Termination w.r.t. Q of the following Term Rewriting System could be proven:
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.
↳ QTRS
↳ DirectTerminationProof
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.
We use [23] with the following order to prove termination.
Recursive path order with status [2].
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:
max'2: multiset
++2: [1,2]
or2: multiset
=2: multiset
mem2: multiset
nil: multiset
max1: multiset
true: multiset
g2: multiset
u: multiset
false: multiset
f2: multiset
null1: multiset
not1: multiset