/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 10 ms]
        (11) QCSDP
        (12) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
        (13) TRUE
    (14) QCSDP
        (15) QCSDPSubtermProof [EQUIVALENT, 2 ms]
        (16) QCSDP
        (17) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
        (18) TRUE


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

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

   active(U11(tt, M, N)) -> mark(U12(tt, M, N))
   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
   active(plus(N, 0)) -> mark(N)
   active(plus(N, s(M))) -> mark(U11(tt, M, N))
   active(x(N, 0)) -> mark(0)
   active(x(N, s(M))) -> mark(U21(tt, M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
   active(s(X)) -> s(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(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(U11(tt, M, N)) -> mark(U12(tt, M, N))
   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
   active(plus(N, 0)) -> mark(N)
   active(plus(N, s(M))) -> mark(U11(tt, M, N))
   active(x(N, 0)) -> mark(0)
   active(x(N, s(M))) -> mark(U21(tt, M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
   active(s(X)) -> s(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(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:

U11: {1}
tt: empty set
U12: {1}
s: {1}
plus: {1, 2}
U21: {1}
U22: {1}
x: {1, 2}
0: empty set
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:

   U11(tt, M, N) -> U12(tt, M, N)
   U12(tt, M, N) -> s(plus(N, M))
   U21(tt, M, N) -> U22(tt, M, N)
   U22(tt, M, N) -> plus(x(N, M), N)
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   x(N, 0) -> 0
   x(N, s(M)) -> U21(tt, M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
s: {1}
plus: {1, 2}
U21: {1}
U22: {1}
x: {1, 2}
0: empty set

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(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:

   U11(tt, M, N) -> U12(tt, M, N)
   U12(tt, M, N) -> s(plus(N, M))
   U21(tt, M, N) -> U22(tt, M, N)
   U22(tt, M, N) -> plus(x(N, M), N)
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   x(N, 0) -> 0
   x(N, s(M)) -> U21(tt, M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
s: {1}
plus: {1, 2}
U21: {1}
U22: {1}
x: {1, 2}
0: empty set


Innermost Strategy.

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(5) CSDependencyPairsProof (EQUIVALENT)
Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
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(6)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {s_1, plus_2, x_2, PLUS_2, X_2} are replacing on all positions.
For all symbols f in {U11_3, U12_3, U21_3, U22_3, U12'_3, U11'_3, U22'_3, U21'_3} 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:
   U11'(tt, M, N) -> U12'(tt, M, N)
   U12'(tt, M, N) -> PLUS(N, M)
   U21'(tt, M, N) -> U22'(tt, M, N)
   U22'(tt, M, N) -> PLUS(x(N, M), N)
   U22'(tt, M, N) -> X(N, M)
   PLUS(N, s(M)) -> U11'(tt, M, N)
   X(N, s(M)) -> U21'(tt, M, N)

The collapsing dependency pairs are DP_c:
   U12'(tt, M, N) -> N
   U12'(tt, M, N) -> M
   U22'(tt, M, N) -> N
   U22'(tt, M, N) -> M


The hidden terms of R are:
none

Every hiding context is built from:none

Hence, the new unhiding pairs DP_u are :
   U12'(tt, M, N) -> U(N)
   U12'(tt, M, N) -> U(M)
   U22'(tt, M, N) -> U(N)
   U22'(tt, M, N) -> U(M)

The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, M, N)
   U12(tt, M, N) -> s(plus(N, M))
   U21(tt, M, N) -> U22(tt, M, N)
   U22(tt, M, N) -> plus(x(N, M), N)
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   x(N, 0) -> 0
   x(N, s(M)) -> U21(tt, M, N)

The set Q consists of the following terms:

   U11(tt, x0, x1)
   U12(tt, x0, x1)
   U21(tt, x0, x1)
   U22(tt, x0, x1)
   plus(x0, 0)
   plus(x0, s(x1))
   x(x0, 0)
   x(x0, s(x1))


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

(7) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 5 less nodes.

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

(8)
Complex Obligation (AND)

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

(9)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {s_1, plus_2, x_2, PLUS_2} are replacing on all positions.
For all symbols f in {U11_3, U12_3, U21_3, U22_3, U12'_3, U11'_3} we have mu(f) = {1}.

