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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RRRPoloETRSProof [EQUIVALENT, 153 ms]
(2) ETRS
(3) RRRPoloETRSProof [EQUIVALENT, 42 ms]
(4) ETRS
(5) RRRPoloETRSProof [EQUIVALENT, 34 ms]
(6) ETRS
(7) RRRPoloETRSProof [EQUIVALENT, 1047 ms]
(8) ETRS
(9) EquationalDependencyPairsProof [EQUIVALENT, 100 ms]
(10) EDP
(11) EUsableRulesReductionPairsProof [EQUIVALENT, 19 ms]
(12) EDP
(13) ERuleRemovalProof [EQUIVALENT, 17 ms]
(14) EDP
(15) EDPPoloProof [EQUIVALENT, 27 ms]
(16) EDP
(17) PisEmptyProof [EQUIVALENT, 0 ms]
(18) YES


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

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

   0(S) -> S
   plus(S, x) -> x
   plus(0(x), 0(y)) -> 0(plus(x, y))
   plus(0(x), 1(y)) -> 1(plus(x, y))
   plus(0(x), j(y)) -> j(plus(x, y))
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   plus(1(x), j(y)) -> 0(plus(x, y))
   opp(S) -> S
   opp(0(x)) -> 0(opp(x))
   opp(1(x)) -> j(opp(x))
   opp(j(x)) -> 1(opp(x))
   minus(x, y) -> plus(opp(y), x)
   times(S, x) -> S
   times(0(x), y) -> 0(times(x, y))
   times(1(x), y) -> plus(0(times(x, y)), y)
   times(j(x), y) -> minus(0(times(x, y)), y)

The set E consists of the following equations:

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


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

(1) RRRPoloETRSProof (EQUIVALENT)
The following E TRS is given: Equational rewrite system:
The TRS R consists of the following rules:

   0(S) -> S
   plus(S, x) -> x
   plus(0(x), 0(y)) -> 0(plus(x, y))
   plus(0(x), 1(y)) -> 1(plus(x, y))
   plus(0(x), j(y)) -> j(plus(x, y))
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   plus(1(x), j(y)) -> 0(plus(x, y))
   opp(S) -> S
   opp(0(x)) -> 0(opp(x))
   opp(1(x)) -> j(opp(x))
   opp(j(x)) -> 1(opp(x))
   minus(x, y) -> plus(opp(y), x)
   times(S, x) -> S
   times(0(x), y) -> 0(times(x, y))
   times(1(x), y) -> plus(0(times(x, y)), y)
   times(j(x), y) -> minus(0(times(x, y)), y)

The set E consists of the following equations:

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

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

   plus(1(x), j(y)) -> 0(plus(x, y))
   opp(1(x)) -> j(opp(x))
   opp(j(x)) -> 1(opp(x))
   minus(x, y) -> plus(opp(y), x)
   times(1(x), y) -> plus(0(times(x, y)), y)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = x_1
   POL(1(x_1)) = 3 + x_1
   POL(S) = 0
   POL(j(x_1)) = 3 + x_1
   POL(minus(x_1, x_2)) = 3 + x_1 + 3*x_2
   POL(opp(x_1)) = 2*x_1
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(times(x_1, x_2)) = x_1 + 2*x_1*x_2 + x_2




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

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

   0(S) -> S
   plus(S, x) -> x
   plus(0(x), 0(y)) -> 0(plus(x, y))
   plus(0(x), 1(y)) -> 1(plus(x, y))
   plus(0(x), j(y)) -> j(plus(x, y))
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   opp(S) -> S
   opp(0(x)) -> 0(opp(x))
   times(S, x) -> S
   times(0(x), y) -> 0(times(x, y))
   times(j(x), y) -> minus(0(times(x, y)), y)

The set E consists of the following equations:

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


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

(3) RRRPoloETRSProof (EQUIVALENT)
The following E TRS is given: Equational rewrite system:
The TRS R consists of the following rules:

   0(S) -> S
   plus(S, x) -> x
   plus(0(x), 0(y)) -> 0(plus(x, y))
   plus(0(x), 1(y)) -> 1(plus(x, y))
   plus(0(x), j(y)) -> j(plus(x, y))
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   opp(S) -> S
   opp(0(x)) -> 0(opp(x))
   times(S, x) -> S
   times(0(x), y) -> 0(times(x, y))
   times(j(x), y) -> minus(0(times(x, y)), y)

The set E consists of the following equations:

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

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

   times(S, x) -> S
   times(j(x), y) -> minus(0(times(x, y)), y)
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = x_1
   POL(1(x_1)) = 2 + x_1
   POL(S) = 0
   POL(j(x_1)) = 2 + x_1
   POL(minus(x_1, x_2)) = x_1 + x_2
   POL(opp(x_1)) = 2*x_1
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(times(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2




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

(4)
Obligation:
Equational rewrite system:
The TRS R consists of the following rules:

