YES
proof of Transformed_CSR_04_Ex26_Luc03b_Z.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 1 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) QDP
(5) QDPOrderProof [EQUIVALENT, 59 ms]
(6) QDP
(7) QDPOrderProof [EQUIVALENT, 31 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) AND
    (11) QDP
        (12) UsableRulesProof [EQUIVALENT, 0 ms]
        (13) QDP
        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) QDPOrderProof [EQUIVALENT, 14 ms]
        (18) QDP
        (19) PisEmptyProof [EQUIVALENT, 0 ms]
        (20) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
   dbl(0) -> 0
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z)))
   terms(X) -> n__terms(X)
   add(X1, X2) -> n__add(X1, X2)
   s(X) -> n__s(X)
   dbl(X) -> n__dbl(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X

Q is empty.

----------------------------------------

(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TERMS(N) -> SQR(N)
   TERMS(N) -> S(N)
   SQR(s(X)) -> S(n__add(sqr(activate(X)), dbl(activate(X))))
   SQR(s(X)) -> SQR(activate(X))
   SQR(s(X)) -> ACTIVATE(X)
   SQR(s(X)) -> DBL(activate(X))
   DBL(s(X)) -> S(n__s(n__dbl(activate(X))))
   DBL(s(X)) -> ACTIVATE(X)
   ADD(s(X), Y) -> S(n__add(activate(X), Y))
   ADD(s(X), Y) -> ACTIVATE(X)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
   ACTIVATE(n__terms(X)) -> TERMS(X)
   ACTIVATE(n__add(X1, X2)) -> ADD(X1, X2)
   ACTIVATE(n__s(X)) -> S(X)
   ACTIVATE(n__dbl(X)) -> DBL(X)
   ACTIVATE(n__first(X1, X2)) -> FIRST(X1, X2)

The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
   dbl(0) -> 0
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z)))
   terms(X) -> n__terms(X)
   add(X1, X2) -> n__add(X1, X2)
   s(X) -> n__s(X)
   dbl(X) -> n__dbl(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.
----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SQR(s(X)) -> SQR(activate(X))
   SQR(s(X)) -> ACTIVATE(X)
   ACTIVATE(n__terms(X)) -> TERMS(X)
   TERMS(N) -> SQR(N)
   SQR(s(X)) -> DBL(activate(X))
   DBL(s(X)) -> ACTIVATE(X)
   ACTIVATE(n__add(X1, X2)) -> ADD(X1, X2)
   ADD(s(X), Y) -> ACTIVATE(X)
   ACTIVATE(n__dbl(X)) -> DBL(X)
   ACTIVATE(n__first(X1, X2)) -> FIRST(X1, X2)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
   dbl(0) -> 0
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z)))
   terms(X) -> n__terms(X)
   add(X1, X2) -> n__add(X1, X2)
   s(X) -> n__s(X)
   dbl(X) -> n__dbl(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(5) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVATE(n__dbl(X)) -> DBL(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(SQR(x_1)) =  	[[4A]] 	 +  	[[3A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(activate(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(ACTIVATE(x_1)) =  	[[4A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(n__terms(x_1)) =  	[[-I]] 	 +  	[[5A]] 	* 	x_1
>>>

   <<<
 POL(TERMS(x_1)) =  	[[4A]] 	 +  	[[5A]] 	* 	x_1
>>>

   <<<
 POL(DBL(x_1)) =  	[[4A]] 	 +  	[[3A]] 	* 	x_1
>>>

   <<<
 POL(n__add(x_1, x_2)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(ADD(x_1, x_2)) =  	[[5A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(n__dbl(x_1)) =  	[[5A]] 	 +  	[[5A]] 	* 	x_1
>>>

   <<<
 POL(n__first(x_1, x_2)) =  	[[2A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2
>>>

   <<<
 POL(FIRST(x_1, x_2)) =  	[[4A]] 	 +  	[[3A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(terms(x_1)) =  	[[-I]] 	 +  	[[5A]] 	* 	x_1
>>>

   <<<
 POL(add(x_1, x_2)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(n__s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(dbl(x_1)) =  	[[5A]] 	 +  	[[5A]] 	* 	x_1
>>>