The TRS P consists of the following rules:

   U12'(tt, M, N) -> PLUS(N, M)
   PLUS(N, s(M)) -> U11'(tt, M, N)
   U11'(tt, M, N) -> U12'(tt, M, N)

The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, M, N)
   U12(tt, M, N) -> s(plus(N, M))
   U21(tt, M, N) -> U22(tt, M, N)
   U22(tt, M, N) -> plus(x(N, M), N)
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   x(N, 0) -> 0
   x(N, s(M)) -> U21(tt, M, N)

The set Q consists of the following terms:

   U11(tt, x0, x1)
   U12(tt, x0, x1)
   U21(tt, x0, x1)
   U22(tt, x0, x1)
   plus(x0, 0)
   plus(x0, s(x1))
   x(x0, 0)
   x(x0, s(x1))


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

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


The following pairs can be oriented strictly and are deleted.

   PLUS(N, s(M)) -> U11'(tt, M, N)
The remaining pairs can at least be oriented weakly.

   U12'(tt, M, N) -> PLUS(N, M)
   U11'(tt, M, N) -> U12'(tt, M, N)
Used ordering:  Combined order from the following AFS and order.
PLUS(x1, x2)  =  x2

U12'(x1, x2, x3)  =  x2

U11'(x1, x2, x3)  =  x2


Subterm Order

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(11)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {s_1, plus_2, x_2, PLUS_2} are replacing on all positions.
For all symbols f in {U11_3, U12_3, U21_3, U22_3, U12'_3, U11'_3} we have mu(f) = {1}.

The TRS P consists of the following rules:

   U12'(tt, M, N) -> PLUS(N, M)
   U11'(tt, M, N) -> U12'(tt, M, N)

The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, M, N)
   U12(tt, M, N) -> s(plus(N, M))
   U21(tt, M, N) -> U22(tt, M, N)
   U22(tt, M, N) -> plus(x(N, M), N)
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   x(N, 0) -> 0
   x(N, s(M)) -> U21(tt, M, N)

The set Q consists of the following terms:

   U11(tt, x0, x1)
   U12(tt, x0, x1)
   U21(tt, x0, x1)
   U22(tt, x0, x1)
   plus(x0, 0)
   plus(x0, s(x1))
   x(x0, 0)
   x(x0, s(x1))


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

(12) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.

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

(13)
TRUE

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

(14)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {s_1, plus_2, x_2, X_2} are replacing on all positions.
For all symbols f in {U11_3, U12_3, U21_3, U22_3, U22'_3, U21'_3} we have mu(f) = {1}.

The TRS P consists of the following rules:

   U22'(tt, M, N) -> X(N, M)
   X(N, s(M)) -> U21'(tt, M, N)
   U21'(tt, M, N) -> U22'(tt, M, N)

The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, M, N)
   U12(tt, M, N) -> s(plus(N, M))
   U21(tt, M, N) -> U22(tt, M, N)
   U22(tt, M, N) -> plus(x(N, M), N)
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   x(N, 0) -> 0
   x(N, s(M)) -> U21(tt, M, N)

The set Q consists of the following terms:

   U11(tt, x0, x1)
   U12(tt, x0, x1)
   U21(tt, x0, x1)
   U22(tt, x0, x1)
   plus(x0, 0)
   plus(x0, s(x1))
   x(x0, 0)
   x(x0, s(x1))


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

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


The following pairs can be oriented strictly and are deleted.

   X(N, s(M)) -> U21'(tt, M, N)
The remaining pairs can at least be oriented weakly.

   U22'(tt, M, N) -> X(N, M)
   U21'(tt, M, N) -> U22'(tt, M, N)
Used ordering:  Combined order from the following AFS and order.
X(x1, x2)  =  x2

U22'(x1, x2, x3)  =  x2

U21'(x1, x2, x3)  =  x2


Subterm Order

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(16)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {s_1, plus_2, x_2, X_2} are replacing on all positions.
For all symbols f in {U11_3, U12_3, U21_3, U22_3, U22'_3, U21'_3} we have mu(f) = {1}.

The TRS P consists of the following rules:

   U22'(tt, M, N) -> X(N, M)
   U21'(tt, M, N) -> U22'(tt, M, N)

The TRS R consists of the following rules:

   U11(tt, M, N) -> U12(tt, M, N)
   U12(tt, M, N) -> s(plus(N, M))
   U21(tt, M, N) -> U22(tt, M, N)
   U22(tt, M, N) -> plus(x(N, M), N)
   plus(N, 0) -> N
   plus(N, s(M)) -> U11(tt, M, N)
   x(N, 0) -> 0
   x(N, s(M)) -> U21(tt, M, N)

The set Q consists of the following terms:

   U11(tt, x0, x1)
   U12(tt, x0, x1)
   U21(tt, x0, x1)
   U22(tt, x0, x1)
   plus(x0, 0)
   plus(x0, s(x1))
   x(x0, 0)
   x(x0, s(x1))


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

(17) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.

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

(18)
TRUE