Term Rewriting System R:
[N, X, Y, Z, X1, X2]
active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(terms(N)) -> CONS(recip(sqr(N)), terms(s(N)))
ACTIVE(terms(N)) -> RECIP(sqr(N))
ACTIVE(terms(N)) -> SQR(N)
ACTIVE(terms(N)) -> TERMS(s(N))
ACTIVE(terms(N)) -> S(N)
ACTIVE(sqr(s(X))) -> S(add(sqr(X), dbl(X)))
ACTIVE(sqr(s(X))) -> ADD(sqr(X), dbl(X))
ACTIVE(sqr(s(X))) -> SQR(X)
ACTIVE(sqr(s(X))) -> DBL(X)
ACTIVE(dbl(s(X))) -> S(s(dbl(X)))
ACTIVE(dbl(s(X))) -> S(dbl(X))
ACTIVE(dbl(s(X))) -> DBL(X)
ACTIVE(add(s(X), Y)) -> S(add(X, Y))
ACTIVE(add(s(X), Y)) -> ADD(X, Y)
ACTIVE(first(s(X), cons(Y, Z))) -> CONS(Y, first(X, Z))
ACTIVE(first(s(X), cons(Y, Z))) -> FIRST(X, Z)
ACTIVE(half(s(s(X)))) -> S(half(X))
ACTIVE(half(s(s(X)))) -> HALF(X)
ACTIVE(terms(X)) -> TERMS(active(X))
ACTIVE(terms(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(recip(X)) -> RECIP(active(X))
ACTIVE(recip(X)) -> ACTIVE(X)
ACTIVE(sqr(X)) -> SQR(active(X))
ACTIVE(sqr(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(add(X1, X2)) -> ADD(active(X1), X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(add(X1, X2)) -> ADD(X1, active(X2))
ACTIVE(add(X1, X2)) -> ACTIVE(X2)
ACTIVE(dbl(X)) -> DBL(active(X))
ACTIVE(dbl(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> FIRST(active(X1), X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(first(X1, X2)) -> FIRST(X1, active(X2))
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(half(X)) -> HALF(active(X))
ACTIVE(half(X)) -> ACTIVE(X)
TERMS(mark(X)) -> TERMS(X)
TERMS(ok(X)) -> TERMS(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
RECIP(mark(X)) -> RECIP(X)
RECIP(ok(X)) -> RECIP(X)
SQR(mark(X)) -> SQR(X)
SQR(ok(X)) -> SQR(X)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
ADD(mark(X1), X2) -> ADD(X1, X2)
ADD(X1, mark(X2)) -> ADD(X1, X2)
ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
DBL(mark(X)) -> DBL(X)
DBL(ok(X)) -> DBL(X)
FIRST(mark(X1), X2) -> FIRST(X1, X2)
FIRST(X1, mark(X2)) -> FIRST(X1, X2)
FIRST(ok(X1), ok(X2)) -> FIRST(X1, X2)
HALF(mark(X)) -> HALF(X)
HALF(ok(X)) -> HALF(X)
PROPER(terms(X)) -> TERMS(proper(X))
PROPER(terms(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(recip(X)) -> RECIP(proper(X))
PROPER(recip(X)) -> PROPER(X)
PROPER(sqr(X)) -> SQR(proper(X))
PROPER(sqr(X)) -> PROPER(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(add(X1, X2)) -> ADD(proper(X1), proper(X2))
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(dbl(X)) -> DBL(proper(X))
PROPER(dbl(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> FIRST(proper(X1), proper(X2))
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(half(X)) -> HALF(proper(X))
PROPER(half(X)) -> PROPER(X)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains 12 SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




As we are in the innermost case, we can delete all 57 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 13
             ↳Size-Change Principle
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. CONS(mark(X1), X2) -> CONS(X1, X2)
  2. CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1
2=2
{1, 2} , {1, 2}
1>1
2>2

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1
2=2
{1, 2} , {1, 2}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳Usable Rules (Innermost)
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

FIRST(ok(X1), ok(X2)) -> FIRST(X1, X2)
FIRST(mark(X1), X2) -> FIRST(X1, X2)
FIRST(X1, mark(X2)) -> FIRST(X1, X2)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




As we are in the innermost case, we can delete all 57 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 14
             ↳Size-Change Principle
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

