/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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) CSDependencyPairsProof [EQUIVALENT, 28 ms]
(2) QCSDP
(3) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QCSDP
        (6) QCSDPSubtermProof [EQUIVALENT, 0 ms]
        (7) QCSDP
        (8) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
        (9) TRUE
    (10) QCSDP
        (11) QCSDPSubtermProof [EQUIVALENT, 0 ms]
        (12) QCSDP
        (13) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
        (14) TRUE
    (15) QCSDP
        (16) QCSDPSubtermProof [EQUIVALENT, 0 ms]
        (17) QCSDP
        (18) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
        (19) TRUE


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

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

   U11(tt, V2) -> U12(isNat(V2))
   U12(tt) -> tt
   U21(tt) -> tt
   U31(tt, V2) -> U32(isNat(V2))
   U32(tt) -> tt
   U41(tt, N) -> N
   U51(tt, M, N) -> U52(isNat(N), M, N)
   U52(tt, M, N) -> s(plus(N, M))
   U61(tt) -> 0
   U71(tt, M, N) -> U72(isNat(N), M, N)
   U72(tt, M, N) -> plus(x(N, M), N)
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
   isNat(s(V1)) -> U21(isNat(V1))
   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
   plus(N, 0) -> U41(isNat(N), N)
   plus(N, s(M)) -> U51(isNat(M), M, N)
   x(N, 0) -> U61(isNat(N))
   x(N, s(M)) -> U71(isNat(M), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U31: {1}
U32: {1}
U41: {1}
U51: {1}
U52: {1}
s: {1}
plus: {1, 2}
U61: {1}
0: empty set
U71: {1}
U72: {1}
x: {1, 2}

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

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

(2)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2, U12'_1, U32'_1, PLUS_2, X_2, U21'_1, U61'_1} are replacing on all positions.
For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U11'_2, U31'_2, U52'_3, U51'_3, U72'_3, U71'_3, U41'_2} we have mu(f) = {1}.
The symbols in {isNat_1, ISNAT_1, U_1} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:
   U11'(tt, V2) -> U12'(isNat(V2))
   U11'(tt, V2) -> ISNAT(V2)
   U31'(tt, V2) -> U32'(isNat(V2))
   U31'(tt, V2) -> ISNAT(V2)
   U51'(tt, M, N) -> U52'(isNat(N), M, N)
   U51'(tt, M, N) -> ISNAT(N)
   U52'(tt, M, N) -> PLUS(N, M)
   U71'(tt, M, N) -> U72'(isNat(N), M, N)
   U71'(tt, M, N) -> ISNAT(N)
   U72'(tt, M, N) -> PLUS(x(N, M), N)
   U72'(tt, M, N) -> X(N, M)
   ISNAT(plus(V1, V2)) -> U11'(isNat(V1), V2)
   ISNAT(plus(V1, V2)) -> ISNAT(V1)
   ISNAT(s(V1)) -> U21'(isNat(V1))
   ISNAT(s(V1)) -> ISNAT(V1)
   ISNAT(x(V1, V2)) -> U31'(isNat(V1), V2)
   ISNAT(x(V1, V2)) -> ISNAT(V1)
   PLUS(N, 0) -> U41'(isNat(N), N)
   PLUS(N, 0) -> ISNAT(N)
   PLUS(N, s(M)) -> U51'(isNat(M), M, N)
   PLUS(N, s(M)) -> ISNAT(M)
   X(N, 0) -> U61'(isNat(N))
   X(N, 0) -> ISNAT(N)
   X(N, s(M)) -> U71'(isNat(M), M, N)
   X(N, s(M)) -> ISNAT(M)

The collapsing dependency pairs are DP_c:
   U41'(tt, N) -> N
   U52'(tt, M, N) -> N
   U52'(tt, M, N) -> M
   U72'(tt, M, N) -> N
   U72'(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 :
   U41'(tt, N) -> U(N)
   U52'(tt, M, N) -> U(N)
   U52'(tt, M, N) -> U(M)
   U72'(tt, M, N) -> U(N)
   U72'(tt, M, N) -> U(M)

The TRS R consists of the following rules:

   U11(tt, V2) -> U12(isNat(V2))
   U12(tt) -> tt
   U21(tt) -> tt
   U31(tt, V2) -> U32(isNat(V2))
   U32(tt) -> tt
   U41(tt, N) -> N
   U51(tt, M, N) -> U52(isNat(N), M, N)
   U52(tt, M, N) -> s(plus(N, M))
   U61(tt) -> 0
   U71(tt, M, N) -> U72(isNat(N), M, N)
   U72(tt, M, N) -> plus(x(N, M), N)
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
   isNat(s(V1)) -> U21(isNat(V1))
   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
   plus(N, 0) -> U41(isNat(N), N)
   plus(N, s(M)) -> U51(isNat(M), M, N)
   x(N, 0) -> U61(isNat(N))
   x(N, s(M)) -> U71(isNat(M), M, N)

Q is empty.