   <<<
 POL(first(x_1, x_2)) =  	[[2A]] 	 +  	[[2A]] 	* 	x_1 	 +  	[[2A]] 	* 	x_2
>>>

   <<<
 POL(sqr(x_1)) =  	[[5A]] 	 +  	[[5A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[5A]]
>>>

   <<<
 POL(recip(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(nil) =  	[[4A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X
   sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
   sqr(0) -> 0
   dbl(0) -> 0
   dbl(X) -> n__dbl(X)
   s(X) -> n__s(X)
   terms(X) -> n__terms(X)
   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
   add(0, X) -> X
   add(X1, X2) -> n__add(X1, X2)
   add(s(X), Y) -> s(n__add(activate(X), Y))
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   first(0, X) -> nil
   first(X1, X2) -> n__first(X1, X2)
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z)))


----------------------------------------

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SQR(s(X)) -> SQR(activate(X))
   SQR(s(X)) -> ACTIVATE(X)
   ACTIVATE(n__terms(X)) -> TERMS(X)
   TERMS(N) -> SQR(N)
   SQR(s(X)) -> DBL(activate(X))
   DBL(s(X)) -> ACTIVATE(X)
   ACTIVATE(n__add(X1, X2)) -> ADD(X1, X2)
   ADD(s(X), Y) -> ACTIVATE(X)
   ACTIVATE(n__first(X1, X2)) -> FIRST(X1, X2)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
   dbl(0) -> 0
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z)))
   terms(X) -> n__terms(X)
   add(X1, X2) -> n__add(X1, X2)
   s(X) -> n__s(X)
   dbl(X) -> n__dbl(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(7) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVATE(n__terms(X)) -> TERMS(X)
   ACTIVATE(n__first(X1, X2)) -> FIRST(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( DBL_1(x_1) ) = x_1 + 2
POL( SQR_1(x_1) ) = 2x_1 + 2
POL( activate_1(x_1) ) = x_1
POL( n__terms_1(x_1) ) = 2x_1 + 2
POL( terms_1(x_1) ) = 2x_1 + 2
POL( n__add_2(x_1, x_2) ) = x_1 + x_2
POL( add_2(x_1, x_2) ) = x_1 + x_2
POL( n__s_1(x_1) ) = x_1
POL( s_1(x_1) ) = x_1
POL( n__dbl_1(x_1) ) = max{0, -2}
POL( dbl_1(x_1) ) = 0
POL( n__first_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( first_2(x_1, x_2) ) = x_1 + 2x_2 + 2
POL( sqr_1(x_1) ) = 2x_1 + 1
POL( cons_2(x_1, x_2) ) = x_1 + x_2
POL( recip_1(x_1) ) = max{0, -2}
POL( 0 ) = 0
POL( nil ) = 2
POL( ACTIVATE_1(x_1) ) = x_1 + 2
POL( TERMS_1(x_1) ) = 2x_1 + 2
POL( ADD_2(x_1, x_2) ) = x_1 + 2
POL( FIRST_2(x_1, x_2) ) = x_1 + 2x_2 + 2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X
   sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
   sqr(0) -> 0
   dbl(0) -> 0
   dbl(X) -> n__dbl(X)
   s(X) -> n__s(X)
   terms(X) -> n__terms(X)
   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
   add(0, X) -> X
   add(X1, X2) -> n__add(X1, X2)
   add(s(X), Y) -> s(n__add(activate(X), Y))
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   first(0, X) -> nil
   first(X1, X2) -> n__first(X1, X2)
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z)))


----------------------------------------

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SQR(s(X)) -> SQR(activate(X))
   SQR(s(X)) -> ACTIVATE(X)
   TERMS(N) -> SQR(N)
   SQR(s(X)) -> DBL(activate(X))
   DBL(s(X)) -> ACTIVATE(X)
   ACTIVATE(n__add(X1, X2)) -> ADD(X1, X2)
   ADD(s(X), Y) -> ACTIVATE(X)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
   dbl(0) -> 0
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z)))
   terms(X) -> n__terms(X)
   add(X1, X2) -> n__add(X1, X2)
   s(X) -> n__s(X)
   dbl(X) -> n__dbl(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes.
----------------------------------------