FIRST(ok(X1), ok(X2)) -> FIRST(X1, X2)
FIRST(mark(X1), X2) -> FIRST(X1, X2)
FIRST(X1, mark(X2)) -> FIRST(X1, X2)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. FIRST(ok(X1), ok(X2)) -> FIRST(X1, X2)
  2. FIRST(mark(X1), X2) -> FIRST(X1, X2)
  3. FIRST(X1, mark(X2)) -> FIRST(X1, X2)
and get the following Size-Change Graph(s):
{1, 2, 3} , {1, 2, 3}
1>1
2>2
{1, 2, 3} , {1, 2, 3}
1>1
2=2
{1, 2, 3} , {1, 2, 3}
1=1
2>2

which lead(s) to this/these maximal multigraph(s):
{1, 2, 3} , {1, 2, 3}
1=1
2>2
{1, 2, 3} , {1, 2, 3}
1>1
2=2
{1, 2, 3} , {1, 2, 3}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳Usable Rules (Innermost)
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

TERMS(ok(X)) -> TERMS(X)
TERMS(mark(X)) -> TERMS(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




As we are in the innermost case, we can delete all 57 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
           →DP Problem 15
             ↳Size-Change Principle
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

TERMS(ok(X)) -> TERMS(X)
TERMS(mark(X)) -> TERMS(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. TERMS(ok(X)) -> TERMS(X)
  2. TERMS(mark(X)) -> TERMS(X)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳Usable Rules (Innermost)
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

RECIP(ok(X)) -> RECIP(X)
RECIP(mark(X)) -> RECIP(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




As we are in the innermost case, we can delete all 57 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
           →DP Problem 16
             ↳Size-Change Principle
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

RECIP(ok(X)) -> RECIP(X)
RECIP(mark(X)) -> RECIP(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. RECIP(ok(X)) -> RECIP(X)
  2. RECIP(mark(X)) -> RECIP(X)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳Usable Rules (Innermost)
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

SQR(ok(X)) -> SQR(X)
SQR(mark(X)) -> SQR(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




As we are in the innermost case, we can delete all 57 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
           →DP Problem 17
             ↳Size-Change Principle
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

SQR(ok(X)) -> SQR(X)
SQR(mark(X)) -> SQR(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. SQR(ok(X)) -> SQR(X)
  2. SQR(mark(X)) -> SQR(X)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳Usable Rules (Innermost)
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

S(ok(X)) -> S(X)
S(mark(X)) -> S(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




As we are in the innermost case, we can delete all 57 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
           →DP Problem 18
             ↳Size-Change Principle
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

S(ok(X)) -> S(X)
S(mark(X)) -> S(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. S(ok(X)) -> S(X)
  2. S(mark(X)) -> S(X)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳Usable Rules (Innermost)
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
ADD(X1, mark(X2)) -> ADD(X1, X2)
ADD(mark(X1), X2) -> ADD(X1, X2)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




As we are in the innermost case, we can delete all 57 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
           →DP Problem 19
             ↳Size-Change Principle
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
ADD(X1, mark(X2)) -> ADD(X1, X2)
ADD(mark(X1), X2) -> ADD(X1, X2)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. ADD(ok(X1), ok(X2)) -> ADD(X1, X2)
  2. ADD(X1, mark(X2)) -> ADD(X1, X2)
  3. ADD(mark(X1), X2) -> ADD(X1, X2)
and get the following Size-Change Graph(s):
{1, 2, 3} , {1, 2, 3}
1>1
2>2
{1, 2, 3} , {1, 2, 3}
1=1
2>2
{1, 2, 3} , {1, 2, 3}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{1, 2, 3} , {1, 2, 3}
1>1
2=2
{1, 2, 3} , {1, 2, 3}
1=1
2>2
{1, 2, 3} , {1, 2, 3}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳Usable Rules (Innermost)
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

DBL(ok(X)) -> DBL(X)
DBL(mark(X)) -> DBL(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




As we are in the innermost case, we can delete all 57 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
           →DP Problem 20
             ↳Size-Change Principle
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

DBL(ok(X)) -> DBL(X)
DBL(mark(X)) -> DBL(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. DBL(ok(X)) -> DBL(X)
  2. DBL(mark(X)) -> DBL(X)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳Usable Rules (Innermost)
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

HALF(ok(X)) -> HALF(X)
HALF(mark(X)) -> HALF(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




