/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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) CSRRRRProof [EQUIVALENT, 56 ms]
(2) CSR
(3) CSDependencyPairsProof [EQUIVALENT, 0 ms]
(4) QCSDP
(5) QCSDPInstantiationProcessor [EQUIVALENT, 0 ms]
(6) QCSDP
(7) QCSUsableRulesProof [EQUIVALENT, 0 ms]
(8) QCSDP
(9) QCSDPInstantiationProcessor [EQUIVALENT, 0 ms]
(10) QCSDP
(11) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
(12) TRUE


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

   h(X) -> g(X, X)
   g(a, X) -> f(b, X)
   f(X, X) -> h(a)
   a -> b

The replacement map contains the following entries:

h: {1}
g: {1}
a: empty set
f: {1}
b: empty set

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

(1) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   h(X) -> g(X, X)
   g(a, X) -> f(b, X)
   f(X, X) -> h(a)
   a -> b

The replacement map contains the following entries:

h: {1}
g: {1}
a: empty set
f: {1}
b: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(a) = 1
   POL(b) = 0
   POL(f(x_1, x_2)) = 1 + x_1
   POL(g(x_1, x_2)) = x_1
   POL(h(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a -> b




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

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

   h(X) -> g(X, X)
   g(a, X) -> f(b, X)
   f(X, X) -> h(a)

The replacement map contains the following entries:

h: {1}
g: {1}
a: empty set
f: {1}
b: empty set

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

(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 {h_1, H_1} are replacing on all positions.
For all symbols f in {g_2, f_2, G_2, F_2} we have mu(f) = {1}.

The ordinary context-sensitive dependency pairs DP_o are:
   H(X) -> G(X, X)
   G(a, X) -> F(b, X)
   F(X, X) -> H(a)

The TRS R consists of the following rules:

   h(X) -> g(X, X)
   g(a, X) -> f(b, X)
   f(X, X) -> h(a)

Q is empty.

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

(5) QCSDPInstantiationProcessor (EQUIVALENT)
Using the Context-Sensitive Instantiation[LPAR08,DA_EMMES] Processor

the pair H(X) -> G(X, X)

was transformed to the following new pairs: 
   H(a) -> G(a, a)



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

(6)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {h_1, H_1} are replacing on all positions.
For all symbols f in {g_2, f_2, F_2, G_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   G(a, X) -> F(b, X)
   F(X, X) -> H(a)
   H(a) -> G(a, a)

The TRS R consists of the following rules:

   h(X) -> g(X, X)
   g(a, X) -> f(b, X)
   f(X, X) -> h(a)

Q is empty.

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

(7) QCSUsableRulesProof (EQUIVALENT)
The following rules are not useable [DA_EMMES] and can be deleted:

   h(x0) -> g(x0, x0)
   g(a, x0) -> f(b, x0)
   f(x0, x0) -> h(a)

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

(8)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {H_1} are replacing on all positions.
For all symbols f in {F_2, G_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   G(a, X) -> F(b, X)
   F(X, X) -> H(a)
   H(a) -> G(a, a)

R is empty.
Q is empty.

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

(9) QCSDPInstantiationProcessor (EQUIVALENT)
Using the Context-Sensitive Instantiation[LPAR08,DA_EMMES] Processor

the pair G(a, X) -> F(b, X)

was transformed to the following new pairs: 
   G(a, a) -> F(b, a)



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

(10)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {H_1} are replacing on all positions.
For all symbols f in {F_2, G_2} we have mu(f) = {1}.

The TRS P consists of the following rules:

   F(X, X) -> H(a)
   H(a) -> G(a, a)
   G(a, a) -> F(b, a)

R is empty.
Q is empty.

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

(11) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 3 less nodes.
The rules G(a, a) -> F(b, a) and F(x0, x0) -> H(a) form no chain, because ECap^mu(F(b, a)) = F(b, a) does not unify with F(x0, x0). 
----------------------------------------

(12)
TRUE