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


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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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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) CSRInnermostProof [EQUIVALENT, 0 ms]
(2) CSR
(3) CSDependencyPairsProof [EQUIVALENT, 7 ms]
(4) QCSDP
(5) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
(6) QCSDP
(7) QCSDPSubtermProof [EQUIVALENT, 0 ms]
(8) QCSDP
(9) PIsEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

2nd: {1}
cons1: {1, 2}
cons: {1}
from: {1}
s: {1}

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(1) CSRInnermostProof (EQUIVALENT)
The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
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(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
   from(X) -> cons(X, from(s(X)))

The replacement map contains the following entries:

2nd: {1}
cons1: {1, 2}
cons: {1}
from: {1}
s: {1}


Innermost Strategy.

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(3) CSDependencyPairsProof (EQUIVALENT)
Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
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(4)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {2nd_1, cons1_2, from_1, s_1, 2ND_1, 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:
   2ND(cons(X, X1)) -> 2ND(cons1(X, X1))

The collapsing dependency pairs are DP_c:
   2ND(cons(X, X1)) -> X1


The hidden terms of R are:

   from(s(x0))

Every hiding context is built from:
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@2a208f78
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@996b748

Hence, the new unhiding pairs DP_u are :
   2ND(cons(X, X1)) -> U(X1)
   U(s(x_0)) -> U(x_0)
   U(from(x_0)) -> U(x_0)
   U(from(s(x0))) -> FROM(s(x0))

The TRS R consists of the following rules:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
   from(X) -> cons(X, from(s(X)))

The set Q consists of the following terms:

   2nd(cons1(x0, cons(x1, x2)))
   2nd(cons(x0, x1))
   from(x0)


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(5) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 2 less nodes.
The rules 2ND(cons(z0, z1)) -> 2ND(cons1(z0, z1)) and 2ND(cons(x0, x1)) -> 2ND(cons1(x0, x1)) form no chain, because ECap^mu(2ND(cons1(z0, z1))) = 2ND(cons1(z0, x_1)) does not unify with 2ND(cons(x0, x1)). The rules 2ND(cons(z0, z1)) -> 2ND(cons1(z0, z1)) and 2ND(cons(x0, x1)) -> U(x1) form no chain, because ECap^mu(2ND(cons1(z0, z1))) = 2ND(cons1(z0, x_1)) does not unify with 2ND(cons(x0, x1)). 
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(6)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {2nd_1, cons1_2, from_1, s_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 TRS P consists of the following rules:

   U(s(x_0)) -> U(x_0)
   U(from(x_0)) -> U(x_0)

The TRS R consists of the following rules:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
   from(X) -> cons(X, from(s(X)))

The set Q consists of the following terms:

   2nd(cons1(x0, cons(x1, x2)))
   2nd(cons(x0, x1))
   from(x0)


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(7) 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)
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

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(8)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {2nd_1, cons1_2, from_1, s_1} 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:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
   from(X) -> cons(X, from(s(X)))

The set Q consists of the following terms:

   2nd(cons1(x0, cons(x1, x2)))
   2nd(cons(x0, x1))
   from(x0)


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