As we are in the innermost case, we can delete all 57 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
           →DP Problem 21
             ↳Size-Change Principle
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

HALF(ok(X)) -> HALF(X)
HALF(mark(X)) -> HALF(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. HALF(ok(X)) -> HALF(X)
  2. HALF(mark(X)) -> HALF(X)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳Usable Rules (Innermost)
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

ACTIVE(half(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbl(X)) -> ACTIVE(X)
ACTIVE(add(X1, X2)) -> ACTIVE(X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(sqr(X)) -> ACTIVE(X)
ACTIVE(recip(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(terms(X)) -> ACTIVE(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




As we are in the innermost case, we can delete all 57 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
           →DP Problem 22
             ↳Size-Change Principle
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

ACTIVE(half(X)) -> ACTIVE(X)
ACTIVE(first(X1, X2)) -> ACTIVE(X2)
ACTIVE(first(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbl(X)) -> ACTIVE(X)
ACTIVE(add(X1, X2)) -> ACTIVE(X2)
ACTIVE(add(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(sqr(X)) -> ACTIVE(X)
ACTIVE(recip(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(terms(X)) -> ACTIVE(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. ACTIVE(half(X)) -> ACTIVE(X)
  2. ACTIVE(first(X1, X2)) -> ACTIVE(X2)
  3. ACTIVE(first(X1, X2)) -> ACTIVE(X1)
  4. ACTIVE(dbl(X)) -> ACTIVE(X)
  5. ACTIVE(add(X1, X2)) -> ACTIVE(X2)
  6. ACTIVE(add(X1, X2)) -> ACTIVE(X1)
  7. ACTIVE(s(X)) -> ACTIVE(X)
  8. ACTIVE(sqr(X)) -> ACTIVE(X)
  9. ACTIVE(recip(X)) -> ACTIVE(X)
  10. ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
  11. ACTIVE(terms(X)) -> ACTIVE(X)
and get the following Size-Change Graph(s):
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} , {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} , {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
first(x1, x2) -> first(x1, x2)
cons(x1, x2) -> cons(x1, x2)
sqr(x1) -> sqr(x1)
dbl(x1) -> dbl(x1)
s(x1) -> s(x1)
terms(x1) -> terms(x1)
recip(x1) -> recip(x1)
half(x1) -> half(x1)
add(x1, x2) -> add(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳Usable Rules (Innermost)
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

PROPER(half(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(dbl(X)) -> PROPER(X)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(sqr(X)) -> PROPER(X)
PROPER(recip(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(terms(X)) -> PROPER(X)


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




As we are in the innermost case, we can delete all 57 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
           →DP Problem 23
             ↳Size-Change Principle
       →DP Problem 12
         ↳UsableRules


Dependency Pairs:

PROPER(half(X)) -> PROPER(X)
PROPER(first(X1, X2)) -> PROPER(X2)
PROPER(first(X1, X2)) -> PROPER(X1)
PROPER(dbl(X)) -> PROPER(X)
PROPER(add(X1, X2)) -> PROPER(X2)
PROPER(add(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(sqr(X)) -> PROPER(X)
PROPER(recip(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(terms(X)) -> PROPER(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. PROPER(half(X)) -> PROPER(X)
  2. PROPER(first(X1, X2)) -> PROPER(X2)
  3. PROPER(first(X1, X2)) -> PROPER(X1)
  4. PROPER(dbl(X)) -> PROPER(X)
  5. PROPER(add(X1, X2)) -> PROPER(X2)
  6. PROPER(add(X1, X2)) -> PROPER(X1)
  7. PROPER(s(X)) -> PROPER(X)
  8. PROPER(sqr(X)) -> PROPER(X)
  9. PROPER(recip(X)) -> PROPER(X)
  10. PROPER(cons(X1, X2)) -> PROPER(X2)
  11. PROPER(cons(X1, X2)) -> PROPER(X1)
  12. PROPER(terms(X)) -> PROPER(X)
and get the following Size-Change Graph(s):
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} , {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} , {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
first(x1, x2) -> first(x1, x2)
cons(x1, x2) -> cons(x1, x2)
sqr(x1) -> sqr(x1)
dbl(x1) -> dbl(x1)
terms(x1) -> terms(x1)
s(x1) -> s(x1)
recip(x1) -> recip(x1)
half(x1) -> half(x1)
add(x1, x2) -> add(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳Usable Rules (Innermost)


