YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) EUsableRulesReductionPairsProof [EQUIVALENT, 14 ms]
        (7) EDP
        (8) ERuleRemovalProof [EQUIVALENT, 0 ms]
        (9) EDP
        (10) EDPPoloProof [EQUIVALENT, 4 ms]
        (11) EDP
        (12) PisEmptyProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) EDP
        (15) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
        (16) EDP
        (17) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
        (18) EDP
        (19) PisEmptyProof [EQUIVALENT, 0 ms]
        (20) YES
    (21) EDP
        (22) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
        (23) EDP
        (24) EDPPoloProof [EQUIVALENT, 0 ms]
        (25) EDP
        (26) PisEmptyProof [EQUIVALENT, 0 ms]
        (27) YES


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

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

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))


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

(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:

   MINUS(s(x), s(y)) -> MINUS(x, y)
   MINUS(minus(x, y), z) -> MINUS(x, plus(y, z))
   MINUS(minus(x, y), z) -> PLUS(y, z)
   QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))
   QUOT(s(x), s(y)) -> MINUS(x, y)
   PLUS(s(x), y) -> PLUS(x, y)
   PLUS(plus(s(x), y), ext) -> PLUS(s(plus(x, y)), ext)
   PLUS(plus(s(x), y), ext) -> PLUS(x, y)

The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext)

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

The set E# consists of the following equations:

   PLUS(x, y) == PLUS(y, x)
   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))

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

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

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

   MINUS(s(x), s(y)) -> MINUS(x, y)
   MINUS(minus(x, y), z) -> MINUS(x, plus(y, z))
   MINUS(minus(x, y), z) -> PLUS(y, z)
   QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))
   QUOT(s(x), s(y)) -> MINUS(x, y)
   PLUS(s(x), y) -> PLUS(x, y)
   PLUS(plus(s(x), y), ext) -> PLUS(s(plus(x, y)), ext)
   PLUS(plus(s(x), y), ext) -> PLUS(x, y)

The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext)

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

The set E# consists of the following equations:

   PLUS(x, y) == PLUS(y, x)
   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))

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 2 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

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

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

   PLUS(plus(s(x), y), ext) -> PLUS(s(plus(x, y)), ext)
   PLUS(s(x), y) -> PLUS(x, y)
   PLUS(plus(s(x), y), ext) -> PLUS(x, y)

The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext)

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

The set E# consists of the following equations:

   PLUS(x, y) == PLUS(y, x)
   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))

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

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

No dependency pairs are removed.

The following rules are removed from R:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
No equations are removed from E.

Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(PLUS(x_1, x_2)) = x_1 + x_2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = x_1


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

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

   PLUS(plus(s(x), y), ext) -> PLUS(s(plus(x, y)), ext)
   PLUS(s(x), y) -> PLUS(x, y)
   PLUS(plus(s(x), y), ext) -> PLUS(x, y)

The TRS R consists of the following rules:

   plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext)
   plus(s(x), y) -> s(plus(x, y))

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

The set E# consists of the following equations:

   PLUS(x, y) == PLUS(y, x)
   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))

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

(8) ERuleRemovalProof (EQUIVALENT)
By using the rule removal processor [DA_STEIN] with the following polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this EDP problem can be strictly oriented.

Strictly oriented dependency pairs:

   PLUS(s(x), y) -> PLUS(x, y)
   PLUS(plus(s(x), y), ext) -> PLUS(x, y)


Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(PLUS(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = 1 + x_1


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

(9)
Obligation:
The TRS P consists of the following rules:

   PLUS(plus(s(x), y), ext) -> PLUS(s(plus(x, y)), ext)

The TRS R consists of the following rules:

   plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext)
   plus(s(x), y) -> s(plus(x, y))

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

The set E# consists of the following equations:

   PLUS(x, y) == PLUS(y, x)
   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))

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

(10) 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.


   PLUS(plus(s(x), y), ext) -> PLUS(s(plus(x, y)), ext)
With the implicit AFS we had to orient the following set of usable rules of R non-strictly.


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


   PLUS(x, y) == PLUS(y, x)
   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))
With the implicit AFS we had to orient the following usable equations of E equivalently.


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

   POL(PLUS(x_1, x_2)) = x_1 + x_2
   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
   POL(s(x_1)) = 0


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

(11)
Obligation:
P is empty.
The TRS R consists of the following rules:

   plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext)
   plus(s(x), y) -> s(plus(x, y))

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

The set E# consists of the following equations:

   PLUS(x, y) == PLUS(y, x)
   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))

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

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

(13)
YES

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

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

   MINUS(minus(x, y), z) -> MINUS(x, plus(y, z))
   MINUS(s(x), s(y)) -> MINUS(x, y)

The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext)

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

The set E# consists of the following equations:

   PLUS(x, y) == PLUS(y, x)
   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))

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

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

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

(16)
Obligation:
The TRS P consists of the following rules:

   MINUS(minus(x, y), z) -> MINUS(x, plus(y, z))
   MINUS(s(x), s(y)) -> MINUS(x, y)

The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext)

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

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

(17) EUsableRulesReductionPairsProof (EQUIVALENT)
By using the usable rules and equations with reduction pair processor [DA_STEIN] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules can be oriented non-strictly, the  usable equations and the esharp equations can be oriented equivalently. All non-usable rules and equations are removed, and those dependency pairs and 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(minus(x, y), z) -> MINUS(x, plus(y, z))
   MINUS(s(x), s(y)) -> MINUS(x, y)
The following rules are removed from R:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
No equations are removed from E.

Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(MINUS(x_1, x_2)) = 3*x_1 + 2*x_2
   POL(minus(x_1, x_2)) = 2 + 3*x_1 + 3*x_2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = 3 + x_1


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

(18)
Obligation:
P is empty.
The TRS R consists of the following rules:

   plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext)
   plus(s(x), y) -> s(plus(x, y))

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

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

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

(20)
YES

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

(21)
Obligation:
The TRS P consists of the following rules:

   QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))

The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext)

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

The set E# consists of the following equations:

   PLUS(x, y) == PLUS(y, x)
   PLUS(plus(x, y), z) == PLUS(x, plus(y, z))

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

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

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

(23)
Obligation:
The TRS P consists of the following rules:

   QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))

The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext)

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

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

(24) 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.


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


   minus(minus(x, y), z) -> minus(x, plus(y, z))
   minus(s(x), s(y)) -> minus(x, y)
   minus(x, 0) -> x
There is no equation of E#.


With the implicit AFS there is no usable equation of E.


Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(QUOT(x_1, x_2)) = 2*x_1
   POL(minus(x_1, x_2)) = 1 + 2*x_1
   POL(plus(x_1, x_2)) = 0
   POL(s(x_1)) = 2 + 3*x_1


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

(25)
Obligation:
P is empty.
The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext)

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))

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

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

(27)
YES