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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 492 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 57 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

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

   a__U11(tt, V1, V2) -> a__U12(a__isNat(V1), V2)
   a__U12(tt, V2) -> a__U13(a__isNat(V2))
   a__U13(tt) -> tt
   a__U21(tt, V1) -> a__U22(a__isNat(V1))
   a__U22(tt) -> tt
   a__U31(tt, V1, V2) -> a__U32(a__isNat(V1), V2)
   a__U32(tt, V2) -> a__U33(a__isNat(V2))
   a__U33(tt) -> tt
   a__U41(tt, N) -> mark(N)
   a__U51(tt, M, N) -> s(a__plus(mark(N), mark(M)))
   a__U61(tt) -> 0
   a__U71(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
   a__and(tt, X) -> mark(X)
   a__isNat(0) -> tt
   a__isNat(plus(V1, V2)) -> a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
   a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
   a__isNat(x(V1, V2)) -> a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
   a__isNatKind(0) -> tt
   a__isNatKind(plus(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2))
   a__isNatKind(s(V1)) -> a__isNatKind(V1)
   a__isNatKind(x(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2))
   a__plus(N, 0) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N)
   a__plus(N, s(M)) -> a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
   a__x(N, 0) -> a__U61(a__and(a__isNat(N), isNatKind(N)))
   a__x(N, s(M)) -> a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
   mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
   mark(isNat(X)) -> a__isNat(X)
   mark(U13(X)) -> a__U13(mark(X))
   mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
   mark(U22(X)) -> a__U22(mark(X))
   mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3)
   mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
   mark(U33(X)) -> a__U33(mark(X))
   mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
   mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
   mark(U61(X)) -> a__U61(mark(X))
   mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
   mark(x(X1, X2)) -> a__x(mark(X1), mark(X2))
   mark(and(X1, X2)) -> a__and(mark(X1), X2)
   mark(isNatKind(X)) -> a__isNatKind(X)
   mark(tt) -> tt
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__U11(X1, X2, X3) -> U11(X1, X2, X3)
   a__U12(X1, X2) -> U12(X1, X2)
   a__isNat(X) -> isNat(X)
   a__U13(X) -> U13(X)
   a__U21(X1, X2) -> U21(X1, X2)
   a__U22(X) -> U22(X)
   a__U31(X1, X2, X3) -> U31(X1, X2, X3)
   a__U32(X1, X2) -> U32(X1, X2)
   a__U33(X) -> U33(X)
   a__U41(X1, X2) -> U41(X1, X2)
   a__U51(X1, X2, X3) -> U51(X1, X2, X3)
   a__plus(X1, X2) -> plus(X1, X2)
   a__U61(X) -> U61(X)
   a__U71(X1, X2, X3) -> U71(X1, X2, X3)
   a__x(X1, X2) -> x(X1, X2)
   a__and(X1, X2) -> and(X1, X2)
   a__isNatKind(X) -> isNatKind(X)

Q is empty.

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

(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
a__U11/3(YES,YES,YES)
tt/0)
a__U12/2(YES,YES)
a__isNat/1)YES(
a__U13/1)YES(
a__U21/2(YES,YES)
a__U22/1(YES)
a__U31/3(YES,YES,YES)
a__U32/2(YES,YES)
a__U33/1)YES(
a__U41/2(YES,YES)
mark/1)YES(
a__U51/3(YES,YES,YES)
s/1(YES)
a__plus/2(YES,YES)
a__U61/1)YES(
0/0)
a__U71/3(YES,YES,YES)
a__x/2(YES,YES)
a__and/2(YES,YES)
plus/2(YES,YES)
a__isNatKind/1)YES(
isNatKind/1)YES(
x/2(YES,YES)
and/2(YES,YES)
isNat/1)YES(
U11/3(YES,YES,YES)
U12/2(YES,YES)
U13/1)YES(
U21/2(YES,YES)
U22/1(YES)
U31/3(YES,YES,YES)
U32/2(YES,YES)
U33/1)YES(
U41/2(YES,YES)
U51/3(YES,YES,YES)
U61/1)YES(
U71/3(YES,YES,YES)

