(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
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(minus(x)) → minus(D(x))
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)))
Q is empty.
(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)))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
D(minus(x)) → minus(D(x))
Q is empty.
(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))
(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) YES