Term Rewriting System R:
[X, Y, Z, X1, X2]
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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(dbl(s(X))) -> S(s(dbl(X)))
ACTIVE(dbl(s(X))) -> S(dbl(X))
ACTIVE(dbl(s(X))) -> DBL(X)
ACTIVE(dbls(cons(X, Y))) -> CONS(dbl(X), dbls(Y))
ACTIVE(dbls(cons(X, Y))) -> DBL(X)
ACTIVE(dbls(cons(X, Y))) -> DBLS(Y)
ACTIVE(sel(s(X), cons(Y, Z))) -> SEL(X, Z)
ACTIVE(indx(cons(X, Y), Z)) -> CONS(sel(X, Z), indx(Y, Z))
ACTIVE(indx(cons(X, Y), Z)) -> SEL(X, Z)
ACTIVE(indx(cons(X, Y), Z)) -> INDX(Y, Z)
ACTIVE(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(dbl1(s(X))) -> S1(s1(dbl1(X)))
ACTIVE(dbl1(s(X))) -> S1(dbl1(X))
ACTIVE(dbl1(s(X))) -> DBL1(X)
ACTIVE(sel1(s(X), cons(Y, Z))) -> SEL1(X, Z)
ACTIVE(quote(s(X))) -> S1(quote(X))
ACTIVE(quote(s(X))) -> QUOTE(X)
ACTIVE(quote(dbl(X))) -> DBL1(X)
ACTIVE(quote(sel(X, Y))) -> SEL1(X, Y)
ACTIVE(dbl(X)) -> DBL(active(X))
ACTIVE(dbl(X)) -> ACTIVE(X)
ACTIVE(dbls(X)) -> DBLS(active(X))
ACTIVE(dbls(X)) -> ACTIVE(X)
ACTIVE(sel(X1, X2)) -> SEL(active(X1), X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> SEL(X1, active(X2))
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(indx(X1, X2)) -> INDX(active(X1), X2)
ACTIVE(indx(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbl1(X)) -> DBL1(active(X))
ACTIVE(dbl1(X)) -> ACTIVE(X)
ACTIVE(s1(X)) -> S1(active(X))
ACTIVE(s1(X)) -> ACTIVE(X)
ACTIVE(sel1(X1, X2)) -> SEL1(active(X1), X2)
ACTIVE(sel1(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel1(X1, X2)) -> SEL1(X1, active(X2))
ACTIVE(sel1(X1, X2)) -> ACTIVE(X2)
ACTIVE(quote(X)) -> QUOTE(active(X))
ACTIVE(quote(X)) -> ACTIVE(X)
DBL(mark(X)) -> DBL(X)
DBL(ok(X)) -> DBL(X)
DBLS(mark(X)) -> DBLS(X)
DBLS(ok(X)) -> DBLS(X)
SEL(mark(X1), X2) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
INDX(mark(X1), X2) -> INDX(X1, X2)
INDX(ok(X1), ok(X2)) -> INDX(X1, X2)
DBL1(mark(X)) -> DBL1(X)
DBL1(ok(X)) -> DBL1(X)
S1(mark(X)) -> S1(X)
S1(ok(X)) -> S1(X)
SEL1(mark(X1), X2) -> SEL1(X1, X2)
SEL1(X1, mark(X2)) -> SEL1(X1, X2)
SEL1(ok(X1), ok(X2)) -> SEL1(X1, X2)
QUOTE(mark(X)) -> QUOTE(X)
QUOTE(ok(X)) -> QUOTE(X)
PROPER(dbl(X)) -> DBL(proper(X))
PROPER(dbl(X)) -> PROPER(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(dbls(X)) -> DBLS(proper(X))
PROPER(dbls(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(sel(X1, X2)) -> SEL(proper(X1), proper(X2))
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(indx(X1, X2)) -> INDX(proper(X1), proper(X2))
PROPER(indx(X1, X2)) -> PROPER(X1)
PROPER(indx(X1, X2)) -> PROPER(X2)
PROPER(from(X)) -> FROM(proper(X))
PROPER(from(X)) -> PROPER(X)
PROPER(dbl1(X)) -> DBL1(proper(X))
PROPER(dbl1(X)) -> PROPER(X)
PROPER(s1(X)) -> S1(proper(X))
PROPER(s1(X)) -> PROPER(X)
PROPER(sel1(X1, X2)) -> SEL1(proper(X1), proper(X2))
PROPER(sel1(X1, X2)) -> PROPER(X1)
PROPER(sel1(X1, X2)) -> PROPER(X2)
PROPER(quote(X)) -> QUOTE(proper(X))
PROPER(quote(X)) -> PROPER(X)
S(ok(X)) -> S(X)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
FROM(ok(X)) -> FROM(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 14 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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pair:

