/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

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


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

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(f(X)) -> mark(cons(X, f(g(X))))
   active(g(0)) -> mark(s(0))
   active(g(s(X))) -> mark(s(s(g(X))))
   active(sel(0, cons(X, Y))) -> mark(X)
   active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
   active(f(X)) -> f(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(g(X)) -> g(active(X))
   active(s(X)) -> s(active(X))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   f(mark(X)) -> mark(f(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   g(mark(X)) -> mark(g(X))
   s(mark(X)) -> mark(s(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   proper(f(X)) -> f(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(g(X)) -> g(proper(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   f(ok(X)) -> ok(f(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   g(ok(X)) -> ok(g(X))
   s(ok(X)) -> ok(s(X))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.

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

(1) QTRSToCSRProof (EQUIVALENT)
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:

   active(f(X)) -> mark(cons(X, f(g(X))))
   active(g(0)) -> mark(s(0))
   active(g(s(X))) -> mark(s(s(g(X))))
   active(sel(0, cons(X, Y))) -> mark(X)
   active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
   active(f(X)) -> f(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(g(X)) -> g(active(X))
   active(s(X)) -> s(active(X))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   f(mark(X)) -> mark(f(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   g(mark(X)) -> mark(g(X))
   s(mark(X)) -> mark(s(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   proper(f(X)) -> f(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(g(X)) -> g(proper(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   f(ok(X)) -> ok(f(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   g(ok(X)) -> ok(g(X))
   s(ok(X)) -> ok(s(X))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.
Special symbols used for the transformation (see [GM04]):
top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
The replacement map contains the following entries:

f: {1}
cons: {1}
g: {1}
0: empty set
s: {1}
sel: {1, 2}
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).
----------------------------------------

(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}

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

(3) CSRInnermostProof (EQUIVALENT)
The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
----------------------------------------

(4)
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.

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

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

(6)
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@7013bbf8
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@78dbeafa

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))


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

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

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

(8)
Complex Obligation (AND)

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

(9)
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))


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

(10) 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

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

(11)
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))


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

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

(13)
YES

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

(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 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))


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

(15) 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

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

(16)
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))


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

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

(18)
YES

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

(19)
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))


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

(20) 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

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

(21)
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))


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

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

(23)
YES