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

(3) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 3 SCCs with 17 less nodes.

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

(4)
Complex Obligation (AND)

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

(5)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U11'_2, U31'_2} we have mu(f) = {1}.
The symbols in {isNat_1, ISNAT_1} are not replacing on any position.

The TRS P consists of the following rules:

   ISNAT(plus(V1, V2)) -> U11'(isNat(V1), V2)
   U11'(tt, V2) -> ISNAT(V2)
   ISNAT(plus(V1, V2)) -> ISNAT(V1)
   ISNAT(s(V1)) -> ISNAT(V1)
   ISNAT(x(V1, V2)) -> U31'(isNat(V1), V2)
   U31'(tt, V2) -> ISNAT(V2)
   ISNAT(x(V1, V2)) -> ISNAT(V1)

The TRS R consists of the following rules:

   U11(tt, V2) -> U12(isNat(V2))
   U12(tt) -> tt
   U21(tt) -> tt
   U31(tt, V2) -> U32(isNat(V2))
   U32(tt) -> tt
   U41(tt, N) -> N
   U51(tt, M, N) -> U52(isNat(N), M, N)
   U52(tt, M, N) -> s(plus(N, M))
   U61(tt) -> 0
   U71(tt, M, N) -> U72(isNat(N), M, N)
   U72(tt, M, N) -> plus(x(N, M), N)
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
   isNat(s(V1)) -> U21(isNat(V1))
   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
   plus(N, 0) -> U41(isNat(N), N)
   plus(N, s(M)) -> U51(isNat(M), M, N)
   x(N, 0) -> U61(isNat(N))
   x(N, s(M)) -> U71(isNat(M), M, N)

Q is empty.

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

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


The following pairs can be oriented strictly and are deleted.

   ISNAT(plus(V1, V2)) -> U11'(isNat(V1), V2)
   ISNAT(plus(V1, V2)) -> ISNAT(V1)
   ISNAT(s(V1)) -> ISNAT(V1)
   ISNAT(x(V1, V2)) -> U31'(isNat(V1), V2)
   ISNAT(x(V1, V2)) -> ISNAT(V1)
The remaining pairs can at least be oriented weakly.

   U11'(tt, V2) -> ISNAT(V2)
   U31'(tt, V2) -> ISNAT(V2)
Used ordering:  Combined order from the following AFS and order.
U11'(x1, x2)  =  x2

ISNAT(x1)  =  x1

U31'(x1, x2)  =  x2


Subterm Order

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

(7)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U11'_2, U31'_2} we have mu(f) = {1}.
The symbols in {isNat_1, ISNAT_1} are not replacing on any position.

The TRS P consists of the following rules:

   U11'(tt, V2) -> ISNAT(V2)
   U31'(tt, V2) -> ISNAT(V2)

The TRS R consists of the following rules:

   U11(tt, V2) -> U12(isNat(V2))
   U12(tt) -> tt
   U21(tt) -> tt
   U31(tt, V2) -> U32(isNat(V2))
   U32(tt) -> tt
   U41(tt, N) -> N
   U51(tt, M, N) -> U52(isNat(N), M, N)
   U52(tt, M, N) -> s(plus(N, M))
   U61(tt) -> 0
   U71(tt, M, N) -> U72(isNat(N), M, N)
   U72(tt, M, N) -> plus(x(N, M), N)
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
   isNat(s(V1)) -> U21(isNat(V1))
   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
   plus(N, 0) -> U41(isNat(N), N)
   plus(N, s(M)) -> U51(isNat(M), M, N)
   x(N, 0) -> U61(isNat(N))
   x(N, s(M)) -> U71(isNat(M), M, N)

Q is empty.

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

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

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

(9)
TRUE

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

(10)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2, PLUS_2} are replacing on all positions.
For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U52'_3, U51'_3} we have mu(f) = {1}.
The symbols in {isNat_1} are not replacing on any position.

The TRS P consists of the following rules:

   U51'(tt, M, N) -> U52'(isNat(N), M, N)
   U52'(tt, M, N) -> PLUS(N, M)
   PLUS(N, s(M)) -> U51'(isNat(M), M, N)

The TRS R consists of the following rules:

   U11(tt, V2) -> U12(isNat(V2))
   U12(tt) -> tt
   U21(tt) -> tt
   U31(tt, V2) -> U32(isNat(V2))
   U32(tt) -> tt
   U41(tt, N) -> N
   U51(tt, M, N) -> U52(isNat(N), M, N)
   U52(tt, M, N) -> s(plus(N, M))
   U61(tt) -> 0
   U71(tt, M, N) -> U72(isNat(N), M, N)
   U72(tt, M, N) -> plus(x(N, M), N)
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
   isNat(s(V1)) -> U21(isNat(V1))
   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
   plus(N, 0) -> U41(isNat(N), N)
   plus(N, s(M)) -> U51(isNat(M), M, N)
   x(N, 0) -> U61(isNat(N))
   x(N, s(M)) -> U71(isNat(M), M, N)

Q is empty.

