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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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


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

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

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)

The replacement map contains the following entries:

fib: {1}
sel: {1, 2}
fib1: {1, 2}
s: {1}
0: empty set
cons: {1}
add: {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:

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)

The replacement map contains the following entries:

fib: {1}
sel: {1, 2}
fib1: {1, 2}
s: {1}
0: empty set
cons: {1}
add: {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 {fib_1, sel_2, fib1_2, s_1, add_2, SEL_2, FIB_1, FIB1_2, ADD_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 ordinary context-sensitive dependency pairs DP_o are:
   FIB(N) -> SEL(N, fib1(s(0), s(0)))
   FIB(N) -> FIB1(s(0), s(0))
   ADD(s(X), Y) -> ADD(X, Y)
   SEL(s(N), cons(X, XS)) -> SEL(N, XS)

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


The hidden terms of R are:

   fib1(x0, add(x1, x0))
   add(x0, x1)

Every hiding context is built from:
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@3011d58c
   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@712b0429

Hence, the new unhiding pairs DP_u are :
   SEL(s(N), cons(X, XS)) -> U(XS)
   U(add(x_0, x_1)) -> U(x_0)
   U(add(x_0, x_1)) -> U(x_1)
   U(fib1(x_0, x_1)) -> U(x_0)
   U(fib1(x_0, x_1)) -> U(x_1)
   U(fib1(x0, add(x1, x0))) -> FIB1(x0, add(x1, x0))
   U(add(x0, x1)) -> ADD(x0, x1)

The TRS R consists of the following rules:

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)

The set Q consists of the following terms:

   fib(x0)
   fib1(x0, x1)
   add(0, x0)
   add(s(x0), x1)
   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 4 less nodes.

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

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {fib_1, sel_2, fib1_2, s_1, add_2, ADD_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:

   ADD(s(X), Y) -> ADD(X, Y)

The TRS R consists of the following rules:

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)

The set Q consists of the following terms:

   fib(x0)
   fib1(x0, x1)
   add(0, x0)
   add(s(x0), x1)
   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.

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


Subterm Order

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

(9)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {fib_1, sel_2, fib1_2, s_1, add_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:

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)

The set Q consists of the following terms:

   fib(x0)
   fib1(x0, x1)
   add(0, x0)
   add(s(x0), x1)
   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 {fib_1, sel_2, fib1_2, s_1, add_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(add(x_0, x_1)) -> U(x_0)
   U(add(x_0, x_1)) -> U(x_1)
   U(fib1(x_0, x_1)) -> U(x_0)
   U(fib1(x_0, x_1)) -> U(x_1)

The TRS R consists of the following rules:

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)

The set Q consists of the following terms:

   fib(x0)
   fib1(x0, x1)
   add(0, x0)
   add(s(x0), x1)
   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(add(x_0, x_1)) -> U(x_0)
   U(add(x_0, x_1)) -> U(x_1)
   U(fib1(x_0, x_1)) -> U(x_0)
   U(fib1(x_0, x_1)) -> U(x_1)
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 {fib_1, sel_2, fib1_2, s_1, add_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:

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)

The set Q consists of the following terms:

   fib(x0)
   fib1(x0, x1)
   add(0, x0)
   add(s(x0), x1)
   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

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

(17)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {fib_1, sel_2, fib1_2, s_1, add_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(N), cons(X, XS)) -> SEL(N, XS)

The TRS R consists of the following rules:

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)

The set Q consists of the following terms:

   fib(x0)
   fib1(x0, x1)
   add(0, x0)
   add(s(x0), x1)
   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(N), cons(X, XS)) -> SEL(N, XS)
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 {fib_1, sel_2, fib1_2, s_1, add_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:

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, fib1(Y, add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, XS)

The set Q consists of the following terms:

   fib(x0)
   fib1(x0, x1)
   add(0, x0)
   add(s(x0), x1)
   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