YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given ETRS could be proven:

(0) ETRS
(1) EquationalDependencyPairsProof [EQUIVALENT, 0 ms]
(2) EDP
(3) EDependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) EDP
        (6) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
        (7) EDP
        (8) EUsableRulesReductionPairsProof [EQUIVALENT, 6 ms]
        (9) EDP
        (10) PisEmptyProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) EDP
        (13) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
        (14) EDP
        (15) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
        (16) EDP
        (17) PisEmptyProof [EQUIVALENT, 0 ms]
        (18) YES
    (19) EDP
        (20) EDPPoloProof [EQUIVALENT, 0 ms]
        (21) EDP
        (22) PisEmptyProof [EQUIVALENT, 0 ms]
        (23) YES


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

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

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

The set E consists of the following equations:

   gcd(x, y) == gcd(y, x)


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

(1) EquationalDependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,DA_STEIN] we result in the following initial EDP problem:
The TRS P consists of the following rules:

   LE(s(x), s(y)) -> LE(x, y)
   MINUS(x, s(y)) -> PRED(minus(x, y))
   MINUS(x, s(y)) -> MINUS(x, y)
   GCD(s(x), s(y)) -> IF_GCD(le(y, x), s(x), s(y))
   GCD(s(x), s(y)) -> LE(y, x)
   IF_GCD(true, s(x), s(y)) -> GCD(minus(x, y), s(y))
   IF_GCD(true, s(x), s(y)) -> MINUS(x, y)
   IF_GCD(false, s(x), s(y)) -> GCD(minus(y, x), s(x))
   IF_GCD(false, s(x), s(y)) -> MINUS(y, x)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

The set E consists of the following equations:

   gcd(x, y) == gcd(y, x)

The set E# consists of the following equations:

   GCD(x, y) == GCD(y, x)

We have to consider all minimal (P,E#,R,E)-chains

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

(2)
Obligation:
The TRS P consists of the following rules:

   LE(s(x), s(y)) -> LE(x, y)
   MINUS(x, s(y)) -> PRED(minus(x, y))
   MINUS(x, s(y)) -> MINUS(x, y)
   GCD(s(x), s(y)) -> IF_GCD(le(y, x), s(x), s(y))
   GCD(s(x), s(y)) -> LE(y, x)
   IF_GCD(true, s(x), s(y)) -> GCD(minus(x, y), s(y))
   IF_GCD(true, s(x), s(y)) -> MINUS(x, y)
   IF_GCD(false, s(x), s(y)) -> GCD(minus(y, x), s(x))
   IF_GCD(false, s(x), s(y)) -> MINUS(y, x)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

The set E consists of the following equations:

   gcd(x, y) == gcd(y, x)

The set E# consists of the following equations:

   GCD(x, y) == GCD(y, x)

We have to consider all minimal (P,E#,R,E)-chains
----------------------------------------

(3) EDependencyGraphProof (EQUIVALENT)
The approximation of the Equational Dependency Graph [DA_STEIN] contains 3 SCCs with 4 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

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

(5)
Obligation:
The TRS P consists of the following rules:

   MINUS(x, s(y)) -> MINUS(x, y)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

The set E consists of the following equations:

   gcd(x, y) == gcd(y, x)

The set E# consists of the following equations:

   GCD(x, y) == GCD(y, x)

We have to consider all minimal (P,E#,R,E)-chains
----------------------------------------

(6) ESharpUsableEquationsProof (EQUIVALENT)
We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]:
   GCD(x, y) == GCD(y, x)

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

(7)
Obligation:
The TRS P consists of the following rules:

   MINUS(x, s(y)) -> MINUS(x, y)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

The set E consists of the following equations:

   gcd(x, y) == gcd(y, x)

E# is empty.
We have to consider all minimal (P,E#,R,E)-chains
----------------------------------------

(8) EUsableRulesReductionPairsProof (EQUIVALENT)
By using the improved usable rules and equations with reduction pair processor [DA_STEIN] with a polynomial ordering [POLO], all dependency pairs and the corresponding improved usable rules can be oriented non-strictly, the improved  usable equations and the esharp equations can be oriented equivalently. All non-usable rules and equations are removed, and those dependency pairs and improved usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