Dependency Pairs:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))


Rules:


active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) -> mark(0)
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(first(0, X)) -> mark(nil)
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(half(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(half(s(s(X)))) -> mark(s(half(X)))
active(half(dbl(X))) -> mark(X)
active(terms(X)) -> terms(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(recip(X)) -> recip(active(X))
active(sqr(X)) -> sqr(active(X))
active(s(X)) -> s(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
active(dbl(X)) -> dbl(active(X))
active(first(X1, X2)) -> first(active(X1), X2)
active(first(X1, X2)) -> first(X1, active(X2))
active(half(X)) -> half(active(X))
terms(mark(X)) -> mark(terms(X))
terms(ok(X)) -> ok(terms(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
first(mark(X1), X2) -> mark(first(X1, X2))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
proper(terms(X)) -> terms(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(sqr(X)) -> sqr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(half(X)) -> half(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




As we are in the innermost case, we can delete all 2 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules
           →DP Problem 24
             ↳Narrowing Transformation


Dependency Pairs:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(half(X)) -> half(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(half(dbl(X))) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(half(0)) -> mark(0)
active(sqr(X)) -> sqr(active(X))
active(dbl(0)) -> mark(0)
active(add(0, X)) -> mark(X)
active(sqr(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(terms(X)) -> terms(active(X))
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(first(X1, X2)) -> first(active(X1), X2)
active(dbl(X)) -> dbl(active(X))
active(recip(X)) -> recip(active(X))
active(first(0, X)) -> mark(nil)
active(add(X1, X2)) -> add(X1, active(X2))
active(half(s(s(X)))) -> mark(s(half(X)))
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
dbl(mark(X)) -> mark(dbl(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
terms(ok(X)) -> ok(terms(X))
terms(mark(X)) -> mark(terms(X))
proper(nil) -> ok(nil)
proper(sqr(X)) -> sqr(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(s(X)) -> s(proper(X))
proper(terms(X)) -> terms(proper(X))
proper(half(X)) -> half(proper(X))
proper(0) -> ok(0)


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

TOP(ok(X)) -> TOP(active(X))
24 new Dependency Pairs are created:

TOP(ok(cons(X1', X2'))) -> TOP(cons(active(X1'), X2'))
TOP(ok(half(X''))) -> TOP(half(active(X'')))
TOP(ok(add(X1', X2'))) -> TOP(add(active(X1'), X2'))
TOP(ok(dbl(s(X'')))) -> TOP(mark(s(s(dbl(X'')))))
TOP(ok(half(dbl(X'')))) -> TOP(mark(X''))
TOP(ok(add(s(X''), Y'))) -> TOP(mark(s(add(X'', Y'))))
TOP(ok(half(0))) -> TOP(mark(0))
TOP(ok(sqr(X''))) -> TOP(sqr(active(X'')))
TOP(ok(dbl(0))) -> TOP(mark(0))
TOP(ok(add(0, X''))) -> TOP(mark(X''))
TOP(ok(sqr(0))) -> TOP(mark(0))
TOP(ok(half(s(0)))) -> TOP(mark(0))
TOP(ok(terms(X''))) -> TOP(terms(active(X'')))
TOP(ok(sqr(s(X'')))) -> TOP(mark(s(add(sqr(X''), dbl(X'')))))
TOP(ok(first(s(X''), cons(Y', Z')))) -> TOP(mark(cons(Y', first(X'', Z'))))
TOP(ok(terms(N'))) -> TOP(mark(cons(recip(sqr(N')), terms(s(N')))))
TOP(ok(first(X1', X2'))) -> TOP(first(active(X1'), X2'))
TOP(ok(dbl(X''))) -> TOP(dbl(active(X'')))
TOP(ok(recip(X''))) -> TOP(recip(active(X'')))
TOP(ok(first(0, X''))) -> TOP(mark(nil))
TOP(ok(add(X1', X2'))) -> TOP(add(X1', active(X2')))
TOP(ok(half(s(s(X''))))) -> TOP(mark(s(half(X''))))
TOP(ok(first(X1', X2'))) -> TOP(first(X1', active(X2')))
TOP(ok(s(X''))) -> TOP(s(active(X'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules
           →DP Problem 24
             ↳Nar
             ...
               →DP Problem 25
                 ↳Narrowing Transformation


