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:

D(t) → s(h)
D(constant) → h
D(b(x, y)) → b(D(x), D(y))
D(c(x, y)) → b(c(y, D(x)), c(x, D(y)))
D(m(x, y)) → m(D(x), D(y))
D(opp(x)) → opp(D(x))
D(div(x, y)) → m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
b(h, x) → x
b(x, h) → x
b(s(x), s(y)) → s(s(b(x, y)))
b(b(x, y), z) → b(x, b(y, z))

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
D(x1)  =  D(x1)
t  =  t
s(x1)  =  x1
h  =  h
constant  =  constant
b(x1, x2)  =  b(x1, x2)
c(x1, x2)  =  c(x1, x2)
m(x1, x2)  =  m(x1, x2)
opp(x1)  =  x1
div(x1, x2)  =  div(x1, x2)
pow(x1, x2)  =  pow(x1, x2)
2  =  2
ln(x1)  =  ln(x1)
1  =  1

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

D₁ > h
D₁ > c₂ > b₂
D₁ > pow₂ > [m₂, div₂]
D₁ > 2
D₁ > ln₁ > [m₂, div₂]
D₁ > 1

Status:

ln₁: multiset
t: multiset
b₂: [1,2]
c₂: multiset
m₂: multiset
D₁: [1]
2: multiset
pow₂: multiset
h: multiset
constant: multiset
div₂: multiset
1: multiset

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

D(t) → s(h)
D(constant) → h
D(b(x, y)) → b(D(x), D(y))
D(c(x, y)) → b(c(y, D(x)), c(x, D(y)))
D(m(x, y)) → m(D(x), D(y))
D(div(x, y)) → m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
b(h, x) → x
b(x, h) → x
b(b(x, y), z) → b(x, b(y, z))

(2) Obligation:

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

D(opp(x)) → opp(D(x))
b(s(x), s(y)) → s(s(b(x, y)))

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(D(x₁)) = 2·x₁   
POL(b(x₁, x₂)) = 2·x₁ + x₂   
POL(opp(x₁)) = 2 + 2·x₁   
POL(s(x₁)) = x₁   

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

D(opp(x)) → opp(D(x))

(4) Obligation:

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

b(s(x), s(y)) → s(s(b(x, y)))

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(b(x₁, x₂)) = 2·x₁ + 2·x₂   
POL(s(x₁)) = 2 + x₁   

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

b(s(x), s(y)) → s(s(b(x, y)))

(6) Obligation:

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

(7) RisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE

(9) RisEmptyProof (EQUIVALENT transformation)

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

(10) TRUE