(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:
Recursive path order with status [RPO].
Quasi-Precedence:
f2 > g2 > or2 > not1
f2 > g2 > =2 > not1
f2 > g2 > u > [nil, true, max1, max'2] > false > not1
++2 > g2 > or2 > not1
++2 > g2 > =2 > not1
++2 > g2 > u > [nil, true, max1, max'2] > false > not1
mem2 > null1 > [nil, true, max1, max'2] > false > not1
mem2 > or2 > not1
mem2 > =2 > not1
Status:
null1: multiset
++2: [1,2]
=2: multiset
f2: multiset
or2: multiset
true: multiset
mem2: [1,2]
max'2: multiset
g2: multiset
u: multiset
not1: multiset
max1: [1]
false: multiset
nil: 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) TRUE