Dependency Pairs:

TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(first(X1', X2'))) -> TOP(first(X1', active(X2')))
TOP(ok(half(s(s(X''))))) -> TOP(mark(s(half(X''))))
TOP(ok(add(X1', X2'))) -> TOP(add(X1', active(X2')))
TOP(ok(first(0, X''))) -> TOP(mark(nil))
TOP(ok(recip(X''))) -> TOP(recip(active(X'')))
TOP(ok(dbl(X''))) -> TOP(dbl(active(X'')))
TOP(ok(first(X1', X2'))) -> TOP(first(active(X1'), X2'))
TOP(ok(terms(N'))) -> TOP(mark(cons(recip(sqr(N')), terms(s(N')))))
TOP(ok(first(s(X''), cons(Y', Z')))) -> TOP(mark(cons(Y', first(X'', Z'))))
TOP(ok(sqr(s(X'')))) -> TOP(mark(s(add(sqr(X''), dbl(X'')))))
TOP(ok(terms(X''))) -> TOP(terms(active(X'')))
TOP(ok(half(s(0)))) -> TOP(mark(0))
TOP(ok(sqr(0))) -> TOP(mark(0))
TOP(ok(add(0, X''))) -> TOP(mark(X''))
TOP(ok(dbl(0))) -> TOP(mark(0))
TOP(ok(sqr(X''))) -> TOP(sqr(active(X'')))
TOP(ok(half(0))) -> TOP(mark(0))
TOP(ok(add(s(X''), Y'))) -> TOP(mark(s(add(X'', Y'))))
TOP(ok(half(dbl(X'')))) -> TOP(mark(X''))
TOP(ok(dbl(s(X'')))) -> TOP(mark(s(s(dbl(X'')))))
TOP(ok(add(X1', X2'))) -> TOP(add(active(X1'), X2'))
TOP(ok(half(X''))) -> TOP(half(active(X'')))
TOP(ok(cons(X1', X2'))) -> TOP(cons(active(X1'), X2'))
TOP(mark(X)) -> TOP(proper(X))


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(half(X)) -> half(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(half(dbl(X))) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(half(0)) -> mark(0)
active(sqr(X)) -> sqr(active(X))
active(dbl(0)) -> mark(0)
active(add(0, X)) -> mark(X)
active(sqr(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(terms(X)) -> terms(active(X))
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(first(X1, X2)) -> first(active(X1), X2)
active(dbl(X)) -> dbl(active(X))
active(recip(X)) -> recip(active(X))
active(first(0, X)) -> mark(nil)
active(add(X1, X2)) -> add(X1, active(X2))
active(half(s(s(X)))) -> mark(s(half(X)))
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
dbl(mark(X)) -> mark(dbl(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
terms(ok(X)) -> ok(terms(X))
terms(mark(X)) -> mark(terms(X))
proper(nil) -> ok(nil)
proper(sqr(X)) -> sqr(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(s(X)) -> s(proper(X))
proper(terms(X)) -> terms(proper(X))
proper(half(X)) -> half(proper(X))
proper(0) -> ok(0)


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

TOP(mark(X)) -> TOP(proper(X))
11 new Dependency Pairs are created:

TOP(mark(nil)) -> TOP(ok(nil))
TOP(mark(sqr(X''))) -> TOP(sqr(proper(X'')))
TOP(mark(first(X1', X2'))) -> TOP(first(proper(X1'), proper(X2')))
TOP(mark(dbl(X''))) -> TOP(dbl(proper(X'')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(add(X1', X2'))) -> TOP(add(proper(X1'), proper(X2')))
TOP(mark(recip(X''))) -> TOP(recip(proper(X'')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(terms(X''))) -> TOP(terms(proper(X'')))
TOP(mark(half(X''))) -> TOP(half(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
       →DP Problem 5
         ↳UsableRules
       →DP Problem 6
         ↳UsableRules
       →DP Problem 7
         ↳UsableRules
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules
           →DP Problem 24
             ↳Nar
             ...
               →DP Problem 26
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