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


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 64 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 15
             ↳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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pair:

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


Rule:

none


Strategy:

innermost




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

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

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

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


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 64 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
           →DP Problem 16
             ↳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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳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
         ↳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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pair:

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


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 64 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
           →DP Problem 17
             ↳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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pair:

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


Rule:

none


Strategy:

innermost




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

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
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:
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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

DBLS(ok(X)) -> DBLS(X)
DBLS(mark(X)) -> DBLS(X)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 64 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
       →DP Problem 2
         ↳UsableRules
       →DP Problem 3
         ↳UsableRules
       →DP Problem 4
         ↳UsableRules
           →DP Problem 18
             ↳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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

DBLS(ok(X)) -> DBLS(X)
DBLS(mark(X)) -> DBLS(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. DBLS(ok(X)) -> DBLS(X)
  2. DBLS(mark(X)) -> DBLS(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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

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


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 64 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 19
             ↳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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
  2. SEL(X1, mark(X2)) -> SEL(X1, X2)
  3. SEL(mark(X1), X2) -> SEL(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
         ↳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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

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


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 64 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 20
             ↳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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. INDX(ok(X1), ok(X2)) -> INDX(X1, X2)
  2. INDX(mark(X1), X2) -> INDX(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
         ↳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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

S1(ok(X)) -> S1(X)
S1(mark(X)) -> S1(X)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 64 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 21
             ↳Size-Change Principle
       →DP Problem 8
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

S1(ok(X)) -> S1(X)
S1(mark(X)) -> S1(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. S1(ok(X)) -> S1(X)
  2. S1(mark(X)) -> S1(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
         ↳Usable Rules (Innermost)
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

DBL1(ok(X)) -> DBL1(X)
DBL1(mark(X)) -> DBL1(X)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 64 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 22
             ↳Size-Change Principle
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

DBL1(ok(X)) -> DBL1(X)
DBL1(mark(X)) -> DBL1(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. DBL1(ok(X)) -> DBL1(X)
  2. DBL1(mark(X)) -> DBL1(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
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

SEL1(ok(X1), ok(X2)) -> SEL1(X1, X2)
SEL1(X1, mark(X2)) -> SEL1(X1, X2)
SEL1(mark(X1), X2) -> SEL1(X1, X2)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 64 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 23
             ↳Size-Change Principle
       →DP Problem 10
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

SEL1(ok(X1), ok(X2)) -> SEL1(X1, X2)
SEL1(X1, mark(X2)) -> SEL1(X1, X2)
SEL1(mark(X1), X2) -> SEL1(X1, X2)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. SEL1(ok(X1), ok(X2)) -> SEL1(X1, X2)
  2. SEL1(X1, mark(X2)) -> SEL1(X1, X2)
  3. SEL1(mark(X1), X2) -> SEL1(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
         ↳UsableRules
       →DP Problem 9
         ↳UsableRules
       →DP Problem 10
         ↳Usable Rules (Innermost)
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

QUOTE(ok(X)) -> QUOTE(X)
QUOTE(mark(X)) -> QUOTE(X)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 64 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 24
             ↳Size-Change Principle
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

QUOTE(ok(X)) -> QUOTE(X)
QUOTE(mark(X)) -> QUOTE(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. QUOTE(ok(X)) -> QUOTE(X)
  2. QUOTE(mark(X)) -> QUOTE(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
         ↳UsableRules
       →DP Problem 11
         ↳Usable Rules (Innermost)
       →DP Problem 12
         ↳UsableRules
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pair:

FROM(ok(X)) -> FROM(X)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 64 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 25
             ↳Size-Change Principle
       →DP Problem 12
         ↳UsableRules
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pair:

