/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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, 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
    (17) QCSDP
        (18) QCSDPSubtermProof [EQUIVALENT, 2 ms]
        (19) QCSDP
        (20) PIsEmptyProof [EQUIVALENT, 0 ms]
        (21) YES


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

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

   f(X) -> cons(X, f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

The replacement map contains the following entries:

f: {1}
cons: {1}
g: {1}
0: empty set
s: {1}
sel: {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:

   f(X) -> cons(X, f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

The replacement map contains the following entries:

f: {1}
cons: {1}
g: {1}
0: empty set
s: {1}
sel: {1, 2}


Innermost Strategy.

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

(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 {f_1, g_1, s_1, sel_2, G_1, SEL_2, F_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:
   G(s(X)) -> G(X)
   SEL(s(X), cons(Y, Z)) -> SEL(X, Z)

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


The hidden terms of R are:

   f(g(x0))
   g(x0)

Every hiding context is built from:
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@338c564
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@1a0180a4

Hence, the new unhiding pairs DP_u are :
   SEL(s(X), cons(Y, Z)) -> U(Z)
   U(g(x_0)) -> U(x_0)
   U(f(x_0)) -> U(x_0)
   U(f(g(x0))) -> F(g(x0))
   U(g(x0)) -> G(x0)

The TRS R consists of the following rules:

   f(X) -> cons(X, f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

The set Q consists of the following terms:

   f(x0)
   g(0)
   g(s(x0))
   sel(0, cons(x0, x1))
   sel(s(x0), cons(x1, x2))


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

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

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

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {f_1, g_1, s_1, sel_2, G_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:

   G(s(X)) -> G(X)

The TRS R consists of the following rules:

   f(X) -> cons(X, f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

The set Q consists of the following terms:

   f(x0)
   g(0)
   g(s(x0))
   sel(0, cons(x0, x1))
   sel(s(x0), cons(x1, x2))


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

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


The following pairs can be oriented strictly and are deleted.

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


Subterm Order

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

(9)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {f_1, g_1, s_1, 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:
none

The TRS R consists of the following rules:

   f(X) -> cons(X, f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

The set Q consists of the following terms:

   f(x0)
   g(0)
   g(s(x0))
   sel(0, cons(x0, x1))
   sel(s(x0), cons(x1, x2))


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

(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 {f_1, g_1, s_1, sel_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(g(x_0)) -> U(x_0)
   U(f(x_0)) -> U(x_0)

The TRS R consists of the following rules:

   f(X) -> cons(X, f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

The set Q consists of the following terms:

   f(x0)
   g(0)
   g(s(x0))
   sel(0, cons(x0, x1))
   sel(s(x0), cons(x1, x2))


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

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


The following pairs can be oriented strictly and are deleted.

   U(g(x_0)) -> U(x_0)
   U(f(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

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

(14)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {f_1, g_1, s_1, 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:
none

The TRS R consists of the following rules:

   f(X) -> cons(X, f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

The set Q consists of the following terms:

   f(x0)
   g(0)
   g(s(x0))
   sel(0, cons(x0, x1))
   sel(s(x0), cons(x1, x2))


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

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

(16)
YES

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(17)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {f_1, g_1, s_1, sel_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(X), cons(Y, Z)) -> SEL(X, Z)

The TRS R consists of the following rules:

   f(X) -> cons(X, f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

The set Q consists of the following terms:

   f(x0)
   g(0)
   g(s(x0))
   sel(0, cons(x0, x1))
   sel(s(x0), cons(x1, x2))


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

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


The following pairs can be oriented strictly and are deleted.

   SEL(s(X), cons(Y, Z)) -> SEL(X, Z)
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

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

(19)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {f_1, g_1, s_1, 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:
none

The TRS R consists of the following rules:

   f(X) -> cons(X, f(g(X)))
   g(0) -> s(0)
   g(s(X)) -> s(s(g(X)))
   sel(0, cons(X, Y)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, Z)

The set Q consists of the following terms:

   f(x0)
   g(0)
   g(s(x0))
   sel(0, cons(x0, x1))
   sel(s(x0), cons(x1, x2))


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

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

(21)
YES