YES
proof of SK90_4.28.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 70 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   mem(x, max(x)) -> not(null(x))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
f/2(YES,YES)
nil/0)
g/2(YES,YES)
++/2(YES,YES)
null/1(YES)
true/0)
false/0)
mem/2(YES,YES)
or/2(YES,YES)
=/2(YES,YES)
max/1)YES(
not/1)YES(
max'/2(YES,YES)
u/0)

Quasi precedence:
f_2 > nil > g_2 > or_2 > =_2
f_2 > nil > g_2 > u > max'_2 > =_2
++_2 > g_2 > or_2 > =_2
++_2 > g_2 > u > max'_2 > =_2
mem_2 > [null_1, false] > true > =_2
mem_2 > or_2 > =_2


Status:
f_2: multiset status
nil: multiset status
g_2: multiset status
++_2: multiset status
null_1: multiset status
true: multiset status
false: multiset status
mem_2: multiset status
or_2: multiset status
=_2: multiset status
max'_2: multiset status
u: multiset status

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

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   mem(x, max(x)) -> not(null(x))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)




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(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(4)
YES