(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
D(x1)  =  D(x1)
t  =  t
1  =  1
constant  =  constant
0  =  0
+(x1, x2)  =  +(x1, x2)
*(x1, x2)  =  *(x1, x2)
-(x1, x2)  =  -(x1, x2)
minus(x1)  =  x1
div(x1, x2)  =  div(x1, x2)
pow(x1, x2)  =  pow(x1, x2)
2  =  2
ln(x1)  =  ln(x1)

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

[D₁, pow₂, ln₁] > 1 > +₂
[D₁, pow₂, ln₁] > [*₂, -₂, div₂, 2] > +₂
t > 1 > +₂
constant > 0 > +₂

Status:

D₁: multiset
t: multiset
1: multiset
constant: multiset
0: multiset
+₂: multiset
*₂: [2,1]
-₂: [1,2]
div₂: [1,2]
pow₂: multiset
2: multiset
ln₁: multiset

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

D(t) → 1
D(constant) → 0
D(+(x, y)) → +(D(x), D(y))
D(*(x, y)) → +(*(y, D(x)), *(x, D(y)))
D(-(x, y)) → -(D(x), D(y))
D(div(x, y)) → -(div(D(x), y), div(*(x, D(y)), pow(y, 2)))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y)))

(3) QTRSRRRProof (EQUIVALENT transformation)

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

D₁ > minus₁

and weight map:

D_1=2
minus_1=1

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(minus(x)) → minus(D(x))

(5) RisEmptyProof (EQUIVALENT transformation)

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