Quasi precedence:
[tt, 0] > [a__U22_1, U22_1]
[tt, 0] > [a__U32_2, U32_2]
[tt, 0] > [a__U41_2, U41_2]
[a__U71_3, a__x_2, x_2, U71_3] > [a__U31_3, U31_3] > [a__U32_2, U32_2]
[a__U71_3, a__x_2, x_2, U71_3] > [a__U51_3, a__plus_2, plus_2, U51_3] > [a__U11_3, a__U12_2, U11_3, U12_2]
[a__U71_3, a__x_2, x_2, U71_3] > [a__U51_3, a__plus_2, plus_2, U51_3] > [a__U41_2, U41_2]
[a__U71_3, a__x_2, x_2, U71_3] > [a__U51_3, a__plus_2, plus_2, U51_3] > [s_1, a__and_2, and_2] > [a__U21_2, U21_2] > [a__U22_1, U22_1]


Status:
a__U11_3: multiset status
tt: multiset status
a__U12_2: multiset status
a__U21_2: multiset status
a__U22_1: multiset status
a__U31_3: multiset status
a__U32_2: multiset status
a__U41_2: multiset status
a__U51_3: [3,2,1]
s_1: multiset status
a__plus_2: [1,2]
0: multiset status
a__U71_3: [2,3,1]
a__x_2: [2,1]
a__and_2: multiset status
plus_2: [1,2]
x_2: [2,1]
and_2: multiset status
U11_3: multiset status
U12_2: multiset status
U21_2: multiset status
U22_1: multiset status
U31_3: multiset status
U32_2: multiset status
U41_2: multiset status
U51_3: [3,2,1]
U71_3: [2,3,1]

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a__U11(tt, V1, V2) -> a__U12(a__isNat(V1), V2)
   a__U12(tt, V2) -> a__U13(a__isNat(V2))
   a__U21(tt, V1) -> a__U22(a__isNat(V1))
   a__U22(tt) -> tt
   a__U31(tt, V1, V2) -> a__U32(a__isNat(V1), V2)
   a__U32(tt, V2) -> a__U33(a__isNat(V2))
   a__U41(tt, N) -> mark(N)
   a__U51(tt, M, N) -> s(a__plus(mark(N), mark(M)))
   a__U71(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N))
   a__and(tt, X) -> mark(X)
   a__isNat(plus(V1, V2)) -> a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
   a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1)
   a__isNat(x(V1, V2)) -> a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)
   a__isNatKind(plus(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2))
   a__isNatKind(s(V1)) -> a__isNatKind(V1)
   a__isNatKind(x(V1, V2)) -> a__and(a__isNatKind(V1), isNatKind(V2))
   a__plus(N, 0) -> a__U41(a__and(a__isNat(N), isNatKind(N)), N)
   a__plus(N, s(M)) -> a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
   a__x(N, 0) -> a__U61(a__and(a__isNat(N), isNatKind(N)))
   a__x(N, s(M)) -> a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)




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

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a__U13(tt) -> tt
   a__U33(tt) -> tt
   a__U61(tt) -> 0
   a__isNat(0) -> tt
   a__isNatKind(0) -> tt
   mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
   mark(isNat(X)) -> a__isNat(X)
   mark(U13(X)) -> a__U13(mark(X))
   mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
   mark(U22(X)) -> a__U22(mark(X))
   mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3)
   mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
   mark(U33(X)) -> a__U33(mark(X))
   mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
   mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
   mark(U61(X)) -> a__U61(mark(X))
   mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
   mark(x(X1, X2)) -> a__x(mark(X1), mark(X2))
   mark(and(X1, X2)) -> a__and(mark(X1), X2)
   mark(isNatKind(X)) -> a__isNatKind(X)
   mark(tt) -> tt
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__U11(X1, X2, X3) -> U11(X1, X2, X3)
   a__U12(X1, X2) -> U12(X1, X2)
   a__isNat(X) -> isNat(X)
   a__U13(X) -> U13(X)
   a__U21(X1, X2) -> U21(X1, X2)
   a__U22(X) -> U22(X)
   a__U31(X1, X2, X3) -> U31(X1, X2, X3)
   a__U32(X1, X2) -> U32(X1, X2)
   a__U33(X) -> U33(X)
   a__U41(X1, X2) -> U41(X1, X2)
   a__U51(X1, X2, X3) -> U51(X1, X2, X3)
   a__plus(X1, X2) -> plus(X1, X2)
   a__U61(X) -> U61(X)
   a__U71(X1, X2, X3) -> U71(X1, X2, X3)
   a__x(X1, X2) -> x(X1, X2)
   a__and(X1, X2) -> and(X1, X2)
   a__isNatKind(X) -> isNatKind(X)

Q is empty.

