(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:
Lexicographic path order with status [LPO].
Precedence:
f2 > nil > or2
f2 > g2 > mem2 > false > or2
f2 > g2 > mem2 > =2 > or2
f2 > g2 > max1 > null1 > true > or2
f2 > g2 > max1 > null1 > false > or2
f2 > g2 > max1 > not1 > or2
f2 > g2 > max'2 > or2
++2 > g2 > mem2 > false > or2
++2 > g2 > mem2 > =2 > or2
++2 > g2 > max1 > null1 > true > or2
++2 > g2 > max1 > null1 > false > or2
++2 > g2 > max1 > not1 > or2
++2 > g2 > max'2 > or2
u > max1 > null1 > true > or2
u > max1 > null1 > false > or2
u > max1 > not1 > or2
u > max'2 > or2
Status:
null1: [1]
++2: [1,2]
=2: [1,2]
f2: [1,2]
or2: [1,2]
true: []
mem2: [1,2]
max'2: [1,2]
g2: [1,2]
u: []
not1: [1]
max1: [1]
false: []
nil: []
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