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


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


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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination of the given CSR could be proven:

(0) CSR
(1) CSDependencyPairsProof [EQUIVALENT, 17 ms]
(2) QCSDP
(3) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QCSDP
        (6) QCSDPSubtermProof [EQUIVALENT, 0 ms]
        (7) QCSDP
        (8) PIsEmptyProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QCSDP
        (11) QCSDPReductionPairProof [EQUIVALENT, 20 ms]
        (12) QCSDP
        (13) PIsEmptyProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QCSDP
        (16) QCSDPSubtermProof [EQUIVALENT, 0 ms]
        (17) QCSDP
        (18) PIsEmptyProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QCSDP
        (21) QCSDPSubtermProof [EQUIVALENT, 0 ms]
        (22) QCSDP
        (23) PIsEmptyProof [EQUIVALENT, 0 ms]
        (24) YES


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

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

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

The replacement map contains the following entries:

from: {1}
cons: {1}
s: {1}
sel: {1, 2}
0: empty set
minus: {1, 2}
quot: {1, 2}
zWquot: {1, 2}
nil: empty set

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

(1) CSDependencyPairsProof (EQUIVALENT)
Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
----------------------------------------

(2)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2, SEL_2, MINUS_2, QUOT_2, ZWQUOT_2, FROM_1} are replacing on all positions.
For all symbols f in {cons_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:
   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
   MINUS(s(X), s(Y)) -> MINUS(X, Y)
   QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))
   QUOT(s(X), s(Y)) -> MINUS(X, Y)
   ZWQUOT(cons(X, XS), cons(Y, YS)) -> QUOT(X, Y)

The collapsing dependency pairs are DP_c:
   SEL(s(N), cons(X, XS)) -> XS


The hidden terms of R are:

   from(s(x0))
   zWquot(x0, x1)

Every hiding context is built from:
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@7a99a5cd
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@601699a1
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@7768edf7

Hence, the new unhiding pairs DP_u are :
   SEL(s(N), cons(X, XS)) -> U(XS)
   U(s(x_0)) -> U(x_0)
   U(from(x_0)) -> U(x_0)
   U(zWquot(x_0, x_1)) -> U(x_0)
   U(zWquot(x_0, x_1)) -> U(x_1)
   U(from(s(x0))) -> FROM(s(x0))
   U(zWquot(x0, x1)) -> ZWQUOT(x0, x1)

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

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

(3) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 5 less nodes.

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2, MINUS_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   MINUS(s(X), s(Y)) -> MINUS(X, Y)

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

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

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


The following pairs can be oriented strictly and are deleted.

   MINUS(s(X), s(Y)) -> MINUS(X, Y)
The remaining pairs can at least be oriented weakly.
none
Used ordering:  Combined order from the following AFS and order.
MINUS(x1, x2)  =  x1


Subterm Order

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

(7)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

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

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

(9)
YES

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

(10)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2, QUOT_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

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

(11) QCSDPReductionPairProof (EQUIVALENT)
Using the order

Polynomial interpretation with max and min functions [POLO,MAXPOLO]:

   POL(0) = 0
   POL(QUOT(x_1, x_2)) = x_1
   POL(cons(x_1, x_2)) = 1 + x_1
   POL(from(x_1)) = 1 + x_1
   POL(minus(x_1, x_2)) = 0
   POL(nil) = 1
   POL(quot(x_1, x_2)) = 1 + x_1
   POL(s(x_1)) = 1 + x_1
   POL(zWquot(x_1, x_2)) = 1 + x_1


the following usable rules


   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   from(X) -> cons(X, from(s(X)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))


could all be oriented weakly.

Furthermore, the pairs


   QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))


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


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

(12)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

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

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

(14)
YES

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

(15)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
For all symbols f in {cons_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(from(x_0)) -> U(x_0)
   U(zWquot(x_0, x_1)) -> U(x_0)
   U(zWquot(x_0, x_1)) -> U(x_1)

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

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

(16) 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(from(x_0)) -> U(x_0)
   U(zWquot(x_0, x_1)) -> U(x_0)
   U(zWquot(x_0, x_1)) -> U(x_1)
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

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

(17)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

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

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

(19)
YES

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

(20)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2, SEL_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   SEL(s(N), cons(X, XS)) -> SEL(N, XS)

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

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

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


The following pairs can be oriented strictly and are deleted.

   SEL(s(N), cons(X, XS)) -> SEL(N, XS)
The remaining pairs can at least be oriented weakly.
none
Used ordering:  Combined order from the following AFS and order.
SEL(x1, x2)  =  x1


Subterm Order

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

(22)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {from_1, s_1, sel_2, minus_2, quot_2, zWquot_2} are replacing on all positions.
For all symbols f in {cons_2} we have mu(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), zWquot(XS, YS))

Q is empty.

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

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

(24)
YES