Termination w.r.t. Q of the given QTRS could be proven:

(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 > nil
f₂ > [g₂, max₁] > false
f₂ > [g₂, max₁] > not₁
f₂ > [g₂, max₁] > max'₂
++₂ > [g₂, max₁] > false
++₂ > [g₂, max₁] > not₁
++₂ > [g₂, max₁] > max'₂
[null₁, mem₂] > true
[null₁, mem₂] > false
[null₁, mem₂] > or₂
[null₁, mem₂] > =₂
[null₁, mem₂] > not₁
u > [g₂, max₁] > false
u > [g₂, max₁] > not₁
u > [g₂, max₁] > max'₂

Status:

null₁: [1]
++₂: [2,1]
=₂: multiset
f₂: [2,1]
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

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(6) TRUE

(7) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(8) TRUE