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

(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:mark_1 > a__U11_3 > a__U33_1 > s_1 > a__U61_1 > 0 > a__and_2 > U61_1 > a__U51_3 > U33_1 > a__U22_1 > a__U21_2 > U21_2 > a__U12_2 > a__x_2 > x_2 > a__isNat_1 > a__plus_2 > U12_2 > isNat_1 > plus_2 > a__U32_2 > U32_2 > a__isNatKind_1 > tt > a__U13_1 > U13_1 > isNatKind_1 > and_2 > a__U71_3 > U71_3 > U51_3 > a__U41_2 > U41_2 > a__U31_3 > U31_3 > U22_1 > U11_3

and weight map:

   tt=2
   0=1
   a__U13_1=1
   a__U33_1=1
   a__U61_1=1
   a__isNat_1=1
   a__isNatKind_1=1
   mark_1=0
   isNat_1=1
   U13_1=1
   U22_1=1
   a__U22_1=1
   U33_1=1
   U61_1=1
   isNatKind_1=1
   s_1=1
   U11_3=0
   a__U11_3=0
   U12_2=0
   a__U12_2=0
   U21_2=0
   a__U21_2=0
   U31_3=0
   a__U31_3=0
   U32_2=0
   a__U32_2=0
   U41_2=0
   a__U41_2=0
   U51_3=0
   a__U51_3=0
   plus_2=0
   a__plus_2=0
   U71_3=0
   a__U71_3=0
   x_2=0
   a__x_2=0
   and_2=0
   a__and_2=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a__U13(tt) -> tt
   a__U33(tt) -> tt
   a__U61(tt) -> 0
   a__isNat(0) -> tt
   a__isNatKind(0) -> tt
   mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3)
   mark(U12(X1, X2)) -> a__U12(mark(X1), X2)
   mark(isNat(X)) -> a__isNat(X)
   mark(U13(X)) -> a__U13(mark(X))
   mark(U21(X1, X2)) -> a__U21(mark(X1), X2)
   mark(U22(X)) -> a__U22(mark(X))
   mark(U31(X1, X2, X3)) -> a__U31(mark(X1), X2, X3)
   mark(U32(X1, X2)) -> a__U32(mark(X1), X2)
   mark(U33(X)) -> a__U33(mark(X))
   mark(U41(X1, X2)) -> a__U41(mark(X1), X2)
   mark(U51(X1, X2, X3)) -> a__U51(mark(X1), X2, X3)
   mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2))
   mark(U61(X)) -> a__U61(mark(X))
   mark(U71(X1, X2, X3)) -> a__U71(mark(X1), X2, X3)
   mark(x(X1, X2)) -> a__x(mark(X1), mark(X2))
   mark(and(X1, X2)) -> a__and(mark(X1), X2)
   mark(isNatKind(X)) -> a__isNatKind(X)
   mark(tt) -> tt
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   a__U11(X1, X2, X3) -> U11(X1, X2, X3)
   a__U12(X1, X2) -> U12(X1, X2)
   a__isNat(X) -> isNat(X)
   a__U13(X) -> U13(X)
   a__U21(X1, X2) -> U21(X1, X2)
   a__U22(X) -> U22(X)
   a__U31(X1, X2, X3) -> U31(X1, X2, X3)
   a__U32(X1, X2) -> U32(X1, X2)
   a__U33(X) -> U33(X)
   a__U41(X1, X2) -> U41(X1, X2)
   a__U51(X1, X2, X3) -> U51(X1, X2, X3)
   a__plus(X1, X2) -> plus(X1, X2)
   a__U61(X) -> U61(X)
   a__U71(X1, X2, X3) -> U71(X1, X2, X3)
   a__x(X1, X2) -> x(X1, X2)
   a__and(X1, X2) -> and(X1, X2)
   a__isNatKind(X) -> isNatKind(X)




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

(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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

(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(6)
YES