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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 160 ms]
(2) ETRS
(3) RRRPoloETRSProof [EQUIVALENT, 45 ms]
(4) ETRS
(5) EquationalDependencyPairsProof [EQUIVALENT, 0 ms]
(6) EDP
(7) EDependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) EDP
        (10) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
        (11) EDP
        (12) EUsableRulesReductionPairsProof [EQUIVALENT, 4 ms]
        (13) EDP
        (14) PisEmptyProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) EDP
        (17) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
        (18) EDP
        (19) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
        (20) EDP
        (21) EDPProblemToQDPProblemProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (24) YES
    (25) EDP
        (26) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
        (27) EDP
        (28) ERuleRemovalProof [EQUIVALENT, 0 ms]
        (29) EDP
        (30) EDPPoloProof [EQUIVALENT, 0 ms]
        (31) EDP
        (32) PisEmptyProof [EQUIVALENT, 0 ms]
        (33) YES
    (34) EDP
        (35) ESharpUsableEquationsProof [EQUIVALENT, 0 ms]
        (36) EDP
        (37) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms]
        (38) EDP
        (39) PisEmptyProof [EQUIVALENT, 0 ms]
        (40) YES


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

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

   sum(x, y) -> S(int(x, y))
   S(nil) -> 0
   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, 0) -> cons(0, nil)
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), 0) -> nil
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(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) RRRPoloETRSProof (EQUIVALENT)
The following E TRS is given: Equational rewrite system:
The TRS R consists of the following rules:

   sum(x, y) -> S(int(x, y))
   S(nil) -> 0
   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, 0) -> cons(0, nil)
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), 0) -> nil
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(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 following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:

   sum(x, y) -> S(int(x, y))
   int(0, 0) -> cons(0, nil)
   int(s(x), 0) -> nil
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(S(x_1)) = x_1
   POL(cons(x_1, x_2)) = 2*x_1 + x_2
   POL(int(x_1, x_2)) = 1 + x_1 + x_2
   POL(intlist(x_1)) = x_1
   POL(nil) = 0
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = x_1
   POL(sum(x_1, x_2)) = 2 + x_1 + x_1*x_2 + x_2




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

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

   S(nil) -> 0
   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(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))


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

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

   S(nil) -> 0
   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(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 following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly by a polynomial ordering:

   S(nil) -> 0
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(S(x_1)) = 2*x_1
   POL(cons(x_1, x_2)) = 2*x_1 + x_2
   POL(int(x_1, x_2)) = 2*x_1 + x_2
   POL(intlist(x_1)) = x_1
   POL(nil) = 1
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = x_1




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

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

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(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))


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

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

   S^1(cons(x, xs)) -> PLUS(x, S(xs))
   S^1(cons(x, xs)) -> S^1(xs)
   PLUS(x, s(y)) -> PLUS(x, y)
   INT(0, s(y)) -> INT(s(0), s(y))
   INT(s(x), s(y)) -> INTLIST(int(x, y))
   INT(s(x), s(y)) -> INT(x, y)
   INTLIST(cons(x, y)) -> INTLIST(y)
   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)

The TRS R consists of the following rules:

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   plus(plus(x, s(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)
Obligation:
The TRS P consists of the following rules:

   S^1(cons(x, xs)) -> PLUS(x, S(xs))
   S^1(cons(x, xs)) -> S^1(xs)
   PLUS(x, s(y)) -> PLUS(x, y)
   INT(0, s(y)) -> INT(s(0), s(y))
   INT(s(x), s(y)) -> INTLIST(int(x, y))
   INT(s(x), s(y)) -> INT(x, y)
   INTLIST(cons(x, y)) -> INTLIST(y)
   PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext)
   PLUS(plus(x, s(y)), ext) -> PLUS(x, y)

The TRS R consists of the following rules:

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   plus(plus(x, s(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
----------------------------------------

(7) EDependencyGraphProof (EQUIVALENT)
The approximation of the Equational Dependency Graph [DA_STEIN] contains 4 SCCs with 2 less nodes.
----------------------------------------

