/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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) PIsEmptyProof [EQUIVALENT, 0 ms]
        (16) YES


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

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

   and(tt, X) -> X
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)

The replacement map contains the following entries:

and: {1}
tt: empty set
plus: {1, 2}
0: empty set
s: {1}
x: {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:

   and(tt, X) -> X
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)

The replacement map contains the following entries:

and: {1}
tt: empty set
plus: {1, 2}
0: empty set
s: {1}
x: {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 {plus_2, s_1, x_2, PLUS_2, X_2} are replacing on all positions.
For all symbols f in {and_2, 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:
   PLUS(N, s(M)) -> PLUS(N, M)
   X(N, s(M)) -> PLUS(x(N, M), N)
   X(N, s(M)) -> X(N, M)

The collapsing dependency pairs are DP_c:
   AND(tt, X) -> X


The hidden terms of R are:
none

Every hiding context is built from:none

Hence, the new unhiding pairs DP_u are :
   AND(tt, X) -> U(X)

The TRS R consists of the following rules:

   and(tt, X) -> X
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)

The set Q consists of the following terms:

   and(tt, x0)
   plus(x0, 0)
   plus(x0, s(x1))
   x(x0, 0)
   x(x0, s(x1))


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

(5) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 1 less node.

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

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {plus_2, s_1, x_2, PLUS_2} are replacing on all positions.
For all symbols f in {and_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   PLUS(N, s(M)) -> PLUS(N, M)

The TRS R consists of the following rules:

   and(tt, X) -> X
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)

The set Q consists of the following terms:

   and(tt, x0)
   plus(x0, 0)
   plus(x0, s(x1))
   x(x0, 0)
   x(x0, s(x1))


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

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


The following pairs can be oriented strictly and are deleted.

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


Subterm Order

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

(9)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {plus_2, s_1, x_2} are replacing on all positions.
For all symbols f in {and_2} we have mu(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   and(tt, X) -> X
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)

The set Q consists of the following terms:

   and(tt, x0)
   plus(x0, 0)
   plus(x0, s(x1))
   x(x0, 0)
   x(x0, s(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 {plus_2, s_1, x_2, X_2} are replacing on all positions.
For all symbols f in {and_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   X(N, s(M)) -> X(N, M)

The TRS R consists of the following rules:

   and(tt, X) -> X
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)

The set Q consists of the following terms:

   and(tt, x0)
   plus(x0, 0)
   plus(x0, s(x1))
   x(x0, 0)
   x(x0, s(x1))


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

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


The following pairs can be oriented strictly and are deleted.

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


Subterm Order

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

(14)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {plus_2, s_1, x_2} are replacing on all positions.
For all symbols f in {and_2} we have mu(f) = {1}.

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

   and(tt, X) -> X
   plus(N, 0) -> N
   plus(N, s(M)) -> s(plus(N, M))
   x(N, 0) -> 0
   x(N, s(M)) -> plus(x(N, M), N)

The set Q consists of the following terms:

   and(tt, x0)
   plus(x0, 0)
   plus(x0, s(x1))
   x(x0, 0)
   x(x0, s(x1))


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

(15) PIsEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
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(16)
YES