TOP(mark(half(X''))) -> TOP(half(proper(X'')))
TOP(mark(terms(X''))) -> TOP(terms(proper(X'')))
TOP(mark(recip(X''))) -> TOP(recip(proper(X'')))
TOP(mark(add(X1', X2'))) -> TOP(add(proper(X1'), proper(X2')))
TOP(mark(dbl(X''))) -> TOP(dbl(proper(X'')))
TOP(ok(first(X1', X2'))) -> TOP(first(X1', active(X2')))
TOP(ok(half(s(s(X''))))) -> TOP(mark(s(half(X''))))
TOP(ok(add(X1', X2'))) -> TOP(add(X1', active(X2')))
TOP(ok(recip(X''))) -> TOP(recip(active(X'')))
TOP(ok(dbl(X''))) -> TOP(dbl(active(X'')))
TOP(ok(first(X1', X2'))) -> TOP(first(active(X1'), X2'))
TOP(ok(terms(N'))) -> TOP(mark(cons(recip(sqr(N')), terms(s(N')))))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(ok(first(s(X''), cons(Y', Z')))) -> TOP(mark(cons(Y', first(X'', Z'))))
TOP(ok(sqr(s(X'')))) -> TOP(mark(s(add(sqr(X''), dbl(X'')))))
TOP(ok(terms(X''))) -> TOP(terms(active(X'')))
TOP(mark(first(X1', X2'))) -> TOP(first(proper(X1'), proper(X2')))
TOP(ok(add(0, X''))) -> TOP(mark(X''))
TOP(ok(sqr(X''))) -> TOP(sqr(active(X'')))
TOP(ok(add(s(X''), Y'))) -> TOP(mark(s(add(X'', Y'))))
TOP(mark(sqr(X''))) -> TOP(sqr(proper(X'')))
TOP(ok(half(dbl(X'')))) -> TOP(mark(X''))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(ok(dbl(s(X'')))) -> TOP(mark(s(s(dbl(X'')))))
TOP(ok(add(X1', X2'))) -> TOP(add(active(X1'), X2'))
TOP(ok(half(X''))) -> TOP(half(active(X'')))
TOP(ok(cons(X1', X2'))) -> TOP(cons(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(half(X)) -> half(active(X))
active(add(X1, X2)) -> add(active(X1), X2)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(half(dbl(X))) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(half(0)) -> mark(0)
active(sqr(X)) -> sqr(active(X))
active(dbl(0)) -> mark(0)
active(add(0, X)) -> mark(X)
active(sqr(0)) -> mark(0)
active(half(s(0))) -> mark(0)
active(terms(X)) -> terms(active(X))
active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
active(first(X1, X2)) -> first(active(X1), X2)
active(dbl(X)) -> dbl(active(X))
active(recip(X)) -> recip(active(X))
active(first(0, X)) -> mark(nil)
active(add(X1, X2)) -> add(X1, active(X2))
active(half(s(s(X)))) -> mark(s(half(X)))
active(first(X1, X2)) -> first(X1, active(X2))
active(s(X)) -> s(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
recip(mark(X)) -> mark(recip(X))
recip(ok(X)) -> ok(recip(X))
half(mark(X)) -> mark(half(X))
half(ok(X)) -> ok(half(X))
first(X1, mark(X2)) -> mark(first(X1, X2))
first(ok(X1), ok(X2)) -> ok(first(X1, X2))
first(mark(X1), X2) -> mark(first(X1, X2))
dbl(ok(X)) -> ok(dbl(X))
dbl(mark(X)) -> mark(dbl(X))
add(ok(X1), ok(X2)) -> ok(add(X1, X2))
add(mark(X1), X2) -> mark(add(X1, X2))
add(X1, mark(X2)) -> mark(add(X1, X2))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))
sqr(mark(X)) -> mark(sqr(X))
sqr(ok(X)) -> ok(sqr(X))
terms(ok(X)) -> ok(terms(X))
terms(mark(X)) -> mark(terms(X))
proper(nil) -> ok(nil)
proper(sqr(X)) -> sqr(proper(X))
proper(first(X1, X2)) -> first(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(add(X1, X2)) -> add(proper(X1), proper(X2))
proper(recip(X)) -> recip(proper(X))
proper(s(X)) -> s(proper(X))
proper(terms(X)) -> terms(proper(X))
proper(half(X)) -> half(proper(X))
proper(0) -> ok(0)


Strategy:

innermost



The Proof could not be continued due to a Timeout.
Innermost Termination of R could not be shown.
Duration:
1:02 minutes