(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)  =  opp(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, opp₁, 1] > [h, constant]
[D₁, opp₁, 1] > [b₂, c₂, m₂, pow₂, ln₁] > div₂
[D₁, opp₁, 1] > 2

Status:

D₁: multiset
t: multiset
h: multiset
constant: multiset
b₂: [1,2]
c₂: multiset
m₂: multiset
opp₁: multiset
div₂: multiset
pow₂: multiset
2: multiset
ln₁: 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))

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Knuth-Bendix order [KBO] with precedence:

b₂ > s₁ > D₁ > opp₁

and weight map:

D_1=2
opp_1=1
s_1=1
b_2=0

The variable weight is 1With 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))
b(s(x), s(y)) → s(s(b(x, y)))

(5) RisEmptyProof (EQUIVALENT transformation)

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