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].
Quasi-Precedence:
f2 > [g2, false, max'2]
nil > [g2, false, max'2]
++2 > [g2, false, max'2]
mem2 > [null1, max1, not1] > true > [g2, false, max'2]
mem2 > or2 > [g2, false, max'2]
mem2 > =2 > [g2, false, max'2]
u > [null1, max1, not1] > true > [g2, false, max'2]
Status: max'2: [2,1]
++2: multiset
or2: multiset
=2: multiset
mem2: multiset
nil: multiset
max1: multiset
true: multiset
g2: multiset
u: multiset
false: multiset
f2: [2,1]
null1: multiset
not1: multiset