FROM(ok(X)) -> FROM(X)


Rule:

none


Strategy:

innermost




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

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

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
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
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳Usable Rules (Innermost)
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

ACTIVE(quote(X)) -> ACTIVE(X)
ACTIVE(sel1(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel1(X1, X2)) -> ACTIVE(X1)
ACTIVE(s1(X)) -> ACTIVE(X)
ACTIVE(dbl1(X)) -> ACTIVE(X)
ACTIVE(indx(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbls(X)) -> ACTIVE(X)
ACTIVE(dbl(X)) -> ACTIVE(X)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 64 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 26
             ↳Size-Change Principle
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

ACTIVE(quote(X)) -> ACTIVE(X)
ACTIVE(sel1(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel1(X1, X2)) -> ACTIVE(X1)
ACTIVE(s1(X)) -> ACTIVE(X)
ACTIVE(dbl1(X)) -> ACTIVE(X)
ACTIVE(indx(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbls(X)) -> ACTIVE(X)
ACTIVE(dbl(X)) -> ACTIVE(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. ACTIVE(quote(X)) -> ACTIVE(X)
  2. ACTIVE(sel1(X1, X2)) -> ACTIVE(X2)
  3. ACTIVE(sel1(X1, X2)) -> ACTIVE(X1)
  4. ACTIVE(s1(X)) -> ACTIVE(X)
  5. ACTIVE(dbl1(X)) -> ACTIVE(X)
  6. ACTIVE(indx(X1, X2)) -> ACTIVE(X1)
  7. ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
  8. ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
  9. ACTIVE(dbls(X)) -> ACTIVE(X)
  10. ACTIVE(dbl(X)) -> ACTIVE(X)
and get the following Size-Change Graph(s):
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10} , {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
1>1

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

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
s1(x1) -> s1(x1)
dbls(x1) -> dbls(x1)
indx(x1, x2) -> indx(x1, x2)
quote(x1) -> quote(x1)
dbl(x1) -> dbl(x1)
sel(x1, x2) -> sel(x1, x2)
sel1(x1, x2) -> sel1(x1, x2)
dbl1(x1) -> dbl1(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
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules
       →DP Problem 13
         ↳Usable Rules (Innermost)
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

PROPER(quote(X)) -> PROPER(X)
PROPER(sel1(X1, X2)) -> PROPER(X2)
PROPER(sel1(X1, X2)) -> PROPER(X1)
PROPER(s1(X)) -> PROPER(X)
PROPER(dbl1(X)) -> PROPER(X)
PROPER(from(X)) -> PROPER(X)
PROPER(indx(X1, X2)) -> PROPER(X2)
PROPER(indx(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(dbls(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(dbl(X)) -> PROPER(X)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 64 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 13
         ↳UsableRules
           →DP Problem 27
             ↳Size-Change Principle
       →DP Problem 14
         ↳UsableRules


Dependency Pairs:

PROPER(quote(X)) -> PROPER(X)
PROPER(sel1(X1, X2)) -> PROPER(X2)
PROPER(sel1(X1, X2)) -> PROPER(X1)
PROPER(s1(X)) -> PROPER(X)
PROPER(dbl1(X)) -> PROPER(X)
PROPER(from(X)) -> PROPER(X)
PROPER(indx(X1, X2)) -> PROPER(X2)
PROPER(indx(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(dbls(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(dbl(X)) -> PROPER(X)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. PROPER(quote(X)) -> PROPER(X)
  2. PROPER(sel1(X1, X2)) -> PROPER(X2)
  3. PROPER(sel1(X1, X2)) -> PROPER(X1)
  4. PROPER(s1(X)) -> PROPER(X)
  5. PROPER(dbl1(X)) -> PROPER(X)
  6. PROPER(from(X)) -> PROPER(X)
  7. PROPER(indx(X1, X2)) -> PROPER(X2)
  8. PROPER(indx(X1, X2)) -> PROPER(X1)
  9. PROPER(sel(X1, X2)) -> PROPER(X2)
  10. PROPER(sel(X1, X2)) -> PROPER(X1)
  11. PROPER(cons(X1, X2)) -> PROPER(X2)
  12. PROPER(cons(X1, X2)) -> PROPER(X1)
  13. PROPER(dbls(X)) -> PROPER(X)
  14. PROPER(s(X)) -> PROPER(X)
  15. PROPER(dbl(X)) -> PROPER(X)
and get the following Size-Change Graph(s):
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} , {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
1>1

which lead(s) to this/these maximal multigraph(s):
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} , {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
1>1