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

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


The following pairs can be oriented strictly and are deleted.

   PLUS(N, s(M)) -> U51'(isNat(M), M, N)
The remaining pairs can at least be oriented weakly.

   U51'(tt, M, N) -> U52'(isNat(N), M, N)
   U52'(tt, M, N) -> PLUS(N, M)
Used ordering:  Combined order from the following AFS and order.
U52'(x1, x2, x3)  =  x2

U51'(x1, x2, x3)  =  x2

PLUS(x1, x2)  =  x2


Subterm Order

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

(12)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2, PLUS_2} are replacing on all positions.
For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U52'_3, U51'_3} we have mu(f) = {1}.
The symbols in {isNat_1} are not replacing on any position.

The TRS P consists of the following rules:

   U51'(tt, M, N) -> U52'(isNat(N), M, N)
   U52'(tt, M, N) -> PLUS(N, M)

The TRS R consists of the following rules:

   U11(tt, V2) -> U12(isNat(V2))
   U12(tt) -> tt
   U21(tt) -> tt
   U31(tt, V2) -> U32(isNat(V2))
   U32(tt) -> tt
   U41(tt, N) -> N
   U51(tt, M, N) -> U52(isNat(N), M, N)
   U52(tt, M, N) -> s(plus(N, M))
   U61(tt) -> 0
   U71(tt, M, N) -> U72(isNat(N), M, N)
   U72(tt, M, N) -> plus(x(N, M), N)
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
   isNat(s(V1)) -> U21(isNat(V1))
   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
   plus(N, 0) -> U41(isNat(N), N)
   plus(N, s(M)) -> U51(isNat(M), M, N)
   x(N, 0) -> U61(isNat(N))
   x(N, s(M)) -> U71(isNat(M), M, N)

Q is empty.

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

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

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

(14)
TRUE

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

(15)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2, X_2} are replacing on all positions.
For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U72'_3, U71'_3} we have mu(f) = {1}.
The symbols in {isNat_1} are not replacing on any position.

The TRS P consists of the following rules:

   U72'(tt, M, N) -> X(N, M)
   X(N, s(M)) -> U71'(isNat(M), M, N)
   U71'(tt, M, N) -> U72'(isNat(N), M, N)

The TRS R consists of the following rules:

   U11(tt, V2) -> U12(isNat(V2))
   U12(tt) -> tt
   U21(tt) -> tt
   U31(tt, V2) -> U32(isNat(V2))
   U32(tt) -> tt
   U41(tt, N) -> N
   U51(tt, M, N) -> U52(isNat(N), M, N)
   U52(tt, M, N) -> s(plus(N, M))
   U61(tt) -> 0
   U71(tt, M, N) -> U72(isNat(N), M, N)
   U72(tt, M, N) -> plus(x(N, M), N)
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
   isNat(s(V1)) -> U21(isNat(V1))
   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
   plus(N, 0) -> U41(isNat(N), N)
   plus(N, s(M)) -> U51(isNat(M), M, N)
   x(N, 0) -> U61(isNat(N))
   x(N, s(M)) -> U71(isNat(M), M, N)

Q is empty.

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

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


The following pairs can be oriented strictly and are deleted.

   X(N, s(M)) -> U71'(isNat(M), M, N)
The remaining pairs can at least be oriented weakly.

   U72'(tt, M, N) -> X(N, M)
   U71'(tt, M, N) -> U72'(isNat(N), M, N)
Used ordering:  Combined order from the following AFS and order.
X(x1, x2)  =  x2

U72'(x1, x2, x3)  =  x2

U71'(x1, x2, x3)  =  x2


Subterm Order

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

(17)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2, X_2} are replacing on all positions.
For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U72'_3, U71'_3} we have mu(f) = {1}.
The symbols in {isNat_1} are not replacing on any position.

The TRS P consists of the following rules:

   U72'(tt, M, N) -> X(N, M)
   U71'(tt, M, N) -> U72'(isNat(N), M, N)

The TRS R consists of the following rules:

   U11(tt, V2) -> U12(isNat(V2))
   U12(tt) -> tt
   U21(tt) -> tt
   U31(tt, V2) -> U32(isNat(V2))
   U32(tt) -> tt
   U41(tt, N) -> N
   U51(tt, M, N) -> U52(isNat(N), M, N)
   U52(tt, M, N) -> s(plus(N, M))
   U61(tt) -> 0
   U71(tt, M, N) -> U72(isNat(N), M, N)
   U72(tt, M, N) -> plus(x(N, M), N)
   isNat(0) -> tt
   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
   isNat(s(V1)) -> U21(isNat(V1))
   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
   plus(N, 0) -> U41(isNat(N), N)
   plus(N, s(M)) -> U51(isNat(M), M, N)
   x(N, 0) -> U61(isNat(N))
   x(N, s(M)) -> U71(isNat(M), M, N)

Q is empty.

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

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

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

(19)
TRUE