   MINUS(x, s(y)) -> MINUS(x, y)
The following rules are removed from R:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
The following equations are removed from E:

   gcd(x, y) == gcd(y, x)
Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(MINUS(x_1, x_2)) = x_1 + 3*x_2
   POL(s(x_1)) = 3*x_1


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

(9)
Obligation:
P is empty.
R is empty.
E is empty.
E# is empty.
We have to consider all minimal (P,E#,R,E)-chains
----------------------------------------

(10) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
----------------------------------------

(11)
YES

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

(12)
Obligation:
The TRS P consists of the following rules:

   LE(s(x), s(y)) -> LE(x, y)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

The set E consists of the following equations:

   gcd(x, y) == gcd(y, x)

The set E# consists of the following equations:

   GCD(x, y) == GCD(y, x)

We have to consider all minimal (P,E#,R,E)-chains
----------------------------------------

(13) ESharpUsableEquationsProof (EQUIVALENT)
We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]:
   GCD(x, y) == GCD(y, x)

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

(14)
Obligation:
The TRS P consists of the following rules:

   LE(s(x), s(y)) -> LE(x, y)

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

The set E consists of the following equations:

   gcd(x, y) == gcd(y, x)

E# is empty.
We have to consider all minimal (P,E#,R,E)-chains
----------------------------------------

(15) EUsableRulesReductionPairsProof (EQUIVALENT)
By using the improved usable rules and equations with reduction pair processor [DA_STEIN] with a polynomial ordering [POLO], all dependency pairs and the corresponding improved usable rules can be oriented non-strictly, the improved  usable equations and the esharp equations can be oriented equivalently. All non-usable rules and equations are removed, and those dependency pairs and improved usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

   LE(s(x), s(y)) -> LE(x, y)
The following rules are removed from R:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
The following equations are removed from E:

   gcd(x, y) == gcd(y, x)
Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(LE(x_1, x_2)) = 3*x_1 + 3*x_2
   POL(s(x_1)) = x_1


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

(16)
Obligation:
P is empty.
R is empty.
E is empty.
E# is empty.
We have to consider all minimal (P,E#,R,E)-chains
----------------------------------------

(17) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
----------------------------------------

(18)
YES

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

(19)
Obligation:
The TRS P consists of the following rules:

   IF_GCD(false, s(x), s(y)) -> GCD(minus(y, x), s(x))
   GCD(s(x), s(y)) -> IF_GCD(le(y, x), s(x), s(y))
   IF_GCD(true, s(x), s(y)) -> GCD(minus(x, y), s(y))

The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

The set E consists of the following equations:

   gcd(x, y) == gcd(y, x)

The set E# consists of the following equations:

   GCD(x, y) == GCD(y, x)

We have to consider all minimal (P,E#,R,E)-chains
----------------------------------------

(20) EDPPoloProof (EQUIVALENT)
We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented.


   IF_GCD(false, s(x), s(y)) -> GCD(minus(y, x), s(x))
   GCD(s(x), s(y)) -> IF_GCD(le(y, x), s(x), s(y))
   IF_GCD(true, s(x), s(y)) -> GCD(minus(x, y), s(y))
With the implicit AFS we had to orient the following set of usable rules of R non-strictly.


   minus(x, s(y)) -> pred(minus(x, y))
   minus(x, 0) -> x
   pred(s(x)) -> x
We had to orient the following equations of E# equivalently.


   GCD(x, y) == GCD(y, x)
With the implicit AFS there is no usable equation of E.


Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(GCD(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(IF_GCD(x_1, x_2, x_3)) = 2*x_2 + 2*x_3
   POL(false) = 0
   POL(le(x_1, x_2)) = 0
   POL(minus(x_1, x_2)) = x_1
   POL(pred(x_1)) = x_1
   POL(s(x_1)) = 2 + x_1
   POL(true) = 0


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

(21)
Obligation:
P is empty.
The TRS R consists of the following rules:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   gcd(0, y) -> y
   gcd(s(x), 0) -> s(x)
   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))

The set E consists of the following equations:

   gcd(x, y) == gcd(y, x)

The set E# consists of the following equations:

   GCD(x, y) == GCD(y, x)

We have to consider all minimal (P,E#,R,E)-chains
----------------------------------------

(22) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
----------------------------------------

(23)
YES