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:

concat(leaf, y) → y
concat(cons(u, v), y) → cons(u, concat(v, y))
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(concat(x₁, x₂)) = x₁ + x₂   
POL(cons(x₁, x₂)) = 1 + x₁ + x₂   
POL(false) = 0   
POL(leaf) = 1   
POL(less_leaves(x₁, x₂)) = x₁ + x₂   
POL(true) = 0   

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

concat(leaf, y) → y
less_leaves(x, leaf) → false
less_leaves(leaf, cons(w, z)) → true
less_leaves(cons(u, v), cons(w, z)) → less_leaves(concat(u, v), concat(w, z))

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

concat(cons(u, v), y) → cons(u, concat(v, y))

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive path order with status [RPO].
Quasi-Precedence:

concat₂ > cons₂

Status:

trivial

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

concat(cons(u, v), y) → cons(u, concat(v, y))

(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(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