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

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
↳7 QTRSRRRProof (⇔)
↳8 QTRS
↳9 QTRSRRRProof (⇔)
↳10 QTRS
↳11 QTRSRRRProof (⇔)
↳12 QTRS
↳13 QTRSRRRProof (⇔)
↳14 QTRS
  ↳15 RisEmptyProof (⇔)
    ↳16 TRUE
  ↳17 RisEmptyProof (⇔)
    ↳18 TRUE
  ↳19 RisEmptyProof (⇔)
    ↳20 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(2nd(cons1(X, cons(Y, Z)))) → mark(Y)
active(2nd(cons(X, X1))) → mark(2nd(cons1(X, X1)))
active(from(X)) → mark(cons(X, from(s(X))))
active(2nd(X)) → 2nd(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
2nd(mark(X)) → mark(2nd(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
2nd(ok(X)) → ok(2nd(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(2nd(x1)) = x1   
POL(active(x1)) = 2·x1   
POL(cons(x1, x2)) = 2·x1 + x2   
POL(cons1(x1, x2)) = x1 + x2   
POL(from(x1)) = 2·x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = 1 + 2·x1   
POL(proper(x1)) = x1   
POL(s(x1)) = x1   
POL(top(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
top(ok(X)) → top(active(X))


(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(2nd(cons1(X, cons(Y, Z)))) → mark(Y)
active(2nd(cons(X, X1))) → mark(2nd(cons1(X, X1)))
active(from(X)) → mark(cons(X, from(s(X))))
active(2nd(X)) → 2nd(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
2nd(mark(X)) → mark(2nd(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
2nd(ok(X)) → ok(2nd(X))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(2nd(x1)) = 2 + 2·x1   
POL(active(x1)) = 2·x1   
POL(cons(x1, x2)) = x1 + x2   
POL(cons1(x1, x2)) = 2·x1 + 2·x2   
POL(from(x1)) = 2·x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = 1 + x1   
POL(proper(x1)) = x1   
POL(s(x1)) = x1   
POL(top(x1)) = 2·x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

active(2nd(cons1(X, cons(Y, Z)))) → mark(Y)
active(2nd(cons(X, X1))) → mark(2nd(cons1(X, X1)))
active(2nd(X)) → 2nd(active(X))
2nd(ok(X)) → ok(2nd(X))


(4) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
2nd(mark(X)) → mark(2nd(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(2nd(x1)) = 1 + 2·x1   
POL(active(x1)) = 2·x1   
POL(cons(x1, x2)) = 2·x1 + x2   
POL(cons1(x1, x2)) = 2·x1 + 2·x2   
POL(from(x1)) = 1 + 2·x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = x1   
POL(s(x1)) = x1   
POL(top(x1)) = 2·x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

active(from(X)) → mark(cons(X, from(s(X))))
active(from(X)) → from(active(X))


(6) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
2nd(mark(X)) → mark(2nd(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(2nd(x1)) = x1   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = x1 + x2   
POL(cons1(x1, x2)) = x1 + x2   
POL(from(x1)) = x1   
POL(mark(x1)) = 1 + x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = x1   
POL(s(x1)) = x1   
POL(top(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

top(mark(X)) → top(proper(X))


(8) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
2nd(mark(X)) → mark(2nd(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
s(ok(X)) → ok(s(X))

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(2nd(x1)) = 1 + 2·x1   
POL(active(x1)) = 1 + 2·x1   
POL(cons(x1, x2)) = 1 + x1 + 2·x2   
POL(cons1(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(from(x1)) = 1 + x1   
POL(mark(x1)) = 1 + x1   
POL(ok(x1)) = 1 + x1   
POL(proper(x1)) = 2·x1   
POL(s(x1)) = 2 + 2·x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
2nd(mark(X)) → mark(2nd(X))
s(mark(X)) → mark(s(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
s(ok(X)) → ok(s(X))


(10) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(active(x1)) = 2 + 2·x1   
POL(cons(x1, x2)) = x1 + x2   
POL(cons1(x1, x2)) = 1 + x1 + x2   
POL(from(x1)) = 2 + 2·x1   
POL(mark(x1)) = 1 + x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
from(mark(X)) → mark(from(X))


(12) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

cons(mark(X1), X2) → mark(cons(X1, X2))

Q is empty.

(13) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(cons(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(mark(x1)) = 1 + x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

cons(mark(X1), X2) → mark(cons(X1, X2))


(14) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(15) RisEmptyProof (EQUIVALENT transformation)

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

(16) TRUE

(17) RisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) RisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE