(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.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Combined order from the following AFS and order.
f(
x1,
x2) =
f(
x1,
x2)
nil =
nil
g(
x1,
x2) =
g(
x1,
x2)
++(
x1,
x2) =
++(
x1,
x2)
null(
x1) =
null(
x1)
true =
true
false =
false
mem(
x1,
x2) =
mem(
x1,
x2)
or(
x1,
x2) =
or(
x1,
x2)
=(
x1,
x2) =
=(
x1,
x2)
max(
x1) =
x1
not(
x1) =
x1
max'(
x1,
x2) =
max'(
x1,
x2)
u =
u
Recursive path order with status [RPO].
Quasi-Precedence:
f2 > nil > g2 > or2 > =2
f2 > nil > g2 > u > max'2 > =2
++2 > g2 > or2 > =2
++2 > g2 > u > max'2 > =2
mem2 > [null1, false] > true > =2
mem2 > or2 > =2
Status:
f2: multiset
nil: multiset
g2: multiset
++2: multiset
null1: multiset
true: multiset
false: multiset
mem2: multiset
or2: multiset
=2: multiset
max'2: multiset
u: multiset
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)
(2) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(3) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(4) YES