   0(S) -> S
   plus(S, x) -> x
   plus(0(x), 0(y)) -> 0(plus(x, y))
   plus(0(x), 1(y)) -> 1(plus(x, y))
   plus(0(x), j(y)) -> j(plus(x, y))
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   opp(S) -> S
   opp(0(x)) -> 0(opp(x))
   times(0(x), y) -> 0(times(x, y))

The set E consists of the following equations:

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


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

(5) RRRPoloETRSProof (EQUIVALENT)
The following E TRS is given: Equational rewrite system:
The TRS R consists of the following rules:

   0(S) -> S
   plus(S, x) -> x
   plus(0(x), 0(y)) -> 0(plus(x, y))
   plus(0(x), 1(y)) -> 1(plus(x, y))
   plus(0(x), j(y)) -> j(plus(x, y))
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   opp(S) -> S
   opp(0(x)) -> 0(opp(x))
   times(0(x), y) -> 0(times(x, y))

The set E consists of the following equations:

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

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

   0(S) -> S
   plus(0(x), 0(y)) -> 0(plus(x, y))
   plus(0(x), 1(y)) -> 1(plus(x, y))
   plus(0(x), j(y)) -> j(plus(x, y))
   opp(S) -> S
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = 1 + 3*x_1
   POL(1(x_1)) = 2*x_1
   POL(S) = 0
   POL(j(x_1)) = 2*x_1
   POL(opp(x_1)) = 1 + 3*x_1
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(times(x_1, x_2)) = x_1 + 3*x_1*x_2 + x_2




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

(6)
Obligation:
Equational rewrite system:
The TRS R consists of the following rules:

   plus(S, x) -> x
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   opp(0(x)) -> 0(opp(x))
   times(0(x), y) -> 0(times(x, y))

The set E consists of the following equations:

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


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

(7) RRRPoloETRSProof (EQUIVALENT)
The following E TRS is given: Equational rewrite system:
The TRS R consists of the following rules:

   plus(S, x) -> x
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   opp(0(x)) -> 0(opp(x))
   times(0(x), y) -> 0(times(x, y))

The set E consists of the following equations:

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

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

   opp(0(x)) -> 0(opp(x))
   times(0(x), y) -> 0(times(x, y))
Used ordering:
Polynomial interpretation [POLO]:

   POL(0(x_1)) = 2 + x_1
   POL(1(x_1)) = 2*x_1
   POL(S) = 0
   POL(j(x_1)) = 2*x_1
   POL(opp(x_1)) = 2 + 2*x_1^2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(times(x_1, x_2)) = 2 + 3*x_1 + 3*x_1*x_2 + 3*x_2




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

(8)
Obligation:
Equational rewrite system:
The TRS R consists of the following rules:

   plus(S, x) -> x
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))

The set E consists of the following equations:

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


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

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

   PLUS(1(x), 1(y)) -> PLUS(1(S), plus(x, y))
   PLUS(1(x), 1(y)) -> PLUS(x, y)
   PLUS(j(x), j(y)) -> PLUS(j(S), plus(x, y))
   PLUS(j(x), j(y)) -> PLUS(x, y)
   PLUS(plus(1(x), 1(y)), ext) -> PLUS(j(plus(1(S), plus(x, y))), ext)
   PLUS(plus(1(x), 1(y)), ext) -> PLUS(1(S), plus(x, y))
   PLUS(plus(1(x), 1(y)), ext) -> PLUS(x, y)
   PLUS(plus(j(x), j(y)), ext) -> PLUS(1(plus(j(S), plus(x, y))), ext)
   PLUS(plus(j(x), j(y)), ext) -> PLUS(j(S), plus(x, y))
   PLUS(plus(j(x), j(y)), ext) -> PLUS(x, y)

The TRS R consists of the following rules:

   plus(S, x) -> x
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   plus(plus(1(x), 1(y)), ext) -> plus(j(plus(1(S), plus(x, y))), ext)
   plus(plus(j(x), j(y)), ext) -> plus(1(plus(j(S), plus(x, y))), ext)

The set E consists of the following equations:

   plus(x, y) == plus(y, x)
   times(x, y) == times(y, x)
   plus(plus(x, y), z) == plus(x, plus(y, z))
   times(times(x, y), z) == times(x, times(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)
Obligation:
The TRS P consists of the following rules:

   PLUS(1(x), 1(y)) -> PLUS(1(S), plus(x, y))
   PLUS(1(x), 1(y)) -> PLUS(x, y)
   PLUS(j(x), j(y)) -> PLUS(j(S), plus(x, y))
   PLUS(j(x), j(y)) -> PLUS(x, y)
   PLUS(plus(1(x), 1(y)), ext) -> PLUS(j(plus(1(S), plus(x, y))), ext)
   PLUS(plus(1(x), 1(y)), ext) -> PLUS(1(S), plus(x, y))
   PLUS(plus(1(x), 1(y)), ext) -> PLUS(x, y)
   PLUS(plus(j(x), j(y)), ext) -> PLUS(1(plus(j(S), plus(x, y))), ext)
   PLUS(plus(j(x), j(y)), ext) -> PLUS(j(S), plus(x, y))
   PLUS(plus(j(x), j(y)), ext) -> PLUS(x, y)

The TRS R consists of the following rules:

   plus(S, x) -> x
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   plus(plus(1(x), 1(y)), ext) -> plus(j(plus(1(S), plus(x, y))), ext)
   plus(plus(j(x), j(y)), ext) -> plus(1(plus(j(S), plus(x, y))), ext)

The set E consists of the following equations:

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

(11) 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.