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
from(x1) -> from(x1)
s1(x1) -> s1(x1)
dbls(x1) -> dbls(x1)
indx(x1, x2) -> indx(x1, x2)
quote(x1) -> quote(x1)
cons(x1, x2) -> cons(x1, x2)
dbl(x1) -> dbl(x1)
sel(x1, x2) -> sel(x1, x2)
s(x1) -> s(x1)
sel1(x1, x2) -> sel1(x1, x2)
dbl1(x1) -> dbl1(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
         ↳UsableRules
       →DP Problem 11
         ↳UsableRules
       →DP Problem 12
         ↳UsableRules
       →DP Problem 13
         ↳UsableRules
       →DP Problem 14
         ↳Usable Rules (Innermost)


Dependency Pairs:

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


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl1(0)) -> mark(01)
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) -> mark(X)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(0)) -> mark(01)
active(quote(s(X))) -> mark(s1(quote(X)))
active(quote(dbl(X))) -> mark(dbl1(X))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl1(X)) -> dbl1(active(X))
active(s1(X)) -> s1(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(quote(X)) -> quote(active(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
quote(mark(X)) -> mark(quote(X))
quote(ok(X)) -> ok(quote(X))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
proper(dbl1(X)) -> dbl1(proper(X))
proper(01) -> ok(01)
proper(s1(X)) -> s1(proper(X))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(quote(X)) -> quote(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(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 13
         ↳UsableRules
       →DP Problem 14
         ↳UsableRules
           →DP Problem 28
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rules:


active(sel1(X1, X2)) -> sel1(X1, active(X2))
active(dbls(X)) -> dbls(active(X))
active(sel1(X1, X2)) -> sel1(active(X1), X2)
active(sel1(s(X), cons(Y, Z))) -> mark(sel1(X, Z))
active(quote(s(X))) -> mark(s1(quote(X)))
active(dbl1(s(X))) -> mark(s1(s1(dbl1(X))))
active(indx(X1, X2)) -> indx(active(X1), X2)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(s1(X)) -> s1(active(X))
active(dbl(0)) -> mark(0)
active(quote(0)) -> mark(01)
active(dbl1(0)) -> mark(01)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(sel1(0, cons(X, Y))) -> mark(X)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(quote(X)) -> quote(active(X))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(quote(sel(X, Y))) -> mark(sel1(X, Y))
active(quote(dbl(X))) -> mark(dbl1(X))
active(dbls(nil)) -> mark(nil)
active(indx(nil, X)) -> mark(nil)
active(dbl1(X)) -> dbl1(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
sel1(mark(X1), X2) -> mark(sel1(X1, X2))
sel1(X1, mark(X2)) -> mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) -> ok(sel1(X1, X2))
dbls(ok(X)) -> ok(dbls(X))
dbls(mark(X)) -> mark(dbls(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
quote(ok(X)) -> ok(quote(X))
quote(mark(X)) -> mark(quote(X))
s1(mark(X)) -> mark(s1(X))
s1(ok(X)) -> ok(s1(X))
dbl1(mark(X)) -> mark(dbl1(X))
dbl1(ok(X)) -> ok(dbl1(X))
dbl(ok(X)) -> ok(dbl(X))
dbl(mark(X)) -> mark(dbl(X))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
from(ok(X)) -> ok(from(X))
proper(nil) -> ok(nil)
proper(dbls(X)) -> dbls(proper(X))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(sel1(X1, X2)) -> sel1(proper(X1), proper(X2))
proper(s1(X)) -> s1(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(dbl(X)) -> dbl(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(01) -> ok(01)
proper(dbl1(X)) -> dbl1(proper(X))
proper(from(X)) -> from(proper(X))
proper(s(X)) -> s(proper(X))
proper(quote(X)) -> quote(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:
1:00 minutes