/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
        (11) TRUE
    (12) QCSDP
        (13) QCSDPSubtermProof [EQUIVALENT, 2 ms]
        (14) QCSDP
        (15) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
        (16) TRUE


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

(0)
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

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

(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:

   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.

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

(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 {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))


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

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

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

(6)
Complex Obligation (AND)

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

(7)
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))


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

(8) 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

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

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

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

(11)
TRUE

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

(12)
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))


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

(13) 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

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

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

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

(16)
TRUE