(8)
Complex Obligation (AND)

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

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

   INTLIST(cons(x, y)) -> INTLIST(y)

The TRS R consists of the following rules:

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   plus(plus(x, s(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
----------------------------------------

(10) 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))

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

(11)
Obligation:
The TRS P consists of the following rules:

   INTLIST(cons(x, y)) -> INTLIST(y)

The TRS R consists of the following rules:

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   plus(plus(x, s(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
----------------------------------------

(12) 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:

   INTLIST(cons(x, y)) -> INTLIST(y)
The following rules are removed from R:

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
The following equations are removed from E:

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

   POL(INTLIST(x_1)) = 3*x_1
   POL(cons(x_1, x_2)) = x_1 + 3*x_2


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

(13)
Obligation:
P is empty.
R is empty.
E is empty.
E# is empty.
We have to consider all minimal (P,E#,R,E)-chains
----------------------------------------

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

(15)
YES

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

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

   INT(0, s(y)) -> INT(s(0), s(y))
   INT(s(x), s(y)) -> INT(x, y)

The TRS R consists of the following rules:

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   plus(plus(x, s(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
----------------------------------------

(17) 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))

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

(18)
Obligation:
The TRS P consists of the following rules:

   INT(0, s(y)) -> INT(s(0), s(y))
   INT(s(x), s(y)) -> INT(x, y)

The TRS R consists of the following rules:

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   plus(plus(x, s(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
----------------------------------------

(19) 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:

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
The following equations are removed from E:

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

   POL(0) = 0
   POL(INT(x_1, x_2)) = x_1 + 3*x_2
   POL(s(x_1)) = x_1


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

(20)
Obligation:
The TRS P consists of the following rules:

   INT(0, s(y)) -> INT(s(0), s(y))
   INT(s(x), s(y)) -> INT(x, y)

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

(21) EDPProblemToQDPProblemProof (EQUIVALENT)
The EDP problem does not contain equations anymore, so we can transform it with the EDP to QDP problem processor [DA_STEIN] into a QDP problem.
----------------------------------------

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   INT(0, s(y)) -> INT(s(0), s(y))
   INT(s(x), s(y)) -> INT(x, y)

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

(23) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*INT(s(x), s(y)) -> INT(x, y)
The graph contains the following edges 1 > 1, 2 > 2


*INT(0, s(y)) -> INT(s(0), s(y))
The graph contains the following edges 2 >= 2


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

(24)
YES

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

(25)
Obligation:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   plus(plus(x, s(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
----------------------------------------

(26) 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:

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(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


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

(27)
Obligation:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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


Used ordering: POLO with Polynomial interpretation [POLO]:

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


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

(29)
Obligation:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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


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(31)
Obligation:
P is empty.
The TRS R consists of the following rules:

   plus(x, s(y)) -> s(plus(x, y))
   plus(plus(x, s(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
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(32) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
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(33)
YES

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(34)
Obligation:
The TRS P consists of the following rules:

   S^1(cons(x, xs)) -> S^1(xs)

The TRS R consists of the following rules:

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   plus(plus(x, s(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
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(35) 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))

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(36)
Obligation:
The TRS P consists of the following rules:

   S^1(cons(x, xs)) -> S^1(xs)

The TRS R consists of the following rules:

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   plus(plus(x, s(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
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(37) 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:

   S^1(cons(x, xs)) -> S^1(xs)
The following rules are removed from R:

   S(cons(x, xs)) -> plus(x, S(xs))
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   int(s(x), s(y)) -> intlist(int(x, y))
   intlist(nil) -> nil
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext)
The following equations are removed from E:

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

   POL(S^1(x_1)) = 3*x_1
   POL(cons(x_1, x_2)) = x_1 + 3*x_2


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(38)
Obligation:
P is empty.
R is empty.
E is empty.
E# is empty.
We have to consider all minimal (P,E#,R,E)-chains
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(39) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,E#,R,E) chain.
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(40)
YES