No rules are removed from R.

The following equations are removed from E:

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

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


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

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

   PLUS(1(x), 1(y)) -> PLUS(1(S), plus(x, y))
   PLUS(1(x), 1(y)) -> PLUS(x, y)
   PLUS(j(x), j(y)) -> PLUS(j(S), plus(x, y))
   PLUS(j(x), j(y)) -> PLUS(x, y)
   PLUS(plus(1(x), 1(y)), ext) -> PLUS(j(plus(1(S), plus(x, y))), ext)
   PLUS(plus(1(x), 1(y)), ext) -> PLUS(1(S), plus(x, y))
   PLUS(plus(1(x), 1(y)), ext) -> PLUS(x, y)
   PLUS(plus(j(x), j(y)), ext) -> PLUS(1(plus(j(S), plus(x, y))), ext)
   PLUS(plus(j(x), j(y)), ext) -> PLUS(j(S), plus(x, y))
   PLUS(plus(j(x), j(y)), ext) -> PLUS(x, y)

The TRS R consists of the following rules:

   plus(S, x) -> x
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   plus(plus(1(x), 1(y)), ext) -> plus(j(plus(1(S), plus(x, y))), ext)
   plus(plus(j(x), j(y)), ext) -> plus(1(plus(j(S), plus(x, y))), ext)

The set E consists of the following equations:

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

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

(13) 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(1(x), 1(y)) -> PLUS(1(S), plus(x, y))
   PLUS(1(x), 1(y)) -> PLUS(x, y)
   PLUS(j(x), j(y)) -> PLUS(j(S), plus(x, y))
   PLUS(j(x), j(y)) -> PLUS(x, y)
   PLUS(plus(1(x), 1(y)), ext) -> PLUS(1(S), plus(x, y))
   PLUS(plus(1(x), 1(y)), ext) -> PLUS(x, y)
   PLUS(plus(j(x), j(y)), ext) -> PLUS(j(S), plus(x, y))
   PLUS(plus(j(x), j(y)), ext) -> PLUS(x, y)


Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(1(x_1)) = 3 + x_1
   POL(PLUS(x_1, x_2)) = x_1 + x_2
   POL(S) = 0
   POL(j(x_1)) = 3 + x_1
   POL(plus(x_1, x_2)) = x_1 + x_2


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

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

   PLUS(plus(1(x), 1(y)), ext) -> PLUS(j(plus(1(S), plus(x, y))), ext)
   PLUS(plus(j(x), j(y)), ext) -> PLUS(1(plus(j(S), plus(x, y))), ext)

The TRS R consists of the following rules:

   plus(S, x) -> x
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   plus(plus(1(x), 1(y)), ext) -> plus(j(plus(1(S), plus(x, y))), ext)
   plus(plus(j(x), j(y)), ext) -> plus(1(plus(j(S), plus(x, y))), ext)

The set E consists of the following equations:

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

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) 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(1(x), 1(y)), ext) -> PLUS(j(plus(1(S), plus(x, y))), ext)
   PLUS(plus(j(x), j(y)), ext) -> PLUS(1(plus(j(S), plus(x, y))), ext)
With the implicit AFS we had to orient the following set of usable rules of R non-strictly.


   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   plus(plus(1(x), 1(y)), ext) -> plus(j(plus(1(S), plus(x, y))), ext)
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(plus(j(x), j(y)), ext) -> plus(1(plus(j(S), plus(x, y))), ext)
   plus(S, x) -> x
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(1(x_1)) = 3
   POL(PLUS(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(S) = 0
   POL(j(x_1)) = 2
   POL(plus(x_1, x_2)) = 2 + x_1 + x_2


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

(16)
Obligation:
P is empty.
The TRS R consists of the following rules:

   plus(S, x) -> x
   plus(1(x), 1(y)) -> j(plus(1(S), plus(x, y)))
   plus(j(x), j(y)) -> 1(plus(j(S), plus(x, y)))
   plus(plus(1(x), 1(y)), ext) -> plus(j(plus(1(S), plus(x, y))), ext)
   plus(plus(j(x), j(y)), ext) -> plus(1(plus(j(S), plus(x, y))), ext)

The set E consists of the following equations:

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

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

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

(18)
YES