(10)
Complex Obligation (AND)

----------------------------------------

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ADD(s(X), Y) -> ACTIVATE(X)
   ACTIVATE(n__add(X1, X2)) -> ADD(X1, X2)

The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
   dbl(0) -> 0
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z)))
   terms(X) -> n__terms(X)
   add(X1, X2) -> n__add(X1, X2)
   s(X) -> n__s(X)
   dbl(X) -> n__dbl(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(12) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   ADD(s(X), Y) -> ACTIVATE(X)
   ACTIVATE(n__add(X1, X2)) -> ADD(X1, X2)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(14) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*ACTIVATE(n__add(X1, X2)) -> ADD(X1, X2)
The graph contains the following edges 1 > 1, 1 > 2


*ADD(s(X), Y) -> ACTIVATE(X)
The graph contains the following edges 1 > 1


----------------------------------------

(15)
YES

----------------------------------------

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SQR(s(X)) -> SQR(activate(X))

The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
   dbl(0) -> 0
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z)))
   terms(X) -> n__terms(X)
   add(X1, X2) -> n__add(X1, X2)
   s(X) -> n__s(X)
   dbl(X) -> n__dbl(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(17) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   SQR(s(X)) -> SQR(activate(X))
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
SQR(x1)  =  SQR(x1)

s(x1)  =  s(x1)

activate(x1)  =  x1

n__terms(x1)  =  n__terms

terms(x1)  =  terms

n__add(x1, x2)  =  n__add(x1, x2)

add(x1, x2)  =  add(x1, x2)

n__s(x1)  =  n__s(x1)

n__dbl(x1)  =  n__dbl(x1)

dbl(x1)  =  dbl(x1)

n__first(x1, x2)  =  n__first(x1)

first(x1, x2)  =  first(x1)

sqr(x1)  =  sqr(x1)

0  =  0

cons(x1, x2)  =  x2

recip(x1)  =  recip

nil  =  nil


Recursive path order with status [RPO].
Quasi-Precedence: SQR_1 > [s_1, n__s_1, recip]
[n__terms, terms] > [s_1, n__s_1, recip]
[n__dbl_1, dbl_1, sqr_1] > [n__add_2, add_2] > [s_1, n__s_1, recip]
[n__dbl_1, dbl_1, sqr_1] > 0 > nil > [s_1, n__s_1, recip]
[n__first_1, first_1] > [s_1, n__s_1, recip]

Status: SQR_1: [1]
s_1: multiset status
n__terms: []
terms: []
n__add_2: multiset status
add_2: multiset status
n__s_1: multiset status
n__dbl_1: [1]
dbl_1: [1]
n__first_1: [1]
first_1: [1]
sqr_1: [1]
0: multiset status
recip: multiset status
nil: multiset status


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X
   sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
   sqr(0) -> 0
   dbl(0) -> 0
   dbl(X) -> n__dbl(X)
   s(X) -> n__s(X)
   terms(X) -> n__terms(X)
   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
   add(0, X) -> X
   add(X1, X2) -> n__add(X1, X2)
   add(s(X), Y) -> s(n__add(activate(X), Y))
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   first(0, X) -> nil
   first(X1, X2) -> n__first(X1, X2)
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z)))


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(18)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   terms(N) -> cons(recip(sqr(N)), n__terms(s(N)))
   sqr(0) -> 0
   sqr(s(X)) -> s(n__add(sqr(activate(X)), dbl(activate(X))))
   dbl(0) -> 0
   dbl(s(X)) -> s(n__s(n__dbl(activate(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(n__add(activate(X), Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(activate(X), activate(Z)))
   terms(X) -> n__terms(X)
   add(X1, X2) -> n__add(X1, X2)
   s(X) -> n__s(X)
   dbl(X) -> n__dbl(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__terms(X)) -> terms(X)
   activate(n__add(X1, X2)) -> add(X1, X2)
   activate(n__s(X)) -> s(X)
   activate(n__dbl(X)) -> dbl(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(19) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
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(20)
YES