(x0, x1)
   and(tt, x0)
   fst(pair(x0, x1))
   head(cons(x0, x1))
   natsFrom(x0)
   sel(x0, x1)
   snd(pair(x0, x1))
   splitAt(0, x0)
   splitAt(s(x0), cons(x1, x2))
   tail(cons(x0, x1))
   take(x0, x1)


----------------------------------------

(13) QCSDPSubtermProof (EQUIVALENT)
We use the subterm processor [DA_EMMES].


The following pairs can be oriented strictly and are deleted.

   SPLITAT(s(N), cons(X, XS)) -> U11'(tt, N, X, XS)
The remaining pairs can at least be oriented weakly.

   U11'(tt, N, X, XS) -> SPLITAT(N, XS)
Used ordering:  Combined order from the following AFS and order.
SPLITAT(x1, x2)  =  x1

U11'(x1, x2, x3, x4)  =  x2


Subterm Order

----------------------------------------

(14)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {splitAt_2, pair_2, afterNth_2, snd_1, fst_1, head_1, natsFrom_1, s_1, sel_2, tail_1, take_2, SPLITAT_2} are replacing on all positions.
For all symbols f in {U11_4, U12_2, cons_2, and_2, U11'_4} we have mu(f) = {1}.

The TRS P consists of the following rules:

   U11'(tt, N, X, XS) -> SPLITAT(N, XS)

The TRS R consists of the following rules:

   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
   afterNth(N, XS) -> snd(splitAt(N, XS))
   and(tt, X) -> X
   fst(pair(X, Y)) -> X
   head(cons(N, XS)) -> N
   natsFrom(N) -> cons(N, natsFrom(s(N)))
   sel(N, XS) -> head(afterNth(N, XS))
   snd(pair(X, Y)) -> Y
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
   tail(cons(N, XS)) -> XS
   take(N, XS) -> fst(splitAt(N, XS))

The set Q consists of the following terms:

   U11(tt, x0, x1, x2)
   U12(pair(x0, x1), x2)
   afterNth(x0, x1)
   and(tt, x0)
   fst(pair(x0, x1))
   head(cons(x0, x1))
   natsFrom(x0)
   sel(x0, x1)
   snd(pair(x0, x1))
   splitAt(0, x0)
   splitAt(s(x0), cons(x1, x2))
   tail(cons(x0, x1))
   take(x0, x1)


----------------------------------------

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

----------------------------------------

(16)
TRUE